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 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.