Introduction to Aircraft Aeroelasticity and Loads
Description
Reviews / Votes
"I strongly recommend this textbook to under-graduates and researchers, not only due to how principles and concepts are explained, but also because it clearly shows the multidisciplinary nature of modern engineering techniques." (The Aeronautical Journal, 1 November 2015)More details
Other editions
Additional editions
Persons
Content
Series Preface xxi
Preface to the Second Edition xxiii
Preface to the First Edition xxv
Abbreviations xxix
Introduction 1
PART I BACKGROUND MATERIAL 7
1 Vibration of Single Degree of Freedom Systems 9
1.1 Setting up Equations of Motion for SDoF Systems 9
1.2 Free Vibration of SDoF Systems 11
1.3 Forced Vibration of SDoF Systems 13
1.4 Harmonic Forced Vibration - Frequency Response Functions 14
1.5 Transient/Random Forced Vibration - Time Domain Solution 17
1.6 Transient Forced Vibration - Frequency Domain Solution 21
1.7 Random Forced Vibration - Frequency Domain Solution 23
1.8 Examples 24
2 Vibration of Multiple Degree of Freedom Systems 27
2.1 Setting up Equations of Motion 27
2.2 Undamped Free Vibration 29
2.3 Damped Free Vibration 31
2.4 Transformation to Modal Coordinates 34
2.5 Two-DoF Rigid Aircraft in Heave and Pitch 38
2.6 'Free-Free' Systems 40
2.7 Harmonic Forced Vibration 41
2.8 Transient/Random Forced Vibration - Time Domain Solution 43
2.9 Transient Forced Vibration - Frequency Domain Solution 44
2.10 Random Forced Vibration - Frequency Domain Solution 44
2.11 Examples 45
3 Vibration of Continuous Systems - Assumed Shapes Approach 49
3.1 Continuous Systems 49
3.2 Modelling Continuous Systems 49
3.3 Elastic and Flexural Axes 51
3.4 Rayleigh-Ritz 'Assumed Shapes' Method 52
3.5 Generalized Equations of Motion - Basic Approach 53
3.6 Generalized Equations of Motion - Matrix Approach 58
3.7 Generating Whole Aircraft 'Free-Free' Modes from 'Branch' Modes 61
3.8 Whole Aircraft 'Free-Free' Modes 64
3.9 Examples 65
4 Introduction to Steady Aerodynamics 69
4.1 The Standard Atmosphere 69
4.2 Effect of Air Speed on Aerodynamic Characteristics 71
4.3 Flows and Pressures Around a Symmetric Aerofoil 73
4.4 Forces on an Aerofoil 74
4.5 Variation of Lift for an Aerofoil at an Angle of Incidence 76
4.6 Pitching Moment Variation and the Aerodynamic Centre 77
4.7 Lift on a Three-dimensional Wing 78
4.8 Drag on a Three-dimensional Wing 82
4.9 Control Surfaces 83
4.10 Transonic Flows 84
4.11 Examples 85
5 Introduction to Loads 87
5.1 Laws of Motion 88
5.2 D'Alembert's Principle - Inertia Forces and Couples 90
5.3 External Loads - Applied and Reactive 94
5.4 Free Body Diagrams 95
5.5 Internal Loads 96
5.6 Internal Loads for a Continuous Member 96
5.7 Internal Loads for a Discretized Member 101
5.8 Intercomponent Loads 103
5.9 Obtaining Stresses from Internal Loads - Structural Members with Simple Load Paths 103
5.10 Examples 104
6 Introduction to Control 109
6.1 Open and Closed Loop Systems 109
6.2 Laplace Transforms 110
6.3 Modelling of Open and Closed Loop Systems using Laplace and Frequency Domains 112
6.4 Stability of Systems 114
6.5 PID Control 121
6.6 Examples 122
PART II INTRODUCTION TO AEROELASTICITY AND LOADS 123
7 Static Aeroelasticity - Effect of Wing Flexibility on Lift Distribution and Divergence 125
7.1 Static Aeroelastic Behaviour of a Two-dimensional Rigid Aerofoil with a Torsional Spring Attachment 126
7.2 Static Aeroelastic Behaviour of a Fixed Root Flexible Wing 130
7.3 Effect of Trim on Static Aeroelastic Behaviour 133
7.4 Effect of Wing Sweep on Static Aeroelastic Behaviour 137
7.5 Examples 142
8 Static Aeroelasticity - Effect of Wing Flexibility on Control Effectiveness 143
8.1 Rolling Effectiveness of a Flexible Wing - Fixed Wing Root Case 144
8.2 Rolling Effectiveness of a Flexible Wing - Steady Roll Case 147
8.3 Effect of Spanwise Position of the Control Surface 151
8.4 Full Aircraft Model - Control Effectiveness 152
8.5 Effect of Trim on Reversal Speed 153
8.6 Examples 153
9 Introduction to Unsteady Aerodynamics 155
9.1 Quasi-steady Aerodynamics 156
9.2 Unsteady Aerodynamics related to Motion 156
9.3 Aerodynamic Lift and Moment for an Aerofoil Oscillating Harmonically in Heave and Pitch 161
9.4 Oscillatory Aerodynamic Derivatives 162
9.5 Aerodynamic Damping and Stiffness 163
9.6 Approximation of Unsteady Aerodynamic Terms 164
9.7 Unsteady Aerodynamics related to Gusts 164
9.8 Examples 168
10 Dynamic Aeroelasticity - Flutter 171
10.1 Simplified Unsteady Aerodynamic Model 172
10.2 Binary Aeroelastic Model 173
10.3 General Form of the Aeroelastic Equations 176
10.4 Eigenvalue Solution of the Flutter Equations 176
10.5 Aeroelastic Behaviour of the Binary Model 177
10.6 Aeroelastic Behaviour of a Multiple Mode System 185
10.7 Flutter Speed Prediction for Binary Systems 185
10.8 Divergence of Dynamic Aeroelastic Systems 188
10.9 Inclusion of Unsteady Reduced Frequency Effects 189
10.10 Control Surface Flutter 193
10.11 Whole Aircraft Model - Inclusion of Rigid Body Modes 199
10.12 Flutter in the Transonic Regime 202
10.13 Effect of Non-Linearities - Limit Cycle Oscillations 202
10.14 Examples 204
11 Aeroservoelasticity 207
11.1 Mathematical Modelling of a Simple Aeroelastic System with a Control Surface 208
11.2 Inclusion of Gust Terms 209
11.3 Implementation of a Control System 210
11.4 Determination of Closed Loop System Stability 211
11.5 Gust Response of the Closed Loop System 213
11.6 Inclusion of Control Law Frequency Dependency in Stability Calculations 214
11.7 Response Determination via the Frequency Domain 215
11.8 State Space Modelling 216
11.9 Examples 217
12 Equilibrium Manoeuvres 219
12.1 Equilibrium Manoeuvre - Rigid Aircraft under Normal Acceleration 221
12.2 Manoeuvre Envelope 226
12.3 Equilibrium Manoeuvre - Rigid Aircraft Pitching 227
12.4 Equilibrium Manoeuvre - Flexible Aircraft Pitching 235
12.5 Representation of the Flight Control System (FCS) 250
12.6 Examples 250
13 Dynamic Manoeuvres 253
13.1 Aircraft Axes 255
13.2 Motion Variables 257
13.3 Axes Transformations 257
13.4 Velocity and Acceleration Components for Moving Axes in 2D 259
13.5 Flight Mechanics Equations of Motion for a Rigid Symmetric Aircraft in 2D 262
13.6 Representation of Disturbing Forces and Moments 265
13.7 Modelling the Flexible Aircraft 267
13.8 Solution of Flight Mechanics Equations for the Rigid Aircraft 272
13.9 Dynamic Manoeuvre - Rigid Aircraft in Longitudinal Motion 273
13.10 Dynamic Manoeuvre - Flexible Aircraft Heave/Pitch 279
13.11 General Form of Longitudinal Equations 287
13.12 Dynamic Manoeuvre for Rigid Aircraft in Lateral Motion 288
13.13 Bookcase Manoeuvres for Rigid Aircraft in Lateral Motion 289
13.14 Flight Control System (FCS) 293
13.15 Representation of the Flight Control System (FCS) 295
13.16 Examples 295
14 Gust and Turbulence Encounters 299
14.1 Gusts and Turbulence 300
14.2 Gust Response in the Time Domain 301
14.3 Time Domain Gust Response - Rigid Aircraft in Heave 303
14.4 Time Domain Gust Response - Rigid Aircraft in Heave/Pitch 310
14.5 Time Domain Gust Response - Flexible Aircraft 316
14.6 General Form of Equations in the Time Domain 321
14.7 Turbulence Response in the Frequency Domain 321
14.8 Frequency Domain Turbulence Response - Rigid Aircraft in Heave 324
14.9 Frequency Domain Turbulence Response - Rigid Aircraft in Heave/Pitch 329
14.10 Frequency Domain Turbulence Response - Flexible Aircraft 330
14.11 General Form of Equations in the Frequency Domain 333
14.12 Representation of the Flight Control System (FCS) 334
14.13 Examples 334
15 Ground Manoeuvres 337
15.1 Landing Gear 337
15.2 Taxi, Take-Off and Landing Roll 342
15.3 Landing 351
15.4 Braking 359
15.5 Turning 360
15.6 Shimmy 361
15.7 Representation of the Flight Control System (FCS) 363
15.8 Examples 363
16 Aircraft Internal Loads 367
16.1 Limit and Ultimate Loads 368
16.2 Internal Loads for an Aircraft 368
16.3 General Internal Loads Expressions - Continuous Wing 370
16.4 Effect of Wing-mounted Engines and Landing Gear 372
16.5 Internal Loads - Continuous Flexible Wing 373
16.6 General Internal Loads Expressions - Discretized Wing 379
16.7 Internal Loads - Discretized Fuselage 384
16.8 Internal Loads - Continuous Turbulence Encounter 387
16.9 Loads Generation and Sorting to yield Critical Cases 388
16.10 Aircraft Dimensioning Cases 390
16.11 Stresses derived from Internal Loads - Complex Load Paths 391
16.12 Examples 391
17 Vibration of Continuous Systems - Finite Element Approach 395
17.1 Introduction to the Finite Element Approach 395
17.2 Formulation of the Beam Bending Element 397
17.3 Assembly and Solution for a Beam Structure 401
17.4 Torsion Element 406
17.5 Combined Bending/Torsion Element 407
17.6 Concentrated Mass Element 408
17.7 Stiffness Element 408
17.8 Rigid Body Elements 409
17.9 Other Elements 410
17.10 Comments on Modelling 411
17.11 Examples 413
18 Potential Flow Aerodynamics 415
18.1 Components of Inviscid, Incompressible Flow Analysis 415
18.2 Inclusion of Vorticity 420
18.3 Numerical Steady Aerodynamic Modelling of Thin Two-dimensional Aerofoils 422
18.4 Steady Aerodynamic Modelling of Three-Dimensional Wings using a Panel Method 425
18.5 Unsteady Aerodynamic Modelling of Wings undergoing Harmonic Motion 429
18.6 Aerodynamic Influence Coefficients in Modal Space 432
18.7 Examples 436
19 Coupling of Structural and Aerodynamic Computational Models 437
19.1 Mathematical Modelling - Static Aeroelastic Case 438
19.2 2D Coupled Static Aeroelastic Model - Pitch 439
19.3 2D Coupled Static Aeroelastic Model - Heave/Pitch 440
19.4 3D Coupled Static Aeroelastic Model 441
19.5 Mathematical Modelling - Dynamic Aeroelastic Response 446
19.6 2D Coupled Dynamic Aeroelastic Model - Bending/Torsion 447
19.7 3D Flutter Analysis 448
19.8 Inclusion of Frequency Dependent Aerodynamics for State-Space Modelling - Rational Function Approximation 450
PART III INTRODUCTION TO INDUSTRIAL PRACTICE 455
20 Aircraft Design and Certification 457
20.1 Aeroelastics and Loads in the Aircraft Design Process 457
20.2 Aircraft Certification Process 459
21 Aeroelasticity and Loads Models 465
21.1 Structural Model 465
21.2 Aerodynamic Model 471
21.3 Flight Control System 473
21.4 Other Model Issues 474
21.5 Loads Transformations 474
22 Static Aeroelasticity and Flutter 475
22.1 Static Aeroelasticity 475
22.2 Flutter 478
23 Flight Manoeuvre and Gust/Turbulence Loads 481
23.1 Evaluation of Internal Loads 481
23.2 Equilibrium/Balanced Flight Manoeuvres 481
23.3 Dynamic Flight Manoeuvres 485
23.4 Gusts and Turbulence 489
24 Ground Manoeuvre Loads 495
24.1 Aircraft/Landing Gear Models for Ground Manoeuvres 495
24.2 Landing Gear/Airframe Interface 496
24.3 Ground Manoeuvres - Landing 496
24.4 Ground Manoeuvres - Ground Handling 497
24.5 Loads Processing 498
25 Testing Relevant to Aeroelasticity and Loads 501
25.1 Introduction 501
25.2 Wind Tunnel Tests 501
25.3 Ground Vibration Test 502
25.4 Structural Coupling Test 503
25.5 Flight Simulator Test 504
25.6 Structural Tests 504
25.7 Flight Flutter Test 505
25.8 Flight Loads Validation 507
Appendices 509
A Aircraft Rigid Body Modes 511
B Table of Longitudinal Aerodynamic Derivatives 513
C Aircraft Symmetric Flexible Modes 517
D Model Condensation 527
E Aerodynamic Derivatives in Body Fixed Axes 531
References 535
Index 539
Preface to the First Edition
Aeroelasticity is the study of the interaction of aerodynamic, elastic and inertia forces. For fixed wing aircraft there are two key areas: (a) static aeroelasticity, where the deformation of the aircraft influences the lift distribution can lead to the statically unstable condition of divergence and will normally reduce the control surface effectiveness, and (b) dynamic aeroelasticity, which includes the critical area of flutter where the aircraft can become dynamically unstable in a condition where the structure extracts energy from the air stream.
Aircraft are also subject to a range of static and dynamic loads resulting from flight manoeuvres (equilibrium/steady and dynamic), ground manoeuvres and gust/turbulence encounters. These load cases are responsible for the critical design loads over the aircraft structure and hence influence the structural design. Determination of such loads involves consideration of aerodynamic, elastic and inertia effects and requires the solution of the dynamic responses; consequently there is a strong link between aeroelasticity and loads.
The aircraft vibration characteristics and response are a result of the flexible modes combining with the rigid body dynamics, with the inclusion of the Flight Control System (FCS) if it is present. In this latter case, the aircraft will be a closed loop system and the FCS affects both the aeroelasticity and loads behaviour. The interaction between the FCS and the aeroelastic system is often called aeroservoelasticity.
This book aims to embrace the range of basic aeroelastic and loads topics that might be encountered in an aircraft design office and to provide an understanding of the main principles involved. Colleagues in industry have often remarked that it is not appropriate to give some of the classical books on aeroelasticity to new graduate engineers as many of the books are too theoretical for a novice aeroelastician. Indeed, the authors have found much of the material in them to be too advanced to be used in the undergraduate level courses that they have taught. Also, the topics of aeroelasticity and loads have tended to be treated separately in textbooks, whereas in industry the fields have become much more integrated. This book is seen as providing some grounding in the basic analysis techniques required which, having been mastered, can then be supplemented via more advanced texts, technical papers and industry reports.
Some of the material covered in this book developed from undergraduate courses given at Queen Mary College, University of London and at the University of Manchester. In the UK, many entrants into the aerospace industry do not have an aerospace background, and almost certainly will have little knowledge of aeroelasticity or loads. To begin to meet this need, during the early 1990s the authors presented several short courses on Aeroelasticity and Structural Dynamics to young engineers in the British aerospace industry, and this has influenced the content and approach of this book. A further major influence was the work by Hancock, Simpson and Wright (1985) on the teaching of flutter, making use of a simplified flapping and pitching wing model with strip theory aerodynamics (including a simplified unsteady aerodynamics model) to illustrate the fundamental principles of flutter. This philosophy has been employed here for the treatment of static aeroelasticity and flutter, and has been extended into the area of loads by focusing on a simplified flexible whole aircraft model in order to highlight key features of modelling and analysis.
The intention of the book is to provide the reader with the technical background to understand the underlying concepts and application of aircraft aeroelasticity and loads. As far as possible, simplified mathematical models for the flexible aircraft are used to illustrate the phenomena and also to demonstrate the link between these models, industrial practice and the certification process. Thus, fairly simple continuum models based upon a small number of assumed modes (so avoiding partial differential equations) have been used. Consequently, much of the book is based upon strip theory aerodynamics and the Rayleigh-Ritz assumed modes method. By using this approach, it has been possible to illustrate most concepts using a maximum of three degrees of freedom. Following on from these continuum models, basic discretized structural and aerodynamic models are introduced in order to demonstrate some underlying approaches in current industrial practice. The book aims to be suitable for final year undergraduate or Masters level students, or engineers in industry who are new to the subject. For example, it could provide the basis of two taught modules in aeroelasticity and loads. It is hoped that the book will fill a gap in providing a broad and relatively basic introductory treatment of aeroelastics and loads.
A significant number of different topics are covered in order to achieve the goals of this book, namely structural dynamics, steady and unsteady aerodynamics, loads, control, static aeroelastic effects, flutter, flight manoeuvres (both steady/equilibrium and dynamic), ground manoeuvres (e.g. landing, taxiing), gust and turbulence encounters, calculation of loads and, finally, Finite Element and three-dimensional panel methods. In addition, a relatively brief explanation is given as to how these topics might typically be approached in industry when seeking to meet the certification requirements. Most of the focus is on commercial and not military aircraft, though of course all of the underlying principles, and much of the implementation, are common between the two.
The notation employed has not been straightforward to define, as many of these disciplines have tended to use the same symbols for different variables and so inevitably this exercise has been a compromise. A further complication is the tendency for aeroelasticity textbooks from the USA to use the reduced frequency k for unsteady aerodynamics, as opposed to the frequency parameter ? that is often used elsewhere. The reduced frequency has been used throughout this textbook to correspond with the classical textbooks of aeroelasticity.
The book is split into three parts. After a brief introduction to aeroelasticity and loads, Part A provides some essential background material on the fundamentals of single and multiple degree of freedom (DoF) vibrations for discrete parameter systems and continuous systems (Rayleigh-Ritz and Finite Element), steady aerodynamics, loads and control. The presentation is not very detailed, assuming that a reader having a degree in engineering will have some background in most of these topics and can reference more comprehensive material if desired.
Part B is the main part of the book, covering the basic principles and concepts required to provide a bridge to begin to understand current industry practice. The chapters on aeroelasticity include static aeroelasticity (lift distribution, divergence and control effectiveness), unsteady aerodynamics, dynamic aeroelasticity (i.e. flutter) and aeroservoelasticity; the treatment is based mostly on a simple two-DoF flapping/pitching wing model, sometimes attached to a rigid fuselage free to heave and pitch. The chapters on loads include equilibrium and dynamic flight manoeuvres, gusts and turbulence encounters, ground manoeuvres and internal loads. The loads analyses are largely based on a three-DoF whole aircraft model with heave and pitch rigid body motions and a free-free flexible mode whose characteristics may be varied, so allowing fuselage bending, wing bending or wing torsional deformation to be dominant. Part B concludes with an introduction to three-dimensional aerodynamic panel methods and simple coupled discrete aerodynamic and structural models in order to move on from the Rayleigh-Ritz assumed modes and strip theory approaches to more advanced methods, which provide the basis for much of the current industrial practice.
The basic theory introduced in Parts I and II provides a suitable background to begin to understand Part III, which provides an outline of industrial practice that might typically be involved in aircraft design and certification, including aeroelastic modelling, static aeroelasticity and flutter, flight manoeuvre and gust/turbulence loads, ground manoeuvre loads and finally testing relevant to aeroelastics and loads. A number of MATLAB/SIMULINK programs are available on a companion website for this book at http://www.wiley/go/wright&cooper.
The authors are grateful to the input from a number of colleagues in the UK university sector: John Ackroyd, Philip Bonello, Grigorios Dimitriadis, Zhengtao Ding, Dominic Diston, Barry Lennox and Gareth Vio. The authors greatly valued the input on industrial practice from Mark Hockenhull, Tom Siddall, Peter Denner, Paul Bruss, Duncan Pattrick, Mark Holden and Norman Wood. The authors also appreciated useful discussions with visiting industrial lecturers to the University of Manchester (namely Rob Chapman, Brian Caldwell, Saman Samarasekera, Chris Fielding and Brian Oldfield). Some of the figures and calculations were provided by Colin Leung, Graham Kell and Gareth Vio. Illustrations were provided with kind agreement of Airbus, Messier-Dowty, DLR, DGA/CEAT, ONERA, Royal Aeronautical Society and ESDU. Use of software was provided by MATLAB.
The authors would also like to acknowledge the encouragement they have received over the years in relation to research and teaching activities in the areas of structural dynamics, aircraft structures, loads and aeroelasticity, namely Alan Simpson (University of Bristol),...
