Introduction to Nonlinear Aeroelasticity E-Book

Grigorios Dimitriadis

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Aerospace Series (PEP)
• Sprache: Englisch

Seitenzahl: 829

Veröffentlichungsjahr: 2017

Beschreibung

Introduction to Nonlinear Aeroelasticity

Introduces the latest developments and technologies in the area of nonlinear aeroelasticity

Nonlinear aeroelasticity has become an increasingly popular research area in recent years. There have been many driving forces behind this development, increasingly flexible structures, nonlinear control laws, materials with nonlinear characteristics and so on. Introduction to Nonlinear Aeroelasticity covers the theoretical basics in nonlinear aeroelasticity and applies the theory to practical problems.

As nonlinear aeroelasticity is a combined topic, necessitating expertise from different areas, the book introduces methodologies from a variety of disciplines such as nonlinear dynamics, bifurcation analysis, unsteady aerodynamics, non-smooth systems and others. The emphasis throughout is on the practical application of the theories and methods, so as to enable the reader to apply their newly acquired knowledge

Key features:

• Covers the major topics in nonlinear aeroelasticity, from the galloping of cables to supersonic panel flutter
• Discusses nonlinear dynamics, bifurcation analysis, numerical continuation, unsteady aerodynamics and non-smooth systems
• Considers the practical application of the theories and methods
• Covers nonlinear dynamics, bifurcation analysis and numerical methods
• Accompanied by a website hosting Matlab code

Introduction to Nonlinear Aeroelasticity is a comprehensive reference for researchers and workers in industry and is also a useful introduction to the subject for graduate and undergraduate students across engineering disciplines.

Leseprobe

Table of Contents

Cover

Title Page

Preface

Dimitriadis: Nonlinear Aeroelasticity – Series Preface Oct 2016

About the Companion Website

1 Introduction

1.1 Sources of Nonlinearity

1.2 Origins of Nonlinear Aeroelasticity

References

2 Nonlinear Dynamics

2.1 Introduction

2.2 Ordinary Differential Equations

2.3 Linear Systems

2.4 Nonlinear Systems

2.5 Stability in the Lyapunov Sense

2.6 Asymmetric Systems

2.7 Existence of Periodic Solutions

2.8 Estimating Periodic Solutions

2.9 Stability of Periodic Solutions

2.10 Concluding Remarks

References

3 Time Integration

3.1 Introduction

3.2 Euler Method

3.3 Central Difference Method

3.4 Runge–Kutta Method

3.5 Time‐Varying Linear Approximation

3.6 Integrating Backwards in Time

3.7 Time Integration of Systems with Multiple Degrees of Freedom

3.8 Forced Response

3.9 Harmonic Balance

3.10 Concluding Remarks

References

4 Determining the Vibration Parameters

4.1 Introduction

4.2 Amplitude and Frequency Determination

4.3 Equivalent Linearisation

4.4 Hilbert Transform

4.5 Time‐Varying Linear Approximation

4.6 Short Time Fourier Transform

4.7 Pinpointing Bifurcations

4.8 Limit Cycle Study

4.9 Poincaré Sections

4.10 Stability of Periodic Solutions

4.11 Concluding Remarks

References

5 Bifurcations of Fundamental Aeroelastic Systems

5.1 Introduction

5.2 Two‐Dimensional Unsteady Pitch‐Plunge‐Control Wing

5.3 Linear Aeroelastic Analysis

5.4 Hardening Stiffness

5.5 Softening Stiffness

5.6 Damping Nonlinearity

5.7 Two‐Parameter Bifurcations

5.8 Asymmetric Nonlinear Aeroelastic Systems

5.9 Concluding Remarks

References

6 Discontinuous Nonlinearities

6.1 Introduction

6.2 Piecewise Linear Stiffness

6.3 Discontinuity‐Induced Bifurcations

6.4 Freeplay and Friction

6.5 Concluding Remarks

References

7 Numerical Continuation

7.1 Introduction

7.2 Algebraic Problems

7.3 Direct Location of Folds

7.4 Fixed Point Solutions of Dynamic Systems

7.5 Periodic Solutions of Dynamic Systems

7.6 Stability of Periodic Solutions Calculated from Numerical Continuation

7.7 Shooting

7.8 Harmonic Balance

7.9 Concluding Remarks

References

8 Low‐Speed Aerodynamic Nonlinearities

8.1 Introduction

8.2 Vortex‐Induced Vibrations

8.3 Galloping

8.4 Stall Flutter

8.5 Concluding Remarks

References

9 High‐Speed Aeroelastic Nonlinearities

9.1 Introduction

9.2 Piston Theory

9.3 Panel Flutter

9.4 Concluding Remarks

References

10 Finite Wings

10.1 Introduction

10.2 Cantilever Plate in Supersonic Flow

10.3 Three‐Dimensional Aerodynamic Modelling by the Vortex Lattice Method

10.4 Concluding Remarks

References

Appendix A: Aeroelastic Models

A.1 Galloping Oscillator

A.2 Two‐Dimensional Pitch‐Plunge‐Control Wing Section with Unsteady Aerodynamics

A.3 Two‐Dimensional Pitch‐Plunge‐Control Wing Section with Quasi‐Steady Aerodynamics

A.4 Two‐Dimensional Pitch‐Plunge Wing Section with Quasi‐Steady Aerodynamics

A.5 Two‐Dimensional Pitching Wing Section with Quasi‐Steady Aerodynamics

A.6 Two‐Dimensional Pitch‐Plunge Wing with Leishman–Beddoes Aerodynamic Model

A.7 Two‐Dimensional Pitch‐Plunge Wing with ONERA Aerodynamic Model

A.8 Two‐Dimensional Pitch‐Plunge‐Control Wing Section with Supersonic Aerodynamics

A.9 Two‐Dimensional Pitch‐Plunge Wing Section with Supersonic Aerodynamics

References

Index

End User License Agreement

List of Tables

Chapter 02

Table 2.1 Summary of fixed points

Chapter 04

Table 4.1 Floquet exponents calculated by the Euler method and Runge–Kutta methods

Chapter 07

Table 7.1 Hopf conditions on each of the fixed points of equations 7.118

Chapter 08

Table 8.1 Summary of Leisman–Beddoes states

Chapter 10

Table 10.1 Variation of flutter speed with number of chordwise and spanwise out‐of‐plane modes

Table 10.2 Variation of flutter speed and frequency with number of chordwise and spanwise panels

Table 10.3 Variation of limit cycle amplitude at two airspeeds with number of modes

List of Illustrations

Chapter 01

Figure 1.1 Linearised load‐displacement diagrams

Chapter 02

Figure 2.1 Stable oscillatory response of harmonic oscillator

Figure 2.2 Neutral oscillatory response of harmonic oscillator

Figure 2.3 Unstable oscillatory response of harmonic oscillator

Figure 2.4 Two‐dimensional rectangular cylinder with a plunge degree of freedom

Figure 2.5 Stable non‐oscillatory response of harmonic oscillator

Figure 2.6 Phase plane plot of many response trajectories attracted by a stable node

Figure 2.7 Phase plane plot of many response trajectories around a saddle

Figure 2.8 Two‐dimensional wing section with a pitch degree of freedom

Figure 2.9 Supercritical (a) and subcritical (b) bifurcations of the pitching wing section with cubic stiffness

Figure 2.10 Real and imaginary parts of eigenvalues around all three fixed points

Figure 2.11 Vector field of pitching wing section with cubic stiffness for

m/s (a) and

m/s (b)

Figure 2.12 Fold bifurcation of the asymmetric 2D pitching wing section with cubic stiffness

Figure 2.13 Eigenvalues of Jacobian around

and

for increasing airspeed

Figure 2.14 Fold and transcritical bifurcations of the asymmetric 2D pitching wing section with cubic stiffness

Figure 2.15 Vector field of asymmetric 2D pitching wing with cubic stiffness at airspeeds before (a) and after (b) the transcritical bifurcation

Figure 2.16 Three possibilities according to the Poincaré–Bendixson theorem

Figure 2.17 Two possibilities according to Poincaré–Bendixson theorem applied to an annular region

Figure 2.18 Plot of

f

(

x

1

) for the galloping equation at two different values of the wind speed

V

Figure 2.19 Variation of limit cycle radius of galloping oscillator with airspeed

Figure 2.20 Hopf bifurcation

Figure 2.21 Stable and unstable limit cycles

Figure 2.22 Stability criterion for the limit cycles of the galloping oscillator

Figure 2.23 Supercritical and subcritical Hopf bifurcation

Figure 2.24 Fold bifurcation

Figure 2.25 Variation of limit cycle radius with airspeed for galloping oscillator with

and

Chapter 03

Figure 3.1 Time response of galloping oscillator calculated by Euler method with three values of Δ

t

Figure 3.2 Time response of the 2D pitching wing section with cubic stiffness calculated by central differences with three values of Δ

t

Figure 3.3 Time response of galloping oscillator calculated by implicit central differences for three different values of Δ

τ

Figure 3.4 Time response of the 2D pitching wing section calculated by Runge–Kutta–Fehlberg approach, for two different initial conditions

Figure 3.5 Variation of Δ

t

with time for the two solutions plotted in Figure 3.4

Figure 3.6 Time response of pitching wing section calculated by linear time‐varying approximation and Runge–Kutta–Fehlberg

Figure 3.7 Percentage difference between linear time‐varying approximation with variable time step and Runge–Kutta–Fehlberg solution

Figure 3.8 Variation of the eigenvalues of the Jacobian with time

Figure 3.9 Backwards and forward time integration of the galloping oscillator by the Runge–Kutta–Fehlberg method at

Figure 3.10 Variation of the unstable limit cycle amplitude of a galloping oscillator calculated by Runge–Kutta–Fehlberg and Harmonic Balance

Figure 3.11 Pitch‐plunge wing section

Figure 3.12 Time response of the 2D pitch‐plunge wing with cubic pitch stiffness at

U

=10 m/s

Figure 3.13 Time response of the 2D pitch‐plunge wing with cubic pitch stiffness at

U

=23 m/s

Figure 3.14 Two‐dimensional projections of the response trajectory of Figure 3.13

Figure 3.15 Two‐dimensional pitching wing section with control surface

Figure 3.16 Response of 2D pitching wing section at

m/s to sinusoidal control surface excitation

Figure 3.17 Response of 2D pitching wing section at

m/s to sinusoidal control surface excitation

Figure 3.18 Complete response amplitude variation with frequency

Figure 3.19 Nonlinear resonance of 2D pitching wing section at

m/s for increasing excitation frequency and amplitude

Figure 3.20 Harmonic Balance solution of 2D pitching wing section at

m/s for increasing excitation frequency

Figure 3.21 Fifth order Harmonic Balance solution of 2D pitching wing section at

m/s for increasing excitation frequency

Chapter 04

Figure 4.1 Mismatch between true maximum and sampled signal maximum

Figure 4.2 Time, amplitude and frequency response of pitching wing section with cubic stiffness at

m/s

Figure 4.3 Time, amplitude and frequency response of pitching wing section with cubic stiffness at

m/s

Figure 4.4 Backbone from Example 4.1

Figure 4.5 Equivalent linearised and numerical backbones of the pitching wing section

Figure 4.6 Variation of centre of oscillation with time

Figure 4.7 Equivalent linearised frequency surface (grid) and numerical backbone (black line) of the pitching wing section

Figure 4.8 Section of displacement response of pitching wing section, showing the instantaneous amplitude calculated by the Hilbert Transform

Figure 4.9 Amplitude and frequency response of pitching wing section with cubic stiffness, calculated using the Hilbert Transform

Figure 4.10 Natural frequency and damping variation with amplitude for pitching wing section with friction

Figure 4.11 Wind‐off response of pitch‐plunge wing with cubic stiffness in plunge

Figure 4.12 Instantaneous amplitudes and frequencies of the plunge and pitch responses

Figure 4.13 Short Time Fourier Transform of the plunge and pitch responses

Figure 4.14 Hopf and pitchfork test function variation with airspeed

Figure 4.15 Limit cycle amplitude and period against airspeed

Figure 4.16 Construction of bifurcation diagram for more complex limit cycles

Figure 4.17 Poincaré sections of 2D and 3D limit cycles

Figure 4.18 Response of 2D pitching wing section with cubic stiffness and sinusoidal control surface excitation,

Figure 4.19 Response of 2D pitching wing section with cubic stiffness and sinusoidal control surface excitation,

Figure 4.20 Poincaré section of chaotic response of 2D pitching wing section with cubic stiffness

Figure 4.21 Variation of the eigenvalues of matrix

B

with airspeed

Chapter 05

Figure 5.1 Pitch‐plunge‐control wing section

Figure 5.2 Mode shapes of the linear unsteady pitch‐plunge‐control system at two different airspeeds

Figure 5.3 Variation of natural frequencies and damping ratios of the linear unsteady pitch‐plunge‐control system with airspeed

Figure 5.4 Variation of real and imaginary parts of eigenvalues (a) and static divergence criteria (b) with airspeed

Figure 5.5 Time response of pitch plunge control wing with cubic stiffness in pitch at two different airspeeds

Figure 5.6 Time‐varying eigenvalues over a complete cycle of oscillation,

m/s

Figure 5.7 LCO amplitude and frequency variation with airspeed

Figure 5.8 Flutter airspeed and frequency variation with equivalent linear stiffness

Figure 5.9 LCO amplitude and frequency variation with airspeed by equivalent linearisation and time integration

Figure 5.10 Comparison of true limit cycle shapes and equivalent linearisation predictions at two different airspeeds

Figure 5.11 Time response of pitch plunge control wing with cubic stiffness in plunge at two different airspeeds and initial conditions

Figure 5.12 Effect of airspeed and initial condition on the response of the pitch‐plunge‐control wing with cubic plunge; a ‘x’ denotes decaying response, a ‘o’ denotes a LCO

Figure 5.13 Equivalent linearisation of pitch‐plunge‐control wing with hardening plunge stiffness

Figure 5.14 Natural frequencies and damping ratios of the linear pitch‐plunge‐control system with varying plunge stiffness at two different airspeeds

Figure 5.15 Limit cycle amplitude and frequency variation with airspeed, showing the two folds

Figure 5.16 Comparison of natural frequencies and damping ratios of three‐DOF wing with very high pitch stiffness and two‐DOF wing

Figure 5.17 Time response of pitch‐plunge‐control wing with cubic pitch stiffness at

m/s

Figure 5.18 Natural frequencies and damping ratios of time‐varying linearised system over a complete cycle of oscillation,

m/s

Figure 5.19 Limit cycle amplitude against airspeed for

and

cases

Figure 5.20 Equivalent linearised estimate of the complete bifurcation

Figure 5.21 Variation of LCO waveform in pitch as the airspeed is increased

Figure 5.22 Frequency content of limit cycle with increasing airspeed

Figure 5.23 Natural frequencies of time varying linear system and harmonics

Figure 5.24 Number of peaks in

α

waveform at two different airspeeds

Figure 5.25 Bifurcation diagram of period‐doubling route to chaos

Figure 5.26 Variation of natural frequencies and damping ratios of the linear unsteady pitch‐plunge‐control system with airspeed

Figure 5.27 Equivalent linearisation of pitch‐plunge‐control wing with hardening control stiffness

Figure 5.28 Time trajectories of pitch plunge control wing with cubic stiffness in the control DOF at two different airspeeds

Figure 5.29 Demonstration of the difference between a trajectory following a limit cycle and one following a torus

Figure 5.30 Phase plane trajectories of pitch plunge control wing with cubic stiffness in the control DOF at two different airspeeds

Figure 5.31 Poincaré section variation with airspeed for torus bifurcation

Figure 5.32 Nonlinear static divergence boundary for system with softening stiffness

Figure 5.33 Results of equivalent linearisation of pitch‐plunge‐control wing with softening stiffness in pitch

Figure 5.34 Response of pitch‐plunge‐control wing with softening stiffness in pitch – supercritical Hopf case

Figure 5.35

plot for a pitch‐plunge‐control wing

Figure 5.36 Equivalent linearisation of pitch‐plunge‐control wing with quadratic damping in control DOF

Figure 5.37 Variation of natural frequencies and real parts of eigenvalues of the linear unsteady pitch‐plunge‐control wing with airspeed

Figure 5.38 Time response of system with quadratic damping in pitch at two different airspeeds

Figure 5.39 Equivalent linearisation of pitch‐plunge‐control wing undergoing static divergence of limit cycles

Figure 5.40 Pitchfork bifurcation

Figure 5.41 Time response of system with quadratic damping and cubic stiffness in pitch at

m/s

Figure 5.42 Bifurcation diagram of system undergoing pitchfork bifurcation of cycles

Figure 5.43 Three‐dimensional view of the pitchfork bifurcation of cycles (the unstable limit cycle is not plotted)

Figure 5.44 Equivalent linearisation of pitch‐plunge‐control wing undergoing pitchfork bifurcation of cycles

Figure 5.45 Equivalent linearisation of pitch‐plunge‐control wing undergoing generalised Hopf bifurcation

Figure 5.46 Variation of flutter speed with

ω

α

(a) and generalised Hopf bifurcation (b)

Figure 5.47 Time responses at the generalised Hopf point

Figure 5.48 Two‐parameter pitchfork‐Hopf bifurcation

Figure 5.49 Equivalent linearisation of pitch‐plunge‐control wing undergoing Hopf–Hopf bifurcation

Figure 5.50 Bifurcation diagram of pitch‐plunge‐control wing undergoing Hopf–Hopf bifurcation

Figure 5.51 Poincaré sections of pitch‐plunge‐control wing undergoing Hopf–Hopf bifurcation

Figure 5.52 Fold bifurcation

Figure 5.53 Time response of system after fold bifurcation,

m/s

Figure 5.54 Time response of system after fold bifurcation,

m/s

Figure 5.55 Bifurcation diagram of pitch‐plunge‐control wing undergoing fold bifurcation

Figure 5.56 Equivalent linearisation of pitch‐plunge‐control wing undergoing fold bifurcation of cycles

Figure 5.57 Two‐ and three‐dimensional projections of stable and unstable limit cycles at

m/s

Figure 5.58 Fold bifurcation followed by transcritical bifurcation

Figure 5.59 System response at three airspeeds around the transcritical bifurcation of fixed points

Figure 5.60 Transcritical bifurcation of limit cycles

Figure 5.61 Equivalent linearisation of pitch‐plunge‐control wing undergoing transcritical bifurcation of cycles

Figure 5.62 Fold‐Hopf bifurcation

Chapter 06

Figure 6.1 Different types of freeplay

Figure 6.2 Asymmetric freeplay and bilinear stiffness

Figure 6.3 Natural frequencies (top) and damping ratios (bottom) of nominal system

Figure 6.4 Natural frequencies (top) and damping ratios (bottom) of system with

Figure 6.5 Natural frequencies (top) and damping ratios (bottom) of system with

Figure 6.6 Natural frequencies (top) and damping ratios (bottom) of system with

Figure 6.7 Position of fixed points of pitch‐plunge‐control wing with freeplay in pitch

Figure 6.8 Position of fixed points of pitch‐plunge‐control wing with freeplay in control degree of freedom

Figure 6.9 Limit cycles traversing two or three piecewise linear domains

Figure 6.10 Sinusoidal displacement (top) and corresponding bilinear load (bottom)

Figure 6.11 Equivalent linearisation of bilinear spring stiffness for three‐domain limit cycles

Figure 6.12 Sinusoidal displacement (top) and corresponding bilinear load (bottom), two‐domain cycle case

Figure 6.13 Variation of limit cycle amplitude and frequency with airspeed

Figure 6.14 Equivalent linearisation of pitch‐plunge‐control wing with freeplay in pitch

Figure 6.15 Equivalent linearisation of pitch‐plunge‐control wing with freeplay in the control degree of freedom

Figure 6.16 Amplitude and centre of two‐domain limit cycles

Figure 6.17 All limit cycles and fixed points of pitch‐plunge‐control wing with freeplay in pitch

Figure 6.18 Idealised diagram of all limit cycles and fixed points in the phase plane

Figure 6.19 Double crossing of the freeplay region near a local maximum

Figure 6.20 Time response of pitch‐plunge‐control wing with freeplay in different degrees of freedom at

m/s

Figure 6.21 Purely linear grazing trajectory attracted by fixed point at

m/s

Figure 6.22 Nonlinear trajectory attracted by limit cycle at

m/s

Figure 6.23 Grazing trajectories attracted by

at decreasing airspeeds

Figure 6.24 Fixed points of pitch‐plunge‐control wing with freeplay in pitch versus

δ

Figure 6.25 Distance of

fixed points from freeplay boundary

δ

, plotted against

δ

and

U

Figure 6.26 Response of nonlinear system at

for different values of

κ

Figure 6.27 Complete bifurcation with respect to parameter

κ

Figure 6.28 Limit cycle amplitude in pitch as a function of

U

and

δ

Figure 6.29 Limit cycles of the pitch‐plunge‐control wing with freeplay in pitch near the grazing bifurcation point

Figure 6.30 Limit cycles of the pitch‐plunge‐control wing with freeplay in pitch near the flutter point of the overlying linear system

Figure 6.31 Grazing of an asymmetric limit cycle of the pitch‐plunge‐control wing with freeplay in the control DOF

Figure 6.32 Time response of a system with freeplay and friction in the control surface

Chapter 07

Figure 7.1 Comparison of exact (solid) and numerical continuation solution (circles) of a quadratic equation

Figure 7.2 Definition of arclength and direction vector

Figure 7.3 Demonstration of prediction‐correction scheme

Figure 7.4 Solution of quadratic equation by numerical continuation with prediction‐correction

Figure 7.5 Demonstration of arclength continuation

Figure 7.6 Solution of quadratic equation by arclength continuation

Figure 7.7 Variation of

and d

U

/d

s

along the branch

Figure 7.8 Demonstration of pseudo‐arclength continuation

Figure 7.9 Direct fold location of quadratic equation

Figure 7.10 Variation of branch point test function with airspeed for

solution

Figure 7.11 Complete estimation of fixed points using numerical continuation

Figure 7.12 Complete estimation of fixed points using numerical continuation and step‐size control

Figure 7.13 Response waveform approximated by equivalent linearisation and refined by continuation

Figure 7.14 Complete continuation results for pitch‐plunge‐control wing with cubic stiffness in plunge

Figure 7.15 Approximate (initial) and accurate (final) arclength positions of folds and break points

Figure 7.16 Demonstration of branch switching around a pitchfork bifurcation

Figure 7.17 Initial and final limit cycles after long time simulation at post‐BP conditions

Figure 7.18 Limit cycle amplitude of both branches for pitch‐plunge‐control wing with cubic stiffness in plunge

Figure 7.19 Limit cycle period of both branches for pitch‐plunge‐control wing with cubic stiffness in plunge

Figure 7.20 Floquet multipliers of the two branches

Figure 7.21 Stable and unstable parts of the two limit cycle branches

Figure 7.22 Demonstration of Poincaré‐type phase‐fixing on a planar system

Figure 7.23 Response waveform approximated by equivalent linearisation and refined by continuation

Figure 7.24 Limit cycle amplitude and period for pitch‐plunge‐control wing with freeplay in the control DOF

Figure 7.25 Result of long time simulation just after branch point

Figure 7.26 Maximum pitch of cycles lying on branches 1 and 2

Figure 7.27 Zoom inside the two rectangles of Figure 7.26

Figure 7.28 Failure of grazing test function

Figure 7.29 Grazing points plotted on results of Figure 7.27

Figure 7.30 Nearby grazing and folding orbits (grazing point 5 in Figure 7.29b)

Figure 7.31 Two limit cycle branch of the pitch‐plunge‐control wing with quadratic damping, cubic stiffness and preload in pitch

Figure 7.32 All three limit cycle branches of the pitch‐plunge‐control wing with quadratic damping, cubic stiffness and preload in pitch

Chapter 08

Figure 8.1 Snapshots of unsteady flow behind static circular cylinder

Figure 8.2 Snapshots of unsteady flow over pitching rectangle

Figure 8.3 Two‐dimensional circular cylinder with a plunge degree of freedom

Figure 8.4 Eigenvalues of underlying linear system for airspeeds between 0 and 10 m/s

Figure 8.5 Natural frequencies and damping ratios of underlying linear system for airspeeds between 0 and 20 m/s

Figure 8.6 Time response of wake oscillator model at

m/s

Figure 8.7 Limit cycle amplitude and frequency for wake oscillator model

Figure 8.8 Limit cycle amplitude and frequency for wake oscillator model with high structural damping

Figure 8.9 Bluff body undergoing galloping oscillations

Figure 8.10 Curve fit of experimental

curve

Figure 8.11 Limit cycle branch of galloping oscillator for different values of coefficient

A

1

Figure 8.12 Airflow components seen by pitching and plunging wing

Figure 8.13 Lift and moment coefficients acting on a NACA 0012 airfoil undergoing dynamic stall (data reproduced from McAlister et al (1982))

Figure 8.14 Conceptual drawing of boundary layer at the leading edge of a static (a) and pitching up (b) airfoil

Figure 8.15 Different LEV formation mechanisms

Figure 8.16 Definition of total normal and chordwise flow components at the quarter chord

Figure 8.17 Comparison between potential Leishman–Beddoes and Wagner model predictions at

Figure 8.18 Comparison between potential Leishman–Beddoes and Wagner model predictions at

Figure 8.19 Static separation point (left) and corresponding lift coefficient predicted by Kirchoff theory (right)

Figure 8.20 Comparison between Leishman–Beddoes predictions and experimental results

Figure 8.21 Comparison between Leishman–Beddoes predictions with parameter adjustments and experimental results

Figure 8.22 Definition of Δ

c

l

(

W

0

/

U

)

Figure 8.23 Definition of Δ

c

m

(

W

0

/

U

)

Figure 8.24 Comparison between ‘mean airfoil’ and NACA 0012 static lift and moment curves

Figure 8.25 Comparison between ONERA model predictions and experimental results

Figure 8.26 Aerodynamic state response against time during pre‐simulation to set the initial values of the aerodynamic states

Figure 8.27 Results of Leishman–Beddoes aeroelastic simulation

Figure 8.28 Results of ONERA aeroelastic simulation

Chapter 09

Figure 9.1 Linear flutter and static divergence boundaries of pitch‐plunge wing

Figure 9.2 Flutter test function for the underlying linear system

Figure 9.3 Limit cycle amplitude and frequency of pitch‐plunge wing with nonlinear piston theory aerodynamics

Figure 9.4 Limit cycle amplitude and frequency of pitch‐plunge wing obtained from airspeed‐based iterations

Figure 9.5 Simple panel flutter system

Figure 9.6 Eigenvalues of aeroelastic panel around

for different

R

x

and

V

values

Figure 9.7 Bifurcation of an unstable node into an unstable focus

Figure 9.8 Pitchfork, Hopf and buckling‐to‐flutter dynamic pressure and frequency of the linearised panel around

Figure 9.9 Wind‐off fixed points for various values of the external compression load

Figure 9.10 Variation of fixed point with

V

for

and

Figure 9.11 Buckled shapes of the panel

Figure 9.12 Eigenvalues of the linearised panel around

for

and different

V

values

Figure 9.13 All fixed point bifurcation dynamic pressures for

Figure 9.14 Time response of

W

(0.75, 

τ

) at low post‐critical conditions,

Figure 9.15 Limit cycle oscillation amplitude at

W

(0.75, 

τ

) and period,

Figure 9.16 Limit cycle branches for

Figure 9.17 Primary and secondary limit cycle branches at

Figure 9.18 Response trajectory starting on a period‐doubling point of limit cycle branch 2 and ending up aperiodic,

Figure 9.19 Response trajectory starting on an unstable point of limit cycle branch 2 and ending on a fixed point,

Figure 9.20 Response trajectory starting on an unstable point of limit cycle branch 2 and ending on a previously undetected limit cycle,

Figure 9.21 Isolated limit cycle branch, plotted along with branches 1 and 2,

Figure 9.22 All detected solution branches of aeroelastic panel system at

Figure 9.23 Time response of

W

(0.75, 

τ

) at high post‐critical conditions,

Figure 9.24 Limit cycle oscillation amplitude at

W

(0.75, 

τ

) and period,

Figure 9.25 All detected solution branches of aeroelastic panel system at

Chapter 10

Figure 10.1 Plate geometry and deformation

Figure 10.2 First four out‐of‐plane mode shapes

Figure 10.3 Variation of eigenvalues of underlying linear system with

V

Figure 10.4 Limit cycle amplitude and period variation with dynamic pressure for different numbers of modes

Figure 10.5 Camber surface of a wing

Figure 10.6 Vortex ring

Figure 10.7 Complete wing and wake discretisation scheme

Figure 10.8 Grid selected for VLM computation

Figure 10.9 Lift calculation for rectangular wing with aspect ratio 3 at

Figure 10.10 First four out‐of‐plane mode shapes for aerodynamic grid

Figure 10.11 Eigenvalues of time domain aeroelastic system at different airspeeds for two different curve‐fit orders

Figure 10.12 Fixed points of cantilever plate in incompressible flow for

Figure 10.13 Eigenvalues of fixed points of cantilever plate in incompressible flow for

Figure 10.14 Fixed point of cantilever plate in incompressible flow for

Figure 10.15 Displacement time history of wingtip trailing edge at two airspeeds

Figure 10.16 Limit cycle oscillation amplitude and period

appendix

Figure A.1 Two‐dimensional rectangular cylinder with a plunge degree of freedom

Figure A.2 Pitch‐plunge wing section with control surface

Figure A.3 Pitch‐plunge wing section

Figure A.4 Two‐dimensional flat plate wing with a pitch degree of freedom

Guide

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INTRODUCTION TO NONLINEAR AEROELASTICITY

Grigorios Dimitriadis

University of LiègeBelgium

This edition first published 2017© 2017 John Wiley & Sons Ltd

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Names: Dimitriadis, Grigorios, 1972– author.Title: Introduction to nonlinear aeroelasticity / Dr. Grigorios Dimitriadis.Description: 1 edition. | Chichester, West Sussex, UK : John Wiley & Sons, Inc., [2017] | Includes bibliographical references and index.Identifiers: LCCN 2016046386 (print) | LCCN 2016059904 (ebook) | ISBN 9781118613474 (cloth : alk. paper) | ISBN 9781118756454 (Adobe PDF) | ISBN 9781118756461 (ePub)Subjects: LCSH: Aeroelasticity. | Nonlinear theories.Classification: LCC TL574.A37 D56 2017 (print) | LCC TL574.A37 (ebook) | DDC 629.132/362–dc23LC record available at https://lccn.loc.gov/2016046386

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Preface

Aeroelasticity is the study of the static and dynamic interaction between airflow and flexible structures. Classical aeroelasticity deals with linearised problems; all displacements are small, all springs are perfect, all contacts are smooth and all flows are attached. Nonlinear aeroelasticity does exactly the opposite; it studies the static and dynamic interaction between airflow and flexible structures in the presence of large deformations, friction, freeplay in actuators, backlash in gears, nonlinear control laws, flow separation, oscillating shock waves and other nonlinear phenomena. The combined output of researchers in the field has now reached the level of maturity necessary to make nonlinear aeroelasticity an important and useful branch of engineering.

This book is an introduction to nonlinear aeroelasticity, which means that it aims to present the phenomena of interest and the most common analysis methodologies. The emphasis of the discussion is on application, so that all theories are accompanied by practical examples solved by means of Matlab codes. The latter are available to the reader on the Wiley website; they have been tested on Matlab versions 2013 and 2014 but could also be compatible with earlier versions. The reader should note that the purpose of the codes is to illustrate the examples and the underlying theories. They solve the particular problems for which they were written but they should not be seen as general nonlinear dynamic analysis codes that can be directly applied to different problems.

Chapter 1 is a brief introduction to nonlinear aeroelasticity, summarising some of the main advances accomplished since the 1940s. Chapter 2 is an introduction to nonlinear dynamics, discussing issues such as fixed point stability and limit cycle oscillations on very simple aeroelastic systems. The solution methods used at this stage are still either qualitative or approximate; Chapter 3 presents in detail the numerical solution methods for calculating the time response of nonlinear systems that will be used throughout the rest of the book. Additional numerical methodologies for analysing nonlinear system responses are presented in Chapter 4.

The bulk of the dynamic phenomena typically encountered in nonlinear aeroelastic systems are demonstrated on a fundamental model in Chapter 5, while Chapter 6 presents additional phenomena caused by non‐smooth nonlinearities. Numerical continuation approaches are introduced in Chapter 7 and are used throughout the rest of the book. The focus of Chapter 8 is low‐speed aerodynamic nonlinearities caused by unsteady flow separation, while high‐speed nonlinear aeroelastic phenomena are discussed in Chapter 9. Finally, Chapter 10 introduces nonlinear structural and linear aerodynamic modelling techniques for finite wings.

I have worked on nonlinear dynamic and aeroelastic research since 1995 and I would like to take this opportunity to thank all the people with whom I have collaborated on the subject over the years. In particular, I would like to thank Jonathan Cooper who introduced me to aeroelasticity and supervised my doctoral research. I would also like to thank Earl Dowell, Bob Kielb, Gareth Vio, Xavier Amandolese and Pascal Hemon who welcomed me in their departments during my sabbatical year.

Dimitriadis: Nonlinear Aeroelasticity – Series Preface Oct 2016

The field of aerospace is multi‐disciplinary and wide ranging, covering a large variety of products, disciplines and domains, not merely in engineering but in many related supporting activities. These combine to enable the aerospace industry to produce innovative and technologically advanced vehicles. The wealth of knowledge and experience that has been gained by expert practitioners in the various aerospace fields needs to be passed onto others working in the industry and also researchers, teachers and the student body in universities.

The Aerospace Series aims to be a practical, topical and relevant series of books aimed at people working in the aerospace industry, including engineering professionals and operators, engineers in academia and allied professions, such as commercial and legal executives. The range of topics is intended to be wide ranging, covering design and development, manufacture, operation and support of aircraft, as well as topics such as infrastructure operations and current advances in research and technology.

Aeroelasticity is the scientific discipline that arises from the interaction of aerodynamic, elastic and inertial forces, and has a significant effect upon the design and performance of all aircraft. The influence of nonlinearities, appearing in structures, aerodynamics and control systems, can have a major influence upon aeroelastic behaviour; for instance, phenomena such as Limit Cycle Oscillations can only occur in nonlinear systems.

This book, Introduction to Nonlinear Aeroelasticity, provides an excellent introduction to the effects of structural and aerodynamic nonlinearities on aeroelastic behaviour and describes a number of methodologies to predict the resulting behaviour. The text is complemented with a comprehensive set of Matlab codes that will enable the reader to readily apply the methods themselves. This book makes a strong addition to the Wiley Series’ existing content in aeroelasticity and related topics.

Peter Belobaba, Jonathan Cooper and Alan Seabridge

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1Introduction

Nonlinear aeroelasticity is the study of the interactions between inertial, elastic and aerodynamic forces on engineering structures that are exposed to an airflow and feature non‐negligible nonlinearity. There exist several good textbooks on linear aeroelasticity for aircraft (Bisplinghoff et al. 1996; Fung 1993; Hodges and Alvin Pierce 2002; Wright and Cooper 2015). Dowell (2004) even includes chapters on nonlinear aeroelasticity and stall flutter, while Paidoussis et al. (2011) discusses a number of nonlinear aeroelastic phenomena occurring in civil engineering structures. However, there is no introductory text that presents the methodologies of nonlinear dynamics and applies them to a wide range of nonlinear aeroelastic systems. The present book aims to fill this gap to a certain degree. The subject area is vast and mutlidisciplinary and it would be impossible to fit every aspect of it in a textbook. The main omission is high fidelity numerical simulation using Computational Fluid Dynamics and Computational Structural Dynamics solvers; these methodologies are already the subject of a dedicated text (Bazilevs et al. 2013). The aerodynamic models used in this book are analytical, empirical or based on panel methods while the structural models are either analytical or make use of series solutions.

The book is introductory but it assumes knowledge of structural dynamics, aerodynamics and some linear aeroelasticity. The main linear aeroelastic phenomena of flutter and static divergence are discussed in detail because they can affect nonlinear behaviour, but the present work is by no means a complete text on linear aeroelasticity. Unsteady aerodynamic modelling is used throughout the book and discussed in Chapters 8, 10 and in the Appendix. However, again this book is not a complete reference on unsteady aerodynamics, linear or nonlinear. On the other hand, nonlinear dynamics and bifurcation analysis are presented in great detail as they do not normally feature in most undergraduate or even graduate Aerospace and Mechanical Engineering courses. The emphasis of all discussions is on the application rather than the rigorous derivation of the theorems; there already exist several classic textbooks for the latter (Kuznetsov 1998; Guckenheimer and Holmes 1983). More application‐based works on nonlinear dynamics also exist (e.g. Strogatz 1994) but they address a wide range of physical, chemical, biological, accounting models, to name a few, whereas the present book concentrates exclusively on aeroelastic phenomena.

Nonlinear aeroelasticity has become an increasingly popular research area over the last 30 years. There have been many driving forces behind this development, including faster computers, increasingly flexible structures, automatic control systems for aircraft and other engineering products, new materials, optimisation‐based design methods and others. Aeroelasticians have acquired expertise from many different fields in order to address nonlinear aeroelastic problems, mainly nonlinear dynamics, bifurcation analysis, control theory, nonlinear structural analysis and Computational Fluid Dynamics. The main applications of nonlinear aeroelasticity lie in aeronautics and civil engineering but other types of structure are also concerned, such as bridges and wind turbines.

In classical linear aeroelasticity, the relationships between the states of a system and the internal forces acting on them are always assumed to be linear. Force‐displacement diagrams for the structure and lift or moment curves for the aerodynamics are always assumed to be linear, while friction is neglected and damping is also linear. As an example, consider a torsional spring that provides a restoring moment M when twisted through an angle ϕ. Figure 1.1a plots experimentally measured values of ϕ and M. Clearly, the function M(ϕ) is not linear but, if we concentrate in the range , the curve is nearly linear and we can curve fit it as the straight line , where K is the linear stiffness of the spring.

Figure 1.1 Linearised load‐displacement diagrams

Figure 1.1b plots the aerodynamic lift coefficient acting on a wing placed at an angle α to a free stream of speed U, defined as

where l is the lift force per unit length, ρ is the air density and c is the chord. The curve cl(α) is by no means linear but, again, if we focus in the range , we can curve fit the lift coefficient as the straight line , where is the lift curve slope. An aeroelastic system featuring the spring of Figure 1.1a and the wing of Figure 1.1b will be nonlinear but, if we ensure that ϕ and α never exceed their respective linear ranges for all operating conditions, then we can treat the system as linear and use linear analysis to design it. In nonlinear aeroelasticity, the angles ϕ and α will always exceed their linear ranges and therefore we must use nonlinear analysis, both static and dynamic, in order to design the system.

Nonlinear dynamics is the field of study of nonlinear ordinary and partial differential equations, which in this book model aeroelastic systems. Unlike linear differential equations, nonlinear equations have no general analytical solutions and, in some cases, several different solutions may coexist at the same operating conditions. Furthermore, nonlinear systems can have many more types of solution than linear ones. The operating conditions of an aeroelastic system are primarily the free stream airspeed and the air density (or flight altitude), while the Reynolds number, Mach number and mean angle of attack can also be important. As these system parameters vary, the number and type of solutions of the nonlinear equations of motion can change drastically. The study of the changing nature of solutions as the system parameters are varied is known as bifurcation analysis. In this book we will use almost exclusively local bifurcation analysis, which means that we will identify individual solutions and track their nature and their intersections with other solutions for all the parameter values of interest.

A wide variety of nonlinear aeroelastic phenomena will be investigated, from the galloping of cables to the buckling and flutter of panels in supersonic flow and from stall flutter to the limit cycle oscillations of finite wings. We will also briefly discuss transonic aeroelastic phenomena but we will not analyse them in detail because such analysis requires high fidelity computational fluid and structural mechanics and is still the subject of extensive research. The equations of motion treated in this book are exclusively ordinary differential equations; whenever we encounter partial differential equations we will first transform them to ordinary using a series solution. It is hoped that the book will contribute towards the current trend of taking nonlinear aeroelasticity out of the research lab and introducing it into the classroom and in industry.

1.1 Sources of Nonlinearity

Traditionally, a lot of effort has been devoted to designing and building engineering structures that are as linear as possible. Despite this effort, nonlinearity, weak or strong, has always been present in engineering systems. In recent years, increasing amounts of nonlinearity have been tolerated or even purposefully included in many applications, since nonlinear analysis methods have progressed sufficiently to allow the handling of nonlinearity at the design stage. Furthermore, nonlinearity can have significant beneficial effects, for example in shock absorbers and suspension systems.

In this book we will only consider nonlinearities that are present in aeroelastic systems. Since aeroelasticity is of particularly importance to the fields of aeronautics, civil engineering and energy harvesting, we will limit the discussion of nonlinearity to these application areas. The nonlinear functions that are most often encountered in these systems have three main sources:

the structure,

the aerodynamics and

the control system.

The structural nonlinearities of interest occur during the normal operation of the underlying engineering system. Nonlinearities appearing in damaged, cracked, plastically deformed and, in general, off‐design systems are beyond the scope of this book. The most common forms of nonlinearity appearing in structures are geometric (caused by large deformations), clearance (i.e. freeplay, contact and other non‐smooth phenomena), dissipative (i.e. friction or other nonlinear damping forces) and inertial (of particular interest in rotors and turbomachinery).

Aerodynamic nonlinearities arise from the existence of either unsteady separated flow or oscillating shock waves or a combination of the two (e.g. shock‐induced separation). Separation‐induced nonlinearity can affect all aeroelastic systems, although bluff bodies such as bridges, towers and cables are always exposed to it. Shock‐induced nonlinearity is of interest mostly to the aeronautical industry. It should be noted that aerodynamic nonlinearity is inertial, dissipative and elastic.

Engineering structures are increasingly designed to feature passive and/or active control systems. These systems can either aim to stabilise the structure (e.g. suppress or mitigate unwanted vibrations) or to control it (e.g. aircraft automatic flight control systems). Passive systems can be seen as parts of the structure and therefore included in the structural nonlinearity category (if they are nonlinear). Active systems, however, can feature a number of prescribed and incidental nonlinearities that can be turned off by running the structure in open loop mode. These nonlinear functions are in a category of their own and can take many forms, such as deflection and rate limits on actuators or nonlinear control laws. Furthermore, control actuators always feature a certain amount of freeplay, which is usually strictly limited by airworthiness regulations.

One more source of nonlinearity can be external stores on aircraft that carry them (mainly military aircraft). Stores such as external fuel tanks, bombs and missiles can cause store‐induced oscillations, particularly at transonic flight conditions. However, the mechanisms behind these oscillations are still not fully understood and the relevant analyses usually involve computational fluid‐structure interaction. Consequently, these phenomena will not be discussed further in this book. Human operator‐related nonlinearities (pilot, driver, rider etc.) will not be considered either.

1.2 Origins of Nonlinear Aeroelasticity

Some of the first investigations of nonlinear aeroelasticity concerned stall flutter and started just after WWII. For example, Victory (1943) reported that the airspeed at which wings undergo flutter decreases at high incidence angles, while Mendelson (1948) attempted to model this phenomenon. Rainey (1956) carried out a range of wind tunnel experiments of aeroelastic models of wings and noted the parameters that affect their stall flutter behaviour. It was quickly recognised that, in order to analyse stall flutter, the phenomenon of unsteady flow separation known as dynamic stall needed to be isolated and studied in detail. Bratt and Wight (1945) and Halfman et al. (1951) carried out two of the first experimental studies of the unsteady aerodynamic loads acting on 2D airfoils oscillating at high angles of attack. They were to be followed by a significant number of increasingly sophisticated experiments, covering a wide range of airfoil geometries, Reynolds numbers, Mach numbers and oscillation amplitudes and frequencies. The phenomena of dynamic stall and stall flutter are discussed in Chapter 8.

The effects of structural nonlinearity were first investigated by Woolston et al. (1955, 1957) and Shen (1959). They both set up aeroelastic systems with structural nonlinearity and solved them using analog computers. The systems included 2D airfoils with nonlinear springs, wings with control surfaces and buckled panels in supersonic flow. Such systems have been explored ever since, using increasingly sophisticated mathematical and experimental methods. They are in fact the basis of nonlinear aeroelasticity and will be discussed in detail in the present book. Two‐dimensional airfoils with nonlinear springs will be analysed in Chapters 2 to 7, panels in supersonic flow will be presented in Chapter 9 and 3D wings in Chapter 10.

Wind tunnel experiments on nonlinear aeroelastic systems with nonlinear springs have been carried out since the 1980s, notably by McIntosh Jr. et al. (1981); Yang and Zhao (1988); Conner et al. (1997). These works provided both valuable insights into the phenomena that can be encountered in nonlinear aeroelasticity and a basis for the validation of various modelling and analysis methods. The focus of the present book is the application of nonlinear dynamic analysis to nonlinear aeroelasticity. Modelling will be discussed in the last three chapters, as well as in the Appendix.

Shen (1959) was one of the first works to apply the Harmonic Balance method to nonlinear aeroelasticity. This method was first presented in the West by Kryloff and Bogoliuboff (1947) and has since become one of the primary analysis tools for nonlinear dynamic systems undergoing periodic oscillations. We will use several different versions of the Harmonic Balance technique throughout this book.

One of the first studies to apply elements of bifurcation theory to nonlinear aeroelastic systems was carried out by Price et al. (1994). They used stability boundaries, Poincaré sections and bifurcation diagrams to analyse the behaviour of a simple 2D mathematical nonlinear aeroelastic system with structural nonlinearity. Aside from the Hopf bifurcation, they also observed period‐doubling bifurcations and chaotic responses. Bifurcation analysis is used throughout the present book but most of the bifurcations typically encountered in nonlinear aeroelasticity are discussed in detail in Chapter 5.

Alighanbari and Price (1996) were the first to use numerical continuation in nonlinear aeroelasticity. Numerical continuation (Allgower and Georg 1990) is a set of mathematical methods for solving nonlinear problems that have static or periodic dynamic solutions. Continuation methods are strongly linked to bifurcation analysis, as they very often start evaluating solutions at bifurcation points. Such methods will be presented in detail in Chapter 7 and used in all subsequent chapters.

Towards the end of the 1990s, Friedmann (1999) identified nonlinear aeroelasticity as a major research direction in his paper on the future of aeroelasticity. Lee et al. (1999) published a lengthy and authoritative review of past and current nonlinear aeroelastic research, describing all major advances in both understanding and methodologies. A few years later, the nonlinear aeroelasticity chapter by Dowell (2004) provided an extensive description of nonlinear aeroelastic phenomena encountered in flight and in benchmark aeroelastic wind tunnel models and summarised the state of the art.

Thirteen years later, there has been a significant increase in the research and application of nonlinear aeroelasticity. Transonic aeroelastic phenomena, the highly flexible structures of High Altitude Long Endurance aircraft, aeroelastic tailoring, gust loads acting on nonlinear aircraft, wind turbine aeroelasticity and high‐fidelity fluid structure interaction have all become major areas of research. Major national and international research projects have addressed such issues and the results are slowly starting to be applied in industry. Given this wealth of activity in the field, it was felt that an introductory text in nonlinear aeroelasticity is missing from the literature. It is hoped that the present book will come to fill this gap, providing a basis for understanding nonlinear aeroelastic phenomena and methodologies on relatively simple systems and preparing the reader for more advanced work in state‐of‐the‐art applications.

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