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### Aircraft Aeroelasticity and Loads Introduction

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Aircraft
Aeroelasticity and Loads Introduction

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Authors:

Jan R. Wright

University of Manchester and

J2W Consulting Ltd, UK

Jonathan E. Cooper

University of Liverpool, UK

559 Pages

Open: Aircraft Aeroelasticity and Loads Introduction

Introduction

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.

TOC

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

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

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

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 ofWing 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

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 Axes transformations 248

14.4 Velocity and acceleration components for moving axes 250

14.5 Flight mechanics equations of motion for a rigid aircraft 252

14.6 Representation of disturbing forces and moments 255

14.7 Equations for flexible aircraft in longitudinal motion 257

14.8 Solution of flight mechanics equations 262

14.9 Flight control system (FCS) 263

15 Dynamic Manoeuvres 265

15.1 Dynamic manoeuvre – rigid aircraft heave/pitch due to elevator input 266

15.2 Dynamic manoeuvre – flexible aircraft heave/pitch due to elevator input 271

15.3 General form of longitudinal equations 278

15.4 Dynamic manoeuvre – rigid aircraft roll due to aileron input 279

15.5 Dynamic manoeuvre – flexible aircraft roll due to aileron input 283

15.6 Flexible corrections to flight mechanics equations 290

15.7 Representation of the flight control system (FCS) 290

15.8 Examples 290

16 Gust and Turbulence Encounters 293

16.1 Gusts and turbulence 294

16.2 Gust response in the time domain 295

16.3 Time domain gust response – rigid aircraft in heave 297

16.4 Time domain gust response – rigid aircraft in heave/pitch 303

16.5 Time domain gust response – flexible aircraft 306

16.6 General form of equations in the time domain 312

16.7 Turbulence response in the frequency domain 314

16.8 Frequency domain turbulence response – rigid aircraft in heave 317

16.9 Frequency domain turbulence response – rigid aircraft in heave/pitch 320

16.10 Frequency domain turbulence response – flexible aircraft 323

16.11 General form of equations in the frequency domain 325

16.12 Representation of the flight control system (FCS) 326

16.13 Examples 326

17 Ground Manoeuvres 329

17.1 Landing gear 329

17.2 Taxi, take-off and landing roll 333

17.3 Landing 340

17.4 Braking 345

17.5 ‘Spin-up’ and ‘spring-back’ condition 348

17.6 Turning 350

17.7 Shimmy 350

17.8 Representation of the flight control system (FCS) 353

17.9 Examples 353

18 Aircraft Internal Loads 355

18.1 Limit and ultimate loads 356

18.2 Internal loads for an aircraft 356

18.3 General internal loads expressions – continuous wing 358

18.4 Effect of wing-mounted engines/landing gear 360

18.5 Internal loads – continuous flexible wing 361

18.6 General internal loads expressions – discretized wing 366

18.7 Internal loads – discretized fuselage 370

18.8 Internal loads – continuous turbulence encounter 373

18.9 Loads generation and sorting to yield critical cases 374

18.10 Aircraft dimensioning cases 376

18.11 Stresses from internal loads – complex load paths 377

18.12 Examples 377

19 Potential Flow Aerodynamics 381

19.1 Elements of inviscid, incompressible flow analysis 381

19.2 Inclusion of vorticity 386

19.3 Numerical steady aerodynamic modelling of thin two-dimensional aerofoils 388

19.4 Steady aerodynamic modelling of three-dimensional wings using a
panel method 391

19.5 Unsteady aerodynamic modelling of wings undergoing harmonic motion 394

19.6 AICs in modal space 397

19.7 Examples 400

20 Coupling of Structural and Aerodynamic Computational Models 401

20.1 Mathematical modelling – static aeroelastic case 401

20.2 Two-dimensional coupled static aeroelastic model – pitch 403

20.3 Two-dimensional coupled static aeroelastic model – heave/pitch 404

20.4 Three-dimensional coupled static aeroelastic model 405

20.5 Mathematical modelling – dynamic aeroelastic response 409

20.6 Two-dimensional coupled dynamic aeroelastic model – bending
and torsion 410

20.7 Three-dimensional flutter analysis 411

20.8 Inclusion of frequency-dependent aerodynamics for state space
modelling – rational fraction approximation 412

Part III Introduction to Industrial Practice 417

21 Aircraft Design and Certification 419

21.1 Aeroelastics and loads in the aircraft design process 419

21.2 Aircraft certification process 421

22 Aeroelasticity and Loads Models 427

22.1 Structural model 427

22.2 Aerodynamic model 432

22.3 Flight control system 435

22.4 Other model issues 435

22.5 Loads transformations 436

23 Static Aeroelasticity and Flutter 437

23.1 Static aeroelasticity 437

23.2 Flutter 439

24 Flight Manoeuvre and Gust/Turbulence Loads 443

24.1 Evaluation of internal loads 443

24.2 Equilibrium/balanced flight manoeuvres 443

24.3 Dynamic flight manoeuvres 446

24.4 Gusts and turbulence 449

25 Ground Manoeuvre Loads 455

25.1 Aircraft/landing gear models for ground manoeuvres 455

25.2 Landing gear/airframe interface 456

25.3 Ground manoeuvres – landing 456

25.4 Ground manoeuvres – ground handling 457

25.5 Loads processing 458

26 Testing Relevant to Aeroelasticity and Loads 461

26.1 Introduction 461

26.2 Wind tunnel tests 461

26.3 Ground vibration test 462

26.4 Structural coupling test 463

26.5 Flight simulator test 464

26.6 Structural tests 464

26.7 Flight flutter test 465

26.8 Flight loads validation 466

Appendices 467

A Aircraft Rigid Body Modes 469

A.1 Rigid body translation modes 469

A.2 Rigid body rotation modes 469

B Table of Longitudinal Aerodynamic Derivatives 471

C Aircraft Symmetric Flexible Modes 473

C.1 Aircraft model 473

C.2 Symmetric free–free flexible mode 474

D Model Condensation 481

D.1 Introduction 481

D.2 Static condensation 481

D.3 Dynamic condensation – Guyan reduction 482

D.4 Static condensation for aeroelastic models 483

D.5 Modal condensation 483

D.6 Modal reduction 484

E Aerodynamic Derivatives in Body Fixed Axes 485

E.1 Longitudinal derivative Zw 485

E.2 Lateral derivatives L p, Lξ 486

F Aircraft Antisymmetric Flexible Modes 489

F.1 Aircraft model 489

F.2 Antisymmetric free–free flexible modes 489

References 491

Programs Accessible (on the CompanionWebsite) via the Internet

G MATLAB/SIMULINK Programs for Vibration

G.1 Forced response of an SDoF system

G.2 Modal solution for an MDoF system

G.3 Finite element solution

H MATLAB/SIMULINK Programs for Flutter

H.1 Dynamic aeroelastic calculations

H.2 Aeroservoelastic system

I MATLAB/SIMULINK Programs for Flight/Ground Manoeuvres and
Gust/Turbulence Encounters

I.1 Rigid aircraft data

I.2 Flexible aircraft data

I.3 Flight case data

I.4 Aerodynamic derivative calculation

I.5 Equilibrium manoeuvres

I.6 Dynamic manoeuvres

I.7 Gust response in the time domain

I.8 Gust response in the frequency domain

I.9 Ground manoeuvres

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