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 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 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 A ij , B ij ijth generalised inertia and stiffness coefficients A n nth coefficient in periodic or finite series A j Blade pitch jth input weighting (active vibration control) A 1 , B 1 Lateral and longitudinal cyclic pitch A 1 , B 1c , C 1 , D 1 , E 1 Coefficients in longitudinal characteristic equation A 2 , B 2 , C 2 , D 2 , E 2 Coefficients in lateral characteristic equation AB ij ij , Normalised generalised coefficients = A ij , B ij /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) a g Acceleration of blade c.g. a T Tailplane lift curve slope a 0 Acceleration of origin of moving frame = a x i + a y j + a z k a 0 Coning angle a 1 , b 1 Longitudinal and lateral flapping coefficients a 0 , a 1 , a 2 , b 1 , b 2 Sine and cosine coefficients in equation for C m Analogous to a 0 , a 1 , b 1 for hingeless rotor B Tip-loss factor (Prandtl) = R e /R B Vector of background vibration responses aab 011 , , [...]... assumed that lagging and feathering do not occur 6 Bramwell s Helicopter Dynamics Feathering bearing assembly Flexible flap element Flexible lag element Lag damper Rotor shaft Spider Pitch control rod Control spindle 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... 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 ˙˙ Bβ + Ω2(B cos β + MbexgR2) sin β = MA k Ω j i eR β O R Fig 1.7 Single flapping blade (1.1) 8 Bramwell s Helicopter Dynamics where MA = – M is the aerodynamic moment in the sense of positive flapping and... 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 Lag hinge Flapping Lagging Flapping hinge Feathering Pitch change (or feathering) hinge Fig 1.1 Typical hinge arrangement In this figure, the flapping and lag hinges intersect, i.e... the Coriolis root δ3 Fig 1.3 The δ3-hinge (a) Flapping β0 (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... 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 Ω +ξ eR Fig 1.9 Blade lagging ξ 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... 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 of a revolution Thus, the flapping 4 motion is very heavily damped It has already been remarked that the centrifugal moment acts like... cos αs – l sin αs Second moment of area of blade section Blade moment of inertia in both flap and lag (Southwell) The unit matrix Non-dimensional inertia factors (Coleman and Stempin) Second moments of inertia of blade section for lagwise and flapwise bending Blade flap and pitch moments of inertia Unit vectors fixed in blade Non-dimensional rolling, pitching and yawing inertias of helicopter Non-dimensional... rolling, pitching and yawing inertias of helicopter Non-dimensional roll-yaw inertia product term for helicopter J J J J 0, J 1 j 1, j 2, j 3, j 4 Modal error squared integral (Duncan) Polar second moment of area of blade section Performance index (active vibration control) Bessel functions of first and second kinds (Miller) Quantities dependent on first blade flapping mode shapes of hingeless blade K... 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... u/ΩR, v /ΩR, w/ΩR Unit force constants relating to nth blade element (Myklestad) Unit moment constants relating to nth blade element (Myklestad) Forward velocity of helicopter or relative velocity far upstream of rotor Forward velocity vector of helicopter Forward speed normalised on thrust velocity = V/v0 Forward speed normalised on tip speed = V/ΩR Total velocity at the rotor Rate of climb in axial flight, . 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. components (Coleman) p l , p u Pressure on lower and upper sides of disc plane, or aerofoil surfaces p 1 , p 2 Pressure just ahead of actuator disc, and pressure in far wake p ∞ Ambient pressure ′′ nn vp ,. half wing span s Vortex axis vector = s 1 i + s 2 j + s 3 k s p Spacing of vortex sheets s t Tail rotor solidity Normalised tail rotor solidity = s t A t ( R) t /sA R T Rotor thrust T Periodic