Table Of ContentBramwell’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
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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
viii Preface to the second edition
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 re-
arranged 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
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.
x Preface to the first edition
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
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
A , B ijth generalised inertia and stiffness coefficients
ij ij
A nth coefficient in periodic or finite series
n
A Blade pitch jth input weighting (active vibration control)
j
A , B Lateral and longitudinal cyclic pitch
1 1
A , B , C , D , E Coefficients in longitudinal characteristic equation
1 1c 1 1 1
A , B , C , D , E Coefficients in lateral characteristic equation
2 2 2 2 2
A , B Normalised generalised coefficients = A , B /0.5mΩ2R3
ij ij ij ij
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)
a Acceleration of blade c.g.
g
a Tailplane lift curve slope
T
a Acceleration of origin of moving frame = a i + a j + a k
0 x y z
a Coning angle
0
a , b Longitudinal and lateral flapping coefficients
1 1
a , a , a , b , b Sine and cosine coefficients in equation for C
0 1 2 1 2 m
a , a , b Analogous to a , a , b for hingeless rotor
0 1 1 0 1 1
B Tip-loss factor (Prandtl) = R /R
e
B Vector of background vibration responses
