Introduction to Aircraft Aeroelasticity and Loads 2e

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