Linear Aeroelastic Stability of Beams and Plates in Three-Dimensional Flow

Linear Aeroelastic Stability of Beams and Plates in

Three-Dimensional Flowby

Samuel Chad Gibbs IV

Department of Mechanical Engineering and Materials ScienceDuke University

Date:

Approved:

Earl H. Dowell, Supervisor

Kenneth C. Hall

Donald B. Bliss

Thomas P. Witelski

Thesis submitted in partial fulfillment of the requirements for the degree ofMaster of Science in the Department of Mechanical Engineering and Materials

Science in the Graduate School of Duke University2012

Abstract

Linear Aeroelastic Stability of Beams and Plates in

Three-Dimensional Flowby

Samuel Chad Gibbs IV

Department of Mechanical Engineering and Materials ScienceDuke University

Date:

Approved:

Earl H. Dowell, Supervisor

Kenneth C. Hall

Donald B. Bliss

Thomas P. Witelski

An abstract of a thesis submitted in partial fulfillment of the requirements for thedegree of Master of Science in the Department of Mechanical Engineering and

Materials Science in the Graduate School of Duke University2012

Copyright c 2012 by Samuel Chad Gibbs IVAll rights reserved except the rights granted by the

Creative Commons Attribution-Noncommercial Licence

Abstract

The aeroelastic stability of beams and plates in three-dimensional flows is explored

as the elastic and aerodynamic parameters are varied. First principal energy meth-

ods are used to derive the structural equations of motion. The structural models

are coupled with a three-dimensional linear vortex lattice model of the aerodynam-

ics. An aeroelastic model with the beam structural model is used to explore the

transition between different fixed boundary conditions and the effect of varying two

non-dimensional parameters, the mass ratio and aspect ratio H, for a beam with

a fixed edge normal to the flow. The trends matched previously published theoreti-

cal and experimental data, validating the current aeroelastic model. The transition

in flutter velocity between the clamped free and pinned free configuration is a non-

monotomic transition, with the lowest flutter velocity coming with a finite size spring

stiffness. Next a plate-membrane model is used to explore the instability dynamics

for different combinations of boundary conditions. For the specific configuration of

the trailing edge free and all other edges clamped, the sensitivity to the physical

parameters shows that decreasing the streamwise length and increasing the tension

in the direction normal to the flow can increase the onset instability velocity. Finally

the transition in aeroelastic instabilities for non-axially aligned flows is explored for

the cantilevered beam and three sides clamped plate. The cantilevered beam con-

figuration transitions from an entirely bending motion when the clamped edge is

normal to the flow to a typical bending/torsional wing flutter when the clamped

iv

edge is aligned with the flow. As the flow is rotated the transition to the wing flutter

occurs when the flow angle is only 10 deg from the perfectly normal configuration.

With three edges clamped, the motion goes from a divergence instability when the

free edge is aligned with the flow to a flutter instability when the free edge is normal

to the flow. The transition occurs at an intermediate angle. Experiments are carried

out to validate the beam and plate elastic models. The beam aeroelastic results are

also confirmed experimentally. Experimental values consistently match well with the

theoretical predictions for both the aeroelastic and structural models.

v

Contents

Abstract iv

List of Tables x

List of Figures xi

List of Abbreviations and Symbols xv

1 Introduction and Literature Review 1

2 Structural Model 10

2.1 Beam Structural Model Derivation . . . . . . . . . . . . . . . . . . . 11

2.1.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.3 Normalized Equations of Motion . . . . . . . . . . . . . . . . 15

2.1.4 Bending Separation of Variables . . . . . . . . . . . . . . . . . 16

2.1.5 Torsional Separation of Variables . . . . . . . . . . . . . . . . 17

2.2 Specific Beam Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Clamped-Free . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Pinned-Free . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.3 Clamped-Clamped . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.4 Free-Free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Pinned Edge Torsional Spring Model . . . . . . . . . . . . . . . . . . 29

2.4 Plate Structural Model . . . . . . . . . . . . . . . . . . . . . . . . . . 33

vi

2.4.1 Plate Structural Analysis Typical Results . . . . . . . . . . . . 38

2.5 Forced System Modification . . . . . . . . . . . . . . . . . . . . . . . 44

3 Aerodynamic Model 47

3.1 Aerodynamic Theory Introduction . . . . . . . . . . . . . . . . . . . . 47

3.2 Vortex Lattice Aeroelastic Model . . . . . . . . . . . . . . . . . . . . 55

3.2.1 Downwash State Relations . . . . . . . . . . . . . . . . . . . . 56

3.2.2 Non-dimensional Generalized Force . . . . . . . . . . . . . . . 58

3.2.3 Governing Aeroelastic Matrix Equations . . . . . . . . . . . . 60

3.3 Code Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3.1 Matrix Definition . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3.2 Flutter Speed and Eigenvalue Determination . . . . . . . . . . 62

3.3.3 Generating Time Histories from Eigenanalysis . . . . . . . . . 67

3.4 Inclusion of Fixed Support Structures . . . . . . . . . . . . . . . . . . 68

3.5 Mirroring to Simulate Wind Tunnel Walls . . . . . . . . . . . . . . . 69

3.6 Using ANSYS Structural Modes . . . . . . . . . . . . . . . . . . . . . 70

3.7 Rotated Wing Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.7.1 Generalized Force Calculation . . . . . . . . . . . . . . . . . . 75

3.7.2 Downwash Calculation . . . . . . . . . . . . . . . . . . . . . . 75

4 Results from Aeroelastic Simulations 77

4.1 Dimensional Beam Simulations . . . . . . . . . . . . . . . . . . . . . 77

4.1.1 Time History Analysis vs. Eigenanalysis . . . . . . . . . . . . 78

4.1.2 Fixed Leading Airfoil Effect . . . . . . . . . . . . . . . . . . . 79

4.2 Wind Tunnel Wall Confinement Effects . . . . . . . . . . . . . . . . . 79

4.2.1 Out-of-Plane Normal to Flow Confinement . . . . . . . . . . . 80

4.2.2 In-Plane Normal to Flow Wind Tunnel Wall Confinement . . . 81

vii

4.3 Non-dimensional Simulations (Modified from Journal of Fluids andStructures Journal Submission) . . . . . . . . . . . . . . . . . . . . . 82

4.3.1 Leading Edge Spring Simulations . . . . . . . . . . . . . . . . 83

4.3.2 Aspect Ratio Variation Simulations . . . . . . . . . . . . . . . 85

4.3.3 Mass Ratio Variation Simulations . . . . . . . . . . . . . . . . 86

4.4 Plate Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.4.1 NASA Simulations (Configuration 6) . . . . . . . . . . . . . . 90

4.4.2 Increasing the Flutter Velocity . . . . . . . . . . . . . . . . . . 94

4.4.3 Additional Plate Boundary Configurations . . . . . . . . . . . 98

4.4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.5 Axially Misaligned Analysis . . . . . . . . . . . . . . . . . . . . . . . 108

4.5.1 Axially Misaligned Beam Simulations . . . . . . . . . . . . . . 108

4.5.2 Axially Misaligned Plate Simulations . . . . . . . . . . . . . . 115

5 Experiments 120

5.1 Experiments to Validate Beam Model . . . . . . . . . . . . . . . . . . 120

5.1.1 Beam Structural Experiments . . . . . . . . . . . . . . . . . . 122

5.1.2 Beam Aeroelastic Experiments . . . . . . . . . . . . . . . . . . 123

5.2 Experiments to Validate Plate Model . . . . . . . . . . . . . . . . . . 125

5.2.1 Design of Experimental Setup . . . . . . . . . . . . . . . . . . 126

5.2.2 Static Structural Experiments . . . . . . . . . . . . . . . . . . 128

5.2.3 Dynamic Structural Experiments . . . . . . . . . . . . . . . . 130

5.2.4 Plate Aeroelastic Experiments . . . . . . . . . . . . . . . . . . 138

5.3 Configuration 1 Aeroelastic Experiments . . . . . . . . . . . . . . . . 139

5.4 Configuration 6 Aeroelastic Experiments . . . . . . . . . . . . . . . . 142

6 Conclusion and Future Work 145

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

viii

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.2.1 Theoretical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.2.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

A Beam Aeroelastic Experimental Data Points 150

B Configuration 2 Raw Data 152

Bibliography 154

ix

List of Tables

2.1 Non-Dimensional Natural Frequencies for a Single Edge Fixed Beam . 30

2.2 NASA Membrane Properties . . . . . . . . . . . . . . . . . . . . . . . 38

4.1 Plate Aeroelastic Simulation Summary (s = 0.01) . . . . . . . . . . . 98

4.2 Plate Aeroelastic Simulation Summary (s = 0.05) . . . . . . . . . . . 99

4.3 Rotated Wing Properties . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.1 Beam Experimental Parameters . . . . . . . . . . . . . . . . . . . . . 121

5.2 Equipment Used in the Ground Vibration Experiment . . . . . . . . . 131

5.3 First Three Modal Damping Ratios with No Tension . . . . . . . . . 137

5.4 Equipment Used in the Flutter Experiment . . . . . . . . . . . . . . . 139

5.5 Plate Aeroelastic Experimental Results . . . . . . . . . . . . . . . . . 141

5.6 Configuration 1 Experimental Results . . . . . . . . . . . . . . . . . . 141

5.7 Flutter Speed and Frequency for the Un-Tensioned Specimen: Theoryand Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

A.1 Experimental Datapoints for a Clamped-Free Plate . . . . . . . . . . 150

A.2 Experimental vs Theoretical Error . . . . . . . . . . . . . . . . . . . 151

x

List of Figures

1.1 Continuous Mold-Line Link . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Plate Configurations to Explore . . . . . . . . . . . . . . . . . . . . . 7

2.1 Clamped Free Schematic . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Clamped Free Frequencies . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Clamped Free Bending Mode Shapes . . . . . . . . . . . . . . . . . . 21

2.4 Pinned Free Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Pinned Free Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.6 Pinned Free Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 Clamped Clamped Schematic . . . . . . . . . . . . . . . . . . . . . . 25

2.8 Clamped Clamped Frequencies . . . . . . . . . . . . . . . . . . . . . 26

2.9 Clamped Clamped Mode Shapes . . . . . . . . . . . . . . . . . . . . . 27

2.10 Free Free Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.11 Free Free Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.12 Pinned-Free, Clamped-Free and Large K Mode Shapes . . . . . . . . 31

2.13 Structural Frequency Evolution with Leading Edge Torsional Spring . 32

2.14 Configuration 1 Natural Frequencies and Mode Shapes . . . . . . . . 40





xi


2.20 Configuration 2 Natural Frequency Evolution for Chord Variation . . 43

2.21 Configuration 2 Natural Frequency Evolution for Tension Variation . 44

3.1 Visualization of Structural Mode Shapes with Vortex Lattice Wake . 48

3.2 Expanded Schematic of Vortex Lattice Mesh . . . . . . . . . . . . . . 50

3.3 Aeroelastic Simulation Model . . . . . . . . . . . . . . . . . . . . . . 60

3.4 Typical Near Flutter Time History . . . . . . . . . . . . . . . . . . . 63

3.5 Near Flutter Time History Modal FFT . . . . . . . . . . . . . . . . . 64

3.6 Near Flutter Time History Modal Damping . . . . . . . . . . . . . . 65

3.7 Typical Velocity Sweep . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.8 Typical Root Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.9 Mirrored Wall Schematic . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.10 Cantilevered Wing Configuration Schematic . . . . . . . . . . . . . . 71

3.11 Aerodynamic and Elastic Coordinate Systems . . . . . . . . . . . . . 73

3.12 Rotated Wing Mesh Visualization . . . . . . . . . . . . . . . . . . . . 74

4.1 Eigenanalyis vs Time History Analyis Root Locus . . . . . . . . . . . 78

4.2 Eigenanalyis vs Time History Analysis Damping vs. Velocity . . . . . 78

4.3 Leading Airfoil Root Locus . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4 Leading Airfoil Damping vs. Velocity . . . . . . . . . . . . . . . . . . 79

4.5 Impact of Out-of-Plane Confinement on Flutter Frequency Prediction 80

4.6 Impact of In-Plane Confinement on Flutter Velocity Prediction . . . . 81

4.7 Impact of In-Plane Confinement on Flutter Frequency Prediction . . 82

4.8 Flutter Frequency and Velocity vs. K . . . . . . . . . . . . . . . . . 84

4.9 Flutter Velocity as a function of the Aspect Ratio . . . . . . . . . . . 86

4.10 Flutter Velocity and Frequency vs. . . . . . . . . . . . . . . . . . . 87

xii

4.11 Growing Mass Ratio Simulation . . . . . . . . . . . . . . . . . . . . . 88

4.12 Configuration 6 Aeroelastic Results . . . . . . . . . . . . . . . . . . . 92

4.13 Plate Structural Model Convergence Plots . . . . . . . . . . . . . . . 93

4.14 Plate Structural Model: Support Structure Influence Plots . . . . . . 95

4.15 Configuration 2 Aspect Ratio Variation Flutter Boundary . . . . . . . 96

4.16 Configuration 2 Aspect Ratio Variation Flutter Boundary Mode Shapes 96

4.17 Configuration 2 Aspect Tension Variation Flutter Boundary . . . . . 97

4.18 Configuration 2 Tension Variation Flutter Boundary Mode Shapes . . 97






4.24 Rotating Beam Flutter Boundary . . . . . . . . . . . . . . . . . . . . 110

4.25 Rotation Angle=0, One Period Flutter Motion . . . . . . . . . . . . . 111

4.26 Rotation Angle=6.92, One Period Flutter Motion . . . . . . . . . . . 112

4.27 Rotation Angle=11.53, One Period Flutter Motion . . . . . . . . . . 113

4.28 Rotation Angle=90, One Period Flutter Motion . . . . . . . . . . . . 114

4.29 Rotated Plate Aeroelastic Boundary . . . . . . . . . . . . . . . . . . 115

4.30 Rotated Plate Aeroelastic Boundary Mode Shapes . . . . . . . . . . . 115

4.31 Rotation Angle=0, One Period Flutter Motion . . . . . . . . . . . . . 117

4.32 Rotation Angle=45 deg, One Period Flutter Motion . . . . . . . . . . 118

4.33 Rotation Angle=60 deg, One Period Flutter Motion . . . . . . . . . . 119

5.1 Experiment Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.2 Natural Frequency Experimental Results . . . . . . . . . . . . . . . . 122

xiii

5.3 Mass Ratio Variation with Experiment . . . . . . . . . . . . . . . . . 124

5.4 CAD Rendering of Bae . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.5 Close up of the Connector . . . . . . . . . . . . . . . . . . . . . . . . 127

5.6 Different Strain Settings Allowing for Varying Span-wise Tension . . . 128

5.7 Stress Strain Curve and Estimation of Elastic Modulus . . . . . . . . 129

5.8 Estimated Poissons Ratio . . . . . . . . . . . . . . . . . . . . . . . . 130

5.9 Photographs of the Experimental Setups . . . . . . . . . . . . . . . . 131

5.10 Configuration 1 Dynamic Experimental Results . . . . . . . . . . . . 132


5.12 The (1,2) Mode Visualization for Configuration 2 . . . . . . . . . . . 133


5.14 Ground Vibration Test Setup for Configuration 4 . . . . . . . . . . . 135

5.15 Laser Readout and Shaker Excitation Locations . . . . . . . . . . . . 136

5.16 Natural Frequency Results for 4 Levels of Tension: Theory and Ex-periment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.17 Photograph of Bae Inside the Wind Tunnel . . . . . . . . . . . . . . 139

5.18 Configuration 1 Aeroelastic Experimental Results . . . . . . . . . . . 140

5.19 Example Waterfall Plot for the Un-Tensioned Specimen . . . . . . . . 142

B.1 Configuration 2 Sample Spectrum Analyzer Output . . . . . . . . . . 153

xiv

List of Abbreviations and Symbols

Symbols

As Number of airfoil elements in normal to flow(~y) direction

Ac Number of airfoil elements in chordwise(~x) direction

At Total number of airfoil elements

E Youngs modulus of the structure

G Shear modulous of the structure

h Structure thickness

I Area moment of inertia of the structure

Iea Moment of inertial around the elastic axis

Kk,l Kernel function for the influence of the kth discrete on thelth panel

K Stiffness matrix

L,Lx Structure streamwise length

M Mass matrix

m Mass per unit length of the structure

p(x, y, t) Aerodynamic pressure at the panel location (x,y) at time (t)

~Q Generalized aerodynamic force

~rt Distance from circulation element to point in space (t)

S, Ly Structure normal to flow length

Ss Number of structure elements in normal to flow(~y) direction

xv

Sc Number of structure elements in streamwise(~x) direction

St Total number of structure elements

T Structure kinetic energy

Ty, Tx Elastic tension in the subscript direction

U Free stream fluid velocity

V Structure potential energy

Vd Vertical velocity of the elastic structure at collocation points

Ws Number of wake elements in normal to flow(~y) direction

Wc Number of wake elements in streamwise(~x) direction

Wt Total Number of structure elements

w(x, y, t) Displacement at Structure location (x,y) at time (t)

x , y Streamwise and span wise direction respectively

(x , y) (x,y) location of the (, ) panel

vortex lattice relaxation factor

W Virtual work

~ Discrete circulation values

(x, y) Continuous circulation at (x,y)

Natural frequency of the structure

Index of column in vortex mesh

Vector of position and velocity coordinates of natural modes

Density

~ Vector with and

Index of row in vortex mesh or aerodynamic damping ratio de-pending on contextitem[s] Structural damping ratio

xvi

Superscripts

Non-dimensional

Time derivative

Spatial derivative~ Vector quantity

Matrix quantity

Abbreviations

VLM Vortex Lattice Method

xvii

1Introduction and Literature Review

This thesis is to outlines a technique to predict the aeroelastic instability bound-

ary for one-dimensional beams and two-dimensional rectangular plates due to three-

dimensional aerodynamic forces. Specifically the linear aeroelastic instability bound-

ary for a wide variety of configurations and parameters is explored. The most com-

mon aeroelastic instability encountered is a flutter instability. Flutter is the dynamic

instability of a structure in a moving fluid that exhibits unsteady oscillations due to

the interaction between the structure and the fluid. Such systems tend to exhibit

limit cycle oscillations (LCO) that persist even if the free stream velocity falls below

the flutter onset velocity creating what is called a hysteresis band, the possibility of

multiple states at a given velocity. However, because all of the analysis conducted in

this paper is linear, the origins of this hysteresis behavior is not explicitly discussed.

Historically, the majority of flutter research has been focused on suppressing flut-

ter because it is catastrophic in many structures including aircraft, bridges, and

turbomachinery. Recently, attention has been refocused to gaining a better un-

derstanding of flutter, especially for the cantilevered beam configuration, due to a

growing interest in small scale energy harvesting systems. In addition to energy

1

harvesting applications, the configurations explored throughout this thesis can also

be used to understand the dynamics of human snoring [19] and to reduce the noise

generated during landing by subsonic fixed wing aircraft [26]. For this thesis, the

aeroelastic models are specifically used to:

Analyze the aeroelastic instabilities for a cantilevered beam in the transitionbetween pinned and clamped leading fixed edge

Analyze the aeroelastic instabilities for a beam with a clamped leading edge asthe governing non-dimensional parameters are varied

Analyze the aeroelastic instabilities for a plate with three sides fixed, a proposedconfiguration to reduce airframe noise on low subsonic aircraft during landing

Analyze the aeroelastic instabilities that occur for plates as the boundary con-ditions are varied

Analyze the aeroelastic instabilities for axially mis-aligned flows for a one-sideclamped beam and three-sides clamped plate

Generally the motivation for the research stems from a desire to continue to

advance the understanding of the aeroelastic instabilities that occur in rectangular

structures. Developing an aeroelastic model requires developing models for both the

structural dynamics and the aerodynamics. Once an aeroelastic model is created the

model is used to analyze configurations of interest.

The first problem explored is the interaction between a cantilevered flexible elastic

beam and a uniform axial flow, a canonical fluid-structure interaction problem. It is

well known that this system exhibits a flutter instability in low subsonic flow as the

free stream velocity is increased above a critical velocity. The structure then enters a

large and violent limit cycle oscillation (LCO). Since the experimental observations

2

of the flapping flag by Taneda [30] in 1968, many scholars have explored the stability

of this system experimentally and theoretically. Although extensively explored in

the literature, a full understanding of the dynamics of this relatively simple fluid-

structure interaction remains elusive. In addition to the problems inherent physical

significance, Doare and Michelin [7], Dunnmon et al. [11] and Giacomello and Porfiri

[16] have recently proposed using the phenomena for energy harvesting applications

and Eloy and Schouveiler [12] and Hellum et al. [18] have explored the potential

of using this flutter for propulsion. Furthermore, Balint and Lucey [3], Huang [20]

and Howell et al. [19] have shown that flutter in the human soft palette can explain

snoring and Watanabe et al. [38] has explored flutter in the printing industry.

Many structural and aerodynamic models have been developed or applied to

improve the understanding of the dynamics of this system. The initial models looked

at the limiting cases where either the streamwise or normal to flow dimension of the

elastic member is assumed to be infinite. For the first case, the problem approaches a

two-dimensional limit. In the two-dimensional limit the potential flow equations have

been solved to determine the aerodynamic forces using the continuous equation with

the appropriate boundary conditions [20, 22, 17, 39] and or discrete approximations.

The discrete approximations can be split into the discrete vortex models [31, 34,

35, 25, 1, 19] or numerical simulations solving the Navier-Stokes equations [3, 39].

In the latter limit, where the length is much larger than the span, a slender body

approximation has been used by Lemaitre et al. [23] to explore the dynamics. For the

two-dimensional case, Howell et al. [19] explored the influence of spatial confinement

and Michelin and Smith [24] and Tang and Padoussis [36] have modeled the influence

of cascades.

In addition to these two-dimensional aerodynamic models, researchers have cou-

pled different structural models when exploring the response of the system. The

structural models have largely consisted of linear and non-linear models of beams

3

with simple out of plane displacements. In general linear structural models are used

to explore the stability boundary as parameters are varied. Non-linear models have

been used by Michelin et al. [25], Tang and Padoussis [35], Tang et al. [32], Tang and

Padoussis [34] and Dunnmon et al. [11] to explore the post critical behaviors such

as LCO amplitude and hysteresis loops which are observed experimentally. Recently

interest in piezoelectric energy harvesting has motivated detailed exploration of the

non-linear post critical behavior because predicting the amplitude and frequency of

the limit cycle is vital to optimizing the energy harvested from the system [11, 16, 7].

The critical velocities predicted by the two-dimensional models are remarkably

similar to each other regardless of the solution technique used. Unfortunately their

collective agreement does not match published experimental results reported by

Taneda [30], Kornecki et al. [22], Watanabe et al. [38], Yamaguchi et al. [40], Tang

et al. [32], Eloy et al. [14] and Dunnmon et al. [11]. In fact, across the range of

parameters tested the two-dimensional model predicted flutter boundaries are signif-

icantly below the experimentally observed values. Even when Huang [20] attempted

to create a two-dimensional experimental model by having test pieces span the wind

tunnel, the experimentally observed critical velocities are still much higher than the

theoretical predictions.

This discrepancy has motivated the application of three-dimensional aerodynamic

models. Many of the initial three-dimensional aerodynamic models were used to ex-

plore the flutter characteristics of a single configuration. For example Tang et al.

[32] used an unsteady three-dimensional vortex lattice model(VLM) and a non-linear

structural model to explore the flutter boundary and post critical behavior of a sin-

gle aluminum plate. The success of initial three-dimensional simulations to match

the flutter boundary between theory and experiment has prompted the most recent

explorations of the stability boundary in parameter space with three-dimensional

aerodynamic models by Eloy et al. [13] and Eloy et al. [14]. In general these simula-

4

tions have shown much better agreement with the experimental results. Furthermore

an exploration of the three-dimensional effects of in-plane normal to the flow confine-

ment by Doare et al. [8] demonstrates that the small distance between wind tunnel

walls and experimental specimen required to produce the two-dimensional limit ex-

perimentally would be prohibitively difficult to achieve. Three-dimensional effects

are believed to explain the systemic discrepancies between strictly two-dimensional

theoretical predictions and experimental observations for the critical flutter velocity.

With the new understanding of the importance of three-dimensional effects on the

quantitative behavior of this fluid-structure system there is a need to analyze the im-

pact of different influences such as structural boundary conditions, confinement and

experimental support structure with a three-dimensional aerodynamic model. The

three-dimensional unsteady vortex lattice model remains a versatile means to ex-

plore the aforementioned influences. Numerical simulations have the benefit of being

able to model the effect of different configurations without changing the framework

of the analysis. The work presented for this configuration is a continuation of the

work done by Tang et al. [32]. The VLM aerodynamic model is generalized and used

to explore the stability boundary for the cantilevered beam in the non-dimensional

parameter space. Specifically the critical flow velocity as a function of mass ratio

and aspect ratio is explored and compared with new experimental results as well as

experimental and theoretical results found in the literature. In general the qualita-

tive trends and quantitative values match the existing three-dimensional theoretical

and experimental results.

Additionally the analysis of this configuration explores the effect of the leading

edge boundary condition on the critical flutter velocity. Using a leading edge tor-

sional spring the transition between the two limiting cases is presented, including a

surprising, non-monotonic transition in the critical flutter velocity. Finally normal to

the flow confinement in both the in plane and out of plane directions are presented.

5

Next, an aeroelastic model is created to analyze the aeroelastic stability of two-

dimensional rectangular plates. The project was initially motivated by a desire to

analyze a plate configuration similar to one created by NASAs proposed aircraft

noise reduction effort is explored. NASA, as a part of its strategic plan in 2000,

defined goals for designing the next generation of commercial transport aircraft with

several performance requirements, one of which is noise reduction.[26] Experimental

and numerical studies have shown that a large portion of aircraft noise during landing

is generated by the interaction of shed vortices and wing structure at the discontinuity

between the wing and the trailing edge flap.[6, 28] The noise reduction potential of

several geometries and mechanisms have been studied, but experiments showed that

the most effective method for significant noise reduction is to introduce a continuous

mold-line link (CML), a fairing surface that smoothly connects the edge of the flap

to the wing.[29] This is shown in Fig. 1.1. The experiments are performed using

a rigid fairing, but to actually implement this method on an aircraft the fairing

must be deformable. Therefore, a flexible plate, or a plate-membrane structure, is

an ideal material for the fairing structure because it can be hidden for most of the

time and extended when the trailing edge flaps are deployed. A plate has stiffness

in bending, while a plate-membrane has both bending stiffness and stiffness due to

applied tension. Both types of structures will herein be referred to as plates for

simplicity.

Despite significant progress in reducing noise from other sources, such as airframe

and propulsive devices, an assessment of the overall progress toward the next gen-

eration of aircraft showed that additional research in CMLs may be necessary for

meeting the noise reduction goal.[4] Because these structures are flexible and would

be designed to be light-weight, it is important to analyze their aeroelastic behavior

to prevent structural failure due to divergence or flutter. Rectangular panel prob-

lems have been studied extensively in the past, specifically the aircraft structural

6

Figure 1.1: Continuous Mold-Line Link

panel problem with all edges clamped[9], and the flag flutter problem described ear-

lier. However, there is less existing research on the aeroelastic behavior of panels for

non-traditional applications, where the more physically correct boundary conditions

are not necessarily those that have been extensively studied. NASAs CML project

is just one of many problems that may require the use of novel plate structure de-

signs. As the design of aerospace structures focuses more on lighter materials and

novel configurations, analytical and experimental results for unexplored boundary

conditions and different materials will important in determining viable designs.

1 2 3

4 5 6

X

X

X X X

X

XX

X

X

X

XX

X

X

X X

X

X

X

Figure 1.2: Combinations of boundary conditions and flow directions explored inthis paper. The diagonal marks indicate a clamped boundary and other boundariesare free with no restraint. The arrows indicate different fluid flow directions thatare considered. The x symbols indicate the presence of a bae next to the plateboundary instead of free space. Each configuration considers a single fluid flowdirection.

This section analyzes the structural dynamics and linear aeroelastic instabilities

of a plate using five different sets of boundary conditions in addition to the NASA

7

CML configuration. The boundary conditions are shown schematically in Figure

1.2, in which the diagonal marks indicate clamped boundary, the absence of marks

indicate free boundary, the x symbols indicate the presence of a bae near the plate

boundary instead of free space, and the flow direction is from left to right. The bae

is necessary in the experimental set up - all clamped boundaries are baed because

there must be a structure with which the clamping is applied. However, some free

boundaries are also baed to provide structural support to the entire experimental

set up. The theory models the structural dynamics using a plate-membrane model

that accounts for flexural rigidity of the material (fourth order derivative) and tension

applied to the material (second order derivative). The structural model is coupled

to an unsteady vortex lattice aerodynamic model that accounts for the plate as well

as any bae structure surrounding the plate. A modular bae system is designed

around the plate and is able to apply either clamped or free boundary conditions

at any of the four edges of the plate. The bae design and experimental data are

presented.

Next, the transition between configurations is explored as the axial alignment of

the flow is varied. This exploration is motivated by the quantitative and qualitative

transition in flutter boundary and motion as the orientation of boundary conditions

relative to the flow is changed. For example, for a plate with three sides free, if

the trailing edge is free the system becomes unstable in a flutter instability, but

if the system is rotated 90 deg so the free edge is aligned with the flow then the

dynamic flutter becomes a static divergence. For this section the appropriate mesh

and coordinate transformations are presented to analyze structures which are not

aligned with the flow. The aeroelastic stability is then solved for as the flow angle is

varied.

Experimental results are then presented to validate the theoretical models. Fi-

nally there are concluding remarks about the research conducted to this point as well

8

a brief discussion of future work.

9

2Structural Model

In this thesis both a one-dimensional beam and a two-dimensional plate structural

model will be derived and discussed. The first structural model developed is that of

a beam in bending and torsion. Although the derivation of the governing structural

equations and natural mode shapes is straight forward, finding a single source that

contains the equations of motion derivation as well as the natural mode shapes for all

boundary conditions is difficult. Because the natural modes for a uniform property

beam are used for the analysis of the plate, it is convenient to have a complete

reference for a beam with all possible combinations of boundary conditions required

for the analysis in this thesis.

The following section outlines the steps, starting with the energies of a beam,

using these energies to derive the unforced equations of motion and the associated

natural boundary conditions for a beam, applying a separation of variables technique

to determe the spatial mode shapes.

10

2.1 Beam Structural Model Derivation

In order to derive the equations of motion for this structure, the first step is to

define the potential and kinetic energy equations for the system. Assuming that the

motion of the beam can be described as the linear combination of an out of plane

displacement w(x, t) and a rotation around the elastic axis of the beam (x, t) the

expression for the potential energy of the beam can be written as, where x is the

axis which runs along the length of the beam:

V =1

2

L0

EI

(2w

x2

)2dx+

1

2

L0

GJ

(

x

)2dx (2.1)

Similarly, the kinetic energy for this system can be written as:

T =1

2

L0

m

(w

t

)2dx+

1

2

L0

Iea

(

t

)2dx (2.2)

with m being the mass per unit length and Iea the moment of inertial around the

elastic axis per unit length. Now that the kinetic and potential energy expressions

have been written, the next step is to apply Hamiltons Principal. The principle as

stated in Dowell and Tang [10] for a conservative system, is that the time integral

of the virtual change in kinetic energy minus the virtual change in potential energy

must equal zero. This can be expressed mathematically as: t2t1

[T V ] dt = 0 (2.3)

The next step is to rewrite the virtual changes in kinetic and potential energy in

terms of a virtual change in w(x, t), ((w)) and (x, t), (()).

Starting with the equation for potential energy and applying the virtual change

operator:

t2t1

V dt =

t2t1

{1

2

L0

EI

(2w

x2

)2dx+

1

2

L0

GJ

(

x

)2dx

}dt (2.4)

11

The operator may be treated like the differential operation:

t2t1

[1

2

L0

2EI

(2w

x2

)

(2w

x2

)dx+

1

2

L0

2GJ

(

x

)

(

x

)dx

]dt (2.5)

Knowing that the final result must end up multiplying w and it is clear that the

next step is to integrate by parts. For this equation integrate by parts with respect

to x for the EI term.

Let: u = EI2w

x2and v =

(2w

x2

)dx (2.6a)

u =

x

[EI

2w

x2

]dx and v =

(w

x

)(2.6b)

Using the integration by parts relationshipudv = vu

vdu (2.7)

and the transformations given in Equations 2.6, The EI portion of Equation 2.5 can

be rewritten as:

t2t1

EI 2wx2

(w

x

) L

0

L

0

x

[EI

2w

x2

]

(w

x

)dx

dt (2.8)Integrating by parts once more: t2

t1

EI 2wx2

(w

x

) L

0

x

[EI

2w

x2

] (w)

L

0

dt+

t2t1

L0

2

x2

[EI

2w

x2

] (w) dx dt

(2.9)

Equation 2.9 is in a form that can be directly included into Equation 2.3. A similar

exercise can be conducted for the GJ portion of the equation. This yields

t2t1

[GJ

x

L

0

L

0

GJ

(2

x2

) dx

]dt (2.10)

12

Next a similar analysis must be done for the kinetic energy. Again integration

by parts is used until there is an integral statement which multiplies w and another

statement which multiplies . Substituting the kinetic energy (T) from Equation

2.2 into Equation 2.3.

t2t1

{1

2

L0

m

(w

t

)2dx+

1

2

L0

Iea

(

t

)2dx

}dt (2.11)

Now applying the operator: t2t1

1

2

L0

2m

(w

t

)

(w

t

)+ 2Iea

(

t

)

(

t

)dx dt (2.12)

At this point it is important to note that is operating on(wt

)and

(t

). In order

to reduce this to w and it is clear that one must integrate by parts with respect

to t. Integrating by parts for the w term yields the following result for the time

integral of the virtual change in kinetic energy.

L0

[m

(w

t

)w

t2

t1

]dx

t2t1

L0

m

(2w

t2

)w dx dt (2.13)

Similarly, integrating by parts for the term yields the following result for the time

integral of the virtual change in kinetic energy:

L0

[Iea

(

t

)

t2

t1

]dx

t2t1

L0

Iea

(2

t2

) dx dt (2.14)

However Equations 2.13 and 2.14 can be made even simpler by invoking a re-

lationship that is commonly used with Hamiltons Principle. Namely it is assumed

that w and at t = t1 and t = t2 are both known and identically equal to zero.

This allows one to rewrite the virtual change in kinetic energy as: t2t1

T dt = t2t1

L0

m

(2w

t2

)w dx dt

t2t1

L0

Iea

(2

t2

) dx dt (2.15)

13

Now that the individual components of the virtual changes in kinetic and potential

energies for Hamiltons principle have been calculated, Equations 2.9, and 2.15 can

be substituted into Equation 2.3 to yield the following result.

0 =

t2t1

[T V ] dt

= t2t1

EI 2wx2

(w

x

) L

0

x

[EI

2w

x2

] (w)

L

0

+GJ

x

L

0

dt+

t2t1

L0

[m

(2w

t2

)

2

x2

[EI

2w

x2

]]w dx dt

+

t2t1

L0

[Iea

(2

t2

)+GJ

(2

x2

)] dx dt

(2.16)

2.1.1 Boundary Conditions

Equation 2.16 represents the governing equation for a beam. Equation 2.16 contains

information about the boundary conditions and the equations of motion for the

system. Because the system has both out of plane and rotational degrees of freedom,

there are two sets of natural boundary conditions and two equations of motion.

Starting with the boundary terms multiplying w.

t2t1

EI 2wx2

(w

x

) L

0

x

[EI

2w

x2

] (w)

L

0

dt = 0 (2.17)In order for Equation 2.17 to be satisfied both of the terms inside the integral

must be equal to zero. Moreover because each term is made up of a product of two

terms, at least one term in each product must be equal to be zero. The boundary

conditions must be satisfied at both x = 0 and x = L. Mathematically this can be

14

stated as:

EI2w

x2= 0 or

(w

x

)= 0

and

x

[EI

2w

x2

]= 0 or (w) = 0

(2.18)

A similar analysis for the natural boundary conditions for the torsional coordinate

yields the following boundary conditions at both x = 0 and x = L.

x= 0 or = 0 (2.19)

2.1.2 Equations of Motion

Equation 2.16 also contains information about the elastic equations of motion for

the system. Once the natural boundary conditions are satisfied, in order for the

integral portion of Equation 2.16 to be satisfied for every w and the fundamental

theorem of calculus of variation requires that the following differential equations must

be satisfied.

m(2w

t2

)

2

x2

[EI

2w

x2

]= 0 (2.20)

and

Iea(2

t2

)+GJ

(2

x2

)= 0 (2.21)

2.1.3 Normalized Equations of Motion

In order to present a more general form of the analysis, it is common to normalize the

equations of motion into their scale invariant forms. The equations are normalized

using the characteristic length L and a characteristic time T equal to L2

mEI

for

the bending equation and L

IeaGJ

for the torsion equation. These normalizing factors

15

will also be used in the aeroelastic analysis. Substituting these normalizing factors

in and assuming the beam characteristics are constant along the beam allows the

equations of motion to be written as:

2w

t2

4w

x4= 0 (2.22)

and

2

t2+2

x2= 0 (2.23)

The boundary conditions remain the same except the scaling factors are removed

and the boundary conditions are satisfied at x = 0 and x = 1.

2.1.4 Bending Separation of Variables

The solution to the homogeneous equation for w(x, t) gives the bending natural

frequencies and the mode-shapes for the system. The equation of motion is solved

using the method of separation of variables. The following substitution is used.

w(x, t) = q(t)(x) (2.24)

Substituting Equation 2.24 into the homogeneous equation of motion yields:

2

t2[q(t)(x)

]+

4

x4[q(t)(x)

]= 0 (2.25)

Evaluating the derivatives and dividing by q(t), and (x)

q(t)

q(t)+(x)(x)

= 0 (2.26)

This can only be satisfied if both q(t)q(t)

and (x)(x)

are equal to a constant of opposite

sign. With this definition the two equations can be solved separately and the value

of the constant 2 and the equations for q(t) and (x) can be determined.

16

Looking first at the equation for q(t) and setting it equal to 2 yields:

q(t) + 2q(t) = 0 (2.27)

This equation can be solved by assuming a solution of the form:

q(t) = A cos(t) +B sin(t) (2.28)

Because there are no initial conditions in the time domain, this is the closest to a

solution for the time function that can be determined. Equation 2.28 also clearly

shows that the s are the natural frequencies of the system.

The next step is to look at the equation for (x):

(x) 2(x) = 0 (2.29)

This equation can best be solved by assuming a solution that is a linear combi-

nation of trigonometric and hyperbolic trigonometric functions. For convenience the

following constant is defined:

k2n = (2.30)

Thus the assumed solution becomes:

(x) = C sinh(knx) +D cosh(knx) + E sin(knx) + F cos(knx) (2.31)

At this point the specific choice of boundary conditions determines the values of

the A, B, C and D, up to an arbitrary constant and the specific values for kn.

2.1.5 Torsional Separation of Variables

The solution to the homogeneous equation for (x, t) will give the torsional natu-

ral frequencies and mode-shapes for the system. The homogeneous version of the

equation of motion is also solved using the method of separation of variables. The

following substitution is used.

(x, t) = A(t)(x) (2.32)

17

Substituting Equation 2.32 into the homogeneous equation of motion yields:

2

t2

[A(t)(x)

] 2x2

[A(t)(x)

]= 0 (2.33)

Evaluating and dividing by A(t), and (t)

A(t)

A(t)

(x)(x)

= 0 (2.34)

This can only be satisfied if both A(t)A(t)

and (x)(x)

are equal to a constant of the same

sign. With this definition the two equations can be solved separately and the value

of the constant 2 and the equations for A(t) and (x) can be determined.

Looking first at the equation for q(t) and setting it equal to 2 yields:

A(t) + 2A(t) = 0 (2.35)

This equation can be solved by assuming a solution of the form:

A(t) = G cos(t) +H sin(t) (2.36)

Again, because there are no initial conditions for the time domain, this is the closest

to a solution for the time function that can be determined. Equation 2.36 also clearly

shows that the s are again the natural frequencies of the system.

The next step is to look at the equation for (x). Setting the equation equal to

(2) and rearranging gives:

(x) + 2(x) = 0 (2.37)

This equation can best be solved by assuming a solution that is a linear combi-

nation of trigonometric functions. For convenience the following constant is defined:

j2n = 2 (2.38)

18

Thus the assumed solution becomes:

(x) = I sin(jnx) + J cos(jnx) (2.39)

At this point the specific choice of boundary conditions determines the values for

I and J up to an arbitrary constant and the specific values for jn.

2.2 Specific Beam Mode Shapes

For the analysis that will be conducted throughout this paper the following mode

shapes and natural frequencies will be used.

2.2.1 Clamped-Free

Bending Mode Shapes

For the clamped-free configuration the boundary conditions are:

x=0,t

= 0

x

x=0,t

= 0

2

x2

x=1,t

= 0

3

x3

x=1,t

= 0

(2.40)

Figure 2.1 shows the diagram of the clamped-free configuration. Applying the bound-

ary conditions at x = 0 yields:

D + F = 0

C + E = 0(2.41)

Applying the boundary conditions at x = 1 yields:

C sinh kn +D cosh kn E sin kn F cos kn = 0

C cosh kn +D sinh kn E cos kn + F sin kn = 0(2.42)

19

xL

Figure 2.1: Clamped Free Schematic

Using Equations 2.41 to simplify Equations 2.42 yields:

C(sinh kn + sin kn) +D(cosh kn + cos kn) = 0

C(cosh kn + cos kn) +D(sinh kn sin kn) = 0(2.43)

Using Equations 2.43 to solve for kn by setting the determinate of the coefficients C

and D equal to zero yields:

cos(kn) = 1cosh(kn)

(2.44)

Figure 2.2 shows the intersection of the two sides of Equation 2.44. The first non-

Figure 2.2: Clamped Free Frequencies

dimensional frequency is 1 = (.597)2pi2 and the second frequency is 2 = (1.49)

2pi2.

20

For the nth frequency where n is larger than two the natural frequency is approxi-

mately n (n 1/2)2pi2. Furthermore the mode shapes can be written in terms ofan arbitrary constant D as:

(x) = D

[sin(kn) sinh(kn)cos(kn) + cosh(kn)

(sinh(knx) sin(knx)) + (cosh(knx) cos(knx))]

(2.45)

The mode shapes are shown in Figure 2.3.

0 0.2 0.4 0.6 0.8 11

01

0 0.2 0.4 0.6 0.8 11

01

0 0.2 0.4 0.6 0.8 11

01

0 0.2 0.4 0.6 0.8 11

01

0 0.2 0.4 0.6 0.8 11

01

Figure 2.3: Clamped Free Bending Mode Shapes

Torsional Mode Shapes

For the clamped-free configuration the boundary conditions are:

x=0,t

= 0

x

x=1,t

= 0

(2.46)


J = 0 (2.47)

21


cos jn = 0 (2.48)

Equation 2.48 has the solution jn =2n1

2pi. Furthermore the torsional mode shapes

can be described by:

(x) = I sin jnx (2.49)

2.2.2 Pinned-Free

Bending Mode Shapes

For the pinned-free configuration the boundary conditions are:

x=0,t

= 0

2

x2

x=0,t

= 0

2

x2

x=1,t

= 0

3

x3

x=1,t

= 0

(2.50)

Figure 2.4 shows the diagram of the pinned-free configuration. Applying the bound-

ary conditions at x = 0 yields:

D + F = 0

D F = 0(2.51)

These two relationships require that D = F = 0. Knowing that D and F are equal

to zero allows on to simplify the form of the solution to:

(x) = C sinh knx+ E sin knx (2.52)


C sinh kn E sin kn = 0

C cosh kn E cos kn = 0(2.53)

22

xL

Figure 2.4: Pinned Free Schematic

Using Equations 2.53 to solve for kn yields:

cos(kn) tanh(kn) = sin(kn) (2.54)

Figure 2.5 shows the intersection of the two sides of Equation 2.54. The first natural

Figure 2.5: Pinned Free Frequencies

frequency occurs at 0. This corresponds to the rigid body motion which has a mode

shape given as 1(x). The other frequencies all take the form of (n 3/4)2pi2 for thenth frequency for every n larger than 1. The mode shapes are described by:

1 = C1x

n(x) = E

(cos kncosh kn

sinh knx+ sin knx

) (2.55)23


0 0.2 0.4 0.6 0.8 11

01

0 0.2 0.4 0.6 0.8 11

01

0 0.2 0.4 0.6 0.8 11

01

0 0.2 0.4 0.6 0.8 11

01

0 0.2 0.4 0.6 0.8 11

01

Figure 2.6: Pinned Free Mode Shapes


For the pinned-free configuration the boundary conditions are:

x=0,t

= 0

x

x=1,t

= 0

(2.56)

These are the same boundary conditions as the clamped free configuration, so the

torsional mode shapes are exactly the same as discussed in the previous section.

24

2.2.3 Clamped-Clamped

Bending Mode Shapes

For the clamped-clamped configuration the boundary conditions are:

x=0,t

= 0

x

x=0,t

= 0

x=1,t

= 0

x

x=1,t

= 0

(2.57)

Figure 2.7 shows the diagram of the clamped-clamped configuration.

x

L

Figure 2.7: Clamped Clamped Schematic


D + F = 0

C + E = 0(2.58)


C sinh kn +D cosh kn + E sin kn + F cos kn = 0

C cosh kn +D sinh kn + E cos kn F sin kn = 0(2.59)

25


C(sinh kn sin kn) +D(cosh kn cos kn) = 0

C(cosh kn cos kn) +D(sinh kn + sin kn) = 0(2.60)

Equations 2.60 can be rearranged to solve for kn:

cos(kn) =1

cosh(kn)(2.61)

Figure 2.8 shows the intersection of the two sides of Equation 2.61. The ith fre-

Figure 2.8: Clamped Clamped Frequencies

quency from this plot can be written as (i+ .5)2pi2.

Finally solving for C in terms of D and plugging into Equation 2.60 yields the

following equation for the mode shapes:

(x) = D

[cosh(kn) cos(kn)

sinh(kn) sin(kn) (sinh(knx) sin(knx)) + (cosh(knx) cos(knx))]

(2.62)


26

0 0.2 0.4 0.6 0.8 11

01

0 0.2 0.4 0.6 0.8 11

01

0 0.2 0.4 0.6 0.8 11

01

0 0.2 0.4 0.6 0.8 11

01

0 0.2 0.4 0.6 0.8 11

01

Figure 2.9: Clamped Clamped Mode Shapes


For the clamped-clamped configuration the boundary conditions are:

x=0,t

= 0

x=1,t

= 0

(2.63)


J = 0 (2.64)


sin jn = 0 (2.65)

Equation 2.65 has the solution jn = npi. Therefore the torsional mode shapes are

described by:

(x) = I sin jnx (2.66)

27

2.2.4 Free-Free

Bending Mode Shapes

For the free-free configuration the boundary conditions are:

2

x2

x=0,t

= 0

3

x3

x=0,t

= 0

2

x2

x=1,t

= 0

3

x3

x=1,t

= 0

(2.67)


D F = 0

C E = 0(2.68)


C sinh kn D cosh kn + E sin kn F cos kn = 0

C cosh kn +D sinh kn + E cos kn F sin kn = 0(2.69)


C(sinh kn sin kn) +D(cosh kn cos kn) = 0

C(cosh kn cos kn) +D(sinh kn + sin kn) = 0(2.70)

Equations 2.70 can be rearranged to solve for kn:

cos(kn) =1

cosh(kn)(2.71)

Figure 2.10 shows the intersection of the two sides of Equation 2.71. There is an

28

0 5 10 151

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

X: 14.14Y: 0.006977

kn

X: 10.99Y: 3.371e005

X: 7.853Y: 0.0007773

X: 0Y: 1

X: 4.685Y: 0.01847

Figure 2.10: Free Free Frequencies

intersection at a frequency equal to zero. This crossing corresponds to the rigid body

and rigid rotation modes which can be written as:

(x) = C1 + C2(1

2 x) (2.72)

where C1 and C2 are arbitrary constants. The ith frequency from this plot can be

written as (i .5)2pi2.Finally solving for C in terms of D and plugging into Equation 2.70 yields the

following equation for the mode shapes:

(x) = D

[cosh(kn) cos(kn)

sinh(kn) sin(kn) (sinh(knx) + sin(knx)) + (cosh(knx) + cos(knx))]

(2.73)


2.3 Pinned Edge Torsional Spring Model

Next a structural model which includes a torsional spring at the leading edge is

derived. This model is presented because it is used to model the transition between

the two fixed boundary conditions, clamped and pinned. A summary of the non-

dimensional (radians/non-dimensional time) natural frequencies for the pinned-free

29

0 0.2 0.4 0.6 0.8 11

01

0 0.2 0.4 0.6 0.8 11

01

0 0.2 0.4 0.6 0.8 11

01

0 0.2 0.4 0.6 0.8 11

01

0 0.2 0.4 0.6 0.8 11

01

Figure 2.11: Free Free Mode Shapes

and clamped-free beams is given in Table 2.1. Furthermore the normalized spatial

mode shapes can be seen in Figure 2.12

Mode Pinned-Free Clamped-FreeNumber Frequency Frequency

1 0 (.517)2pi2

2 (2 34)2pi2 (1.49)2pi2

3 (3 34)2pi2 (3 1

2)2pi2

......

...n (n 3

4)2pi2 (n 1

2)2pi2

Table 2.1: Non-Dimensional Natural Frequencies for a Single Edge Fixed Beam

The leading edge spring can either be modeled by incorporating the potential

energy due to the spring into the equations of motion or modifying the boundary

conditions to include the restoring moment due to the torsional spring. For this

thesis the boundary condition method is used because the resulting mode shapes are

the natural modes of the spring system and therefore the elastic portion of the aeroe-

lastic equations remain uncoupled. This minimizes the number of modes required to

30

0 0.2 0.4 0.6 0.8 10

1

2

1(x)

0 0.2 0.4 0.6 0.8 12

0

2

2(x)

0 0.2 0.4 0.6 0.8 12

1

0

1

23(x)

x

0 0.2 0.4 0.6 0.8 12

1

0

1

2

4(x)

x

Figure 2.12: The solid line corresponds to the clamped-free mode shapes, dashedline to the pined-free mode shapes, and the dotted line to the K = 1000 modeshapes. All mode shapes normalized to a generalized mass of one.

capture the dynamics in the aeroelastic simulations. The boundary conditions at the

pinned edge with the torsional spring can be determined by applying a force balance

at x = 0. Here the torsional force applied by the spring modeled by hooks law must

be identically equal to bending moment. Mathematically this can be written as:

2w(0, t)

x2= K

w(0, t)

x(2.74)

where K = K(L/EI). To ensure the mode shapes satisfy this boundary condition

as well as the three other natural boundary conditions, the assumed solution is

plugged into the boundary condition equations. This process yields the following

matrix equation.

1 1 1 1

K 1 K 1cosh

sinh

cos sin

sinh cosh

sin cos

CDEF

=

0000

(2.75)

The set of four coupled equations captured in Equation 2.75 can be used to

solve for the natural frequencies by determining the values of which make the

31

determinate of the matrix equal to zero. There are an infinite number of frequencies

that will satisfy this requirement. Depending on the number of mode shapes desired,

the Nullspace of the matrix can be used to determine the values for C, D, E, F up

to an arbitrary constant for each of the s which satisfy the determinant equation.

A common choice for the constant is one that normalizes the generalized mass to

one.

102 100 1020

10

20

30

40

50

60

70

K

n

[rad

ians]

Figure 2.13: Structural Frequency Evolution. The solid lines are the natural fre-quencies of the beam with a torsional spring, the dotted lines are the pinned-freenatural frequencies and the dashed lines are the clamped-free natural frequencies.

Before moving on to the aeroelastic analysis, it is important to demonstrate the

ability of a leading edge spring to model the transition between pinned-free and

clamped-free structural modes. To evaluate the effectiveness of the method one can

look at the convergence of the frequencies as a function of non-dimensional torsional

spring stiffness (K). First, it is reassuring to see that the frequencies do converge

to the clamped-free frequencies at high values of K. The second observation that

can be made from Figure 2.13 is that the non-dimensional spring stiffness range

where the frequencies move from the pinned-free to the clamped-free values shifts to

32

higher values of K as the mode number increases. Physically, this arises because the

stiffness of a given mode increases in proportion the natural frequency squared, so in

order for the torsional spring to affect the larger modes, its stiffness must be larger.

This also means that lower frequencies close the gap between the higher frequency

modes before these higher frequencies begin to move and restore the gap. This is

a result that could explain why increasing the torsional spring stiffness can initially

lower the flutter velocity, a result which will be shown later.

The results of this structural analysis suggest that the pinned edge torsional

spring will be an effective way to model the transition between pinned-free and

clamped-free flutter. Furthermore, being able to model both of the boundary con-

ditions simply by varying the parameter K is an elegant way to create the model

of the system with arbitrary boundary conditions. In fact it is clear that modifying

the terms in Equation 2.75 will allow you to model any arbitrary boundary beam

boundary conditions.

2.4 Plate Structural Model

Although the beam model is useful for plates with aspect ratios far from unity,

when this is not the case and boundary conditions in both the directions need to

be accounted for, a more complex structural model must be used. In order to do

this a two-dimensional plate structural model is used. Instead of relying on a finite

element model simulation to determine the mode shapes and natural frequencies, an

analytical approach is implemented. Because a direct solution of the plate equation

PDE with the appropriate natural boundary conditions is difficult without specific

boundary conditions and the use of special functions, a Raleigh-Ritz method is used.

The basis functions are a product of beam modes in each of the two plate dimensions

are used. Because the Raleigh-Ritz method only requires the geometric boundary

conditions and not the natural boundary conditions to be satisfied, the beam modes

33

are a viable set of assumed solutions.

The Raleigh-Ritz method begins with expressing the assumed form of the dis-

placement.

w(x, y, t) =n

qn(t)jk(x, y) (2.76)

In Equation 2.76, the nth structural mode is labeled jk because the mode shape

can be broken in to two components.

jk(x, y) = j(x)k(y) (2.77)

The energies of the system need to be derived and then placed into Lagranges

equations. The energies for a plate in tension can be written as[9]:

T =1

2 h

Lx0

Ly0

(w

t

)2dx dy (2.78)

V =1

2

Lx0

Ly0

[Tx

(w

x

)2+ Ty

(w

y

)2+Dx

(2w

x2

)2+Dy

(2w

y2

)2

+2D(

4w

x2y2

)+ 4Dxy

(2w

xy

)2 ]dx dy

(2.79)

It is at this point that the assumed form of the solution is plugged into Equations

2.78 and 2.79. The inertial term in the kinetic energy equation can be rewritten as:

T =1

2h

Lx0

Ly0

n

m

qn(t)qm(t)jk(x, y)pq(x, y) dx dy (2.80)

where jk is the nth mode shape and pq is the mth mode shape.

Because the generalized coordinates do not vary with position they can be pulled

outside of the integral. Furthermore, it is useful to use vector notation to represent

the double sum. The displacement w(x, y, t) can be written as a multiplication of

two column vectors as shown here:

w(x, y, t) = ~q T ~ = ~T~q (2.81)

34

Using this relationship, Equation 2.80 can be rewritten as:

T = ~q TM~q (2.82)

Where the mass matrix M is equal to:

M =1

2h

Lx0

Ly0

~ ~T dx dy (2.83)

Because the terms in Equation 2.83 are products of beam mode shapes integrated

in each direction, the orthogonality of the beam modes means that only when both

the indices of are equal is the integral not equal to zero. Because of this the mass

matrix is a diagonal matrix.

The next term to explore is the first term in the potential energy expression.

Before witting down the substitution for w(x, y, t) it is important to discuss what

happens when a spatial derivative of the assumed mode is taken. For example the

first derivative with respect to x of the displacement is given by:

w

x= ~q T

x

(~)

(2.84)

Equation 2.84 can be further simplified by expanding out the assumed modes into

its x and y components. This will be captured in the description of ~ using the

following notation:

~ = ~ (2.85)

where the previous equation shows that the structural mode shape vector has two

components. Plugging this into Equation 2.84 and using the prime () notation to

indicate a spatial derivative with respect to the direction of the mode shape yields

the following relationship.

w

x= ~q T ~ (2.86)

35

Plugging in this relationship into the tension in the x-direction portion of the

potential energy expression yields:

Vtx =1

2~q T Lx

0

Ly0

Tx ~ ~ T dy dx~q (2.87)

To simplify the previous equation, the orthogonality of the y mode shapes can be

used to cancel terms where the index of the y mode shapes are not equal. Furthermore

the following notation will be used to define the integral portion of the previous

equation.

Ktx =1

2

Lx0

Ly0

Tx ~ ~ T dy dx (2.88)

Using a similar method, the y direction tension term can be written as:

Vty = ~qT Kty~q (2.89)

where Kty is the y tension stiffness matrix which can be written as:

Kty =1

2

Lx0

Ly0

Ty ~ ~ T dy dx (2.90)

Furthermore by inspection the following stiffness matrices can be defined for the

additional potential energy terms. Once the stiffness matrices are defined the form

of the associated potential energy is the same as shown in Equation 2.89. Starting

with the potentials associated with the Dx and Dy terms.

KDx =1

2Dx

Lx0

Ly0

~ ~ T dy dx (2.91)

KDx =1

2Dy

Lx0

Ly0

~ ~ T dy dx (2.92)

Because both the mode shapes and their second derivatives are orthogonal, the KDx

and KDy matrices are diagonal.

36

Finally the last two stiffness matrices can be written as:

KD = D Lx

0

Ly0

~ ~ T dy dx (2.93)

KDxy = 2Dxy

Lx0

Ly0

~ ~ T dy dx (2.94)

for these two terms there is no modal orthogonality so all of the integrations of the

mode shapes must be conducted.

Now that the useful definitions have been made, the potential and kinetic energy

relationships can be rewritten in a simplified form as:

T =

t(~q T )M

t(~q) (2.95)

V = ~q T[Ktx + Kty + KDx + KDy + KD + KDxy

]~q (2.96)

With the potential and kinetic energies defined they can be plugged into La-

granges equation to yield the equations of motion. The familiar form of Lagranges

equation is:

L

qn ddt

[L

qn

]= 0 (2.97)

where L is the Lagrangian and equal to the kinetic energy minus the potential en-

ergy. After plugging in the energies given in Equations 2.95 and 2.96 the following

equations of motion in matrix form is produced.

[M

2

t2~q +

(Ktx + Kty + KDx + KDy + KD + KDxy

)~q

]= 0 (2.98)

The equation of motion given in Equation 2.98 can be solved in many different

ways. For example the equation could be placed into state space and solved using

numerical integration techniques. The author choose to use eigenanalysis of the

37

system. For this analysis a solution of the form q(t) = qeit is assumed. Plugging

this solution into equation 24 yields:

(2M + K) ~q = 0 (2.99)where K is composed of the sum of the individual stiffness terms. Equation 2.99 is

in the form of a generalized eigenvalue problem which can be solved using available

eigenvalue solvers. The resulting eigenvalues can be used to reconstruct the natural

frequencies and mode shapes of the plate system. This model allows a variation of

boundary conditions by changing the assumed solution with the appropriate beam

modes.

2.4.1 Plate Structural Analysis Typical Results

Initial plate simulations are done for a material provided by NASA for the use in

noise reduction between control surfaces and wings on the configurations outlined in

the introduction. The material properties are given in Table 2.2.

Table 2.2: NASA Membrane Properties

Property Symbol ValueDensity s 1230 kg/m

3

Youngs Modulus E 18.4 MPaPoissons Ratio .5Thickness h 1.74 mmChord 152.4 mmSpan 114.3 mm

The results of the elastic simulation for each of the configurations are given in

Figures 2.14-2.19. The natural frequencies have been sorted by their y direction

mode number which is determined from the system eigenvector. Below the plot

of the natural frequencies are plots of the first four modes with the lowest natural

38

frequencies. For each mode shape, a thick line along a boundary represents a clamped

boundary condition. The title above each mode shape gives the natural frequency

followed by the mode number organized as (x Mode, y Mode). For the aeroelastic

setup corresponding to a given configuration the flow is assumed to flow along the

x-axis.

A discussion about the structural model can occur at this point. The first and sec-

ond beam mode shapes for a free-free beam are a rigid body rotation and translation

and share a natural frequency of zero so it is not surprising that there is overlap in the

natural frequencies for cases where there is at least one free-free boundary condition.

Next, the more edges fixed, the higher the natural frequencies are. This is intuitive

because the structural modes constructing the plate which increase in their natural

frequency the more fixed edges they have. By looking at the mode shape figures, it

appears that the beam mode basis function assumption are a good assumption for

the natural modes of the system. This can be seen by looking at the construction

of each of the plate modes which are clearly combinations of an assumed mode in

each of the directions with only small contributions from additional modes. More

discussion of the agreement with experiment is given in the experiments section.

An alternative method to the plate model presented here would have been to use

ANSYS or another finite element package to determine the modes shapes and natural

frequencies. However, this method would have required running an external simula-

tion any time a parameter or boundary condition is changed. Using the beam mode

combination basis functions and building this elastic model into the aeroelastic anal-

ysis allowed the author to vary the tension, dimensions and boundary conditions on

the fly which makes exploring the flutter boundary as a function of these parameters

easier.

Additionally, elastic simulations for different streamwise lengths and tension in

the normal to flow direction are conducted. These simulations are run because it

39

1 2 30

10

20

30

40

50Normal to Flow Mode 1

Nat

ural

Fre

quen

cy [H

z]

1 2 30

20

40

60

Normal to Flow Mode 2

1 2 30

50


Nat

ural

Fre

quen

cy [H

z]

Streamwise Mode1 2 3

0

50

100


Streamwise Mode

0

0.1

0

0.05

0.1

2

1

0

1

2

1.689 Hz (1,1)

0

0.1

0

0.05

0.1

2

1

0

1

2

4.929 Hz (1,2)

0

0.1

0

0.05

0.1

2

1

0

1

2

10.524 Hz (2,1)

0

0.1

0

0.05

0.1

2

1

0

1

2

16.374 Hz (2,2)

Figure 2.14: Configuration 1 Natural Frequencies and Mode Shapes

1 2 30

5

10

15

20


Nat

ural

Fre

quen

cy [H

z]

1 2 30

10

20

30


1 2 30

20

40

60


Nat

ural

Fre

quen

cy [H

z]


0

50

100


Streamwise Mode

0

0.1

0

0.05

0.1

2

1

0

1

2

3.004 Hz (1,1)

0

0.1

0

0.05

0.1

2

1

0

1

2

5.580 Hz (2,1)

0

0.1

0

0.05

0.1

2

1

0

1

2

14.587 Hz (3,1)

0

0.1

0

0.05

0.1

2

1

0

1

2

18.709 Hz (1,2)


40

1 2 30

20

40

60


Nat

ural

Fre

quen

cy [H

z]

1 2 30

20

40

60


1 2 30

50


Nat

ural

Fre

quen

cy [H

z]


0

50

100

150


Streamwise Mode

0

0.1

0

0.05

0.1

2

1

0

1

2

10.808 Hz (1,1)

0

0.1

0

0.05

0.1

2

1

0

1

2

13.329 Hz (1,2)

0

0.1

0

0.05

0.1

2

1

0

1

2

28.245 Hz (1,3)

0

0.1

0

0.05

0.1

2

1

0

1

2

29.627 Hz (2,1)


1 2 30

10

20


Nat

ural

Fre

quen

cy [H

z]

1 2 30

20

40

60


1 2 30

50

100


Nat

ural

Fre

quen

cy [H

z]


0

50

100

150


Streamwise Mode

0

0.1

0

0.05

0.1

2

1

0

1

2

19.215 Hz (1,1)

0

0.1

0

0.05

0.1

2

1

0

1

2

20.756 Hz (2,1)

0

0.1

0

0.05

0.1

2

1

0

1

2

28.303 Hz (3,1)

0

0.1

0

0.05

0.1

2

1

0

1

2

52.669 Hz (1,2)


41

1 2 30

20

40


Nat

ural

Fre

quen

cy [H

z]

1 2 30

20

40

60


1 2 30

50

100


Nat

ural

Fre

quen

cy [H

z]


0

50

100

150


Streamwise Mode

0

0.1

0

0.05

0.1

2

1

0

1

2

12.078 Hz (1,1)

0

0.1

0

0.05

0.1

2

1

0

1

2

26.358 Hz (1,2)

0

0.1

0

0.05

0.1

2

1

0

1

2

31.049 Hz (2,1)

0

0.1

0

0.05

0.1

2

1

0

1

2

45.597 Hz (2,2)


1 2 30

20

40


Nat

ural

Fre

quen

cy [H

z]

1 2 30

20

40

60


1 2 30

50

100


Nat

ural

Fre

quen

cy [H

z]


0

50

100

150


Streamwise Mode

0

0.1

0

0.05

0.1

2

1

0

1

2

19.816 Hz (1,1)

0

0.1

0

0.05

0.1

2

1

0

1

2

26.611 Hz (2,1)

0

0.1

0

0.05

0.1

2

1

0

1

2

43.183 Hz (3,1)

0

0.1

0

0.05

0.1

2

1

0

1

2

53.443 Hz (1,2)


42

is hypothesized that these two variations may be able to largely impact the flutter

boundary for Configuration 6, the configuration of interest for the NASA noise sup-

pression research. Figure 2.20 shows the frequency evolution as the streamwise chord

is varied. Interestingly there are natural frequency crossings. This occurs because

the natural frequency of the normal to flow direction mode remains the same, while

the streamwise frequency varies. Another trend that is observed is for a given mode

in the normal to flow direction, as the chord increases all the frequencies which share

the same normal to the flow mode number begin to converge. In the limit as the

chord goes to infinity the system appears to converge to the beam natural frequen-

cies in the normal to flow direction. This arises because, as the streamwise length

increases, the local response at any cross section in the normal to flow direction

does not depend on the boundary conditions in the streamwise direction, essentially

turning the cross section into a beam with the normal to flow direction boundary

conditions.

0.05 0.1 0.15 0.2 0.25 0.30

20

40

60

80

100

120

140

160

180

200

Streamwise Dimension

[Hz]

Figure 2.20: Natural frequency evolution as the streamwise chord is varied forConfiguration 2. Solid lines correspond to first mode in the normal to flow direction,dashed lines to the second mode in the normal to flow direction and the dotted linescorrespond to the first two beam natural frequencies for the normal to flow directionmode shapes.

Finally, the natural frequency dependence on tension is also explored. Figure

43

2.21 clearly shows that frequencies evolve differently than others. This arises due

to the fact that for a given mode in the normal to flow direction, the direction of

the applied tension, the effect of the tension is multiplied by the natural frequency

squared because the tension term is not attached to a time derivative in the equations

of motion. This means that the tension has a larger effect on the higher normal to

flow direction mode numbers. This can be seen by comparing the evolution of the

two lowest frequency solid lines, to the two dotted lines which are the two lowest

frequencies that are comprised of the second mode in the tension direction.

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

Tension in the Normal to Flow Direction [N/m]

[Hz]

Figure 2.21: Natural frequency evolution as the normal to the flow tension isvaried for Configuration 2. Solid lines correspond to first mode in the normal to flowdirection, dotted lines to the second mode in the normal to flow direction.

2.5 Forced System Modification

All of the analysis done up to this point is done on an unforced elastic structures.

Before moving on to the discussion of the aerodynamic theory it is usefully to iden-

tify how the structural dynamics equations are modified to include external forcing.

Regardless of how the equations of motion are derived, whether through Hamiltons

principle for the beam equations or Lagranges equation for the beam, the final un-

forced equations can be written in the following form:

44

M~q + K~q = 0 (2.100)

Where M is the generalized mass matrix, K is the generalized stiffness matrix, and

~q are the modal coordinates for the included mode shapes.

If the system is forced Equation 2.100 is modified by adding a generalized forcing

term to the other side of the equation.

M~q + K~q = ~Q (2.101)

This ith element in the generalized force vector is determined by taking the

real force applied to the system multiplying it by the ith generalized mode shape

and integrating the result over the plate. This is a classical result that is found

throughout the literature.

Qi =

A

Fi(x, y) dA (2.102)

In solving the aeroelastic equations the goal is to determine the aerodynamic

force and then to solve Equation 2.101 to determine the structural response. What

makes the problem interesting is that the aerodynamic forces are tightly coupled to

the structures displacement and motion. The next section will outline in more detail

the specifics of the aerodynamic modeling which is used to model the aerodynamic

forcing due to the dynamic response of the structure.

Before moving on the the aerodynamic theory it is useful to discretize the struc-

tural equations of motion because the vortex lattice aerodynamic equations are dis-

crete. First the elastic equations of motion are placed into state space and time

discretized. The best way to illustrate this process is to start by looking at the ith

equation for the relationship defined in Equation 2.102. This relationship can be

45

expressed as

Nj=1

[Mj,i qi + Kj,i qi

]= ~Qi (2.103)

As is common for transforming an equation into state space, it is necessary to define

two state variables y1 = qi and y2 = qi and discretized the variables as follows:

y2n+1/2 =

yn+12 yn2t

(2.104a)

yn+1/21 =

yn+11 + yn1

2(2.104b)

y2n+1/2 =

yn+12 + yn2

2=yn+11 yn1

t= y1

n+1/2 (2.104c)

The last equation is just a discrete relationship between y1 and y2. Moving both of

the discrete representations of the half time step to one side and setting equal to

zero one can obtain the following relationship.

yn+11 yn1t

yn+12 + y

n2

2= 0 (2.105)

Furthermore, the definitions in Equation 2.104 can be used to re-write Equation

2.103 as:

Nj=1

[Mj,i

(~q n+1i ~q ni

t

)+ Kj,i

(~q n+1i + ~q

ni

2

)]= ~Q

n+1/2i (2.106)

46

3Aerodynamic Model

3.1 Aerodynamic Theory Introduction

As mentioned earlier, the forcing on the elastic model is due to the flow of the

surrounding fluid. For this application the aerodynamic forces are calculated using

a vortex lattice method. This method is a lattice method of accounting for discrete

vortex filaments (tubes of constant circulation) as they progress through time. For

this specific application, a certain type of vortex filament called a horseshoe vortex is

used. The reason to track the vortex filaments is that the strength of the circulation

inside the filament corresponds to the applied forces. The general explanation of

the method is that a set of vortex elements are fixed to stationary points on the

structure. These elements allow the structure to interact with the fluid. Their

strength is governed by a requirement that the downwash they create satisfy the no-

flow through boundary condition at what are known as the collocation points on the

structure. Additional vortex elements that are free to move are introduced behind

the structure and are used to account for the influence of the unsteady wake. All of

the models used in this thesis include a flat prescribed wake which makes tracking

47

the convected vorticity in the wake significantly easier. The flat prescribed wake

behind a rectangular structure is shown in Figure 3.1.

Figure 3.1: Visualization of Structural Mode Shapes with Vortex Lattice Wake

Before discussing the specifics of the VLM, it is usef

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Linear Aeroelastic Stability of Beams and Plates in Three-Dimensional Flow

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