Introduction to Aircraft Aeroelasticity and Loads

Jan R. WrightUniversity of Manchester, UKJonathan E. CooperUniversity of Bristol, UK

“I strongly recommend this textbook to under­graduates 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)

• Series Preface xxiPreface to the Second Edition xxiiiPreface to the First Edition xxvAbbreviations xxixIntroduction 1PART I BACKGROUND MATERIAL 71 Vibration of Single Degree of Freedom Systems 91.1 Setting up Equations of Motion for SDoF Systems 91.2 Free Vibration of SDoF Systems 111.3 Forced Vibration of SDoF Systems 131.4 Harmonic Forced Vibration – Frequency Response Functions 141.5 Transient/Random Forced Vibration – Time Domain Solution 171.6 Transient Forced Vibration – Frequency Domain Solution 211.7 Random Forced Vibration – Frequency Domain Solution 231.8 Examples 242 Vibration of Multiple Degree of Freedom Systems 272.1 Setting up Equations of Motion 272.2 Undamped Free Vibration 292.3 Damped Free Vibration 312.4 Transformation to Modal Coordinates 342.5 Two-DoF Rigid Aircraft in Heave and Pitch 382.6 ‘Free–Free’ Systems 402.7 Harmonic Forced Vibration 412.8 Transient/Random Forced Vibration – Time Domain Solution 432.9 Transient Forced Vibration – Frequency Domain Solution 442.10 Random Forced Vibration – Frequency Domain Solution 442.11 Examples 453 Vibration of Continuous Systems – Assumed Shapes Approach 493.1 Continuous Systems 493.2 Modelling Continuous Systems 493.3 Elastic and Flexural Axes 513.4 Rayleigh–Ritz ‘Assumed Shapes’ Method 523.5 Generalized Equations of Motion – Basic Approach 533.6 Generalized Equations of Motion – Matrix Approach 583.7 Generating Whole Aircraft ‘Free–Free’ Modes from ‘Branch’ Modes 613.8 Whole Aircraft ‘Free–Free’ Modes 643.9 Examples 654 Introduction to Steady Aerodynamics 694.1 The Standard Atmosphere 694.2 Effect of Air Speed on Aerodynamic Characteristics 714.3 Flows and Pressures Around a Symmetric Aerofoil 734.4 Forces on an Aerofoil 744.5 Variation of Lift for an Aerofoil at an Angle of Incidence 764.6 Pitching Moment Variation and the Aerodynamic Centre 774.7 Lift on a Three-dimensional Wing 784.8 Drag on a Three-dimensional Wing 824.9 Control Surfaces 834.10 Transonic Flows 844.11 Examples 855 Introduction to Loads 875.1 Laws of Motion 885.2 D’Alembert’s Principle – Inertia Forces and Couples 905.3 External Loads – Applied and Reactive 945.4 Free Body Diagrams 955.5 Internal Loads 965.6 Internal Loads for a Continuous Member 965.7 Internal Loads for a Discretized Member 1015.8 Intercomponent Loads 1035.9 Obtaining Stresses from Internal Loads – Structural Members with Simple Load Paths 1035.10 Examples 1046 Introduction to Control 1096.1 Open and Closed Loop Systems 1096.2 Laplace Transforms 1106.3 Modelling of Open and Closed Loop Systems using Laplace and Frequency Domains 1126.4 Stability of Systems 1146.5 PID Control 1216.6 Examples 122PART II INTRODUCTION TO AEROELASTICITY AND LOADS 1237 Static Aeroelasticity – Effect of Wing Flexibility on Lift Distribution and Divergence 1257.1 Static Aeroelastic Behaviour of a Two-dimensional Rigid Aerofoil with a Torsional Spring Attachment 1267.2 Static Aeroelastic Behaviour of a Fixed Root Flexible Wing 1307.3 Effect of Trim on Static Aeroelastic Behaviour 1337.4 Effect of Wing Sweep on Static Aeroelastic Behaviour 1377.5 Examples 1428 Static Aeroelasticity – Effect of Wing Flexibility on Control Effectiveness 1438.1 Rolling Effectiveness of a Flexible Wing – Fixed Wing Root Case 1448.2 Rolling Effectiveness of a Flexible Wing – Steady Roll Case 1478.3 Effect of Spanwise Position of the Control Surface 1518.4 Full Aircraft Model – Control Effectiveness 1528.5 Effect of Trim on Reversal Speed 1538.6 Examples 1539 Introduction to Unsteady Aerodynamics 1559.1 Quasi-steady Aerodynamics 1569.2 Unsteady Aerodynamics related to Motion 1569.3 Aerodynamic Lift and Moment for an Aerofoil Oscillating Harmonically in Heave and Pitch 1619.4 Oscillatory Aerodynamic Derivatives 1629.5 Aerodynamic Damping and Stiffness 1639.6 Approximation of Unsteady Aerodynamic Terms 1649.7 Unsteady Aerodynamics related to Gusts 1649.8 Examples 16810 Dynamic Aeroelasticity – Flutter 17110.1 Simplified Unsteady Aerodynamic Model 17210.2 Binary Aeroelastic Model 17310.3 General Form of the Aeroelastic Equations 17610.4 Eigenvalue Solution of the Flutter Equations 17610.5 Aeroelastic Behaviour of the Binary Model 17710.6 Aeroelastic Behaviour of a Multiple Mode System 18510.7 Flutter Speed Prediction for Binary Systems 18510.8 Divergence of Dynamic Aeroelastic Systems 18810.9 Inclusion of Unsteady Reduced Frequency Effects 18910.10 Control Surface Flutter 19310.11 Whole Aircraft Model – Inclusion of Rigid Body Modes 19910.12 Flutter in the Transonic Regime 20210.13 Effect of Non-Linearities – Limit Cycle Oscillations 20210.14 Examples 20411 Aeroservoelasticity 20711.1 Mathematical Modelling of a Simple Aeroelastic System with a Control Surface 20811.2 Inclusion of Gust Terms 20911.3 Implementation of a Control System 21011.4 Determination of Closed Loop System Stability 21111.5 Gust Response of the Closed Loop System 21311.6 Inclusion of Control Law Frequency Dependency in Stability Calculations 21411.7 Response Determination via the Frequency Domain 21511.8 State Space Modelling 21611.9 Examples 21712 Equilibrium Manoeuvres 21912.1 Equilibrium Manoeuvre – Rigid Aircraft under Normal Acceleration 22112.2 Manoeuvre Envelope 22612.3 Equilibrium Manoeuvre – Rigid Aircraft Pitching 22712.4 Equilibrium Manoeuvre – Flexible Aircraft Pitching 23512.5 Representation of the Flight Control System (FCS) 25012.6 Examples 25013 Dynamic Manoeuvres 25313.1 Aircraft Axes 25513.2 Motion Variables 25713.3 Axes Transformations 25713.4 Velocity and Acceleration Components for Moving Axes in 2D 25913.5 Flight Mechanics Equations of Motion for a Rigid Symmetric Aircraft in 2D 26213.6 Representation of Disturbing Forces and Moments 26513.7 Modelling the Flexible Aircraft 26713.8 Solution of Flight Mechanics Equations for the Rigid Aircraft 27213.9 Dynamic Manoeuvre – Rigid Aircraft in Longitudinal Motion 27313.10 Dynamic Manoeuvre – Flexible Aircraft Heave/Pitch 27913.11 General Form of Longitudinal Equations 28713.12 Dynamic Manoeuvre for Rigid Aircraft in Lateral Motion 28813.13 Bookcase Manoeuvres for Rigid Aircraft in Lateral Motion 28913.14 Flight Control System (FCS) 29313.15 Representation of the Flight Control System (FCS) 29513.16 Examples 29514 Gust and Turbulence Encounters 29914.1 Gusts and Turbulence 30014.2 Gust Response in the Time Domain 30114.3 Time Domain Gust Response – Rigid Aircraft in Heave 30314.4 Time Domain Gust Response – Rigid Aircraft in Heave/Pitch 31014.5 Time Domain Gust Response – Flexible Aircraft 31614.6 General Form of Equations in the Time Domain 32114.7 Turbulence Response in the Frequency Domain 32114.8 Frequency Domain Turbulence Response – Rigid Aircraft in Heave 32414.9 Frequency Domain Turbulence Response – Rigid Aircraft in Heave/Pitch 32914.10 Frequency Domain Turbulence Response – Flexible Aircraft 33014.11 General Form of Equations in the Frequency Domain 33314.12 Representation of the Flight Control System (FCS) 33414.13 Examples 33415 Ground Manoeuvres 33715.1 Landing Gear 33715.2 Taxi, Take-Off and Landing Roll 34215.3 Landing 35115.4 Braking 35915.5 Turning 36015.6 Shimmy 36115.7 Representation of the Flight Control System (FCS) 36315.8 Examples 36316 Aircraft Internal Loads 36716.1 Limit and Ultimate Loads 36816.2 Internal Loads for an Aircraft 36816.3 General Internal Loads Expressions – Continuous Wing 37016.4 Effect of Wing-mounted Engines and Landing Gear 37216.5 Internal Loads – Continuous Flexible Wing 37316.6 General Internal Loads Expressions – Discretized Wing 37916.7 Internal Loads – Discretized Fuselage 38416.8 Internal Loads – Continuous Turbulence Encounter 38716.9 Loads Generation and Sorting to yield Critical Cases 38816.10 Aircraft Dimensioning Cases 39016.11 Stresses derived from Internal Loads – Complex Load Paths 39116.12 Examples 39117 Vibration of Continuous Systems – Finite Element Approach 39517.1 Introduction to the Finite Element Approach 39517.2 Formulation of the Beam Bending Element 39717.3 Assembly and Solution for a Beam Structure 40117.4 Torsion Element 40617.5 Combined Bending/Torsion Element 40717.6 Concentrated Mass Element 40817.7 Stiffness Element 40817.8 Rigid Body Elements 40917.9 Other Elements 41017.10 Comments on Modelling 41117.11 Examples 41318 Potential Flow Aerodynamics 41518.1 Components of Inviscid, Incompressible Flow Analysis 41518.2 Inclusion of Vorticity 42018.3 Numerical Steady Aerodynamic Modelling of Thin Two-dimensional Aerofoils 42218.4 Steady Aerodynamic Modelling of Three-Dimensional Wings using a Panel Method 42518.5 Unsteady Aerodynamic Modelling of Wings undergoing Harmonic Motion 42918.6 Aerodynamic Influence Coefficients in Modal Space 43218.7 Examples 43619 Coupling of Structural and Aerodynamic Computational Models 43719.1 Mathematical Modelling – Static Aeroelastic Case 43819.2 2D Coupled Static Aeroelastic Model – Pitch 43919.3 2D Coupled Static Aeroelastic Model – Heave/Pitch 44019.4 3D Coupled Static Aeroelastic Model 44119.5 Mathematical Modelling – Dynamic Aeroelastic Response 44619.6 2D Coupled Dynamic Aeroelastic Model – Bending/Torsion 44719.7 3D Flutter Analysis 44819.8 Inclusion of Frequency Dependent Aerodynamics for State–Space Modelling – Rational Function Approximation 450PART III INTRODUCTION TO INDUSTRIAL PRACTICE 45520 Aircraft Design and Certification 45720.1 Aeroelastics and Loads in the Aircraft Design Process 45720.2 Aircraft Certification Process 45921 Aeroelasticity and Loads Models 46521.1 Structural Model 46521.2 Aerodynamic Model 47121.3 Flight Control System 47321.4 Other Model Issues 47421.5 Loads Transformations 47422 Static Aeroelasticity and Flutter 47522.1 Static Aeroelasticity 47522.2 Flutter 47823 Flight Manoeuvre and Gust/Turbulence Loads 48123.1 Evaluation of Internal Loads 48123.2 Equilibrium/Balanced Flight Manoeuvres 48123.3 Dynamic Flight Manoeuvres 48523.4 Gusts and Turbulence 48924 Ground Manoeuvre Loads 49524.1 Aircraft/Landing Gear Models for Ground Manoeuvres 49524.2 Landing Gear/Airframe Interface 49624.3 Ground Manoeuvres – Landing 49624.4 Ground Manoeuvres – Ground Handling 49724.5 Loads Processing 49825 Testing Relevant to Aeroelasticity and Loads 50125.1 Introduction 50125.2 Wind Tunnel Tests 50125.3 Ground Vibration Test 50225.4 Structural Coupling Test 50325.5 Flight Simulator Test 50425.6 Structural Tests 50425.7 Flight Flutter Test 50525.8 Flight Loads Validation 507Appendices 509A Aircraft Rigid Body Modes 511B Table of Longitudinal Aerodynamic Derivatives 513C Aircraft Symmetric Flexible Modes 517D Model Condensation 527E Aerodynamic Derivatives in Body Fixed Axes 531References 535Index 539