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