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Flight Dynamics Principles This Page Intentionally Left Blank Prelims-H6927.Tex 11/7/2007 11: 13 Page Iii Prelims-H6927.tex 11/7/2007 11: 13 Page i Flight Dynamics Principles This page intentionally left blank Prelims-H6927.tex 11/7/2007 11: 13 Page iii Flight Dynamics Principles M.V. Cook BSc, MSc, CEng, FRAeS, CMath, FIMA Senior Lecturer in the School of Engineering Cranfield University AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Butterworth-Heinemann is an imprint of Elsevier Prelims-H6927.tex 11/7/2007 11: 13 Page iv Butterworth-Heinemann is an imprint of Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA First edition 1997 Second edition 2007 Copyright © 2007, M.V.Cook. Published by Elsevier Ltd. All rights reserved The right of Michael Cook to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress ISBN: 978-0-7506-6927-6 For information on all Butterworth-Heinemann publications visit our web site at http://books.elsevier.com Typeset by Charontec Ltd (A Macmillan Company), Chennai, India www.charontec.com Printed and bound in Great Britain 07080910 10987654321 Prelims-H6927.tex 11/7/2007 11: 13 Page v Contents Preface to the first edition ix Preface to the second edition xi Acknowledgements xiii Nomenclature xv 1. Introduction 1 1.1 Overview 1 1.2 Flying and handling qualities 3 1.3 General considerations 4 1.4 Aircraft equations of motion 7 1.5 Aerodynamics 7 1.6 Computers 8 1.7 Summary 10 References 11 2. Systems of axes and notation 12 2.1 Earth axes 12 2.2 Aircraft body fixed axes 13 2.3 Euler angles and aircraft attitude 18 2.4 Axes transformations 18 2.5 Aircraft reference geometry 24 2.6 Controls notation 27 2.7 Aerodynamic reference centres 28 References 30 Problems 30 3. Static equilibrium and trim 32 3.1 Trim equilibrium 32 3.2 The pitching moment equation 40 3.3 Longitudinal static stability 44 3.4 Lateral static stability 53 3.5 Directional static stability 54 3.6 Calculation of aircraft trim condition 57 References 64 Problems 64 4. The equations of motion 66 4.1 The equations of motion of a rigid symmetric aircraft 66 4.2 The linearised equations of motion 73 v Prelims-H6927.tex 11/7/2007 11: 13 Page vi vi Contents 4.3 The decoupled equations of motion 79 4.4 Alternative forms of the equations of motion 82 References 95 Problems 96 5. The solution of the equations of motion 98 5.1 Methods of solution 98 5.2 Cramer’s rule 99 5.3 Aircraft response transfer functions 101 5.4 Response to controls 108 5.5 Acceleration response transfer functions 112 5.6 The state space method 114 5.7 State space model augmentation 128 References 134 Problems 134 6. Longitudinal dynamics 138 6.1 Response to controls 138 6.2 The dynamic stability modes 144 6.3 Reduced order models 147 6.4 Frequency response 158 6.5 Flying and handling qualities 165 6.6 Mode excitation 167 References 170 Problems 171 7. Lateral–directional dynamics 174 7.1 Response to controls 174 7.2 The dynamic stability modes 183 7.3 Reduced order models 188 7.4 Frequency response 195 7.5 Flying and handling qualities 200 7.6 Mode excitation 202 References 206 Problems 206 8. Manoeuvrability 210 8.1 Introduction 210 8.2 The steady pull-up manoeuvre 212 8.3 The pitching moment equation 214 8.4 Longitudinal manoeuvre stability 216 8.5 Aircraft dynamics and manoeuvrability 222 References 223 9. Stability 224 9.1 Introduction 224 9.2 The characteristic equation 227 9.3 The Routh–Hurwitz stability criterion 227 Prelims-H6927.tex 11/7/2007 11: 13 Page vii Contents vii 9.4 The stability quartic 231 9.5 Graphical interpretation of stability 234 References 238 Problems 238 10. Flying and handling qualities 240 10.1 Introduction 240 10.2 Short term dynamic models 241 10.3 Flying qualities requirements 249 10.4 Aircraft role 251 10.5 Pilot opinion rating 255 10.6 Longitudinal flying qualities requirements 256 10.7 Control anticipation parameter 260 10.8 Lateral–directional flying qualities requirements 263 10.9 Flying qualities requirements on the s-plane 266 References 271 Problems 272 11. Stability augmentation 274 11.1 Introduction 274 11.2 Augmentation system design 280 11.3 Closed loop system analysis 283 11.4 The root locus plot 287 11.5 Longitudinal stability augmentation 293 11.6 Lateral–directional stability augmentation 300 11.7 The pole placement method 311 References 316 Problems 316 12. Aerodynamic modelling 320 12.1 Introduction 320 12.2 Quasi-static derivatives 321 12.3 Derivative estimation 323 12.4 The effects of compressibility 327 12.5 Limitations of aerodynamic modelling 335 References 336 13. Aerodynamic stability and control derivatives 337 13.1 Introduction 337 13.2 Longitudinal aerodynamic stability derivatives 337 13.3 Lateral–directional aerodynamic stability derivatives 350 13.4 Aerodynamic control derivatives 371 13.5 North American derivative coefficient notation 377 References 385 Problems 385 Prelims-H6927.tex 11/7/2007 11: 13 Page viii viii Contents 14. Coursework Studies 390 14.1 Introduction 390 14.2 Working the assignments 390 14.3 Reporting 390 Assignment 1. Stability augmentation of the North American X-15 hypersonic research aeroplane 391 Assignment 2. The stability and control characteristics of a civil transport aeroplane with relaxed longitudinal static stability 392 Assignment 3. Lateral–directional handling qualities design for the Lockheed F-104 Starfighter aircraft. 396 Assignment 4. Analysis of the effects of Mach number on the longitudinal stability and control characteristics of the LTV A7-A Corsair aircraft 401 Appendices 1 AeroTrim – A Symmetric Trim Calculator for Subsonic Flight Conditions 405 2 Definitions of Aerodynamic Stability and Control Derivatives 412 3 Aircraft Response Transfer Functions Referred to Aircraft Body Axes 419 4 Units, Conversions and Constants 425 5 A Very Short Table of Laplace Transforms 426 6 The Dynamics of a Linear Second Order System 427 7 North American Aerodynamic Derivative Notation 431 8 Approximate Expressions for the Dimensionless Aerodynamic Stability and Control Derivatives 434 9 The Transformation of Aerodynamic Stability Derivatives from a Body Axes Reference to a Wind Axes Reference 438 10 The Transformation of the Moments and Products of Inertia from a Body Axes Reference to a Wind Axes Reference 448 11 The Root Locus Plot 451 Index 457 Prelims-H6927.tex 11/7/2007 11: 13 Page ix Preface to the first edition When I joined the staff of the College of Aeronautics some years ago I was presented with a well worn collection of lecture notes and invited to teach Aircraft Stability and Control to postgraduate students. Inspection of the notes revealed the unmistakable signs that their roots reached back to the work of W.J. Duncan, which is perhaps not surprising since Duncan was the first Professor of Aerodynamics at Cranfield some 50 years ago. It is undoubtedly a privilege and, at first, was very daunting to be given the opportunity to follow in the footsteps of such a distinguished academic. From that humble beginning my interpretation of the subject has continuously evolved to its present form which provided the basis for this book. The classical linearised theory of the stability and control of aircraft is timeless, it is brilliant in its relative simplicity and it is very securely anchored in the domain of the aerodynamicist. So what is new? The short answer is; not a great deal. However, today the material is used and applied in ways that have changed considerably, due largely to the advent of the digital computer. The computer is used as the principal tool for analysis and design, and it is also the essential component of the modern flight control system on which all advanced technology aeroplanes depend. It is the latter development in particular which has had, and continues to have, a major influence on the way in which the material of the subject is now used. It is no longer possible to guarantee good flying and handling qualities simply by tailoring the stability and control characteristics of an advanced technology aeroplane by aerodynamic design alone. Flight control systems now play an equally important part in determining the flying and handling qualities of an aeroplane by augmenting the stability and control characteristics of the airframe in a beneficial way. Therefore the subject has had to evolve in order to facilitate integration with flight control and, today, the integrated subject is much broader in scope and is more frequently referred to as Flight Dynamics.
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