Grigorios Dimitriadis, University of Liège, Belgium

• Preface xiDimitriadis: Nonlinear Aeroelasticity – Series Preface Oct 2016 xiiiAbout the Companion Website xv1 Introduction 11.1 Sources of Nonlinearity 31.2 Origins of Nonlinear Aeroelasticity 5References 62 Nonlinear Dynamics 92.1 Introduction 92.2 Ordinary Differential Equations 92.3 Linear Systems 112.3.1 Stable Oscillatory Response 132.3.2 Neutral Oscillatory Response 152.3.3 Unstable Oscillatory Response 172.3.4 Stable Non-oscillatory Response 192.3.5 Unstable Non-oscillatory Response 212.3.6 Fixed Point Summary 232.4 Nonlinear Systems 242.4.1 Linearisation Around Fixed Points 252.4.2 The Pitching Wing Section with Cubic Stiffness 282.4.3 The Pitchfork Bifurcation 302.5 Stability in the Lyapunov Sense 342.6 Asymmetric Systems 372.6.1 The Fold Bifurcation 382.6.2 The Transcritical Bifurcation 412.7 Existence of Periodic Solutions 452.7.1 Nonlinear Aeroelastic Galloping 472.8 Estimating Periodic Solutions 492.8.1 Periodic Solutions of the Nonlinear Galloping Oscillator 502.8.2 The Hopf Bifurcation 522.9 Stability of Periodic Solutions 532.9.1 Stability of Galloping Oscillations 552.9.2 Supercritical and Subcritical Hopf Bifurcations 562.9.3 The Fold Bifurcation of Cycles 562.10 Concluding Remarks 61References 613 Time Integration 633.1 Introduction 633.2 Euler Method 643.2.1 Linear Systems 653.2.2 Nonlinear Systems 663.3 Central Difference Method 683.3.1 Explicit Solution of Nonlinear Systems 693.3.2 Implicit Solution of Nonlinear Systems 723.4 Runge–Kutta Method 743.5 Time-Varying Linear Approximation 803.6 Integrating Backwards in Time 863.7 Time Integration of Systems with Multiple Degrees of Freedom 883.8 Forced Response 923.9 Harmonic Balance 993.9.1 Newton–Raphson 1033.9.2 Discrete Fourier Transform Techniques 1063.10 Concluding Remarks 110References 1114 Determining the Vibration Parameters 1134.1 Introduction 1134.2 Amplitude and Frequency Determination 1134.2.1 Event Detection 1174.3 Equivalent Linearisation 1204.4 Hilbert Transform 1254.5 Time-Varying Linear Approximation 1294.6 Short Time Fourier Transform 1314.7 Pinpointing Bifurcations 1374.7.1 Newton–Raphson 1414.7.2 Successive Bisection 1424.8 Limit Cycle Study 1434.9 Poincaré Sections 1464.10 Stability of Periodic Solutions 1494.10.1 Floquet Analysis 1524.11 Concluding Remarks 156References 1565 Bifurcations of Fundamental Aeroelastic Systems 1595.1 Introduction 1595.2 Two-Dimensional Unsteady Pitch-Plunge-Control Wing 1605.3 Linear Aeroelastic Analysis 1615.4 Hardening Stiffness 1705.4.1 Supercritical Hopf Bifurcation 1705.4.2 Subcritical Hopf Bifurcation 1805.4.3 Fold Bifurcation of Cycles 1835.4.4 Flutter of Nonlinear Systems 1895.4.5 Period-Doubling Bifurcation 1935.4.6 Torus Bifurcation 2015.5 Softening Stiffness 2095.6 Damping Nonlinearity 2145.6.1 Subcritical Hopf Bifurcation 2165.6.2 Static Divergence of Cycles 2205.6.3 Pitchfork Bifurcation of Cycles 2245.7 Two-Parameter Bifurcations 2335.7.1 Generalised Hopf Bifurcation 2335.7.2 Pitchfork–Hopf Bifurcation 2375.7.3 Hopf-Hopf Bifurcation 2405.8 Asymmetric Nonlinear Aeroelastic Systems 2425.8.1 Fold Bifurcation of Fixed Points and Cycles 2435.8.2 Transcritical Bifurcation of Fixed Points and Cycles 2515.8.3 Fold-Hopf Bifurcation 2565.9 Concluding Remarks 257References 2596 Discontinuous Nonlinearities 2616.1 Introduction 2616.2 Piecewise Linear Stiffness 2626.2.1 Underlying and Overlying Linear Systems 2646.2.2 Fixed Points and Boundary Equilibrium Bifurcations 2696.2.3 Equivalent Linearisation of Piecewise Linear Stiffness 2726.2.4 Three-Domain Limit Cycles 2786.2.5 Two-Domain Limit Cycles 2856.2.6 Time Domain Solutions 2896.3 Discontinuity-Induced Bifurcations 2976.3.1 The Boundary Equilibrium Bifurcation 2976.3.2 The Grazing Bifurcation 3026.4 Freeplay and Friction 3096.5 Concluding Remarks 310References 3107 Numerical Continuation 3137.1 Introduction 3137.2 Algebraic Problems 3147.2.1 Prediction Correction 3167.2.2 Arclength Continuation 3217.2.3 Pseudo-Arclength Continuation 3277.3 Direct Location of Folds 3287.4 Fixed Point Solutions of Dynamic Systems 3327.4.1 Branch Points 3327.4.2 Arclength Step Control 3377.5 Periodic Solutions of Dynamic Systems 3427.5.1 Starting the Continuation Scheme 3487.5.2 Folds and Branch Points 3517.5.3 Branch Switching 3557.6 Stability of Periodic Solutions Calculated from Numerical Continuation 3587.7 Shooting 3647.7.1 Starting the Continuation Scheme 3677.7.2 Arclength Continuation 3687.7.3 Stability Analysis 3707.7.4 Branch Point Location and Branch Switching 3727.7.5 Grazing 3757.8 Harmonic Balance 3797.9 Concluding Remarks 387References 3878 Low-Speed Aerodynamic Nonlinearities 3898.1 Introduction 3898.2 Vortex-Induced Vibrations 3938.3 Galloping 4028.4 Stall Flutter 4118.4.1 Dynamic Stall 4138.4.2 Leishman–Beddoes Model 4178.4.3 ONERA Model 4348.4.4 Aeroelastic Simulations using Dynamic Stall Models 4428.5 Concluding Remarks 449References 4499 High-Speed Aeroelastic Nonlinearities 4539.1 Introduction 4539.2 Piston Theory 4539.3 Panel Flutter 4689.3.1 Buckling 4709.3.2 Limit Cycle Oscillations 4849.4 Concluding Remarks 501References 50110 Finite Wings 50310.1 Introduction 50310.2 Cantilever Plate in Supersonic Flow 50410.3 Three-Dimensional Aerodynamic Modelling by the Vortex Lattice Method 51910.3.1 Aeroelastic Coupling 52810.3.2 Transforming to the Time Domain 53610.3.3 Nonlinear Response 54210.4 Concluding Remarks 552References 552Appendix A: Aeroelastic Models 555Index 571

Grigorios Dimitriadis, University of Liège, Belgium

• Preface xiDimitriadis: Nonlinear Aeroelasticity – Series Preface Oct 2016 xiiiAbout the Companion Website xv1 Introduction 11.1 Sources of Nonlinearity 31.2 Origins of Nonlinear Aeroelasticity 5References 62 Nonlinear Dynamics 92.1 Introduction 92.2 Ordinary Differential Equations 92.3 Linear Systems 112.3.1 Stable Oscillatory Response 132.3.2 Neutral Oscillatory Response 152.3.3 Unstable Oscillatory Response 172.3.4 Stable Non-oscillatory Response 192.3.5 Unstable Non-oscillatory Response 212.3.6 Fixed Point Summary 232.4 Nonlinear Systems 242.4.1 Linearisation Around Fixed Points 252.4.2 The Pitching Wing Section with Cubic Stiffness 282.4.3 The Pitchfork Bifurcation 302.5 Stability in the Lyapunov Sense 342.6 Asymmetric Systems 372.6.1 The Fold Bifurcation 382.6.2 The Transcritical Bifurcation 412.7 Existence of Periodic Solutions 452.7.1 Nonlinear Aeroelastic Galloping 472.8 Estimating Periodic Solutions 492.8.1 Periodic Solutions of the Nonlinear Galloping Oscillator 502.8.2 The Hopf Bifurcation 522.9 Stability of Periodic Solutions 532.9.1 Stability of Galloping Oscillations 552.9.2 Supercritical and Subcritical Hopf Bifurcations 562.9.3 The Fold Bifurcation of Cycles 562.10 Concluding Remarks 61References 613 Time Integration 633.1 Introduction 633.2 Euler Method 643.2.1 Linear Systems 653.2.2 Nonlinear Systems 663.3 Central Difference Method 683.3.1 Explicit Solution of Nonlinear Systems 693.3.2 Implicit Solution of Nonlinear Systems 723.4 Runge–Kutta Method 743.5 Time-Varying Linear Approximation 803.6 Integrating Backwards in Time 863.7 Time Integration of Systems with Multiple Degrees of Freedom 883.8 Forced Response 923.9 Harmonic Balance 993.9.1 Newton–Raphson 1033.9.2 Discrete Fourier Transform Techniques 1063.10 Concluding Remarks 110References 1114 Determining the Vibration Parameters 1134.1 Introduction 1134.2 Amplitude and Frequency Determination 1134.2.1 Event Detection 1174.3 Equivalent Linearisation 1204.4 Hilbert Transform 1254.5 Time-Varying Linear Approximation 1294.6 Short Time Fourier Transform 1314.7 Pinpointing Bifurcations 1374.7.1 Newton–Raphson 1414.7.2 Successive Bisection 1424.8 Limit Cycle Study 1434.9 Poincaré Sections 1464.10 Stability of Periodic Solutions 1494.10.1 Floquet Analysis 1524.11 Concluding Remarks 156References 1565 Bifurcations of Fundamental Aeroelastic Systems 1595.1 Introduction 1595.2 Two-Dimensional Unsteady Pitch-Plunge-Control Wing 1605.3 Linear Aeroelastic Analysis 1615.4 Hardening Stiffness 1705.4.1 Supercritical Hopf Bifurcation 1705.4.2 Subcritical Hopf Bifurcation 1805.4.3 Fold Bifurcation of Cycles 1835.4.4 Flutter of Nonlinear Systems 1895.4.5 Period-Doubling Bifurcation 1935.4.6 Torus Bifurcation 2015.5 Softening Stiffness 2095.6 Damping Nonlinearity 2145.6.1 Subcritical Hopf Bifurcation 2165.6.2 Static Divergence of Cycles 2205.6.3 Pitchfork Bifurcation of Cycles 2245.7 Two-Parameter Bifurcations 2335.7.1 Generalised Hopf Bifurcation 2335.7.2 Pitchfork–Hopf Bifurcation 2375.7.3 Hopf-Hopf Bifurcation 2405.8 Asymmetric Nonlinear Aeroelastic Systems 2425.8.1 Fold Bifurcation of Fixed Points and Cycles 2435.8.2 Transcritical Bifurcation of Fixed Points and Cycles 2515.8.3 Fold-Hopf Bifurcation 2565.9 Concluding Remarks 257References 2596 Discontinuous Nonlinearities 2616.1 Introduction 2616.2 Piecewise Linear Stiffness 2626.2.1 Underlying and Overlying Linear Systems 2646.2.2 Fixed Points and Boundary Equilibrium Bifurcations 2696.2.3 Equivalent Linearisation of Piecewise Linear Stiffness 2726.2.4 Three-Domain Limit Cycles 2786.2.5 Two-Domain Limit Cycles 2856.2.6 Time Domain Solutions 2896.3 Discontinuity-Induced Bifurcations 2976.3.1 The Boundary Equilibrium Bifurcation 2976.3.2 The Grazing Bifurcation 3026.4 Freeplay and Friction 3096.5 Concluding Remarks 310References 3107 Numerical Continuation 3137.1 Introduction 3137.2 Algebraic Problems 3147.2.1 Prediction Correction 3167.2.2 Arclength Continuation 3217.2.3 Pseudo-Arclength Continuation 3277.3 Direct Location of Folds 3287.4 Fixed Point Solutions of Dynamic Systems 3327.4.1 Branch Points 3327.4.2 Arclength Step Control 3377.5 Periodic Solutions of Dynamic Systems 3427.5.1 Starting the Continuation Scheme 3487.5.2 Folds and Branch Points 3517.5.3 Branch Switching 3557.6 Stability of Periodic Solutions Calculated from Numerical Continuation 3587.7 Shooting 3647.7.1 Starting the Continuation Scheme 3677.7.2 Arclength Continuation 3687.7.3 Stability Analysis 3707.7.4 Branch Point Location and Branch Switching 3727.7.5 Grazing 3757.8 Harmonic Balance 3797.9 Concluding Remarks 387References 3878 Low-Speed Aerodynamic Nonlinearities 3898.1 Introduction 3898.2 Vortex-Induced Vibrations 3938.3 Galloping 4028.4 Stall Flutter 4118.4.1 Dynamic Stall 4138.4.2 Leishman–Beddoes Model 4178.4.3 ONERA Model 4348.4.4 Aeroelastic Simulations using Dynamic Stall Models 4428.5 Concluding Remarks 449References 4499 High-Speed Aeroelastic Nonlinearities 4539.1 Introduction 4539.2 Piston Theory 4539.3 Panel Flutter 4689.3.1 Buckling 4709.3.2 Limit Cycle Oscillations 4849.4 Concluding Remarks 501References 50110 Finite Wings 50310.1 Introduction 50310.2 Cantilever Plate in Supersonic Flow 50410.3 Three-Dimensional Aerodynamic Modelling by the Vortex Lattice Method 51910.3.1 Aeroelastic Coupling 52810.3.2 Transforming to the Time Domain 53610.3.3 Nonlinear Response 54210.4 Concluding Remarks 552References 552Appendix A: Aeroelastic Models 555Index 571

Grigorios Dimitriadis, University of Liège, Belgium

• Preface xiDimitriadis: Nonlinear Aeroelasticity – Series Preface Oct 2016 xiiiAbout the Companion Website xv1 Introduction 11.1 Sources of Nonlinearity 31.2 Origins of Nonlinear Aeroelasticity 5References 62 Nonlinear Dynamics 92.1 Introduction 92.2 Ordinary Differential Equations 92.3 Linear Systems 112.3.1 Stable Oscillatory Response 132.3.2 Neutral Oscillatory Response 152.3.3 Unstable Oscillatory Response 172.3.4 Stable Non-oscillatory Response 192.3.5 Unstable Non-oscillatory Response 212.3.6 Fixed Point Summary 232.4 Nonlinear Systems 242.4.1 Linearisation Around Fixed Points 252.4.2 The Pitching Wing Section with Cubic Stiffness 282.4.3 The Pitchfork Bifurcation 302.5 Stability in the Lyapunov Sense 342.6 Asymmetric Systems 372.6.1 The Fold Bifurcation 382.6.2 The Transcritical Bifurcation 412.7 Existence of Periodic Solutions 452.7.1 Nonlinear Aeroelastic Galloping 472.8 Estimating Periodic Solutions 492.8.1 Periodic Solutions of the Nonlinear Galloping Oscillator 502.8.2 The Hopf Bifurcation 522.9 Stability of Periodic Solutions 532.9.1 Stability of Galloping Oscillations 552.9.2 Supercritical and Subcritical Hopf Bifurcations 562.9.3 The Fold Bifurcation of Cycles 562.10 Concluding Remarks 61References 613 Time Integration 633.1 Introduction 633.2 Euler Method 643.2.1 Linear Systems 653.2.2 Nonlinear Systems 663.3 Central Difference Method 683.3.1 Explicit Solution of Nonlinear Systems 693.3.2 Implicit Solution of Nonlinear Systems 723.4 Runge–Kutta Method 743.5 Time-Varying Linear Approximation 803.6 Integrating Backwards in Time 863.7 Time Integration of Systems with Multiple Degrees of Freedom 883.8 Forced Response 923.9 Harmonic Balance 993.9.1 Newton–Raphson 1033.9.2 Discrete Fourier Transform Techniques 1063.10 Concluding Remarks 110References 1114 Determining the Vibration Parameters 1134.1 Introduction 1134.2 Amplitude and Frequency Determination 1134.2.1 Event Detection 1174.3 Equivalent Linearisation 1204.4 Hilbert Transform 1254.5 Time-Varying Linear Approximation 1294.6 Short Time Fourier Transform 1314.7 Pinpointing Bifurcations 1374.7.1 Newton–Raphson 1414.7.2 Successive Bisection 1424.8 Limit Cycle Study 1434.9 Poincaré Sections 1464.10 Stability of Periodic Solutions 1494.10.1 Floquet Analysis 1524.11 Concluding Remarks 156References 1565 Bifurcations of Fundamental Aeroelastic Systems 1595.1 Introduction 1595.2 Two-Dimensional Unsteady Pitch-Plunge-Control Wing 1605.3 Linear Aeroelastic Analysis 1615.4 Hardening Stiffness 1705.4.1 Supercritical Hopf Bifurcation 1705.4.2 Subcritical Hopf Bifurcation 1805.4.3 Fold Bifurcation of Cycles 1835.4.4 Flutter of Nonlinear Systems 1895.4.5 Period-Doubling Bifurcation 1935.4.6 Torus Bifurcation 2015.5 Softening Stiffness 2095.6 Damping Nonlinearity 2145.6.1 Subcritical Hopf Bifurcation 2165.6.2 Static Divergence of Cycles 2205.6.3 Pitchfork Bifurcation of Cycles 2245.7 Two-Parameter Bifurcations 2335.7.1 Generalised Hopf Bifurcation 2335.7.2 Pitchfork–Hopf Bifurcation 2375.7.3 Hopf-Hopf Bifurcation 2405.8 Asymmetric Nonlinear Aeroelastic Systems 2425.8.1 Fold Bifurcation of Fixed Points and Cycles 2435.8.2 Transcritical Bifurcation of Fixed Points and Cycles 2515.8.3 Fold-Hopf Bifurcation 2565.9 Concluding Remarks 257References 2596 Discontinuous Nonlinearities 2616.1 Introduction 2616.2 Piecewise Linear Stiffness 2626.2.1 Underlying and Overlying Linear Systems 2646.2.2 Fixed Points and Boundary Equilibrium Bifurcations 2696.2.3 Equivalent Linearisation of Piecewise Linear Stiffness 2726.2.4 Three-Domain Limit Cycles 2786.2.5 Two-Domain Limit Cycles 2856.2.6 Time Domain Solutions 2896.3 Discontinuity-Induced Bifurcations 2976.3.1 The Boundary Equilibrium Bifurcation 2976.3.2 The Grazing Bifurcation 3026.4 Freeplay and Friction 3096.5 Concluding Remarks 310References 3107 Numerical Continuation 3137.1 Introduction 3137.2 Algebraic Problems 3147.2.1 Prediction Correction 3167.2.2 Arclength Continuation 3217.2.3 Pseudo-Arclength Continuation 3277.3 Direct Location of Folds 3287.4 Fixed Point Solutions of Dynamic Systems 3327.4.1 Branch Points 3327.4.2 Arclength Step Control 3377.5 Periodic Solutions of Dynamic Systems 3427.5.1 Starting the Continuation Scheme 3487.5.2 Folds and Branch Points 3517.5.3 Branch Switching 3557.6 Stability of Periodic Solutions Calculated from Numerical Continuation 3587.7 Shooting 3647.7.1 Starting the Continuation Scheme 3677.7.2 Arclength Continuation 3687.7.3 Stability Analysis 3707.7.4 Branch Point Location and Branch Switching 3727.7.5 Grazing 3757.8 Harmonic Balance 3797.9 Concluding Remarks 387References 3878 Low-Speed Aerodynamic Nonlinearities 3898.1 Introduction 3898.2 Vortex-Induced Vibrations 3938.3 Galloping 4028.4 Stall Flutter 4118.4.1 Dynamic Stall 4138.4.2 Leishman–Beddoes Model 4178.4.3 ONERA Model 4348.4.4 Aeroelastic Simulations using Dynamic Stall Models 4428.5 Concluding Remarks 449References 4499 High-Speed Aeroelastic Nonlinearities 4539.1 Introduction 4539.2 Piston Theory 4539.3 Panel Flutter 4689.3.1 Buckling 4709.3.2 Limit Cycle Oscillations 4849.4 Concluding Remarks 501References 50110 Finite Wings 50310.1 Introduction 50310.2 Cantilever Plate in Supersonic Flow 50410.3 Three-Dimensional Aerodynamic Modelling by the Vortex Lattice Method 51910.3.1 Aeroelastic Coupling 52810.3.2 Transforming to the Time Domain 53610.3.3 Nonlinear Response 54210.4 Concluding Remarks 552References 552Appendix A: Aeroelastic Models 555Index 571