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Bramwell’s Helicopter Dynamics
Bramwell’s
Helicopter Dynamics
Second edition
A. R. S. Bramwell
George Done
David Balmford
Oxford Auckland Boston Johannesburg Melbourne New Delhi
Butterworth-Heinemann
Linacre House, Jordan Hill, Oxford OX2 8DP
225 Wildwood Avenue, Woburn, MA 01801-2041
A division of Reed Educational and Professional Publishing Ltd
A member of the Reed Elsevier plc group
First published by Edward Arnold (Publishers) Ltd 1976
Second edition published by Butterworth-Heinemann 2001
© A. R. S. Bramwell, George Done and David Balmford 2001
All rights reserved. No part of this publication may be reproduced in
any material form (including photocopying or storing in any medium by
electronic means and whether or not transiently or incidentally to some
other use of this publication) without the written permission of the
copyright holder except in accordance with the provisions of the Copyright,
Designs and Patents Act 1988 or under the terms of a licence issued by the
Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London,
England W1P OLP. Applications for the copyright holder’s written
permission to reproduce any part of this publication should be
addressed to the publishers
British Library Cataloguing in Publication Data
Bramwell, A.R.S.
Bramwell’s helicopter dynamics. – 2nd ed.
1 Helicopters – Aerodynamics
I Title II Done, George III Balmford, David IV Helicopter
dynamics
629.1′33352
Library of Congress Cataloguing in Publication Data
Bramwell, A.R.S.
Bramwell’s helicopter dynamics / A.R.S. Bramwell, George Done,
David Balmford.
–2nd ed.
p. cm.
Rev. ed. of: Helicopter dynamics. c1976
Includes index
ISBN 0 7506 5075 3
1 Helicopter–Dynamics 2 Helicopters–Aerodynamics I Done, George Taylor
Sutton II Balmford, David III Bramwell, A.R.S. Helicopter dynamics
IV Title
TL716.B664 2001
629.133′352--dc21 00-049381
ISBN 0 7506 5075 3
Typeset at Replika Press Pvt Ltd, 100% EOU, Delhi 110 040, India
Printed and bound in Great Britain by Bath Press, Avon.
Contents
Preface to the second edition vii
Preface to the first edition ix
Acknowledgements xi
Notation xiii
1. Basic mechanics of rotor systems and helicopter flight 1
2. Rotor aerodynamics in axial flight 33
3. Rotor aerodynamics and dynamics in forward flight 77
4. Trim and performance in axial and forward flight 115
5. Flight dynamics and control 137
6. Rotor aerodynamics in forward flight 196
7. Structural dynamics of elastic blades 238
8. Rotor induced vibration 290
9. Aeroelastic and aeromechanical behaviour 319
Appendices 360
Index 371
Preface to the second edition
At the time of publication of the first edition of the book in 1976, Bramwell’s
Helicopter Dynamics was a unique addition to the fundamental knowledge of dynamics
of rotorcraft due to its coverage in a single volume of subjects ranging from
aerodynamics, through flight dynamics to vibrational dynamics and aeroelasticity. It
proved to be popular, and the first edition sold out relatively quickly. Unfortunately,
before the book could be revised with a view to producing a second edition, Bram (as
he was known to his friends and colleagues) succumbed to a short illness and died.
As well as leaving a sudden space in the helicopter world, his death left the publishers
with their desire for further editions unfulfilled. Following an approach from the
publishers, the present authors agreed, with considerable trepidation, to undertake
the task of producing a second edition.
Indeed, being asked was an honour, particularly so for one of us (GD), since we
had been colleagues together at City University for a short period of two years.
However, although it may be one thing to produce a book from one’s own lecture
notes and published papers, it is entirely a different proposition to do the same when
the original material is not your own, as we were to discover. It was necessary to try
to understand why Bram’s book was so popular with the helicopter fraternity, in
order that any revisions should not destroy any of the vital qualities in this regard.
One of the characteristics that we felt endeared the book to its followers was the way
explanations of what are complicated phenomena were established from fundamental
laws and simple assumptions. Theoretical expressions were developed from the basic
mathematics in a straightforward and measured style that was particular to Bram’s
way of thinking and writing. We positively wished and endeavoured to retain his
inimitable qualities and characteristics.
Long sections of the book are analytical, starting from fundamental principles, and
do not change significantly in the course of time; however, we have tried to eradicate
errors, printer’s and otherwise, and improve explanations where considered necessary.
There are also many sections that are largely descriptive, and, over the space of 25
years since the first edition, these had tended to become out of date, both in terms of
the state-of-the-art and supporting references; thus, these have been updated.
Opportunities, too, have been taken to expand the treatment of, and to include additional
information in, the vibrational dynamics area, with both the additional and updated
content introduced, hopefully, in such a way as to be compatible with Bram’s style.
Another change which has taken place in the past quarter century is the now
greater familiarity of the users of books such as this one with matrices and vectors.
Hence, Chapter 1 of the first edition, which was aimed at introducing and explaining
the necessary associated matrix and vector operations, has disappeared from the
second edition. Also, some rather fundamental fluid dynamics that also appeared in
this chapter was considered unnecessary in view of the material being readily available
in undergraduate textbooks. What remained from the original Chapter 1 that was
thought still necessary now appears in the Appendix. Readers familiar with the first
edition will notice the inclusion of a notation list in the present edition. This became
an essential item in re-editing the book, because there were many instances in the
first edition of repeated symbols for different parameters, and different symbols for
the same parameters, due to the fact that the much of the material in the original book
was based on various technical papers published at different times. As far as has been
possible, the notation has now been made consistent throughout all chapters; this has
resulted in some of the least used symbols being changed.
Apart from the removal of the elementary material in the original Chapter 1, the
overall structure of the book has not changed to any great degree. The order of the
chapters is as before, although there has been some re-titling and compression of two
chapters into one. Some of the sections in the last three chapters have been rearranged to provide a more natural development.
Since publication of the first edition, there have appeared in the market-place
several excellent scientific textbooks on rotorcraft which cover some of the content
of Bram’s book to a far greater depth and degree of specialisation, and also other
texts which are aimed at a broad coverage but at a lower academic level. However,
the comprehensive nature of the subject matter dealt with in this volume should
continue to appeal to those helicopter engineers who require a reasonably in-depth
and authoritative text covering a wide range of topics.
Sherborne David Balmford
Kew George Done
2001
viii Preface to the second edition
Preface to the first edition
In spite of the large numbers of helicopters now flying, and the fact that helicopters
form an important part of the air strength of the world’s armed services, the study of
helicopter dynamics and aerodynamics has always occupied a lowly place in aeronautical
instruction; in fact, it is probably true to say that in most aeronautical universities in
Great Britain and the United States the helicopter is almost, if not entirely, absent
from the curriculum. This neglect is also seen in the dearth of textbooks on the
subject; it is fifteen years since the last textbook in English was published, and over
twenty years have passed since the first appearance of Gessow and Myer’s excellent
introductory text Aerodynamics of the Helicopter, which has not so far been revised.
The object of the present volume is to give an up-to-date account of the more
important branches of the dynamics and aerodynamics of the helicopter. It is hoped
that it will be useful to both undergraduate and postgraduate students of aeronautics
and also to workers in industry and the research establishments. In these days of fast
computers it is a temptation to consign a problem to arithmetical computer calculation
straightaway. While this is unavoidable in many complicated problems, such as the
calculation of induced velocity, the important physical understanding is thereby often
lost. Fortunately, most problems of the helicopter can be discussed adequately without
becoming too involved mathematically, and it is usually possible to arrive at relatively
simple formulae which are not only useful in preliminary design but which also
enable a physical interpretation of the dynamic and aerodynamic phenomena to be
obtained. The intention throughout this book, therefore, has been to try to arrive at
useful mathematical results and ‘working formulae’ and at the same time to emphasize
the physical understanding of the problem.
The first chapter summarizes some essential mechanics, mathematics, and
aerodynamics which find application in later parts of the book. Apart from some
recent research into the aerodynamics of the hovering rotor, discussed in Chapter 3,
the next six chapters are really based on the pioneer work of Glauert and Lock of the
1920s and its developments up to the 1950s. In these chapters only simple assumptions
about the dynamics and aerodynamics are made, yet they enable many important
results to be obtained for the calculation of induced velocity, rotor forces and moments,
performance, and the static and dynamic stability and control in both hovering and
forward flight.
Chapter 8 considers the complicated problem of the calculation of the induced
velocity and the rotor blade forces when the vortex wakes from the individual blades
are taken into account. Simple analytical results are possible in only a few special
cases and usually resort has to be made to digital computation. Aerofoil characteristics
under conditions of high incidence and high Mach number for steady and unsteady
conditions are also discussed.
Chapter 9 considers the motion of the flexible blade (regarded up to this point as
a rigid beam) and discusses methods of calculating the mode shapes and frequencies
for flapwise, lagwise, and torsional displacements for both hinged and hingeless
blades.
The last three chapters consider helicopter vibration and the problems of aeroelastic
coupling between the modes of vibration of the blade and between those of the blade
and fuselage.
I should like to thank two of my colleagues: Dr M. M. Freestone for kindly
reading parts of the manuscript and making many valuable suggestions, and Dr R. F.
Williams for allowing me to quote his method for the calculation of the mode shapes
and frequencies of a rotor blade.
A.R.S.B.
South Croydon, 1975
x Preface to the first edition
Acknowledgements
The authors would like to thank the persons and organisations listed below for
permission to reproduce material for some of the figures in this book. Many such
figures appeared in the first edition, and do so also in the second, the relevant
acknowledgements being to: American Helicopter Society for Figs 3.25 to 3.32, 6.40,
6.47, 6.48, and 9.16; American Institute for Aeronautics and Astronautics for Figs
6.50, 6.51, and 6.52; Her Majesty’s Stationery Office for Figs 3.6, 3.9, 4.7, 4.9, 4.10,
and 6.11; A. J. Landgrebe for Figs 2.24 and 2.33; National Aeronautics and Space
Administration for Figs 3.10, 3.11, 6.41, and 9.12; R.A. Piziali for Figs 6.24 and
6.25; Royal Aeronautical Society for Figs 4.15, 4.20, 6.19, 6.21, and 6.22; Royal
Aircraft Establishment (now Defence Evaluation and Research Agency) for Figs 3.8,
6.31, 6.32, 6.33, 6.40, 6.42, 6.46, 7.3, 7.28, 8.30, and 8.31.
For figures that have appeared for the first time in the second edition,
acknowledgements are also due to: GKN Westland Helicopters Ltd. for Figs 1.5(a),
1.5(b), 1.6(a), and 1.6(b), 6.37, 6.38, 7.28, 8.3 to 8.9, 8.12 to 8.18, 8.20 to 8.32, 9.13,
9.17 and 9.23; Stephen Fiddes for Fig. 2.37; Gordon Leishman of the University of
Maryland for Figs 6.28 and 6.30; Jean-Jacques Philippe of ONERA for Figs 6.34,
6.35, and 6.36. In a few cases, the figure is an adaptation of the original.
We are also indebted to several other friends and colleagues for contributions
provided in many other ways, ranging from discussions on content and provision of
photographic and other material, through to highlighting errors, typographical and
otherwise, arising in the first edition. These are Dave Gibbings and Ian Simons,
formerly of GKN Westland Helicopters, Gordon Leishman of the University of Maryland
and Gareth Padfield of the University of Liverpool.
Notation
A Rotor disc area
A Blade aspect ratio = R/c
A, B Constants in solution for blade torsion mode
A, B, C Moments of inertia of helicopter in roll, pitch and yaw,
or of blade in pitch, flap and lag
A, B, C, D, E, F, G Coefficients in general polynomial equation
A′, B′, C′ Moments of inertia of teetering rotor with built-in pitch
and coning
Aij, Bij ijth generalised inertia and stiffness coefficients
An nth coefficient in periodic or finite series
Aj Blade pitch jth input weighting (active vibration control)
A1, B1 Lateral and longitudinal cyclic pitch
A1, B1c, C1, D1, E1 Coefficients in longitudinal characteristic equation
A2, B2, C2, D2, E2 Coefficients in lateral characteristic equation
A B ij ij , Normalised generalised coefficients = Aij , Bij /0.5mΩ
2
R
3
a Lift curve slope of blade section
a Distance from edge of vortex sheet
a Offset of fixed pendulum point from rotor centre of rotation
(bifilar absorber)
a, b, c, d, e Square matrices, and column matrix (e) (Dynamic FEM)
a*, b*, c* Subsidiary square matrices (Dynamic FEM)
ag Acceleration of blade c.g.
aT Tailplane lift curve slope
a0 Acceleration of origin of moving frame = axi + ayj + azk
a0 Coning angle
a1, b1 Longitudinal and lateral flapping coefficients
a0, a1, a2, b1, b2 Sine and cosine coefficients in equation for Cm
Analogous to a0, a1, b1 for hingeless rotor
B Tip-loss factor (Prandtl) = Re/R
B Vector of background vibration responses
a a b 0 1 1 , ,
xiv Notation
Coefficients B1c, C1 with speed derivatives neglected
Laplace transform of B1 (cyclic pitch)
b Number of blades
b Aerofoil semi-chord
b Effective pendulum length (bifilar absorber)
C Torsional moment of inertia per unit blade length
C, S Cosine and sine multi-blade summation terms
C, F, G, H, S Coefficients in solution for normal acceleration
C(k) Theodorsen’s function
CD Drag coefficient
CH H-force coefficient = H/ρAΩ
2
R
2
CL Lift coefficient
CLT Tailplane lift coefficient
CT Thrust coefficient based on disc area = stc = T/ρAΩ
2
R
2
CM Pitching moment coefficient
CN Normal force coefficient
CL Equivalent CL = 3
0
1
∫
x
2
CLdx
Cl Rolling moment coefficient of blade
Cm Pitching moment coefficient of blade
Cmf Pitching moment coefficient of fuselage = Mf /ρsAΩ
2
R
3
Cms Pitching moment coefficient due to hinge offset
= Ms/ρsAΩ
2
R
3
Cp Pressure coefficient
C1, C2, S1, S2 Coefficients in less usual solution for normal acceleration
C1, D1, F1, G1 Integrals of blade flapping mode shape functions (first
and second moments, and powers)
C F . .
ξ β
, Flap-lag cross coupling damping coefficients
c Blade or aerofoil chord
c Viscous damping coefficient
c Offset of c.g. of oscillatory mass from pivot point (bifilar
absorber)
ccrit Critical damping coefficient = 2(k/m)
–1/2
ce Equivalent chord
cl Linkage ratio on Bell stabilising bar
cn, dn, en, fn, gn, hn, jn, kn Coefficients relating to Sn, Mn, αn, Zn at blade station n
(Myklestad)
c0, cn Downwash factors (Mangler and Squire)
c1, c2, c3, c4 Constants determined from initial conditions
D Drag of fuselage, or local blade section
D Diameter of holes in fixed arm and oscillatory mass (bifilar
absorber)
D, E, F Blade or helicopter products of inertia
′ ′ B C 1c 1 ,
B1
Notation xv
Denominator in integral for ωt
d Drag factor, where blade drag = dΩ
2
d Diameter of pin connecting fixed arm and oscillatory
mass (bifilar absorber)
d0 Fuselage drag ratio = SFP /sA
d1, d2 Bobweight arm lengths (DAVI)
E Young’s modulus
Es Modulus of rigidity or shear modulus
E1 Generalised inertia of first flapping mode
E1, E2 Wake energy contributions
e Rotor blade hinge offset, as fraction of R
eA Distance between blade centroid and elastic axis
e1, e2, e3 Orthogonal unit vectors fixed in hub
F Aerodynamic force on blade or helicopter, or general
external force vector
= Xi + Yj + Z k
Ratio of Lock number equivalents for hingeless blade
= γ2 /γ1
FR, FI Real and imaginary parts of L/Lq
Fy, Fz Lagwise and flapwise forces acting on a blade section
Lag damping coefficient
f Lateral distance of c.g. from shaft, as fraction of R
f Function affecting the k correction factor
= 0.5b(1 – x)/sin φ (Prandtl vortex sheet model)
f Bending flexibility matrix
f (n), g(n) Generalised inertias for nth flap and lag bending modes
fb Factor dependent on number of blades
fin Forcing term assumed constant for ith step in nth mode
G Centrifugal tension in blade
g Gravitational constant
H Rotor force component perpendicular to thrust axis
(positive to rear) (H-force)
H Total head pressure
H Absolute angular momentum vector = h1i + h2j + h3k
Non-dimensional quantities (air resonance)
HB, HD Flap and pitch damping coefficients (Coleman and
Stempin)
HD H-force referred to disc axes
HP H-force due to profile drag
Hi H-force due to induced drag
H0, H1, H2 Coefficients used in longitudinal response solution
Aerodynamic damping terms (Coleman and Stempin)
D
F
H H J , ,
ˆ ˆ
′ ′ H H 1 5
,
F.
ξ
xvi Notation
h Height of hub above c.g. as fraction of R
h Vertical spacing between vortex sheets (Loewy and Jones)
h Relative angular momentum vector
h′ Tail rotor height above c.g. based on wind axes
= ht cos αs – lt sin αs
hc H-force coefficient = CH/s
ht Tail rotor height above c.g., as fraction of R
h1 Height of hub above c.g. based on wind axes
= h cos αs – l sin αs
I Second moment of area of blade section
I Blade moment of inertia in both flap and lag (Southwell)
I The unit matrix
IA, IB Non-dimensional inertia factors (Coleman and Stempin)
Iy, Iz Second moments of inertia of blade section for lagwise
and flapwise bending
Iβ, Iθ Blade flap and pitch moments of inertia
i, j, k Unit vectors fixed in blade
iA, iB, iC Non-dimensional rolling, pitching and yawing inertias
of helicopter
iE Non-dimensional roll-yaw inertia product term for
helicopter
J Modal error squared integral (Duncan)
J Polar second moment of area of blade section
J Performance index (active vibration control)
J0, J1 Bessel functions of first and second kinds (Miller)
j1, j2, j3, j4 Quantities dependent on first blade flapping mode shapes
of hingeless blade
K Induced velocity gradient (Glauert)
K Stiffness between gearbox and fuselage (DAVI)
Hingeless rotor blade constant = γ2F1/2
K(x) Elliptic integral
Kθ′ Stiffness of pitch control (Coleman and Stempin)
K0 (ik), K1(ik) Bessel functions of the second kind (Theodorsen)
k Correction factor to induced velocity for number of blades
(Prandtl and Goldstein)
k Incremental correction factor to induced power relative
to that for constant induced velocity
k Frequency parameter = nc/2V (Theodorsen),
= ωb/ΩR (Miller)
k Blade structural constant = EI/mΩ
2
R
4
k Spring stiffness
kA, kB Non-dimensional pitching and flapping radii of gyration
= c(A/M)
1/2
, R(B/M)
1/2
kT Correction factor to trim due to tailplane
K
Notation xvii
ki Induced velocity ratio (axial flight) = vi/v2
ks Equivalent flap hinge stiffness for hingeless blade
kβ, kξ Pitch/flap bending and pitch/lag bending coupling
coefficients
kθ Stiffness of control system about feathering axis
Non-dimensional artificial lag damping
k1, k2 Wake constants (Landgrebe)
k1, k2 Constants associated with transient motion
kA, kB Effective pitching and rolling stiffnesses (air resonance)
L Blade sectional lift force
L Lagrangian = T – U
L, M, N Moments about i, j, k for a rigid body, or of helicopter in
roll, pitch and yaw, or of blade in pitch, flap and lag
Non-dimensional quantity (air resonance) = 2 0 a J
LA Aerodynamic torsional moment
Lb Lift due to bound circulation
Le Elastic moment in flap plane
Lq Quasi-steady lift
Lv, Lp etc. Rolling moment derivatives
L0 Steady lift, and at instantaneous incidence
l Distance forward of c.g. from shaft in terms of R
l Position vector to vortex = l1 i + l2 j + l3 k
l′ Tail rotor arm based on wind axes = lt cos αs + ht sin αs
lT Tailplane arm, as fraction of R
lb Blade inertia to mass moment ratio (ground resonance)
= I/Mbrg
ln Length of nth beam element (Myklestad)
lt Tail rotor arm, as fraction of R
lv, lp etc. Non-dimensional normalised rolling moment derivatives
etc. Non-dimensional rolling moment derivatives
l1 Distance forward of c.g. from hub based on wind axes
= l cos αs + h sin αs
M Mass (general), chassis mass (ground resonance)
M Bending moment
M Column vector of blade bending moments
Rotor figure of merit = Tvi/P
MA Aerodynamic moment about flapping hinge, or hub
MT Pitching moment due to tailplane
Mb Blade mass
Mc Blade root bending moment coefficient = M/ρbcΩ
2
R
4
Me Elastic moment in lag plane
Mf Pitching moment of fuselage
Mr Moment of rotor forces about c.g.
k.
ξ
L
M
′ ′ l l v p ,
xviii Notation
Ms Pitching moment per unit tilt of all the blades due to
hinge offset
Mu, Mq, etc. Pitching moment derivatives
M1 Unit load bending moment
M1, M2 Combined rotor/gearbox, and fuselage mass (DAVI)
m Mass, or mass per unit length
m Frequency ratio (Miller) = ω/Ω
mbob Bobweight mass (DAVI)
mu, mq, etc. Non-dimensional normalised pitching moment derivatives
′ ′ m m u q , , etc. Non-dimensional pitching moment derivatives
NA Aerodynamic lagging moment on a blade
Ni Inertia lagging moment about real or virtual hinge
Nv, Np, etc. Yawing moment derivatives
N1, P1, Q1, R1, S1, T1 Relate to B1c, C1, D1, E1
N2, P2, Q2, R2, S2, T2 Relate to A2, B2, C2, D2, E2
n Offset hinge factor = (1 – e)
3
(1 + e/3)
n Static load factor (‘g’)
n Frequency of oscillation (Theodorsen), frequency/Ω
(Miller)
n Laplace transform of n (static load factor)
nv, np, etc. Non-dimensional normalised yawing moment derivatives
etc. Non-dimensional yawing moment derivatives
P Power to drive blade, or rotor
Pi Induced power
Pi, Qi, Si, Ti, Ui, Vi Coefficients of periodic terms in expressions for lateral
hub force components
Pin Inertia force acting on chassis
Pi0 Induced power for constant induced velocity
Pp Profile drag power
Pt Tail rotor power
P0 Induced power for constant induced velocity
P1(ψ), P2(ψ) Periodic functions
p Pressure
p Roll angular velocity
p Laplace variable
p Chassis frequency, and aerofoil stall flutter frequency
Non-dimensional roll velocity = p/Ω
p(t), q(t) Forcing function components (Coleman)
pl, pu Pressure on lower and upper sides of disc plane, or aerofoil
surfaces
p1, p2 Pressure just ahead of actuator disc, and pressure in far
wake
p∞ Ambient pressure
′ ′ n n v p ,
ˆ p
Notation xix
Q Rotor torque
Volume flow through control volume sides, or flux
Q(x) Blade torsion mode shape
QP Torque due to rotor profile drag
Qi Induced rotor torque
Q1(x) First blade torsion mode shape
q Pitching velocity
q Local fluid velocity
q Induced velocity vector at a point on blade
ˆ q Non-dimensional pitching velocity = q/Ω
qc Torque coefficient = Q/ρsAΩ
2
R
3
qr Radial velocity component
qz Local fluid velocity in axial direction
qψ Tangential velocity component
R Rotor radius
R Routh’s discriminant
R Reaction forces at hinge = R1e1 + R2e2 + R3e3
RD Radius of blade drag centre from hub
Reff Effective blade radius (Prandtl)
R0 Far wake radius
R1, R2 Control surface and far wake radii
r Distance of blade element from hinge or axis of rotation
r Radial wake coordinate
r Position vector = xi + y j + z k
Tip vortex radial coordinate (Landgrebe)
rg Position vector of blade or system c.g. = xgi + ygj + zgk
r1 Radial position of vortex filament on blade
S Centrifugal force of blade
S Shear force
S(x) Flap bending mode shape
SB Projected side area of fuselage
SFP Fuselage equivalent flat plate area
ST Tailplane area
S1(x) First flap bending mode shape
s Rotor solidity = bc/πR
s Half width of vortex sheet, equivalent to half wing span
s Vortex axis vector = s1i + s2j + s3k
sp Spacing of vortex sheets
st Tail rotor solidity
Normalised tail rotor solidity = stAt (ΩR)t /sAΩR
T Rotor thrust
T Periodic time
T Kinetic energy
T Moment of resultant external forces about O
Q
r
st
xx Notation
T Rotor/fuselage transfer matrix (active vibration control)
T(x) Lag bending mode shape
T1(x) First lag bending mode shape
TD Thrust referred to disc axes
Td Time to double amplitude
Tf Following time (inversely proportional to viscous damping)
– (Bell bar)
Th Time to half amplitude
Tt Tail rotor thrust
T0 Thrust for constant vi
t Time
Time non-dimensionalising factor = W/gρsAΩR
tc Thrust coefficient based on total blade area =T/ρsAΩ
2
R
2
tcD Thrust coefficient referred to disc axes
U Velocity of wake normal to axis
U Strain energy
U, V, W Initial flight velocity components along x, y, z axes
UB Strain energy due to bending
UG Strain energy due to centrifugal tension
UP Component of air velocity relative to blade element
perpendicular to plane of no-feathering
UT Component of air velocity relative to blade element
tangential to plane of no-feathering
U0, U1, U2 Coefficients used in longitudinal response solution
u, v, w Perturbational velocities
u, v Coleman coordinates
u′, v′ Wake velocity components
Laplace transforms of u, v, w (perturbational velocities)
Non-dimensional perturbational velocities = u/ΩR, v /ΩR,
w/ΩR
uFn, vFn Unit force constants relating to nth blade element
(Myklestad)
uMn, vMn Unit moment constants relating to nth blade element
(Myklestad)
V Forward velocity of helicopter or relative velocity far
upstream of rotor
V Forward velocity vector of helicopter
Forward speed normalised on thrust velocity = V/v0
Forward speed normalised on tip speed = V/ΩR
V′ Total velocity at the rotor
VC Rate of climb in axial flight, or axial velocity
Climb speed normalised on induced velocity (actuator
disc) = VC/vi
Tail volume ratio = STlT/sA
ˆ
t
u w , , v
ˆ ˆ ˆ u w , , v
V
ˆ
V
VC
VT
Notation xxi
Vdes Descent velocity
v Absolute velocity vector
vi General induced velocity at rotor
viT Downwash at blade tip (with linear distribution)
vi0 Mean induced velocity
vrel Relative velocity vector
Induced velocity normalised on thrust velocity = vi/v0
v0 Mean induced velocity in hover (thrust velocity)
v0 Velocity of origin of moving frame
v2 Slipstream velocity in far wake
v3/4 Induced velocity component at 3/4 chord point
W Helicopter weight
W Total relative flow velocity at blade section
W Work done
W Torque on blade element
W Total velocity vector at blade section
Wi Vibration measurement weighting (active vibration
control)
W0, W1, W2 Coefficients used in longitudinal response solution
w Velocity of wake sheets near rotor
w Wake velocity
w Induced velocity normal to disc, or aerofoil
wD Disc loading = T/A
wP, wQ Induced velocity components normal to rotor at P, Q
wb Downward component of induced velocity due to bound
vortices
wc Weight coefficient = W/ρsAΩ
2
R
2
ws Induced velocity component for shed part of wake
wt Downward component of induced velocity due to trailing
vortex
X, Y, Z General or aerodynamic force components
X Vector of higher harmonic control (HHC) inputs
Mean hub force components
Xu, Xw, etc. X force derivatives
x, y, z Position coordinates (dimensional, or non-dimensionalised
on R)
Distance of datum point on aerofoil from mid-chord
xk General variable measured with respect to rotating kth
blade
xkg, ykg Coordinates of the kth blade relative to centre of hub
(ground resonance)
xrg, yrg Coordinates of rotor c.g. (ground resonance)
xst Static deflection of single degree of freedom system
= F0/k
vi
x
X Y Z , ,
xxii Notation
xu, xw, etc. Non-dimensional X force derivatives
x1 Non-dimensional position of vortex filament on blade
= r1/R
Y, Z Displacement of point on blade relative to axes rotating
with blade
Y Vector of measurement vibration components (active
vibration control)
Yf Fuselage side-force
Yv, Yp, etc. Y force derivatives
Y0, Y1 Bessel functions of first and second kinds (Miller)
yv, yp, etc. Non-dimensional Y force derivatives
Z Blade bending deflection vector
ZE Elastic deflection vector of blade bending
ZR Rigid body rotation deflection vector (about flapping
hinge)
Zu, Zw, etc. Z force derivatives
z Distance along rotor axis
Tip vortex axial coordinate (Langrebe)
zu, zw, etc. Non-dimensional Z force derivatives
z0 Wake coordinate
α Incidence of blade section
α Blade torsional stiffness constant = ω0(CR/EsJ)
1/2
Equivalent lag damping coefficient (Ormiston and Hodges)
αD Disc incidence
αT Tailplane incidence
αT0 No lift setting of tailplane (with respect to fuselage)
αi Downwash angle relative to blade
αi Stiffening effect due to rotation = ( – )/
i
2
nr
2 2
ω ω Ω
αi Spanwise slope at RH end of ith element (Myklestad)
αnf Incidence with respect to plane of no-feathering
αs Rotor hub incidence (i.e. shaft tilt)
α0 Incidence in the absence of induced velocity
α0, α1 Coefficients in polynomial expression for α
α1, α2 Amplitudes (Floquet)
α1, α2, α3 Lag hinge projected angles
β Blade flapping angle, at hinge
Analogous to β for hingeless rotor blade
βs Blade flapping, relative to shaft
βss Side-slip angle
β0 Built-in coning angle
χ Wake angle
Flap bending frequency difference term (air resonance)
= λ1
2
– 1
zT
α
β
χ
Notation xxiii
χi(t) ith generalised coordinate for lagwise bending
ξ Blade lagging angle
ξ Distance of vortex element from centre of aerofoil, based
on semi-chord b
ξ, ζ Aft and downwards position coordinates based on on R
and aligned with mean downwash angle (for defining
tailplane position)
ξk Lag angle of kth blade (ground resonance)
∆ Stability quartic in Laplace variable p
∆ Function of κβ, κξ
δ Profile drag coefficient = δ0 + δ1α + δ2α
2
δ Lateral deflection at a point on a beam
δ, δc Blade lag and chassis damping coefficients (ground
resonance)
δ1, δ2, δ3 Flapping hinge projected angles
ε Blade hinge offset factor = MbexgR
2
/B = 3e/2(1 – e)
ε Downwash angle at tailplane
ε Phase angle
ε0 Mean downwash angle at rotor = vi0 /V
ε(x) Modal error function (Duncan)
φ Shaft angle to vertical (roll of fuselage)
φ Velocity potential, or real part of velocity potential
φ Inflow angle at blade element = tan
–1
(UP/UT)
φ Blade azimuth angle when vortex was shed
φ (x, z) Potential for plane steady flow past a cylinder (Sears)
φi(t) ith generalised coordinate for flapwise bending
Γ Circulation, vortex strength
Γ Blade rotating lag frequency in absence of Coriolis force
coupling (ground resonance)
Γn Amplitude of bound circulation (Miller)
Γnc, Γns In and out of phase components of Γn
Γq Quasi-static circulation
Γ1 Function of derivatives = – mq + µmB1/zB1
γ Lock’s inertia number = ρacR
4
/B
γ Vorticity (Theodorsen)
γ Angular displacement of pendulum arm (bifilar absorber)
γ (x) Assumed general blade bending mode shape (Lagrange)
γ1, γ2 Lock number equivalents for flexible blade
η Contraction ratio of slipstream in hover
η Transformed radial position coordinate (Mangler and
Squire) = (1–x
2
)
1/2
xxiv Notation
η Non-dimensional chordwise position (thin aerofoil theory)
η, ζ Coleman coordinates (ground resonance)
κ, κc General blade lag, and chassis frequencies in terms of Ω
κβ, κξ Functions of κβH, κβB and κξH, κξB
κβH, κξH Flap and lag stiffnesses of ‘hub springs’
κβB, κξB Remainder of above stiffnesses outboard of feathering
hinge
κ1 First blade uncoupled natural rotating lag frequency in
terms of Ω
Λ Local wake helix angle
Wake constant (Landgrebe)
Λb Bobweight arm length ratio = d1/d2 (DAVI)
Λ0 Far wake helix angle = w/ΩR0
λ Mean inflow ratio relative to plane of no-feathering
= sin αnf – λi
λ Rotating flap bending frequency in terms of Ω
λ, λn General and nth eigenvalue in characteristic equation
λ′ General inflow ratio (function of ψ, r)
= (V sin αnf – vi)/ΩR
λD Mean inflow ratio relative to disc plane = sin αD – λi
λc Climb inflow ratio = Vc/ΩR (axial flight), ≈ sin τc (forward
flight)
λi vi0/ΩR, or vi/ΩR for hovering flight
λiT viT/ΩR
λre, λim Real and imaginary parts of eigenvalue λ
λ1 First blade uncoupled natural rotating flap frequency in
terms of Ω
µ Constant determining natural undamped frequency of a
non-rotating beam, from standard published results
= ( /EI) nr
2 1/4
mω
µ Mass ratio (ground resonance) = 0.5bMb/(M + bMb)
µ, µD Advance ratios = ˆ
V cos αnf,
ˆ
V cos αD
Magnification factor = x0/xst
µ* Relative density parameter = W/gρsAR
µb Bobweight mass ratio = mbob/M1 (DAVI)
ν Helicopter pitching frequency ratio in terms of rotor
revolutions
ν Far wake velocity ratio
ν Factor depending on disc tilt (Mangler and Squire)
= (1 – sin αD)/(1 + sin αD)
ν Lag bending frequency ratio
Air resonance factor = γE1/2
Λ
µ
ν
Notation xxv
˜
ω
ˆ
ν Incremental frequency term (Floquet) = γ /16
ν1 First blade uncoupled natural rotating torsional frequency
in terms of Ω
ν1, ν2 Exponent constants (Floquet)
θ Blade pitch or feathering angle
θ Fuselage pitch attitude (shaft angle to vertical)
θ Laplace transform of θ (fuselage pitch)
θbar Angular displacement of Bell stabilising bar
θn Amplitude of blade pitch variation at circular frequency
n
θt Tail-rotor collective pitch
θ0 Collective pitch angle
θ1 Blade twist (washout)
ρ Ambient air density, or material density
ρ Component of inflow angle = tan
–1
(vi/W)
ρe Degree of elastic coupling = κβ/κβB = κξ/κξB
ρm Fuselage mass ratio = M2/M1 (DAVI)
σ Solidity based on local radius = bc/π r
σg Distance of c.g. of blade elemental strip behind flexural
axis in terms of c
σ1, σ–1 Functions of lag frequency (ground resonance)
τ Period of one rotor revolution
τ Non-dimensional aerodynamic unit of time = t t /ˆ
τc Climb angle
τdes Angle of descent
Ω Rotor or blade angular velocity
Ω Angular velocity vector = ω1i + ω2j + ω3k
ω Total wake swirl velocity
ω Circular frequency
Normalised excitation frequency
ωb Component of ω due to bound circulation
ωn Natural frequency
ωnr Natural frequency of non-rotating blade
ωt Component of ω due to trailing vortices
ωβ Uncoupled rotating flap natural frequency
ωξ Uncoupled rotating lag natural frequency
ωθ Rotating torsional natural frequency
ω0 Non-rotating torsional natural frequency
ψ Azimuthal angular position of blade, or general angular
coordinate
xxvi Notation
ψ Imaginary part of velocity potential
ψ Yaw displacement of helicopter (from steady state)
ψn Angle between adjacent blades
ψw Wake azimuth relative to blade
ζ Non-dimensional damping factor = c/ccrit
ζmean Weighted mean damping (stall flutter)
ζi(t) ith generalised coordinate for blade torsion
Suffices
The following suffices refer to:
A Aerodynamic
A, B, D, E Inertia moments and products
D Rotor disc (tip-path plane)
D Drag
L Lift
M Moment
N Normal force
P Perpendicular
P Profile drag
T Tip of blade
T Thrust
T Tailplane
T Tangential
b Bound vorticity
c Climbing
c Coefficient
c Chassis
e Effective
f Fuselage
g Blade c.g., or c.g. of system of particles
h On matrices indicates row is used to correspond to hinge
i Induced
l, u Lower and upper surfaces
kg C.g. of kth blade relative to hub (ground resonance)
nf No feathering
nr Non-rotating
p Pressure
r Radial direction
r Root of blade
r Rotor
rg Rotor c.g. (ground resonance)
s Shaft
Notation xxvii
s Due to centrifugal force at blade hinge
s Setting angle of tailplane
s Shed part of wake
ss Sideslip
t Trailing vortices
t Tail rotor
w Wake
z In z direction
β Flap
ξ Lag
θ Pitch
ψ In tangential direction
0 Generally modulus, or amplitude of
∞ At infinity
1
Basic mechanics of rotor systems
and helicopter flight
1.1 Introduction
In this chapter we shall discuss some of the fundamental mechanisms of rotor systems
from both the mechanical system and the kinematic motion and dynamics points of
view. A brief description of the rotor hinge system leads on to a study of the blade
motion and rotor forces and moments. Only the simplest aerodynamic assumptions
are made in order to obtain an elementary appreciation of the rotor characteristics. It
is fortunate that, in spite of the considerable flexibility of rotor blades, much of
helicopter theory can be effected by regarding the blade as rigid, with obvious
simplifications in the analysis. Analyses that involve more detail in both aerodynamics
and blade properties are made in later chapters. The simple rotor system analysis in
this chapter allows finally the whole helicopter trimmed flight equilibrium equations
to be derived.
1.2 The rotor hinge system
The development of the autogyro and, later, the helicopter owes much to the introduction
of hinges about which the blades are free to move. The use of hinges was first
suggested by Renard in 1904 as a means of relieving the large bending stresses at the
blade root and of eliminating the rolling moment which arises in forward flight, but
the first successful practical application was due to Cierva in the early 1920s. The
most important of these hinges is the flapping hinge which allows the blade to flap,
i.e. to move in a plane containing the blade and the shaft. Now a blade which is free
to flap experiences large Coriolis moments in the plane of rotation and a further
hinge – called the drag or lag hinge – is provided to relieve these moments. Lastly,
the blade can be feathered about a third axis, usually parallel to the blade span, to
enable the blade pitch angle to be changed. A diagrammatic view of a typical hinge
arrangement is shown in Fig. 1.1.
2 Bramwell’s Helicopter Dynamics
In this figure, the flapping and lag hinges intersect, i.e. the hinges are at the same
distance from the rotor shaft, but this need not necessarily be the case in a particular
design. Neither are the hinges always absolutely mutually perpendicular.
Consider the arrangement shown in Fig. 1.2. Let OX be taken parallel to the bladespan axis and OZ perpendicular to the plane of the rotor hub. Let OP represent either
the flapping hinge axis or the lag hinge axis. The flapping hinge is referred to as the
δ-hinge and the lag hinge as the α-hinge. We then define:
– the angle between OZ and the projection of OP onto the plane OYZ as δ1 or α1,
– the angle between OZ and the projection of OP onto the plane OXZ as δ2 or α2,
– the angle between OY and the projection of OP onto the plane OXY as δ3 or α3.
These are the definitions in common use in industry. The most important angles in
practice are α2, which leads to pitch resulting from lagging of the blade, and δ3,
which couples pitch and flap, as follows.
Lag hinge
Flapping hinge
Pitch change
(or feathering) hinge
Flapping
Lagging
Feathering
Fig. 1.1 Typical hinge arrangement
α1, δ1
Z
α2, δ2
α3, δ3
Y
O
P
Fig. 1.2 Blade hinge angles
X
Basic mechanics of rotor systems and helicopter flight 3
Referring to Fig. 1.3, when δ3 is positive, positive blade flapping causes the blade
pitch angle to be reduced. It will also be appreciated that, if the drag hinge is mounted
outboard of the flapping hinge, movement about the lag hinge produces a δ3 effect.
If the blade moves through angle ξ0 and flaps through angle β relative to the hub
plane, the change of pitch angle ∆θ due to flapping is found to be
∆θ = –tan β tan(δ3 – ξ0)
or for small angles
∆θ = –β tan(δ3 – ξ0)
so ∆θ is proportional to β.
The dynamic coupling of blade motions will be dealt with in more detail in
Chapter 9.
The blades of two-bladed rotors are usually mounted as a single unit on a ‘seesaw’ or ‘teetering’ hinge, Fig. 1.4(a). No lag hinges are fitted, but the Coriolis root
δ3
Fig. 1.3 The δ3-hinge
(a)
β0
Flapping
(b)
Fig. 1.4 (a) Teetering or see-saw rotor. (b) Underslung rotor, showing radial components of velocity on
upwards flapping blade
4 Bramwell’s Helicopter Dynamics
bending moments may be greatly reduced by ‘underslinging’ the rotor Fig. 1.4(b). It
can be seen from the figure that, when the rotor flaps, the radial components of
velocity of points on the upwards flapping blade below the hinge line are positive
while those above are negative. Thus the corresponding Coriolis forces are of opposite
sign and, by proper choice of the hinge height, the moment at the blade root can be
reduced to second order magnitude. This assumes that a certain amount of pre-cone
or blade flap, β0, is initially built in.
Although, as stated earlier, the adoption of blade hinges was an important step in
the evolution of the helicopter, several problems are posed by the presence of hinges
and the dampers which are also fitted to restrain the lagging motion. Not only do the
bearings operate under very high centrifugal loads, requiring frequent servicing and
maintenance, but when the number of blades is large the hub becomes very bulky and
may contribute a large proportion of the total drag. Figure 1.5(a) shows a diagrammatic
view of the Westland Wessex hub, on which, as may be observed, the flapping and
lag hinges intersect. Figure 1.5(b) is a photograph of the same rotor hub, showing
also the swash plate mechanism that enables the cyclic and collective pitch control
(discussed in section 1.7).
Fig. 1.5 (a) Diagrammatic view of Westland Wessex hub
Lag hinge
Feathering
bearing
assembly
Flapping hinge
Basic mechanics of rotor systems and helicopter flight 5
More recently, improvements in blade design and construction enabled rotors to
be developed which dispensed with the flapping and lagging hinges. These ‘hingeless’,
or less accurately termed ‘semi-rigid’, rotors have blades which are connected to the
shaft in cantilever fashion but which have flexible elements near to the root, allowing
the flapping and lagging freedoms. Such a design is shown in Fig. 1.6(a) which is that
of the Wesland Lynx helicopter. In this case, the flexible element is close in to the
rotor shaft, with the feathering hinge between it and the flexible lag element, which
is the furthest outboard. The diagram also indicates the attached lag dampers mentioned
previously, and discussed in relation to ground resonance in Chapter 9, and a different,
but less common, mechanism for changing the cyclic and collective pitch on the
blades. A photograph of the same hub design, but with five blades, is shown in Fig.
1.6(b). Rotor hub designs for current medium to large helicopters commonly use a
high proportion of composite material for the main structural elements, with elastomeric
elements providing freedoms where only low stiffnesses are required (e.g. to allow
blade feathering).
We now derive the equations of flapping, lagging, and feathering motion of the
hinged blade – but assuming it to be rigid, as mentioned in the introduction. The
motion of the hingeless blade will be considered in Chapter 7. Fortunately, except for
the lagging motion, the equations can be derived with sufficient accuracy by treating
each degree of freedom separately, e.g. in considering flapping motion it can be
assumed that lagging and feathering do not occur.
Fig. 1.5 (b) Photograph of Westland Wessex hub
6 Bramwell’s Helicopter Dynamics
Flexible flap element
Feathering bearing assembly
Flexible lag element
Pitch control rod
Rotor shaft
Control spindle
Spider
Lag damper
Fig. 1.6 (a) Diagrammatic view of Westland Lynx hub
Fig. 1.6 (b) Photograph of Westland Lynx five-bladed hub
Basic mechanics of rotor systems and helicopter flight 7
1.3 The flapping equation
Consider a single blade as shown in Fig. 1.7 and let the flapping hinge be mounted
a distance eR from the axis of rotation. The shaft rotates with constant angular
velocity Ω and the blade flaps with angular velocity
.
β . Take axes fixed in the blade,
parallel to the principal axes, origin at the hinge, with the i axis along the blade span,
the j axis perpendicular to the span and parallel to the plane of rotation, and the k axis
completing the right-hand set. To a very good approximation the blade can be treated
as a lamina.
Then, if A is the moment of inertia about i, and B the moment of inertia about j,
the moment of inertia C about k is equal to A + B. The angular velocity components
ω1, ω2, ω3 about these axes are
ω1 = Ω sin β, ω2 = –
.
β, ω3 = Ω cos β
The acceleration, a0, of the origin is clearly Ω
2
eR and perpendicular to the shaft.
Along the principal axes the components are
{–Ω
2
eR cos β, 0, Ω
2
eR sin β}
The position vector of the blade c.g. is rg = xgRi, so that the components of rg × a0
are
{0, exg Ω
2
R
2
sin β, 0}
The flapping motion takes place about the j axis, so putting the above values in the
second of the ‘extended’ Euler’s equations derived in the appendix (eqn A.1.15), and
using A + B = C, gives
B
..
β + Ω
2
(B cos β + MbexgR
2
) sin β = MA (1.1)
β
Ω
k
eR
j
O
Fig. 1.7 Single flapping blade
i
R
8 Bramwell’s Helicopter Dynamics
where MA = – M is the aerodynamic moment in the sense of positive flapping and Mb
is the blade mass. For small flapping angles eqn 1.1 can be written
..
β + Ω
2
(1 + ε) β = MA/B (1.2)
where ε = MbexgR
2
/B.
If the blade has uniform mass distribution, it can easily be verified that
ε = 3e/2 (1 – e). A typical value of e is 0.04, giving ε as approximately 0.06.
The flapping equation (eqn 1.1) could also have been derived by considering an
element of the blade of mass dm, and at a distance r from the hinge, to be under the
action of a centrifugal force (eR + r cos β) Ω
2
dm directed outwards and perpendicular
to the shaft. The integral of the moment of all such forces along the blade is found to
be the second term of eqn 1.1, i.e. Ω
2
(B cos β + MbexgR
2
) sin β. Regarding it as an
external moment like MA, this centrifugal moment (for small β) acts like a torsional
spring tending to return the blade to the plane of rotation.
The other two extended Euler’s equations (eqns A.1.17 and A.1.19) give
L = 0 and N = –2BΩ
.
β sin β
These are the moments about the feathering and lag axes, respectively, which are
required to constrain the blade to the flapping plane, or, in other words, –L and –N are
the couples which the blade exerts on the hub due to flapping only. It can be seen that
flapping produces no feathering inertia moment, but the in-plane moment 2BΩ
.
β sin β
is often so large that it is usually relieved by the provision of a lag hinge or equivalent
flexibility, as mentioned in section 1.2. This moment is the moment of the Coriolis
inertia forces acting in the in-plane direction.
More generally, if the rotor hub is pitching with angular velocity q, Fig. 1.8, the
angular velocity components of the blade are
{q sin ψ cos β + Ω sin β, q cos ψ –
.
β , –q sin ψ sin β + Ω cos β}
Ω
k
q
eR
j
i
β
ψ
p
Fig. 1.8 Blade influenced by rotor hub pitching velocity q and rolling velocity p
Basic mechanics of rotor systems and helicopter flight 9
where ψ is the azimuth angle of the blade, defined as the angle between the blade
span and the rear centre line of the helicopter. The absolute accelerations of the hinge
point O are the centripetal acceleration Ω
2
eR acting radially inwards and eR( . q cos ψ
– 2qΩ sin ψ) acting normal to the plane of the rotor hub.
Inserting these values into eqn A.1.15 and neglecting q
2
, which is usually very
small compared with Ω
2
, we finally obtain after some manipulation
..
β + Ω
2
(1 + ε)β = MA/B – 2Ωq(1 + ε) sin ψ + . q(1 + ε) cos ψ (1.3)
The second term on the right is the gyroscopic inertia moment due to pitching
velocity, and the third term is due to the pitching acceleration.
We now find that the feathering moment L is
L = A(2Ωq cos ψ + . q sin ψ)
which means that the pitching motion produces a moment tending to twist the blade.
The moment about the lag axis is hardly affected by the pitching motion, so that N
remains as before.
When the rotor hub is rolling with angular velocity p, Fig. 1.8, the equivalent
equation to 1.3 may be derived in like manner, and in this case the extra terms on the
right-hand side can be shown to be (1 + ε) (2Ωp cos ψ + . p sin ψ).
1.4 The equation of lagging motion
We assume the flapping angle to be zero and that the blade moves forward on the lag
hinge through angle ξ (Fig. 1.9). The angular velocity of the blade is (Ω +
.
ξ )k and
a0 = – Ω
2
eR(cos ξi – sin ξ j). Then the third of Euler’s extended equations gives
C
..
ξ + MexgΩ
2
R
2
sin ξ = N (1.4)
or, for small ξ,
..
ξ + Ω
2
εξ = N/C (1.5)
where ε, in this case, is MbexgR
2
/C, eR being the drag-hinge offset distance.
It can easily be verified that if flapping motion is also included, the only important
term arising is the moment 2BΩβ
.
β calculated in the previous section. With a lag
Ω
eR
ξ
Ω +
.
ξ
Fig. 1.9 Blade lagging
10 Bramwell’s Helicopter Dynamics
hinge fitted, this moment can be regarded as an inertia moment and considered as
part of N. Then, if NA is taken as the aerodynamic lagging moment, together with any
artificial damping which may be, and usually is, added, eqn 1.5 can be written finally
as
..
ξ + Ω
2
εξ – 2Ωβ
.
β = NA/C (since B ≈ C) (1.6)
The lagging motion produces no moment about the feathering axis, but the
instantaneous angular velocity Ω +
.
ξ will affect the centrifugal and aerodynamic
flapping moments and may have to be taken into account when considering coupled
motion (see section 9.7).
1.5 Feathering motion
It is assumed that the flapping and lagging angles are zero and that the blade feathers
through angle θ (Fig. 1.10). The angular velocity components about i, j, k are
.
θ , Ω sin θ, Ω cos θ. The first of Euler’s extended equations gives
A .
θ + AΩ
2
sin θ cos θ = L (1.7)
and, for the small feathering angles which normally occur, we can write
..
θ + Ω
2
θ = L/A (1.8)
The second and third of Euler’s equations show that the feathering motion produces
no flapping moment but a lagging moment of –2A .
θ Ω sin θ. This latter moment is
extremely small compared with the flapping Coriolis moment and can be neglected.
1.6 Flapping motion in hovering flight
The equation of blade flapping (1.2) is
d
2
β/dt
2
+ Ω
2
(1 + ε)β = MA/B
It is convenient to change the independent variable from time to blade azimuth
angle by means of ψ = Ωt. Then since
d/dt = Ωd/dψ and d
2
/dt
2
= Ω
2
d
2
/dψ
2
k
j
θ i
·
θ
Fig. 1.10 Blade feathering
Basic mechanics of rotor systems and helicopter flight 11
eqn 1.2 becomes
d
2
β/dψ
2
+ (1 + ε)β = MA/BΩ
2
(1.9)
This equation is valid for any case of steady rectilinear flight including hovering.
The problem is to express MA as a function of ψ and then to solve the equation. We
now consider some simple but important examples in hovering flight.
1.6.1 Disturbed flapping motion at constant blade pitch angle
We suppose that the blades are set at a constant blade pitch angle relative to the shaft
and that the rotor is rotating steadily with angular velocity Ω. Since we are interested
only in the character of the disturbed motion, the aerodynamic moment corresponding
to the constant pitch angle will be ignored and attention will be concentrated on the
aerodynamic moments arising from disturbed flapping motion.
Now, when a blade flaps with angular velocity
.
β , there is a relative downwash of
velocity r
.
β at a point on the blade distance r from the hinge. Assuming cos β = 1, the
chordwise component of wind velocity is Ω(r + eR), so that the local change of
incidence ∆α due to flapping is
∆α
β β ψ
=
–
( + )
=
– d /d
+
r
r eR
x
x e
.
Ω
where x = r/R.
Assuming a constant lift slope a for the blade section, the lift on an element of
blade is
dL = –
1
2 ρacΩ
2
R
3
(x + e)x(dβ/dψ)dx
The moment of this lift about the flapping hinge is rdL and the total aerodynamic
moment, assuming the blade chord c to be constant, is
M r L ac R x x e x
R e
A
0
(1– )
1
2
2 4
0
(1– e)
2
= d = – ( + )(d /d )d
∫ ∫
Ω ρ β ψ
giving
MA/BΩ
2
= – (γ /8)(1 – e)
3
(1 + e/3)dβ/dψ
where γ = ρacR
4
/B is called Lock’s inertia number.
Writing n for (1 – e)
3
(1 + e/3), the flapping equation becomes
d
2
β/dψ
2
+ (nγ/8)dβ/dψ + (1 + ε)β = 0 (1.10)
Equation 1.10 is the equation of damped harmonic motion with a natural undamped
frequency Ω√(1 + ε). If ε is zero (no flapping hinge offset), the natural undamped
frequency is exactly equal to the shaft frequency. Normally ε is about 0.06, giving an
undamped flapping frequency about 3 per cent higher than the shaft frequency.
Taking a typical value of γ of 6 gives a value for nγ /8 of about 0.7. This means that
the damping of the motion is about 35 per cent of critical, or that the time-constant
12 Bramwell’s Helicopter Dynamics
in terms of the azimuth angle is about 90° or
1
4
of a revolution. Thus, the flapping
motion is very heavily damped. It has already been remarked that the centrifugal
moment acts like a spring, and we now see that flapping produces an aerodynamic
moment proportional to flapping rate, i.e. in hovering flight the blade behaves like a
mass–spring–dashpot system. In forward flight the damping is more complicated and
includes a periodic component, but the notion of the blade as a second order system
is often a useful one in a physical interpretation of blade motion.
1.6.2 Flapping motion due to cyclic feathering
Suppose that, in addition to a constant (collective) pitch angle θ0, the blade pitch is
veried in a sinusoidal manner relative to the hub plane. The blade pitch θ can then be
expressed as
θ = θ0 – A1 cos ψ – B1 sin ψ (1.11)
To simplify the calculations we will take e = 0, since the small values of flapping
hinge offset normally employed have little effect on the flapping motion.
In calculating the flapping moment MA, the induced velocity, or rotor downwash,
to be discussed in Chapter 2, will be ignored. By a similar analysis to that above, the
flapping moment is easily found to be given by
MA/BΩ
2
= γ (θ0 – A1 cos ψ – B1 sin ψ)/8
Substituting in eqn 1.9 leads to the steady-state solution
β = γθ0/8 – A1 sin ψ + B1 cos ψ (1.12)
The term γθ0/8 represents a constant flapping angle and corresponds to a motion
in which the blade traces out a shallow cone, and for this reason the angle is called
the coning angle. If the induced velocity had been included, the coning angle would
have been reduced somewhat. For our present purpose the exact calculation of the
coning angle is unimportant. The terms –A1 sin ψ + B1 cos ψ represent a tilt of the
axis of the cone away from the shaft axis. Since ψ is usually measured from the
rearmost position of the blade, i.e. along the axis of the rear fuselage, a positive value
of B1 denotes a forward (nose down) tilt of the cone, Fig. 1.11, while a positive value
of A1 denotes a sideways component of tilt in the direction of ψ = 90°. The blade tips
trace out the ‘base’ of the cone, which is often referred to as the tip path plane or as
the rotor disc, Fig. 1.11.
In steady flight the blade motion must be periodic and is therefore capable of
being expressed in a Fourier series as
β = a0 – a1 cos ψ – b1 sin ψ – a2 cos 2ψ – b2 sin 2ψ – … (1.13)
For the case in question,
a0 = γθ0/8, a1 = – B1, b1 = A1
a2 = b2 = … etc. = 0
When the flight condition is steady, eqn 1.9 can always be solved by assuming the
Basic mechanics of rotor systems and helicopter flight 13
form of eqn 1.13, substituting in the flapping equation, and equating coefficients of
the trigonometric terms. This is a method we shall be forced to adopt when the
flapping equation contains periodic coefficients, as will be the case in forward flight.
In terms of eqn 1.13, a0 represents the coning angle and a1 and b1 represent
respectively, a backward and sideways tilt of the rotor disc, the sideways tilt being in
the direction of ψ = 90°. The higher harmonics a2, b2, a3,…, etc., which will have
non-zero values in forward flight, can be interpreted as distortions or a ‘crinkling’ of
the rotor cone. But although these harmonics can be calculated, the blade displacements
they represent are only of the same order as those of the elastic deflections which, so
far, have been neglected. Thus, it is inconsistent to calculate the higher harmonics of
the rigid blade mode of motion without including the other deflections of the blade.
Stewart
1
has shown that the higher harmonics are usually about one tenth of the
values of those of the next order above.
Comparison of eqns 1.11 and 1.12 shows that the amplitude of the periodic flapping
is precisely the same as the applied cyclic feathering and that the flapping lags the
cyclic pitch by 90°. The phase angle is exactly what we might have expected, since
the aerodynamic flapping moment forces the blade at its undamped natural frequency
and, as is well known, the phase angle of a second order dynamic system at resonance
is 90° whatever the damping. Further, the fact that the amplitude of flapping is
exactly the same as the applied feathering has a simple physical explanation. Suppose
that initially no collective or cyclic pitch were applied; the blades would then trace
out a plane perpendicular to the rotor shaft. If cyclic pitch were then applied, and the
blades remained in the initial plane of rotation, they would experience a cyclic
variation of incidence and, hence, of aerodynamic moment. The moment would
cause the blades to flap and, since, as we have found, blade flapping motion is stable,
the blades must seek a new plane of rotation such that the flapping moment vanishes.
This is clearly a plane in which there is no cyclic feathering and it follows from
Fig. 1.12 that this plane makes the same angle to the shaft as the amplitude of the
cyclic pitch variation. It is also obvious that the effect of applying cyclic pitch is
precisely the same as if cyclic pitch had been absent but the shaft had been tilted
through the same angle. Tilting the rotor shaft or, more precisely, the rotor hub
plane, is the predominant method of controlling the rotor of an autogyro. Tilting
the shaft of a helicopter is impossible if it is driven by a fuselage mounted engine,
and the rotor must be controlled by cyclic feathering.
Tip path plane
a0
B1
Fig. 1.11 Interpretation of flapping and feathering coefficients
14 Bramwell’s Helicopter Dynamics
The above discussion illustrates the phenomenon of the so-called ‘equivalence of
feathering and flapping’; the interpretation is a purely geometric one. If flapping and
feathering are purely sinusoidal, the amplitude of either depends entirely upon the
axis to which it is referred. In Fig. 1.12, aa′ is the shaft axis, bb′ is the axis perpendicular
to the blade chord, cc′ the axis perpendicular to the tip path plane. If Fig. 1.12 shows
the blade at its greatest pitch angle, bb′ is clearly the axis relative to which the cyclic
feathering vanishes and is called the no-feathering axis. Similarly cc′ is the axis of no
flapping.
Let a1s be the angle between the shaft and the tip path plane and B1 the angle
between the shaft and the no-feathering axis. Viewed from the no-feathering axis the
cyclic feathering is, by definition, zero but the angle of the tip path plane is a1s – B1.
On the other hand, viewed from the tip path plane, the flapping is zero but the
feathering amplitude is B1 – a1s. Thus feathering and flapping can be interchanged
and either may be made to vanish by the appropriate choice of axis. The ability to
select an axis relative to which either the feathering or flapping vanishes is useful in
simplifying the analysis of rotor blade motion and for interpreting rotor behaviour.
The coning angle a0 and collective pitch θ0 play no part in the principle of equivalence.
Strictly speaking, the principle of equivalence fails if the flapping hinges are
offset, because the angle of the tip path plane will then no longer be the same as the
amplitude of blade flapping, as a sketch will easily show. However, the size of the
offset is usually so small that the equivalence idea can be generally applied. Offset
hinges, as will be seen later, make an important contribution to the moments on the
helicopter.
Another important feature of blade flapping motion can be deduced from the
flapping equation. Assuming ε to be negligible, the flapping equation (eqn 1.9) can
be written
d
2
β/dψ
2
+ β = MA/BΩ
2
in which β is defined relative to a plane perpendicular to the shaft axis.
Now, assuming that higher harmonics can be neglected, steady blade flapping can
be expressed in the form
β = a0 – a1 cos ψ – b1 sin ψ
a1s B1
ab c
b′ c′
a′
Tip path plane
ψ = 90°
θ = B1 sin ψ
ab
b′
a′
–B1
ψ = 270°
Fig. 1.12 Equivalence of flapping and feathering
Basic mechanics of rotor systems and helicopter flight 15
and on substitution for β into the flapping equation above
MA = BΩ
2
a0 = constant
Thus, for first harmonic motion, the blade flaps in such a way as to maintain a
constant aerodynamic flapping moment. This does not necessarily mean that the
blade thrust is also constant, since, except in hovering flight, the blade loading
distribution varies with azimuth angle and the centre of pressure of the loading
moves along the blade.
1.6.3 Flapping motion due to pitching or rolling
An important hovering flight case for which the response of the rotor can be calculated
is pitching or rolling. Consider first the case of pitching at constant angular velocity
q. The equation of motion, eqn 1.3, with ε = 0, is
d
2
β/dt
2
+ Ω
2
β = MA/B – 2Ωq sin ψ (1.14)
Due to pitching and flapping, the velocity component normal to the blade at a
point distance r from the hub is r(q cos ψ –
.
β ); with cos β = 1 and neglecting a very
small term in q, the chordwise velocity is Ωr. The corresponding change of incidence
∆α is therefore
∆α = (q cos ψ –
.
β )/Ω = ˆ q cos ψ – dβ/dψ
where ˆ q = q/Ω.
The contribution to the flapping moment of the flapping velocity
.
β has already
been considered in section 1.6.1; by a similar calculation the moment due to the
pitching velocity q is found to be
(MA)pitching = ρacΩ
2
R
4
ˆ q cos ψ/8 (1.15)
Equation 1.14 now becomes
d
d
+
8
d
d
+ =
8
cos – 2 sin
2
2
β
ψ
γ β
ψ
β
γ
ψ ψ ˆ ˆ q q (1.16)
Assuming a steady-state solution, β = a0 – a1 cos ψ – b1 sin ψ gives
a q b q 1 1 = – 16 / , = – ˆ ˆ γ (1.17)
Hence, when the shaft has a steady positive rate of pitch, the rotor disc tilts
forward by amount 16ˆ q/γ and sideways (towards ψ = 270°) by amount ˆ q. The
longitudinal a1 tilt is due to the gyroscopic moment on the blade, and the lateral b1
tilt to the aerodynamic moment due to flapping. For typical values of γ, the lateral tilt
is roughly half the longitudinal tilt.
The same result can be obtained in a somewhat different way by focusing attention
on the rotor disc. If steady blade motion is assumed to occur, each blade behaves
identically and the rotor can be regarded as a rigid body rotating in space with
angular velocity components Ω about the shaft and q perpendicular to the shaft.
16 Bramwell’s Helicopter Dynamics
According to elementary gyroscopic theory, the rotor will experience a precessing
moment bCΩq tending to tilt it laterally towards ψ = 90°, bC being the moment of
inertia of all the blades in the plane of rotation. In addition, there is the aerodynamic
moment on the rotor due to its pitching rotation. Using eqn 1.15, we find that the total
moment for all the blades is bCρaΩ
2
R
4
ˆ q/16 and is in the nose down sense. Now,
these two moments must be in equilibrium with an aerodynamic moment produced
by a cyclic pitch variation in the tip path plane, and the rotor achieves this by
appropriate tilts a1 and b1 relative to the shaft. This cyclic pitch variation, by the
arguments of section 1.6.1, is easily seen to be – a1 sin ψ + b1 cos ψ. By comparing this
with eqn 1.11, the aerodynamic moment on one blade is seen to be – BΩ
2
γ (a1 sin ψ
– b1 cos ψ)/8. For all blades there would therfore be a steady moment bBΩ
2
γb1/16
acting in the nose down sense and a moment bBΩ
2
γa1/16 in the direction ψ = 90°. For
these moments to be equal and opposite to those above, we must have, as before,
a1 = – 16ˆ q/γ and b1 = – ˆ q
where we have taken B = C, since flapping and in-plane moments of inertia of the
blade are almost identical – typically, C ≈ 1.003B.
When the pitching rate q is not constant, eqn 1.3 becomes (e = 0)
d
d
+
8
d
d
+ =
8
( ) cos – 2 ( ) sin +
d
d
cos
2
2
β
ψ
γ β
ψ
β
γ
ψ ψ ψψ
ψ
ψ ˆ ˆ
ˆ
q q
q
(1.18)
According to the theory of differential equations there is a solution in the form
β = a0(ψ) – a1(ψ) cos ψ – b1(ψ) sin ψ (1.19)
where, as indicated, the flapping coefficients are no longer constants, as in the previous
cases, but functions of time or azimuth angle.
The case of sinusoidally varying pitching velocity, which is important in stability
investigations, has been analysed by Sissingh
2
and Zbrozek
3
. Taking q = q0 sin vψ
and substituting eqn 1.19 into eqn 1.18 gives, after equating coefficients of sin ψ and
cos ψ,
γ
ψ
γ
ψ ψ
ψ
8
+ 2
d
d
–
8
d
d
–
d
d
= – 2 sin 1
1 1
2
1
2 0 a
a b b
q v ˆ
γ
ψ ψ
γ
ψ
ψ
8
d
d
+ 2
d
d
+
8
+ 2
d
d
= cos
1
2
1
2 1
1
0
a a
b
b
q v v ˆ
The solutions for a1(ψ) and b1(ψ) are straightforward, but rather lengthy. Sissingh
has shown that the tip path plane oscillates relative to the shaft, performing a beat
motion out of phase with the shaft oscillation. Now, v is the ratio of the pitching
frequency to the rotational frequency of the shaft and in typical disturbed motion is
usually much less than 0.1. On this basis Zbrozek has shown that, to good
approximations, the lengthy expressions for a1 and b1 can be reduced to
a1 ≈ –16 ˆ q/γ + [(16/γ)
2
– 1] dˆ q/dψ
Basic mechanics of rotor systems and helicopter flight 17
To pilot’s control To pilot’s control
Fig. 1.13 Swash plate mechanism
b1 ≈ – ˆ q + (24 /γ)dˆ q/dψ
Since a typical lateral or longitudinal stability oscillation is about 10 seconds, and
the period of the rotor is about
1
4
second (240 rev/min), v is about 0.025. With ˆ q =
ˆ
q0 sin vψ, the second terms of a1 and b1 are quite small and by neglecting them
Zbrozek’s expressions for a1 and b1 become the same as for the steady case. Thus, in
disturbed motion, both a1 and b1 are proportional to q, and the rotor responds as if the
instantaneous values were steady. This is the justification for the ‘quasi-steady’ treatment
of rotor behaviour in which the rotor response is calculated as if the continuously
changing motion were a sequence of steady conditions. This assumption greatly
simplifies stability and control investigations. The ‘quasi-static’ behaviour of the
rotor might also have been expected from its response as a second order system. The
impressed motion considered above corresponds to forcing at a very low frequency
ratio, and it is well known that the response is almost the same as if the instantaneous
value of the forcing function were applied statically.
If the rolling case is considered, with constant angular velocity and ε = 0 as in the
pitching case, it can be shown that the equivalent to eqn 1.16 is
d
d
+
8
d
d
+ =
8
sin + 2 cos
2
2
β
ψ
γ β
ψ
β
γ
ψ ψ ˆ ˆ p p (1.16a)
in which ˆ p = p/Ω.
1.7 The cyclic and collective pitch control
The development of a satisfactory feathering mechanism was the last link in the
creation of a successful helicopter and it enabled the rotor to be controlled without
tilting the hub or shaft, as had been possible with the free-wheeling autogyro rotor.
A brief description of the feathering mechanism is given below.
The principal feature of the feathering system is the swash plate mechanism, Fig.
1.13. This consists of two plates, of which the lower plate does not rotate with the
shaft but can be tilted in any direction by the pilot’s cyclic control. The upper plate
rotates with the shaft but is constrained to remain parallel to the lower plate. It can
be seen that if the swash plates are tilted the blade chord remains parallel to the swash
plate and, as the blade rotates with the shaft, cyclic feathering takes place relative to
18 Bramwell’s Helicopter Dynamics
the plane perpendicular to the shaft. The swash plate is, of course, a plane of nofeathering, and the axis through the centre of the hub and perpendicular to the swash
plates is the no-feathering axis. Figure 1.5(b) shows a typical swash plate control
mechanism.
Other feathering mechanisms have been employed such as that in Fig. 1.6(a), but
the one described above is used a majority of helicopters.
Collective (constant) pitch is applied by the collective lever which effectively
raises or lowers the swash plate without introducing further tilt; this alters the pitch
angle of all the blades by the same amount.
1.8 Lagging motion
The lagging-motion equation, eqn 1.6, is
.. .
ξ εξββ + – 2 = /
2
Ω Ω N C
or
d
2
ξ /dψ
2
+ εξ – 2β dβ/dψ = N/CΩ
2
(1.20)
where, as explained in section 1.4, the term 2Ωβ
.
β represents the Coriolis moment
due to blade flapping.
In finding the free lagging motion, we assume the flapping motion to be absent
and take N to be the aerodynamic (drag) moment of the blade about the lag hinge.
Let dΩ
2
be the drag of the blade when it rotates at steady angular velocity Ω. The
lagging motion increases the instantaneous angular velocity of the blade to Ω +
.
ξ
and the drag can be assumed to be d(Ω +
.
ξ )
2
. Since
.
ξ is small compared with Ω, the
drag is approximately dΩ
2
+ 2dΩ
.
ξ . If RD is the distance of the centre of drag of the
blade from the hub, and assuming the lag hinge offset to be small,
N = – dRD(Ω
2
+ 2Ω
.
ξ )
But dRDΩ
3
is the power, P say, required to drive one blade. Therefore
N = – (P/Ω)(1 + 2
.
ξ /Ω)
so that eqn 1.20 can be written
d
2
ξ/dψ
2
+ (2P/CΩ
3
)dξ/dψ + εξ = – P/CΩ
3
(1.21)
Equation 1.21 is the equation of damped harmonic motion about a steady value
ξ = – P/CΩ
3
ε. Typical values of P/CΩ
3
and ε are 0.006 and 0.075 respectively, giving
a steady value of ξ of about 4
1
2°.
The frequency of oscillation is 0.27Ω and the
damping is only about 2 per cent of critical. The much lower damping of the lagging
mode is due to the fact that the blade motion in this case is governed by the changes
of drag, and not of incidence. This low natural damping is usually augmented by
hydraulic or elastomeric damping to avoid potential instability problems.
Basic mechanics of rotor systems and helicopter flight 19
1.9 Lagging motion due to flapping
It will be assumed that the lag hinges are parallel to the rotor shaft so that ξ represents
a change of blade angle in the plane of the hub, as in the previous section. The
flapping motion must also be taken relative to this plane, and we will assume it takes
the form β = a0 – a1 cos ψ. In some later work we will have to distinguish flapping
relative to the shaft from that relative to the no-feathering axis by writing βs = a0s +
a1s cos ψ.
Regarding the Coriolis moment due to flapping as a forcing function, and ignoring
the damping, we rewrite eqn 1.20 as
d
2
ξ/dψ
2
+ εξ = 2β dβ/dψ + N/CΩ
2
(1.20a)
and we have to consider the meaning of the moment N in this case. Since the blade
lift is perpendicular to the flow direction, the axis of the aerodynamic flapping
moment must lie in the tip path plane, because this is the plane in which the blades
move steadily. Thus there must be a component of the flapping moment about the lag
hinge, and it is clear from Fig. 1.14 that this component is – MAa1 sin ψ and is equal
to N.
But, we have seen that, for first harmonic flapping, and assuming zero flapping
hinge offset, MA = BΩ
2
a0, as explained in section 1.6.2. Therefore
N/CΩ
2
= – a0a1 sin ψ
and, using the assumed form of β = a0 – a1 cos ψ, then
2β dβ/dψ + N/CΩ
2
= a0a1 sin ψ – a1
2
sin 2ψ
where, again, we have taken B = C. The solution to eqn 1.20a is
ξ = – [a0a1/(1 – ε)] sin ψ + [a1
2
/(4 – ε)] sin 2ψ
The second term is generally smaller than the first and, since, ε is very small, to
a fair approximation ξ can be written
Lag hinge axis
MA
Hub axis
a1
Fig. 1.14 Aerodynamic flapping moment component about lag hinge
a0
20 Bramwell’s Helicopter Dynamics
ξ = – a0a1 sin ψ (1.22)
Thus, the flapping motion forces a lagging motion which lags the flapping motion
by 90°.
Equation 1.22 has a simple physical explanation. It can be seen from Fig. 1.15 that
the blade movement about the lag hinge, i.e. in the hub plane, is simply steady
motion of the blade in the tip path plane projected onto the hub plane. In other words,
the ξ motion calculated above corresponds to uniform motion in the tilted rotor cone,
and this is a result we should expect, for, since the flapping angle is constant relative
to the tip path plane, there can be no Coriolis moments in this plane and the blade
must rotate with constant angular velocity.
1.10 Feathering motion
For small pitch angles the blade feathering equation can be written
d
2
θ/dψ
2
+ θ = L/AΩ
2
(1.23)
For free motion, L = 0, and the blade oscillates with shaft frequency. If θ is held
constant, corresponding to collective pitch application, there will be a moment AΩ
2
θ
trying to feather the blade into fine pitch. This moment is called the ‘feathering
moment’ and must be resisted by the feathering mechanism. The fact that the natural
a1
a0
ξ
Fig. 1.15 Blade lagging motion due to flap
Basic mechanics of rotor systems and helicopter flight 21
frequency of the feathering motion is exactly one cycle per revolution of the shaft –
exactly as required – means that, except to overcome friction, no forces are necessary
in the control links to maintain the motion.
The feathering moment AΩ
2
θ can be explained in terms of the centrifugal forces
acting on the blade. In Fig. 1.16, AA′ is a chord of the blade. Consider elementary
masses at the leading and trailing edges of the blade. The centrifugal forces acting on
these masses are inclined outwards and, therefore, have components in opposite
directions. But the centrifugal force directions both lie in planes perpendicular to the
shaft so that when the blade is pitched the opposite directed components exert a
couple tending to feather the blade into fine pitch. Integrating this moment in the
chordwise and spanwise directions can be shown to lead to AΩ
2
θ.
As with the flapping motion, the centrifugal moment acts like a spring giving a
frequency exactly equal to that of the shaft, but, again like the flapping motion, if the
feathering motion is viewed from a plane passing through the chord, the feathering
motion vanishes and the centrifugal moment in this plane also vanishes, Fig. 1.17.
1.11 Rotor forces and moments
So far we have derived the equations of blade flapping, lagging, and feathering and
have considered some simple cases of blade motion to illustrate some of its dynamic
properties. We now have to consider the effect of this blade motion on the helicopter
as a whole. We shall derive expressions for the forces and moments on the helicopter
A
A′
A
A′
Fig. 1.16 Feathering moment
Positive
incidence
Negative
incidence
Tilt of
shaft
Fig. 1.17 Feathering in tip path plane due to rotor tilt
Rotor
plane
22 Bramwell’s Helicopter Dynamics
and consider requirements for trimmed flight. These requirements will appear as the
control angles necessary to establish a given flight condition analogous to the static
stability analysis of the fixed wing aircraft.
In order to be able to write down the equations of motion of the helicopter in
steady and accelerated flight, it is necessary to calculate the forces exerted by the
blade on the hub. To do this we shall have to relate the motion expressed in terms of
axes fixed in the blade to axes fixed in the rotating hub and then to axes fixed in the
helicopter.
As before, let, i, j, k be the set of unit axes fixed in the blade. Let e1, e2, e3 be a
set of unit axes fixed in the rotating hub, Fig. 1.18.
When the blade is in its undeflected position, i.e. when there is no flapping or
lagging, the blade axes coincide with the hub axes. Now, suppose the blade flaps
through angle β about e2, bringing the blade axes into a position whose unit vectors
are i1, j1, k1. The relationships between e1, e2, e3 and i1, j1, k1 are related through a
rotation matrix transformation as
i
j
k
e
e
e
1
1
1
1
2
3
=
cos 0 sin
0 1 0
– sin 0 cos
β β
β β
(1.24)
The blade now rotates about the lag axis through angle ξ, bringing the unit vectors
of the blade into their final positions i, j, k. The relationships between i, j, k and i1,
j1, k1, are, in matrix form
i
j
k
i
j
k
=
cos sin 0
– sin cos 0
0 0 1
1
1
1
ξ ξ
ξ ξ (1.25)
The relationships between e1, e2, e3 and i, j, k, are on multiplying the transformation
matrices in eqns l.24 and 1.25 together
Ω
e3
e2
k
j i
β
ξ
ψ
Fig. 1.18 Deflected rotating blade
Basic mechanics of rotor systems and helicopter flight 23
i
j
k
e
e
e
=
cos cos sin cos sin
– sin cos cos – sin sin
– sin 0 cos
1
2
3
ξ β ξ ξβ
ξ β ξ ξβ
β β
(1.26)
The above relationships enable us to express quantities measured in one set of
axes in terms of another set, and we shall need them for calculating the forces and
moments on the helicopter.
Let the distance of the centre of gravity of the blade measured from the hinge be
rg. In terms of axes fixed in the blade the position vector of the c.g. is rg = rgi, and
in terms of hub axes the position vector is
rg + eRe1 = rg(cos β cos ξe1 + sin ξe2 + sin β cos ξe3) + eRe1
Expressing the absolute acceleration ag of the c.g. as ag = a1e1 + a2e2 + a3e3, the
components, by applying the standard equations of the kinematics of a rigid body, are
found to be
a r
t
r
t
r eR g 1
2
2 g
2
g =
d
d
(cos cos ) – 2
d
d
(sin ) – ( cos cos + ) β ξ ξ βξ Ω Ω (1.27)
a r
t
r
t
r g 2
2
2 g
2
g =
d
d
(sin ) + 2
d
d
(cos cos ) – sin ξ β ξξ Ω Ω (1.28)
a r
t
g 3
2
2
=
d
d
(sin cos ) β ξ (1.29)
Now, let the aerodynamic force on the blade be F and let R be the force exerted
by the hinge on the blade. If Mb is the blade mass, the equation of motion is
F + R = Mbag (1.30)
If F = F1e1 + F2e2 + F3e3 and R = R1e1 + R2e2 + R3e3,
R M a F
R M a F
R M a F
1 b 1 1
2 b 2 2
3 b 3 3
= –
= –
= –
(1.31)
We now wish to resolve these rotating force components along axes fixed in the
helicopter. In order to comply with the usual stability axes, we take a set of unit axes
x, y, z, with the z axis pointing downwards along the negative direction of e3, Fig.
1.19.
If X, Y, Z are the hub force components along the fixed axes, then, remembering
that the force the blade exerts on the hub is – R,
X = R1 cos ψ – R2 sin ψ
Y = – R1 sin ψ – R2 cos ψ
Z = R3
These forces are time dependent, not only because of the sin ψ and cos ψ terms,
24 Bramwell’s Helicopter Dynamics
but because the aerodynamic force will also be a function of the azimuth angle. For
performance and stability calculations we are interested in the time-averaged values.
Consider the average value of X taken over a complete revolution. We have
0
b
0
1
0
1 d = cos d – cos d
τ τ τ
ψ ψ
∫ ∫ ∫
X t M a t F t
– sin d + sin d b
0
2
0
2 M a t F t
τ τ
ψ ψ
∫ ∫
where τ is a complete period.
Now, from eqn 1.27, the first term of a1 is rg d
2
(cos β cos ξ)/dt
2
. Integrating twice
by parts,
0
2
2
0
d
d
(cos cos ) cos d =
d
d
(cos cos ) cos
τ τ
β ξ ψ βξ ψ
∫
t
t
t
+ cos cos sin – cos cos cos d
0 0
Ω
Ω
∫
β ξ ψ βξ ψ
τ τ
t
If the other terms involving a1 and a2 are integrated in a similar way, we find that,
if the motion is periodic, i.e. if the flight condition is steady, all the terms in the
brackets vanish at the limits and all the remaining integrals cancel identically.
In other words, the mean values of all the inertia forces are zero and the only
forces which remain are the aerodynamic forces. The vanishing of the inertia forces
in steady unaccelerated flight might have been expected on physical grounds, but it
is a common mistake to believe that this is not necessarily the case; the reason for this
is that small angle approximations for the flapping and lagging angles are often made
when resolving the inertia forces, and considerable residuals may remain, particularly
as the centrifugal force is extremely large
4
.
The mean forces are therefore
(1/ ) d = = – (1/ ) cos d + (1/ ) sin d
0 0
1
0
2 τ τ ψτψ
τ τ τ
∫ ∫ ∫
X t X F t F t (1.32)
e3
y
e2
z
ψ
x
e1
Fig. 1.19 Blade and helicopter axes
Basic mechanics of rotor systems and helicopter flight 25
(1/ ) d = = (1/ ) sin d + (1/ ) cos d
0 0
1
0
2 τ τ ψτψ
τ τ τ
∫ ∫ ∫
Y t Y F t F t (1.33)
(1/ ) d = = (1/ ) d
0 0
3 τ τ
τ τ
∫ ∫
Z t Z F t (1.34)
The force components X and Y in the plane of the hub give rise to pitching and
rolling moments about the helicopter’s centre of gravity. Further, the force at the
offset hinge, in the direction of the shaft, also exerts pitching and rolling moments.
This force has the same magnitude as Z, so that, if hR is the height of the hub above
the c.g. and eR is the distance of the hinge from the shaft axis, the average rolling
moment per blade on the helicopter is
L YhR
eR
Z t = + sin d
0
τ
ψ
τ
∫
= +
d
d
(sin cos ) sin d – sin d
b g
2
0
2
2
0
3 YhR
M ex R
t
t
eR
F t
τ
β ξ ψ
τ
ψ
τ τ
∫ ∫
= –
2
sin cos sin d –
2
sin d
b g
2 2
0
2
0
2
3 YhR
M ex R eR
F
Ω
∫ ∫ π
β ξ ψψ
π
ψ ψ
π π
(1.35)
in which xgR = rg.
Similarly, the pitching moment M per blade is
M XhR
M ex R eR
F = – –
2
sin cos cos d –
2
cos d
b g
2 2
0
2
0
2
3
Ω
∫ ∫ π
β ξ ψ ψ
π
ψ ψ
π π
(1.36)
The first integrals in L and M are the inertia couples which arise when the plane
of a rotor with offset hinges is tilted relative to the shaft. If the flapping motion
relative to the shaft is βs = a0 – a1s cos ψ – b1s sin ψ, then, for small β and ξ, these
integrals are
1
2 MbexgΩ
2
R
2
b1s and
1
2 MbexgΩ
2
R
2
a1s respectively.
The second integrals in L and M are found to be very much smaller and can be
neglected, so that to a good approximation
L YhR M ex R b = +
1
2 b g
2 2
1s Ω
= +
1
2 1s YhR SeRb (1.37)
and M XhR SeRa = – +
1
2 1s (1.38)
where S = MbxgΩ
2
R is the centrifugal force of the blade for zero offset, and
approximately so for small offset.
For the small hinge offsets that are usual, the second terms of eqns 1.37 and 1.38
can be interpreted, in the case of two opposing blades, as the couple due to the
displaced centrifugal force vectors, and these are parallel to the tip path plane which
is the plane of steady rotation, Fig. 1.20. For small offsets, the inclination of the tip
path plane is approximately equal to the flapping angle.
26 Bramwell’s Helicopter Dynamics
1.12 Rotor forces and choice of axes
In the previous section, the rotor force components were expressed in terms of a set
of axes fixed in the helicopter. As explained there, such a formulation is necessary to
study the forces and moments on the whole helicopter. However, it is more usual, and
natural, when considering the rotor as a lifting device, to regard it as producing a
thrust, defined along some convenient direction, together with small components of
force in the other two perpendicular directions. For this purpose, three axes systems
are in common use, as follows. The question as to which axis is the most useful
depends upon the problem being considered, and will become apparent in the
applications dealt with later, although some indications are given below.
1.12.1 The no-feathering or control axis
As explained in section 1.6.2, this is the axis normal to the plane of the swash plate.
By definition, no cyclic feathering occurs relative to this axis, the blade pitch being
the constant value supplied by the collective pitch application. Since the pitch angle
is constant, the only other blade motion contribution to the local blade incidence is
that due to the flapping. The no-feathering axis is often used to express the blade
flapping, especially when blade aerodynamic forces are being established, since
constant blade pitch at a section eases the mathematical development. The rotor
aerodynamics and dynamics established in Chapter 3 are expressed using this axis
system.
1.12.2 The tip path plane or disc axis
The tip path plane axis is the axis perpendicular to the plane through the blade tips
and, for zero offset flapping hinges, it is therefore the axis of no flapping. The
definition applies only to first harmonic motion since, when there are higher harmonics,
the blades no longer trace out a plane. Of the higher harmonics, only the odd values
affect the tilt of the disc, and these are usually extremely small compared with the
first harmonics, as explained in section 1.6.2. Now, although there is no first harmonic
flapping relative to the tip path plane, there will be cyclic feathering and the amount
of feathering is exactly equal to the flapping relative to the no-feathering axis. Thus,
in this case, the blade incidence will be determined from the collective pitch and the
apparent feathering motion in the tip path plane.
Fig. 1.20 Centrifugal force couple on tilted rotor with offset hinges
Basic mechanics of rotor systems and helicopter flight 27
When the flapping hinges are offset, the tip path plane axis is no longer the axis
of no flapping, as can be easily seen from a diagram like Fig. 1.20 with exaggerated
hinge offset. Strictly speaking, both feathering and flapping occur relative to the tip
path plane but, provided the offset is small, as it usually is, the error in assuming that
there is no flapping is negligible.
1.12.3 The shaft or hub plane axis
This axis is usually less convenient for calculating the rotor forces, as the blade
incidence must be expressed in terms of both feathering and flapping. It is, nevertheless,
a useful axis for dealing with hingeless rotors, since blade flapping relative to the hub
is of prime importance. Indeed, the blade mechanics developed so far in the current
chapter have been with reference to the shaft axis.
1.13 The rotor disc
In all the three cases discussed above, it is usual to call the force component along the
axis, whichever of the above it is, the thrust T. The component perpendicular to this
axis and pointing rearward is called the H force, and the third component, pointing
sideways to starboard, is the Y force. Usually the Y force is very small and attention
is mainly focused on the thrust and the H force, i.e. the longitudinal force components.
Calculations and measurements show that the resultant rotor force is almost
perpendicular to the tip path plane, usually pointing backwards slightly. It is for this
reason that the tip path plane axis is useful since the resultant force is almost exactly
equal to the thrust; the H force can be regarded as a kind of rotor drag.
Let us denote the thrust and the H force relative to the no-feathering axis by T and
H respectively, and use the subscript D to denote the tip path plane (disc) axis and s
to denote the shaft axis. Now the flapping and feathering angles are usually small –
larger than 10° would be regarded as extreme values – so that, referring to Fig. 1.21,
the approximate relations between the thrust and forces referred to the different axes
are
T
TD
Resultant
H
Tip path plane
HD
a1s
a1 B1
No-feathering
axis
Fig. 1.21 Thrust and H-force vectors
28 Bramwell’s Helicopter Dynamics
T ≈ TD ≈ Ts (1.39)
H ≈ HD + TDa1 ≈ Hs + TsB1 (1.40)
where B1 is the amplitude of the longitudinal cyclic pitch.
1.14 Longitudinal trim equations
The foregoing analysis in this chapter has provided the main features of rotor behaviour,
and demonstrated the dependence of rotor forces on the various different parameters
involved. By extending the analysis to include the helicopter fuselage whilst maintaining
the simplified rotor model, it is possible to derive the equations of equilibrium in
steady, uniform, trimmed flight. From this, the attitude of the fuselage may be
determined. These exercises are done for longitudinal flight in this and the following
section, and lateral control to trim is studied in the final section 1.16.
Referring to Fig. 1.22, and resolving forces in the vertical direction,
W + D sin τc = T cos (θ – B1) – H sin (θ – B1) (1.41)
and, resolving horizontally,
D cos τc = – T sin (θ – B1) – H cos (θ – B1) (1.42)
where θ is the angle between the vertical and shaft, positive nose up, and τc is the
angle of climb. Since θ and B1 are small angles, eqns 1.41 and 1.42 can be written
approximately as
W + D sin τc = T (1.43)
H + D cos τc = T(B1 – θ) (1.44)
Since D sin τc is very much smaller than W, then T ≈ W.
The origin of moments O is defined as the point on the shaft met by the perpendicular
from the c.g. If hR is the height of the hub above this point, and lR the distance of the
T
a1
B1
a1s
eR
Horizon
Fuselage axis
Mf
Shaft
Horizon
θ
V
W
Fig. 1.22 Forces and moments in longitudinal plane
τc
lR
O
MS
H
hR
Drag
Basic mechanics of rotor systems and helicopter flight 29
c.g. forward of this point, taking moments about O and making the small angle
assumption gives
– WlR – ThRB1 + HhR + Mf – Ms(B1 – a1) = 0 (1.45)
where Mf is the fuselage pitching moment and Ms = fbSeR is the centrifugal moment
per unit tilt of all the blades (eqn 1.38), and fb is a factor depending on the number
of blades.
Solving eqn 1.45 for B1 gives
B1 = (Mf – WlR + HhR + Msa1)/(ThR + Ms) (1.46)
Equation 1.46 gives the longitudinal cyclic pitch required for trim. For very small
or zero offset, we can put Ms = 0 and, since T ≈ W, we have
B1 = Mf/WhR – l/h + H/W (1.47)
and, if the fuselage pitching moment Mf = 0,
B1 = – l/h + H/W (1.48)
The denominator of the right-hand side of eqn 1.46 represents the control moment
for unit displacement of the rotor disc relative to the hub axis. The term ThR is the
moment due to the tilt of the thrust vector, which is the only control moment acting
if the hinges are centrally located (e = 0) or the rotor is of the see-saw type. Ms is the
centrifugal couple due to the flapping hinge offset, section 1.11. We shall see in
Chapter 4 that for a typical offset distance, e = 0.04 say, the total moment is more
than doubled by the offset hinge contribution.
The importance of the offset hinge moment is that it not only augments the control
power but is also independent of the thrust and can be designed to provide adequate
control power in those flight conditions where the thrust is temporarily reduced, e.g.
the ‘push-over’ manoeuvre or the transition from powered flight to autorotation.
Equation 1.48 has a simple physical interpretation: the longitudinal cyclic pitch
B1 must be such as to make the resultant rotor force pass through the helicopter’s
centre of gravity, Fig. 1.23; the figure shows that B1 + l/h = tan
–1
(H/T) ≈ H/W, which
is eqn 1.48.
If Ms and Mf are not zero, the resultant force vector no longer passes through the
c.g. but must exert a moment about it in order to balance the Ms and Mf moments.
Resultant
T
H
N.F.A.
hR
c.g.
B1
Fig. 1.23 Resultant rotor force vector
lR
tan
–1
(HIT)
30 Bramwell’s Helicopter Dynamics
Expressed in terms of the tip path plane, eqn 1.47 becomes
B1 = a1 + HD/W – l/h + Mf/WhR (1.49)
Now HD, the force component in the plane of the rotor disc, is usually quite small,
so that for given values of l/h and Mf, the cyclic pitch to trim is roughly that required
to eliminate the backward flapping of the rotor. This is a convenient way of interpreting
the cyclic pitch to trim and shows the advantage of using the tip path plane as a
reference plane in this case. With Mf = 0, Fig. 1.23 can be redrawn as Fig. 1.24.
1.15 The attitude of the helicopter
Since T ≈ W, eqn 1.44 can be written
(D/W) cos τc + H/W = B1 – θ (1.50)
Eliminating B1 by means of eqn 1.46 gives
θ
τ
= –
cos
– +
– + +
+
c f s 1
s
D
W
H
W
M WlR HhR M a
WhR M
(1.51)
If Ms is negligible,
θ = – (D/W) cos τc – l/h + Mf/WhR
Thus, for a given c.g. position and supposing Mf to be constant, the helicopter
fuselage attitude is directly proportional to the drag and, hence, the square of the speed.
Another important conclusion is that, unlike a fixed wing aircraft, the attitude depends
very little on the angle of climb for, since the angle of climb is contained only in
cos τc even quite steep climbs have little effect on (D/W) cos τc and therefore on θ.
1.16 Lateral control to trim
Referring to Fig. 1.25, resolving horizontally, with T ≈ W and ignoring the sideways
pointing Y force,
W(A1 + b1 + φ) + Tt = 0 (1.52)
where Tt is the tailrotor thrust.
Taking moments about O, and for small angles of bank,
WfR + WhR(A1 + b1) + Ms(A1 + b1) + TthtR = 0 (1.53)
where fR is the lateral displacement of the c.g. and htR is the tailrotor height.
Solving eqn 1.53 for A1 gives
A b
WfR T h R
WhR M
1 1
t t
s
= – –
+
+
(1.54)
which is the lateral cyclic pitch to trim.
Basic mechanics of rotor systems and helicopter flight 31
Eliminating A1 from eqn 1.54 gives the trimmed bank attitude:
φ = – +
+
+
t t t
s
T
W
Wf R T h R
WhR M
(1.55)
If Ms = 0 (no offset hinge) and ht = h, which is usually approximately true, then
φ ≈ f/h
which means that the c.g. lies vertically below the rotor, for the rotor thrust vector
T
A1
Tt
φ
W
Fig. 1.26 Helicopter lateral attitude
Fig. 1.25 Forces and moments in lateral plane
T = TD
Resultant
c.g.
a1
B1
N.F.A.
tan
–1
(HD/TD)
HD
Fig. 1.24 Rotor force components in tip path (disc) plane
b1
A1
Tt
htR
hR
fR
W
φ
O
N.F.A.
32 Bramwell’s Helicopter Dynamics
must be tilted relative to the vertical to balance the tailrotor side force and tilted away
from the c.g. to balance the tailrotor moment, Fig. 1.26.
References
1. Stewart, W., ‘Higher harmonics of flapping on the helicopter rotor’, Aeronautical Research
Council CP 121, 1952.
2. Sissingh, G. J., ‘The frequency response of the ordinary rotor blade, the Hiller servo-blade and
the Young-Bell stabilizer’, Aeronautical Research Council R&M 2860, 1950.
3. Zbrozek, J. K., ‘The simple harmonic motion of a helicopter rotor with hinged blades’, Aeronautical
Research Council R&M 2813, 1949.
4. Correspondence in Aircraft Engineering, September and November 1955.
2
Rotor aerodynamics in
axial flight
2.1 Introduction
One of the most important aerodynamic problems of the helicopter is the determination
of the loading of the rotor blades. For this purpose it is essential to know the local
components of airflow at any station along the blade, and this in turn requires a
knowledge of the air velocity induced by the lift of the blades. In developing the
analysis, reference is made to well known and fundamental theorems, laws and
equations in aerodynamics, which may be found in standard texts, such as Houghton
and Carpenter
1
.
An element of a rotor blade can be regarded as an elementary aerofoil and, in
accordance with the Kutta–Zhukowsky theorem, there is a bound vortex of circulation
Γ about the aerofoil which, in general, varies along the span. Now Helmholtz’s
theorem implies that a vortex cannot terminate in the interior of a fluid, and the
vortex bounding the element continues as free vortex lines springing from the trailing
edge of the element, Fig. 2.1(a). These free vortex lines are called trailing vortices.
If Γ is the strength of the bound vortex of an element, and if Γ + dΓ is the vortex
strength of the neighbouring element, the neighbouring trailing vortices are in the
(b)
Γ Γ + dΓ
Γ Γ Γ+ dΓ
dΓ
Γ + dΓ
Γ
Γ
Γ
(a)
Fig. 2.1 Bound and trailing vortices
34 Bramwell’s Helicopter Dynamics
opposite sense and the resultant trailing-vortex strength is dΓ, Fig. 2.1(b). Thus,
when the circulation varies along the span there is an associated distribution of
trailing vortices forming a vortex sheet springing from the blade’s trailing edge. In
principle, once the distribution of vortex lines trailing from the rotor is determined,
the induced velocity at a given point of the flow can be calculated by applying the
Biot–Savart law to an element of the sheet and integrating over the sheet to obtain its
total effect. The velocity distribution induced by the bound vortices and the vortex
sheets of all the blades constitutes the rotor slipstream. Unfortunately, the geometry
of the vortex sheet is extremely difficult to calculate, especially for the important
case of the hovering rotor, since the flow through the rotor is determined largely by
the velocities induced by the sheet itself; i.e. the sheet geometry and the velocity field
it gives rise to are interdependent. In contrast, for a propeller operating under normal
flight conditions, the velocities induced by the trailing vortices are found to be small
compared with the relatively high axial velocity and can usually be regarded merely
as perturbations to be superimposed on the otherwise uniform axial and rotational
flow components.
We shall leave a detailed discussion of the flow pattern induced by the vortex
wake until later in the chapter, since much useful information about the performance
of the rotor can be gained from a simple flow pattern which can be treated by
momentum methods. The method is known as the classical actuator disc theory.
2.2 Actuator disc theory
In the actuator disc analysis, the following assumptions are made.
(i) The thrust is uniformly distributed over the rotor disc across which there is a
sudden jump of pressure ∆p. The uniform thrust distribution can be interpreted
as an assumption that the rotor has an infinite number of blades.
(ii) No rotation or ‘swirl’ is imparted to the flow. This is not a necessary restriction,
since the effects of rotation can be included in the analysis
2
. However, the
problem becomes more complicated than is really justified, particularly as the
swirl velocities in typical helicopter operation, as will be shown later, are
usually negligible.
(iii) The slipstream of the rotor is a clearly defined mass of moving air outside
which the air is practically undisturbed.
A further assumption of the classic actuator disc theory is that the pressure in the
ultimate slipstream is the same as the pressure of the surrounding undisturbed air.
This assumption implies that the slipstream is like a jet whose velocity is unrelated
to that of surrounding air, but the above description of the vortex wake generated by
the rotor requires us to look at the assumption more critically. Since an element of the
vortex sheet moves with the local velocity generated by the rest of the sheet, there is
no normal flow component relative to the sheet itself. But this is also the boundary
Rotor aerodynamics in axial flight 35
condition for a solid surface moving through the fluid, so we deduce that the velocity
field generated by the vortex sheet is the same as if the sheet were a rigid membrane
whose constituent elements move with the local induced velocity. An important
particular case has been considered by Betz
3
who showed that for a certain blade
loading, called the ‘ideal loading’, the power loss is a minimum and the trailing
vortices lie on a helical surface of constant pitch which moves axially at constant
velocity. This case is analogous to that for the minimum induced drag of a wing for
which the span loading is known to be elliptic and the induced velocity along the
span is constant. As mentioned above, the velocity field caused by the motion of the
helical surface relative to the surrounding air constitutes the rotor ‘slipstream’. The
formidable hydrodynamical problem of calculating this velocity field for a propeller
having a finite number of blades was solved by Goldstein
4
in 1929.
Actually, as will be discussed later, the ‘ideal’ blade loading does not occur in
practice, and the vortex wake does not conform to the simple helical surface referred
to above. Nevertheless, let us suppose that, close to the rotor at least, the wake
consists of well defined sheets moving away from the rotor with constant velocity w;
Fig. 2.2 shows the sheet springing from one blade. Imagine an observer stationed at
the point P and at rest relative to the undisturbed air. As the vortex sheets leave the
blade and move downwards, the observer would be aware of a periodic flow. Now
Bernoulli’s equation for unsteady flow is
p q t + + / = constant
1
2
2
ρ ρ φ ∂ ∂
where q is the local fluid velocity and φ is the velocity potential of the flow. At a great
distance from the wake both q and φ tend to zero and p approaches the ambient value
p∞, therefore
p q t p + + / =
1
2
2
ρ ρ φ ∂ ∂ ∞ (2.1)
This equation holds throughout the flow field, including points between the sheets,
since only the sheets themselves represent regions for which the flow is not irrotational.
If z denotes the distance along the rotor axis, r the radial distance, and ψ the angular
co-ordinate of the point P, the periodicity of the flow enables us to write
φ = f (z – wt, r, ψ)
°
P
Fig. 2.2 Vortex sheet leaving blade
36 Bramwell’s Helicopter Dynamics
because φ is constant for a point z0, r, ψ which moves with the wake, and z0 = z – wt.
Hence
∂φ/∂t = –w∂φ /∂z
But
∂φ/∂ z = qz
where qz is the fluid velocity component in the axial direction, therefore
p q wq p z + – =
1
2
2
ρ ρ ∞ (2.2)
It will be shown shortly that the flow component in the axial direction is small
compared with the rotor tip velocity, and this means that the vortex sheets are nearly
parallel to the rotor plane and are also fairly close together. The flow about these
sheets, as seen by the stationary observer, is indicated in Fig. 2.3.
Except near the edges of the sheets, where there may be a considerable radial flow,
the velocity between the sheets is very nearly equal to the velocity of the sheets
themselves, i.e. qz and q are both approximately equal to w. Hence, for the rotor case,
eqn 2.2 becomes
p p w = +
1
2
2
∞ ρ (2.3)
showing that the pressure in the wake is generally higher than the ambient value p∞.
The above result can be obtained in another way. If the observer is moving with
the sheets, the total head pressure at a point a great distance from the sheets is clearly
p∞ +
1
2
2
ρw . Within the sheets, which are assumed to be almost parallel and close
together, the flow is relatively at rest, but since the total head is constant, whether
within the sheets or without, we have
p p w = +
1
2
2
∞ ρ
as before.
The fact that the vortex sheet theory gives rise to an ‘overpressure’ in the wake has
been remarked upon by Theordorsen,
5
who derived eqn 2.2, but since his work was
concerned with propellers in their normal operating state, for which the ‘overpressure’
is extremely small, its significance in relation to the helicopter rotor seems to have
been overlooked.
Blade
w
Fig. 2.3 Flow about adjacent vortex sheets
Rotor aerodynamics in axial flight 37
In practice, the helicopter blade is usually designed to give a favourable loading
in forward flight and, as a result, the ‘ideal’ loading and helical wake is not achieved
in hovering and axial flight. It appears
6
that the wake pressure is somewhat overestimated
by eqn 2.3, and one should expect a value which is somewhere between that given by
eqn 2.3 and the ambient value p∞.
In the application of the classical actuator disc theory, we shall take an arbitrary
value of the pressure in the final wake at first and then investigate the special case (i)
where the wake pressure is equal to the ambient pressure p∞ and (ii) the value given
by eqn 2.3 above.
Let us take a cylindrical control surface surrounding a control volume whose
radius is R1, which encloses the rotor, radius R, and its slipstream, Fig. 2.4. Although
the slipstream does not extend upstream of the rotor, it is convenient to imagine that
it does so for the purpose of applying momentum principles. Far upstream of the
Fig. 2.4 Control volume for rotor in axial flight
Control Volume
R2
p∞
p2 Vc + v2
Vc
p + ∆p Vc + vi
p
R
R1
p∞ Vc
38 Bramwell’s Helicopter Dynamics
rotor, the air velocity relative to the rotor is the rate of climb Vc and the pressure is
p∞. As the air approaches the rotor, the airspeed increases to Vc + vi at the rotor itself.
Because the airflow is continuous there is no sudden change of velocity at the rotor,
but there is a jump of pressure ∆p which accounts for the rotor thrust T = ∆pA, A
being the rotor disc area πR
2
. The slipstream velocity continues to increase downstream
of the rotor, reaching a value in the ultimate wake of Vc + v2, where the slipstream
radius is R2 and the pressure p2.
Since the slipstream velocity is higher than the undisturbed axial velocity Vc, it is
clear that the mass of fluid leaving the bottom end of the control volume exceeds that
entering at the top. There must therefore be some flow through the cylindrical sides
of the control surface. If this flux is denoted by Q, we have
Q R R V R V R V = ( – ) + ( + ) – c c 2 c π π π 1
2
2
2
2
2
1
2
v
= 2 π R2
2
v
Thus the total mass per unit time entering the control surface is
ρπ ρπ R V R
1
2
2
2
c 2 + v
and the total mass leaving the surface is
ρπ ρπ ( – ) + ( + ) c c 2 R R V R V
1
2
2
2
2
2
v
Since the flux entering the control surface consists of air having velocity Vc, the
momentum per unit time entering the surface is
ρ π π V R V R c c 2 ( + )
1
2
2
2
v
and the momentum per unit time leaving the surface is
ρπ ρπ ( R R V R V
1
2
2
2
c
2
2
2
c 2
2
– ) + ( + ) v
Hence, the rate of change of momentum in the axial direction is
ρπ ρπ ρπ ρπ ( R R V R V R V R V
1
2
2
2
c
2
2
2
c 2
2
1
2
c
2
2
2
c – ) + ( + ) – – v v2
= ( + )
2
2
c 2 ρπR V v v 2
The total force in the axial direction acting on the control surface consists of the
rotor thrust plus the pressure forces on the ends of the cylinder. Equating this force
to the rate of change of momentum, we get
T R p R R p R p R V + – ( – ) – = ( + )
1
2
1
2
2
2
2
2
2 2
2
c 2 π π πρπ ∞ ∞ v v 2
or T R V R p p = ( + ) + – )
2
2
c 2 2
2
2 ρπ π v v 2 ( ∞ (2.4)
Continuity of the flow requires that
ρ(Vc + vi)A = ρ(Vc + v2)πR2
2
(2.5)
Rotor aerodynamics in axial flight 39
so that eqn 2.4 can be written
T/A = ∆p = ρ(Vc + vi)v2 + (p2 – p∞)(Vc + vi)/(Vc + v2) (2.6)
Applying Bernoulli’s equation to points upstream of the rotor gives
p V p V ∞ + = + ( + )
1
2 c
2 1
2 c i
2
ρ ρ v (2.7)
and for points downstream of the rotor
p p V p V + + ( + ) = + ( + )
1
2 c i
2 1
2 c 2
2
∆ ρ ρ v v 2 (2.8)
Subtracting eqn 2.7 from eqn 2.8 gives
∆p p p V = – + ( + ) c
1
2 2 2 2 ∞ ρ v v (2.9)
and equating eqns 2.6 and 2.9 we have
ρv v v v v v 2 i
1
2 2 2 2 i c 2 ( – ) = ( – )( – )/( + ) p p V ∞ (2.10)
Let us assume, as for the classical actuator disc, that the pressure in the final wake
is the same as the ambient pressure, i.e. that p2 – p∞ = 0. Then, from eqn 2.10,
v v i
1
2 2 =
irrespective of the axial velocity Vc of the rotor. Thus the increment of velocity at the
rotor disc, which we usually refer to as the ‘induced’ velocity, is half the value in the
ultimate wake. Putting this relationship, and p2 = p∞, in eqn 2.6 we have
T = 2ρA(Vc + vi)vi (2.11)
from which the induced velocity may be calculated when the thrust is known. In
particular, in hovering flight, Vc = 0 and
vi = v0 = √(T/2ρA) (2.12)
in which v0 is termed the ‘thrust velocity’.
If wD is the disc loading, T/A, in N/m
2
, and ρ has the International Standard
Atmosphere (ISA) value corresponding to sea-level,
vi = v0 = 0.64√wD
A typical value of wD is 250 N/m
2
, giving an induced velocity (thrust velocity) of
10.2 m/s.
To calculate the power being supplied by the rotor, we must consider the rate at
which kinetic energy is being imparted to the air. The rate at which kinetic energy
enters the control surface is
1
2 c 2 c
2
( + ρπ ρπ R V R V
1
2
2
2
v )
and the rate at which kinetic energy leaves the control surface is
1
2 c
3
c 2 [ ( – ) + ( + ) ] ρπ ρπ R R V R V
1
2
2
2
2
2 3
v
40 Bramwell’s Helicopter Dynamics
The power P delivered by the rotor is found to be
P A V V p p A V = ( + )( + ) + ( – ) ( + ) c i c
1
2 2 2 2 c i ρ v v v v ∞ (2.13)
in which the first term on the right-hand side is total rate of change of kinetic energy,
the second term derives from rate of doing work by the pressures on the ends of the
control volume, and the continuity relation, eqn 2.5, has been used.
With p2 = p∞ and vi =
1
2 2 v , we see from eqn 2.11 that
P = T(Vc + vi) (2.14)
The first term on the right of eqn 2.14 is the useful work done in climbing at speed
Vc. The term Tvi is the induced power, i.e. the work done producing the (unwanted)
slipstream. In hovering flight,
P = Tvi = T
3/2
/√(2ρA)
If the thrust of the helicopter is 45000 N, with the disc loading of 250 N/m
2
referred to above, the induced power Pi is
Pi =
45 000 10.2
kW
×
1000
= 453 kW
and this would represent about 60 per cent of the total power in hovering flight, the
rest being used to overcome the blade drag and tailrotor and transmission losses. The
induced power calculated above is a rather optimistic value because it has been
assumed that the induced velocity is uniformly distributed over the disc and that this
can be shown to be the optimum distribution. As we shall see shortly, for the induced
velocity distributions likely to occur in practice, the induced power may be 10–15 per
cent larger than the ‘ideal’ value just calculated.
The contraction ratio is the ratio of the radius of the final wake to that of the disc.
The continuity equation gives at once
R2/R = √(vi/v2)
= 1/√2
when p2 = p∞.
Now let us assume that the pressure in the ultimate slipstream is given from eqn
2.3 as
p p 2
1
2 2
2
= + ∞ ρv
where we have taken the wake velocity w to be the same as the local velocity
increment v2. Then from eqn 2.10 we get
v v v v v v 2
2
2
2
= + ( + )/( + ) i 2
1
2 c i c 2 V V (2.15)
If we define ki = vi/v2 and V V c c i
= / , v eqn 2.15 can be written
V k k c i i = (3 – 2/ )/(1 – 2 )
Rotor aerodynamics in axial flight 41
In hovering flight, Vc = 0, we find ki
2
3
= , i.e. the final slipstream velocity is only
3
2
times the induced velocity instead of twice the induced velocity when p2 = p∞. As
the axial velocity Vc increases, ki varies as shown in Fig. 2.5.
It can be seen that ki →
1
2
as the axial velocity increases indefinitely. In practice,
however, Vc is unlikely to exceed 2.
In general, the thrust is, from eqn 2.6,
T A V A V V = ( + ) + ( + )/( + ) c i 2
1
2 c i c 2 ρ ρ v v v v v 2
2
= ( + ) c 2 2 ρA V v v (2.16)
from eqn 2.15.
In particular, in hovering flight we find from eqn 2.16 that
vi
2
3
= ( ) √ T/ A ρ
or, at sea-level, vi = 0.604 √wD m/s when the disc loading wD is given in N/m
2
.
This result shows that the induced velocity is about 6 per cent lower, for a given
disc loading, than when p2 = p∞.
From eqn 2.13 the thrust power P is
P A V V A V = ( + )( + ) + ( + ) c i c
1
2 2 2
1
2 c i ρ ρ v v v v v 2
2
= ( + )( + ) c i c 2 2 ρA V V v v v
= ( + ) c i
T V v
as in the previous case. However, as we have just seen, the induced velocity is less
than for the case p2 = p∞, so the induced power is correspondingly lower. In hovering
flight we see that the induced power is about 6 per cent lower.
It was stated earlier that, in practice, it appears that the wake ‘overpressure’ is
somewhat less than the ‘ideal’ value given by eqn 2.3; consequently the difference
between the induced velocities and the induced powers for the two cases considered
Fig. 2.5 Variation of final slipstream velocity factor with axial velocity factor
0.65
0.60
0.55
0.50
0 1 2 3 4 5 6
ki
Vc
42 Bramwell’s Helicopter Dynamics
would be expected to be less than the 6 per cent calculated above. In what follows,
it will be assumed that the wake pressure and the ambient pressure are equal, since
the considerable simplification it affords justifies the acceptance of the fairly small
inaccuracies just mentioned, particularly as it is not certain what the wake pressure
should be. This assumption conforms, of course, to the classical actuator disc theory.
One should be aware, therefore, that some of the quantities calculated by this theory
differ by a few per cent from those calculated with a wake ‘overpressure’. However,
more exact rotor analyses, which require a knowledge of the geometry of the vortex
wake and which will be discussed later in this chapter, will depend on the contraction
ratio and the ratio of the velocity in the final wake to that at the rotor disc. As we have
seen, the effect of the ‘overpressure’ on these quantities is considerable, although it
is usually taken into account only indirectly through wake visualisation methods.
2.3 Vertical descent and the vortex ring state
The results obtained so far have been made possible only because it has been assumed
that there has been a definite flow through the rotor with a well-defined slipstream.
In vertical descent, however, it is clear that the relative upward flow will, if it
becomes large enough, prevent a slipstream from forming, and some of the air will
recirculate the rotor in what is known as the vortex ring state, Figs 2.6(a) and (b). The
vortex ring state occurs when the rate of descent is of the same order as the induced
velocity in hovering flight. It can also occur in forward flight and in either case leads
to very high descent rates together with uncommanded pitch and roll excursions.
Recovery is by reducing the collective pitch and attaining a forward flight velocity
component, thereby moving the rotor into unrecirculated air. Flight tests describing
the condition have been made by Brotherhood.
7,8
At higher rates of descent the recirculation ceases and a well defined slipstream
develops again, but the wake widens after passing through the rotor and the vortex
wake develops on the upper side of the rotor, Fig. 2.7(a). In contrast to the verticalclimb case, the air slows down on passing through the rotor and the condition is
(b) (a)
Fig. 2.6 Vortex ring flow in vertical descent: (a) slow rate of descent; (b) faster rate of descent
Rotor aerodynamics in axial flight 43
known as the windmill brake state. There is a transitional state between this and the
vortex ring state in which the rotor acts rather like a bluff body, producing a turbulent
wake downstream (i.e. on the upper side of the rotor). This is generally known as the
turbulent wake state and is shown in Fig. 2.7(b).
In the absence of a well-defined slipstream, the momentum theory can no longer
be readily applied, since the mass flow and velocity changes cannot be easily defined.
A simple theoretical relationship between the induced velocity and the axial velocity
of the rotor in such cases is no longer possible. We are then forced to obtain the
induced velocity experimentally by inferring it from the results of the blade element
theory, to be discussed in section 2.3, in connection with the measured rate of descent
and blade collective pitch angle.
To calculate the induced velocity in hovering and climbing flight, we make use of
eqn 2.11. Then, if v0 is the induced velocity in hovering flight, or ‘thrust velocity’,
we define
v v v v i i 0 c c 0 = / and = / V V
so that eqn 2.11 can be written
v v i c i
( + ) = 1 V (2.17)
For vertical descent velocities which are large enough for a slipstream to be
developed again, i.e. the windmill-brake state, eqn 2.11 must be written as
2ρA|Vc + vi|vi = T
the modulus sign indicating that the mass flow, represented by the term Vc + vi, must
be positive (which it is certain to be in climbing flight). The correct result for descending
flight can be expressed as
(a) (b)
Fig. 2.7 (a) Windmill brake state; (b) Turbulent wake state
44 Bramwell’s Helicopter Dynamics
2ρA(Vc + vi)vi = – T
or, in non-dimensional form, as
v v i c i
( + ) = –1 V (2.18)
From eqns 2.17 and 2.18, and using values of the induced velocity obtained from
flight and wind tunnel tests for the vortex ring state, we can describe the complete
curve of vi as a function of Vc, Fig. 2.8.
The broken lines of Fig. 2.8 are the continuations of eqns 2.17 and 2.18 into
regions for which the vortex ring state renders them invalid. Of particular interest is
the state of ‘ideal autorotation’ in which there is zero mean flow through the rotor so
that Vc = – vi. This is given by the intersection of the curve of vi against Vc with the
line vi c = – V , shown chain-dotted in Fig. 2.8. We find that this occurs when
Vc = – 1.8. The condition of ‘ideal autorotation’ is equivalent to the motion of a
circular plate broadside on to the stream which destroys the momentum of the air
approaching it. The thrust of the rotor in this condition can be equated to the drag of
such a disc, so that, if CD is the drag coefficient,
T C V A A = = 2 D c
2
0
2 1
2 ρ ρ v
or
C V D c
2
= 4/
Substituting the value of Vc found above, we have
CD = 4/(1.8)
2
= 1.23
which is close to the drag coefficient of a circular plate. Thus, in ‘ideal autorotation’
the rotor behaves rather like a parachute.
2.4 The swirl velocity
So far the rotational or ‘swirl’ velocity has been omitted from the calculations. Since
Fig. 2.8 Variation of induced velocity in vertical flight
Vc
Turbulent
wake state
Vortex
ring state
2
1
Experimental
induced velocity
Normal
working state
–4 –3 –2 –1 0 1 2
Windmill
brake state
vi
Rotor aerodynamics in axial flight 45
the flow upstream of the rotor is irrotational, there can be no swirl ahead of the rotor
disc. Behind the rotor disc there are two contributions to the swirl velocity: that due
to the bound circulation about the blades and that due to the spiral vortex lines
forming the slipstream, Fig. 2.9. Let the swirl angular velocity contributions be
denoted by ω b and ωt respectively. Ahead of the rotor we have ωb + ω t = 0 or
ω b = – ω t. On passing through the disc the contribution from the bound circulation
changes sign, while the contribution from the trailing vortices remains the same;
hence, behind the disc the total angular velocity is ω = –ωb + ω t = 2ω t. This value
remains constant in the slipstream because no extra circulation is added; we should
also expect the swirl velocity to remain constant from the fact that the bound vorticity
contribution diminishes as we go away from the disc but the trailing vortex contribution
steadily increases from its value at the disc to twice that value in the ultimate slipstream
since, for any point there, the vortex lines are doubly infinite.
To relate the swirl velocity to the thrust on the rotor disc, consider axes fixed in the
advancing and rotating rotor blade. With respect to these axes the flow is steady and,
since the flow is irrotational, except at the vortex lines leaving the blade, the constant
in Bernoulli’s equation must be the same everywhere. Let qz, qr, qψ be the velocity
components of the air relative to fixed axes situated in the rotor disc, and let the rotor
rotate with angular velocity Ω, Fig. 2.10. The velocity components relative to a given
point on the blade are qz, qr, qψ – Ωr. At a great distance ahead of the rotor, the
velocity components relative to the same point of the blade are qz = Vc, qr = 0,
qψ = – Ωr. Also, p = p∞. Bernoulli’s equation for the flow which applies everywhere, is
p V p q q q r z r ∞ + = + + + ( – )
1
2 c
2 1
2
2 1
2
2 1
2
2
ρ ρ ρρψ Ω
Let p1 be the pressure just in front of the disc; since there is no swirl in front of the
disc qψ = 0, hence
p V p q q r z r ∞ + = + ( + + )
1
2 c
2 1
2
2 2 2
ρ ρ 1
2
Ω (2.19)
Just behind the disc the pressure will have jumped to p1 + ∆p, the axial velocity
will be unchanged, and the radial velocity will have changed sign since we shall have
qr
qz
qψ
Ω
Fig. 2.9 Spiral vortices in axial flight Fig. 2.10 Velocity components relative to blade
r
46 Bramwell’s Helicopter Dynamics
passed through the vortex sheet leaving the blade. (For an infinite number of blades,
the radial velocity will be zero.) Thus, behind the disc we have
p V p p q q q q r + r z r ∞ + = + + ( + + – 2 )
1
2 c
2 1
2
2 2 2 2
ρ ρ ψ ψ 1
2
∆ Ω Ω(2.20)
Subtracting eqn 2.19 from eqn 2.20 gives
∆ Ω p q r q = ( – )
1
2
ρ θ ψ (2.21)
If we write qψ = ωr, eqn 2.21 can be expressed as
∆ Ω p r = ( – )
1
2
2
ρω ω (2.22)
The total head pressure just ahead of the disc and relative to axes fixed in the disc
is
H p q q z r = + ( + )
1
2
2 2
1 ρ
Just behind the disc we have
H H p p q q q z r + = + + ( + + )
1
2
2 2 2
∆ ∆ 1 ρ ψ
giving
∆ ∆ H p q = +
1
2
2
ρ ψ
= +
1
2
2 2
∆p r ρω (2.23)
Thus, the change in total head pressure exceeds the jump in static pressure across
the disc by a term representing the kinetic energy of the swirl of the slipstream.
To get some idea of the swirl velocity, qψ = ωr, in a typical case, let us note that
eqn 2.22 can be expressed in terms of the disc loading as
∆ Ω p w r = = ( – D
1
2
2
ρω ω ) (2.24)
and take the values wD = 250 N/m
2
, Ω = 25 rad/s and r = 6 m. For sea-level density
we find ω = 0.23 rad/s and qψ = 1.38 m/s. Since the induced velocity in hovering for
this disc loading has been found to be 10.2 m/s, the angle of flow relative to the rotor
axis is 7.8°. We also see that the second term in eqn 2.23 is only about
1
2
per cent
larger than ∆p; this justifies the neglect of the swirl velocity in the earlier analysis.
2.5 Blade element theory in vertical flight
The relationship developed in the previous sections between the thrust and the induced
velocity requires that either the thrust or the induced velocity is known. We now
consider the lift characteristics of the blade regarded as an aerofoil to obtain a further
relationship between thrust and induced velocity, thereby enabling both to be evaluated.
The calculations follow closely the standard methods of aerofoil theory but the rotor
Rotor aerodynamics in axial flight 47
analysis is simplified considerably because the blade incidence and inflow angles are
usually so small that the familiar small angle approximations may be made.
Consider an element of blade of chord c with width dr at a radius r from the axis
of rotation. The geometric pitch angle of the blade element relative to the plane of
rotation is θ, the climbing speed is Vc, and the local induced velocity is vi. The
direction of the flow relative to the blade makes an angle φ (usually called the inflow
angle) with the plane of rotation, Fig. 2.11, and φ is given by
tan φ = (Vc + vi)/Ωr
or, for small φ,
φ = (Vc + vi)/Ωr
The lift on the blade elements is
d = d
1
2
2
L W C c r L ρ
≈ d
1
2
2 2
ρΩ r C c r L
since, for small φ, W
2
≈ Ω
2
r
2
.
Let us suppose that the lift slope a of the section is constant so that, if the section
incidence α is measured from the no-lift line, we can write
CL = aα = a(θ – φ)
Empirical data suggests a lift slope of about 5.7. The elementary lift is now
d = ( – ) d
1
2
2 2
L r a c r ρ θ φ Ω
Since φ is usually a small angle, we can write dL ≈ dT, where dT is the elementary
thrust, the force perpendicular to the plane of rotation. The total thrust is therefore
T ab c r r
R
= ( – ) d
1
2
2
0
2
ρ θ φ Ω
∫
(2.25)
where b is the number of blades.
dT dL
dD
φ
W
Vc + vi
φ
θ
Ωr
Fig. 2.11 Force components on blade
48 Bramwell’s Helicopter Dynamics
Defining
λ λ c c i i
= / , = / , = / V R R x r R Ω Ω v
eqn 2.25 can be written
T ab R c x x x = [ – ( + ) ] d
1
2
2 3
0
1
2
c i ρ θ λλ Ω
∫
(2.26)
If the chord, induced velocity, and ‘collective’ pitch angle θ are constant along the
blade, eqn 2.26 can be integrated easily to give
T acb R = [ – ( + )]
1
2
2 3 1
3
1
2 c i ρ θ λλ Ω 0 (2.27)
where θ0 is the constant (collective) pitch angle.
Defining a thrust coefficient by
tc = T/ρsAΩ
2
R
2
where s = bc/πR is the rotor solidity, eqn 2.27 gives
tc = (a/4)[2θ0/3 – (λc + λi)] (2.28)
In American work, the thrust coefficient is usually defined by
CT = T/ρAΩ
2
R
2
so that the two thrust coefficients are related by tc = CT/s.
From the momentum theory, the induced velocity and the thrust are related by
T = 2ρA(Vc + vi)vi (2.11)
which can be written in non-dimensional form as
λ λ λ i
2
c i
1
2 c + – = 0 st (2.29)
the positive root being the correct one to take. With λc being given, eqns 2.28 and
2.29 can be solved for tc if θ0 is known, or the required pitch angle θ0 can be
calculated if tc is given. In hovering flight we have simply
tc = (a/4)(2θ0/3 – λi) (2.30)
and
stc i
2
= 2λ (2.31)
Equations 2.28 and 2.29 have been obtained on the assumption that the blade pitch
and chord were constant along the blade and that the downwash velocity had the
constant ‘momentum’ value given by eqn 2.11. Modern helicopter blades usually
have constant chord and approximately linear twist, and, if we assume that a linear
variation of induced velocity is quite a good approximation to that obtaining in
practice, eqn 2.26 can again be integrated quite easily. Let us write the local blade
pitch as θ0 – θ1x and the local induced velocity as vi = viTx, where θ1 is the blade
Rotor aerodynamics in axial flight 49
‘washout’ angle and viT is the downwash velocity at the blade tip. Then eqn 2.26
integrates to give
t
a
c 0 1 c iT
=
4
2
3
–
3
4
– –
2
3
θ θ λλ
( )
(2.32)
where λiT = viT/ΩR.
Now it is generally accepted that eqn 2.11 can be expressed in differential form as
dT = 4πrρvi(Vc + vi) dr (2.33)
where 2πr dr is the area of the annulus of width dr over which the thrust dT is
distributed. It can be shown
9
that eqn 2.33 is not strictly valid but it has given
successful results in airscrew work and may be regarded as sufficiently accurate for
most purposes. It appears to be true
10
for the linearised problem in which we take
Vc + vi ≈ Vc and dT = 4πrρviVc dr. Then putting v1 = viTx in eqn 2.33 and integrating
gives
T = ρπR
2
(viT
2
+ 4viTVc/3)
or, in coefficient form,
λiT
2
+ 4λiTλc/3 – stc = 0 (2.34)
Numerical solutions of eqn 2.34 show that the values of λiT are very nearly equal
to √2λi (λi being the constant momentum value of eqn 2.29) for a wide range of λc
and is exactly equal to √2λi for the hovering condition (λc = 0). Thus, when we
assume the induced velocity is linear, which, as we have said, is good approximation
to real conditions, viT can be replaced with good accuracy by √2λi. Substituting for
λiT in eqn 2.32 gives
t
a
c 0 1 c i
=
4
2
3
–
3
4
– –
2 2
3
θ θ λ λ
( )
√
(2.35)
But θ0 –
3
4 θ1 is the blade pitch angle at
3
4 R and 2√2/3 = 0.943; hence, if we take
θ0 as the value of θ at the
3
4
radial position and approximate 2√2/3 by unity, we can
use the simple equations 2.28 and 2.29 or 2.30 and 2.31 for all cases. These
approximations mean that the thrust will be underestimated by about 2 or 3 per cent
relative to eqns 2.32 and 2.34, but, since the blade lift slope and the actual induced
velocity will not be known precisely, further refinement is hardly justified.
It can easily be verified that if the blade planform also has linear taper, eqn 2.28
still holds, with the exception of some very small terms, if the chord is taken as that
at
3
4 R as well as the blade pitch angle.
A useful relationship between the thrust coefficient and the blade lift coefficient
can be obtained since, for constant blade chord,
T bc R x C x L = d
1
2
2 3
0
1
2
ρ Ω
∫
50 Bramwell’s Helicopter Dynamics
or t x C x L c
1
2
0
1
2
= d
∫
= /6 CL (2.36)
where C x C x L L = 3 d
0
1
2
∫
If the lift coefficient is constant along the blade, then
tc = CT/s = CL/6
Usually the rotor operates at a mean CL of between 0.35 and 0.6, giving typical
values of tc within the range of 0.06 to 0.1.
The rotor torque can be calculated in a similar way to the rotor thrust. From
Fig. 2.11, the torque dQ of a blade element about the axis of rotation is
dQ = r(dD + φ dL)
= ( + )d
1
2
2 3
ρ δ φ Ω r c C r L (2.37)
where δ is the local blade section drag coefficient. If δ is assumed to be constant, eqn
2.37 can be integrated to give
Q bc R bc R x C x L = /8 + d
2 4 1
2
2 4
0
1
3
δρ ρ φ Ω Ω
∫
(2.38)
Defining a torque coefficient qc by
qc = Q/ρsAΩ
2
R
3
eqn 2.37 can then be written in coefficient form as
q x C x L c
1
2
0
1
3
= /8 + d δ φ
∫
(2.39)
Assuming constant induced velocity, φ = (λc + λi)/x, so that eqn 2.39 becomes, on
using eqn 2.36,
qc = δ/8 + (λc + λi)tc (2.40)
For the special case of hovering flight, λc = 0,
qc = δ/8 + λitc
= /8 + ( /2) c
3/2
δ √ s t (2.41)
The first term of eqn 2.41 represents the torque required to overcome the profile
drag; the second represents the torque to overcome the induced drag of the blades. It
can be seen that the second term is the non-dimensional form of the hovering power
calculated in section 2.1 from energy and momentum considerations.
Using momentum principles we can find the effect of a non-uniform induced
Rotor aerodynamics in axial flight 51
velocity distribution on the induced power. Let us assume that eqn 2.14 holds in
differential form; then in hovering flight we can write
dP = dTvi = 4 d
3
π ρ r r i
v
where vi is the local induced velocity. If we take the linear induced velocity distribution
vi = viTx, we have
d = 4 d
2
iT
3 4
P R x x π ρv
so that
P R = 4 /5
2
iT
3
π ρv (2.42)
The thrust from momentum considerations is
Τ πρ = 4 d
2
0
1
iT
2 3
R x x
∫
v
=
2
iT
2
ρπR v (2.43)
If the induced velocity vi is constant, we have, for the corresponding thrust T0,
T0 = 2ρπR
2
vi
2
(2.44)
Comparing eqns 2.43 and 2.44 we see that for the thrusts to be the same we must
have
v v iT
2
i
2
= 2
Then
P = 8√2ρπR
2
vi
3
/5
and, if P0 is the induced power when the induced velocity is constant,
P0 = 2ρπR
2
vi
3
Hence
P/P0 = 4√2/5 = 1.131
that is, when the induced velocity is linear, the induced power is about 13 per cent
higher than if the induced velocity were constant; the latter condition corresponding
to the least induced power for a given thrust. For the linear induced velocity, the
torque coefficient would be
qc = δ/8 + 1.13√(s/2)tc
3/2
(2.45)
A typical value assumed for δ is 0.012. With typical values of 0.05 and 0.08 for
the solidity and thrust coefficient respectively, the two terms of qc are 0.0015 and
0.00403, showing that the induced power is more than two and a half times the
profile drag power.
52 Bramwell’s Helicopter Dynamics
Tests on aerofoils with rotor blade type of construction show that δ depends
considerably on incidence and can be represented in the form
δ = δ0 + δ1α + δ2α
2
(2.46)
Bailey
11
has suggested the values
δ = 0.0087 – 0.0216α + 0.4α
2
(α in radians)
and has used them in the calculation of thrust, H-force, and torque coefficients in
hovering and vertical flight. The expressions which had to be calculated were very
lengthy and the results were given in tabular form. They are to be found in the book
by Gessow and Myers.
12
Since, however, Bailey used constant induced velocity in
his calculations, it is rather doubtful whether the results he obtained would have been
much better than if δ had been assumed constant because, in forward flight especially,
the induced velocity differs considerably from the constant mean value, with
correspondingly large variations in local blade incidence.
Another parameter of great importance is Mach number, especially for current
helicopters which operate at higher tip speeds than formerly. With Mach number and
induced velocity properly taken into account, the calculations of thrust and torque
become more complicated; consideration of Mach number effects is provided in
Chapter 6. However, equations 2.28 and 2.45 give acceptable accuracy for many
performance problems.
2.6 Calculation of the inflow angle
When making rotor calculations, it is often useful to know the inflow angle when the
rotor geometry and operating conditions are given. We saw in the last section that the
elementary thrust dT on an annulus of the rotor disc, when there are b blades, is
d = – )d
1
2
2 2
T abc r r ρ θ φ Ω ( (2.47)
where it has been supposed that the local lift coefficient is given by CL = aα.
Now
φ = (Vc + vi)/Ωr
so that eqn 2.47 can be written
d = – ( + )/ ]d
1
2
2 2
c i T abc r V r r ρ θ Ω Ω [ v (2.48)
Momentum theory applied to the annulus gives
dT = 4πρ(Vc + vi)vir dr
and on eliminating dT from eqn 2.48 we have
v v i
2
c i
c
+ + – – = 0 V
abc abc r V
r
Ω Ω
Ω 8 8
2
π π
θ
(2.49)
Rotor aerodynamics in axial flight 53
Writing λi = vi/ΩR and λc = Vc/ΩR, as before, and putting σ = bc/πr, where σ is
the solidity based on the local radius, eqn 2.49 becomes
λ λ
σ
λ
σ
θ
λ
i
2
c i
c
+ + – – = 0
a x a x
x 8 8
(2.50)
In eqn 2.50 σ and θ are variables, so that variable twist and taper can be taken into
account. In hovering flight λc = 0 and eqn 2.50 reduces to
λi
2
+ (aσx/8)λi – (aσx
2
/8)θ = 0 (2.51)
Since φ = vi/Ωr = λi/x, eqn 2.51 can be written as
φ
2
+ (aσ/8)φ – (aσ/8)θ = 0
or
φ
2
= (aσ/8)(θ – φ) (2.52)
Hence, given the local blade pitch angle and solidity, the local value of φ can be
calculated and then used in eqns 2.25 and 2.38 to obtain the thrust and torque.
Further, θ – φ is the local blade incidence and a(θ – φ) the local blade lift coefficient.
As an example of the use of eqn 2.52, let us consider a three-bladed rotor whose
pitch angle at the blade root is 12° and whose blades have a washout* of 5°. The
blade has a radius of 25 ft (7.6 m) and a constant chord of 1.5 ft (0.46 m). The lift
slope of the blade section is assumed to be 5.7. Table 2.1 shows how the required
quantities vary along the span, φ being obtained as the solution of eqn 2.52.
From eqn 2.36 it can be seen that we can calculate the thrust coefficient by the
integration of x
2
CL, which is proportional to the blade aerodynamic loading. The
variation of x
2
CL along the blade span is shown in Fig. 2.12. On integration we find
that tc = 0.0639.
Let us compare this value with the thrust coefficient calculated from eqns 2.30 and
2.31. Eliminating λi gives the following quadratic in tc
1/2
:
tc = (a/4)[2θ0/3 – √(stc/2)] (2.53)
and the pitch angle to be used is the value of θ at
3
4
R, i.e. 7.5°as discussed in
* A twisted rotor blade or wing is said to have ‘washout’ when the incidence of the tip section is
less than that of the root.
Table 2.1 Variation of φ, α and CL with blade section radius
x = r/R 0.3 0.5 0.7 0.8 0.9 1
θ rad 0.178 0.158 0.136 0.126 0.115 0.105
σ 0.191 0.114 0.082 0.0715 0.0636 0.0573
φ 0.102 0.0795 0.0639 0.0585 0.0531 0.0483
α = (θ – φ)° 4.36 4.49 4.13 3.86 3.54 3.24
CL = a(θ – φ) 0.434 0.447 0.411 0.385 0.353 0.324
54 Bramwell’s Helicopter Dynamics
section 2.4. Solving eqn 2.53, with s = 0.0573, gives tc = 0.0638, which agrees
extremely well with the previous result and shows that the simple analysis gives an
accuracy well within that of the assumed value for the lift slope.
2.7 The optimum rotor
It was stated in the last section that the lowest induced power occurs when the
induced velocity is uniform over the disc. The optimum rotor would be one designed
so that this state was achieved and, in addition, the angle of attack would be chosen
so that the section would be operating at the most efficient lift coefficient, which is
not necessarily at the highest CL/CD ratio.
In hovering flight, the pitch angle of a blade element is
θ = α + vi/Ωr
= α + λi/x (2.54)
where vi is constant. The angle of attack α is also the constant value chosen as the
most efficient. Thus the pitch angle can be considered as consisting of a constant part
and a part which varies inversely with blade radius.
Now the thrust on an annulus of the rotor from the blade element theory is
d = d
1
2
2 2
T r a c r ρ α Ω
and from momentum theory
dT = 4 i
2
πρrv dr
Equating these differential thrusts shows that to ensure constant induced velocity
the chord must vary inversely with the radius. Thus, the optimum rotor must be
twisted in accordance with eqn 2.54 and tapered inversely as the radius. The latter
requirement would result in an unusual blade shape and one that would be difficult
to construct. Departures from the optimum blade, which usually means only that the
0.4
0.3
0.2
0.1
0 0.2 0.4 0.6 0.8 1.0
x
x
2
CL
Fig. 2.12 Non-dimensional blade loading as a function of span
Rotor aerodynamics in axial flight 55
chord is kept constant, do not result in a serious loss of efficiency; usually the amount
is about 2 to 3 per cent more power for a given thrust. The subject is dealt with in
some detail by Gessow and Myers
12
. The reader is recommended to compare an
optimum rotor with one of the same solidity having, say, constant chord and twist
differing from the optimum.
The equivalent chord ce of a rotor on a thrust basis is defined as
c
cx x
x x
e
0
1
2
0
1
2
=
d
d
∫
∫
= 3 d
0
1
2
∫
cx x (2.55)
and on a torque basis
c cx x e
0
1
3
= 4 d
∫
(2.56)
These are the values of the chord for which constant-chord blades would yield the
same thrust and torque as a tapered blade, for the same radius and incidence distribution.
2.8 The efficiency of a rotor
The efficiency of any device should indicate the measure of the success with which
that device performs its duty. It is reasonable to want a hovering rotor to produce the
most thrust for the least power; that is, to make the ratio T/P as large as possible. This
simple criterion has been objected to on the grounds that T/P is not a dimensionless
quantity. The standard measure of efficiency adopted in helicopter work is the figure
of merit M defined by
M = Tvi/P
where vi is the mean momentum induced velocity in hover. Since Tvi is the ideal
induced power, the figure of merit is the ratio of the induced power to the total power.
Since P = Tvi + Pp, where Pp is the profile drag power, the figure of merit can also
be written as
M = Tvi/(Tvi + Pp)
and, in non-dimensional form, as
M s t q = ( /2) / c
3/2
c √
It could well be argued that the figure of merit so defined is even less satisfactory
than the ratio T/P, because for constant thrust a high value can be achieved by
increasing the induced velocity (by reducing the radius, say), thereby increasing the
56 Bramwell’s Helicopter Dynamics
total power, which is the opposite of the desired effect. The reason for this rather
unsatisfactory feature of the figure M is that the induced power is regarded rather
like the ‘useful work’ of standard airscrew theory and takes no account of the fact
that induced power is no more desirable than profile power.
The difficulty of defining an efficiency factor for the helicopter is that many
parameters are involved and some of them cannot be arbitrarily varied because of
structural and mechanical as well as aerodynamic limitations. Nevertheless, some
useful conclusions can be drawn from an examination of the thrust/power ratio. Let
us write this ratio as
T
P
T
T A sA R
=
/ ) + /8
3/2 3 3
√(2ρ δρΩ
(2.57)
The first term in the denominator of eqn 2.57 is the induced power, and the second
term is the profile drag power.
Suppose the rotor radius is kept constant and the thrust is kept constant in such a
way as to keep the incidence at a favourable value. This means that the mean lift
coefficient and, hence, tc is kept constant. But since T = tcρsAΩ
2
R
2
, we must have
sΩ
2
constant, so the only variable term in T/P is the profile drag power, which must
therefore be proportional to Ω. Thus T/P can be increased by reducing Ω, which also
requires s to increase; or, in other words, we need a low rotor speed and high solidity
if the radius is to be kept constant.
Suppose now we fix the thrust, solidity, and tip speed ΩR and vary the rotor
radius. Differentiating eqn 2.57 with respect to A gives
∂
∂
√
√
A
T
P
T
s R T A
s R T A
T P =
/8 – /2 2
/8 + / (2 )]
= 0, for maximum /
3 3/2
3 3 3/2 2
δρ ρ
δρ ρ
Ω
Ω
3
[
i.e. δρ ρ s A R T A /8 = / (2 )
3 1
2
Ω
3
√
that is, the profile power is half the induced power for maximum T/P. The figure of
merit for this condition is
2
3
.
Finally, for a given tip speed, solidity, disc area, and drag coefficient, we can write
T
P R
t
s t
=
1
/8 + ( /2)
c
c
Ω δ √
3 2 /
To find the optimum thrust coefficient we have
∂
∂
√ √
√
t
T
P R
s t t s
s t c
c
3
2 c
c
2
=
1 /8 + ( /2) – ( /2)
/8 + ( /2) ]
= 0
Ω
δ
δ
3 2 3 2
3 2
/ /
/
[
or
1
2 c ( /2) = /8 √ s t
3 2 /
δ
and, as above, the profile power is half the induced power and the figure of merit is
again
2
3
. If we take as typical values s = 0.05 and δ = 0.012, we find the optimum
value of tc to be 0.072.
Rotor aerodynamics in axial flight 57
2.9 Ground effect on the lifting rotor
When a rotor hovers near the ground, the presence of the ground has a considerable
effect on the induced-velocity distribution over the rotor and, hence, on the thrust and
power. At the ground the vertical component of velocity must vanish, and we can
expect that over the rotor the induced velocity would be less than in free air. The
reduction of induced velocity results in a proportionate reduction of induced power
for a given thrust and since, as we saw earlier, the induced power may be at least two
thirds of the total power, the improvement in performance may be quite remarkable;
indeed, some of the earlier, underpowered, helicopters could hover only with the help
of the ground.
A theoretical treatment of ground effect has been made by Knight and Hefner
13
.
It was assumed that the circulation along the blade was constant, so that a vortex
trailed from the blade tip having the same circulation as that round the chord. The
spiral vortices from each blade were assumed to form a uniform vortex cylinder
reaching the ground. Now, it is well known that the presence of a plane ground can
be represented by an appropriate image system such that the flow normal to the plane
vanishes, which is the required boundary condition. In this case the image system is
a cylinder of opposite vorticity, Fig. 2.13. It is clear that the effect of the image
system is to produce an upflow tending to reduce the induced velocity at the rotor.
The treatment is very similar to that for a fixed wing flying near the ground, which
may be represented by a simplified horse-shoe vortex and its image system.
The ratio of the induced velocity to that which would have occurred in free air is
shown in Fig. 2.14 as a function of the radial position and the ratio of rotor height,
h, to rotor radius. The ratio of the corresponding powers is given in Fig. 2.15 as a
function of thrust coefficient and rotor height. A typical value of the ratio of the rotor
height to rotor radius when the helicopter is on the point of taking off is about 0.3,
Ground
plane
Fig. 2.13 Reflection of tip vortex in ground plane
58 Bramwell’s Helicopter Dynamics
1.0
0.8
0.6
0.4
0.2
0 0.2 0.4 0.6 0.8 1.0
3
2
3/2
1
1/2
h/R =
1
/4
r/R
vi/vi∞
Fig. 2.14 Ground effect on mean induced velocity
1.0
0.8
0.6
0.4
0.2
0
1 2 3
tc/s = 3.5
1.5
0.5
h/R
P/P∞
Fig. 2.15 Ground effect on induced power
and it can be seen from Fig. 2.15 that the induced power is about half that which
would have occurred in free air, representing a rduction of about a third of the total
power.
The same improvement in performance can be presented in another way. From a
number of tests on model rotors, Zbrozek
14
has derived curves of the thrust that can
be produced for a given power and has expressed the results as the ratio of the thrust
in ground effect to the thrust in free air as a function of the rotor height and thrust
coefficient, Fig. 2.16.
Hovering in ground effect (IGE) confers considerable operational benefits at high
altitude when the power available may not be sufficient to hover out of ground effect
(OGE). A take-off at altitude, for example, may be initiated followed by transition to
forward flight IGE until a speed is reached such that the power required becomes less
than the power available (see Chapter 4) and a climb out may be performed.
Rotor aerodynamics in axial flight 59
2
1.5
1
0 1 2
0.2
tc = 0.05
T/T∞
h/R
0.1
Fig. 2.16 Ground effect on thrust
2.10 Rotor wake models
As stated at the beginning of this chapter, since the only flow through the rotor in
hovering flight is due to the velocity field created by the bound vortices and the
trailing vortex sheets, and since the distribution of the trailed vorticity is determined
by this velocity field, the problem of calculating the flow becomes extremely complex
and a purely analytical solution is not feasible.
A hierarchy of methods of analysis has developed of which the simplest is the
uniform or rigid wake model, mentioned at the start of this chapter. This consists of
a helical surface representing a vortex sheet trailed from each blade and moving
axially at constant velocity. From the point of view of an observer on the rotating
blade, the wake configuration remains fixed. A more refined model is that of the
prescribed wake which embodies improvements to the wake description and velocity
field, including those that have been observed from experiment. The most refined of
all models is that of the free wake, whose configuration interacts with and is consistent
with the velocity field. These are now considered in more detail.
2.10.1 The rigid wake and the methods of Goldstein, Lock, and Theodorsen
We referred earlier to the theorem of Betz which states that the induced power of an
airscrew or rotor is least when the vortex wake springing from the blades moves
axially as if it were a rigid helical surface. A proof of Betz’s theorem has been given
by Theodorsen
5
.
To appreciate the implications of this result we consider the analogous but much
simpler problem of a fixed wing and its vortex wake. It is well known that the
induced drag of a wing is least when the induced velocity of the wing is constant
along the span, in which case the spanwise loading is elliptical. Sufficiently far
behind the wing the induced velocity becomes independent of the rearward distance,
and such a flow can be produced by a two-dimensional strip whose width is equal to
the span and which moves perpendicular to its plane with velocity w, Fig. 2.17. The
circulation about the wing, corresponding to this optimum case, can be evaluated by
60 Bramwell’s Helicopter Dynamics
calculating the line integral C1 joining the points P and P′ which are, respectively,
just above and just below the wing, Fig. 2.17. Now, since the flow is irrotational, the
line integral C1 is equal to the line integral C2, and this integral is equal to the
difference of potential between the points P and P′, which are equidistant from the tip
of the strip, Fig. 2.18. The velocity potential, and hence circulation Γ, for a twodimensional strip moving normal to itself is found to be elliptical with a maximum
value Γ0 = 2ws at the wing centre, 2s being the width of the sheet (and the wing span).
For the wing itself, however, the maximum circulation should be given by Γ0 = 4ws,
since the wing and its trailing wake represent only half the assumed two-dimensional
strip, and the induced velocity for a given circulation is half the two-dimensional
value.
The corresponding problem for the rotor is to find the velocity potential for a
series of helical vortex sheets, Fig. 2.19, one for each blade, moving with constant
velocity w relative to the otherwise undisturbed air. Once this problem has been
solved, the effect of the number and relative spacing of the sheets on the blade
circulation can be investigated. It is also possible to determine the relationship between
the induced velocity at the blade and the mean velocity between the sheets.
The formidable problem presented by this motion was eventually solved by Goldstein
4
in 1929, but here we shall merely describe a simpler method due to Prandtl
15
, especially
as Prandtl’s result has been used to calculate the rotor blade ‘tip loss’. Prandtl’s
method, which contains all the essentials of Goldstein’s problem, was to replace the
curved sheets of the helical surface by a series of two-dimensional sheets on the
C2
P′
P
C1
Fig. 2.17 Induced velocity distribution behind elliptically loaded wing
w
C2
P′
Fig. 2.18 Circulation about wing
w
P
Rotor aerodynamics in axial flight 61
w
Fig. 2.19 Vortex sheet arrangement near rotor blade
• P′
• P
y
x
w
Fig. 2.20 Two-dimensional flow about vortex sheets
d Td L
W
vi
Vc
Ωr φ
Fig. 2.21 Velocity and force components at a blade element
assumption that the radius of curvature of the outer parts of the sheets is so large that
they can be considered as doubly infinite straight strips, Fig. 2.20.
Before considering the flow about these sheets, let us rewrite the thrust equations
for an element of a blade in terms of the local circulation about the element. Let the
induced velocity at the blade element be vi, the axial velocity be Vc, and the overall
resultant velocity be W, Fig. 2.21.
For an infinite number of blades the thrust on an annulus of radius r and width dr
is, from momentum considerations,
dT = 2πrρ(Vc + vi cos φ) 2vi cos φ dr (2.58)
where we have assumed that the induced velocity in the far wake is twice that at the
disc.
From blade element theory we also have
d = d cos = cos d
1
2
2
T L bW C c r L φ ρ φ (2.59)
α
ρ
62 Bramwell’s Helicopter Dynamics
where dL is the lift on the element, b is the number of blades, and CL is the local lift
coefficient.
Now, from the Kutta–Zhukowsky theorem,
1
2
2
= ρ ρ W C c W L Γ (2.60)
where Γ is the circulation about the element. Then from eqns 2.58, 2.59, and 2.60 we
obtain
dT = ρbWΓ cos φ dr
But from Fig. 2.21 we see that
W sin φ = Vc + vi cos φ
so that on eliminating dT we finally get
vi = bΓ/4πr sin φ (2.61)
which gives a relationship between the local induced velocity and the circulation
when the number of blades is infinite.
Returning now to Prandtl’s representation of the vortex sheets, Fig. 2.20, let the
infinite array of sheets move relative to the surrounding air with velocity w. The
complex potential of such a flow is known* to be
φ ψ
π
π
+ = cos e
–1 p
i
ws
z
s
(2.62)
where sp is the spacing of the sheets, which is easily seen to be given by
sp = (2πr/b) sin φ (2.63)
It can be verified that eqns 2.62 and 2.63 satisfy the requirements of the problem.
Now consider the points P and P′ just above and just below one of the sheets
(x < 0), Fig. 2.20. The required circulation, as explained earlier, is equal to the
difference of potential between these points. Then taking, for simplicity, the sheet on
the x axis, on which ψ = 0, the potential difference across the sheet at a distance a
from the edge of the sheet is easily found from eqn 2.62 to be
φ φ
π
π
P P
–1
–
– = =
2
cos e
p
′ Γ
ws
a
s
= wsk
where k
a
s
=
2
cos e
–1
–
p
π
π
* φ in eqn 2.62 is the standard symbol for velocity potential and should not be confused with the
inflow angle.
Rotor aerodynamics in axial flight 63
Hence,
w = Γ/spk
Now, if we can take the induced velocity at the blade to be half this value, i.e. if
we suppose vi =
1
2 w, we have, on substituting from eqn 2.63 for sp
vi = bΓ/4πrk sin φ (2.64)
This is precisely the same expression as eqn 2.61 except for the factor k, which can
be regarded as a correction factor for the number of blades. Since k is always less
than unity, the induced velocity for a given circulation is always larger the fewer the
number of blades. Put in another way, there is a loss of circulation near the blade tips
when the number of blades is finite.
If a is interpreted as the distance R – r from the blade tips and s is based on the
value at the blade tip, k can be written as
k
b x
=
2
cos e
–1
– (1 – )
sin
π
φ
=
2
cos e
–1
π
– f
(2.65)
where f b x = (1 – )/sin
1
2 φ . This relationship is shown plotted in Fig. 2.22.
Goldstein’s more exact analysis for the helical vortex sheets resulted in an equation
identical with eqn 2.64 but k did not have the simple form of eqn 2.65. Goldstein’s
values of k, which are a function of the number of blades, the radial position of the
element in question, and the inflow angle φ, are given in Fig. 2.23.
Proceeding with the rotor analysis, we see from Fig. 2.21, that
tan ρ = vi/W
= bcCL/8πrk sin φ
from eqns 2.60 and 2.64, or
tan ρ = σ CL/8k sin φ (2.66)
1.0
0.8
0.6
0.4
0.2
0 0.5 1.0 1.5 2.0 2.5 3.0
f
Fig. 2.22 Variation of circulation factor as function of f
k
64 Bramwell’s Helicopter Dynamics
In hovering flight ρ = φ and, since CL = a(θ – φ), eqn 2.66 becomes
8kφ
2
= σa(θ – φ) (2.67)
which is precisely the same as eqn 2.52 except for the factor k.
The value of k in eqn 2.64 has been derived on the assumption that the ideal wake
conditions exist, and this, in turn, implies a certain range of blade loadings depending
on the inflow angle and number of blades. Hence, any calculations using these values
of k are strictly valid only for these particular loadings. Lock
16
, however, widened the
range of application by assuming that the values of k would apply with reasonable
accuracy to any practical load distributions.
The rigid wake analysis of Goldstein implies a certain variation of the inflow
angle φ with respect to the radius; Lock’s assumption allows us to use φ freely, just
as we did for the calculations shown in Table 2.1.
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.0
0.8
0.6
0.4
0.2
b = 2
0.3
0.1
0.2
1.0
0.8
0.6
0.4
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1
0.2
0.3
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.0
0.8
0.6
0.4
k
k
0.05
r/R
φ = 0.05
0.3
0.2
0.1
r/R
φ = 0.05
Fig. 2.23 Goldstein circulation factors
r/R
b = 3
b = 4
k
Rotor aerodynamics in axial flight 65
Let us apply the Goldstein–Lock analysis to the rotor whose induced velocity and
local lift coefficient were calculated in section 2.5. This time we solve eqn 2.67 with
the appropriate value of k instead of taking it as unity. Actually, as φ is unknown, the
correct value of k cannot be found immediately, but let us first calculate φ with k =
1 (corresponding to an infinite number of blades) and then find k from Fig. 2.22; if
necessary we can use the new value of φ to obtain a better value of k, and so on. The
convergence is usually very rapid. For the case under consideration, k becomes worth
considering only for radial distances greater than 0.9 R. The results of the calculations
are shown in Table 2.2.
For the small inflow angles, i.e. the small vortex sheet spacings, which normally
occur in practice, it might be expected that Prandtl’s much simpler analysis would be
adequate. The same calculations were made using Prandtl’s formula and the results
are compared with those of the Goldstein–Lock analysis, Table 2.2, from which it can
be seen that the differences are indeed very small.
The effect on the blade loading distribution x
2
CL is shown in Fig. 2.24, and the
loss of thrust, or ‘tip loss’, amounts to about 4
1
2
per cent of the total thrust. The loss
would be larger for two blades and less for four or more blades. The figure also
shows clearly that the difference between the Prandtl and the Goldstein–Lock analyses
is very small.
2.10.2 The tip loss factor
The above calculations show how we can estimate the loss of thrust near the blade
b = ∞
Goldstein–Lock
Prandtl
0.4
0.3
0.2
0.1
0
0.2 0.4 0.6 0.8 1.0
r/R
x
2
CL
Fig. 2.24 Calculations of tip effect
Table 2.2 Variation of factor k and inflow angle φ near blade tip
x 0.9 0.95 0.96 0.97
Goldstein k 0.97 0.85 0.79 0.68
φ 0.0536 0.0532 0.0540 0.0567
Prandtl k 0.97 0.85 0.81 0.73
φ 0.0536 0.0532 0.0537 0.0551
66 Bramwell’s Helicopter Dynamics
tips. It would clearly be desirable to have a means of calculating the tip loss which
is simpler than the strip analysis described above. For this purpose Prandtl devised a
tip loss factor which gave the ratio of the mean induced velocity over the rotor to an
effective velocity at the blades themselves.
If the number of blades were infinite, the vortex sheets would be indefinitely close
together and all the air between them would be carried down with the sheets. The air
outside the sheets would remain at rest. When the number of blades is finite, however,
the spaces between the sheets allows some of the air to escape upwards round the
edges so that, for a given velocity of the sheets, the average downwash velocity is
somewhat less than at the sheets themselves, and this latter velocity corresponds to
the induced velocity at the blades. Expressed in another way, if a finite-bladed rotor
carries a given thrust, the mean induced velocity at the blades is higher than the value
of the induced velocity calculated on the basis of infinite blades.
Prandtl regarded the defect of mean velocity as equivalent to a shortening of the
lengths of the sheets, i.e. of a reduction of the radius of the blades from R to an
effective value Reff. By finding the ratio between the mean velocity between the
sheets and the velocity of the sheets themselves, Prandtl showed
17
that
R – Reff = (1.386/b)xR sin φ
or Reff/R ≈ 1 – (1.386/b) sin φ (2.68)
since, near the tip, x can be taken as unity.
For hovering flight it is usual to take
sin φ = λi = √(CT/2) (2.69)
so that to a good approximation eqn 2.68 can be written as
Reff/R = B = 1 – √CT/b (2.70)
This expression, and others similar to it, has often been used to determine the ‘tip
loss factor’ B. For the three-bladed rotor considered earlier, B is about 0.980.
Now the idea of Prandtl’s tip loss factor is to represent the increased induced
velocity at the blades by calculating the mean induced velocity from the simple
momentum theory but based on a reduced radius Reff = BR. The tip loss factor is
intended to apply only to the calculation of the induced velocity, but in most textbooks
and in many technical papers it has been interpreted as meaning that the outer portion
of the blade R – Reff is incapable of carrying lift. This means that the thrust integral
would be written
T T r r T x x
BR B
= (d /d ) d = (d /d ) d
0 0
∫ ∫
Clearly this interpretation is quite different from the one intended by Prandtl. If the
upper limit of the integral of eqn 2.26 is B instead of unity, we have, for constant
induced velocity and hovering flight,
T b ac R B B = ( /3 – /2)
1
2
2 3 3
0
2
i ρ θ λ Ω
Rotor aerodynamics in axial flight 67
giving
tc = (aB
2
/4)(2Bθ0/3 – λi) (2.71)
instead of
tc = (a/4) (2θ0/3 – λi) (2.28)
Using the values for θ0 and λi of section 2.5, we find that eqn 2.71 leads to a
reduction of thrust of about 7.7 per cent compared with the thrust given by eqn 2.28.
Thus the use of B as an upper limit to the thrust integral considerably overestimates
the ‘tip loss’. In addition, the expressions for thrust and other rotor quantities become
very complicated when the upper limit B is applied to the forward flight cases. A
much simpler method of accounting for the tip loss is to apply the reduced radius to
the calculation of the induced velocity in the manner conceived by Prandtl. Thus the
induced velocity is simply increased by 1/B
2
and the expression for the thrust coefficient
becomes
tc = (a/4)(2θ0/3 – λi/B
2
) (2.72)
This expression gives a thrust reduction of 4 per cent, which is quite close to the
value of 4
1
2 per cent from the strip theory.
It is suggested that eqn 2.72 should be used as the simplest and most accurate
method of allowing for tip loss. In forward flight the value of B given by eqn 2.70 no
longer applies, as the vortex wake is skewed relative to the rotor disc which makes
it necessary to adopt a different analysis.
2.10.3 Theodorsen’s theory
The analyses of Prandtl and Goldstein–Lock described above can be justly criticised
in that the wake contraction has not been taken into account. It was assumed that the
induced velocity in the wake was twice that at the rotor disc but that the wake was
a uniform helix having the same diameter as the rotor. Because of this restriction, the
analysis was assumed to apply to ‘light loadings’. Theodorsen
5
was the first to make
an attempt to take the wake contraction into account. On the assumption that the ideal
helical wake was being created, Theodorsen used Goldstein’s circulation results to
establish relationships between the power (or thrust) of the propeller and the far wake
parameters. By considering the efficiency of a blade element, the ratio of the induced
velocity in the wake to that at the propeller could be found and the slipstream
contraction could also be calculated. Thus, with the helix angle and diameter of the
final wake known, the corresponding values of the induced velocity and helix angle
at the blade could be found. From Goldstein’s results the ideal circulation and, hence,
the required blade geometry could then be calculated. Actually, Theodorsen’s method
works in the reverse sense to that of Goldstein–Lock, as he begins with the final wake
and calculates the (ideal) propeller which generates it.
Later, in 1969, Theodorsen
18
extended his method to the analysis of static propellers
and hovering rotors. The full theory is given in his paper and the book previously
referred to, but the results for hovering flight are interesting and are given in the
68 Bramwell’s Helicopter Dynamics
figures below. In these figures the common parameter is Λ0, the tangent of the wake
helix angle, defined by
Λ0 = w/ΩR0
where w is the velocity of the far wake and R0 is its radius. With CT (= stc) assumed
known for the rotor, Λ0 can be read off from Fig 2.25. Then, from Figs 2.26 and 2.27
we can find the contraction ratio η and ν, the ratio of the final wake velocity to that
at the rotor. Since for most helicopters CT is not likely to exceed 0.01, it can be seen
that η will vary only between about 0.816 and 0.835 and ν from about 1.5 to 1.65.
The theory depends on the assumption that the ideal wake exists; as we noted in
section 2.1, however, these values may be different for the non-uniform wakes likely
to occur in practice.
In his 1969 paper, Theodorsen does not explain how his results for hovering flight
were to be used, but it is reasonable to assume that they would be applied in the same
way as for the conventional propeller described in his book. Theodorsen’s consideration
of the wake contraction is not complete, however. The implication of his method is
that, although the local wake helix angle and induced velocity are correctly estimated,
the wake at the propeller is still assumed to be cylindrical since the theory makes use
0.020
0.015
0.010
0.005
0 0.1 0.2 0.3
6
4
b = 2
CT
Λ0
Fig. 2.25 Thrust coefficient as a function of wake helix angle (i.e. velocity)
0.9
0.85
0.80
0 0.1 0.2 0.3
Λ0
b = 2
6
∞
Fig. 2.26 Contraction ratio as a function of wake velocity
η
4
Rotor aerodynamics in axial flight 69
2.0
1.8
1.6
1.4
ν
0 0.1 0.2 0.3
Λ0
b = 2
10
Fig. 2.27 Induced velocity ratio as a function of wake velocity
of Goldstein’s results, Fig. 2.28. This is probably quite justified for the conventional
propeller, since not only does the contraction amount to only a few per cent at most
but also the high axial velocity means that the contraction is complete only at a
considerable distance from the disc. For the hovering rotor, however, it is known that
the contraction is usually complete within a rotor diameter, and the curved part of the
contracting wake lies close enough to the blades to have a considerable influence on
the incidence near the tips, Fig. 2.29.
Unfortunately, although the amount of contraction, as we have just seen, can be
obtained quite easily, the calculation of the rate of contraction in the neighbourhood
of the rotor is very complicated and certainly cannot be found in closed form as with
Theodorsen’s other results. To obtain reliable information we need to make use of
experimental data, to be combined, if possible, with theory. These methods are described
in a later section.
2.10.4 The prescribed wake
The Goldstein–Lock theory assumes that the wake is a uniform or rigid helix. In
hovering and vertical flight of low axial velocity, the flow through the rotor is
dominated by the induced velocity which, as we have seen, is not generally uniform
over the rotor and in the wake. The vortex elements springing from the blade are
R
R0
tan
–1
Λ0
tan
–1
Λ
R
R0
Fig. 2.28 Wake parameters in Theodorsen’s
calculations (propeller theory)
Fig. 2.29 Wake contraction for a hovering rotor
4
6
70 Bramwell’s Helicopter Dynamics
therefore transported downwards at different rates, rather than at a constant rate, as
assumed by the rigid wake model.
The reduction in bound circulation about the blade towards the tip as shown in
Fig. 2.23, which is quite rapid for the lower inflow angles, implies a concentration of
trailed vorticity here, that quickly rolls up to form a concentrated vortex emanating
from near the tip. Typically, the bound circulation also reduces inboard, but at a lesser
rate
20
(in radial terms) and this leads to a trailing vortex sheet of opposite sign to that
of the tip vortex. The wake model shown in Fig. 2.30 consisting of an assembly of
discrete vortex elements can be considered to form a suitable representation, which
may be used to evaluate the induced velocity.
An initial calculation of the vertical induced velocity is made using momentum
theory. The vertical displacements of the vortex elements are then calculated, consistent
with the assumed induced velocity, and the pattern of the vortex wake is therefore
defined. Then, by applying the Biot–Savart law to the individual elements, the induced
velocity at the rotor disc can be calculated by summing the contributions of all the
elements.
Langrebe
19
has made such calculations and compared them with those obtained
from the simple momentum theory, Fig. 2.31. It can be seen that the difference
Blade
Discrete
tip-vortex
filament
Discrete
inboard-vortex
filament
Fig. 2.30 Vortex wake representation (after Landgrebe)
0
10
20
30
Momentum (B = 0.97)
Prescribed wake
0 20 40 60 80 100
Axial induced velocity
m/s
Blade radial co-ordinate, r/R, per cent
Fig. 2.31 Comparison of induced velocity by momentum and prescribed wake methods
Rotor aerodynamics in axial flight 71
between the methods is quite small, in spite of the fact that the uniform helical wake
assumes quite a different loading distribution from those occurring in practice. The
indications are that Lock’s assumption, which is that Goldstein’s results for a uniform
helix can still be applied when the blade loading is not ideal, gives quite accurate
results.
The wake geometries of the theoretical methods described so far fail to take into
account completely the contraction of the wake and other distortions of the wake
when the blade loading does not conform to the ideal distribution. In the mid-1950s,
Gray
21,22
, from the results of smoke studies, concluded that the wake from a blade
consisted of strong tip vortex and an inner vortex sheet of opposite sense. This
arrangement is shown diagrammatically in Fig. 2.32.
It was observed that the outer part of the sheet moves faster than the inner part,
with the result that the sheet becomes more and more inclined to the rotor plane, and
that the outer part of the sheet moves faster than the tip vortex.
Landgrebe’s
19
later series of smoke tests confirmed Gray’s results and also confirmed
that cross-sections of the tip vortices do not necessarily occur at the ends of the
corresponding sheet. It was also observed that the tip vortex from a blade moves
downwards relatively slowly until it passes beneath the following blade, from which
point it moves down more rapidly.
These results were confirmed by the experiments of Tangler et al.
23
, who investigated
the wake pattern by methods of flow visualisation and hot-wire anemometry. These
tests indicated very clearly the movement of the tip vortices and the way in which
they eventually interact and diffuse after maximum wake contraction has occurred.
The ultimate wake appears to be unstable and moves downstream in a confused
manner. The instability of a helicopter wake was implied by the results of Levy and
Forsdyke
24
, who investigated the motion of a helical vortex. They showed that the
Ω
Γ
Γ
Γ
Tip vortex
Vortex sheet
Fig. 2.32 Landgrebe’s calculation of wake velocity in hovering flight
Blade
72 Bramwell’s Helicopter Dynamics
vortex would be stable only if the tangent of the helix pitch angle were greater than
0.3; in general this is not satisfied by a hovering helicopter rotor.
The axial and radial co-ordinates normalised on R of the tip vortex, derived from
Landgrebe’s results for a particular case, are shown in Fig. 2.33 and for two radial
positions of the inner sheet in Fig. 2.34. The change of downward velocity of the tip
velocity just referred to is seen in Fig. 2.33 as a sudden change of slope of the axial
displacement.
Landgrebe reduced these results to formulae giving the radial and axial co-ordinates
of a tip vortex in terms of the azimuth angle, and corresponding results for the inner
sheets. For example, the axial displacement of the tip vortex is given by
z k b Τ ψ ψ π = , for 0 2 / 1 w w ≤ ≤
= ( ) + ( – 2 / ), for 2 /
w = 2 b 2 w w z k b b Τ ψ π ψ π ψπ / >
where ψw is the wake azimuth angle relative to the blade,
0.2
0.4
0.6
0.8
1.0
0 90° 180° 270° 360° 450° 540° 630° 720°
Tip-vortex co-ordinates
Wake azimuth angle, ψw
Fig. 2.33 Tip vortex co-ordinates as a function of wake azimuth angle
0
0.2
0.4
0.6
0.8
0° 180° 360° 540° 720°
At = 1.0 r
At = 0 r
Inboard vortex
sheet co-ordinates
Wake azimuth co-ordinate, ψw
Fig. 2.34 Inboard vortex sheet co-ordinates (after Landgrebe)
r
–zT
Tip vortex co-ordinates and – r zT
Axial co-ordinate, –zT
Rotor aerodynamics in axial flight 73
k1 = – 0.25(tc + 0.001θ1),
and k2 = – (1 + 0.01θ1)√CT,
where θ1 is the blade twist in degrees, and CT = stc in which s is the solidity.
The formula for the radial co-ordinate of the tip vortex is
r = 0.78 + 0.22 e
– w Λψ
where Λ = 0.145 + 27CT
The two formulae define the boundary of the wake, at least for the part near the
rotor which remains stable, and form the basis for an experimentally prescribed
wake. It is interesting to note that Landgrebe’s results indicate a final slipstream
contraction ratio of 0.78, which is closer to the value of 0.816 predicted by Theodorsen’s
ideal wake theory than to the 0.707 of the classical momentum theory.
These experiments show how rapidly the wake contracts under the rotor. Using the
results given in Fig. 2.33 we can draw the wake boundary for that case, as shown
below in Fig. 2.35. It can be seen that the contraction is practically complete within
only about half a rotor radius, most of it occurring within a distance of 20 per cent of
the radius.
The importance of these figures is that they show that the vortices at the boundary
of the slipstream are displaced well inboard of the blade tip while still close to the
rotor. This means that the inwardly displaced vortices induce an upwash in the tip
region instead of the strong downwash which would occur if the wake contraction
were small or occurred relatively slowly as with the conventional propeller.
Landgrebe calculated the blade incidence distribution using the wake geometry
deduced from the smoke tests and found that the upwash results in a sharp rise of
incidence just inboard of the tip, Fig. 2.36. Because of the high Mach numbers
occurring at the tip, even in hovering flight, locally high incidences are very undesirable,
since they may lead to shock stall with corresponding loss of lift and increase of drag.
Further, the locally high tip incidence increases the spanwise loading gradient and
intensifies the already strong tip vortex.
z/R
r/R
Fig. 2.35 Contraction of rotor wake using Fig. 2.33 results
74 Bramwell’s Helicopter Dynamics
2.10.5 Free wake analysis
Amongst the earliest work on free wakes in the hover is that of Clark and Leiper.
25
An initial wake geometry was assumed which is based on the mean induced velocity
calculated from the simple momentum theory. The wake is broken into a number of
straight vortex filaments whose strengths are determined by the bound circulation
distribution. Using the Biot–Savart law, the induced velocity components due to
these filaments are then computed and the initially assumed velocity field is modified
accordingly. The positions of the vortex elements, which ‘float’ with the fluid, are
modified in turn to conform with the velocity field. The induced velocity at the blade
and the bound-circulation distribution are also recalculated. The process is allowed to
iterate until satisfactory convergence has been achieved, indicating that the wake
geometry is consistent with the velocity field it induced. The calculations were
considerably simplified by defining a ‘far wake’ as that part of the wake beyond a
distance corresponding to two rotor revolutions and representing this far wake by a
stack of stepped vortex rings approximating to a helix whose spacing is determined
by the number of blades and mean local induced velocity.
The calculations of Clark and Leiper clearly identified the wake features found
experimentally by Landgrebe
19
, namely, that the outer trailing vortices roll up quickly
to form a strong tip vortex, while the inner vortices move downwards as a vortex
sheet which becomes progressively more inclined to the rotor plane. Their model
also showed that the initial position of the tip vortex depends strongly on the number
of blades; if the number of blades is high, the tip vortex remains roughly in the plane
of the rotor until it becomes close to the succeeding blade, when it is convected
downwards. As stated before, this feature had been noted by Landgrebe and is indicated
by the change of slope of ZT in Fig. 2.33.
The later model of Favier et al.
26
was similar to that of Clark and Leiper in that a
rigid far wake was assumed (semi-infinite circular vortex cylinder beyond ψ = 10π/
b, and a constant pitch helix between ψ = 5π/b and 10π/b). A free wake was assumed
from the blade to ψ = 2π/b and a prescribed wake based on Landgrebe
19
from here
to the start of the rigid wake. This confirmed the experimental observations previously
mentioned.
8°
6°
4°
2°
0
0.2 0.4 0.6 0.8 1.0
Root
cutout
Section angle of attack
θ0.75 = 12°
Radial co-ordinate, r/R
Goldstein–Lock analysis
Experimental prescribed wake
Fig. 2.36 Effect of experimental prescribed wake (Landgrebe) on incidence distribution
Rotor aerodynamics in axial flight 75
The free wake model of Brown and Fiddes
27
develops a three turn free wake from
a constant pitch helical wake using a relaxation process, with the influence of the far
wake being accounted for. An interesting feature is the merging of adjacent vortices
according to a criterion in order to prevent mutual orbiting. The method was validated
against the experimental results of Carradonna and Tung
28
on a constant chord,
untwisted, two-bladed rotor. The converged wake geometry is shown in Fig. 2.37. In
this case, a panel method was used on the blade itself, rather than a lifting line or
single bound vortex.
References
1. Houghton, E. L. and Carpenter P. W., Aerodynamics for engineering students, London, Edward
Arnold, 1993.
2. Durand, W. F. (ed)., Aerodynamic theory, vol. IV, section L, New York, Dover Publications,
1963.
3. Betz, A., Schraubenpropellers mit geringstem Energieverlust, Gottinger Nachrichten, 1919,
p. 193.
4. Goldstein, S., ‘On the vortex theory of screw propellers’, Proc. Roy. Soc., 123, 1929.
5. Theodorsen, T., Theory of propellers, New York, McGraw-Hill, 1948.
6. Bramwell, A. R. S., ‘A note on the static pressure in the wake of a hovering rotor’, Res. Memo.
City Univ. Lond. Aero. 73/3, 1973.
Fig. 2.37 Converged wake geometry for Caradonna and Tung
28
rotor (from Brown and Fiddes
27
)
76 Bramwell’s Helicopter Dynamics
7. Brotherhood, P., ‘Flow through a helicopter rotor in vertical descent’, Aeronautical Research
Council R&M 2735, 1949.
8. Brotherhood, P., ‘Flight measurements of the stability and control of a Westland “Whirlwind”
helicopter in vertical descent’, RAE TR 68021, 1968.
9. Goorjian, P. M., ‘An invalid equation in the general momentum theory of the actuator disc’,
AIAAJ, 10(4) 1972.
10. Bramwell, A. R. S., ‘Some remarks on the induced velocity field of a lifting rotor and on
Glauert’s formula’, Aeronautical Research Council CP 1301, 1974.
11. Bailey, F. R., jnr, ‘A simplified theoretical method of determining the characteristics of a
lifting rotor in forward flight’, NACA Rep. 716, 1941.
12. Gessow, A. and Myers, Garry C., jnr, Aerodynamics of the helicopter, New York, Ungar, 1952.
13. Knight, M. and Hefner, R. A., ‘Analysis of ground effect on the lifting airscrew’, NACA TN
835, 1941.
14. Zbrozek, J. K., ‘Ground effect on the lifting rotor’, Aeronautical Research Council R&M
2347, 1947.
15. Prandtl, L., Appendix to Betz, A., ibid.
16. Lock, C. N. H., ‘The application of Goldstein’s theory to the practical design of airscrews’,
Aeronautical Research Council R&M 1377, 1931.
17. Durand, W. F., ibid, pp. 264, 265.
18. Theodorsen, T., ‘Theory of static propellers and helicopter rotors’, Proc. 25th Annual National
Forum American Helicopter Society 236, 1969.
19. Landgrebe, A. J., ‘Analytical and experimental investigation of helicopter rotor and hover
performance and wake geometry characteristics’, USAAMRDL TR 71–24, 1971.
20. Johnson, W., ‘Airloads and wake models for a comprehensive helicopter analysis,’ Vertica,
14(3), pp. 255–300, 1990.
21. Gray, R. B., ‘An aerodynamic analysis of a single-bladed rotor in hovering and low speed
forward flight as determined from smoke studies of the vorticity distribution in the wake’,
Princeton Univ. Aeronaut. Eng. Rep. 356, 1956.
22. Gray, R. B., ‘Vortex modeling for rotor aerodynamics – the Alexander A. Nikolsky Lecture’,
J. Amer. Helicopter Soc., 37(1), 1992.
23. Tangler, James L., Wohlfield, Robert M. and Miley, Stan J., ‘An experimental investigation of
vortex stability, tip shapes, compressibility and noise for hovering model rotors’, NASA Contractor
Rep. NASA CR – 2305, 1973.
24. Levy, M. A. and Forsdyke, A. C., ‘The steady motion and stability of a helical vortex’, Proc.
Roy. Soc. Series A, vol. 120, 1928.
25. Clark, D. R. and Leiper, A. C., ‘The free wake analysis – a method for the prediction of
helicopter hovering performance’, J. Amer. Helicopter Soc., 15(1), pp. 3–11, Jan. 1970.
26. Favier, D., Nsi Mba, M., Barbi, C. and Maresca, C., ‘A free wake analysis for hovering rotors
and advancing propellors’, Vertica, 11(3), pp. 493–511, 1987.
27. Brown, K. D. and Fiddes, S. P., ‘New developments in rotor wake methodology’, Paper No.
14, 22nd European Rotorcraft Forum, 17–19 Sept. 1996, Brighton, U.K.
28. Carradonna, F. X. and Tung, C., ‘Experimental and analytical studies of a model helicopter
rotor in hover’, Vertica, 5, pp. 149–161, 1981.
3
Rotor aerodynamics and
dynamics in forward flight
3.1 Introduction
In this chapter we examine firstly the aerodynamics of the rotor in forward flight and
then the dynamics. In order to be able to determine blade lift, drag and flapping
moment, and, ultimately, rotor performance, it is necessary, as with axial flight, to
determine the induced velocity in forward flight. Fortunately, for many important
problems a detailed description of the induced velocity distribution is not necessary,
and much useful work can be done by treating the rotor as a lifting surface with an
infinite number of blades. The dynamics of the blades and rotor are equally susceptible
to simple approaches based on straightforward ideas of induced velocity, aerofoil
characteristics and blade modelling. The effects of blade flexibility and more
comprehensive induced velocity distributions are dealt with in later chapters, as is a
discussion of the peculiarities of blade-section characteristics undergoing continually
changing conditions.
3.2 Induced velocity in forward flight
The first proposal for calculating the induced velocity for a rotor carrying a given
thrust was due to Glauert
1
. He regarded the rotor as an elliptically loaded circular
wing to which lifting-line wing theory could be applied, and proposed that a mean
induced velocity vi0 could be obtained from the formula
vi0 = T/2ρAV′ (3.1)
where, if V is the relative velocity far upstream of the rotor, the total velocity at the
rotor V′ = √{V
2
cos
2
αD + (V sin αD – vi0)
2
}, A is the rotor area, and αD is the rotor
disc incidence.
Although a general proof of this formula has never been given, its validity was
78 Bramwell’s Helicopter Dynamics
justified on the grounds that in hovering flight it reduces to the momentum formula,
eqn 2.12, while in forward flight, when V′ ≈ V, it assumes the same form as for the
induced velocity of an elliptically loaded wing. The formula appears to be true,
however, for all loading distributions in the ‘high-speed case’
2
.
Glauert’s eqn 3.1 can be interpreted by imagining a circular jet of air of velocity
V having the same diameter as the rotor, flowing past and around the rotor and being
deflected downwards so that far downstream it has an induced velocity component
normal to the rotor disc of 2vi, Fig. 3.1. This interpretation, and the fact that eqn 3.1
reduces to the ‘momentum’ formula of hovering flight, may lead one to think that the
formula is obvious from momentum considerations: it should be appreciated, however,
that the flow depicted in Fig. 3.1 is quite fictitious and that eqn 3.1 is far from
obvious. More will be said about this point later in section 3.5.
Equation 3.1 can be expressed as a unique curve if, as in Chapter 2, we define
v0 = √(T/2ρA) as the ‘thrust velocity’. Then, for αD = 0 and if
v v v v i0 i0 0 0 = / and = / V V
eqn 3.1 can be written
v v i0
4
i0
2
– 1 = 0 + V
2
(3.2)
This equation is shown plotted in Fig. 3.2.
The dotted line shows the relations vi0 = 1/V, which for V > 1 lies extremely
close to the true result. This implies that, for forward speeds greater than about
10 m/s, the induced velocity is much smaller than the forward speed and is equivalent
to an approximation to eqn 3.1 in the form
vi0 = T/2ρAV (3.3)
Although eqn 3.1 provides a very simple and useful formula for the estimation of
the mean induced velocity, it was appreciated by Glauert that the induced velocity
over the rotor is far from uniform. From aerofoil and wing theory one would expect
V
V sin(– α D)
vio
V
2vio
V′
V
V cos αD
Fig. 3.1 Flow interpretation of Glauert’s formula (showing disc incidence αD as negative for normal helicopter
flight case)
– αD
Rotor aerodynamics and dynamics in forward flight 79
an upwash at the leading edge of the rotor and an increase of induced velocity
towards the trailing edge. Accordingly, Glauert proposed a second formula:
vi = vi0(1 + Kx cos ψ) (3.4)
where vi is the general induced velocity and vi0 is the induced velocity at the rotor
centre (and also the mean induced velocity), taken as the value given by eqn 3.1,
x = r/R, and K is a factor chosen to be slightly greater than unity. A typical value
taken for K is 1.2, so that on the longitudinal axis (ψ = 0 or π) there is an upwash at
the leading edge and a linear increase of induced velocity towards the trailing edge;
in fact, the value of K denotes the slope of the induced velocity distribution along the
longitudinal axis.
An attempt to calculate the longitudinal induced velocity distribution and to find
a theoretical value of K was made by Coleman, Feingold, and Stempin. Their analysis
was based on the fact that a rotor carrying a uniform load and moving through the air
at an angle to its plane leaves behind a vortex wake in the form of an elliptical
cylindrical shell, Fig. 3.3.
This cylindrical shell can be regarded as a continuous distribution of vortex rings
whose planes are parallel to the rotor plane and whose ‘strength’ is determined by the
magnitude of the load and the forward speed of the rotor. By applying the Biot–
Savart law to the cylindrical wake, the induced velocity could be obtained in the form
of a double integral. The integral could not be evaluated at a general point of the rotor
disc, but an exact expression was obtained for points on the longitudinal axis. However,
even this result was quite complicated and expressible only in the form of elliptical
1.0
0.8
0.6
0.4
0.2
0 1 2 3 4 5 6
V
Fig. 3.2 Non-dimensional induced velocity as a function of forward speed
vi0
80 Bramwell’s Helicopter Dynamics
χ
V
αD
Fig. 3.3 Vortex ring representation of rotor wake in forward flight
3
2
1
–3 –2 –1 0 1 2 3
3
2
1
χ = 26.6°
3
2
1
0
3
2
1
0
–1
–1
x
vi/vi0
χ = 0°
χ = 63.4°
χ = 90°
Fig. 3.4 Longitudinal induced velocity distribution for uniformly loaded rotor
integrals. A number of longitudinal distributions depending on the wake angle χ are
shown in Fig. 3.4. It can be seen that the induced velocity distribution is not exactly
linear but can be taken as approximately so over most of the rotor diameter. The slope
of the ratio vi/vi0 at the rotor centre was found from the analysis to be tan (χ/2), so
that Glauert’s formula, eqn 3.4, could be written as
vi = vi0(1 + x tan (χ/2) cos ψ) (3.5)
Since χ ≤ 90°, this approximation to the slope fails to give the required upwash at
the leading edge of the rotor, although giving some dependence on the wake angle.
Another result obtained by Coleman et al. is that, since the vortex rings composing
the wake lie at an angle to the wake axis, there will be a component of induced
velocity causing the wake to move downwards normal to its axis and that, in the
ultimate wake, this component of wake velocity is 2vi0 tan (χ/2). Also, if the disc
incidence is αD, the above result leads to the following relationship between the
mean induced velocity and the ultimate wake angle
vi0/V = cos (χ + αD)/2 tan (χ/2) (3.6)
Rotor aerodynamics and dynamics in forward flight 81
3.3 The method of Mangler and Squire
One of the more complete induced velocity calculations in which the rotor is treated
as a lifting surface with a pressure jump is that of Mangler and Squire.
4
They considered
a rotor loading distribution which closely resembled that of a typical rotor and succeeded
in obtaining an exact solution for the induced velocity for any point on the rotor.
Their method was quite different from Coleman’s and it will be instructive to discuss
it in some detail.
If the induced velocity field is regarded as a small perturbation superimposed
upon an otherwise uniform velocity field, and if the x axis is taken along the direction
of the uniform velocity V, Euler’s equations of motion for an elemental volume of
fluid (e.g. Ref. 1, Chapter 2, p. 150) can be linearised to read
V∂u/∂x = – (1/ρ) ∂p/∂x (3.7)
V∂v/∂x = – (1/ρ) ∂p/∂y (3.8)
V∂w/∂x = – (1/ρ) ∂p/∂z (3.9)
Differentiating eqns 3.7 to 3.9 with respect to x, y, z respectively and then adding
gives
V
x
u
x y
w
z
p
x
p
y
p
z
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
+ + = –
1
+ +
2
v
ρ 2
2
2
2
2
But, by continuity,
∂u/∂x + ∂v /∂y + ∂w/∂z = 0
so that
∂
2
p/∂x
2
+ ∂
2
p/∂y
2
+ ∂
2
p/∂z
2
= 0 (3.10)
Hence, we have the rather interesting result that for small disturbances the pressure
field satisfies Laplace’s equation. The first part of the problem, then, is to find a
solution of Laplace’s equation which also satisfies the given pressure discontinuity
across the rotor disc. Secondly, having obtained such a solution for p, any one of the
induced velocity components can be calculated by integrating the appropriate one of
eqns 3.7 to 3.9. Thus, if the disturbance far in front of the rotor is assumed to be zero,
the velocity component w normal to the flight direction is given by
w V p/ z x
x
= – (1/ ) ( ) d ρ
∞
∫
∂ ∂ (3.11)
where x corresponds to the chosen point P in the field. The integration, in this case,
is performed with the values of y and z appropriate to the path of integration; e.g. if
the path of integration were in the plane of the rotor, z would be zero and y would be
constant and have the value corresponding to the field point P.
The other two velocity components, u and v, would be obtained by similar
integrations.
82 Bramwell’s Helicopter Dynamics
Although the method has been illustrated here by the use of Cartesian co-ordinates
for the sake of simplicity, Mangler and Squire used elliptical co-ordinates since they
allowed the boundary conditions to be satisfied more easily and because these functions
give the necessary discontinuity across the disc. In boundary problems of this kind,
the solution appears as a multiply infinite series; in this case the solution for the
pressure at the rotor disc appears as an infinite series of pressure mode ‘shapes’
which have to be chosen to match some prescribed pressure distribution corresponding
to the rotor loading. It was found that the first two pressure shapes at the disc plane
were given by
( – = (1 – ) u
3
4
2 2
p p V C x l T )1 ρ √
( – = – (1 – 5 /2) (1 – ) u
1
2
2 2 2
p p V C x x l T )2 ρ √
and a particular linear combination gives
(pl – pu)/ρV
2
CT = 15x
2
√(1 – x
2
)/8 (3.12)
This distribution is shown in Fig. 3.5 and is such that the load vanishes at the edge
and the middle of the rotor, thereby representing a typical rotor loading very well. It
was indeed fortunate that a reasonable loading shape could be obtained by using only
the first two pressure functions as, even with these, the calculations of the induced
velocity were very lengthy. Examples of the results of the calculations are shown in
Figs 3.6 and 3.7. It can be seen that the longitudinal variation of induced velocity is
far from being linear and that there is actually an upwash over a small portion of the
rear half of the rotor as well as at the leading edge. It can also be seen that the
downwash is very strong near the outer edges of the rear half of the rotor, reaching
values three or four times greater than the mean. The downwash distribution is
symmetric about the longitudinal axis.
Mangler and Squire also succeeded in expressing their results in the form of a
Fourier series, i.e. as
v v i i0
1
2 0
=1
D = 4 [ – ( , ) cos ] c c n
n
n Σ
∞
η α ψ (3.13)
where
c0 = 15η(1 – η
2
)/8 (3.14)
and for n ≥ 2, and even,
r/R
Blade
loading
0 1
Fig. 3.5 Rotor loading assumed by Mangler
Rotor aerodynamics and dynamics in forward flight 83
1.2
1.6
2.0 2.4 2.8
Relative
flow velocity
–1.44 0.4 – 0.4 0 2.24
vi/vi0
3.2
Fig. 3.6 Induced velocity distribution according to Mangler’s theory, αD = 15°
c
n
n
n
n n
n
n
= (– 1)
15
8
+
– 1
9 + – 6
– 9
+
3
– 9
( – 2)/2
2
2 2
2 2
η η η
⋅
×
1 –
1 +
1 – sin
1 + sin
/2
D
D
/2
η
η
α
α
n n
(3.15)
and c1
2 2 1/2 D
D
1/2
= –
15
256
(5 – 9 )(1 – )
1 – sin
1 + sin
π
η η
α
α
c3
2 3/2 D
D
3/2
=
45
256
(1 – )
1 – sin
1 + sin
π
η
α
α
where η
2
= 1 – x
2
.
For odd values of n ≥ 5, cn = 0. The use of the mean induced velocity in eqn 3.13 (as
given by eqn 3.3) should not be taken to imply that eqn 3.13 may be used at the hover
or for slow speeds, since the Mangler and Squire theory assumed that vi << V.
Downwash
4
3
2
1
0
–1
–2
αD = 90° 30°
45°
15°
0°
90°
30°
45°
15°
0°
Trailing
edge
Leading
edge
Upwash
vi/vi0
Fig. 3.7 Longitudinal induced velocity distribution according to Mangler’s theory
–0.5 1
0 0.8 0.8
–1 x
84 Bramwell’s Helicopter Dynamics
3.4 Flight and wind tunnel tests
A series of pioneering attempts to measure the induced velocity in forward flight was
made in 1948 by the UK Royal Aircraft Establishment
5
(now part of DERA, the
Defence Evaluation and Research Agency). Smoke generators were suspended from
a slow flying fixed-wing aircraft in front of the helicopter, a Sikorsky Hoverfly, and
Fig. 3.8 Flow pattern as revealed by smoke filaments in forward flight. Speed 53 mph. Plane of smoke
filaments 0.4R to starboard
were arranged to emit a band of smoke filaments which lay in a vertical plane. Figure
3.8 shows a typical example photograph taken by a third aircraft flying in formation
with the other two. Tests were conducted at a number of different speeds, and with
the plane of the filaments intersecting the rotor at various radii between 0.25R and
0.4R on the advancing side.
The smoke filaments gave a broad picture of the flow through the rotor but were
not really distinct enough to show the velocity distribution in detail. Figure 3.9
shows the induced velocity distribution along the longitudinal axis as deduced from
the deflection of the smoke filaments. The results would appear to indicate that the
distribution is roughly linear, but it is also clear from the photographs that the
filaments become too diffused to show the details predicted by Mangler and Squire’s
theory. The photograph clearly shows, however, an interesting phenomenon:
the series of ‘whorls’ appearing behind the tailrotor are the cross-sections of the
trailing vortices shed by the individual rotor blades cut by the plane of the smoke
filaments.
A comprehensive and more easily controlled series of tests was conducted in a
wind tunnel by Heyson and Katzoff
6
. The method used was to place a grid of wool
tufts in a number of positions in the vicinity of the rotor and to record the tuft
deflections. An analysis of the deflections gave the induced-velocity field and it was
found that the theoretical distribution of Mangler and Squire was largely confirmed.
Heyson and Katzoff also made numerical calculations of the induced velocity field
by using the same basic uniform loading model as Coleman et al. but superimposing
them linearly to obtain symmetrical loadings having arbitrary radial distributions.
Examples of the induced velocity distributions along the longitudinal axis found by
Rotor aerodynamics and dynamics in forward flight 85
20°
15°
10°
5°
0
Non-dimensional longitudinal position x
Angle between local flow
direction and tip-path plane
Forward Aft
–0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0
Note: Points were measured at lateral distances
between 0.25R and 0.4R on advancing
side of rotor
20°
15°
10°
5°
0
–1.0 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0
Forward Aft
15°
10°
5°
0
–1.0 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0
Forward Aft
x
x
µ = 0.188
µ = 0.167
µ = 0.138
Fig. 3.9 Longitudinal induced velocity distribution obtained from smoke photographs
Heyson and Katzoff and compared with that of Mangler and Squire are shown in
Fig. 3.10.
Heyson and Katzoff also investigated the velocity field behind the rotor and
found it to be remarkably similar to that of a circular wing, showing two distinct
trailing vortices. An example of the tuft pattern at a distance of about one diameter
behind the rotor, as seen by an observer looking forward through the grid, is shown
in Fig. 3.11.
Katzoff has obtained some useful symmetry relationships between the induced
velocity components for a uniform load. By superimposing a skew-symmetric wake
on the original one, Fig. 3.12, a two-dimensional elliptic wake is created by means of
which the following relationships can be deduced.
(i) If P and Q are two points within the rotor disc and symmetrically located on
either side of the lateral axis, the sum of the induced velocity components wP and
wQ, normal to the rotor plane, is equal to the normal component of the induced
velocity within the wake. Since this is constant, it follows that wP + wQ is
86 Bramwell’s Helicopter Dynamics
–1 –0.5 0 0.5 1 1.5 2 2.5 3
3
2
1
χ = 0.75°
µ = 0.095
–1
–1 –0.5 0 1 1.5 2 2.5 3
3
2
1
–1
χ = 82.3°
µ = 0.14
χ = 83.9°
µ = 0.232
–1 –0.5 0 1 1.5 2 2.5 3
3
2
1
–1
vi/vi0
Triangular loading
Measured
Fig. 3.10 Longitudinal induced velocity distribution obtained from wind tunnel tests for different values of µ
Uniform loading
Mangler loading
Fig. 3.11 Vortex pattern behind rotor
0.3
0.2
0.1
0
–0.1
–0.2
–0.3
–0.4
–0.5
–0.6
–0.7
–0.8
–0.9
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –1.0 –1.2 –1.4
Advancing side Retreating side
Non-dimensional lateral distance, y
Distance behind hub, x = 1.07
Vertical distance below hub, z
x
Rotor aerodynamics and dynamics in forward flight 87
Fig. 3.12 Symmetry relations for induced velocity in forward flight
+ =
constant everywhere over the disc and, therefore, that the induced velocity
distribution at the rotor is skew-symmetric with respect to the lateral axis.
(ii) If P and Q are symmetrically located about the lateral axis but lie outside the
disc, we have
wP + wQ = v′ sin χ
where v′ is the longitudinal component of velocity in the ellipse plane at the
point corresponding to either P or Q, Fig. 3.13.
The components of velocity u′ and v′ about the elliptic wake arise from the selfinduced motion of the wake itself through the surrounding air. If the velocity of the
wake normal to its axis is U, we have seen that Coleman et al. have found U to be
given by
U = 2vi0 tan (χ/2)
where vi0 is the induced velocity at the rotor centre (which is also the value along the
lateral axis) and χ is the angle between the wake and the rotor plane. On the lateral
axis, wP = wQ = w, say, and, by calculating the flow about the ellipse
2
, we find that
on the lateral axis outside the rotor
w = vi0 [1 – x(x
2
– sin
2
χ)
–1/2
] (3.16)
where x = r/R. This variation is shown in Fig. 3.14.
On the lateral axis, therefore, there is a strong upwash outside the rotor disc as
opposed to the downwash on the inside. The upwash also exists forward of the lateral
axis and for some distance behind it, but a downwash appears further to the rear at
a point depending on the incidence of the rotor.
v ′
°Q
u′
°P
Fig. 3.13 Symmetry relations for induced velocity in forward flight
P• •Q Q′• • ′ P+Q′• • ′+Q
88 Bramwell’s Helicopter Dynamics
3.5 General remarks on the forward flight case
We have seen that the mean induced velocity in forward flight can be given by
Glauert’s formula
vi = T/2ρAV′ (3.1)
and that it can be expressed in a form which resembles the ‘momentum’ formula of
climbing and hovering flight. Now, we assumed in Chapter 2 that this relationship
could be expressed in the differential form
dT = 4πrρ (Vc + vi)vi dr
and that from this expression the induced velocity for non-uniform loadings could be
derived. However, the axial flight case, which includes the conventional propeller, is
a special one for two reasons, namely,
(i) the induced velocity is in the same direction as the general flow, and
(ii) because of the symmetry of this, the flow is confined to well defined concentric
shells for which the mass flow can easily be calculated.
Under these circumstances, momentum principles can be confidently applied and
have been used already to obtain a number of results. But in forward flight neither of
these two conditions applies. We have already seen that, in the linearised problem of
the lifting rotor, the pressure field satisfies Laplace’s equation. In particular, it can be
shown
2
that for the uniformly loaded rotor the pressure at any point is proportional
to the solid angle which the rotor disc subtends at that point. Further, the acceleration
and pressure gradient fields, for the linearised case, both have precisely the same
form as the velocity field of a vortex ring. Suppose, for simplicity, that the rotor is
moving in its own plane and that we wish to find the component of induced velocity
w normal to the rotor disc. As we have already seen, w can be calculated by the
integral in eqn 3.11. The interpretation of this integral is that we are integrating the
1
2
0 1 1
2
χ = 30°
60°
75°90°
r/R
Fig. 3.14 Induced velocity near uniformly loaded rotor on the lateral axis
w
vi0
Rotor aerodynamics and dynamics in forward flight 89
component of pressure gradient normal to the rotor along a path in the plane of the
disc. The shape of this pressure gradient component along the longitudinal axis of the
rotor is shown in Fig. 3.15.
Starting from a point to the left of Fig. 3.15, and a great distance from the leading
edge, we see that the pressure gradient is upward so that, if w is zero a long way in
front of the rotor, integration results in a gradual increase of upwash as the rotor is
approached. Just behind the leading edge the pressure gradient reverses and the
upwash diminishes until a downwash develops at a point between the leading edge
and the rotor centre, gradually increasing as the trailing edge is approached. Proceeding
further to the right, the pressure gradient becomes positive again as the trailing edge
is passed, gradually reducing the downwash until it becomes exactly twice that at the
rotor centre.
What has just been described is the qualitative evaluation of the downwash
corresponding to the case χ = 90° calculated by Coleman et al. and presented in Fig.
3.4. In fact, the integration for this case can be performed analytically and gives
simply
w
p
V
xK x = –
4
[2 4 ( )]
∆
πρ
π m
1 > > 0
– 1 < < 0
x
x
w
p
V
K x = –
4
[2 (1/ )]
∆
πρ
π m
1 < <
– 1 > > –
x
x
∞
∞
where ∆p is the pressure step across the disc, x is normalised with respect to the rotor
radius, and K(x) is the complete elliptical integral of the first kind. This agrees with
the more general result of Coleman et al.
It is clear that, unlike the axial case, the induced velocity at a particular point in
forward flight depends not merely on the local loading but on the way in which the
pressure gradient varies along the path of integration, which for the linearised analysis
is determined only by the rotor-disc incidence. Of course, in any given flight condition,
Pressure
gradient
x
–2 –1 0 1 2
Fig. 3.15 Pressure gradient along longitudinal axis of uniformly loaded rotor
90 Bramwell’s Helicopter Dynamics
the rotor thrust must be accountable in terms of the momentum changes in the air, but
care must be taken to ensure that all the forces acting on the air are properly taken
into account when applying the principle.
3.6 Induced power in forward flight
We saw in Chapter 2 that for a uniformly loaded rotor carrying a thrust T the
induced power is Tvi, where vi is the mean induced velocity. When the loading is
non-uniform, the induced power is always higher than this ‘ideal’ value. In particular,
for an axially symmetric loading the increase is about 13 per cent. A figure of
about 15 per cent greater than the ‘ideal’ value is generally accepted to account for
the non-uniformities of loading and induced velocity in both axial and forward
flight.
We now inquire as to whether this factor for forward flight is valid when we note
from Figs 3.6 and 3.7 the considerable distortion of the induced velocity field for
these cases.
As in the axial case, to find the induced power we calculate the energy being
supplied to the slipstream by the passage of a uniformly loaded rotor. In forward
flight the energy of the slipstream consists of two distinct contributions:
(i) the energy of the flow in the cylindrical wake (this is the only source of energy
in the axial case);
(ii) the energy imparted to the air outside the cylindrical wake due to the movement
of the wake through it. (We saw earlier in this chapter that the wake moves
normal to itself with velocity 2vi0 tan (χ/2).)
If E1 and E2 are the two contributions per unit length of wake, the power expended,
P, is V(E1 + E2), and this is the induced power. The mass flow through the wake at
the rotor is ρπR
2
vi cos χ, since πR
2
cos χ is the cross-sectional area of the wake. The
absolute velocity of the air in the ultimate wake is 2vi sec
2
(χ/2), as discussed earlier,
so that
E R 1 i
2 2 2
= 2 cos sec ( /2) ρ π χ χ v
By considering the kinetic energy of the air outside the elliptic wake, it can be
shown
2
that
E R 2 i
2 2 2
= 2 tan ( /2) ρ π χ v
for which we find that
P = Tvi
as in hovering and axial flight.
The extension of the above calculation to arbitrary loadings is much more difficult,
even when the relationship between the loading and the induced velocity is known
Rotor aerodynamics and dynamics in forward flight 91
completely, as in the work of Mangler and Squire referred to earlier. The case of the
rotor in high speed flight (χ = 90°) carrying Mangler’s loading can be calculated
quite easily
2
, however, and the result shows that the induced power is about 1.17
times greater than if the induced velocity were uniform.
One can also calculate the induced power by considering the in-plane
component of the blade thrust vector which is tilted backwards by the induced
velocity, as is considered in classical blade element theory, Chapter 2. If ∆p is the
axisymmetrical radial pressure distribution, the thrust carried on an annulus of
width dr is
dT = 2πr ∆p dr
and this thrust is shared by the b blades of the rotor. The elementary induced torque
is
dQi = r dTvi/Ωr = (vi/Ω) dT
where vi is the induced velocity at a given blade. The corresponding induced power
is
dPi = Ω dQi = vi dT
= 2πr∆pvi dr
From Mangler and Squire’s work, and also the work of Coleman et al., for
axisymmetrical pressure loadings the induced velocity can be expressed as
v v i i0
1
2 0
=1
= 4 [ – cos ] c c n
n
n Σ
∞
ψ
as in eqn 3.13 and where 2vi0c0 is the same as the induced velocity in axial flight for
the same loading. Because of the cosine terms, the mean value of the power with
respect to azimuth depends only on the first term of the series and we have
P pc r r
R
i i0
0
0 = 4 d π v
∫
∆
= 4 ( ) ( ) d i0
2
0
1
0 π v R p x c x x x
∫
∆
Thus, when the rotor loading is axisymmetrical, the induced power can be calculated
from a knowledge only of the induced velocity distribution in axial flight fast enough
for linearisation to apply. For the particular case of Mangler and Squire’s loading, we
find
P T x x x i i0
0
1
= (225/8) (1 – ) d v
∫
5 2
= (75/64)Tvi0 = 1.172Tvi0
= (1 + k)倀
92 Bramwell’s Helicopter Dynamics
say, where Pi0 is the induced power for constant induced velocity. This value
agrees with that given earlier. It should be noted that the rotor disc loading is
proportional to x
2
√(1 – x
2
) which implies a blade thrust loading proportional to
x
3
√(1 – x
2
).
If a similar analysis for hovering flight is made, i.e. if we assume that ∆p = 2ρvi
2
instead of 2ρVcvi as in the calculations above, we find that Mangler and Squire’s
distribution gives 1 + k = 1.11. Hence, the induced power is 11 per cent higher
than the ideal power in hovering flight, rising to about 17 per cent at high forward
speed.
If ∆p = Cx
n
(implying a blade thrust loading proportional to x
n+1
), we easily find
that
1 + = (1 + ) / (1 + ) when = 0
1
2
3/2 3
4
k n n µ
and
1 + = (1 + ) /(1 + ) at high
1
2
2
k n n µ
These relationships are shown in Fig. 3.16.
1.4
1.3
1.2
1.1
1
Mangler-hovering
Mangler-forward flight
Forward flight
Hovering
0 1 2
n
1 + k
3.7 Velocity components at the blade
Before being able to calculate the forces and moments on a blade, it is necessary to
know the velocity components of the air relative to any point of the blade. In the
following sections the blade will be assumed to be a rigid beam with a flapping
hinge, and only simple ideas of induced velocity and aerofoil characteristics will be
used. The analysis follows closely the classical work of Glauert
1
and Lock
7
, and,
because the mathematical development is eased, the no-feathering axis system will
be used.
We take as our final reference axes a set of right-hand axes fixed in the blade as
Fig. 3.16 Induced power factor as a function of radial pressure distribution exponent
Rotor aerodynamics and dynamics in forward flight 93
was shown in Fig. 1.18. It is sufficient to assume for the calculation of the aerodynamic
forces that the flapping hinge offset is zero. The only velocity component affected by
the flapping hinge offset is that due to blade flapping, but, since the hinge offset is
usually only a few per cent of the blade radius, the error in assuming it to be zero is
negligible. Taking the upward direction as positive, Fig. 3.17, and taking unit vectors
i1, j1, k1, with k1 along the no-feathering axis and i1 sideways, the forward velocity
V of the helicopter can be expressed as
i1
j1
v
k1
No-feathering axis
Fig. 3.17 Blade axes
αnf
ψ
Forward
j1
j2
i1
Fig. 3.18 View in plane of no feathering
Ω
V = V cos αnf j1 – V sin αnfk1
The blade is itself rotating with angular velocity Ω and it lies at an azimuth angle
ψ with respect to the rearward direction of the helicopter, Fig. 3.18. The blade, at
present, lies in the no-feathering plane. Taking a new unit axes system i2, j2, k2, with
i2 and j2 defined as in Fig. 3.18, the helicopter’s forward velocity can now be written
as
V = –V cos αnf cos ψi2 + V cos αnf sin ψj2 – V sin αnfk2
since k1 = k2.
i2
94 Bramwell’s Helicopter Dynamics
The rotational motion of the blade will add a velocity component Ωr at a blade
section radius r, in the direction j2; then, writing W for the total velocity vector at this
section, we have
W = – V cos αnf cos ψi2 + (V cos αnf sin ψ + Ωr) j2 – V sin αnfk2
Now let the blade flap through angle β about j2 into the final blade position
represented by the axes i, j, k, Fig. 3.19. The relationship between the sets of axes i2,
j2, k2 and i, j, k is the same as in eqn 1.24, i.e.
k
i
i2
k2
N.F.A.
Fig. 3.19 Flapped blade
β
=
cos 0 sin
0 1 0
– sin 0 cos
2
2
2
i
j
k
i
j
k
β β
β β
Using this transformation, W can be written
W = – (V cos αnf cos ψ cos β + V sin αnf sin β)i + (V cos αnf sin ψ + Ωr) j
+ (V cos αnf cos ψ sin β – V sin αnf cos β)k
To complete the calculation of the velocity components we must add the contributions
due to flapping r
.
j β and the relative wind due to the induced velocity – vik. Thus, the
components of relative wind along and perpendicular to the blade section are:
i direction: V cos αnf cos ψ cos β + V sin αnf sin β (3.17)
j direction: –V cos αnf sin ψ – Ωr (3.18)
k direction: –V cos αnf cos ψ sin β + V sin αnf cos β – r
.
β – vi (3.19)
It is generally assumed that the spanwise component of velocity in the i direction
can be neglected. It is also usual to denote the component that is tangential to the
plane of no feathering, or j direction (eqn 3.18), by UT, taken as positive when it
blows from leading to trailing edge, and the k direction or perpendicular component
in eqn 3.19 by UP. Then for small values of β, UP and UT can be written
Rotor aerodynamics and dynamics in forward flight 95
UP = – Vβ cos αnf cos ψ + V sin αnf – r
.
β – vi (3.20)
UT = V cos αnf sin ψ + Ωr (3.21)
We now define
λ′ = (V sin αnf – vi)/ΩR
where vi may be a function of azimuth and radius, and
µ = (V cos αnf)/ΩR
UP and UT can then be written as
U R x P = ( – / – cos ) Ω Ω ′ λ β µβψ
.
= ΩR(λ′ – x dβ/dψ – µβ cos ψ) (3.22)
and
UT = ΩR (x + µ sin ψ) (3.23)
Since UT lies in the no-feathering plane, the local incidence α of the blade can be
written (Fig. 3.20)
α = θ + φ
where
φ = tan
–1
(UP/UT) ≈ UP/UT (3.24)
since φ is a small angle except, perhaps, near the blade root.
When 0 < ψ < 180° the blade is said to be ‘advancing’, and the half of the rotor
disc defined by this range of azimuth angle is referred to as the ‘advancing’ side of
the rotor disc; in this region the relative wind due to the rotational speed of the blade
is increased by a component of the forward speed. Similarly, in the azimuth range
180° < ψ < 360° the blade is said to be ‘retreating’ and to lie in the ‘retreating’ half
of the disc; in this region the forward speed component reduces the relative chordwise
wind.
It is clear that over some part of the retreating blade the forward speed component
will be greater than that due to the rotational speed, i.e. the relative flow will be from
the trailing edge to the leading edge. Referring to eqn 3.23, this occurs when
Fig. 3.20 Velocity components at a blade section
UT
UP
W
φ
θ
96 Bramwell’s Helicopter Dynamics
x + µ sin ψ < 0 and this inequality defines a circular region whose diameter is µR,
Fig. 3.21. This region is known as the ‘reverse flow region’. Since the incidence of
the blade is defined in relation to leading edge to trailing edge flow, it is obvious that
the calculation of the lift and flapping moment in this region must be treated with
some care. Fortunately, the contribution from this region is usually very small, e.g. for
µ = 0.3 the area of the reverse flow region is only 2
1
4
per cent of the total rotor area
and the velocities there will also be small. The dynamic pressure in the reverse flow
region is low; thus, below values of µ of about 0.4, and bearing in mind that an
advance ratio of 0.5 represents a practical maximum for current helicopters, the
effect of the reverse flow on rotor thrust and other performance contributors may be
neglected, as is done in the following sections.
180°
270°
0°
µR
90° UT
UP
Forward
ψ
Fig. 3.21 Reversed flow region
θ
3.8 Calculation of the rotor thrust
The lift dL on an element of the blade is given by
d = d
1
2
2
L W C c r L ρ (3.25)
and, if the thrust dT is taken as the component of the resultant force along the nofeathering axis (Fig. 3.22),
Fig. 3.22 Lift and drag components at a blade section
UP
dD
w
dL
UT θ
φ
Rotor aerodynamics and dynamics in forward flight 97
dT = (dL cos φ + dD sin φ) cos β
≈ dL
Also, having ignored the spanwise component of the induced velocity,
W U U U
2
P
2
T
2
T
2
= + ≈
since UP is usually much smaller than UT.
Assuming constant lift slope a, we can write
CL = aα
as in Chapter 2, where a generally has a value of about 5.7.
The elementary thrust is therefore
d = ( + / ) d
1
2 T
2
P T T a U U U c r ρ θ
= ( + ) d
1
2 T
2
P T a U U U c r ρ θ
= [ ( + sin ) + ( – d /d – cos )
1
2
2 2 2
a R x x ρ θ µ ψ λβψµβψ Ω ′
× (x + µ sin ψ)]c dr (3.26)
Before eqn 3.26 can be integrated, β and λ′ must first be expressed as functions of
ψ. Assuming steady flight conditions, β can be expressed as
β = a0 – a1 cos ψ – b1 sin ψ – a2 cos 2ψ – b2 sin 2ψ – ... (3.27)
so that
dβ/dψ = a1 sin ψ – b1 cos ψ + 2a2 sin 2ψ – 2b2 cos 2ψ – ... (3.28)
Let us take Mangler and Squire’s series for the induced velocity, i.e. let
v v i i0
1
2 0
=1
n D = 4 [ + ( , )cos ] c c x n
n
Σ
∞
α ψ (3.29)
In this, we assume that the expression holds equally for the no-feathering plane as for
the plane it actually applies to, which most nearly corresponds to the tip path plane.
To find the total rotor thrust we calculate the average thrust of a blade taken round
the disc and multiply by the number of blades. To do this it is easier first to average
the elementary thrust, given by eqn 3.26, with respect to azimuth and then integrate
along the blade. The average value of dT over the azimuth range 0 < ψ < 360° is
found to be
d = [ ( + ) + sin – 2 + ]d
1
2
2 3 1
2
2
nf 0 i
1
2
2
2 T ac R x xV xc b x ρ θ µ α λµ Ω
2 ˆ (3.30)
where λi = vi0/ΩR, vi0 being the mean induced velocity, and ˆ
V = V/ΩR.
Assuming for simplicity that the chord c and pitch angle θ are constant along the
blade, integration of eqn 3.30 along the blade gives
98 Bramwell’s Helicopter Dynamics
T ac R V b = [ ( + 3 /2) + sin – + ]
1
4
2 3 2
3 0
2
nf i
1
4
2
2 ρ θ µ α λµ Ω 1 ˆ
since, from eqn 3.14
0
1
0
2
d = (15/8) (1 – ) d
∫ ∫
c x x x x
0
1
η η
= (15/8) 1 – d
3 2
0
1
∫
x x x
=
1
4
It is easy to show that we could have obtained exactly the same result if eqns 3.1
or 3.4 had been used for the induced velocity. Then defining
λ α λ = sin – nf i
ˆ
V (3.31)
the thrust for b blades becomes
T b ac R = [ ( + 3 /2) + ]
1
4
2 3 2
3 0
2
ρ θ µ λ Ω 1 (3.32)
where we have neglected the very small term in b2.
Defining a thrust coefficient by
tc = T/ρsAΩ
2
R
2
eqn 3.42 gives
t
a
c 0
2
=
4
2
3
( + 3 /2) + θ µ λ 1
[ ] (3.33)
Current blades are usually without taper so that it is justifiable to integrate
eqn 3.30 on the assumption that the chord is constant. Most blades have considerable
twist, however, but in Chapter 2 we found that for linear twist the thrust equation in
hovering flight was still valid provided the pitch angle θ0 at
3
4
radius was taken.
Since the thrust equation in forward flight involves only a small extra term in µ
2
,
taking the
3
4
-radius pitch angle for θ0 should be a very good approximation in this
case too. For a tapering blade the chord can also be taken as that at
3
4
radius when
defining the rotor solidity.
3.9 The in-plane H-force
Referring to Fig. 3.23, the force component dH perpendicular to the no-feathering
axis and in the rearward direction is
dH = (dD cos φ – dL sin φ) sin ψ – (dL cos φ + dD sin φ) sin β cos ψ (3.34)
Rotor aerodynamics and dynamics in forward flight 99
For small φ and β, eqn 3.34 becomes
dH = dD sin ψ – dL(β cos ψ + φ sin ψ) (3.35)
The first term of eqn 3.35 can be regarded as the profile drag contribution to H,
while the bracketed term can be regarded as the ‘induced’ component arising from
the inclination of the lift vector.
The profile drag term HP, found by averaging the first term of eqn 3.35 round the
disc, is
H b U c r
R
P
0 0
2
1
2 T
2
= ( /2 ) sin d d π ρ δψψ
π
∫ ∫
(3.36)
Assuming the profile drag coefficient δ and chord c to be constant, we find
H bc R P
1
4
2 3
= ρ δµΩ
The elementary ‘induced’ component is
dHi = – dL(β cos ψ + φ sin ψ)
= – ( + / ) [ cos + ( / ) sin ] d
1
2 T
2
P T P T ρ θ βψ ψ aU U U U U c r 0
= – [( + ) cos + ( + ) sin ] d
1
2 0 T
2
P T 0 P T P
2
ρ θ β ψ θ ψ a U U U U U U c r
The mean value for b blades is
H b ac U U U
R
i
0 0
2
1
2 0 T
2
P T = – ( /2 ) [( + ) cos π ρ θ βψ
π
∫ ∫
+ ( + ) sin ]d d 0 P T P
2
θ ψ ψ U U U r (3.37)
Mangler and Squire’s expression for the induced velocity can be used here but the
analysis becomes rather complicated and, since the terms involving the induced
velocity are fairly small, it is convenient to assume it to be constant. Then substituting
for UP, UT and β from eqns 3.22, 3.23, and 3.27, in which harmonic terms of higher
order than one have been ignored in β, and replacing λ′ by λ we find for H
No-feathering axis
dL cos φ + dD sin φ
sin β (dD sin φ
+
dL cos φ)
dD cos φ – dL sin φ
ψ
Fig. 3.23 Force components in plane of rotor
β
100 Bramwell’s Helicopter Dynamics
H abc R =
1
2
2 3
ρ Ω
×
+
1
3
+
3
4
–
1
2
+
1
4
–
1
6
+
1
4
1 0 1 0 1
2
0 1
µδ
θ λ µλθµ µ
2
0
2
a
a a a a b a (3.38)
We shall see in section 3.12 that, when the induced velocity is constant,
b
a
1
0
1
2
2
=
4 /3
1 +
µ
µ
Ignoring the small term in µ
2
in the denominator, the last two terms in eqn 3.38
reduce to µa0
2
/36, which is very small compared with the other terms and may be
neglected. The coefficient form of the H-force can then be written finally as
h C s
H
sA R
a
a
a a a H c 2 2 1 0 1 0 1
2
= / = =
2
+
1
3
+
3
4
–
1
2
+
1
4
ρ
µδ
θ λ µθλµ
Ω 2
(3.39)
3.10 The rotor torque, Q
The torque dQ about the no-feathering axis on a blade element is, Fig. 3.23,
dQ = r(dD cos φ – dL sin φ)
or d = d – d
1
2 T
2 1
2 T
2
Q U cr r U C cr r L ρ δ ρ φ
The first term denotes the torque due to the profile drag. Calling this QP we have,
for b blades,
Q b U cr r
R
P
0 0
2
1
2 T
2
= ( /2 ) d d π ρ δψ
π
∫ ∫
= ( /2 ) ( + sin ) d d
2 4
0
1
0
2
2
b c R x x x π ρ δ µ ψ ψ
π
Ω
∫ ∫
= ρbcδ Ω
2
R
4
(1 + µ
2
)/8 (3.40)
assuming the chord and drag coefficient to be constant.
The mean induced torque Qi is
Q b U C cr r
R
L i
0 0
2
1
2 T
2
= – ( /2 ) d d π ρ φψ
π
∫ ∫
= – ( /4 ) ( + ) d d
0 0
2
0 P T P
2
ρ π θ ψ
π
abc U U U r r
R
∫ ∫
The integrand can be expanded and averaged as for the H-force but this is not
Rotor aerodynamics and dynamics in forward flight 101
necessary and it is more convenient and instructive to proceed as follows. The integrand,
including the scalar term
1
2 ρac, can be written
1
2 0 P T P
2 1
2 0 P T P
2
T nf
( + ) = ( / )( + )( – cos sin ) ρ θ ρ θ α ψ ac U U U r ac U U U U V Ω
since, from eqn 3.21,
Ωr = UT – V cos αnf sin ψ
or r = (UT – V cos αnf sin ψ)/Ω
Hence
1
2 0 P T P
2
( + ) d ρ θ ac U U U r r
=
1
2
( + ) d –
1
2
cos
( + ) sin d
P
0 T
2
P T
nf
0 P T P
2
ρ
θ
ρ α
θ ψ
acU
U U U r
acV
U U U r
Ω Ω
= d –
1
2
cos
( + ) sin d
P nf
0 P T P
2
U
T
acV
U U U r
Ω Ω
ρ α
θ ψ
But, from the previous sections,
1
2 0 P T P
2
i
1
2 0 T
2
P T ( + ) sin d = – d – ( + ) cos d ρ θ ψ Η ρ θ β ψ ac U U U r ac U U U r
= – d – d cos i Η β ψ T
so that
1
2 0 P T P
2
( + ) d ρ θ ac U U U r r
= d + (d cos + d )
cos P
i
nf U
T T H
V
Ω Ω
β ψ
α
=
d
sin – –
d
d
+ d
cos
nf i0 i
nf T
V r H
V
Ω
Ω
Ω
α
β
ψ
α
v
(3.41)
since UP = V sin αnf – vi0 – Ωr dβ/dψ – V cos αnf β cos ψ.
Now
0
A d d /d = d /d
R
r T M
∫
β ψ β ψ
and from eqn (1.2) for zero flapping hinge offset
MA = B Ω
2
(d
2
βs/dψ
2
+ βs) = BΩ
2
(d
2
β/dψ
2
+ β)
in which βs is used temporarily to indicate that eqn (1.2) was derived for flapping
defined relative to a plane perpendicular to the shaft axis. The second equality is
easily derived using a1 = a1s + B1 and b1 = bis + A1.
Therefore the mean value of – d d /d
0
R
r T
∫
β ψ with respect to azimuth is
102 Bramwell’s Helicopter Dynamics
–
2
d
d
+
d
d
d
2 2 2
BΩ
π
β
ψ
β
β
ψ
ψ
π
0
2
∫
= –
4
d
d
d
d
d +
d
d
( )d
2 2 2 2
2 BΩ
π ψ
β
ψ
ψ
ψ
β ψ
π π
0 0
∫ ∫
= –
4
d
d
+
2
2
0
2
2
0
2
BΩ
π
β
ψ
β
π
π
[ ]
= 0
Since the terms in the brackets are periodic and are therefore identical at the limits.
The other terms multiplying dT and dHi in eqn 3.41 are constants so that
Q
T
V H
V
i nf i0 i
nf
= – ( sin – ) +
cos
Ω Ω
α
α
v
= – (Tλ + Hiµ)R
and the total torque is
Q = QP + Qi
= ρbcδ Ω
2
R
4
(1 + µ
2
)/8 – (Tλ + Hiµ)R
= ρbcδ Ω
2
R
4
(1 + 3µ
2
)/8 – (Tλ + Hµ)R (3.42)
The torque coefficient is
qc = Q/ρbcRΩ
2
R
3
= Q/ρsAΩ
2
R
3
= δ(1 + 3µ
2
)/8 – λtc – µhc (3.43)
The term – (λtc + µhc) might have been expected on physical grounds since it is
the scalar product, in non-dimensional form, of the resultant rotor force and resultant
flow through the rotor, i.e. it represents the work done by the rotor in producing the
rotor force. The first term of eqn 3.43 represents, of course, the torque required to
overcome the profile drag.
Equation 3.43 has been derived on the assumption that the spanwise velocity
component can be neglected. While this is a reasonable assumption for the calculation
of the lift, Bennett
8
has argued that this does not apply to the drag which depends on
the resultant velocity over the blade. From Bennett’s calculations, the profile drag
contribution to the torque should be written as δ (1 + nµ
2
)/8, where n has the following
values
µ 0 0.3 0.6 1
n 4.5 4.58 4.66 4.67
Rotor aerodynamics and dynamics in forward flight 103
From similar calculations, Stepniewski
9
proposes the expression δ (1 + 4.7µ
2
)/8.
We should also consider the increase of induced power due to the non-uniformity
of the induced velocity by introducing the factor k as discussed in Chapter 3. The
expression for the torque coefficient then becomes
qc = δ (1 + 4.7µ
2
)/8 – λtc – µhc + kλitc (3.44)
3.11 Blade flapping
The aerodynamic flapping moment dMA about the hinge due to the elementary lift is
d = d = + d A
1
2 T
2 P
T
M r L aU
U
U
cr r ρ θ
=
1
2
2 4
ρac R Ω
× ′
( + sin ) + –
d
d
– cos ( + sin ) d
2
θ µ ψ λ
β
ψ
µβ ψ µ ψ x x x x x
(3.45)
Expanding eqn 3.45 and integrating, assuming constant pitch angle, gives
M ac R A
1
2
2 4
= ρ Ω
×
( ) ′
∫
1
4
+
2
3
sin +
1
2
sin + d –
1
4
d
d
–
1
3
cos 0
2 2
0
1
2
θ µ ψ µ ψ λ
β
ψ
µβ ψ x x
+ sin d –
1
3
d
d
sin –
1
2
sin cos
0
1
2
µ ψ λ µ
β
ψ
ψ µ β ψψ
∫
′
x x (3.46)
It can easily be shown that for hovering flight the assumption of linear taper leads
to the same flapping moment as eqn 3.46, provided the pitch angle is taken as that at
0.8R (cf. 0.75R in the calculation of the thrust). This value will be changed slightly
when variable induced velocity is included, but it will be assumed that in calculations
involving a twisted blade it will be sufficiently accurate to use eqn 3.46 with the
collective pitch angle taken as that at 0.8R.
To evaluate the inflow integrals in eqn 3.46 we again use Mangler and Squire’s
induced velocity distribution (eqn 3.13), considering the first harmonic only. Then
0
1
2
0
1
2
nf i
0
1
0
2
i
0
1
1
2
d = sin d – 2 d + 4 cos d
∫∫ ∫ ∫
′ λ α λλψ x x x V x c x x c x x
ˆ
=
1
3
sin –
15
4
(1 – ) d nf i
0
1
4 2
ˆ
V x x x α λ
∫
√
–
15
64
cos (9 – 4) d i
1/2
0
1
2 3 π
λ ν ψ
∫
x x x
104 Bramwell’s Helicopter Dynamics
=
1
3
sin –
15
128
–
15
128
cos nf i i
1/2
ˆ
V α
π
λ
π
λ ν ψ
in which ν = (1 – sin αD)/(1 + sin αD).
To a reasonable approximation this integral may be expressed as
0
1
2 1/2
i d = ( – 1.1 cos )/3
∫
′ λ λ νλψ x x
The second integral in eqn 3.46 is
0
1
0
1
nf i
0
1
3 2
d = sin d – (15/4) (1 – ) d
∫ ∫ ∫
′ √ x x xV x x x x λ α λ
ˆ
–
15
64
cos (9 – 4) d i
1/2
0
1
2 2 π
λ ν ψ
∫
x x x
= sin – –
7
64
cos
1
2 nf
1
2 i i
ˆ /
V α λ
π
λ ν ψ
1 2
= ( – .69 cos )
1
2 i λ λ νψ 0
1 2 /
Hence,
M ac R A
2 4
0
2 2
=
1
8
1 +
8
3
sin + 2 sin ρ θ µ ψ µ ψ Ω
( )
+
4
3
– 1.46 cos –
d
d
–
4
3
cos + 2 sin
1/2
i λ ν λψ
β
ψ
µβ ψµλψ
–
4
3
d
d
sin – sin2 – 0.69 sin 2
2
i
1/2
µ
β
ψ
ψ µ β ψ µλν ψ (3.47)
The flapping equation of a centrally hinged blade relative to a plane perpendicular
to the shaft axis has been found in Chapter 1 to be
d
2
βs/dψ
2
+ (1 + ε)βs = MA/BΩ
2
(1.9)
As before, with β defined from the no-feathering plane, the equation can be shown
to be exactly similar in form, i.e.
d
2
β/dψ
2
+ (1 + ε)β = MA/BΩ
2
Then, on using eqn 3.47 and rearranging, we obtain the differential equation of
flapping in the form
d
d
+
8
1 +
4
3
sin
d
d
+ 1 + +
4
3
cos + sin 2
2
2
2
β
ψ
γ
µ ψ
β
ψ
ε
γ
µ ψ µ ψβ
8
Rotor aerodynamics and dynamics in forward flight 105
=
8
+
8
3
sin + 2 sin +
4
3
– 1.46 cos 0
2 2 1/2
i
γ
θ µ ψ µ ψλνλψ 1
+ 2 sin – 0.69 sin 2 i µλ ψ µλν ψ
1 2 /
(3.48)
Equation 3.48 is a linear equation with periodic coefficients and there is no known
solution in closed form. Moreover, as discussed in section 3.7, it is valid only for the
advancing region 0 < ψ < 180°, since in the reverse flow area the lift and flapping
moment are incorrectly evaluated. For the speeds typical of present day helicopters,
however, this results in negligible error, as was stated earlier.
The free motion of the blade is found by putting the terms on the right-hand side
of eqn 3.48 equal to zero. The flapping equation is then
d
d
+
8
1 +
4
3
sin
d
d
+ + +
4
3
cos + sin 2 = 0
2
2
2
β
ψ
γ
µ ψ
β
ψ
ε
γ
µ ψ µ ψβ
( ) ( )
1
8
(3.49)
Considerable attention has been given to this equation, since its solution answers
the important question of the stability of the flapping motion. In hovering flight,
µ = 0, as we have seen already, the equation reduces to one with constant coefficients,
eqn 1.10, and it was found that the corresponding motion is heavily damped. It is
reasonable to expect that the damping would remain high for low values of µ
but further investigation is required to find the effect of the periodic terms for high
values of µ. As we shall see shortly, the denominator of one of the flapping coefficients
of steady motion is the term 1 – µ
2
/2, indicating that infinite flapping amplitudes
might be expected to occur at µ = √2.
Several attempts to solve eqn 3.49 analytically have been made, notably by Glauert
and Shone,
10
Bennett,
11
Horvay,
12
Shutler and Jones,
13
and Lowis.
14
Of these, only
Lowis has attempted to take the reverse flow area into account. The others indicate
that the flapping motion appears to be stable for µ < 1 but their results are not really
valid for values of µ greater than about 0.7 because of the neglect of the reverse-flow
region. The last three authors make use of Floquet’s theorem, which states that an
equation of the type eqn 3.49 has a solution of the form
β α ψ α ψ
ν ψ νψ
= e ( ) + e ( ) 1 1 2 2
1 2
P P (3.50)
where α1, α2, ν1, ν2 are constants and P1(ψ), P2(ψ) are periodic functions of period
2π. Hovering flight is a special case of eqn 3.50 with solution
β α νψα νψ
νψ
= e [ sin (1 – ) + cos (1 – ) ] 1
2 1/2
2
2 1/2 –ˆ
ˆ ˆ
where ˆ
ν = γ /16.
The stability of the motion is determined by the values of ν1 and ν2, which may not
be real, and the investigations mentioned have been directed to finding their values.
106 Bramwell’s Helicopter Dynamics
An exact analytical determination is not possible, and ν1 and ν2 must be evaluated
numerically for a range of flight parameters. Lowis found an approximate method of
taking the reverse flow region into account which amounted merely to changing the
sing of γ in eqn 3.49 over a range of ψ in the retreating region depending on the value
of µ. His results showed that flapping instability occurs for µ in the range 2.2 to 2.8,
depending on the inertia number γ.
Another method of dealing with eqn 3.49 is to use computational methods. The
reversed-flow area and periodic coefficients can be easily and exactly taken into
account. Of course, the output gives the flapping response to a given set of conditions,
which does not provide as much information as the values of ν1 and ν2. Nevertheless,
by suitably choosing the initial conditions, the response can give an adequate picture
of the flapping behaviour and stability. Investigations of this kind have been made by
Wilde and Bramwell
15
and Sissingh
16
. The results obtained by Wilde and Bramwell
for γ = 6 and ε = 0 are shown in Fig. 3.24; they agree closely with the results of Lowis
and Sissingh.
3.12 The flapping coefficients
In the previous section the free blade motion was discussed; we now wish to find the
forced blade motion, that is, the steady motion in forward flight corresponding to a
given collective pitch angle, tip speed ratio, and inflow ratio – these conditions
completely define the operating state of the rotor. To do this we assume that the
transient response is stable and that, as in previous sections, the forced blade motion
is periodic and can be expressed as
β = a0 – a1 cos ψ – b1 sin ψ – a2 cos 2ψ – b2 sin 2ψ – …
remembering that β is defined relative to the plane of no-feathering.
This expression is substituted into eqn 3.48 and, assuming that it represents a
solution, the coefficients of the terms in sin ψ, cos ψ, … on the left- and right-hand
sides of eqn 3.48 can be equated. If we consider only the constant term and the two
first harmonic terms, we obtain, after manipulation of terms such as sin ψ, sin 2ψ,
etc.,
Fig. 3.24 Blade flapping response at high tip speed ratios
µ = 0.5
β
Time
µ=2.25
Rotor aerodynamics and dynamics in forward flight 107
a0 0
2
=
8(1 + )
(1 + ) +
4
3
γ
ε
θ µ λ
[ ] (3.51)
a b 1
0
2 2 1 =
2 (4 /3 + )
1 – /2
+
8
.
– /2
µ θ λ
µ γ
ε
µ 1
(3.52)
b
a
a 1
0
1/2
i
2 2 1 =
4( + 1.1 )/3
1 + /2
–
8
.
+ /2
µ ν λ
µ γ
ε
µ 1
(3.53)
If Glauert’s formula, eqn 3.4, for the induced velocity had been used instead of
Mangler and Squire’s, the values of a0 and a1 would be unaltered but it would have
been found that
b
a K
a 1
0 i
2 2 1 =
4( + 1.7 )/3
1 + /2
–
8
.
+ /2
µ λ
µ γ
ε
µ
5
1
(3.54)
For typical values of flapping hinge offset the terms in ε are usually negligible
except, perhaps, in the equations for b1 (eqns 3.53 and 3.54) at high values of µ.
When ε = 0, the formulae reduce to their classical forms:
a0 0
2
=
8
(1 + ) +
4
3
γ
θ µ λ
[ ] (3.55)
a1
0
2
=
2 (4 /3 + )
1 – /2
µ θ λ
µ
(3.56)
b
a
1
0
1/2
i
2
=
4( + 1.1 )/3
1 + /2
µ ν λ
µ
(3.57)
The case of constant induced velocity is found by putting vi = vi0 in eqn 3.13,
whence it is seen that the term involving ν
1/2
in eqns 3.53 and 3.57 is absent.
Alternatively, letting K = 0 in eqn 3.54 leads to the same result.
Higher order flapping coefficients can be obtained by considering higher harmonics
of the flapping motion, but it is found that the coefficients appear explicitly as the
solutions of an infinite chain of simultaneous equations and cannot be evaluated
easily. Stewart
17
has extracted the coefficients up to the fourth harmonic and has
shown that their magnitudes decrease very rapidly with order of harmonic. As a
rough rule it was found that the magnitude of a coefficient was about one tenth of the
value of that of the next lower harmonic. The calculation of the higher harmonics, as
has been mentioned in Chapter 1, is somewhat academic since the effects of the
higher modes of blade bending are at least as great as these harmonics of the first
(rigid) blade mode.
However, it should be noted here that the higher harmonic terms do become
important when considering rotor induced vibration (see Chapter 8). The simple rigid
blade flapping model used in the current analysis may be seen to give rise to first and
108 Bramwell’s Helicopter Dynamics
second harmonic forcing terms on the right-hand side of eqn 3.48 (or 1 Ω and 2 Ω
terms, to use the terminology adopted in Chapter 8), but in a more general case with
a flexible blade flapping model, forcing terms are generated over a wide range of
harmonics.
3.13 Force and torque coefficients referred to disc axes
It is useful to obtain expressions for the force and torque coefficients when referred
to the tip path plane axis (or rotor-disc axis) instead of the no-feathering axis. In
Chapter 1, section 1.13, we saw that, since the angle between these two axes is the
small flapping angle a1, we have the relationships
TD ≈ T
HD ≈ H – Ta1
For the tip-speed and inflow ratios we also have
µ α µ D D = cos
ˆ
V ≈
λ α λλµ D D i = sin – + ,
ˆ
V a ≈
where TD, HD, µD, and λD are referred to the tip path plane.
Substituting for λ in eqns 3.55 to 3.57 for the flapping coefficients, we have for a1
a
a
1
0 D 1
2
=
2 (4 /3 + – )
1 – /2
µ θ λ µ
µ
remembering that this is still with reference to the plane of no-feathering. On solving
for a1 we get
a1
0 D
2
=
2 (4 /3 + )
1 + 3 /2
µ θ λ
µ
(3.60)
The coning angle a0 becomes
a a 0 0
2
D 1 =
8
(1 + ) +
4
3
–
4
3
γ
θ µ λ µ
and on substituting eqn 3.60 we have
a0 0
2 4
2 D
2
2
=
8
– 19 /18 + 3 /2
1 + 3 /
+
4
3
– /2
+ 3 /2
γ
θ
µ µ
µ
λ
µ
µ
1
2
1
1
(3.61)
The lateral flapping coefficient b1 remains unaltered:
b
a
1
0
1/2
i
2
=
4( + 1.1 )/3
1 + /2
µ ν λ
µ
(3.57)
Rotor aerodynamics and dynamics in forward flight 109
The thrust coefficient can be written
t t
a
a c c 0
2
D 1 D
= =
4
2
3
(1 + 3 /2) + – θ µ λ µ
(3.62)
and on substituting for a1 we have
t
a
c 0
2 4
2 D
2
2 D
=
4
2
3
– + 9 /4
1 + 3 /2
+
– /2
+ 3 /2
θ
µ µ
µ
λ
µ
µ
1 1
1
(3.63)
The H-force coefficient takes the simple form
h
a
a c
1
4
D 1
2 1 0 D
= +
4
[ – ] µδ
λ
µθ (3.64)
which can also be written as
h
a
c
1
4
D 0
2
2
D
2 D
= +
4
( /3)(1 – 9 )
1 + 3 /2
+
1 + 3 /2
µδ
µλ θ µ
µ
λ
µ
/2
(3.65)
Finally, the torque coefficient can easily be seen to have the same form as
eqn 3.43, that is
q t h c
2
D c c = (1 + 3 )/8 – –
D D
δ µ λ µ (3.66)
On considering the resultant blade flow velocity and induced power coefficient we
also have the same result as eqn 3.44, i.e.
q t h k t c
2
D c c i c = (1 + 4.7 )/8 – – +
D D
δ µ λ µ λ (3.67)
3.14 Comparison with experiment
In order to arrive at fairly simple formulae for the force and flapping coefficients, a
number of simplifying assumptions were made in the analysis and it is important to
test the accuracy of the results by comparisons with experimental data. Squire et al.
18
have conducted wind tunnel tests on a 3.65 m diameter rotor and compared the
results with theoretical values. Generally speaking, the agreement was found to be
good. Since the ranges of parameters in Squire’s tests were arbitrary, many of the
combinations were outside the range of normal helicopter operations. Much more
recently, Harris
19
has conducted tests on a 1.53 m diameter rotor which contains a
large number of cases in which the collective pitch was adjusted so that the thrust
coefficient was kept constant at a value typical of steady flight. Thus, the variation of
µ in the tests corresponded to a helicopter changing its forward speed under conditions
of trim. Harris also measured the coning angle and other quantities not considered in
Squire’s tests.
The force and flapping coefficients contain two quantities, namely, the blade lift
110 Bramwell’s Helicopter Dynamics
slope a and the drag coefficient δ, which are not known for a particular rotor although,
of course, in the absence of data, reasonable assumptions can be made. In the tests
described by Harris, the thrust and torque were measured over a range of collective
pitch angles at a nominal tip speed ratio of 0.08. Now we saw earlier in this chapter
that the mean induced velocity can be found from
vi0 = T/2ρAV
which can be expressed as
λi = stc/2µ
provided V > 1, or as µ > λi, hov. This inequality is satisfied for the tests made at
µ = 0.08 referred to above.
The thrust coefficient can be written from eqn 3.33 as
t
a st
c 0
2
nf
c
=
4
2
3
(1 + 3 /2) + –
2
θ µ µα
µ
or t
as a
c 0
2
nf
1 +
8
=
4
2
3
(1 + 3 /2) +
µ
θ µ µα
[ ]
Differentiating with respect to θ0 gives
∂
∂
t a
+ as
c
0
2
=
(1 + 3 /2)/6
1 /8 θ
µ
µ
With µ and solidity s known, the slope of the thrust coefficient with collective
pitch depends only on the blade lift slope a. Thus a can be determined from the slope
of a graph of thrust coefficient against collective pitch. Figure 3.25 shows such data
from Harris’s tests, from which we obtain ∂tc/∂θ0 = 0.523 and a = 5.5. The variation
of qc against collective pitch is shown in Fig. 3.26. When tc = 0 we have qc = δ/8
(since µhc is extremely small) and from the figure we find that δ/8 = 0.0018, giving
δ = 0.0144.
With a and δ deduced from the tests we can now calculate force, torque, and
o Experimental
— Theoretical
0.1
0.05
0 5° 10° 15°
θ0.75
Fig. 3.25 Thrust coefficient as a function of collective pitch
tc
Rotor aerodynamics and dynamics in forward flight 111
0.01
0.005
0 5° 10° 15°
θ0.75
αnf = –1.4°
µ = 0.08
qc = 0.0018 + 1.13tcλ
Fig. 3.26 Torque coefficient as a function of collective pitch
flapping coefficients. These are shown in Figs 3.27 to 3.31, together with the
corresponding measured values.
It can be seen from Figs 3.27 and 3.26 that the agreement between theoretical and
measured values of thrust and torque is very good; the slight discrepancy at larger
values of collective pitch in the case of qc probably indicates that the value of the
profile drag coefficient δ should be higher in this region. The theoretical values of
hc, on the other hand, show less good agreement with experiment, Fig. 3.28, but it
should be noted that, unlike tc, hc represents only a small component of the resultant
rotor force and that the expression for hc does not take into account the effects of the
variable induced velocity since, as was mentioned in section 3.9, this would result in
a very complicated analysis. It is easy to account for the effect of the induced velocity
on the total power, however, since we need only apply a factor to the induced power,
as discussed earlier in this chapter. As can be seen from Fig. 3.29, the agreement
between the estimated and measured torque coefficients is excellent.
Agreement between the theoretical and experimental values of the flapping angles
is less satisfactory. The theoretical value of the coning angle a0, as one would expect,
is zero at the value of collective pitch angle for which the thrust also vanishes, and
it is not understood why the measured values show a significant coning angle at this
point or why the slope with collective-pitch angle is less than the theoretical value,
Fig. 3.30. Fortunately, since the coning angle plays very little part in performance
and stability estimations, the discrepancy is not serious.
Comparison between theoretical and experimental values of the backward flapping
angle a1, Fig. 3.31, displays a tendency previously observed
20
in connection with
–8 –6 –4 –2 0 2 4 6
0.1
0.06
0.04
tc
αnf
µ = 0.08
θ0.75 = 8.97°
Fig. 3.27 Thrust coefficient as a function of shaft angle
qc
112 Bramwell’s Helicopter Dynamics
0.01
0.005
0 0.1 0.2 0.3
hc
µ
tc = 0.08
Fig. 3.28 H-force coefficient as a function of tip speed ratio
0.008
0.006
0.004
0.002
0 0.1 0.2 0.3
qc
µ
Fig. 3.29 Torque coefficient as a function of tip speed ratio
0 5° 10° 15°
5°
4°
3°
2°
1°
θ 0.75
a0
Fig. 3.30 Coning angle as a function of collective pitch
Rotor aerodynamics and dynamics in forward flight 113
µ
5°
4°
3°
2°
1°
0 0.1 0.2 0.3
a1
Squire’s results, namely, that the theoretical values underestimate the actual flapping
angle by about 10 to 20 per cent. This is probably due to the simplifying assumptions
made for the induced velocity distribution.
The longitudinal induced velocity distribution has a very pronounced influence on
the sideways flapping angle b1. As was discussed in section 3.12, the sideways
flapping can be attributed to two effects: the incidence variation due to coning and
that due to the longitudinal induced velocity distribution. As can be seen from Fig.
3.32, nearly all of the sideways flapping at low speeds is due to the longitudinal
induced velocity distribution, and calculations depend strongly on the assumptions
made. Harris
19
has considered a number of simple expressions which have been used
for representing the longitudinal induced velocity distribution but none of them
shows the ‘peakiness’ evinced by the measured values. Equation 3.53 appears to give
the best agreement of those expressions examined.
Fig. 3.31 Longitudinal flapping as a function of tip speed ratio
4°
3°
2°
1°
0 0.1 0.2 0.3
b1
eqn 3.53
eqn 3.54 (K = 1.2)
4µa0/3
µ
Fig. 3.32 Lateral flapping as a function of tip speed ratio
114 Bramwell’s Helicopter Dynamics
References
1. Glauert, H., ‘A general theory of the autogyro’, Aeronautical Research Council R&M 1111,
1926.
2. Bramwell, A. R. S., ‘Some remarks on the induced velocity field of a lifting rotor and on
Glauert’s formula’, Aeronautical Research Council CP 1301, 1974.
3. Coleman, R. P., Feingold, A. M. and Stempin, C. W., ‘Evaluation of the induced velocity field
of an idealized helicopter rotor’, NACA ARR L5E10, 1947.
4. Mangler, K. W. and Squire, H. B., ‘The induced velocity field of a rotor,’ Aeronautical
Research Council R&M 2642, 1950.
5. Brotherhood, P. and Stewart, W., ‘An experimental investigation of the flow through a helicopter
in forward flight’, Aeronautical Research Council R&M 2734, 1949.
6. Heyson, H. H. and Katzoff, S., ‘Induced velocities near a lifting rotor with non-uniform disc
loading’, NACA Rep. 1319, 1958.
7. Lock, C. N. H. ‘Further developments of autogyro theory’, Aeronautical Research Council
R&M 1127, 1927.
8. Bennett, J. A. J., ‘Rotary wing aircraft’, Aircraft Engineering, March 1940.
9. Stepniewski, W. Z., ‘Basic aerodynamics and performance of the helicopter’, AGARD Lect.
Ser. 63, 1973.
10. Glauert, H. and Shone, G., ‘The disturbed motion of the blades of a gyroplane’, Aeronautical
Research Council Paper 993, 1933.
11. Bennett, J. A. J., ‘Rotary wing aircraft’, Aircraft Engineering, May 1940.
12. Horvay, G., ‘Rotor blade flapping motion’, Q. Appl. Math., July 1947.
13. Shutler, A. G. and Jones, J. P., ‘The stability of rotor blade flapping motion’, Aeronautical
Research Council R&M 3178, 1958.
14. Lowis, O. J., ‘The effect of the reverse flow on the stability of rotor blade flapping motion at
high tip speed ratios’, Aeronautical Research Council Paper 24 431, 1963.
15. Wilde, E., Bramwell, A. R. S. and Summerscales, R., ‘The flapping behaviour of a helicopter
rotor at high tip speed ratios’, Aeronautical Research Council CP 877, 1966.
16. Sissingh, G. J., ‘Lifting rotors operating at high speeds and advance ratios’, AGARD Conf.
Proc. CP–22, paper 5 (part II), 1967.
17. Stewart, W., ‘Higher harmonics of flapping on the helicopter rotor’, Aeronautical Research
Council CP 121, 1952.
18. Squire, H. B., Fail, R. A. and Eyre, R. C. W., ‘Wind tunnel tests on a 12 ft helicopter rotor’,
Aeronautical Research Council CP 2695, 1949.
19. Harris, F. D., ‘Articulated rotor blade flapping motion at low advance ratio’, J. Amer. Helicopter
Soc., January 1972.
20. Bramwell, A. R. S., ‘Stability and control of the single rotor helicopter’, Aeronautical Research
Council R&M 3104, 1959.
4
Trim and performance in axial
and forward flight
4.1 Introduction
Having obtained formulae for the rotor forces and blade flapping motion in Chapter
3, we are now in a position to solve the trim equations derived in Chapter 1. In
Chapter 1 it was possible to draw some general conclusions relating to trimmed flight
with little specific reference to the values of the flight parameters such as collective
pitch, inflow ratio, and so on. We now seek a method which enables us to calculate
these quantities for a given helicopter under given steady-flight conditions. These
values are necessary, not only because one wishes to know the control displacements
required to maintain a given flight condition, but because these, and other, parameters
are needed for performance and stability calculations.
Having solved the trim equations, one can then calculate the corresponding power
and establish relationships between a given set of flight conditions and the power
required to achieve them. It is also of interest to calculate such quantities as the
maximum speed, maximum rate of climb, etc., and for this purpose it is easier to
consider the maximum power available to the rotor and to solve an energy equation.
If the trim and energy equations are based on the same set of assumptions, it is, of
course, merely a matter of convenience as to which ones are used to obtain the
desired quantities.
The following analysis and discussion will be based on the comparatively simple
formulae derived in Chapter 3. The reader is reminded that these formulae are only
approximate and that their derivation has been made possible only by making a
number of simplifying assumptions, the chief of which are that the lift slope and
drag coefficient of the blade section are constant, and that blade stall does not occur.
The introduction of more complicated aerodynamic data, in which the lift and drag
are both arbitrary functions of incidence and Mach number, requires the use of
computational methods for solution of the trim equations and calculation of performance.
However, the simplified equations are often adequate for all but the most advanced
116 Bramwell’s Helicopter Dynamics
design work and have the important advantage that they enable a physical interpretation
of helicopter flight to be made easily.
4.2 Helicopter trim in forward flight
In Chapter 1 the longitudinal and lateral trim equations were derived. Because of the
asymmetry of the helicopter, e.g. the presence of a tailrotor in a single rotor helicopter,
the longitudinal and lateral equations should, strictly speaking, be solved simultaneously;
indeed, in a very thorough analysis of helicopter trim, Price
1
finds that the various
trim parameters are related through no less than fourteen equations. However, the
complicated process of having to satisfy such a large number of equations simultaneously
is not necessary in practice, especially as the accuracy of some of the aerodynamic
data would hardly justify such detail. The longitudinal and lateral equations will
therefore be treated as separate groups and solved independently of one another.
Now it was stated in Chapter 1 that the resultant rotor force is almost perpendicular
to the rotor tip path plane, i.e. the H-force is small when the rotor force components
are referred to the tip path plane axes. Because of this fact these axes are very useful
for investigating helicopter trim, as it is much easier, when using the corresponding
force coefficients, to establish the rotor incidence αD and thence to obtain the other
parameters.
4.2.1 Longitudinal trim
Referring to Fig. 4.1, we can write the trim equations given in Chapter 1, eqns 1.41
and 1.42, with reference the tip path plane or disc axes.
Resolving forces vertically and horizontally we have
TD cos (αD + τc) – HD sin (αD + τc) = W + D sin τ (4.1)
TD sin (αD + τc) + HD cos (αD + τc) = – D cos τ (4.2)
Now the angle αD + τ is the inclination of the rotor-disc plane to the horizontal and
HD
V
αD
TD
a1– B1
D
Horizon
W
V
Fig. 4.1 Forces and moments in longitudinal plane
τc
Trim and performance in axial and forward flight 117
is usually a small angle in steady flight. Thus the usual small angle assumptions can
be applied to eqns 4.1 and 4.2 which then become, approximately,
TD ≈ T = W + D sin τc (4.3)
T(αD + τc) + HD = – D cos τc (4.4)
where the term HD sin (αD + τc) has been neglected. The angle of climb τc might not
be a small angle.
Let the helicopter fuselage drag D be written as
D V S =
1
2
2
FP ρ (4.5)
where SFP is the so-called equivalent flat plate area.
Then expressing eqns 4.3 and 4.4 in coefficient form by dividing through by
ρsAΩ
2
R
2
gives
t w V d c c
1
2
2
0 c D
= + sin
ˆ
τ (4.6)
and
t h V d c D c c
1
2
2
0 c D D
( + ) + = – cos α τ τ
ˆ (4.7)
where wc is a weight coefficient, d0 = SFP/sA, and ˆ
V V R. = /Ω
Solving eqn 4.7 for αD gives
α τ τ D
1
2
2
0 c c c c = – ( cos + ) / –
D D
ˆ
V d h t (4.8)
tcD having been obtained from eqn 4.6. Unless the rotor-disc incidence is unusually
large, it is usual to take t w c c D
= . With this approximation, the only unknown quantity
in eqn 4.8 is hcD . Now, from numerical calculations, it appears that the most important
term of hcD is the first one of eqns 3.64 or 3.65, i.e. the term representing the rotor
profile drag. Therefore as a good first approximation to hcD we can write
hc
1
4 D
= µδ (4.9)
and obtain, from eqn 4.8, the first approximation to αD.
The inflow ratio can now be calculated from
λ α λ D D i
= sin –
ˆ
V
= µD tan αD – λi
or, approximately,
λD = (µα)D – λi (4.10)
To obtain λi we can use the values of vi of Fig. 3.2 which applies when αD ≈ 0. Now
λi i
0
= v
v
ΩR
(4.11)
118 Bramwell’s Helicopter Dynamics
and V V
R R
=
0
D
0
ˆ Ω Ω
v v
≈ µ (4.12)
where v0 = √(T/2ρA) is the mean induced velocity in hovering flight or ‘thrust
velocity’, and µD = ˆ
V cos D α . Then, if v0/ΩR is computed, V can be obtained from
eqn 4.12 and, with vi being read off from Fig. 3.2, λi can be finally calculated from
eqn 4.11.
With λD now known, or rather a first approximation to it, and taking t w c c D
= ,
eqn 3.63 can be solved for the collective pitch angle θ0. These values of λD and θ0
enable a better approximation to hcD to be calculated from eqn 3.64 (or 3.65) and
new values of αD and λD to be obtained from eqns 4.8 and 4.10 respectively. In most
cases, however, the improved value of hcD makes very little difference to αD, as
indicated by the numerical example below, which also indicates the usefulness of the
disc axes for these calculations. Finally, the flapping coefficients can be calculated.
Example. To illustrate the procedure just described, we shall calculate the trim of a
four-bladed helicopter in level flight at sea level at tip speed ratio 0.3. The helicopter
is represented by the following data:
W = 45 000 N, solidity s = 0.05, R = 8 m, h = 0.25
δ = 0.013, ΩR = 208 m/s, SFP = 2.3 m
2
, b = 4, a = 5.7
Blade data: Mb = 74.7 kg; in terms of R, xg = 0.45, e = 0.04
It will be assumed that the fuselage pitching moment, Mf, including, possibly, a
tailplane, is zero.
The above data give
sA d w d = 10 m , = 0.23, = 0.085, = 0.0104 = 9.6 m/s
2
0 c
1
2
2
0 0 µ v
The first approximation to hcD is
1
4 = 0.000 975, µδ and hence
α µ D
1
2
2
0 c c = – ( + )/
D
d h w
= – 0.134 = – 7.67°
Now v0/ΩR = 0.0462, so that V R = / = 6.50. 0 µΩ v From Fig. 3.2 or eqn 3.2 we
find vi0 = 0.154,from which λi = v v i0 0/ ΩR = 0.0071.
Then, from eqn 4.10,
λD = – 0.0473
Solving eqn 3.63 for θ0 gives
θ0 = 0.1824 = 10.5°
To obtain a better value of hcD we need an estimate of a1. From eqn 3.60, and
using the above values of θ0 and λD, we find
a1 = 0.104 = 5.93°
Trim and performance in axial and forward flight 119
and, using eqn 3.64 or 3.65, we calculate hcD , giving
hcD = 0.001 172
This value is about 20 per cent greater than the first approximation,
1
4 , µδ but the
new value of λD using this revised value of hcD is
λD = – 0.0479
which is close to one per cent of the first value. The small difference in λD using the
recalculated value of hcD is also found at other values of µ, suggesting that the
original approximation to hcD leads to sufficiently accurate values of λD and θ0. In
any case, it is very unlikely that the fuselage drag, upon which λD and αD depend
strongly, would be known accurately enough to justify ‘exact’ calculations of the
H-force.
From eqns 3.61, 3.57, and 3.66, we find
a0 = 0.066 = 3.78°, b1 = 0.037 = 2.1°
qc = 0.00579
The total power required at this speed is therefore 638 kW.
To calculate the cyclic pitch to trim, we write eqn 1.45 in terms of force components
related to disc axes, i.e.
–WfR – TDhRB1 + (HD + TDa1)hR + Mf – Ms(B1 – a1) = 0
Putting TD = W and solving for B1 gives
B1 = a1 + (Mf + HDhR – WfR)/(WhR + Ms) (4.13)
= + ( + – )/( + ) 1 c c c f D s
a C h h w f w h C m m (4.14)
where Cmf = Mf/ρsAΩ
2
R
3
and C M sA R bSeR sA R
ms = / = / s
2 3 1
2
2 3
ρ ρ Ω Ω
or C bM x e sAR
ms = /2 b g ρ
since S M r = . b g
2
Ω
The pitching moment Mf of the basic fuselage is not likely to be known very
accurately. The rather unstreamlined shape of most helicopter fuselages and the fact
that the fuselage lies in the complicated downwash pattern of the rotor makes it
difficult to make estimates of the pitching moment. Also, because of the problem of
correctly representing the downwash, it is difficult to obtain reliable wind tunnel
measurements of the pitching moment. The contribution from a tailplane, however,
can be estimated comparatively accurately, although it depends on the fuselage incidence,
which is not known in advance. A method for calculating the tailplane moment will
be given later in the chapter.
120 Bramwell’s Helicopter Dynamics
For the purpose of illustrating the calculation of the trim, we assume Cmf = 0.
From the data given we find also that
t h w h Cm c c D s
= = 0.0214 and = 0.0274
The two quantities above represent the control moment contributions, expressed
in non-dimensional form, referred to in section 1.14, i.e. the thrust moment and the
centrifugal couple due to the flapping hinge offset, which has been taken as e = 0.04.
It can be seen that this typical value of flapping hinge offset results in a total control
moment which is more than double that due to the thrust alone.
Equation 4.14 provides the longitudinal cyclic pitch angle B1 to trim for different
c.g. positions and to calculate the fuselage attitude θ, we can write eqn 3.50 in terms
of the disc axes as
(D/W) cos τc + HD/W + TDa1/W = B1 – θ
or, in non-dimensional form, as
θ µ = – – / – d / 1 1 c c
1
2
2
0 c D
B a h w w
A negative sign for θ indicates that the fuselage is in a nose-down attitude. Except
at low speeds, by far the largest term in the equation for θ is the fuselage drag term
1
2
2
0 c / . µ d w
Values of B1 and θ for three different values of f (c.g. position) are shown in Table 4.1.
The variation of longitudinal trim parameters with tip speed ratio in the range 0 to
0.35 are shown in Figs 4.2 to 4.6. The effects of hinge offset and c.g. position are
shown in Figs 4.5 and 4.6 for B1 and fuselage attitude θ.
The variation of λ with µ, Fig. 4.2, displays the reduction of the thrust velocity
part of the induced velocity at the higher speeds, shown by the broken line, and then
the gradually increasing total inflow due to the forward tilt of the rotor disc required
to overcome the helicopter drag (full line).
Table 4.1 Values of cyclic pitch B1 and fuselage pitch attitude θ for different c.g. positions
f 0 0.01 0.02
C.g. position On shaft 7.9 cm fwd 15.8 cm fwd
B1° 6.32 5.31 4.31
θ° –7.45 –8.45 –9.45
– 0.08
– 0.06
– 0.04
– 0.02
0 0.1 0.2 0.3
µ
λ
λi
λ
Fig. 4.2 Inflow ratio in trimmed flight
Trim and performance in axial and forward flight 121
0 0.1 0.2 0.3
µ
14°
12°
10°
8°
6°
4°
2°
θ0
Fig. 4.3 Collective pitch variation in trimmed flight
10°
8°
6°
4°
2°
0 0.1 0.2 0.3
µ
a1
a0
bi
Fig. 4.4 Flapping angles in trimmed flight
12°
10°
8°
6°
4°
2°
0
–2°
–4°
e = 0
e = 0.04
f = 0
0.01
0.02
0.2
0.3 µ
Long-cyclic pitch, B1
Fig. 4.5 Longitudinal pitch angle to trim
In order that the thrust should be kept constant the collective pitch angle θ0, as
follows from eqn 3.33, must vary in almost exactly the same way as λ, and this can
also be seen in Fig. 4.3.
As might be expected, at constant rotor thrust the coning angle a0 is practically
constant over the entire speed range, Fig. 4.4, while the longitudinal flapping angle
a1 varies roughly linearly with speed.
The variation of longitudinal cyclic pitch B1 required for trim is shown in Fig. 4.5
for three c.g. positions and for the flapping offsets represented by e = 0 and e = 0.04.
122 Bramwell’s Helicopter Dynamics
Referring to eqn 4.14, we see that if Cmf = 0 and if the c.g. is on the shaft (f = 0),
B1 differs from a1 only by the small term in hcD , and that the difference becomes
smaller the larger the hinge offset or hub couple. Our previous calculations show
that, at µ = 0.3, the difference between B1 and a1 is only 0.36° for the offset e = 0.04;
for zero offset, e = 0 = Cms and the difference is 0.79°.
With zero hinge offset, the effect of moving the c.g. is merely to change the
amount of cyclic pitch to trim by the ratio of c.g. distances f /h, as explained in section
1.14. According to eqn 4.14, with h = 0.25 a forward c.g. displacement of 7.9 cm
(f = 0.01) requires a change of cyclic pitch of –2.3° in hovering flight, and since the
H-force term is always small, this change of cyclic pitch is roughly constant over
the entire speed range, as can be seen in Fig. 4.5.
With a flapping hinge offset of 31.5 cm (e = 0.04), the cyclic pitch change for the
same c.g. shift is much smaller since the control moment, as we have seen earlier, is
more than doubled by the addition of the offset hinge moment so that a given external
moment requires a correspondingly smaller amount of cyclic pitch movement to
maintain trim, resulting in a lessened fuselage attitude, Fig. 4.6. Expressed in another
way, and as clearly illustrated in Fig. 4.5, the existence of a hub couple, such as
would be obtained from offset hinges or hingeless blades, allows a much larger c.g.
travel for a given cyclic pitch range. The amount of cyclic pitch available is usually
limited by the rotor tilt allowed by the helicopter geometry, and early helicopters,
which had little or no flapping hinge offset, had only a small c.g. travel. For this
reason the fuselages of these helicopters tended to be rather wide, since the load
carried had to be confined within limited longitudinal dimensions. The larger c.g.
range of more recent helicopters with comparatively large hinge offsets, or with
hingeless blades, allows a more slender fuselage design.
Since wind-tunnel measurements of rotor forces and flapping motion generally
show good agreement with theoretical values, any discrepancy between measured
and theoretical values of the longitudinal cyclic pitch to trim is usually attributed to
incorrect estimates of the fuselage pitching moment, since this is the only moment
contribution which is likely to be seriously in error. An example of theoretical and
flight test values of the longitudinal cyclic pitch angle to trim for the Sikorsky S–51
is shown in Fig. 4.7.
0 0.1 0.2 0.3
–2°
–4°
–6°
–8°
–10°
–12°
–14°
–16°
Fuselage attitude, θ
f = 0
f = 0.01
f = 0.02
e = 0
e = 0.04
µ
Fig. 4.6 Fuselage angle in trimmed flight
Trim and performance in axial and forward flight 123
4.2.2 Effect of tailplane on trim
Let us suppose that the pitching moment of the tailplane can be calculated in isolation
from the fuselage on which it is mounted. The fuselage datum line for referring
angles is a line perpendicular to the rotor hub axis. If the no-lift line of the tailplane
makes an angle αT0 to the datum line it can be seen from Fig. 4.8 that the tailplane
incidence αT is given by
αT = αD + B1 – a1 + αT0 – ε = θ – τc + αT0 – ε
where ε is the downwash angle at the tailplane relative to the undisturbed flow.
If ST is the tailplane area, aT is the lift slope and lTR is the tail arm, the pitching
moment MT is
M V S l Ra B a T
1
2
2
T T T D 1 1 T = – ( – + + – )
0
ρ α αε
If this moment can be added independently to the basic fuselage pitching moment
Mf, the equation for the longitudinal cyclic pitch to trim, eqn 4.13, can be modified
to become
B a
M H hR Wf R M
WhR M
1 1
f D T
s
= +
+ – +
+
= +
+ – – ( + – + – )
+
1
c c
1
2
2
T T D 1 1 T
c
f D 0
s
a
C h h w f V a B a
w h C
m
m
µ α αε
8°
6°
4°
2°
0
–2°
–4°
0.1 0.2 0.3 µ
c.g. 3.0 cm forward of shaft
c.g. 7.4 cm forward of shaft
c.g. 12.7 cm forward of shaft
Theory (no fuselage moment)
B1
Fig. 4.7 Measured longitudinal control to trim; Sikorsky S–51, level flight 172 rev/min
V
a1 – B1
Vertical
τc ε
Rotor
hub axis
αD
θ
αT0
Fig. 4.8 Determination of tailplane incidence
124 Bramwell’s Helicopter Dynamics
where V S l sA T T T = / is the tail volume ratio.
This equation can be arranged to give
B a
C h h w f V a
k w h C
m
1 1
c c
1
2
2
T T D T
T c m
= +
+ – – ( + – )
( + ).
f D 0
s
µ α αε
(4.15)
where k V a w h Cm T
1
2
2
T T c = 1 + /( + ).
s
µ The equation may be compared with eqn
4.14 for the no-tailplane case.
To see the effect of the tailplane on the trim of our example helicopter, let us take
α T0 T = 12 , = 0.1, ° V aT = 3.5, and let the distance of the tailplane below the hub be
1.25 m (= 0.156R). For the appropriate values of the downwash angle ε, we use the
charts below, Figs 4.9 and 4.10, taken from Reference 2, which summarise the results
of Heyson and Katzoff referred to in Chapter 3. As described there, the downwash
pattern behind the rotor consists of two distinct trailing vortices. The mean downwash
angle ε0, based on the mean induced velocity, is used to define the axis of reference
for longitudinal and normal position coordinates ξand ζ (positive aft and downwards
from the rotor hub, and non-dimensionalised on R) of which the downwash, vi, is a
function. Vertical downwash distributions are provided for two positions aft of the
rotor hub.
Taking the case µ = 0.3, as before, we have
ε λ λµ 0 i0 i i
= / = v V V /ˆ / ≈
= 0.024 = 1.38°
and ξ = lT = 1.2
Referring to Fig. 4.8, the vertical displacement of the trailing vortices at the nondimensional distance ξ = 1.2 behind the rotor is 1.2(ε0 – αD)R = 1.2(0.024 + 0.134) R
= 0.19R below the rotor plane. Since the distance of the tailplane below the rotor is
0.142R (having taken account of the small angle a1 – B1), the tailplane is 0.048R
above the ζ axis. Using the ξ = 1.07 case in Fig. 4.9 as being the closest to the desired
ξ = 1.2, with ζ = + 0.048, we find vi/vi0 = 1.8. Thus, the downwash angle at the
tailplane is 1.8 × 1.38° = 2.48°.
3
1
vi/vi0
–0.3 –0.2 –0.1 0 0.1 0.2 0.3
ζ
ξ = 1.07
Fig. 4.9 Vertical distribution of induced velocity at ξ = 1.07 behind rotor
Trim and performance in axial and forward flight 125
From the data given previously, we find that α α ε D T + – = 1.85
0 ° and kT = 1.32,
so that, from eqn 4.15,
B1 – a1 = – 0.19°
whereas, without the tailplane,
B1 – a1 = 0.36°
Figure 4.11 shows the effect of the given tailplane on the longitudinal trim. The
full line shows the previously calculated tailplane case with the c.g. on the shaft and
with e = 0.04, as shown in Fig. 4.5. The two broken lines show the cyclic pitch to trim
for two tailplane settings of 12° and 15°. At low speeds the large downwash angle
causes the tailplane incidence to be negative and the consequent download requires
a small forward application of the stick to trim, relative to the tailless case. At higher
speeds the downwash angle increases, the tailplane incidence tends to become positive
and a backward stick movement is required. At still higher speeds the increasingly
nose down attitude acquired by the fuselage causes the tailplane incidence to reduce
again, and the slope of the trim curve rapidly increases. For a fixed tailplane the trend
indicated in Fig. 4.11 is quite general.
4.2.3 Lateral control to trim
One of the forces contributing to the lateral trim of the helicopter is the tailrotor
thrust Tt. Assuming that the tailrotor thrust moment is the only moment balancing the
main rotor torque, we have
Tt = Q/ltR
vi/vi0
ξ = 2.07
–0.3 –0.2 –0.1 0 0.1 0.2 0.3
ζ
2
1
Fig. 4.10 Vertical distribution of induced velocity at ξ = 2.07 behind rotor
10°
8°
6°
4°
2°
0 0.1 0.2 0.3
B1
µ
No tail, f = 0, e = 0.04
Fig. 4.11 Effect of tailplane on longitudinal control to trim
α T0 = 15°
α T0 = ㄲ
126 Bramwell’s Helicopter Dynamics
where ltR is the distance of the tailrotor from the c.g. of the helicopter. Calculation of
the torque coefficient at µ = 0.3 from eqn 3.67 gives
qc = 0.00632 or Q = 25800 Nm
If the tailarm is 11 m, the tailrotor thrust is
Tt = 2340 N
From the parameters already calculated, and from eqn 3.53 we find
b1 = 2.24°
For the lateral cyclic pitch to trim, eqn 1.53 can be written in non-dimensional
form as
A b
w f T W t h
t h Cm
1 1
c 1 t c t
c
= – –
+ ( / )
+ s
(4.16)
and the lateral tilt φ of the fuselage is
φ = –b1 – A1 – Tt/W (4.17)
If we take f1 = 0, i.e. no lateral displacement of the c.g., and ht = 0.2, we find
A1 = – 3.34° and φ = – 1.98°
The lateral trim quantities b1, A1, and φ are shown in Fig. 4.12.
Although there may be a contribution from the fin and fuselage, it will be supposed,
for illustration, that the tailrotor thrust is the only agency balancing the main rotor
torque. The calculation of the rotor torque is discussed in the next section.
If the solidity, rotor area, and tip speed of the tailrotor are denoted by st, At and
(ΩR)t respectively, the corresponding thrust coefficient is defined by
t T s A R c t t t t
2
t
= / ( ) ρ Ω
Let us suppose that the tailrotor axis is perpendicular to the flight direction, i.e. the
incidence of the no-feathering axis of the tailrotor is zero. The only contribution to
the inflow ratio λt is the induced velocity ratio λit, which can be calculated from
4°
3°
2°
1°
0
–1°
–2°
–3°
0.1 0.2 0.3
–A1
b1
µ
φ
Fig. 4.12 Lateral control angles to trim
Trim and performance in axial and forward flight 127
λit t c = ( /2),
t
√ s t in hovering flight
and λ µ it t c t = /2 ,
t
s t for µt > 0.05
Usually, the tip speeds of the main rotor and the tailrotor are the same, i.e. µt can
be taken as µ.
To calculate the collective pitch we can use eqn 3.33, where tc is referred to the nofeathering axis, and obtain
θ
µ
λ 0t 2 c it
=
3
2(1 + 3 /2)
4
–
t
a
t
(4.18)
Taking st = 0.1, Rt = 1.4 m, (ΩR)t = ΩR = 208 m/s, the tailrotor collective-pitch
angle to trim has been calculated and is shown in Fig. 4.13. The high values at
hovering and low speed are partly due to the high solidities typical of tailrotors
resulting in somewhat higher induced velocities than the main rotor.
4.3 Helicopter performance in forward flight
It is now possible to estimate the performance of the helicopter in forward flight, this
being the performance at a specific flight condition, or point on the flight envelope.
This should not be confused with the mission performance, which is aimed at assessing
the overall ability of the helicopter to complete a particular operational mission that
consists of a series of inter-related tasks.
The trim calculations of the previous sections give all the information needed for
calculating the power required for a given flight condition; in fact, using eqn 3.66,
the torque and power were calculated from the values of θ0 and λ was obtained from
the trim equations. For the performance alone, however, calculation of the trim
parameters is not necessary. A form of the torque equation for performance calculations
more convenient than eqn 3.66 can be obtained by considering the balance of forces
along the flight path in conjunction with eqn 3.66.
Referring to Fig. 4.1, we see that
TD sin αD + HD cos αD + W sin τc + D = 0
20°
16°
12°
8°
4°
0
0.1 0.2 0.3
θ0t
µ
Fig. 4.13 Tailrotor pitch angle to trim
128 Bramwell’s Helicopter Dynamics
Multiplying by ˆ
V V R = /Ω and remembering that ˆ
V sin = + D D i α λ λgives
( + ) + + sin + = 0 D i D D c λ λ µ τ T H WV DV
ˆ ˆ
which can be written in non-dimensional form as
λ µ λ τ D c c i c c c
1
2
3
0 D D D
+ = –( + sin + ) t h t w V V d
ˆ ˆ
Substituting for λ µ D c c D D
+ t h in eqn 3.66 gives
q t w V V d c
2
i c c c
1
2
3
0 = (1 + 3 )/8 + + sin +
D
δ µ λ τ
ˆ ˆ (4.19)
This expression for the torque coefficient can be regarded as the non-dimensional
form of an energy equation; the first term represents the power required to overcome
the profile drag of the blades, the second represents the induced power, the third
is the power required for climbing, and the last term is the power required to
overcome the fuselage drag. Of course, eqn 4.19 could have been derived from
energy considerations directly, but it is instructive to derive it from the balance of
forces.
Equation 4.19 has been derived from eqn 3.66 on the assumption that the induced
velocity was constant. Since the induced power in eqn 4.19 appears as a separate
term, it is a simple matter to include the effect of non-uniform induced velocity, as
mentioned in the previous chapter. Now, as we saw in Chapter 3, the induced power
can be expressed as (1 + k)Pi0, where Pi0 is the ‘ideal’ induced power for a constant
induced velocity distribution defined by vi0T and which, in non-dimensional form, is
represented by the second term of eqn 4.19. Values of k for the Mangler and Squire
induced velocity distribution were given in Fig. 3.16. Thus, the contribution of the
induced power to the torque coefficient can be expressed more accurately by
(1 + k)λitcD . Further, we have yet to include the torque which must be provided to
the tailrotor. The tailrotor is driven by a shaft geared to the main rotor, but the torque
supplied to the shaft depends on the inclination of the tailrotor axis to the fuselage.
Thus, for example, it is possible to incline the axis so that the tailrotor autorotates and
for no power to be necessary at the tailrotor, causing a drag force which, in turn,
would require a forward tilt of the main rotor to trim it, with a corresponding increase
of power to be developed at the main rotor shaft. It can easily be verified that the
amount of power required at the tailrotor shaft, plus the work which must be done to
overcome the tailrotor force, is independent of the tailrotor shaft angle. As we have
found with the main rotor, the power absorbed can be expressed simply as that which
would be needed to overcome the profile drag of the blades and the induced power.
Hence, the power Pt required for the tailrotor is
P t s A R t
t
t
2
it c t t t
3
=
8
(1 + 3 ) + ( )
t
δ
µ λ ρ
Ω
and the effective increment to the mainrotor torque coefficient is
Trim and performance in axial and forward flight 129
q t
s A R
sA R
c
t
t
2
it c
t t t
2
3 t t
=
8
(1 + 3 ) +
( )
( )
δ
µ λ
Ω
Ω
Now it is reasonable to assume that, as the tip speeds of the tailrotor and the main
rotor are usually equal, the terms in the square bracket have roughly the same values
as those of the main rotor, although, as we saw earlier, λit may be rather higher than
in hovering flight. Hence, the power to be attributed to the tailrotor is, to a good
approximation, stAt/sA times that of the main rotor. Thus, a simple way to calculate
the tailrotor power is merely to increase the mainrotor power by the fraction stAt/sA
whose value is typically about 0.06. As a percentage of the total power, the tailrotor
power varies from about 6 per cent in hovering to about 3 per cent at high speed.
The torque coefficient can finally be written as
q k t
s A
sA
w V V d i c
2
c
t t
c c
1
2
3
0 =
8
(1 + 3 ) + (1 + ) 1 + + sin +
D
δ
µ λ τ
ˆ ˆ (4.20)
The required power P is calculated from ΩQ = qcρsAΩ
3
R
3
and is shown for the
example helicopter in level flight in Fig. 4.14. The four contributions to the power are
shown by the broken lines, the value of the induced power factor k being taken as
0.17.
Suppose the maximum installed power of our example helicopter is 900 kW. It can
be seen from Fig. 4.14 that the maximum excess power occurs at µ = 0.154 (32 m/s)
and is 496 kW. This gives a maximum rate of climb of 11 m/s.
The maximum forward speed occurs when the installed power and the required
power are equal; the intersection of the two curves in Fig. 4.14 occurs at µ = 0.358,
i.e. at 74.8 m/s.
Fig. 4.14 Variation of power with forward speed
800
600
400
200
kW
Power
Maximum installed power
Total
Parasite
Blade profile
Induced
Tailrotor
0 0.1 0.2 0.3 0.4
µ
130 Bramwell’s Helicopter Dynamics
4.3.1 Fuselage parasite drag
The figure for the parasite drag of our example helicopter is a value, typical for its
weight, of production helicopters. The flat plate parasite drag of a number of helicopters
as a plot of equivalent flat plate area is shown against gross weight in Fig. 4.15. The
points define fairly well a typical curve of drag against weight. A second curve is
shown which is based on aerodynamically clean experimental helicopters. This latter
curve represents the lowest drag which can reasonably be achieved in helicopter
design, although it falls far short of best fixed wing practice. It is clear that the
particular basic shape which must be adopted by helicopter fuselages, and the fact
that the helicopter is normally expected to fulfil a variety of roles, means that it is
unable to reach the degree of aerodynamic refinement which is possible in fixed wing
practice. In fact, both helicopter drag curves are roughly proportional to W
1/2
, instead
of W
2/3
as might have been expected, which is an indication of a large amount of
separation drag. The drag curve of the much cleaner fixed wing aircraft is more
nearly proportional to W
2/3
.
A breakdown of the fuselage parasite drag is shown in the table below.
4
3
2
1
0 20 40 60 80 100 120 140 160 180 kN
Clean fixed wing aircraft
XH51A
SA 341
UH1
(research)
Lynx (Navy)
Lynx (Army)
Experimental helicopters
S52
Scout
UH1B
S61
Belvedere
S58
Super Frelon
Typical production
helicopters
EH101(Civil)
CH 53A
CH 47
HH (X)
EH101 (Mil)
CH 46
Equivalent flat plate area SFP m
2
Gross weight
S67
Fig. 4.15 Parasite drag of helicopters
Component Percentage drag
Basic fuselage with protuberances 20 to 40
Landing gear or fairing 6 to 25
Rotor pylon and hub 35 to 50
Tailrotor and tail surfaces 5 to 15
The drag of the rotor pylon and hub represents a high proportion of the overall
drag, and this is therefore an area where drag reduction leads to considerable benefit;
hence the appearance of hub and pylon fairings on the larger and faster helicopters.
S55
Trim and performance in axial and forward flight 131
Interference drag plays a significant role, because on a helicopter there are a number
of separate aerodynamic ‘shapes’ in close proximity whose pressure distributions
and boundary layers can interact with each other. Hub and pylon fairings are designed
to minimise interference drag in addition to reducing the basic parasitic drag contribution
of these components, the upper cambered shape of the former being a result
3
.
Also, larger and faster helicopters tend to utilise retractable landing gear, which
leads to the lower figure in the above table.
4.3.2 Analytical estimation of performance
Except at very low speeds (when the disc incidence may not always be small) we can
put ˆ
V = µ and λi = stc/2µ; also, writing λc for ˆ
V sin τc, eqn 4.20 can be expressed
as
q k
st s A
sA
t d c
2 1
2
c
2
t t
c c
1
2
3
0 =
8
(1 + 3 ) + (1 + ) 1 + + +
δ
µ
µ
λ µ
(4.21)
where qc is the torque coefficient corresponding to the given power.
The expression for qc can be used to calculate either the torque and power for a
given flight condition or, as described below, the maximum speed and rate of climb
for a given torque.
To find the maximum level speed (λc = 0) for a given power, i.e. given qc, we have
to solve the quartic in µ expressed by eqn 4.21. Now at high speed we note that the
induced power is small; therefore, neglecting this term and the term 3µ
2
of the profile
drag, we find as a first approximation to µ, µ1 say,
µ
δ
1
3 c t t
0
=
2[ – (1 + / )/8] q s A sA
d
The value of qc corresponding to the maximum power (900 kW) is 0.00834. Then,
with the previously given values of δ and d0, we find
µ1
3
= 0.0561 or µ1 = 0.383
With this value of µ, we calculate the terms previously neglected to give the
second approximation µ2 as
µ2
3
= 0.0468 or µ2 = 0.36
which is extremely close to the correct value. Thus, the iteration provides the required
maximum value in two steps.
To find the maximum rate of climb we must satisfy the condition
∂λc/∂µ = 0
This condition leads to
6 + [3 – 2(1 + ) ] 1 + = 0 0
4 3
c
2 t t
d k st
s A
sA
µ δµ
(4.22)
132 Bramwell’s Helicopter Dynamics
To solve this equation for µ we note from Fig. 4.14 that the blade profile drag
contributes little to the slope of the power curve (which led to eqn 4.22), so that for
a first aproximation µ1 we can neglect the second term of eqn 4.22 and obtain
µ1
4 c
2
0
t t
=
(1 + )
3
1 +
k st
d
s A
sA
From our data we find
µ1
4
= 0.000642
or
µ1 = 0.159
For the second approximation, a value for the second term is calculated using µ1,
giving
µ2
4
= 0.000642 – 0.000117 = 0.000525
or
µ2 = 0.152
This agrees with the value obtained graphically and, again, the iteration leads to a
satisfactory answer in two steps. This value of µ is substituted into eqn 4.21 and the
equation is solved for λc, giving the required rate of climb.
4.3.3 Autorotative forward flight
Autorotation is defined as self-sustained rotation of the rotor in the absence of applied
torque, i.e. when Q = qc = 0. The work to be done to overcome the rotor and fuselage
drag must be obtained at the expense of the potential energy of the helicopter. Level
flight autorotation is impossible, and steady flight can be achieved only by descending.
To find the rate of descent at a given forward speed we simply put qc = 0 in eqn 4.21
and solve for λc at the appropriate value of µ. Thus
λ
δ
µ
µ
µ c
c
2 1
2
c 1
2
3 0
c
= –
8
(1 + 3 ) + (1 + ) +
t
k
st d
t
(4.23)
and the rate of descent Vdes is given by
Vdes = – λcΩR
The angle of descent τdes is clearly
τdes = tan
–1
(Vdes/V) (4.24)
The rate and angle of descent of our example helicopter is shown in Figs 4.16 and
4.17.
It can be seen from eqns 4.21 and 4.23 that the rate of descent is proportional to
the torque coefficient in level flight; in fact, the rate of descent curve, Fig. 4.16, is
Trim and performance in axial and forward flight 133
250
200
150
100
50
0 0.1 0.2 0.3
µ
Fig. 4.16 Rate of descent in autorotation
0
–10°
–20°
–30°
–40°
–50°
–60°
Angle of descent
0.1 0.2 0.3
µ
Fig. 4.17 Angle of descent in autorotation
merely the power curve, Fig. 4.14, drawn to a different scale. Thus, the minimum rate
of descent occurs at the same speed as the minimum power in level flight.
From eqn 4.24 the condition for least angle of descent is given by
d
d
=
cos d
d
– = 0
des
2
des
2
des
des
τ τ
V V
V
V
V
V
i.e. dVdes/dV = Vdes/V
Except at low speeds, when the disc angle may be quite large, this condition can
be written as
dλc/dµ = λc/µ
(------- indicates tangent to curve,
providing least angle of descent)
Rate of descent m/s
134 Bramwell’s Helicopter Dynamics
and the solution can be found by the point at which the line drawn from the origin
makes a tangent to the curve of λc against µ or of Vdes against µ, as shown in
Fig. 4.16.
In autorotation there must be a flow up through the rotor disc so that the total
moment, or torque, of the blade forces is zero. Figure 4.18 shows the forces on a
blade section with the resultant force dR perpendicular, or nearly perpendicular, to
the plane of rotation. It can be seen that the resultant velocity vector W must be
inclined upwards relative to the plane of rotation in order that there should be a
component of lift to balance the blade drag.
It is clear that in autorotation the collective pitch will be lower than in forward
flight. To find the collective pitch angle to trim it is best to use eqn 3.66, putting
qc = 0 and neglecting the small term in µhcD , giving
λ δ µ D
2
c = (1 + 3 )/8 D
t (4.25)
Since t w c c D D
= , λ can easily be calculated from eqn 4.25 and then substituted in
eqn 3.63 to obtain θ0. The collective pitch variation with µ is shown in Fig. 4.19. The
fact that it is practically constant follows from the need for an almost constant flow
through the rotor to maintain zero torque, as can be seen from an inspection of
eqn 4.25.
4.3.4 General remarks on performance estimation
The performance estimations discussed in this chapter have been based on very
simple assumptions, particularly with regard to the aerodynamic properties of the
Fig. 4.18 Forces on aerofoil in autorotation
dD
dL dR
Plane of rotation
W
5°
4°
3°
2°
1°
0
θ0
0.1 0.2 0.3
µ
Fig. 4.19 Collective pitch to trim in autorotation
Trim and performance in axial and forward flight 135
blades. One of the most important, and which has allowed a particularly simple
analysis, is the assumption of constant blade section drag coefficient even though, as
we shall discuss in detail in Chapter 6, the local incidence may vary over a wide
range and enter the stall region.
An early attempt to consider the dependence of the drag coefficient δ on the
incidence α was that of Bailey (1941)
4
, who assumed that
δ = 0.0087 – 0.0216α + 0.4α
2
This expression was inserted into the same sort of analysis as presented in Chapter
3, using a tip loss factor B = 0.97, γ = 15, and an arbitrary amount of linear twist. The
induced velocity was assumed to be constant.
As might be expected, the expressions for the force, torque, and flapping coefficients
were quite complicated, partly on account of the presence of the tip-loss factor B.
For performance estimation, Bailey and Gustafson
5
calculated the induced, fuselage,
and tailrotor power contributions in a manner similar to that described in this chapter,
but for the profile power Bailey’s results were used by expressing them in chart form
for zero blade twist. However, in order to use the charts it was still necessary to find
the trim values of θ0 and λ and also to interpolate between charts. Although Bailey’s
analysis would appear to contain a more accurate representation of the blade drag, it
is doubtful if it justifies the extra complexity or even gives a more reliable value of
the profile power; for example, the inclusion of the tip loss factor leads to many terms
in B
4
and B
5
so that a bad choice of the value of B can clearly make a considerable
difference to the final result. In any case, the value of B normally assumed is based
on hovering flight theory and is not applicable to forward flight.
This illustrates the case against too great an expenditure of effort in estimating the
performance of the rotor, as can be seen also by referring to Fig. 4.20. The figure
shows the effective L /D ratio of the complete helicopter plotted against the L /D ratio
10
8
6
4
2
0
Total L/D
Rotor L/D
2 4 6 8 10 12 14 16
Clean helicopters
Aerodynamically unrefined
M = 20 000 kg
10 000
5000
2500
2500
5000
20 000
10 000
Fig. 4.20 Effect of L /D of rotor on L/D of complete helicopter
136 Bramwell’s Helicopter Dynamics
of the rotor alone. The effective drag has been calculated from the power expended,
P, by
D = P/V
giving
L /D = VW/P = VW/(Pp + Pi + Pt + Pf)
where Pp, Pi, Pt, and Pf are, respectively, the blade profile drag and the induced,
tailrotor, and fuselage power contributions.
At cruising speeds, i.e. for tip speed ratios of between, say, 0.25 and 0.35, it can
be calculated from the data of Fig. 4.14 that the L /D ratio for the rotor alone of our
example helicopter varies from about 7 to 10. Figure 4.20 shows that, at these values,
a comparatively large increase of the L/D ratio of the rotor would be needed to
produce a significant increase in the L /D ratio of the complete helicopter, especially
at low values of the gross weight.
Thus, there is a limit to the expenditure of effort that ought reasonably to be spent
in either making calculations of the rotor power or effecting real improvements in
rotor performance through aerodynamic refinement.
What has been said above applies strictly to the calculation of the performance of
the helicopter, by which we mean the estimation of the power for a given flight
condition or the flight range possible for a given installed power. The high speed
performance of modern helicopters, however, is far more likely to be restricted by the
vibration and increase of control loads due to blade stall and compressibility than
through lack of power. It is in this area that the aerodynamics of the rotor must be
considered in sufficient detail to be able to design a rotor in which these undesirable
effects are reduced to a minimum. The simple rotor force and flapping analysis of the
previous chapter is no longer adequate, and more advanced methods are necessary.
The complicated aerodynamics which need to be considered in these flight conditions
will be described in Chapter 6.
References
1. Price, H. L., ‘Rotor dynamics and helicopter stability’, Aircraft Engineering March to July
1963.
2. Bramwell, A. R. S., ‘Part I – the longitudinal stability and control of the tandem rotor helicopter.
Part II – the lateral stability and control of the tandem rotor helicopter,’ Aeronautical Research
Council R&M 3223, 1961.
3. Stroub, R. H., Young, L. A., Graham, D. R. and Louie, A. W., ‘Investigation of generic hub
fairing and pylon shapes to reduce hub drag, Paper No. 2.9, 13th European Rotorcraft Forum,
Arles, 8–11 Sept. 1987.
4. Bailey, F. J., jnr, ‘A simplified theoretical method of determining the characteristics of a lifting
rotor in forward flight’, NACA Rep. 716, 1941.
5. Bailey, F. J., jnr and Gustafson, F. B., ‘Charts for the estimation of the characteristics of a
helicopter rotor in forward flight. I – profile drag–lift ratio for untwisted rectangular blades’,
NACA ACR L4H07, 1944.
5
Flight dynamics and control
5.1 Introduction
At first sight, the study of helicopter flight dynamics and control may seem very
complicated, since each blade possesses degrees of freedom in addition to those of
the fuselage. Fortunately, apart from some special cases of helicopter dynamic stability
such as the phenomenon of air resonance to be considered in Chapter 9 a knowledge
of the motion of the individual blades is not required, and for calculating the forces
and moments in disturbed flight it is sufficient to consider only the behaviour of the
rotor as a whole.
The simplifying assumptions which have enabled helicopter dynamics of flight
and dynamic stability to be handled in ‘classical’ form are due mainly to the original
work of Hohenemser
1
and Sissingh
2
. These assumptions are as follows.
(i) In disturbed flight the rotor behaves as if the motion were a sequence of steady
conditions, i.e. the accelerations of the helicopter are small enough to have a
negligible effect on the rotor response. This assumption was justified in Chapter 1
where it was shown that the rotating blade could be represented by a second
order system having a natural frequency which is the same as the rotor angular
frequency; typical disturbed motion corresponds to forcing the blade at a very
low frequency ratio so that the rotor responds as if the instantaneous disturbance
were being applied steadily.
The rotor can thus be regarded as responding instantaneously to speed and
angular rates, just as is generally assumed for the fixed wing aircraft.
(ii) The rotor speed remains constant. This assumption is justified because not only
is the rotor speed controlled by the engine, but the changes of torque under
normal steady flight conditions are quite small. In autogyro flight neither of
these two conditions applies, and the rotor angular velocity variations may be
quite considerable.
(iii) Lateral and longitudinal motions are uncoupled and can be treated independently
of one another, as is normally the case with the fixed wing aircraft. Now, we
138 Bramwell’s Helicopter Dynamics
have seen that the rotor tilts sideways with forward speed, and we shall meet
other examples in which the lateral and longitudinal responses are coupled.
Nevertheless, it is assumed that the effects of coupling are quite small, and, for
the present purpose of studying the flight dynamics at a particular speed and
configuration, may be ignored.
Before dealing with the flight dynamics and the dynamic stability problem analytically,
let us consider the physical effects of velocity and angular rate disturbances on the
helicopter.
5.1.1 Forward speed disturbance
We have seen in Chapters 3 and 4 that, for constant collective pitch and inflow ratio
λ, the backward flapping angle a1 of the disc is almost exactly proportional to
forward speed. Figure 3.31, for example, shows that, even when θ0 and λ vary in
trimmed flight, a1 is still roughly linear with forward speed, and it follows that when
the forward speed is increased the rotor tilts back by an amount which is almost
independent of the original flight speed; a typical value is 1° for about 10 m/s. It is
found that the in-plane H-force, whose steady value is already very small, changes
very little so it can be assumed that the rotor thrust force tilts back with the disc,
Fig. 5.1.
The tilt of the thrust vector gives a backward force component, relative to the
original flight direction, and a nose up pitching moment. The thrust also changes but,
unlike the fixed wing aircraft, the change may be positive or negative depending on
the flight speed; for example, at high forward speed, when the disc will be tilted
forward at quite a large angle, a change of forward speed increases the flow through
the disc, reducing the blade incidence and causing a loss of thrust. Of course, if in
trimmed flight the thrust vector does not pass through the helicopter c.g., the change
of thrust will also contribute to the pitching moment, but it is usually found that the
total moment is dominated by the thrust tilt just described. With offset hinges or
hingeless blades there is an additional moment due to the disc tilt alone, as discussed
in Chapter 1. The fuselage drag also provides a substantial contribution to the backward
force, particularly at high speed.
H + δH
δa1
T + δT
Fig. 5.1 Rotor force and flapping in disturbed flight
c.g.
Flight dynamics and control 139
5.1.2 Vertical speed (incidence) disturbance
An upward (positive) velocity of the surrounding air mass imposed on the helicopter
increases the incidence of all the blades, and there is a consequent increase of the
total lift. The helicopter can be regarded as having a lift slope like a fixed wing
aircraft, but, as we saw in Chapter 3, the local blade lift is proportional to the term
UpUT and, since the vertical velocity increases UP by a constant amount, the change
of lift depends on the chordwise velocity component UT which is laterally asymmetric
and has a maximum on the advancing side of the disc. Because of the 90° phase lag,
this results in a backward tilt of the rotor disc producing a backward force component
and a nose up pitching moment. The effect increases roughly linearly with speed and
is zero in hovering flight. Thus, the helicopter rotor is unstable with respect to
incidence and becomes progressively more so as forward speed increases. A tailplane
is often fitted to provide positive incidence stability, but it is really effective only in
the upper half of the speed range.
5.1.3 Pitching angular velocity disturbance
Imagine the helicopter to be pitching with constant angular velocity q and that the
rotor is in equilibrium and pitching in space at the same rate. The rotor can then
be regarded as a gyroscope which will therefore be subjected to a precessing moment
tending to tilt it sideways. Because of the 90° response lag the rotor actually tilts
in the longitudinal plane, causing longitudinal forces and moments. Now we know
that a tilt of the rotor relative to the shaft produces cyclic pitch variations in the tip
path (disc) plane and a consequent aerodynamic moment. The rotor disc will therefore
tilt such that the aerodynamic moment is in equilibrium with the precessing moment.
It is found that the nose up pitching of the helicopter produces a nose down tilt of
the rotor. The corresponding nose down moment is in a favourable sense, i.e. it
opposes the disturbance. We find that in this angular motion the rotor force does not
remain perpendicular to the disc: the change of H-force is quite large and tilts the
resultant force vector in the opposite direction to that of the disc, thereby reducing
the moment. Under conditions of large inflow ratio, such as rapid climb, the sign
of the moment may even be reversed. The destabilising effect of the H-force is less
important when the moment is augmented by the presence of offset hinges or
hingeless blades.
In addition to the precessing moment mentioned above, the pitching motion causes
aerodynamic incidence changes which result in a lateral tilt of the rotor and therefore
of lateral forces and moments (section 1.6.3). Typically, the lateral tilt is about half
the longitudinal tilt. This is another example of the coupling between lateral and
longitudinal motion.
The above discussion applies equally well to rolling motion and we see that
rolling causes a moment in opposition to the motion, i.e. there is ‘damping in roll’,
as with a fixed wing aeroplane, but for a given size of aircraft the helicopter roll
damping is usually much weaker.
140 Bramwell’s Helicopter Dynamics
5.1.4 Sideslip disturbance
When the helicopter sideslips, i.e. when there is a component of wind relative to the
undisturbed flight direction, it appears to the rotor as if the relative wind speed were
unchanged but that it comes from a different direction. Thus the direction of maximum
flapping of the rotor is merely rotated through the angle of sideslip and, as we shall
see later, the change of sideways rotor flapping angle b1 depends directly on the
backward flapping angle a1 and vice-versa. Since a1 is usually much larger than b1,
the main result is a sideways tilt of the rotor away from the side wind. There is
therefore a rolling moment opposing the sideslip, corresponding to the dihedral effect
of the fixed wing aircraft. Further, the blades of the tailrotor experience a change of
incidence and the tailrotor acts like a fin producing favourable ‘weathercock’ stability.
5.1.5 Yawing disturbance
Yawing velocity causes a change of incidence at the tailrotor and, again, produces a
favourable ‘fin’ effect – or ‘damping in yaw’.
Summarising the above physical description of the effects of the disturbances, we
can expect that the nose up pitching moments which arise with forward and vertical
speed changes will result in longitudinal characteristics very different from those of
the fixed wing aeroplane. On the other hand it is rather remarkable that, although the
lateral force and moments arise in quite a different way from those of the conventional
aeroplane, they have similar signs and we shall see that the stability characteristics
are similar.
The dynamic stability analysis which now follows is formulated at the same level
as the equivalent fixed wing analysis in standard undergraduate texts, and is subject
to similar assumptions and approximations. For a fully comprehensive text on helicopter
flight dynamics the reader is referred to Padfield
3
. The analysis begins with the
longitudinal motion.
5.2 The longitudinal equations of motion
The equations of motion of a rigid body referred to axes fixed in the body are derived
in Appendix A2. We found that the assumption of small disturbances enabled the
inertia terms to be linearised and the lateral and longitudinal dynamic terms to be
uncoupled. We have now to choose a suitable initial orientation of the axes in undisturbed
flight. Several axes systems were discussed in Chapter 1, but none of them appears
to have any particular advantage as an initial set for describing the dynamic stability.
At the same time, the rotor forces have no special relationship with the undisturbed
flow direction, as is the case with the conventional aeroplane, so the initial flight
direction (wind axes) offers no advantage in expressing the aerodynamic forces.
However, wind axes at least remove the terms qW0 and pW0 from the force equations
A.2.12 and A.2.13 and have the advantage of being thoroughly established in fixed
wing aircraft work.
Flight dynamics and control 141
We shall therefore use wind axes to describe the stability equations, i.e. in undisturbed
flight the x axis is directed parallel to the flight path, with the z axis pointing downwards
and the y axis pointing to starboard.
The linearised equations of longitudinal motion are therefore
( / ) = – cos + c W g u W X . θ τ ∆ (5.1)
( / ) = –( / ) = – sin + c W g w W g V W Z . .
θ θ τ∆ (5.2)
B M
..
θ = ∆ (5.3)
where ∆X, ∆Z, ∆M are the aerodynamic force and moment increments in disturbed
flight, Fig. 5.2, .
θ = , q and τc is the climb angle.
Since the disturbances in u, w, and q are supposed to be small, the force and
moment increments can be written as the first terms of a Taylor series, i.e. ∆X can be
written as
∆X
X
u
u
X
w
w
X
q
q
X
B
B
X
= + + + +
1
1
0
0
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂θ
θ
= + + + +
1 0
1 0 X u X w X q X B X u w q B θ θ (5.4)
where u, w, … , etc. are understood to be differential quantities. B1 and θ0 are the
cyclic and collective pitch control terms respectively. Similar expressions can be
written for ∆Z and ∆M. The terms Xu, Xw, … , Xq are called aerodynamic derivatives
in fixed wing aircraft work, but the term is less appropriate here because the force
and moment increments are due to rotor disc tilt as well as to changes in aerodynamic
forces. The derivatives are regarded as constants in disturbed motion. The variables
B1 and θ0 are due to pilot (or possibly autostabiliser) action and are specified functions
of time or of the other variables.
The stability eqns 5.1 to 5.3 can be written as
( / ) – – – + cos = + c 1 0 1 0
W g u X u X w X q W X B X u w q B
. θ τ θ θ
(5.5)
– + ( / ) – – – ( / ) + sin = + c 1 0 1 0
Z u W g w Z w Z q W g V W Z B Z u w q B
. .
θ θ τ θ θ (5.6)
– – – + – = +
1 0
1 0 M u M w M w B M q M B M u w w q B . . ..
θ θ θ (5.7)
V
∆X
u θ
τc
w
Horizontal
∆M
W ∆Z
Fig. 5.2 Longitudinal force components in longitudinal plane
142 Bramwell’s Helicopter Dynamics
Apart from the form of the control terms, the equations are identical to those of the
fixed wing aircraft. The term M w w . . allows for the effect of ‘downwash lag ’ on the
tailplane, if fitted, as in the fixed wing aircraft work.
5.2.1 Non-dimensionalisation of the equations
The fixed wing scheme of non-dimensionalisation can conveniently be used for the
helicopter, but the following reference quantities are more useful:
(i) the rotor blade radius R is the unit of length,
(ii) the rotor tip speed ΩR is the unit of speed,
(iii) the blade area sπR
2
= sA is the reference area, where s = bc/πR is the rotor
solidity.
Let us define the following non-dimensional quantities:
ˆ u u R = /Ω
ˆ w w R = /Ω
ˆ q q = /Ω
The non-dimensional aerodynamic unit of time is defined by
τ = / t tˆ
where ˆ t W g sA R = / . ρ Ω
Note that
ˆ ˆ
ˆ
q q
t
d /d ; =
1 d
d
≠ θ τ
θ
τ Ω
The helicopter longitudinal relative density parameter µ* is defined by
µ ρ * = / = W g sAR t Ωˆ
and the non-dimensional moment of inertia iB is defined by
iB = B/(WR
2
/g)
Finally we define the non-dimensional derivatives as
xu = Xu/ρsAΩR, xw = Xw/ρsAΩR, ′ x X sA R q q = /
2
ρ Ω
zu = Zu/ρsAΩR, zw = Zw/ρsAΩR, ′ z Z sA R q q = /
2
ρ Ω
′ m M sA R u u = / ,
2
ρ Ω ′ m M sA R w w = / ,
2
ρ Ω ′ m M sA R q q = /
3
ρ Ω
x X sA R B B 1 1
= / ,
2 2
ρ Ω ′ m M sAR w w . . = /
2
ρ ′ m M sA R B B 1 1
= /
2 3
ρ Ω
x X sA R θ θ ρ 0 0
= / ,
2 2
Ω z Z sA R B B 1 1
= / ,
2 2
ρ Ω ′ m M sA R θ θ ρ 0 0
= /
2 3
Ω
z Z sA R θ θ ρ 0 0
= / ,
2 2
Ω
Flight dynamics and control 143
Then dividing the force eqns 5.5 and 5.6 by ρsAΩ
2
R
2
and the moment eqn 5.7 by
ρsAΩ
2
R
3
we have the non-dimensional form of the stability equations as
d
d
– – –
*
d
d
+ cos = + c c 1 0 1 0
ˆ
ˆ ˆ
u
x u x w
x
w x B x u w
q
B
τ µ
θ
τ
θ τ θ θ
′
(5.8)
– +
d
d
– – +
*
d
d
+ sin = + c c 1 0 1 0
z u
w
z w V
z
w z B z u w
q
B
ˆ
ˆ
ˆ ˆ
τ µ
θ
τ
θ τ θ θ
′
(5.9)
–
*
–
*
–
d
d
+
d
d
– d
2
2
µ µ
τ
θ
τ
θ
i
m u
i
m w
m
i
w m
i B
u
B
w
w
B
q
B
′ ′
′ ′
ˆ ˆ
ˆ .
=
*
+
*
1 0
1 0
µ µ
θ θ
i
m B
i
m
B
B
B
′ ′ (5.10)
in which wc = W/ρsAΩ
2
R
2
.
The above system of non-dimensionalisation, based on the original work of Bryant
and Gates
4
, was intended to display the mass and inertia parameters represented by
µ* and iB. However, in practice, the slight advantage does not justify the somewhat
unwieldy notation and we propose here to write mu for µ µ * / , = / * ′ ′ m i x x u B q q etc. It
was for this reason that a ‘dash’ was applied to some of the non-dimensional derivatives
above, indicating that the final forms for these derivatives had yet to be defined.
Since, also, the non-dimensional variables ˆ ˆ u w and appear only in combination
with the non-dimensional derivatives, which are written in lower case letters, there
should be no ambiguity if ˆ ˆ u w and are written simply as u and w. Then the final nondimensional form of the equations can be written as
d
d
– – –
d
d
+ cos = + c c 1 0 1 0
u
x u x w x w x B x u w q B
τ
θ
τ
θ τ θ θ (5.11)
– +
d
d
– – ( + )
d
d
+ sin = + c c 1 0 1 0
z u
w
z w V z w z B z u w q B
τ
θ
τ
θ τ θ θ
ˆ (5.12)
– – –
d
d
+
d
d
–
d
d
= +
2
2 1 0 1 0
m u m w m
w
m m B m u w w q B .
τ
θ
τ
θ
τ
θ θ (5.13)
5.3 Longitudinal dynamic stability
To study the longitudinal dynamic stability, the controls are assumed fixed. It is
worth noting here that in comparing the stability of a helicopter with that of a fixed
wing aircraft, the concept of static stability is not as meaningful because of the
relative lack of importance of the c.g. position in determining the longitudinal behaviour
of the helicopter. Neither is the concept of stick-free stability, due to the inherent
instability with respect to incidence at low speeds (section 5.1.2), and lack of a
natural force feedback to the pilot’s controls that relates to longitudinal control (as in
the case for fixed wing aircraft).
144 Bramwell’s Helicopter Dynamics
Thus we put B1 = θ0 = 0, remembering that, in the equations, B1 and θ0 are
variations from the trim values. We also find that zq is always zero and that xq is
negligibly small.
Equations 5.11, 5.12, and 5.13 are linear differential equations with constant
coefficients, and to solve them we put*
u = u0e
λτ
, w = w0e
λτ
, θ = θ0e
λτ
where u0, w0, θ0, and λ are constants.
Substituting in eqns 5.11, 5.12, and 5.13 and cancelling throughout by e
λτ
we
obtain
(λ – xu) u0 – xww0 + wcθ0 cos τc = 0 (5.14)
– zuu0 + (λ – zw) w0 – ( ˆ
Vλ – wc sin τc)θ0 = 0 (5.15)
– – ( + ) + ( – ) = 0 0 0
2
0 m u m m w m u w w q λ λ λθ . (5.16)
For non-trivial and consistent values of u0, w0, θ0 it is necessary for the determinant
of the coefficients in eqns 5.14, 5.15, and 5.16 to be zero, i.e. for
– – cos
– – –( – sin )
– –( + ) –
= 0
c c
c c
2
λ τ
λ λ τ
λ λ λ
x x w
z z V w
m m m m
u w
u w
u w w q
ˆ
.
Expanding this determinant leads to the characteristic equation
A1λ
4
+ B1cλ
3
+ C1λ
2
+ D1λ + E1 = 0 (5.17)
where
A1 = 1
B N m Vm q w 1c 1 = – – †
ˆ
. (5.18)
C P N m Q m Vm q w w 1 1 1 1 = – – – ˙
ˆ (5.19)
D S m P m R m Qm u q w w 1 1 1 1 = – – – . (5.20)
E1 = T1mu – R1mw (5.21)
and
N1 = – xu – zw (5.22)
* Care must be taken not to confuse θ0 here with the collective pitch angle; from this point down
to eqn (5.16) θ0 refers to the maximum amplitude of pitch θ of the whole aircraft.
†B1c as a coefficient of the characteristic equation is distinct from B1, the longitudinal cyclic pitch
angle.
Flight dynamics and control 145
P1 = xuzw – xwzu (5.23)
Q V x w u 1 c c = – – sin
ˆ τ (5.24)
R1 = – wc(zu cos τc – xu sin τc) (5.25)
S w V xw 1 c c = cos – τ ˆ (5.26)
T1 = – wc(zw cos τc – xw sin τc) (5.27)
the suffix 1 denoting longitudinal coefficients.
Equation 5.17 will have four roots λ1, λ2, λ3, λ4 which may be either real or
complex. Thus the general solution for u can be written
u c c c c = e + e + e + e 1 2 3 4
1 2 3 4 λ τ λτ λτ λτ
(5.28)
where the constants c1, c2, c3, and c4 can be determined from the initial conditions.
Solutions for w and θ can be written in the same way.
Since we are concerned here only with the stability of the motion, we need consider
only the values of λ.
When λ is real and positive, e
λτ
increases without limit and the motion is known
as a divergence, Fig. 5.3.
When λ is real and negative, e
λτ
decreases steadily to zero and the motion is
known as a subsidence, Fig. 5.3.
When λ is complex it can be written as
λ = λre ± iλim
since the complex roots always appear as conjugate pairs.
The mode of motion corresponding to this pair of roots can be expressed as
u k k = e sin + e cos 1 im 2 im
re re λ τ λ τ
λ τ λ τ (5.29)
If λre is positive, the motion is a divergent oscillation, Fig. 5.4; if λre is negative,
the motion is a convergent or damped oscillation.
The rate at which these motions subside or diverge is determined from the real
values or real parts of λ. It is usual to express this rate as the time to halve or to
double the amplitude of the motion. It can easily be seen that the time to half
amplitude Τh is given by
λ positive
(divergence)
λ negative
(subsidence)
τ
e
λτ
Fig. 5.3 Divergence and subsidence
146 Bramwell’s Helicopter Dynamics
e
re λ τ
Damped oscillation (λre negative)
Divergent oscillation (λre positive)
Fig. 5.4 Damped and divergent oscillations
T t t h = ln 2 /(– ) = 0.693 /(– ) ˆ ˆ
λ λ (5.30)
when λ is real and negative or
T t h re = 0.693 /(– ) ˆ λ (5.31)
when λ is complex and λre is negative.
The time to double amplitude, Td, is given by
T t d = 0.693 / ˆλ (5.32)
for real and positive λ
or T t d re = 0.693 / ˆλ (5.33)
when λ is complex and λre positive.
From eqn 5.29 it can be seen that the periodic time, T, is given by
λ τ λ π im im
= / = 2 T tˆ
i.e. T t = 2 / im π λ
ˆ (5.34)
τ
τ
Flight dynamics and control 147
TD
HD
Z
V
X
αD
αD
δα1
Disturbed flight
TD + δTD
HD + δHD
V + u
w
Steady flight
Fig. 5.5 Longitudinal forces and flapping in disturbed flight
5.4 Calculation of the derivatives
The main contributions to the incremental forces and moments arise from the rotor,
as discussed at the beginning of this chapter. We have also assumed, with good
justification, that the rotor forces and moments, like those of the fixed wing aircraft,
depend only on the instantaneous values of speed, incidence, and rate of pitch. To
calculate the derivatives, then, it is necessary only to resolve the rotor forces onto the
chosen axes and perform straightforward differentiation on the expressions for force
and flapping in Chapter 3. Some of the fuselage contributions can be calculated with
fair confidence, particularly those of the tailplane.
As far as the rotor calculations are concerned it does not matter whether tip path
plane (disc) axes or no-feathering axes are used but, as we found in Chapter 3, the Hforce is usually very small when the rotor forces are expressed in terms of either set
of axes, and this is an aid in the physical appreciation of the problem. We shall
therefore use disc axes in general in our analysis but occasionally use no-feathering
axes when it makes the differentiation easier.
Referring to Fig. 5.5, let αD be the disc incidence in steady flight. In disturbed
flight the longitudinal flapping increases by amount δa1 and the incidence of the wind
axes is α ≈ w/V. Resolving the thrust and in-plane force along the wind axes, we get
δX = –(TD + δTD) sin (αD + δa1) – (HD + δHD) cos (αD + δa1) – TD sin αD
– HD cos αD
≈ – TDδa1 – δTDαD – δHD
since αD and δa1 are small angles.
Also
δZ = – (TD + δTD) cos (αD + δa1) + (HD + δHD) sin (αD + δa1)
+ TD cos αD – HD sin αD
≈ – + + D D D D 1 δ δ α δ T H H a
≈ – δ吀
148 Bramwell’s Helicopter Dynamics
since the terms in HD are very small.
Then
X
X
u
T
a
u
T
u
H
u
u = = – – – D
1
D
D ∂
∂
∂
∂
∂
∂
∂
∂
α (5.35)
Z
Z
u
T
u
u = = –
D ∂
∂
∂
∂
(5.36)
X
X
w
T
a
w
T
w
H
w
w = = – – – D
1
D
D D ∂
∂
∂
∂
∂
∂
∂
∂
α (5.37)
Z
Z
w
T
w
w = = –
D ∂
∂
∂
∂
(5.38)
X
X
q
T
a
q
T
q
H
q
q = = – – – D
1
D
D D ∂
∂
∂
∂
∂
∂
∂
∂
α (5.39)
Z
Z
q
T
q
q = = –
D ∂
∂
∂
∂
(5.40)
Now, since the disc makes a small angle to the x axis, we can write
d
d
=
1 d
d
1 d
d
1 d
d D u R u R R Ω Ω Ω ˆ
≈ ≈
µ µ
Then
x
X
sA R
t
a t h
u
u
= = – – – c
1
D
c cD
ρ µ
α
µ µ Ω
∂
∂
∂
∂
∂
∂
(5.41)
and similarly
z
t
u = –
c ∂
∂µ
(5.42)
x t
a
w
t
w
h
w
w = – – – c
1
D
c cD ∂
∂
∂
∂
∂
∂ ˆ ˆ ˆ
α (5.43)
z
t
w
w = –
c ∂
∂ ˆ
(5.44)
x t
a
q
t
q
h
q
q = – – – c
1
D
c cD ∂
∂
∂
∂
∂
∂ ˆ ˆ ˆ
α (5.45)
z
t
q
q = –
c ∂
∂ ˆ
(5.46)
To calculate the moment derivatives, we see from Fig. 5.6 that the moment of the
rotor forces δMr about the c.g. is
δMr = (–δX cos αs + δZ sin αs)lR + (δZ cos αs + δX sin αs)hR
= – h1RδX + l1RδZ
Flight dynamics and control 149
where
h1 = h cos αs – l sin αs
≈ h – lαs
and
l1 = l cos αs + h sin αs
≈ l + hαs
αs being the incidence of the rotor hub axis in trimmed flight.
In addition to the moment of the forces there is the hub moment due to offset
hinges or hingeless blades, Msδa1, and the fuselage pitching moment δMf. Hence the
total moment increment is
δM = – h1RδX + l1RδZ + Msδa1 + δMf
from which
M h RX l RZ M
a
u
M u u u u = – + + + ( ) 1 1 s
1
f
∂
∂
(5.47)
M h RX l RZ M
a
w
M w w w w = – + + + ( ) 1 1 s
1
f
∂
∂
(5.48)
M h RX l RZ M
a
q
M q q q q = – + + + ( ) 1 1 s
1
f
∂
∂
(5.49)
which, in non-dimensional form, become
′ m h x l z C
a
m u u u m u = – + + + ( ) 1 1
1
f s
∂
∂µ
(5.50)
′ m h x l z C
a
w
m w w w m w = – + + + ( ) 1 1
1
f s
∂
∂ ˆ
(5.51)
′ m h x l z C
a
q
m q q q m q = – + + + ( ) 1 1
1
f s
∂
∂ ˆ
(5.52)
hR
h1R
c.g.
δZ
δX
αs
l1R
Hub plane
Hub axis
lR
Fig. 5.6 Force components contributing to longitudinal moment
150 Bramwell’s Helicopter Dynamics
with
m
i
m m
i
m m
m
i
u
B
u w
B
w q
q
B
= , =
*
, = –
µ µ *
′ ′
′
The moment derivatives can also be expressed in terms of the thrust and in-plane
forces. It can easily be verified that
δM = – (l – ha1s)RδT + hR(Tδa1 + δH) + Msδa1 + δMf
where a1s = a1 – B1, which is a small angle.
Then
′
m l ha
t
h t
a h
C
a
m u m u = – ( – ) + + + + ( ) 1s
c
c
1 c 1
f
D
s
∂
∂
∂
∂
∂
∂
∂
∂ µ µ µ µ
(5.53)
′
m l ha
t
w
h t
a
w
h
w
C
a
w
m w m w = – ( – ) + + + + ( ) 1s
c
c
1 c 1
f
D
s
∂
∂
∂
∂
∂
∂
∂
∂ ˆ ˆ ˆ ˆ
(5.54)
′
m l ha
t
q
h t
a
q
h
q
C
a
q
m q m q = – ( – ) + + + + ( ) 1s
c
c
1 c 1
f
D
s
∂
∂
∂
∂
∂
∂
∂
∂ ˆ ˆ ˆ ˆ
(5.55)
5.4.1 The rotor force and flapping derivatives
To complete the calculation of the sets of force and moment derivatives above, we
need the basic rotor force and flapping derivatives ∂tc/∂µ, ∂a1/∂ ˆ w, … , etc. Now one
of the important variables in the expressions for tc, hc and a1 is the inflow ratio λ (or
λD), and it will be useful to find its derivatives first.
In order to do this, it will be assumed that the inflow remains constant for the
purpose of subsequent differentiation. In fact, the thrust coefficient, for example, has
an effect on the inflow via the lift deficiency function familiar in fixed wing aircraft
aerodynamic theory. This can be developed (see Johnson
5
) to provide a relationship
between local (elemental) momentum theory and local elemental lift which leads to
expressions for pitch and roll. These are dependent on the coefficients of an expression
for the local induced velocity coefficient which depends on radial position and
harmonically on azimuth angle. Pitt and Peters
6
, and Peters and Ha Quang
7
have
formalised this ‘dynamic inflow’ approach and demonstrated its applicability in the
flight mechanics of manoeuvring flight.
However, the present development is aimed at a more elementary level, and thus
the inflow, as mentioned above, will be held constant whilst differentiating to obtain
the derivatives. Since tcD and tc are almost identical, it is convenient to use eqn 3.33
for the relation between the thrust coefficient and the inflow ratio; i.e. starting from
t
a
c 0
2
=
4
2
3
(1 + 3 /2) + θ µ λ
(3.33)
where λ is referred to the no-feathering axis, we differentiate with respect to µ to
obtain
Flight dynamics and control 151
(4/a)∂tc/∂µ = 2µθ0 + ∂λ/∂µ (5.56)
Now, for small αnf
λ = µαnf – λi
where λi is the mean ‘momentum’ inflow ratio, therefore
∂λ/∂µ = αnf – ∂λi/∂µ (5.57)
since αnf remains constant with changes of µ.
In non-dimensional form, eqn 3.1 can be written (for small αD)
λ λ i st V = /2( + ) c
2
i
2 1/2
ˆ
so that
∂
∂
∂
∂
∂
∂
λ
µ λ µ λ
λ
λ
µ
i
2
i
2 1/2
c c
2
i
2 3/2
i
=
2( + )
–
2( + )
+
s
V
t st
V
V
i
ˆ ˆ
ˆ
since ∂/∂µ ≈ ∂/∂ ˆ
V, or
∂
∂
∂
∂
∂
∂
λ
µ
λ
µ
λ λ λ
µ
i i
c
c i
3
2
c
2
i
4
2
c
2
i
= –
4
–
4
t
t V
s t s t
ˆ
(5.58)
Further, the induced velocity in hovering flight, or ‘thrust velocity’ v0 (Chapter 2)
is related to the thrust coefficient by
v0
2 1
2 c
2 2
= st R Ω
so that eqn 5.58 can also be written as
∂λ
∂µ
λ ∂
µ
λ
µ
i i
c
c
i
3
i
4 i
= – –
t
t
V
∂
∂
∂
v v (5.59)
where V v = / 0 V and v v v i i 0 = / , and vi can be taken from the chart, Fig. 3.2.
Then, from eqns 5.56, 5.57 and 5.59, we find,
∂
∂
λ
µ
µθ α λ
λ
i 0 nf c i i
3
c i i
4
=
2 + – (4 / )
1 + (4 / )(1 + )
t a V
t a
v
v
(5.60)
For µ > 0.08 (See sections 3.2 and 3.14)
λ µ i c i
4
/2 and 1 ≈ st v
and eqn 5.60 can be written
∂
∂
λ
µ
µθ α µ
µ
i 0 nf c
=
2 + – 4 /
1 + 8
t a
as /
(5.61)
The incidence, αnf can be written alternatively as αD – a1.
With ∂λc/∂µ known, it is now possible to obtain ∂tc/∂µ. Eliminating ∂λi/∂µ from
eqn 5.59 using eqn 5.60 gives
152 Bramwell’s Helicopter Dynamics
∂
∂
t a V
a t
c 0 D 1 i
3
i
4
i c i
4
=
2 + – + /(1 + )
4/ + ( / )/(1 + ) µ
µθ α
λ
v v
v
(5.62)
and for µ > 0.08 this simplifies to
∂
∂
t a
as
a st
c
0 D 1 c
2
=
2
8 +
(2 + – + /2 )
µ
µ
µ
µθ α µ (5.63)
We note that ∂λi/∂µ and ∂tc/∂µ are both zero in hovering flight.
To find ∂a1/∂µ it is convenient to use the expression for a1 with λ referred to the
no-feathering axis, eqn 3.56
a1
0
2
=
2 (4 /3 + )
1 – /2
µ θ λ
µ
(3.56)
Differentiating with respect to µ and rearranging gives
∂
∂
∂ a a 1 1
2
= –
2
1 – /2 d µ µ
µ
µ
λ
µ
(5.64)
When µ is zero, eqn 5.64 reduces to
∂a1/∂µ = 8θ0/3 + 2λ (5.65)
To calculate ∂ ∂ hcD / µ we differentiate eqn 3.64; numerical examples show, however,
that only the profile drag term is of any importance, and we have simply
∂ ∂ hc
1
4 D
/ = µ δ (5.66)
When considering derivatives with respect to the vertical velocity w, the component
of flow through the rotor must be expressed as λi – ˆ w before differentiating. This
expression of the flow component is not valid if the flight path is steep, for then the
z axis, i.e. the w direction, makes a considerable angle to the rotor axis. However,
steep flight paths are possible in only a narrow range at low forward speed and
overall provide no practical restriction. The induced velocity ratio must then be
written
λ λ i c
2
i
2 1/2
= /2[ + ( – ) ] st V w
ˆ ˆ (5.67)
where ˆ w is made zero after differentiation.
Differentiating eqn 5.67 with respect to ˆ w gives
∂
∂
∂
∂
∂ ∂ λ
λ
λ λ
λ
i
2
i
2 1/2
c c i i
2
i
2 3/2
=
2( + )
–
( / – 1)
2[ + ( – ) ]
ˆ ˆ ˆ
ˆ
ˆ ˆ w
s
V
t
w
st w
V w
= +
4
1 –
i
c
c i
4
2
c
2
i λ λ λ
t
t
w s t w
∂
∂
∂
∂ ˆ ˆ
= + 1 –
i
c
c
i
4 i λ λ
t
t
w w
∂
∂
∂
∂ ˆ ˆ
v
(5.68)
Flight dynamics and control 153
Writing λ α λ = sin – nf i
ˆ
V
∂
∂
∂
∂
∂
∂
λ
α
α λ
ˆ
ˆ
ˆ ˆ w
V
w w
= cos – nf
nf i
But, except for steep flight paths, it is clear from Fig. 5.7 that the change of
incidence δαnf of the no-feathering axis due to the disturbance w is
δαnf = w/V
i.e. ∂ ∂ α nf/ = 1/ ˆ ˆ
w V
Thus, providing αnf is not too large (so that cos αnf ≈ 1),
∂ ∂ ∂ ∂ λ λ / = 1 – / i
ˆ ˆ w w (5.69)
Since
λD = λ + µa1
the derivative of λD is
∂ ∂ ∂∂ ∂∂ λ λ µ D 1 / = / + / ˆ ˆ ˆ w w a w (5.70)
Differentiating eqn 3.33
∂
∂
∂
∂
t
w
a
w
c
=
4 ˆ ˆ
λ
=
4
1 –
i a
w
∂
∂
λ
ˆ
(5.71)
Hence, from eqn 5.68,
∂
∂
λ λ
λ
i i c i
4
i c i
4
=
( /4) / +
1 + ( /4) / + ˆ w
a t
a t
v
v
(5.72)
so that
∂
∂
λ
λ
ˆ w a t
=
1
1 + ( /4) / + i c i
4
v
(5.73)
αnf
No-feathering axis
δαnf
V
w
Fig. 5.7 Change of incidence of no-feathering axis
154 Bramwell’s Helicopter Dynamics
and
∂
∂
t
w
a
a t
c
i c i
4
=
4
1
1 + ( /4) / + ˆ λ v
(5.74)
Differentiating eqn 3.56 gives
∂ ∂ a
w w
1
2
d
=
2
1 – /2d ˆ ˆ
µ
µ
λ
=
2
(1 – /2)(1 + ( /4) / + )
2
i c i
4
µ
µ λ a t v
(5.75)
Then, from eqn 5.70,
∂
∂
λ
λ
µ
µ
D
i c i
4
2
2
=
1
1 + ( /4) / +
1 + 3 /2
1 – /2 ˆ w a t v
⋅ (5.76)
and so we have from eqn 3.64
∂
∂
h
w
a
a t
a
i
c
c i
4
1
2 1 0
D
2
D
=
4(1 + ( /4) / + )
– +
1 – /2 ˆ λ
µθ
µλ
µ v
(5.77)
The derivatives 5.74, 5.75, and 5.77 can be simplified for most of the speed range.
For µ λ = 0, = 1 and = 2 , i c i
2
v st and we have
∂ ∂ t w z a as w c i i
/ = – = 2 /(16 + ) ˆ λ λ (5.78)
and ∂ ∂ ∂ ∂ a w h w 1 c / = / = 0
D
ˆ ˆ
For µ > 0.08, vi ≈ 0 and stc = 2µλi, and we have
∂
∂
t
w
z
a
as
w
c
= – =
2
8 + ˆ
µ
µ
(5.79)
∂
∂
a
w as
1
2
2
=
16
(1 – /2)(8 + ) ˆ
µ
µ µ
(5.80)
and
∂
∂
h
w
a
as
c
2
0
2
D
2
D
=
4
8 +
(1 – 9 /2)/6 +
1 – /2 ˆ
µ
µ
θ µ λ
µ
(5.81)
5.4.2 The q-derivatives
In the equation for the thrust coefficient,
t
a
c 0
2
=
4
2
3
(1 + 3 /2) + θ µ λ
[ ] (3.33)
all the terms are independent of the pitch rate q; therefore
Flight dynamics and control 155
∂ ∂ t q zq c/ = – = 0 ˆ
Now a rate of pitch applied to the rotor shaft gives rise to extra aerodynamic and
inertia terms in the flapping equation. According to eqn 1.16 we must add the terms
γ ψ ψ ˆ ˆ q q cos /8 – 2 sin to the right-hand side of eqn 3.48. Then we find, from
examination of the various harmonic terms, that the longitudinal flapping ∆a1 due to
the rate of pitch is
∆
γ µ
a
q
1 2
= –
16
1 – /2
⋅
ˆ
or
∂
∂
a
q
1
2
= –
16 1
1 – /2 ˆ γ µ
⋅ (5.82)
We find that the aerodynamic incidence changes represented by the term 2 ˆ q sin ψ
cause sideways flapping given by
∂
∂
b
q
1
2
= –
1
(1 + /2) ˆ µ
(5.83)
Also, from eqn 3.51, it follows that
∂ ∂ a q 0/ = 0 ˆ (5.84)
Due to the pitching motion, the normal velocity at the rotor blade becomes modified to
U R a xq P D 0 = ( – cos + cos ) Ω λ µ ψ ψ ˆ
Using this expression in the calculation of the in-plane force HD, we find for its
coefficient
h
a
a b a a q a b q c 0 1 0
2
0 1 D 0 D 1 D
=
1
4
–
4
1
3
–
1
2
+
1
3
–
1
2
+ +
1
8
µδ µ λ µθ λ µ ˆ ˆ
(5.85)
or, in terms of λ (referred to the no-feathering axis),
h
a
a b a a q a a a b q c 0 1 0
2
0 1 1
2
0
2
0 1 1 D
=
1
4
–
4
1
3
–
1
2
+
1
3
–
1
2
–
1
2
+ + +
1
8
µδ µ λ µ µθ λ µ θ µ ˆ ˆ
(5.86)
In these expressions for hcD we have not cancelled the first two terms in the
brackets, as in eqn 3.39. Then differentiating eqn 5.86, using eqns 5.83 and 5.84, and
remembering that ∂λ/∂ ˆ q = 0, gives
∂
∂
∂
∂
∂
∂
∂
∂
h
q
a a
a
a
q
a
a
q
a
q
c 0
2 0
1
1
1 2
0
1 D
= –
4
– /3
(1 + /2)
+
1
3
–
1
2
– +
ˆ ˆ ˆ ˆ
µ
λ µ µ θ
+
1
8
–
8(1 + /2)
1 2
µ
µ
µ
b
qˆ
Typically ˆ q is very small compared with a0, a1, b1 and the final term in this
156 Bramwell’s Helicopter Dynamics
expression may be neglected. Assuming that ˆ q does not alter the induced velocity
unduly, then the same assumptions that followed eqn 3.38 may be made, leading to
∂
∂
∂
∂
h
q
a
a
a
q
a a c
1
2
0
1
2
0
2
D
=
4
1
2
+ – –
12(1 + /2) ˆ ˆ
λ µ µ θ
µ
µ [ ]
Again, the final term involving µ
2
can be shown to be small compared with the
bracketed term and may be neglected, leading to
∂
∂
h
q
a
a
c
2 1
2
0
D
= –
4
(1 – /2)
1
2
+ –
ˆ γ µ
λ µ µ θ
(5.87)
Since ∂ ∂ t q c/ ˆ is zero, eqn 5.55 becomes
′
[ ]
m
ah
a
C
m q
m
q = –
4
(1 – /2)
2
3
+
3
2
+ –
16
(1 – /2)
+ ( )
2 0 1 2 f
s
γ µ
θ λ µ
γ µ
(5.88)
Equation 5.87 displays the ‘Amer effect’
8
, i.e. the considerable and destabilising
change of the in-plane force during pitching, the terms in the square bracket of eqn
5.87 adding to those of tc, eqn 3.33. In hovering flight these extra terms reduce the
effect of the thrust tilt by about 25–30 per cent; in climbing flight (large λ) the
reduction is much greater.
5.4.3 The tailplane derivatives
The derivative (mu)T If MT is the pitching moment due to the tailplane, then, in the
notation of Chapter 4, section 4.2.2
M V S l RCL T
1
2
2
T T = – T
ρ (5.89)
and
∂
∂
∂
∂
M
u
VS l R C V
C
u
L
L T
T T
1
2
= – +
T
T
ρ
The tailplane lift coefficient can be expressed as
C a
LT 0
= ( + – – ) T T α θ τε
from section 4.2.2. Then
V
C
u
a V
u
L ∂
∂
∂
∂
T
= – T
ε
and, since ε = vi/V,
V
u V V
∂
∂
∂
∂
ε
= –
i i v v
= –
i i ∂
∂
λ
µ
λ
µ
Flight dynamics and control 157
where vi is evaluated at the tailplane by the method discussed in Chapter 4.
Hence
∂
∂
∂
∂
M
u
VS l R C a L
T
T T
1
2 T
i i
= – + –
T
ρ
λ
µ
λ
µ
which in non-dimensional form is
( ) = – + – T T
1
2 T
i i
T
m V C a u L ′
µ
λ
µ
λ
µ
∂
∂
(5.90)
The derivative (mw)T Differentiating eqn 5.89 with respect to w gives
∂
∂
∂
∂
M
w
V S l R
C
w
L T 1
2
2
T T = –
T
ρ
and
∂
∂
∂
∂
∂
∂
C
w
a
V
a
V w
LT
= 1 – = 1 –
T T i ε
α
λ
Hence
( ) = – 1 – T
1
2 T T
i
m V a
w
w ′
µ
λ ∂
∂ ˆ
(5.91)
Here again, the downwash term is evaluated at the tailplane.
The derivative (mq)T For a steady pitching rate q the change of incidence
at the tailplane is
∆αT = lTRq /V
and the moment change is
∆ ρ M a VS l R q = –
1
2 T T T
2 2
Therefore
( ) = – T
1
2 T T T
2 2
M a VS l R q ρ
In non-dimensional form,
( ) = – T
1
2 T T T m a V l q ′ µ (5.92)
The derivative (m ) w T ˙ This is the moment derivative arising from the time taken for
the changes of downwash to reach the tailplane, and may be calculated in the same
manner as for the fixed wing aircraft. It appears that for the small tailplanes typical
of most helicopters this derivative is of little importance. According to Bramwell
9
the
derivative is
( ) = – T
1
2 T T T
i
m a V l
w
w ˙
ˆ
′
∂
∂
λ
(5.93)
158 Bramwell’s Helicopter Dynamics
5.4.4 Summary of longitudinal derivatives
For convenience, all the longitudinal derivatives are collected together below:
x t
a t h
u = – – – c
1
D
c cD ∂
∂
∂
∂
∂
∂ µ
α
µ µ
(5.41)
z
t
u = –
c ∂
∂µ
(5.42)
x t
a
w
t
w
h
w
w = – – – c
1
D
c cD ∂
∂
∂
∂
∂
∂ ˆ ˆ ˆ
α (5.43)
z
t
w
w = –
c ∂
∂ ˆ
(5.44)
Since ∂tc/∂ ˆ q is identically zero, zq = 0 and
x t
a
q
h
q
q = – – c
1 cD ∂
∂
∂
∂ ˆ ˆ
but xq is negligible as a force derivative, although it should be included in the
moment derivative ′ mq.
′ m h x l z C
a
m u u u m u = – + + + ( ) 1 1
1
f s
∂
∂µ
(5.50)
= –( – ) + + + + ( ) 1s
c
c
1 c 1
f
D
s
l ha
t
h t
a h
C
a
m m u
∂
∂
∂
∂
∂
∂
∂
∂ µ µ µ µ
(5.53)
′ m h x l z C
a
w
m w w w m w = – + + + ( ) 1 1
1
f s
∂
∂ ˆ
(5.51)
= –( – ) + + + + ( ) 1s
c
c
1 c 1
f
D
s
l ha
t
w
h t
a
w
h
w
C
a
w
m m w
∂
∂
∂
∂
∂
∂
∂
∂ ˆ ˆ ˆ ˆ
(5.54)
′ m h x C
a
q
m q q m q = – + + ( ) 1
1
f s
∂
∂ ˆ
(5.52)
= + + + ( ) c
1 c 1
f
D
s
h t
a
q
h
q
C
a
q
m m q
∂
∂
∂
∂
∂
∂ ˆ ˆ ˆ
(5.55)
The tailplane derivatives, which may be the only contributions capable of being
calculated for the fuselage, are
( ) = – + – T T
1
2 T
i i
T
m V C a u L ′
µ
λ
µ
λ
µ
∂
∂
(5.90)
( ) = – 1 – T
1
2 T T
i
m a V w ′
µ
λ
µ
∂
∂
(5.91)
Flight dynamics and control 159
( ) = – T
1
2 T T T m a V l q ′ µ (5.92)
( ) = – T
1
2 T T T
i
m a V l
w
w ˙
ˆ
′
∂
∂
λ
(5.93),
Other derivative terms which are used in the above are
∂
∂
∂
∂
∂
∂
λ
µ
α
λ
µ
α α
λ
µ
= – = – – nf
i
D 1
i
(5.57)
∂
∂
λ
µ
µθ α λ
λ
i 0 nf c i i
3
c i i
4
=
2 + – (4 / )
1 + (4/ )( / )(1 + )
t a V
a t
v
v
(5.60)
∂
∂
t V
a t
c 0 nf i
3
i
4
i c i
4
=
2 + + /(1 + )
4/ + ( / )/(1 + )
(= 0 when = 0)
µ
µθ α
λ
µ
v v
v
(5.62)
∂
∂
t a
as
st c
0 nf
c
2
2
8 +
2 + +
2
(for > 0.08)
µ
µ
µ
µθ α
µ
µ ≈
(5.63)
∂
∂
∂
∂
a a 1 1
2
= –
2
1 – /2 µ µ
µ
µ
λ
µ
⋅ (5.64)
∂
∂
hc 1
4
D
=
µ
δ (5.66)
∂
∂
t
w
a
a t
c
i c i
4
=
4
1
1 + ( /4) / + ˆ
⋅
λ v
(5.74)
∂
∂
t
w
a
as
c i
i
=
2
16 + ˆ
λ
λ
when = 0 µ (5.78)
∂
∂
t
w as
c 2
8 + ˆ
≈
µ
µ
(for > 0.08) µ (5.79)
∂
∂
a
w a t
1
2
i c i
4
=
2
(1 – /2)(1 + ( /4) / + ) ˆ
µ
µ λ v
(5.75)
∂
∂
a
w as
1
2
2
16
(1 – /2)(8 + ) ˆ
≈
µ
µ µ
(for > 0.08) µ (5.80)
∂
∂
h
w
a
a t
a
c
i c i
4
1
2 1 0
D
2
D
=
4(1 + ( /4) / + )
– +
1 – /2 ˆ λ
µθ
µλ
µ v
(5.77)
∂
∂
h
w
a
as
c
2
0
2
D
2
D
=
4
8 +
(1 – 9 /2)/6 +
1 – /2
(for > 0.08)
ˆ
µ
µ
θ µ λ
µ
µ (5.81)
∂
∂
t
q
c
= 0,
160 Bramwell’s Helicopter Dynamics
–1.2
–1.0
–0.8
–0.6
–0.4
–0.2
zw
xw
xu
zu
0.1 µ
0.2
0.02
0.01
0
–0.01
–0.02
0.1 0.2 0.3 µ
′ m f w, = 0
′ m f u, = 0.02
′ m f u, = 0
′ m f w, = 0.02
′ mq/10
Fig. 5.8 Typical variation of helicopter derivatives with speed
∂
∂
a
q
1
2
= –
16 1
1 – /2 ˆ γ µ
⋅ (5.82)
∂
∂
∂
∂
h
q
a
a
a
q
c 1
2 1
2
0
1 D
=
4
+ –
ˆ ˆ
λ µ µ θ
[ ] (5.87)
5.5 The longitudinal stability characteristics
The longitudinal derivatives for the example helicopter of Chapter 4 have been
calculated and are shown in Fig. 5.8. But, before discussing the stability over the
whole speed range, let us examine the hovering case since, although the analysis is
very much simplified by the absence of a number of derivatives, the stability
characteristics are typical of most of the flight range.
The stability derivatives for the hovering case (c.g. on shaft axis) are
xu = – 0.032 xw = 0 xq = 0
zu = 0 zw = – 0.52 zq = 0
′ mu = 0.016 ′ mw = 0 ′ mq = – 0.099
0.3 0
Flight dynamics and control 161
The relative density parameter µ* = 47.6 and ˆ t = 1.82 seconds.
Taking iB = 0.11 as the non-dimensional longitudinal moment of inertia gives
mu = 6.8, mw = 0, mq = – 0.90
In hovering flight ˆ
V = = 0 c τ and, since zu = 0 also, we see at once from eqn 5.15
that the vertical motion is uncoupled from the pitching and fore-and-aft motion, and
eqn 5.15 gives
λ – zw = 0
or λ = – 0.52
indicating a heavily damped subsidence. The other modes of motion are determined
from
–
– –
= 0
c
2
λ
λ λ
x w
m m
u
u q
or λ
3
– (xu + mq)λ
2
+ xumqλ + muwc = 0 (5.94)
This is the characteristic equation for hovering flight; it could also have been
derived from the characteristic quartic, of course, which would have given the root
λ = zw. Inserting the numerical values above gives
λ
3
+ 0.93λ
3
+ 0.029λ + 0.58 = 0
which has the solution
λ1 = – 1.26, λ2,3 = 0.165 ± 0.65i
The real root represents a heavily damped subsidence, whose amplitude is halved
in one second, and the complex roots represent a divergent oscillation with a period
of 17.5 seconds and whose amplitude doubles in 7.1 seconds.
The motions corresponding to the roots of the characteristic cubic, eqn 5.94,
involve attitude and speed changes only; the vertical motion in hovering flight, as we
have seen, is independent of these two degrees of freedom. To get some idea of the
nature of these modes of motion we substitute the values of λ back into the equations
of motion from which the roots originated, i.e. eqns 5.14 and 5.16, with w absent and
τc = 0, giving
(λ – xu) u0 + wcθ0 = 0
–muu0 + (λ
2
– mqλ)θ0 = 0
Since the characteristic equation expresses the consistency of these equations,
either can be used to find the ratio u0/θ0 for the root in question. Then from eqn 5.16
we have, using the numerical values of the derivatives,
u0/θ0 = (λ
2
+ 0.9λ)/6.8
Taking the real root λ = – 1.26 gives
162 Bramwell’s Helicopter Dynamics
u0/θ0 = 0.0655
In dimensional terms this represents a subsidence in which there is a forward
speed change of 1 m/s for every 4.3° of nose up attitude. It is difficult to attach a
physical meaning to this mode.
The complex root λ = 0.165 + 0.65i gives
u0/θ0 = – 0.036 + 0.117i
Reverting to dimensional values, this result shows that speed changes of about
0.5 m/s accompany attitude changes of 1°. The complex ratio u0/θ0 can be represented
by rotating vectors, and we see that the speed leads the attitude by about 107°. The
physical interpretation of this motion is as follows, Fig. 5.9. Imagine the hovering
helicopter to experience a small horizontal velocity disturbance Fig. 5.9(a). The
relative airspeed causes the rotor to tilt backwards and exert a nose up pitching
moment on the helicopter. A nose up attitude then begins to develop, and the backward
component of rotor thrust decelerates the helicopter until its forward motion is arrested.
At this point (b) the disc tilt and rotor moment vanish but the nose up attitude remains
so that backward motion begins, causing the rotor to tilt forwards and exert a nose
down moment (c). Following this, a nose down attitude is attained (d) which accelerates
the helicopter forward and returns it to the situation (a). The cycle then begins again
but, as we have found analytically, the motion is unstable and its amplitude increases
steadily.
The unstable motion described above is due entirely to the characteristic backward
flapping of the rotor with forward speed, although the rate of divergence is reduced
by the favourable damping in pitch mq. If it were possible for mu to be negative, i.e.
for the rotor to flap forward with speed, it follows that the last term of eqn 5.94 would
be negative, implying a positive real root and a pure divergence, which is even less
desirable. Zbrozek
10
investigated the effects of configuration changes on the dynamic
stability but found that no reasonable departure from the conventional helicopter
shape would significantly improve the stability. In particular, the c.g. position, which
is of great importance in the stability of the fixed wing aircraft, has no effect on the
stability of the hovering helicopter. For the helicopter with zero offset hinges and
zero fuselage moment, the rotor force vector must pass through the c.g., as we saw
in Chapter 1, so that moving the c.g. merely has the effect of changing the fuselage
attitude without altering the pitching moments. When the flapping hinges are offset,
or if the blades are hingeless, a hub moment can be exerted and it is no longer
T
T
T T
M
M
V (a) V = 0 (b)
V (c) V = 0 (d)
Fig. 5.9 Disturbed longitudinal motion of helicopter
Flight dynamics and control 163
necessary for the rotor force to pass through the c.g., so that changes of rotor force
can contribute to the pitching and rolling moments. But, as we saw earlier, in hovering
flight the thrust changes due to forward speed, ∂tc/∂µ, and to pitching rate, ∂ ∂ t q c/ , ˆ
are both zero and, although ∂ ∂ t w c/ 0 ˆ ≠ , inspection of the coefficients of the quartic,
eqns 5.18 to 5.21, shows that when ˆ
V = 0 the mw derivative can make no contribution.
Thus, in hovering flight, it is true for all types of helicopter that movement of the c.g.
has no effect on the dynamic stability.
5.5.1 Forward flight
The roots of the stability quartic, eqn 5.17, have been calculated for the speed range
µ = 0 to µ = 0.35 and are shown in root-locus form in Fig. 5.10 for the two cases, c.g.
on the shaft (l = 0) and forward c.g. (l = 0.02). On the scale of this figure, the large
negative roots (corresponding to λ = – 1.26 of the hovering case) cannot be shown
but, since they represent the most stable mode, they are of least significance. The
most important roots are those representing the unstable oscillation and it can be seen
that, for the case l = 0, the destabilising effect of the positive mw becomes more
important as the speed increases; if higher speeds had been considered it would have
been found that the two complex branches of the curve would have met on the real
axis and then moved in opposite directions along this axis, implying two real roots
and at least one divergent mode.
The effect of setting the c.g. forward of the shaft is shown by the case l = 0.02
(lR = 13 cm). Here the moment generated by the rotor thrust is stabilising and
opposes the moment due to the rotor tilt. Although the aircraft remains unstable, the
λim
Unstable
Stable
0.4
0.3
0.2
0.1
0.05
0.6
0.2
0.4
µ=0
0.25
0.35
–0.2
–0.6
–0.4
µ= 0.1 0.15 0 0.3 0.35 0.2 0.4 0.6 λre
–0.8 –0.6 –0.4 –0.2
l = 0.02
l = 0
Fig. 5.10 Root-locus plot for typical single rotor helicopter
164 Bramwell’s Helicopter Dynamics
deterioration of stability with increase of speed is much reduced. As we have noted
before, to obtain a moment contribution due to change of thrust requires a continuous
hub moment to be exerted on the fuselage in order for the rotor force vector to be
displaced from the c.g., and this can be achieved with offset hinges or hingeless
blades. But a constant moment exerted on the fuselage implies a fluctuating load on
the rotating hub, and problems of fatigue limit the amount that can be tolerated. Thus,
although setting the c.g. forward of the shaft with offset or hingeless rotors improves
the stability, the improvement is restricted by the necessity to keep loads low enough
to avoid fatigue failure.
5.5.2 The effect of a tailplane
The effect of a tailplane on the stability of the helicopter has been calculated by
considering the tailplane referred to in Chapter 4, i.e. one having a tail volume of 0.1
and a lift slope of 3.5. The derivatives ( ) and ( ) T T m m w q ′ ′ were calculated from eqns
5.91 and 5.92 respectively; for the purpose of illustration, (mu)T was taken as zero
since it depends on the rigging angle, which can be arbitrarily chosen, and ( )T mw . ′
was neglected. The results for the case l = 0 are shown in Fig. 5.11, together with the
tailless case for comparison. It can be seen that in the upper half of the speed range
the beneficial effects of the tailplane become large enough to confer positive stability.
As the speed increases, the two negative roots coalesce to form two complex branches
whose values indicate a well-damped rapid oscillation.
λim
–0.6 –0.4 – 0.2 0 0.2 0.4 0.6 λre
–0.2
–0.4
–0.6
0.2
0.4
0.2
0.8
0.35
0.3
0.2
0.1
µ = 0
0.6
0.3
0.35
Tailless, e = 0.04, l = 0
With tailplane
Fig. 5.11 Effect of tailplane on stability roots
Flight dynamics and control 165
5.5.3 The effect of hingeless rotors
The full analysis of the hingeless rotor will be made in Chapter 7, but to discuss its
effect on helicopter stability it need only be assumed that, like the rotor with offset
flapping hinges, the hingeless rotor can exert a longitudinal moment proportional to
the tilt of the disc a1s (relative to the shaft) and a lateral moment proportional to the
sideways tilt b1s. For illustration we shall take a hingeless rotor which, for a given
rotor tilt, can exert a hub moment five times greater than that of the 4 per cent offset
hinges of our example helicopter. In Chapter 7 it will be found that, under a given set
of conditions, the flapping of a hingeless rotor is almost identical to that of a hinged
one. Thus, all the rotor forces and flapping derivatives will be the same as before and
it is necessary only to increase terms such as C a
ms 1/ ∂ ∂µ (eqn 5. 53) to 5 /
s 1 C a m ∂ ∂µ
to represent the hub moments of a hingeless rotor. This has been applied to the
moment derivatives calculated earlier (and shown in Fig. 5.8).
For hovering flight, the characteristic cubic of our example helicopter becomes
λ
3
+ 3.41λ
2
+ 0.11λ + 1.95 = 0
whose roots are
λ1 = – 3.54 and λ2,3 = 0.065 ± 0.74i
The subsidence is even more heavily damped than previously. Then divergence of
the oscillation is a little milder, and the period has decreased from 17.5 seconds to
15.4 seconds.
The roots of the quartic have been calculated for forward flight and are shown in
Fig. 5.12. Also shown is the same hingeless helicopter fitted with the tailplane of the
previous example. It can be seen that the instability, as might have been expected, is
intensified by the hingeless blades and that at the top speeds the unstable oscillation
degenerates into two purely divergent motions, indicated by the two positive real
roots. The tailplane reduces the severity of the instability but for our case is unable
to provide positive stability. Setting the c.g. forward would further improve the
stability, but we are again faced with the objection that this would involve the fluctuating
hub moments and associated fatigue problems.
The longitudinal dynamic stability of the hingeless rotor helicopter is therefore
generally inferior to that of the helicopter with flapping hinges of small offset, and
worsens at high speed. Autostabilisation is generally incorporated to bring about an
improvement. In mitigation of the inferior stability is the fact that the larger hub
moments allow much greater control power to be achieved.
As is well known, the longitudinal stability quartic of the fixed wing aircraft splits
up into two quadratics whose coefficients are related in a very simple way to the
coefficients of the quartic, leading to a simple physical interpretation of the motion.
Unfortunately this is not so for the helicopter. We have already seen that in hovering
flight the interaction of the pitching and horizontal motions leads to a characteristic
cubic equation, and this is further complicated by coupling with the vertical motion
as speed increases.
166 Bramwell’s Helicopter Dynamics
5.6 Lateral dynamic stability
5.6.1 The equations of motion
Writing the lateral equations of motion A.2.13, A.2.15 and A.2.17 in the same manner
as for the longitudinal equations, we have
( / ) – – + ( / ) – – cos – sin c c W g Y Y p W g Vr Y r W W p r
. v v v φ τ ψτ
= +
1 t
1 t Y A Y A θ θ (5.95)
– + – – – = +
1 t
1 t L Ap L p Er L r L A L p r A vv . . θ θ (5.96)
– – – + – = +
1 t
1 t N Ep N p Cr N r N A N p r A vv . . θ θ (5.97)
The variables are defined as shown in Fig. 5.13; A1 and θt are the lateral cyclic and
tailrotor collective pitch angles respectively.
The non-dimensional derivatives are defined by:
yv = Yv/ρsAΩR, yp and yr are found to be negligibly small.
′ l L sA R v v
= / ,
2
ρ Ω ′ n N sA R v v
= /
2
ρ Ω
′ l L sA R p p = / ,
3
ρ Ω ′ l L sA R r r = /
3
ρ Ω
′ n N sA R p p = / ,
3
ρ Ω ′ n N sA R r r = /
3
ρ Ω
λim
0.8
0.6
0.4
0.3
0.35
0.2
0.1
0.2
0.3
0.35
0.2
0.3
0.1
–0.6 –0.4 – 0.2 0 0.2 0.4 0.6 0.8 1.0
λre
–0.2
–0.4
–0.6
µ = 0.35
Fig. 5.12 Stability roots of helicopter with hingeless blades
Hingeless
helicopter
Hingeless
helicopter
with
tailplane
Flight dynamics and control 167
′ l L sA R A A 1 1
= /
2 3
ρ Ω ′ l L sA R θ θ ρ t t
= /
2 3
Ω
′ n N sA R A A 1 1
= / ,
2 3
ρ Ω ′ n N sA R θ θ ρ t t
= /
2 3
Ω
y Y sA R A A 1 1
= / ,
2 3
ρ Ω γ ρ θ θ t t
= /
2 2
Y sA R Ω
The non-dimensional moments and products of inertia are defined by
iA = A/mR
2
, ic = C/mR
2
, iE = E/mR
2
and µ* and ˆ t are the same as for the longitudinal case.
The non-dimensional forms of the equations of motion are then
d
d
– – cos +
d
d
– sin = + c c c c 1 t 1 t
v
v v
τ
φ τ
ψ
ψ τ θ θ
y w V
t
w y A y A
ˆ (5.98)
– +
d
d
–
d
d
–
d
d
–
d
d
= +
2
2
2
2 1 t 1 t
l l
i
i
l l A l p
E
A
r A vv
φ
τ
φ
τ
ψ
τ
ψ
τ
θ θ (5.99)
– –
d
d
–
d
d
+
d
d
–
d
d
= +
2
2
2
2 1 1 t
n
i
i
n n n A n
E
C
p r A t vv
φ
τ
φ
τ
ψ
τ
ψ
τ
θ θ (5.100)
where l l i l l i n n i n n i A p p A C p p C v v v v
= * / , = / , , = * / , = / , , µ µ ′ ′ … ′ ′ …
5.6.2 Stick-fixed dynamic stability
Putting the control displacements A1 and θt to zero and assuming solutions of the
form
v = v0e
λτ
, φ = φ0e
λτ
, … , etc.
T
Y
V
p
βss
ψ
δb1
Tt
y
W
φ
Tt
v
Fig. 5.13 Nomenclature diagram for lateral stability
r
168 Bramwell’s Helicopter Dynamics
leads to the vanishing of the determinant
– – cos – sin
– – –( / ) –
– –( / ) – –
= 0
c c c c
2 2
2 2
λ τ λτ
λ λ λ
λ λ λλ
y w V w
l y l i i l
n i i n n
p E A r
E C p r
v
v
v
ˆ
and to the characteristic frequency equation
λ(A2λ
4
+ B2λ
3
+ C2λ
2
+ D2λ + E2) = 0 (5.101)
where
A i i i E A C 2
2
= 1 – / (5.102)
B y i i i N E A C 2
2
2 = – (1 – / ) – v (5.103)
C y N P l i i n V E C 2 2 2 = + + ( / ) + v v v V
ˆ ˆ (5.104)
D2 = yvP2 + lvQ2 – nvR2 (5.105)
E2 = lvS2 – nvT2 (5.106)
and
N2 = lp + nr + (iE/iC)lr + (iE/iA)np (5.107)
P2 = lpnr – lrnp (5.108)
Q n V w p 2 c c = – cos
ˆ τ (5.109)
R l V w p 2 c c = – sin
ˆ τ (5.110)
S2 = npwc sin τc + nrwc cos τc (5.111)
T2 = lpwc sin τc + lrwc cos τc (5.112)
the suffix 2 denoting the lateral coefficients.
The zero root of eqn 5.101 implies that the aircraft has no preference for a particular
heading.
5.6.3 The lateral stability derivatives
Referring to Fig. 5.13 the side force ∆Y in disturbed lateral motion will be
∆Y = Tδb1 + δTt + δYf
where δTt and δYf are the incremental tailrotor and fuselage forces respectively. We
assume that in lateral motion the main rotor thrust remains constant and that, apart
from rolling motion, the force in the plane of the rotor disc remains very small.
In general, the helicopter longitudinal axis will be inclined to the wind axes, Fig.
5.14, and the effective tailrotor height ′ h R t and the rearward distance ′ l R t from the
c.g. are related to the datum distances by
Flight dynamics and control 169
′ h R h l R t t s t s = ( cos – sin ) α α (5.113)
≈ (ht – ltαs)R
and
′ l R l h R t t s t s = ( cos + sin ) α α (5.114)
≈ ltR
The rolling moment will consist of the moment of the main rotor thrust and side
force, the hub moment due to rotor tilt, and the moment of the tailrotor thrust, i.e.
∆L = h1RTδb1 + h1RδY + Msδb1 + ′ h R T t t δ
It will be assumed that the contribution of the fuselage to the rolling moment is
negligible.
The yawing moment contributions will arise from changes in the tailrotor thrust
and the fuselage and fin. Thus
∆N l R T N = – + t t f ′ δ δ
The sideslip, rolling, and yawing disturbances all give rise to axial velocity
components at the tailrotor, and the associated thrust changes can be calculated from
the relations already obtained from the main rotor. Thus, if we denote any of these
velocity components by w, we have
∂Tt/∂v = –∂T/∂w
In the rolling motion, w ≡ – ′ h Rp t , and
∂
∂
∂
∂
T
p
h R
T
w
t
t = – ′
and for yawing, since w l Rr , t ≡ ′
hR
h1R
V l1R
htR
ltR
lR
′ l R t ′ h R t
Fig. 5.14 Moment arms for calculating lateral moments
αs
170 Bramwell’s Helicopter Dynamics
∂
∂
∂ T
r
l R
T
w
t
t =
d
′
Then
L h R T
b Y
h R
T
w
M
b
v
v v v
= + – + 1
1
t s
1 ∂
∂
∂
∂
∂
∂
∂
∂
′ (5.115)
L h R T
b
p
Y
p
h R
T
w
M
b
p
p = + – + 1
1
t
2 2
s
1 ∂
∂
∂
∂
∂
∂
∂
∂
′ (5.116)
L h l R
T
w
r = t t
2
′ ′
∂
∂
(5.117)
N l R
T
w
N v v
= + ( ) t f ′
∂
∂
(5.118)
N h l R
T
w
N p t p = + ( ) t
2
f ′ ′
∂
∂
(5.119)
N l R
T
w
N r r = – + ( ) t
2 2
f ′
∂
∂
(5.120)
To calculate ∂b1/∂v we recall that, when a rotor is placed in a stream of air of
velocity V, the rotor tilts backwards with angle a1 and sideways with angle b1 with
reference to the plane of no-feathering (Chapter 3). The resultant tilt is therefore
( + )
1
2
1
2 1/2
a b at angle ψ0 = tan
–1
(b1/a1) towards the advancing blade, ψ0 being
measured from the rearmost position of the blade. Then, when a small side wind
blows, the relative wind appears to come from a new direction, making sideslip angle
βss = v /(V cos αnf) to the original direction, in the no-feathering plane. Thus, the
resultant flapping will be of the same magnitude but rotated through angle βss so that
the new sideways flapping will be
b b a b 1 1 1
2
1
2 1/2
0 ss + = ( + ) sin ( – ) δ ψ β
= ( + ) (sin cos – sin cos )
1
2
1
2 1/2
0 ss ss 0 a b ψ β βψ
= – , for small . 1 1 ss ss b a β β
Therefore
δb1 = – a1βss = – a1v/(V cos αnf)
and so
∂ ∂ b a 1 1 / = – / ˆ v µ (5.121)
where ˆ v v = / . ΩR
Flight dynamics and control 171
It should be noted that the sign of ∂ ∂ b1/ ˆ v depends on the sense of rotation of the
rotor, and this should be allowed for in the derivation of any formulae; but a little
consideration shows that the rotor always tilts away from the component of relative
wind so that there should be no confusion about the appropriate sign of the forces and
moments due to sideways flapping. In this work the rotor is supposed to be rotating
in an anticlockwise direction when viewed from above, i.e. positive sideways flapping
is directed to starboard.
By the same reasoning, the side wind will always cause a sideways component of
the longitudinal in-plane H-force, and it is easy to see that
∂Y/∂v = – HD/V cos αD
remembering that we are considering forces in the plane of the disc.
The non-dimensional form is
∂ ∂ y h c c
1
4
/ = – / – ˆ v µ δ ≈
in which yc = Y/ρsAΩ
2
R
2
and the inflow contribution to hc is ignored (see eqn 3.64).
The calculation of the rolling moment due to the rate of roll p follows a similar
procedure to that described for the pitching moment, and we find that the nondimensional rolling moment derivative due to the rotor forces is
( ) = ( + /8) / 1 c D 1 l h t a b p p r λ ∂ ∂ˆ (5.123)
in which ∂yc/δb1 ≡ ∂hc/∂a1 = aλD/8 from eqn 3.64. Also, if the extra terms due to ˆ p
from eqn 1.16a are added to the right-hand side of eqn 3.48, and an examination is
made of the various harmonic terms (as in the pitching case), then it can be shown
that
∂ ∂ b p 1
2
/ = – 16/ (1 + /2) ˆ γ µ (5.124)
It has been shown
11
that the fuselage contribution to the side force can be expressed
approximately as
(yv)f = – 0.3µSB/sA (5.125)
where SB is the projected side area of the fuselage.
The non-dimensional forms of all the lateral derivatives are
y t
a
s
t
w
S
sA
B
v = – – – – 0.3 c
1 1
4 t
c
µ
δ µ
∂
∂ ˆ
(5.126)
′ ′ l h t C
a
h s
t
w
m v = – ( + ) – 1 c
1
t t
c
s
µ
∂
∂ ˆ
(5.127)
′ ′ l
h t a C
h s
t
w
p
m
= –
16[ ( + /8) + ]
1 + /2
–
1 c D
2 t
2
t
c s
γ
λ
µ
∂
∂ ˆ
(5.128)
′ ′ ′ l h l
t
w
r = t t
c ∂
∂ ˆ
(5.129)
172 Bramwell’s Helicopter Dynamics
′ ′ ′ n l s
t
w
n v v
= + ( ) t t
c
f
∂
∂ ˆ
(5.130)
′ ′ ′ ′ n h l s
t
w
n p p = + ( ) t t t
c
f
∂
∂ ˆ
(5.131)
′ ′ ′ n l s
t
w
n r r = – + ( ) t
2
t
c
f
∂
∂ ˆ
(5.132)
where s s A R sA R t t t t = ( ) / . Ω Ω But, usually, (ΩR)t ΩR; hence s s A sA t t t = / .
The fuselage derivatives are not likely to be known accurately except, perhaps, the
contribution of the fin. In the example below, the fuselage moment derivatives have
been taken as zero.
5.7 The lateral stability characteristics
The lateral stability derivatives for the example helicopter have been calculated and
are shown in Fig. 5.15, taking
lt = 1.2 and ht = 0.1
and considering level flight, τc = 0.
The non-dimensional moments of inertia are taken to be
iA = 0.033, iC = 0.11, iE = 0
0.08
0.06
0.04
0.02
0
–0.02
–0.04
–0.06
–0.08
–0.10
0.1 0.2 0.3 µ 0.4
′ nv
′ ′ l n r p and
1
2 yv
′ nr
′ lp
′ lv
Fig. 5.15 Lateral derivatives for typical single rotor helicopter
Derivative values
Flight dynamics and control 173
Now we found for the longitudinal hovering case that, in addition to the speed
being zero, some of the longitudinal derivatives were also zero, and this led to a great
simplification. Strictly speaking this is not so for the lateral case, because the yawing
and rolling motions are coupled by the tailrotor, and this coupling is represented by
the derivatives lr and np. However, if we assume that lr is negligible – if, for example,
the tailrotor shaft were on the roll axis – the resulting motion would be analogous to
the longitudinal case, with the corresponding characteristic equation
(λ – nr)[λ
3
– (yv + lp)λ
2
+ yvlpλ – lvwc] = 0 (5.133)
The root λ = nr indicates that the yawing motion is independent of the sideways
and rolling motion. The cubic is analogous to the longitudinal equation, eqn 5.94, but
we should note that the moment of inertia in roll is much lower than in pitch, with a
consequent increase in the numerical values of the coefficients.
The numerical values of the lateral derivatives are
nr = – 0.25, yv = – 0.052
′ ′ l m l p q p = = – 0.099, = –3.0
′ ′ l m l u v v
= – = – 0.016, = –23
The characteristic cubic is
λ λ λ
3 2
+ 3.05 + 0.16 + 1.96 = 0
with roots
λ = –3.19 and λ = 0.07 ± 0.78i
Substitution of the real root λ = – 3.19 back into the equations of motion shows
that it corresponds to an almost pure rolling motion, i.e. it can be regarded as the
‘damping-in-roll’ root. This motion is heavily damped with a time to half amplitude
of less than half a second. The complex root represents a divergent oscillation of
period 14.8 seconds which doubles amplitude in 18 seconds. The yawing motion
previously mentioned has a time to half amplitude of about 5 seconds.
5.7.1 Forward flight
The quartic eqn 5.101 has been solved for a range of µ, and the roots are shown in
the root-locus plot of Fig. 5.16. It can be seen that the mildly unstable oscillation in
hovering flight very soon becomes stable and becomes progressively more so as the
speed increases. The increase of the imaginary part of the root indicates that the
period of the oscillation steadily becomes shorter. The mode shape of the oscillation
at the higher speeds shows that ψ ≈ – βss = – ˆ v/ , µ i.e. the helicopter ‘weathercocks’
with very little sideways translation, and the motion is similar to the ‘Dutch roll’
oscillation of the conventional aeroplane. By neglecting the rolling that occurs and
using the above approximation, it follows from eqn 5.100 that the time of oscillation
of the ‘Dutch roll’ oscillation is approximately
T t n = 2 / ( ) π µ
ˆ √ v
174 Bramwell’s Helicopter Dynamics
S
µ =0
0.3
0.2
0.1
–0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.2
4% offset hinge
Hingeless
r
0.3
0.2
0.1
–1
2
3
1
0.35
Spiral roots
Fig. 5.16 Root-locus plot of lateral stability
We also find that the quartic has a small root given approximately by
λ = – E2/D2
which corresponds to the ‘spiral root’ and, as we have already seen, a large negative
root which corresponds to almost pure damping in roll. Thus, except for fairly low
speeds, where the characteristic hover mode is dominant, the lateral stability modes
of the helicopter are similar to those of the conventional aircraft. This is rather
remarkable when it is considered that, apart from the tailrotor, which would be
expected to act like a fin, the forces and moments from a helicopter rotor arise in
quite a different manner to those of the aerodynamic surfaces of a fixed wing aircraft.
5.7.2 The effect of hingeless rotors
As with the longitudinal case, we represent the hingeless rotor by increasing the hub
moment of the 4 per cent offset hinge of our example helicopter by a factor of five.
This increases the moment derivatives lp and iv, since they arise almost entirely from
the main rotor, but not the ‘weathercock’ derivative nv.
The characteristic equation for hovering flight becomes
λ
3
+ 11.2λ
2
+ 0.59λ + 6.48 = 0
whose roots are
λ1 = – 11.2 and λ2,3 = ± 0.76i
The subsidence is now very heavily damped and the previously unstable oscillation
has now become neutrally stable with a period of 15.2 seconds, which is almost
identical to the period of the longitudinal motion.
µ= 0.05
–2
–3
Flight dynamics and control 175
It is interesting to note that increasing the rotor hub ‘stiffness’ (hub moment per
unit rotor tilt) is equivalent to decreasing the moment of inertia. It happens that
increasing the hub moment five times in the longitudinal case produces almost the
same effect as changing from the longitudinal to lateral moments of inertia, and we
see that the cubic equations for these cases are numerically very similar.
Let us decrease the moment of inertia so that it approaches zero. It can easily be
seen that the cubic characteristic equation, eqn 5.133, then degenerates to the quadratic
λ
2
– yvλ – (lv/lp)wc = 0
whose roots are
λ = [ + ( / ) ]
1
2
1
4
2
c y y l l w p v v v ± √
This represents a lightly damped oscillation whose period, ignoring the yv term, is
T t l l w p = 2 ( / ) c π ˆ√ v
This, in our notation, is Hohenemser’s formula
1
for the period of oscillation in
hovering flight. For the longitudinal case the period is
T t m m w q u = 2 (– / ) c π ˆ√
For the case considered,
Time of oscillation (seconds)
Exact Hohenemser
Longitudinal (4% offset) 17.5 14.2
Longitudinal (hingeless) 15.4 15.1
Lateral (4% offset) 14.8 14.2
Lateral (hingeless) 15.2 15.0
Thus, for high control power or low inertia, Hohenemser’s formula gives quite
accurate results.
The stability in forward flight with the hingeless rotor, Fig. 5.16, shows that,
unlike the longitudinal case, the lateral stability is generally improved compared with
the aircraft with 4 per cent offset hinges.
5.8 Autostabilisation
We have seen that the helicopter is unstable both laterally and longitudinally in
hovering flight and that the longitudinal instability becomes worse with increase of
forward speed, particularly when the rotor has hingeless blades. We have also seen
that a tailplane is really effective only in the upper half of the speed range and that,
since the unstable characteristics of the rotor also deteriorate with speed, the tailplane
may be incapable of making the machine completely stable.
176 Bramwell’s Helicopter Dynamics
Although adequate control power is usually available to correct disturbances, an
unstable aircraft will require continuous correction and will be tiring to fly for long
periods, even in quite calm weather. Furthermore, in some conditions, such as flying
on instruments, an unstable aircraft could be quite dangerous and it is clearly desirable
to provide some form of artificial stabilisation to make good the inherent deficiencies.
The stabilisation devices in common use fall into two categories:
(i) a mechanical/gyro device which is an integral part of the rotor system, as used on
Bell and Hiller helicopters;
(ii) automatic flight control systems using feedback control based on signals from
sensors such as attitude or rate gyros.
5.8.1 Mechanical/gyro devices
The simple Bell stabilising bar embodies the essentials of all mechanical/gyro devices.
The Bell bar is basically a bar pivoted to the rotor shaft, Fig. 5.17, and provided with
viscous damping. The bar behaves like a gyroscope with lag damping.
The bar is linked to the blade so that a tilt of the bar relative to the shaft causes a
change of pitch of the rotor blade.
Since the bar can pivot relative to the shaft, its equation of motion is precisely the
same form as that of the blade under the excitation of gyroscopic and inertia moments,
eqn 1.16. Thus, if θbar is the angular displacement of the bar, its equation of motion
is
.. . . θ θ θ ψψ bar f bar
2
bar
+ (2/ ) + = – 2 sin + cos T q q Ω Ω
in which the viscous damping is conveniently represented by the ‘following’ time Tf,
or the time taken for a sudden displacement of the bar to diminish by 63 per cent. The
last term of eqn 1.16 is negligibly small in practice and so the equation above can
also be written
d
2
θbar/dψ
2
+ (2/TfΩ)dθbar/dψ + θbar = – 2ˆ q sin ψ (5.134)
Let cl be the linkage ratio such that
θ = clθbar (5.135)
θ
Control
Damper
Bar
θbar
Fig. 5.17 Bell stabilising bar
Flight dynamics and control 177
so that, if cl is unity, one degree of the tilt of the bar produces one degree of blade
pitch. Since the bar does not affect the collective pitch, eqn 5.135 can be written in
the form
clθbar = – A1 sin ψ – B1 cos ψ (5.136)
where the cyclic pitch amplitudes A1 and B1 are functions of time.
Substituting eqn 5.136 in 5.134 and equating coefficients of sin ψ and cos ψ gives
′′ ′ ′ A T A B T B c q l 1 f 1 1 f 1 + (2/ ) + 2 + (2/ ) = 2 Ω Ω ˆ (5.137)
and
2 + (2/ ) – – (2/ ) = 0 1 f 1 1 f 1 ′ ′′′ A T A B T B Ω Ω (5.138)
where the dashes denote differentiation with respect to ψ.
The free motion corresponding to eqns 5.137 and 5.138 is found to consist of a
high frequency ‘nutation’ mode, of no practical interest, and a significant low frequency
mode. An approximation to this latter mode can be obtained by ignoring the A1 terms,
giving
d /d + (1/ ) = 1 f 1 B T B c q l ψ Ω ˆ (5.139)
A similar relationship exists between the lateral cyclic pitch A1 and the rate of
roll p.
In terms of aerodynamic time, eqn 5.139 can be written as
d /d + ( / ) = d /d 1 f 1 B t T B cl τ θ τ
ˆ (5.140)
Then considering hovering flight, the equations of stick fixed longitudinal motion
are
du/dτ – xuu + wcθ = 0
– + d /d – d /d – = 0
2 2
1 1
m u m m B u q B θ τ θτ
c B t T B ld /d – d /d – (/ ) = 0 1 f 1 θ τ τ ˆ
The characteristic equation of this motion can be written as
( + / ){hovering cubic} – ( – ) = 0 f 1
λ λ λ
ˆ
t T c m x l B u (5.141)
where the ‘hovering cubic’ is the uncontrolled characteristic equation, eqn 5.94. The
control derivative mB1 is the moment coefficient for unit cyclic pitch change; and
therefore, for our example helicopter, mB1 = – 0.0214(zero offset).
The quartic, eqn 5.141, has been solved for a large range of values of the following
time T t f/ˆ and the two linkage ratios c c l l = and = 1.
1
2 The roots indicate a heavily
damped oscillation of short period and a lightly damped oscillation of long period,
which can be regarded as the original undamped stick-fixed motion modified by the
presence of the bar. The roots of this latter oscillation are shown in Fig. 5.18. It can
be seen that the amount of stabilisation provided by the bar is rather limited.
178 Bramwell’s Helicopter Dynamics
The Hiller stabilising bar is similar to the Bell bar, except that small aerofoils on
the bar provide aerodynamic damping in place of the viscous damping. A further
difference is that the pilot controls the bar directly, which acts as a servo control
between the stick and the rotor blades. By their nature, Bell and Hiller stabilising bars
are well adapted to the two-bladed teetering rotors which are a characteristic of these
helicopters.
5.8.2 Automatic flight control systems
Some automatic flight control systems (AFCS) are specifically designed to deal with
the basic instability inherent in conventional helicopters. The latter implies continuous
pilot activity on the controls in order to fly uncomplicated manoeuvres or even
simply straight and level, which can be tiring over long periods. The stability
augmentation system (SAS) described briefly in (a) below is designed to address this
problem. The automatic stabilisation system (ASE) described in (b) below is aimed
at maintaining a desired manoeuvre attitude and therefore normally operates over a
shorter time period. More detailed descriptions of these and other types of AFCS
fitted to helicopters may be found in McLean
12
and Pallett and Coyle
13
. Generally,
these systems are fitted to the larger and more expensive machines, the application of
mechanical gyro devices such as the Bell and Hiller bars being confined to the
smaller helicopters. The latter tend to confer only a limited improvement of stability
and can add considerably to the drag of the rotor head, whilst the former offers a
more flexible means of stabilisation and is accommodated completely within the
airframe.
S
No bar
T t f/ = 0
ˆ
cl=1
–0.2
3
2
cl =1/2
2
3
4
–0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3
–0.6
–0.4
0.2
0.4
0.8
0.6
4
r
1
Fig. 5.18 Effect of Bell stabilising bar on lateral stability roots
T t f/ =1
ˆ
Flight dynamics and control 179
(a) SAS (stability augmentation system)
The SAS is designed to maintain the helicopter at the datum to which it has been
trimmed. It uses a simple feedback control in which a rate gyro senses pitch rate, for
example, which, on integration, provides a correcting input at the swash plate (if this
is the means of rotor control). Within the feedback loop, however, there is a so-called
‘leaky integrator’ path parallel with the output from the rate gyro, the effect of these
parallel paths being equivalent to that of a phase-lag network. The leaky integrator
path leads to a signal proportional to the angle through which the helicopter has been
disturbed; this initially provides an input to the swash plate that counteracts the
angular disturbance. In the longer term, the input decays or ‘leaks away’, so that if
the helicopter does not respond to the corrective action, or the pilot holds the new
attitude, then the final angular position becomes the new equilibrium state.
(b) ASE (automatic stabilisation equipment)
The ASE is designed to control a desired attitude, e.g. pitch or roll. The attitude
angles are sensed from a gyro and these, and their rates which are found by
differentiation, are summed in appropriate proportions, together with signals from
the cyclic stick (control) position and the c.g. trim system. The latter centres the gyro
to a datum corresponding to successive new flight conditions. The signal from the
stick is used to cancel the gyro signal when the pilot demands a manoeuvre, otherwise
the attitude control would hold the original datum and oppose the manoeuvre demanded.
5.9 Control response
We now consider the behaviour of the helicopter in response to the pilot’s control
input and to vertical gusts. A detailed account of the motion following these disturbances
would be out of place here, and the discussion will be limited to those features of
helicopter response which are usually regarded as being the most important. These
are
(a) the normal acceleration in response to a cyclic pitch displacement,
(b) the normal acceleration in response to a vertical gust,
(c) the pitching and rolling response to cyclic pitch displacements.
Response to collective pitch changes will not be considered, since it is not normally
regarded as a ‘short-term’ control, although it may be possibly used in an autostabilisation system.
Detailed knowledge of the control response is essential for determining the flying
qualities of a helicopter, and subsequent assessment of how these relate to the separately
identifiable tasks within a flying mission. An extensive treatment of the topic is
provided in Padfield
3
.
In order to proceed with this limited study of the control response, the force and
moment derivatives with respect to cyclic and collective pitch applications are now found.
180 Bramwell’s Helicopter Dynamics
5.9.1 The B1 control derivatives
The application of longitudinal cyclic pitch tilts the no-feathering axis in the longitudinal
plane and produces precisely the same effect as if the cyclic stick had been kept fixed
and the incidence of the helicopter had been reduced by the same amount, i.e.
∆ ∆ ∆ B w 1 = – = –(1/ ) α µ ˆ
or
∂
∂
∂
∂ B w 1
= – µ
ˆ
Thus
∂
∂
∂
∂
t
B
t
w
c
1
c
= – µ
ˆ
(5.142)
∂
∂
∂
∂
h
B
h
w
c
1
c D D
= – µ
ˆ
(5.143)
∂
∂
∂
∂
a
B
a
w
1
1
1
= –µ
ˆ
(5.144)
the w derivatives being given by eqns 5.79 to 5.81.
In obtaining the force and moment derivatives we have to allow for the fact that
the no-feathering axis moves relative to the shaft when cyclic pitch is applied. Thus,
the X force must be written
X = – T sin αD – HD cos αD
= – T sin (αs + a1 – B1) – HD cos(αs + a1 – B1)
where αs is the incidence of the shaft (rotor hub axis)
Then
∂
∂
∂
∂
∂
∂
X
B
X
T
B
T
a
B
B
1 1
D D
1
1
= = – sin – cos – 1
1 α α
+ cos + sin – 1
D
1
D D D
1
1
∂
∂
∂
∂
H
B
H
a
B
α α
(5.145)
≈
– + 1 + –
1
D
1 D
1
∂
∂
∂
∂
∂
∂
T
B
T
a
w
H
B
α µ
ˆ
(5.146)
In non-dimensional form,
x
X
sA R
t
B
t
a
w
h
B
B
B
1
1 D
= = + 1 + –
2 2
c
1
D c
1 c
1 ρ
α µ
Ω
∂
∂
∂
∂
∂
∂ ˆ
(5.147)
Similarly
zB1 = – ∂tc/∂B1 = – uzw (5.148)
Flight dynamics and control 181
The moment derivative is
′
m h x l z C
a
w
B B m B 1 1
1
1 1 1 s
= – + – 1 + µ
∂
∂ ˆ
(5.149)
which can also be written as
′
m l ha
t
B
t h C
a
w
h
h
B
m B 1s
c
1
c
1 c
1
1 s
D
= –( – ) – ( + ) 1 + +
∂
∂
∂
∂
∂
∂
µ
ˆ
(5.150)
Numerical examples show that the terms in hcD can be neglected. The force and
moment derivatives for our example helicopter have been calculated and are shown
in Fig. 5.19.
5.9.2 The θ0 control derivatives
Differentiating eqn 3.33 with respect to θ0 gives
∂
∂
∂
∂
t a c
0
2
0
=
4
2
3
(1 + 3 /2) +
θ
µ
λ
θ
But λ = µαnf – λi
therefore
∂λ/∂θ0 = – ∂λi/∂θ0
since µ and αnf are independent of θ0.
Taking as before
λ λ i c
2
i
2
= /2 ( + ) st V √ ˆ
Fig. 5.19 Control derivatives (cyclic pitch)
0.6
0.5
0.4
0.3
0.2
0.1
0 0.1 0.2 0.3 µ
0.12
0.1
0.08
0.06
0.04
0.02
xB1
– 1
mB
zB1
zB1
xB1
and
– 1
mB
182 Bramwell’s Helicopter Dynamics
we have
∂
∂
∂
∂
∂
∂
λ
θ λ θ
λ
λ
λ
θ
i
0
2
i
2
c
0
c i
2
i
2 3/2
i
0
=
2 ( + )
–
2( + )
s
V
t st
V √ ˆ ˆ
= –
i
c
c
0
i
4 i
0
λ
θ
λ
θ t
t ∂
∂
∂
∂
v
or
(1 + ) =
i
4 i
0
i
c
c
0
v
∂
∂
∂
∂
λ
θ
λ
θ t
t
(5.151)
giving finally
∂
∂
t a
a t
c
0
2
i c i
4
=
6
1 + 3 /2
1 + ( /4 )(1 + ) θ
µ
λ
⋅
v
(5.152)
For ˆ
V = = 0 µ we have approximately
∂
∂
t a
as
c
0
i
i
=
4
3 8 + θ
λ
λ
⋅ (5.153)
and for µ > 0.08
∂
∂
t
a
as
c
0
2
=
4
3
1 + 3 /2
8 + θ
µ
µ
µ
(5.154)
From
a1
0
2
=
2 (4 /3 + )
1 – /2
µ θ λ
µ
∂
∂
∂
∂
a1
0
2
0
=
2
1 – /2
4
3
+
θ
µ
µ
λ
θ
=
2
1 – /2
4
3
–
2
i
0
µ
µ
λ
θ
∂
∂
Since ∂a1/∂θ0 is obviously zero when µ = 0, it is sufficient to consider only the
simpler case µ > 0.08. Then from eqns 5.151 and 5.152 we find
∂
∂
λ
θ
µ
µ
i
0
2
=
2
3
1 + 3 /2
8 +
as
as
(5.155)
giving
∂
∂
a as
as
1
0
2
1
2
2
=
8 /3
1 – /2
1 –
(1 + 3 /2)
8 + θ
µ
µ
µ
µ
(5.156)
Flight dynamics and control 183
Finally, differentiating eqn 3.64
∂
∂
∂
∂
∂
∂
h a a a c
0 0
D 1
0
D 0 D
=
8
–
4 θ θ
λ
θ
µλ θ
=
8
+ – 2 + 1
D
0
D
1
0
D 0
D
0
a
a
a ∂
∂
∂
∂
∂
∂
λ
θ
λ
θ
µ λ θ
λ
θ
(5.157)
Since
λD = µ(αnf + a1) – λi
∂
∂
∂
∂
∂
∂
λ
θ
µ
θ
λ
θ
D
0
1
0
i
0
= –
a
so that eqn 5.157 can be calculated in conjunction with eqns 5.155 and 5.156.
The force and moment derivatives are
x t
a t h
θ
θ
α
θ θ
0
D
= – – – c
1
0
D
c
0
c
0
∂
∂
∂
∂
∂
∂
(5.158)
z
t
θ
θ
0
= –
c
0
∂
∂
(5.159)
′ m h x l z C
a
m θ θ θ
θ
0 0 0 s
= – + – 1 1
1
0
∂
∂
(5.160)
which can also be expressed as
′ m l ha
t
t h C
a
h
h
m θ
θ θ θ
0 s
D
= –( – ) + ( + ) + 1s
c
0
c
1
0
c
0
∂
∂
∂
∂
∂
∂
(5.161)
Again, the terms in hcD are negligibly small. The θ0 derivatives for our example
helicopter are shown in Fig. 5.20.
5.9.3 Control response in forward flight
As mentioned earlier, response to collective pitch variation will not be considered, so
the non-dimensional equations of motion reduce to
d /d – – + cos = ( ) c c 1 1
u x u x w w x B u w B τ θ ττ(5.162)
– + d /d – – d /d + sin = ( ) c c 1 1
z u w z w V w z B u w B τ θ τθττ
ˆ (5.163)
– – – d /d + d /d – d /d = ( )
2 2
1 1
m u m w m w m m B u w w q B . τ θ τ θτ τ (5.164)
in which θ here is the fuselage pitch attitude change.
The cyclic pitch displacement B1 is a function of time, either as a prescribed
control input by the pilot or, as we discussed earlier in the section on autostabilisation,
184 Bramwell’s Helicopter Dynamics
related to some of the other variables through a control law. We have already considered
an example of the latter case in the section on stabilising bars.
Many of the control displacements which result in responses giving useful information
enable the above equations to be conveniently solved by the Laplace transformation.
Denoting the Laplace transforms of u, w, and θ by u w , and θ and supposing the
aircraft to be initially in trim, the transformed equations of motion become
( – ) – + cos = c c 1 1
p x u x w w x B u w B θ τ (5.165)
– + ( – ) – ( – sin ) = c c 1 1
z u p z w Vp w z B u w B
ˆ
τ θ (5.166)
– – ( + ) + ( – ) =
2
1 1
m u pm m w p m p m B u w w q B . θ (5.167)
Solving for u w , and θ gives
u B x p U p U U B P / = /( + + + )/ 1
3
2
2
1 0 1 ∆ (5.168)
w B z p W p W p W B / = ( + + + )/ 1
3
2
2
1 0 1 ∆ (5.169)
θ/ = ( + + )/ 1 2
2
1 0 1
B m H p H p H B ∆ (5.170)
where
∆ = p
4
+ B1cp
3
+ C1p
2
+ D1p + E1 (the stability quartic)
and
U z m Vm x
z
x
w q w w
B
B
2 = – ( + + ) +
1
1
ˆ
.
U z m Vm m w V x w
m
x
w q w w w
B
B
1 c c c c = – + sin + ( – cos )
1
1
ˆ ˆ
˙ τ τ
–( + cos ) c c
1
1
x m m w
z
x
w q w
B
B
. τ
1.2
1.0
0.8
0.6
0.4
0.2
0
0.05
0.04
0.03
0.02
0.01
0.2 0.3 µ
0.1
– 0
zθ
c.g. on shaft (l =0)
– 0
zθ
and
′ mθ0
′ mθ0
xθ0
Fig. 5.20 Control derivatives (collective pitch)
xθ0
Flight dynamics and control 185
U m w m w
z
x
z x w
m
x
w w
B
B
w w
B
B
0 c c c c c c c = sin – cos + ( cos – sin )
1
1
1
1
τ τ ττ
W x m z
x
z
V
m
z
u q u
B
B
B
B
2 = – – + +
1
1
1
1
ˆ
W x m z m Vm
x
z
Vx w
m
z
u q u q w
B
B
u
B
B
1 c c = – ( – ) – ( + sin )
1
1
1
1
ˆ ˆ τ
W m w m w
x
z
z w x w
m
z
u u
B
B
u u
B
B
0 c c c c c c c c = cos – sin + ( cos – sin )
1
1
1
1
τ τ ττ
H m
z
m
w
B
B
2 = 1 +
1
1
.
H x z m z m
x
m
m x m
z
m
u w u u w
B
B
w w w
B
B
1 = – – + ( + ) + ( – )
1
1
1
1
. .
H x z x z z m z m
x
m
x m x m
z
m
u w w u u w w u
B
B
w u u w
B
B
0 = – + ( – ) + ( – )
1
1
1
1
Equations 5.168, 5.169, and 5.170 are the transfer functions of the variables u, w,
and θ in relation to the control input B1.
Let us use the equations to find the normal acceleration due to a sudden increase
of cyclic pitch B1. The increment of acceleration can be expressed as ‘g’ units and
ng V
t
w
t
=
d
d
–
d
d
θ
or, in non-dimensional form, taking ˆ
V = , µ as
n
w
w
=
1 d
d
–
d
d c
µ
θ
τ τ
(5.171)
The Laplace transform of eqn 5.171 is
n
p
w
w = ( – )
c
µθ
and for the input of cyclic pitch
B B p 1 1 = /
so that
n
B
w B
w
B
= –
1
c 1 1
µ
θ
(5.172)
Usually the solution of ∆ = 0 consists of two real roots and a complex pair, and the
inverse of eqn 5.172 then takes the form
186 Bramwell’s Helicopter Dynamics
n F G H C S = + e + e + e [ cos + sin ]
1 2 re
im im
λ τ λτ λ τ
λ τ λ τ (5.173)
with similar solutions for u, w, and θ. For the less usual solution of two complex
roots, the inverse of n takes the form
n F C S C S = + e ( cos + sin ) + e ( cos + sin )
re1
1 1
re2
2 2
1 im 1 im 2 im 2 im
λ τ λ τ
λ τ λ τ λτ λτ
In the above two equations, F, G, H, C, C1, etc. are constants that are determined
by the initial conditions.
The normal acceleration in response to a step input of longitudinal cyclic pitch has
been calculated for our example helicopter (e = 0.04), for µ = 0.3, and is shown by
the full line of Fig. 5.21. Now, in a manouevre of this kind it would not be expected
that the stick would be held fixed for more than about three seconds, because the pilot
would want to retrim the aircraft into a new steady state. During such a relatively
short time interval, the forward speed would be expected to change very little and it
would seem reasonable to omit the speed terms, i.e. the u- and X-force derivatives,
from the response equations. The result of doing this is shown by the broken line of
Fig. 5.21, and it can be seen that the error in omitting these terms is quite small. The
simplification to the response equations is, however, considerable. The normal
acceleration can then be written as
n
w
w
=
1 d
d
–
d
d c
µ
θ
τ τ
= –
1
( + )
c
1 1
w
z B z w B w (5.174)
from, the simplified eqn 5.163.
Then, for a step input of cyclic pitch,
No speed terms
Full solution
0 1 2 3 4
t seconds
2.5
2.0
1.5
1.0
0.5
– / 1 nB
o
Fig. 5.21 Time history of normal acceleration in response to sudden change of cyclic pitch
Flight dynamics and control 187
n
z B
w p
z
p
p B p C
B
w = –
1
+
+
+ +
1 1
c
1
2
1 1
Γ
′ ′
(5.175)
where Γ1 = – mq + µm z B B 1 1
/ and ′ ′ B C 1c 1 and are the quartic coefficients B1c and C1
with the speed derivatives neglected.
The response calculations presented in Figs 5.22 to 5.24 have been made using
eqn 5.175.
Figure 5.22 compares the response of our example helicopter (e = 0.04) with a
hingeless helicopter which, as before, is assumed to have a hub moment five times
larger than that of the 4 per cent offset flapping hinge. It can be seen that the greater
control power and the increased instability result in a rapid increase of acceleration,
and the roots of p B p C
2
1c 1 + + = 0 ′ ′ indicate this growth to be rapidly divergent.
A peculiarity of helicopter normal acceleration is the sudden jump of acceleration
which occurs with the initial application of cyclic pitch. This is due to the force
produced by the sudden change of rotor disc incidence which is roughly proportional
to speed.
Fitting the tailplane considered earlier to both helicopters, Figs 5.23 and 5.24,
shows that the rate of increase of acceleration reaches a maximum and then begins to
decrease.
5.9.4 Control response in hovering flight
Another important flying quality of the helicopter is the rolling response in hovering
flight, particularly in relation to accurate manoeuvring near the ground. We shall
assume that we need consider only pure rolling, in which case the equations of
motion reduce simply to
d
d
–
d
d
=
2
2 1 1
φ
τ
φ
τ
l l A p A (5.178)
0 1 2 3 4
t seconds
2.5
2.0
1.5
1.0
0.5
– / 1 nB
o
e = 0.04
Hingeless
Fig. 5.22 Time history of normal acceleration for hingeless helicopter
188 Bramwell’s Helicopter Dynamics
0 1 2 3 4
t seconds
– / 1 nB
o
2.0
1.5
1.0
0.5
Tailless, e = 0.04
With tailplane
Fig. 5.23 Effect of tailplane on longitudinal response (articulated rotor)
0 1 2 3 4
t seconds
2.0
1.5
1.0
0.5
– / 1 nB
o
Hingeless, no tailplane
With tailplane
The rolling response to a sudden application of cyclic pitch A1 is
d
d
= – (1 – e )
1
1
φ
τ
τ
l
l
A
A
p
lp
Numerical values for our example helicopter with 4 per cent hinges and of helicopters
with zero offset and with a hingeless rotor are shown in Fig. 5.25.
It can be seen that the high roll damping of the hingeless helicopter enables it to
reach a constant rate of roll within less than a second, whereas the helicopter with
zero flapping hinge offset has not reached a steady rate by even four seconds. It is
interesting to note that, for a given amount of cyclic pitch, the final rates of roll are
the same. This is because the control power, represented by lA1, and the roll damping
lp vary in roughly the same proportion, and it is the ratio between them that determines
the final rate of roll. Pilots have described the response of the hingeless helicopter as
‘crisp’, whereas some would say of the helicopter with zero hinge offset that it ‘lags’
the application of control. The probable explanation for this is that the roll damping
of the latter helicopter is so weak that, roughly speaking, control application demands
Fig. 5.24 Effect of tailplane on longitudinal response (hingeless rotor)
Flight dynamics and control 189
roll acceleration which is, so to speak, two stages in advance of the roll angle which
is usually the required response. Thus, since a steady state is not reached in a fairly
short time, considerable anticipation is required of the pilot in order to check the
rolling motion and prevent overshooting the required bank angle. The control of the
hingeless helicopter is said to command roll rate because of the high damping.
An indication of the pilot’s action required to achieve a 20° angle of bank in 2
seconds is shown in Fig. 5.26 for the hingeless helicopter and for one with zero
flapping hinge offset. The roll manoeuvre demanded is assumed to be given by
φ = 1.25[cos(3π /2)t – 9 cos(π /2)t + 8]°
The quantities d
2
φ /dτ
2
and dφ /dτ are therefore known and inserted in eqn 5.178 to
obtain the lateral control A1 directly.
It can be seen that for the helicopter with zero offset hinges the manoeuvre is only
about one quarter completed when the stick begins to move in the opposite direction,
and is barely half completed before considerable opposite stick has to be applied to
check the rolling motion and settle into the required bank angle. The hingeless
helicopter, on the other hand, requires very little opposite stick to achieve the same
manoeuvre. The amount of stick movement required is not only much less but is
smoother than that of the helicopter with zero offset. This response would be regarded
as very satisfactory, and is one of the benefits of the hingeless rotor.
0 1 2 3 4 5
t seconds
15
10
5
Rate of roll, deg/s
Hingeless e = 0.04
e = 0
Fig. 5.25 Rate of roll in response to sudden application of lateral control
Hingeless
e = 0
Roll angle
demanded
20°
10°
2 1
t seconds
Angle of bank
Lateral cyclic pitch
6°
5°
4°
3°
2°
1°
0°
–1°
–2°
–3°
–4°
Fig. 5.26 Lateral cyclic pitch to achieve given roll manoeuvre
190 Bramwell’s Helicopter Dynamics
5.9.5 Response to vertical gusts
In this section we shall describe briefly the response of a helicopter to a sharp vertical
gust. A thorough investigation taking into account the time taken for the rotor to enter
the gust and the subsequent blade bending and flapping response would be very
lengthy. We shall assume that the rotor takes a negligible time to enter the gust
completely and, as in the section on forward flight response, we shall assume that
changes of forward speed are negligible, at least for the important initial response.
We shall also neglect the flapping motion of the blades.
The effect of meeting an up-gust, as far as the calculation of the blade forces and
moments is concerned, is identical to the effect of a downward velocity of the complete
helicopter; i.e. if the gust has velocity wg, the vertical force due to the gust alone is
Zwwg and the pitching moment is Mwwg.
The non-dimensional equations of motion corresponding to level flight are therefore
d
d
– –
d
d
= g
w
z w z w w w
τ
µ
θ
τ
(5.179)
–
d
d
– +
d
d
–
d
d
=
d
d
+
2
2
g
g m
w
m w m m
w
m w w w q w w . .
τ
θ
τ
θ
τ τ
(5.180)
The increment of normal acceleration ng is given, as before, by
ng
g
w
w
=
d
d
–
d
d c
µ
θ
τ τ
or
n
z
w
w w
w
= – ( + )
c
g (5.181)
from eqn 5.179.
By comparing eqn 5.181 with eqn 5.174, the Laplace transformation of the response
to a sharp-edged gust is
n
z w
w p
z
p
p B p C
w
w = –
1
+
+
+ +
g
c
1
2
1c 1
Γ
′ ′
(5.182)
Equation 5.182 has been solved for the cases of the helicopter with 4 per cent
flapping hinge offset and one with a hingeless rotor, both with and without a tailplane,
and the results for µ = 0.3 are shown in Figs 5.27 and 5.28.
Both tailless helicopters are unstable, as has been shown before, and we see that,
apart from a slight initial drop, the normal acceleration increases indefinitely, the
hingeless helicopter diverging more rapidly due to the larger, positive, value of mw.
When a tailplane is fitted, the response is improved and the normal acceleration of
the helicopter with offset hinges subsides quite quickly. The hingeless helicopter at
this forward speed is just about neutrally stable, and the normal acceleration remains
practically constant. At higher forward speeds, the hingeless helicopter becomes
unstable and the normal acceleration diverges.
Flight dynamics and control 191
2
1
0
Normal acceleration (g units)
1 2 3
e = 0.04
Hingeless
t seconds
Fig. 5.27 Longitudinal acceleration in response to sharp-edged vertical gust for a tailless helicopter
*If the centre of lift coincides with the centre of percussion of the blade the initial hinge reaction
will be zero.
It is sometimes said that, because helicopter blades can flap or bend, the rotor is
effectively self-alleviating with regard to the gust loads. There is a grain of truth in
this as far as the initial load is concerned because, while the blades are accelerating,
the reaction at the hinge will usually be less than the sudden change of lift*. But we
know that the blade flapping response is extremely rapid and can be regarded as
practically complete within about
1
4
second, so that after that time the lift load will
be fully transmitted to the hub and airframe. This extremely short time interval is
insignificant compared with the total response time, and justifies the assumption that
the blade flapping motion can be ignored. Furthermore, the rather academic example
of a sharp-edged gust represents the worst case; the more realistic ‘ramp’ type of gust
would give the blades even more time to reach equilibrium before the maximum gust
velocity is reached.
It is interesting to compare the initial acceleration response of the helicopter with
that of the fixed wing aircraft. If wg is the initial gust velocity, the normal acceleration
is given by
1 2 3
t seconds
e = 0.04
Hingeless
1
0.5
0
–0.5
Normal acceleration (g units)
Fig. 5.28 Effect of tailplane on gust response
192 Bramwell’s Helicopter Dynamics
ng
Z w
W g
w
=
–
/
g
for the helicopter and for the fixed-wing aircraft.
For the helicopter,
n
z sA Rw
w sA R
z w
w R
w w
= – = –
g
c
2 2
g
c
ρ
ρ
Ω
Ω Ω
and for the fixed-wing aircraft
n z
S
W
Vw w = – g ρ
For the helicopter we can take zw = –2aµ/(8µ + as), and for the fixed wing aircraft
z a w = – ,
1
2
where a in the latter case is the lift slope of the complete aircraft.
Let us compare our example helicopter (hinged or hingeless) with a fixed wing
aircraft whose wing loading W/S is 1450 N/m
2
and whose lift slope is 4. Then, if wg
is 10.5 m/s (standard gust velocity), the initial acceleration for both aircraft is as
shown in Fig. 5.29.
That the normal acceleration of the helicopter is practically constant is easily seen
physically since, for even quite high forward speeds, the rotor lift is dominated by the
(constant) rotational speed of the rotor, so the effect of a given vertical gust is almost
independent of speed. On the other hand, the response of the fixed wing aircraft
increases with speed.
5.9.6 The longitudinal characteristic quartic coefficients E 1 and C1
The coefficient E1. Consider a helicopter changing from one trimmed speed to
another, the collective pitch remaining fixed. For small changes of speed from level
flight, the resulting angle of climb or descent will be quite small, corresponding to
the ‘gliding flight’ of classical fixed wing stability theory.
Fixed wing
W/S = 1430 N/m
2
Helicopter
W/A = 270 N/m
2
2.0
1.5
1.0
0.5
n
0 20 40 60 80 100 m/s
Speed
Fig. 5.29 Comparison of helicopter and fixed wing aircraft response to vertical gust
Flight dynamics and control 193
The pitching moment under these conditions will be a function of the tip speed
ratio µ, the (non-dimensional) vertical velocity w, and the longitudinal cyclic pitch
B1. Since the helicopter is supposed to be in trim, we must have Cm(µ, w, B1) = 0 at
all speeds. Therefore
d
d
= +
d
d
+
d
d
= 0
1
1 C C C
w
w C
B
B m m m m
µ µ µ µ
∂
∂
∂
∂
∂
∂
Since the thrust will be practically constant, we also have
d
d
= +
d
d
+
d
d
= 0
c c c c
1
1 t t t
w
w t
B
B
µ µ µ µ
∂
∂
∂
∂
∂
∂
In terms of the aerodynamic derivatives, these two equations can be written
approximately as
m m
w
m
B
u w B +
d
d
+
d
d
= 0
1
1
µ µ
and
z z
w
z
B
u w B +
d
d
+
d
d
= 0
1
1
µ µ
Eliminating dw/dµ gives
d
d
=
–
–
1
1 1
B m z m z
m z m z
w u u w
B w w B µ
Now, when calculating the control derivatives, we found that z z B w 1
= – µ , so that
d
d
=
–
( + )
1
1
B m z m z
z m m
w u u w
w B w µ µ
=
( + )
1
c 1
E
z t m m w B w µ
from eqns 5.21, 5.25, and 5.27, with τc = 0.
Thus, the rate of change of cyclic pitch to trim with forward speed is proportional
to the constant term in the stability quartic, i.e. it is a measure of the static stability
of the helicopter. The result is analogous to that of the fixed wing aircraft. Now, in
the theory of the static stability of subsonic aircraft it is usual to regard mu as zero,
so that the static stability is entirely determined by the sign of mw, which is proportional
to ∂Cm/∂CL. But ∂Cm/∂CL is proportional to the distance of the c.g. from the neutral
point, and for this reason – ∂Cm/∂CL is called the ‘static margin’. Thus the static
stability of the fixed wing aircraft is related simply to a quantity which has a clear
physical meaning. Unfortunately, as we have seen, in helicopter work there is no
such simple parameter which directly controls the static and dynamic stability. Both
types of stability are affected by a number of quantities over which the designer has
only limited powers of variation. At low speeds mw is small, and the static stability
is dominated by mu which, although positive, is responsible for the helicopter’s
characteristic divergent oscillation. At high speeds the static stability is often improved,
because mu has changed very little and mw, although being much larger and positive
194 Bramwell’s Helicopter Dynamics
(i.e. in the unstable sense), is usually associated with a positive zu. When a tailplane
is fitted to improve the dynamic stability, at the high speeds for which zu is usually
positive, the combination zumw is such as to decrease the value of E1. Thus the
desirability of a positive static stability is not as straight/forward for the helicopter as
it is with a fixed wing aircraft.
The coefficient C1. Consider a helicopter performing a pull-up manoeuvre at constant
speed in a vertical plane, Fig. 5.30. It will be assumed that the flight path is circular
and the helicopter is in trim. The pitching moment will be a function of the vertical
velocity w, the rate of pitch q, and the cyclic pitch B1; i.e. in the trimmed manoeuvre
we have Cm(w, q, B1) = 0 with w and q being dependent on ng, the excess normal
acceleration. Therefore
d
d
=
d
d
+
d
d
+
d
d
= 0
1
1 C
n
C
w
w
n
C
q
q
n
C
B
B
n
m m m m ∂
∂
∂
∂
∂
∂
or
d
d
= –
( / )d /d + ( / )d /d
/
1
1
B
n
C w w n C q q n
C B
m m
m
∂ ∂ ∂ ∂
∂ ∂
(5.183)
where, in eqn 5.183, w and q have been non-dimensionalised (section 5.2). Then, if
∆T is the increase of thrust in the manoeuvre, we have
ng
T
W g
=
/
∆
or
n
T w w R
W
z
w
w
w
=
( / )
= –
c
∂ ∂ Ω
so that
d
d
= –
c
w
n
w
zw
Now
q
ng
V
=
Ω
therefore
d
d
= =
*
c
q
n
g
V
w
Ω µµ
Substituting in eqn 5.183 and expressing in derivative form gives
Fig. 5.30 Helicopter in vertical pull-up
V
q
Flight dynamics and control 195
d
d
=
/ – / *
1 c c
1
B
n
m w z m w
m
w w q
B
′ ′
′
µµ
=
–
1
µ
µ
m z m
z m
w w q
w B
∝ ′ 1 C
Hence, as with the fixed wing aircraft, the rate of change of control angle to trim
with normal acceleration is proportional to ′ C1, i.e. the value of C1 when the forward
velocity terms have been neglected. Now we have seen that the normal acceleration
response to either a longitudinal control input or a vertical gust depends on the value
of ′ C1; if it is positive the motion is stable (since the coefficient ′ B1c is always
positive), and if it is negative the response is divergent. Thus the slope of the control
angle to trim in a pull-up gives a direct indication of the normal acceleration response.
Again, as with the static stability, the c.g. position has little influence on the slope of
the control angle with acceleration, but we note that ′ C1 is not confounded by the
presence of forward velocity derivatives and its value can be directly affected by a
tailplane. Thus, with a large enough tailplane, one can always ensure that ′ C1 is
positive, although the tailplane may have to be inconveniently large to suppress the
inherent instability of the hingeless rotor at high speeds. However we should note
that a large tailplane (negative mw) with a positive zu may lead to static instability, as
indicated in the discussion relating to E1.
References
1. Hohenemser, K., ‘Dynamic stability of a helicopter with hinged rotor blades’, NACA Tech.
Memo. 907, 1939.
2. Sissingh, G. J., ‘Contributions to the dynamic stability of rotary wing aircraft with articulated
blades’, Air Material Command Trans. F–TS–690–RE, August 1946.
3. Padfield, G. D., Helicopter flight dynamics, Blackwell Science, 1996.
4. Bryant, L. W. and Gates, S. B., ‘Nomenclature for stability coefficients’, Aeronautical Research
Council R&M 1801, 1937.
5. Johnson, W., Helicopter theory, Princeton Univ. Press, Princeton NJ, 1980.
6. Pitt, D. M. and Peters, D. A., ‘Rotor dynamic inflow derivatives and time constants from
various inflow models’, Paper No. 55, 9th European Rotorcraft Forum, Stresa, Italy, 13–15
Sept. 1983.
7. Peters, D. A. and HaQuang, N., ‘Dynamic inflow for practical applications’, J. Amer. Helicopter
Soc., 33(4), pp. 64–68, Oct. 1988.
8. Amer, K. B., ‘Theory of helicopter damping in pitch or roll and a comparison with flight
measurements’, NACA Tech. Note 2136, 1950.
9. Bramwell, A. R. S., ‘Longitudinal stability and control of the single rotor helicopter’, Aeronautical
Research Council R&M 3104, 1959.
10. Zbrozek, J. K., ‘Introduction to the dynamic longitudinal stability of the single rotor helicopter’,
RAE Rep. Aero. 2248, 1948.
11. Bramwell, A. R. S., ‘The lateral stability and control of the tandem-rotor helicopter’, Part II
Aeronautical Research Council R&M 3223, 1961.
12. McLean, D., Automatic flight control systems, Prentice-Hall, 1990.
13. Pallett, E. H. J. and Coyle, S., Automatic flight control, Blackwell Science, 1993 (4th edition).
6
Rotor aerodynamics in forward
flight
6.1 Introduction
In Chapter 3 we gave some methods for calculating the induced velocity in forward
flight on the assumption that the rotor could be regarded as a lifting surface. These
methods give simple expressions for the induced velocity which can be incorporated
into the equations for calculating the rotor forces and blade flapping and which
eventually lead to fairly simple formulae for these forces and moments. We have so
far used only ‘linear’ aerofoil characteristics, e.g. a linear lift slope without stall and
a constant drag coefficient.
We now consider the fact that, as in hovering flight dealt with in Chapter 2, vortex
wakes spring from the individual blades and form a complicated downwash pattern
as the vortex elements spiral downwards below the rotor plane. We also consider the
aerofoil characteristics under conditions of high Mach number and changing incidence.
6.2 The vortex wake
Consider the blade in steady forward flight. As in hovering flight the circulation, in
general, varies along the span and, in consequence, vortex lines leave the trailing
edge and spiral downwards beneath the rotor. In addition, however, and as we saw in
Chapter 3, both the incidence and chordwise velocity vary over wide ranges in
forward flight causing timewise changes of incidence at a given radial position. In
accordance with Kelvin’s theorem each change of circulation at the blade must result
in a counter vortex being shed in the wake. Thus, in forward flight the vortex wake
also includes vortex lines which lie in the spanwise direction. These are referred to
as ‘shed’ vortices, to distinguish them from the ‘trailing’ vortex lines arising from the
spanwise circulation variation. The trailing vortices themselves are of varying strength
due to the timewise variation in the bound circulation in forward flight. The vortex
wake from a blade can be represented as in Fig. 6.1.
Rotor aerodynamics in forward flight 197
Shed vortex
Trailing vortex
Let us look at a small portion of the wake in more detail. Consider an element of
the blade of span dr. The local circulation is Γ, and trailing vortices of this strength
leave the ends of this element, Fig. 6.2. If the rate of change of circulation is ∂Γ/∂t,
the circulation at an instant dt before would have been Γ – (∂Γ/∂t)dt, and the change
will have resulted in a spanwise vortex element or filament of strength (∂Γ/∂t)dt
being shed behind the element, as shown in Fig. 6.2. Since the strength of the bound
circulation at the earlier instant was Γ – (∂Γ/∂t)dt, the arrangement of vortex filaments
can be represented in the form shown in Fig. 6.3, i.e. the vortex wake can be regarded
as an assembly of infinitesimal vortex rings. The distribution of vortex rings is
equivalent to a layer of doublets whose axes are perpendicular to the vortex sheet,
and Baskin
1
has used this to develop an induced velocity theory.
Fig. 6.1 Shed and trailing vortices from lifting blade
dΓ
Γ
Γ + dΓ
Γ
Γ
Γ + dΓ
Γ+ dΓ
Fig. 6.2 Spanwise change of bound circulation
Γ
Γ
Γ
Γ
d
d
d
Γ
t
t
Γ
Γ
– d
∂
∂t
t
Γ
Γ
– d
∂
∂t
t
Γ
Γ
Fig. 6.3 Modelling of vortex wake
198 Bramwell’s Helicopter Dynamics
The total induced velocity at a point P on a given blade of the rotor will be the
combined effect of the velocities induced by the trailing and shed vortices of all the
blades and of the bound vortices of all the blades except the one in question. If Γ is
the strength of a vortex element whose length is ds, and l is the displacement vector
between the element and the point P, Fig. 6.4, the Biot–Savart law gives for the
corresponding increment of induced velocity
d =
d
3
q
l s Γ
4π
⋅
×
l
(6.1)
Using this formula it would appear straightforward, in principle, to integrate the
contributions mentioned above. Unfortunately, as we also saw in the hovering case,
the vortices interact with one another and the wake pattern becomes very complicated.
This means that we are unsure of the positions of the vortex elements, particularly in
the more distant parts of the wake, or, in other words, l in eqn 6.1 is not known for
certain. Due to the forward flight case not having the axial symmetry of vertical
flight, the pattern of vortex interaction is different because, except at low speeds, the
flow through the rotor is determined largely by the component of forward flight
speed which is, of course, constant over the disc. Let us suppose that the induced
velocity is small compared with the forward speed (corresponding to a lightly loaded
rotor at high forward speed) and ignore the distortions due to vortex interaction. To
include partially the effect of the induced velocity we can calculate the ‘momentum’
value and add it to the component of the forward speed perpendicular to the rotor
disc. The trailing vortex lines then lie on the surfaces of skewed cylinders. This
represents, in effect, a rigid wake model by analogy with that for hovering flight. The
plan and side views of a trailing vortex line are shown in Figs 6.5 and 6.6.
If the x axis lies along the direction of motion and in the plane of the rotor, and the
z axis points upwards perpendicular to the rotor plane, as shown in Figs 6.5 and 6.6,
the co-ordinates of a trailing vortex element which was shed when the blade azimuth
angle was φ are
x = – r1 cos φ – Vt cos αD, y = x1R sin φ, z = Vt sin αD – vit
where t is the time taken for the blade to rotate from φ to ψ. Since ψ – φ = Ωt, we have
x = – r1 cos ψ – (V cos αD/Ω)(ψ – φ) = – r1 cos ψ – µDR(ψ – φ)
y = x1R sin φ
z = (V sin αD – vi)(ψ – φ)/Ω = λDR(ψ – φ)
l
Γ
P
ds
Fig. 6.4 Induced velocity at P
Rotor aerodynamics in forward flight 199
where µD = V cos αD/ΩR, λD = (V sin αD – vi)/ΩR, x1 = r1/R
It can be seen that the plan view of a trailing vortex filament, under the given
assumptions, is a cycloid, and the side view shows the constant downward displacement
of the cycloid.
Returning to eqn 6.1, let
l = l1i + l2j + l3k
and ds = ds1i + ds2 j + ds3k
where i, j, k are unit vectors along the x, y, z, axes. Then, if wt denotes the downward
component of induced velocity due to a trailing vortex of strength Γ
d =
d – d
t
2 1
3/2
w
l s l s
l
Γ
4
1 2
π
(6.2)
where l l l l
2
1
2
2
2
3
2
= + +
Then, measuring ds in the direction away from the blade,
ds1 = – dx = – R(x1 sin φ + µD)dφ
ds2 = – dy = – x1R cos φ dφ
From Figs 6.5 and 6.6 we see that, relative to the blade from which the vortex
element originated,
l1 = r cos ψ – r1 cos φ + µDR(ψ – φ)
y
r1
x
z
V
–αD
φ
ds
r
l
Γ
Figs 6.5 and 6.6 Plan and side views of vortex filament
ψ
200 Bramwell’s Helicopter Dynamics
l2 = –R(x sin ψ – x1 sin φ)
l3 = λD R(ψ – φ)
The blade in question will also be affected by the trailing vortices of the preceding
blades. The above equations still apply provided we allow for the appropriate phase
shift in the translational terms of l1, l2, l3. Thus, more generally,
l1 = R[x cos ψ – x1 cos φ + µD(ψ – φ + (2π/b)n)]
l2 = –R(x sin ψ – x1 sin φ) (6.3)
l3 = λDR(ψ – φ + (2π/b)n)
where n = 0, 1, 2, …, b – 1, and b is the number of blades.
Now the appropriate value of Γ of the trailing vortex filament is the change of
circulation (Fig. 6.2) over an element of the span dr, i.e. (∂Γ/∂r)dr. Then, integrating
along the trailing vortices from infinity up to the blade and along the span and
summing for all the blades, we have
w
R
x xx x x
D
n
b n
t
=0
0
1
–
1 1 D 1
3/2
=
1
4
+ cos( – ) + ( sin – sin )
π
ψ φ µ ψ φ
ψ
Σ
∫ ∫∞
+
{ – – (2 ) } cos
d d
D
3/2
1
1
µ ψ φ π φ
φ
/b n
D x
x
∂
∂
Γ
(6.4)
where ψn = (2π/b)n, n = 0, 1, 2, …, b – 1
and D x x xx b n = + – 2 cos ( – ) + ( + ){ – + (2 / ) }
2
1
2
1 D
2
D
2 2
ψ φ µ λ ψ φ π
+ 2 { – – (2 / ) }( cos – cos ) D 1 µ ψ φ π ψ φ b n x x
To calculate the induced velocity wb from the bound vortices of each blade, consider
two blades separated by angle ψn, as shown in Fig. 6.7.
The co-ordinates of a bound vortex element Γn are
x1 = – r1 cos (ψ + ψn), y1 = r1 sin (ψ + ψn)
Hence, for the elementary line
ds1 = – dr1 cos (ψ + ψn), ds2 = dr1 sin (ψ + ψn)
y
l
Γn
r1
ψn
ds
ψ
x
P
Fig. 6.7 Effect of bound circulation on succeeding blade
r
Rotor aerodynamics in forward flight 201
and the components of the line joining the line element to a point on the reference
blade are
l1 = – r cos ψ + r1 cos (ψ + ψn)
l2 = r sin ψ – r1 sin (ψ + ψn)
Then, noting that this calculation does not include the reference blade, n = 0, the
total contribution of the bound vorticity to the downwash at P is
w
R
x x
x x xx
x
n
b
n n
n
b
= 1
– 1
0
1
1
2
1
2
1
3/2 1 =
1
4
sin ( )
( + – 2 cos )
d
π
ψ
ψ
Σ
∫
Γ
(6.5)
According to Miller
2
, the contribution of the bound vortices is usually extremely
small.
Finally, calculating the induced velocity component ws for the shed part of the
wake, we have, see Fig. 6.8,
ds1 = – dr1 cos ψ0, ds2 = dr1 sin ψ0
The values of l1, l2, l3 are the same as those of eqn 6.3; hence, integrating in both
the azimuthwise and spanwise directions gives
w
R
x R
D
x
n
b
n
n
s
= 0
– 1
0
1
–
D
3/2 1 =
1
4
sin( – ) – ( – ) sin
d d
π
ψ φ µ ψφ φ
φ
φ
ψ
Σ
∫ ∫∞
∂
∂
Γ
(6.6)
The upper limit of the inner integral in eqn 6.6, must be treated with care. In
integral eqn 6.4, the azimuthwise integration is taken up to the span axis of the blade
since, in calculating the effect of the trailing vortices, it is reasonable to neglect
chordwise velocity variations and apply the idea of lifting line theory to the high
aspect ratio blade. The shed wake integral, on the other hand, really demands lifting
surface techniques since the chordwise vorticity distribution behind the aerofoil induces
a considerable chordwise velocity distribution as well as difficulties of singularities
in the region of the trailing edge. This matter will be dealt with in more detail in
section 6.2.2.
The induced velocity contributions given by eqns 6.4, 6.5, and 6.6 must now be
related to the circulation which gave rise to them. If W is the resultant chordwise
velocity at the blade, the local loading is
ψ φ
Fig. 6.8 Shed vortex element
∂
∂
∂
∂φ
φ
Γ Γ
t
t d = d
202 Bramwell’s Helicopter Dynamics
d /d =
1
2
2
L r W C c L ρ (6.7)
and the Kutta–Zhukowsky relation is
dL/dr = ρWΓ
Equating eqns 6.7 and 6.8 gives
Γ =
1
2 WC c L
Let us suppose that the lift slope is constant. Then
CL = aα
and writing
α = α0 + αi
where α0 is the incidence in the absence of the induced velocity and αi is the downwash
angle relative to the blade, we have
Γ = ( + )
1
2 0 i
a Wc α α
But
αi ≈ w/W
therefore
Γ = + ( + + )
1
2 0
1
2 t b s aW c ac w w w α (6.9)
where wt, wb, ws are given by eqns 6.4, 6.5, and 6.6.
Equation 6.9 is the integral equation for the circulation of the reference blade.
Although the equation has been simplified by the assumption of a rigid wake, no
standard method exists for solving it. One possible way which suggests itself is to
assume a simple induced velocity distribution, such as a uniform or Glauert type, and
make a first approximation to the circulation. From this approximation the integrals
eqns 6.4 to 6.6 are calculated, and new values of Γ are obtained from eqn 6.9. The
process is repeated until satisfactory convergence has been obtained. It cannot be
certain that convergence will always occur.
It is not necessary, of course, to develop an integral equation such as eqn 6.9.
Some methods begin by assuming a prescribed wake model based upon experimental
data or reasonable physical arguments and performing piecewise calculations of their
contributions to the induced velocity, as discussed in the following section.
6.2.1 Prescribed wake model development
One of the first attempts to depart from the idea of the rotor as a lifting surface, and
to use reasonable physical arguments to model the vortex wakes from individual
blades in such a way as to make the problem tractable, was that of Willmer
3
. The
main feature that allowed effective computation, particularly in view of the speed of
Rotor aerodynamics in forward flight 203
computers at that time, was his principle of ‘rectangularisation’. Consider the trailing
wake from two successive blades, Fig. 6.9. Willmer argued that the radius of curvature
of the sheets is large enough (especially for the important outer parts of the blade) for
the sheets to be regarded as straight. Actually this is the same assumption as the one
made by Prandtl in his method of calculating ‘tip loss’ in hovering flight, section
2.10.1.
Under ‘rectangularisation’, the trailing wakes, Fig. 6.9, are replaced by those
shown in Fig. 6.10. The wake attached to the reference blade is assumed to be a
straight sheet extending back to infinity, while those of the other blades are assumed
to be doubly infinite sheets placed so that the straight vortex lines are tangential to
the curved wakes where they pass under the reference blade. Part of the problem in
Willmer’s work was to determine the appropriate positions of the wakes of the
blades. Any number of parts of the original curved wake could be included, and the
choice made depended on the required accuracy. Once the numbers and positions of
the sheets had been chosen, the induced velocity at the reference blade could be
calculated by an extension of Glauert’s wing theory. It should be noted that this does
not imply that the vortex wake is regarded as a number of discrete vortices, as is
sometimes thought, but that the vortex wake is continuous and conditions are satisfied
at a number of points along the span (method of collocation). Unfortunately, because
the loading of a rotor blade is very different from the elliptic, or near elliptic, loading
of the conventional fixed wing, a larger number of spanwise points are needed for
acceptable accuracy. A comparison of the calculation of the blade thrust by Willmer’s
method with that of experiment and of the theory using constant induced velocity for
Fig. 6.9 Plan view of successive vortex sheets
Fig. 6.10 ‘Rectangularisation’ of vortex sheet
204 Bramwell’s Helicopter Dynamics
µ = 0.08 is shown in Fig. 6.11. Although the agreement with experiment can only be
described as fair, it is clear that the new model provided a great improvement on the
theory using constant induced velocity and went far to explain the large thrust variations
which often lead to large vibration at low values of µ.
Two important defects of the work were that it took no account of shed vorticity
and, related to this, it assumed that the strength of the trailing vortex elements were
constant along their length.
6.2.2 The shed vorticity
The question of shed vorticity and investigations into suitable representations of the
vortex wake have been considered by Miller
2,4
. Since Miller and others have given
considerable attention to the effect of the shed wakes, it may be useful to give a short
account of the methods of calculation here.
Consider an aerofoil, Fig. 6.12, whose incidence is changing. Any change of
circulation about the aerofoil must be accompanied by a corresponding vortex or
vorticity of the opposite sense in the wake, since Kelvin’s theorem requires that the
total circulation in a circuit containing the aerofoil and the wake must remain constant
with time. Thus, an aerofoil whose incidence is continually changing must deposit a
sheet of vorticity in the wake whose local strength is related to the time history of rate
of change of circulation. Now, the presence of vorticity in the wake plays a part in
satisfying the Kutta–Zhukowsky condition at the trailing edge of the aerofoil, so that
the circulation about the aerofoil is different from what would have been the case had
the vorticity been absent. Also, since the motion we are discussing is unsteady, there
1.75
1.50
1.25
1.0
0.75
0.50
Experimental
Willmer
Constant induced velocity
90° 180° 270° 360°
Lift
Mean lift
Fig. 6.11 Variation of blade loading with azimuth angle
Γ
Fig. 6.12 Vortices shed by aerofoil changing its incidence
Rotor aerodynamics in forward flight 205
is an extra pressure, or ‘virtual mass’, term which relates to the acceleration of the air
particles in the flow about the aerofoil.
The most important case of unsteady motion is a sinusoidal variation of incidence.
The first complete theory was given by Theodorsen
5
, who showed that for an aerofoil
whose incidence is changing by performing a sinusoidal vertical (heaving) motion,
the lift L as a fraction of the steady lift L0 at the instantaneous incidence is given by
L L C k k / = ( ) + i( ) 0
1
2
(6.10)
where C(k) is Theodorsen’s function and k is the frequency parameter nc/2V, n being
the frequency of oscillation and c the aerofoil chord. Theodorsen’s function is defined
by
C(k) = K1(ik)/[K0(ik) + K1(ik)]
where K0(ik) and K1(ik) are Bessel functions of the second kind.
For an aerofoil oscillating about its mid-chord, the corresponding result is
L L k C k / = [i + (i + 2/ ) ( )] 0
1
2
(6.11)
These results can be expressed in vector form as shown in Figs 6.13 and 6.14.
The results eqns 6.10 and 6.11 can be obtained in a number of ways
6
but, since the
‘method of vortices’ is closely related to the ideas being discussed in this chapter, it
will be described briefly below.
Corresponding to changes of circulation dΓ about the aerofoil, vortex elements of
strength γ bdξ, equal and opposite to the circulation, are deposited in the wake, γ
being the vorticity and ξ the distance of the element from the centre of the aerofoil,
non-dimensionalised in terms of the semi-chord b, Fig. 6.15.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
–0.1
–0.2
–0.3
–0.4
Component in quadrature with w
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1
1.6
1.2
k = 2
1.0
0.8
0.6
0.4 Component in phase with w
1.8
1.2
0.4
0.2 0.1
0.04
L
Vw 2πρ
M
Vw πρ
0.2
0.1
0.04
Fig. 6.13 Amplitude and phase of lift and moment of aerofoil oscillating in heave
206 Bramwell’s Helicopter Dynamics
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
–0.1
–0.2
–0.3
–0.4
–0.5
0.1 0.2 0.3 0.4 0.5 0.6
0.7 0.8 0.9 1.0
k = 1.4
1.2
1.0
0.8
0.6
0.4
Component in phase with α
0.2
0.1
0.04
0.04
0.1
1.0
0.4
0.2
1.4
1.8
Component in quadrature with α
L
V 2
2
πρ α
M
V πρ α
2
Fig. 6.14 Amplitude and phase of lift and moment of aerofoil oscillating in pitch
V
–1
x
η
α
1
ξ
0
Fig. 6.15 Shed vortex elements
In particular, for a given increment of time dt, the change of circulation is dΓ and
a vortex element – Vγ dt is shed at the trailing edge. Since γ is a function of both time
and distance, we can write
dΓ/dt = –Vγ (l, t) (6.12)
and since the total amount of vorticity shed in the wake must be equal to the total
circulation about the aerofoil, we also have the relation
Γ( ) = – ( , )d
1
t b t
∞
∫
γ ξ ξ (6.13)
From thin aerofoil theory
7
, the chordwise vorticity distribution can be expressed
as
γ η η ( ) = tan ( cos ) + sin (cos ) 0
1
2
–1
=1
–1
x A A n
n
n Σ
∞
(6.14)
Γ
γ dη
γ dξ
Rotor aerodynamics in forward flight 207
The assumed form, eqn 6.14, automatically satisfies the Kutta–Zhukowsky condition
at the trailing edge. The induced velocity at x due to a vortex element on the aerofoil
at η is
d ( ) =
( , ) d
2 ( – )
v x
t
x
γ η η
π η
(6.15)
Substituting the series eqn 6.14 for γ and integrating across the chord from –1 to +
1, gives
v( ) = – – cos (cos )
1
2 0
1
1
2
–1
x A A n n Σ
∞
η
where we make use of the well known result
0
π
φ
φ η
φ π
η
η ∫
cos
cos – cos (cos )
d =
sin (cos )
sin (cos )
–1
–1
–1
n n
(6.16)
Let us suppose that, in addition to its forward velocity V, the aerofoil has a vertical
(heaving) velocity of . z at a point about which the aerofoil is also rotating, Fig. 6.16.
The relative air velocity normal to the chord at x due to the motion is
u x V z b x x ( ) = + – ( – ) α α
. . .
The induced velocity at x due to a vortex element in the wake is
d ( ) =
d
2 ( – )
w x
x
γ ξ
π ξ
and the velocity at x induced by the entire wake is
w x
x
( ) =
d
2 ( – ) 1
∞
∫
γ ξ
π ξ
Since there can be no flow perpendicular to the chord, we must have
u(x) + v(x) + w(x) = 0
or
– – cos (cos ) + [ + – ( – )] +
d
2 ( – )
= 0
1
2 0
1
2 =1
–1
1
A A n V z b x x
x
n
n Σ
∞ ∞
∫
η α α
γ ξ
π ξ
˙ ˙
(6.17)
α
0
–x
– x
. α
. z
Fig. 6.16 Aerofoil oscillating in heave
208 Bramwell’s Helicopter Dynamics
Now eqn 6.16 can also be expressed as
0
π
φ
ξ φ
φ
π ξ ξ
ξ ∫
√
√
cos
– cos
d =
[ – ( – 1)]
( – 1)
2
2
n
n
so that the Fourier coefficients An occurring in eqn 6.17 are easily found to be
A =
1 d
( – 1)
+ 2( + – )
A =
2
( – 1)
– 1 d + 2
A =
2 – ( – 1)
( – 1)
d
0
1
2
1
1
2
1
2
2
π
γ ξ
ξ
α α
π
ξ
ξ
γ ξ α
π
ξ ξ
ξ
γ ξ
∞
∞
∞
∫
∫
∫
√
√
√
√
V z bx
b
n
. .
.
(6.18)
The linearised Bernoulli equation gives for the pressure difference between the
upper and lower surfaces
p p p
t
V
x
l – = 2 + u
∂
∂
∂
∂
φ φ
(6.19)
In terms of the vorticity on the aerofoil, the velocity potential is
φ γ ( ) =
1
2
d
–1
x b x
x
∫
so that
∂
∂
φ
γ
x
b =
1
2
and
∂
∂
∂
∂
φ
γ
t
b
t
x
x
= d
1
2
–1
∫
The lift on the aerofoil is
L b p p x l u = ( – )d
–1
+1
∫
Substituting for the pressure difference from eqn 6.19 and the vorticity from
eqn 6.14 leads to
L b b
t
A A b
t
A A V A A = ( – ) + ( + ) + ( + )
1
2 0
1
2 2 0
1
2 1 0
1
2 1 ρπ
∂
∂
∂
∂
(6.20)
The circulation about the aerofoil is
Γ = d = ( + )
–1
+1
0
1
2 1 b x b A A
∫
γ π (6.21)
Rotor aerodynamics in forward flight 209
From eqns 6.18 and 6.21, the first two terms in the bracket of eqn 6.20 can be
written as
1
2 0
1
2 2
1
2 2
2
( – ) +
1
=
1 – + ( – 1)
( – 1)
d b
t
A A
t
b
t
∂
∂
∂
∂
∂
∂ π π
ξ ξ ξ
ξ
γ ξ
Γ
∞
∫
√
√
+ ( + + ) +
1
b V z bx
t
. .. .
α α
π
∂
∂
Γ
= d – ( – 1) d
1 1
2 b
t
b
t π
γ ξ ξ
π
ξ γ ξ
∂
∂
∂
∂
∞ ∞
∫ ∫
√
– ( + + ) +
1
b V z bx
t
. .. .
α α
π
∂
∂
Γ
(6.22)
Now, each element of vorticity in the wake is carried backwards with velocity V,
so the vorticity must be of the form
γ (ξ, t) = f (t – bξ /V)
Since, for a given element of vorticity, t – bξ /V is constant, it follows that
d = d + d = 0 γ
γ γ
ξ
ξ
∂
∂
∂
∂ t
t
and d – d = 0 t
b
V
ξ
i.e.
∂
∂
∂
∂
γ γ
ξ t
V
b
= – (6.23)
in the wake.
Differentiating under the integral sign and making use of eqn 6.23, we find
∂
∂
∂
∂ t t
d – ( – 1) d
1 1
2
∞ ∞
∫ ∫
√ γ ξ ξ ξ γ ξ
= – d + ( – 1) d
1 1
2 V
b
V
b
∞ ∞
∫ ∫
√
∂
∂
∂
∂
γ
ξ
ξ ξ ξ
γ
ξ
ξ
= – + d + ( – 1)
1
1
2
1
V
b
V
b
V
b
γ ξ γ ξ γ ξ
[ ] √
[ ]
∞
∞
∞
∫
–
( – 1)
d
1
2
V
b
∞
∫ √
γ ξ
ξ
ξ
= – [ – ( – 1)] ( , ) + d
2
1
V
b
t
V
b
ξ ξ γξ γξ √
{ }
∞
∞
∫
1
–
( – 1)
d
1
2
V
b
∞
∫ √
γ ξ
ξ
ξ,
on integrating by parts.
210 Bramwell’s Helicopter Dynamics
Now ξ – √(ξ
2
– 1) = 1/[ξ + √(ξ
2
– 1)], from which it follows that, since γ(ξ, t) is
finite, the term in the bracket vanishes at the upper limit. At the lower limit the term
in the bracket is equal to (V/b)γ (1, t). This term cancels identically with the last term
of the eqn 6.22, so that
1
2 0
1
2 2 0
1
2 1 ( – ) +
1
+ ( + ) b
t
A A
t
V A A
∂
∂
∂
∂ π
Γ
= d –
( – 1)
d + ( + – ) + ( + )
1 1
2
0
1
2 1
V V
b V z bx V A A
π
γ ξ
π
γξ
ξ
ξ α α
∞ ∞
∫ ∫ √
˙ ˙˙ ˙
= + + ( + – ) 0 VA Vb b V z bx . . .. ..
α α α
on eliminating the integrals by means of eqn 6.18. The lift can then be written as
L Vb V z b x Vb = 2 [ + + ( – )] +
d
( – 1)
1
2
1
2
πρ α α ρ
γ ξ
ξ
. .
∞
∫ √
+ ( + – ) πρ α α b V z bx
2
. ..
(6.24)
The first term is the ‘quasi-steady’ lift which alone would occur if the motion were
steady, the second term represents the lift due to the vortex wake, and the third term
is the lift due to the fluid inertia or ‘apparent mass’. Denoting the quasi-steady lift by
Lq we have
L Vb V z b x q
1
2
= 2 [ + + ( – )] πρ α α . .
=
+ 1
( – 1)
– 1 d
1
2
∞
∫ √
ξ
ξ
γ ξ
on eliminating Γ and A A 0
1
2 1 + from eqns 6.13, 6.18, and 6.21. It then follows that
eqn 6.24 can be written
L L b V z bx =
d
( – 1)
d
( – 1)
+
d
( – 1)
+ ( + – ) q
1
2
1
2
1
2
⋅
√
√ √
∞
∞ ∞
∫
∫ ∫
γ ξ ξ
ξ
γ ξ ξ
ξ
γ ξ
ξ
πρ α α
2
˙ ˙˙ ˙˙ (6.25)
For sinusoidal motion, the lift can be expressed as L = L0 e
iω t
and the vorticity in
the wake, as was discussed earlier, must be of the form
γ (ξ, t) = γ0 e
iω (t–bξ/v)
= γ0 e
iω t
e
–ikξ
where k = ωb/V is the ‘reduced frequency’.
The ratio of integrals occurring in eqn 6.25 then becomes
Rotor aerodynamics in forward flight 211
1
2
1
2
1
2
1
2
1
d
( – 1)
d
( – 1)
+
d
( – 1)
=
d
( – 1)
+ 1
– 1
∞
∞ ∞
∞
∞
∫
∫ ∫
∫
∫
√
√ √
√
γ ξ ξ
ξ
γξ ξ
ξ
γ ξ
ξ
γ ξ ξ
ξ
ξ
ξ
=
( – 1)
e d
+ 1
– 1
e d
1
2
– i
1
– i
∞
∞
∫
∫
√
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
k
k
= C(k), Theodorsen’s function, referred to earlier.
The lift can now be written
L C k L b V z bx = ( ) + ( + – ) q πρ α α
2
. .. .. (6.26)
The pitching moment about the point of rotation x = x is found to be
M bC k x L b xz V x b x = ( )( + ) + [ – ( – ) – ( + )]
1
2 q
1
2
1
8
2
πρ α α
3
.. . .. (6.27)
In particular, the lift in purely rotational harmonic motion about the mid-chord
can be written as
L Vb V b k C k = + 2 (1 + i /2) ( )
2 2
πρ α πρ α
. (6.28)
and for heaving motion as
L b z VbC k z = + 2 ( )
2
πρ πρ .. . (6.29)
In both cases the term proportional to C(k) is due to the bound circulation.
If L0 is the steady lift at the instantaneous incidence the values of L/L0 can easily
be shown to reduce to the forms of eqns 6.10 and 6.11.
It is interesting now to consider the velocity component v3/4 at the
3
4
-chord point
due to the aerofoil motion. For aerofoil rotations about the mid-chord, we have
v3/4
1
2
= + α α V b .
For harmonic motion, α = α0 e
iωt
, and
v3/4 = 1 + i /2) . ( αV k
If Lb denotes the lift due to the bound circulation, i.e. that part of the lift proportional
to C(k), we see from eqn 6.28 that
Lb = 2πρVbC(k)v3/4
For the heaving motion we have simply
v3/4 = . z
212 Bramwell’s Helicopter Dynamics
and, again, from eqn 6.29,
Lb = 2πρVbC(k)v3/4
Thus, in both cases, it appears that the part of the lift arising from the bound circulation
is proportional to the ‘downwash’ at the
3
4
-chord point. For this reason, the
3
4
-chord
point is known as the ‘rear aerodynamic centre’ and is the appropriate chordwise
position to consider when lifting-line techniques are used in cases such as those
described above, where the chordwise variation of velocity may be important.
The effect of shed vorticity in the special case of hovering flight has been investigated
by Loewy
8
and Jones
9
. Although hovering flight is usually regarded as a symmetrical
flight state, any disturbed blade motion will give rise to periodic lift forces and the
generation of a shed vortex wake in addition to the trailing vortices. The work of
Loewy and Jones was based on two-dimensional analysis which one would expect to
overestimate the effects somewhat, since the shed vortex lines in practice extend only
the length of the span. The mathematical model used is as indicated in Fig. 6.17. The
shed wake pattern is imagined to consist of the semi-infinite wake attached to the
blade and a vertical array of doubly infinite wakes representing those of other blades
and of the previous passages of the reference blade. The vertical spacing h between
each sheet is determined from the mean flow through the rotor. If the effect of the
lower sheets is averaged over the blade chord (lifting-line assumption), the lift on the
blade is then
L Vb Vb
n h
L zb
n
=
d
( – 1)
+
d
+
+ +
1
2 =1
–
2 2 2 0
2
ρ
γ ξ
ξ
ρ
γ ξ ξ
ξ
ρπ
∞ ∞
∞
+ ∝
∫ ∫ √
Σ ˙˙
which is the same as eqn 6.24 except for the addition of the second term which
denotes the contribution of those sheets below the blade. For oscillations which are
integer frequencies of the rotor speed, Miller found that
L
L
C k m,h
J Y
J Y Y J
F iF
hk m
q
1 1
1 1 0 0
2 i R I = ( , ) =
– i
– i + +i +[2i/e e –1)]
+
π
≡ (6.30)
where J0, Y0, J1, Y1 are Bessel functions of the first and second kinds, of orders zero
and unity and argument k; m = ω /Ω and k = ω b/ΩR. Figure 6.18 shows the comparison
of eqn 6.30 with Loewy’s exact lifting-surface results. The agreement is extremely
good for FR but there is poor agreement for the phase shift represented by FI. However
as pointed out by Miller, typical values of h are greater than unity and the error is
fairly small.
h
h
Fig. 6.17 Vortex sheets shed by oscillating aerofoil in hover
Rotor aerodynamics in forward flight 213
–FI
0.5
0
The case of an infinite number of blades leads to a particularly simple result. The
shed vorticity is then uniformly distributed vertically, and Miller found that, if the
frequency is nΩ, the induced velocity at the aerofoil becomes
w
nbc
R z
z
n
n t nc R
=
i e
8
e
+
d d
i
2
0 –
i / 2
2 2
Γ
Ω
π λ ξ
ξ
ξ
∞
∞
+ ∝
∫ ∫
where λ is the mean inflow ratio and Γn is the amplitude of the bound circulation.
From the blade element theory, if θ = θn e
inΩt
is the blade pitch variation we have
Γ = πΩRc(θ – w/ΩR)
since the quasi-static circulation is Γq = π ΩRcθn.
Elimination of w and writing s = bc/πR leads finally to
C = 1/(1 + sπ/4λ)
Thus, in this approximation, C is independent of the frequency. The analysis is
unable to predict the phase shift. Comparison of this method with Loewy’s results
shows very good agreement for values of h below 5, i.e. for typical values of h
occurring in practice.
Miller extended the approximate analysis just discussed to the three-dimensional
vertical flight case. It was assumed that the circulation was constant along the blade,
so that the trailing vortex wake consisted only of a tip vortex and a vortex along the
rotor axis. Applying the above approximations to the integrals 6.4 and 6.6 leads to the
result
w = (b/4πλR)(Γns sin nψ + Γnc cos nψ)
when the bound circulation is given by
Loewy
Eqn 6.30
h
k = 0.1
1.0
0.5
0 h
k = 0.5
1.0
0.5
0 5 10
0.5
0
k = 0.1
5 10
h 5 10 h 5 10
k = 0.5
Fig. 6.18 Comparison between exact and approximate oscillatory parameters
FR FR
–FI
214 Bramwell’s Helicopter Dynamics
Γn = Γns sin nψ + Γnc cos nψ
The lift deficiency function C turns out to be exactly that of the previous analysis,
namely,
C = 1/(1 + sπ/4λ)
It is interesting to note that, although the contribution of the shed wake in this case
is smaller on account of the finite length of the vortex lines, this is exactly compensated
for by the effect of the trailing vortex lines which, in the two-dimensional case, were
omitted.
The values of the lift deficiency function indicate that the effective lift slope of the
blade is much reduced and, under some conditions, could approach zero. This implies
correspondingly reduced damping of the blade flapping motion.
Because of the good agreement between Miller’s approximate two-dimensional
analysis and the exact analysis of Loewy, Miller argued that for calculations of the
forward flight case it is reasonable to consider both the shed and trailing wakes in
two parts: a ‘near’ shed wake which corresponds to the first quadrant of the wake
and a ‘far’ wake consisting of the remainder. Furthermore, the ‘near’ wake could
be regarded as straight and extended back to infinity, as was assumed by Willmer.
It was supposed that the classical two-dimensional unsteady theory of Theodorsen
could be applied to account for the shed vorticity in this section of the wake.
For the treatment of the shed vorticity in the far wake, it was assumed that the
chordwise variation of velocity at the blade could be neglected and that only a mean
velocity need be considered, as in Miller’s two-dimensional analysis. Calculations
showed that, above µ = 0.2, the far wake has little effect and the values of FR
and FI approach those of the classical two-dimensional theory. The results
suggested that a good approximation could be obtained by simply ‘fairing’ the
results found for the hovering case to those of the two-dimensional aerofoil theory,
Fig. 6.19.
1.0
0.5
0
0.1 0.2 0.3 µ
Complete shed and trailing wake
Shed wake only
Near shed wake only
– FI
FR
Fig. 6.19 Variation of oscillatory parameters in forward flight
Third harmonic
Rotor aerodynamics in forward flight 215
Calculations made by assuming infinite straight vortex filaments for the spiral
trailing vortices in the integral 6.4 showed little loss of accuracy but a great reduction
of computer time.
Calculations also showed that good accuracy was retained even when the trailing
wake was assumed to consist of only two discrete vortices, one springing from the tip
and another a little inboard of the mid-span. Such a vortex system has indeed been
found by Tararine
10
from smoke tests, Fig. 6.20.
Comparison of Miller’s calculations with experimental air loads is shown in Figs
6.21 and 6.22.
The peak loading at an azimuth angle of about 90° is due to the tip vortex of the
preceding blade. This peak travels down the blade as it advances from 90° to 180°,
and can be detected in Figs 6.21 and 6.22. It might be thought that the peak could be
reduced by increasing the number of blades, but, although the vortex strength is
reduced, so are the blade spacings, with the result that the vortices pass closer to the
blade in question. According to Miller, the peak loading is influenced very little by
the number of blades.
A further conclusion of Miller’s work is that almost all the harmonic content of the
air loads is contained in harmonics above the second, and they are derived from the
effects of the far wake. These harmonics are practically independent of blade flapping
motion.
As was stated earlier, Miller’s work was based on the assumption of a rigid wake.
Fig. 6.20 Representation of shed and trailing vortices (after Tararine)
5
4
3
2
1
0 90° 180° 270° 360°
Theory
Experiment
x = 0.95
µ = 0.2
ψ
Fig. 6.21 Blade loading variation with azimuth angle
kN/m
216 Bramwell’s Helicopter Dynamics
x = 0.85
µ = 0.2
Theory
Experiment
5
4
3
2
1
0 90° 180° 270° 360°
ψ
Fig. 6.22 Blade loading variation with azimuth angle
6.2.3 Free wake model development
In the 1960s, Cornell Aeronautical Laboratory carried out a programme of work
aimed at improving the understanding of, and the ability to model, the wake of a
helicopter in steady state flight. As part of this, Piziali
11
proposed a wake model that
embodied the shed and trailing vorticity in the form of a mesh of straight line vortex
filaments, as shown in Fig. 6.23. This occupied the near wake adjacent to the blade,
whilst the far wake was modelled simply by concentrated root and tip vortices, the
former being deleted for some studies. The mesh part of the wake could be truncated
at any chosen position downstream. Initial studies in the forward flight case placed
the wake mesh points on an undistorted skewed helix (i.e. a rigid wake), but later
models allowed distortions from this to occur for a fixed length of time, the distorted
wake being maintained thereafter. Blade flapping bending modes were included in
the programme, and the calculated loading, blade response, and corresponding induced
velocity were interdependent and were calculated iteratively until all quantities were
self-consistent.
Like Miller, Piziali gave great attention to the unsteady aerodynamic characteristics
due to the shed wakes. To test the validity of the finite element representation of the
shed vorticity, the calculations were made for the two-dimensional aerofoil and compared
Trailing-vortex
filaments
Shed-vortex
filaments
Root and tip
vortices
Fig. 6.23 Piziali’s wake model
kN/m
Rotor aerodynamics in forward flight 217
with the classical exact theory discussed in the previous section. It was found that,
unless the interval between successive vortex lines was much smaller than was really
practical for the computational capabilities that were then current, agreement was
rather poor, but by advancing the shed wake towards the aerofoil by about 70 per cent
of a complete interval the agreement was greatly improved. In a later paper by
Piziali, a smoothing routine was employed to simulate a continuous wake from a
number of finite elements. Comparisons of measured loadings with those from Piziali’s
method are shown in Figs 6.24 and 6.25.
In the free wake model of Landgrebe
12
, the basic rotor wake is similar to that
described above, but the vortex elements are allowed to take up positions determined
by the free stream velocity and the induced velocity of all the other elements. Only
the bound and trailing vortices were considered in the calculations; the shed vorticity
was accounted for by using two-dimensional unsteady aerofoil data.
One of the difficulties of representing the wake by discrete vortex elements is that
infinite velocities occur at the vortex itself. To overcome this, Landgrebe supposed
that the vortex had a core within which the induced velocity can be neglected. The
core size assumed by Landgrebe was 1 per cent of the rotor radius. Other investigators
have assumed that the velocity within the core varies linearly with radial distance
4
2
0
–2
–4
0 80 160 240 320
Blade loading
ψ
4
2
0
–2
–4
0 80 160 249 240
Blade loading
r/R = 0.90
HU – 1A, µ = 0.26
r/R = 0.75
HU – 1A, µ = 0.26
ψ
Fig. 6.25 Blade loading variation with azimuth angle
0 80 160 240 320
r/R = 0.9
H – 34, µ = 0.18
ψ
0 80 160 240 320
r/R = 0.75
H – 34, µ = 0.18
ψ
Fig. 6.24 Blade loading variation with azimuth angle
2
1
0
–1
–2
Blade loading
kN/m
2
1
0
–1
–2
Blade loading
kN/m
kN/m
kN/m
218 Bramwell’s Helicopter Dynamics
(i.e. as if the core were solid). Measurements conducted to investigate the structure
of the tip vortex have been described by Cook
13
.
In order to reduce the computational time and expense, not all the vortex elements
were assumed to be free to convect in accordance with the local velocities. Computation
was greatly reduced by assuming that the positions of the vortex elements beyond a
certain distance from the point of interest remained the same as in the original
prescribed wake which formed the starting point of the calculations. An example of
Landgrebe’s calculations is given in Fig. 6.26, which shows the axial displacement of
the tip vortex filament compared with the ‘classical’ rigid wake uniform displacement.
Another example is the theoretical wake boundary at low µ compared with that from
a smoke visualisation study, Fig. 6.27. It can be seen that the vortex filaments at the
front of the disc lie very close to the disc. This might have been expected from the
Glauert or Mangler distributions, since they predict a slight upwash at the leading
edge of the disc. Landgrebe’s calculations also showed that the wake boundary rolls
up, eventually forming two large vortices very similar to those observed by Heyson
and Katzoff, as described in Chapter 3.
Classical wake
Complete vortex interaction
Approx. vortex interaction
0.1
0.2
0.3
0.4
0.5
0 1 2 3
Z/R
Fig. 6.26 Displacement of tip vortex in forward flight
V V/ ΩR = 0.05 Rotor plane
(Like symbols indicate positions of
vortex elements deposited in wake by
blades at same instant of time. The time
interval between spaces corresponds
to 90° of blade rotation.)
Experiment
Classical wake
Distorted-wake analysis
Fig. 6.27 Displacement of wake boundary
0
Rotor aerodynamics in forward flight 219
The pioneering studies of Piziali, Landgrebe and associates, together with those of
Clark and Leiper, for hovering flight, and others of that era laid the foundations for
much subsequent wake modelling research. The main difficulties that had to be
overcome, once computational power became sufficient, were largely numerical,
such as instability and/or poor convergence. In order to avoid such problems, Bagai
and Leishman
14
used a relaxation method, as opposed to a time-stepping approach,
which enforces periodicity as a boundary condition and specifies that the trailing
vortex filaments be attached to the blades as an initial condition. The basic model
was similar to that in Fig. 6.23. An improvement over an existing free wake modelling
code was demonstrated and comparisons made with existing experimental data.
Perspective views of the computed rotor wake in the hover and in forward flight
shown in Fig. 6.28; the rolling up of the wake boundaries downstream at the faster
forward flight speeds is clearly seen.
The advent of computational fluid dynamics, or CFD, has brought with it a steady
advance in the complexity of flows that can be analysed. However, the computation
of the complete flow field around a helicopter in forward flight, which may include
interaction with the fuselage and tail surfaces, remains an outstanding problem with
current computing capabilities. A brief account of the difficulties of using the various
CFD approaches is given by Coppens et al.
15
in the introduction to an account of a
time-marching method of computing the rotor wake.
Figure 6.28 for the forward flight cases shows the rotor blades passing close to tip
vortices shed by preceding blades; this proximity, which is termed blade–vortex
interaction, or BVI, leads to impulsive loading and noise problems. It is particularly
severe in descending forward flight, such as during an approach to landing, when the
rotor tends to fly through its own wake. The plan view of a typical pattern of BVI is
shown in Fig. 6.29 (from Tangler
16
) for a two-bladed rotor at an advance ratio of
0.145. Four interactions are shown on the advancing side and three on the retreating
Hover µ = 0.05
µ = 0.075 µ = 0.1
Fig. 6.28 Perspective views of computed rotor wake (Bagai and Leishman
14
)
220 Bramwell’s Helicopter Dynamics
side. The strength of each interaction is dependent on the strength of the tip vortex,
the vortex core size, the instantaneous angle between the blade and the vortex at the
interaction, and the normal separation between the two, or the ‘miss’ distance. When
the rotor is in descending flight, the miss distance can become quite small.
On the advancing side, the approach of a blade section to a preceding vortex is as
shown in Fig. 6.30, with a typical resulting lift coefficient history indicated (adapted
from Leishman
17
). On the retreating side, the rotation direction of the vortex with
respect to the blade is reversed, and the lift coefficient history is similarly affected.
Since the lift change occurs over a very small period of time, it is practically impulsive,
leading to two predominant effects. One is that of blade loading, which possesses a
high frequency content, and the other is that of acoustic radiation, or noise.
The introduction of the improved wake modelling described earlier has allowed
the problems of BVI to be defined more accurately and solutions proposed with a
greater degree of confidence. The impulsive blade loading problem is a problem
because it leads to undesired structural vibration, but a possible solution can be
provided by use of higher harmonic control, or HHC. In this approach, higher than
180°
V
4
90°
6
7
5
2
3
1
0°
Fig. 6.29 Plan view of blade-vortex interactions (from Tangler
16
)
Rotor aerodynamics in forward flight 221
once-per-revolution input is introduced to the pitch control by means of subsidiary
signals to the actuators below the swash plate. An alternative scheme allows for the
signals to be fed to supplementary actuators positioned between the swash plate and
the blades, this being termed individual blade control, or IBC. Practical tests on both
arrangements applied to a four-bladed wind tunnel rotor model are described by
Kube and Schultz
18
, and these cover both vibration and noise aspects.
Considerable effort has been expended in studying noise propagation from
helicopters, since commercial viability is strongly affected by environmental
acceptabilty. Schmitz and Yu
19
provided a comprehensive review of theoretical and
experimental studies as an aid to understanding the problems, and Lowson
20
suggested
the way forward towards achieving quieter helicopters. From the point of view of the
observer on the ground, the most recognisable component of sound radiation is that
of ‘blade slap’, which tends to occur at slow to medium speeds with the helicopter in
descending flight. The interaction which normally provides the predominant effect is
number 4 in Fig. 6.29, for which the miss distance is small and the intersection angle
is highly oblique. For a four-bladed rotor, this occurs at an azimuth angle of about
60°, and leads to a directional preference for the radiated sound pressure level, as
described by Ehrenfried et al.
21
in experimental tests and by Lowson
22
from fundamental
considerations.
6.3 Aerofoil characteristics in forward flight
We saw in Chapter 3 that the blade encounters a wide range of conditions in forward
0.1
0
–0.1
–0.2
Lift coefficient CL
4 –2 0 2 4
Vortex position, xv (chords)
xv
Γ
Fig. 6.30 Lift coefficient time history (adapted from Leishman
17
)
222 Bramwell’s Helicopter Dynamics
flight. The incidence of the retreating blade increases with tip speed ratio so that the
stalling angle may be reached, particularly on the important outer section of the
blade, while on the advancing blade the Mach number may reach values at which the
compressibility drag rise begins. These varying conditions may be conveniently
shown in a ‘figure-of-eight’ diagram, as in Fig. 6.31, which shows the α–Mach
number relationship at the radial position x = 0.913 for a ‘Wessex’ rotor flying at
µ = 0.32. The hovering case would be represented by one point on the diagram.
As µ increases, the ‘figure-of-eight’ gradually expands, extending into regions of
higher CL and higher M. To test whether the aerofoil will encounter stall and
compressibility drag rise, one can plot on the same diagram the α–M boundaries for
these conditions. For the NACA 0012, which is the aerofoil on the ‘Wessex’, the
boundaries are shown in Fig. 6.32, together with the α–M variation of Fig. 6.31. It
can be seen that the α–M loop passes well into the stall region on the retreating blade
and well into the drag-rise region on the advancing blade. The effects on the pressure
distribution of the NACA 0012 aerofoil have been examined in detail by Pearcey et
al.
23
and they show that the flow is supersonic over some part of the aerofoil for almost
the complete azimuth cycle of the blade. This is indicated in Fig. 6.32 by the fact that
the α–M loop lies entirely beyond the supercritical boundary. Pressure distributions
at the azimuth angles ψ = 112°, and ψ = 237° are shown in Fig. 6.33. Pearcey found
that the stall following ψ = 237° is precipitated by shock induced separation, the Mach
Fig. 6.32 Stall and compressibility boundary
16°
14°
12°
10°
8°
6°
4°
2°
0
0.2 0.4 0.6 0.8
M
α Boundary for
supercritical
flow
Onset of rapid
drag rise
CL max.
boundary
Fig. 6.31 ‘Figure-of-eight’ variation of incidence and Mach number
16°
14°
12°
10°
8°
6°
4°
2°
0
0.2 0.4 0.6 0.8
ψ = 303°
µ = 0.32
x = 0.913
M
340°
355°
10°
155°
112°
42°
199°
α
Rotor aerodynamics in forward flight 223
number near the leading edge being about 1.4 although the free stream Mach number
is only 0.4. There is then a collapse of the pressure distribution until reattachment
occurs when the blade reaches its most rearmost position (ψ = 360°).
The conditions described above relate to a section near to the blade tip, and
impose constraints on the performance of the helicopter. Improvements can be gained
through appropriate design of the tip to alleviate the adverse effects on both advancing
and retreating sides, and through design of the aerofoil shape itself in the tip region
and inboard of the tip. The NACA 0012 aerofoil was not designed with helicopter use
in mind and is not ideal for operating under the extreme conditions described above.
In pursuit of the alleviation of the most undesirable transonic effects on the advancing
side, tip shapes other than straight rectangular have been investigated and are currently
in use. A comprehensive study by Desopper et al.
24
was based on a series of
computational and experimental studies at ONERA. Preliminary work (including
that of other researchers) indicated that the intensity of the transonic flow was reduced
for a large azimuthal sector of the advancing blade side by using a constant 30 degree
sweptback tip, and hence the power required to drive the rotor was also decreased.
However, a strong expansion observed on the outboard part of this tip limited the
total benefit to be gained. A sweptback tip having a progressively increasing angle of
sweep, i.e. a parabolic leading edge, was found to eliminate, or at least delay, this
expansion, hence providing a further gain.
A systematic study of blade tip shapes was then undertaken covering analytical
and experimental studies, which provided ample verification. The tip shapes considered
in the former are shown in Fig. 6.34 and the pressure contours computed using a
three-dimensional unsteady transonic small perturbation method are shown in Fig.
6.35. The parabolic swept tip (PF2) indicates a significant decrease in supercritical
flow intensity, this being borne out by experiment as indicated in Fig. 6.36 (the FL5
tip is similar to PF2 except that there is no discontinuity in sweepback angle at the
start of the parabolic tip section).
On the retreating side of the rotor disc, the main performance limiting factor is that
of stall, or to use a more appropriate term, dynamic stall. The stalling characteristics of
an aerofoil subject to rapidly changing or dynamic conditions is markedly different
from that operating in steady conditions, as will be described later. The maximum
value of CL under dynamic conditions that can be attained determines the stall boundary,
and blade design aimed at increasing this value can lead to a significant improvement
in performance.
Research conducted by the UK Royal Aircraft Establishment (RAE), now the Defence,
1.4
1.2
1.0
ψ = 112°
0 0.5 1.0
ML
x/c
1.4
1.2
1.0
ψ = 237°
0 0.5 1.0
ML
x/c
Fig. 6.33 Chordwise pressure distribution (showing local Mach number)
224 Bramwell’s Helicopter Dynamics
Rectangular F30
0.1R
0.05R
F-30
0.1R
0.1R
Anhedral effect
δ = 10°
PF2
Fig. 6.34 Different tip shapes, ONERA non-lifting unsteady calculations
24
ISO-Mach Lines
µ = 0.5
ψ = 90°
NACA 00.11
F-30
MωR = 0.64
Fig. 6.35 Study of different tip shapes, non-lifting unsteady calculations, pressure contours
24
RECT F30 PF2
Rotor aerodynamics in forward flight 225
V0 = 91 m/s
r/R = 0.9
Rect.
FL5
0.5
0
0.25 0.5 0.75
– Cp Thrust level
CT/σ = 0.0665
r/R = 0.95
0.5
0
0.25 0.5 0.75
– Cp
Rect.
FL5
x x
Fig. 6.36 Measured pressure distributions on the upper side of rectangular and parabolic swept (FL5) blade tips
24
Evaluation and Research Agency (DERA), with GKN Westland Helicopters led to the
BERP (British Experimental Rotor Programme) rotor design
25
. The blade has a sweptback
paddle-shaped tip, as shown in Fig. 6.37, which is its most obvious observable
characteristic. However, the design of the blade as a whole and the aerofoil sections
selected are also vital features. Flight tests on rectangular blades had shown that the
high incidence performance required on the retreating side was not needed over the whole
span, but the requirements could not be relaxed between 65 and 95 per cent radius.
Thus, if a thin section were used near the tip to avoid advancing blade Mach number
constraints, this would have an adverse effect on the retreating blade stall performance,
even though a more appropriate, i.e. thicker, aerofoil section were used inboard.
As will be seen in a later paragraph, the BERP tip confers a high incidence
capability independent of these considerations, and allows the conflicting requirements
on the advancing and retreating side of the rotor to be met independently. The BERP
blade uses an aerofoil (RAE 9645) between 65 and 85 per cent of blade radius that
produces a high dynamic CLmax for good stall performance. However, it also gives
rise to a high nose down pitching moment which would be undesirable were it not
counteracted by using a different section, RAE 9648, inboard of 65 per cent radius.
This has a reflexed trailing edge and a nose up pitching moment characteristic; being
inboard, its lower stalling angle is not so important. The BERP tip is swept and its
increased chord provides it with a lower thickness to chord ratio than just inboard
(without decreasing the actual blade thickness), thereby avoiding the shock on the
upper side and consequent pitching moment change when the blade is advancing.
When the blade is retreating, a tip vortex is initiated from the outermost part of the
tip at low angles of incidence, but when at high angles of incidence, the vortex starts
at the point where the chord changes, i.e. at the the start of the parabolic leading edge.
It acts rather in the same manner as the bound vortex on a narrow delta wing aircraft,
e.g. Concorde, maintaining a flow on the upper surface of the aerofoil at a high
incidence and delaying stall (see Fig. 6.38).
The forward extension of the leading edge at the start of the parabolic sweepback
allows the pitching moment caused by a normal geometric sweep of the tip to be minimised.
6.4 Aerofoil characteristics when oscillating at conditions of
high incidence
So far the discussion of rotor characteristics and the calculations made have assumed
226 Bramwell’s Helicopter Dynamics
Fig. 6.37 The BERP blade
that the characteristics of the aerofoil sections of the blade are the same as those
measured in two-dimensional steady flow. Provided the theoretical values of incidence
do not exceed about 12° anywhere over the rotor disc, theoretical and experimental
values of rotor forces and flapping motion are generally in good agreement. However,
it was observed in, for example, the wind tunnel tests of Squire et al.
26
that, for a
given collective pitch angle, the slope of the curve of the thrust coefficient with shaft
incidence decreased when the shaft angle exceeded a certain value. It was assumed
that, in this region, stall was occurring on the retreating blade and that the stall area
Rotor aerodynamics in forward flight 227
Fig. 6.40 Thrust coefficient as a function of shaft angle (Boeing-Vertol tests)
Fig. 6.38 Tip vortex at high incidence on BERP tip
20°
12°
0.16
0.12
0.08
0.04
–16° –8° 0° 8°
tc
‘Linear’
aerodynamics CH-47C
8ft model rotor
‘Nonlinear’ steady
2-D aerodynamics
αnf
0.12
0.11
0.10
0.09
0.08
0.07
0.06
0.05
–20° –18°–16°–14°–12°–10°–8° –6°
12ft rotor tests
Linear aerodynamics
Nonlinear aerodynamics
Shaft angle, αnf
Thrust coefficient, tc
Fig. 6.39 Thrust coefficient as a function of shaft angle (RAE tests)
was increasing with increase of shaft incidence. At that time the introduction of
aerodynamic data which included the stall was not possible, as it required computer
facilities which were not available until the 1960s. When such data was included in
rotor theory, it was rather suprising to find that the slope of the thrust coefficient in
the stalled region was much less than that given by experiment
27
, Fig. 6.39. The
phenomenon had also been reported by Harris et al.
28
, Fig. 6.40.
228 Bramwell’s Helicopter Dynamics
120°
90°
60°
30°
ψ = 0°
330°
240°
0.2
0.3
0.2
0.3
0.5
0.7
1.1
1.3
180°
150°
300°
270°
210°
0.5
µ = 0.3
θ0 = 8°
αnf = –5°
Reversed
flow
Fig. 6.41 CL contours derived from wind tunnel tests (Meyer and Falabella)
150°
120°
90°
60°
30°
ψ = 0°
330°
300°
270°
240°
210°
180°
0.95
0.9
0.85
0.85
0.85
0.85
0.9
0.95
0.95
µ = 0.3
θ0 = 12°
αnf = –10°
Reversed
flow
Fig. 6.42 Computed CL contours using two-dimensional steady aerofoil data
0.7
0.9
1.3 1.1
0.9
0.95
It was also found that the calculated torque was considerably higher than the
experimental values. It was therefore concluded that the rotor blade did not stall in
the same manner as when under two-dimensional steady conditions. This notion was
reinforced by analysis of the wind tunnel tests of Meyer and Falabella
29
and the
flight-test data of Scheiman
30
which indicated values of lift coefficient far in excess
of the maximum steady values, as in Fig. 6.41, which shows the results from the tests
of Meyer and Falabella. The CL contours for calculations made with the steady twodimensional data and uniform induced velocity distribution are shown in Fig. 6.42
where it can be seen that the magnitudes and shapes of the CL values and contours are
quite different from those of Meyer and Falabella.
Now, it has been known for some considerable time
31
that, when a wing is changing
its incidence, the stalling angle and associated lift may be different from that in
steady flow. In particular, when the aerofoil is oscillating about a mean incidence
above that of the steady-state stall, the lift coefficient may exceed the steady-state
Rotor aerodynamics in forward flight 229
Fig. 6.44 Effect of applying oscillating aerofoil data to rotor calculations (RAE)
Unsteady
‘synthesised’
aerofoil data
0.13
0.12
0.11
0.10
0.09
0.08
0.07
0.06
0.05
‘Nonlinear’
steady
2-D aerodynamics
–20° –18°–16°–14°–12° –10° –8° –6° –4°
tc
αnf
Fig. 6.43 Hysteresis loop for aerofoil oscillating above stall
2.0
1.5
1.0
0.5
0 5° 10° 15° 20°
NACA 0012
f = 4Hz
Steady
6°
αmean = 12°
CL
α
6°
value when the incidence is increasing and fall below it when the incidence is decreasing.
Further results have been given by Halfman et al.
32
for flutter investigations, but
valuable data for helicopter applications were first given by Carta
33
, who reported
oscillation tests on a NACA 0012 aerofoil. A typical result of Carta’s tests is shown
in Fig. 6.43. The aerofoil in this case was oscillated at 4 Hz (typical rotor frequency)
with an amplitude of 6° about a mean incidence of 12°.
It can be seen that under these conditions the lift coefficient varies by as much as
0.5 from the steady values, and the difference may often be far greater. In the papers
by Bramwell et al.
27
and Harris et al.
28
, attempts were made to express the oscillating
aerofoil data of the kind shown in Fig. 6.42 in numerical or mathematical form and
apply it to rotor force calculations. In Bramwell’s paper the difference between the
steady and unsteady lift coefficients was superimposed on the experimentally derived
hovering aerofoil characteristics (‘synthesised’ data); in Harris’s paper the values
were superimposed on two-dimensional steady aerofoil data in which the CL was
corrected for the local sweep angle. The improvement in the calculated performance
is shown in Figs 6.44 and 6.45. The former also paid attention to the CL contours over
the disc. Figure 6.46 shows the contours obtained by superimposing oscillating aerofoil
230 Bramwell’s Helicopter Dynamics
0.16
0.12
0.08
0.04
–16° –8° 0° 8° 16°
Effect of including
unsteady aerodynamics
Effect of including
spanwise flow
µ = 0.35
αnf
tc
Fig. 6.45 Effect of applying oscillating aerofoil data to rotor calculations (Boeing-Vertol)
30°
180°
150°
120°
90°
60°
330°
300°
270°
240°
210°
0.3
0.3
0.7
1.1
0.9
Reversed
flow
1.7
ψ = 0°
µ = 0.3
θ0 = 12°
αnf = –10°
Fig. 6.46 Effect of applying oscillating aerofoil data on CL contours
1.5
1.3
0.7
0.5
0.9
0.5
0
data on the synthesised aerofoil characteristics; these should be compared with those
of Figs 6.41 and 6.42. Although the test conditions were not exactly the same as those
of the calculations, the figures show that the use of oscillating aerofoil data results in
contours whose appearance resembles those of the rotor tests far more closely than
the steady two-dimensional calculations.
Since the work of Carta, many others have investigated the unusual aerofoil behaviour
which occurs at high incidence under unsteady conditions, notably Ham and Garelick
34
,
and McCroskey and Fisher
35
. From the analysis of chordwise pressure distributions
on the blade, it is seen that, as the blade moves into the retreating region, the rapid
rate of increase of incidence allows the aerofoil section to exceed the normal steady
stall incidence without signs of a flow breakdown; i.e. there is still a large suction
peak near the leading edge. As pointed out by Carta, this may be due, in part at least,
to the apparent increase of camber due to the rotation of the aerofoil. As the incidence
increases further, a vortex is formed and shed from the leading edge and moves
backwards across the aerofoil at a speed somewhat less than the local flow speed.
The effect of the vortex is to produce a suction peak in the mid-chord region, resulting
Rotor aerodynamics in forward flight 231
remarkably similar. Figure 6.48 shows the variation of CN and CM with blade azimuth
angle for the point r/R = 0.75 on the rotor blade, and the corresponding values from
the two-dimensional tests for similar values of the incidence α and of .
α .
From the similarity of the curves, it can be inferred that the performance of the
blade section at high incidence has little to do with the three-dimensional rotating
environment and that the section characteristics can be obtained with sufficient accuracy
from unsteady two-dimensional aerofoil tests.
The particular flow states mentioned above can be identified on Fig. 6.48; thus, for
the model rotor results (full line), for ψ increasing from 90°, the normal force coefficient
increases to beyond the maximum steady state value (about 1.5) until ψ = 210°.
in a large nose down pitching moment. The lift continues to grow even after the
leading edge suction has started to collapse. After this, the aerofoil moves into a
condition of deep stall and the tests show that torsion flutter may occur. When the
blade reaches approximately the rear of the disc, where the blade incidence is greatly
reduced, the flow returns to the steady pattern of low incidence. The aerofoil’s chordwise
pressure distributions in the sequence described are sketched in Fig. 6.47, with typical
values of CN and incidence at four azimuth angles on the retreating blade.
McCroskey and Fisher
35
have observed that the relationship between the normal
force and pitching moment coefficients in their model helicopter measurements and
those of the two-dimensional aerofoil measurements of Ham and Garelick
34
are
Forward flight
Hover or non-rotating data
ψ = 180°
α = 9°
CN = 1.2
6
4
2
0
–2
0.2 0.4 0.6 0.8 1
x/c
–Cp
8
6
4
2
0
–2
0.2 0.4 0.6 0.8 1
x/c
–Cp
6
4
2
0
–2
–Cp
6
4
2
0
–2
0.2 0.4 0.6 0.8 1
x/c
–Cp
0.2 0.4 0.6 0.8 1
x/c
ψ = 270°
α = 34°
CN = 2.5
ψ = 240°
α = 24°
CN = 1.2
ψ = 210°
α = 17°
CN = 2.1
Fig. 6.47 Chordwise pressure distributions at high incidence
232 Bramwell’s Helicopter Dynamics
Beyond this azimuth angle, CN continues to increase but accompanied by a significant
nose down pitching moment; this is associated with the shedding and movement
rearwards of the concentrated vorticity from the leading edge region. After ψ = 240°,
CN decreases corresponding to the aerofoil being fully stalled, and the vortex having
passed clear of the trailing edge. At about 360°, the flow re-attaches.
This sequence of events has been modelled by Leishman and Beddoes
36
and
validated through comparison with experiment following initial work by Beddoes
37
to produce a practical and versatile design tool. For the attached flow in the initial
phase, the changing normal force and moment are modelled indicially using lift and
moment deficiency functions, rather as for fixed wing theory in relation to gusts and
other time-dependent conditions. Leading edge separation occurs when CN achieves
a critical value which is dependent on local Mach number; however, there is a lag or
time delay in unsteady flow which allows CN to reach higher than normal static
values. This delay is determined empirically and has been found to be largely
independent of aerofoil shape.
Subsequently, the vortex which separates from the leading edge is transported
downstream causing the centre of pressure also to move rearwards. Meanwhile, the
vortex itself generates lift which dissipates (exponentially) as fast as it accumulates,
until the vortex passes clear of the trailing edge, whereupon the lift (or CN) decays
rapidly to a value appropriate to fully separated flow, assuming the angle of incidence
is still sufficiently high. The speed of convection of the vortex downstream is derived
from a large body of experimental data involving dynamic stall over a wide range of
Mach numbers.
The indicial approach has been further implemented by Leishman
38
to account for
arbitrary motion of a rotor blade section, as well as encounters with gusts or interactions
with vortices shed by other blades.
0
–0.2
–0.4
–0.6
CM
0 1 2 3
McCroskey and
Fisher (model rotor)
Ham and Garelick
(unsteady aerofoil)
ψ = 90°
160°
180°
200°
210°
220°
270°
250°
240°
360°
330°
300°
CN
Fig. 6.48 Variation of pitching moment and normal force coefficients with azimuth angle
Rotor aerodynamics in forward flight 233
6.5 The boundary layer on a rotating blade
The first attempt at calculating the boundary layer on a rotating blade was made by
Fogarty
39
in 1951 for the case of hovering flight. Fogarty made use of a theorem of
Sears
40
which states that if φ1(x, z) is the potential for plane steady flow past the
cylinder in a parallel stream at unit speed, then the potential φ for flow about the
cylinder when rotating at angular velocity Ω is
φ = Ωy[φ1(x, z) – x] (6.31)
It was also shown that the velocity components u1, v1, w1, relative to the blade are,
Fig. 6.49,
u1 = Ωy ∂φ1/∂x
v1 = Ω[φ1(x, z) – 2x]
w1 = Ωy ∂φ1/∂z
Thus the velocity components u1 and w1 in the plane of the cylinder are the same
as those of steady plane flow about the cylinder in a stream of velocity Ωy. The
spanwise component v1, which is independent of the radial distance, can be found
directly from the plane flow potential φ1. These relationships establish the appropriate
boundary conditions outside the boundary layer.
The equations used are the familiar Navier–Stokes equations referred to co-ordinates
rotating with the blade, Fig. 6.49. If the usual boundary layer approximations are
made, and certain terms are neglected on account of the high aspect ratio of the blade,
we have for the boundary layer equations
u
u
x
w
u
z
x
p
x
u
z
+ – = –
1
+
2
2
∂
∂
∂
∂
∂
∂
∂
∂
Ω
2
ρ
ν
u
x
w
z
u y
p
y z
+ + 2 – = –
1
+
2
2
2
∂
∂
∂
∂
∂
∂
∂
∂
v v v
Ω Ω
ρ
ν
∂p/∂z = 0
together with the continuity equation
∂u/∂x + ∂w/∂z = 0
To find the pressure gradients ∂p/∂x and ∂p/∂y we use Bernoulli’s equation for the
external potential flow, i.e.
Ω
z
w
u
y
v
0
x
Fig. 6.49 Co-ordinates of rotating blade
234 Bramwell’s Helicopter Dynamics
p u x y / + + ) = ( + ) + constant
1
2 1
2
1
2 1
2
2 2 2
ρ ( v Ω
Then, neglecting the small terms v1∂v1/∂x and v1∂v1/∂y, we have approximately
1
= –
2
1
ρ
∂
∂
∂
∂
p
x
x u
u
x
Ω
1
1
= –
2
1
ρ
∂
∂
∂
∂
p
y
y u
u
y
Ω
1
so that, on substituting for the pressure gradients, the boundary layer equations are
u
u
x
w
u
z
u
u
x
u
z
+ = + 1
∂
∂
∂
∂
∂
∂
∂
∂
1
2
2
ν (6.32)
u
x
w
z
u u
u
y z
+ + 2 = + 1
∂
∂
∂
∂
∂
∂
∂
∂
v v v
Ω
1
2
2
ν (6.33)
∂u/∂x + ∂w/∂ z = 0
together with the boundary conditions
u = v = w = 0 for z = 0
u → u1 = Ωy ∂φ1/∂x, v → v1 = Ω[φ1(x, z) – 2x]
w → w1 = Ωy ∂φ1/∂z, for z → ∞
We notice that eqn 6.32 is the same as the boundary layer equation for twodimensional plane flow and may therefore be solved by any of the known methods.
Having solved this equation for u and w, we can obtain v from the solution of
eqn 6.33. This has been done by Fogarty for the case of a flat plate and the section
defined by z = kx(1 – x
2
). The results for the flat plate are shown in Fig. 6.50.
The importance of Fogarty’s results is that the spanwise flow, or ‘centrifugal pumping’,
is usually very small. As an example, let us take a point at a radial distance of 6 m
from the hub. Since the chord will be about 0.5 m at most, let us take x = 0.3 m.
5
4
3
2
1
0 0.2 0.4 0.6 0.8 1.0
Vr/Ωx
Vθ / Ωy
Ωx or Ωy
1
/2z√(Ωy/νx)
Fig. 6.50 Velocity distributions in laminar boundary layer
Rotor aerodynamics in forward flight 235
Then, if Ω is 25 rad/s, Ωx = 7.5 m/s and the largest spanwise velocity is about
7.5 × 0.3 = 2.25 m/s, compared with the free stream chordwise velocity of 150 m/s.
The spanwise flow becomes considerable only near the hub, where x/y is no longer
a small quantity. Thus, for the conventional rotor blade the spanwise, or secondary,
flow appears to be insignificant, although it might be important for wide blades such
as marine propellers.
McCroskey et al.
41
have calculated the details for a turbulent boundary layer, the
results of which are shown in Fig. 6.51. We see that the spanwise flow is even
smaller, due mainly to the larger stresses in the turbulent layer.
McCroskey and Yaggy
42
have extended the theorem of Sears to the forward flight
case and have made calculations on the basis of Fogarty’s laminar flow theory in
hovering flight. The calculations show that the spanwise flow is dominated by the
spanwise component of forward speed, as the example in Fig. 6.52 shows.
The chordwise flow is affected by several terms which are additional to those
arising in the hovering case. Generally, the effects represented by these terms are
favourable with regard to the delay of laminar separation, particularly in the retreating
quadrant 180° to 270°. This may be another reason for the improved performance of
the rotor blade compared with steady two-dimensional aerofoil characteristics although,
as we saw in the previous section, it seems that it is the unsteady motion which is
largely responsible for the peculiar behaviour of the aerodynamic characteristics at
high incidence.
7
6
5
4
3
2
1
–0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4 0.5
y √(ν/ Ωyx)
180°
135°
225°
90°
270°
45°
ψ = 0°
315°
V∞/Ωy = 0.3
x/y = 0.1
V /Ωy
Fig. 6.52 Laminar boundary profiles in forward flight
1
0.5
Turbulent
Laminar
0 0.1 0.2
y/δ
Vr/ Ωx
Fig. 6.51 Velocity distribution in turbulent boundary layer
236 Bramwell’s Helicopter Dynamics
References
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1964.
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– including comparisons with experimental data’, J. Amer. Helicopter Soc., 40(3), 1995.
15. Coppens, G., Costes, M., Leroy, P. and Devinant, P., ‘Computation of helicopter rotor wake
using a high order panel method’, Paper No. AE11, 24th European Rotorcraft Forum, Marseilles,
France, Sept. 1998.
16. Tangler, J. L., ‘Schlieren and noise studies of rotors in forward flight’, Paper No. 33-05, 33rd
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the acoustic analogy’, Paper No. 80, 22nd European Rotorcraft Forum, Brighton, UK, 17–19
Sept. 1996.
18. Kube, R. and Schultz, K.-J., ‘Vibration and BVI noise reduction by active rotor control: HHC
compared to IBC’, Paper No. 85, 22nd European Rotorcraft Forum, Brighton, UK, 17–19
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experimental investigations of BVI-noise generation and radiation from the HART-test campaign’,
Paper No. I–2, 21st European Rotorcraft Forum, St Petersburg, 30 Aug–1 Sept. 1995.
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Rotorcraft Forum, St Petersburg, 30 Aug–1 Sept. 1995.
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of a new rotor profile on the basis of flight phenomena; aerofoil research and flight tests’,
AGARD Conf. Proc. CP–111, 1972.
24. Desopper, A., Lafon, P., Ceroni, P. and Philippe, J. J., ‘Ten years of rotor flow studies at
ONERA’, J. Amer. Helicopter Soc., 34(1), 1989.
Rotor aerodynamics in forward flight 237
25. Perry, F. J., Wilby, P. G. and Jones, A. F., ‘The BERP rotor – how does it work, and what has
it been doing lately?’ Vertiflite, Spring, 1998.
26. Squire, H. B., Fail, R. A. and Eyre, R. C. W., ‘Wind tunnel tests on a 12 ft diameter helicopter
rotor’, Aeronautical Research Council R&M 2695, 1949.
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determination of helicopter rotor characteristics with particular reference to high blade incidence
conditions’, RAE Rep. 66139, 1966.
28. Harris, F. D., Tarzanin, F. T. and Fisher, R. K., ‘Rotor high speed performance, theory vs. test’,
J. Amer. Helicopter Soc., July 1970.
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a model helicopter rotor blade’, NACA TN 2953, 1953.
30. Scheiman, James, ‘A tabulation of helicopter rotor blade differential pressures, stresses and
motions in forward flight’, NASA TM X-952, March 1964.
31. Farren, W. S., ‘Reaction on a wing whose angle of incidence is changing rapidly’, Aeronautical
Research Council R&M 1648, 1935.
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derivative data and stall flutter prediction technique’, NACA TN 2533, 1951.
33. Carta, F. O., ‘Experimental investigation of the unsteady aerodynamic characteristics of NACA
0012 airfoil’, United Aircr. Lab. Rep. M–1283–1, August 1960.
34. Ham, N. D. and Garelick, M. S., ‘Detailed stall oscillations in helicopter rotors’, J. Amer.
Helicopter Soc., 13(2), 1968.
35. McCroskey, W. J., and Fisher, R. K., jnr, ‘Detailed aerodynamic measurements on a model
rotor in the blade stall regime’, J. Amer. Helicopter Soc., Vol. 17, no. 1, 1972.
36. Leishman, J. G. and Beddoes, T. S., ‘A semi-empirical model for dynamic stall’, J. Amer.
Helicopter Soc., 34(3), 1989.
37. Beddoes, T. S., ‘A synthesis of unsteady aerodynamic effects including stall hysteresis’, Paper
No. 17, 1st European Rotorcraft Forum, Southampton, UK, Sept. 1975.
38. Leishman, J. G., ‘Modeling of subsonic unsteady aerodynamics for rotary wing applications’,
J. Amer. Helicopter Soc., 35(1), 1990.
39. Fogarty, L. E., ‘The laminar boundary layer on a rotating blade’, J. Aeronaut. Sci., 18(3),
1951.
40. Sears. W. R., ‘Potential flow around a rotating cylinder’, J. Aeronaut. Sci., Vol. 17, no. 3, 1950.
41. McCroskey, W. J., Nash, J. F., and Hicks, J. G., ‘Turbulent boundary layer flow over a rotating
flat plate blade’, AIAA J. 9(1) 1971.
42. McCroskey, W. J. and Yaggy, P. F., ‘Laminar boundary layers on helicopter rotors in forward
flight’, AIAA J. 6(10), 1968.
7
Structural dynamics of elastic
blades
7.1 Introduction
In previous chapters the rotor blade has been assumed to be rigid, and this assumption
has been adequate to provide the solution to a number of important helicopter problems
with acceptable accuracy. A rotor blade is of course very flexible, and a number of
problems arise which make it necessary to study the effects of flexibility on blade
motion. For example, we need to know whether the motion of the blade significantly
affects the estimated performance and rotor blade loading; what stresses occur in the
deformed blade; and whether the frequencies of the blade motion coincide with the
frequencies of the aerodynamic forcing loads. The problem of blade flexibility is a
very complex one, for not only has the blade several degrees of freedom but the
aerodynamic loading depends strongly on the blade shape. However, for the purpose
of illustration, we shall consider the flapwise, torsional, and lagwise motions of the
blade separately, and try to draw some general conclusions. For the far more complicated
problem of coupled motion between the degrees of freedom only a brief discussion
will be given and the reader will be referred to specialist papers.
7.2 Free vibrations of rotor blades
7.2.1 Flapwise bending
We define flapwise bending as deflection of the blade in a plane perpendicular to the
rotor hub plane, Fig. 7.1.
Let Z be the displacement of an element of the blade above the flapping plane and
r the distance from the axis of rotation. Consider the motion of this element under the
forces acting on it, Fig. 7.2; S is the local shear force, M the bending moment, and G
the centrifugal tension in the blade.
Structural dynamics of elastic blades 239
The equilibrium of the blade element requires that
dG + mΩ
2
r dr = 0 (7.1)
d + d = 0
2
2
S m r
Z
t
∂
∂
(7.2)
G dZ + S dr – dM = 0 (7.3)
From eqn 7.1 we get at once
G m r r
r
R
= d
2
∫
Ω
and eqns 7.2 and 7.3 give
∂
∂
∂
∂
S
r
m
Z
t
= –
2
2
(7.4)
∂
∂
∂
∂
M
r
G
Z
r
S = + (7.5)
Differentiating eqn 7.5 and substituting eqn 7.4 we get
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
2
2
2
2
= + = –
M
r r
G
Z
r
S
r r
G
Z
r
m
Z
t
But elementary bending theory gives
M EI
Z
r
=
2
2
∂
∂
so that we have, finally, for the equation of bending
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
2
2
2
2
2
2
– + = 0
r
EI
Z
r r
G
Z
r
m
Z
t
(7.6)
Equation 7.6 represents the free motion of the blade in vacuo.
Let the solution of eqn 7.6 take the form
Fig. 7.1 Blade flapwise bending
r
Z
Fig. 7.2 Forces on blade element
S
M
G
dr
G + dG
M + dM
S + dS
dZ
240 Bramwell’s Helicopter Dynamics
Z = S(r)φ(t) (7.7)
where S(r) is a function of r alone and φ(t) is a function of t alone. Substituting eqn
7.7 into eqn 7.6 gives
d ( d /d )/d – d( d /d )/d
( )
=
d /d
( )
2 2 2 2 2 2
EI S r r G S r r
mS r
t
t
φ
φ
(7.8)
Now, the left-hand side of eqn 7.8 is a function of r only and the right hand side
is a function of t only. Since the two sides are always equal they must be equal to a
constant. The constant has dimensions T
–2
and it is convenient to write it as λ
2
Ω
2
.
Thus, we have
d
d
d
d
–
d
d
d
d
+ = 0
2
2
2
2
2 2
r
EI
S
r r
G
S
r
m S
λ Ω (7.9)
and
d
d
= 0
2
2
2 2
φ
λ φ
t
Ω (7.10)
Let us write z = Z/R, x = r/R, and ψ = Ωt. Then eqns 7.9 and 7.10 become
d
d
d
d
–
d
d
d
d
– = 0
2
2
2
2
2 2 2 4
x
EI
S
x
R
x
G
S
x
m R S
λ Ω (7.11)
and
d
d
+ = 0
2
2
2
φ
ψ
λ φ (7.12)
The boundary conditions to be satisfied are as follows.
(a) Hinged blade (including flapping hinge offset)
At x = e (at flapping hinge)
S = 0
d
2
S/dx
2
= 0 (zero bending moment)
At x = 1 (blade tip)
d
2
S/dx
2
= 0 (zero bending moment)
d
3
S/dx
3
= 0 (zero shear force)
(b) Hingeless blade
At x = 0
S = 0
dS/dx = 0 (zero slope)
Structural dynamics of elastic blades 241
At x = 1
d
2
S/dx
2
= 0
d
3
S/dx
3
= 0
Equation 7.7 will be a solution of eqn 7.6 provided S(x) and φ (ψ) satisfy eqns 7.11
and 7.12 and the appropriate boundary conditions.
There is an infinite number of solutions of eqn 7.11 since λ is not a fixed number
but can be adjusted to satisfy the equation in association with the appropriate boundary
conditions. These solutions represent the blade shape and are called normal modes on
account of the orthogonal property to be proved in section 7.3.
Equation 7.12 is the equation of simple harmonic motion, and the infinite number
of discrete values of λ determine the frequencies ω = λΩ of the corresponding mode
shape. λn is therefore the ratio of the blade natural frequency ωn to the shaft rotational
frequency Ω for the nth mode shape.
It can easily be verified that when there is no flapping offset, i.e. e = 0, a solution
of eqn 7.11 is S(x) = x, with λ = 1, that is, the first normal mode shape of the centrally
hinged blade is a straight line whose flapwise frequency is exactly equal to the
rotational frequency of the shaft. This is the ‘rigid’ blade shape already assumed in
the previous chapters. If e ≠ 0, the mode shape is not exactly a straight line, although
for typical values of e it is very close to it.
Methods for calculating the blade mode shapes and frequencies are dealt with
below. Figure 7.3 shows the first four shapes for a typical helicopter blade.
Fig. 7.3 Mode shapes of typical helicopter blade
1
0
–1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
0
–1
S3(x), λ3 = 4.60
S4(x), λ4 = 7.18
S1(x), λ1 = 1
r/R
r/R
S2(x), λ2 = 2.58
242 Bramwell’s Helicopter Dynamics
7.2.2 Calculation of blade mode shapes and frequencies
We must now attempt to solve eqn 7.11 in order to obtain the blade mode shapes Sn(x)
and associated frequencies λnΩ. In general, both the blade elastic stiffness EI and
mass distribution m will be complicated functions of the radial station x, and it is
obvious that a simple analytical solution of the blade bending equation is out of the
question. Typical spanwise variations of stiffness and mass for a uniform blade are
shown in Fig. 7.4.
It can be seen that over the greater part of the blade the stiffness and mass distributions
are practically constant, but that near the blade root large and often discontinuous
changes occur because of the root attachment.
Now it is well known that for a uniform non-rotating beam there is an exact closed
solution of the mode equation and that other exact solutions exist for some particular
mass and stiffness distributions. However, there is no known closed solution for the
rotating beam, even for the apparently simple case of constant mass and stiffness, the
equation of which is given below, eqn 7.13:
k
z
x x
x
z
x
z n
2
4
4
2 2 d
d
–
1
2
d
d
(1 – )
d
d
– = 0
[ ] λ (7.13)
where k
2
= EI/mΩ
4
is constant.
The mode shapes and frequencies of a rotating uniform blade for k
2
= 0.0055 have
been calculated and are shown in Fig. 7.5. The broken lines show the mode shapes
and frequencies for a non-rotating blade of the same thickness (standard results for
a uniform vibrating beam). It is perhaps a little surprising to note that the effect of
rotation is quite small, which suggests that the non-rotating mode shapes might serve
as useful approximations in the calculation of the rotating modes.
The lack of an exact solution of eqn 7.13 is unfortunate because, although the
mass and stiffness distributions might not be representative of a practical blade, it
would be very useful for testing the accuracy of approximate methods of solution.
Flapwise stiffness EI
Mass per unit length
Spanwise co-ordinate Spanwise co-ordinate
Fig. 7.4 Mass and stiffness distributions of a typical uniform blade
Structural dynamics of elastic blades 243
The approximate methods usually used fall under two main headings:
(i) method of assumed modes,
(ii) method of lumped parameters.
7.2.2.1 Method of assumed modes
This method consists of choosing a finite sequence of functions which, preferably,
approximate to the expected mode shapes and satisfy the appropriate boundary
conditions. This latter requirement is not essential, however, since it is always possible
to include restraint conditions in the analysis.
We shall consider three methods employing assumed modes:
(a) Lagrange’s equations
(b) Rayleigh–Ritz procedure,
(c) Galerkin’s method.
(a) Lagrange’s equations
Let the sequence of functions which are to be used to approximate to the blade shape
be
γ1(x), γ2(x), … , γi(x), …, γn(x)
and let the displacement be
z x x
i
n
i i ( , ) = ( ) ( )
=1
ψ γ φψ Σ
1
0.5
0
–0.5
S1(x)
0.2 0.4 0.6 0.8 1 r/R 0.2 0.4 0.6 0.8 1
1
0.5
0
–0.5
–1
S3(x)
0.2 0.4 0.6 0.8 1
1
0.5
0
–0.5
–1
S2(x)
λ1 = 1
λ3 = 5.59
λ2 = 2.75
λ2 = 1.15
r/R 0.2 0.4 0.6 0.8 1
1
0.5
0
–0.5
–1
S4(x)
r/R
λ4 = 9.79
Rotating (k
2
= 0.0055)
Non-rotating
r/R
Fig. 7.5 Mode shapes of rotating and non-rotating uniform blade
λ3 = 3.72–
λ4 = 㜮㜷
244 Bramwell’s Helicopter Dynamics
It will be assumed that the functions γi(x) satisfy the boundary conditions.
Then
∂ ∂ψ γ φ z x x
i
n
i i / = ( ) ( )
=1
Σ ′
and the kinetic energy T in term of blade axes (see section 7.3.2) is therefore
T R m x
i
n
j
n
i j i j =
1
2
d
2 3
=1 =1 0
1
Ω Σ Σ ′ ′
∫
φ φ γ γ (7.14)
=
=1 =1
Σ Σ ′′ ′′
i
n
j
n
ij i j A φ φ
where A R m x ij i j =
1
2
d
2 3
0
1
Ω
∫
γ γ
is a generalised mass or inertia coefficient.
It should be noted that the orthogonal properties do not apply to the integral of
eqn 7.14 since the functions γi(x) themselves are not exact solutions of the blade
bending equation, eqn 7.11.
The strain and potential energy U is, as will be seen from eqn 7.92.
U
R
El
d
x x
R G
x x
x
B
i
n
j
n
i j
i j i j
i
n
j
n
ij i j
=
1
2 d
d
d
+
d
d
d
d
d
=
=1 =1 0
1 2
2
2
2
2
=1 =1
Σ Σ
Σ Σ
∫
⋅
φ φ
γ γ γ γ
φ φ
where B
R
EI
x x
R G
x x
x ij
i j i j
=
1
2
d
d
d
d
+
d
d
d
d
d
0
1 2
2
2
2
2
∫
γ γ γ γ
is a generalised stiffness coefficient (including both structural and centrifugal stiffening
effects.
Then applying Lagrange’s equations eqn 7.95, with ∂W/∂φn = 0, gives
Σ Σ ′′
j
n
ij j
j
n
ij j A B
=1 =1
+ = 0 φ φ (7.15)
If φj = φj0 sin λjψ, eqn 7.15 becomes
Σ
j
n
ij j ij j B A
=1
2
0 ( – ) = 0 λ φ (7.16)
As a simple example, let us calculate the first two mode shapes and frequencies of
a centrally hinged uniform blade and suppose that the deflection amplitude can be
expressed as
γ0 = φ10γ1 + φ20γ2
where
γ1(x) = x and γ2(x) = 10x
3
/3 – 10x
4
/3 + x
5
Structural dynamics of elastic blades 245
It can easily be seen that the functions satisfy the boundary conditions of a centrally
hinged blade. Then evaluating the integrals Aij and Bij we get
A A A A
B B B B k
11 12 21 22
11 12 21 22
2
=
1
3
, = =
16
63
, =
1304
6237
=
1
3
= =
16
63
, =
80
21
+
1850
6237
,
the bars denoting that the integrals have been divided by
1
2
2 3
m R Ω , and where k
2
=
EI/mΩ
2
R
4
. In this case the tension G is
1
2
(1 – ).
2 3 2
m R x Ω substituting in eqn 7.16
gives, in matrix form,
1
3
16
63
16
63
+
1850
6237
–
1
3
16
63
16
63
1304
6237
= 0
2
10
20
2
2
2
2
10
20
80
21
k
φ
φ
λ
λ
λ
λ
φ
φ
or
1
3
( – 1)
16
63
( – 1)
16
63
( – 1)
–
80
21
–
1850
6237
= 0
2
2
2
2 2
10
20
λ
λ
λ
λ
φ
φ
1304
6237
k
The frequency equation is found by putting the determinant of the square matrix
to zero. If we take k
2
= 0.004 as a typical value, the determinant gives
– 1 0.7619( – 1)
– 1 0.8232 – 1.2279
= 0
2 2
2 2
λ λ
λ λ
or (λ
2
– 1)(λ
2
– 7.5988) = 0
This equation gives λ = 1, a value we should expect since γ1 = x is known to be
an exact solution, and λ = 2.757. We cannot find the absolute amplitudes of the
modes but only the ratio of the amplitudes. Taking the second equation of the matrix
form of eqn 7.16 gives
(λ φ λ φ 1
2
10
2
20 – 1) + (0.8232 – 1.2279) = 0
When λ = 1, we have φ20/φ10 = 0, i.e. if φ10 ≠ 0, φ20 = 0 indicating that in the
oscillation whose frequency is Ω the mode shape consists entirely of the function
γ = x. This is, again, just what we should expect, as γ = x is an exact solution of the
flapping equation. The higher frequency 2.757Ω gives φ10/φ20 = – 0.759 and the
corresponding mode shape can be written
S2(x) = –3.15x + 4.15x
3
(10/3 – 10x/3 + x
2
)
The two modes are shown in Fig. 7.6. The accuracy of the mode shapes and
frequencies improves with the number of functions chosen, the lower modes being
improved the most. In general, the stiffness and mass distribution will be such as to
require numerical integration of the integrals Aij and Bij.
246 Bramwell’s Helicopter Dynamics
(b) The Rayleigh–Ritz procedure
It can be shown
1
from the calculus of variations that a function satisfying the fourth
order linear differential equation
d
d
d
d
–
d
d
d
d
– = 0
2
2
2
2
2 2 2 4
x
EI
S
x
R
x
G
S
x
m R S
λ Ω (7.11)
in association with either set of boundary conditions for the hinged or hingeless
blade, is a function which gives a stationary value to the integral
L
R
m R S EI
S
x
R G
S
x
x =
1
2
–
d
d
–
d
d
d
0
1
2 2 4 2
2
2
2
2
2
∫
λ Ω (7.17)
It will be seen from energy considerations (eqns 7.87, 7.89 and 7.91) that L, termed
the Lagrangian, is the difference between the kinetic and strain energies in a given
mode of motion, i.e.
L = T – U
The Rayleigh–Ritz procedure is to assume a finite series of approximation functions
Sn(x) = A1γ1(x) + A2γ2(x) + … + Aiγi + … + Anγn
where the terms γi satisfy the boundary conditions, and substitute the series into
eqn 7.17. Since A1, A2, …, etc. can be arbitrarily varied, the condition that L should
have stationary values is
∂
∂
∂
∂
∂
∂
∂
∂
L
A
L
A
L
A
L
A i n 1 2
= = = = = … …
giving n equations for the evaluation of the terms Ai.
1
0.5
0 0.2 0.4 0.6 0.8 1.0
S1(x)
x
0.2 0.4 0.6 0.8 1.0
x
1
0.5
0
–0.5
–1
S2(x)
Fig. 7.6 Normal mode shapes
Structural dynamics of elastic blades 247
We find that this gives exactly the same set of equations as we obtained by the
Lagrange method of the previous section. It is worth mentioning, however, that the
Rayleigh–Ritz method can be applied to problems, e.g. static equilibrium, to which
the Lagrange equations would not normally be applicable.
The Rayleigh–Ritz method is a generalization of the application of Rayleigh’s
principle
2
that the frequencies corresponding to the solutions of eqn 7.11 have stationary
values. This means that, if the chosen functions differ from the exact solutions Si by
small quantities of the first order, the calculated frequencies will be in error by only
small quantities of the second order. Thus, it can be expected that quite a poor
approximation to the true mode shape will give a good approximation to the frequency.
Now, when a system oscillates in a normal mode with frequency ωi = λiΩ, each
part of the system oscillates in phase or in antiphase with every other part of the
system. Thus we can write for a typical displacement in a normal mode
Z = γi(x) sin (ωit + ε)
= γi(x) sin (λiψ + ε)
Then the kinetic energy T of the system, as will be seen from eqn 7.87, is
T R m x =
1
2
d
2 3
0
1 2
Ω
∫
∂γ
∂ψ
=
1
2
cos ( + ) ( )d
2 2 3 2
0
1
2
λ λ ψε i i R my x x Ω
∫
and from eqns 7.89 and 7.91 the potential energy of the rotating blade is
U = UB + UG
=
1
2
sin ( + )
d
d
d +
1
2
sin ( + )
d
d
d
2
0
1 2
2
0
1 2
R
EI
x
x R G
x
x i
i
i
i
λ ψ ε
γ
λ ψ ε
γ
∫ ∫
In this simple harmonic motion the maximum kinetic energy must equal the maximum
potential energy. Thus the coefficients of cos
2
(λiψ + ε) and sin
2
(λiψ + ε) can be
equated giving
λ
γ γ
γ
i
i i
i
EI x x R G x x
R m x
2 2 0
1
2 2 2 2
0
1
2
4
0
1
2
=
(d /d ) d + (d /d ) d
d
Ω
∫ ∫
∫
(7.18)
Then, if a mode shape γi(x) is assumed which is reasonably near an exact mode,
eqn 7.18 gives the corresponding frequency. The use of the Rayleigh method is
usually confined to the calculation of the lowest frequency, since the corresponding
mode shape will probably be comparatively simple and a good guess can usually be
made.
Equation 7.18 has been used by Southwell
3
to obtain a relationship to the non
248 Bramwell’s Helicopter Dynamics
rotating frequency of the blade and its natural frequency under rotation. Equation
7.18 can be written
ω λ ω
γ
γ
i i nr
i
i
R
G x x
m x
2 2 2 2
2
0
1
2
0
1
2
= = +
1
(d /d ) d
d
Ω ⋅
∫
∫
(7.19)
where ωnr is the non-rotating frequency of the blade.
Now
G m r r
R mx x
r
R
x
= d
= d
2
2 2
1
∫
∫
Ω
Ω
therefore
0
1 2
2 2
0
1 1 2
d
d
d = d
d
d
d
∫ ∫ ∫
G
x
x R mx x
x
x
i
x
i γ γ
Ω
The integral on the right-hand side can be transformed to read
Ω
2 2
0
1
0
2
d
d
d d R mx
x
x x
x
i
∫ ∫
γ
and eqn 7.19 can finally be expressed as
ω ω α i i
2
nr
2 2
= + Ω (7.20)
where ωnr is the natural frequency of the non-rotating blade and
α
γ
γ
i
x
i
i
mx x x x
m x
=
(d /d ) d d
d
0
1
0
2
0
1
2
∫ ∫
∫
(7.21)
Equation 7.20 is Southwell’s formula.
To consider a specific case, let us use Southwell’s formula to calculate the first
bending frequency of a uniform blade hinged at the root (k
2
= 0.004) having the mode
shape found earlier, namely
S2(x) = –3.15x + 4.15x
3
(10/3 – 10x/3 + x
2
)
Substituting in eqn 7.21 with m constant we find
α2 = 8.241
α1 being unity and corresponding to the rigid blade mode. The frequency of the non-
rotating blade is obtained from the standard results
4
for a pinned-free non-rotating
uniform beam, which give
Structural dynamics of elastic blades 249
ω µ
µ
nr
2 4
4 4 2 2
= /
=
EI m
R k Ω
where µ is a constant given in the above reference.
If we take Ω = 25 rad/s, then k
2
Ω
2
= 2.5 and the value of (µR)
4
is 237.7.
Hence
ω nr
2
= 594.3
and Southwell’s formula for this case is therefore
ω 2
2 2
= 594.3 + 8.241Ω
The frequency at Ω = 25 rad/s is 3.031 Ω, which is higher than the value we
calculated before but this is to be attributed to the fact that the assumed mode shape,
which was based on only two functions, differs considerably from the more exact
value, and the departure from the true shape is equivalent to the imposition of constraints
which effectively increase the stiffness.
Southwell’s formula is very useful for showing the effect of the rotor speed on the
natural frequency of the blade. Strictly speaking, αi is not constant because the mode
shape would be expected to change slightly with Ω. However, assuming αi constant,
we see from eqn 7.20 that as the rotational speed becomes very large we have
ωi → Ω√αi
The curves of eqn 7.20 for the various modes are often plotted in conjunction with a
‘spoke’ diagram, Fig. 7.7. One can see at a glance if there are any natural frequencies
of the blade which coincide with a harmonic of the rotor speed, indicating the possibility
of resonance.
(c) The Galerkin method
We start with the blade bending equation
ωi
Ω√α3
5Ω
4Ω
3Ω
2Ω
1Ω
Ω√α2
Rigid-blade mode
2nd bending
mode
1st
bending
mode
Ω
Fig. 7.7 ‘Spoke’ diagram of bending frequencies
250 Bramwell’s Helicopter Dynamics
d
d
d
d
–
d
d
d
d
– = 0
2
2
2
2
2 2 2 4
x
EI
S
x
R
x
G
S
x
m R S
n n
n n
λ Ω (7.11)
and suppose as before, that
Sn(x) = A1γ1(x) + A2γ2(x) + … + Aiγi(x) + … + Anγn(x) (7.22)
We now substitute for Sn(x) in eqn 7.11, multiply each term by γi, integrate from
0 to 1, and equate the result to zero. When i is made to take the successive values
1, 2, …, n, we obtain n simultaneous equations which determine the coefficients Ai.
Duncan has shown
5
that this process is equivalent to finding the error ε (x) which
results from substituting the approximate solution eqn 7.22 into eqn 7.11 and finding
the stationary values of the integral
J x d
0
1
2
≡
∫
ε
by satisfying the condition
∂J/∂Ai = 0
The Galerkin process can be regarded as a ‘least-squares’ fit of the assumed
solution eqn 7.22 to the exact solution of eqn 7.11.
Duncan has also shown
6
that the Galerkin and Rayleigh–Ritz methods, and therefore
the Lagrange method, are all equivalent in linear problems, but notes that the Galerkin
method can be extended to non-linear and non-conservative problems.
The application of the Galerkin method differs from the other two in that the
integrals to be calculated are those derived from the terms in the differential equation
7.11, instead of those to be found in eqns 7.88 and 7.90. This means that in the
Galerkin method the integrand of the elastic terms involves differential coefficients
of the stiffness, and this may be a disadvantage if the stiffness undergoes sudden
changes.
Figure 7.8 shows the rigid mode and the first bending mode when the first four
functions
1
0
–1
0.5 1
S1(x), λ1 = 1
S2(x), λ2 = 2.68
Fig. 7.8 Rigid and first bending modes and frequencies of a uniform beam
r/R
Structural dynamics of elastic blades 251
γ = x, γi = (i + 2)(i + 3)x
i+1
/6 – i(i + 3)x
i+2
/3 + i(i + 1)x
i+3
/6 i = 2, 3, 4
are used to calculate the mode shapes and frequencies of a uniform beam.
7.2.2.2 Method of lumped parameters
We now consider two methods of analysis which are described as ‘lumped-parameter’
methods. In this technique the continuous blade is represented by a number of discrete
segments, so that the partial differential equation of blade bending is replaced by a
set of simultaneous ordinary differential equations.
The two methods to be described are
(a) the Myklestad method,
(b) the dynamic finite element method.
(a) The Myklestad method
This is a development of the Holzer
7
method which was originally used for the
calculation of torsional oscillations but was later extended and modified by Myklestad
for the calculation of lateral beam vibrations.
Consider the deflected blade, Fig. 7.9 divided into a number of concentrated
masses between which the elastic properties remain unaltered. The nth element and
the forces acting on it are shown in Fig. 7.10.
With reference to Figs 7.11 and 7.12 we also define the following elastic coefficients
in relation to the application of a unit force and of a unit moment.
Assuming the elastic properties are constant over the element, the four coefficients
can be expressed as
u l EI
u l EI
l EI
Mn ln EI
Fn n n
Mn n n
Fn n n
n
= /6( )
= /2( )
= /2( )
= /( )
3
2
2
v
v
We now imagine the blade to be forced to vibrate with harmonic motion of frequency
ω. This adds an inertia force mn+1ω
2
Zn+1 to the element, and the equilibrium of the
element leads to the following relations:
Gn+1 = Gn + mn+1Ω
2
rn+1 (7.23)
Z
rn
rn +1 αn+1
mn
αn
Zn+1
Z
Z1
m1
r
Gn+1
Zn+1
Sn+1
ln
Zn
Gn
αn
Sn
αn+1
mn +1
Mn +1
Fig. 7.9 Distributed-mass representation of helicopter
blade (Myklestad)
Fig. 7.10 Forces on blade element (Myklestad
method)
252 Bramwell’s Helicopter Dynamics
Sn+1 = Sn – mn+1ω
2
Zn+1 (7.24)
Mn+1 = Mn – Snln + Gn(Zn – Zn+1) (7.25)
αn+1 = αn(1 + GnvFn) – SnvFn + MnvMn (7.26)
Zn+1 = Zn – (ln + uFnGn)αn + uFnSn – uMnMn (7.27)
We obtain at once from eqn 7.23
G m r n
i
n
i i =
=1
2
Σ Ω (7.28)
and eliminating Zn – Zn+1 between eqns 7.25 and 7.27 gives
Mn+1 = Mn(1 + uMnGn) – Sn(ln + uFnGn) + Gn(ln + uFnGn) (7.29)
For a given value of ω, these recurrence relations enable us to calculate S, M, α
and Z at any point along the blade in terms of the corresponding values at one end.
Thus, for example, the boundary conditions at the blade tip are
S1 = m1ω
2
; M1 = 0; α1 = α10, say; Z1 = 1
It may seem that the first boundary condition contradicts the requirement of zero
shear force at the tip, but it must be remembered that the mass of the last element is
concentrated at the tip so that it really represents an average condition for the element.
The last two boundary conditions are arbitrary and merely define the scale of the
blade displacement. It can easily be seen that, at any given station, S, M, α, and Z
must be of the form
Sn = cn + dnα10 (7.30)
Mn = en + fnα10 (7.31)
αn = gn + hnα10 (7.32)
Zn = jn + knα10 (7.33)
where cn, dn, …, etc. are numbers* which depend on the blade station in question.
uMn
vMn
M = 1
ln
UFn
vFn
F = 1
ln
Fig. 7.11 Definition of unit load coefficients Fig. 7.12 Definition of unit moment coefficients
* Recurrence relations expressing these numbers in terms of the original constants and tabulation
methods for the calculation procedure are given in detail in reference 7 and need not be given here.
Structural dynamics of elastic blades 253
Now at the blade root the displacement is zero. Thus, if the suffix r denotes the
root,
Zr = jr + krα10
or α10 = –jr/kr
The remaining root boundary condition to be satisfied is either
Mr = 0, for the hinged blade
or
αr = 0, (or, possibly, a known non-zero value) for the hingeless blade.
For the first case, we have from eqn 7.31, after substituting for α10,
Mr = (erkr – jrfr)/kr = 0
Now the term in the bracket is a function of the assumed value of the frequency
ω. Thus, if a range of values of ω is taken, a natural frequency occurs every time the
function becomes zero, Fig. 7.13. For such a value of ω, the mode shape, defined by
the values of Zn, can be determined from eqn 7.33.
Similarly, for the hingeless blade, we have from eqn 7.32
αr = (grkr – hrjr)/kr = 0
and, again, we seek the values of ω which make the function in the bracket zero.
Now, in the methods described earlier, if it is required to calculate, say, the fifth
mode and frequency, it may be necessary to take at least ten functions and solve the
resulting ten simultaneous equations to obtain reasonable accuracy for the fifth and
lower modes. In the Myklestad method, however, each mode can be calculated
independently of the others, although a certain amount of ‘searching’ may be necessary
to locate the appropriate zero.
The Myklestad method described above refers only to pure flapwise bending. An
extension of the method has been given by Isakson and Eisley
8
which includes the
effect of coupling between the bending and torsional modes of vibration. Their
analysis, however, is far too lengthy to be included here, and the reader is referred to
the original paper.
(b) The dynamic finite element method
9
As in the Myklestad method described in the previous section, the blade is divided
into a number of elements, not necessarily of equal length. In this case, however, the
erkr – jrfr
ω
Fig. 7.13 Variation of boundary function with frequency
254 Bramwell’s Helicopter Dynamics
mass of an element is not concentrated at its ends but is imagined to be distributed
uniformly along its length. The system of forces and moments acting along the
element is as shown in Fig. 7.14, and its equilibrium in harmonic motion leads to the
following equations
G G rm r G m r r n n
r
r
n n n
n
n
+1
2 2 2
+1
2
= + d = +
1
2
( – )
+1 ∫
Ω Ω (7.34)
S S Zm r S m l Z Z n n
r
r
n n n n
n
n
+1
2 2
+1 = – d = –
1
2
( + )
+1 ∫
ω ω (7.35)
M M G Z Z S l r r mZ r n n n n n n n
r
r
n
n
n
+1 +1 +1
2
= – ( – ) – + ( – ) d
+1 ∫
ω
– ( – ) d
+1
+1
2
r
r
n n
n
n
Z Z rm r
∫
Ω
= – ( – ) – + (2Z + Z )/6 +1
2 2
+1 M G Z Z S l l m n n n n n n n n m ω
– (2 + )( – )/6
2
+1 +1 Ω l m r r Z Z n n n n n (7.36)
These three equations are analogous to eqns 7.23, 7.24, and 7.25 of Myklestad’s
method. Inspection of the above equations shows that the bending moments
M0, M1, …, Mn, … along the blade can be expressed in matrix form as
M = ω
2
aZ + Ω
2
bZ (7.37)
where a and b are square matrices, and functions of r and m, Z is the column vector
of the element displacements, and M is the column vector of bending moments.
The deformation of the blade represented by Z may be split into two components
Z = ZE + ZR
where ZE is the elastic deformation of blade bending and ZR is the rigid-body rotation
about the flapping hinge. If there is no flapping hinge, ZR = 0.
To obtain a relationship expressing the blade deformation as a function of the
applied moment distribution, we use the unit load method
10
. This states that, if M1 is
the bending moment distribution due to the application of a unit load at a point at
which we wish to calculate the deflection, and if M is the actual moment distribution
then the required deflection δ is given by
Sn+1
Mn+1
Gn+1
Zn+1
ln Zn
Sn
Gn
Mn
Fig. 7.14 Forces on blade element
Structural dynamics of elastic blades 255
δ = ( / )d
0
R
MM EI r
∫
1
Let us consider the contribution to this deflection due to one of the blade elements.
Let MA and MB be the moments at the ends of the element due to the applied loading,
and let MA1 and MB1 be the moments due to the unit load, Fig. 7.15, at which it is
required to know the deflection. If we suppose that these bending moments vary
linearly across the element, the contribution δAB to the deflection can easily be
calculated and expressed conveniently in matrix form as
δ AB
AB
MM EI r = ( / )d 1
∫
= [ , ]
3 6
6 3
1 1 M M
l
EI
l
EI
l
EI
l
EI
M
M
A B
A
B
where l is the length of the element.
If a number of successive elements is now considered, with the corresponding
moment distributions MA1, MB1, …, MN1, … and MA, MB, … , MN, … , the deflection
due to all these elements is
δ = [ , ; , , .] 1 1 1 M M M A B N … …
·
3 6
0 0 0
6 3
+
3 6
6 3
+
3 6
:
l
EI
l
EI
l
EI
l
EI
l
EI
l
EI
l
EI
l
EI
l
EI
l
EI
M
M
M
A
B
N
0 0
0 0
0 0 0
M (7.38)
To define the deformation of the complete blade, we need to know the deflections
MA1
MA
MB1
MB
1
Fig. 7.15 Moment due to application of unit load
.............................................................................
................................
256 Bramwell’s Helicopter Dynamics
at the ends of each of the elements. This will require repeating the above calculations
using the bending moment distributions corresponding to unit loads applied at each
station. When this is done, the row matrix [MA1, MB1], …] then becomes a square
matrix M1, say, the columns of which give the bending moment distributions due to
each of the applied unit loads. The centre matrix of eqn 7.38 is called a flexibility
matrix and will be denoted by f. The total set of deflections is then given by
Z M f M
T
E 1
= (7.39)
where ZE is the matrix of the values of δ. Eliminating M between eqns 7.37 and 7.39
gives
ZE = ω
2
a*(ZE + ZR) + Ω
2
b*(ZE + ZR) (7.40)
where a* and b* are matrices resulting from the addition and multiplication of
previously defined matrices. Rearranging eqn 7.40 gives
(I – Ω
2
b*)ZE = ω
2
a*(ZE + ZR) + Ω
2
b*ZR
giving ZE = ω
2
[I – Ω
2
b*]
–1
a*(ZE + ZR) + Ω
2
[I – Ω
2
b*]
–1
b*ZR
or ZE = ω
2
c(ZE + ZR) + Ω
2
dZR (7.41)
where c and d are square matrices.
The hingeless rotor
If no flapping hinge is present, ZR = 0, giving
ZE = ω
2
cZE
or (c – I/ω
2
)ZE = 0 (7.42)
The eigenvalues and eigenvectors of eqn 7.42 give the required frequencies and
mode shapes.
The hinged rotor
When there is a flapping hinge, the moment at the hinge line must be zero and from
eqn 7.37 we have
Mh = ω
2
ah(ZE + ZR) + Ω
2
bh(ZE + ZR) = 0 (7.43)
the suffix h indicating that the row of the matrix corresponding to the hinge line is
used. Substituting the value of ZE given by eqn 7.41 into the second bracket of eqn
7.43 gives
ω
2
ah(ZE + ZR) + Ω
2
bh[ω
2
c(ZE + ZR) + dZR + ZR] = 0
or, on rearranging,
ω
2
(ah + Ω
2
bhc)(ZE + ZR) + Ω
2
bh(I + d)ZR = 0 (7.44)
Now the unknowns in ZR are linearly related through an unknown rotation α about
the flapping hinge, i.e.,
Structural dynamics of elastic blades 257
ZR = eα (7.45)
where e is a column matrix giving the distances of the ends of the blade elements
from the flapping hinge. Then substituting eqn 7.45 into eqn 7.44 gives
Ω
2
bh(I + d)eα = – ω
2
(ah + Ω
2
bhc)(ZE + ZR)
or
α = – ω
2
[Ω
2
bh(I + d)e]
–1
(ah + Ω
2
bhc)(ZE + ZR)
so that eqn 7.45 can be written
ZR = – ω
2
e[Ω
2
bh(I + d)e]
–1
(ah + Ω
2
bhc)(ZE + ZR) (7.46)
Substituting for ZR in eqn 7.41 gives
ZE = ω
2
[c – de{Ω
2
bh(I + d)e}
–1
(ah + Ω
2
bhc)](ZE + ZR) (7.47)
and on adding eqns 7.46 and 7.47 we get
(ZE + ZR) = ω
2
[c – (I + d)e{Ω
2
bh(I + d)}
–1
(ah + Ω
2
bhc)](ZE + ZR) (7.48)
Defining c* to be the matrix in the square brackets, we have
ZE + ZR = ω
2
c*(ZE + ZR)
or
[c* – I/ω
2
](ZE + ZR) = 0 (7.49)
The eigen values and eigen vectors of c* yield the required frequencies and mode
shapes of the hinged blade.
The above method need not be confined to straight elements. For a given number
of elements, the accuracy can be greatly improved by taking curved elements – for
example, a cubic variation – the curvature being determined by the deformation of
neighbouring elements. The relationships eqns 7.37 and 7.39 have to be modified
accordingly, but the net result is a considerable increase of accuracy for only a slight
increase of computer time.
7.2.2 Lagwise bending
When considering motion of the blade in the lagging plane*, i.e. in the plane of
rotation, we have to note that the centrifugal force on a blade element is directed
radially outwards from the hub and therefore, unlike the flapping case, the direction,
as well as the magnitude of the force, varies along the blade. Consider the forces,
inertial and aerodynamic, acting on a given element of the blade and their moment
about another point P of the blade, Fig. 7.16. If r1 is the radial distance of the element
and r the radial distance of the point P from the hub, the moment of all the forces
about P is
*Lagging and chordwise motion are the same only if the blade chord is parallel to the plane of
rotation, i.e. when the pitch angle is zero.
258 Bramwell’s Helicopter Dynamics
M EI
Y
r
m r r r r m r Y Y r
r
R
r
R
= = ( – ) sin d – ( – ) cos d
2
2
2
1 1 1 1
2
1 1 1 1
∂
∂
α α
∫ ∫
Ω Ω
– ( – )d
2
1
2 1 1
r
R
m
Y
t
r r r
∫
∂
∂
(7.50)
Differentiating with respect to r we have, after cancelling certain terms,
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
M
r
m Y r
Y
r
m r r m
Y
t
r
m Y r
Y
r
G r m
Y
t
r
r
R
r
R
r
R
r
R
r
R
= – d + d + d
= – d + ( ) + d
2
1 1
2
1 1
2
1
2 1
2
1 1
2
1
2 1
∫ ∫ ∫
∫ ∫
Ω Ω
Ω
Differentiating again gives
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
2
2
2
2
2
2
2
2
2
= = – –
M
r r
EI
Y
r r
G
Y
r
m
Y
r
Y
Ω
or
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
2
2
2
2
2
2
2
– + – = 0
r
EI
Y
r r
G
Y
r
m
Y
r
Y
Ω (7.51)
where it is understood that the value of EI refers to lagwise bending.
To discuss the free lagging motion of the blade, assume a solution
Y = RT(x)χ(t)
where T(x) is a function of x alone and χ(t) is a function of t alone. By the same
argument as that used for the flapping equation, we arrive at the mode shape equation
d
d
d
d
–
d
d
d
d
– ( + 1) = 0
2
2
2
2
2 2 2 4
x
EI
T
x
R
x
G
T
x
m R T
ν Ω (7.52)
Y
α1
r
r1
Y
X
Y1
Ω
2
r1dm
∂
∂
2
1
2
d
Y
t
m
Fig. 7.16 Lagwise forces acting on blade
P
Structural dynamics of elastic blades 259
and the frequency equation
d
2
χ/dψ
2
+ ν
2
χ = 0 (7.53)
The boundary conditions relating to eqn 7.52 are the same as those of flapping
motion:
(a) Hinged blade (including lag hinge offset)
At x = e, y = 0, d
2
y/dx
2
= 0 (where y = Y/R)
At x = 1, d
2
y/dx
2
= 0, d
3
y/dx
3
= 0
(b) Hingeless blade
At x = 0, y = 0, dy/dx = 0
At x = 1, d
2
y/dx
2
= 0, d
3
y/dx
3
= 0
The lagging mode shape equation, eqn 7.52, is identical in form to the flapping
equation except that the frequency ratio ν appears as ν
2
+ 1 in the mode equation.
This is explained physically by the fact that the centrifugal force field in the lagging
plane is radial instead of parallel as in the flapping plane, and the relationship between
the mode shape and the frequency is different.
The only exact solution of eqn 7.52 is T(x) = x for e = 0, which gives v
2
+ 1 = 1
or ν = 0. Thus, the first mode shape is a straight line from the hub associated with
zero lagging frequency. When the lag hinge offset is not zero there is no exact
solution to the lag equation but we would expect a close approximation to be a
straight line moving about the hinge point, i.e. like a rigid blade, with a frequency of
approximately √(3/2e)Ω, e being the non-dimensional lag hinge offset. As we mentioned
in Chapter 1, a typical value of e is about 0.05, in which case ν 1 = 0.274.
The calculation of the lag mode shapes and frequencies generally can be achieved
by the methods described for the flapping equation.
For hingeless blades the first mode shape is determined largely by the stiffness
near the root; in fact, the structural element at the root is designed to give suitable
flap and lag frequencies. Since the stiffening effect of the centrifugal field is much
less than in the flapping motion, the first lag frequency is much more sensitive to
changes of root stiffness. The lagging and flapping stiffnesses are ‘matched’ in such
a way as to minimize torsional moments when lagging and flapping motion occurs.
This point will be discussed in more detail in Chapter 9 on aeroelastic coupling, but
the typical first mode frequencies arising from such a choice of stiffnesses are usually
in the region of 0.55 Ω to 0.7 Ω, i.e. much higher than for the hinged blade.
When considering the higher modes, it will be appreciated that the lag stiffness
will be much greater than the flapping stiffness; typically, over most of the blade the
lagwise stiffness is about ten times greater. Calculations show that the shapes of the
lagging modes have the same general appearance as those of the flapping motion but,
whereas the first lag frequency is usually much smaller than the first flap frequency,
the reverse is true for all the corresponding higher modes since the curvature of the
blades invokes the much greater stiffness.
260 Bramwell’s Helicopter Dynamics
7.2.3 Torsional deflections
Consider a portion of the blade of span dr under the action of torsional moments. If
C is the torsional moment of inertia per unit length of span, the inertia moment due
to angular acceleration ..
θ is – C ..
θ dr and that due to the ‘propeller moment’ section
1.10, is – CΩ
2
θ dr.
Then the elementary moment dL tending to twist the blade in the nose-up sense is
dL = – C ..
θ dr – CΩ
2
θ dr (7.54)
Now consider the torques acting on the sides of the element and let us define the
positive value of the torque when its sense agrees with the positive direction of r. If
W is the torque on the element, Fig. 7.17, the value is –W on the left-hand side and
W + (∂W/∂r) dr on the right-hand side. For equilibrium we must have
– + d + + d = 0 W
L
r
r W
W
r
r
∂
∂
∂
∂
or
∂W/∂r + ∂L/∂r = 0 (7.55)
Now the relationship between the angle of twist θ and the torque on the element
is
W E J
r
s =
∂θ
∂
where EsJ is the torsional rigidity of the element, Es being the modulus of rigidity or
shear modulus and J being the polar second moment of area.
Hence, from eqns 7.54 and 7.55, the blade torsion equation is
∂
∂
∂
∂ r
E J
r
C C s – – = 0
2 θ
θ θ
..
Ω (7.56)
Assuming solutions of the form
θ = Q(x)ζ(t)
we obtain the mode-shape equation
d
d
d
d
+ ( – ) = 0
2 2
r
E J
Q
r
C Q s
ωθ Ω (7.57)
and
d /d + = 0
2 2 2
ζ ω ζ θ
t
∂
∂
L
r
m d
W
W
r
r + d
∂
∂
–W
Fig. 7.17 Torsional moments acting on element
Structural dynamics of elastic blades 261
where ωθ is a natural frequency of the free motion of the rotating blade.
Now the pitch control usually has a considerable degree of flexibility, so that the
boundary condition to be satisfied at the root is given by
[EsJ dθ/dr] = kθθ0 (7.58)
where θ0 is the change of blade pitch at the feathering hinge and kθ is the stiffness of
the control system. Actually, the flexibility of the control system may be such as to
allow as much as 70 to 80 per cent of the deflection at the tip to occur at the
feathering hinge. The boundary condition at the tip is
dθ/dr = 0 at r = R (7.59)
which expresses the fact that the moment vanishes at the tip.
Consideration of eqn 7.57 shows that, since the torsional stiffness EsJ is unaffected
by rotor speed, the torsional mode shapes are also independent of the rotor speed, i.e.
the mode shapes of a blade are the same whether it is rotating or not. It therefore
follows that
ωθ
2 2
– = constant Ω
When Ω = 0, ωθ is equal to the non-rotating torsional frequency, ω0 say; hence
ω ω θ
2
0
2 2
= + Ω
i.e. the square of the torsional frequency is simply the sum of the squares of the nonrotating frequency and of the rotor speed.
Unlike the mode shape equations of flapping and lagging, the simpler form of the
torsional equation, eqn 7.57, enables some exact solutions to be obtained.
Consider the case of constant torsional stiffness (except for flexibility at the root)
and zero rotor rotational speed. Equation 7.57 can be written
d
2
Q/dx
2
+ α
2
Q = 0 (7.60)
where α
2
= ω 0
2 2
s / CR E J (7.61)
The solution of eqn 7.60 is
Q = A cos α x + B sin α x (7.62)
The boundary conditions, eqns 7.58 and 7.59, give
Bα = (Rkθ /EsJ)A (7.63)
and
tan α = B/A = Rkθ /EsJα (7.64)
or
α tan α = Rkθ /EsJ (7.65)
Solutions of eqn 7.64 in conjunction with eqn 7.61 give the frequencies of the nonrotating blade for given values of kθ and EsJ.
262 Bramwell’s Helicopter Dynamics
The displacement at the blade tip is given by
Q(1) = A cos α + B sin α
= A[cos α + (B/A) sin α]
= A sec α, from eqn 7.64
The displacement at the root due to control flexibility is
Q(0) = A
Thus the ratio of the root deflection to that at the tip is
Q(0)/Q(1) = cos α (7.66)
Let us suppose that in the first mode of motion the control stiffness is such that
the root deflection is half that at the tip. Then, for this case, we see from eqn 7.66 that
cos α =
1
2
or α = π/3. Then, if we take EsJ/CR = 4000 as a typical value, we have
from eqn 7.61 that
ω0 = 10.5 Hz
which is about two and a half times the typical rotor frequency. Thus, at normal rotor
speed (4 Hz), the torsional frequency of our rotor blade is about 3.35 Ω.
The choice of α for the first mode fixes the value of Rkθ /EsJ in eqn 7.65, and the
values of α for the higher modes are given by the solutions of
α tan α = (π /3) tan (π /3) = π /√3 = 1.82
The next two solutions are easily found to be α = 3.62 and 6.54, giving ω0 = 36.2
and 65.4 Hz respectively. The corresponding mode shape outboard of the feathering
hinge and normalised with respect to the tip deflection is
Q(x) = A[cos α x + (B/A) sin α x]
= cos α cos α x + sin α sin α x
= cos [α (1 – x)] (7.67)
The mode shapes are found by giving α its appropriate values. The first shapes are
shown in Fig. 7.18.
It is usual that the first mode frequency is several times the rotor frequency and
may be as high as 6Ω or even more.
7.2.4 Intermodal coupling
In the previous sections we have examined the flapping and lagging and torsional
mode shapes and frequencies on the assumptions that the motions are independent of
one another. In general, however, a certain amount of elastic and inertial coupling
exists when there is built-in twist and if the elastic and mass axes do not coincide.
The complete coupled equations have been given by Houbolt and Brooks
11
and, more
recently, by Sobey
12
.
Structural dynamics of elastic blades 263
Aerodynamic and inertia coupling of the first-mode motions relating to classical
blade flutter will be considered in Chapter 9.
To give an indication of how the torsional motion of the blade modifies the
flapping equation, consider a blade with zero built-in twist. Referring to Figs 7.19
and 7.20 we see that twist introduces the following flapping moments and shears.
(i) A component GeA sin θ of the moment GeA, GeA being the moment of the
blade tension about the elastic axis when the latter is displaced from the
section centroid.
(ii) The moment of the centrifugal force due to the displacement of the centre of
gravity of the blade section relative to the elastic axis. The moment is easily
seen to be equal to
r
R
mr e r
∫
( )d
2
1 1 1 1 Ω θ .
(iii) The change of shear force dS due to the inertia loading – me x
..
θ d resulting
from the twisting motion.
1
0
–1
Q1(x)
Q2(x)
Q3(x)
1 r/R
Fig. 7.18 Torsional mode shapes of a uniform blade
Elastic axis
Centroid
c.g.
θ
eA
e
Elastic axis
Centroid
c.g.
Fig. 7.19 Blade section geometry
Fig. 7.20 Blade section forces and moments
GeA
( + ) d
.. ..
Z e m θ
θ
264 Bramwell’s Helicopter Dynamics
Then, if M′ is the sum of the moments due to the effects of twist, we have, for
small θ,
∂
∂
∂
∂
∂
∂
θ
∂
∂
θ θ
2
2
2
2 A
2
= = ( ) + ( ) –
′ M
r
S
r r
Ge
r
rme me Ω
..
Adding these terms to the uncoupled flap-bending equation, eqn 7.6 we get
∂
∂
∂
∂
θ
∂
∂
∂
∂
∂
∂
θ θ
2
2
2
2 A
2
– – – ( ) + ( + ) = 0
r
EI
Z
r
Ge
r
G
Z
r r
rme m Z e
Ω
.. ..
This agrees with the equation of flapwise bending given by Houbolt and Brooks
for the special case of zero built-in twist. By similar arguments we could derive the
coupled lag and torsional equations of motion.
When the blade also has built-in twist, there is elastic coupling between the flapping
and lagging motion. The full equations are given by Houbolt and Brooks, but a
simple model used by Ormiston and Hodges
13
to investigate flap-Jag instability will
serve here to give a simple illustration of the effects of elastic coupling on the
flapping and lagging motion. The flexibility of the blade is represented in the diagram
of Fig. 7.21. Part of the flexibility is contained in the hub springs, which have
flapping and lagging stiffnesses of κ κ β ξ H H
and respectively. The remaining part of
the blade flexibility lies just outboard of the feathering hinge so that the associated
spring system, with stiffnesses κ βB
and κξB
, rotates when blade pitch is applied.
It is clear that, when pitch is applied (i.e. θ ≠ 0), flapping motion, i.e displacements
of the blade perpendicular to the plane of rotation, causes moments in the plane of
rotation, and vice versa. It is not difficult to show that for flap and lag displacements
β and ξ the elastic moments Me and Le in the flap and lag planes are given by
M e e
2 e
= – [ + ( – ) sin ] –
2
( – ) sin 2
β
κ ρ κκ θ
ξρ
κ κ θ β ξ β ξ β
∆ ∆
(7.68)
Ω
Kξ H
Kβ H
KξB
.
θ
Fig. 7.21 Representation of stiffness coupling
Kβ B
Structural dynamics of elastic blades 265
Le e
2 e
= – [ – ( – ) sin ] –
2
( – ) sin 2
ξ
κ ρ κ κ θ
βρ
κ κ θ ξ ξ β ξ β
∆ ∆
(7.69)
where
∆ = 1 + (1 – )
+
sin e e
2
ρ ρ
κ κ
κ κ
θ
ξ β
ξ β
(7.70)
κ
κ κ
κ κ
κ
κ κ
κ κ
β
β β
β β
ξ
β ξ
ξ ξ
=
( + )
, =
( + )
B H
B H
B H
B H
and ρe is the degree of elastic coupling defined by
ρ κ κ κκ β β ξ ξ e = / = /
B B
(7.71)
When ρe = 0, the hinge system is contained entirely at the hub, i.e. the outboard
springs are infinitely stiff, and no elastic coupling is possible. When ρe = 1, all the
blade flexibility exists outboard of the feathering hinge and there is full elastic
coupling.
To illustrate the effect of elastic coupling using the above model, let us consider
a fully coupled hingeless blade (ρe = 1) with a given flapping stiffness but whose lag
stiffness is to be varied. Let the collective-pitch angle be 15°.
Suppose that the uncoupled flapping and lagging frequencies ωβ and ωξ are given
by the following Southwell formulae:
ω κ β
β β
2 2
= ( / ) + 1.12 I Ω (7.72)
ω κ ξ
ξ ξ
2 2
= ( / ) + 0.23 I Ω (7.73)
where I is the (same) moment of inertia in flapping and lagging and Ω is the rotor
angular velocity. Let κβ /I be fixed and equal to 0.13 Ω
2
. If, for simplicity, we suppose
that the coning angle is zero, so that the Coriolis moments can be ignored, and also
that the change of lag stiffness κξ does not alter the constant in eqn 7.73 then, using
eqns 7.68 and 7.69, the flapping and lagging equations are
d
d
+ 1.185 + 0.067 +
1
4
– 0.13 = 0
2
2 2 2
β
ψ
κ
β
κ
ξ
ξ ξ
I I Ω Ω
(7.74)
and
d
d
+ 0.239 + 0.933 +
1
4
– 0.13 = 0
2
2 2 2
ξ
ψ
κ
ξ
κ
β
ξ ξ
I I Ω Ω
(7.75)
Substituting the assumed solutions β = β0 e
λψ
and ξ = ξ0 e
λψ
leads to the characteristic
equation
λ
κ
λ
κ ξ ξ
2
2
2
2
+ 1.185 + 0.067 + 0.239 + 0.933
I I Ω Ω
–
1
16
– 0.13 = 0
2
2
κξ
IΩ
(7.76)
266 Bramwell’s Helicopter Dynamics
Solution of this equation for a range of lag stiffnesses κξ gives the flap and lag
frequencies shown in Fig. 7.22. The figure shows an interesting phenomenon. When
the blades are elastically uncoupled, the curves of flap and lag frequencies cross
over as the lag stiffness is increased. However, when elastic coupling exists, the
frequencies lie close to the uncoupled values when the frequencies are fairly widely
separated, but as the cross-over point is approached the curves begin to diverge
from one another and each becomes asymptotic to the frequency curve of the other
mode.
It is instructive to examine the flap–lag ratio of each mode as the lag stiffness is
varied. The flap–lag ratio can be obtained from either of eqns 7.74 and 7.75. Thus,
from eqn 7.74, after substituting β = β0 e
λψ
and ξ = ξ0e
λψ
, we get
β
ξ
κ
λ κ
ξ
ξ
0
0
2 2
2 2
=
( / – 0.13)
16[ + (1.185 + 0.067 / )]
I
I
Ω
Ω
Inserting values of λ corresponding to the lower curve shows that β0/ξ0 is very
small for low values of κξ, indicating that the motion consists almost entirely of lag,
agreeing with the corresponding uncoupled motion, but that for large values of κξ we
find that β0/ξ0 is also large, indicating that the motion now consists mainly of flapping,
in agreement with the uncoupled frequency curve which lies close to it. Thus, along
either of the frequency curves of coupled motion the mode changes its character in
the region of the uncoupled cross-over point.
A similar situation arises when flap and lag stiffness are kept constant but the
frequencies change with rotor speed such that there is an intersection of a flap and a
lag mode. For example, the second lag frequency might intersect the frequency curve
of the fourth flap mode, as indicated in Fig. 7.23. Again, we would find that the
lagging motion at low speeds would become mainly flapping motion after passing
the uncoupled cross-over point, and vice versa.
It appears, therefore, that the coupling is important in the region where the frequency
curves of two uncoupled modes would intersect one another.
Fig. 7.22 Effect of stiffness coupling on flap and lag frequencies
Uncoupled lag frequency
Uncoupled flap frequency
Mainly flap
Mainly lag
Mainly lag
Mainly flap
ωβ, ωξ
√(κξ/IΩ
2
)
Structural dynamics of elastic blades 267
50
40
30
20
10
Natural frequency, Hz
1st lag
1st flap
1st torsion
2nd flap
3rd flap
2nd lag
4th flap
0 50 100 150 200 250 300
10Ω
9Ω
8Ω
4th flap
7Ω
6Ω
1st torsion
5Ω
3Ω
2Ω
3rd flap
2nd lag
4Ω
2nd flap
1Ω
1st flap
1st lag
Rotor speed, rev/min
Fig. 7.23 Variation of blade bending frequencies with rotor speed
7.3 Forced response of rotor blades
7.3.1 Orthogonal property of the normal modes
The mode shapes possess a very important orthogonal property which can be deduced
from the flapwise bending equation, eqn 7.11. Let Sm(x) and Sn(x) be two different
solutions of eqn 7.11, with their associated frequency ratios λm and λn.
Thus, we have
d
d
d
d
–
d
d
d
d
– = 0
2
2
2
2
2 2 2 4
x
EI
S
x
R
x
G
S
x
m R S
m m
m m
λ Ω (7.77)
and
d
d
d
d
–
d
d
d
d
– = 0
2
2
2
2
2 2 2 4
x
EI
S
x
R
x
G
S
x
m R S
n n
n n
λ Ω (7.78)
Now multiply eqn 7.77 by Sn and eqn 7.78 by Sm. Consider the first equation:
S
x
EI
S
x
R S
x
G
S
x
m R S S n
m
n
m
m m n
d
d
d
d
–
d
d
d
d
– = 0
2
2
2
2
2 2 2 4
λ Ω (7.79)
Integrating the first term by parts from x = 0 to x = 1 gives
0
1 2
2
2
2
2
2
0
1
d
d
d
d
d =
d
d
d
d
∫
S
x
EI
S
x
x S
x
EI
S
x
n
m
n
m
–
d
d
d
d
d
d
d
0
1 2
2
∫
S
x x
EI
S
x
x
n m
268 Bramwell’s Helicopter Dynamics
The term in the square brackets vanishes on account of the boundary conditions
for both the hinged and hingeless blade. Further,
0
1 2
2
2
2
0
1
0
1 2
2
2
2
0
1 2
2
2
2
d
d
d
d
d
d
=
d
d
d
d
–
d
d
d
d
d
= –
d
d
d
d
d
∫ ∫
∫
S
x x
EI
S
x
dx
S
x
EI
S
x
EI
S
x
S
x
x
EI
S
x
S
x
x
n m n m m n
m n
on applying the boundary conditions again.
Similarly integrating the second term of eqn 7.79
0
1
0
1
d
d
d
d
d = –
d
d
d
d
d
∫ ∫
S
x
G
S
x
x G
S
x
Sn
x
x m
n m
Thus, the term by term integration of eqn 7.79 yields
0
1 2
2
2
2
2
0
1
2 2 4
0
1
d
d
d
d
d +
d
d
d
d
d – d = 0
∫ ∫ ∫
EI
S
x
S
x
x R G
S
x
S
x
x R mS S x
m n m n
m m n λ Ω
(7.80)
Treating eqn 7.78 in the same way gives
0
1 2
2
2
2
2
0
1
2 2 4
0
1
d
d
d
d
d +
d
d
d
d
d – d = 0
∫ ∫ ∫
EI
S
x
S
x
x R G
S
x
S
x
x R mS S x
m n m n
n m n λ Ω
(7.81)
Subtracting eqn 7.81 from eqn 7.80 gives
(λ λ n m m n mS S x
2 2
0
1
– ) d = 0
∫
We conclude that, since λn ≠ λm when m ≠ n,
0
1
d = 0,
= ( ), say, when =
∫
≠ mS S x m n
f n m n
m n
(7.82)
Equation 7.82 expresses the orthogonal properties of the modes in relation to the
weighting function m(x) (mass distribution). This means, for example, that if we
integrate the product of any two dissimilar mode shapes together with the mass
distribution m(x), the result is zero. This fact serves as a useful check on the accuracy
of computed mode shapes.
The orthogonal properties do not hold for a hingeless blade with a built-in coning
angle β0, say, for in this case the boundary conditions are dS/dx = tan β0 at x = 0 and
some of the terms in the integration of eqn 7.79 do not vanish at the limits. However,
this is not really a restriction since the reference axes can be tilted so that the new dS/
dx becomes zero at the blade root. There will then be a component of centrifugal
Structural dynamics of elastic blades 269
force – mΩ
2
r tan β0 normal to the undeflected blade which can be treated as a spanwise
loading to be added to ∂F/∂r. Thus, the orthogonal properties can be made to apply
to the blade with built-in coning by using the boundary conditions (b) and applying
an extra spanwise loading –mΩ
2
r tan β0.
We should also note the rather interesting fact that the teetering rotor satisfies both
sets of boundary conditions. Figures 7.24(a) and 7.24(b) show two examples of the
way the blade can bend. In Fig. 7.24(a) the shape of the blade is anti-symmetric about
the hinge, so that there is a point of inflection there; i.e. d
2
S/dx
2
= 0 at x = 0 and the
boundary conditions are those of the hinged blade. In Fig. 9.4(b) the bending is
symmetrical and each half behaves as if the root were rigidly fixed to the shaft; the
boundary conditions therefore are those of the hingeless blade. Thus, the teetering rotor
has twice as many mode shapes and frequencies as a single hinged or hingeless blade.
7.3.2 Forced response equations
Equation 7.6 refers to the free vibration of the rotor blade in the flapwise sense. The
corresponding equation, taking into account the presence of external loading F, is
∂
∂
∂
∂
∂
∂
∂ ∂
∂
∂
∂
2
2
2
2
2
2
– + =
r
EI
Z
r r
G
Z
r
m
Z
t
F
r
∂
(7.83)
We now make use of the orthogonal properties of the normal mode shapes to simplify
the problem of determining the blade motion under the action of applied forces.
Let us suppose that the solution of eqn 7.83 to a known blade loading can be
expressed as
Z R S x
n
n n = ( ) ( )
=1
Σ
∞
φ ψ (7.84)
where the terms Sn(x) are the blade mode shapes calculated from solving eqn 7.11 but
the terms φn(ψ) have yet to be determined. Substituting eqn 7.84 into eqn 7.83 gives
Σ
∞
n
n
n n
x
EI
S
x
R
x
G
S
x =1
2
2
2
2
2
( )
d
d
d
d
–
d
d
d
d
φ ψ
+
d
d
=
2 4
=1
2
2
2
m R S R
F
x n
n Ω Σ
∞
φ
ψ
∂
∂
(a) (b)
Fig. 7.24 Possible deflections of the teetering rotor
270 Bramwell’s Helicopter Dynamics
But from eqn 7.77 the terms in the large bracket can be replaced by
m R S
n
n n Ω Σ
∞
2 4
=1
2
λ
giving
Σ
∞
Ω
n
n
n n n S x
m R
F
x =1
2
2
2
2 4
d
d
+ ( ) ( ) =
1 φ
ψ
λ φ ψ
∂
∂
(7.85)
Multiplying eqn 7.85 through by mSm(x), integrating from 0 to 1, and applying the
orthogonal properties as defined by eqn 7.82, gives
d
d
+ =
1
( )
( )d
2
2
2
2 2
0
1
φ
ψ
λ φ
∂
∂
n
n n n
R f n
F
x
S x x
Ω ∫
(7.86)
where f(n) =
0
1
2
( )d
∫
mS x x n is the generalised mass or inertia term for the nth mode.
Thus, the terms φn of eqn 7.84 are determined from the solutions of eqn 7.86.
Unfortunately, although the orthogonal properties ‘decouple’ the inertia and elastic
terms, represented by the left-hand side of eqn 7.83, the blade loading ∂F/∂r depends
on the blade deflections which will, in general, contain all the blade modes of motion.
Otherwise, each azimuth co-ordinate φn could be calculated individually from
eqn 7.86. We shall discuss means of solving eqn 7.86 later in the chapter.
We may also derive the forced response equation from energy considerations.
It is convenient to express the energy forms in terms of the blade axes, which are
non-Newtonian due to the rotation present. However, energy forms relative to these
axes may be used in the same way as those for absolute energy forms, provided the
inertia forces resulting from the rotation are treated as normal external forces
14,15
in
developing energy forms. In the present case, the inertia forces acting on a mass
element in the blade are the centrifugal force and the Coriolis force. The former is
normally considered to provide a potential energy term arising from the centrifugal
effect, which can then be added to the potential or strain energy from elastic forces;
the latter is usually removed to the right-hand side of the equations of motion,
together with the other external, i.e. aerodynamic, forces.
The kinetic energy T of the mass elements of the blade relative to axes rotating
with the blade is
Τ
∂
∂
=
1
2
d
0
2 R
m
Z
t
r
∫
=
1
2
d
2 3
0
1 2
Ω
∫
R m
z
x
∂
∂ψ
(7.87)
m being, as usual, the mass per unit length of the blade.
Assuming as before
Structural dynamics of elastic blades 271
z S x
n
n n = ( ) ( )
=1
Σ
∞
φ ψ
we have
∂
∂ψ
φ
ψ
z
S x
n
n
n
= ( )
d
d =1
Σ
∞
so that substitution in eqn 7.87 we find
Τ
φ
ψ
φ
ψ
=
1
2
d
d
d
d
d
2 3
=1 =1
0
1
Ω Σ Σ
∞ ∞
∫
R mS S x
m n
m n
m n
=
1
2
d
d
( )
2 3
=1
2
Ω
Σ
∞
R f n
n
n φ
ψ
(7.88)
from the orthogonal relations eqn 7.82
The strain energy UB due to bending is
10
U EI
Z
r
r B
R
=
1
2
d
0
2
2
2
∫
∂
∂
=
1
2
d
0
1 2
2
2
R
EI
z
x
x
∫
∂
∂
(7.89)
=
1
2
d
d
d
d
d
=1 =1
0
1 2
2
2
2
R
EI
S
x
S
x
x
m n
m n
m n
Σ Σ
∞ ∞
∫
φ φ (7.90)
The potential energy UG due to the centrifugal stiffening effect arises from the centrifugal
force G acting on a mass element doing negative work when the radius of its point
of application decreases as a result of blade bending (the ‘shortening effect’). Then
U G r r Z r r G
R
= { – [1 – ( / ) ] } d
0
2 1/2
∫
∂ ∂
=
1
2
d
0
1 2
R G
z
x
x
∫
∂
∂
(7.91)
for small deflections. Substituting for ∂z/∂x we have
U R G
S
x
S
x
x G
m n
m n
m n
=
1
2
d
d
d
d
d
=1 =1
0
1
Σ Σ
∞ ∞
∫
φ φ
The total strain and potential energy is
U U U
R
EI
S
x
S
x
R G
S
x
S
x
x B G
m n
m n
m n m n
= + =
1
2
d
d
d
d
+
d
d
d
d
d
=1 =1
0
1 2
2
2
2
2
Σ Σ
∞ ∞
∫
φ φ
272 Bramwell’s Helicopter Dynamics
=
1
2
d
2 2 3
=1 =1
2
0
1
λ φ n
m n
n m n R mS S x Ω Σ Σ
∞ ∞
∫
(7.92)
from eqn 7.81, so that
U R f n n
n
n =
1
2
( )
2 2 3
=1
2
λ φ Ω Σ
∞
(7.93)
on using the orthogonal properties again.
The external work done by the elementary force (∂F/∂r)dr in an arbitrary displacement
is
δ δ
∂
∂
δφ ( ) = d ( )
=1
W R
F
x
x S x
n
n n Σ
∞
and for the whole blade
δ δφ
∂
∂
W R
F
x
S x
n
n n = d
=1
0
1
Σ
∞
∫
so that
∂
∂φ
∂
∂
W
R
F
x
S x
n
n = d
0
1
∫
(7.94)
Note that since only blade bending is being considered, the Coriolis force, which acts
in the lead–lag sense, does not contribute to this work form.
Now, Lagrange’s equations for small displacements are
d
d
– + =
t
T T U W
n n n n
∂
∂φ
∂
∂φ
∂
∂φ
∂
∂φ
or, with ψ = Ωt,
d
d
– + =
ψ
∂
∂φ
∂
∂φ
∂
∂φ
∂
∂φ
T T U W
n n n n ′
(7.95)
Then, using eqns 7.88, 7.93, and 7.94, Lagrange’s equations give
d
d
+ =
1
( )
( )d
2
2
2
2 2
0
1
φ
ψ
λ φ
∂
∂
n
n n n
R f n
F
x
S x x
Ω ∫
as before (7.96)
The response equations of the blade in lagging motion are of the same form as for
the flapping motion, i.e. they take the form
d
d
+ =
1
( )
( )d
2
2
2
2 2
0
1
χ
ψ
ν χ
∂
∂
n
n
y
n
R g n
F
x
T x x
Ω ∫
where
g n mT x x n ( ) = ( )d
0
1
2
∫
Structural dynamics of elastic blades 273
is the generalised inertia for the nth lag bending mode, and Fg is the lagwise external
(aerodynamic) loading.
7.3.3 Solution of the forced response equation
We shall now consider the solution of the forced response equation, eqn 7.86 above.
For a rigid blade, φ1 can be identified with the flapping angle β and the right hand
side with the aerodynamic moment MA. As mentioned before, the orthogonal properties
of the normal modes decouple the inertia and stiffness terms so that the terms on the
left hand side of eqn 7.86 are independent of the other modes. Unfortunately, the
aerodynamic loading contains all the modes, since the complete blade motion is
required to determine the aerodynamic incidence. This means that the right hand side
is a function of φ1, φ2, …, etc., so that the mode equations cannot be solved independently.
We are therefore faced with the problem of integrating these equations in the most
efficient way.
The usual procedure is to assume a starting value for the blade motion– for example,
we may take the classical rigid blade flapping – and compute the aerodynamic loading
and the integral on the right hand side of eqn 7.86 for the modes being considered.
The value of the integral is then imagined to remain constant over a small azimuth
variation ∆ψ and the corresponding changes of φn and dφn/dψ are computed from
eqn 7.86. The loading integral is then re-calculated and φn and dφn/dψ are computed
for the next azimuth step ∆ψ. These stepwise integrations are continued until the
required values of φn converge to within an acceptable limit. The integrations may
extend over several resolutions before satisfactory convergence has been achieved.
Part of the process of convergence concerns the transient blade motion, because of
the inevitability of some error between the assumed blade motion, taken as the
starting value, and the true motion; and part will be due to the inherent features in the
integration process. There must, of course, be as many integrations as there are
modes taken to represent the blade motion.
Of the methods of integration, the most common is probably the fourth order
Runge-Kutta, but it appears that convergence difficulties arise if the step lengths are
too small, particularly for the higher modes.
A simpler and yet more accurate method is to take advantage of the fact that, if the
right hand side of eqn 7.86 is kept constant, the equation can be solved exactly over
the step length. Let us denote the right hand side of eqn 7.86 by fi, where fi is the
constant value corresponding to the ith step. Then eqn 7.86 can be written as
d /d + =
2 2 2
φ ψ λφ φ n n n in (7.96)
To avoid confusion, we shall drop the suffix n from φn and let φ represent any mode
displacement. Let φi and ′ φi be the values of φ and dφ /dψ at the start of the ith step;
then eqn 7.96 can be written as
d
2
φ/dψ
2
+ λ
2
φ = fi
whose solution is easily found to be
274 Bramwell’s Helicopter Dynamics
φ
λ
λψ φ λψ
φ
λ
λψ = (1 – cos ) + cos + sin
2
fi
i
i′
If ∆ψ is the step change of ψ, the values of φ and dφ/dψ at the start of the (i + 1)th
step are
′
′
φ
λ
λ ψ φ λψ
φ
λ
λ ψ i
i
i
i f
+1 2
= (1 – cos ) + cos + sin ∆ ∆ ∆(7.97)
and
′ ′ φ
λ
λ ψ λφ λ ψ φ λψ i
i
i i
f
+1 = sin – sin + cos ∆ ∆ ∆ (7.98)
These are the initial values at the start of the next interval, at the end of which
φ
λ
λ ψ φ λψ
φ
λ
λ ψ i
i
i
i f
+2
+1
2 +1
+1
= (1 – cos ) + cos + sin ∆ ∆ ∆
′
(7.99)
Eliminating ′ ′ φ φ i i +1 and by means of eqns 7.97 and 7.98 leads to the recurrence
relationship
φ φ λψφ
λ ψ
λ
i i i i i f f +2 + 1 2 +1 = 2 cos – –
1 – cos
( + ) ∆
∆
(7.100)
connecting the mode displacement with the values at the two previous intervals and
the corresponding loading integrals.
Wilkinson and Shilladay
16
have obtained the same result by appealing to sampled
data theory and using the Z-transform. The values of fi can be represented as discrete
input functions occurring at regular intervals, as in Fig. 7.25.
The use of constant values of fi over the interval is referred to as a zero-order hold,
Fig. 7.26. Improved accuracy can be obtained by assuming that fi varies linearly over
an interval according to the slope of the straight line connecting the values of fi – 2 and
fi–1, Fig. 7.27. This is called a first-order hold. The differential equation for this case
is
Fig. 7.25 Discrete input frequency
f
ψ
∆ψ ∆ψ
f
ψ
Fig. 7.26 Zero-order hold forcing
Structural dynamics of elastic blades 275
d
d
+ = +
–
2
2
2 –1 φ
ψ
λ φ
ψ
ψ f
f f
i
i i
∆
(7.101)
with the solution
φ
λ
λ ψ
λ ψ λψ
λψ φ λψ
φ
λ
λψ = (1 – cos ) +
–
–
–
sin + cos + sin
2
–1
2
–1
3
f f f f f
i i i i i
i
i
∆
∆ ∆
′
(7.102)
By proceeding as before, we obtain the recurrence relationship
φi+2 = 2φi+1 cos λ∆ψ – φi +
1
( – 2 + ) cos –
sin
+ 2 (1 – cos )
2 –1 +1 +1
λ
λ ψ
λ ψ
λ ψ
λ ψ f f f f i i i i ∆
∆
∆
∆
(7.103)
Wilkinson and Shilladay have tested the recurrence relationships eqns 7.100 and
7.103 for cases with known exact solutions and have found that they are more
accurate and allow faster computation than the Runge–Kutta method.
We can also take advantage of the fact that in steady flight the aerodynamic
loading, and the consequent blade motion, is periodic so that the right hand side of
the modal equation 7.86 is expressible in the form of a Fourier series. If this is done,
the complete solution of eqn 7.86 can be written down at once, since it represents the
well known response of a second-order system forced by a series of harmonic functions,
Appendix A.4. Of course, the loading integral on the right hand side of eqn 7.86 will
not, initially, be known exactly, since it depends in a complicated way on the motion
being calculated. An integrative solution must be adopted, and the classical rigid
blade flapping can be used as a first approximation. The loading integral is then
calculated, expressed as a Fourier series, and the values of the mode displacements
φn are obtained. The values of φn and dφn/dψ are then used to obtain a second
approximation to the loading ∂F/∂x, and the process is repeated until satisfactory
convergence has been achieved. There are, of course, a number of numerical methods
for calculating the Fourier series of a set of periodic values.
If, in the method just described, the modal equation is used as it appears in
f
ψ
Fig. 7.27 First-order hold forcing
276 Bramwell’s Helicopter Dynamics
eqn 7.86, we shall encounter the difficulty of resonance
16
for those modes with a
natural frequency equal or close to the rotor frequency. This can be avoided by noting
that the aerodynamic loading contains damping which, although non-linear on account
of the non-linear aerodynamic data which will undoubtedly be used, will nevertheless
be roughly proportional to dφn/dψ.
It such a term is removed from the loading integral of eqn 7.86 and is transferred
to the left hand side of the equation, we shall have the case of a harmonically forced
system with quite high linear damping, and the large amplitudes near resonant frequency
will not occur. Since dφn/dψ will not be known when transferred, the calculation of
the damping terms remaining in the integral must be based on the blade motion of the
previous approximation, as has been described for the case earlier.
7.4 Blade deflections in flight
The deflection of the rotor blade in flight can be calculated by the method of the
previous section in conjunction with the estimated blade mode shapes and frequencies
and the aerodynamic loading. These calculations can also be used to obtain the
stresses in the blade and at the hub. For illustration, however, we give below examples
of the measurement of blade deflection obtained by photographing the blade in flight.
In this experiment, a 16 mm camera taking 240 frames per second (about 60 frames
per rotor revolution) was mounted on the hub of an experimental Westland ‘Scout’
with a hingeless rotor, and was directed to look along the span of one of the blades.
A typical photograph (ψ = 290°, 54 m/s, level flight) is shown in Fig. 7.28. It was
assumed that the contributions of the first three modes were sufficient to account for
the measured blade deflection. The calculated mode shapes and their associated
frequencies are shown in Fig. 7.29.
The flapping deflection of any part of the blade would then be expressible in the
form
Z(x, ψ) = R[S1(x)φ1(ψ) + S2(x)φ2(ψ) + S3(x)φ3(ψ)]
With S1, S2 and S3 known, the generalised co-ordinates φ1, φ2, φ3 in terms of
azimuth could be determined from analysis of the photographs. The results, expressed
in terms of the displacement of the blade tip, are shown in Fig. 7.30.
It should be emphasised that the displacements are measured relative to the hub
axis. The amount of first-mode displacement which occurs, is of course, directly
dependent on the amount of cyclic pitch applied and will be such as to trim the
longitudinal and lateral moments about the hub, including the hub moments due to
blade deflection in the second and third modes.
It can be seen that the contributions of the second and third modes to the blade
deflection are quite small and would be still smaller if the first mode deflection were
measured relative to the no-feathering axis. This justifies the assumption of the rigid
blade which is used in much of helicopter analysis. Calculations show that, for the
flight case considered, neglect of the second and third modes results in only about a
7 per cent error in the hub moment.
Structural dynamics of elastic blades 277
Fig. 7.28 Typical photograph of blade in flight (hingeless Westland ‘Scout’, 53.5 m/s, ψ = 290°)
7.5 Basic features of the hingeless rotor
Many helicopters now employ hingeless rotor systems. The design of such systems
has become possible due to two main factors: the increased understanding of the
aeroelastic and dynamic hebaviour of rotor systems and the application of composite
material to critical components in the aeronautical field.
The initial development was the replacement of the conventional flapping and
lagging hinges by flexible structural elements. Such a system is normally referred to
as the semi-rigid rotor. Subsequent developments led to the totally bearingless rotor
where, in addition to the elimination of the flapping and lagging hinges, blade feathering
is accommodated by the torsional flexibility of the hub elements.
The main advantages claimed for hingeless rotors are:
278 Bramwell’s Helicopter Dynamics
1.0
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
0.2 0.4 0.6 0.8 1
S3
S1
λ1 = 1.087
r/R
λ2 = 2.587
λ = 4.553
S2
Blade deflection
Fig. 7.29 Blade-flapping mode shapes
100
50
0
–50
–100
–120
0° 90° 180° 270° 360° ψ
3rd mode
2nd mode
1st mode
Flap-mode components
(Tip deflection in mm relative to built-in coning angle)
Fig. 7.30 Modal content of blade flapwise deflection
Structural dynamics of elastic blades 279
(i) the elimination of hinges leads to a great simplification of hub design, with
consequent reduction of manufacture and maintenance costs, and also reduced
aerodynamic drag;
(ii) much greater hub moments, leading to improved control power which is virtually
independent of the level of rotor thrust, and increased angular rate damping.
Both these features contribute to an improvement in handling qualities.
In this section we shall consider only the flapping mode shapes and frequencies of
the hingeless blade and the hub moments exerted when the blade is deflected. The
problems of the various forms of intermodal coupling and the particular problem of
air resonance are considered in Chapter 9. A comprehensive review of the benefits
and problems associated with hingeless rotors has been given by Hohenemser
17
.
The mode shape analysis of the hingeless blade can be dealt with by the methods
described earlier in this chapter, noting that the root boundary conditions are z = 0
and dz/dx = 0, or perhaps some fixed value (‘built-in’ coning).
Typical lower mode shapes and frequencies are given in Fig. 7.31.
The first flapping frequency ratio of hingeless rotors usually lies within the range
1.08 to 1.17 and, as can be seen from Fig. 7.31, the curvature of the blade is confined
mainly to the root region, most of the outer part of the blade being almost straight.
It is this local curvature, however, when combined with a typical distribution of
bending stiffness in this region, which produces the large hub moments characteristic
of the hingeless blade.
In order to investigate the second of the benefits of the hingeless rotor listed
above, a method is needed for calculating the moments transmitted to the hub due to
1.0
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
S1(x), λ1 = 1.092
S2(x), λ2 = 3.157
S3(x), λ3 = 5.57
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Fig. 7.31 Typical flapping mode shapes for hingeless blade
280 Bramwell’s Helicopter Dynamics
the deflection of the blades. The first analysis for such calculations was made by
Young
18
who produced a very simple expression for the hub moment. Fig. 7.32
refers.
Young calculated the moment at the blade root M(0, t) of the aerodynamic, centrifugal,
and inertia loadings as
(0, ) = – – d
0
2
2
2
M t
F
r
m Z m
Z
t
r r
R
∫
Ω
∂
∂
∂
∂
(7.104)
By means of the blade bending equation
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
2
2
2
2
2
2
– + =
r
EI
Z
r r
G
Z
r
m
Z
t
F
r
the terms ∂F/∂r – m∂
2
Z/∂t
2
can be eliminated from eqn 7.104 giving
M t
r
EI
Z
r r
G
Z
r
m Z r r
R
(0, ) = – – d
0
2
2
2
2
2
∫
Ω
∂
∂
∂
∂
∂
∂
∂
∂
Writing the blade deflection as
Z R S r t
n
n n = ( ) ( )
=1
Σ
∞
φ
the root moment can be written
M t R
r
EI
S
r r
G
S
r
m S r r
R
n
n
n n
n (0, ) =
d
d
d
d
–
d
d
d
d
– d
0 =1
2
2
2
2
2
∫
Σ
∞
Ω
φ
But the blade mode shapes satisfy the equation
d
d
d
d
–
d
d
d
d
– = 0
2
2
2
2
2
r
EI
S
r r
G
S
r
m S
n n
n n
Ω λ
Z
Ω
0
dF
Ω
2
r dm
Z
X
r
Fig. 7.32 Flapwise forces acting on blade
∂ Ζ
∂
2
2
d
t
m
Structural dynamics of elastic blades 281
so that the root moment finally becomes
M t R mrS r
n
n n
R
n (0, ) = ( – 1) d
2
=1
2
0
Ω Σ
∞
∫
φ λ (7.105)
Since, in the case of the rotor with a flapping hinge we only need to use the first rigid
blade mode for considerations of performance and stability and central, let us assume
that we need only consider the first bending mode of the hingeless rotor for the same
purposes.
Then if only the first mode is considered, the moment is
M t R t mxS x i (0, ) = ( – 1) ( ) d
3
1
0
1
λ φ
2
∫
(7.106)
This is a slightly more general form of the equation given by Young, who calculated
only the increment due to a cyclic pitch change in hovering flight.
It has been shown by Simons
19
and by Curtiss and Shupe
20
that the derivation of
eqn 7.106 implies that the aerodynamic loading distribution has the same shape as
the first bending mode, i.e. that
∂
∂
F
x
S x m x ( ) ( ) 1 ∝ ⋅
This implicit assumption may not be seriously in error at low values of µ, but one
might expect that at high values the loading might depart considerably from the first
mode shape and throw doubt on the validity of eqn 7.106.
To avoid this difficulty, another expression for the root moment can be derived.
References to eqn. 7.104, the root moment can be written as
M t r
F
r
r mt
Z
Z r
R R
(0, ) = d – + d
0
2
0
2
2
∫ ∫
Ω
∂
∂
∂
∂ψ
(7.107)
Writing, as before
Z R S x
n
n n = ( ) ( )
=1
Σ
∞
φ ψ
eqn 9.110 becomes
M t r
F
r
r R Mx S x
R
n
n
n n (0, ) = d –
d
d
+ d
0
2 3
=1
2
2
0
1
∫ ∫
Ω
Σ
∞
∂
∂
φ
ψ
φ
Equation 7.106 has been used by Bramwell
2
to obtain stability derivatives of a
hingeless rotor.
Defining a root moment coefficient by
Mc = M/ρbcΩ
2
R
4
282 Bramwell’s Helicopter Dynamics
eqn 7.106 can be written in non-dimensional form as
M
a
b
c
1
2
1
1 =
( – 1)
( )
λ
γ
φ ψ (7.108)
where γ
ρ
1
0
1
1
=
d
acR
mx S x
∫
which is equivalent to the lock number for the hinged rigid blade.
Since, for considerations of performance and stability and control, we may consider
only the 1st harmonic response, the flapping of the hingeless blade can be written in
the form
ϕ ψ β ψ ψ 1 0 1 1 ( ) = = – cos – sin a a b
when β , , , 0 a a b 1 , are analogous to the rigid blade flapping angle and flapping
coefficients. Then the pitching and rolling moment coefficients Cm and Cl of a hingeless
rotor with b blades are
C
a a
z
m
1
2
1
1
=
( – 1) λ
γ
(7.109)
C
a b
l =
( – 1)
2
1
2
1
1
λ
γ
(7.110)
The flapping coefficients a a b 0 1 1 , , can be calculated by a similar analysis to that for
the hinged rigid blade. The equations, although quite straight forward, are rather
cumbersome because unlike those of the centrally hinged blade, the integrals involving
the mode shapes do not lend to simple fractions and the equations do not give the
coefficients explicitly. The equations have been given by Bramwell
21
and Curtiss and
Shupe
20
. The calculations show that the amplitude of flapping is only a little less than
that if the blade were hinged.
Returning to the determination of the hub moment, and since we are interested
only in the pitching and rolling moments on the helicopter, only the first harmonic
motion need be considered, in which case, for the nth mode
φ ψ ψ n n n n a a b = ( ) – ( ) cos – ( ) sin 1 1 0
so that
d
d
+ = ( )
2
2
φ
ψ
φ ω
n
n n a
hence, substituting in the modified version of eqn 7.110, we have
M r
F
r
r R mxS x
n
n n (0, ) = d + ( ) d
2 3
=1
0
0
1
ψ
∂
∂
φ Ω Σ
∞
∫
Structural dynamics of elastic blades 283
When all the blades are taken together to form the total pitching and rolling
moments, the last term on the right-hand side vanishes, so that the hub moment
reduces to
M r
F
r
r
R
(0, ) = d ψ
∂
∂ ∂ ∫
(7.111)
In this equation, given by Simons and by Curtiss and Shupe, no restriction need be
placed on the loading, although, in practice, the loading will depend on the number
of modes chosen to represent the blade motion. According to Curtiss and Shupe, only
one mode is necessary over a well range of conditions, but for high values of µ two
modes ought to be used for better accuracy. On the other hand, if the general equation
in terms of blade flapping is used, eqn 7.106, we have to include that number of
modes which, in superposition, approximate to the true loading shape. In general,
this is longer than that for eqn 7.111.
Let us now consider modelling the hingeless rotor by defining an equivalent
hinged rigid blade system.
Although Young derived eqn 7.106 for the particular case of a cyclic pitch application
in hovering flight, instead of proceeding with the calculations in terms of mode
shape, as has been discussed above, he replaced the hingeless blade with a rigid blade
with an offset hinge and a torsional spring. The amount of offset and the torsional
spring strength were chosen to match the rotating and non-rotating natural frequencies
of the actual blade. This model of the hingeless blade has received general acceptance
and serves very well for calculating the flapping motion and hub moment; indeed it
is quite usual to express the moment of a hingeless blade in terms of an ‘equivalent
hinge offset’.
An interesting point is that, if the offset blade and torsional spring are taken as a
mechanical equivalent to the hingeless blade, the hub moment of the hingeless blade,
when calculated from first principles, approximates more closely to that given by
eqn 7.111 than to that given by eqn 7.106. For, taking the spring strength as ks per unit
flapping angle, Fig. 7.33, the equations of motion of the blade are
Ω Ω
∫ ∫
2 3
2
2
1
2 2 3
1
s A
d
d
( – ) d + ( – )d + = R m x e x R mx x e x k M
e e
β
ψ
β β (7.112)
and
S
r
S
eR
rg
β
ks β
Fig. 7.33 Equivalent offset hinge
284 Bramwell’s Helicopter Dynamics
M r F S b
2
g
2
2
d
d
= – Ω
β
ψ
(7.113)
where MA is the moment of the aerodynamic force about the hinge Mb is the blade
mass. F is the aerodynamic force on the blade, and S is the shear force at the hinge.
β, as usual, is assumed to be a small angle. The moment at the root is clearly
M SeR k r
F
r
r
eR
(0, ) = + + d s
0
ψ β
∂
∂
∫
(7.114)
Now, for simple harmonic motion, and omitting the constant coning angle, we have
d
d
= –
2
2
β
ψ
β
so that eqn 7.112 can be written
M R mx x e x m x e x xk
e e
A
2 3
1 1
2
s = ( – ) d – ( – ) d Ω
∫ ∫
β β
= ( – ) d +
2 3
1
s Ω
∫
R e m x e x k
e
β β (7.115)
Similarly eqn 7.113,
F S M r – =
d
d
b
2
g
2
2
Ω
β
ψ
can be written
F S m r eR r
eR
R
– = ( – )d
2
–Ω
∫
β
= ( – ) d
2 2
1
–Ω
∫
R m x e x
e
β (7.116)
Eliminating the integral between eqns 7.116 and 7.115 gives
SeR + ksβ = FeR + MA
so that substituting in eqn 7.114 gives for the hub moment,
M FeR M r
F
r
r
e
eR
(0, ) = + + d A ψ
∂
∂
∫
(7.117)
= d + ( – ) d + d
0
eR
F
r
r r eR
F
r
r r
F
r
r
eR
R
eR
R eR
∫ ∫ ∫ ∂
∂ ∂
∂
∂
∂
= d
0
R
r
F
r
r
∫
⋅
∂
∂
Structural dynamics of elastic blades 285
which is identical to eqn 7.111, except, of course, that in the evaluation of
∂
∂
F
r
the
incidence would be determined by the rigid blade flapping.
If now, we ignore the aerodynamic shear force at the hinge, i.e. we put F = 0, then
we have from eqn 7.116
δ β = ( – ) d
2 2
1
Ω
∫
R m x e x
e
and, if we ignore the moment at the root due to aerodynamic forces between the
hinge and the root, then we have from eqn. 7.114
M(0, ψ) = SeR + ksβ
giving
M e R M x e x k
e
s (0, ) = ( – )d +
2 3
1
ψ β β Ω
∫
(7.118)
If we ignore the aerodynamic shear force at the hinge and the moment of aerodynamic
forces about the hinge i.e. we put MA = 0 in addition, then the blade flapping motion
is unforced and the motion will take place at the natural frequency of rigid blade
flapping.
This implies
d
d
+ = 0
2
2 1
2
β
ψ
λ β
Substituting for
d
d
2
2
β
ψ
into eqn 7.112 gives
– ( – ) d + ( – )d + = 0
2 3
1
2
1
2 2 3
1
s Ω Ω
∫ ∫
R m x e x R Mx x e x k
e e
λ β β β
hence, λ1
2
3
1
s
2
3
1
2
=
( – )d + /
( – ) d
R Mx x e x k
R m x e x
e
e
∫
∫
Ω
giving λ1
2
3
1
s
2
3
1
2
– 1 =
( – )d + /
( – ) d
eR M x e x k
R M x e x
e
e
∫
∫
Ω
Substituting the numerator of this expression with eqn. 7.118 finally gives
Μ ψ λ β (0, ) = ( – 1) ( – ) d
2 3
1
2
1
2
Ω
∫
R m x e x
e
(7.119)
286 Bramwell’s Helicopter Dynamics
This expression is equavalent to eqn 7.106 with the flapping angle β in place of φ1,
with the integral term corresponding to the mode shape integral.
In fact, the integral R m x e x
e
3
1
2
( – ) d
∫
is the moment of inertia of the blade
about the flapping hinge. If we denote this by Tβ, we can write eqn. 7.119 as
M I (0, ) = ( – 1)
2
1
2
ψ λ β β
Ω (7.120)
Thus, Young’s model of the hingeless blade is seen, by comparison of eqns 7.106 and
7.119, to be equivalent to the model utilising the modal approach (and considering
only the first mode of bending), providing that the effect of the aerodynamic forces
are ignored.
It might appear that the offset hinge rigid blade model, as depend by eqn 7.119,
might be a very convenient way of handling the rigid blade problem. However, it is
easily seen that calculation will involve integrals whose lower limit is e and, as this
gives rise to a large number of terms in powers of e up to e
4
, the algebra becomes
rather unwieldy. In practice, an analysis based directly on the true mode shapes is no
less convenient, and will be followed in order to determine the pitching and rolling
moment coefficients in terms of rotor and aerodynamic parameters. It is the case that,
although eqn 7.107 is formally very simple and appears to require no more than two
modes for its accurate evaluation, it has some disadvantages compared with eqn
7.106. When eqn 7.110 is evaluated in terms of the aerodynamic parameters θ0, λ,
and the flapping and central coefficients, the expressions are found to be quite lengthing.
Defining the integrals
C x S x F xS x G S x 1
0
1
2
1 1
0
1 1
0
1
1 = d , =
1
d , = d
∫ ∫ ∫
we find for the pitching and rolling coefficients
C
a
C G b a G F M =
4
+
1
4
(1 – ) – (1 – ) –
1
2
1
2
1 0 1 –
{ } { }
µ µ
–
1
4
1 +
1
2
–
1
4
+
1
4
+ 4 / 1
2
1 A q i i µ κ λ ργ
′
ˆ ˆ (7.121)
and
C
a
B C G a l = –
4
–
1
4
1 + 3
2
– –
1
4
(1 – ) +
2
2
1 1
2
1
0
µ
µ
µθ
3
+
1
2
6 +
1
2
+
1
4
–
4
1
1
µ µλ
γ
ˆ
ˆ
p
q
(7.122)
where A1 and B1 are the cyclic control angles, λi is the mean induced velocity ratio,
κ the induced velocity gradient, θ1 the blade twist workout, and ˆ ˆ p q and one the nondimensional pitching and rolling velocities. Similar equations have been given by
Curtiss and Shupe
20
.
Structural dynamics of elastic blades 287
Equations 7.121 and 7.122 should be compared with the particularly simple results
indicated by eqns 7.109 and 7.110. Further, numerical examples show that the ‘coupling’
moments, e.g. the rolling moment arising from a longitudinal central displacement,
appear as the small difference between two nearly equal quantities, i.e. they are ‘illconditioned’ and the results could be husbandry
22
. If we denote by (Mc)a the hub
moment coefficient based on the aerodynamic loading, eqn 7.111, and if (Mc)f is the
coefficient based on the flapping displacement eqn 7.106, Simons
19
has shown that
when one mode is considered
( ) = ( ) + ( /2 ) ( – ) d c f
0
1
1
2
M M a b x FS x x a c
∫
α (7.123)
where α is the local blade incidence, F = / , 2 1 γ γ and the equivalent Lock’s number
for the hingeless blade, γ2 = ρ acR/E1, where E mS x 1
0
1
1
2
= d
∫
is the generalised
mass or inertia of the blade first flapping mode.
Thus, the hub moment of eqn 7.111 can be found in terms of eqn 7.106, together
with a ‘correction’ term represented by the integral of eqn 7.123. Then to a very good
approximation
22
it is found that
( ) = ( /2 )( – 1) + m 1 1
2
1 0 C a a j a a i γ λ µ
and
(C a b j B j j l i ) = ( /2 )( – 1) + –
1
2
– – a 1 1
2
1 2 1 3 0 4 γ λ µθ µθ µλ
(7.125)
where j a G F F D 1 1 1 1 = ( /2) 1 – –
1
2
–
1
2
(1 – )
j FC
j F F
j FG
2 1
3 1
4 1
=
1
4
–
=
2
3
– 2
=
1
2
–
and D S x 1
0
1
1
2
= d
∫
Equations 7.124 and 7.125 are only a little more complicated than eqns 7.109 and
7.110 but, for an analysis based on one mode, are just as accurate as eqns 7.121 and
7.122 and avoid the ‘ill-conditioning’ feature.
The stability and control derivatives for a hingeless rotor have been given by
Bramwell
21,22
. Generally speaking, it is found that the force derivatives are only
slightly defferent from those calculated for the hinged rotor and presented in Chapter
5. The flapping motion of the blade, in spite of the elastic moment at the root, is also
very slightly different from that of the rigid hinged blade under similar conditions, so
288 Bramwell’s Helicopter Dynamics
that the principal difference between the two rotors is the moment they exert on the
hub. For the centrally hinged rotor, the pitching moment coefficient for a unit of the
rotor has already been shown to be teh and, of course, the same thrust moment applies
to the hingeless rotor.
Let us use eqn. 7.108 to find the additional elastic moment coefficient for a typical
case.
For unit flapping we have
M a c 1 1
2
= ( /2 )( – 1) γ λ
Typical values for a hingeless rotor are a = 5.7, γ1 = 7.5, λ1
2
= 1.24, so that Cm =
0.091. For teh we have already used the value 0.0214 (Chapter 4), so that the total
coefficient for the hingeless rotor is 0.1124, giving a ratio of 5.26 to 1 when compared
to the value for the centrally hinged rotor.
Thus quite a good approximation to the derivatives of a hingeless rotor helicopter
is to take the force derivatives calculated in Chapter 5 and increase the flapping
moments in some ratio determined from a calculation similar to that above.
References
1. Hildebrand, F. B., Methods of applied mathematics, Englewood Cliffs NJ, Prentice-Hall,
1956.
2. Temple, G. and Bickley, W. G., Rayleigh’s principle, New York, Dover Publications, 1956.
3. Southwell, R. V. and Gough, Barbara S., ‘On the free transverse vibrations of airscrew blades’,
Rep. Memo. aeronaut. Res. Coun. 766, 1921.
4. Bishop, R. E. D. and Johnson, D. C., The mechanics of vibration, London, Camb. Univ. Press,
1960.
5. Duncan, W. J., ‘Galerkin’s method in mechanics and differential equations’, Rep. Memo.
aeronaut. Res. Coun. 1798, 1937.
6. Duncan, W. J., ‘Principles of the Galerkin method’, Rep. Memo. aeronaut. Res. Coun. 1948,
1938.
7. Holzer, H., Die Berechnung der Drehschwingungen, Berlin, Springer Verlag, 1921.
8. Isakson, G. and Eisley, J. G., ‘Natural frequencies in coupled bending and torsion of twisted
rotating and non-rotating blades’, NASA CR–65, July, 1964.
9. Williams, R. F., Unpublished material, City Univ. London.
10. Scanlan, Robert H. and Rosenbaum, Robert, Introduction to aircraft vibration and flutter,
New York, Macmillan, 1960.
11. Houbolt, John C. and Brooks, George W., ‘Differential equations of motion for combined
flapwise bending, chordwise bending and torsion of twisted non-uniform rotor blades’, NACA
Rep. 1346, 1958.
12. Sobey, A. J., ‘Dynamical analysis of the shaft-fixed blade’, R. Aircr. Establ. tech. Rep. 73175,
1974.
13. Ormiston, Robert A. and Hodges, Dewey, H. ‘Linear flap–lag dynamics of hingeless rotor
blades in hover’, J. Am. Helicopter Soc., April 1972.
14. Done, G. T. S. and Simpson, A., ‘Dynamic instability of certain conservative and non-conservative
systems’, J. Mech. Eng. Sci., 19(6), 1977.
15. Done, G. T. S., ‘Relative energy concepts in rotating system dynamics’, 2nd Int. Conf. on
Vibrations in Rotating Machinery, Cambridge, UK, 2–4 Sept. 1980.
Structural dynamics of elastic blades 289
16. Wilkinson, R. and Shilladay, J. D., ‘A comparison of the azimuthwise integration methods
used in the rotor performance computer programs’, Westland Helicopters Res. Memo. 55,
1969.
17. Hohenemser, K. H., ‘Hingeless rotorcraft flight dynamics’, AGAR Dograph 197, 1974.
18. Young, M. I., ‘A simplified theory of hingeless rotors with application to tandem helicopters’,
Proc. 18th Annual natn. Forum Am. Helicopter Soc., 1962.
19. Simons, I. A., ‘Some thoughts on a stability and control research programme with special
reference to hingeless rotor helicopter’, Westland Helicopters Res. Memo. 79, 1970.
20. Curtiss, H. C., jnr and Shupe, N. K., ‘A stability and control theory for hingeless rotors’, 27th
Annual natn. Forum Am. Helicopter Soc., Preprint 541, 1971.
21. Bramwell, A. R. S., ‘A method for calculating the stability and control derivatives of helicopters
with hingeless rotors’, Res. Memo. City Univ. Lond. Aero. 69/4, 1969.
22. Bramwell, A. R. S., ‘Further note on the calculation of the hub moments of a hingeless
helicopter rotor’, Res. Memo. City Univ. Lond. Aero. 71/2, 1971.
8
Rotor induced vibration
8.1 Introduction
We have seen in Chapter 3 that the aerodynamic loads on a helicopter rotor blade
vary considerably as it moves round the rotor disc, and in steady flight these loads are
periodic. The rotor forces and moments causing fuselage vibration are transmitted
from the blades to the rotor hub and then by means of the main rotor drive shaft into
the main rotor gearbox bearings and hence into the gearbox casing, and finally into
the fuselage at the gearbox attachment points.
Again, as we saw in Chapter 3, these loads arise from the aerodynamic forces on
the rotor blades, together with the inertia forces produced by the flapping and lagging
motions of the blade.
In addition to the main rotor loads, tail rotor force fluctuations may also be of
concern but in the vast majority of cases the main rotor forcing is the prime cause of
unwanted vibration.
The control of vibration is important for four main reasons:
(i) to improve crew efficiency, and hence safety of operation;
(ii) to improve the comfort of passengers;
(iii) to improve the reliability of avionic and mechanical equipment;
(iv) to improve the fatigue lives of airframe structural components.
Hence it is very important to control vibration throughout the design, development
and in-service stages of a helicopter project.
It may be appreciated from section 3.11 that the generation of oscillatory aerodynamic
loads at frequencies which are integral multiples of the rotational speed is fundamental
to the edgewise operation of the rotor in forward flight, and hence forced vibrations
of the helicopter cannot be entirely eliminated. Therefore the efforts of the design and
development organisations must be devoted to the minimisation of the vibratory
loads and to the minimisation of the fuselage response.
Subsequent sections of this chapter will indicate how this is achieved, although it
Rotor induced vibration 291
should be recognised that it is not yet possible to predict accurately the vibration
level of a helicopter in a specific flight condition. However, much valuable information
can be generated which can lead to an acceptable standard of vibration.
8.2 The exciting forces
The hub forces and moments from each blade can be resolved into force components
X, Y, Z, and moment components L, M, N relative to fixed axes in the helicopter,
Fig. 8.1. The axes are taken to conform with standard stability notation, i.e. with
the Z axis pointing downwards along the hub axis and the Y axis pointing to starboard.
The moment N can be disregarded since it forms part of the rotor torque.
In Chapter 1, section 1.11, we defined the rotating reaction forces along and
perpendicular to the hub axes as R1, R2, R3, Fig. 8.2.
If ψ is the azimuth angle of the zeroth (reference) blade, the azimuth angle of the
kth blade is ψk = ψ + 2πk/b. The corresponding components in the fixed X, Y, Z
directions are given by
Xk = – R1k cos ψk + R2k sin ψk
M
y
z
x
L
N
Fig. 8.1 Force components resolved along helicopter body axes
X
Y
Z
ψ
R1
R2
R3
Fig. 8.2 Reactions at blade hinge
292 Bramwell’s Helicopter Dynamics
Yk = R1k sin ψk + R2k cos ψk
Zk = – R3k
Now, in steady flight the forces R1k, R2k, R3k will be periodic and therefore expressible
in the form of a Fourier series as
R1k = P0 + P1 cos ψk + P2 cos 2ψk + P3 cos 3ψk + …
+ Q1 sin ψk + Q2 sin 2ψk + Q3 sin 3ψk + … (8.1)
R2k = S0 + S1 cos ψk + S2 cos 2ψk + S3 cos 3ψk + …
+ T1 sin ψk + T2 sin 2ψk + T3 sin 3ψk + … (8.2)
R3k = U0 + U1 cosψk + U2 cos 2ψk + U3 cos 3ψk + …
+ V1 sin ψk + V2 sin 2ψk + V3 sin 3ψk + … (8.3)
Considering the force components along the shaft first, the total force in the Z
direction is
Z R bU U n V n
k
b
k
n k
b
n k
n k
b
n k
= – = – – cos – sin
=0
–1
3 0
=1 =0
–1
=1 =0
–1
Σ Σ ΣΣΣ
∞ ∞
ψ ψ
From the results of Appendix A.3, the summations can be simplified to give
Z = – b[U0 + Ub cos bψ + U2b cos 2bψ + …
+ Vb sin bψ + V2b sin 2bψ + …] (8.4)
Thus, the only harmonics which remain are those which are multiples of the number
of blades. If, for example, the rotor has four blades, only harmonics of frequency 4Ω,
8Ω, …, etc. contribute to the total vertical force, apart from the steady load bU0. This
result assumes, of course, that all the blades are perfectly matched (track and balance).
If one or more blades are not matched, other frequencies Ω, 2Ω, …, etc. may arise.
Considering now the force components in the plane of the hub, the X force is of
the form
X P S
k
b
k
k
b
k
= – cos + sin
=0
–1
0
=0
–1
0 Σ Σ ψ ψ
– cos cos – sin cos
=1 =0
–1
=1 =0
–1
Σ Σ ΣΣ
∞ ∞
n k
b
n k k
n k
b
n k k
P n Q n ψ ψ ψ ψ
+ cos sin + sin sin
–1 – 0
–1
=1 =0
–1
Σ Σ Σ Σ
∞ ∝
n k
b
n k k
n k
b
n k k
S n T n ψ ψ ψ ψ (8.5)
If the blades are perfectly matched, P0 and S0 are the same for all blades and the
first two terms vanish. Using the well known formulae for the products of sines and
cosines, the remaining sums can be written as
Rotor induced vibration 293
X P T n P T n
n k
b
n n k
n k
b
n n k
= – ( + ) cos ( + 1) – ( – ) cos ( – 1)
1
2 =1 =0
–1
1
2 =1 =0
–1
Σ Σ Σ Σ
∞ ∞
ψ ψ
– ( – ) sin ( + 1) – ( + ) sin ( – 1)
1
2 =1 =0
–1
1
2 =1 =0
–1
Σ Σ Σ Σ
∞ ∞
n k
b
n n k
n k
b
n n k Q S n Q S n ψ ψ
(8.6)
Using the results of Appendix A.3, the first and third sums reduce to b cos mbψ and
b sin mbψ, respectively, when n + 1 = mb, i.e. when n = mb – 1, (m = 1, 2, 3, …), and
the second and fourth sums to b cos mbψ and b sin mbψ when n = mb + 1. Therefore,
X becomes
X b P T P T mb
m
mb – mb mb mb
= – [ + + – ]cos
1
2 = 1
1 – 1 + 1 + 1 Σ
∞
ψ
– [ – + + ]sin
1
2 = 1
1 – 1 + 1 – 1 b Q S Q S mb
m
mb – mb mb mb Σ
∞
ψ (8.7)
Similarly
Y b S Q S Q mb
m
mb – mb mb mb
= [ – + + ]cos
1
2 = 1
1 – 1 + 1 + 1 Σ
∞
ψ
+ [ + – + ]sin
1
2 = 1
1 – 1 + 1 + 1
b P T P T mb
m
mb – mb mb mb Σ
∞
ψ (8.8)
Thus, if a blade of a three-bladed rotor produces a force F = R2 cos 2ψ on the hub,
the corresponding X and Y force components are – R2 cos 2ψ cos ψ and R2 cos 2ψ
sin ψ respectively. By expressing the trigonometrical products as the sums of sines
and cosines, these components are –
1
2 R2 cos 3ψ and –
1
2 R2 cos ψ in the X direction
and
1
2 R2 sin 3ψ and –
1
2 R2 sin ψ in the Y direction. But since n + 1 = b, for this case,
the terms in 3ψ add, whereas the terms in ψ cancel. If, however, F = R3 cos 3ψ, the
X and Y components for the individual blade would consist of terms in 2ψ and 4ψ, but
for the complete three-bladed rotor these terms would vanish since neither (n + 1) nor
(n – 1) is equal to b.
The type of hub influences the vibratory loading systems applied to the fuselage.
The vibratory pitch and roll moments generated by an articulated rotor are significantly
smaller than those produced by a semi-rigid or hingeless rotor.
The choice of the number of blades is also important. This is primarily due to two
effects. First, the level of aerodynamic oscillatory loading tends to decrease as the
harmonic order increases, and since the bΩ forcing is the prime cause of fuselage
vibration, the larger the number of blades, the smaller will be the basic aerodynamic
vibratory input.
Figure 8.3 shows the predicted rotor head loadings for a typical medium weight
helicopter with a semi-rigid hub and four or five blades through the forward speed
range. It indicates a very significant reduction in hub vibratory moments when five
blades are used, which is primarily due to the fact that the 3Ω aerodynamic loading
is not transmitted to the fuselage with a five bladed rotor.
294 Bramwell’s Helicopter Dynamics
100 120 140 160
Forward speed, knots
600
400
200
100 120 140 160
Forward speed, knots
600
400
200
100 120 140 160
Forward speed, knots
600
400
200
4 Blades
5 Blades
80000
60000
40000
20000
80000
60000
40000
20000
100 120 140 160
Forward speed, knots
100 120 140 160
Forward speed, knots
Pitch moment
lbs ins
Roll moment
lbs ins
Longitudinal
shear, lbs
Vertical
shear, lbs
Lateral
shear, lbs
Fig. 8.3 Comparison of four and five blade hub loads
8.3 The dynamic design of rotor blades
The successful design of a helicopter rotor system is concerned with meeting the
exacting dynamic requirements which it is necessary to satisfy if acceptable behaviour
is to be achieved in the areas of handling qualities, fuselage vibration levels, acceptable
airframe and rotor blade fatigue lives, and rotor system aeroelastic stability. The
trend towards increasing mechanical simplification combined with increased
aerodynamic and structural efficiency has intensified the difficulties experienced in
achieving the optimum dynamic characteristics. However, the particular properties
of composite materials, often used in a hybrid glass and carbon form of construction,
have more readily enabled the design aims to be met.
The correct dynamic design of rotor blades is essential for two main reasons:
(i) the minimisation of the amplification of the rotor blade aerodynamic vibratory
loading which is transmitted to the fuselage; and
(ii) the minimisation of the total vibratory loading of the blade to provide an
acceptable fatigue life.
For many years, helicopters utilised rotor blades of essentially constant radial
Rotor induced vibration 295
distributions of mass and stiffness, and their dynamic characteristics, when associated
with a rotor hub having a low offset of flapping hinge, and a modest helicopter cruise
speed, led to a not very satisfactory but acceptable situation. However, the advent of
rotor systems with higher actual or effective offsets of flapping hinge, together with
helicopter cruise speeds in excess of 150 knots, has led to the necessity for refining
the structural design of the rotor blade to offset the increase in vibratory loading due
to the above effects.
Figure 8.4 indicates the unsatisfactory positioning of natural frequencies which
occurs if rotor blades of constant radial structural properties are associated with a
rotor hub having a significant effective offset of flapping hinge.
The proximity of the second and third flapping mode frequencies to the 3Ω and
5Ω aerodynamic forcing loads will lead to high vibratory fatigue loading of the
blades and, for the case of a four-bladed rotor, large 4Ω pitch and roll moments
transmitted to the airframe.
Figure 8.5 indicates the situation that could be achieved utilising the design flexibility
10Ω L3
2Ω
F4
F3
L2
9Ω
8Ω
7Ω
6Ω
5Ω
4Ω
3Ω
F2
F1 1Ω
R.P.M
L1
L3
L2
F2
F4
F3
F1
L1
Frequency
Rotor speed
Blade radius
Blade mass distribution Blade flapwise stiffness Blade chordwise stiffness
50 100 150 200 250 300
Fig. 8.4 Frequency spectrum for blade with constant radial properties
→
296 Bramwell’s Helicopter Dynamics
possible using a hybrid combination of composite materials to produce a significant
degree of mass and stiffness taper along the blade span. This improvement is
characterised by the separation of the second and third flapping mode frequencies
from the 3Ω and 5Ω aerodynamic forcing frequencies respectively. Calculations
indicate that the 4Ω moment applied to the fuselage is reduced by 47 per cent for the
improved design.
The ratio of material properties that strongly influences the dynamic characteristics
of a blade is the modulus of the material divided by the density, i.e. E /ρ and Es/ρ. As
the values of these are essentially constant for all normally used metals, the scope for
the dynamic tuning of metal blades is severely limited.
However, for fibre reinforced composite materials, the values of E/ρ can vary
from 50 per cent to 250 per cent of the typical metal values, and the value of Es/ρ can
vary from 20 per cent to 200 per cent of the metal values, dependent upon fibre type
and orientation.
Modern composite blades can use a mixture of glass and carbon fibre, and the
Blade weight distribution Blade flapwise stiffness
10Ω
9Ω
8Ω
L3
F4
7Ω
5Ω
6Ω
L2
4Ω
F3
3Ω
F2
2Ω
F1
1Ω
L1
R.P.M
Rotor speed
Frequency
L3
F4
F3
F2
F1
L1
L2
Blade radius
Blade chordwise stiffness
50 100 150 200 250 300
Fig. 8.5 Frequency spectrum for blade with tapered radial properties
→
Rotor induced vibration 297
required values of flapwise, lagwise and torsional stiffnesses can be achieved almost
independently.
Figure 8.6 indicates a possible method of achieving satisfactory dynamic
characteristics.
Flapwise stiffness can be adjusted without affecting lagwise or torsional stiffness
by introducing discrete layers of carbon fibre on the upper and lower surfaces of the
glass fibre D-spar. The lagwise stiffness can be adjusted by introducing carbon fibre
in a spanwise sense at the extreme trailing edge of the blade without affecting flapwise
or torsional stiffness. Control of torsional stiffness is best achieved by selecting the
appropriate fibre type and orientation for the trailing edge skins.
The desirable characteristics of spanwise taper of mass and stiffness can be readily
obtained by the selective introduction of unidirectional carbon fibre in the spanwise
sense on the upper and lower walls of the main spar section.
8.4 Main rotor gearbox mounting systems
In order to minimise the vibratory forces fed into the fuselage, various methods of
mounting the main rotor gearbox have been used.
The following descriptions form a representative selection of the main passive
systems which have been used with varying degrees of success.
(i) Simple soft mounting of the rotor/gearbox/engine system (Fig. 8.7). Soft
suspension systems derive from the principle illustrated by the simple single
degree of freedom sprung mass undergoing forced vibration that may be found
in student textbooks on vibrations, e.g. Thomson
1
. The force transmitted through
the spring to the support is seen to be reduced the lower the ratio of natural
frequency to excitation frequency becomes.
Since the quasi-static deflections of the suspended system would be
unacceptably large if conventional soft mountings are placed at the gearbox
Blade section
Trailing edge skins
Glass cloth, uni. glass (±45°)
or uni carbon (±45°) to
control torsional stiffness
Uni. glass or carbon
to control lag stiffness
Discrete layers
of carbon to
control flap
stiffness
‘D’-SPAR
Uni. glass to
give basic strength
to blade
Metal erosion shield
Fig. 8.6 Composite blade construction
Chord line
1/4 chord
298 Bramwell’s Helicopter Dynamics
feet, it becomes necessary to isolate as large a mass as possible. Hence it is
attractive to mount the gearbox and engines on a raft, and then attach the raft
to the fuselage using the chosen mountings. By this means, an adequately low
natural frequency of the isolated system can be achieved without unacceptably
large displacements of the rotor, gearbox, engines and controls under trim and
manoeuvre loads. However, attenuation of the vibratory forces by this means
is only modest and certain loadings may be transmitted without any reduction.
(ii) A rotor/gearbox/engine mounting system designed to respond to the bΩ forcing
frequency in such a manner that the spring and inertia forces generated by the
response cancel, as far as possible, the effects of the rotor generated forcing
(Fig. 8.8).
For example, a possible design criterion for such a system could be the
minimisation of the total bΩ frequency pitching moment at the helicopter
centre of gravity. The attenuation efficiency of such a system would only be
optimal for a specific flight condition having a particular relationship between
the rotor hub forces and moments.
(iii) Systems employing the Dynamic Anti-Resonant Vibration Isolator (DAVI)
1
.
Originally developed for the isolation of crew seats by the Kaman company,
this principle has been applied very successfully to the mounting of the gearbox
2
.
Fig. 8.8 Flexibly mounted gearbox and engine system
Fig. 8.7 Westland W-30 raft mounting system
Elastomeric mounts
M
F
C.G.
Engine
Main Gearbox
Rotor induced vibration 299
Figure 8.9 illustrates its use and response characteristics. An appreciation of
how the vibration reduction is achieved may be obtained by study of the
simplified but equivalent model shown in Fig. 8.10. In this, the upper mass M1
represents the gearbox and rotor, and M2 the helicopter fuselage, and between
them is mounted a rigid arm, as shown, which carries a small mass mbob (the
‘bobweight’) at its tip.
The equations of motion when there is no damping are easily obtained as
– [ + (1+ ) ] + (1+ ) –
(1 + ) – – [ + (1+ ) ]+
2
1 bob b
2 2
bob b b
2
bob b
2
2 bob b
2
1
2
ω Λ ωΛΛ
ω Λ Λ ω Λ
M m K m K
m K M m K
x
x
b
=
0
0
i
F e
t ω
DAVI unit
Blade passing frequency
Fuselage
response
Fig. 8.9 The DAVI gearbox mounting system
F0e
iωt
X1
d1 d2
X2
Bobweight
m
M1
M2
Spring
K
Fig. 8.10 Simplified DAVI model
300 Bramwell’s Helicopter Dynamics
where F0 is the amplitude of the exciting force and Λb = d2/d1 is the ratio of
lengths of the rigid arm either side of the mounting point on M1. The system
is seen to be ‘free-free’ so the net linear momentum is zero at all instants of
time. This leads to a relationship between x1 and x2 which is used to eliminate
x1 from one of the equations, allowing the following expression for x2K/F0
(the normalised fuselage response) to be formed.
x K
F
2
0
2
b b b
2
b b b
2 2
b b
2 2
m b b
2
=
[ (1 + ) – 1]
[ (1+ ) –1] –[1– (1+ )][1– ( + (1+ ) )]
˜
˜ ˜ ˜
ω µ Λ Λ
ω µ Λ Λ ω µ Λ ω ρ µ Λ
The main parameters of the system are the fuselage to gearbox and rotor mass
ratio ρm = M2/M1, the bobweight mass ratio µb = mbob/M1, the bobweight arm
length ratio Λb, and the normalised excitation frequency ˜
ω = ω(K/M1)
–1/2
.
The denominator of the response expression is the characteristic equation for
the system, the roots of which provide the natural frequencies (one of which
is zero, since the system is ‘free-free’), and the numerator similarly provides
the zero response frequency. The latter shows that the bobweight mass is
reduced as Λb is increased. However, there is a practical upper limit, since the
greater the ratio is, the stiffer, and hence the heavier, the arm becomes. Figure
8.11 shows the variation of the response amplitude with excitation frequency
for the simple undamped DAVI model for a particular set of chosen parameters.
The DAVI unit can provide a very high attenuation over a narrow frequency
range, whilst utilising a value of spring stiffness which overcomes the problem
of excessive quasi-static deflections. The system is also relatively insensitive
to variations in the isolated mass. The balance between the degree of attenuation
0.6
0.4
0.2
0.5 1.0 1.5 2.0
X10
Normalised fuselage response | x2K/F0 |
˜ ω
Normalised excitation frequency
Fig. 8.11 Response amplitude of simplified DAVI model as a function of excitation frequency
(Λb = ρm = 10, µb = 0.01). Note: amplitudes on right-hand section of curve are multiplied by ten.
Rotor induced vibration 301
Fig. 8.12 Principle of the nodal beam mounting system
and the ability to cater for variations in rotor speed can be controlled by the
amount of damping built into the DAVI units.
(iv) The Bell ‘Nodamatic’ gearbox mounting system
3
This method (Fig. 8.12) interposes a beam mounting arrangement between the
gearbox and the fuselage and is configured such that the fuselage is effectively
suspended from the node points of the beam system when it is vibrating in
response to the rotor hub forcing loads and moments. In fact, the system is
mathematically equivalent to the DAVI principle, with the beam system providing
the stiffness and inertia properties of the DAVI units.
8.5 Dynamic response of the fuselage
Due to the fundamental design requirements of the helicopter primarily in terms of
the requirements of crew and equipment, and aerodynamic drag, the basic shape of
the fuselage will be determined by considerations other than its vibrational
characteristics. In the early stages of design, the fuselage can be represented by an
assemblage of beams of defined bending and torsional stiffness properties (Fig. 8.13)
together with the appropriate mass distribution.
Calculations can then be performed which will:
(i) indicate the rough proximity of any major beam bending mode frequency to
the bΩ rotor forcing frequency; and
(ii) indicate the sensitivity of the forced response to changes in the stiffness of
structural components which may be amenable to significant alteration, e.g.
DETAIL FROM A
TYPICAL DESIGN
Elastomeric Sandwich
(Rubber/Steel Sandwich)
Gearbox Mounting
Strut
Vibratory
Loads
Beam Nodes
Location on Helicopter
Airfame
Spherical Bearing
Elastomeric
Bearing
302 Bramwell’s Helicopter Dynamics
Excitation
8
7
13 12
11
6
5 4 9
19 20
14
16
17
2
1
Pilot’s
position
P
18 3
10
15
Fig. 8.13 Fuselage beam (stick) model
Fig. 8.14 Army Lynx with struts on tailcone
main rotor head stiffness relative to the fuselage, and tail boom stiffness.
Thus, if ground vibration or flight tests indicate evidence of a major forced
response problem, it may be possible to significantly reduce the response by
making an acceptable change to the structural stiffness in a particular area
(Figs 8.14 and 8.15).
Rotor induced vibration 303
Fig. 8.15 Lynx cockpit 4R vibration as a function of strut stiffness
Fig. 8.16 Structural manipulation
An extension of this approach referred to as ‘structural manipulation’ has been developed
whereby an order of ranking of areas of structural sensitivity is determined (Fig.
8.16). The method
4
then optimises the values of stiffness changes required in a
chosen number of structural elements to minimise a defined ‘index of performance’
which could be in the form of the vibrational response in a designated area of the
fuselage due to specified forcing components applied at a defined location.
AREA OF EACH STIFFENING STRUT (IN
2
)
0.1 0.2 0.3 0.4 0.5 0.6
0.01
0.0075
0.005
0.0025
0
VIBRATION
LEVEL (IN)
BASIS:
The locus of a single
point response to
a single point
excitation when a
single structural
parameter is
varied is a circle.
This property
forms the basis of
optimisation
routine.
Locus of Response
Vector as K Varies
Sensitivity Using
‘Circle’ Criteria
Response
Lynx Stick Model
K
Excitation
Imag.
K
Real
Fuselage Location
Gearbox Area
Engines
Sides,
Roof
Floor
Tail Boom
304 Bramwell’s Helicopter Dynamics
When the structural design of the fuselage is at a more advanced stage, a finite
element method of dynamic analysis such as NASTRAN
5
will be used to predict
more accurately the dynamic response characteristics of the airframe. Figure 8.17
indicates a typical finite element model.
Due to difficulties in assessing the stiffness properties of structural joints, the
results of such calculations will be used to predict trends in the dynamic behaviour
of the fuselage as a result of possible changes in stiffness and mass distribution.
Following manufacture and vibration test of a prototype airframe, the finite element
model would be empirically adjusted to match the test results. The use of the adjusted
model will increase confidence in subsequent analytical predictions, from which
beneficial design changes may result.
8.6 Vibration absorbers
Having made all the beneficial stiffness and mass distribution changes which are
possible without major redesign, it may still be the case that the levels of vibration
experienced in flight at crew stations or passenger locations are unacceptable.
In this case, it will be necessary to have recourse to additional methods of vibration
reduction which can be incorporated with minimum disturbance to the existing design.
The absorbers described in this section are all based on the principle of the simple
sprung mass vibration absorber described and analysed in vibration text-books
1
. This
shows that by attaching an extra small mass by means of a spring or elastic mount to
a body undergoing forced vibration, the amplitude of vibration can be reduced to zero
(if the system is undamped, as in the ideal case), or nearly zero for small damping,
if the natural frequency of the sprung mass by itself is made to coincide with that of
the forcing frequency.
The available absorbers may be conveniently divided into two categories:
(i) those designed to reduce overall levels of vibration throughout the fuselage;
(ii) those designed to produce a reduction in vibration in a local area of the
fuselage.
Fig. 8.17 Fuselage finite element model
Rotor induced vibration 305
Passive methods which fall into the first category are:
(a) The rotor head mounted vibration absorber, of which two distinct types have
been successfully employed:
(i) The centrifugal pendulum (bifilar) absorber
6
as developed by Sikorsky
(Fig. 8.18.)
In the bifilar type, the required spring stiffness is provided by centrifugal force
and hence the natural frequency of the device varies with rotor speed, as does
the forcing frequency. Thus the bifilar absorber exhibits a degree of selftuning with respect to changes in rotor speed.
Since the natural frequency also depends on the square root of the ratio of
mounting radius to pendulum length, the small value of the latter which the
bifilar geometry confers allows a practical design which can be mounted close
in to the centre of the hub.
Figure 8.19 illustrates the basic geometry of the bifilar absorber, and also
the model from which the following equation of motion can be derived:
b ..
γ + (a + c) Ω
2
γ = 0
Fig. 8.18 Bifilar absorber fitted to Lynx rotor head
306 Bramwell’s Helicopter Dynamics
Oscillating mass, m
c b a
Ω
d
D
X
ψ
m
c
a
Fig. 8.19 Geometry and model of the bifilar
where γ is the angular motion of the pendulum arm
a is the offset of the fixed pendulum point from the rotor centre of
rotation
b is the effective length of the pendulum
c is the offset of the centre of gravity of the oscillating mass from its
pivot point
d is the diameter of the pin connecting the fixed arm and the oscillating
mass
D is the diameter of the holes in the fixed arm and the oscillating mass
Thus b = D – d
It should be noted that in the model, the arm of length c always remains
parallel to the arm of length a.
For the case of small amplitudes of oscillation of the moving mass, we may
write ..
γ ω γ =
2
n , where ωn is the natural frequency of oscillation of the moving
mass, giving
w a c
D d
n
Ω
2
=
+
–
Assuming some typical values of absorber geometry
a = 75 cm c = 6 cm D = 10 cm
then, for an absorber tuned to 3Ω forcing frequency, the pin diameter, d, is
equal to 1 cm. The equivalent pendulum length, (D – d), is equal to 9 cm, thus
confirming its very small value.
However, due to the large oscillatory amplitude necessary to provide adequate
force from a device of acceptable mass, the absorber exhibits a tuning nonlinearity with respect to amplitude and hence flight condition. Other features
of this type of absorber include the fact that a single bifilar can only reduce the
γ
Y
Ω
b
CL of
rotation
Rotor induced vibration 307
Fig. 8.20 Flexispring absorber
magnitude of either the (b – 1)Ω or the (b + 1)Ω components in the rotating
system, the relatively low ratio of mass producing the cancelling force to the
total installed mass, and the significant maintenance requirements necessary
to maintain minimum damping of the moving mass.
Applications include the S-61 series, the Black Hawk and the S-76 Sikorsky
helicopters. The S-76 uses two bifilars tuned to 3Ω and 5Ω forcing frequencies.
(ii) The fixed frequency ‘flexispring’ absorber
7
as developed by Westland (Fig. 8.20).
This type of absorber utilises glass fibre reinforced plastic for the spring
material, and operates efficiently over the small variation of rotor speed about
the governed datum typical of the modern rotor speed control system. The rate
of force-cancelling mass to total installed mass is high, it responds to the total
bΩ frequency forcing, and requires no maintenance. Applications include the
Westland Lynx and W.30 helicopters.
Figure 8.21 indicates the effect of the Flexispring absorber on the cockpit
vibration levels of the Westland 30 as a function of forward speed.
It should be noted that, although both the above types of absorber can only
generate oscillatory forces in the plane of the rotor system, they will respond
beneficially to any forced response mode which involves in-plane motion of
the rotor head, resulting from the total applied rotor forcing system.
(b) A centrifugal pendulum type of absorber mounted on the rotor blade
8
. This
type of absorber has been used on the Bolkow Bo 105 and Hughes 500
helicopters. Figure 8.22 shows the Hughes installation which consists of
Lower
cover
Tuning
weights
Outer
ring
Rotor hub
Springs
Top cover
Spindle
308 Bramwell’s Helicopter Dynamics
Without absorber
With absorber
2
1
40 80 120
2
1
40 80 120
Forward speed, knots Forward speed, knots
Velocity, ins/sec
Velocity, ins/sec
Co-pilot’s seat, floor
vertical
Pilot’s seat, floor
vertical
Fig. 8.21 Effect of the vibration absorber on Westland 30 4R vibration
Fig. 8.22 Hughes blade mounted pendulum absorber
Rotor induced vibration 309
absorbers tuned to the 3Ω and 5Ω excitation frequencies for the four-bladed
rotor version, where their purpose is to reduce the response of the second and
third flapwise bending modes of the blade to the 3Ω and 5Ω frequency oscillatory
air loads in the rotating system.
Passive methods which fall into the second category are:
(a) The fuselage mounted classical mass-spring absorber.
This involves the mounting of a suitably heavy mass, usually situated in the
local region of the operating crew and passengers, tuned to the bΩ forcing
frequency. Use may be made of an existing heavy item such as the aircraft
battery, as in the Sea-King helicopter (Fig. 8.23 shows the installation), or the
mass may be parasitic, as in certain models of the Boeing Vertol Chinook
helicopter. In the former case, the natural frequency of the absorber is constant,
and to be effective the inherent damping of the moving mass must be minimal
so that the amplification at resonance is large. However, under such circumstances
the efficiency of the absorber will decline with excursions of rotor speed away
from the nominal governed value. This particular disadvantage is overcome in
the case of the Chinook helicopter by the use of an electrically actuated
system which changes the effective value of the sprung mass in accordance
with changes in rotor speed.
Fig. 8.23 Sea King battery vibration absorber
310 Bramwell’s Helicopter Dynamics
8.7 Active control of vibration
Recent developments in the reduction of helicopter vibration by the use of active
control systems have produced encouraging results. These developments have taken
place along two quite different approaches.
8.7.1 The application of higher harmonic pitch control to the rotor
blades (HHC)
9
Since airframe vibration originates with the rotor blade oscillatory air loads, a potentially
attractive method of vibration control is the application of blade pitch at frequencies
greater than the rotor rotational frequency, in order to produce a forcing system
generating an airframe response which would oppose that arising from the basic rotor
forcing.
Figure 8.24 shows diagrammatically the concept of HHC. The rotor generates
oscillatory forces which cause the fuselage to vibrate. Transducers mounted at key
locations in the fuselage measure the vibration, and this data is analysed by an
onboard computer. Based upon this data, the computer generates, using optimal
control techniques, signals which are transmitted to a set of actuators which, typically,
vary the blade pitch at frequencies of (b – 1)Ω, bΩ, and (b + 1)Ω. Oscillatory aerodynamic
loads are thus produced which modify the total bΩ frequency response of the fuselage,
and the whole cycle of measurement of the modified vibration, data analysis and
resultant actuator response is repeated.
Using the appropriate control strategy, the process can be made to converge to
minimise the fuselage vibration. Key elements in the successful implementation of
HHC are hydraulic actuators with an adequate high-frequency response characteristic
and a successful control algorithm.
Flight condition
Main rotor
Helicopter
structure
Airframe
vibration
Sensors
Measured
vibration
Adaptive
control
unit
Blade
control
actuators
Controlled
blade pitch
signals
Fig. 8.24 Concept of HHC
Rotor induced vibration 311
The oscillatory blade pitch motions can be applied either through a conventional
swashplate or spider control system, or by means of individual blade actuation in the
rotating system (Fig. 8.25). In the latter case, the input can be applied near the root
of the blade, or by the use of an outboard aerodynamic servo tab.
In defining the control algorithm, self-adaptive techniques are used in order to
cater for change in fuselage dynamics (due to loading changes, for example) and
rotor behaviour with flight condition.
The control algorithm depends on the assumption of a linear relationship between
measured vibration at bΩ frequency and the higher harmonic rotor forcing of the
form
Y = TX + B
where Y is a vector consisting of the bΩ sine and cosine Fourier components of the
measured vibration at a number of fuselage locations. The HHC input X is a vector
consisting of Fourier sine and cosine components of blade pitch at (b – 1)Ω, bΩ and
(b + 1)Ω frequencies. T denotes the rotor/fuselage transfer matrix, and B is the
background uncontrolled vibration. Since these parameters vary with flight condition,
a statistical estimator (Kalman filter) is used to track them during flight.
The final part of the control system is an optimal controller which uses the estimated
T and B parameters to minimise the index of performance J, where
J W Y A X
i
m
i i
j
n
j j
= +
= 1
2
= 1
2
Σ Σ
where Wi are the relative weightings of the m vibration measurements Yi and Aj are
the relative weightings of the n HHC blade pitch inputs Xj. This index allows for the
weighting of the various vibration measuring positions according to helicopter role,
Fig. 8.25 HHC blade pitch actuation
312 Bramwell’s Helicopter Dynamics
and the ability to limit the authority of the HHC inputs when constrained by the
proximity of rotor aerodynamic limits.
The estimates of the T and B parameters are continuously updated, which allows
the system to adapt to changes in rotor aerodynamic and fuselage dynamic states.
Figure 8.26 indicates a typical blade pitch waveform with and without HHC. A
feature of some concern is the increase in blade pitch at the 270° retreating blade
azimuth position. This could, near the flight envelope boundary, have the effect of
introducing a premature onset of blade stall.
It is possible that additional blade area may have to be provided to prevent this.
8.7.2 The active control of structural response (ACSR)
10
Subsequent to the development of HHC, an alternative approach to the active control
of helicopter vibration has evolved. This consists of connecting a number of actuators
between convenient points on the airframe to apply oscillatory forces to the structure.
The magnitude and phase of the loads generated by the actuators are determined by
a control algorithm which minimises the vibrational response of the fuselage at a
number of key positions. Figure 8.27 shows the basic concept of ACSR.
The basis of ACSR is that, if a force F is applied to a structure at a point P and an
equal and opposite force (the reaction) is applied at a point Q, then the effect will be
to excite all the modes of vibration of the structure which possess relative motion
between points P and Q. This requirement for relative motion in the modal response
between the points where the actuator forces are applied is an essential feature of
ACSR.
Experience to date has indicated that positioning actuators across the main rotor
Fig. 8.26 HHC blade pitch waveform
With HHC
90 180 270 360
Rotor azimuth degs
Normal trim-no HHC
8
4
0
–4
– 8
Blade
cyclic
pitch
degs
Rotor induced vibration 313
Vibration
source
Structure
dynamics
Adaptive
control unit
Actuators Sensors
Cancellation forces
Actuator demand
Vibratory forcing
Measured vibration
Actuator control
manifold
Actuator
Compliant element
Integral ACSR
actuator/struts
Airframe top structure
Torque struts
Main rotor gearbox
Fig. 8.28 ACSR gear box mounting struts with integral actuators
gearbox and fuselage interface is very effective, and Fig. 8.28 shows a method of
gearbox attachment which combines the mounting struts with the actuators.
The basic equation relating the various parameters is identical in form to that for
HHC,
i.e. Y = TX + B
where Y is the measured fuselage vibration
X is the vector of actuator forces
Airframe vibration
Fig. 8.27 Concept of ACSR
314 Bramwell’s Helicopter Dynamics
Fig. 8.29 ACSR frequency domain control strategy
T is the transfer matrix relating the actuator forces to the fuselage vibration
B is the background, uncontrolled, vibration
In general, with N control forces, the response at N locations in the fuselage can
be reduced to zero, provided that the T matrix is non-singular. However, it is
considered preferable to attempt to reduce the vibration at a larger number of
locations to acceptably low levels rather than to attempt to achieve zero vibration
at a few positions.
Two categories of control algorithm are applicable to the implementation of
ACSR. These may be classified as either frequency or time domain in nature.
Emphasis has been placed on the frequency domain approach in the early applications
of this technique.
The general arrangement for the frequency domain control strategy is shown in
Fig. 8.29. This indicates that the primary functions of the controller are signal processing,
parameter estimation and control.
As a very broad statement, it appears that active control techniques can produce
vibration levels which are in the region of about one-half of the levels achieved by
passive methods, and this has indicated that the long-term goal of the ‘jet smooth
ride’ helicopter may at last be a possibility.
Figure 8.30 shows a comparison of the vibration levels of the Westland W30
helicopter without a vibration reduction system, and when fitted with a Flexispring
rotor head absorber, and an ACSR system.
8.8 Vibration at frequencies other than bΩ
The minimisation of forcing frequency components from the rotor system which are
Helicopter flight conditions Main
rotor
Rotor head forces
Airframe vibration
Fuselage
dynamics
Controlling forces
Actuators Accelerometers
Measured
vibration
Actuator
demand
Actuator
loop
closure
Frequency/
time
Optimal
controller
Dynamics
estimator
Signal
processor
time/
frequency
Feedback
force
Adaptive controller
Rotor induced vibration 315
normally self-cancelling within the rotor is very much related to the ability to
manufacture identical blades and lag dampers. The major residual is usually at 1Ω
frequency, and this is minimised by blade balancing and tracking procedures on a
whirl tower and on the helicopter during ground running, and if necessary also in
hovering and high-speed forward flight.
Reasonably identical lag plane damper performance characteristics are required so
that undesirable vibration under conditions of 1Ω blade flapping (which results in
large lag plane oscillations) can be avoided.
A quite distinct type of vibration problem which is of aerodynamic origin is due
to the effects of vortices shed from the region near the main rotor head which can,
under certain flight conditions, strike the tail rotor and the fixed vertical and horizontal
tail surfaces. This phenomenon has been observed mainly as a response at the fuselage
fundamental lateral bending frequency and is particularly dependent on the sideslip
angle in flight. This problem may be intensified by the addition of excrescences such
as a radome to the upper surface of the fuselage aft of the rotor.
This problem is often referred to as the ‘lateral shakes’ or the ‘shuffle’. Solutions
to this problem have been found by the addition of suitable fairings to the main rotor
head and to the airframe aft of the head. Figures 8.31 and 8.32 indicate the origin of
Head absorber
Baseline
ACSR
Forward speed (knots)
0.5
0.4
0.3
0.2
0.1
0
Average cabin/cockpit vibration (g)
Fig. 8.30 Comparison of Westland W30 vibration levels
40 60 80 100
Turbulent wake from pylon/rotor head
area affects fin/tail rotor
Fig. 8.31 Origin of the ‘shuffle’ problem
316 Bramwell’s Helicopter Dynamics
Flow curvature caused by beanie
Fig. 8.32 Effect of ‘beanie’ on the turbulent wake
Fig. 8.33 Spectra of vibration amplitudes
0.5
0.4
0.3
0.2
0.1
0
mm
1 2 3 4 5 6
Lateral
0.3
0.2
0.1
0
Longitudinal
Vertical
Harmonic of rotor speed
1 2 3 4 5 6
1 2 3 4 5 6
0.6
0.5
0.4
0.3
0.2
0.1
0
Amplitude
the problem, and the beneficial effect of the rotor head fairing (often referred to as the
‘beanie’).
There can also be vibration problems due to mechanical excitation arising from
the transmission system. The effects of these sources of vibration can be held to
acceptable limits by the correct positioning of shaft whirling speeds, an adequate
standard of static and dynamic balancing of main and tail rotor hubs and transmission
shafts, and the accurate machining of gear teeth profiles.
8.9 Measurement of vibration in flight
As an example of the vibration amplitudes and forces occurring on a helicopter in
Rotor induced vibration 317
*Now the Defence Evaluation and Research Agency (DERA).
0.4
0.3
0.2
0.1
0
0.2
0.1
0
0.2
0.1
0
10 20 30 40 50
10 20 30 40 50
10 20 30 40 50
m/s
m/s
Airspeed
m/s
lateral
Amplitude
Longitudinal
Vertical
Fig. 8.34 Vibration amplitude as a function of airspeed
Each type of symbol refers to a separate flight.
flight, we discuss below some results from measurements made at the Royal Aircraft
Establishment.* The measurements of the amplitudes and forces in three perpendicular
directions were made near the hub of a three-bladed single rotor helicopter over the
full range of flight speeds.
The results showed that the amplitude of the first-harmonic vibrations were quite
large, as can be seen from the harmonic spectrum, Fig. 8.33. However, the first
harmonic measurements showed a large amount of scatter, particularly from flight to
flight, and this indicates that the vibration was probably due to a variable amount of
rotor blade imbalance. The next largest component of vibration was that of the third
harmonic, but the amount of scatter in this case was very small, Fig. 8.34, indicating
excitation from aerodynamic and inertia forces of the complete three-bladed rotor, as
discussed in section 8.2.
The results shown in Fig. 8.34 show two features in common with those of other
helicopter vibration measurements. The first is that the level of vibration generally
increases with speed. This is to be attributed, of course, to the increasing asymmetry
of the rotor thrust loading and the corresponding increase of the harmonic content in
mm
318 Bramwell’s Helicopter Dynamics
the blade flapping. The other feature is the pronounced ‘hump’ at about 13 m/s (µ ≈
0.07). It is in this region that the influence of the trailing vortices from the blades is
greatest. As was discussed in Chapter 6, at low speeds the upwash at the front of the
rotor is quite large in relation to the forward speed, and this has the effect of keeping
the trailing vortices close to the rotor. Consequently, there will be large velocity
gradients near the front part of the rotor, giving rise to large higher harmonics in the
flapping motion. It is of interest to note that large lateral blade flapping also occurs
in this range of flight speeds, and this was found to be due to the asymmetry of the
induced-velocity distribution with respect to the lateral axis.
References
1. Thomson, W. T., Theory of vibrations with applications, New Jersey, Prentice-Hall Inc., 1981.
2. Flannelly, W. G., ‘The Dynamic Anti-Resonant Vibration Isolator’, 22nd Annual Forum of the
American Helicopter Society, Washington, May 1976.
3. Gaffey, T. M. and Balke, R. W., ‘Isolation of Rotor Induced Vibration with the Bell Focal
Pylon Nodal Beam System’, Paper 760892, SAENAEM Meeting, November 1976.
4. Done, G. T. S. and Hughes, A. D., ‘Reducing vibration by structural modification’, Vertica, 1,
31–38, 1976.
5. McCormick, C. W., ‘NASTRAN users’ manual’, NASA SP–22, 1969.
6. Paul, W. F. ‘The Main Rotor Bifilar Pendulum Vibration Absorber’, Vertiflite, February 1970.
7. White, R. W. ‘A Fixed Frequency Rotor Head Vibration Absorber Based upon GFRP Springs’,
Fifth European Rotorcraft Forum, Amsterdam, 1979.
8. Amer, K. B. and Neff, J. R., ‘Vertical-Plane Pendulum Absorbers for Minimising Helicopter
Vibratory Loads’, AHS/NASA Specialists Meeting on Rotorcraft Dynamics, Moffett Field,
Calif., February 1974.
9. Wood, E. R., Powers, R. W., Cline, J. H. and Hammond, C. E., ‘On Developing and Flight
Testing a Higher Harmonic Control System’, 39th Annual Forum of the American Helicopter
Society, St Louis, Mo., May 1983.
10. Staple, A. E., ‘An Evaluation of Active Control of Structural Response as a Means of Reducing
Helicopter Vibration’, Fifteenth European Rotorcraft Forum, Amsterdam, September 1989.
9
Aeroelastic and aeromechanical
behaviour
9.1 Introduction
In previous chapters, the effect on the motion of the blade of coupling between its
degrees of freedom has been ignored. For performance and stability and control
aspects, this neglect is, in general, entirely justified, but possible blade and rotor
system instabilities due to coupling must be considered.
There are also important instabilities involving coupling of rotor blade and airframe
motion, and these must also be taken into account.
Rotor blade instabilities to be considered are:
(i) main rotor pitch–lag instability;
(ii) main rotor pitch–flap flutter;
(iii) main rotor stall flutter;
(iv) main rotor blade weaving;
(v) tail rotor pitch–flap (‘umbrella mode’) instability;
(vi) main and tail rotor flap–lag instability;
(vii) tail rotor pitch–flap–lag instability.
A review of the aeroelastic problems of helicopter and V/STOL aircraft has been
given by Loewy
1
.
The coupled rotor blade and airframe instabilities to be considered are:
(i) ground resonance;
(ii) air resonance.
320 Bramwell’s Helicopter Dynamics
9.2 Main rotor pitch–lag instability
Although referred to as pitch–lag instability, the degrees of freedom participating in
the motion are flap and lag, with the instability arising from the presence of a
kinematic coupling between pitch and lag, or a torsional moment which twists the
blade when it is deflected in the flapping and lagging senses.
The effect of these couplings is to induce aerodynamic lift loads in response to
blade lag deflections which cause the blade to flap and the resulting Coriolis loads
induce more lag motion.
One aspect which can cause a degradation in pitch–lag stability in certain flight
conditions is the change in kinematic pitch–lag coupling as a function of steady
coning angle, steady lag deflection and impressed blade pitch due to changes in the
orientation of the pitch control system track rods. This problem will be exacerbated
by the use of short track rods.
With semi-rigid and bearingless rotors the steady flap and lag deflections tend to
be much smaller than those for articulated rotors, hence this type of instability is less
likely to occur for these types of rotor system.
Pitch–lag instability is generally experienced as a limit cycle oscillation of the
rotor blades phased in a manner which transmits a stirring motion to the airframe,
due to the oscillatory shear forces generated by the blades in the plane of rotation.
On some helicopters, this type of instability occurs only under conditions of
large amplitude forced oscillation of the blades in the lag plane (due to the Coriolis
forces generated by large amplitude cyclic flapping), so that the effectiveness of
any lag plane damper which may be present is reduced. The frequency of the
oscillation of the blades relative to the rotating hub is that of the fundamental lag
plane mode.
The first cause of coupling referred to in the first paragraph of this section may be
regarded as the α2 effect (see section 1.2), and the relationship between the blade
pitch and lagging angle is, for small angles,
∆θ = – α2ξ
The second cause of coupling can be understood with reference to Fig. 9.1. Forces
dFy and dFz are capable of exerting torques about the blade span axes when the blade
Z
Y dFz
dFy
Fig. 9.1 Blade bending deformation
Aeroelastic and aeromechanical behaviour 321
is deformed in bending. To calculate these torques, consider the moments exerted by
these force components, with reference to the projections of the deformed blade onto
the XY and XZ axes, Figs 9.2 and 9.3. The torque exerted by the sum of the elementary
forces dFy and dFz about a point P at a distance r from the axis of rotation is
L r Z Z r r
Z
r
F
r
r
r
r
y
( ) = – ( – ) – ( – )
d
d
d
d
d
1
1 1
1
1
∫ [ ]
+ ( – ) – ( – )
d
d
d
d
d
1
1 1
1
1
r
r
z
Y Y r r
Y
r
F
r
r
∫ [ ] (9.1)
Differentiating eqn 9.1 with respect to r and cancelling terms results in
d
d
= –
d
d
( – ) d +
d
d
( – ) d
2
2 1
2
2 1
1 1
L
r
Z
r
r r F
Y
r
r r F
r
r
y
r
r
z
∫ ∫
= –
d
d
+
d
d
2
2 A
2
2 A
Z
r
N
Y
r
M (9.2)
since the integrals are the external lagging and flapping moments respectively. But,
if EIy and EIz are the lagwise and flapwise blade stiffnesses, we also have
d
2
Z/dr
2
= MA/EIz and d
2
Y/dr
2
= NA/EIy
so that eqn 9.2 can be written
d
d
=
1
–
1
A A
L
r
M N
EI EI y z
(9.3)
It is clear that the torque due to blade deflection will be zero if EIy = EIz at every
point of the blade. If this is satisfied the blade is called a ‘matched-stiffness’ blade.
For a hingeless blade the structural element near the root, which allows most of the
lag bending of the blade, can be ‘matched’ relatively easily to achieve almost zero
Z
X
r
r1
P
Z
Z1
(Perpendicular
to paper)
dFy
Y
X
Y
Y1
dFz
(Perpendicular
to paper)
r
r1
P
Fig. 9.2 Projection of the deformed blade onto the
XY plane
Fig. 9.3 Projection of the deformed blade onto the
XZ plane
322 Bramwell’s Helicopter Dynamics
torsional moment along the entire blade. It is obvious that the torque exerted on a
hinged blade will also be zero if the feathering hinge lies outboard of the lagging and
flapping hinges, Fig. 9.4, for then the axis about which the torque is calculated
follows the blade when it lags and flaps.
Hansford and Simons
2
have shown that by neglecting the pitching inertia of the
blade, which is justifiable because of the high torsional stiffness, it is possible to
write the torsional deflection θ as
θ ψ βψξψ
λ κ
ν
β
θ
( ) = ( ) ( )
1 – +
1
2
1
2
1
2
I
I
(9.4)
where Iβ and Iθ are the blade flapping and pitching moments of inertia, and λ1Ω, κ1Ω
and v1Ω are the rotating blade uncoupled natural frequencies in flap, lag and torsion
respectively.
If, for example, the flapping frequency is 1.1Ω, the required lagging frequency for
zero twist is found to be 0.458Ω. Thus, a condition of zero or very small blade twist
can be achieved by suitably choosing the flapping and lagging frequencies. It can be
seen from eqn 9.4 that the relationship is non-linear in β and ξ, but for small variations
about steady flapping and lagging angles β0 and ξ0 the twist can be expressed as
∆θ
λ κ
ν
β ζ ζβ
β
θ
=
1 – +
( + )
1
2 0 0
1
2
1
2
⋅
I
I
(9.5)
This relationship between the pitch angle ∆θ and the flapping and lagging angles
can be expressed as
∆θ = kββ + kξξ (9.6)
The same relationship applies to the hinged blade. The coefficient kβ represents the
δ3 hinge effect and kξ the α2 hinge effect, as mentioned at the beginning of this
section. The inclusion of these terms merely adds – γ (kββ + kξξ)/8 to the left-hand
side of the appropriate equation of motion (see eqn 9.25). The usual solution procedure
leading to a quartic characteristic equation may be followed, and the stability can be
discussed in terms of the roots of the equation. This has been done by Pei
3
, who finds
for an approximate stability criterion that
Z
X
β
Fig. 9.4 Outboard position of feathering hinge
Aeroelastic and aeromechanical behaviour 323
F
k
k
C .
/
ξ
ξ
β
β θ
β
θ
+
2
1 – ( )
> 0
0 0 0
⋅
0
2
Ω (9.7)
where F.
ξ
is the lag damping coefficient in the equation of motion involving lagging
motion (see, for example eqn 9.26) and C is the moment of inertia in lagging motion.
However, Pei also shows that the above criterion can be deduced from simple physical
arguments. Since the lagging motion is generally of much lower frequency than that
of the natural flapping motion, the flapping response can be calculated as if the
lagging excitation at any instant were being steadily applied. Therefore, the change
of lift moment due to the lagging and flapping motion can be written as
∆M
M M
k k = = ( + )
∂
∂
∂
∂ θ
∆θ
θ
ξ β ξ β
The change in flapping angle β, if ξ is at sufficiently low frequency, can be
expressed as (change of lift moment)/(centrifugal moment) or,
β
ξ β θ ξ β
=
( + ) /
2
k k M
B
∂ ∂
Ω
But in steady motion, with lift moment ∆M, the coning angle β0 is given by
β
θ θ
0
0
2
=
/ ∂ ∂ M
BΩ
so that
β = (β0/θ0) (kξξ + kββ)
or β
β θ ξ
β θ
ξ
β
=
( )
1 – ( / )
0 0
0 0
/ k
k
(9.8)
As a result of the flapping motion, the Coriolis moment N causing lagging is, for
small disturbances,
N C C
k
k
= 2 = 2
1 – ( / )
0
0 0 0
Ω Ω β β
β
θ
ξ
β θ
ξ
β
.
.
0
2
⋅
from eqn 9.8. Since this moment is proportional to
.
ξ , it may be regarded as a viscous
damping moment. Then, including the damping of the lag damper and the drag
moment, represented by F˙
ξ
, the total damping will be positive if
F
k
k
C .
ξ
ξ
β
β θ
β
θ
+
2
1 – ( / )
> 0
0 0 0
⋅
0
2
Ω
as given by eqn 9.7.
For the helicopter with hinged blades, the criterion can be expressed in terms of
the α2 and δ3 hinges as
324 Bramwell’s Helicopter Dynamics
F C .
ξ
α
β θ δ
β
θ
+
2 tan
1 – ( / )tan
> 0
2
0 0 3 0
⋅
0
2
Ω
Since F.
ξ
represents the lag damping, and is therefore positive, instability can
occur only if the second term is a sufficiently large negative number. The sign of this
term can be regarded as depending only on α2, since δ3 would have to be unusually
large and positive to change the sign of the denominator. Thus, instability is possible
when α2 is negative, i.e. when the blade pitch increases as the blade moves forward
in lagging motion.
9.3 Main rotor pitch–flap flutter
Essential for this form of instability is a mechanism coupling blade pitch and flap.
The type of coupling normally encountered is due to adverse offsets of the chordwise
centre of gravity of the blade section from the blade feathering axis which is normally
coincident with the
1
4
-chord point. This instability is very similar in nature to the
fixed wing bending torsion flutter problem, and has been avoided on almost all
helicopter rotor blades by mass balancing the blade so that the chordwise position of
the centre of gravity of each spanwise element is forward of the blade section
1
4
-
chord point. This is usually accomplished by the use of non-load-carrying mass
balance weights along the full length of the blade.
In the region of the blade root end attachment, local reinforcing usually has the
effect of moving the centre of gravity well aft of the
1
4 -chord point. However, analysis
indicates that for a blade which is reasonably stiff in flatwise bending and torsion, the
product of inertia of the blade about axes coincident with the flapping and feathering
hinge lines is the significant parameter. Therefore, the effect of an aft movement of
the centre of gravity in the root region can be counteracted by the addition of a
relatively small mass positioned near the leading edge of the blade close to the tip.
For the case where complete decoupling is not achieved by mass balancing, then
the blade flatwise and torsional stiffnesses, and the effective torsional stiffness of the
pitch control circuit at the blade root, are important parameters.
The presence of blade flutter would normally be detected by the presence of high
oscillatory loads in the blade pitch control circuit.
Since on a conventional rotor blade the amount of mass balance material is of the
order of 9 to 12 per cent of total blade mass, then considerable weight saving could
be achieved if it were possible to relax the mass balance requirements. However, a
reduction in blade mass implies an increase in blade coning angle for a given value
of lift, and this can lead to problems involving increased lag plane loading, and more
severe damping requirements in lag to suppress certain types of instability. A reduction
in blade mass would decrease the rotational moment of inertia of the total rotor
system, thus adversely influencing the autorotational characteristics of the rotor and,
in the case of partial power failure, the ‘fly-away’ manoeuvre.
Compared to the classical bending torsion flutter of aircraft wings, the motion of
Aeroelastic and aeromechanical behaviour 325
the helicopter blade is modified by the powerful centrifugal action which effectively
increases the stiffness of the flapping motion.
The mechanism of pitch-flap flutter is that, as the blade flaps, inertial and aerodynamic
moments arise which twist the blade and, in turn, modify the aerodynamic flapping
moment. The inertial effect of pitching on the flapping motion is very small.
Let us derive the flapping and torsional equations of motion. It will be sufficient
to consider a rigid blade hinged at the root but with arbitrary flapping and torsional
frequencies. We shall take the elastic axis of the blade as one of our reference axes
which, since it will not in general be a principal axis of the blade, requires the
equations of motion to be derived with a little care. If the axes are fixed in the blade,
Fig. 9.5 the angular velocity components about these axes when the blade flaps and
twists are found to be
ω θ βθ β 1 = + sin +
. .
Ω Ω ≈
ω β θ θββθ 2 = – cos + sin cos + Ω Ω ≈
.
ω β θ θβ 3 = sin + cos cos Ω Ω ≈
Since the blade is very thin in relation to its chord and span, the products of inertia
D and E are negligible, but F, which represents the chordwise distribution of mass
relative to the elastic axis, should be retained. Then, from eqn A.1.7 the components
of angular momentum are
h1 = Aω1 – Fω2
h2 = Bω2 – Fω1
h3 = Cω3
Differentiating with respect to time and retaining only first-order terms, we obtain
. .. ..
h A A F F 1
2 2
= + + + θ θ ββ Ω Ω
. .. ..
h B B F F 2
2 2
= – – – – β θ θ θ Ω Ω
z
x
Ω
θ
β
θ
β
.
θ
.
β
y
θ
Fig. 9.5 Blade deflection in combined flapping and torsion
326 Bramwell’s Helicopter Dynamics
The above expressions represent the inertia terms of the torsional and flapping
motion. If the elastic stiffnesses about these axes are included, the equations of
motion can be written
d
d
+ +
d
d
+ =
2
2 1
2
2
2 2
θ
ψ
ν θ
β
ψ
β
F
A
L
A
A
Ω
(9.9)
d
d
+ +
d
d
+ =
2
2 1
2
2
2 2
β
ψ
λ β
θ
ψ
θ
F
B
M
B
A
Ω
(9.10)
where ν1Ω and λ1Ω are the uncoupled torsional and flapping frequencies and LA and
MA are the aerodynamic torsional and flapping moments.
In calculating the flapping and feathering moments, we should consider the unsteady
aerodynamic coefficients discussed in Chapter 6 and represented by the function
C(k). It will be assumed that the flexural axis coincides with the aerodynamic centre.
Then, from eqn 6.26, with x = –
1
2
and replacing the theoretical lift slope 2π by a
general value a, the spanwise thrust distribution is
d
d
=
1
2
– +
1
2
( ) +
1
8
– +
1
4
2 2 2 T
r
ac r
c
r
C k ac r r c ρ α
β α
ρ α βα Ω
Ω Ω
Ω
.
.
. .. ..
[ ]
where .
α is the component of angular velocity about the blade span, which includes
the component of shaft angular velocity due to flapping, i.e. .
α = .
θ + Ωβ and
z = – rβ. The flapping moment MA is found by integrating rdT/dr over the span,
which gives
M ac R
c
R
C k A
2 4
=
1
8
– +
2
3
( ) ρ α
β α
Ω
Ω Ω
.
.
+
1
8
1
3
–
1
3
+
1
8
2 4
ρ
α β α
ac R
c
R
c
R
c
R
Ω
Ω Ω Ω
.
..
..
2 2
The term c/R is the reciprocal of the aspect ratio of the blade and is typically about
1/20. Since the terms in ..
α and
..
β are usually negligible, the expression for MA
reduces approximately to
M ac R C k B C k A
2 4 2
=
1
8
–
d
d
( ) =
8
–
d
d
( ) ρ α
β
ψ
γ
α
β
ψ
Ω Ω
Similarly, the aerodynamic pitching moment about the flexural axis is approximately
L ac R ac R A
2 2 2 2
=
1
32
d
d
=
1
32
d
d
+ ρ
α
ψ
ρ
θ
ψ
β
3 3
Ω Ω
=
1
32
d
d
+
2
2
γ θ
ψ
β
B
A
Ω
Aeroelastic and aeromechanical behaviour 327
where A is the blade aspect ratio R/c. The equations of coupled pitch-flapping motion
are therefore
d
d
+ +
d
d
+ = –
1
32
d
d
+
2
2 1
2
2
2 2
θ
ψ
ν θ
β
ψ
β
γ θ
ψ
β
F
A
B
A
A
and
d
d
+ +
d
d
+ =
8
–
d
d
( )
2
2 1
2
2
2
β
ψ
λ β
θ
ψ
θ
γ
θ
β
ψ
F
B
C k
where θ represents changes of α from the steady state.
If the displacement of the c.g. of a blade elemental strip σgc (positive when the c.g.
is behind the flexural axis) is constant along the span, it is easy to show that F =
– σgcrg M, where rg is the spanwise position of the c.g. of the blade; and, if we also
write A = Mk c
A
2 2
, the equations above become
d
d
+
1
32
d
d
+ –
d
d
+ +
1
32
= 0
2
2 2 1
2 g g
2
2
2 2
θ
ψ
γ θ
ψ
ν θ
σ β
ψ
β
γ
β
B
A
r
ck
B
A
A
A A
d
d
+
8
( )
d
d
+ +
d
d
+ –
8
( ) = 0
2
2 1
2
2
2
β
ψ
γ β
ψ
λ β
θ
ψ
θ
γ
θ C k
F
B
C k
The term proportional to F/B in the flapping equation and the second term involving
the aspect ratio in the pitching equation can be ignored. Then, writing for the flapping
moment of inertia B = Mk R B
2 2
, the final equations in hovering flight are
d
d
+
8
1
4
d
d
+ –
d
d
+ = 0
2
2 1
2 g g
2
2
θ
ψ
γ θ
ψ
ν θ
σ β
ψ
β ⋅
k
k
r
ck
B
A A
2
2
(9.11)
d
d
+
8
( )
d
d
+ –
8
( ) = 0
2
2 1
2
β
ψ
γ β
ψ
λ β
γ
θ C k C k (9.12)
The frequency equation corresponding to the equations of motion 9.11 and 9.12 is
easily found to be
λ
γ
λ
4
2
3
+
8
( ) +
1
4
C k
k
k
B
A
+ + –
8
( ) + ( )
1
2
1
2 g g
2
2 2
2
λ ν
γ σ γ
λ
r
ck
C k
k
k
C k
B
A 252
+
8
( ) +
1
4
+ –
8
( ) = 0
1
2
2
1
2
1
2 g g
2
γ
ν λ νλ
γ σ
C k
k
k
r
ck
C k
B
A
(9.13)
328 Bramwell’s Helicopter Dynamics
Of the above constants, the only two which can readily be varied are the chordwise
blade c.g. position and the non-rotating torsional frequency. Divergence occurs when
the term independent of λ is equal to zero, i.e. when
ν
γ σ
λ
1
2 g g
2
1
2
=
8
( ) r
ck
C k
(9.14)
For C(k) = 1, i.e. when unsteady aerodynamic effects are neglected, eqn 9.14
defines a straight line relating the torsional frequency to the chordwise c.g. location.
It is clear that the boundary exists for positive values of σg, i.e. a rearward location
of the c.g. relative to the
1
4 -chord point. The physical interpretation of divergence is
simply that, if the c.g. of the blade is located sufficiently far behind the
1
4
-chord
point, the component of centrifugal force about the blade-span axis, which arises
when the blade flaps, exerts a nose up moment which is greater than the torsional
restoring moment.
To find the flutter boundary we put λ = iω, the condition for undamped oscillations,
and equate the real and imaginary parts to zero. We then find
ν ω
ω
1
2 2
2
g
2
= +
– 1)( / )
4 ( )
( x k
C k
(9.15)
and
σ
γ
ω λ
γ ω
ω
g
2 2
g
2
2
2 2 2
2
= 2
( )
– +
( ) /64
– 1
k
k
ck
r C k
C k B
A
A
1
2
(9.16)
Equations 9.15 and 9.16 provide the relationship between ν1 and σg for which
undamped oscillations occur with ω
2
, the square of the flutter frequency ratio, as
parameter. Thus, if ν1 is given, eqn 9.15 can be solved for ω
2
, which can then be
inserted into eqn 9.16 to obtain σg.
A sketch of typical divergence and flutter boundaries as functions of σg and ν1
2
for
C(k) = 1 is shown in Fig. 9.6. The two curves correspond to the four roots of the
characteristic frequency quartic, eqn 9.14. If the torsional frequency is fixed at the
value corresponding to the point A of Fig. 9.6 and the chordwise c.g. position is
moved aft, the line AA′ is traced. The motion is completely stable until the intersection
with the divergence boundary is reached. At this point, A′, one of the roots of the
frequency quartic becomes zero, and just to the right of A′ the root becomes positive,
denoting the unstable divergence. The remaining roots between A′ and A″ are a
negative real root (stable) and a complex pair denoting a damped oscillation. To the
right of the point A″ the oscillation becomes undamped also. Moving along BB″ the
unstable flutter oscillation is met first, and then the divergence.
Analysis of the equations 9.13, 9.15 and 9.16 shows that the flutter and divergence
boundaries exist only for positive values of σg, which shows that pitch-flap motion
must be stable for σg < 0, i.e. when the centre of gravity of the blade is ahead of the
flexural axis. Since the torsional frequency must have a positive value, Fig. 9.6
shows that stable motion is possible even when the centre of gravity is behind the
Aeroelastic and aeromechanical behaviour 329
flexural axis. Inspection of eqns 9.15 and 9.16 shows that for flapping frequencies
greater than Ω, i.e. λ1 > 1, the flutter boundary is shifted to the left. This suggests that
offset or hingeless blades are rather less stable than centrally hinged blades.
The effect of taking unsteady aerodynamic coefficients (C(k) ≠ 1) into account is
quite small. The appropriate value of C(k) must be obtained iteratively, since the
flutter frequency ω must be known before C(k) can be estimated. The flutter boundary
is modified as sketched in Fig. 9.7.
Stammers
4
has shown that forward flight has a stabilising influence on flutter
which tends to occur at half-integer frequencies.
9.4 Main rotor stall flutter
This is a single degree of freedom flutter problem involving torsional oscillation of
the blade. It normally takes the form of a limit cycle oscillation giving rise to high
B
B″
B′
A A′
Flutter
Unstable
Divergence
Stable
σg
A″
Fig. 9.6 Stability boundaries for torsional motion
C(k) ≠ 1
C(k) = 1
σg
Fig. 9.7 Stability boundaries for torsional motion (C(k) ≠ 1)
ν1
2
ν1
2
330 Bramwell’s Helicopter Dynamics
oscillatory loads in the blade pitch control circuit, and occurs over a stalled region of
the retreating blade side of the rotor disc. This region must cover a sufficient proportion
of the total disc area for the instability to allow one or more cycles of torsional
oscillation before the blade moves into a stable (non-stalled) region of the rotor disc.
Consider an aerofoil oscillating periodically in pitch in an airstream of velocity V.
The pitching moment coefficient can be expressed as
C
M
V c
a a pt a pt b pt m =
( )
= + sin + sin 2 + + cos +
1
2
2 2 0 1 2 1
α
ρ
… …
where α is the instantaneous incidence, p is the angular frequency, and c is the chord.
The work done in one cycle is
W =
∫
M dα
If the oscillations are harmonic, the incidence can be written as
α = α0 + α1 sin pt
where α0 is the mean incidence and α1 is the amplitude of the oscillations.
We then find that
W V c p b pt t
p
= cos d
1
2
2 2
0
2 /
1 1
2
ρ α
π
∫
=
1
2
2 2
1 1 πρ α V c b
The work is therefore proportional to the coefficient b1; if b1 is negative, work is
dissipated and the damping is positive. Thus, the damping depends on the out-ofphase component of the pitching moment.
The basic mechanism of the instability is negative damping in pitch due to
aerodynamic pitching moment hysteresis caused by the periodic shedding of intense
vorticity at a blade angle of attack instantaneously greater than the static stalling
angle
5
.
Hovering rotor tests of these oscillations have been described by Ham and Young
6
.
Explanation of the phenomenon has already been alluded to in Chapter 6. We saw
there that at high incidence a suction peak occurs over the rear part of the aerofoil,
resulting in a large nose down pitching moment, and this gives rise to a substantial
out-of-phase moment of negative damping, particularly at the comparatively low
values of the reduced frequency typical of helicopter operation. The variation of
damping with incidence and reduced frequency is graphically displayed in the threedimensional diagram taken from Carta’s paper, Fig. 9.8. The ‘hollow’ indicates the
region in which negative damping occurs.
Now, as we saw in Chapter 6, the aerofoil undergoes a large variation of incidence,
and therefore of rate of change of incidence, at high tip speed ratios. It will be assumed
that the rotor blade responds to the changing conditions at its fundamental torsional
frequency ωθ, which, in association with the local chordwise velocity ΩR(x + µ sin ψ),
Aeroelastic and aeromechanical behaviour 331
enables us to define the instantaneous value of the reduced frequency k, i.e. we assume
that the appropriate value of k is given by
k(x, ψ) = ωθb/ΩR (x + µ sin ψ)
where b is the semi-chord. Thus, for a given radial position, the variations of incidence
and reduced frequency trace out a closed path on the surface as shown in Fig. 9.8. In
the range, or ranges, of azimuth angle which lie in the region of negative damping,
torsional flutter can be expected to occur. It has been assumed that the damping at
any azimuth angle is the same as if the incidence and reduced frequency were fixed,
i.e. that the damping corresponding to given values of α and k applies instantaneously
at the given azimuth angle. For the blade as a whole we can define a ‘weighted-mean’
damping by integrating the local two-dimensional values in association with the first
torsional mode shape Q1(x) to obtain
ζ ζ α mean
0
1
1
2
= ( ) ( , ) d
∫
Q x k x
where the reduced frequency to be used is that defined above.
The torsional damping defined in this way has been calculated by Carta for a case
in which µ = 0.17 and CT/σ = 0.111, and the result is shown in Fig. 9.9. We see that
negative damping occurs in the region 225° to 10°, and in this region we can expect
stall flutter to occur. To test the validity of this assumption, the torsional stress and
pitch link loads for the same flight case, Fig. 9.10, have been examined. It can be
seen that large values of torsional stress and pitch link loads occur in this region,
demonstrating that the calculation of the damping gives a good indication of the
possibility of stall flutter.
The high pitch link loads which occur in stall flutter may be a serious obstacle to
increased flight speeds. According to Ham
7
, the most profitable way of reducing
these loads is to use aerofoil sections which have high dynamic stall angles, so that
they can operate in the retreating blade region below the stall angle as much as
possible.
0
4
8 12
16
20
24
28 32
Unstable
region
1.2
0.8
0.4
0
0.4
0.3
0.2
0.1
0
Reduced frequency, k
Aerodynamic
damping
0.5
Incidence, α
Fig. 9.8 Contours of blade torsional damping
332 Bramwell’s Helicopter Dynamics
9.5 Main rotor blade weaving
Blade weaving is the name given to a form of instability which can affect two-bladed
‘teetering’ rotors, i.e. rotors with two blades rigidly connected together at their root
ends with a built-in non-zero coning angle, and suspended on a central flapping or
‘teetering’ hinge. The phenomenon involves the same basic coupling mechanism as
the main rotor pitch–flap flutter problem described in section 9.3, but since both
blades move as a single entity, extra terms arise in addition to those of eqns 9.11 and
9.12.
Consider the two-bladed teetering rotor of Fig. 9.11. The blades are joined at angle
2β0 (twice the built-in coning angle) and are feathered so that the steady pitch angle
between the blades is 2θ0. The two blades are then imagined to move as a rigid body
except that torsional flexibility of the control system will allow some mutual feathering.
The latter motion will have a negligible effect on the moments of inertia of the whole
0 50 100 150 200 250 300 350
80
60
40
20
0
ω0/Ω 6
Aerodynamic damping moment
Azimuth angle ψ
Fig. 9.9 Aerodynamic torsional damping as a function of azimuth
Pitch link load
Torsion strain
at 90% radius
Torsion strain
at 46% radius
60° 180° 300° 60°
120° 240° 360°
ψ
Fig. 9.10 Variation of pitch link loads with azimuth angle
Nm
Aeroelastic and aeromechanical behaviour 333
rotor system, so the assumption of complete rigidity is justified. If the lines of the
centre of mass intersect at the hub, Euler’s equations, eqns A.1.11 to A.1.13 can be
applied to the rotor as a whole. Coleman and Stempin
8
, who first investigated this
motion, have shown that, if A, B, C are the principal moments of inertia of the
individual blades, the corresponding moments of inertia of the rotor with coning
angle β0 and collective pitch setting θ0 are
A′ = 2A cos
2
β0 + 2B sin
2
θ0 sin
2
β0 + 2C cos
2
θ0 sin
2
β0 ≈ 2(A + Bβ 0
2
)
B′ = 2B cos
2
θ0 + 2C sin
2
θ0 ≈ 2B
C′ = 2A sin
2
β0 + 2B sin
2
θ0 cos
2
β0 + 2C cos
2
θ0 cos
2
β0 ≈ 2(C – Bβ 0
2
)
assuming A + B = C for the individual blades.
As might be expected, the flapping and lagging moments of inertia are very little
different from those of the original blades, but the pitching moment of inertia is
greatly increased by coning angle. The angular velocity components of the rotor are
the same as those of the previous section, and the linearised Euler equations are
found to be
.. .
θ θ β +
–
+
+ –
=
2 A
′ ′
′
′ ′ ′
′
′
C B
A
A B C
A
L
A
Ω Ω
.. .
β β θ +
–
–
+ –
=
2 A
′ ′
′
′ ′ ′
′ ′
C A
B
A B C
B
M
B
Ω Ω
The important difference between these equations and those of the pitch–flap
coupling of section 9.3 is that, since the rotor can no longer be regarded as a lamina,
A′ + B′ – C′ ≠ 0 so that the coefficients of Ω
.
θ and Ω
.
β do not vanish as they did in
the latter case. Let us write the equations in the non-dimensional form originally used
by Coleman and Stempin. They are
d
d
+
d
d
+ ( + ) – (1 – )
d
d
– = 0
2
2 A A
θ
ψ
θ
ψ
θ Ι
β
ψ
β θ
H I K H D D ′ (9.17)
Ω
β
2β0
θ
Fig. 9.11 Teetering rotor with combined flapping and feathering
2θ0
334 Bramwell’s Helicopter Dynamics
d
d
+
d
d
+ + (1 – )
d
d
+ = 0
2
2 A
β
ψ
β
ψ
β Ι
θ
ψ
θ H I H B B B (9.18)
In the above equations, HB and HD represent damping of the pitching and flapping
motion; IA and IB are defined by
Ι A
C B
A
H
A
=
–
+
1
′ ′
′
′
′
and
Ι B
C A
B
H
B
=
–
–
5
′ ′
′
′
′
and we have approximately
1 – =
2
+
0
2 1
I
B
A
H
A
A β
′
′
and
1 – = –
0
2 5
I
H
B
B 2β
′
′
where the terms in ′ H1 and ′ H5 represent aerodynamic damping. It can be seen that,
since B/A is usually a very large ratio, 1 – IA is strongly dependent on coning angle
whereas 1 – IB varies little. ′ Kθ is the stiffness of the pitch control. Coleman and
Stempin state that HD depends strongly on coning angle but that HB is practically
unaffected.
The equations of motion 9.17 and 9.18 are similar in form to those of section 9.3
(eqns 9.11 and 9.12) except for the presence here of the terms in dβ /dψ and dθ/dψ.
(Since the motion has been referred to principal axes through the centre of gravity of
the blade, the term in
..
β is absent.) Coleman and Stempin’s calculations show that
these extra terms are destabilising and that the instability depends mainly on the
coning angle and to a lesser extent on the pitch setting angle θ0. An important result
of their investigations is that instability can occur even when the centre of mass of the
blade is ahead of the
1
4 -chord point. A sketch of the variation of the stability boundaries
with control stiffness, coning angle, and chordwise c.g. location is given in Fig. 9.12.
When instability occurs, the blade tips trace out a wavy or ‘weaving’ path which
gives rise to the name of the phenomenon.
9.6 Tail rotor pitch–flap (‘umbrella mode’) instability
9
This type of instability depends on the existence of some form of coupling between
blade pitching and flapping motions, and is very similar to the main rotor blade
pitch–flap flutter problem described in section 9.3.
In the case of the tail rotor, the coupling usually arises from the large δ3 coupling
introduced to reduce tail rotor blade flapping and stresses in forward flight. This
Aeroelastic and aeromechanical behaviour 335
coupling is normally of the order of one degree of reduction in blade pitch per degree
of (‘upward’) blade flapping.
This form of instability is usually experienced as a limit cycle oscillation with all
blades moving in phase. Hence the description ‘umbrella mode’ applied to this type
of motion.
The stiffness and damping of the pitch control circuit between the blades and the
pitch control actuator have a fundamental influence on this phenomenon.
In general, stable solutions can be found with both high and low pitching mode
frequencies, in the absence of pitch circuit damping.
Where the possibility of instability arises, the onset of this is difficult to predict
due to the presence of backlash and friction damping effects in the pitch control
circuit, and in fact the occurrence in practice tends to be somewhat erratic. Disturbances
below a certain threshold will subside, but beyond this divergence will occur, rapidly
reaching a constant level.
The frequency of the oscillation is at or close to the fundamental flapping frequency
of the tail rotor blade which is typically 1.2Ω. The reason that it is significantly
greater than 1Ω is primarily due to the additional aerodynamic stiffness arising from
the δ3 coupling.
Figure 9.13 indicates a typical stability boundary as a function of pitch control
circuit stiffness and damping.
9.7 Main and tail rotor flap–lag instability
We have already seen in Chapter 1 that blade flapping produces large Coriolis moments
in the plane of the rotor. For the articulated rotor, a drag hinge is provided to relieve
0.30
0.25
0.20
0.15
0.10
0.05
0
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
30 25 20 15 % chordwise c.g.
Stable
Unstable
Coning angle, β0 radians
AA 1/K / ′⋅ ′ θ
B/A = 1000
IB = 1, θ0 = 0
Fig. 9.12 Stability boundaries of blade weaving motion
336 Bramwell’s Helicopter Dynamics
the blade of these moments and allow it to move in the plane of rotation (lagging).
For semi-rigid and totally bearingless rotors, and for the majority of tail rotors, the
lagwise flexibility provided at the root end of the blade permits the corresponding
lagwise movement, and is designed to accommodate the resulting moments.
The lagging motion means that the instantaneous angular velocity of the blade in
the plane of rotation is slightly different from the (assumed) constant angular velocity
of the shaft, and this in turn means that the centrifugal flapping moment depends on
the lagging motion. The variations of relative airspeed due to lagging also affect the
aerodynamic flapping moment. Thus, the flapping and lagging motions are clearly
coupled, but to investigate the stability of these motions we must derive the appropriate
equations of motion. Referring to Fig. 9.14, if the lagging angular velocity is
.
ξ , then
the instantaneous angular velocity is Ω +
.
ξ and, by the arguments of Chapter 1, the
centrifugal flapping moment is –B(Ω +
.
ξ )
2
β, tending to restore the blade to the plane
of rotation. Neglecting the term in
.
ξ
2
, the first-order flapping moment is
–B(Ω
2
+ 2Ω
.
ξ ) β and the equations of flapping and lagging motion are
B B B M
.. .
β λ β βξ + + 2 =
2
A 1
2
Ω Ω (9.19)
Total damping
Total control circuit stiffness
Unstable
Fig. 9.13 Stability boundary for coupled pitch–flap motion
Aeroelastic and aeromechanical behaviour 337
Ω
β
ξ
Fig. 9.14 Simple flap–lag blade model
C C C N
.. .
ξ κ ξ ββ + – 2 =
2
A 1
2
Ω Ω (9.20)
where we have written λ1Ω and κ1Ω for the undamped natural flapping and lagging
frequencies which can apply to both hinged and hingeless blades. The aerodynamic
moments MA and NA will contain aerodynamic coupling terms. To calculate them we
consider the force on a blade element under conditions of flapping and lagging. For
simplicity we consider only hovering flight. Referring to Fig. 9.15, the elementary
flapping and lagging forces are, respectively,
dZ = dL cos φ + dD sin φ ≈ dL
dY = dL sin φ dD cos φ ≈ dLφ – dD
Now
d = ( + )d
1
2
2
L W ca r ρ θ φ
≈ ( + ) d
1
2 T
2
ρ θ φ U ca r
and
d d
1
2 T
2
D U cC r D ≈ ρ
Also,
tan φ ≈ φ = UP/UT
dL dZ
UT
UP
W
dD
dY
φ
θ
φ
Fig. 9.15 Forces on blade element
338 Bramwell’s Helicopter Dynamics
therefore,
d = ( + )
1
2 T
2
P T Z ac dr U U U ρ θ (9.21)
and
d = ( + ) – d
1
2 P T P
2 1
2 T
2
Y ac dr U U U U C c r D ρ θ ρ (9.22)
The velocity components UP and UT are
U r U r T P i
= ( + ) ; = – – Ω
. .
ξ β v
Expanding eqns 9.21 and 9.22 and neglecting squares and products of
.
β and
.
ξ
gives
d = [ ( + 2 ) – – – ]d
1
2
2 2 2 2
i i Z ac r r r r r ρ θ ξ β ξ Ω Ω ΩΩΩ
˙ ˙ ˙
v v
d = – ( + 2 )d
1
2
2 2 2
Y cC r r r D ρ ξ Ω Ω
.
– ( + + – – )d
1
2
2
i i i
2
i ρ θ β θ θξ β ac r r r r r Ω Ω
˙ ˙ ˙
v v v v 2
Integrating rdZ and rdY, assuming CD, θ, and vi to be constant, we obtain for the
aerodynamic flapping and lagging moments
M ac R A
2 4
i i
=
1
8
–
4
3
– + 2 –
4
3
ρ θ λ
β
θ λ
ξ
Ω
Ω Ω
. .
(9.23)
N ac R
C
a
D
A
2 4
i i
2
i
= –
1
8
+
4
3
– 2 + –
8
3
ρ θλλθλ
β
Ω
Ω
.
+ +
4
3
i
2C
a
D
θλ
ξ
.
Ω
(9.24)
The constant terms on the right-hand sides of eqns 9.23 and 9.24 give the steady
state flapping and lagging angles β0 and ξ0. Since we are concerned only with
perturbations from the steady state, these terms can be omitted; but, since the last two
terms of eqns 9.19 and 9.20 representing the Coriolis acceleration are products, they
must be written in first order form as 2BΩβ0
.
ξ and – 2CΩβ0
.
β .
Finally, if it is assumed that the flapping and lagging moments of inertia, B and C,
are equal, the equations of disturbed motion become
d
d
+
d
d
+ +
d
d
= 0
2
1
2
β
ψ
γ β
ψ
λ β
ξ
ψ
ξ 2
8
C. (9.25)
F F . .
β ξ
β
ψ
ξ
ψ
ξ
ψ
κ ξ
d
d
+
d
d
+
d
d
+ = 0
2
1
2
2
(9.26)
Aeroelastic and aeromechanical behaviour 339
where
C.
ξ
β
γ
θ λ = 2 – 2 –
4
3
0 i
8
F.
β
γ
θ λ β = –
8
3
– 2 i 0
8
F k
C
a
D
. .
ξ ξ
γ
θλ = +
2
+
4
3
i
8
and where k.
ξ
is the non-dimensional artificial lag damping, if any. The characteristic
equation of this motion is
( + /8 + )( + + ) – = 0
2
1
2 2
1
2 2
λ γλλλ λκ λ
ξ ξ β
F C F . . .
which is of the form
A B C D E λ λ λλ
4 2
+ + + + = 0
3
where
A = 1
B = γ /8 + F.
ξ
C = λ κ γ
ξ ξ β
1
2
1
2
+ + /8 – F C F . . .
D = γκ λ
ξ
1
2
1
2
/8 + F.
E = λ κ 1
2
1
2
To find the neutral stability boundaries we equate Routh’s discriminant to zero,
i.e. we put
R = BCD – D
2
– B
2
E = 0
It has been shown by Ormiston and Hodges
10
that this expression can be put into
the form
( – 4 /3) =
2( – 1)(2 – )
2
+ +
64 ( – )
(1 + )( + )
i
2 1
4
1
2
1
2
1
2
1
2
2
1
2
1
2
θ λ
λ
λ λ
α λ κ
γ α καλ
ξ
C
a
k
D
.
(9.27)
where
α = k.
ξ
+ 2CD/a + 4θλi/3
Equation 9.27 includes artificial damping omitted by Ormiston and Hodges.
The relationship, eqn 9.27 shows clearly that instability, if it occurs at all, does so
only if 1 < λ1
2
< 2, i.e. when the flapping frequency is between Ω and Ω√2. Further,
for a given value of λ1, the lowest possible value of θ for instability occurs when κ1
= λ1, i.e. when the lagging and flapping frequencies are identical, in which case the
corresponding value of θ is given by
340 Bramwell’s Helicopter Dynamics
( – 4 /3) =
(2 / )
2( – 1)(2 – )
i
2
1
4
1
2
1
2
θ λ
λ
λ λ
ξ
C a k D .
The absolute minimum value of collective pitch occurs when λ1 = κ1 = √(4/3) =
1.153, from which we easily find that
θ = 4λi/3 + 2√(2CD/a + k.
ξ
) (9.28)
The stability boundaries for a case in which k.
ξ
= 0, taken from Ormiston and
Hodge’s paper, is shown if Fig. 9.16. It is clear that, unless the lag frequency is higher
than about 0.95Ω, instability would not be expected to occur.
The vast majority of rotor systems, both articulated and hingeless, with more than
two blades have fundamental lag frequencies significantly below this value and are
therefore not likely to be susceptible to this form of instability. (The case of the twobladed teetering rotor with built-in coning angle is treated separately in section 9.5.)
An exception to the general rule was the gyro-controlled ‘rigid’ rotor developed
by Lockheed, where the fundamental lag frequency was greater than 1Ω and the
design of such a system would have to include consideration of this type of instability.
However, the fundamental lag frequency of tail rotors is normally greater than 1Ω
(but significantly less than 2Ω in order to avoid excessive blade and hub vibratory
loading), and tail rotors must be designed to avoid this form of instability, which is
generally referred to as tail rotor ‘buzz’ and is usually of a mild ‘limit cycle’ nature.
Since the tendency to instability increases as the fundamental flap and lag frequencies
approach the same value, it is instructive to examine the factors which influence
these frequencies. It is usually the case that the elastic stiffness of the root region of
the blade is significantly less in the flatwise direction compared to the chordwise
direction. Hence, as blade pitch is increased, the lower flatwise stiffness reduces the
lag frequency from its value at zero pitch. Furthermore, due to the large value of
1.6
1.5
1.4
1.3
1.2
1.1
1.0
0.9
0.8
1.0 1.1 1.2 1.3 1.4 1.5 1.6
Flap frequency ratio, λ1
θ = 0.2
0.3
0.4
0.5
κ1 = λ1
Lag frequency ratio, κ1
Fig. 9.16 Stability boundaries for coupled flap–lag motion
Aeroelastic and aeromechanical behaviour 341
pitch–flap (δ3) coupling typically employed on tail rotors (in order to reduce flapping
in forward flight), the fundamental flapping frequency will be well above lΩ.
Consequently it is very probable that the flap and lag frequencies will coalesce at
high values of blade pitch.
Figure 9.16 clearly indicates the increasing area of instability as blade pitch is
increased.
9.8 Tail rotor pitch–flap–lag instability
9
The analysis describing the tail rotor ‘buzz’ phenomenon referred to in section 9.7
does not require the inclusion of the torsion, or pitch, degree of freedom, and hence
may be considered to represent the case of a torsionally stiff blade and pitch control
circuit.
If this additional degree of freedom is included in the analysis, an additional
region of instability appears which represents the much more severe instability known
as tail rotor ‘bang’.
This form of instability is associated with the coalescence of the flap, lag and
torsion mode frequencies.
Figure 9.17 illustrates the position of the stability boundaries as a function of the
4
3
2
1
1.0 1.5 2.0
Lag mode frequency, Ω
Unstable
Stable
Torsion mode frequency, Ω
Fig. 9.17 Stability boundaries for coupled pitch–flap–lag motion
A
B
342 Bramwell’s Helicopter Dynamics
frequencies of the torsion and lag modes, for typical values of uncoupled flap frequency,
δ3 coupling, and pitch angle setting.
Region A of the figure represents the ‘buzz’ instability appropriate to high values
of torsional stiffness, and Region B represents the ‘bang’ instability.
One aspect which has to be given careful consideration is the placement of the
torsion mode frequency. Because the blades will experience different stiffnesses at
the pitch change track rod depending upon the phase relationship between the torsional
motion of each blade, a number of torsion mode frequencies will exist.
Each of these frequencies must be chosen so as to avoid the unstable region.
This is an example of a case where the behaviour of the tail rotor and control
system must be considered in combination in order to define the correct data for the
appropriate single blade analysis.
In practice, increasing the bending stiffness of the pitch change ‘spider’ arms, and
of the pitch change rod attached to the centre of the ‘spider’, has produced freedom
from both the ‘buzz’ and ‘bang’ instabilities.
9.9 Ground resonance
Any helicopter with a main rotor blade fundamental lag mode frequency less than 1Ω
is susceptible in principle to the instability known as ground resonance.
The degrees of freedom involved are the lead–lag motion of the rotor blades and
any fuselage mode containing motion of the rotor hub in the rotor plane.
If the blades lag in phase, the centre of gravity of the rotor system remains on the
axis of rotation, but out-of-phase oscillatory motion of the blades will cause the rotor
system centre of gravity to move off the axis of rotation and describe a circle, giving
rise to inertia forces which will subject the fuselage and its chassis to an oscillatory
force.
The mode of blade motion in which the direction of rotation of the centre of
gravity of the rotor system is in the opposite sense to the direction of rotation of the
rotor system is known as the ‘regressive’ lag mode, whereas the ‘progressive’ lag
mode refers to the case where the centre of gravity moves in the same direction as the
direction of rotor rotation.
The potential for instability occurs in the vicinity of a frequency coalescence
between the ‘regressive’ lag mode and a fuselage mode, provided that the fundamental
lag frequency is less than the rotational speed of the rotor. The phase relationships
between the couplings are such that a frequency coalescence when the lag mode
frequency is greater than the rotor speed does not produce an instability, neither does
a coalescence with the ‘progressive’ lag mode.
The important parameters with respect to ground resonance are blade lag mode
frequency and damping, fuselage frequency and damping, and fuselage mode shape.
Of lesser significance are flap mode stiffness and aerodynamic loads. Ground resonance
is basically a purely mechanical instability which could exist in vacuo.
In practice, a stability augmentation system can influence ground resonance, and
if a significant response of such a system can be anticipated at the frequencies
Aeroelastic and aeromechanical behaviour 343
associated with ground resonance, then its characteristics must be included in the
analysis. An alternative approach is to ensure that the feedback systems are filtered
in such a way so as not to respond at the frequencies of potential ground resonance
oscillations.
Let us calculate the displacement of the rotor centre of gravity for an arbitrary
motion of the blades. Let xkg and ykg be the co-ordinates of the centre of gravity of
the kth blade relative to the centre of the hub, Fig. 9.18. This figure can represent
either a hinged blade or a hingeless blade (as an offset rigid blade with hinge restraint,
Chapter 7).
We easily find that
xkg = – eR cos ψk – rg cos (ψk + ξk)
ykg = eR sin ψk + rg sin (ψk + ξk)
Since ξk is a small angle, these relationships can be written approximately as
xkg = – (eR + rg) cos ψk + rgξk sin ψk
ykg = (eR + rg) sin ψk + rgξk cos ψk
Summing over the b blades of the rotor the co-ordinates of the centre of gravity of
the rotor are
x
eR r
b
r
b
r
k
b
k
k
b
k k g
= –
+
cos + sin
g
=0
–1
g
=1
–1
Σ Σ ψ ξ ψ
or
x r b r
k
b
k k g
= ( / ) sin g
=0
–1
Σ ξ ψ (9.29)
and, similarly,
y r b r
k
b
k k g
= ( / ) cos g
=0
–1
Σ ξ ψ (9.30)
x
y
eR
rg
ψk
ξk
Fig. 9.18 Blade displacement in lagging motion
344 Bramwell’s Helicopter Dynamics
Now, suppose that the blades oscillate in lagging motion with frequency κΩ such
that
ξk = ξ0 cos κψk
Then substituting in eqn 9.29
x r b r
k
b
k k g
= ( / ) cos sin g 0
=0
–1
ξ κψψ Σ
= ( /2 ) [sin ( + 1) – sin ( – 1) ] g 0
=0
–1
r b
k
b
k k ξ κ ψκψ Σ
Since κ is arbitrary we find from eqn 9.29 that
x
r
b
b
b
b
b
rg =
2
sin( + 1) +
– 1
– sin( – 1) +
– 1 g 0
1 –1
ξ
σκψ π σκψ π
(9.31)
where σ
κ π
κ π
σ
κ π
κ π
1 –1 =
sin ( + 1)
sin[( + 1) / ]
and =
sin ( – 1)
sin[( – 1) / ] b b
Similarly we find that
y
r
b
b
b
b
b
rg =
2
cos( + 1) +
– 1
+ cos( – 1) +
– 1 g 0
1 –1
ξ
σκψ π σκψ π
(9.32)
The interpretation of eqns 9.31 and 9.32 is that the centre of gravity of the rotor
whirls round the hub with a displacement consisting of two modes, one with frequency
(κ + 1)Ω and amplitude rgξ0σ1/2b and the other with frequency (κ – 1)Ω and amplitude
rgξ0σ-1/2b. The motion corresponding to the upper frequency (κ + 1)Ω is in the same
direction as the rotor rotation and is described as a ‘progressive’ motion; the motion
corresponding to the lower frequency (κ – 1)Ω is in the same or opposite direction to
the rotor rotation according to whether κ – 1 is positive or negative. When negative,
the motion is described as ‘regressive’.
The whirling centre of gravity produces periodic inertia forces which excite motion
of the whole airframe on its undercarriage (‘chassis’ mode). If one of the frequencies
of the oscillating inertia forces coincides with a chassis frequency, the potential for
the occurrence of ground resonance exists.
This coincidence of frequencies can be represented in diagrammatic form, Fig.
9.19, which applies to an articulated rotor with no lag hinge restraint. The comparable
diagram for a rotor with hinge restraint, or utilising an elastic hub element is shown
in Fig. 9.20.
For a blade without hinge restraint, i.e. without a drag-hinge spring or elastic
element, the lag frequency will always be less than the rotor speed and, therefore,
κ – 1 will always be negative. In Figs 9.19 and 9.20 the horizontal line represents the
(constant) chassis frequency ωc.
Aeroelastic and aeromechanical behaviour 345
In Fig. 9.20 the point A represents the blade frequency when the rotor hub is
stationary and is assumed here to be higher than the chassis frequency.
As the rotor speed increases, the branch corresponding to the whirl frequency
(κ – 1)Ω intersects the chassis line at B and again at C when (κ – 1) < 0.
If the chassis frequency were higher than the frequency of the non-rotating blade
there would have been an intersection with the (κ + 1)Ω branch.
The corresponding intersection for the case of the articulated rotor is point D on
Fig. 9.19. It will be shown later that this intersection and the one corresponding to the
point B cannot lead to instability. Ground resonance, if it occurs, is associated only
with intersection C.
Let us now derive the equations of motion of the chassis–rotor system. It will be
assumed that the chassis motion is confined to a single degree of freedom in the plane
of the rotor – say, the lateral direction. The extension to two degrees of chassis
motion is quite straightforward, but the analysis becomes rather too involved for
simple results to be obtained. In practice, however, the chassis frequencies in the two
directions are often far enough apart for the single-degree-of-freedom analysis to be
applied with reasonable approximation to either direction separately.
D
C
(κ + 1)Ω, progressive
(κ – 1)Ω, regressive
ωc
Rotor speed Ω
Blade frequency, κΩ
Fig. 9.19 Uncoupled chassis and rotor frequencies – articulated rotor, no hinge restraint
(κ – 1)Ω, regressive
C
B
ωc
Rotor speed Ω
Blade frequency, κΩ
(κ + 1)Ω, progressive
A
Fig. 9.20 Uncoupled chassis and rotor frequencies – rotor with drag hinge spring or elastic element
346 Bramwell’s Helicopter Dynamics
The equations of blade lagging motion, relative to an unaccelerated hub, have
already been obtained in Chapters 1 and 7. We must now include the inertia moment
acting on the blade due to lateral oscillations of the chassis mode. Referring to Fig.
9.21 the inertia force on a blade element due to hub acceleration is .. y dm in the
negative y direction. Hence, the inertia lagging moment Ni about the real or virtual
hinge is
N ry m
eR
R
k k i = – cos ( + )d
∫
˙˙ ψ ξ
For small lagging angle ξk, and neglecting the product .. y k ξ , we have, approximately,
N y r m k
eR
R
i = – cos d ˙˙ ψ
∫
= – Mbrg ˙˙ y cos ψk (9.33)
where Mb is the mass of the blade. The equation of motion of the kth blade can then
be written
˙˙ ˙ ˙ ˙˙ ξ κ δξκξ ψ k k k k M r y I + 2 + = – cos /
2 2
b g Ω Ω
= – ( / ) cos b
˙˙ y l k ψ (9.34)
where δ is the damping coefficient, which may include both aerodynamic and artificial
damping, κΩ is the natural undamped frequency, and lb = I/Mb rg.
The displacement of the centre of gravity of the rotor in the y direction has already
been found to be
y r b
k
b
k k rg g
=0
–1
= ( / ) cos Σ ξ ψ (9.30)
so that the equation of motion of the airframe and chassis is
( + ) = – – ( + ) – 2 ( + ) b b rg b c
2 2
c c b M bM y bM y M bM y M bM y ˙˙ ˙˙ κ κ δ Ω Ω
= cos – ( + ) g b
=0
–1
b c
2 2
r M M bM y
k
b
k k Σ ξ ψ κΩ
– 2κcΩδc (M + Mb)˙ y (9.35)
ψk
ξk
.. y
.. y m d
r
Fig. 9.21 Inertial force on blade due to fuselage motion
Aeroelastic and aeromechanical behaviour 347
where δc is the damping coefficient of the chassis and κcΩ is its undamped natural
frequency, M being the effective mass of the fuselage in the chassis mode.
The periodic terms in eqns 9.34 and 9.35 can be removed by using the Coleman
co-ordinates, Appendix A.3. Let
η ξ ψ = – (2/ ) sin
=0
–1
b
k
b
k k Σ
and
ζ ξ ψ = – (2/ ) cos
=0
–1
b
k
b
k k Σ
from which we get
Σ Σ
k
b
k k
k
b
k k b b
=0
–1
1
2 =0
–1
1
2
sin = ( – ); cos = – ( – )
˙ ˙ ˙ ˙
ξ ψ ζη ξψ ηζ Ω Ω
Σ
k
b
k k b
=0
–1
1
2
sin = – ( – 2 – )
˙˙ ˙˙ ˙
ξ ψ η ζ η Ω Ω
2
Σ
k
b
k
=0
–1
1
2
cos = – ( + 2 – )
˙˙ ˙˙
ξ ψ ζ η ζ Ω Ω
2
Then, multiplying eqn 9.34 by sin ψk and summing over all the blades, we get
˙˙ ˙ ˙
η δηκ ηζ δζ + 2 + ( – 1) – 2 – 2 = 0
2
Ω Ω ΩΩ
2 2
(9.36)
Repeating the process with cos ψk gives
˙˙ ˙ ˙ ˙˙ ζ δζκ ζ η δη + 2 + ( – 1) + 2 + 2 – / = 0
2
b Ω Ω ΩΩ
2 2
y l (9.37)
and eqn 9.35 can be written
˙˙ ˙ ˙˙
y y y r + 2 + – = 0 c c
2
g Ω Ω δ κ µ ζ
2
(9.38)
where µ = /( + )
1
2 b b bM M bM is the ‘mass ratio’, i.e. the ratio of half the blade mass
to the total mass.
Equations 9.36, 9.37 and 9.38 are three simultaneous differential equations with
constant coefficients in the variables η, ζ, and y. The solutions of these equations can
be obtained in the same way as the stability equations of Chapter 5 by assuming that
η = η0e
λt
, ζ = ζ0 e
λt
, y = y0 e
λt
Substituting in the equations and expanding the determinant of the co-efficients
leads to a frequency equation of the form
Aλ
6
+ Bλ
5
+ Cλ
4
+ Dλ
3
+ Eλ
2
+ Fλ + G = 0 (9.39)
where the coefficients are functions of the non-dimensional chassis and blade natural
frequencies κcΩ and κΩ. Unfortunately, the above sextic cannot be solved in general
terms and, even when solved numerically, the many parameters involved make
348 Bramwell’s Helicopter Dynamics
interpretation difficult. One method of obtaining useful results is to note that the
regions of instability will be bounded by curves representing undamped (neutral)
oscillations. These can be found by assuming a solution to eqn 9.39 in the form
λ = iω, where ω is real. Inserting this solution into the sextic and equating real
and imaginary parts leads to the equations
Aω
6
– Cω
4
+ Eω
2
– G = 0 (9.40)
Bω
4
– Dω
2
+ F = 0 (9.41)
in which A = 1.
To solve these equations we take a range of values of κc, calculate the values of a,
c, and e (which are functions of κc) and solve the biquadratic equation 9.41 for ω.
These values of ω are inserted into the left-hand side of eqn 9.40 which is then
plotted against ω. The values of ω which give zero values to the left-hand side of eqn
9.40 are solutions of eqns 9.40 and 9.41. We therefore have the values of the chassis
frequency at which undamped oscillations occur. Coleman and Feingold
11
and Price
12
have presented these boundaries in the form of charts which enable the ranges of
rotor speed, if any, for which the oscillations are unstable to be found.
Although the method described above gives a complete solution to the equations,
it is possible to derive some valuable conclusions by simpler approaches. We shall
describe first an approach given by Mil
13
.
Let us suppose that the chassis mode consists of lateral oscillations of amplitude
y0 and frequency p in the form
y = y0 sin pt (9.42)
Then, according to eqn 9.34 the motion of the blade is given by
.. .
ξ κ δξκξ ψ k k k k
y
l
p pt + 2 + = sin cos
0
b
2
Ω Ω
2 2
=
1
2
sin ( + ) +
2
+ sin ( – ) –
2 0 y
l
p p t
k
b
p t
k
b
Ω Ω
π π
{ } { }
(9.43)
Equation 9.43 represents a second order system excited by a pair of simpleharmonic forcing functions. The solution must therefore be of the form
ξk = ξ1 sin [(p + Ω)t + ψ1] + ξ2 sin [(p – Ω)t + ψ2] (9.44)
where ξ1, ξ2, ψ1, ψ2 are amplitudes and phase angles which can be determined from
the known response of such systems, Appendix A.4. Now, the blade motion would be
expected to reach large amplitudes if one of the forcing frequencies lies close to the
natural frequency of the lagging motion of the blade. Let us first consider resonance
in the case
κ Ω = |p – Ω|
The amplitude of the blade oscillations will then be determined almost entirely by
the second of the forcing terms on the right-hand side of eqn 9.43 and the response
can be written approximately as
Aeroelastic and aeromechanical behaviour 349
ξk = ξ0 cos [(p – Ω)t – 2πk/b] (9.45)
where
ξ0 = – y0p
2
/4δκΩ(p – Ω)
Now, from eqn 9.30 the centre of gravity of the blades when the chassis mode
consists of oscillations in the form given by eqn 9.41 is
y y r b t k b
k
b
k rg g
=0
–1
= + ( / ) cos ( + 2 / ) Σ ξ π Ω
Substituting for ξk from eqn 9.45 and expressing the products of the cosine terms
as sums gives
y y r b pt p t k b
k
b
rg g 0
=0
–1
= + ( /2 ) [cos + cos {(2 – ) + 4 / }] ξ π Σ Ω
From an analysis similar to that of Appendix A.3, we can easily show that
Σ
k
b
p t k b
=0
–1
cos [(2 – ) + 4 / ] = 0 Ω π
so that
y r pt rg
1
2 g 0 = cos ξ
The inertia force Pin acting on the chassis due to the oscillating centre of gravity
of the blades is given by
P bM y in b rg = – ˙˙
= – –
8 ( – )
cos b
b 0
4
g
b
bM y
bM y p r
l p
pt ˙˙
δ κ Ω Ω
and from eqn 9.35 the equation of motion of the chassis is
( ˙˙ ˙ M bM y M bM y M bM + ) + 2 ( + ) + ( + ) b c c b b c
2 2
δ κ κ Ω Ω
= –
8 ( – )
cos
b 0
4
b
bM y p r
l p
pt
g
δ κ Ω Ω
or
˙˙ ˙ y y y
y p r
l p
pt + 2 + = –
8 ( – )
cos c c c
2 2 0
4
g
b
δ κ κ
µ
δ κ
Ω Ω
Ω Ω
But the solution to this equation must correspond to the motion assumed originally,
eqn 9.41 and this requires that
p = κcΩ
and
δδ
µκ
κ κ
c
c
2
c
g
b
=
8 (1 – )
⋅
r
l
(9.46)
350 Bramwell’s Helicopter Dynamics
remembering also that
κΩ = | κc – 1| Ω
From these three relationships the critical rotor speed may be calculated. To ensure
that the oscillations at resonance will be no worse than neutrally damped, the product
of the dampings of the chassis and of the blade should satisfy the relationship eqn 9.46.
If we now consider the other possible resonant state represented by
κ Ω = | p + Ω |
we find that the above formulae remain the same but that Ω is everywhere replaced
by – Ω. Then, according to eqn 9.46 it appears that for undamped oscillations δδc
would have to be negative, i.e. that either δ and δc would have to be negative. Since
this is impossible, we conclude that ground resonance does not occur for κ Ω =
| p + Ω |.
Referring to eqn 9.46 again, and since δδc cannot be negative, undamped vibrations
can only occur if Ω > p. In Figs 9.19 and 9.20 this is satisfied only by point C, i.e.
the intersection of the branch of negative (regressive) values of κ – 1 with the chassis
frequency.
An interesting method of simplifying the ground resonance problem has been
given by Done
14
. In this method, details of which can be found in the original paper,
it has been assumed that, by a suitable choice of co-ordinates, the higher frequency
(progressive) mode branching from the point A in Fig. 9.20 can be neglected, so that
the simplified equations relate only to the chassis mode and to the lower frequency
rotor mode.
By means of this approximation and using eqns 9.36 to 9.38 Done arrives at a
characteristic quartic equation of the form
( + 2 + )[ + 2 + ( – ) ] +
( – )
2
2
c c c
2 2 2
c c
2 4
λ δ κλκ λδκ
µ
λ Ω Ω ΩΩΓ
Ω Γ
Γ
+
( – )
2
= 0 c c
2 3
µ
δ κ λ
Ω Γ
Γ
Ω
2
(9.47)
where Γ is the blade rotating lag frequency in the absence of Coriolis force coupling.
The roots λ = λre ± iλim, where λim ≡ κ, of eqn 9.47 can be plotted as shown in
Fig. 9.22. The lower diagram, showing the imaginary parts of the roots, corresponds
to the diagram of Fig. 9.20 but with the degrees of freedom coupled. As can be seen
the effect of coupling is to modify the shape of the diagram in the neighbourhood of
the (uncoupled) crossing points. If unstable oscillations occur, the branches fail to
meet at C since real roots occur in this region, as is shown by the small loop in the
λre diagram. The extent of the instability is indicated by the size of the gap between
the branches.
The amount of damping required which just gives harmonic oscillations can be
found by substituting λ = ± iκcΩ = ± i(Ω – Γ) into eqn 9.47. Done shows that this
leads to
δδ µκ κ κ c c
2
c = /8 (1 – )
Aeroelastic and aeromechanical behaviour 351
as given before in eqn 9.45 except that, since in Done’s analysis the blade is represented
by a point mass, the value of rg/lb is equal to unity.
The damping requirement to suppress ground resonance is strongly influenced by
the lag mode frequency.
Figure 9.23 indicates the variation of the damping requirement for a range of lag
mode frequencies. The closer the lag mode frequency is to 1Ω, the smaller the
amount of damping that is required to ensure stability.
As previously indicated, ground resonance is completely eliminated if the blade
lag frequency is greater than 1Ω. A rotor with this characteristic is termed ‘supercritical’.
This solution is found in the two-bladed teetering rotor and the Lockheed gyrocontrolled multi-blade ‘rigid rotor‘ design.
However, unless the lag mode frequency is sufficiently greater than 1Ω, e.g. 1.5Ω,
there may be fatigue strength problems due to high amplification of the first harmonic
air and Coriolis loads. Flap–lag instability is also a possibility for a rotor with a
‘supercritical’ lag frequency, as described in section 9.7.
A significant point to note when designing for, and demonstrating in practice,
freedom from ground resonance is that the full available range of rotor lift must be
considered. This is because chassis geometry, oleo stiffness, tyre stiffness, and damping
rates are all functions of the reaction at the ground contact position. The non-linear
effects of oleo stiffness are very significant, and due to oleo ‘sticking’ when the load
becomes less than the ‘break-out’ load in the high lift condition, the oleo cannot
dissipate any energy. Hence, apart from the effects of stiffness change, the only
significant contribution to chassis mode damping in this condition is provided by the
hysteresis effects of the tyres.
Also to be considered are the full range of tyre pressures (deflated to overinflated),
λre
κ
Rotor speed Ω
Fig. 9.22 Effect of coupling on fuselage and whirling frequencies
352 Bramwell’s Helicopter Dynamics
tie-down situation and wheel orientation (particularly relevant to Naval helicopters),
wheel brake situation, and surface contact conditions.
The various defined loading states of the helicopter will also have to be considered,
since these will influence the chassis mode frequencies and shapes, e.g. the roll/
lateral mode frequency will be reduced by the presence of external stores.
Viscous damping in both the airframe and rotor is normally assumed in the
analysis, and once stability boundaries have been established, a conversion to the
particular form of damping employed in the rotor can be made. For example, a
common design of blade hydraulic damper utilises a very high rate of ‘V
2
’ damping
followed by a constant force cut-off. The conversion of the viscous damping
requirement to another form of damping is based on equating the energy dissipation
per cycle of oscillation. This leads to the concept of an allowable blade ‘swing
angle’ in the ground resonance mode, above which the motion of the helicopter will
be divergent. Disturbances of the helicopter producing blade ‘swing angles’ below
this value will subside.
This situation implies that it is necessary to know the levels of hub acceleration in
the plane of the rotor which will be experienced in service, and which may force the
blades to oscillate in the ground resonance mode.
In order to ensure stability over the full range of the many variables involved, it
has become the practice to define a chassis mode case (which may be realistic or
artificial) that will lead to the worst possible instability.
It can be shown that more blade lag damping is required as the chassis mode frequency
increases (provided that the frequency coalescence still occurs below the maximum
possible rotor speed). Thus an assumption is often made in the design stage that the
frequency coalescence occurs at the maximum operating rotor speed, and sufficient
60
40
20
Fuselage damping, % critical
Lag frequency
0.2Ω
0.3Ω
0.4Ω
0.5Ω
0.6Ω 0.7Ω
0 20 40 60
Lag damping, % critical
Fig. 9.23 Damping requirement as a function of lag mode frequency
Aeroelastic and aeromechanical behaviour 353
damping is then provided to ensure stability in this worst case. Thus, freedom from
ground resonance will be established for the full range of operating conditions.
9.10 Air resonance
In Chapter 5, the dynamic stability of the helicopter was discussed on the assumption
that the motion was ‘quasi-steady’, i.e. the acceleration of the helicopter could be
ignored and the response of the rotor depended only in the instantaneous translational
and angular velocities of the helicopter. As we shall see, in making this assumption
a certain mode of motion is suppressed which is of little significance as a conventional
stability mode, but may, in the case of hingeless rotors, couple with the blade ‘regressive’
lag mode to produce an instability in flight which is closely related to ground resonance.
For such rotor systems, the regressive cyclic flapping mode can couple with the
fuselage roll and pitch motions to produce ‘slow gyroscopic’ modes (sometimes also
referred to as ‘pendulum’ modes) of the helicopter at frequencies which are close to
the ‘regressive’ lag mode frequency at normal operating rotor speed.
Let us suppose that the hingeless helicopter is pitching and rolling with a frequency
high enough to prevent significant translational velocities. As we shall be concerned only
with the first flapping mode, let the displacement of a point of the blade be given by
Z = RS1(x)β (ψ)
where we write β (ψ) as the azimuth co-ordinate of the blade by analogy with the
flapping angle β of the rigid blade, as in section 7.5. The mode response equation is,
by eqn 7.86.
d
d
+ =
1
(1)
( )d
2
2 2 2 1
β
ψ
λ β 1
2
0
1
Ω R f
F
x
S x x
∫
∂
∂
where λ1Ω is the flapping frequency and S1(x) is the first mode bending shape.
For pitching and rolling motion only, the blade loading ∂F/∂x can easily be shown
to be
∂
∂
F
x
ac R xS xp xq
2
= –
d
d
+ sin + cos
1
2
3
1 ρ
β
ψ
ψ ψ Ω ˆ ˆ
+ 2 ( cos – sin ) +
d
sin +
d
cos
2 2 2 2
m R xp xq m R x
p
d
x
q
d
Ω Ω ˆ ˆ
ˆ ˆ
ψ ψ
ψ
ψ
ψ
ψ
The first bracket represents the change of aerodynamic incidence due to the angular
motion and flapping; the second and third brackets denote the gyroscopic and angular
acceleration inertia forces. The flapping equation of the kth blade can then be written
d
d
+
2
d
d
+ =
2
( sin + cos )
2
2
2 1
1
2 2
1 1
β
ψ
γ β
ψ
λ β
γ
ψ ψ
k k
k
E
F p F q ˆ ˆ
+ 2 cos – 2 sin +
d
d
sin +
d
d
cos
2
1
γ
γ
ψ ψ
ψ
ψ
ψ
ψ ˆ ˆ
ˆ ˆ
p q
p q
(9.48)
354 Bramwell’s Helicopter Dynamics
where E1, F1, γ1 and γ2 have been defined in section 7.5.
We now define the Coleman co-ordinates of flapping motion by
a
b
b
b k
b
k k
k
b
k k 1
=0
–1
1
=0
–1
= –
2
cos ; = –
2
sin Σ Σ β ψ βψ (9.49)
Then, adopting the same procedure as for the lagging equations, the result of summing
eqn 9.48 over all the blades leads to
′′ ′ ′
a a a b b
F
q F p
q
1 1 1
2
1 1 1
2 1
+ + ( – 1) + 2 + = –
2
– 2 +
d
d
ν λ ν
γ
ψ
ˆ ˆ
ˆ
(9.50)
′′ ′ ′
b b b a a
F
p F q
p
1 1 1
2
1 1 1
2 1
+ + ( – 1) – 2 – = –
2
+ 2 +
d
d
ν λ ν
γ
ψ
ˆ ˆ
ˆ
(9.51)
where ν γ γγ = /2, = / 2 1 2 1 E F , and the dashes denote differentiation with respect
to ψ.
From eqn 7.106 of Chapter 7, the moment exerted on the hub by the flapping
deflection of the kth blade is
M R mxS x x k k = ( – 1) ( ) d
1
2 2
0
1
1 λ β Ω
3
∫
Resolving about the rolling and pitching axes and summing over all the blades
gives for the rolling and pitching moments
L b ac R b = ( – 1)
2 4
1
2
1 1 ρ λ γ Ω /2
M b ac R a = ( – 1)
2 4
1
2
1 1 ρ λ γ Ω /2
Including the thrust moment, the rolling and pitching equations of the helicopter
are
A
p
t
b ac R
b Thb
d
d
=
( – 1)
+
2 4
1
2
1
1 1
ρ λ
γ
Ω
2
(9.52)
B
q
t
b ac R a
Tha
d
d
=
( – 1)
+
2 4
1
2
1
1
1
ρ λ
γ
Ω
2
(9.53)
Non-dimensionalising eqns 9.52 and 9.53 by dividing by ρsAΩ
2
R
3
gives
d
d
=
( – 1)
*
+
*
1
2
1
1
c
1
ˆ p a
i
b
t h
i
b
A A ψ
λ
µ γ µ 2
(9.54)
d
d
=
( – 1)
*
+
*
1
2
1
1
c
1
ˆ q a
i
a
t h
i
a
B B ψ
λ
µ γ µ 2
(9.55)
where µ*, iA, and iB are the mass parameter and inertia coefficients defined in
Chapter 5.
Aeroelastic and aeromechanical behaviour 355
The rolling and pitching motions when the blade flapping acceleration is included
are represented by the four equations
′′ ′ ′ a a a b b Fp F
q
Kq 1 1 1 1 1 + + + 2 + + 2 +
d
d
+ = 0 ν χ ν
ψ
ˆ
ˆ
ˆ (9.56)
– 2 – + + + +
d
d
+ – 2 = 0 1 1 1 1 1 ′ ′′′ a a b b b F
p
Kp Fq ν ν χ
ψ
ˆ
ˆ ˆ (9.57)
k b
p
A 1 –
d
d
= 0
ˆ
ψ
(9.58)
k a
q
B 1 –
d
d
= 0
ˆ
ψ
(9.59)
where
k
a
i
t h
i
k
a
i
t h
i
A
A A
B
B B
=
( – 1)
*
+
*
; =
( – 1)
*
+
*
1
2
1
c 1
2
1
c λ
µ γ µ
λ
µ γ µ 2 2
K F = /2, = – 1 2 1 1
2
γ χ λ
The usual form for the solution, ˆ ˆ p p = e 0
λτ
… etc., leads to a sextic characteristic
equation
Aλ
6
+ Bλ
5
+ Cλ
4
+ Dλ
3
+ Eλ
2
+ Fλ + G = 0
where
A = 1
B = 2ν
C F k k A B = 4 + 2 + + ( + ) χ ν
2
D k k K F A B = 2 (2 + ) + ( + ) ( + ) ν χ ν
E k k F K k k F A B A B = + + ( + ) [ (4 + ) + ] + χ ν χ ν
2 2 2
F k k K F FKk k A B A B = ( + ) ( + 2 ) + 2 χ ν
G k k K F A B = ( + 4 )
2 2
Taking as typical values
ν χ = 0.836, = 0.245, = 1.08, = 1.146 F K
k k Α Β
= 0.102, = 0.0204,
the roots of the sextic are found to be
λ1,2 = – 0.408 ± 2.03i
λ3,4 = – 0.215 ± 0.246
λ5 = – 0.356, λ6 = – 0.07
If the complex roots are substituted back into the equations of motion and the
356 Bramwell’s Helicopter Dynamics
ratios of the variables are examined, as was done in Chapter 5, it is found that the first
complex pair corresponds to a tilt of the rotor disc whose axis moves in the same
direction as the rotor with a frequency just over twice that of the rotor; the second
pair of roots corresponds to a rotor tilt which is a retrograde, or backward, motion,
with a frequency about a quarter that of the rotor. It is this latter mode of motion
whose frequency is likely to lie close to the lagging frequency of the blade.
Let us suppose that the shaft is fixed, i.e. that the pitching and rolling motion is
now absent. This assumption can be expressed mathematically by making kA and kB
infinitely large. The sextic is found to degenerate into a quartic which, with the other
constants having the same values as before, has the roots
λ1,2 = – 0.42 ± 2.03i
λ3,4 = – 0.42 ± 0.21i
It can be seen that the fast, forward, motion is hardly affected by constraining the
angular motion of the fuselage and that only the damping of the slow, backward,
mode is changed significantly. It can easily be shown
15
that the motion described
above is exactly the same as that of a damped spring-restrained gyroscope whose
polar moment of inertia is the same as that of the rotor, and the spring stiffness is the
same as the hub moment due to the rotor tilt.
The motion of the helicopter consists, therefore, of a fast rotor precession, which
is practically independent of the motion of the fuselage, and a slow precession in
which the fuselage motion, mainly rolling, has some influence on the damping and
frequency. Because the fuselage moves considerably in this latter mode, and appears
to rock under the rotor, it is often referred to as the ‘pendulum’ mode, but this is an
incorrect description since, as we have seen, the mode is still present even when the
fuselage is fixed. The ‘slow gyroscopic mode’ is suggested as a more apt description.
The two real roots of the sextic correspond to the damping in pitch and roll, as
already discussed in Chapter 5.
We now have to investigate the interaction of this blade flapping and fuselage
motion with the lagging motion of the blades. Due to the rolling and pitching of the
fuselage, the associated acceleration of the hub contributes inertia terms to the lagging
equations as in the case of ground resonance. However, blade flapping also occurs in
this motion, and the corresponding Coriolis inertia terms must be included. Also, the
inertia of the lagging blades causes further pitching and rolling moments on the
helicopter.
The Coriolis terms due to blade flapping are proportional to β β ψ d /d , which, for
small perturbations, can be linearised to a0 d /d β ψ , a0 being assumed constant at the
value for steady hovering flight. There is also a lagwise aerodynamic force due to the
change of direction of the local flow when the blade flaps, and this is clearly proportional
to the local induced velocity and to the flapping rate d /d β ψ . Thus, both the coning
angle and the induced velocity are parameters of the motion.
If the co-ordinate of a point of the lagging blade is expressed as
Y = RT1 (x) ξ(ψ)
Aeroelastic and aeromechanical behaviour 357
the following non-dimensional quantities may be defined
15
, as
ˆ
H
bg R
W
mT x x J
mS T S x x
mT x
=
2
( )d ; =
(d /d )d
d
0
1
1
1 1 1
∫
∫
∫
0
1
0
1
1
2
L a J H
mT x
mT x
= 2 ; =
d
d
0
1
1
2
∫
∫
0
1
0
1
and in addition to eqns 9.50 and 9.51 we have the roll, pitch, and lagging equations
in the form
hH
i
k b
p
A
A
ˆ ˆ
+ –
d
d
= 0 1 ′′ ζ
ψ
(9.60)
– + –
d
d
= 0 1
hH
i
k a
q
B
B
ˆ ˆ
′′ η
ψ
(9.61)
′′ ′ ′ ′ ζ δζκ ζηδη
ψ
+ 2 + ( – 1) + 2 + 2 + – –
d
d
= 0
2
1 1 La Lb Hh
p ˆ
(9.62)
′′ ′ ′ ′ η δηκ ηζδζ
ψ
+ 2 + ( – 1) – 2 – 2 + + +
d
d
= 0
2
1 1 La Lb Hh
qˆ
(9.63)
The derivation of these equations is given in Bramwell’s paper
15
. As can be seen,
we now have six second order differential equations which lead to a polynomial
characteristic equation of the tenth degree when the usual method of solution is
adopted. Lytwyn et al.
16
have considered a mathematical model with eighteen degrees
of freedom which includes flexibility of the fuselage and of the feathering mechanism.
The problem is too complicated for simple results to be obtained as was found
possible with ground resonance. Numerical methods have to be used for calculating
the roots of the equations for ranges of values of the parameters occurring in a
particular case.
The results of such calculations, though, have been rather inconsistent. There
appears to be no ‘resonance’ of the kind occurring in the ground resonance problem,
i.e. there is no sudden deterioration of damping near the coincidence of blade lagging
and fuselage rolling mode frequencies. There may be an unstable lagging motion
17
when only the natural air damping (about 2 per cent of critical) and structural damping
are present, and this instability remains fairly constant over a wide range of rotor
speeds. This may be due to the fact that the flap damping is very high, which would
tend to ‘flatten out’ the response curve so that resonance is not apparent. Increasing
the lag damping to about 5 per cent of critical seems to place this mode in the stable
region. Typical roots from such calculations for the case of 2 per cent damping are
λ1,2 = + 0.008 ± 0.37i (slow)
λ3,4 = – 0.03 ± 1.7i (fast)
the unit of time being that corresponding to one radian of rotor revolution.
358 Bramwell’s Helicopter Dynamics
Figure 9.24 indicates the real part of the slow lag mode root as a function of rotor
speed for a range of values of fundamental lag mode frequency and 2 per cent critical
lag damping.
Since the lag natural frequency for semi-rigid and hingeless rotor systems is
typically greater than 0.6Ω, it is seen that stability is ensured for values of rotor speed
throughout the normal range and well beyond.
Because the fuselage ‘stiffness’ term arises from blade flapping, which is
aerodynamically heavily damped, the effective damping in the ‘slow gyroscopic’
modes is high. However, we have noted that the important parameter in the suppression
of ground resonance is the product of the blade lag mode and chassis mode dampings.
Indeed, analysis shows that with high fuselage damping and low lag damping the
width of the unstable region is large
17
.
Thus, the possibility of air resonance remains if the lag damping is very low. On
the other hand, analysis shows that the amount of lag damping required to suppress
the instability is quite small. Consequently air resonance will only be a potential
problem for rotors with relatively high flap stiffness and low lag damping, a possible
combination for both semi-rigid and hingeless rotor systems.
However, the amount of lag damping required is much less than is typically required
to suppress ground resonance. Therefore, a helicopter which is stable under the
conditions most likely to give rise to ground resonance is unlikely to be subject to an
air resonance problem.
Reference 19 provides useful additional information on aspects of air resonance.
References
1. Loewy, R. G., ‘Review of rotary-wing V/STOL dynamic and aeroelastic problems’, J. Amer.
Helicopter Soc., July 1969.
2. Hansford, R. E., and Simons, I. A., ‘Torsion–flap–lag coupling on helicopter rotor blades’, J.
Amer. Helicopter Soc., October 1973.
0.02
0.01
– 0.01
– 0.02
0.8
0.9 1.0 1.1
1.2
0.6
0.4
0.2
ωb
Ω
0.8
Ω
Slow lag mode
Fig. 9.24 Real part of slow lag mode root, 2 per cent critical lag damping
Aeroelastic and aeromechanical behaviour 359
3. Pei, Chi Chou, ‘Pitch-lag instability of helicopter rotor blades’, J. Amer. Helicopter Soc.,
July 1958.
4. Stammers, C. W., ‘The flutter of a helicopter rotor blade in forward flight’, Aeronaut. Q.,
February 1970.
5. Carta, F. O., ‘An analysis of the stall flutter instability of helicopter rotor blades’, J. Amer.
Helicopter Soc., October 1967.
6. Ham, N. D. and Young, M. I., ‘Torsional oscillations of rotor blades due to stall’, J. Aircraft,
May to June 1966.
7. Ham, N. D. ‘Helicopter blade flutter’, AGARD Rep. 607, January 1973.
8. Coleman, Robert P., and Stempin, Carl W., ‘A preliminary theoretical study of aerodynamic
instability of a two-bladed helicopter rotor’, NACA Res. Memo. L6H 23, 1946.
9. Balmford, D. E. H., Hansford, R. E. and King, S. P. ‘Helicopter dynamics’. Westland Helicopters
Ltd. Report, March 1985.
10. Ormiston, Robert A. and Hodges, Dewey H., ‘Linear flap–lag dynamics of hingeless rotor
blades in hover’, J. Amer. Helicopter Soc., April 1972.
11. Coleman, R. P. and Feingold, A. M., ‘Theory of self-excited mechanical oscillations of helicopter
rotors with hinged blades’, NACA Rep. 1351, 1958.
12. Price, H. L., ‘Simplified helicopter ground resonance stability boundaries’, Aircraft Engineering,
October and November 1962.
13. Mil, M. L., et al., Helicopters – calculation and design, vol. II, ‘Vibrations and dynamic
stability’, NASA Tech. Transl. NASA TT F–519, 1968.
14. Done, G. T. S., ‘A simplified approach to helicopter ground resonance’, Aeronaut. J., May 1974.
15. Bramwell, A. R. S., ‘An introduction to helicopter air resonance’, Aeronautical Research
Council R & M. 3777, 1975.
16. Lytwyn, R. T., Miao, W., and Woitsch, W., ‘Airborne and ground resonance of hingeless
rotors’ J. Amer. Helicopter Soc., April 1971.
17. Baldock, J. C. A., ‘Some calculations for air resonance of a helicopter with non-articulated
rotor blades’, Aeronautical Research Council R & M 3743, 1974.
18. Ormiston, R. A., ‘Aeromechanical stability of soft in-plane hingeless rotor helicopters,’ Paper
25, 3rd European Rotorcraft Forum, Aix-en-Provence, September 1977.
19. King, S. P. ‘Theoretical and experimental investigations into helicopter air resonance,’ 39th
Annual Forum of American Helicopter Society, St Louis, MO, May 1983.
Appendices
A.1 Euler’s equations
A.1.1 Angular momentum and the equations of angular motion
We define the relative angular momentum, h, of a system of particles comprising a
body by
h = ∑r × (mvr) (A.1.1)
where r is the position vector of a particle of mass m and vrel is the velocity of the
particle relative to the origin, which may be that of a moving frame. The summation
is taken over all the particles of the system. Then
dh/dt = ∑(dr/dt) × (mvrel) + ∑r × (mdvrel/dt) (A.1.2)
Now, if v is the absolute velocity of the particle and v0 the velocity of the origin
of the moving frame,
vrel = dr/dt = v – v0
and dvrel/dt = dv/dt – dv0/dt
Since
(dr/dt) × vrel = (dr/dt) × (dr/dt)
the first term of eqn A.1.2 is zero; therefore,
dh/dt = ∑ r × (mdv/dt) – ∑r × (mdv0/dt)
But mdv/dt = F, where F is the resultant external force acting on the particle.
Hence
∑r × (mdv/dt) = ∑r × F = T
where T is the moment of the external forces about the origin.
Also, since dv0/dt is constant over the system of particles,
Appendices 361
∑r × (mdv0/dt) = Mrg × a0
since ∑mr is the mass moment of the system relative to the origin, M being the total
system mass, rg the position vector of the centre of gravity of the system, and a0 the
acceleration of the origin of the moving frame. Thus, finally,
dh/dt = T – Mrg × a0 (A.1.3)
We can obtain an alternative formula by defining the absolute angular momentum
by
H = ∑r × (mv) (A.1.4)
where, as above, v is the absolute velocity of the particle. By an argument similar to
that above we obtain the equation
dH/dt = T – v0 × Mvg (A.1.5)
where vg is the velocity of the centre of gravity of the system of particles.
We now expand the general vector eqns A.1.3 and A.1.5 in terms of components
measured in the chosen frame of reference.
From the defining equations A.1.1 and A.1.2 we find that
H = h + ∑r × mv0 = h + Mrg × v0
For subsequent applications we shall choose axes which are fixed in the body, as
it is clearly convenient that the inertial properties of the body should remain constant
with time. We can then write for the particle velocity vr
vr = × r
where is the angular velocity of the body and axes, so that, from eqn A.1.1
h = ∑r × (m × r)
= ∑m(r · r) – ∑mr(· r) (A.1.6)
Now let us write
r = xi + yj + z k
= ω1i + ω2j + ω3k
where i, j, k are a set of orthogonal unit vectors fixed in the body. If we also write
h = h1i + h2 j + h3k
we find on expanding eqn A.1.6 that the the components of angular momentum are
given by
h A F E
h B D F
h C E D
1 1 2
2 2 3
3 3 1
= – –
= – –
= – –
ω ω ω
ω ω ω
ω ω ω
3
1
2
(A.1.7)
where A, B, C, D, E, F are the moments and products of inertia defined by
362 Bramwell’s Helicopter Dynamics
A = ∑m(y
2
+ z
2
), B = ∑m(x
2
+ z
2
) , C = ∑m(x
2
+ y
2
)
D = ∑myz, E = ∑mxz, F = ∑mxy
The relations given by eqn 1.12 can be expressed conveniently in matrix form as
h
h
h
A F E
– F B D
– E – D C
1
2
3
1
2
3
=
– –
–
ω
ω
ω
where the square matrix is sometimes referred to as the inertia tensor.
The inertia tensor can be regarded as an operator which transforms the angular
velocity vector into the angular-momentum vector.
It is always possible to choose axes through any point in the body such that the
products of inertia D, E, F vanish. These axes are called principal axes, in which case
h = Aω1i + Bω2 j + Cω3k (A.1.8)
If the origin of the axes is a fixed point or the centre of gravity, we find that
dh/dt = dH/dt = T (A.1.9)
since, in both cases, rg × a0 = 0 and either v0 = 0 or v0 = vg so that v0 × vg = 0.
Since, in general, the axes will be moving
dh/dt = ∂h/∂t + × h (A.1.10)
and taking the special case of rotation about a fixed point and referring the motion to
principal axes, eqns A.1.8, A.1.9 and A.1.10 give
A B C L .
ω ω ω 1 2 – ( – ) = 3 (A.1.11)
B C A M .
ω ω ω 2 3 – ( – ) = 1 (A.1.12)
C A B N .
ω ω ω 3 1 – ( – ) = 2 (A.1.13)
where L, M, N are the components of the torque T.
Equations A.1.11, A.1.12 and A.1.13 are known as Euler’s dynamical equations
and can easily be remembered from their cyclic form.
A.1.2. Euler’s equations for a rigid blade
In dealing with blade motion, however, we often wish to regard the blade as a rigid
body moving about a hinge system which is offset from the rotor axis. To simplify
matters we can assume that the hinge system is effectively concentrated at a point.
Thus, the blade moves about a point which is not fixed, Fig. A.1.1, and Euler’s
equations no longer apply.
On the other hand, it would be inconvenient to take the centre of gravity as origin
for, in calculating the torque, we would have to consider the unknown reactions at the
hinge. We can, however, extend Euler’s equations by making use of eqn A.1.3. Let us
write the position vector of the c.g. relative to the hinge as
Appendices 363
rg = xgi + yg j + zgk
and the absolute acceleration of the hinge as
a0 = axi + ayj + azk
Then the term rg × a0 in eqn A.1.3 becomes
rg × a0 = (ygaz – zgay)i + (zgax – xgaz)j + (xgay – ygax)k
and, taking principal axes as before, eqn A.1.3 can be expanded to give
A .
ω 1 – (B – C)ω2ω3 – Mb(ygaz – zgay) = L (A.1.14)
B .
ω 2 – (C – A)ω3ω1 – Mb(zgax – xgaz) = M (A.1.15)
C .
ω 3 – (A – B)ω1ω2 – Mb(xgay – ygax) = N (A.1.16)
in which Mb is the blade mass. Equations A.1.14, A.1.15 and A.1.16 are referred to
as the ‘extended’ Euler equations. Actually, the centre of gravity of the blade can be
considered to lie practically on the i axis, so that we can write rg = xgi and the above
equations can be simplified to
A .
ω 1 – (B – C)ω2ω3 = L (A.1.17)
B .
ω 2 – (C – A)ω3ω1 + Mbxgaz = M (A.1.18)
C .
ω 3 – (A – B)ω1ω2 – Mbxgay = N (A.1.19)
If the hinge offset distance is eR and the rotor shaft rotates with angular velocity
Ω, then a0 is clearly of magnitude Ω
2
eR and is directed from the hinge to the shaft
axis.
A.2 The stability equations
The theory of Appendix A.1 can be applied to obtain the stability equations which are
discussed in detail in Chapter 5. We take an orthogonal set of axes, fixed in the
helicopter and whose origin is located at the helicopter’s centre of gravity. It can be
supposed, as is usual with the fixed wing aircraft, that the helicopter has a longitudinal
plane of symmetry, although this is rather less true of the single rotor helicopter
k
i
j
Ω
c.g.
o
Fig. A.1.1 Blade pivot point
364 Bramwell’s Helicopter Dynamics
because of its tailrotor. The axes will be chosen so that the x and z directions lie in
the longitudinal plane with the y axis pointing to starboard, Fig. A.1.2.
The equations of motion of the helicopter, treating it as a rigid body, are
F = mdv/dt (A.2.1)
and T = dh/dt (A.2.2)
where F is the resultant external force on the helicopter and m its mass, and T is the
moment of this force.
We shall suppose that in trimmed flight there is no sideslip, so that the initial flight
velocity components along the x, y, z axes are U, 0, W. During disturbed flight the
increments of velocity will be denoted by u, v, w, so that the velocity vector v can be
written as
v = (U + u)i + vj + (W + w)k (A.2.3)
It will be supposed that the disturbance velocity components are small compared
with the steady components U and W. In trimmed flight the angular velocity of the
helicopter can be written as
= pi + qj + rk (A.2.4)
Then if X, Y, Z are the components of the force F, the force equations, from eqn
A.2.1 are
m[ . u + q(W + w) – vr] = X (A.2.5)
m[ . v + r(U + u) – p(W + w)] = Y (A.2.6)
m[ . w + pv – q(U + u)] = Z (A.2.7)
The angular momentum equation (eqn A.2.2) can be written
T = ∂h/∂t + × h (A.2.8)
y
x
z
Fig. A.1.2 Axis system for the helicopter
Appendices 365
and the components of momentum are, from eqn A.1.7,
h1 = Ap – Fq – Er
h2 = Bq – Dr – Fp
h3 = Cr – Ep – Dq
With T = Li + Mj + Nk, the expansion of eqn A.2.8 gives
A . p – (B – C)qr + D(r
2
– q
2
) – E(pq + . r) + F(pr – . q) = L (A.2.9)
B . q – (C – A)rp + E(p
2
– r
2
) – F(qr + . p) + D(qp – . r) = M (A.2.10)
C . r – (A – B)pq + F(q
2
– p
2
) – D(rp + . q) + E(rq – . p) = N (A.2.11)
The two sets of equations A.2.5 to A.2.7 and A.2.9 to A.2.11 can be simplified
considerably by assuming that the disturbance velocity and angular velocity components
are so small that squares and products of them can be neglected. Further, if we
assume that the helicopter has a plane of symmetry, the products of inertia D and F
both vanish.
Then the force and moment equations simplify to
mu . + qW = X (A.2.12)
m. v+ rU – pW = Y (A.2.13)
mw . – qU = Z (A.2.14)
and Ap Er L . . – = (A.2.15)
Bq . = M (A.2.16)
Cr Ep N . . – = (A.2.17)
In particular, if ‘wind axes’ are chosen, i.e. if the x axis is initially taken to lie
parallel to the flight direction, then W = 0 and the terms containing W vanish in eqns
A.2.12 and A.2.13.
A.3 Multiblade summations
In some helicopter problems it is necessary to calculate the total force or moment on
the helicopter by adding the contributions of the individual blades. In steady motion
these blade contributions are periodic, and a typical term for a given blade would be
An cos nψ. If there are b equally spaced blades, the contribution of the neighbouring
blades will be An cos n (ψ ± 2π/b) and the total effect of all the blades is therefore
An cos nψ + An cos n(ψ + 2π/b) + … + An cos n[ψ + 2π (b – 1)/b]
= cos ( + 2 / )
0
–1
A n k b n
k=
b
Σ ψ π
366 Bramwell’s Helicopter Dynamics
Let C n k b
k=
b
= cos ( + 2 / )
0
–1
Σ ψ π
and S n k b
k=
b
= sin ( + 2 / )
0
–1
Σ ψ π
so that
C S
k=
b
n k b n
k=
b
kn b
+ i = e = e e
0
–1
i ( + 2 / ) i
0
–1
2 i/
Σ Σ
ψ π ψ π
The terms in the summation are a geometric series, and we easily find that
C S
n n
n b
+ i =
e (e – 1)
e – 1
i 2 i
2 i/
ψ π
π
If n is not an integer, C + iS can be written
C S
n n n n
n b n b n b
+ i =
e e (e – e )
e (e – e )
i i i –i
i / i –i
ψ π π π
π π π / /
=
sin
sin ( / )
cos +
– 1
+ i sin +
– 1 π
π
ψ π ψ π
n
n b
n
b
b
n
b
b
giving
C
n
n b
n
b
b
=
sin
sin ( / )
cos +
– 1 π
π
ψ π
(A.3.1)
and
S
n
n b
n
b
b
=
sin
sin ( / )
sin +
– 1 π
π
ψ π
(A.3.2)
If n is an integer but not a multiple of the number of blades b, we see that C +
iS = 0. If n is a multiple of b
C + iS = 0/0
By L’Hospital’s theorem,
C S
n
n
n
n
n b
n b
+ i = e
d(e – 1)/d
d(e – 1)/d
i
2 i
2 i/
= multiple of
ψ
π
π
=
e [2 i e ]
2 i e
i 2 i
2 i/
b
n n
n b
ψ π
π
π
π
= b e
inψ
giving
C = b cos nψ and S = b sin nψ (A.3.3)
Appendices 367
Thus
C
n
n b
n
b
b
n =
sin
sin ( / )
cos +
– 1
, if is not an integer
π
π
ψ π
= 0, if n is not a multiple of b
= b cos nψ, if n is a multiple of b
S
n
n b
n
b
b
n =
sin
sin ( / )
sin +
– 1
, if is not an integer
π
π
ψ π
= 0, if n is not a multiple of b
= b sin nψ, if n is a multiple of b
The Coleman co-ordinates
In ground and air resonance problems (Chapter 9) there are equations of the form
.. . x kx x p t q t k k k k k + 2 + = ( ) sin + ( ) cos +
2
Ω Ω … ψ ψ (A.3.4)
where p(t) and q(t) are functions of time.
The equation can be taken to represent a variable quantity xk which is measured
with respect to the rotating kth blade. We wish to find the total effect of all the blades.
To do this we define new co-ordinates u and v, say, such that
u b x
k=
b
k k = – (2/ ) cos
0
–1
Σ ψ (A.3.5)
v b x
k=
b
k k = – (2/ ) sin
0
–1
Σ ψ (A.3.6)
where, as in the previous section, ψk takes the values ψ, ψ + 2π/b, …, ψ + 2π(b – 1)/b.
Differentiating eqns A.3.5 and A.3.6, we easily find, remembering dψ/dt = Ω,
Σ Σ Ω Ω
k=
b
k k
k=
b
k k x b u x b u
0
–1
0
–1
sin = ( – )/ 2; cos = – ( + )/ 2 . . . . ψ ψ v v
Σ Ω Ω
k=
b
k k x b u
0
–1
sin = – ( – 2 – )/2 .. .. . ψ v v
2
Σ Ω Ω
k=
b
k k x b u u
0
–1
cos = – ( + 2 – )/2 .. .. . ψ v
2
Also, we can show that
Σ Σ
k=
b
k
k=
b
k b
0
–1
2
0
–1
2
sin = cos = /2 ψ ψ
and Σ
k=
b
k k
0
–1
sin cos = ψ ψ 0
368 Bramwell’s Helicopter Dynamics
Then, multiplying eqn A.3.4 by cos ψk, summing over the blades, and using the
above relationships, we get
.. . . u ku k q t + 2 + 2 + 2 = – ( ) Ω Ω Ω v v
2
Similarly, performing the same procedure with sin ψk,
.. . . v v + 2 – 2 – 2 = – ( )
2
Ω Ω Ω k u ku p t
The transformations represented by eqns A.3.5 and A.3.6, called the Coleman
transformations, have removed the periodic terms from eqn A.3.4. They effectively
resolve a rotating quantity into components along axes fixed in the helicopter body.
A.4 The frequency response of a second order system
It has been seen that the rotor blade, whether supposed rigid or flexible, can be
treated as a spring–mass–dashpot, or second order, system when the blade displacement
from equilibrium is small. Further, in steady flight, the loads forcing the blade are
periodic and can be expressed in the form of a Fourier series. The blade motion can
therefore be derived from the response of a second order system to a harmonic
forcing function and it is useful to present the main results in this Appendix.
The differential equation of the motion can be represented typically as
mx cx kx F t .. . + + = cos 0 ω (A.4.1)
where m is the mass of the system, c is the viscous damping coefficient, k is the
spring stiffness, and F0 is the amplitude of the applied force.
When F0 = 0 we have the case of free motion; when the damping is zero the free
motion is sinusoidal with natural angular frequency ωn given by ωn = √(k/m).
Let us write the critical damping coefficient (just suppresses oscillatory motion)
ccrit = 2 √(k/m) = 2k/ωn and define a non-dimensional damping factor ζ by ζ = c/ccrit.
The equation of motion, eqn A.4.1, can then be written
.. . x x x F m t + 2 + = ( / ) cos n n 0 ζω ω ω
2
The solution of this equation is known to be
x = x0 cos (ω t – ε) (A.4.2)
where x0 = F0/ √[(k – mω
2
)
2
+ c
2
ω
2
]
and ε = tan
–1
[cω/(k – mω
2
)]
Defining a frequency ratio ˜
ω by ˜
ω = ω/ωn, x0 and ε can be written
x0 = F0/[k √{(1 – ˜
ω
2
)
2
+ 4ζ
2
˜
ω
2
}] (A.4.3)
and
ε = tan
–1
[2ζ ˜
ω /(1 – ˜
ω
2
)]
Appendices 369
Now F0/k is the static deflection, i.e. the displacement of the system under a steady
load F0. Denoting this deflection by xst we can represent the amplitude x0 of the
oscillating displacement as the static deflection multiplied by a magnification factor
µ, where
µ = x0/xst = 1/√[(1 – ˜
ω
2
)
2
+ 4ζ
2
˜
ω
2
] (A.4.4)
Thus the response of the system can be completely expressed by the quantities µ
and ε as functions of the frequency ratio ˜
ω and the damping ratio ζ, these quantities
being shown plotted in Figs A.4.1 and A.4.2.
It can be seen from the solution of eqn A.4.2 that the response frequency is always
0.5 1.0 1.5 2.0 2.5 3.0
180°
135°
90°
45°
0
0.1
0.25
0.5
1
2
4
4
2
0.5
1
0.1
˜ ω
ε
0.25
ξ = 0.01
Fig. A.4.2 Phase relationship between response of second order system and forcing function
5
4
3
2
1
0
0.5 1.0 1.5 2.0 2.5
0.1
0.15
0.25
0.5
1
4
ξ = 0
ξ = 0
µ
˜ ω
2
Fig. A.4.1 Amplitude of second order system to harmonic forcing function
370 Bramwell’s Helicopter Dynamics
the same as the forcing frequency. When ˜
ω = 1, i.e. when ω = ωn, the system is said
to be forced at its resonant frequency. It might be thought that the response amplitude
is greatest at this frequency, but examination of Fig. A.4.1 shows that the maximum
amplitude occurs at rather less than the resonant frequency. In fact, the aerodynamic
damping factor of a flapping blade is typically of the order ζ = 0.4, from which we
see that the maximum amplitude occurs at ˜
ω = 0.85. However, at resonance the
phase angle is always 90° however great the damping.
Index
Rayleigh–Ritz method, 246
Southwell formula, 248
flapwise bending modes and frequencies,
238
intermodal coupling, 262
lagwise bending modes and frequencies,
257
forced response equation, 269
torsional modes and frequencies, 260
Biot–Savart law, 198
Blade deflections measured in flight, 276
Blade element theory:
forward flight, 92
vertical flight, 46
Blades, bending, see bending of blades
Blades, dynamic design, 294
Blade–vortex interactions (BVI), 219–221
Boundary layer on rotating blade:
forward flight, 235
hovering flight, 233
Circulation:
effect of finite number of blades,
62
Coleman co-ordinates, 367
Coleman et al:
ground resonance, 342
induced velocity distribution, 79
Coning angle, equation for, 107
Control derivatives, 180–183
Control response, 180–192
to cyclic pitch, 185
to vertical gusts, 190
to lateral cyclic pitch, 188
to longitudinal cyclic pitch, 183
Cyclic pitch:
helicopter response to, 183
Absorbers, see vibration
Actuator disc, 34, 37
Aeroelastic behaviour:
main rotor
blade weaving, 332
flap-lag instability, 335
pitch-flap flutter, 324
pitch-lag instability, 320
stall flutter, 329
tail rotor
flap-lag instability, 335
pitch-flap-lag instability, 341
pitch-flap (umbrella mode) instability,
334
Aerofoil characteristics:
in forward flight, 221–232
effect of Mach number, 222
oscillating at low incidence, 204–212
oscillating at high incidence, 225–232
Aeromechanical behaviour:
air resonance, 352
ground resonance, 342
Air resonance, 352
Angular momentum, 360
Attitude of helicopter:
lateral, 30
longitudinal, 28, 116
Autorotation, 132
Autostabilization, 175–179
Bell stabilising bar, 176
Bending of blades, 238–289
calculation of modes and frequencies, 242
finite-element method, 253
Galerkin method, 249
Lagrange’s equations, 243
Myklestad method, 251
372 Index
Cyclic pitch (continued)
lateral cyclic pitch to trim, 30
longitudinal cyclic pitch to trim, 28
Damping in pitch and roll, 139
Downwash velocity:
forward flight, 77
hovering flight, 39
Dynamic stability, 137–195
equations, 363
lateral, 166–175
longitudinal, 140–166
Eigenvalues and eigenvectors of flapping
blade, 241
Euler’s equations, 362
extended equations, 363
Feathering:
equation of motion, 10
torsional vibrations, 260
Figure of merit, 55
Flapping:
coefficients, 106
in terms of disc axes, 108
equation, 7
forward flight, 105
at high µ, 105
in hovering flight, 10
response to pitching and rolling, 15
Flutter, 324
stall, 329
Free wake model 74, 216
Fuselage drag, 130
variation with weight, 130
Goldstein circulation factors, 63
Goldstein–Lock theory, 59
Ground effect, 57
Ground resonance, 342
H-force:
definition, 27
forward flight, 98
Hingeless rotor:
basic features, 277
hub moment, 279
effect on stability, 165, 174, 189
Hovering flight:
actuator disc theory, 34
blade element theory, 46
figure of merit, 55
induced velocity in, 39
inflow angle, 52
optimum rotor, 54
performance in, 50
Induced power:
forward flight, 90
hovering flight, 40
Induced velocity:
flight and wind tunnel tests, 84
forward flight, 77
hovering flight, 39
vertical climb and descent, 43
Inflow angle:
forward flight, 95
vertical flight, 52
In-plane motion:
equation of motion for, 9
effect of drag hinge distance on, 9
geometric interpretation, 19
Lagging:
equation of motion, 9
motion due to flapping, 19
Lateral trim equations, 30
solution of, 125
Lift coefficient:
contours of, 228
in terms of thrust coefficient, 49
Longitudinal trim equations, 28, 116
solution of, 1, 117–123
stability, 140
Mangler and Squire induced velocity
distribution, 81
Manoeuvrability, 186
pull-up manoeuvre, 194
Multiblade summations, 365
Offset hinges:
effect on flapping frequency, 7
effect on lag frequency, 9
moment due to, 25
Optimum rotor, 54
figure of merit, 55
Index 373
Orthogonal property of normal modes, 267
Performance:
autorotation, 132
calculation of, 128
forward flight, 127–136
hovering flight, 51
Power calculations:
forward flight, 128
hovering flight, 51
induced power, 51
Prandtl wake theory, 60
Prescribed wake model, 69, 202
Response to vertical, gust, 190
Reverse flow area, 106
Rotating frame of reference:
angular momentum in, 360
Rotor:
axes systems, 26
forces and moments, 21
hinge system, 1
response to pitching and rolling, 15
Rotor wake, see vortex wake
Second order system, frequency response, 368
Solidity, definition, 48, 55
Stability characteristics:
forward flight, 163, 173
effect of hingeless rotor on, 165
hovering flight, 160, 173
effect of tailplane on, 164
Stability derivatives:
lateral, 168–172
longitudinal, 143–147
Stability equations, 363
lateral, 166–168
longitudinal, 140–143
Stall flutter, 329
Stall of oscillating aerofoil, 228
Static stability, 193
Swash plate, 17
Swirl velocity, 44
Tailplane:
effect on control response, 187
effect on dynamic stability, 164
effect on gust response, 190
effect on longitudinal trim, 123
Tailrotor loss in forward flight, 128
Theodorsen’s wake theory, 67
Thrust:
coefficient:
definition, 48
forward flight, 96
hovering flight, 47
definition, 27
equations:
forward flight, 97
hovering, 48
effect on high incidence on, 225
Tip loss factor, 65
Tip shapes, 223
Tip vortex, 70
Torque coefficient, 50
forward flight, 102
vertical flight, 50
Turbulent wake state, 43
Velocity components at blade, 92–96
Vibration:
active control of vibration, 310–314
active control of structural response
(ACSR), 312
higher harmonic control (HHC), 310
dynamic design of rotor blades, 294
exciting forces, 291
frequencies other than hΩ, 314
fuselage response, 301
main rotor gearbox mounting systems, 297
DAVI mounting system, 298
measurement in flight, 316
vibration absorbers, 304
battery, 309
bifilar, 305
Flexispring, 307
pendulum, 307
Vortex ring state, 42
Vortex wake:
forward flight, 196–221
prescribed wake, 202–216
free wake, 216–221
hovering flight, 59–75
rigid wake, 59–69
prescribed wake, 69–74
free wake analysis, 74–75
Vortices
shed, 196
trailing, 196
Weaving, blade, 332
Wind axes, 365
Windmill brake state, 42 |
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