Aeroelastic Studies on a High Aspect Ratio Wing by Linda M. Mendenhall B. S., Aeronautics and Astronautics B. S., Astronomy and Astrophysics, The Pennsylvania State University, University Park, Pennsylvania (1997) Submitted to the Department of Aeronautics and Astronautics in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN AERONAUTICS AND ASTRONAUTICS at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2003 @ Massachusetts Institute of Technology 2003. All rights reserved. Author Department of Aeronautics and Astronautics January 31, 2003 Certified by Carlos E. S. Cesnik Visiting Associate Professor of Aeronautics and Astronautics Thesis Supervisor Accepted by / Edward M. Greitzer H.N. Slater Professor of Aeronautics and Astronautics Chair, Committee on Graduate Students MASSACHUSETTS INSTITUTE OF TECHNOLOGY AERO SEP 1 0 2003 LIBRARIES Aeroelastic Studies on a High Aspect Ratio Wing by Linda M. Mendenhall Submitted to the Department of Aeronautics and Astronautics on January 31, 2003, in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics Abstract A high aspect ratio (10.56) composite wing was designed and fabricated in order to verify a new analysis code developed to predict dynamic behavior of high-aspect-ratio wings. The chosen wing design was manufactured and tested (both bench-top and wind tunnel testing), and its dynamic behavior was characterized. Strain gauges and accelerometers were embedded within the wing at various distances along the span to record wing natural frequency information during bench-top 'tap' testing, and frequency changes as a function of wind speed and root angle of attack during wind tunnel testing. Bench-top natural frequencies and mode shapes for the 1 St and 2 nd Bending, 1 st Chordwise Bending, and 1 st Torsion modes are in good agreement with the analytical model. Mode shapes with frequencies exceeding 100 Hz were not considered. Within the tunnel, the wing was studies at 10, 20, and 50 root angles of attack. The 2' root angle of attack configuration was flown to the flutter point by observing of the wing behavior. This was within 99% of the predicted flutter speed. Thesis Supervisor: Carlos E. S. Cesnik Title: Visiting Associate Professor of Aeronautics and Astronautics 2 Acknowledgments The completion of this research required the assistance of many people and institutions. I would first like to thank MIT Lincoln Laboratory for allowing me to go back to school to further my education. In particular, the Lincoln Scholars Commitee for the financial support and time away from work necessary to complete such an endevor. Also my Group leaders, Bob Davis and Steve Forman, who were extremely understanding about my absence. A special thank-you to my Lincoln Tor-mentor (I mean Mentor), John Sultana, who always provided something to laugh about! Also, thanks to all the people in my group and through out the laboratory who have helped, especially Dennis Burianek, Vinny Cerrati, Charlie Nickerson, and Anne Vogel. A giant thank you goes to my family for putting up with the neglect and all of my moodiness. Especially my husband, Dr Jeffrey Mendenhall, who took over all the household and farm duties and still found time to pick me up at the train station. Special thanks to Rajah, Angie, Jack, Ian, Betty, and Sierra, for providing the devotion and unconditional love that every graduate student needs. To my parents, Charles and Marion Baker, for instilling in me the importance of education. Thanks to all my friends who allowed me to drop out of existance for awhile! I would also like to thank many people on MIT campus, who provided necessary assitance along the way. John Kane, Dave Roberts, and Donald Weiner for providing invaluable assistance during the manufacturing process. Also Dick Perdichizzi for his assistance in the tunnel. A special thanks to Professor John Dugundji for taking such an interest in my work and helping me through all of the difficult times. Thanks also to my advisor, Carlos Cesnik, for providing me with such an interesting topic. Partial funding for this work was provided by NASA Langley Research Center, under grant NAG-1-01102, and monitored by Dr. W. Keats Wilkie. Such support does not constitute an endorsement by NASA of the views expressed herein. This work was also performed for the Air Force under Air Force Contract F19628-00-C002. The opinions, interpretations, conclusions, and recommendations are solely those of the author and are not necessarily endorsed by the United States Government. 3 " I know where the pieces fit " -Schism, Tool 2001 4 Contents Abstract 2 Acknowledgments 3 19 1 Introduction 2 1.1 M otivation . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2 O bjective . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3.1 Previous Work . . . . . . . . . . . . . . . . . . . 20 1.3.2 Present Work . . . . . . . . . . . . . . . . . . . . 21 Design and Analysis of High Aspect Ratio Wing 22 2.1 Design Criteria . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Two-Layer Design . . . . . . . . . . . . . . . . . . . . . . 23 2.2.1 Basic Characteristics . . . . . . . . . . . . . . . . 23 2.2.2 Wing Non-linear Characteristics . . . . . . . . . . 26 Three-Layer Design . . . . . . . . . . . . . . . . . . . . . 51 2.3.1 Basic Characteristics . . . . . . . . . . . . . . . . 51 2.3.2 Wing Nonlinear Characteristics . . . . . . . . . . 52 . . . . . . . . . . . . . . . . . . . . 78 2.4.1 Case 1 - Maximum Vertical Displacement . . . . . 80 2.4.2 Case 2 - Maximum Pitch Rotation . . . . . . . . 80 Design Selection . . . . . . . . . . . . . . . . . . . . . . . 84 2.3 2.4 2.5 Nastran Comparison 5 3 88 Experimental Procedure Wing Manufacture ...... . . . . . . . . . 88 3.1.1 Foam Core ...... . . . . . . . . . 88 3.1.2 Instrumentation ... . . . . . . . . . 89 3.1.3 Lay-Up . . . . . . . . . . . . . . . . 93 3.1.4 Post Cure . . . . . . . . . . . . . . . 94 3.1.5 Clamp . . . . . . . . . . . . . . . . . 94 3.2 Data Acquisition Equipment . . . . . . . . . 97 3.3 Wind Tunnel Load Cell . . . . . . . . . . . . 97 3.1 99 4 Experimental Studies Bench-top Tests . . . . . . . . . . . . . . . . . 99 4.1.1 Strain Gauge Calibration . . . . . . . . 99 4.1.2 Tap Tests for Dynamic Properties . . . 104 4.1.3 Shaker Tests for Dynamic Properties . 108 4.2 Re-work of Analytical Model . . . . . . . . . . 109 4.3 Wind Tunnel Tests . . . . . . . . . . . . . . . 136 4.1 5 4.3.1 Wing Calibration in the Wind Tunnel. 136 4.3.2 Wind Tunnel Tests . . . . . . . . . . . 146 180 Concluding Remarks 5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 5.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.3 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . 181 A Mathematical Formulation 185 . . . . . . . . . . . . . . . . . . . . . . 185 . . . . . . . . . . . . . . . . . . . . . . 185 A.3 Structural Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 A.4 Aerodynamic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 A.1 Introduction ................ A.2 Geometrical Lay-out ........ A.5 Aeroelastic Formulation 6 B Material Properties 190 C Active Wing Design 192 C .1 A ctive D esign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 C.1.1 Basic Static and Dynamic Properties .................. 193 C.1.2 Wing Non-linear Characteristics .......................... 195 7 List of Figures 2-1 Cross section of two-layer wing (NACA 0012) 2-2 First six normal modes for two-layer wing (in vacuum) . . . . . . . . . . . . 25 2-3 Static tip deflection for increasing speed at different root angles of attack . . 26 2-4 Elastic tip twist for increasing speed at different root angles of attack . . . . 27 2-5 Frequency change due to root angle of attack change at U 30 m/s . . . . . 28 2-6 Frequency change due to change in speed for 20 root angle of attack . . . . . 30 2-7 Frequency change due to change in speed for 50 root angle of attack . . . . . 31 2-8 Flutter speeds for two-layer design 32 2-9 Root locus plot for 0' root angle of attack . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 33 2-10 Mode shapes for two-layer design at 0' root angle of attack and its corresponding flutter speed of 55.8 m/s . . . . . . . . . . . . . . . . . . . . . . . . 2-11 Root locus plot for 10 root angle of attack . . . . . . . . . . . . . . . . . . . 2-12 Magnification of root locus plot for 1 root angle of attack . . . . . . . . . . 34 35 36 2-13 Mode shapes for two-layer design at 10 root angle of attack and its corresponding flutter speed of 50.2 m/s . . . . . . . . . . . . . . . . . . . . . . . . 37 2-14 Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (10 root angle of attack) for the two-layer design . . . . . . . 38 2-15 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (10 root angle of attack) for the two-layer design . . . . . . . . . . . . . . . . 39 2-16 Nonlinear time simulation of the wing tip displacement motion at 10% above its flutter speed (1 root angle of attack) for the two-layer design . . . . . . . 8 39 2-17 Maximum ply strain reached at 10% above flutter speed of 10 root angle of attack for the two-layer design . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2-18 Root locus plot for 20 root angle of attack . . . . . . . . . . . . . . . . . . . 41 2-19 Mode shapes for two-layer design at 2' root angle of attack and at its corresponding flutter speed of 47.5 m/s . . . . . . . . . . . . . . . . . . . . . . . . 42 2-20 Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (2' root angle of attack) for the two-layer design . . . . . . . 43 2-21 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (2' root angle of attack) for the two-layer design . . . . . . . . . . . . . . . . 44 2-22 Nonlinear time simulation of the wing tip motion at 10% above its flutter speed (2' root angle of attack) for the two-layer design . . . . . . . . . . . . 44 2-23 Maximum ply strain reached at 10% above flutter speed of 20 root angle of attack for the two-layer design . . . . . . . . . . . . . . . . . . . . . . . . . . 2-24 Root locus plot for 5' root angle of attack . . . . . . . . . . . . . . . . . . . 45 46 2-25 Mode shapes for two-layer design at 50 root angle of attack and at its corresponding flutter speed of 38.8 m/s . . . . . . . . . . . . . . . . . . . . . . . . 47 2-26 Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (50 root angle of attack) for the two-layer design . . . . . . . 48 2-27 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (50 root angle of attack) for the two-layer design . . . . . . . . . . . . . . . . 49 2-28 Nonlinear time simulation of the wing tip motion at 10% above its flutter speed (50 root angle of attack) for the two-layer design . . . . . . . . . . . . 49 2-29 Maximum ply strain at reached at 10% above flutter speed of 50 root angle of attack for the two-layer design . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2-30 Cross section of three-layer wing (NACA 0012) . . . . . . . . . . . . . . . . . 51 2-31 First six modes shapes for the three-layer wing (in vacuum) . . . . . . . . . 53 2-32 Static tip deflection for increasing speed at different root angles of attack . . 54 2-33 Elastic tip twist for increasing speed at different root angles of attack . . . . 55 2-34 Frequency change due to root angle of attack change at U = 30 m/s ..... 57 2-35 Frequency change due to change in speed for 2' root angle of attack . . . . . 58 9 2-36 Frequency change due to change in speed for 50 root angle of attack . . . . . 59 2-37 Flutter speeds for three-layer design . . . . . . . . . . . . . . . . . . . . . . . 60 2-38 Root locus plot for 0' root angle of attack . . . . . . . . . . . . . . . . . . . 61 2-39 Mode shapes for three-layer design at 00 root angle of attack and at its corresponding flutter speed 64.6 m/s . . . . . . . . . . . . . . . . . . . . . . . . . 2-40 Root locus plot for 10 root angle of attack . . . . . . . . . . . . . . . . . . . 62 63 2-41 Mode shapes for three-layer design at 10 root angle of attack and at its corresponding flutter speed of 62.1 m/s . . . . . . . . . . . . . . . . . . . . . . . . 64 2-42 Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (10 root angle of attack) for the three-layer design . . . . . . 65 2-43 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (10 root angle of attack) for the three-layer design . . . . . . . . . . . . . . . 66 2-44 Nonlinear time simulation of the wing tip motion at 10% above its flutter speed (10 root angle of attack) for the three-layer design . . . . . . . . . . . 66 2-45 Maximum ply strain reached at 10% above flutter speed of 10 root angle of attack for the three-layer design . . . . . . . . . . . . . . . . . . . . . . . . . 2-46 Root locus plot for 20 root angle of attack . . . . . . . . . . . . . . . . . . . 67 68 2-47 Mode shapes for three-layer design at 20 root angle of attack and at its corresponding flutter speed of 56.9 m/s . . . . . . . . . . . . . . . . . . . . . . . . 69 2-48 Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (20 root angle of attack) for the three-layer design . . . . . . 70 2-49 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (20 root angle of attack) for the three-layer design . . . . . . . . . . . . . . . 71 2-50 Nonlinear time simulation of the wing tip motion at 10% above its flutter speed (20 root angle of attack) for the three-layer design . . . . . . . . . . . 71 2-51 Maximum ply strain reached at 10% above flutter speed of 20 root angle of attack for the three-layer design . . . . . . . . . . . . . . . . . . . . . . . . . 72 2-52 Root locus plot for 50 root angle of attack . . . . . . . . . . . . . . . . . . . 73 2-53 Mode shapes for three-layer design at 5' root angle of attack and at its corresponding flutter speed of 46.4 m/s . . . . . . . . . . . . . . . . . . . . . . . . 10 74 2-54 Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (50 root angle of attack) for the three-layer design . . . . . . 75 2-55 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (50 root angle of attack) for the three-layer design . . . . . . . . . . . . . . . 76 2-56 Nonlinear time simulation of the wing tip motion at 10% above its flutter speed (50 root angle of attack) for the three-layer design . . . . . . . . . . . 76 2-57 Maximum ply strain reached at 10% above flutter speed of 50 root angle of attack ....... .. ....................................... 77 2-58 Finite element mesh of the three-layer design used in Nastran (5550 elements, 13617 degrees of freedom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2-59 Nastran normal (chordwise) strain for 5' root angle of attack . . . . . . . . . 81 2-60 Nastran normal (spanwise) strain for 5' root angle of attack . . . . . . . . . 82 2-61 Nastran in-plane shear strain for 5' root angle of attack . . . . . . . . . . . . 83 2-62 Nastran normal (chordwise) strain for 5' root angle of attack . . . . . . . . . 85 2-63 Nastran normal (spanwise) strain for 5' root angle of attack . . . . . . . . . 86 2-64 Nastran in-plane shear strain for 50 root angle of attack . . . . . . . . . . . . 87 3-1 Four sections of the foam core pieced together in mold . . . . . . . . . . . . 89 3-2 Location of Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3-3 General wiring diagram for Wheatstone bridges . . . . . . . . . . . . . . . . 91 3-4 Wiring diagram for Wheatstone bridges in bending 91 3-5 Wiring diagram for Wheatstone bridges in forward/aft bending . . . . . . . 91 3-6 Wiring diagram for Wheatstone bridges in twist . . . . . . . . . . . . . . . . 91 3-7 Strain gauge at mid-span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3-8 Strain gauges wired at the root . . . . . . . . . . . . . . . . . . . . . . . . . 92 3-9 Top of root clam p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3-10 Bottom of root clamp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3-11 Pin diagram for full bridges 97 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-12 Photo of accelerometer and strain gauge conditioners 4-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Experimental set-up for bench-top tests . . . . . . . . . . . . . . . . . . . . . 100 11 4-2 Experimental instrumentation for bench-top tests . . . . . . . . . . . . . . . 100 4-3 Time trace of data from bench-top bending calibration . . . . . . . . . . . . 101 4-4 Calibration of strain gauge voltage output for bending . . . . . . . . . . . . 102 4-5 Time trace of data from bench-top twist calibration . . . . . . . . . . . . . . 103 4-6 Calibration of strain gauge voltage output for tip twist . . . . . . . . . . . . 104 4-7 Tap test for bending frequencies-root bending gauge . . . . . . . . . . . . . 105 4-8 Tap test for torsion frequencies-root torsion gauge . . . . . . . . . . . . . . 106 4-9 Tap test for chordwise bending frequencies-forward/aft gauge . . . . . . . . 107 4-10 Shaker at the wing leading edge for the natural frequency bench test . . . . 108 4-11 Fundamental mode shapes with adjusted weight and length . . . . . . . . . . 110 4-12 Predicted fundamental mode shapes based on added stiffness corrections . . 112 4-13 Root locus plot for 10 root angle of attack based on modified wing stiffness properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4-14 Magnified root locus plot for 10 root angle of attack based on modified wing stiffness properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4-15 Root locus plot for 2' root angle of attack based on modified wing stiffness properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4-16 Magnified root locus plot for 20 root angle of attack based on modified wing stiffness properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4-17 Root locus plot for 50 root angle of attack based on modified wing stiffness properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4-18 Predicted flutter speeds for various root angles of attack based on modified wing stiffness properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4-19 Predicted mode shapes after final modifications of different wing properties . 120 4-20 Root locus plot for 00 root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4-21 Magnified root locus plot for 00 root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . 122 4-22 V-g (frequency part) for 0' root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 12 4-23 V-g (damping part) for 00 root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4-24 Root locus plot for 10 root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4-25 Magnified root locus plot for 10 root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . 126 4-26 V-g (frequency part) for 10 root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4-27 V-g (damping part) for 10 root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4-28 Root locus plot for 20 root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4-29 Magnified root locus plot for 2' root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . 130 4-30 V-g plot (frequency part) for 20 root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . 131 4-31 V-g plot (damping part) for 20 root angle of Attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . 132 4-32 Root locus plot for 50 root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4-33 V-g (frequency part) for 50 root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4-34 V-g (damping part) for 50 root angle of attack based on final modifications of different wing properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4-35 Predicted flutter speed for various root angles of attack . . . . . . . . . . . . 137 4-36 Wing set-up in the Wright Brothers wind tunnel . . . . . . . . . . . . . . . . 138 4-37 Set-up for hanging weights to calibrate strain gauges 4-38 Time trace for elastic axis determination . . . . . . . . . . . . . 139 . . . . . . . . . . . . . . . . . . . . 140 4-39 Time trace for wing loaded at 50% chord in wind tunnel . . . . . . . . . . . 141 4-40 Time trace for wing loaded at 90% chord in wind tunnel . . . . . . . . . . . 142 13 4-41 Calibration of root bending gauge from both 50% and 90% chord load cases 143 4-42 Calibration of root torsion gauge from both 50% and 90% chord load cases 144 4-43 Calibration of other bending gauge (located at 35% span) from both 50% and 90% chord load cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4-44 Other bending gauge readings from undisturbed wing mounted in tunnel . . 147 . . . . 148 4-45 Root bending gauge readings from tap test-wing mounted in tunnel 4-46 Root torsion gauge readings from tap test-wing mounted in tunnel . . . . . 149 4-47 Forward/Aft bending gauge readings from tap test-wing mounted in tunnel 150 4-48 Lift curve for wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4-49 Frequency plot from root bending gauge at 20.1 m/s for 10 root angle of attack152 4-50 Frequency plot from root torsion gauge at 20.1 m/s for 10 root angle of attack 153 4-51 Frequency plot from forward/aft gauge at 20.1 m/s for 10 root angle of attack 154 4-52 Frequency plot from the other bending gauge at 20.1 m/s for 10 root angle of attack ......... 155 ....................................... 4-53 Frequency plot from root bending gauge at 36.7 m/s for 10 root angle of attack156 4-54 Frequency plot from root torsion gauge at 36.7 m/s for 1' root angle of attack 157 4-55 Frequency plot from forward/aft gauge at 36.7 m/s for 10 root angle of attack 158 4-56 Frequency plot from the other bending gauge at 36.7 m/s for 10 root angle of attack . . .. ....... 159 ....... ........................... 4-57 Time trace of data from root bending gauge at 20.1 m/s for 2' root angle of attack ......... 160 ....................................... 4-58 Frequency plot from root bending gauge at 20.1 m/s for 20 root angle of attack161 4-59 Frequency plot from root torsion gauge at 20.1 m/s for 20 root angle of attack 162 4-60 Frequency plot from forward/aft gauge at 20.1 m/s for 2' root angle of attack 163 4-61 Frequency plot from the other bending gauge at 20.1 m/s for 20 root angle of attack . . ....... .................. .. .......... ..... .164 4-62 Time trace of data from root torsion gauge at 52.3 m/s for 2' root angle of attack . . . ........ ............. 166 ....... .............. 4-63 Time trace of data from root torsion gauge at 52.3 m/s for 20 root angle of attack . .. . ....... . ................... 14 ............ .. 167 4-64 Frequency plot from root torsion gauge at 52.3 m/s for 20 root angle of attack 168 4-65 Frequency plot from forward/aft gauge at 52.3 m/s for 2' root angle of attack 169 4-66 Frequency plot from the other bending gauge at 52.3 m/s for 2' root angle of attack ......... 170 ....................................... 4-67 Frequency plot from root torsion gauge at 20.1 m/s for 50 root angle of attack 171 4-68 Frequency plot from forward/aft gauge at 20.1 m/s for 5' root angle of attack 172 4-69 Frequency plot from the other bending gauge at 20.1 m/s for 5' root angle of attack ........ .... 173 ......... ....................... 4-70 Frequency plot from root torsion gauge at 33.5 m/s for 5' root angle of attack 174 4-71 Frequency plot from forward/aft gauge at 33.5 m/s for 5' root angle of attack 175 4-72 Frequency plot from the other bending gauge at 33.5 m/s for 5' root angle of attack .......... 176 ......... .......................... 4-73 Root bending gauge readings from tap test-wing mounted in tunnel . . . . 177 4-74 Root torsion gauge readings from tap test-wing mounted in tunnel . . . . . 178 4-75 Average tip accelerometer readings from tap test-wing mounted in tunnel . 179 A-i Coordinate systems for the wing . . . . . . . . . . . . . . . . . . . . . . . . . 186 A-2 Variables Defining Airfoil Motion . . . . . . . . . . . . . . . . . . . . . . . . 188 C-1 Cross section of active wing . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 C-2 First six normal modes for active wing design (in vacuum) . . . . . . . . . . 194 C-3 Static tip deflection for increasing speed at different root angles of attack . . 195 C-4 Elastic tip twist for increasing speed at different root angles of attack . . . . 196 C-5 Flutter speeds for active wing design . . . . . . . . . . . . . . . . . . . . . . 197 C-6 Root locus plot for 10 root angle of attack . . . . . . . . . . . . . . . . . . . 198 C-7 Magnification of root locus plot for 10 root angle of attack . . . . . . . . . . 199 C-8 Mode Shapes for active wing design at 10 root angle of attack and its corresponding flutter speed of 79.3 m/s . . . . . . . . . . . . . . . . . . . . . . . . 200 C-9 Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (1 root angle of attack) for the active wing design 15 . . . . . 201 C-10 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (10 root angle of attack) for the active wing design . . . . . . . . . . . . . . . 202 C-11 Nonlinear time simulation of the wing tip displacement motion at 10% above its flutter speed (1' root angle of attack) for the active wing design C-12 Root locus plot for 20 root angle of attack . . . . . 202 . . . . . . . . . . . . . . . . . . . 203 C-13 Magnification of root locus plot for 2' root angle of attack . . . . . . . . . . 204 C-14 Mode shapes for active wing design at 2' root angle of attack and its corresponding flutter speed of 77.3 m/s . . . . . . . . . . . . . . . . . . . . . . . . 205 C-15 Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (2' root angle of attack) for the active wing design . . . . . 206 C-16 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (20 root angle of attack) for the active wing design . . . . . . . . . . . . . . . 207 C-17 Nonlinear time simulation of the wing tip displacement motion at 10% above its flutter speed (20 root angle of attack) for the active wing design . . . . . 207 C-18 Root locus plot for 5' root angle of attack . . . . . . . . . . . . . . . . . . . 208 C-19 Mode shapes for active wing design at 50 root angle of attack and its corresponding flutter speed of 68 m/s . . . . . . . . . . . . . . . . . . . . . . . . . 209 C-20 Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (50 root angle of attack) for the active wing design . . . . . 210 C-21 Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (50 root angle of attack) for the active wing design . . . . . . . . . . . . . . . 211 C-22 Nonlinear time simulation of the wing tip displacement motion at 10% above its flutter speed (50 root angle of attack) for the active wing design 16 . . . . . 211 List of Tables 2.1 Non-zero stiffness matrix terms for two-layer design . . . . . . . . . . . . . . 24 2.2 Non-zero inertial matrix terms for two-layer design . . . . . . . . . . . . . . 24 2.3 Natural frequencies of the two-layer design in vacuum . . . . . . . . . . . . . 24 2.4 Dynamic properties at U = 30 m/s . . . . . . . . . . . . . . . . . . . . . . . 29 2.5 Dynamic properties at angle of attack = 2 . . . . . . . . . . . . . . . . . . . . 29 2.6 Dynamic properties at angle of attack = 5 . . . . . . . . . . . . . . . . . . . 32 2.7 Non-zero stiffness matrix terms for three-layer design . . . . . . . . . . . . . 51 2.8 Non-zero inertial matrix terms for two-layer design . . . . . . . . . . . . . . 51 2.9 Natural frequencies of the three-layer design in vacuum . . . . . . . . . . . . 52 2.10 Dynamic properties at U = 30 m/s . . . . . . . . . . . . . . . . . . . . . . . 56 2.11 Dynamic properties at root angle of attack = 20 . . . . . . . . . . . . . . . . 56 2.12 Dynamic properties at root angle of attack = 50 . . . . . . . . . . . . . . . . 60 2.13 Summary of Strain Analysis Results . . . . . . . . . . . . . . . . . . . . . . . 78 2.14 Nastran Case 1: Displacements and Rotations . . . . . . . . . . . . . . . . . 80 2.15 Nastran case 1: summary of results . . . . . . . . . . . . . . . . . . . . . . . 80 2.16 Nastran case 2: displacements and rotations . . . . . . . . . . . . . . . . . . 81 2.17 Nastran case 2: summary of results . . . . . . . . . . . . . . . . . . . . . . . 84 3.1 Accelerometers used in the wing model . . . . . . . . . . . . . . . . . . . . . 93 3.2 Load cell used for the wing tunnel tests . . . . . . . . . . . . . . . . . . . . . 98 4.1 Natural Frequencies From Bench-top Tap Tests 4.2 Natural frequencies with adjusted weight and length . . . . . . . . . . . . . . 109 17 . . . . . . . . . . . . . . . . 106 4.3 Intermediate predicted natural frequencies based on added stiffness corrections 111 4.4 Intermediate predicted flutter speeds based on corrected structural properties 111 4.5 Final values of adjusted parameters . . . . . . . . . . . . . . . . . . . . . . . 119 4.6 Final natural frequencies from wing analysis . . . . . . . . . . . . . . . . . . 120 4.7 Predicted flutter speeds with final model adjustments . . . . . . . . . . . . . 136 4.8 Frequencies detected from strain gauges at 20.1 m/s for 2' root angle of attack165 B.1 Properties of the materials used in this study . . . . . . . . . . . . . . . . . . 191 C. 1 Non-zero stiffness matrix terms for the active wing design . . . . . . . . . . . 193 C.2 Non-zero inertial matrix terms for active wing design . . . . . . . . . . . . . 193 C.3 Natural frequencies of the active wing design in vacuum . . . . . . . . . . . . 194 18 Chapter 1 Introduction 1.1 Motivation Many industries consider using High Altitude Long Endurance (HALE) aircraft to perform autonomous sensing (environmental, urban, and reconnaissance), or to assist in civilian or military communication as a relay [1]. The wings of these vehicles are typically of high aspect ratio. Long slender wings result in large deflections during flight, especially when flown at high angles of attack. These large deflections cannot be ignored when analyzing the wing's flight characteristics. The change in the wing's stiffness, due to nonlinear geometric behavior, requires new methods of modeling [2]. This change in stiffness results in changes to the wing's dynamic behavior. There has been considerable research in this area by numerous groups, resulting in new modeling techniques and codes which need to be validated through experimentation. One such computational modeling tool is being developed by Cesnik and Brown [3]. This research is a follow-on of work performed by Cesnik and Ortega-Morales [4]. The ground work for this was set forth by Patil, Hodges, and Cesnik [1], [5], and [6]. 1.2 Objective The primary objective of this research is to design and test an experimental wind tunnel model wing. This test wing should present nonlinear aeroelastic behavior. The collected 19 data will then support the validation of the MATLAB computer code being developed by Cesnik and Brown [3]. Originally this would encompass all portions of the code, including the ability to model active materials embedded within the layers of the composite wing. However, due to reasons beyond the control of the author, no active structural tests are included in this study. 1.3 Background As mentioned above, there has been substantial effort towards developing analytical models which accurately depict the behavior of high aspect ratio wings undergoing large deflections. This work is summarized in the next two sections. 1.3.1 Previous Work In the late 1980s, Minguet and Dugundji studied the effects of deflection on dynamic behavior of thin composite strips [7]. In their study, it was shown that deflection has significant effect on the torsion and chordwise bending frequencies. Their beam analysis obtained a set of linearized equations about a deflected position. The experiments for this research utilized flat composite beams of various composite orientations and thicknesses. This study showed that deflections of as little as 3% of the total length affected the torsional and chordwise bending modes, while the normal bending modes remained relatively uneffected. They also showed that tip bending deflections between 5% and 10% of the total beam length resulted in a mix between the 1 st torsion and 1 "t chordwise bending modes. Patil, Hodges and Cesnik [8] have worked to improve modeling capabilities for high aspect ratio wings, which include geometrical non-linearities for structural and aerodynamic analyses. Their research has shown that accurate modeling of geometric non-linearities is necessary to fully understand the torsion/bending coupling observed during flutter of wings with large deflections [8]. Their modeling of nonlinear aeroelasticity is based on a mixed variational formulation for the dynamics of beams in moving reference frames and finitestate airloads to accommodate deformable airfoil effects [1]. 20 Tang and Dowell have studied various ways of modeling high aspect ratio wing aeroelastic behavior and correlated their work with wind tunnel tests. In the late 1990's, they built a 45-cm wing with an aspect ratio of 8.89 [9). The wing was flown into the flutter domain to study limit cycle oscillation behavior. They also used this wing to study the modeling of gust response [10]. Ortega-Morales and Cesnik created a framework for implementing active aeroelastic tailoring of high aspect ratio wings, as well as gust mitigation [4]. The active tailoring was implemented through embedded piezoelectric actuators within the composite skin of the wing. This work implemented many of the analysis techniques studied in [1], [8], [2], [5], and [6], and consisted of an asymptotically correct active cross-section formulation, geometrically-exact mixed formulation for dynamics of moving beams, and finite-state unsteady aerodynamics. 1.3.2 Present Work Cesnik and Brown are continuing the work done in [4]. This work formulated the MATLAB code used within this thesis, which employs a strain-based finite element representation of a nonlinear, large deflection beam model experiencing finite-state unsteady airloads [11]. This program has the capability to model active materials embedded within the composite lay-up of the wing and to take into account effects from the fuselage and tail [3]. The basic formulation behind this modeling effort is summarized in Appendix A. Two different wing designs were generated using the code from [11] and are presented in Chapter 2 of this thesis. These designs were chosen for their torsional flexibility and interesting flutter characteristics. Chapter 3 discusses the manufacture and instrumentation of the wing. The benchtop and wind tunnel experiments, as well as the experimental results are discussed in Chapter 4. Along with these results is a modified numerical model which replicates the results of the tests. Finally, conclusions and suggestions for further work are presented in Chapter 5. This thesis contains three Appendices. Appendix A provides a summary of the mathematical formulation for the code, Appendix B contains relevant material properties, and Appendix C provides the preliminary studies for a future active wing build-up. 21 Chapter 2 Design and Analysis of High Aspect Ratio Wing This chapter describes the design and analysis of the static flexible wing that was built to validate the model of [3] for subsonic speeds. Some of the design parameters were set by the wind tunnel limitations and others by ease of fabrication. 2.1 Design Criteria Since this study targets large deflection aeroelastic response, a high aspect ratio flexible wing was needed. To reduce the risk of damage to the wind tunnel during the dynamic portion of testing, the design flutter speed for this wing should be in the range of 40 m/s. This relatively low flutter speed reduces the amount of momentum the wing would have in case it becomes unstable and brakes away from the mounting fixture. As a result, a very flexible wing design, soft in torsion, is expected. Although the wing needed to be as long as possible to increase the aspect ratio, the span of the wing was also limited by the wind tunnel test section dimension. The minimum chord of the wing also has limitations. It could not be too small, as that would significantly increase the complexity of manufacturing. With these constraints in mind, the length of the wing was set to 1.13 meters (44.5 inches) and the chord length was set to 0.107 meters (4.22 inches). This yielded an aspect ratio of 10.56. 22 The airfoil shape chosen is the NACA 0012. Safety to the wing tunnel was a major concern, so the wing needed to possess margins of safety for strain within the E-glass/epoxy of at least 1. For added safety this requirement needed to be met when the wing was analyzed at 10% above the predicted flutter speed. Also, the wing needed to exhibit limit cycle oscillation at speeds up to 10% above the predicted flutter values. After several preliminary layup considerations, two basics ones were selected for detailed studies: a so-called "two-layer" design and a "three-layer" design. They represent flexible designs, with flutter speeds within the appropriate range. They also demonstrate interesting flutter characteristics. 2.2 2.2.1 Two-Layer Design Basic Characteristics The two-layer design is composed of two-layers of E-glass/epoxy fabric oriented at 0' around an inner foam core. A table of the material properties is provided in Appendix B. There is no spar in this design. This design provides a very flexible wing while still maintaining closed cell properties. A cross-sectional representation is given in Figure 2-1. [021 foam Figure 2-1: Cross section of two-layer wing (NACA 0012) The cross-sectional properties for the two-layer design are provided below. The stiffness matrix terms for this cross-section are given in Table 2.1. Detailed definition of the elements of this matrix are provided in Appendix A. The inertia matrix for this cross-section is provided in Table 2.2. The center of gravity for the cross-section is located 0.0182 m aft of the reference line (located at 30% chord). This places the center of gravity at 47% of the chord. 23 Table 2.1: Non-zero stiffness matrix terms for two-layer design: 1 = Extension, 2 = Torsion, 3 = Flatwise Bending, 4 = Chordwise Bending Kn1 K 14 = K 4 1 9.55 * 105 N -1.68 * 104 N*m 14.24 N*m 2 22.25 N*m 2 1.18 * 103 N*m 2 K2 2 K33 K44 Table 2.2: Non-zero inertial matrix terms for two-layer design I, I22 133 0.10 * 10-3 m4 0.22 * 10-5 m 4 0.98 * 10-4 m 4 The dynamic properties of the wing are important to its aeroelastic response. The first six natural frequencies of the wing are summarized in Table 2.3. The corresponding mode shapes are provided in Figure 2-2. Table 2.3: Natural frequencies of the two-layer design in vacuum Mode Frequency Mode Shape 1 2 8 Hz 50 Hz 3 4 5 52 Hz 95 Hz 153 Hz Bending 1 st Chordwise Bending 2 "d Bending 6 292 Hz 1 st 1" Torsion 3 rd Bending 2 nd 24 Torsion Speed = 0.00 m/s @ Freq = 7.97 Hz Speed = 0.00 m/s @ Freq = 49.9 Hz 0.8 Speed = 0.00 m/s @ Freq = 51.5 Hz Speed = 0.00 m/s @ Freq = 95.1 Hz 0.2 0.4 0.6 0.8 0.1 Speed = 0.00 m/s @ Freq = 153 Hz Speed = 0.00 m/s @ Freq = 292 Hz 0.2 0.2 0.4 -0.1 -0.05 0.4 0.6 0.6 0.8 0.8 0 1 1 Figure 2-2: First six normal modes for two-layer wing (in vacuum) 25 2.2.2 Wing Non-linear Characteristics Different root angles of attack of the wing result in different wing tip deflections once the wing is exposed to airloads. The change in tip deflection is graphed in Figure 2-3 for the range of angles of attack of interest: 0' to 5'. The tip deflection increases with increased angle of attack. This is to be expected, since the aerodynamic load is directly proportional to the angle of attack. 0.6 0.5 0.4 E 0 0.3 a a) 0 ca. 0.2 0*0 -0.11 0 5 10 15 25 20 30 35 40 45 Speed (m/s) Figure 2-3: Static tip deflection for increasing speed at different root angles of attack The wing tip twist also changes with increased root angle of attack. This data is provided in Figure 2-4. To better illustrate this effect, the initial root angle of attack has been subtracted from the total tip twist, yielding only the elastic angle change due to gravitational 26 50 and aero-loading. 0.9 0.8 0.7 -.-.-. AO 0 . . . .... . - . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . 0.63 . . .. .. -. -. -. F 0 .4 -. 0.13 .-.- . .. . 5 5-0 5-0 0 . .. . . . . 30-5 40 5 45 -0 01 Speed (mis) Figure 2-4: Elastic tip twist for increasing speed at different root angles of attack The changes in tip deflection and twist result in a change in the dynamic behavior of the wing, and ultimately a change in the wing stability. This is illustrated by the change in the mode shapes and their corresponding frequencies for the wing at different root angles of attack at 30 in/s. Table 2.4 gives the frequencies of the first six modes for 00, 1 , 2~ and 50. Figure 2-5 contains the first four modes graphically. It is interesting to note that as the wing is bent upwards due to the force of air pressure, the natural frequencies of the second mode decrease and the fourth mode increase. effect will cause some different behaviors in the wing as the different modes interact. 27 This 110 1T 100 90 80 70 60 2B Cr 50 LL. 1CB 40 30 20 1B 10 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Root Angle of Attack (deg) Figure 2-5: Frequency change due to root angle of attack change at U 28 = 30 m/s 5 Table 2.4: Dynamic properties at U = 30 m/s Angle of Attack Tip Deflection 00 -0.005 m 10 0.03 n 20 0.06 m 50 0.16 n 8 8 8 8 1" Bending 50 49 48 40 2" Bending 1 st Torsional 3 rd Bending 52 95 153 52 96 153 52 98 153 52 107 153 2 "d 292 293 295 303 1" Mode Shapes Chordwise Bending Torsional A similar behavior is also seen when the speed increases for a set angle of attack. This behavior is illustrated in Table 2.5 and Table 2.6. The first four modes for each of these examples is presented graphically in Figure 2-6 and Figure 2-7, respectively. Table 2.5: Dynamic properties at angle of attack = 20 Speed Tip Deflection Mode Shapes 30 m/s 0.06 rn 35 m/s 0.09 m 40 m/s 0.12 rn 45 m/s 0.16 rn 1" Bending 1 st Chordwise Bending 2 " Bending 8 48 52 8 46 52 8 43 52 8 40 52 1' Torsional 3 rd Bending 2 nd Torsional 98 153 295 100 153 297 103 153 300 107 153 303 This change in the dynamic behavior of the wing results in changes in its flutter speed. This is shown in Figures 2-9, 2-11, 2-18 and 2-24, which present the root locus plots for the wing at root angles of attack of 0', 10, 20 and 5'. The flutter speeds as function of root angle of attack determined from these plots are summarized in Figure 2-8. 29 120 1T 80 z U C 0) 60 0. U- 2B 40 1CB 20 1B 0 30 32 34 36 38 40 42 44 46 48 Speed (m/s) Figure 2-6: Frequency change due to change in speed for 2' root angle of attack 30 50 IT 120 100 80 Cr 0~ 60 UL- 2B 40 1CB 20 1B 0 30 32 34 36 38 40 42 44 46 48 Speed (m/s) Figure 2-7: Frequency change due to change in speed for 50 root angle of attack 31 50 Table 2.6: Dynamic properties at angle of attack = 50 Speed Tip Deflection Mode Shapes 30 m/s 0.16 m 35 m/s 0.22 m 40 m/s 0.30 m 45 m/s 0.38 m 8 40 52 107 8 35 52 114 8 30 51 119 8 25 51 122 153 303 153 300 152 293 151 284 1" Bending 1 st Chordwise Bending 2 " Bending 1 " Torsional Bending _ 2 " Torsional 3 rd 60 50 40 E a> . 30 C') a> i20 10 O'L 0 1 2 3 4 Root Angle of Attack (deg) 5 Figure 2-8: Flutter speeds for two-layer design 32 6 7 Zero-Degree Root Angle of Attack The flutter speed for zero degree root angle of attack is determined to be 55.8 m/s from the root locus plot, given in Figure 2-9. The unstable mode is the fourth mode, which is primarily the first torsion mode. There is some chordwise bending present as well. The mode shapes for the wing at this speed are given in Figure 2-10. Root locus for Root AOA = 0* 200 180 - ------ --------- ---------- -- 40 30 * ---- 140 - 20 ---- - -- - - ------------- - - - --- - - -- -- 3B 10 :0 --------------------------- ---------- 120 U 100 r ---------- ------------ 0~ 'I) LI~ -- - -- T --------- --- --- - - --- - --- ------------- ---------------------- I -------------------------------- - -- --- - - -- 80 ---- - - -- ----60 -------------------- ---------------- - -- - - - ---- ------- -2B ~-- 1CB -931 40 - 20 ------ - - - - - - - - - - - - - -- - - - - - - - - - - - ---- - ---- - ---- - - ----- - - - - - - - - - --- - ----- -- -- - - - - - - --- - --- - - - - - - - - - - - - - - -~ -20 0L 50 40 30 10 0 0 10 Real Part of Root Figure 2-9: Root locus plot for 00 root angle of attack 33 - - 1B 1e 20 - 20 Speed = 55.80 m/s @ Freq = 7.97 Hz Speed = 55.80 m/s @ Freq = 49.9 Hz -0.6 -0.4 Speed = 55.80 m/s @ Freq = 51.5 Hz Speed = 55.80 m/s @ Freq = 95 Hz 0.2 0.4 0.6 0.8 1 Speed = 55.80 m/s @ Freq = 153 Hz Speed = 55.80 m/s @ Freq = 292 Hz 0.2 0.2 -0.1 -0.05 0.4 0.4 - 0.6 0.6 0.8 0.8 0 1 1 Figure 2-10: Mode shapes for two-layer design at 0' root angle of attack and its corresponding flutter speed of 55.8 m/s 34 One-Degree Root Angle of Attack The flutter speed for one-degree root angle of attack is determined to be 50.2 m/s, shown in Figure 2-11. The mode of instability is the second mode. A magnification of this mode is provided in Figure 2-12. This mode is primarily the first chordwise bending. However, the forces due to air pressure on the tilted wing result in some torsion being present as well. The first six mode shapes at this speed are given in Figure 2-13. Root locus for Root AOA = 1* 200 160 3B ifP] AfAf 140 ------------- ----------100 H- ----------- 1T Cr !0 4) LL ----------- 60 2B 14A lCB 40 20 H- - - - - - - - - - - ------------1B U 50 50 40 30 10 20 0 10 Real Part of Root Figure 2-11: Root locus plot for 10 root angle of attack 35 20 Root locus for Root AOA = 1 55 -- -- - -- -- I-- -- I--- I-- - - 2B 5 0 - --400 8 -- - - - C 4 5 - - - - --- ------- 1 ... . - . ..-..-- - -. - .. .- ..-- - cr2 1GB LL- ..... 04. 40 -- - - -- - - 10 8 6 I -- - 4 2 0 2 4 -- - - - - 6 8 10 Real Part of Root Figure 2-12: Magnification of root locus plot for 10 root angle of attack 36 -- - - - Speed = 50.20 m/s @ Freq = 7.97 Hz Speed = 50.20 m/s @ Freq = 44.7 Hz -0.05 0. 0.05 0.1 Speed = 50.20 m/s @ Freq = 51.9 Hz 1 Speed = 50.20 m/s @ Freq = 101 Hz 0.2 0.4 0.6 1 Speed = 50.20 m/s @ Freq = 153 Hz Speed = 50.20 m/s @ Freq = 298 Hz 0.2 0.2 -0.1 -0.05 0.4 0.4 0.6 0.6 0.8 0.8 0 1 1 Figure 2-13: Mode shapes for two-layer design at 1' root angle of attack and its corresponding flutter speed of 50.2 m/s 37 When the wing is flown at 10% above the flutter speed for a 10 root angle of attack, the wing enters into a Limit Cycle Oscillation (LCO). The tip deflections and the tip twist are plotted for this case up to 1 second. These plots are given in Figure 2-14 thru Figure 2-16. Speed = 55.20 m/s with a Time step = 0.0005 s 0.14 0.12 0.1 E 0) Ca) CI) F0 0.08 0.06 D_ Fr C.D 0.04 0.02 0 -0.02' 0 0.2 0.4 0.6 0.8 1 Time (s) Figure 2-14: Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (10 root angle of attack) for the two-layer design 38 Speed = 55.20 m/s with a Time step = 0.0005 s 5 4 3 2 0) 0_0 1 0 1 -2-3 0 0.2 0.4 0.6 0.8 1 Time (s) Figure 2-15: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (10 root angle of attack) for the two-layer design Speed = 55.20 m/s with a Time step = 0.0005 s 0.13 0.125 0.12 E U) 0.1 15 [ 0 N 0.11 0. 105 F 0.1 '-0.015 -0.01 -0.005 0 0.005 Y Displacement (m) 0.01 0.015 Figure 2-16: Nonlinear time simulation of the wing tip displacement motion at 10% above its flutter speed (10 root angle of attack) for the two-layer design 39 A strain analysis was performed on the wing when it was at its maximum deflection. This occurs during the initial rise shown in Figure 2-14. The maximum tensile strain is 2660 pm/m and the maximum shear strain is -769 pm/m. Both occur at the bottom of the root of the wing in the outer ply. These results are summarized in Figure 2-17. From Appendix B, the maximum allowable strain for E-glass/epoxy is 10000 pm/m in tension and 15000 pum/m in shear. This results in a margin of safety (Equation 2.1) of 2.76 for tension and 18.5 for shear. Allowable MS(%) = Actual - 1 Actual (2.1) eps1 (longitudinal fiber strain), ply 3 eglass 120: MAX = 0.0026595 eps2 (transverse fiber strain), ply 3 eglass 120: MAX = -0.00039362 eps3 (inplane shear strain), ply 3 eglass 120: MAX = -0.00076936 Figure 2-17: Maximum ply strain reached at 10% above flutter speed of 10 root angle of attack for the two-layer design 40 Two-Degree Root Angle of Attack The flutter speed for two-degree root angle of attack is determined to be 47.5 m/s from the root locus plot shown in Figure 2-18. The instability mode is the same as for 10, except the amount of torsion present has increased. Also, by the time the angle of attack is 2', the effects of the large deflections on the wing cause the natural frequency of the fourth mode to increase with increasing speed instead of decrease as it did in the 00 and 10 cases. The drop in the second natural frequency due to the increased upward bend also causes the second mode to turn over faster and amplifies the effects of flow over the wing. The first six modes at flutter speed are provided in Figure 2-19. Root locus for Root AOA = 2* 200 180 -- - --- --- - ------ - ---- --- --- - 160 3B 30 40 2n 1 140 120 V- ---- - - - -- - - - . .- ------ - - N = U C 100 --- _- -- - ------------------T - --- -- - 0 ~1) LI~ - 80 ------- 60 . - -44 -- -- - --- - - - - - -- --- - - - -- - - - - -~- - - -- ------ - --2B-' 140 1CB 40 -- - -- - - - - - -- - - - - - --- *~ 20 --- - - - - --- V B 0 - ------ -_ . ..-.. --. -- ------- ---.- .- -. -- .- 1B -20E 50 40 30 10 20 10 Real Part of Root 0)i 0 10 Figure 2-18: Root locus plot for 2' root angle of attack 41 20 Speed = 47.50 m/s @ Freq = 7.98 Hz Speed = 47.50 m/s @ Freq = 38.2 Hz Speed = 47.50 m/s @ Freq = 51.8 Hz 1 Speed = 47.50 m/s @ Freq = 109 Hz Speed = 47.50 m/s @ Freq = 153 Hz Speed = 47.50 m/s @ Freq = 302 Hz 0.2 0.2 0.4 0.4 0.6 -.. 0.6 0.8 0.8 1 Figure 2-19: Mode shapes for two-layer design at 2' root angle of attack and at its corresponding flutter speed of 47.5 m/s 42 For 2' angle of attack, the wing also enters a LCO when flown at 10% above its flutter speed. The tip deflections and the tip twist are plotted for this case up to 1 second. These plots are given in Figure 2-20 thru Figure 2-22. Notice from Figure 2-22 that the LCO now has more than one frequency (no single ellipse on the plot), indicating potentially chaotic response. Speed = 52.30 m/s with a Time step = 0.0005 s 0.3 0.25 0.2 E a) 0 CO 0.15 0.1 0 0.05 0 -0.05- 0 0.2 0.4 0.6 0.8 1 Time (s) Figure 2-20: Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (20 root angle of attack) for the two-layer design 43 Speed = 52.30 m/s with a Time step = 0.0005 s 15 10 5 0) U, V 0 I0. R -5 -10 -15 0 0.2 0.4 0.6 1 0.8 Time (s) Figure 2-21: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (20 root angle of attack) for the two-layer design Speed = 52.30 m/s with a Time step = 0.0005 s 0.16 0.15 0.14 E 0.13 U> E (D 0 0.12 CN 0 N 0.11 0.11- 0.09 0.08' -0.04 -0.03 -0.02 0.01 0 -0.01 Y Displacement (m) 0.02 0.03 0.04 Figure 2-22: Nonlinear time simulation of the wing tip motion at 10% above its flutter speed (20 root angle of attack) for the two-layer design 44 A strain analysis was performed on the wing when it was at its maximum deflection. This occurs during the transient rise shown in Figure 2-20. The maximum tensile strain is 4484 pm/m and the maximum shear strain is -2726 pm/m. Both occur at the bottom of the root of the wing in the outer ply. These details are summarized in Figure 2-23. These strains result in a margin of safety of 1.23 and 4.5, respectively. eps1 (longitudinal fiber strain), ply 3 eglass 120: MAX = 0.0044837 eps2 (transverse fiber strain), ply 3 eglass 120: MAX = -0.00066361 eps3 (inplane shear strain), ply 3 eglass 120: MAX = -0.0027264 Figure 2-23: Maximum ply strain reached at 10% above flutter speed of 2' root angle of attack for the two-layer design Five-Degree Root Angle of Attack The flutter speed for five-degree root angle of attack is determined to be 38.8 m/s from the root locus plot provided in Figure 2-24. As with the 1 and 2' cases, the mode of instability is the second mode, which is the 1" chordwise bending with some torsional effects. As was seen in the 20 case, the changes in the dynamic behavior due to the increased upward bend result in the second mode rolling over faster to the positive dampening axis and the fourth 45 mode to further curve upward. Although with the fourth mode, we see an interaction with the fifth mode (3 rd bending) starting to cause the fourth mode to turn towards the instability line as well. The first six mode shapes for this case are provided in Figure 2-25. Root locus for Root AOA = 5* 200 180 - - - -- - - - - --- - - - - ------ - -- - - - -- - ----- -- 160 3B 140 120 I Li 100 4 4024 --------------3 20 30 0 -- 13 0 -- 1T 10 ~LI LL 80 --- - - - - - - - ----- - - - - - - - - - - - --- - - - - - -- ---. 60 . . . . . . - . .- - -- - --20- - - - - -- - --- 2B 40 --- ------ --------- ----- ---- - - ----- - - - ---- - .- - - 0- -- - 1 - -- - - - -- 20 -20 0 50 40 30 10 20 10 1B 0 10 Real Part of Root Figure 2-24: Root locus plot for 50 root angle of attack 46 20 Speed = 38.80 m/s @ Freq = 7.99 Hz Speed = 38.80 m/s @ Freq = 30.8 Hz Speed = 38.80 m/s @ Freq = 51.5 Hz Speed = 38.80 m/s @ Freq = 118 Hz Speed = 38.80 m/s @ Freq = 152 Hz Speed = 38.80 m/s @ Freq = 294 Hz 0.1 00 - - - -- -00. 0.8 Figure 2-25: Mode shapes for two-layer design at 5' root angle of attack and at its corresponding flutter speed of 38.8 m/s 47 For 5' root angle of attack, the wing also enters a LCO when flown at 10% above the flutter speed. The tip deflections and the tip twist are plotted for this case up to 1 second. These plots are given in Figure 2-26 thru Figure 2-28. Speed = 42.70 m/s with a Time step = 0.0005 s 0.4 0.35 0.3 E 0 0.25 0.2 CO CO, 0.15 0.1 0.05 0 -0.05 L 0 0.2 0.4 0.6 0.8 1 Time (s) Figure 2-26: Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (50 root angle of attack) for the two-layer design A strain analysis was performed on the wing when it was at its maximum deflection. This occurs during the initial rise shown in Figure 2-26. The maximum tensile strain is 7405 pm/m and the maximum shear strain is -5018 pm/m. Both occur at the bottom of the root of the wing in the outer ply. These details are summarized in Figure 2-29. These strains result in a margin of safety of 0.35 and 1.99, respectively. 48 Speed = 42.70 m/s with a Time step = 0.0005 s 20 15 10 aO CL 5 P, 0 -5 - -10 0 0.2 0.4 0.6 0.8 1 Time (s) Figure 2-27: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (50 root angle of attack) for the two-layer design Speed = 42.70 m/s with a Time step = 0.0005 s 0.3 0.28- 0.26C a) E C, 0.24C., .O 0 N 0 .22 F 0.2 F 0.18' -0.05 -0.04 -0.03 -0.02 -0.01 Y Displacement (m) 0 0.01 0.02 Figure 2-28: Nonlinear time simulation of the wing tip motion at 10% above its flutter speed (50 root angle of attack) for the two-layer design 49 eps1 (longitudinal fiber strain), ply 3 eglass 120: MAX = 0.0074051 eps2 (transverse fiber strain), ply 3 eglass 120: MAX = -0.001096 eps3 (in plane shear strain), ply 3 eglass 120: MAX = -0.0050181 .. . .. . . I [IWI, MULL,,, IDIUWI lirnIT Figure 2-29: Maximum ply strain at reached at 10% above flutter speed of 5' root angle of attack for the two-layer design 50 2.3 Three-Layer Design 2.3.1 Basic Characteristics The three-layer design resembles the previous one with the exception of an added layer of E-glass/epoxy fabric, also oriented at 00 (Figure 2-30. [03] foam Figure 2-30: Cross section of three-layer wing (NACA 0012) The cross sectional stiffness properties for this design are provided in Table 2.7. A detailed definition of the terms for this matrix are provided in Appendix A. The inertia matrix for Table 2.7: Non-zero stiffness matrix terms for three-layer design: 1 = Extension, 2 = Torsion, 3 = Flatwise Bending, 4 = Chordwise Bending Ku1 K14= K41 1.39 *106 N -2.35 * 10 4 N*m K2 2 19.66 N*m 2 K33 33.19 N*m 2 K44 1.63 * 103 N*m 2 this cross-section is provided in Table 2.8. The center of gravity for the cross-section is Table 2.8: Non-zero inertial matrix terms for two-layer design Ill I22 133 0.13 * 10-3 m 4 0.32 * 10- 5 m 4 4 0.13 * 10-3 m 51 located 0.0173 m aft of the reference line (located at 30% chord). This places the center of gravity at 46% of the chord. The first six normal modes (in vacuum) and their corresponding frequencies are provided in Table 2.9. It is interesting that the values for the normal modes have changed very little from the two layer case. This can be explained by the fact that the increase in stiffness is comparable to the increase in inertia. The corresponding mode shapes are provided in Figure 2-31. Table 2.9: Natural frequencies of the three-layer design in vacuum 2.3.2 Mode 1 Frequency 8 Hz 2 3 4 5 6 51 Hz 54 Hz 97 Hz 161 Hz 299 Hz Mode Shape 1" Bending 1 st Chordwise Bending 2 "d Bending 1 st Torsion 3 rd Bending 2 nd Torsion Wing Nonlinear Characteristics The effects from changes in root angle of attack are similar to those seen in the two-layer design. The tip deflection again increases with increase root angle of attack. This is plotted in Figure 2-32. The tip twist also increases with increased root angle of attack, as shown in Figure 2-33. As with the two-layer data, the root angle of attack was subtracted from the tip angle to better illustrate the effects to do gravity and aero-loading only. It should also be noted that although the trends are the same, the values of deflection and twist are smaller. This is expected as the stiffness of the wing is increased with the additional layer of E-glass/epoxy. Deflection and twist changes versus angle of attack also result in changes in the dynamic behavior of the wing. The wing's mode shapes and natural frequencies were computed for a wind speed of 30 m/s. The results are summarized in Table 2.10. This data is presented 52 Speed = 0.00 m/s @ Freq = 8.37 Hz Speed = 0.00 m/s @ Freq = 50.9 Hz 0.8 Speed = 0.00 m/s @ Freq = 54.1 Hz Speed = 0.00 m/s @ Freq = 97.1 Hz 0.2 0.4 0.6 0.8 1 Speed = 0.00 m/s @ Freq = 161 Hz Speed = 0.00 m/s @ Freq = 299 Hz 0.2 0.2 -0.1 -0.05 0.4 0.4 - 0.6 0.6 0.8 0.8 0 1 1 Figure 2-31: First six modes shapes for the three-layer wing (in vacuum) 53 0.35 50 0 .3 - -..... -.-.-.-- 5- -1- - 0.25 - -.-.-. 203 1 - - - 5 4- -- .. . .. .. . .. .. . - - - - - - - - -.-- - - - -.-- -.-. ....... - - - - --.. --.. 0 .2 0 .1 5 - --..-. .- -.- --..- .- -.-. 2* -0.05 0 5 10 15 20 25 30 35 40 45 Speed (m/s) Figure 2-32: Static tip deflection for increasing speed at different root angles of attack 54 50 0.8 50 -..- --.. 0 .6 - - - 0.6 ... . IQ . . . . . . . . . . . .. .. . . . .. . . . . .. . . . . . . . . . . . _20 4 S 0.3 - ... - - ---- - .... . . . . .. . . . .. . . . ... . . . ..- -. . ... ..-. ... . -.. - 0.2 - - - - -. - - -~ 0.1 --0* 0 0 5 10 15 25 20 30 35 40 45 Speed (mis) Figure 2-33: Elastic tip twist for increasing speed at different root angles of attack 55 50 graphically for the first four modes in Figure 2-34. Table 2.10: Dynamic properties at U 30 m/s 00 -0.005 m 8 10 0.02 m 8 20 0.04 m 8 50 0.10 m 8 1" Chordwise Bending 51 51 50 45 2 "d Bending 1 " Torsional 54 97 54 97 54 98 54 104 Bending _ 2 " Torsional 161 299 161 299 161 300 161 305 Angle of Attack Tip Deflection 1 "t Bending Mode Shapes = 3 rd The three-layer design also shows a change in the chordwise bending and torsional modes for changes in root angle of attack. As the root angle of attack is increased the frequencies for the 1I"chordwise bending mode decrease and the frequencies for the 1" torsional mode increase. The bending modes remain unchanged. A similar behavior is seen when the speed is varied for a constant root angle of attack. This is illustrated in Table 2.11 and Table 2.12. This data is also provided graphically for the first four modes in Figure 2-35 and Figure 2-36. Table 2.11: Dynamic properties at root angle of attack Speed Tip Deflection Mode Shapes 30m/s 35m/s 40m/s 0.04 m 0.06 m 0.08 m = 20 45m/s 0.10 m 1" Bending 8 8 8 8 1" Chordwise Bending 50 49 48 46 2" Bending " Torsional 1 54 98 54 99 54 101 54 103 Bending 2nd Torsional 161 300 161 301 161 302 161 305 3 rd The flutter speeds for varying angles of attack are given in Figure 2-37. As expected, the addition of the third layer of E-glass resulted in an increase in the flutter speed. 56 1*10 00 - 1T 90 80 70 60 2B NU -~ 50-- - --- LL. 1CB 40 30 20 1B 10 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Root Angle of Attack (deg) Figure 2-34: Frequency change due to root angle of attack change at U = 30 m/s 57 5 120 100 80 IR C 6.. 60 U- 2B 1CB 40 20 * II 18 I 30 32 34 36 38 40 42 44 46 48 Speed (m/s) Figure 2-35: Frequency change due to change in speed for 2' root angle of attack 58 50 120 1T 80 07 60 2B U- 40 1CB 20 11B 0 30 32 34 36 40 38 42 44 46 48 Speed (m/s) Figure 2-36: Frequency change due to change in speed for 5' root angle of attack 59 50 Table 2.12: Dynamic properties at root angle of attack = 5' Speed 30m/s 35m/s 40m/s 45m/s Tip Deflection 1 " Bending 0.10 m 8 0.15 m 8 0.20 m 8 0.25 m 8 45 54 42 54 38 54 33 54 Torsional 104 108 114 119 Bending 2" Torsional 161 305 161 308 161 308 160 303 Chordwise Bending 2" Bending 1 st Mode Shapes 1" 3 rd 70 60 50 E40 U, 0 a ~30 20- 10- 01 0 1 2 4 3 Root Angle of Attack (deg) 5 Figure 2-37: Flutter speeds for three-layer design 60 6 7 Zero-Degree Root Angle of Attack The flutter speed for 0' root angle of attack is determined to be 64.6 m/s from the root locus plot provided in Figure 2-38. The unstable mode is the fourth mode, which cooresponds to the mode. This is the same as was seen in the two-layer case. The first six 1"t torsional mode shapes at flutter speed for this wing are shown in Figure 2-39. Root locus for Root AOA = 00* 200 180 K----------- 160 0 .1. 44 Iflii 4 -2010 !0 140 F------- 120 K- - - - - - - - - - - - - - - - -- - - I U C LI~ --T 100 |-------- 80 --- K- - 60 2B -1 -4 -6 --44 4 --------,8 ICB -- - - --- - 40 - -- --- 20 --- ---- 1B iB 0 50 I 40 30 I I 10 20 Real Part of Root I 0 10 Figure 2-38: Root locus plot for 0' root angle of attack 61 20 Speed = 64.60 m/s @ Freq = 8.37 Hz Speed = 64.60 m/s @ Freq = 50.9 Hz 0 Speed = 64.60 m/s @ Freq = 54.1 Hz Speed = 64.60 m/s @ Freq = 97 Hz 0.1 0.2 0.4 -0.1 0.6 -0.05 0.8 1 0 Speed = 64.60 m/s @ Freq = 161 Hz Speed = 64.60 m/s @ Freq = 299 Hz 0.2 0.2 0.4 0.4 -0.1 -0.05 0 0.6 0.6 0.8 0.8 1 1 Figure 2-39: Mode shapes for three-layer design at 0' root angle of attack and at its corresponding flutter speed 64.6 m/s 62 One-Degree Root Angle of Attack The flutter speed for the 10 root angle of attack is 62.1 m/s. This can be seen on the root locus plot provided in Figure 2-40. The mode of instability is the second mode. This mode is primarily the first chordwise bending. However, the forces due to air pressure on the tilted wing result in some torsion being present as well. The first six mode shapes for this case are provided in Figure 2-41. Root locus for Root AOA = 1* 200 180 F-------- 160 - 3B- - ------------------------ ------ -u 140 |- U :3 120 -- 10 -- 0 U_ 80 - ...- - 60 -4M 44 '4" -30 -.- ----.-. -23 1 - 2B CB 40 F-------- - -- - - -- - - - 20 - -.-. -.- -. . 01 50 40 30 20 10 Real Part of Root -- 1B 10 B 0 10 Figure 2-40: Root locus plot for 1 root angle of attack 63 20 Speed = 62.10 m/s @ Freq = 8.38 Hz Speed = 62.10 m/s @ Freq = 45.1 Hz -0.05 00 0.05 0.1 1 Speed = 62.10 m/s @ Freq = 54.5 Hz Speed = 62.10 m/s @ Freq = 104 Hz 0.2 0.4 0.6 1 Speed = 62.10 m/s @ Freq = 161 Hz Speed = 62.10 m/s @ Freq = 305 Hz 0.2 0.2 -0.1-0.05 0.4 0.4 0.6 0.6 0.8 0.8 0 1 1 Figure 2-41: Mode shapes for three-layer design at 1 root angle of attack and at its corresponding flutter speed of 62.1 m/s 64 When this wing is flown at 10% above the predicted flutter speed, the wing enters into a LCO. The tip deflections and the tip twist are plotted for this case up to 1 second. These plots are provided in Figure 2-42 thru Figure 2-44. Speed = 68.20 m/s with a Time step = 0.0005 s 0.16 0.14 E 0.12 C CD) 0.1 E a) 0 0.08 Q_ 0.06 CO) 0.04 0.02 0 -0.02'0 0.2 0.4 0.6 0.8 1 Time (s) Figure 2-42: Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (10 root angle of attack) for the three-layer design A strain analysis was performed on the wing when it was at its maximum deflection. This occurs during the initial rise shown in Figure 2-42. The maximum tensile strain is 2697 pm/m and the maximum shear strain is -1765 pm/m. Both occur at the outer ply of the bottom part of the airfoil at the root of the wing. These results are summarized in Figure 2-45. These strains result in a margin of safety of 2.71 for tension and 7.50 for torsion. 65 Speed = 68.20 m/s with a Time step = 0.0005 s 10 8 6 CD 4 2 a. C~ 0 -2-4 -6 0 0.2 0.4 0.6 0.8 1 Time (s) Figure 2-43: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (10 root angle of attack) for the three-layer design Speed = 68.20 m/s with a Time step = 0.0005 s 0.13 0.125 F 0. 12 E 0.1 15- a> 4- E CD 0. CN 0.11 0. 105 I N 0.1 0.095 0.092 -0.02 -0.015 -0.01 0.005 0 -0.005 Y Displacement (m) 0.01 0.015 0.02 Figure 2-44: Nonlinear time simulation of the wing tip motion at 10% above its flutter speed (10 root angle of attack) for the three-layer design 66 eps1 (longitudinal fiber strain), ply 4 eglass 120: MAX = 0.0026968 eps2 (transverse fiber strain), ply 4 eglass 120: MAX = -0.00039914 eps3 (in plane shear strain), ply 4 eglass 120: MAX = -0.0017653 . . . . . . . Figure 2-45: Maximum ply strain reached at 10% above flutter speed of 10 root angle of attack for the three-layer design 67 Two-Degree Root Angle of Attack The flutter speed for the 2' root angle of attack is 56.9 m/s. This can be seen on the root locus plot provided in Figure 2-46. The second mode is again the mode which crosses the stability axes. The larger tip deflections and tip twists observed for the high angles of attack cause this mode to turn over and go unstable faster. The first six mode shapes for this case are provided in Figure 2-47. Root locus for Root AOA = 20 200 180 160 -------------------- --------- -- - -I- - - - - - - - - I - - - - - - - - - - -- ---- ---- --- - - - - -- - ----------- - -- --- -- - - - - - -- --- 3 B - - - --- --- -- - -- -- -- - -- ---------------140 ---------- --------- --------- 120 - ---- - - --- - - --- - 0 U 80- -- - 60 -- q480 4 4 --- - - 3E 2B 2E - -- - - - - --- - - ----- ------ --- - -- - - -- - - -- - - - - - - - - - --- -- - - -_55 -52 40 - - - - - - - --- -- - - - - - - - - - - - - - ------ - - - -- - - -- -- 8: - - - ----- ---20 -- - - - - - - -- - - ------ * 30 0' 50 40 30 20 20 0 0 10 Real Part of Root Figure 2-46: Root locus plot for 2' root angle of attack 68 - - - -- 1 10 10 ---- - --- - 20 Speed = 56.90 m/s @ Freq = 8.38 Hz Speed = 56.90 m/s @ Freq = 39.3 Hz Speed = 56.90 m/s @ Freq = 54.4 Hz 1 Speed = 56.90 m/s @ Freq = 111 Hz Speed = 56.90 m/s @ Freq = 161 Hz Speed = 56.90 m/s @ Freq = 309 Hz 0.2 -0.1-0.05 0.2 0.4 0.4 0.6 0.6 0.8 0 0.8 1 1 Figure 2-47: Mode shapes for three-layer design at 2' root angle of attack and at its corresponding flutter speed of 56.9 m/s 69 When this wing is flown at 10% above the predicted flutter speed, the wing enters into a LCO. The tip deflections and the tip twist are plotted for this case up to 1 second. These plots are provided in Figure 2-48 thru Figure 2-50. Speed = 62.60 m/s with a Time step = 0.0005 s 0.3 0.25 E 0.2 E a> -Z 0.15 0,, C 0.1 C.L F_: F5 0.05 0 1 -0.05 0 0.2 0.4 0.6 0.8 1 Time (s) Figure 2-48: Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (20 root angle of attack) for the three-layer design A strain analysis was performed on the wing when it was at its maximum deflection. This occurs during the initial rise shown in Figure 2-48. The maximum tensile strain is 4421 ptm/m and the maximum shear strain is -4079 pm/m. Both occur at the outer ply of the bottom part of the airfoil at the root of the wing. These results are summarized in Figure 2-51. These strains result in a margin of safety of 1.26 and 2.68, respectively. 70 Speed = 62.60 m/s with a Time step = 0.0005 s 10 5 CD _0 0 (jO -5 - 10 I -15 0.2 0 0.4 0.6 0.8 1 Time (s) Figure 2-49: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (20 root angle of attack) for the three-layer design Speed = 62.60 m/s with a Time step = 0.0005 s 0.16 0.15 0.14 E 0.13 CD, 0. CU, N 0.12 0.11 0.1 0.09 -0.03 -0.02 -0.01 0 0.01 Y Displacement (m) 0.02 0.03 Figure 2-50: Nonlinear time simulation of the wing tip motion at 10% above its flutter speed (2' root angle of attack) for the three-layer design 71 eps1 (longitudinal fiber strain), ply 4 eglass 120: MAX = 0.0044209 eps2 (transverse fiber strain), ply 4 eglass 120: MAX = -0.00065431 eps3 (in plane shear strain), ply 4 eglass 120: MAX = -0.0040786 Figure 2-51: Maximum ply strain reached at 10% above flutter speed of 20 root angle of attack for the three-layer design 72 Five-Degree Root Angle of Attack The flutter speed for the 50 root angle of attack is 46.4 m/s. This can be seen on the root locus plot provided in Figure 2-52. This plot shows the mode of instability is again the 1 " chordwise bending mode (mode 2). Again the increase in angle of attack results in the wing becoming unstable at lower speeds. The 1 st torsion (mode 4) and 3 rd bending (mode 5) are also starting to interact with each other, causing the torsion mode to head to the instability axes. The first six mode shapes for this case are provided in Figure 2-53. 200I 180 I Root locus for Root AOA = I I H-------- 160 F--- - 20 140 120 5* - ------ 38 10 0 ------- F- 8: ----- ----- 100 :0 U 80 -- 60 -- 40 -- ---- ---- --------- -------- --- --------- ------- 2B 0. 0 1 ------ c B 20 - -- - -- - -- -- -- - - -B 1B EU J~J 50 50 40 30 10 20 Real Part of Root -u 0 10 Figure 2-52: Root locus plot for 50 root angle of attack 73 20 Speed = 46.40 m/s @ Freq = 8.39 Hz Speed = 46.40 m/s @ Freq = 31.8 Hz Speed = 46.40 m/s @ Freq = 54.1 Hz Speed = 46.40 m/s @ Freq = 120 Hz -0.1 -0.05 0 0.6 0.8 1 Speed = 46.40 m/s @ Freq = 160 Hz Speed = 46.40 m/s @ Freq = 302 Hz 0.2 0.1 --- 0 - -0.05 -0 ..--- - .-0.4 0.6 0.8 0.2 1 Figure 2-53: Mode shapes for three-layer design at 5' root angle of attack and at its corresponding flutter speed of 46.4 m/s 74 When this wing is flown at 10% above the predicted flutter speed, the wing enters into a LCO. The tip deflections and the tip twist are plotted for this case up to 1 second. These plots are provided in Figure 2-54 thru Figure 2-56. Speed = 51.20 m/s with a Time step = 0.0005 s 0.4 0.35 0.3 E E 0.25 (D 0. CU) 0.2 0.15 F 0.1 U 0.05 0 -0.05'0 0.2 0.4 0.6 0.8 1 Time (s) Figure 2-54: Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (5' root angle of attack) for the three-layer design A strain analysis was performed on the wing when it was at its maximum deflection. This occurs during the initial rise shown in Figure 2-54. The maximum tensile strain is 7374 tm/m and the maximum shear strain is -6799 pm/m. Both occur at the outer ply of the bottom part of the airfoil at the root of the wing. These details are summarized in Figure 2-57. These strains result in a margin of safety of 0.36 and 1.21, respectively. 75 Speed = 51.20 m/s with a Time step = 0.0005 s 20 15 10 a) F_ 5 0 -5- -10 0 0.2 0.4 0.6 0.8 1 Time (s) Figure 2-55: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (50 root angle of attack) for the three-layer design Speed = 51.20 m/s with a Time step = 0.0005 s 0.32 0.31 - 0.30.290.28E 0) C.~0.27ND 0 N 0.260.250.240.230.22 -0.06 -0.04 -0.02 0 Y Displacement (m) 0.02 0.04 Figure 2-56: Nonlinear time simulation of the wing tip motion at 10% above its flutter speed (50 root angle of attack) for the three-layer design 76 eps1 (longitudinal fiber strain), ply 4 eglass 120: MAX = 0.0073741 / eps2 (transverse fiber strain), ply 4 eglass 120: MAX = -0.0010914 eps3 (in plane shear strain), ply 4 eglass 120: MAX = -0.0067985 Figure 2-57: Maximum ply strain reached at 10% above flutter speed of 50 root angle of attack 77 The following is a table to summarize the strain analysis for the different cases. Table 2.13: Summary of Strain Analysis Results Angle of Attack 1 tension torsion 2' tension torsion 50 2.4 tension torsion Two-Layer Results 2660 p m/m -769 p m/m 4484 p m/m -2726 p m/m 7405 y m/m -5018 p m/m Three-Layer Results 2697 p m/m -1765 y m/m 4421 y m/m -4079 p m/m 7374 p m/m -6799 p m/m Nastran Comparison To ensure that the wing would survive the loadings in the tunnel, a stress analysis was performed using MSC.Nastran finite element code. Since the MATLAB code from [3] was not yet validated experimentally, this would provide added assurance that the stress and strain data predicted by the MATLAB code is correct. Figure 2-58 provides a close-up of the finite element model mesh. The wing was discretized using 5550 elements. The foam was modeled as 4-noded solid elements and the composite skin was modeled as 4-noded shell elements. The worse stress case scenerio for the three-layer design is the 5' root angle of attack case. The MATLAB code was used to provide static deflection values for the wing at 51.15 m/s, which is 10% above the predicted flutter speed. The x,y,z displacements and pitch rotation for the wing were matched for two different cases: maximum vertical (z) displacement and maximum pitch rotation. This was achieved by applying varing forces distributed along the span of the wing at 25% of the chord and then forces to the leading and trailing edges to establish the desired rotation. 78 Figure 2-58: Finite element mesh of the three-layer design used in Nastran (5550 elements, 13617 degrees of freedom) 79 2.4.1 Case 1 - Maximum Vertical Displacement This case matched the maximum vertical displacement predicted by the MATLAB code during the limit cycle oscillation. The original predicted values and the values determined in Nastran are provided in Table 2.14. Table 2.14: Nastran Case 1: Displacements and Rotations chordwise deflection spanwise deflection vertical deflection pitch rotation MATLAB Prediction Nastran Result 0.0034 m 0.0603 m 0.3159 m 0.0199 m 0.0539 m 0.3126 m -9.110 -8.850 The MATLAB and Nastran displacement results match within 11%, except in the chordwise direction. The overall magnitude of the displacements are equivalent and the major contributors, vertical displacement and pitch rotation, are within 1% and 3%, respectively. Since the results show such high margins of safety (Equation 2.1), the analysis was considered adequate. The results from this case are summarized in Table 2.15 and fringe plots of the stress profiles are provided in Figure 2-59 through Figure 2-61. Table 2.15: Nastran case 1: summary of results chordwise spanwise chord-span shear 2.4.2 Maximum Strain 2570 pm/m 5430 pm/m -5740 pm/m Margin of Safety 6.4 2.5 8.0 Case 2 - Maximum Pitch Rotation This case matched the maximum pitch rotation predicted by the MATLAB code during the limit cycle oscillation. The original predicted values and the values determined in Nastran are provided in Table 2.16. 80 MSC.Patran 2001 r3 13-Oct-02 20:38:45 2.57-00 Fringe: lift. PW Linear: 100. % of Load_2: Nonlinear Strains, Strain Tensor-(NON-LAYERED) 2.08-00 Deform: lift, PW Linear: 100. %of Load_2: Displacements. Translational 1.60-003 1.12 6.40-00 aximum value (2570 rn/rn) at the wing root on the top-side of the wing. 1.58-00 -3.23-00 -8.05-00 4 -1.29-00 -1.77-00 -2.25-00 -2.73-00 -3.21-00 -3.69-00 -4.18-00 -4.66-00 defaultFringe: Max 2.57-003 @Nd 865 Min -4.66-003 @Nd 461 defaultDeformation: Max 1.25+001 @Nd 1836 xY Figure 2-59: Nastran normal (chordwise) strain for 5' root angle of attack Table 2.16: Nastran case 2: displacements and rotations chordwise deflection spanwise deflection vertical deflection pitch rotation MATLAB Prediction 0.0502 m 0.0370 m 0.2391 m Nastran Result 0.0176 m 0.0348 m 0.2520 m 9.930 10.50 81 MSC.Patran 2001 r3 13-Oct-02 20:38:27 5.43-00 Fringe: lift PW Linear:100. % of Load_2: Nonlinear Strains. Strain Tensor-(NON-LAYERED) (ZZ) Deform: lift PW Linear: 100. % of Load_2: Displacements. Translational 4.71-00 3.99-00 3.27-00 2.54-003 1.82-00 1.10-003 3.72-004 -3.51-00 -1.07-003 -1.80-00 -2.52-00 -3.24-00 -3.97-00 Maximum value (5430 pim/m) at the wing root, on the underside of the wing Y .Z X -4.69-00 -5.41-00 default_- Fringe : Max 5.43-003 @Nd 461 Min -5.41-003 @Nd 865 defaultDeformation : Max 1.25+001 @Nd 1836 Figure 2-60: Nastran normal (spanwise) strain for 5' root angle of attack 82 MSC.Patran 2001 r3 13-Oct-02 20:38:55 2.06-00 Fringe: lift, PW Linear: 100. % of Load_2: Nonlinear Strains. Strain Tensor-(NON-LAYERED) (Z> 1.54-003 Deform: lift PW Linear: 100. % of Load-2: Displacements, Translational 1.02-003 5.04-004 Maximum value (2060 gm/m) at the wing root at the trailing edge -1.58-005 -5.36-00 -1.06-00 -1.58-00 -2.10-00 -2.62-00 -3.14-003 -3.66-003 -4.18-00 -4.70-003 -5.22-003 Y -5.74-003 defaultFringe: Max 2.06-003 @Nd 2 Min -5.74-003 @Nd 1685 defaultDeformation: Max 1.25+001 @Nd 1836 Figure 2-61: Nastran in-plane shear strain for 50 root angle of attack 83 Similar to the previous case, the MATLAB and Nastran displacements match within 6%. The overall magnitude of displacement is equivalent and the main contributors, vertical displacement and pitch rotation, are over-predicted. Again, although the Nastran displacements do not match the MATLAB exactly, the margins of safety are high enough to consider the values sufficient. The results from this case are summarized in Table 2.17 and fringe plots of the stress profiles are provided in Figure 2-62 through Figure 2-64. Table 2.17: Nastran case 2: summary of results chordwise spanwise chord-span shear 2.5 Maximum Strain Margin of Safety 2390 pum/m 4960 pm/m 7950 pm/m 6.9 2.8 5.5 Design Selection The three-layer design was chosen for the wind tunnel model. This decision was based on the resulting predicted flutter speeds and on the resulting strains. The flutter speeds for the two-layer case were on average 16% lower than the three-layer design. However, the deflections experienced by the two-layer design ranged from 41% to 61% higher than the three-layer design. This resulted in an increase in the strain levels for the two-layer wing. To be safe, the increase in deflection and strain was considered more detrimental than an increase in the flutter speed. Therefore, the three layer design was chosen. 84 MSC Patran 2001 r3 13-Oct-02 19:37:40 2.39-00 Fringe: lift PW Linear: 100. % of Load: Nonlinear Strains, Strain Tensor-(NON-LAYERED)M 1.99-00 Deform: lift, PW Linear: 100. % of Load: Displacements. Translational 1.59-00 1.19-00 7.86-00 Maximum value (2390 pm/m) at the wing root on the top-side of the wing 3.85-004 -1.51-00 -4.16-00 -8.16-00 -1.22-00 -1.62-003 -2.02-00 -2.42-00 -2.82-003 -3.22-00 Y -3.62-003 defaultFringe: Max 2.39-003 @Nd 865 Min -3.62-003 @Nd 206 defaultDeformation: Max 1.08+001 @Nd 1225 Figure 2-62: Nastran normal (chordwise) strain for 5' root angle of attack 85 MSC.Patran 2001 r3 13-Oct-02 19:38:44 4.96-00 Fringe: lift PW Linear: 100. % of Load: Nonlinear Strains, Strain Tensor-(NON-LAYERED) (ZZ) 4.30-003 Deform: lift PW Linear: 100. % of Load: Displacements, Translational 3.63-00 2.97-00 2.31-00 1.65-00 9.82-00 3.19-004 -3.44-00 -1.01-003 -1.67-00 Maximum value (4960 pm/m) at the wing root on the under-side of the wing -2.33-003 -3.00-00 -3.66-00 -4.32-003 y -4.98-0031 defaultFringe: Max 4.96-003 @Nd 206 Min -4.98-003 @Nd 865 defaultDeformation: Max 1.08+001 @Nd 1225 Figure 2-63: Nastran normal (spanwise) strain for 5* root angle of attack 86 7.95-00 MSC.Patran 2001 r3 13-Oct-02 19:38:36 Fringe: lift. PW Linear: 100. % of Load: Nonlinear Strains. Strain Tensor-(NON-LAYERED) (Z7.21-00 Deform: lift PW Linear: 100. % of Load: Displacements. Translational 6.47-003 5.73-003 4.98-003 Maximum value (7950 gm/m) at the wing root at the trailing edge 4.24-003 3.50-00 2.76-00 2.01-003 1.27-00 5.28-004 -2.14-004 -9.57-00 -1.70-003 -2.44-003 Y -3.18-003 defaultFringe: Max 7.95-003 @Nd 1786 Min -3.18-003 @Nd 1071 default Deformation: Max 1.08+001 @Nd 1225 Figure 2-64: Nastran in-plane shear strain for 50 root angle of attack 87 Chapter 3 Experimental Procedure This chapter describes the experimental work that took place as part of this project. The manufacturing methods are presented, as are details for instrumentation location and wiring. The chapter also contains the description of the experimental tests: bench top and wind tunnel. 3.1 Wing Manufacture In this section, different components that make the instrumented wind tunnel model are discussed. The foam cores were machined in the Aeronautics and Astronautics machine shop on the TRAK K3 mill machine with the help of Ricky Watkins and Donald Weiner. The wing lay-up and cure were performed within the TELAC facilities with the support of John Kane. The root clamps for holding the wing during the bench top and wind tunnel characterization were machined at MIT Lincoln Laboratory. The wing mold was provided by NASA Langley. 3.1.1 Foam Core The foam core of the wing was manufactured out of Rohacell 31. This material was selected to ensure survival of the foam at the elevated temperatures of the wing cure process. The foam for each wing was machined in four seperate pieces, consisting of the top and bottom 88 of the wing from the root to 0.5715 m and the top and bottom of the wing from 0.5715 m to the tip at 1.13 m. The core is slightly oversized by 0.254 mm (10 mil) to ensure adequate backpressure within the mold during the cure process. After the accelerometers are embedded within the foam at the tip of the wing, the individual pieces are glued together with five minute epoxy. The foam pieced together in the mold is shown in Figure 3-1. Figure 3-1: Four sections of the foam core pieced together in mold 3.1.2 Instrumentation Seven full strain gauge bridges and two accelerometers were added on the foam core. The locations of the sensors are provided in Figure 3-2. An additional torsional bridge at the root was attached after the wing was cured and thus is located on the outer surface of the wing. The accelerometers were placed at the tip of the wing where maximum deflection happens. Since the root of the wing experiences the highest strain, one of each of the strain gauge bridge types was placed there. The locations of the other strain gauges along the span of the wing correspond to points of high strain for the higher normal modes, i.e., bending and 2 "d torsion. 89 2 "d and 3rd 35% 45% 55% 70% CL 0 oC 80% + Leading Edge Figure 3-2: Location of sensors: bending gauges are represented by squares, forward/aft gauges by diamonds, torsion gauge by a star, and accelerometers by circles Strain Gauges The strain gauges used to make-up the Wheatstone bridges were from the Micro Measurements Division of Measurements Group, Inc. The gauges for the bending bridges were model EA-06-125AD-120 and the gauges for the torsional and forward/aft bending bridges were model CEA-06-125UT-120. Both models consisted of gauges with 120 Q resistance. The gauges were wired to record spanwise bending and torsion. The four spanwise bending bridges consisted of a pair of gauges oriented in the 00 direction on the top and bottom of the wing. The four torsional bridges consisted of a pair, one in the +45' direction and one in the -45' direction on the top and bottom of the wing [12]. A general Wheatstone bridge wiring diagram is provided in Figure 3-3. The wire diagrams for the specific types of bridges use the same numbering convention. A map of the proper wiring scheme for a spanwise bending bridge is presented in Figure 3-4 and Figure 3-5 details the wiring for the torsional bridges. After some discussion with Professor John Dugundji, MIT, it is believed that the torsional wiring presented in [12] actually refers to a forward and aft motion, as opposed to the twisting motion originally thought. A twist bridge was wired to the outer skin of the wing at the root to measure the twist motion of the wing. The wiring for this bridge is provided in Figure 3-6. 90 -Vout Figure 3-3: General wiring diagram for Wheatstone bridges 1,- a .c Leading .b 4 'c .a 2- Edge d Figure 3-4: Wiring diagram for Wheatstone bridges in bending <&6 Leading Edge tK Figure 3-5: Wiring diagram for Wheatstone bridges in forward/aft bending Leading o' Edge &6 Figure 3-6: Wiring diagram for Wheatstone bridges in twist 91 The internal strain gauges where attached to a single square layer (25 mm by 25 mm) of pre-cured E-glass/epoxy oriented at 00/900. A photo of the internal strain gauges on the wing is provided in Figure 3-7. The squares were glued down with five minute epoxy to the foam. The strain gauges were then also afixed with five minute epoxy. Magnet wire was used to make all of the required electrical connections. The two bridges at the root, with the wiring of both types of bridges, is shown in Figure 3-8. Figure 3-7: Strain gauge at mid-span Figure 3-8: Strain gauges wired at the root 92 Accelerometers The accelerometers used for this experiment were the Endevco Model 22 piezoelectric accelerometers. These sensors were embedded within the foam core at the wing tip, 14.3 mm from the leading edge and 39.7 mm from the leading edge. The specifications for the accelerometers are provided in Table 3.1. The wires were run parallel to the leading edge in channels between the two halves of foam. The pair of accelerometers is used in combination to calculate the twist and bending frequencies at the wing tip. When the output of one is subtracted from the other, the tip twist response is obtained. The average of the two accelerometers produces the bending response. Table 3.1: Accelerometers used in the wing model Gauge CW18-22 CW13-22 3.1.3 Location 14.3 mm from LE 39.7 mm from LE Charge Sensitivity 0.374 pC/g 0.385 pC/g II Lay-Up In preparation to lay-up the composite on the instrumented foam core, the foam core was put in the oven at 266'F for 24 hours to remove any residual moisture. Moisture has a plasticizer effect on the foam, which causes a decrease in its compressibility strength [13]. If this step is not performed, the foam will collapse under the high temperature and pressure of the curing process. The required three layers of E-glass/epoxy were cut to 0.305 m by 1.143 m and stacked flat on the table. They were then gathered up and placed into the bottom half of the mold as one piece of cloth. The long edge was placed slightly behind the trailing position and the short end was aligned with the root. At this point, the additional layers of E-glass/epoxy around the root section where wrapped around the foam core. These consisted of a 76-mm strip, a 64-mm strip, a 51-mm strip, and a 38-mm strip, each being two plies thick. These additional plies were added to the root area to provide additional strength where the bolts 93 from the clamp would be passing through the wing. Once this was done, the foam core was inserted. The rest of the lay-up was then wrapped around the leading edge of the foam and pulled tightly toward the trailing edge. Once the lay-up was smooth on both sides of the wing, a straight edge was placed at the wing trailing edge and the excess was trimmed off. Care was taken to ensure the wires passing out of the root would not interfere with the root clamp. The top of the mold was lifted into position and then the two pieces were clamped together. A thermocouple was attached between the two halves of the mold, as the temperature of the mold is the one which needs to follow the cure profile. With the mold assembly completed, the wing was placed into the autoclave and cured at 250'F for 90 minutes. It was then allowed to cool slowly over 48 hours. 3.1.4 Post Cure The wing was removed from the mold after slowly cooling and only minor cosmetic touch-ups were required. Some resin had been pulled between the two mold pieces at the leading edge. This resin was scrapped off and then the leading edge was sanded for a smooth finish. This resin was also pulled at the trailing edge, but this is desirable as it creates a smooth taper to the trailing edge, so no modifications were made to the trailing edge. A band saw was used to remove about 6-mm from the wing tip to provide a smooth edge to that side of the wing as well. Finally three holes were drilled into the root of the wing, 28.5-mm behind the leading edge, to accomadate the three bolts from the clamp. 3.1.5 Clamp The root clamp was made from Aluminum 6061 and was NC machined at MIT Lincoln Laboratory. It consists of two seperate pieces which bolt together through the wing. Drawings of the top and bottom are provided in Figure 3-9 and Figure 3-10. The clamp is designed such that a gap between the two pieces exists at the leading and trailing edge. This allows for some shape mismatch between the final wing and the clamp. 94 Units: Inches Figure 3-9: Top of root clamp 95 F r 31/2 NACA 0012 Airfoil Units: Inches Figure 3-10: Bottom of root clamp 96 3.2 Data Acquisition Equipment The strain gauge and accelerometer signals need to be processed through conditioners before they can be sent to a computer to store. Since all of the bridges were full bridges they needed to be connected to the conditioner with the pin readout provided in Figure 3-11. The little letters represent the connections to the strain gauges and the capital letters refer to the pins on the connector. The conditioner box converts the signal to an amplified voltage which is B C F a+ -c b+ -d A D F Figure 3-11: Pin diagram for full bridges sent to the computer via a BNC cable. The accelerometer signal was also processed through a conditioner before the signal voltage was transmitted to the computer. A photo of both conditioners is provided in Figure 3-12. These are the conditioners used for the bench top tests. The wind tunnel tests used similar strain gauge conditioners, but in a different box. 3.3 Wind Tunnel Load Cell The wind tunnel tests required the use of a load cell. The load cell is the interface between the wing root clamp and the fixed stand in the tunnel. It provides three axis of load data, as well as the yaw moment. For this wing, a 75-lb Multi-Axes Load Cell from JR3, Inc. was used. The load settings and ratings are provided in Table 3.2. To determine the loads, the voltage channel coming from the load cell must be multiplied by a calibration matrix. The equation for this load cell is: 97 Accelerometer conditioners Strain Gauge conditioners Figure 3-12: Photo of accelerometer and strain gauge conditioners Table 3.2: Load cell used for the wing tunnel tests Load Cell Channel F Fv Fz MX MV M2 Maximum Load Measurement 75 lb 75 lb 150 lb 0 in-lb 0 in-lb 225 in-lb Drag (ib) 9.2758 Axial (ib) 0.2977 Lift (ib) Yaw (in-lb) -0.1770 9.1166 0.2176 -0.1894 -0.1918 0.0131 98 -0.0589 Sensor Load Rating 75 lb 75 lb 150 lb 225 in-lb 225 in-lb 225 in-lb -0.1796 (V) 0.0861 0.0407 (V) 18.2251 0.6773 (V) 27.7896 (V) -0.0766 (3.1) Chapter 4 Experimental Studies This chapter discusses the bench top and the wind tunnel test results. There is also a section on adjustments made to the model to cause the natural frequencies to match those seen in the bench top tests. 4.1 Bench-top Tests A series of bench top tests were conducted to determine the characteristics of the manufactured wing and to ensure operation of all the sensors. It was discovered that the forward/aft strain gauge bridge at the root was electrically shorted during manufacture. It was not possible to be repaired, so it was not monitored during the tests. Aside from simple functionality tests, only the gauges at the root for bending and torsion were monitored during benchtop tests. These tests were performed in TELAC. A picture of the experimental set-up is provided in Figure 4-1 and a picture of the monitoring equipment is provided in Figure 4-2. 4.1.1 Strain Gauge Calibration A static calibration of the strain gauges was performed to determine the deflection-voltage relationship. This was performed with the strain gauge conditioners within TELAC. So these results do not correspond with voltages from the wind tunnel tests. This calibration was performed to establish a process to be used later with the tunnel instrumentation. The wing 99 Figure 4-1: Experimental set-up for bench-top tests Figure 4-2: Experimental instrumentation for bench-top tests 100 was displaced in the vertical direction by various pre-established tip displacement values in both directions. This was done by holding the wing tip by hand and aligning it with pre-drawn tick marks on a reference scale. The wing was then held to these locations for a few seconds. This provided data on bending deflections versus voltage. A time trace of this calibration for the root bending strain gauge bridge is provided in Figure 4-3. When 8 7.5 - 6 23 = 7 6.5 M 8=18 6 - 5-5 3.5 0 0 5 4.5 4- 3.5 0 20 40 60 80 100 120 140 160 180 200 Time (s) Figure 4-3: Time trace of data from bench-top bending calibration this data is plotted with the known deflections, a curve of voltage versus tip deflection is generated. This is shown in Figure 4-4. The wing was also twisted to various amounts at the tip, which provided a calibration for tip twist. For this, the leading edge of the wing was again deflected by hand to pre-drawn tick marks, with the trailing edge held at the original position. A time trace of the data from the root torsion strain gauge bridge is provided in Figure 4-5. This data was plotted 101 8 5c5- 4 604 3-30 -20 -10 0 10 Wing Tip Displacement (mm) 20 30 residuals 0.3 - Linear: norm of residuals = 0.41744 0.2 0.1 - --- 0 -0.1 -0.2 -0.3 -20 -15 -10 -5 0 5 10 15 20 Figure 4-4: Calibration of strain gauge voltage output for bending 102 25 30 7 6.5 a=2.12 0 6 a =00 0 a =-2.12 0 -1.060 5 a =-3.720 4.51 0 20 40 60 Time (s) 80 100 Figure 4-5: Time trace of data from bench-top twist calibration 103 120 against the known twist values to provide a curve for the voltage output generated through tip twist. This curve is given in Figure 4-6. 7 > 6.5 () C S6 0 0) 0 5.5 t- 4.5'-4 -3 -1 -2 0 1 Wing Tip Twist (deg) 2 3 4 residuals I I 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 F -4 Linear: norm of residuals = 0.28536 I I I -3 -2 -1 I I I 0 1 2 1 3 4 Figure 4-6: Calibration of strain gauge voltage output for tip twist 4.1.2 Tap Tests for Dynamic Properties Various places on the wing were tapped to excite the different normal modes, so that its natural frequencies could be identified. This data was collected via the strain gauges. The data from various tap tests is given below in Figure 4-7 through Figure 4-9. The top graph of each figure is a time trace of the voltage output from the strain gauges and the bottom is a Fast-Fourier Transform (FFT) of the output. The first tap was to excite the bending modes. The tap was applied to the tip of the wing around 30% chord line. The frequencies 104 excited were at 6 Hz and at 38 Hz (1st and 2 nd bending). The second tap was at the tip but off the elastic axis, at the trailing edge of the wing tip, thus the torsional modes were excited. This plot shows dynamics at 70 Hz (1st torsion), as well as the 6-Hz and 38-Hz bending frequencies. The final tap was on the leading edge of the wing tip in the forward/aft direction. This excited the chordwise bending mode. This plot shows the 1" chordwise bending frequency at 45 Hz. The 6-Hz 1" bending was also excited. Volts 5.40- 5.250.00 dBVrns 0 05 0,10 0.15 0.20 0.25 0.30 0.35 0.40 Peak Power 28.14 Vrms^"'2 Peak Frequency: 0,45 0.50 0.00 Hz 14.50.0-20.0-40.0-uA 60.0-Hz 0,0 20.0 40.0 60.0 80.0 100.0 120.0 Figure 4-7: Tap test for bending frequencies-root bending gauge It can be seen through these graphs that tap tests are not perfect, as exciting purely one type of mode is extremely difficult. However, the first four modes of vibration can be determined along with their mode shape. This data is summarized in Table 4.1. 105 Volts 4.30 - I I1 4,00 RAA :3.50 A Ov 325 AAAK Am; IN A) w v VMWITV we 5, 0,00 dBVrms 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.00 Hz Peak Power :12.09 Vrms^2 Peak Frequency : 10.8 0.0-20.0-40.0- Hz -57.5 -. 20.0 40.0 60.0 80.0 Figure 4-8: Tap test for torsion frequencies-root torsion gauge Table 4.1: Natural Frequencies From Bench-top Tap Tests Mode Frequency Mode Shape 1 6.4 Hz 1" 2 38 Hz 2 "d Bending 3 4 45.0 Hz 70.0 Hz 1 Bending " Chordwise Bending 1 st Torsional 106 100.0 Volts 6.54 - 650 -iv 6.48 0.00 dBVrms - - - 6.52 - - 1 - 1 1/ 1 -i -- V 1v 1 1 I 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Peak Power 42.31 Vrms^,2 Peak Frequency: 0.00 Hz 16.30.0- -20.0-40.0-60.00.0 10.0 20.0 30.0 40.0 50.0 60.0 Figure 4-9: Tap test for chordwise bending frequencies-forward/aft gauge 107 4.1.3 Shaker Tests for Dynamic Properties Finally, a mini shaker type 4810 from Briiel & Kjaer was applied to different spots on the wing to validate the tap tests and to fully understand the dynamics of the wing. The shaker was first applied to 30% of the chord at about 0.125-m out from the root clamp. The shaker was excited with sine waves of frequencies varing from 1 Hz to 110 Hz. This test was followed by the shaker exciting the leading edge of the wing, as shown in Figure 4-10. Again the shaker was excited with frequencies varing from 1 Hz to 110 Hz. With the shaker Figure 4-10: Shaker at the wing leading edge for the natural frequency bench test at the leading edge, the first torsion mode could be excited. The shaker test confirmed the frequencies and mode shapes determined through the tap test, as the node line for the wing could be observed and felt through a light touch. These results support the use of the tap test in the tunnel, where a shaker test would be difficult. 108 4.2 Re-work of Analytical Model During the bench top tests the wing was weighed to determine the true cross sectional weight and the final length of the wing. The wing cross-sectional weight is 0.2176 kg/m. The length of the wing outside the clamp is 1.038 m. With these adjusted within the model, the originally calculated natural frequncies for the wing (Table 2.9) changed and the 1 st bending mode matches that from the bench-top tests. The first six natural frequencies are given in Table 4.2. A diagram of all the mode shapes is provided in Figure 4-11. The %-error value is calculated with Equation 4.1. %error = Calculated- Measured Measured - 100% (4.1) Table 4.2: Natural frequencies with adjusted weight and length Mode 1 2 3 4 5 Calculated Frequency 6.44 Hz 39.1 Hz 41.7 Hz 78.4 Hz 124 Hz 6 241 Hz Measured Frequency 6.4 Hz 45 Hz 38 Hz 70 Hz -_-_- % Error 0% 13% 9.7% 12% Mode Shape 1 st Bending st Chordwise Bending 1 2"d Bending 1st Torsional 3rd Bending 2 "d Torsional Some adjustment of material properties was made to further modify the wing characterists to match the bench-top results. The shear stiffness, Q66 (defined in Appendix B), of the E-glass/epoxy was modified from 4.1 GPa to 3.25 GPa. This changed the first torsional frequency of the modeled wing to 70 Hz without changing the other frequencies. The Young's Modulus of the foam in the chordwise direction was also adjusted from 36.0 MPa to 558 MPa. This changed the first chordwise bending mode to 45 Hz. The discrepancy between the true second bending and the predicted second bending is believed due to the non-uniform distribution of mass along the span of the wing. The cross sectional wing weight takes the weight of the accelerometers, strain gauges, and epoxy which was used to attach the separate 109 Speed = 0.00 m/s @ Freq = 6.44 Hz Speed = 0.00 m/s @ Freq = 39.1 Hz -0.6 -0.2 0 0.8 0.84 0.6 Speed = 0.00 m/s @ Freq = 41.7 Hz Speed = 0.00 m/s @ Freq =78.4 Hz 0.2 0.4 0.6 0.8 1 1 Speed = 0.00 m/s @ Freq = 124 Hz Speed = 0.00 m/s @ Freq = 241 Hz 0.2 0.2 0.4 0.4 -0.1 -0.05 0.6 0.6 0 0.8 0.8 1 1 Figure 4-11: Fundamental mode shapes with adjusted weight and length 110 foam pieces and spreads them evenly across the entire wing. This un-even distribution of mass will cause slight deviations in the second bending, causing the 6.25 times the first bending frequency rule, value for isotropic homogeneous cantilever beams, to not apply. With these modifications to the wing properties, the model predictions for the wing are close to the measured frequencies. The natural frequencies are given in Table 4.3 and the mode shapes are provided in Figure 4-12. Table 4.3: Intermediate predicted natural frequencies based on added stiffness corrections Mode Calculated Frequency Measured Frequency % Error Mode Shape 1 6.44 Hz 6.4 Hz 0% 2 3 41.5 Hz 45.0 Hz 38 Hz 45 Hz 9.2% 0% 4 5 6 70.0 Hz 123 Hz 216 Hz 70 Hz 0% - 1" - - 2 "d 1 "t Bending Bending 1 "t Chordwise Bending 2 nd Torsional 3 rd Bending Torsional These new changes in the model were used to calculate new flutter speeds. The flutter speeds are given in Table 4.4 along with their corresponding root locus plots in Figure 4-13 through Figure 4-17. A portion of the root locus was magnified for 10 and 20. This data is provided in Figures 4-14 and 4-16. Table 4.4: Intermediate predicted flutter speeds based on corrected structural properties Angle of Attack 10 Flutter Speed 33.0 m/s 20 52.7 m/s 42.75 m/s 50 A more indepth study of flutter speed versus angle of attack has provided a graph depicting some strange behavior of the wing between 1 and 20 angle of attack (Figure 4-18). Similar behavior of flutter speed versus angle of attack was discussed in [1]. There, a discontinous jump in the flutter speed was seen between 00 and 1 angle of attack. For this 111 Speed = 0.00 m/s @ Freq = 6.44 Hz Speed = 0.00 m/s @ Freq = 41.5 Hz 0.1 0 -0.1 -01-0.05 0 Speed = 0.00 m/s @ Freq = 45 Hz Speed = 0.00 m/s @ Freq = 70 Hz 0.01 -0.4 0.2 0.4 \ -0.2 0.6 0.. 0.8 1 Speed = 0.00 m/s @ Freq = 123 Hz Speed = 0.00 m/s @ Freq = 216 Hz 0.2 0.2 0.4 0.4 -0.1 -0.05 0 0.6 0.6 - 0.8 0.8 1 1 Figure 4-12: Predicted fundamental mode shapes based on added stiffness corrections 112 Root locus for Root AOA = 1 * 200 180 |- 140 F -------- 120 - 100 - 80 - U " 7" 3B - 1 .- - --0--- -20 I U C 0~ 'J) LL 0 1T 60 ---- ------- - 2B ~~ 40 -- * 7Y0 20 1- - - - - -- - -- -- - 0 50 -rtr--- - - 1B . -Rn 40 ,n 4 . I 30 1n a n I I 20 10 0 10 20 Real Part of Root Figure 4-13: Root locus plot for 10 root angle of attack based on modified wing stiffness properties 113 Root locus for Root AOA = 1* 50 48 46 - - - - -- -- - -- ------ - - - - - - - - - - - - - - - -- - - -- - - -- -- -- - 1CB 44 - - --- - - -- - - -- - - - - - - 42 C LL~ 2B 10A C -- -- - --- 40 --- - - - - 38 6 -36 - - - ---- 34 -- - - - - -- - - - - --- --- -- - -- -- - ---- - - 32 -- --- - - 30 10 8 -- -- -- - - -- - - 6 4 2 0 2 4 6 8 10 Real Part of Root Figure 4-14: Magnified root locus plot for 1' root angle of attack based on modified wing stiffness properties 114 Root locus for Root AOA = 2* 200 180 -- - 160 -- 140 -- --------- 3B: 120 ----------- 100 F---------- LL - 80 F- 0 60 F - - --- ----- - - - - 02B 40 F- - - ------- 1CB --------20-- F" 0' 50 0 - - - - - - -- - -- - - 1B: -Q A 40 30 10 20 Real Part of Root 0 10 20 Figure 4-15: Root locus plot for 2' root angle of attack based on modified wing stiffness properties 115 Root locus for Root AOA = 2* 50 45 40 Q) LI- 35 30 25 10 8 6 4 2 2 0 Real Part of Root 4 6 8 10 Figure 4-16: Magnified root locus plot for 2' root angle of attack based on modified wing stiffness properties 116 Root locus for Root AOA 5* = 200 -- - - - - - - - -0- 2T -- - 180 ---- -- ----- 160 - -- - - --- - ---- ------- 140 ---F - - 3B 120 - 0 1- - -- NU :3 100 -- - --- U- - ---- -- ------ 80 0 1T: 60 -- 2B 0 0 1CB 20 o1B "0r 0 50 40 30 10 20 Real Part of Root 0 10 20 Figure 4-17: Root locus plot for 50 root angle of attack based on modified wing stiffness properties 117 55 50- 45- E CD40 a,) U, 35- 30- 25 - 0 1 2 4 3 Root Angle of Attack (deg) 5 6 7 Figure 4-18: Predicted flutter speeds for various root angles of attack based on modified wing stiffness properties 118 current wing, however, the behavior of the wing at 10 is not believed to be accurate, as is proved during the wind tunnel tests. As will be discussed in more detail later, the wing was flown in the tunnel up to 82 m/s at 1' angle of attack, which is 250% above this predicted flutter speed. For this reason the analytical model was again adjusted. A point mass was placed at 42% of the span to simulate the line of epoxy which joined the foam. The size of the point mass was adjusted until the second bending frequency matched the measured 38 Hz. The density of the foam was adjusted so that the first bending frequency returned to 6.44 Hz after the addition of this new mass. Table 4.5 provides a summary of the final values for the adjusted parameters. Table 4.5: Final values of adjusted parameters Parameter wing span p foam Q66 E-glass/epoxy Et foam Point mass @ 46% span Original Value 1m 30 kg/m 3 4.10 GPa 36 MPa 0 kg Adjusted Value 1.038 m 94.5 kg/m 3 3.265 GPa 558 MPa 0.0286 kg With this last adjustment, the first four predicted frequencies match those measured. Table 4.6 shows the frequencies values while Figure 4-19 shows the corresponding mode shapes. These new dynamic characteristics resulted in a change in the aeroelastic behaviour of the wing. The new stability characteristics are shown in the root locus plots provided in Figure 4-20, Figure 4-24, Figure 4-28, and Figure 4-32. A magnified portion of these graphs for 0', 10, and 2' root angles of attack is provided in Figures 4-21, 4-25, and 4-29. The data is also presented in V-g plot format, which is provided in Figure 4-22, Figure 4-23, Figure 4-26, Figure 4-27, Figure 4-30, Figure 4-31, Figure 4-33, and Figure 4-34. 119 Table 4.6: Final natural frequencies from wing analysis Mode Calculated Frequency Measured Frequency % Error Mode Shape 1 6.44 Hz 6.4 Hz 0% 2 3 4 38.0 Hz 45.0 Hz 70.0 Hz 38 Hz 45 Hz 70 Hz 0% 0% 0% 2 "3 Bending 1 " Chordwise Bending 1 st Torsional 5 6 123 Hz 216 Hz - - - - Bending ,d Torsional 2 Speed = 0.00 m/s @ Freq = 6.44 Hz 1 st Bending 3 rd Speed = 0.00 m/s @ Freq = 38 Hz -0. Speed = 0.00 m/s @ Freq = 45 Hz Speed = 0.00 m/s @ Freq = 70 Hz 0.2 0.4 0.6 0.8 1 Speed = 0.00 m/s @ Freq = 120 Hz Speed = 0.00 m/s @ Freq = 208 Hz 0.2 0.2 0.4 0.4 -0.1 -0.05 0.6 0.6 0 0.8 0.8 1 Figure 4-19: Predicted mode shapes after final modifications of different wing properties 120 Root locus for Root AOA = 0* 140 120 48 44 4E) BE) 2 10 0 H- 100 80 1T U Cr a.) U-. 60 F- - 8 1CB 6 450 40 2B 20 K iR - - 40: 0' 50 40 I 30 -I30 20: I 20 10 Real Part of Root 13 --------- 0 0 10 20 Figure 4-20: Root locus plot for 0' root angle of attack based on final modifications of different wing properties 121 Root locus for Root AOA = 0* 45.7 45.6 45.5 45.4 -- - ------ 1T 45.3 LL -- -\0 45.1 45 --------- 1CB 44.9 F- 44.8 F------------- 44.7 0. 5 I 0 I I 0.5 1 Real Part of Root I 1.5 2 Figure 4-21: Magnified root locus plot for 00 root angle of attack based on final modifications of different wing properties 122 0 Degree Angle of Attack 120 100 80 N I C.) 0) * * * * * * 60 H 0~ 0) LL 401 ** 20 - * * 0 0 * * 10 20 30 40 Speed (m/s) 50 60 70 Figure 4-22: V-g (frequency part) for 00 root angle of attack based on final modifications of different wing properties 123 0 Degree Angle of Attack 0.1 -* 0.05 * * %I, - * * * 0.0 - ** - - - - - - - - - - * * * * * * * * * * - - - --W IF W * * ** * * * * * * * * * * * -0.1 F * * * * * -0.15 I 0 10 20 I I I I 30 40 50 60 Speed (m/s) Figure 4-23: V-g (damping part) for 00 root angle of attack based on final modifications of different wing properties 124 70 1* Root locus for Root AOA = 140 120 ------- 100 - - - - - - -- - -- -- 3B --- ------------------ ---- - I E) 0 -------- 80 1T: 70 :3 U- - -- - - -- - - - 60 F- -- 1CB 40 . 213 ... .... ...... . . -0 20 H4----- ----- -- - - - -- - - - - -z_ %j 01 50 40 30 20 10 Real Part of Root -- -- -Iv -- - -V 0 10 20 Figure 4-24: Root locus plot for 10 root angle of attack based on final modifications of different wing properties 125 Root locus for Root AOA = 1* 50 48 - 46 -- -------- 1CB 44 -- 42 1-- - - ---------C LL~ - 38 -- - - -- --28 ,10 ----- 36 1- 34 H --- --- 6 4 -- -- 32 F - 30' 10 8 2 0 2 4 6 8 10 Real Part of Root Figure 4-25: Magnified root locus plot for 10 root angle of attack based on final modifications of different wing properties 126 1 Degree Angle of Attack 120 100 80 I C., * * * * * * * * 10 20 601- U- 40 20 F 0 0 30 40 Speed (m/s) 50 60 70 Figure 4-26: V-g (frequency part) for 10 root angle of attack based on final modifications of different wing properties 127 1 Degree Angle of Attack 0.1 0Y * * 0 ** * E * * * * * * -0.1 I* -0.2 0 10 * 20 30 40 Speed (m/s) 50 * * 60 Figure 4-27: V-g (damping part) for 10 root angle of attack based on final modifications of different wing properties 128 70 Root locus for Root AOA = 2* 140 - -- 120 h -- --- - --- - ---- -3 & ---- -3E) -3- -2E)3le20 100 80 l ---- --- -------- - - - - - - - - - - -- - - -- ------.0 - - - - Cr U- 1CB 40 r 20 -0-- 1B: 40 0 50 2B - 40 30-20 30 10 20 10 Real Part of Root 0 0 10 20 Figure 4-28: Root locus plot for 20 root angle of attack based on final modifications of different wing properties 129 Root locus for Root AOA = 20* 50 40 1CB ''04 45 F- - - -- - - - - - - - - -- F- -- - - - - - --- - - -- - --- - ---- - --0 20 2B N V U) 6 - 35 0~ U) 30 k - 25 k 20' 10 8 6 -- ---------------- 4 - 2 0 2 4 6 8 10 Real Part of Root Figure 4-29: Magnified root locus plot for 2' root angle of attack based on final modifications of different wing properties 130 2 Degree Angle of Attack 120 100 80 - - I Cr a) 601 U) 40, * * * ** 201 **** & 0 0 10 20 40 30 50 60 70 Speed (m/s) Figure 4-30: V-g plot (frequency part) for 2' root angle of attack based on final modifications of different wing properties 131 2 Degree Angle of Attack 0.2 0.15 - 0.1 * 0 -* 0.05 * 0) E 0 ** - * -0.05 ** -0.1 -0.15 0 10 20 30 40 50 60 Speed (m/s) Figure 4-31: V-g plot (damping part) for 2' root angle of Attack based on final modifications of different wing properties 132 70 Root locus for Root AOA = 50* 140 120 F 100 F -- - 6 * 1T: C U- 0 60F 1CB 2B 4 ---20 -- - - - - -- -- 40 0 50 - - ----------- ---- - - 20: 30 40 30 10 20 Real Part of Root - - - -- - - 1B 10 0 0 10 20 Figure 4-32: Root locus plot for 5' root angle of attack based on final modifications of different wing properties 133 5 Degree Angle of Attack 120 100 h 80 I N I * * * * C.) a) 60 0) U- 40 * *** * ** ** ** * 20P 0 0 * * 10 20 30 Speed (m/s) 40 * 50 60 Figure 4-33: V-g (frequency part) for 50 root angle of attack based on final modifications of different wing properties 134 5 Degree Angle of Attack 0.1 0.05 - * * * * 0 0 E :: * . -0.05 ** -* -0.1 -0.15 ** 0 10 30 20 40 50 Speed (m/s) Figure 4-34: V-g (damping part) for 50 root angle of attack based on final modifications of different wing properties The strange behaviour observed at 10 root angle of attack in the stiffness adjustments has shifted and is now occuring at 2'. By plotting the data in V-g plots, it was determined that the wing is experiencing hump flutter at the lower speed. The mode would not be seen experimentally, as a very small structural damping within the system would have negated the weak flutter seen at the first crossing of the instability axis. The second crossing, at 51.8 m/s for 20 angle of attack, is a much stronger flutter and would manifest itself during experimentation. The predicted flutter speeds for this wing are provided in Table 4.7. 135 60 Table 4.7: Predicted flutter speeds with final model adjustments Angle of Attack 10 20 Flutter Speed 42.5 m/s 51.8 m/s 50 42.7 m/s The curve for flutter speed versus root angle of attack is provided in Figure 4-35. The solid line represents the final predicted flutter speed. The dotted line shows the speed of the first crossing of the instability axis, or the hump mode speed, and is ignored for reasons mentioned above. The three individual points represent the maximum wind tunnel tested speeds. 4.3 Wind Tunnel Tests The experimental portion of this thesis culminated with the wind tunnel tests. A photo of the wing in the tunnel is provided in Figure 4-36. Once the set-up was completed, calibration tests were performed before starting the actual tunnel tests. During the wind tunnel tests, four strain gauges were monitored-root bending, root torsion, forward/aft (located at 45% of span) and another bending (located at 35% of span). 4.3.1 Wing Calibration in the Wind Tunnel As with the bench-top tests, a calibration of the strain gauges with the wing in the tunnel was performed to determine the deflection-to-voltage ratio of the bridges as different conditioners were being used. The deflection was produced in a slightly different manner for the tunnel tests. Weights were suspended from the wing at various locations along the chord at 2/3 the span of the wing (Figure 4-37). Three different tests were performed with these weights, which were suspended from the wing via a hanger (17 gm). The first test used 217 gm (2.13 N) from the wing at points along the wing chord. This test was to locate the elastic axis. The location of the elastic axis 136 60 1 1 1 predicted flutter speeds -e- predicted flutter speeds (disregarding hump instability) * highest wind tunnel tested speed -0- 55- 5045 E U) 40- '0 35- '0.* o 30' 0 1 2 0 3 4 Root Angle of Attack (deg) 5 6 Figure 4-35: Predicted flutter speed for various root angles of attack 137 7 Figure 4-36: Wing set-up in the Wright Brothers wind tunnel 138 Figure 4-37: Set-up for hanging weights to calibrate strain gauges 139 was determined to be at 73% of the chord from the leading edge. This result contradicts the analytical expectation of 30% of the chord. This is due to the location of the strain gauge used to calculate this result. The root torsion gauge was used and is located about 10 mm from the clamp. The area near the clamp is subjected to end effects and does not behave according to St. Venant's principle. If the gauge was located at least a chord-length away from the clamp, the results would more accurately reflect the analytical value. A plot of the resulting strain gauge output is provided in Figure 4-38. 0.05 0.04 0.03 a, 0 0.02 0 0 0 0.01 0 0.01 0.02 0 100 200 300 400 500 600 700 Time (s) Figure 4-38: Time trace for elastic axis determination The second test was to hang weights(3.109 N)- 117 gm (1.147 N), 217 gm (2.128 N), and 317 gm at 50% of the chord to calibrate the strain gauge output due to bending. The locations of the leading and trailing edges at the wing tip were recorded for each weight as well. This was accomplished by first measuring the height of the unloaded wing tip (leading 140 and trailing edge) and then the height for all subsequent loadings. A time trace of the strain gauge data is provided in Figure 4-39. 7 6 5 0 324- 2 00 50 100 150 200 250 Time (s) Figure 4-39: Time trace for wing loaded at 50% chord in wind tunnel Finally this second test was repeated, except the weights were hung from 90% of the chord. This provided a second calibration for the gauges. A time trace of this data is shown in Figure 4-40. A calibration curve for the root bending gauge as a function of voltage-to-bending moment at the root is provided in Figure 4-41. A curve for the root torsion gauge depicting the voltage-to-pitch moment at the root relation is given in Figure 4-42. To get the data from both loading cases to agree, the moment was found about an axis at 63.81% chord line. A curve for the other bending gauge provides the voltage-to-bending moment at 35% span (gauge location) and is shown in Figure 4-43. 141 6 5 0 CC 350 0 50 Time (S) wing loaded at 90% chord in wind tunnel Figure 4-40: Time trace for 142 Root Bending Gauge: Weight at 50% and 90% Chord 6 5 y = 2.1035 x + 0.47736 0, C -a I- 0 I- 0 2 ME 1 - 0- 0 0.5 1 1.5 Bending Moment at Gauge (N*m) 2 2.5 Figure 4-41: Calibration of root bending gauge from both 50% and 90% chord load cases 143 Root Torsion: Weight at 50% and 90% Chord -0.015 -0.016- -0.017- -0.0180) y = -0.138 x + -0.022918 Cz -0.019C 0 -0.02- 0 0 -0.021 - -0.022- -0.023- -0.024' -0.0 5 I -0.045 I I -0.04 I -0.035 I I I -0.02 -0.03 -0.025 -0.015 Pitch Moment at Gauge (N*m) I I -0.01 -0.005 Figure 4-42: Calibration of root torsion gauge from both 50% and 90% chord load cases 144 0 Another Bending Gauge: Weight at 50% and 90% Chord -0.15 -0.16 F * -0.17 F 0 C a>) -0.18 F m, 0) C > -0.19 F y = -0.038802 x + -0.15481 -0.2 F -0.21 0 0.2 0.4 0.8 0.6 Bending Moment at Gauge (N*m) 1 1.2 Figure 4-43: Calibration of other bending gauge (located at 35% span) from both 50% and 90% chord load cases 145 1.4 In addition to these static tests, the tap tests were also repeated. From these tap tests, the frequencies of the wing were confirmed with the new mounting set-up to be the same as in the bench, with the addition of three unexpected frequencies detected. A 15-Hz chorwise bending mode was introduced, due to flexibility of the load cell mounting. A mode at 20 Hz is present in most of the tunnel data, this is also believed to be associated with the flexibility of the load cell mounting. Another possibility is that some larger motors produce a 20-Hz dynamic disturbance. The 60-Hz mode is often found in dynamic data and is due to electricity noise. Data from the undisturbed system is provided in Figure 4-44. The 100 Hz dynamics is also believed to be either noise from the tunnel environment, as its magnitude is quite large, even in the undisturbed wing data. The graphs for the bending, torsion, and forward/aft taps are provided in Figure 4-45 through Figure 4-47. The data from the root bending gauge shows the expected 6-Hz 1" bending mode and the 38-Hz 2 nd bending mode. It also shows significant dynamics at 15 Hz and 20 Hz. The data from the root torsion gauge shows the 6-Hz lst bending mode and the 38-Hz 2 nd bending mode again, but also shows the 70-Hz 1 " torsion mode. Again, an unexpected dynamics at 15 Hz, 20 Hz, and 100 Hz can be seen from the data. The data from the forward/aft strain gauge shows the first two bending modes as well, and also shows the 45 Hz 1 " chordwise bending mode. The dynamics at 15 Hz was again detected. 4.3.2 Wind Tunnel Tests With the natural frequencies of the wing confirmed, the wind tunnel tests were conducted. The first wind tunnel test was to determine the lift curve for the wing. The wing was then flown at three different root angles of attack (10, 2', and 50). The starting speed for each of these root angles of attack tests was 20.1 m/s (45 mph) with the upper limits varing according to the behavior of the wing. 146 10 10 10 N 10 E CMJ z ) 10- 10 10- - 10 0 10 20 30 40 50 60 Frequency (Hz) 70 80 90 100 Figure 4-44: Other bending gauge readings from undisturbed wing mounted in tunnel 147 110 10 102 103 10 I E_ C\j 10- z c. C) U) 10 108- 10-9 L -iiI 10-9 - 0 10 20 30 40 60 50 Frequency (Hz) 70 80 90 100 Figure 4-45: Root bending gauge readings from tap test-wing mounted in tunnel 148 110 10-3 10-- 10 N c'J E z ci) 1 -6 10 10- 10-0 10 20 30 40 60 50 Frequency (Hz) 70 80 90 100 Figure 4-46: Root torsion gauge readings from tap test-wing mounted in tunnel 149 110 10-3 10 10-- Cn 10 _ 10- 10- 0 10 20 30 40 50 60 Frequency (Hz) 70 80 90 100 Figure 4-47: Forward/Aft bending gauge readings from tap test-wing mounted in tunnel 150 110 Wing Lift Curve The lift curve was generated by flying the wing at 20 m/s (33.6 mph) and incrementing the angle of attack from 00 to 12'. The lift generated was monitored through the load cell. Both the theoretical and experimental lift curves are provided in Figure 4-48. The symbols represent the experimental data and the solid line is the theoretical curve generated from [3]. In addition to the calibration matrix coefficient, the load cell data was multiplied by a constant factor of 10. The exact reason for this multiplier is unknown to the author. A further calibration of the load cell is necessary to properly identify the correct calibration term. 30- y 3*x+1.4 25' 20S15 -theory 10- experiment * linear fit 5-- 00 0 1 2 3 4 5 6 Root Angle of Attack (deg) 7 8 9 10 residuals 6- - 42 -0 U -2 -4- Linear: norm of residuals =8.3112 -60 1 2 3 4 5 6 Figure 4-48: Lift curve for wing 151 1 7 1 8 9 1 d10 1' Root Angle of Attack Tests The 1' test were conducted from 20.1 m/s (45 mph) to 36.7 m/s (82 mph). Since this was the first round of tests, the flight speed was not driven up near flutter to ensure that data would be collected for all three angles of attack. The data from the sensors at 20.1 m/s (45 mph) show many of the natural frequencies and is provided in Figure 4-49 through Figure 4-52. This data was processed through a Welsh averaging scheme with a 125 data point window and an overlap of 63 data points. At a minimum, all flight data was processed through this scheme. 10 1 10 10 10 H I I I I 30 40 I I I I I 70 80 90 100 -2 -3 -4 I C' E CMJ z 10~ CD) 10 107 108 0 10 20 60 50 Frequency (Hz) 110 Figure 4-49: Frequency plot from root bending gauge at 20.1 m/s for 10 root angle of attack 152 103 10 E z 0 C/) -6 10- - 10- 108 0 10 20 30 40 50 60 Frequency (Hz) 70 80 90 100 110 Figure 4-50: Frequency plot from root torsion gauge at 20.1 m/s for 1' root angle of attack 153 10 10-- 10-N N U 106 10- 10-8 0 10 20 30 40 50 60 Frequency (Hz) 70 80 90 100 110 Figure 4-51: Frequency plot from forward/aft gauge at 20.1 m/s for 1 root angle of attack 154 10-3 10 10 1~ 10- E 10-5- 10-6 10 10- 10100 10 20 30 40 60 50 Frequency (Hz) 70 80 90 100 110 Figure 4-52: Frequency plot from the other bending gauge at 20.1 m/s for 1 root angle of attack 155 The data from the sensors at 36.7 m/s (82 mph) is noisier and the natural frequencies are harder to detect. These results are shown in Figure 4-53 through Figure 4-56. 10 10-2 10-3 'N 10 E z -5 al 10 -6 10 10~ 0 10 20 30 40 50 60 Frequency (Hz) 70 80 90 100 110 Figure 4-53: Frequency plot from root bending gauge at 36.7 m/s for 1 root angle of attack 156 103 10 E N '. C') 10~-- 10- 0 10 20 30 40 50 60 Frequency (Hz) 70 80 90 100 110 Figure 4-54: Frequency plot from root torsion gauge at 36.7 m/s for 10 root angle of attack 157 10-3 10 \I 10-5- 10-6 10- 10II|1 0 10 I 20 30 40 I I I 60 50 Frequency (Hz) 70 80 Figure 4-55: Frequency plot from forward/aft gauge at 36.7 m/s for 1 158 90 100 110 root angle of attack 10-3 104 - 10- -6 10 E z 10 108 10- I 0 10 I 20 I 30 I 40 I 50 I 60 I I 70 80 I 90 100 110 Frequency (Hz) Figure 4-56: Frequency plot from the other bending gauge at 36.7 m/s for 10 root angle of attack 20 Root Angle of Attack Tests The 2' tests were conducted from 20.1 m/s (45 mph) to 36.7 m/s (82 mph) and then from 20.1 m/s (45 mph) to 52.3 m/s (117 mph) after the 5' case was completed. The natural frequencies at 20.1 m/s were fairly straight forward to determine. The added turbulence from the wind tunnel did not interfere with this 10 s data collect, so this data did not require extensive processing or filtering. The time trace from the root bending gauge is provided in Figure 4-57 to show how clean the data looks. 159 -2.5 -3- E CMJ z CD m--3.5 CO C E 0 2 -4- C -4.5- -5' 0 1 2 3 4 5 Time (s) 6 7 8 9 Figure 4-57: Time trace of data from root bending gauge at 20.1 m/s for 2' root angle of attack 160 10 The data from the four strain gauges with the aforementioned averaging technique are provided in Figure 4-58 through Figure 4-61. 10 102 103 104 N E N,~ z 0n 10 a-) 106 10 10 0 10 20 30 40 60 50 Frequency (Hz) 70 80 90 100 110 Figure 4-58: Frequency plot from root bending gauge at 20.1 m/s for 2' root angle of attack 161 103 10 10-5 E z CMJ 10~ 10- 0 10 20 30 40 50 60 Frequency (Hz) 70 80 90 100 110 Figure 4-59: Frequency plot from root torsion gauge at 20.1 m/s for 2' root angle of attack 162 10-3 10-~ 10 -7 U)10-6- 10 10-8 0 10 20 30 40 60 50 Frequency (Hz) 70 80 90 100 110 Figure 4-60: Frequency plot from forward/aft gauge at 20.1 m/s for 2' root angle of attack 163 10 10 10 -6 10 E Cvi z 0 10 - 108 10- 0 10 20 30 40 60 50 Frequency (Hz) 70 80 90 100 110 Figure 4-61: Frequency plot from the other bending gauge at 20.1 m/s for 20 root angle of attack 164 All four bridges detected a 20-Hz disturbance as well as one at 37-Hz. The additional bending gauge only slightly detected the 37 Hz signal. A summary of which gauge detected which frequencies is provided in Table 4.8. Table 4.8: Frequencies detected from strain gauges at 20.1 m/s for 2' root angle of attack Gauge 6 Hz 20 Hz 37 Hz Root Bending Root Torsion Chordwise Bending Additional Bending X X X X X X X X X 70 Hz 100 Hz X X X A data set of importance from the 2' root angle of attack tests is at 52.3 m/s. Through visual observation of the wing behavior, it is believed that the wing was operating just below the flutter speed. Two root torsion gauge time traces of the wing operating at this speed are provided in Figure 4-62 and Figure 4-63. The data from the sensors is provided in Figure 4-64 through Figure 4-66. 165 2.2 2- 1.8C'J E N'. z " 1.6 - ~1.4 E 0 0) 1.2 C, 0.8 0.6' 0 1 2 3 4 5 Time (s) 6 7 8 9 Figure 4-62: Time trace of data from root torsion gauge at 52.3 rn/s for 2' root angle of attack 166 10 2 1.8 C\J E N 1.6 z Cz 1.4 CD E 0 ~1.2 CL 1 0.8 0 1 2 3 4 5 Time (s) 6 7 8 9 Figure 4-63: Time trace of data from root torsion gauge at 52.3 m/s for 2' root angle of attack 167 10 10-3 10-4N z C,) IIf V I iiJ 10- 0 10 20 30 40 60 50 Frequency (Hz) 70 80 90 100 110 Figure 4-64: Frequency plot from root torsion gauge at 52.3 m/s for 2' root angle of attack 168 10~3 10 10-- 10-6 10 10-- 0 10 20 30 40 60 50 Frequency (Hz) 70 80 90 100 110 Figure 4-65: Frequency plot from forward/aft gauge at 52.3 m/s for 2' root angle of attack 169 103 10-- 10-- N -6 10 E z 10- 10 10-- | 0 10 | 20 I 30 40 I I I I 70 80 90 100 I 1 60 50 Frequency (Hz) 110 Figure 4-66: Frequency plot from the other bending gauge at 52.3 m/s for 2' root angle of attack 50 Root Angle of Attack Tests The 50 tests were conducted from 20.1 m/s (45 mph) to 33.5 m/s (75 mph). The wing at this root angle of attack produced very large tip deflections. Therefore, the wing was not flown up to 36.7 m/s (82 mph) as originally desired because of fear that the wing would break and damage the tunnel. The data from a 20.1-m/s case is provided in Figure 4-67 through Figure 4-69. The data from the root bending gauge is not provided as the signal exceeded the recording limits. 170 10-3 10-- 10-5 E z 0 Cl) 10- 0 10 20 30 40 60 50 Frequency (Hz) 70 80 90 100 110 Figure 4-67: Frequency plot from root torsion gauge at 20.1 m/s for 50 root angle of attack 171 10-3 - 10~4 10- 10 10-- 10 - 0 10 20 30 40 60 50 Frequency (Hz) 70 80 90 100 110 Figure 4-68: Frequency plot from forward/aft gauge at 20.1 m/s for 50 root angle of attack 172 10 10 10 N 1-6 110 E 10 10-8 10 0 I I I I 10 20 30 40 I I 50 60 Frequency (Hz) I I 70 80 I 90 110 100 Figure 4-69: Frequency plot from the other bending gauge at 20.1 m/s for 50 root angle of attack The data from the 33.5-m/s (75-mph) case shows some interesting frequency results. This data is provided in Figure 4-70 through Figure 4-72. There are three dynamics present in the root torsion gauge that were not targeted as noise from the tunnel environment: 34 Hz, 42 Hz, and 46 Hz. The forward/aft gauge and the bending gauge at 35% chord have dynamics at 34 Hz, 38 Hz, and 40 Hz. This range of frequencies is where the root locus plot (Figure 4-32) predicts both the 2 nd bending and the 1 st chordwise bending to be interacting. 173 10- 3 10-- E z C,) 10- 10- 0 10 20 30 40 50 60 Frequency (Hz) 70 80 90 100 110 Figure 4-70: Frequency plot from root torsion gauge at 33.5 m/s for 5' root angle of attack 174 10-3 10 10-N \I U) 10 1 6 10-- 108 0 10 20 30 40 50 60 Frequency (Hz) 70 80 90 100 110 Figure 4-71: Frequency plot from forward/aft gauge at 33.5 m/s for 50 root angle of attack 175 10-3 10-- 10- S-6 10 E z o -7 10 10 101- 10 0 20 30 40 50 60 Frequency (Hz) 70 80 90 100 110 Figure 4-72: Frequency plot from the other bending gauge at 33.5 m/s for 5' root angle of attack Post-Tunnel Tap Tests After the tests in the tunnel were completed, a follow-up tap test was conducted to determine if anything changed during flight. The results from a bending excitation tap are provided in Figure 4-73. This tap test captured the 6-Hz 1 " bending mode, the 38-Hz the 45-Hz 1 " chordwise bending mode, and the 70-Hz 1 2 nd bending mode, " torsion mode. The torsion mode was excited through a twisting tap. The data from the torsion strain gage is provided in Figure 4-74. This graph shows the 60-Hz 18' bending mode and the 70-Hz 1 " torsion mode. A mode at 100 Hz is also being excited. The forward/aft tap test is not provided as the tap 176 10 10-2 10 10-N E z 10 0 10-6 10 10 0 10 20 30 40 50 60 Frequency (Hz) 70 80 90 100 Figure 4-73: Root bending gauge readings from tap test-wing mounted in tunnel 177 110 was not recorded at the correct time. However, the chordwise bending mode was seen in the bending tap. The chordwise bending mode was excited through a tap on the leading edge. 10-3 E z 10 100 10 20 30 40 60 50 Frequency (Hz) 70 80 90 100 110 Figure 4-74: Root torsion gauge readings from tap test-wing mounted in tunnel The data from the forward/aft gauge is shown in Figure 4-75. This data yields the 45 Hz 1 1' chordwise bending mode. This dynamic is masked quite a bit by the noise from the tunnel, especially the 15-Hz and 60-Hz signal. 178 10 10-2 10-3 ,.0 C') 10~4 10- 0 I I 10 20 I 30 40 I iII I 50 60 Frequency (Hz) 70 80 90 100 Figure 4-75: Average tip accelerometer readings from tap test-wing mounted in tunnel 179 110 Chapter 5 Concluding Remarks This chapter presents a summary of the findings from the analytical and experimental portions of this thesis. Along with the summary are sections on the conclusions drawn from the research as well as recommendations for future work. The recommendations section suggests improvements to the manufacture of the wing and the test set-up. 5.1 Summary The aeroelastic behavior of high-aspect ratio wings was studied via a new numerical formulation [11] which encompasses a non-linear, large deflection beam model with strain-based finite element representation and finite-state unsteady airloads. Using this formulation, a wing was designed, built, and its structural behavior evaluated under a variety of conditions. Bench-top tests were performed to understand the structural dynamic behavior of the wing. The first four natural frequencies at zero wind speed measured during bench-top 'tap' testing were in good agreement with the analytical model. The wing was then tested within the Wright Brothers Wind Tunnel to further understand the changes in dynamic behavior due to flight speed and root angle of attack. The wing was flown at speeds ranging from 20 m/s to 51.8 m/s and at 1*, 2 , and 50 root angles of attack. The tip deflections and natural frequencies were recorded for these cases. 180 5.2 Conclusions The results presented herein validate the analytical model of the wing's aeroelastic behavior under a variety of angle of attacks and wind speeds. Few parameter changes were introduced to the model to reproduce the actual dynamic behavior. The need for these changes can be associated with complexities resulting from the manufacturing process. With these changes implemented, the predicted flutter speed is within 99% of the estimated flutter speed from observations in the wind tunnel. Experimental validation of the analytical model provides confidence in the ability of the code to predict the behavior of other systems under similar conditions. This capability would be useful for predicting the theoretical performance of high aspect ratio wings currently under consideration for upcoming remote sensing and reconnaissance vehicle programs. 5.3 Recommendations for Future Work The manufacture and test of the active wing presented in Appendix C would further validate the capabilities of the analysis code. This would also demonstrate the capabilities of embedded piezoelectric actuators (in the composite wing skin) to increase the flight envelope of HALE vehicles. Note that a stress analysis, similar to the one conducted for the passive wing, will still have to be performed for the active wing design. Several areas of improvement in the wing manufacturing and the test set-up have been identified that would facilitate future evaluations. First, although it would be more difficult, the foam core should be manufactured as a single piece, rather than four seperate ones. This would provide for more uniform cross-sectional properties. Second, the use of tip accelerometers was of limited value and could have been eliminated without a large impact on the results. The omission of the accelerometers would aleviate one of the reasons for using four seperate pieces of foam for the core-there would no longer be a need to run wires through the center of the wing. Omitting the accelerometers also helps with the homogeneity of the cross section. Finally, the load cell added unwanted noise into the data and had little practical value in the experiment. If a lift curve is desired it could be attained via a separate 181 set-up. During the dynamic tests of the wing, where mode shapes and frequencies are important, the flexibility of the load cell mounting fixture added undesirable dynamics to the system. 182 References [1] Patil, M. J., D. H. Hodges, and C. E. S. Cesnik, "Nonlinear Aeroelasticity and Flight Dynamics of High-Altitude Long-Endurance Aircraft," Journal of Aircraft, Vol. 38, No. 1, 2001, pp. 88-94. [2] Patil, M. J., D. H. Hodges, and C. E. S. Cesnik, "Characterizing the Effects of Geometrical Nonlinearities on Aeroelastic Behavior of High Aspect-Ratio Wings," in Proceedings of the InternationalForum on Aeroelasticity and Structural Dynamics, 1999. [3] Cesnik, C. E. S. and E. L. Brown, "Modeling of High Aspect Ratio Active Flexible Wings for Roll Control," in Proceedings of AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, 43rd, AIAA-2002-1719, 2002. [4] Ortega-Morales, M. and C. E. S. Cesnik, "Modeling and Control of the Aeroelastic Responce of Highly Flexible Active Wings," Tech. Rep. AMSL # 00-2, Massachusetts Institute of Technology, February 2000. [5] Patil, M. J., D. H. Hodges, and C. E. S. Cesnik, "Nonlinear Aeroelasticity Analysis of High Aspect Ratio Wings," in Proceedings of the 39th Structures, Structural Dynamics, and Materials Conference, 1998. [6] Cesnik, C. E. S., D. H. Hodges, and M. J. Patil, "Aeroelastic Analysis of Composite Wings," in Proceedings of the 37th Structures, Structural Dynamics, and Materials Conference, 1996. 183 [7] Minguet, P. and J. Dugundji, "Experiments and Analysis for Composite Blades Under Large Deflections Part II: Dynamic Behavior," AIAA Journal, Vol. 28, No. 9, 1990, pp. 1580-1588. [8] Patil, M. J. and D. H. Hodges, "On the Importance of Aerodynamic and Structural Geometrical Nonlinearities in Aeroelastic Behavior of High-Aspect-Ratio Wings," in Proceedings of AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, 41st, AIAA-2000-1448, 2000. [9] Tang, D. and E. H. Dowell, "Experimental and Theoretical Study on Aeroelastic Response of High-Aspect-Ratio Wings," AIAA Journal, Vol. 39, No. 8, 2001, pp. 14301441. [10] Tang, D. and E. H. Dowell, "Experimental and Theoretical Study on Gust Response of High-Aspect-Ratio Wings," AIAA Journal,Vol. 40, No. 3, 2002, pp. 419-429. [11] Brown, E. L., Integrated Strain Actuation in Aircraft with Highly Flexible Composite Wings, Ph.D. thesis, Massachusetts Institute of Technology, January 2003. [12] Rodgers, J. P., Development of an Integral Twist Actuated Rotor Blade for Individual Blade Control, Ph.D. thesis, Massachusetts Institute of Technology, October 1998. [13] Juhl, B., "Roehm America, Inc. - Rohacell." email, 2002. personal communication. [14] Peters, D. A. and M. J. Johnson, "Finite-State Airloads for Deformable Airfoils on Fixed and Rotating Wings," in Symposium on Aeroelasticity and Fluid/StructureInteraction, Proceedings of the Annual Winter Meeting, November 1994. [15] Peters, D. A., S. Karunamoorthy, and W. M. Cao, "Finite State Induced Flow Models Part I: Two-Dimensional Thin Airfoil," Journalof Aircraft, Vol. 32, No. 2, 1995, pp. 313322. [16] Jones, R. M., Mechanics of Composite Materials, Hemisphere Publishing Corporation, 1975. 184 Appendix A Mathematical Formulation A.1 Introduction This appendix provides a summary of the mathematical modeling used in this research. A brief description of the geometrical, structural, and aerodynamic issues is presented. A detailed description of the mathematical formulation can be found in [3] and [11]. The main features of this code are: " nonlinear, large deflection beam model , " strain-based finite element representation , and " finite-state unsteady airloads . A.2 Geometrical Lay-out There are three reference frames governing the wing and its interactions with the environment. The first being the global coordinate system through which the wind speed and gravity affect the wing. The undeformed reference frame of the wing is measured with respect to the cross section area centroid along the 30% chord line of the wing. Finally the deformed reference frame follows the deformation of the beam reference line. A diagram of these coordinate frames with respect to the wing is provided in Figure A-1. 185 UY(s) p(s) x Figure A-1: Coordinate systems for the wing A.3 Structural Formulation The structural analysis of the wing is based on the principles of virtual work. These principles yield the following general equation: 6W = fy(f(x, y, z)6u(x, y, z))dV where 6W is the virtual work done by a force, f(x, y, z), to move an object a virtual distance, 6u(x, y, z). Separating this equation into individual forces yields the general equation: 6W = F1uou + F2,6u + -- -+ Funu For an object to be in equilibrium, the above equation must be zero, that is, 6W = 0. Within the program, the wing is allowed to move through three-dimensional bending and twisting deformations, while extension and shear deformations are assumed negligible and thus ignored. With this assumption, the virtual work equation becomes [3]: 6W = In this equation, opT =o(6pTF + 6)M 6kTMintjdu is the change in the position of the reference line, p(s), see Figure A-1. The term 66f represents the virtual rotation about the reference line, and JT denotes the virtual curvature of the wing. The variable F stands for the external forces on the wing, M 186 represents the external moments, and M"' represents the internal moments acting on the wing. When the vector q is defined to be the curvatures,rs in (x,y,z) and the variable R is defined as all the generalized loads, the equation becomes [3]: SW = 6qT (-Mqa - Cqe - Kqq + Rq) In this equation, Mq denotes the mass matrix, Cq symbolizes the damping matrix, and Kq is the stiffness matrix. Setting this system into equilibrium, yields: Mqi + Cqj+ Kqq = R The stiffness matrix, Kq, is defined as: K, = Ku1 K 12 K 13 K 14 K K 12 K 22 K 23 K 24 13 K 23 K33 K 34 14 K 24 K K K 34 44 where K 11 is the extension stiffness, K 22 is the torsional stiffness, K 33 is the bending stiffness about y, and K 44 is the bending stiffness about z (or the chordwise bending stiffness). The off-diagonal terms are the cross-coupling terms, e.g., K 14 is the extension-chordwise-bending stiffness term. A.4 Aerodynamic Formulation The aerodynamic model used in this aeroelastic formulation comes from the theory derived in [14] and [15]. A full discussion on this is provided in [11] and [3]. The theory accounts for thin deformable airfoil cross section undergoing large deformations at subsonic speeds. It also includes small deformations about this large deflected state. The lift, L, moment, M, and drag, D, are defined by the following equations: 187 Ao] - lb2 - 'bd&) L = 2rpb(p[({b - d)d M 2rpb(Q[-(d + lb)i - (d + !b)AO - d2 d] D = lbdz - {b(d 2 + 1b2 )d,) -27rpb(z2 + d 2&2 + A2 + 2dds + 2AOs + 2ddAo) Within these equations, b is the semichord, d is the distance between the reference axis and the mid-chord, and Ao represents the inflow term due to free vorticity [3]. denotes the air density. The variables which represent the motion: y, y, z, i, The term, p z, a, d, and & are defined in Figure A-2. z)z -c,-a Figure A-2: Variables defining airfoil motion [3] These equations are linearized with respect to time, ta, by the following equations [3]: Y 6- = (Wa + AW' (a A61\) o = (Ao,a + AAo) Applying these perturbations to the equation of virtual work yields: JW = A.5 f O(Loz + D6y + M6a)ds Aeroelastic Formulation The structural and aerodynamic formulations combine to produce the aeroelastic equations [3]. M4+Cq+Kq+DA=F A = F 14+ F 2 4+ F3 A 188 In this equation, the mass, M, and damping, C, matrices are functions of the strains. The structural stiffness, K, matrix is constant for a given geometry and material distribution. The force vector, F, contain the inertial and aerodynamic forces and is a function of initial values of t, q, and A. It also contains the effects of the active material layers in the wing construction. See [11] for further details. 189 Appendix B Material Properties The material file associated with the MATLAB code needs material properties to be converted from engineering moduli (El, Et, vu, Gut) in the local ply coordinate system to the plane-stress stiffness constants, Q [16]. Equations B.1 through B.5 show the relations. The subscripts '1' and 't' represent longitudinal and transverse directions, respectively. The two Poisson's ratios are related via Equation B.6. The material properties for the foam core are provided in the standard nomenclature. Q11 = Qi2 = Q22 = Q16 El (B.1) vEt (B.2) 1 - Vit uti 1- vuura 1- vuti = Q26 = Q66 = vU = Gu Eivut 190 0 (B.3) (B.4) (B.5) (B.6) Table B.1: Properties of the materials used in this study E-glass/epoxy Q11 Q12 Q22 Q66 19.7 2.92 19.7 4.10 p GPa GPa GPa GPa 309 pm/V -129 pm/V n/a 0.1143 mm 0.0011 m 0.127 mm d12 thickness 33.6 7.54 16.6 5.13 GPa GPa GPa GPa n/a n/a di, telectrode Piezocomposite 1716 kg/m 3 4060 kg/m 3 Ei, e22 10000 pm/m 4000 pm/m eY2 15000 pm/m 5500 pm/m E G p Ci122 e12 Rohacell 31 36 MPa 14 kPa 30 kg/m 3 27000 pm/m 30000 pm/m 191 Appendix C Active Wing Design This appendix describes the design and analysis of the active flexible wing that is proposed as a follow-on to the passive wing built for this thesis. Stress analysis will still have to be conducted for this design. C.1 Active Design The active wing design is composed of three layers of E-glass/epoxy fabric oriented at 0' around an inner foam core, with two active layers embedded within the E-glass. There is no spar in this design. The goal is to have a flexible wing with low flutter speeds, for the reasons mentioned during the passive design. This design provides a flexible wing while still maintaining closed cell properties. A drawing of the wing lay-up is provided in Figure C-1. [0 /+ 45 MFC/ 0 /- 4 5 MFC/ 0 ] foam Figure C-1: Cross section of active wing 192 C.1.1 Basic Static and Dynamic Properties The cross-sectional properties for the active design are provided below. The stiffness matrix terms for this cross section are given in Table C.1. Detailed definitions of the elements of the matrix are provided in Appendix A. The inertia matrix for this cross-section is provided in Table C.2. The center of gravity for the cross-section is located 0.0214 m aft of the reference line (located at 30% chord). This places the center of gravity at 49.9% of the chord. Table C.1: Non-zero stiffness matrix terms for the active wing design: 1 = Extension, 2 = Torsion, 3 = Flatwise Bending, 4 = Chordwise Bending Ku1 K 14 = K4 1 K22 K33 K44 2.27 * 106 N -4.78 * 104 N*m 57.72 N*m 2 57.00 N*m 2 2.69 * 10 3 N*m 2 Table C.2: Non-zero inertial matrix terms for active wing design Ill '22 133 0.244 * 10-3 m 4 0.998 * 10~ m 4 4 0.234 * 10-3 m The dynamic properties of the wing are important to its aeroelastic response. The first six natural frequencies of the wing are summarized in Table C.3. The corresponding mode shapes are provided in Figure C-2. 193 Table C.3: Natural frequencies of the active wing design in vacuum Mode Frequency Mode Shape 1 2 6.8 Hz 36.8 Hz " Bending 1 " Chordwise Bending 3 44.1 Hz 2 nd Bending 4 121. Hz 1" Torsional 5 6 133. Hz 232. Hz Bending 2 "d Torsional 1 3 rd Speed = 0.00 m/s @ Freq = 36.8 Hz Speed = 0.00 m/s @ Freq = 6.8 Hz -0.6 -0.4 Speed = 0.00 m/s @ Freq = 44.1 Hz Speed = 0.00 m/s @ Freq = 121 Hz 0.2 -:1 0.4 0.6 0.8 1 Speed = 0.00 m/s @ Freq = 232 Hz Speed = 0.00 m/s @ Freq = 133 Hz 0.2 -0.1 0.4 0.6 0.8 1 1 Figure C-2: First six normal modes for active wing design (in vacuum) 194 C.1.2 Wing Non-linear Characteristics As seen in the passive wing designs, different root angles of attack of the wing result in different wing tip deflections once the wing is exposed to airloads. The change in tip deflection is graphed in Figure C-3 for the range of root angles of attack of interest: 1' to 5'. The tip deflection increases with increased angle of attack. 0.18 0.16 0.14 0.12 0.1 C 0 0.08 .) 0- 0.06 0.04- 0.02- 0-0.02 - 0 5 10 15 25 20 30 35 40 45 Speed (m/s) Figure C-3: Static tip deflection for increasing speed at different root angles of attack The wing tip twist also changes with increased root angle of attack. This data is provided in Figure C-4. To better illustrate this effect, the initial root angle of attack has been subtracted from the total tip twist, yielding only the elastic angle change due to gravitational and aero-loading. 195 50 0.22 0.8 . -- -- 0.18 ... ). .. . . .. . ..... . . . ... . . . ... . . . .. . . ... I0l2 0.1 -0.12----4 -.- .. .. . . . .. . . . .--- 2-0--------.-.-.- . --- -.----. - 0.08 12 . - 0.14--6 . .. . . - - . ... .. . .. 0.0 0 5 10 15 20 25 30 35 40 45 Speed (mis) Figure C-4: Elastic tip twist for increasing speed at different root angles of attack 196 50 The changes in tip deflection and twist result in a change in the dynamic behavior of the wing, and ultimately a change in the wing's flutter speed. This is shown in the root locus plots for the wing at root angles of attack of 10, 2' and 5'. The flutter speeds as a function of root angle of attack determined from these plots are provided in Figure C-5. 90 80- 70- 6042) E 50 - _0 CD (D CL. 40 - 30 20 10 0 0 1 2 4 3 Root Angle of Attack (deg) 5 Figure C-5: Flutter speeds for active wing design 197 6 7 One-Degree Root Angle of Attack The flutter speed for one-degree root angle of attack is determined to be 79.3 m/sec, shown in Figure C-6. The mode of instability is the second mode. A magnified plot of this mode's behavior is provided in Figure C-7. This mode is primarily the first chordwise bending. However, the forces due to air pressure on the tilted wing result in some torsion being present as well. The first six mode shapes at this speed are given in Figure C-8. Root locus for Root AOA = 140 1 1 1 837576 60 so 4@ go -2e 13E)0 3B 120 F- 100 80 - ---- - - - - --- -- - --- T - - -I --- -- - -- ---- -- - -- U - - --- -- - - - - - - - - ------------0 60 2B - -- 40 20 -- - - - - - - - -- - - - - - - - - - --- ---1CB -- - -- -- ---- -- - - - - -- - - -- - - - - -- - - - --------1B 0' 50 40 30 20 10 0 10 20 Real Part of Root Figure C-6: Root locus plot for 10 root angle of attack 198 Root locus for Root AOA = 1* 38 37.5 --------- 37 ----- 1CB --- - -- 36.5 -- -----N I 0 -. .. .-... 36 - ------ --- --- - - -- - 0* 0 LL - 35.5 ----.-.- --- --- - - 5 35 -- -- - -- - ---- - ----34.5.- 34' 0. 2 ---. 0. 1 0 0.1 0.2 Real Part of Root 0.3 0.4 0.5 Figure C-7: Magnification of root locus plot for 1 root angle of attack 199 Speed = 79.30 m/s @ Freq = 35.5 Hz Speed = 79.30 m/s @ Freq = 6.81 Hz Speed = 79.30 m/s @ Freq = 44.1 Hz Speed = 79.30 m/s @ Freq = 124 Hz 0.2 0.1) 0.2 0) Speed = 79.30 m/s @ Freq = 232 Hz Speed = 79.30 m/s @ Freq = 132 Hz 0.2 0.4 0.6 0 0.8 0.1 1 0.8 Figure C-8: Mode Shapes for active wing design at 1 root angle of attack and its corresponding flutter speed of 79.3 m/s 200 When the wing is flown at 10% above the flutter speed for a 10 angle of attack, the wing enters into a Limit Cycle Oscillation (LCO). The tip deflections and the tip twist are plotted for this case up to 1 second. These plots are given in Figure C-9 thru Figure C-11. Speed = 87.23 m/s with a Time step = 0.0005 s 0.12 0.118 .-I.11 E _) E 0.114 U 0.112 0.108' 0 0.2 0.6 0.4 0.8 1 Time (s) Figure C-9: Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (10 root angle of attack) for the active wing design Two-Degree Root Angle of Attack The flutter speed for two degrees angle of attack is determined to be 77.3 m/s from the root locus plot shown in Figure C-12. The instability mode is the same as for 10, except the amount of torsion present is increased. The drop in the second natural frequency due to the increased upward bend, also causes the second mode to turn over faster and amplifies the effects of flow over the wing. The first six modes at flutter speed are provided in Figure C-14. 201 Speed = 87.23 m/s with a Time step = 0.0005 s 3 2.5 2 0) a, ~0 C', 1.5 H0~ r 1 0.51 0 0 0.2 0.4 0.6 1 0.8 Time (s) Figure C-10: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (10 root angle of attack) for the active wing design Speed = 87.23 m/s with a Time step = 0.0005 s 0.115 1 1 I 0.11 45 F E 0.114 E ND 0. CU) C0 0.1 135 I 0.1 13 - 0.1125' -3.5 -3 -2.5 -2 Y Displacement (m) -1.5 -1 x 10 3 Figure C-11: Nonlinear time simulation of the wing tip displacement motion at 10% above its flutter speed (10 root angle of attack) for the active wing design 202 Root locus for Root AOA = 2* 140 0-i 3B - - --- ------------------ 120 - -I --- - - - - - ---100 - -- -- -- -- - --- ----- ------ - -- - -- -- -- - 80 ---- --- ------- U Cr 4) U_ 60 -- ---- -- 40 20 W00 2B ---- -- --- 1CB 20 F 664 QL 50 40 30 3: 20 10 2S 1e 0 1B 0 10 20 Real Part of Root Figure C-12: Root locus plot for 20 root angle of attack 203 Root locus for Root AOA = 2* 38 37 - . ...... .. . -. .. . 1CB 0 I 36 - . 35 - .-- N 0: a, 34 0* a, U- ---- - -- 33 -- 32 - -. . ..-... ..-. ..-. 5 ------ --- -- -. .. .. 31 30 1 - -- -- - 0. 8 0. 6 0. 4 0. 2 0 0.2 0.4 0.6 0.8 1 Real Part of Roc Figure C-13: Magnification of root locus plot for 20 root angle of attack 204 Speed = 77.30 m/s @ Freq = 32.6 Hz Speed = 77.30 m/s @ Freq = 6.81 Hz Speed = 77.30 m/s @ Freq = 129 Hz Speed = 77.30 m/s @ Freq = 44.1 Hz 0.15 0.1 0.05 -0.05 0 Speed = 77.30 m/s @ Freq = 232 Hz Speed = 77.30 m/s @ Freq = 132 Hz 0 0.1 0.8 1 Figure C-14: Mode shapes for active wing design at 20 root angle of attack and its corresponding flutter speed of 77.3 m/s 205 For 20 angle of attack, the wing also enters a LCO when flown at 10% above the flutter speed. The tip deflections and the tip twist are plotted for this case up to 3 seconds. These plots are given in Figure C-15 thru Figure C-17. Speed = 85.03 m/s with a Time step = 0.0005 s 0.3 0.25 a) E 0) 0 0z 0.2 0.15 CL 0.1 0 CD) 0.05 0 -0.05' 0 0.5 1 1.5 Time (s) 2 2.5 3 Figure C-15: Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (20 root angle of attack) for the active wing design Five-Degree Root Angle of Attack The flutter speed for five degrees angle of attack is shown to be 68 m/s in the root locus plot provided in Figure C-18. As with the 1 and 2' cases, the mode of instability is the second mode, which is the 1 "t chordwise bending with some torsional effects. The changes in the dynamic behavior due to the increased upward bend result in the second mode rolling over faster to the positive dampening axis. The fourth mode (1st torsion) has curved upward and now interacts with the fifth mode (3 rd bending mode). This interaction between the fourth and fifth mode has caused the fifth mode to turn towards the instability line as well. The first six mode shapes for this case are provided in Figure C-19. 206 Speed = 85.03 m/s with a Time step = 0.0005 s 8 6 4 CO 2 0 -2 - -4 - -6 - 0 0.5 1 1.5 Time (s) 2 2.5 3 Figure C-16: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (20 root angle of attack) for the active wing design Speed = 85.03 m/s with a Time step = 0.0005 s 0.17 0. 160.15 a) E a) 0. Cz 0. An E N 0.14 0. 13 0. 12 0.11 0.1' -0.0 4 ' -0.03 '-0.02 ' ' '-0.01 0.01 0 Y Displacement (m) '-0.02 0.03 Figure C-17: Nonlinear time simulation of the wing tip displacement motion at 10% above its flutter speed (20 root angle of attack) for the active wing design 207 Root locus for Root AOA = 5 140 0 3B 120 [ -- - - - -- 100 - - - - 1T - 1 -- -- - -- ----- -- --- 80 C Cr U- --------------- ------------- -- 60 -0 - 40 - 2 - - -+G-o 1CB 0 20 ----- - -- - - -- -- - ---- ------- -----i 0' 50 40 006 40 30 3i 20 10 Real Part of Root -20 10-4 0 1 18 10 20 Figure C-18: Root locus plot for 50 root angle of attack 208 Speed = 68.00 m/s @ Freq = 6.83 Hz Speed = 68.00 m/s @ Freq = 26.7 Hz 43.8 Hz Speed = 68.00 m/s @ Freq = 129 Hz 139 Hz Speed = 68.00 m/s @ Freq = 233 Hz 0.6 0.4 0.2 0 -. 00040.30.201 Speed = 68.00 m/s @ Freq 0.2N 0.1T -0. 1 -0.05 - 0 0.8 Speed = 68.00 m/s @ Freq 0.3 0.21 0.1 = -mm -ON 0-- - -- - -- 0.2 -n i -U. - 0.8 Figure C-19: Mode shapes for active wing design at 5' root angle of attack and its corresponding flutter speed of 68 m/s 209 For 50 angle ofattack, the wing also enters a LCO when flown at 10% above the flutter speed. The tip delections and the tip twist are plotted for this case up to 1 second. These plots are given inPigure C-20 thru Figure C-22. Speed = 74.80 m/s with a Time step = 0.0005 s 0.4 0.35 0.3 E C) 0.25 CD) 0~ 0.15 0- as 0.1 0.05 0 -0.05 - 0 0.2 0.4 0.6 0.8 1 Time (s) Figure C-20: Nonlinear time simulation of the wing vertical tip displacement at 10% above its flutter speed (50 root angle of attack) for the active wing design 210 Speed = 74.80 m/s with a Time step = 0.0005 s 14 12 10 8 U, V 6 C,, 4 I0~ 20-2-4 - 0 0.2 0.8 0.6 0.4 1 Time (s) Figure C-21: Nonlinear time simulation of the wing tip twist at 10% above its flutter speed (5' root angle of attack) for the active wing design Speed = 74.80 m/s with a Time step = 0.0005 s 0.38 0.370.360.35E E0.34E a 0.330.32N 0.31 - 0.30.290.28 ' -0.08 -0.06 -0.04 -0.02 0 Y Displacement (m) Figure C-22: Nonlinear time simulation of the wing tip displacement motion at 10% above its flutter speed (50 root angle of attack) for the active wing design 211