JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Introduction to Aircraft Aeroelasticity and Loads Jan R. Wright University of Manchester and J2W Consulting Ltd, UK Jonathan E. Cooper University of Liverpool, UK iii JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Introduction to Aircraft Aeroelasticity and Loads i JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 ii JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Introduction to Aircraft Aeroelasticity and Loads Jan R. Wright University of Manchester and J2W Consulting Ltd, UK Jonathan E. Cooper University of Liverpool, UK iii JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Copyright C 2007 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wileyeurope.com or www.wiley.com All Rights Reserved. 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, scanning or otherwise, except under the terms 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 W1T 4LP, UK, without the permission in writing of the Publisher. 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Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production. iv JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 To our wives, Joy and Sarah v JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 vi JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Contents Preface xv Introduction xix Abbreviations xxv Part I Background Material 1 1 Vibration of Single Degree of Freedom Systems 3 1.1 Setting up equations of motion for single DoF systems 3 1.2 Free vibration of single DoF systems 5 1.3 Forced vibration of single DoF systems 7 1.4 Harmonic forced vibration – frequency response functions 7 1.5 Transient/random forced vibration – time domain solution 10 1.6 Transient forced vibration – frequency domain solution 14 1.7 Random forced vibration – frequency domain solution 16 1.8 Examples 17 2 Vibration of Multiple Degree of Freedom Systems 19 2.1 Setting up equations of motion 19 2.2 Undamped free vibration 21 2.3 Damped free vibration 24 2.4 Transformation to modal coordinates 27 2.5 ‘Free–free’ systems 31 2.6 Harmonic forced vibration 31 2.7 Transient/random forced vibration – time domain solution 33 2.8 Transient forced vibration – frequency domain solution 34 2.9 Random forced vibration – frequency domain solution 34 2.10 Examples 35 3 Vibration of Continuous Systems – Assumed Shapes Approach 37 3.1 Rayleigh–Ritz ‘assumed shapes’ method 38 3.2 Generalized equations of motion – basic approach 39 3.3 Generalized equations of motion – matrix approach 44 3.4 Generating aircraft ‘free–free’ modes from ‘branch’ modes 46 3.5 Whole aircraft ‘free–free’ modes 49 3.6 Examples 50 vii JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 viii CONTENTS 4 Vibration of Continuous Systems – Discretization Approach 53 4.1 Introduction to the finite element (FE) approach 53 4.2 Formulation of the beam bending element 54 4.3 Assembly and solution for a structure with beam elements 58 4.4 Torsion element 63 4.5 Combined bending/torsion element 64 4.6 Comments on modelling 65 4.7 Examples 66 5 Introduction to Steady Aerodynamics 69 5.1 The standard atmosphere 69 5.2 Effect of air speed on aerodynamic characteristics 71 5.3 Flows and pressures around a symmetric aerofoil 72 5.4 Forces on an aerofoil 74 5.5 Variation of lift for an aerofoil at an angle of incidence 75 5.6 Pitching moment variation and the aerodynamic centre 76 5.7 Lift on a three-dimensional wing 77 5.8 Drag on a three-dimensional wing 81 5.9 Control surfaces 82 5.10 Supersonic aerodynamics – piston theory 83 5.11 Transonic flows 84 5.12 Examples 84 6 Introduction to Loads 87 6.1 Laws of motion 87 6.2 D’Alembert’s principle – inertia forces and couples 90 6.3 Externally applied/reactive loads 93 6.4 Free body diagrams 94 6.5 Internal loads 95 6.6 Internal loads for continuous representation of a structure 96 6.7 Internal loads for discretized representation of a structure 100 6.8 Intercomponent loads 102 6.9 Obtaining stresses from internal loads – structural members with simple load paths 103 6.10 Examples 103 7 Introduction to Control 107 7.1 Open and closed loop systems 107 7.2 Laplace transforms 108 7.3 Modelling of open and closed loop systems using Laplace and frequency domains 110 7.4 Stability of systems 111 7.5 PID control 118 7.6 Examples 119 Part II Introduction to Aeroelasticity and Loads 121 8 Static Aeroelasticity – Effect of Wing Flexibility on Lift Distribution and Divergence 123 8.1 Static aeroelastic behaviour of a two-dimensional rigid aerofoil with spring attachment 124 8.2 Static aeroelastic behaviour of a fixed root flexible wing 127 JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 CONTENTS ix 8.3 Effect of trim on static aeroelastic behaviour 129 8.4 Effect of wing sweep on static aeroelastic behaviour 134 8.5 Examples 139 9 Static Aeroelasticity – Effect of Wing Flexibility on Control Effectiveness 141 9.1 Rolling effectiveness of a flexible wing – the steady roll case 141 9.2 Rolling effectiveness of a flexible wing – the fixed wing root case 146 9.3 Effect of spanwise position of the control surface 149 9.4 Full aircraft model – control effectiveness 149 9.5 Effect of trim on reversal speed 151 9.6 Examples 151 10 Introduction to Unsteady Aerodynamics 153 10.1 Quasi-steady aerodynamics 153 10.2 Unsteady aerodynamics 154 10.3 Aerodynamic lift and moment for a harmonically oscillating aerofoil 157 10.4 Oscillatory aerodynamic derivatives 159 10.5 Aerodynamic damping and stiffness 160 10.6 Unsteady aerodynamics related to gusts 161 10.7 Examples 165 11 Dynamic Aeroelasticity – Flutter 167 11.1 Simplified unsteady aerodynamic model 167 11.2 Binary aeroelastic model 168 11.3 General form of the aeroelastic equations 171 11.4 Eigenvalue solution of flutter equations 171 11.5 Aeroelastic behaviour of the binary model 172 11.6 Aeroelastic behaviour of a flexible wing 180 11.7 Aeroelastic behaviour of a multiple mode system 182 11.8 Flutter speed prediction for binary systems 182 11.9 Flutter conic 184 11.10 Divergence of aeroelastic systems 186 11.11 Inclusion of unsteady reduced frequency effects 187 11.12 Control surface flutter 191 11.13 Whole aircraft model – inclusion of rigid body modes 193 11.14 Flutter in the transonic regime 194 11.15 Flutter in the supersonic regime – wing and panel flutter 194 11.16 Effect of nonlinearities – limit cycle oscillations 197 11.17 Examples 198 12 Aeroservoelasticity 201 12.1 Mathematical modelling of a simple aeroelastic system with a control surface 202 12.2 Inclusion of gust terms 203 12.3 Implementation of a control system 204 12.4 Determination of closed loop system stability 204 12.5 Gust response of the closed loop system 205 12.6 Inclusion of control law frequency dependency in stability calculations 206 12.7 Response determination via the frequency domain 208 12.8 State space modelling 208 12.9 Examples 209 JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 x CONTENTS 13 Equilibrium Manoeuvres 211 13.1 Equilibrium manoeuvre – rigid aircraft under normal acceleration 213 13.2 Manoeuvre envelope 217 13.3 Equilibrium manoeuvre – rigid aircraft pitching 218 13.4 Equilibrium manoeuvre – flexible aircraft pitching 225 13.5 Flexible corrections to rigid aircraft pitching derivatives 238 13.6 Equilibrium manoeuvres – aircraft rolling and yawing 239 13.7 Representation of the flight control system (FCS) 243 13.8 Examples 243 14 Flight Mechanics Model for Dynamic Manoeuvres 245 14.1 Aircraft axes 246 14.2 Motion variables 247 14.3