Introduction to Aircraft Aeroelasticity and Loads

Jan R. Wright University of Manchester, UK Jonathan E. Cooper University of Bristol, UK

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