Linear Aeroelastic Stability of Beams and Plates in
Three-Dimensional Flow by Samuel Chad Gibbs IV
Department of Mechanical Engineering and Materials Science Duke 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 of Master of Science in the Department of Mechanical Engineering and Materials Science in the Graduate School of Duke University 2012 Abstract Linear Aeroelastic Stability of Beams and Plates in Three-Dimensional Flow by Samuel Chad Gibbs IV
Department of Mechanical Engineering and Materials Science Duke 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 the degree of Master of Science in the Department of Mechanical Engineering and Materials Science in the Graduate School of Duke University 2012 Copyright c 2012 by Samuel Chad Gibbs IV All 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 µ 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 Tablesx
List of Figures xi
List of Abbreviations and Symbols xv
1 Introduction and Literature Review1
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 and Structures 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 (ζs = 0.01)...... 98
4.2 Plate Aeroelastic Simulation Summary (ζs = 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: Theory and 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 Kα Mode Shapes...... 31 2.13 Structural Frequency Evolution with Leading Edge Torsional Spring. 32
2.14 Configuration 1 Natural Frequencies and Mode Shapes...... 40
2.15 Configuration 2 Natural Frequencies and Mode Shapes...... 40
2.16 Configuration 3 Natural Frequencies and Mode Shapes...... 41
2.17 Configuration 4 Natural Frequencies and Mode Shapes...... 41
2.18 Configuration 5 Natural Frequencies and Mode Shapes...... 42
xi 2.19 Configuration 6 Natural Frequencies and Mode Shapes...... 42
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. Kα ...... 84 4.9 Flutter Velocity as a function of the Aspect Ratio...... 86
4.10 Flutter Velocity and Frequency vs. µ ...... 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
4.19 Configuration 1 Aeroelastic Results...... 101
4.20 Configuration 2 Aeroelastic Results...... 102
4.21 Configuration 3 Aeroelastic Results...... 103
4.22 Configuration 4 Aeroelastic Results...... 105
4.23 Configuration 5 Aeroelastic Results...... 106
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 Baffle...... 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 Poisson’s Ratio...... 130
5.9 Photographs of the Experimental Setups...... 131
5.10 Configuration 1 Dynamic Experimental Results...... 132
5.11 Configuration 2 Dynamic Experimental Results...... 133
5.12 The (1,2) Mode Visualization for Configuration 2...... 133
5.13 Configuration 4 Dynamic Experimental Results...... 134
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 Baffle 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 Young’s 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 k’th discrete Γ on the l’th 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
(xζ , yη) (x,y) location of the (ζ, η) panel α vortex lattice relaxation factor
δW Virtual work
~Γ Discrete circulation values
γ(x, y) Continuous circulation at (x,y)
λ Natural frequency of the structure
η Index of column in vortex mesh
Φ Vector of position and velocity coordinates of natural modes
ρ Density
Θ~ Vector with Γ and Φ
ζ Index of row in vortex mesh or aerodynamic damping ratio de- pending on context item[ζs] Structural damping ratio
xvi Superscripts
˜ Non-dimensional
˙ Time derivative
0 Spatial derivative
~ Vector quantity
¯ Matrix quantity
Abbreviations
VLM Vortex Lattice Method
xvii 1
Introduction 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 transition between pinned and clamped leading fixed edge
• Analyze the aeroelastic instabilities for a beam with a clamped leading edge as the governing non-dimensional parameters are varied
• Analyze the aeroelastic instabilities for a plate with three sides fixed, a proposed configuration 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-side clamped 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 problem’s inherent physical significance, Doar´eand 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 Pa¯ıdoussis[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 Pa¯ıdoussis[35], Tang et al. [32], Tang and Pa¯ıdoussis[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 Doar´eet 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 NASA’s 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 CML’s 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. NASA’s 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 X X
X XXX X
X 4 5 6 X X X
X X X XX X
X X X Figure 1.2: Combinations of boundary conditions and flow directions explored in this paper. The diagonal marks indicate a clamped boundary and other boundaries are free with no restraint. The arrows indicate different fluid flow directions that are considered. The ’x’ symbols indicate the presence of a baffle next to the plate boundary instead of free space. Each configuration considers a single fluid flow direction.
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 baffle near the plate boundary instead of free space, and the flow direction is from left to right. The baffle is necessary in the experimental set up - all clamped boundaries are baffled because there must be a structure with which the clamping is applied. However, some free boundaries are also baffled 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 baffle structure surrounding the plate. A modular baffle 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 baffle 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 2
Structural 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 θ(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:
1 Z L ∂2w2 1 Z L ∂θ 2 V = EI 2 dx + GJ dx (2.1) 2 0 ∂x 2 0 ∂x
Similarly, the kinetic energy for this system can be written as:
1 Z L ∂w2 1 Z L ∂θ2 T = m dx + Iea dx (2.2) 2 0 ∂t 2 0 ∂t
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 Hamilton’s 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:
Z t2 [δT − δV ] dt = 0 (2.3) t1
The next step is to rewrite the virtual changes in kinetic and potential energy in terms of a virtual change in w(x, t), (δ(w)) and θ(x, t), (δ(θ)). Starting with the equation for potential energy and applying the virtual change δ operator:
( 2 ) Z t2 Z t2 1 Z L ∂2w 1 Z L ∂θ 2 δV dt = δ EI 2 dx + GJ dx dt (2.4) t1 t1 2 0 ∂x 2 0 ∂x 11 The δ operator may be treated like the differential operation:
" # Z t2 1 Z L ∂2w ∂2w 1 Z L ∂θ ∂θ 2EI 2 δ 2 dx + 2GJ δ dx dt (2.5) t1 2 0 ∂x ∂x 2 0 ∂x ∂x
Knowing that the final result must end up multiplying δw and δθ 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. ∂2w ∂2w Let: u = EI and ∂v = δ dx (2.6a) ∂x2 ∂x2
∂ ∂2w ∂w ∂u = EI dx and v = δ (2.6b) ∂x ∂x2 ∂x
Using the integration by parts relationship
Z Z udv = vu − vdu (2.7) and the transformations given in Equations 2.6, The EI portion of Equation 2.5 can be rewritten as:
L Z t2 ∂2w ∂w Z L ∂ ∂2w ∂w EI δ − EI δ dx dt (2.8) 2 2 t ∂x ∂x 0 ∂x ∂x ∂x 1 0
Integrating by parts once more: L L Z t2 ∂2w ∂w ∂ ∂2w EI δ − EI δ (w) dt ∂x2 ∂x ∂x ∂x2 t1 0 0 (2.9) Z t2 Z L ∂2 ∂2w + 2 EI 2 δ (w) dx dt t1 0 ∂x ∂x
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
" L # Z t2 ∂θ Z L ∂2θ GJ δθ − GJ δθ dx dt (2.10) 2 t ∂x 0 ∂x 1 0 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 δw and another statement which multiplies δθ. Substituting the kinetic energy (T) from Equation 2.2 into Equation 2.3.
( ) Z t2 1 Z L ∂w2 1 Z L ∂θ2 δ m dx + Iea dx dt (2.11) t1 2 0 ∂t 2 0 ∂t
Now applying the δ operator:
Z t2 1 Z L ∂w ∂w ∂θ ∂θ 2m δ + 2Iea δ dx dt (2.12) t1 2 0 ∂t ∂t ∂t ∂t
∂w ∂θ At this point it is important to note that δ is operating on ∂t and ∂t . In order to reduce this to δw and δθ it is clear that one must integrate by parts with respect to t. Integrating by parts for the δw term yields the following result for the time integral of the virtual change in kinetic energy.
" t2 # Z L ∂w Z t2 Z L ∂2w m δw dx − m δw dx dt (2.13) 2 0 ∂t t1 0 ∂t t1
Similarly, integrating by parts for the δθ term yields the following result for the time integral of the virtual change in kinetic energy:
" t2 # Z L ∂θ Z t2 Z L ∂2θ I δθ dx − I δθ dx dt (2.14) ea ea 2 0 ∂t t1 0 ∂t t1
However Equations 2.13 and 2.14 can be made even simpler by invoking a re- lationship that is commonly used with Hamilton’s Principle. Namely it is assumed
that δw and δθ 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:
Z t2 Z t2 Z L ∂2w Z t2 Z L ∂2θ δT dt = − m 2 δw dx dt − Iea 2 δθ dx dt (2.15) t1 t1 0 ∂t t1 0 ∂t 13 Now that the individual components of the virtual changes in kinetic and potential energies for Hamilton’s principle have been calculated, Equations 2.9, and 2.15 can be substituted into Equation 2.3 to yield the following result.
Z t2 0 = [δT − δV ] dt t1
L L L Z t2 ∂2w ∂w ∂ ∂2w ∂θ = − EI δ − EI δ (w) + GJ δθ dt ∂x2 ∂x ∂x ∂x2 ∂x t1 0 0 0 (2.16) Z t2 Z L ∂2w ∂2 ∂2w + −m 2 − 2 EI 2 δw dx dt t1 0 ∂t ∂x ∂x
Z t2 Z L ∂2θ ∂2θ + −Iea 2 + GJ 2 δθ dx dt t1 0 ∂t ∂x
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 δw.
L L Z t2 ∂2w ∂w ∂ ∂2w EI δ − EI δ (w) dt = 0 (2.17) 2 2 t ∂x ∂x ∂x ∂x 1 0 0
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: ∂2w ∂w EI = 0 or δ = 0 ∂x2 ∂x
and (2.18)
∂ ∂2w EI = 0 or δ (w) = 0 ∂x ∂x2
A similar analysis for the natural boundary conditions for the torsional coordinate θ yields the following boundary conditions at both x = 0 and x = L.
∂θ = 0 or δθ = 0 (2.19) ∂x
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 δw and δθ the fundamental theorem of calculus of variation requires that the following differential equations must be satisfied. ∂2w ∂2 ∂2w − m − EI = 0 (2.20) ∂t2 ∂x2 ∂x2 and ∂2θ ∂2θ − I + GJ = 0 (2.21) ea ∂t2 ∂x2
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
2p m using the characteristic length L and a characteristic time T equal to L EI for q Iea the bending equation and L GJ 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˜ ∂4w˜ − − = 0 (2.22) ∂t˜2 ∂x˜4 and ∂2θ ∂2θ − + = 0 (2.23) ∂t˜2 ∂x˜2
The boundary conditions remain the same except the scaling factors are removed and the boundary conditions are satisfied atx ˜ = 0 andx ˜ = 1.
2.1.4 Bending Separation of Variables
The solution to the homogeneous equation forw ˜(˜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˜)φ(˜x) (2.24)
Substituting Equation 2.24 into the homogeneous equation of motion yields:
∂2 ∂4 q(t˜)φ(˜x) + q(t˜)φ(˜x) = 0 (2.25) ∂t2 ∂x˜4
Evaluating the derivatives and dividing by q(t˜), and φ(˜x)
q¨(t˜) φ0000(˜x) + = 0 (2.26) q(t˜) φ(˜x)
q¨(t˜) φ0000(˜x) This can only be satisfied if both q(t˜) and φ(˜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 λ2 and the equations for q(t˜) and φ(˜x) can be determined.
16 Looking first at the equation for q(t˜) and setting it equal to −λ2 yields:
q¨(t˜) + λ2q(t˜) = 0 (2.27)
This equation can be solved by assuming a solution of the form:
q(t) = A cos(λt˜) + B sin(λt˜) (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 λ’s are the natural frequencies of the system. The next step is to look at the equation for φ(˜x):
φ0000(˜x) − λ2φ(˜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:
2 kn = λ (2.30)
Thus the assumed solution becomes:
φ(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 θ(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.
θ(˜x, t˜) = A(t˜)ψ(˜x) (2.32)
17 Substituting Equation 2.32 into the homogeneous equation of motion yields:
∂2 ∂2 A(t˜)ψ(˜x) − A(t˜)ψ(˜x) = 0 (2.33) ∂t˜2 ∂x˜2
Evaluating and dividing by A(t˜), and ψ(t˜)
A¨(t˜) ψ00(˜x) − = 0 (2.34) A(t˜) ψ(˜x)
A¨(t˜) ψ00(˜x) This can only be satisfied if both A(t˜) and ψ(˜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 λ2 and the equations for A(t˜) and ψ(˜x) can be determined. Looking first at the equation for q(t˜) and setting it equal to −λ2 yields:
A¨(t˜) + λ2A(t˜) = 0 (2.35)
This equation can be solved by assuming a solution of the form:
A(t) = G cos(λt˜) + H sin(λt˜) (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 λ’s are again the natural frequencies of the system. The next step is to look at the equation for ψ(˜x). Setting the equation equal to (−λ2) and rearranging gives:
ψ00(˜x) + λ2ψ(˜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:
2 2 jn = λ (2.38) 18 Thus the assumed solution becomes:
ψ(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:
φ = 0 x˜=0,t˜
∂φ = 0 ∂x˜ x˜=0,t˜ (2.40) 2 ∂ φ 2 = 0 ∂x˜ x˜=1,t˜
3 ∂ φ 3 = 0 ∂x˜ x˜=1,t˜
Figure 2.1 shows the diagram of the clamped-free configuration. Applying the bound- ary conditions atx ˜ = 0 yields: D + F = 0 (2.41) C + E = 0
Applying the boundary conditions atx ˜ = 1 yields:
C sinh kn + D cosh kn − E sin kn − F cos kn = 0 (2.42) C cosh kn + D sinh kn − E cos kn + F sin kn = 0 19 L
x
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 (2.43) C(cosh kn + cos kn) + D(sinh kn − sin kn) = 0
Using Equations 2.43 to solve for kn by setting the determinate of the coefficients C and D equal to zero yields: 1 cos(kn) = − (2.44) cosh(kn)
Figure 2.2 shows the intersection of the two sides of Equation 2.44. The first non-
Figure 2.2: Clamped Free Frequencies
2 2 2 2 dimensional frequency is λ1 = (.597) π and the second frequency is λ2 = (1.49) π .
20 For the n’th frequency where n is larger than two the natural frequency is approxi-
2 2 mately λn ≈ (n − 1/2) π . Furthermore the mode shapes can be written in terms of an arbitrary constant D as: sin(kn) − sinh(kn) φ(˜x) = D (sinh(knx˜) − sin(knx˜)) + (cosh(knx˜) − cos(knx˜)) cos(kn) + cosh(kn) (2.45) The mode shapes are shown in Figure 2.3.
1 0 −1 0 0.2 0.4 0.6 0.8 1 1 0 −1 0 0.2 0.4 0.6 0.8 1 1 0 −1 0 0.2 0.4 0.6 0.8 1 1 0 −1 0 0.2 0.4 0.6 0.8 1 1 0 −1 0 0.2 0.4 0.6 0.8 1
Figure 2.3: Clamped Free Bending Mode Shapes
Torsional Mode Shapes
For the clamped-free configuration the boundary conditions are:
ψ = 0 x˜=0,t˜ (2.46)
∂ψ = 0 ∂x˜ x˜=1,t˜
Applying the boundary conditions atx ˜ = 0 yields:
J = 0 (2.47)
21 Applying the boundary conditions atx ˜ = 1 yields:
cos jn = 0 (2.48)
2n−1 Equation 2.48 has the solution jn = 2 π. Furthermore the torsional mode shapes can be described by:
Ψ(˜x) = I sin jnx˜ (2.49)
2.2.2 Pinned-Free Bending Mode Shapes
For the pinned-free configuration the boundary conditions are:
φ = 0 x˜=0,t˜
2 ∂ φ 2 = 0 ∂x˜ x˜=0,t˜ (2.50) 2 ∂ φ 2 = 0 ∂x˜ x˜=1,t˜
3 ∂ φ 3 = 0 ∂x˜ x˜=1,t˜
Figure 2.4 shows the diagram of the pinned-free configuration. Applying the bound- ary conditions atx ˜ = 0 yields: D + F = 0 (2.51) D − F = 0
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:
φ(˜x) = C sinh knx˜ + E sin knx˜ (2.52)
Applying the boundary conditions atx ˜ = 1 yields:
C sinh kn − E sin kn = 0 (2.53) C cosh kn − E cos kn = 0 22 L
x
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
2 2 shape given as φ1(˜x). The other frequencies all take the form of (n − 3/4) π for the n’th frequency for every n larger than 1. The mode shapes are described by:
φ1 = C1x˜ (2.55) cos kn φn(˜x) = E sinh knx˜ + sin knx˜ cosh kn 23 The mode shapes are shown in Figure 2.6.
1 0 −1 0 0.2 0.4 0.6 0.8 1 1 0 −1 0 0.2 0.4 0.6 0.8 1 1 0 −1 0 0.2 0.4 0.6 0.8 1 1 0 −1 0 0.2 0.4 0.6 0.8 1 1 0 −1 0 0.2 0.4 0.6 0.8 1
Figure 2.6: Pinned Free Mode Shapes
Torsional Mode Shapes
For the pinned-free configuration the boundary conditions are:
ψ = 0 x˜=0,t (2.56)
∂ψ = 0 ∂x x˜=1,t
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:
φ = 0 x˜=0,t˜
∂φ = 0 ∂x˜ x˜=0,t˜ (2.57)
φ = 0 x˜=1,t˜
∂φ = 0 ∂x˜ x˜=1,t˜
Figure 2.7 shows the diagram of the clamped-clamped configuration.
L
x
Figure 2.7: Clamped Clamped Schematic
Applying the boundary conditions atx ˜ = 0 yields: D + F = 0 (2.58) C + E = 0
Applying the boundary conditions atx ˜ = 1 yields:
C sinh kn + D cosh kn + E sin kn + F cos kn = 0 (2.59) C cosh kn + D sinh kn + E cos kn − F sin kn = 0 25 Using Equations 2.58 to simplify Equations 2.59 yields:
C(sinh kn − sin kn) + D(cosh kn − cos kn) = 0 (2.60) C(cosh kn − cos kn) + D(sinh kn + sin kn) = 0
Equations 2.60 can be rearranged to solve for kn:
1 cos(kn) = (2.61) cosh(kn)
Figure 2.8 shows the intersection of the two sides of Equation 2.61. The i’th fre-
Figure 2.8: Clamped Clamped Frequencies
quency from this plot can be written as (i + .5)2π2. Finally solving for C in terms of D and plugging into Equation 2.60 yields the following equation for the mode shapes:
cosh(kn) − cos(kn) φ(˜x) = D − (sinh(knx˜) − sin(knx˜)) + (cosh(knx˜) − cos(knx˜)) sinh(kn) − sin(kn) (2.62) The mode shapes are shown in Figure 2.9.
26 1 0 −1 0 0.2 0.4 0.6 0.8 1 1 0 −1 0 0.2 0.4 0.6 0.8 1 1 0 −1 0 0.2 0.4 0.6 0.8 1 1 0 −1 0 0.2 0.4 0.6 0.8 1 1 0 −1 0 0.2 0.4 0.6 0.8 1
Figure 2.9: Clamped Clamped Mode Shapes
Torsional Mode Shapes
For the clamped-clamped configuration the boundary conditions are:
ψ = 0 x˜=0,t˜ (2.63)
ψ = 0 x˜=1,t˜
Applying the boundary conditions atx ˜ = 0 yields:
J = 0 (2.64)
Applying the boundary conditions atx ˜ = 1 yields:
sin jn = 0 (2.65)
Equation 2.65 has the solution jn = nπ. Therefore the torsional mode shapes are described by:
Ψ(˜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 ∂ φ 2 = 0 ∂x˜ x˜=0,t˜
3 ∂ φ 3 = 0 ∂x˜ x˜=0,t˜ (2.67) 2 ∂ φ 2 = 0 ∂x˜ x˜=1,t˜
3 ∂ φ 3 = 0 ∂x˜ x˜=1,t˜
Applying the boundary conditions atx ˜ = 0 yields:
D − F = 0 (2.68) C − E = 0
Applying the boundary conditions atx ˜ = 1 yields:
C sinh kn − D cosh kn + E sin kn − F cos kn = 0 (2.69) C cosh kn + D sinh kn + E cos kn − F sin kn = 0
Using Equations 2.68 to simplify Equations 2.69 yields:
C(sinh kn − sin kn) + D(cosh kn − cos kn) = 0 (2.70) C(cosh kn − cos kn) + D(sinh kn + sin kn) = 0
Equations 2.70 can be rearranged to solve for kn:
1 cos(kn) = (2.71) cosh(kn)
Figure 2.10 shows the intersection of the two sides of Equation 2.71. There is an
28 1 X: 0 0.8 Y: 1
0.6
0.4
0.2 X: 7.853 X: 14.14 Y: 0.0007773 Y: −0.006977 0 X: 4.685 X: 10.99 Y: 0.01847 −0.2 Y: 3.371e−005
−0.4
−0.6
−0.8
−1 0 5 10 15 k n
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:
1 φ(˜x) = C + C ( − x˜) (2.72) 1 2 2
where C1 and C2 are arbitrary constants. The i’th frequency from this plot can be written as (i − .5)2π2. Finally solving for C in terms of D and plugging into Equation 2.70 yields the following equation for the mode shapes:
cosh(kn) − cos(kn) φ(˜x) = D − (sinh(knx˜) + sin(knx˜)) + (cosh(knx˜) + cos(knx˜)) sinh(kn) − sin(kn) (2.73) The mode shapes are shown in Figure 2.11.
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 1 0 −1 0 0.2 0.4 0.6 0.8 1 1 0 −1 0 0.2 0.4 0.6 0.8 1 1 0 −1 0 0.2 0.4 0.6 0.8 1 1 0 −1 0 0.2 0.4 0.6 0.8 1 1 0 −1 0 0.2 0.4 0.6 0.8 1
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-Free Number Frequency Frequency 1 0 (.517)2π2 3 2 2 2 2 2 (2 − 4 ) π (1.49) π 3 2 2 1 2 2 3 (3 − 4 ) π (3 − 2 ) π ...... 3 2 2 1 2 2 n (n − 4 ) π (n − 2 ) π
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 2 2 ) ) x x (˜ (˜ 1 0 2 1 φ φ
0 −2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
2 2
1 1 ) ) x x (˜ 0 (˜ 0 3 4 φ φ −1 −1
−2 −2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x˜ x˜ Figure 2.12: The solid line corresponds to the clamped-free mode shapes, dashed ˜ line to the pined-free mode shapes, and the dotted line to the Kα = 1000 mode shapes. 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 atx ˜ = 0. Here the torsional force applied by the spring modeled by hook’s law must be identically equal to bending moment. Mathematically this can be written as:
∂2w˜(0, t) ∂w˜(0, t) = K˜ (2.74) ∂x˜2 α ∂x˜
˜ where Kα = Kα(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 C 0 ˜ ˜ −Kα 1 −Kα −1 D 0 √ √ √ √ = (2.75) cosh√λ sinh √λ − cos√λ sin √λ E 0 sinh λ cosh λ − sin λ − cos λ F 0
The set of four coupled equations captured in Equation 2.75 can be used to solve for the natural frequencies λ by determining the values of λ 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 λ’s which satisfy the determinant equation. A common choice for the constant is one that normalizes the generalized mass to one.
70
60
50
40
[radians] 30 n ω 20
10
0 −2 0 2 10 10 10
K˜ α
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-free natural 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 (Kα). First, it is reassuring to see that the frequencies do converge ˜ to the clamped-free frequencies at high values of Kα. 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 Kα 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 Kα 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. X w(x, y, t) = qn(t)Ψjk(x, y) (2.76) n
In Equation 2.76, the n’th structural mode is labeled Ψjk because the mode shape can be broken in to two components.
Ψjk(x, y) = φj(x)θk(y) (2.77)
The energies of the system need to be derived and then placed into Lagrange’s equations. The energies for a plate in tension can be written as[9]:
1 Z Lx Z Ly ∂w2 T = ρ · h dx dy (2.78) 2 0 0 ∂t " 1 Z Lx Z Ly ∂w2 ∂w2 ∂2w2 ∂2w2 V = Tx + Ty + Dx 2 + Dy 2 2 0 0 ∂x ∂y ∂x ∂y (2.79) # ∂4w ∂2w 2 +2D0 + 4D dx dy ∂x2∂y2 xy ∂x∂y
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:
1 Z Lx Z Ly X X T = ρh q˙ (t)q ˙ (t)Ψ (x, y)Ψ (x, y) dx dy (2.80) 2 n m jk pq 0 0 n m where Ψjk is the n’th mode shape and Ψpq is the m’th 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 Ψ~ = Ψ~ T ~q (2.81)
34 Using this relationship, Equation 2.80 can be rewritten as:
T = ~q˙ T M¯ ~q˙ (2.82)
Where the mass matrix M is equal to:
1 Z Lx Z Ly M¯ = ρh Ψ~ · Ψ~ T dx dy (2.83) 2 0 0
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 Ψ 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 ∂ = ~q T Ψ~ (2.84) ∂x ∂x
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 Ψ~ using the following notation: Ψ~ = φθ~ (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 = ~q T φ~0θ (2.86) ∂x
35 Plugging in this relationship into the tension in the x-direction portion of the potential energy expression yields:
L L 1 Z x Z y T T ~0 ~0 Vtx = ~q Txφ θ · φ θ dy dx~q (2.87) 2 0 0
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.
L L 1 Z x Z y T ¯ ~0 ~0 Ktx = Txφ θ · φ θ dy dx (2.88) 2 0 0
Using a similar method, the y direction tension term can be written as:
T ¯ Vty = ~q Kty~q (2.89)
¯ where Kty is the y tension stiffness matrix which can be written as:
L L 1 Z x Z y T ¯ ~ 0 ~ 0 Kty = Tyφθ · φθ dy dx (2.90) 2 0 0
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.
L L 1 Z x Z y T ¯ ~ 00 ~ 00 KDx = Dx φθ · φθ dy dx (2.91) 2 0 0
L L 1 Z x Z y T ¯ ~00 ~00 KDx = Dy φ θ · φ θ dy dx (2.92) 2 0 0 ¯ 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:
L L Z x Z y T ¯ 0 ~00 ~ 00 KD0 = D φ θ · φθ dy dx (2.93) 0 0
L L Z x Z y T ¯ ~0 0 ~0 0 KDxy = 2Dxy φ θ · φ θ dy dx (2.94) 0 0
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 = (~q T )M¯ (~q) (2.95) ∂t ∂t
T ¯ ¯ ¯ ¯ ¯ ¯ V = ~q Ktx + Kty + KDx + KDy + KD0 + KDxy ~q (2.96)
With the potential and kinetic energies defined they can be plugged into La- grange’s equation to yield the equations of motion. The familiar form of Lagrange’s equation is: ∂L d ∂L − = 0 (2.97) ∂qn dt ∂q˙n 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.
2 ∂ − M¯ ~q + K¯ + K¯ + K¯ + K¯ + K¯ 0 + K¯ ~q = 0 (2.98) ∂t2 tx ty Dx Dy D Dxy
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) =qe ¯ iωt is assumed. Plugging this solution into equation 24 yields:
−ω2M¯ + 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 Value 3 Density ρs 1230 kg/m Young’s Modulus E 18.4 MPa Poisson’s Ratio ν .5 Thickness h 1.74 mm Chord 152.4 mm Span 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 Normal to Flow Mode 1 Normal to Flow Mode 2 50 40 60 30 40 20 20 10
Natural Frequency [Hz] 0 0 1 2 3 1 2 3
Normal to Flow Mode 3 Normal to Flow Mode 4 100 150
100 50 50
Natural Frequency [Hz] 0 0 1 2 3 1 2 3 Streamwise Mode Streamwise Mode
1.689 Hz (1,1) 4.929 Hz (1,2) 10.524 Hz (2,1) 16.374 Hz (2,2)
2 2 2 2
1 1 1 1
0 0 0 0
−1 −1 −1 −1
−2 −2 −2 −2 0.1 0.1 0.1 0.1 0.05 0.1 0.05 0.1 0.05 0.1 0.05 0.1 0 0 0 0 0 0 0 0 Figure 2.14: Configuration 1 Natural Frequencies and Mode Shapes
Normal to Flow Mode 1 Normal to Flow Mode 2 25 40
20 30 15 20 10 5 10
Natural Frequency [Hz] 0 0 1 2 3 1 2 3
Normal to Flow Mode 3 Normal to Flow Mode 4 150 60 100 40 50 20
Natural Frequency [Hz] 0 0 1 2 3 1 2 3 Streamwise Mode Streamwise Mode
3.004 Hz (1,1) 5.580 Hz (2,1) 14.587 Hz (3,1) 18.709 Hz (1,2)
2 2 2 2
1 1 1 1
0 0 0 0
−1 −1 −1 −1
−2 −2 −2 −2 0.1 0.1 0.1 0.1 0.05 0.1 0.05 0.1 0.05 0.1 0.05 0.1 0 0 0 0 0 0 0 0 Figure 2.15: Configuration 2 Natural Frequencies and Mode Shapes
40 Normal to Flow Mode 1 Normal to Flow Mode 2
60 60
40 40
20 20
Natural Frequency [Hz] 0 0 1 2 3 1 2 3
Normal to Flow Mode 3 Normal to Flow Mode 4 100 200
150
50 100
50
Natural Frequency [Hz] 0 0 1 2 3 1 2 3 Streamwise Mode Streamwise Mode
10.808 Hz (1,1) 13.329 Hz (1,2) 28.245 Hz (1,3) 29.627 Hz (2,1)
2 2 2 2
1 1 1 1
0 0 0 0
−1 −1 −1 −1
−2 −2 −2 −2 0.1 0.1 0.1 0.1 0.05 0.1 0.05 0.1 0.05 0.1 0.05 0.1 0 0 0 0 0 0 0 0 Figure 2.16: Configuration 3 Natural Frequencies and Mode Shapes
Normal to Flow Mode 1 Normal to Flow Mode 2 30 60 20 40 10 20
Natural Frequency [Hz] 0 0 1 2 3 1 2 3
Normal to Flow Mode 3 Normal to Flow Mode 4 200
100 150
100 50 50
Natural Frequency [Hz] 0 0 1 2 3 1 2 3 Streamwise Mode Streamwise Mode
19.215 Hz (1,1) 20.756 Hz (2,1) 28.303 Hz (3,1) 52.669 Hz (1,2)
2 2 2 2
1 1 1 1
0 0 0 0
−1 −1 −1 −1
−2 −2 −2 −2 0.1 0.1 0.1 0.1 0.05 0.1 0.05 0.1 0.05 0.1 0.05 0.1 0 0 0 0 0 0 0 0 Figure 2.17: Configuration 4 Natural Frequencies and Mode Shapes
41 Normal to Flow Mode 1 Normal to Flow Mode 2 60 60 40 40 20 20
Natural Frequency [Hz] 0 0 1 2 3 1 2 3
Normal to Flow Mode 3 Normal to Flow Mode 4 200
100 150
100 50 50
Natural Frequency [Hz] 0 0 1 2 3 1 2 3 Streamwise Mode Streamwise Mode
12.078 Hz (1,1) 26.358 Hz (1,2) 31.049 Hz (2,1) 45.597 Hz (2,2)
2 2 2 2
1 1 1 1
0 0 0 0
−1 −1 −1 −1
−2 −2 −2 −2 0.1 0.1 0.1 0.1 0.05 0.1 0.05 0.1 0.05 0.1 0.05 0.1 0 0 0 0 0 0 0 0 Figure 2.18: Configuration 5 Natural Frequencies and Mode Shapes
Normal to Flow Mode 1 Normal to Flow Mode 2 60 80
60 40 40 20 20
Natural Frequency [Hz] 0 0 1 2 3 1 2 3
Normal to Flow Mode 3 Normal to Flow Mode 4 150 200
150 100 100 50 50
Natural Frequency [Hz] 0 0 1 2 3 1 2 3 Streamwise Mode Streamwise Mode
19.816 Hz (1,1) 26.611 Hz (2,1) 43.183 Hz (3,1) 53.443 Hz (1,2)
2 2 2 2
1 1 1 1
0 0 0 0
−1 −1 −1 −1
−2 −2 −2 −2 0.1 0.1 0.1 0.1 0.05 0.1 0.05 0.1 0.05 0.1 0.05 0.1 0 0 0 0 0 0 0 0 Figure 2.19: Configuration 6 Natural Frequencies and Mode Shapes
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.
200
180
160
140
120 ]
100 Hz [
ω 80
60
40
20
0 0.05 0.1 0.15 0.2 0.25 0.3 Streamwise Dimension Figure 2.20: Natural frequency evolution as the streamwise chord is varied for Configuration 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 lines correspond to the first two beam natural frequencies for the normal to flow direction mode 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.
300
250
200 ]
150 Hz [ ω
100
50
0 0 20 40 60 80 100 120 140 160 180 200 Tension in the Normal to Flow Direction [N/m] Figure 2.21: Natural frequency evolution as the normal to the flow tension is varied for Configuration 2. Solid lines correspond to first mode in the normal to flow direction, 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 Hamilton’s principle for the beam equations or Lagrange’s 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 i’th element in the generalized force vector is determined by taking the real force applied to the system multiplying it by the i’th generalized mode shape and integrating the result over the plate. This is a classical result that is found throughout the literature.
Z Qi = F φi(x, y) dA (2.102) ∂A
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 i’th equation for the relationship defined in Equation 2.102. This relationship can be
45 expressed as
N X ¯ ¯ ~ Mj,i · q¨i + Kj,i · qi = Qi (2.103) j=1
As is common for transforming an equation into state space, it is necessary to define
two state variables y1 = qi and y2 =q ˙i and discretized the variables as follows:
yn+1 − yn y˙ n+1/2 = 2 2 (2.104a) 2 ∆t yn+1 + yn yn+1/2 = 1 1 (2.104b) 1 2 yn+1 + yn yn+1 − yn y n+1/2 = 2 2 = 1 1 =y ˙ n+1/2 (2.104c) 2 2 ∆t 1
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+1 − yn yn+1 + yn 1 1 − 2 2 = 0 (2.105) ∆t 2
Furthermore, the definitions in Equation 2.104 can be used to re-write Equation 2.103 as:
N " ˙ n+1 ˙ n ! n+1 n # X ~q − ~q ~q + ~q n+1/2 M¯ i i + K¯ i i = Q~ (2.106) j,i ∆t j,i 2 i j=1
46 3
Aerodynamic 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 useful to define some general properties of the vortex filament. A vortex filament is a tube of flow that has a vorticity, which, in the limit as the tube diameter becomes small, contains a fixed circulation Γ. The vortex filament represents a fundamental solution to Laplace’s Equation, the governing equation for a constant density inviscid, irrotational flow. The vortex tubes must also satisfy the Helmholtz Vortex Theorems. First the circu- lation around a vortex filament is constant all along the tube. Mathematically this RR can be stated as c.s ~ω · ~nds = 0 where ~ω is the vorticity. Second, a vortex tube can never end in a fluid, but must close on itself, end at a boundary, or go to infinity. And third a fluid element that is initially irrotational remains irrotational.
48 The velocity induced by a vortex filament at any position in the fluid is governed by the Biot-Savart Law. This vector equation dictates both the direction and the strength of the induced velocity from a vortex filament with circulation strength Γ. Specifically the velocity at point t is: Z Γ ~rt × d~s Vt = − 3 (3.1) 4π c ~rt
It is computationally efficient to create an algebraic kernel function which relates the velocity field of a specific vortex element at other positions in the flow. To do this one must define a carefully constructed vortex filament, such as the horseshoe vortex, that satisfies the three Helmholtz Vortex Theorems, and analytically calculate the induced velocity at a point (x, y, z). The kernel function for common vortex elements (rings, horseshoe vortices, infinite vortex filaments) can be found in the aerodynamics literature. The discussion of the aerodynamic theory begins with a discussion of the mesh on the elastic structure and in the wake. In this derivation the simple square mesh on a rectangular plate as shown in Figure 3.2 is used. The information shown in the figure can be stated in words with the following set of definitions that must be made before proceeding. First it is assumed that the streamwise dimension all of the elements is uniformly ∆x. Second ζ is defined as the index of the row and η is defined as the index of the column in the mesh. Third, the horseshoe vortex Γ(ζ, η)
1 passes through the 4 chord of the element in the ζ’th row and η’th column. Finally, 3 the collocation point (xζ , yη) is at the 4 chord of the element in the ζ’th row and η’th column. It is at this collocation point on the panel that we require that no induced velocity through the panel. For this case the following numbering convention is used: the element in the ζ’th row and η’th column is defined to be the ((ζ − 1) ∗ Sc + η)’th element. This allows the transformation of an η by ζ matrix that contains the strength of the individual
49 Wake Span Plate Chord (Ws Elements) (Sc Elements)
Plate Span (Ss Elements)
Wake Chord i'th Element (Wc Elements)
Colocation Δy X point
Δx
Γi
Figure 3.2: Expanded Schematic of Vortex Lattice Mesh
Γ(ζ, η)’s to a column vector ~Γ.
Solving for the circulation (~Γ) in this method is achieved by solving a matrix equation that relates the circulation at time n+1 to the known circulation at time n. In order to do this a set of equations equal to the number of unknown circulation strengths (St+Wt) must be derived. The set of governing equations for the vortex lattice method can be segmented into four types of equations that govern the circu- lation on a given element. Furthermore, for the square lattice that is constructed for this problem the equations for a given column are the same for all of the elements in that column. For more complicated structures or to model additional effects the same types of fundamental equations are modified as needed. The four types of equations are:
• A11 → Over the structure (St equations)
This set of equations relates the downwash caused by horseshoe vortex elements to the motion of the elastic structure at the collocation points.
• W11 → First column of the wake (Wc equations).
50 This set of equations relates the circulation in this first column of the wake to the change in circulation on the structure.
• W12 → Second to second to last column in the wake ((Wt − 2 · Wc) equations)
For this set of equations by specifying a time step shown in Equation 3.2 the circulation for a given element at a given time step can be assumed to be convected circulation of the element in the same row, but previous column at the previous time step. 1 ∆t = ∆x (3.2) U∞
L Equation 3.2 also sets the scaling factor of T for the velocity. Plugging in the scaling factors determined for the non-dimensional structural analysis, U = 1 ˜ √ m U and the non-dimensional, scale invariant time step definition given in L EI Equation 3.3. 1 ∆t˜= ∆˜x (3.3) ˜ U∞
• W13 → Last column of the wake (Wc equations)
Because there are a finite number of elements in the computational model the last column includes the convection from the previous column at the previ- ous time step plus an added relaxation factor applied to the circulation at the given element from the previous time step.
First: Explore circulation over the wing (A1)
The circulation strength (Γ) over the panel elements is constrained by a boundary
3 condition at the 4 chord of the panels on the structure. As discussed previously, the horseshoe vortex creates a resultant velocity everywhere in the fluid which is
3 described by the kernel function. At the 4 chord of the panels it is required that
51 the sum of the downwash from all the circulations in the vortex mesh be identically equal to the vertical velocity of the panel at the given time, in the reference frame of the fluid. For the ζ’th row and η’th column this is described as:
Sc+Wc Ss+Ws n+1 X X n+1 Vd (ζ, η) = K(ζ,η),(i,j)Γ(i,j) (3.4) i=1 j=1
where K(ζ,η),(i,j) is the kernel function which transforms the downwash caused by the horseshoe vortex at (i, j) at a point (ζ, η) and whose form can be found in many aerodynamic textbooks. The kernel function for horseshoe vorticies can be found in Katz and Plotkin [21]. In this document both dimensional and non-dimensional analysis is conducted. For the non-dimensional case Equation 3.4 also makes it clear that the normalized circulation Γ must normalized by the velocity scaling factor times the length scale. Use of the (˜) notation will denote a non-dimensional parameter or result. At this point it is necessary to use the numbering convention defined earlier to write the governing equation for the normalized circulation strength on the panel.
Let k = ((ζ − 1) · Sc + η)
l = ((i − 1) · Sc + j) (3.5) St+Wt ˜ n+1 X ˜ n+1 Vd (k) = Kk,lΓ(l) l=1
where k goes from 1 to the number of elements on the panel St and l goes from 1 to the total number of elements in the lattice of horseshoe vortices(St + Wt). The matrix form of this equation for all of the elements on the structure can be
52 written as:
˜ Γ1 . . K K ············ K V (1) 1,1 1,2 1,St+Wt . d . . . . . . . . . = . . . . . . . . KSt,1 ··············· KSt,St+Wt Vd(St) . . ˜ ΓSt+Wt
Second: Explore circulation in first column of wake circulation (W1)
The horseshoe elements in the wake left behind the elastic member are treated differ- ently from the elements on the elastic structure. In the wake, there is no boundary condition to govern the strength of the circulation. Instead, in the wake the equa- tions account for the horseshoe elements as they move with the fluid. For the first column in the streamwise direction following the elastic structure the governing equa- tion states that the circulation at time step n + 1 is the change in the strength the circulation in the same row as the given element on the elastic structure between time step n and n + 1. Mathematically this can be written as shown in Equation 3.6 given in the global index notation with j = ((i − 1) · Ss + η).
Sc ˜n+1 X ˜n+1 ˜n Γ (St + η) = − [Γ (j) − Γ (j)] (3.6) j=1
After re-arranging and setting equal to zero this can be written in matrix form as
¯ ~˜n+1 ¯ ~˜n W11Γ + W12Γ = 0 (3.7)
Note, for these relationships the non-dimensional and dimensional forms are the same.
53 Third: Explore circulation in second to second to last column of wake (W2)
The calculation of the vorticity in the wake uses the understanding that the vortex is convected through the wake at the free stream velocity U. Because the time step is defined to be the free stream velocity divided by the chord length of the mesh the vorticity of an element in the ζ’th row is shed a distance U∆t = ∆x in the process of one time step. At this point it is important to note that setting a ∆t that is a given fraction of the fastest structural frequency of structural oscillation effectively sets the streamwise dimension of the mesh. Mathematically the convection relationship can be written as: ˜n+1 ˜n Γ (i) = Γ (i − Ws) (3.8)
˜n+1 ˜n Γ (i) − Γ (i − Ws) = 0 (3.9)
˜n+1 ˜n This creates two matrices, W2,1 for the Γ term and W2,2 for the Γ , which look like the identity matrix and the negative of an offset identity respectively.
Fourth: Explore circulation in the last column of wake (W3)
While the hypothetical wake for the vortex lattice method extends indefinitely, in order to construct an efficient model, a finite wake size is used. To deal with the finite nature of the modeled wake a relaxation factor is introduced to capture the effect of the horseshoe vortices that leave the wake. This relaxation factor is often denoted as α. For the last column in the wake the equation is much the same as the rest of the wake with the addition of the relaxation term. Mathematically this is written as ˜n+1 ˜n ˜n Γ (i) = Γ (i − Ws) + αΓ (i) (3.10)
˜n+1 ˜n ˜n Γ (i) − Γ (i − Ws) − αΓ (i) = 0 (3.11)
˜n+1 ˜n This again creates two matrices, W3,1 for the Γ term and W3,2 for the Γ , which look like the identity matrix and the negative of an offset identity with alpha on the
54 diagonal respectively. Now that the set of relationships have been defined the final step is to create a
final set of matrix equations that represents a set of St + Wt equations which govern the strength of the Γ’s.˜ Using the matrix identifiers defined earlier this can be written as: ~˜n+1 ¯~˜n ~n+1 σ¯Γ + ξΓ = Vd (3.12)
Where A11 0 σ¯ = , ξ¯ = W11 W12 W21 W22 W31 W32
~˜ Finally the downwash vector Vd is zero everywhere except for on the panel where it is equal to the vertical velocity of the elastic member. This represents the coupling term for the aerodynamics and will require determining the convective derivative of the panel velocity.
3.2 Vortex Lattice Aeroelastic Model
Now that the non-dimensional governing aerodynamic equations has been defined it is possible to construct the non-dimensional aeroelastic system. This requires a discussion of the way that the governing equations are coupled. For this method the equations are coupled in two places: the downwash state relationship in the aerodynamics and the generalized force term in the elastic equation of motion.
55 3.2.1 Downwash State Relations
The no flow through boundary condition on the structure allows the VLM aerody- namic model to interact with the structural model. This downwash relationship is governed by the actual movement of the elastic plate. Further complicating the sys- tem is the fact that the coordinate system for the downwash is the coordinate system of the moving vortex, while the global coordinate system is fixed on the plate. The relationship between the moving flow and the fixed plate requires specifying both the reference frame and the quantity that is required for the downwash.
dw˜ V˜ = (3.13) d dt˜ plate moving in the reference frame of flow
Note, this equation would be the same in dimensional form. In order to find this relationship one has to note that the position of the plate (w ˜(˜x, t˜)) is a function of both the position (˜x) and time (t˜). In order to compute the velocity one has to apply the chain rule to the position
∂w˜ ∂w˜ dw˜ = dx˜ + dt˜ (3.14) ∂x˜ ∂t˜ fluid plate plate
The downwash for the vortex lattice derivation is the time rate of change of the vertical locationw ˜ so it is necessary to divide both sides of the equation by dt˜
dw˜ ∂w˜ dx˜ ∂w˜ dt˜ = + (3.15) dt˜ ∂x˜ dt˜ ∂t˜ dt˜ fluid plate plate
Considering the above equation two simplifications can be applied directly. The first
dt˜ dx˜ is to note that dt˜ is identically equal to 1 for all times. Second, the dt˜ is a statement of how fast the plate is moving in the streamwise direction in the reference frame of
dx˜ the fluid flow. From this statement it follows that the dt˜ is equal to the free stream 56 ˜ ˜ velocity of the fluid U∞. Using these relations the downwash Vd is equal to:
dw˜ ∂w˜ ∂w˜ V˜ = = U˜ + (3.16) d dt˜ ∂x˜ ∞ ∂t˜ fluid plate plate
At this point there is a continuous definition (in time and space) for the downwash velocity. However, because the VLM is discrete in both space in time, it is necessary to derive a representation of the downwash at the discrete collocation points and finite time steps. Invoking the separation of variables used to solve the homogeneous P equation as discussed earlier (w ˜(˜x, t˜) = φi(˜x)qi(t˜)) and examining the k’th term the time and space derivatives can be written as:
w˜˙ k(˜x, t˜) = φk(˜x)q ˙k(t˜) (3.17a)
0 ˜ 0 ˜ w˜k(˜x, t) = φk(˜x)qk(t) (3.17b)
Substituting these definitions into Equation 3.16 allows one to write
˜ ˜ 0 ˜ ˜ Vd,k(˜x) = U∞φk(˜x)qk(t) +q ˙k(t)φk(˜x) (3.18)
˜ where Vd,k(˜x) is the downwash due to the motion of the k’th mode at positionx ˜. This relationship can now be written in a matrix form that is useful for solving the system. ˜ n+1 ˜ n+1 ˜ 0 qk(t) Vd,k (˜x) = U∞φk(˜x) φk(˜x) · q˙k(t˜)
There is one of these equations for every element on the structure and for every eigenmode. The complete matrix equation can be written as:
˜~ n+1 ¯~ Vd = βΨ (3.19)
57 where
q1 q˙ 1 β¯(:, i) = ˜ 0 0 , Ψ~ = . (3.20) U∞φk(˜xi) φ1(˜xi) ······ U∞φk(˜xi) φn(˜xi) . qn q˙n
wherex ˜i is thex ˜ location location of the collocation point of the i’th panel.
3.2.2 Non-dimensional Generalized Force
The final coupling equation is the generalized force caused by the aerodynamics. In order to calculate the generalized force a transformation from the circulation to the induced pressure must be defined. An application of Bernoulli’s equation yields the following pressure field due to a continuous circulation field γ(x, t):
∂ Z x p(x, t) = ρ U∞γ(x, y) + γ(xi, t)dxi (3.21) ∂t 0
However, Equation 3.21 is not particularly useful because the VLM requires a discrete description. Discretising the previous equation in terms of Γ(i) for each of the i elements on the panel gives:.
ceil(i/Ss) ∂ X P (i)n = ρ∆y U Γn(i) + Γn(k · S + (i − ceil(i/S )))∆x (3.22) ∞ ∂t s s k=0
Where P is now the pressure force per unit length caused by the aerodynamics. With this definition it is clear that an approximation of the time derivative must be made to allow the equation to become completely discrete. Using the limit definition of a
∂ derivative ∂t Γ(k) can be written as:
∂ Γ(k)n+1 − Γ(k)n Γ(k) = (3.23) ∂t ∆t 58 This approximation is a good approximation for the time derivative centered on the time step n + 1/2. Therefore it is common to define the pressure at the half step using the fact that for a value only defined at discrete time steps the value at the half time step is the average of the values at time n and time n + 1.Using the relationship
in Equation 3.23 and the simplification that c(k) = k ·Ss +(i−ceil(i/Ss)) and noting
∆x that ∆t is equal to U∞, the equation written as:
P (i)(n+1/2) = ceil(i/Ss) (3.24) 1 X ρ∆yU (Γn(i) + Γn+1(i)) + (Γn+1(c(k)) − Γn(c(k)) ∞ 2 k=0
The dimensional form of the pressure force written above in used for the dimen- sional aeroelastic analysis. Before moving on it is important to discuss the non- dimensionalization of Equation 3.24. From the normalizing of the elastic equations of motion previously discussed in the torsional spring section, the non-dimensional EI ˜ pressure per unit length is given by p(x, t) = L2 p˜(˜x, t). Plugging this, plus the other normalizing factors into Equation 3.24 yields two non-dimensional parameters that define this aeroelastic system. The first one is the aspect ratio of the system H∗ which is equal to the normal to flow direction dimension divided by the streamwise dimension. The second is the mass ratio µ which is defined as a ratio of the mass
of the air to the mass of the beam, specifically, µ = ρaL . Using these definitions the ρsh non-dimensional pressure equation becomes:
P˜(i)(n+1/2) = ceil(i/Ss) (3.25) µ 1 X ∆˜yU˜ (Γ˜n(i) + Γ˜n+1(i)) + (Γ˜n+1(c(k)) − Γ˜n(c(k))) H∗ ∞ 2 k=0
59 3.2.3 Governing Aeroelastic Matrix Equations
Finally the pressure defined in Equation 3.25 is used in the governing aeroelastic equations through the generalized force terms Q which are written using a sum in place of the integral in the original definition:
St X ˜ Qn = qn(t) P (i)φn(i) (3.26) i=1
y Top of Rigid Airfoil Wind Tunnel
Flat Plate Vortex Wake Horshoe Vortex Air Flow Span
X
Plate Chord
Bottom of Airfoil Wind Tunnel Chord Figure 3.3: Aeroelastic Simulation Model
Equation 3.26 is then plugged into the matrix structural equation of motion, Equation 2.106 and then combined with the time discretized equations for the aero- dynamics to yield a matrix equation of the form given in Equation 3.27.
κ¯ · Θn+1 + Ω¯Θn = 0 (3.27)
In this caseκ ¯ and Ω¯ are large sparse matrices containing sectors that are described
60 in terms of previously defined sets of equations.
σ¯ β¯ ξ¯ 0 Γ˜ κ¯ = , Ω¯ = , Θ = (3.28) ¯ ¯ ¯ ¯ C1 D1 C2 D2 y
σ¯ and ξ¯ contain the aerodynamic terms, β¯ contains the downwash relationships, D¯1 and D¯2 contain the elastic terms and C¯1 and C¯2 contain the generalized force relationships.
3.3 Code Development
The theory established in the previous sections is incorporated into a code used to produce vortex lattice based aeroelastic simulations. The capability of the code has grown from a simple time history based analysis of a dimensional cantilevered plate with a user observation of the result to deduce the frequency and damping to a fully automated system that can run multiple types of simulations and analysis for frequency and flutter velocity. Specifically the system can be analyzed in either the frequency or the time domain. For both simulations the frequencies and damping for the system are determined automatically. Furthermore analysis can either be run in a velocity sweep method to get a clean response evolution of the system, or an intelligent flutter velocity discovery method when the flutter boundary as a function of a structural or aerodynamic parameter is required. The code is written in Matlab and includes a text based user interface. Depending on the user choices, the code then creates the appropriate data storage structure and runs the analysis. The code is structured in a way that common tasks such as building the aerodynamic and elastic matrices are in self-contained modules which can be called by different types of analysis.
61 3.3.1 Matrix Definition
After the text based interface determines the configuration and type of analysis that will be run, the parameters that are required to create the aeroelastic vortex lattice matrices are generated. At this point a matrix definition code is called. Using the defined parameters and relations established in the theory, the matrix definition subroutine generates theκ ¯ and Ω¯ matrices. This setup is extremely useful for expanding the vortex lattice analysis module to include different configurations. For example, instead of using the analytic mode shapes which are derived in the structural theory derivation section, ANSYS finite element modes can be used without changing the aerodynamic matrix definition code. This is done by modifying the position at a given (x,y) coordinate for a given mode to return the position from an interpolation of the ANSYS result whenever called.
3.3.2 Flutter Speed and Eigenvalue Determination
One of the elegant aspects of the vortex lattice method is that for any similar geom- etry and aerodynamics the motion is governed by Equation 3.27. Because of this, once theκ ¯ and Ω¯ matrices have been defined the analysis methods remain the same regardless of the configuration that is being simulated. Regardless of how the ma- trix equation is built the analysis methods presented proceed in a common manner. Because the system defined is strictly linear, both an eigenvalue solution and an analysis of the time history simulation will yield the same aeroelastic results.
Time History: Frequency
The first analysis that is done is based on the time history method. A time history of the solution can be constructed by time stepping Equation 3.27 from an arbitrary initial condition. The time history can then be analyzed to determine the system
62 damping and frequency for a specific flow velocity. Figure 3.4 shows a typical time history result for a cantilevered beam that is slightly above its flutter velocity.
Figure 3.4: Typical Near Flutter Time History
The frequency of oscillation of the system is determined from a fast Fourier transform (FFT) of the time history. Figure 3.5 shows the FFTs of the individual mode time histories. These FFTs reveal the relative strength of the modes. For example it is clear that the second structural mode drives the aeroelastic instability for this system. The minimal contribution of the 3rd and 4th mode also suggestions that the number of structural modes used to represent this system is adequate. It is also clear at this above flutter speed that the peak frequency for all of the modes has been driven to the flutter frequency of system and the individual modal frequency content has been lost.
63 Figure 3.5: Near Flutter Time History Modal FFT
Time history: Damping
The time history is also used to determine the system damping. This is an important parameter because it indicates when the system goes from a decaying oscillation (positive damping) to a growing oscillation (negative damping). The damping for the system is estimated by first making the assumption that the solution has an oscillatory behavior:
q(t) = eλr+iλi (3.29)
This interpretation leads to the conclusion that the time history is the product of an exponential and oscillatory solutions. To isolate the exponential portion of the solution to discover λr, it is convenient to look at the peak to peak decay near the
λrt end of the time history. Looking at two adjacent peaks and defining qr(t) = e and
λr(t+∆t) qr(t + ∆t) = e and then dividing qr(t + ∆t) by qr(t) yields:
q (t + ∆t) r = eλr∆t (3.30) qr(t)
64 Figure 3.6: Near Flutter Time History Modal Damping
This clearly gives rise to the definition of the damping of the system as:
ln [q (t + ∆t)] − ln [q (t)] λ = r r (3.31) r ∆t
This definition allows the damping (−λr) of the system to be determined by looking at the peak to peak slope of the natural log of the time history. Figure 3.6 includes the natural log of the time history with the slope of the peaks drawn on for each of the modes. It is clear from this plot that the system is above its flutter speed because the second mode has a positive slope which corresponds with a growing exponential and negative damping. This figure also shows that the cleanest way to determine the system damping is to look at the natural log of the time history of the mode which is driving the flutter. Because this signal is growing the fastest it will
65 be the dominating factor in the system response.
Frequency Domain Analysis
For the entirely linear system that is analyzed, all of the information that is gleaned from the time history analysis can be determined from an eigenanalysis without time stepping the solution. The eigenanalysis for this system is conducted on Equation 3.27 assuming a solution for the circulation and the state variables of:
Θ = Θ¯ eλt (3.32)
Substituting this relationship into Equation 3.27 yields:
eλ∆tκ¯ + Ω¯ Θ¯ = 0 (3.33)
It is clear that Equation 3.33 is in the form of a generalized eigenvalue problem once one defines Λ = exp[λ∆t]. As before, the real and imaginary parts of the eigen- values are the values that will determine the stability and frequency of the system. Fortunately the eigenvalues of the system with the largest magnitude correspond to the structural motion found from time marching. After determining λ = ln(Λ)/∆t the values are sorted by their proximity to the eigenvalues of the previous velocity and the real and imaginary parts directly correspond to the damping and frequency of the system, respectively. The flutter velocity is determined by running a series of different velocities and tracking the modal damping and frequency. The veloc- ity at which the damping becomes positive represents the flutter velocity and the corresponding frequency from the eigenvalue is the flutter frequency. The eigenanalysis for the strictly linear system has two advantages over a time history analysis. First, the frequency and damping values found through the eige- nanalysis can be recovered for each of the individual modes for all velocities. This allows a clear definition of the mode which drives the system unstable and the cre- ation of a root locus plot to analyze how frequencies evolve with damping. Second,
66 because only the largest eigenvalues are important for the analysis, Matlab’s eigs function can be used to solve for the largest eigenvalues quickly. The speed of the eigenanalysis allows for the creation of a velocity sweep for the stability of the system for a range of different parameters. Figure 3.7 shows the damping of the structural modes as the velocity is increased from below to above the flutter velocity for a fixed leading edge beam configuration. This velocity sweep clearly shows that the second mode is the mode that drives the system unstable. Figure 3.8 shows how the frequency changes as the damping changes. Again this shows that the second mode drives the system unstable at a frequency that is close the frequency of second structural mode.
Figure 3.7: Typical Velocity Sweep Figure 3.8: Typical Root Locus
3.3.3 Generating Time Histories from Eigenanalysis
Although the eigenanalysis is often chosen for the linear case to speed up the analysis, post-critical visualization of the structure motion can give insight into the nature of an instability. In order to visualize this for the eigenvalue solution, the complex eigen- vectors, specifically the complex eigenvector associated with the least stable mode, is used to reconstruct the motion. Starting from the definition of the displacement:
X w(x, t) = qn(t)φn(x) (3.34) n 67 and using the assumed form of the temporal coordinates:
~q = ~veλt (3.35)
Where λ and ~v are the eigenvalue and corresponding eigenvector determined through the aeroelastic analysis. Both the eigenvalue and the eigenvector can be written in terms of real and imaginary parts:
~v = ~vR + i~vI (3.36) λ = λR + iλI
Plugging this form into the definition of the generalized coordinates and using Euler identities to separate the real and complex parts of Equation 3.35.
λRt λRt ~q = (~vR cos λI t − ~vI sin λI t)e + i(~vR sin λI t + ~vI cos λI t)e (3.37)
Using the just the real part of the generalized coordinates for the eigenvalue with the largest real part allows one to reconstruct the time history of the displacement. Furthermore if the real part is ignored the motion at the flutter boundary can be reconstructed to a good approximation. This process is used to generate time histo- ries and videos from eigenanalysis results. Again the value of using an eigenanalysis can be seen through this exercise. After finding the eigenvalue and eigenvector time histories of arbitrary lengths may be generated by plugging in the desired time vector into Equation 3.37.
3.4 Inclusion of Fixed Support Structures
Because of the way that the vortex lattice theory is developed, it is simple to include the influence of rigid support structures that are used for testing to ensure smooth flow. This is done by including vortex elements on the support structure and re- quiring that the downwash be equal to the zero. In practice, the (0,0) coordinate
68 remains on the bottom leading edge of the elastic panel and circulations elements are added to on the sides of the structure. Therefore any element between 0 and Lx in the streamwise direction and 0 and Ly in the normal to flow direction are known to be on the elastic structure, while elements outside of this range are assumed to be on the support structure or in the wake.
3.5 Mirroring to Simulate Wind Tunnel Walls
Based on observations of the experiments there is a possibility that the wind tunnel wall confinement has a strong influence of the motion of the cantilevered panel. A common technique for simulating the existence of the wind tunnel wall with the vortex lattice method is called mirroring. The basic principal of this technique is that phantom circulations (Γ)ˆ are introduced to ensure that the induced velocity from the circulation elements on the real panel are exactly canceled out by the phantom circulation elements at the wind tunnel wall locations. When implementing the mirrored technique into the vortex lattice analysis mod- ule the first task is to identify where the vortex lattice code will be modified by the inclusion of the wind tunnel wall. It is important that if the mirrored circulation ˆ Γi is the same distance from the wall the original circulation element Γi and has the opposite sign the vertical component of the velocities at the wind tunnel wall (directly between them) will be zero. This result is because the influence function is a function of the circulation strength and the distance implying that points equal distant (in a given direction) but opposite signed and with the same magnitude cir- culations have an induced velocity in the given direction equal to zero. A picture of the mirrored panel configuration for confinement in the normal to the flow out of plane displacements is given in Figure 3.9. Mathematically this effect is captured by modifying the kernel function to include
69 Wind Tunnel Diameter
Figure 3.9: Schematic of the airfoil/panel geometry with the wind tunnel wall simulated by a single pair of mirrored elements the impact of the mirrored circulation elements.
n+1 X n+1 n+1 Vd = K(i,j) + K(i,j)u + K(i,j)d Γ (3.38)
where the subscripts u and d correspond to the upper and lower mirrored influences. This method can also be used to simulate the impact of normal to the flow in plane confinement, a method that is used experimentally to recreate two dimensional aero- dynamics. The model can also be improved by including multiple levels of mirrors that counteract the induced velocity by the mirrored circulations at the far wall. This is an elegant method of simulating the influence of the walls which demonstrates the VLM methods ability to incorporate additional effects with minimal effort.
3.6 Using ANSYS Structural Modes
Another aspect of the vortex lattice method is that the elastic model can be mod- ified to include a different structural models without a large change to the overall
70 Figure 3.10: Cantilevered Wing Configuration Schematic architecture of the analysis. In order to validate the ability of the code to be applied to a system with numerically determined structural modes, the code is modified to include the ANSYS structural modes for a flat plate in a wing like configuration shown in Figure 3.10. This configuration is convenient because there is an extensive library of existing research which allowed for the comparison of the theoretical sim- ulations to existing experimental and analytical results. Because the elastic model remains the same with the only difference being the structural mode shapes, the only modifications to the code required to implement this analysis is a new way to predict the displacement at a point (x,y) for a given mode shape. The modal displacement at the collocation and circulation locations (φ(x, y)) is calculated by interpolating the ANSYS numerical modes. By instructing the code to use the new function to calculate the modal positions for the creation of the gener- ¯ ¯ alized force matrices C1 and C2 and the downwash matrix β, the code is effectively modified from a code that analyzed the cantilevered plates in a flapping flag config- uration with only bending modes to a code that uses ANSYS modes to analyze the panel in the wing configuration. The ability to switch quickly the configuration of the elastic model without large modifications to the code and additional theory derivation is promising for developing
71 the ability to use the code to run analysis on flat plates in fluid flow with a wide range of boundary conditions for which there is less understanding of the flutter characteristics.
3.7 Rotated Wing Analysis
In the study of the aeroelasticity of beams and plates, the flow is usually assumed to be axially aligned with the structure. For example, a simple plate with one edge clamped the two axially aligned configurations have been explored extensively in the literature. Air flowing parallel to the clamped edge has been explored because this configuration looks very similar to an aircraft wing. The flutter motion for this con- figuration is known to be a combination of the first torsion and first bending modes for normal parameters. If the wing is rotated 90 deg, so the airflow is perpendicular to the clamped leading edge the instability is dominated by a second bending mode flutter, an instability which has been labeled flag flutter in the literature. However, even though both of these cases have been discussed in the literature the flutter at flow configurations that occur between the wing flutter and flag flutter have not been explored. This story is also true for the other configurations explored in this thesis. During this process the sensitivity of frequency and flutter velocity to small deviations from the perfectly normal or perfectly axial cases are explored. First, it is clear that the existing structural models can be used, and the changes to the theory to capture this transition are all aerodynamic mesh changes. The first step in developing the aerodynamic theory for a structure that is not axially aligned with the flow is to decide what type of vortex elements are used. For the initial model typical horseshoe elements are used and the mesh is determined by including all horseshoe elements which have a collocation point that are on the elastic structure. In order to implement this model with the existing structural model, two coordinate systems are used as shown in Figure 3.11. For this system the prime notation dictates
72 β
y y' x
x Figure 3.11: Aerodynamic and Elastic Coordinate Systems quantities that are measured in the structure coordinate system. By thinking of the system in two separate coordinate systems, the strictly aerodynamic and strictly elastic portions of the aeroelastic equations can be treated exactly the same. The rotation is captured in the downwash and generalized force equations where a careful accounting of the coordinate system is used. The relationship between the two coordinate systems can be written as:
x = (Ly0 − y0) sin β + x0 cos β (3.39) y = y0 cos β + x0 sin β
A similar transformation can be defined to go from the aerodynamic coordinate system ([x,y]) to the the structure coordinate system ([x’,y’]). In order to use the typical horseshoe elements and a square mesh, the first step is to define a square box aligned with the flow that completely encompasses the rotated structure. A
rectangular grid in the flow coordinate system is then defined that has Ss element
in the normal to flow direction and Sc elements in the flow direction. With this grid defined, the code loops though all of the collocation points and checks if they lie on
73 the structure. Practically this is done by transforming all of the collocation points to
0 0 their coordinates in the structure coordinate system and checking that 0 ≤ xc ≤ Lx
0 0 and 0 ≤ yc ≤ Ly . While looping through the points, if the collocation point is on the structure the [x,y] location of the collocation point and the [x,y] location of the top and bottom of the horseshoe element for that panel are stored as well as the row in the mesh that the element falls in.
3
2.5
2
1.5
1 y
0.5
0
−0.5
−1
0 1 2 3 4 5 x Figure 3.12: Mesh for β = 450. The red x’s are the collocation points, the green .’s are the top and bottom of the circulation elements on the the structure and the black .’s are the top and bottom of the circulation elements in the wake
After the structure circulation elements have been constructed the location of the wake elements are also determined. Figure 3.12 shows the mesh that is generated for a panel at a β = 450. The trailing edge of the wake directly mirrors the trailing edge of the panel to allow simple convection and relaxation relations derived earlier to be
74 used. A concern with this method is that, there is a discretization error associated with keeping the circulation elements square and not having the elements mirror the leading and trailing edge. This fact can lead to sawtooth instability frequency and velocity results, especially at angles near 0 or 90 degs. For the preliminary analysis it is determined that implementing a finer mesh until a smooth boundary is found is more efficient than deriving complex vortex elements that change in shape for every flow angle.
3.7.1 Generalized Force Calculation
The pressure force is again calculated using the application of Bernoulli’s principal stated in Equation 3.24. However the summation limits are different. The summa- tion incorporates the pressure induced by all of the circulations prior to the given circulation, but in the same row. For the axially aligned cases, the elements in the same streamwise row as a given element can be determined by the dimensions of the aerodynamic mesh. For the case of the rotated mesh for this analysis, while the initial aerodynamic mesh is created a separate vector containing the row of bound circulation elements is created, and the pressure is determined by summing up all of the elements that are in the row of the given circulation element, but have an index smaller than the given circulation element. Next it is assumed that the force caused by the circulation acts on the horseshoe vortex halfway between the two trailing elements. The x and y locations in the fluid coordinate system at this point are transformed to the panel coordinate system and the mode shape used to calculate the generalized force is evaluated at this point.
3.7.2 Downwash Calculation
The downwash calculation, like the generalized force calculation, required transform- ing the x and y coordinates of the collocation points for the bound vorticies in the
75 fluid coordinate system to their panel coordinate system equivalents. Once this tran- sition is made the dimensional form of Equation 3.18 is used.
76 4
Results from Aeroelastic Simulations
4.1 Dimensional Beam Simulations
Now that the methodology and analysis techniques have been described the accuracy of the aeroelastic model must be validated. Fortunately there is an existing liter- ature which contains both analytical and experimental results for the cantilevered beam configuration. A paper by Tang et al. [33] contains experimental data for a rectangular aluminum panel that is .39mm thick, has a 266.7mm streamwise dimen- sion, and a 76.2mm normal to the flow dimension. The aluminum panel is made of
3 3 7075 aluminum which has a density(ρs) of 2.84 × 10 kg/m and a stiffness (E) of 72 × 109kg/m2. The density of air is assumed to be 1.2kg/m3 and the vortex lattice relaxation factor (α) is set to .992. With these material properties the first four natural frequencies of the structure found using the presented beam model are 4.46 Hz, 27.79 Hz, 78.24 Hz, and 153.35 Hz which are nearly identical to the published values in Tang et al. [33] of 4.57 Hz, 28.62 Hz, and 80.15 Hz. With these material properties it is expected that the system will have a flutter velocity of 29.5 m/s and a frequency of 22.5 Hz. The figures presented
77 earlier such as Figures 3.7 and 3.8 are the plots that are generated for the system analyzed with 4 structural modes, the number of streamwise elements required for a time step of 1/40th the period of the fastest structural frequency and 10 elements in the normal to the flow direction. The plots show a typical flutter velocity prediction of 27.5 m/s and a flutter frequency of 23.5. Both of these results are well within 10% of the experimental value and represent a good aeoroelastic prediction. For all the simulations that are run, convergence studies are conducted to ensure that the length of the wake, the number of structural modes included, and the aerodynamic mesh are well converged. For the initial dimensional model of the flapping flag 10 elements in the normal to the flow direction, 100 elements in the streamwise direction and 6 structural modes proved to be adequate.
4.1.1 Time History Analysis vs. Eigenanalysis
Figure 4.1: Root Locus Figure 4.2: Damping vs. Velocity
Another study that is conducted to validate the model is to compare the fre- quency and damping values calculated from the eigenanalysis versus a time history. It is expecteted that for the linear model these two results should be exactly the same. Looking at Figures 4.1 and 4.2 it is clear that this assumption is correct. The only divergence between the two curves comes from the fact that the damp- ing and frequency determination from the time history is susceptible to noise which
78 sometimes throws off the damping prediction.
4.1.2 Fixed Leading Airfoil Effect
Figure 4.3: Root Locus Figure 4.4: Damping vs. Velocity
The comparison of the eigenanalyses with and without a fixed leading airfoil for the cantilevered configuration is shown in Figures 4.3 and 4.4 and clearly indicates that including the fixed airfoil does not change the result in any noticeable manner. This result implies that the fixed leading airfoil does not affect the motion of the panel which is what one would intuitively believe. Furthermore, running an analysis which does not include the leading airfoil reduces the number of elements and is therefore more computationally efficient. The support structure impact is explored in more detail for the NASA plate configuration later in this section.
4.2 Wind Tunnel Wall Confinement Effects
The influence of confinement effects in both the in plane and out of plane direction is explored using the non-dimensional aeroelastic model. A beam which is clamped free in the streamwise direction, has an aspect ratio of 0.5 and a mass ratio of .232 is simulated with 100 streamwise, 10 normal elements and 8 structural modes.
79 4.2.1 Out-of-Plane Normal to Flow Confinement
16
14
12
10 ˜
U 8
6
4
2
0 0 1 10 10 Out of Plane Wall Gap
20
18
16
14
] 12
10 Radians [ 8 ω
6
4
2
0 0 1 10 10 Out of Plane Wall Gap Figure 4.5: Impact of Out-of-Plane Confinement on Flutter Frequency Prediction
The distance of the simulated top and bottom wind tunnel walls is varied from 3/10ths to 10 times the streamwise length of the panel. Once the top and bottom wind tunnel get closer than this numerical instabilities occur because of the singular- ities that make up the vortex lattice mesh. The wind tunnel wall influence is clearly very small for the parameter range considered. However, for the actual wind tun- nel wall configuration the spacing between the elastic structure and the wind tunnel wall is usually slightly larger than the streamwise dimension of the structure. At this
80 distance, the influence of the wall is negligible suggesting that it does not influence the linear flutter boundary in experiments. Once the non-linear structural model has been included, it will be interesting to observe if there is a larger influence of the wind tunnel walls on the LCO amplitude because the observed LCO amplitude is quite large, rapidly closing the gap between the structure and the wind tunnel walls. Finally these results confirm that if an experiment is conducted with wind tunnel walls that are closer than the streamwise length of the elastic panel then confinement effects should be included even when calculating the linear stability boundary.
4.2.2 In-Plane Normal to Flow Wind Tunnel Wall Confinement
15
10 ˜ U
5
0 −4 −3 −2 −1 0 1 10 10 10 10 10 10 Wind Tunnel Wall Gap
Figure 4.6: Impact of In-Plane Confinement on Flutter Velocity Prediction
The in-plane axial confinement in the normal to the flow direction can also be modeled using the method of images. This is an interesting case because a method to recreate two-dimensional theoretical results experimentally is to have the structure you are evaluating span the entire cross section of the wind tunnel. Figures 4.6 and 4.7 show that the wind tunnel walls must be less than 1/10th of the streamwise dimension away from the side of the panel before any effects of the wall are felt. It
81 20
18
16
14
] 12
10 Radians [ 8 ω
6
4
2
0 −4 −3 −2 −1 0 1 10 10 10 10 10 10 Wind Tunnel Wall Gap
Figure 4.7: Impact of In-Plane Confinement on Flutter Frequency Prediction is conceivable that you would be able to get this close experimentally. However, in order to reach the confined, two-dimensional asymptote, the wind tunnel wall must be less than 1/1000th the streamwise dimension away from the elastic panel. It is unlikely that the experiments would be able to get this close, as for a typical 1 meter streamwise length this would require less than 1 mm separation which would be almost impossible to achieve experimentally. This result may give some insight into why two-dimensional theory has always significantly under predicted the flutter boundary even when experiments have attempted to simulate a two-dimensional airflow.
4.3 Non-dimensional Simulations (Modified from Journal of Fluids and Structures Journal Submission)
This section contains a theoretical study of the flutter characteristics of clamped-free and pinned-free beams with varying mass ratios and aspect ratios. The simulation configuration is given in Figure 3.3. As shown in the figure the rigid airfoil which is present in the experimental configuration shown in Figure 5.1 is not included in
82 the vortex lattice mesh and therefore not included in the aeroelastic simulations. The influence of the airfoil can be included in the aeroelastic model by including bound circulation on the fixed airfoil which is governed by Equation 3.16, where the downwash on the rigid structure is equal to zero. The airfoil is not included to allow for theoretical results that could be compared to previous theoretical simulations. However, initial simulations done comparing the flutter velocity with and without the airfoil show that the airfoil can act to destabilize the system and lower the theoretical flutter velocity by up to 20 percent for small mass ratio. A further exploration of the influence of the leading edge airfoil, which is present in experiments, will not be explored in this paper, but could be the subject of future research.
4.3.1 Leading Edge Spring Simulations
The first question studied in detail is the change in flutter characteristics from the pinned-free to clamped-free boundary conditions. Using µ = .277, H∗ = .5, and N = 10 appropriate in-vacuum beam modes, a set of simulations is run with a varying magnitude pinned edge torsional spring. The simulation is run using 150 panel elements and 300 wake elements in the streamwise direction and 10 elements in the normal to flow direction. Figure 4.8b shows the transition between pinned-free and clamped-free flutter. The analysis suggests a monotonic transition for the flutter frequency between the pinned-free and clamped-free configurations. This result matches the transition in frequency behavior observed in the structural model, an unsurprising result. Fur- ˜ thermore the critical values for Kα are in the same range as they are for the transition in the natural frequency analysis for the flutter mode. The result also demonstrates that for this configuration flutter arises from an interaction between the first and second modes for both the pinned-free and camped-free case. However the flutter velocity transition from pinned-free to clamped-free does not
83 14
13
12
11 ˜ U 10
9
8
7 −2 −1 0 1 2 3 10 10 10 10 10 10
K˜ α
(a)
25
20
15
[radians] 10 n ω
5
0 −2 −1 0 1 2 3 10 10 10 10 10 10
K˜ α (b) ˜ ˜ Figure 4.8: Flutter Velocity (a) and Frequency (b) vs. Kα. Small values of Kα correspond to a pinned-free case (dotted) and large values correspond to a clamped- free case (dashed). The first and second natural frequency evolution results are also included as the thin lines in (b). The thick lines correspond to the aeroelastic results.
84 monotonically move from the pinned-free case to the clamped-free case, see Figure 4.8a. This unexpected result shows that a small torsional spring at the leading edge of a pinned-free beam will actually drive the flutter velocity below the pinned-free critical velocity. In fact, for small values of torsional spring stiffness, making the spring stiffer will actually drive down the flutter velocity. It is hypothesized that the larger effect on the natural frequencies of the lower modes at small torsional spring stiffnesses initially brings the first two natural frequencies closer which leads to a reduction in the flutter boundary. Because of the inherent torsional stiffness for many pinned systems, this is a significant result because it suggests that using the pinned-free model may not be a conservative estimate, and therefore an effort to quantify the torsional stiffness of the pinned connection must be explored. This result could also be significant for applications in energy harvesting where a lower flutter velocity is desired. Looking closer at the inflection point of the flutter velocity curve, ˜ it appears to occur at a Kα which corresponds with the spring stiffness required to start the transition from pinned-free to clamped-free frequencies from the elastic simulation. It is at this critical point where the torsional spring begins to become strong enough to force the response to behave more like the clamped-free case.
4.3.2 Aspect Ratio Variation Simulations
Another key parameter to explore with this model is the Aspect Ratio (H∗). Figure 4.9 shows the current theoretical prediction for a µ = 0.6 beam as the aspect ratio is varied. Again, for this set of simulations, 150 panel elements and 300 wake elements in the streamwise direction, 10 normal to flow elements and 10 clamped-free modes are used. The result shown in Figure 4.9 matches previous theoretical results published by Eloy et al. [14]. Also shown in the figure are the experimental data points collected by Eloy et al. [14]. For the linear analysis presented here the only experimental data that the theoretical model should be compared to is the unfilled squares because the
85 gap down to the filled in squares represents a hysteretic effect which is not captured by the current linear model.
25
20 ˜
U 15
10
5 −1 0 10 10 H∗ Figure 4.9: Flutter velocity as a function of the aspect ratio at a mass ratio (µ) of 0.6 and clamped-free boundary conditions. The thick line corresponds to the current researchers theoretical predictions, the dashed line is taken from Eloy et al. [14]. The squares are previously published experimental data points [14]. The empty squares correspond to the velocity at which the system becomes unstable as the velocity increases and the filled in squares correspond to the velocity where the response returns from unstable oscillations to stable as the flow velocity is decreased.
4.3.3 Mass Ratio Variation Simulations
The next set of simulations which is conducted provides a comparison between the pinned-free and clamped-free flutter as a function of the mass ratio. This is explored both from a frequency and a flutter velocity perspective. Aspect ratios of 0.5, 1.0 and 1.5 are simulated. This set of simulations is conducted with the same lattice properties as the previous analyses. Figure 4.10a also shows the comparison between the flutter velocities of the pinned-free and clamped-free beams. It is clear from the results that for mass ra- tios between .1 and 1, the pinned-free and clamped-free flutter boundaries are very
86 22
20 Clamped S˜ = 0.5
18 Clamped S˜ = 1.0
16 Pinned S˜ = 0.5 S˜ . 14 Clamped = 1 0 from [25]
12 ˜ U 10
8
6
4
2
0 −1 0 1 10 10 10 µ
(a)
60
50
40 ]
30 radians [ ω 20
10
0 −1 0 1 10 10 10 µ (b) Figure 4.10: Flutter Velocity (a) and Frequency (b) vs. µ. The thick solid line cor- responds to the Clamped-Free beam with S˜ = 1.0, the thin dashed line to Clamped- Free beam with S˜ = 0.5, and the thick dotted line to a Pinned-Free beam with S˜ = 0.5. Also included in the velocity figure is theoretical predictions for a Clamped- Free beam with S˜ = 1.0 from Eloy et al. [13]
87 µ=.537 µ=.9349 µ=1.628 µ=2.834
70 70 70 70
60 60 60 60
50 50 50 50 ]
40 40 40 40 Radians [ ˜ ω 30 30 30 30
20 20 20 20
10 10 10 10
0 0 0 0 −10 0 10 −10 0 10 −10 0 10 −10 0 10 Damping Damping Damping Damping (a)
µ=.537 µ=.9349 µ=1.628 µ=2.834 10 10 10 10
8 8 8 8
6 6 6 6
4 4 4 4
2 2 2 2
0 0 0 0
Damping −2 −2 −2 −2
−4 −4 −4 −4
−6 −6 −6 −6
−8 −8 −8 −8
−10 −10 −10 −10 10 15 10 15 10 15 10 15 U˜ U˜ U˜ U˜ (b) Figure 4.11: The two figures show the evolution as the mass ratio is increased over the range where the flutter characteristics move from second bending to third bending for the clamped-free configuration. (a) shows the root locus evolution and (b) shows the velocity vs damping evolution. The triangles correspond to first bending, the x’s to second bending and the dots to third bending. The solid line is the zero damping and the open circle identifies where a mode becomes unstable.
88 similar. This is also the case for the frequency comparison shown in Figure 4.10b. For both cases the flutter mode, as identified by the frequency is second mode flut- ter (second mode for pinned-free is often called its first bending mode, because the first mode is a rigid body motion). Also for both cases the frequency of the flutter falls below the respective in-vacuum second mode frequency. Another common trend which is observed is that the frequency of oscillation begins to decrease as the mass ratio increases. A phenomenon observed both using the lattice method discussed here and in alternate analysis done with more traditional aerodynamic theory, see Eloy et al. [13] and Guo [17], is a transition in flutter mode to third mode flutter at a higher frequency and velocity as the mass ratio increases above a critical value. At the mass ratios where there is flutter in both modes, there is an interesting behavior in the modal damping evolution. At the lower velocity the second mode goes unstable in its normal manner. However, instead of having an aeroelastic damping value whose magnitude continues to grow, the damping levels out. Simultaneously the third mode begins to become less negatively damped and the frequencies of the second and third mode begin to come together. At the velocity corresponding to the third mode flutter, the third mode becomes unstable and the second mode becomes stable again. This transition is shown in Figures 4.11a and 4.11b If a time marching analysis was done, all that would be observed is the jump in frequency and flutter shape at the upper flutter velocity, while the eigenanalysis allows the tracking of the stability of the individual modes. As with previous works, this transition occurs at a lower mass ratio for the pinned-free case. Unfortunately the current experimental model would not allow for testing of mass ratios where higher mode flutter would be expected to be observed. It is clear from Figures 4.10a and 4.10b that the difference between the pinned- free and clamped-free cases would be more noticeable in the flutter frequency than in
89 the flutter velocity. In fact the difference between the clamped-free and pinned-free flutter velocity values is so small, it may not be observable during experiments. Overall, this implementation of the vortex lattice method for modeling the aero- dynamics produced results similar to the theoretical results of previous researchers. Although the vortex lattice method may take longer to create a simulation, it has value in that it produces results that compare well with experimental data and can be directly modified to capture aerodynamic nonlinearities and other real world consid- erations such as wind tunnel walls and experimental support structures. For example see Preidikman and Mook [27] or Attar [2].
4.4 Plate Simulations
Next aeroelastic simulations using the plate (versus beam) structural model are pre- sented.
4.4.1 NASA Simulations (Configuration 6)
The aeroelastic simulations for the NASA configuration, three sides clamped and the trailing edge free, are done both to predict the nature and onset velocity of the instability as well as explore the sensitivity to factors such as the tension in the structure, the size of the support structure relative to the elastic member and the effect of changing the aspect ratio. Before discussing the sensitivities to these parameters, the results of a typical aeroelastic simulation are presented. The typical analysis is done using the parameters listed in Table 2.2 with the inclusion of a structural damping ratio equal to 0.01. Six structural modes in the streamwise direction and three in the normal direction are used giving the system 18 structural degrees of freedom. The analysis considered rigid support structures on all four sides of the elastic membrane with lengths equal to 1/2 the streamwise length of the elastic membrane. The aerodynamic mesh is comprised of 150 elements in the streamwise
90 direction and 10 elements in the normal to the flow direction. The wake extends 400 elements in the streamwise direction behind the trailing edge of the rigid support structure. As with the elastic simulations, the linear system is analyzed in the frequency domain and the stability of the system is assessed using the aeroelastic eigenvalues at different discrete flow velocities. In Figure 4.12 the typical aeroelastic eigenvalues are presented in three different forms. First is the damping ratio versus the flow velocity which can be used to determine the critical velocity where the aeroelastic system becomes unstable and small perturbations to the system would grow exponentially. Next, is a root locus plot which is used to determine the frequency at which this instability occurs. Finally a plot of the frequency evolution as the velocity changes gives insight into the interactions between the frequencies that occur to cause the instability. In this case the interaction between the first and second frequencies is the cause of the instability. The fourth plot includes snapshots of the mode shape of the instability. The mode shape is reconstructed using the magnitude of the complex eigenvector associated with the unstable eigenvalue. For this configuration the mode shape confirms that the the instability motion is a combination of the first and second structural modes in the streamwise direction and the first mode in the normal or spanwise direction. Before analyzing methods of increasing the stability of the system convergence studies are conducted on the size of the aerodynamic mesh, wake length and the number of structural degrees of freedom included in the model. Specifically the modal convergence as more modes in the streamwise direction is of interest due to previous studies of the system which suggested higher mode flutter [5]. Figure 4.13 clearly shows that the stability boundary for this configuration does not depend on the number of structural modes included provided at leas four modes are included. Although the boundary does increase slightly with the inclusion of more modes there
91 100
0.2 80 0.1
0
] 60
−0.1 Hz [ ζ
ω 40 −0.2
−0.3 20
−0.4 0 −0.5 −10 0 10 8 10 12 14 16 18 20 U[m/s] Real(Eigenvalue) (a) Damping Ratio vs Flow Velocity (b) Root Locus
100
80 0.2
0.1
] 60 0 Hz [
ω 40 −0.1
20 −0.2 0.15 0.1 0.15 0.1 0 0.05 8 10 12 14 16 18 20 0.05 y U[m/s] 0 0 x (c) Frequency vs Flow Velocity (d) Mode Shape Figure 4.12: Configuration 6 aeroelastic results which demonstrates the plots cre- ated during a typical plate aeroelastic simulation is not the jump to a higher mode that was presented in the previous work. The present theory only includes a finite size support structure and therefore includes an aerodynamic wake which the previous author did not account for. Even though the influence of the support structure given in Figure 4.14 suggests that further increasing the support structure dimensions past one chord length would not change the aeroelastic boundary it is conceivable that in the limit of the boundary going to infinity the previous results could be recovered.
92 18 25 16 14 20 12 ] ] 15 10 Hz m/s [
[ 8
ω 10 U 6
4 5 2 0 0 4 5 6 7 8 9 10 4 5 6 7 8 9 10 Streamwise Modes Streamwise Modes
20 25
15 20 ] ] 15 10 Hz m/s [ [ ω
U 10
5 5
0 0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Modes Normal to Flow Modes Normal to Flow
25 16
14 20 12 ] 10 ] 15 Hz m/s
8 [ [
ω 10 U 6 4 5 2 0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 Streamwise Panel Elements Streamise Elements Figure 4.13: Flutter boundary as the number of streamwise modes, normal modes, and streamwise elements included in the structural model is varied
Similar studies are conducted for the aerodynamic mesh size and modes in the normal direction. The studies clearly revealed for this configuration, the solution has converged for the following values:
• Streamwise Mesh Elements on Elastic Structure : 100
• Normal Mesh Elements on Elastic Structure : 10
• Streamwise Modes Included : 6
• Normal Modes Included : 3
The solution is said to be converged if an asymptote is reached when plotting the flutter characteristics versus the parameter. The values determined from this analysis are used for the remainder of the simulations discussed in this section unless otherwise stated.
93 Support Structure Size Variation
One of the reasons for using the VLM method to analyze this system is the capabil- ity to model a finite size support structure and analyze the influence of the size and inclusion of the support structure on the aeroelastic stability. Figure 4.14 shows the results of varying the size of support structure while maintaining the unvaried sup- port structure’s nominal size of 1/2 of the streamwise chord of the elastic structure. In general the aeroelastic results are not sensitive to the inclusion of the support structure which suggests that their inclusion in aeroelastic models of plates may not be necessary. Interestingly it appears that the inclusion of the leading airfoil increases the instability onset velocity slightly while including the top and bottom rigid support structures (varied simultaneously) has the impact of lowering the in- stability onset velocity. The trailing support structure, unless it is small relative to the structure does not appear to impact the velocity in either direction. For all cases the impact on the frequency of the aeroelastic instability is even less pronounced.
4.4.2 Increasing the Flutter Velocity Aspect Ratio Variation
Two methods of increasing the flutter velocity are explored. Higher onset instability velocities are desired for this application because the current predicted flutter ve- locity is sufficiently low that if a similar configuration is implemented to reduce the acoustic signature of an aircraft during landing, flutter would be encountered. The first parameter to explore is the flutter boundary’s dependence on aspect ratio. To visualize this dependence all of the parameters are held constant while the stream- wise dimension is varied. Figure 4.15 shows the instability boundary frequency and velocity as the parameter is varied. The plot demonstrates two interesting behaviors. First, for lengths shorter than 0.15 m, it is possible to raise the flutter velocity and frequency by continuously
94 18 25
16
20 14
12 15 ] 10 ] Hz m/s [ [
8 ω U 10 6
4 5
2
0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Chord Normalized Leading Airfoil Size Chord Normalized Leading Airfoil Size
16 25
14
20 12
] 10 ] 15 Hz m/s [ [ 8 ω U 6 10
4 5 2
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Chord Normalized Trailing Airfoil Size Chord Normalized Trailing Airfoil Size
18
16 25
14 20 12 ] 10 ] 15 Hz m/s [ [ 8 ω U 10 6
4 5 2
0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Chord Normalized Top and Bottom Support Chord Normalized Top and Bottom Support
Figure 4.14: Flutter boundary as the support structure size (chord normalized) is varied. For each plot the size of the support structures not being varied is 1/2 the streamwise chord of the elastic structure
95 40 100
90 35 80 30 70 25 ] 60 ]
20 50 Hz m/s [ [ ω
U 40 15 30 10 20 5 10
0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 Chord [m] Chord [m] Figure 4.15: Flutter boundary as the streamwise chord is varied. In the frequency plot, the solid line without x’s correspond to the first three elastic natural frequencies of the unforced system
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2 0.2 0.1 0.15 0.2 0.1 0.1 0.05 0.1 0.1 0.05 0.1 0.05 0.05 0.05 y 0 0 x y 0 0 x y 0 0 x
(a) Chord = 0.05 m (b) Chord = 0.1475 m (c) Chord = 0.20 m Figure 4.16: Snapshot of the mode shape at the aeroelastic instability for three different streamwise chord lenghts shortening the length. For the CML design this suggests that designs that are able to minimize the streamwise length of flexible structure will be best able to avoid flutter. The other behavior is a jump in flutter frequency and velocity at a length of 0.20 m. From Figure 4.15 it is clear that the instability frequency jumps from being between the first and second natural frequencies to between the second and third natural frequencies. This change can be seen more clearly by comparing (b) and (c) form Figure 4.16 where the shape has clearly gone from a second to third bending mode in the chordwise direction. This transition is very similar to a phenomenon that is observed in the cantilevered beam analysis literature as the mass ratio(ρaLy/ρsh)
96 of the beam is increased for a constant aspect ratio[13, 17].
Tension Variation
50 100
45 90
40 80
35 70
] 30 60 ]
50
25 Hz m/s [ [ ω
U 20 40
15 30
10 20
5 10
0 0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 Ty [N/m] Ty [N/m] Figure 4.17: Flutter Boundary as the tension in the normal to streamwise direction. In the frequency plot, the dashed lines correspond to the first three elastic natural frequencies of the unforced system
0.2 0.2 0.2 0.2
0.1 0.1 0.1 0.1
0 0 0 0
−0.1 −0.1 −0.1 −0.1
−0.2 −0.2 −0.2 −0.2 0.15 0.15 0.15 0.15 0.1 0.15 0.1 0.15 0.1 0.15 0.1 0.15 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 y 0 0 x y 0 0 x y 0 0 x y 0 0 x
(a) Tension = 0 N/m (b) Tension = 90 N/m(c) Tension = 400 N/m(d) Tension = 700 N/m Figure 4.18: Snapshot of the mode shape at the aeroelastic instability for three different normal tension values
The next parameter which could reasonably be varied for the CML design, if the streamwise length is fixed, is the tension in the clamped-clamped normal to flow direction. Furthermore, it is reasonable to expect than when a flap is deployed it will induce a tension in the CML plate/membrane. For the analysis all values are the nominal values for the plate as the tension value (T y) is varied. The results of the simulation are shown in Figure 4.17. The first thing that is clear is that the initial increase in the tension has an impressive capability to rapidly increase the flutter boundary.
97 Another trend is the change in flutter mode and subsequent leveling of the flutter boundary once the tension reaches 90 N/m. The frequency plot in Figure 4.17 and the transition in mode shape from (a) to (b) and (c) in Figure 4.18 clearly demonstrates a qualitative and quantitative transition from a combination of first and second mode to a primarily second mode flutter. More significantly from a design perspective is the leveling off of the flutter boundary once the transition in flutter mode occurs. If this theoretical result is confirmed experimentally, it suggests that once a critical level is reached, flutter suppression cannot be achieved by further increasing the tension. Instead different parameters such as the stiffness of the system must be changed. Additionally this confirms that for the three sides clamped flexible membrane/plate with these parameters, flutter may occur at a velocity that will be encountered during the landing of a transport aircraft.
4.4.3 Additional Plate Boundary Configurations
Table 4.1: Plate Aeroelastic Simulation Summary (ζs = 0.01)
Configuration Type Velocity [m/s] Frequency [hz] 1 Flutter 8.55 8.54 2 Flutter 8.09 3.95 3 Divergence 16.04 4 Flutter 18.89 17.54 5 Divergence 19.25 6 Flutter 15.09 23.21
The plate aeroelastic model that has been developed up to this point, is general enough, that by simply using the correct beam mode shapes which satisfy the ge- ometric boundary conditions, any rectangular configuration may be studied. This capability and the existence of certain configurations which have not been explored extensively in the literature for low subsonic flow (Configurations 3-6 from Figure
98 1.2) has motivated the exploration of the 6 configurations shown. Configuration 1 and 2 are used to validate the model against previous three-dimensional aeroelastic simulations and experiments. For each configuration the instability type, velocity and frequency are of interest. Table 4.1 outlines the aeroelastic results that have been collected using the originally estimated structural damping ratio of 0.01. The structural and aerodynamic parameters remain the same as those used for the pre- vious plate analysis, with the material properties being those outlined in Table 2.2, 6 streamwise and 3 normal to the flow structural modes, and an aerodynamic mesh comprised of 10 elements in the normal direction and 100 in the streamwise direction. A baffle 1/2 the streamwise length is included on all sides with x’s from Figure 1.2.
Table 4.2: Plate Aeroelastic Simulation Summary (ζs = 0.05)
Configuration Type Velocity [m/s] Frequency [hz] 1 Flutter 10.15 7.54 2 Flutter 8.77 3.64 3 Divergence 16.25 4 Divergence 32.45 5 Divergence 19.62 6 Flutter 21.40 21.37
The detailed exploration of the material during the elastic experimental tests for Configuration 6 revealed that the actual structural damping of the material was closer to 0.05 than the 0.01 used in the initial simulations. Table 4.2 shows the updated aeroelastic stability boundaries with the higher structural damping ratio included. For Configuration 1, 2 and 6, the flutter velocity is increased slightly by increasing the structural damping. For the configurations that diverged, the addition of structural damping does not change the instability boundary. Finally the addition of structural damping to Configuration 4 caused the instability to move from a dynamic flutter to a static divergence. This arose because the original flutter mode for this configuration
99 is a hump mode. The addition of the structural damping caused a suppression of the hump mode leaving the first instability as a divergence instability at a significantly higher velocity. The implications of the increased structural damping for the detailed simulations for Configuration 6 includes a shift higher in flutter boundary although qualitatively the trends remain. The discussion of the individual configurations will be limited to a discussion of the instabilities that arise using the initial, lower estimate for the structural damping ratio.
Configuration 1 Flutter Results
Configuration 1 is clamped-free in the streamwise direction and free-free in the nor- mal direction. This is the same configuration that has been described as the flag flutter problem. Not surprisingly, even though a different structural model in the normal direction is used the same aeroelastic behavior is captured by this model. Specifically the root locus given in (b) of Figure 4.19 shows the interaction of two roots that correspond to the first and second bending modes in the streamwise di- rection. Another feature of the aeroelastic instability for this configuration is the rela- tively sharp crossing of the zero axis in (a). This slope signifies that effects such as structural damping, which are at best just approximations, will not largely ef- fect the flutter boundary. This may explain while historically, the VLM theoretical aeroelastic models have matched well the experimental results for this configuration.
Configuration 2 Flutter Results
Configuration 2 has free-free boundary conditions in the streamwise direction and clamped-free boundary conditions in the normal direction. This configuration is of- ten explored in the literature because it is a simplified version of an aircraft wing. This model differs slightly from the normal wing configuration because it has an
100 0.5 40 0.4 35 0.3 30 0.2
0.1 25 ]
ζ 0 20 Hz [
−0.1 ω 15 −0.2 10 −0.3
−0.4 5
−0.5 0 6 6.5 7 7.5 8 8.5 9 9.5 10 −10 −5 0 5 10 U[m/s] Damping (Real(Eigenvalue)) (a) Damping Ratio vs Flow Velocity (b) Root Locus
20
18
16 0.2 14 0.1 12 ]
10 Hz 0 [
ω 8 −0.1 6
4 −0.2 0.15 2 0.1 0.15 0.1 0.05 0 0.05 6 6.5 7 7.5 8 8.5 9 9.5 10 y U[m/s] 0 0 x (c) Frequency vs Flow Velocity (d) Mode Shape Figure 4.19: Aeroelastic results for Configuration 1. The analysis clearly shows a coupling between the first and second bending modes in the streamwise direction aspect ratio which is less than 1, while aircraft traditionally are designed with higher aspect ratios. Nonetheless the instability of this system occurs due to the interaction between the first bending and first torsion mode in the normal to flow direction. Be- cause torsion modes are not explicitly modeled with the employed structural model, the coupling in this model is between the first bending in the normal direction and the rigid body rotation in the streamwise direction. Interestingly, although Configuration 2 can be described as a 90 degree rotation from Configuration 1 with respect to the flow, the differences in the flutter instability
101 0.5
60
50 0
40 ] ζ 30 Hz [ ω −0.5 20
10
−1 0 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 −6 −4 −2 0 2 4 U[m/s] Real(Eigenvalue) (a) Damping Ratio vs Flow Velocity (b) Root Locus
60
50 0.2
40 0.1 ]
30 Hz 0 [ ω −0.1 20
−0.2 10 0.15 0.1 0.15 0.1 0.05 0 0.05 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 y 0 0 x U[m/s] (c) Frequency vs Flow Velocity (d) Mode Shape Figure 4.20: Aeroelastic results for Configuration 2. The analysis clearly shows a coupling between the first bending and first bending mode in the normal to flow direction and the antisymetric rigid body displacement mode in the streamwise di- rection are significant, something that is easily seen by comparing the mode shapes for each of the configurations. In a later section the transition between these two configurations by rotating the flow is explored in detail.
Configuration 3 Flutter Results
Configuration 3 represents a configuration which has not received as much explo- ration in the literature, especially in the context of three-dimensional aerodynamic theories. For this configuration there is a noticeably different governing dynamics
102 0.25 50 0.2 45 0.15 40 0.1 35 0.05 30 ]
ζ 0 25 Hz [
−0.05 ω 20
−0.1 15
−0.15 10
−0.2 5
−0.25 0 13 14 15 16 17 18 19 20 −5 0 5 10 U[m/s] Real(Eigenvalue) (a) Damping Ratio vs Flow Velocity (b) Root Locus
50
45
40 0.2 35 0.1 30 ]
25 Hz 0 [
ω 20 −0.1 15 −0.2 10 0.15 0.15 5 0.1 0.1 0.05 0 0.05 13 14 15 16 17 18 19 20 y 0 0 x U[m/s] (c) Frequency vs Flow Velocity (d) Mode Shape Figure 4.21: Aeroelastic results for Configuration 3. The analysis shows a diverge of the first bending motion in the streamwise direction and rigid body translation in the normal direction for the aeroelastic instability. Figure 4.21 (c) clearly shows that at the instability onset velocity the frequency of the first mode goes to 0. This correlates to a static post-critical response for the system that is called divergence. This figure also shows that this divergence is not the result of the interaction of two of the natural modes of the system. Instead it is a aeroelastic instability dominated by the first mode. Divergence is often considered the more benign form of aeroelastic instability because it does not lead to limit cycle oscillations which can induce life cycle fatigue. Furthermore, non-linear simulations for the dynamics of post buckled beams show
103 that they retain the ability to maintain a load and encounter deflections that are only on the order of the thickness of the plate near the divergence speed. In fact, panels on aircraft which are often clamped on four sides can experience divergence and not threaten the aerodynamic performance or structural integrity of the aircraft. The three-dimensional results confirms the two-dimensional results presented by Guo [17].
Configuration 4 Flutter Results
Configuration 4 is another configuration which has not received as much attention in the literature. The results of the analysis shown in Figure 4.22 demonstrate that flutter is again the dynamic instability that dominates the motion is a flutter insta- bility. For this configuration there are two branches that become unstable near each other. The first branch that becomes unstable appears to do so without interacting with any other frequencies. Furthermore this modes crossing of the zero damping axis is much shallower than the other instabilities encountered so far. This shal- low crossing means that structural damping could largely change the theoretically predicted flutter boundary. The second instability which occurs when the damping ratio of the third mode becomes positive near 20.75 m/s is a more typical coalescence flutter type instability with interactions between the first and second bending modes in the streamwise direction and rigid body translation in the normal to flow direction. Furthermore when the branch corresponding to this instability crosses the zero damping axis in Figure 4.22 (a), it does so with a significantly steeper slope than the earlier instability. The combination of this with the shallow slope of the first instability leads the author to speculate that in an experimental test the second instability may be the dominant instability. Looking at the root locus in 4.22 (b) and (c) it is clear that the first three natural frequencies are incredibly close to begin with. This could lead rise to a mode
104 0.1
0.05
0
−0.05
−0.1
ζ −0.15
−0.2
−0.25
−0.3
−0.35
−0.4 15 16 17 18 19 20 21 U[m/s] (a) Damping Ratio vs Flow Velocity
60 60
50 50
40 40 ] ] 30 Hz 30 Hz [ [ ω ω
20 20
10 10
0 0 15 16 17 18 19 20 21 −6 −4 −2 0 2 U[m/s] Real(Eigenvalue) (b) Frequency vs Flow Velocity (c) Root Locus
0.2 0.2
0.1 0.1
0 0
−0.1 −0.1
−0.2 −0.2 0.15 0.15 0.1 0.15 0.1 0.15 0.1 0.1 0.05 0.05 0.05 0.05 y 0 0 x y 0 0 x
(d) Aeroelastic Mode Shape for the first branch(e) Aeroelastic Mode Shape for the second branch going unstable going unstable Figure 4.22: Aeroelastic results for Configuration 4. The analysis shows two unique aeroelastic instabilities which occur in the same region
105 that theoretically will become unstable very quickly while in practice it will be a benign instability which will be readily damped by either structural or aerodynamic damping.
Configuration 5 Flutter Results
60 0.6
0.4 50
0.2 40 ]
ζ 0 30 Hz [
−0.2 ω 20 −0.4
10 −0.6
17 18 19 20 21 22 23 0 0 5 10 15 20 U[m/s] Real(Eigenvalue) (a) Damping Ratio vs Flow Velocity (b) Root Locus
60
50 0.2
40 0.1 ]
30 Hz 0 [ ω −0.1 20
−0.2 10 0.15 0.1 0.15 0.1 0.05 0 0.05 17 18 19 20 21 22 23 y 0 0 x U[m/s] (c) Frequency vs Flow Velocity (d) Mode Shape Figure 4.23: Aeroelastic results for Configuration 5. This configuration experiences a static divergence
Just like Configurations 3 and 4 can be thought of as being in the same family when rotated, Configuration 5 is a rotation of Configuration 6 which is analyzed in detail when discussing the NASA noise suppression efforts. Similarly to the previous
106 boundary condition pairs, as one moves the free edges from being normal to the flow to being axially aligned with the flow and therefore causing the leading and trailing edges to be fixed, the instability transitions from a dynamic instability observed for Configurations 4 and 6 to the static instability seen for this configuration and Configuration 3. However, unlike the previous result, the static instability in this case occurs at a higher velocity than the rotated dynamic instability for Configuration 6.
4.4.4 Discussion
The aeroelastic model deployed for this set of simulation have the interesting capa- bility of varying the structural boundary conditions as if it is a parameter of the system. By developing this model it is easy to identify the aeroelastic instabilities which will arise for different configurations. This effort both provides insight into the type of instabilities that arise for different boundary conditions as well as identify some interesting trends. First, the model identified that for configurations with both the leading and trailing edge free, a divergence instability is to be expected. Second, looking back at the summary of the results given in Table 4.1, an interest- ing comparison can be made between Configuration 4 and the NASA configuration. The only difference between these two configurations is whether or not the leading edge is clamped. Intuitively one would expect that an extra edge clamped would increase the instability onset velocity. From the table it is clear that the opposite actually occurs. In fact, by moving from a clamped to a free leading edge the flutter boundary increases by more than 20%. This result has consequences in the prelimi- nary design phase for this system as it suggests that constraining as many edges as physically possible may not be the best way to create the most stable configuration.
107 4.5 Axially Misaligned Analysis
As mentioned in the previous section, it is clear that for a fixed number of clamped boundary conditions, rotating the flow by 90 deg will drastically change the type of instability that is experienced. In order the quantitatively look at this transition the aeroelastic model which allows for axially misaligned flows is deployed. The simulations are conducted to determine at what angle the transition occurs and what the transition looks like in terms of the flutter velocity and flutter frequency.
4.5.1 Axially Misaligned Beam Simulations
Table 4.3: Rotated Wing Properties
Property Symbol Value Thickness h 1 mm 3 Density ρs 2700 kg/m Young’s Modulus E 69 GPa Poisson’s Ratio µ .3
Clamped-Free Dimension 600 mm Free-Free Dimension 300 mm 3 Air Density ρa 1.2 kg/m
The first transition that is explored uses a beam model with a single edge clamped. As we have seen before the transition between the leading edge clamped and the side edge clamped corresponds with a transition between a flutter instability that is dominated by a coalescence flutter between the first and second bending modes for the first case and a first bending, first torsion coalescence in the latter case. Because the flutter boundaries for Configuration 1 and 2 are too similar, different structural parameters are used to explore the transition. Furthermore, in order to have a model which could be validated at both extrema positions (clamped leading edge and clamped side edges), values are selected which had existing theoretical predictions
108 for both. These values are given in Table 4.3. Furthermore as mentioned in the theory section, a significantly finer aerodynamic mesh is implemented. Specifically, 44 elements in the normal to the flow direction and 120 elements in the streamwise direction and a wake extending 2.5 times the longer dimension are used. Figure 4.24 shows the flutter velocity and frequency boundaries as the flow is rotated with respect to the beam. The transition is somewhat surprising. As you can clearly see from the frequency transition, the large jump in frequency occurs at a small rotation angle. Physically, this means that the instability begins to look like the wing flutter coupling of the first bending and first torsional modes at this small angle. This can clearly be seen by looking at the snapshots of the mode shape for a rotation angle of 11.53 deg shown in Figure 4.27. In these snapshots from the time history, it is clear that there is a large contribution from the first torsional mode which is indicative of wing flutter. The immediate implications for this low angle transition are for the energy harvesting applications of the cantilevered beam configuration. As energy harvesters are nominally optimized to capture energy at a specific frequency, the precipitous drop in frequency for non-axially aligned flows can drastically reduce the energy captured by the system. Designers of such systems must be certain that the incoming flow will remain axially aligned or attempt to create an energy harvesting system that is able to capture energy over the wider band of frequencies. To the best knowledge of this researcher this is the first time these non-axially aligned configuration has been explored either experimentally or theoretically. There is currently an effort to confirm these theoretical results experimentally. If these trends are confirmed in experiment it suggests that additional care must be taken when conducting aeroelastic analysis for systems which may have deviations from axially aligned flow when deployed.
109 40
35
30
25 )
20 m/s ( U
15
10
5
0 0 10 20 30 40 50 60 70 80 90 Rotation Angle
20
18
16
14
12 ]
10 Hz [ ω 8
6
4
2
0 0 10 20 30 40 50 60 70 80 90 Rotation Angle Figure 4.24: Flutter boundary as the wing is rotated from the flapping flag con- figuration to the wing configuration. In the frequency plot, the solid line without x’s correspond to the first three elastic natural frequencies of the unforced system
110 0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2 0.5 0.5 0.5 0.5 0.5 0.5
y 0 0 x y 0 0 x y 0 0 x
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2 0.5 0.5 0.5 0.5 0.5 0.5
y 0 0 x y 0 0 x y 0 0 x
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2 0.5 0.5 0.5 0.5 0.5 0.5
y 0 0 x y 0 0 x y 0 0 x
Figure 4.25: Rotation Angle=0, One Period Flutter Motion
111 0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2 0.5 0.5 0.5 0.5 0.5 0.5
y 0 0 x y 0 0 x y 0 0 x
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2 0.5 0.5 0.5 0.5 0.5 0.5
y 0 0 x y 0 0 x y 0 0 x
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2 0.5 0.5 0.5 0.5 0.5 0.5
y 0 0 x y 0 0 x y 0 0 x
Figure 4.26: Rotation Angle=6.92, One Period Flutter Motion
112 0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2 0.5 0.5 0.5 0.5 0.5 0.5
y 0 0 x y 0 0 x y 0 0 x
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2 0.5 0.5 0.5 0.5 0.5 0.5
y 0 0 x y 0 0 x y 0 0 x
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2 0.5 0.5 0.5 0.5 0.5 0.5
y 0 0 x y 0 0 x y 0 0 x
Figure 4.27: Rotation Angle=11.53, One Period Flutter Motion
113 0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2 0.5 0.5 0.5 0.5 0.5 0.5
y 0 0 x y 0 0 x y 0 0 x
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2 0.5 0.5 0.5 0.5 0.5 0.5
y 0 0 x y 0 0 x y 0 0 x
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2 0.5 0.5 0.5 0.5 0.5 0.5
y 0 0 x y 0 0 x y 0 0 x
Figure 4.28: Rotation Angle=90, One Period Flutter Motion
114 4.5.2 Axially Misaligned Plate Simulations
35 30
30 25
25 20
] 20 ]
15 Hz m/s [ [ ω U 15
10 10
5 5
0 0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 Rotation Angle Rotation Angle Figure 4.29: Flutter boundary as the plate is rotated from the leading edge clamped to the top edge clamped. In the frequency plot, the solid line without x’s correspond to the first three elastic natural frequencies of the unforced system
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2 0.15 0.15 0.15 0.1 0.15 0.1 0.15 0.1 0.15 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0.05 0.05 y 0 0 x y 0 0 x y 0 0 x
(a) Rotation Angle = 0 deg (b) Rotation Angle = 45 deg (c) Rotation Angle = 90 deg Figure 4.30: Snapshot of the mode shapes at the aeroelastic instability for three different streamwise rotation angles
A trend that is observed when exploring the different plate boundary conditions is that a static divergence is encountered when both the leading and trailing edges are clamped. Again it is of interest to explore how the instability transitions from a dynamic flutter to a static divergence, and at what critical incident flow angle does the transition occur. Figure 4.29 shows this transition. It is clear that the transition
115 occurs near 45 deg and there is a transition range that extends from 30 to 60 deg. This central transition range is what was originally expected for all configurations when the incident flow angle is varied. The images taken from the time simulation of the plate at different rotation angles confirms the transition to a divergence mode once the angle is at 60 deg.
116 0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2
0.1 0.1 0.1 0.1 0.1 0.1
y 0 0 x y 0 0 x y 0 0 x
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2
0.1 0.1 0.1 0.1 0.1 0.1
y 0 0 x y 0 0 x y 0 0 x
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2
0.1 0.1 0.1 0.1 0.1 0.1
y 0 0 x y 0 0 x y 0 0 x
Figure 4.31: Rotation Angle=0, One Period Flutter Motion
117 0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2
0.1 0.1 0.1 0.1 0.1 0.1
y 0 0 x y 0 0 x y 0 0 x
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2
0.1 0.1 0.1 0.1 0.1 0.1
y 0 0 x y 0 0 x y 0 0 x
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2
0.1 0.1 0.1 0.1 0.1 0.1
y 0 0 x y 0 0 x y 0 0 x
Figure 4.32: Rotation Angle=45 deg, One Period Flutter Motion
118 0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2
0.1 0.1 0.1 0.1 0.1 0.1
y 0 0 x y 0 0 x y 0 0 x
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2
0.1 0.1 0.1 0.1 0.1 0.1
y 0 0 x y 0 0 x y 0 0 x
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2
0.1 0.1 0.1 0.1 0.1 0.1
y 0 0 x y 0 0 x y 0 0 x
Figure 4.33: Rotation Angle=60 deg, One Period Flutter Motion
119 5
Experiments
5.1 Experiments to Validate Beam Model
Figure 5.1: Experiment Apparatus
To validate the non-dimensional beam model vibration and aeroelastic experi- ments are conducted on samples of varying sized 3003 aluminum plates. For the .381 mm thick aluminum the length is varied from 200mm to 300mm in increments of 25 mm, and an aspect ratio of 0.5 is used. For the .25 mm thick aluminum the length is varied between 225 mm and 275 mm in 25mm increments and again the aspect
120 ratio of 0.5 is used. The properties for this material are assumed to be the common values for the alloy given in Table 5.1.
Table 5.1: Beam Experimental Parameters
Property Symbol Value Elastic plate properties Alloy 3003 Thickness h .381 mm, .25 mm 3 Density ρs 2840 kg/m Young’s Modulus E 72 GPa
Airfoil Streamwise Length 101 mm Airfoil Normal to Flow Length 550 mm 3 Air Density ρa 1.2 kg/m
Piezoelectric Patch Properties Source Measurement Specialty Series DT Series Patch Size 30mm by 12mm
Spectrum Analyzer Properties Manufacturer Scientific Atlanta Name Spectral Dynamics SD380
To record the frequency content of the plate movements two methods are used. First, a small piezoelectric patch is attached at the root of the plate. The properties of the piezoelectric patch are given in Table 5.1. The piezoelectric patch is chosen to be small enough that it will not affect the motion of the system. This is verified by the ground vibration experiments. For the second method an accelerometer is placed at the root of the plate. Results are not sensitive to the measurement device and they are interchanged throughout the experimental process. For both methods, the sensor signal is collected and analyzed in real time by the Spectral Dynamics SD380 spectrum analyzer for frequency content.
121 5.1.1 Beam Structural Experiments
Vibration testing is done to ensure that the plate frequencies, which are used in the theoretical aeroelastic model, are accurate representations of the actual natural frequencies of the test specimens. Furthermore the structural testing ensures that the test apparatus and frequency measuring piezoelectric patch or accelerometer do not have a large effect on the test specimen’s behavior. Figure 5.1 shows the experimental apparatus, described in the previous section, which is used to measure the frequency of the plate. The natural frequencies of the plate are determined by applying an impulse in force at the tip of the beam and observing the frequency content of the response. Overall the natural frequencies measured in experiment match the expected clamped-free natural frequencies over the range of test specimens as can be seen in Figure 5.2. This experiment also helps validate the time scaling because it is clear that for all of the mass ratios the non-dimensional frequencies do in fact remain constant. Finally this experiment shows that the frequency measuring devices do
70
60
50 ] 40 Clamped−Free Theory Clamped−Free Experiment radians [ 30 ˜ ω
20
10
0 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 µ Figure 5.2: Natural Frequency Experimental Results
122 not change the natural frequencies and therefore do not affect the response of the system. This confirms that the effect of the measuring device does not need to be explicitly dealt with in the structural model.
5.1.2 Beam Aeroelastic Experiments
The aeroelastic experiments are carried out in the Duke University wind tunnel. The specimen is mounted in the wind tunnel using a rigid airfoil that spans the height of the wind tunnel to provide the leading edge clamping of the elastic plate. The leading airfoil is used both to mount the elastic panel, as well as to ensure smooth flow over the elastic panel. As with the structural experiments, the flutter frequency is calculated from the signal of the attached piezoelectric patch or accelerometer during at the flutter velocity. The flutter velocity is measured by slowly incrementing the flow velocity in the wind tunnel up until the specimen entered an oscillation. As the velocity of the wind tunnel came close to the flutter velocity, a peak in the frequency response begins to appear. At this point the increment in the flow velocity is reduced to around .25 m/s per increment. At each flow speed the velocity is held for 2-3 seconds before incrementing again. At a certain velocity, the oscillations grow and the specimen enters a large oscillation. The velocity where the beam entered this oscillation is recorded as the flutter velocity and the frequency at this speed is read from the spectrum analyzer. For each specimen the test is repeated three times and the average flutter velocity and frequency is recorded. The goal of the wind tunnel testing is to validate the theoretical model with experimental data points. Specifically, a study of flutter as a function of the mass ratio for the clamped-free configuration is presented. Good agreement between the clamped-free experiment and theory helps validate the aeroelastic model and suggests that simulations outside of the experimental test suite are also valid.
123 25
20
15 ˜ U 10
5
0 −1 0 10 10 µ
(a)
30
25 ] 20
15 Radians
[ 10 ω
5
0 −1 0 10 10 µ
(b) Figure 5.3: Mass Ratio Variation with Experiment. This figure includes new exper- imental data (x), previous experimental data from Huang [20] for S˜ = .6 to 1.5(*), ˜ ˜ Eloy et al. [14] for S = 1.0 (), and Eloy et al. [15] S = 0.5 (). Also included in the figure are theoretical results for S˜ = 0.5 (thick line) and S˜ = 1.0 (thin line).
124 The experimental testing for the mass ratio variation is done for the clamped-free configuration because the Duke University wind tunnel has an established experimen- tal setup and test protocol for this configuration. As one can see by looking at Figure 5.3, there is very good agreement between theory and experiment. Quantitatively this is shown by a small average error and standard deviation of the error from the experiments given in Table A.2. The averages are calculated by subtracting the the- oretical value from the experimental value and then dividing the difference by the theoretical values. This small error is consistent with previous comparisons with dimensional vortex lattice simulations and experiments carried out by Tang et al. [32] and Dunnmon et al. [11]. For the frequency results presented in Figure 5.3 there is a consistent bias for the experimental values to be under the theoretical values. This may be because a beam is used in the simulations while a plate is used in the experimental setup. An initial exploration of the inclusion of the leading edge airfoil in the theoretical model also suggest that the experimental apparatus may also be a cause of the lower flutter frequencies and lower flutter velocities. This impact would also explain the increasing error as the mass ratio increases which corresponds to a relatively larger support structure compared to the size of the elastic specimen. Regardless, the good agreement between theory and experiment is encouraging and suggests that for the flutter velocity and frequency, the vortex lattice aerodynamic method is an accurate model for the linear response of the system.
5.2 Experiments to Validate Plate Model
The experimental work to validate the plate structural and aeroelastic model is pri- marily conducted to support the NASA continuous mold link project, and therefore the material tested in the experiment is a red polymer plate-membrane that is sup- plied by NASA. Although the CML configuration is three sides clamped, one side free, care is taken to use an experimental setup that is capable of simulating all of
125 the 5 additional boundary conditions explored in the theory section. At this point the author wishes to formally acknowledge fellow graduate student Ivan Wang for his significant contributions to the design of the experimental support structural and the collection of experimental data.
5.2.1 Design of Experimental Setup
The primary experimental apparatus is a modular baffle structure that can imple- ment a clamped boundary condition on one or more sides of a rectangular plate, in addition to providing a means to streamline the flow that goes over the plate. A CAD rendering of the baffle design is shown in Figure 5.4. The figure shows (1) the top baffle, (2) the bottom baffle, (3) the leading edge baffle, (4) the trailing edge baffle, and (5) the connector pieces that link the individual baffle sections. Each baffle section consists of a front and back structure, as well as a clamp that can be screwed on to constrain the plate.
Figure 5.4: CAD Rendering of Baffle
126 Each baffle section also has a flange that can be secured to a stable structure outside the wind tunnel. Therefore, each baffle section can be mounted in the wind tunnel individually, allowing all combinations of boundary conditions to be tested. This modular design revolves around the connector, which is shown in Figure 5.5. The T-shape design and the slot allows the top and bottom baffles to slide relative to the leading and trailing edge baffles, such that plates of different spans (top to bottom dimension) can be tested. Specifically, the T-shape allows an extended back section of the baffle, on which additional bolt holes can be tapped for securing additional clamps. Also, the slot on the connector allows the connector to slide without worrying about matching up bolt holes for securing the connector. To test a plate with a larger span, the only modifications are to make a new clamp piece to extend the boundary and to tap bolt holes on the back side of the baffle to mount the new clamp.
Figure 5.5: Close up of the Connector
It is also not difficult to test plates with larger streamwise lengths by making additional top and bottom baffle sections, which are designed with symmetric edges on the leading and trailing edge sides such that additional sections can be secured together using the same connector design. Finally, the baffle allows the plate to be tensioned by setting the strain. Figure 5.6 shows three different strain settings, each corresponding to a level of tension.
127 Because the top and bottom baffles are designed to be able to slide, the bottom baffle can slide down to a different strain setting in order to tension the plate before the remaining clamps are applied. From a practical point of view, some tensioning is necessary in order to avoid free play nonlinearities, but the material may also have nonlinear stiffness under different tensile loads. This is especially important for the NASA CML project since the structures will be tensioned during flap deployment.
2
2
3
1
1
Figure 5.6: Different Strain Settings Allowing for Varying Span-wise Tension
5.2.2 Static Structural Experiments
The nominal properties from Bloomhardt and Dowell [5] of the red plate-membrane are the values used in the aeroelastic simulations listed in Table 2.2. Before designing the actual wind tunnel experiment, some static tension tests were conducted by Ivan Wang to obtain estimates of the elastic modulus and Poisson’s ratio in order to validate the given material properties. A material sample is secured in an axial load cell, and the load cell is used to pull on the sample to apply a measurable amount of tension. The sample has a length of 0.1135 m and a width of 1.27 cm. The axial strain (change in length) and transverse strain (change in cross sectional width) are then measured to calculate a stress-strain curve as well as estimate Poisson’s ratio. Figure 5.7 (a) shows the stress-strain plot for one of the trials of the tensile test. The results are shown up to a strain of 5%. Some nonlinearity can be observed in the curve.
128 0.25 30
25 0.2
20 0.15
15
0.1 Stress (MPa) 10 Young’s Modulus (MPa) 0.05 5
0 0 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 Strain Strain (a) (b) Figure 5.7: (a) Measured Stress Strain Curve for One Tensile Test Trial. (b) Estimated Elastic Modulus vs Axial Strain
The variation in the elastic modulus with respect to strain is calculated from taking the derivative of the stress-strain curve obtained from the axial load cell data. The results are summarized in Figure 5.7 (b). On average, the stiffness is about 17 MPa for strain less than .3%, and then varies around 6 MPa for higher strains. It is interesting to note that the reported value is less than 2 MPa. It is postulated that there is a typographical error in Bloomhardt and Dowell [5], and that the actual value is 18.4 MPa. This is explored further in the following section in which dynamic (natural frequency) testing results are discussed. The sample length is measured with a set of calipers with 0.03-mm precision after the tensile test and no measurable plastic deformation occurred after reaching at least 5% axial strain. Therefore, the baffle tension mechanism is designed with three built-in strain settings: 2%, 5%. Poisson’s ratio is estimated by measuring the transverse dimension (width) of the sample cross section under tension, and calculating the ratio of transverse strain to axial strain. Three separate trials are conducted. This is a rough estimate because it does not account for the curvature of the sides of the test sample as it is stretched. Nevertheless, the results for Poisson’s ratio, shown in Figure 5.8, suggest that the Poisson’s ratio is near 0.5, which is expected of elastic polymers.
129 Poisson’s Ratio Results − Average 0.486346 StdDev 0.100638
0.8
0.7
0.6
0.5
0.4
Poisson’s Ratio 0.3
0.2 Trial 1 Trial 2 0.1 Trial 3 Mean 0 0 0.02 0.04 0.06 0.08 0.1 Axial Strain Figure 5.8: Estimated Poisson’s Ratio
5.2.3 Dynamic Structural Experiments
Ground vibration experiments are conducted to measure the natural frequencies and confirm the mode shapes for four of the six configurations. Figure 5.9a shows a far view photograph of the experimental set up for the ground vibration test for Configuration 6, including 1) the test specimen, 2) clamps that secure the baffle to a fixed structure, 3) the electromagnetic shaker, and 4) the laser vibrometer. Figure 5.9b shows the excitation mechanism in more detail, specifically showing 1) the test specimen, 2) the shaker, and 3) the aluminum tape onto which the shaker tip is affixed. A laser vibrometer is used to measure the velocity at one point on the membrane. A shaker is used to excite another point on the opposite side of the membrane. The shaker is specified to excite the structure over a sine sweep from 0.25 Hz to 100 Hz, and a spectrum analyzer is used to calculate the frequency response function using the shaker force transducer signal as the input and the laser vibrometer signal as the output. Small pieces of aluminum tape, approximately 1 cm long on each side, are placed at the locations where velocity measurements are desired because the laser vibrometer requires a reflective surface to function properly. Because the mass of the aluminum tape is much less than the mass of the plate-membrane specimen, the added inertial effects of the tape are ignored. Aluminum tape is also used as a mounting surface
130 (a) Far View of Ground Vibration Test(b) Close View of Ground Vibration Test Setup Setup Figure 5.9: Photographs of the Experimental Setups
Table 5.2: Equipment Used in the Ground Vibration Experiment Component Brand Model Signal amplifier Bruel & Kjaer Type 2635 Spectrum analyzer Scientific Atlanta SD380 Shaker Bruel & Kjaer Type 4810 Force transducer Bruel & Kjaer Type 8200 Laser vibrometer Ometron VPI4000 for the shaker tip. The wax used for attaching the shaker tip does not stick to the test specimen, so aluminum tape is first placed on the specimen, and the shaker tip is attached to the aluminum tape with wax. Again, the added inertial effects are ignored. Table 5.2 summarizes the equipment used for conducting the ground vibration experiments. The natural frequencies are the peaks in the frequency response function. To determine the mode shapes, the specimen is excited at the measured natural fre- quencies, and the response of the specimen is observed and compared to theoretical predictions of mode shapes. A strobe light is used to make it easier to visualize the mode shape at higher frequencies.
131 Normal to Flow Mode 1 Normal to Flow Mode 2 Normal to Flow Mode 3 20 40 80
18 35 70 16 30 60 14 25 50 12
10 20 40
8 15 30
Natural Frequency [Hz] 6 10 20 4 5 10 2
0 0 0 1 2 3 1 2 3 1 2 3 Streamwise Mode Streamwise Mode Streamwise Mode Figure 5.10: Configuration 1 ground vibration test theoretical and experimental natural frequencies. The x’s correspond to the theoretical predictions, and the o’s are the average experimental results and error bars are included.
Configuration 1
Configuration 1 has the short edge clamped and all other edges free. Figure 5.10 shows the comparison between the ground vibration experimental data and the the- oretical predictions. The trends and magnitude of the natural frequencies are very good. This experiment confirms that the structural model is a valid model, and that the use of a poison’s ratio of .5 and a stiffness of 18.4 MPa is correct.
Configuration 2
Figure 5.10 shows the predicted and measured natural frequencies, organized by the mode shape for Configuration 2. Each sub-figure lists the natural frequencies versus the streamwise mode number for a fixed normal to the flow mode number. The normal direction is perpendicular to the clamped edge. The results show that the theory is able to predict the natural frequencies within 10% of the measured values. Figure 5.12 shows an image captured durring the mode identification stage of the experiment. After the natural frequencies are determined the plates are forced at
132 Normal to Flow Mode 1 Normal to Flow Mode 2 Normal to Flow Mode 3 20 40 80
18 35 70 16 30 60 14 25 50 12
10 20 40
8 15 30
Natural Frequency [Hz] 6 10 20 4 5 10 2
0 0 0 1 2 3 1 2 3 1 2 3 Streamwise Mode Streamwise Mode Streamwise Mode Figure 5.11: Configuration 2 ground vibration test theoretical and experimental natural frequencies. The x’s correspond to the theoretical predictions, and the o’s are the average experimental results and error bars are included.
Figure 5.12: The (1,2) Mode Visualization for Configuration 2
133 the natural frequencies so the mode shape associated with every frequency can be determined. For this picture the system is forced at 25 Hz and the system responded in the rigid body translation in the streamwise direction and the second bending in the normal direction. It is important to note that again the theoretical calculations use an elastic mod- ulus of 18.4 MPa instead of the 1.84 MPa reported in Bloomhardt and Dowell [5]. As mentioned previously, it is possible that there is a typographical error in the reference.
Configuration 4
Normal to Flow Mode 1 Normal to Flow Mode 2 40 70
35 60
30 50
25 40 20 30 15
Natural Frequency [Hz] 20 10
5 10
0 0 1 2 3 1 2 3 Streamwise Mode Streamwise Mode Figure 5.13: Configuration 4 ground vibration test theoretical and experimental natural frequencies. The x’s correspond to the theoretical predictions, and the o’s are the average experimental results and error bars are included.
Configuration 4 is clamped-clamped in the normal to flow direction and free-free in the streamwise direction. Figure 5.14 shows the setup for this configuration. At- tached to the near side of the membrane is the shaft attached to the shaker and seen in the background is the large box that makes up the laser vibrometer. Setting up Configuration 4 so there is no tension in the streamwise direction and keeping the undisturbed membrane flat proved difficult. The experimental results in Figure
134 Figure 5.14: Ground Vibration Test Setup for Configuration 4
5.13 are uniformly below the theoretical values. This can either be caused by over- estimating the modulus of the material, or the presence of axial compression in the initial setup. Reducing the modulus to 16.4 MPa or including a compression of 4 N/m causes the experimental values to exactly match the theoretical predictions.
Configuration 6
For this series of tests, the measurements were taken at a combination of 2 laser locations and 3 shaker locations, for a total of 6 measurements. Figure 5.15 shows the locations of laser vibrometer readings and shaker excitations. In addition, impact tests were used as another method to obtain natural frequencies because shakers are known to affect the structural dynamics of flexible structures. For this ground vibration test the frequency peaks and the half power bandwidths are recorded for calculating the natural frequencies and the damping ratios. Depend- ing on the location of the shaker, the sine sweep results for the (1,1) mode can vary by 30% from the impact test results, though the frequencies of the other modes vary
135 (a) Laser Point Locations on Front (b) Shaker Tip Locations on Back Side Side Figure 5.15: Laser Readout and Shaker Excitation Locations at most by 10% from the impact test results. This confirms the expectation that the shaker introduces mass and stiffness that affects the overall structural dynamics. Therefore, the experimental data presented in this paper averages the impact test data, but the shaker is still used to determine the mode shapes. The data is collected and averages and standard deviations are calculated for each tension level and each natural mode. The frequency response does not give any information about the mode shapes, so additional testing is done to match each frequency to a mode shape by exciting the specimen at the natural frequency and comparing the resulting response to theoretically predicted mode shapes. Finding the mode shapes allows the natural frequencies to be organized by the mode number in the cross-flow direction, which is the top-bottom direction. The results are presented in Fig. 5.16 in 4 subfigures, one for each tension level. In each subfigure, the left half shows the first 3 natural frequencies that exhibit the first mode in the cross-flow direction - the (1,1), (2,1), and (3,1) modes - and the right half shows the first 3 natural frequencies that exhibit the second mode in the cross-flow direction - the (2,1), (2,2), and (3,2) modes. The damping ratio can be estimated from the transfer function using the half power method[37]. Table 5.3 lists the estimated modal damping for the first 3 modes.
136 Normal to Flow Mode 1 Normal to Flow Mode 2 Normal to Flow Mode 1 Normal to Flow Mode 2 60 80 70 100
90 70 60 50 80 60 50 70 40 50 60 40 30 40 50 30 40 30 20
Natural Frequency [Hz] Natural Frequency [Hz] Experiment Experiment 20 30 20 Theory Theory 20 10 10 10 10
0 0 0 0 1 2 3 1 2 3 1 2 3 1 2 3 Streamwise Mode Streamwise Mode Streamwise Mode Streamwise Mode (a) Natural Frequencies for Ty=0 N/m (b) Natural Frequencies for Ty=56 N/m
Normal to Flow Mode 1 Normal to Flow Mode 2 Normal to Flow Mode 1 Normal to Flow Mode 2 70 110 70 120
100 60 60 90 100
50 80 50 80 70 40 40 60 60 50 30 30 40 40 Natural Frequency [Hz] 20 30 Experiment Natural Frequency [Hz] 20 Experiment Theory Theory 20 20 10 10 10
0 0 0 0 1 2 3 1 2 3 1 2 3 1 2 3 Streamwise Mode Streamwise Mode Streamwise Mode Streamwise Mode (c) Natural Frequencies for Ty=122 N/m (d) Natural Frequencies for Ty=200 N/m Figure 5.16: Natural Frequency Results for 4 Levels of Tension: Theory and Ex- periment
Note that the results do not take multiple-degrees-of-freedom behavior into account and the calculations treat each frequency peak as a single-degree-of-freedom system. Nevertheless, the results suggest high damping ratio compared to typical isotropic elastic materials.
Table 5.3: First Three Modal Damping Ratios with No Tension Mode Number Damping Ratio Std Dev (1,1) 3.6% 1.1% (2,1) 5.9% 1.6% (3,1) 5.8% 2.0%
137 Summary
The experimental results confirmed the validity of the structural model to capture the dynamics of the plate and validated the parameters of the plate. Based on the results from the configurations tested it appears that the 18.4 MPa material modulus is correct. Additionally the more detailed experiments conducted for Configuration 6 suggest that the material has higher structural damping then was assumed for the aeroelastic simulations.
5.2.4 Plate Aeroelastic Experiments
Flutter experiments are conducted in the Duke university wind tunnel, which is a low speed wind tunnel, at a Mach number on the order of 0.1. The air speed is steadily increased and the response of the membrane is measured using a strain gauge that is attached to the specimen. The strain gauge does not protrude from the specimen, so it has little effect on the flow field. The strain gauge signal is analyzed using LabVIEW as well as the spectrum analyzer to obtain frequency content and a time history of the response amplitude. From the data, it is possible to determine the flutter speed and flutter frequency. Table 5.4 summarizes the equipment and software used for conducting the membrane flutter experiments. The laser vibrometer was not used because the reflection of the acrylic wind tunnel door resulted in poor data resolution. Figure 5.17 shows a photograph of the aeroelastic experiment set up, specifically showing the baffle mounted inside the wind tunnel cross section, with air flow from right to left. Table 5.5 gives a summary of the aeroelastic experiments. If there is a range given in the experimental values then a significant hysteresis behavior was observed. The first number given for both the frequency and velocity is the lower flutter boundary as the wind tunnel velocity is decreased from fluttering to stable behavior. The second number is the upper flutter boundary corresponding to the velocity and frequency
138 Table 5.4: Equipment Used in the Flutter Experiment Component Brand Model Signal amplifier Bruel & Kjaer Type 2635 Spectrum analyzer Scientific Atlanta SD380 Strain gauge Micro-Measurements CEA-06-125UN-350 Data acquisition National Instruments NI 9219
Figure 5.17: Photograph of Baffle Inside the Wind Tunnel that the system goes from stable to unstable as the wind tunnel velocity is increased.
5.3 Configuration 1 Aeroelastic Experiments
Configuration 1 is the same configuration as the beam experiments. During the experiments for this configuration significant hysteresis behavior were observed. This resulted in a wide range between the velocity at which the system originally becomes unstable as the wind tunnel velocity is increased and the velocity at which the system returns to being stable as the wind tunnel velocity is decrease. Figure 5.18 (a) shows the time history of the acceleration and the time history of the velocity side by side and clearly demonstrates this behavior.
139 0.1 20
0.05
15
0
10 −0.05 Wind Speed [m/s] Accelerometer Data [V]
−0.1 5
−0.15
0 30 40 50 60 70 30 40 50 60 70 Time [s] Time [s]
(a) Acceleration and Velocity Time History
0.06 0.06
0.05 0.05
0.04 0.04
0.03 0.03 FFT of Accelerometer Data FFT of Accelerometer Data 0.02 0.02
0.01 0.01
0 0 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 Frequency [Hz] Frequency [Hz]
(b) Upper Flutter Boundary FFT (c) Lower Flutter Boundary FFT Figure 5.18: Aeroelastic experimental results for Configuration 1. This configura- tion experiences a significant hysterisis behavior during experiment
140 Table 5.5: Plate Aeroelastic Experimental Results
Theory Experiment Configuration Velocity Frequency Velocity Frequency [m/s] [hz] [m/s] [hz] 1 8.55 8.54 5.94 − 17.50 3.78 − 8.18 2 8.09 3.95 7.77 − 8.90 4.15 − 3.70 6 15.09 23.21 24.05 29.05
Table 5.6: Configuration 1 Experimental Results
Lower Upper Run Velocity Frequency Velocity Frequency [m/s] [hz] [m/s] [hz] 1 5.96 3.66 17.05 7.93 2 5.99 18.29 8.55 3 5.80 3.91 18.24 8.18 4 6.00 3.78 16.43 8.06 Avg 5.94 3.78 17.50 8.18
Interestingly only the frequency at the upper flutter boundary matches the theo- retical predictions while only the velocity at the lower flutter boundary is similar to theoretical predictions. The second result can be explained by the motion seen near the lower flutter boundary. As the velocity decreases the motion of the flag begins to contain a significant torsional component as the flag attempts to buckle under its own weight due to its lack of stiffness. A proposed remedy to this problem is to include a lightweight support string at the trailing edge of the cantilevered system to support the weight of the the structure and avoid the buckling. This sag may also explain the high initial flutter velocity as the sag induces a curvature which may act to stiffen the system before the motion begins.
141 5.4 Configuration 6 Aeroelastic Experiments
The aeroelastic tests are done for the un-tensioned case. For each test, the air speed is increased in increments, and the frequency response is taken at those specific air speeds. Because 3 of the sides are clamped, the aeroelastic response is not easily observed by eye. Instead, the flutter boundary is determined based only on the strain gauge data. A waterfall plot of the frequency response functions over a range of air speeds is created for each test run, and the flutter boundary is the combination of air speed and frequency at which the strain gauge response begins to noticeably increase. An example waterfall plot is shown in Fig. 5.19. For this particular example, the flutter speed is approximately 23 m/s, and the flutter frequency is approximately 28 Hz.
28
26 0.02 24
0.01 22
0 20 40 30 18 20 U (m/s) 10 16 ω (Hz) 0 Figure 5.19: Example Waterfall Plot for the Un-Tensioned Specimen
Five trials are conducted for the un-tensioned specimen. The experimentally measured flutter speed and frequency are compared to the theoretically predicted
142 values in Table 5.7. Two sets of theoretical values, one computed with 1% structural damping, and one computed with 5% structural damping, are shown in the Table. The experimental results agree better with the high damping theoretical results. This corroborates with the high structural damping measurements from the ground vibra- tion tests, and suggests that the flutter boundary is sensitive to structural damping for this configuration. The aeroelastic results for this particular configuration show
Table 5.7: Flutter Speed and Frequency for the Un-Tensioned Specimen: Theory and Experiment Damping Flutter Flutter Ratio Speed (m/s) Frequency (Hz) Low Damping Theory 1.0% 15.5 23.1 High Damping Theory 5.0% 21.6 21.3 Experiment Average 5.1% 24.1 29.1 Experiment Stdev 1.3% 0.47 0.55 High Damping Theory Error -10% -27% the effect of structural damping. The theoretical flutter speed increases by 30% when the structural damping is increased from a typical value of 1% to the experi- mentally estimated value of 5%. The error in flutter speed compared to experiment is 10%. However, the error in flutter frequency is 27%. One possible source is error is unintended tension in the membrane, which if introduced would cause the flutter speed and frequency to increase. The usage of structural damping in the aeroelastic calculations can also be improved. Using a different value for each mode may affect the aeroelastic results. Another effect is static deformation due to initial angle of attack of the baffle structure. It is very difficult to align the baffle perfectly with the flow, and a nonzero static angle of attack leads to aerodynamic loads on the specimen, static displace- ment, and additional tensioning. This behavior is observed when starting the ex- periment as the strain measurement steadily increases as the air speed is increased. However, the extent to which the baffle is misaligned has not been quantified, and
143 will be done in future work. Lastly, the small aeroelastic deformations make it difficult to determine when flutter occurs. Typically for wing or panel flutter, a drastic increase in displacement of the test specimen occurs over a very small range of air speeds, making it easier to determine the flutter speed. Therefore, the low oscillation amplitude as well as the slow rate of increase in amplitude are possible causes of overestimating the flutter speed and frequency when looking at the waterfall data.
144 6
Conclusion and Future Work
6.1 Conclusions
This thesis is a survey of the linear aeroelastic behavior of beams and plate/membranes in low subsonic three-dimensional flow. The aerodynamics are modeled using a lin- ear vortex lattice model, and various beam and plate/membrane structural models are used. The work included the confirmation and expansion of existing theoretical and experimental results as well as an exploration of configurations and parameters which have not been explicitly explored in the aeroelastic literature. While much of the research is motivated by potential applications such as energy harvesters or noise reduction for subsonic transport aircraft, the analysis was conducted for very simplified configurations to try and isolate the fundamental dynamics of the systems that are explored. First the cantilevered beam configuration aeroelastic model was non-dimensionalized and shown to be dependent on two non-dimensional parameters, the aspect ratio H∗ and the mass ratio, µ, a result which was consistent with previous theoretical ex- plorations of the system.[13, 14] Furthermore, the trends as the parameters were
145 varied also matched well with previous theoretical and experimental results as well as new experimental results collected in the Duke University wind tunnel. A novel aeroelastic result was also discovered during this work by exploring the transition between a pinned and clamped leading edge using a torsional spring. The theoretical exploration exposed the non-trivial leading edge spring stiffness corresponded to the lowest flutter velocity. Next the structural model was changed to allow the modeling of plates with arbitrary boundary conditions. Six configurations, including one that is a simplified model of the NASA CML design were explored. For the three sides clamped, trailing edge free configuration, the configuration corresponding to the NASA CML design, an exploration of including tension in the clamped-clamped direction and varying the streamwise dimension is presented. The simulations demonstrate that the flutter velocity can be increased by either decreasing the streamwise dimension or increasing the tension in the normal to flow direction, however doing so yields non monotonic trends as the flutter motion experiences qualitative transitions as the parameters are varied. In addition to a detailed exploration of the NASA configuration, the flutter type and boundary for the five additional configurations was presented. For configurations with more than 1 fixed boundary condition, clamping the leading and trailing edge caused a divergence instability while a free trailing edge boundary condition leads to a flutter instability. Finally the transition between boundary conditions was explored by implementing a vortex lattice model that allowed for axially misaligned flows. The transition for the single side clamped configuration occurred at a low flow angle, much differently than is expected, while the transition between flutter and divergence for the three sides clamped case occurred at an intermediate flow angle. Overall the vortex lattice aerodynamic model coupled with first principals struc-
146 tural model proves to be a powerful tool in analyzing the aeroelastic stability of plates and beams in three-dimensional flow. The body of work presented in this document is a comprehensive review of the aeroelastic instabilities and trends that occur for simple plate and beam like structures subject to aerodynamic flows. While a spe- cific application is not targeted the implications of the research have been discussed throughout the document.
6.2 Future Work
6.2.1 Theoretical
There are many additional theoretical avenues which have yet to be explored. One of the first things that could be explored is the post critical aeroelastic response, specifically the limit cycle oscillations that are seen experimentally. In order to cap- ture this response, non-linear structural or aerodynamic models need to be included. Structural non-linearities could be modeled by including a cubic stiffening in the structure or modeling free play at the fixed edges. Aerodynamic non-linearities can be included by allowing for a free wake evolution, and allowing the bound circulation elements to move with the structure instead of remaining fixed in the plate plane. By including non-linearities, it will no longer be possible to analyze the system in the frequency domain, and instead time simulations will be used. In order to speed up these time simulations if a structural non-linearity is modeled, a reduced order aerodynamic model built around the linear eigenmodes of the aerodynamic matrix equation could be developed. The non-linear model is especially interesting because of hysteresis seen experimentally. Neither, the non-linearity which is responsible for this hysteresis, nor the type of bifurcation underlying this behavior have been satisfactorily explained in the literature, Another theoretical development relates to analysis of non-axially aligned flow. The existing theory approximates the angled edges of the structures as step functions,
147 which allows simple horseshoe vortex elements to be used. In order to model the actual geometry, developing an aerodynamic mesh which uses parallelogram elements instead of square elements would be an improvement. Specifically modeling flow angles near 0 and 90 deg may be accomplished without using a fine mesh.
6.2.2 Experimental
There is a significant amount of experimental work that will be done to support the existing theoretical predictions. First, the leading edge torsional spring result which suggested that a finite strength torsional spring at the leading edge will correspond to the lowest flutter velocity has not been confirmed experimentally. Before the result is fully believed this experiment must be conducted. Next, the ability to conduct experiments on models with axially misaligned flows is desirable. There is currently an undergraduate research project at Duke focused on building an experimental apparatus that will allow the rotation of a one side clamped plate which will attempt to confirm the theoretical results presented earlier in this document. Finally, experiments on beams and plates with additional parameter val- ues for all configurations would be useful as they would help validate the theoretical model over a larger region of parameter space.
6.2.3 Applications
Finally, the author is interested in solving real problems using the methods and techniques developed. The model could be used to improve the understanding of the aeroelastic instabilities that have hindered the development of High Altitude Long Endurance (HALE) aircraft, as exhibited by the high profile crash of NASA’s Helios prototype in 2003. HALE combines the low-cost relocation and storage of aircraft with the persistence and vantage point of a satellite system. However, in order to achieve mission success, these aircraft must have a high fuel fraction leading to flex-
148 ible designs that are susceptible to aeroelastic instabilities. Currently the DARPA program, Project Vulture, has funded the Boeing Company to build an experimental HALE aircraft and my research group is part of this team with significant respon- sibility for nonlinear aeroelastic analysis and testing of wind tunnel models. One could improve the existing Nonlinear Aeroelastic Trim and Stability for HALE air- craft (NATASHA) code, developed by D.H. Hodges and others to study the dynamics of HALE aircraft, by incorporating a nonlinear VLM module. This 3D aerodynamic model will complement the existing geometrically exact nonlinear structural model to provide more accurate aeroelastic predictions required for HALE design and anal- ysis. This will improve the validity of this tool and help HALE aircraft become a vital tool for maintaining an operational advantage for US defense and intelligence operations around the world. Additional applications include conducting more detailed analysis of the NASA CML configuration and providing design support for the project or developing an free wake aeroelastic wind turbine model.
149 Appendix A
Beam Aeroelastic Experimental Data Points
This appendix contains the data collected from the Duke University wind tunnel testing. All velocities are in the units of normalized velocity and all of the frequencies are in the units of radians/non-dimensional time.
Table A.1: Experimental Datapoints for a Clamped-Free Plate
Theory Experiment Error (%) ˜ ˜ ˜ µ Uflutter ω˜flutter Uflutter ω˜flutter Uflutter ω˜flutter
0.185 15.50 17.45 13.11 17.42 15.40 0.21 0.208 14.69 17.42 13.15 16.47 10.50 5.50 0.222 14.27 17.40 13.54 17.40 5.12 0.04 0.231 14.03 17.39 12.60 17.20 10.22 1.11 0.254 13.47 17.35 12.46 17.73 7.44 -2.19 0.277 12.98 17.31 11.98 17.95 7.69 -3.70 0.277 12.98 17.31 12.90 17.43 0.64 -0.68 0.312 12.36 17.25 8.96 14.58 27.53 15.46 0.333 12.05 17.21 12.03 17.43 0.17 -1.28 0.347 11.85 17.18 8.67 13.56 26.89 21.06 0.381 11.44 17.11 8.62 15.66 24.72 8.48
150 Table A.2: Experimental vs Theoretical Error
Velocity Frequency
Error 12.39% 4.0%
Error Standard 9.51% 7.57% Deviation
151 Appendix B
Configuration 2 Raw Data
This appendix shows the type of raw data that is collected from the plate ground vibration tests presented in the experimental section. The transfer function between the shaker input and laser vibrometer output is calculated on the fly by the spec- trum analyzer. After a set of sweeps a plot such as the one presented below for Configuration 2 is generated and used to determine the natural frequencies. For all configurations the location of the shaker and the laser vibrometer is varied and multiple sine sweeps are carried out. Only one data set is shown here as the data collected during different runs is qualitatively the same.
152 Figure B.1: Sample transfer function created by the spectrum analyzer after a sine sweep from 0 Hz to 100 Hz has been conducted.
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157