AEROSERVOELASTIC DESIGN FOR CLOSED-LOOP FLIGHT DYNAMICS OF A MAV

By

JUDSON T. BABCOCK

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2013

c 2013 Judson T. Babcock

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To my beautiful wife and the love of my life, Elisha

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ACKNOWLEDGMENTS

First and foremost, I would like to thank my wife Elisha for her prayers and support

throughout my doctoral program and our marriage. During these years she has excelled

in parenting our children on a daily basis, a task which is often more strenuous and

formidable than aeroservoelasticity.

Secondly, I would like to sincerely thank my advisor, Dr. Rick Lind, for supporting

my education with his guidance and mentorship. I am also grateful to my committee

members Dr. Larry Ukeiley, Dr. Peter Ifju, and Dr. Dave Bloomquist for their time, insight,

and advice.

Im very thankful to the Department of Aeronautics at the United States Air Force

Academy for their sponsorship in this program. I would especially like to extend my

thanks to Dr. Tom Yechout for his advice and encouragement, not only during my time

as one of his undergraduate students but also during my doctoral program.

Im indebted to Dr. Gregg Abate for his mentorship during my years at the Air Force

Research Laboratory. It was under his guidance that I become involved in experimental

aerodynamics and decided to pursue my doctorate. During that time in the wind tunnel,

I gained tremendous insight from Dr. Roberto Albertani and Dr. Larry Ukeiley and the

resulting research was a significant contribution to this dissertation.

I would like to thank my fellow students Abe Pachikara, Ahmed Jorge, and Adam

Hart for their advice and stimulating discussions during our concurrent doctoral

programs. I am also very grateful to Prof. Mark Drela for his advice and the use of

his software, without which this work would not have been possible in its current form.

Finally and most importantly, I would like to thank my personal Savior, Jesus Christ,

not only for His sacrifice for me but also for His guidance and blessings in my life.

Now unto the King eternal, immortal, invisible, the only

wise God, be honor and glory for ever and ever. Amen.

1 Timothy 1:17

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The views expressed in this document are those of the author and do not reflect the

official policy or position of the United States Air Force, Department of Defense, or the

U.S. Government.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.1 Prior Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.1.1 Aeroelastic Model Development . . . . . . . . . . . . . . . . . . . . 261.1.2 Modeling Aeroelastic Flight Dynamics and Control . . . . . . . . . 301.1.3 Modeling and Simulation of Aeroelastic MAVs . . . . . . . . . . . . 32

1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.3 Document Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2 THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.1 Flight Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.1.1 Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.1.1.1 Earth reference frame . . . . . . . . . . . . . . . . . . . . 392.1.1.2 Body reference frame . . . . . . . . . . . . . . . . . . . . 39

2.1.2 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . 412.1.3 Nonlinear Equations of Motion . . . . . . . . . . . . . . . . . . . . . 43

2.1.3.1 Translational dynamics . . . . . . . . . . . . . . . . . . . 442.1.3.2 Rotational dynamics . . . . . . . . . . . . . . . . . . . . . 472.1.3.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . 522.1.3.4 The rigid body equations of motion . . . . . . . . . . . . . 55

2.1.4 Linearized Equations of Motion . . . . . . . . . . . . . . . . . . . . 552.1.5 Solution Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.2 Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.2.1 Single DOF System: The Mass-Spring-Damper . . . . . . . . . . . 602.2.3 Multi-DOF Mass-Spring-Damper System . . . . . . . . . . . . . . . 642.2.5 Structural Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.2.6 Structural Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.3 Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.3.1 Static Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.3.3 Dynamic Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.3.3.1 Unsteady aerodynamics . . . . . . . . . . . . . . . . . . . 732.3.3.2 Aeroelastic equations of motion . . . . . . . . . . . . . . 762.3.3.3 Elastic stability derivatives . . . . . . . . . . . . . . . . . 78

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2.4 Optimal Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3 METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.1 ASWING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.1.1 ASWING Coordinate Systems . . . . . . . . . . . . . . . . . . . . . 843.1.2 Structural Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.1.3 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.1.4 Stall Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.1.5 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.1.6 Validation and Application . . . . . . . . . . . . . . . . . . . . . . . 93

3.2 The GenMAV Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.2.1 Aerodynamic Properties . . . . . . . . . . . . . . . . . . . . . . . . 943.2.3 Structural Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.2.3.1 Wing stiffness . . . . . . . . . . . . . . . . . . . . . . . . 993.2.3.2 Wing elastic axis . . . . . . . . . . . . . . . . . . . . . . . 1013.2.3.3 Wing tension and mass centroid axes . . . . . . . . . . . 102

3.2.4 ASWING Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.2.4.1 Structural . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.2.4.2 Aerodynamic . . . . . . . . . . . . . . . . . . . . . . . . . 1043.2.4.3 Mass properties . . . . . . . . . . . . . . . . . . . . . . . 1053.2.4.4 Control surfaces . . . . . . . . . . . . . . . . . . . . . . . 105

3.2.5 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.2.6 Structural Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4 EXPERIMENTAL AEROELASTICITY OF A MEMBRANE WING . . . . . . . . 115

4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.2 Experimental Set-Up and Procedure . . . . . . . . . . . . . . . . . . . . . 117

4.2.1 Low-speed Wind Tunnel . . . . . . . . . . . . . . . . . . . . . . . . 1184.2.2 2-DOF Motion Rig . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.2.3 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.2.4 Strain Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.3.1 Design of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 1214.3.2 Motion Development and Control . . . . . . . . . . . . . . . . . . . 1224.3.3 MAV Wing Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.4.1 Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.4.2 Pitch Damping Derivatives . . . . . . . . . . . . . . . . . . . . . . . 130

4.5 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . . . . . 131

5 AEROELASTICITY AND FLIGHT DYNAMICS: UNIFORM WING STIFFNESS . 133

5.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.1.1 Design Space Overview . . . . . . . . . . . . . . . . . . . . . . . . 1355.1.2 Modeling Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 135

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5.2 Case 1: Medium to High Stiffness . . . . . . . . . . . . . . . . . . . . . . 1365.2.1 Trim Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.2.2 Flight Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.2.2.1 Lateral-directional dynamics . . . . . . . . . . . . . . . . 1395.2.2.2 Longitudinal dynamics . . . . . . . . . . . . . . . . . . . . 141

5.2.3 Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.3 Case 2: Low to Medium Stiffness . . . . . . . . . . . . . . . . . . . . . . . 1435.3.1 Trim Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455.3.2 Flight Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.3.2.1 Lateral-directional dynamics . . . . . . . . . . . . . . . . 1475.3.2.2 Longitudinal dynamics . . . . . . . . . . . . . . . . . . . . 152

5.3.3 Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 1575.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.4 Case 3: Airspeed Effects on the Rigid Aircraft . . . . . . . . . . . . . . . . 1625.4.1 Trim Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1625.4.2 Flight Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1625.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5.5 Case 4: Airspeed Effects on Flexible Configurations . . . . . . . . . . . . 1665.5.1 Trim Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1665.5.2 Flight Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1675.5.3 Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 1705.5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

5.6 Case 5: Effects of the Elastic Axis . . . . . . . . . . . . . . . . . . . . . . 1715.6.1 Trim Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1715.6.2 Flight Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1725.6.3 Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 1745.6.4 Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1835.6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

5.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

6 AEROELASTICITY AND FLIGHT DYNAMICS: NON-UNIFORM WING STIFFNESS188

6.1 Linearly Varying Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 1896.1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1896.1.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

6.2 Root vs Tip Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1946.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

6.3 Aeroelastic Span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1976.3.1 Stepwise Distributions in the Aeroelastic Span . . . . . . . . . . . . 1986.3.2 Non-Stepwise Distributions in the Aeroelastic Span . . . . . . . . . 202

6.4 Aeroelastic Root Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . 2056.4.1 Aeroelastic Root Stiffness for Decreasing Stiffness . . . . . . . . . 2066.4.2 Aeroelastic Root Stiffness for Increasing Stiffness . . . . . . . . . . 209

6.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

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7 AEROELASTIC EFFECTS OF WING BATTENS . . . . . . . . . . . . . . . . . 216

7.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2177.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

7.2.1 Trim Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 2197.2.2 Sensitivity of Lift and Drag . . . . . . . . . . . . . . . . . . . . . . . 2217.2.3 Static Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2227.2.4 Control Effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . 2277.2.5 Longitudinal Flight Dynamics . . . . . . . . . . . . . . . . . . . . . 2287.2.6 Lateral-Directional Flight Dynamics . . . . . . . . . . . . . . . . . . 2307.2.7 Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 2337.2.8 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2367.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

8 AEROSERVOELASTIC DESIGN USING WING STIFFNESS . . . . . . . . . . 238

8.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2408.1.1 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2408.1.2 ROM Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2438.1.3 Design Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

8.2 Open Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2458.2.1 Frequency Domain Results . . . . . . . . . . . . . . . . . . . . . . 246

8.2.1.1 Longitudinal . . . . . . . . . . . . . . . . . . . . . . . . . 2468.2.1.2 Lateral-directional . . . . . . . . . . . . . . . . . . . . . . 247

8.2.2 Time Domain Results . . . . . . . . . . . . . . . . . . . . . . . . . 2508.2.2.1 Longitudinal . . . . . . . . . . . . . . . . . . . . . . . . . 2508.2.2.2 Lateral-directional . . . . . . . . . . . . . . . . . . . . . . 251

8.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2528.3 Closed Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

8.3.1 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2548.3.2 Longitudinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2568.3.3 Lateral-Directional . . . . . . . . . . . . . . . . . . . . . . . . . . . 2588.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

8.4 Model Following . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2608.4.1 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2628.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2628.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

8.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

9 AEROSERVOELASTIC GUST ALLEVIATION . . . . . . . . . . . . . . . . . . . 267

9.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2709.1.1 Gust Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2709.1.2 GenMAV Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2729.1.3 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

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9.1.4 Design Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2759.2 Gust Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

9.2.1 Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . 2779.2.2 Time Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2849.2.3 Aerodynamic Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 285

9.3 Gust Rejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2879.3.1 Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . 2879.3.2 Time Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

9.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2919.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

10 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

10.1 Research Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29310.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

APPENDIX: EXTENDED THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

A.1 Dyadic Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299A.2 Rate of Change Transport Theorem . . . . . . . . . . . . . . . . . . . . . 299A.3 Lagranges Energy Equations . . . . . . . . . . . . . . . . . . . . . . . . . 300

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

10

LIST OF TABLES

Table page

3-1 Aerodynamic parameters specified along a beam in ASWING . . . . . . . . . . 95

3-2 Experimental data from the two-point bending stiffness test . . . . . . . . . . . 101

3-3 Experimental data from the torsional stiffness test . . . . . . . . . . . . . . . . 101

3-4 User-specified structural parameters in ASWING . . . . . . . . . . . . . . . . . 104

3-5 Aerodynamic parameters specified along a beam in ASWING . . . . . . . . . . 105

3-6 Mass properties of ASWING model compared to AVL model . . . . . . . . . . 106

3-7 Control derivatives for the GenMAV ASWING model . . . . . . . . . . . . . . . 106

3-8 Comparison of stability derivatives between ASWING and AVL . . . . . . . . . 108

3-9 Flight test maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4-1 Experimental factors and their respective ranges . . . . . . . . . . . . . . . . . 122

4-2 Characterization of membrane pre-tension for the three levels tested . . . . . . 125

4-3 Value of constant model parameters (coded units) . . . . . . . . . . . . . . . . 127

5-1 Overview of the run cases and the independent variables in each case . . . . . 136

5-2 Case 2: Normalized eigenvector components of the spiral mode . . . . . . . . 148

5-3 Case 2: Normalized eigenvector components of the roll convergence . . . . . . 150

5-4 Case 2: Natural frequencies of the oscillatory flight modes . . . . . . . . . . . . 152

5-5 Case 2: Damping ratios of the oscillatory flight modes . . . . . . . . . . . . . . 152

5-6 Case 2: Normalized eigenvector components of the dutch roll mode . . . . . . 153

5-7 Case 2: Normalized eigenvector components of the phugoid mode . . . . . . . 155

5-8 Case 2: Normalized eigenvector components of the short period mode . . . . 156

5-9 Case 2: Natural frequencies of the structural modes . . . . . . . . . . . . . . . 160

5-10 Case 2: Damping ratios of the structural modes . . . . . . . . . . . . . . . . . . 161

5-11 Case 4: Range of factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6-1 Stepwise changes in EI on inner/outer 25% span: natural frequency . . . . . . 195

6-2 Stepwise changes in EI on inner/outer 25% span: damping ratio . . . . . . . . 196

11

6-3 Results from stepwise changes in EI on inner/outer 25% span . . . . . . . . . 196

6-4 Comparison of changes in root stiffness to overall stiffness: natural frequencies 197

6-5 Comparison of changes in root stiffness to overall stiffness: damping ratios . . 197

6-6 Comparison of changes in root stiffness to overall stiffness: time constants . . 197

7-1 Normalized eigenvector components of the phugoid mode . . . . . . . . . . . . 229

7-2 Normalized eigenvector components of the short period mode . . . . . . . . . 229

7-3 Normalized eigenvector components of the dutch roll mode . . . . . . . . . . . 231

7-4 Normalized eigenvector components of the spiral convergence . . . . . . . . . 232

7-5 Normalized eigenvector components of the roll convergence . . . . . . . . . . 234

9-1 Percent changes from high to low EI in response to a w gust . . . . . . . . . . 282

9-2 Percent changes from high to low EI in response to a u gust . . . . . . . . . . 283

12

LIST OF FIGURES

Figure page

1-1 Overview of aeroelastic interactions . . . . . . . . . . . . . . . . . . . . . . . . 24

2-1 Earth coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2-2 Body coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2-3 Stability coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2-4 Overview of coordinate transformations . . . . . . . . . . . . . . . . . . . . . . 41

2-5 Angular momentum of a particle P relative to point O . . . . . . . . . . . . . . . 48

2-6 Angular momentum of a differential element on a rigid body . . . . . . . . . . . 49

2-7 Single DOF mass-spring-damper system . . . . . . . . . . . . . . . . . . . . . 60

2-8 Free vibration response of an underdamped mass-spring-damper system . . . 63

2-9 Two DOF mass-spring-damper system . . . . . . . . . . . . . . . . . . . . . . . 64

2-10 Aeroelastic pitching airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2-11 Harmonically oscillating airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3-1 Aircraft body and local beam coordinate systems in ASWING . . . . . . . . . . 85

3-2 A wing represented by a lifting line composed of three horseshoe vortices . . . 89

3-3 The GenMAV aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3-4 The GenMAV airfoil at three locations along the span . . . . . . . . . . . . . . . 95

3-5 Center of pressure location on an airfoil . . . . . . . . . . . . . . . . . . . . . . 96

3-6 Center of pressure along the GenMAV wing for = 0 . . . . . . . . . . . . . . 98

3-7 The GenMAV wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3-8 Two-point bending stiffness test . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3-9 Experimental two-point bending stiffness test setup . . . . . . . . . . . . . . . . 100

3-10 Wing deflection in response to an applied load . . . . . . . . . . . . . . . . . . 100

3-11 Wing twist vs. excitation location . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3-12 Elastic axis of the GenMAV wing . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3-13 Tension axis of the GenMAV wing . . . . . . . . . . . . . . . . . . . . . . . . . 103

13

3-14 Geometry of the GenMAV in ASWING . . . . . . . . . . . . . . . . . . . . . . . 104

3-15 Comparison of flight modes between ASWING and AVL . . . . . . . . . . . . . 107

3-16 Comparison of ASWING model with flight test maneuver A . . . . . . . . . . . 109

3-17 Comparison of ASWING model with flight test maneuver B . . . . . . . . . . . 110

3-18 Comparison of ASWING model with flight test maneuver C . . . . . . . . . . . 110

3-19 Comparison of ASWING model with flight test maneuver D . . . . . . . . . . . 111

3-20 Modal frequencies of the stiff-chord and flexible-chord wing configurations . . . 112

3-21 First bending mode shapes from GVT test . . . . . . . . . . . . . . . . . . . . . 113

3-22 First torsion mode shapes from GVT test . . . . . . . . . . . . . . . . . . . . . 113

3-23 Mode shape of chord at y = 0.65b . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4-1 Wind tunnel setup and model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4-2 Two degrees-of-freedom test rig . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4-3 Kinematic plots of an and motion . . . . . . . . . . . . . . . . . . . . . . . . 121

4-4 Contour plots of the wind-off membrane strain state . . . . . . . . . . . . . . . 124

4-5 Plot of 2 in the two directions as a function of velocity . . . . . . . . . . . . . . 125

4-6 Lift coefficient of the low-tension flexible wing . . . . . . . . . . . . . . . . . . . 126

4-7 Model comparison to static wind tunnel data . . . . . . . . . . . . . . . . . . . . 128

4-8 The medium-tension flexible wing: lift and drag . . . . . . . . . . . . . . . . . . 128

4-9 The rigid wing: lift and drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4-10 Lift-to-drag ratios of the rigid wing and flexible wing . . . . . . . . . . . . . . . . 130

4-11 Pitching moment coefficient of the rigid wing and flexible wing . . . . . . . . . . 131

5-1 Case 1: Design space of bending and torsional stiffness . . . . . . . . . . . . . 137

5-2 Case 1: Aircraft and e at trim . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5-3 Case 1: Wing deformations at trim . . . . . . . . . . . . . . . . . . . . . . . . . 138

5-4 Case 1: All poles corresponding to the flight modes for the design space . . . . 139

5-5 Case 1: Roll convergence time constant . . . . . . . . . . . . . . . . . . . . . . 140

5-6 Case 1: Dutch roll modal characteristics . . . . . . . . . . . . . . . . . . . . . . 140

14

5-7 Case 1: Phugoid modal characteristics . . . . . . . . . . . . . . . . . . . . . . . 141

5-8 Case 1: Short period modal characteristics . . . . . . . . . . . . . . . . . . . . 142

5-9 Case 1: First symmetric bending modal characteristics . . . . . . . . . . . . . . 142

5-10 Case 2: Design space of bending and torsional stiffness . . . . . . . . . . . . . 144

5-11 Case 2: Aircraft and e at trim . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5-12 Case 2: Wing deformations at trim . . . . . . . . . . . . . . . . . . . . . . . . . 145

5-13 Elastic axis and center of pressure along the wing . . . . . . . . . . . . . . . . 146

5-14 Location of the span-averaged xcp and elastic axis versus angle of attack . . . 147

5-15 Case 2: All poles corresponding to the flight modes for the design space . . . . 148

5-16 Case 2: Poles of the flight modes . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5-17 Case 2: Roll convergence time constant . . . . . . . . . . . . . . . . . . . . . . 150

5-18 Case 2: Dutch roll modal characteristics . . . . . . . . . . . . . . . . . . . . . . 151

5-19 Animation of the dutch roll mode with varying EI and GJ=1.0 . . . . . . . . . . 153

5-20 Case 2: Phugoid modal characteristics . . . . . . . . . . . . . . . . . . . . . . . 155

5-21 Case 2: Short period modal characteristics . . . . . . . . . . . . . . . . . . . . 156

5-22 Animation of the short period mode with varying EI and GJ=1.0 . . . . . . . . 157

5-23 Case 2: First symmetric bending modal characteristics . . . . . . . . . . . . . . 158

5-24 Animation of the first bending mode with varying EI and GJ=1.0 . . . . . . . . 159

5-25 Case 2: First symmetric torsion modal characteristics . . . . . . . . . . . . . . 159

5-26 Effect of airspeed on the first bending natural frequency at low EI . . . . . . . 160

5-27 Case 3: Trim results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5-28 Case 3: All poles corresponding to the flight modes . . . . . . . . . . . . . . . 163

5-29 Case 3: Poles of the flight modes . . . . . . . . . . . . . . . . . . . . . . . . . . 164

5-30 Case 3: Natural frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5-31 Case 3: Damping ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5-32 Case 4: Trim results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

5-33 Case 4: Wing deformations at trim . . . . . . . . . . . . . . . . . . . . . . . . . 167

15

5-34 Case 4: Dutch roll natural frequency and damping ratio . . . . . . . . . . . . . 168

5-35 Case 4: Phugoid natural frequency and damping ratio . . . . . . . . . . . . . . 169

5-36 Case 4: Short period natural frequency and damping ratio . . . . . . . . . . . . 169

5-37 Case 4: First bending mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

5-38 Wing tip deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

5-39 Wing twist at y = 0.75b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

5-40 Aircraft trim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

5-41 Aircraft trim e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

5-42 Dutch roll natural frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

5-43 Dutch roll damping ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

5-44 Phugoid natural frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

5-45 Phugoid damping ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

5-46 Short period natural frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

5-47 Short period damping ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

5-48 First symmetric bending natural frequency . . . . . . . . . . . . . . . . . . . . . 181

5-49 First symmetric bending damping ratio . . . . . . . . . . . . . . . . . . . . . . . 181

5-50 First symmetric torsion natural frequency . . . . . . . . . . . . . . . . . . . . . 182

5-51 First symmetric torsion damping ratio . . . . . . . . . . . . . . . . . . . . . . . 182

5-52 Bending-torsion flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6-1 Sample linear stiffness distribution with EI0 = 1,EI = 0.5 . . . . . . . . . . . 189

6-2 Trim results with linearly varying stiffness . . . . . . . . . . . . . . . . . . . . . 190

6-3 Wing tip deflection with linearly varying stiffness . . . . . . . . . . . . . . . . . 190

6-4 Poles of the flight modes with linearly varying stiffness . . . . . . . . . . . . . . 192

6-5 Natural frequencies with linearly varying stiffness . . . . . . . . . . . . . . . . . 193

6-6 Damping ratios with linearly varying stiffness . . . . . . . . . . . . . . . . . . . 193

6-7 First bending mode with linearly varying stiffness . . . . . . . . . . . . . . . . . 194

6-8 Stepwise changes created in the root and tip EI . . . . . . . . . . . . . . . . . 195

16

6-9 Aeroelastic span analysis of trim. . . . . . . . . . . . . . . . . . . . . . . . . . . 200

6-10 Aeroelastic span analysis of the dutch roll mode . . . . . . . . . . . . . . . . . 200

6-11 Aeroelastic span analysis of the phugoid mode . . . . . . . . . . . . . . . . . . 201

6-12 Aeroelastic span analysis of the short period mode . . . . . . . . . . . . . . . . 201

6-13 Aeroelastic span analysis of the first bending mode . . . . . . . . . . . . . . . . 202

6-14 Stepwise, linear, and exponential slopes over a portion of the inner span . . . . 203

6-15 Effects of slope in the aeroelastic span: phugoid and dutch roll . . . . . . . . . 204

6-16 Effects of slope in the aeroelastic span: short period . . . . . . . . . . . . . . . 204

6-17 Effects of slope in the aeroelastic span: first bending mode . . . . . . . . . . . 205

6-18 Aeroelastic root stiffness for decreasing EI: trim conditions . . . . . . . . . . . 207

6-19 Aeroelastic root stiffness for decreasing EI: dutch roll mode . . . . . . . . . . 207

6-20 Aeroelastic root stiffness for decreasing EI: phugoid mode . . . . . . . . . . . 208

6-21 Aeroelastic root stiffness for decreasing EI: short period mode . . . . . . . . . 208

6-22 Aeroelastic root stiffness for decreasing EI: first bending mode . . . . . . . . . 209

6-23 Aeroelastic root stiffness for increasing EI: trim conditions . . . . . . . . . . . 211

6-24 Aeroelastic root stiffness for increasing EI: dutch roll mode . . . . . . . . . . . 212

6-25 Aeroelastic root stiffness for increasing EI: phugoid mode . . . . . . . . . . . . 212

6-26 Aeroelastic root stiffness for increasing EI: short period mode . . . . . . . . . 213

6-27 Aeroelastic root stiffness for increasing EI: first bending mode . . . . . . . . . 213

7-1 GenMAV wing geometry with three battens . . . . . . . . . . . . . . . . . . . . 218

7-2 Profile of torsional stiffness across the half-span for each batten configuration . 219

7-3 Trim and e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

7-4 Span-averaged wing twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

7-5 Effective angle of attack at trim . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

7-6 Aircraft lift and drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

7-7 Longitudinal static stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

7-8 Directional static stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

17

7-9 Lateral static stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

7-10 Primary control derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

7-11 Natural frequency and damping of the longitudinal modes . . . . . . . . . . . . 230

7-12 Natural frequency and damping of the dutch roll mode . . . . . . . . . . . . . . 230

7-13 Time constant of the spiral convergence . . . . . . . . . . . . . . . . . . . . . . 232

7-14 Time constant of the roll convergence . . . . . . . . . . . . . . . . . . . . . . . 233

7-15 Natural frequency and damping of the structural modes . . . . . . . . . . . . . 234

7-16 Twist of batten-reinforced wings compared to a wing with uniform GJ . . . . . . 235

8-1 Comparison of the ROM to the full model (EI = 0.1, GJ = 1.0) . . . . . . . . . 244

8-2 Design space of bending and torsional stiffness . . . . . . . . . . . . . . . . . . 245

8-3 Elevator to angle of attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

8-4 Elevator to pitch rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

8-5 Aileron to bank angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

8-6 Aileron to roll rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

8-7 Rudder to yaw rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

8-8 Open-loop angle of attack response to a step input in elevator . . . . . . . . . . 250

8-9 Open-loop bank angle response to a step input in aileron . . . . . . . . . . . . 251

8-10 Open-loop roll rate response to a step input in aileron . . . . . . . . . . . . . . 252

8-11 Open-loop yaw rate response to a step input in rudder . . . . . . . . . . . . . . 253

8-12 Block diagram of a LQR tracking controller . . . . . . . . . . . . . . . . . . . . 254

8-13 Output of a stiff aircraft tracking a ref = 5.4 command . . . . . . . . . . . . . 257

8-14 Longitudinal tracking performance using fixed weighting matrices . . . . . . . . 258

8-15 Longitudinal tracking performance with fixed modes . . . . . . . . . . . . . . . 259

8-16 Output of a stiff aircraft tracking a = 5.4 command . . . . . . . . . . . . . . . 260

8-17 Lateral-directional tracking performance using fixed weighting matrices . . . . 261

8-18 Block diagram of a model-following controller using LQR . . . . . . . . . . . . . 262

8-19 A stiff aircraft following the response of a flexible aircraft . . . . . . . . . . . . . 263

18

8-20 A flexible aircraft following the response of a stiff aircraft . . . . . . . . . . . . . 265

9-1 Example disturbances for a 1 Hz gust with g = 3 . . . . . . . . . . . . . . . 271

9-2 Block diagram of a gust rejection control design . . . . . . . . . . . . . . . . . . 275

9-3 LQR gains for the GenMAV aircraft with varying levels of stiffness . . . . . . . . 276

9-4 Gust sensitivity in response to a u-gust . . . . . . . . . . . . . . . . . . . . . . 277

9-5 Gust sensitivity in response to a w-gust . . . . . . . . . . . . . . . . . . . . . . 279

9-6 Response of wing tip velocity to the gust disturbance . . . . . . . . . . . . . . . 279

9-7 Gust sensitivity in response to a e-gust . . . . . . . . . . . . . . . . . . . . . . 280

9-8 Gust sensitivity to a w gust with constant phugoid and short period damping . . 282

9-9 Pitch rate response to a u gust at the phugoid modes natural frequency . . . . 285

9-10 Gust sensitivity RMS values for the pitch rate response . . . . . . . . . . . . . 286

9-11 Aerodynamic derivatives relevant to gust sensitivity . . . . . . . . . . . . . . . . 286

9-12 Gust rejection in response to a u-gust . . . . . . . . . . . . . . . . . . . . . . . 288

9-13 Gust rejection in response to a w-gust . . . . . . . . . . . . . . . . . . . . . . . 289

9-14 Response of wing tip velocity to the gust disturbance . . . . . . . . . . . . . . . 289

9-15 Gust rejection RMS values for the pitch rate response . . . . . . . . . . . . . . 290

19

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

AEROSERVOELASTIC DESIGN FOR CLOSED-LOOP FLIGHT DYNAMICS OF A MAV

By

Judson T. Babcock

August 2013

Chair: Rick LindMajor: Aerospace Engineering

Some fixed-wing micro air vehicles (MAVs) have high levels of structural flexibility,

a property which can change the flight dynamics and control characteristics of the

vehicle. However, the exact level of flexibility is typically the result of a trial-and-error

approach instead of being part of a rigorous design framework and may result in

unknown aeroelastic effects on the flight dynamics. The current research investigates

the nature of these aeroservoelastic effects by using a generic MAV configuration. The

main parameter of interest is the stiffness of the wing.

Bending and torsional stiffness of the wing are independently varied from 1.0 Nm2

to 0.07 Nm2 while the trim conditions, flight dynamics, and structural dynamics are

analyzed. Large changes in both the frequencies and damping ratios of the oscillatory

flight modes are seen. The bending stiffness mainly affects the lateral-directional flight

modes through an increase in the effective dihedral angle due to increased wing tip

deflection. The direction and magnitude of the effect varies greatly between modes.

Non-traditional mode shapes resulting from decreased bending stiffness are observed in

the dutch roll mode and phugoid mode.

The effects of torsional stiffness depend on the relative positioning of the elastic axis

and center of pressure. When the elastic axis is near the center of pressure, changing

torsional stiffness has only minor effects on the flight dynamics. Elastic axis locations

which are further away from the center of pressure result in stronger effects from

20

changes in torsional stiffness. In general, the torsional stiffness affects the longitudinal

modes more than the lateral directional modes because of the changing angle of attack

and pitching moment.

Aeroservoelastic effects of wing stiffness on the tracking performance of the aircraft

are investigated. For an LQR controller with fixed weightings, the tracking performance

decreases as stiffness decreases. Changes in the phugoid mode damping and shape at

low bending stiffness are found to have a very strong effect on the longitudinal tracking

performance.

The possibility of virtually changing the stiffness of the wing by using a model-following

control scheme is investigated. It is observed that the stiff aircraft can approximate the

response of the flexible but the flexible aircraft is unable to adequately approximate the

performance of the stiff aircraft.

An important consideration for micro air vehicles is their response to a wind gust. A

frequency-domain approach is used to evaluate the aircrafts longitudinal gust response

in the presence of aeroservoelastic effects. The level of wing bending stiffness is

found to have an important effect on the gust sensitivity and gust rejection properties

of the aircraft. The direction and frequency of the gust can drastically change the

gust sensitivity of the aircraft. Lowering wing stiffness can reduce the gust sensitivity

at low gust frequencies but can increase it at high frequencies. Changes in modal

damping and shape due to decreasing wing stiffness have a strong influence on the gust

sensitivity. For a basic LQR controller with fixed weighting matrices, the gust rejection

properties are very good across the range of stiffness values.

21

CHAPTER 1INTRODUCTION

Aeroelasticity studies the interaction of aerodynamic, inertial, and elastic forces

acting on a flexible structure exposed to air flow. Aeroelasticity is relevant for a variety of

fields, including civil engineering, automotive engineering, and mechanical engineering.

However, the science of aeroelasticity is most commonly applied as a discipline of

aeronautical engineering. Aircraft structures are never perfectly rigid; consequently, their

behavior in flight will always be aeroelastic in nature. When not properly considered in

vehicle or control design, the aeroelastic phenomena that arise are usually undesirable

and can range in severity from benign to catastrophic.

Aeroelasticity has played a major role in aeronautics since the beginning of powered

flight in 1903. Earlier that year, before the Wright brothers made their historic flight,

Samuel Langley made two unsuccessful attempts that resulted in catastrophic failure

of his aircrafts wings. This structural failure was due to insufficient torsional stiffness,

which resulted in the aerodynamic forces overcoming the structural forces [64]. This

aeroelastic phenomenon is known as divergence and was a major concern in early

aircraft design until the 1930s when aircraft structures were designed with metallic skins

capable of providing additional torsional stiffness.

Aeroelasticity was a major concern throughout the early history of aviation [36].

The first documented case of flutter involved the Handley Page O/400 bomber in 1916.

Flutter continued to be a major concern during and after the First World War. After

the Second World War, the prevalence of aeroelastic phenomenon increased further

due to increased flight speeds, thinner shapes, more complex designs, and more

demanding aircraft missions [15, 63]. These trends continued into the modern era

where new aeroelastic applications began to surface, such as flexible airships, missiles,

wind turbines, and rotorcraft [29, 56, 86, 96]. Despite over a century of research and

development, aeroelasticity continues to be a dynamic and challenging field [57, 89].

22

Two frontiers of flight today are hypersonics [62] and micro or nano-sized aircraft [135].

Both of these frontiers will challenge and expand the science of aeroelasticity.

In aircraft, aeroelastic phenomena arise when aerodynamic forces cause structural

deformations, which cause additional changes in the aerodynamic forces. This

cycle is repeated. In certain cases, a steady-state equilibrium is reached where the

elastic forces balance out the aerodynamic forces. These phenomena are grouped

as static aeroelasticity and can have important consequences for the steady-state

flight of an aircraft. The aircraft loads, control effectiveness, trim behavior, and static

stability all depend on the static aeroelastic behavior. Negative consequences of static

aeroelasticity can result in divergence, reduced control effectiveness, or control reversal.

In some situations, oscillations between the aerodynamic forces and structural

forces continue over time. This class of phenomena is called dynamic aeroelasticity.

Dynamic aeroelasticity is concerned with the oscillatory nature of the interaction

between the structure and fluid flow, namely the extraction of energy from the flow

field by the structure [186]. The main interest is the phenomenon of flutter, although

the effects of dynamic aeroelasticity on the flight dynamics of the aircraft are also of

importance.

One pioneer in the field of aeroelasticity was Arthur R. Collar (1908-1986) [18]. He

is credited with forming the aeroelastic triangle [34] which is reproduced in modified form

in Fig. 1-1. The figure illustrates the three-way interaction between the aerodynamic,

structural, and inertial forces which results in dynamic aeroelasticity. It further depicts

static aeroelasticity arising from the interaction of aerodynamic and elastic forces.

Stability and control phenomena result from the interaction of inertial and aerodynamic

forces and vibration results from the interaction of inertial and elastic forces.

The discipline of aeroservoelasticity extends the aeroelastic interactions in Fig. 1-1

to include the interaction of a control system. Since a flight control system is now

a common feature among aircraft, aeroservoelasticity has become very relevant in

23

AerodynamicForces

ElasticForces

InertialForces

DynamicAeroelasticity

Static Aeroelasticity Stability & Control

Structural Vibration

Figure 1-1. Overview of aeroelastic interactions

aircraft design. Most aeroservoelastic problems occur when the aircrafts sensors

detect not only the rigid-body motion of the aircraft but also the motion from the flexible

structure [186]. The sensor measurements from these structural vibrations are fed back

into the control system, which might react in a way that further increases the vibrations.

The result can be reduced control system performance, reduced handling qualities,

increased structural fatigue, or even catastrophic failure.

Flight dynamics is a branch of applied mechanics which deals with the motion

of vehicles flying in the atmosphere [51]. Flight dynamics is a broad field which

incorporates applied mathematics, aerodynamics, rigid body mechanics, aeroelasticity,

and the dynamics of a human pilot. Flight dynamics is unique because it deals with

aerodynamic forces in the absence of kinematic constraints. Flight dynamics problems

can include aircraft performance, motion trajectories, stability, vehicle response to

control inputs, response to turbulence, handling qualities, and aeroelastic concerns.

The primary goal of any flight dynamics analysis is to determine the trajectory and

orientation of the aircraft body over time.

Micro air vehicles (MAVs) have emerged over the past two decades as small,

unmanned aircraft with certain characteristics that enable them to accomplish a unique

24

set of missions. The definition of a MAV can vary but in the current study is defined

as bird-sized or smaller. Two prominent characteristics of MAVs are operation in

low Reynolds number flight regimes and small physical dimensions. MAVs are also

characterized by their agility and low cost. These properties make them ideal to operate

in urban areas, tunnels, caves, or other constrained environments. Military missions

can include surveillance, reconnaissance, communication, detection, tracking, chemical

or biological monitoring, or precision strike [95]. Civilian missions can include disaster

relief, agriculture, mapping, communications, and surveillance activities such as forest

fire monitoring, scientific observation, or monitoring of electric power lines [118].

MAVs operate in a very sensitive Reynolds number regime, typically on the order

of 103 105, which results in a fundamental shift in the aerodynamic behavior. In this

regime, the flow is characterized by complex features and interactions such as unsteady

three-dimensional separation, transition in boundary layers and shear layers, vortical

flows, bluff body flows, and unsteady flight environments [152].

Because of these challenges, MAV designers have taken much inspiration from

biological flight vehicles which successfully operate in the same Reynolds number

regime [150]. Small birds, bats, and flying insects all use flexible wings to achieve flight.

To obtain successful flight with similar flexible designs, researchers and engineers

will have to gain a new understanding of the aeroelastic interactions that occur at this

scale [89, 120].

The single most important factor that has allowed the United States Air Force

to become the most effective and powerful air force in the world is its unmatched

technological advantage [146]. Maintaining this advantage while advanced technologies

proliferate is a prime focus of the Air Force [175]. In particular, the Air Force has a

vested interest in small, micro, and nano-sized air vehicles because of the life-saving

situational awareness capabilities that they can provide to the commander and the

individual soldier. These benefits have been amplified as military operations now

25

frequently consist of small teams operating in non-traditional environments like urban

centers [95]. As a result, these systems are some of the most in-demand capabilities

that the Air Force provides [106].

Micro air vehicles are an integral part of the Air Force strategic vision for unmanned

aircraft systems [145]. These aircraft are envisioned to conduct a wide variety of

challenging indoor and outdoor missions. The Air Force Research Laboratory has a

goal of demonstrating a nano-sized UAV platform performing missions in an urban

environment by the year 2030 [106]. Their research suggests that using airframes with

inherent flexibility may provide some advantages in accomplishing these missions [1].

However, this flexibility poses several technical challenges in the areas of aeroelasticity,

flight dynamics, and flight control. This research is aimed at addressing some of these

challenges and the results will aid the Air Force in the design of future micro air vehicles.

1.1 Prior Research

1.1.1 Aeroelastic Model Development

Much effort has been invested in developing models for investigating the aeroelastic

behavior of flight vehicles. Early efforts derive equations of motion for an unrestrained

flexible vehicle [19]. Solutions to the equations of motion use linearization about an

equilibrium by assuming small perturbations in the elastic and rigid body degrees of

freedom [107].

Further development efforts for the aeroelastic equations of motion are the Waszak

study [176] and the Buttrill study [27]. Both of these developments use a mean axis

body reference frame and utilize Lagranges method where the elastic strain energy

of the vehicle is included in the potential energy terms. Both express the aerodynamic

forces for an elastic aircraft in a stability derivative form. Such a reference frame

removes any inertial coupling between the rigid-body and elastic degrees of freedom.

Both make use of several important assumptions such as small structural deformations

26

(which allows the bodys inertia to be treated as constant), synchronous elastic motion,

and assuming that the structure can be treated as a collection of point masses.

A framework is created for the integration of analytical dynamics, structural

dynamics, aerodynamics, and control for the simulation of dynamic aircraft response [97

104]. The theory uses a reference frame attached to an undeformed aircraft. Such a

reference frame avoids the complications associated with a mean axis reference

frame, such as expressing the aerodynamic forces and enforcing the constraints in

the mean axes. The equations of motion are derived using Lagranges equations

with quasi-coordinates. Aerodynamics are estimated using strip theory, but it is noted

that a new aerodynamic method for computing the whole aircraft time response in a

rapid manner is needed. A perturbation approach is used, separating the problem

into a zero-order problem for the rigid body motion and a first-order problem for the

elastic displacements. An emphasis is placed on low computational cost for on-board

computing.

The aeroelastic stability and response of a nonlinear aeroelastic wing is investigated

using a geometrically exact structural model coupled with a nonlinear aerodynamic

model including stall effects [127]. Finite state aerodynamic theory [133, 134] is used to

obtain a state-space representation of the aerodynamics with a low number of states.

The method accounts for large scale airfoil motion as well as small deformations of the

airfoil such as trailing-edge flap deflection. The method is validated against the Goland

wing [58] via a flutter analysis.

The previous work led to the creation of the Nonlinear Aeroelastic Trim and Stability

of HALE Aircraft (NATASHA) program [30, 65, 125, 126, 189], which was developed

to analyze the aeroelastic characteristics of highly flexible flying wings, specifically the

high-altitude long-endurance (HALE) class of aircraft. Examples of HALE aircraft include

NASAs Pathfinder, Pathfinder Plus, Helios, and the European HeliPlat UAV [50, 142].

HALE aircraft are challenging to model because of time-varying inertia properties,

27

coupled inertial forces due to a rotating coordinate system and relative velocity of flexible

members, and external forces and moments which are no longer based on a rigid body

geometry. Results from NATASHA compared favorably with external data [153].

Anther approach to model HALE aircraft is undertaken by separating the dynamics

into nominal and perturbation dynamics [174]. The nominal dynamics are used to

simulate the large-scale motion of the maneuver and the perturbation dynamics are

used to address the stability of the aircraft along the flight path.

A comparison between models for a very flexible, high aspect-ratio wing is

conducted using a common framework [124]. The models include an intrinsic model,

strain-based model, and finite element model. The velocities and wing tip displacements

of an aircraft are compared. Results show good agreement between the models and the

intrinsic and strain-based formulations are found to have a low computational cost as

compared to the finite element method.

FLEXSTAB was a computer program developed for the analysis of elastic aircraft

configurations at subsonic and supersonic speeds [49, 171]. The program uses linear

methods to evaluate static and dynamic stability, the trim state, aerodynamics, and

elastic deformations with an emphasis on the stability and control characteristics.

An Automated STRuctural Optimization System (ASTROS) was developed to

perform automated preliminary structural design for an aeroelastic vehicle using a

finite-element approach coupled with steady and unsteady aerodynamics. The code

is capable of static aeroelastic analysis, flutter analysis, and limited control response

analysis [114].

The Aeroelastic Design Optimization Program (ADOP) was a similar tool developed

in competition with ASTROS [42]. Although the ADOP bid was unsuccessful, their effort

focused on minimizing structural weight without violating three static aeroelastic design

constraints: lift effectiveness, roll effectiveness, and divergence.

28

ASWING is an integrated tool for aerodynamic, structural, and control law analysis

and design in a fully and nonlinearly coupled manner [44, 45]. The method allows rapid

conceptual analysis and is especially suited to the early phases of aircraft design. The

formulation is based on a nonlinear beam approach with lifting-line aerodynamics which

includes corrections for aerodynamic lag.

The University of Michigans Nonlinear Aeroelastic Simulation Toolbox (UM/NAST)

is a computational framework for the aeroelastic analyses of very flexible aircraft [24,

148, 165, 166]. It uses a reduced order, nonlinear, strain-based beam formulation to

model the structure and incorporates unsteady aerodynamics and nonlinear control

dynamics. It is also able to model composite beam structures with embedded active

piezoelectric materials.

Researchers at Texas A&M developed and tested an experimental aeroelastic test

apparatus which allowed investigation into nonlinear aeroelastic responses [121123].

Their apparatus has nonlinear springs and interchangeable cams which can alter

the linearity of the response. Numerical flutter simulations compare well with the

experiments. Applications are also made for nonlinear control law development [77, 78].

The Boeing X-53 Active Aeroelastic Wing (AAW) development program [21, 33,

39, 73] was undertaken jointly by the Air Force Research Laboratory, Boeing Phantom

Works, and NASA Dryden Flight Research Center to actively control aeroelastic wing

twist for the purpose of aircraft control. The program modified a F/A-18A fighter aircraft

to control the aeroelastic wing twist through multiple leading and trailing edge flaps, thus

obtaining the desired amount of wing control power. The F/A-18A was chosen because

of its relatively high aspect-ratio, thin, flexible wing. In fact, the preproduction aircraft,

based on the original Northrop YF-17 design, showed degraded roll performance

because of low torsional stiffness. Although this degradation was fixed by stiffening the

wing for the production aircraft, the X-53 aircraft was modified to return to the original,

preproduction torsional stiffness levels. The X-53 successfully proved the AAW concept

29

during roll maneuvers in test flights. The AAW technology was also applied to an F-16

aeroelastic model with favorable results for wing control power [131].

1.1.2 Modeling Aeroelastic Flight Dynamics and Control

The subject of aircraft dynamics has often been divided between flight dynamics

and aeroelasticity [101]. In general, flight dynamics has to do with a rigid-body aircraft

undergoing maneuvers. Aeroelasticity is usually concerned with the interactions

between the aerodynamics and structure of a non-maneuvering flexible aircraft. In

this regard, flight dynamics and aeroelasticity have developed separately. However, the

importance of considering their interaction has often been noted [34, 35, 94, 115, 139].

Investigations into the short-period mode of an aircraft with an elastic wing and

varying levels of sweep are conducted [94]. The method is restricted to a longitudinal

investigation to reduce the model size. The authors report a loss of static stability due to

wing flexibility for all wing configurations and a decrease in dynamic stability for the 0

sweep configuration.

A forward-swept configuration is analyzed to show the importance of including

aircraft rigid-body modes in the aeroelastic analysis [181]. Coupling of the elastic

and rigid body modes is found to depend on the inertial, aerodynamic, and stiffness

characteristics of the aircraft. Possible consequences of this coupling are body-freedom

flutter or divergence.

An aeroelastic model developed from Lagranges equations is applied to a high

speed transport with a moderate level of flexibility [176]. Results show an unstable

phugoid mode for the flexible aircraft. The short period frequency and damping of the

aeroelastic model are 55% and 14% different from the rigid model, respectively.

A dynamic aeroelastic model is used to evaluate Cooper-Harper pilot ratings to

demonstrate the important relationship between flight dynamics and aeroelasticity [143].

Both longitudinal and lateral rigid-body dynamics are included in the model. A severe

30

degradation of the handling qualities results as the lowest structural frequency

decreased.

Consideration of the interactions between aeroelasticity and flight dynamics is

very important for HALE aircraft because of their unique characteristics [188]. These

aircraft operate with unusually large wing deflections; consequently, traditional linear

theory will not provide accurate estimations of the flight dynamics. Furthermore, the

low structural frequencies of these aircraft are within the range of the rigid body modes.

These unique characteristics caused a prototype HELIOS aircraft to experience an

in-flight mishap on June 26, 2003 [117]. After encountering turbulence, aeroelastic

effects caused a very high dihedral angle to develop which led to a divergent pitching

mode. Oscillations of increasing amplitude resulted in high airspeed and high dynamic

pressures, which caused the wing structure to fail. The mishap investigation showed

that the lift distribution of such an aircraft can be very sensitive to small amplitude gusts,

especially when undergoing large deformations that involve a high dihedral angle. The

mishap investigation attributed the crash to a lack of adequate analysis methods which

led to an inaccurate risk assessment and an inappropriate decision to fly the aircraft.

Studies using the NATASHA program found a significant change in the flight

dynamics characteristics of HALE aircraft due to wing flexibility, specifically the

phugoid and short period modes [30, 125, 127, 128]. In one study, the pair of complex

short-period roots merges to become two real roots and the phugoid mode goes

unstable when the aircraft is under loading [126]. Nonlinear flight simulation of the

aircraft indicates that the phugoid instability led to catastrophic consequences. The trim

shape is found to be highly dependent on the flight mission and flight conditions. This

result is critical because the flight dynamic response for each trim shape can be quite

different.

The flight dynamics of a very flexible vehicle are successfully characterized using

UM/NAST [147]. It is found that the rigid-body model did not adequately capture the

31

dynamics of the flexible vehicle. For symmetric maneuvers, results showed that a

linearized aeroelastic model is adequate. However, for asymmetric maneuvering, a

nonlinear approach is necessary to capture the vehicle response. This work led to

the development of a specialized UAV for flight testing of very flexible, aspect-ratio

wings [28].

1.1.3 Modeling and Simulation of Aeroelastic MAVs

Increasing effort is being placed on aeroelastic analyses of micro air vehicles in

their unique flight regime. Although highly flexible MAV wings have been used in practice

with notable improvements in the flying qualities [71], little has been done to characterize

them analytically.

A flexible membrane airfoil interacting with the fluid flow is modeled by coupling a

two-dimensional elastic membrane model based on normal and shear stresses with a

two-dimensional computational fluid dynamics (CFD) code [149]. The model is applied

to a rigid wing, flexible wing, and hybrid wing. The results show an increase in peak

aerodynamic performance of the flexible wing.

The previous work is extended to finite membrane wings of varying configuration [159].

A thin membrane model combined with a composite structural model is used in order to

model the mixed membrane/carbon fiber structure. Four different numerical approaches

to modeling the membrane are implemented and reviewed, including high fidelity

nonlinear and hyperelastic membrane models. The CFD code is expanded to a three

dimensional incompressible Navier-Stokes CFD model including a k viscous

turbulence model. The experimental and numerical work show the advantages and

disadvantages of a membrane wing. Some of the advantages are increased camber

and delayed stall. Additionally, improved lift, drag, pitching moments lead to improved

static stability and gust rejection. One disadvantage is a possible roll instability from the

flexible design. The study concludes with a recommendation for a higher fidelity model

in order to capture unsteady phenomena like vortex shedding, vibration.

32

An experimental effort to characterize the static stability of an elastic MAV with

various structural configurations is conducted [72]. The structural configurations consist

of varying the number of battens used to constrain the membrane wing. A delay in

stall is seen due a decrease in the effective angle of attack caused by the passive

deformation of the wing. The flexible wings had weaker wing tip vortices, lower lift to

drag ratios, and extensive membrane vibrations during testing. Overall, the vehicle is

statically stable. A state space simulation model is developed to assess the flight control

characteristics [178]. The simulation models are based on aeroelastic experimental data

but are not fully coupled structural/aerodynamic models.

The flight dynamics of a generic MAV with a various levels of wing flexibility is

analyzed [162164]. Flight test maneuvers show a large difference in the dynamic

response of very flexible configurations compared to very rigid wing configurations. A

reduction in the transient response of the flexible wing configuration as compared to the

rigid wing configuration is noted. Flight test results are compared to rigid body analytical

predictions and show some differences.

1.2 Problem Statement

Typical methods of analyzing the flight dynamics of an aircraft treat the aircraft as

a rigid body. For aircraft which have a large amount of frequency separation between

the flight and structural modes, this assumption has been valid [180]. However, as

aircraft structures become lighter and more flexible, the frequency separation reduces,

especially between the short-period and first bending modes [45]. In such situations,

the flight dynamics can be significantly influenced by the elastic structure [167, 177],

possibly leading to dynamic instability [181].

Aircraft design typically only evaluates changes in conventional design parameters

without considering changes to parameters which would have a direct effect on the

aeroelastic response of the aircraft, such as structural stiffness. As a result, the nature

of the aeroelastic influence on the flight dynamics is not captured.

33

Many engineers and researchers desire to exploit flexibility in micro air vehicles,

but a lack of understanding of the interaction between aeroelasticity and flight dynamics

can lead to sub-optimal designs. The unique structural characteristics and flight regimes

of MAVs implies that the interaction effects at these small scales may not follow the

interaction effects at larger scales. The relatively small amount of research done on

aeroelastic MAVs has focused on static aeroelasticity and does not adequately address

the dynamic interactions between aeroelasticity and flight dynamics from a design

perspective [71, 135, 159].

Consideration of these effects may provide the aircraft designer with a new design

parameter to alter the open-loop behavior of the aircraft in the preliminary design

process. The purpose may be to avoid undesirable effects on the flight dynamics or to

take advantage of effects that are beneficial to the mission. It may also be desirable to

rapidly tailor or morph the structure of an existing MAV to alter the flight dynamics in a

beneficial way. An understanding of the structures influence on the flight dynamics is

critical to achieve these goals, and a thorough study of the effects of aeroelasticity on

the flight dynamics of an MAV is needed.

A unique class of MAV structure is the membrane wing. The interaction of the

membrane with the aerodynamics is difficult to characterize. Some work has been

done to numerically and experimentally characterize the static stability and control

characteristics of membrane wings. No prior research has experimentally investigated

the dynamic stability and control characteristics. Such an investigation could lead to

improved MAV vehicle designs.

Battens are sometimes used to stiffen and constrain a highly flexible wing. Common

design parameters are the number, size, and orientation of the wing battens. Some

limited investigations have been successfully conducted in this area [87, 88, 157,

159], however, no research has investigated the aeroelastic impact of these design

parameters on the vehicles flight dynamics.

34

Aeroservoelastic design synthesis involves examining the interaction of the

structural and control design in the aircraft design process. This approach is uncommon

in aircraft design but has the potential to reveal important trade-offs between the

control system, structure, and aerodynamics to the designer. For a flexible MAV,

these interactions are not well understood but may have a critical impact on aircraft

performance and mission success.

The constrained and cluttered environments in which MAVs often operate present

unique environmental challenges in the form of wind gusts. As an MAV flies down a

street in an urban canyon, for example, it could be subject to gusts whose velocities and

spatial dimensions may be on the order of the aircrafts velocity and dimensions. When

combined with the low inertia of MAVs, these gust-induced forces and moments could

easily upset the aircraft. These gusts become even more important when considering

the aeroelastic effects that are possible in flexible MAVs. As the wing becomes more

flexible, it could absorb some energy of the gust, possibly decreasing the sensitivity of

the overall vehicle to the gust. At the same time, high levels of flexibility could delay a

controller-induced reaction to the gust, which might reduce the gust rejection ability.

However, the flight dynamics of an aeroelastic aircraft can change drastically and it is not

clear how they interact with a MAVs gust sensitivity and gust rejection properties.

The current research proposes to contribute to the body of knowledge by systematically

investigating the effects of aeroelasticity on the flight dynamics of a fixed-wing, flexible

MAV. In particular, the bending and torsional stiffness of the wing are chosen as the two

main parameters of interest. The effects on the trim conditions, rigid body modes, and

structural modes are characterized. Direct correlations between the stiffness and the

flight dynamics are obtained, increasing the ability to utilize or mitigate these effects.

Research is conducted to investigate the possibility of influencing the flight

dynamics through small, specialized changes in the structural configuration of the

aircraft. Knowing the effects of such changes on the flight dynamics could enable

35

designers to easily tailor the MAV structure for a particular goal without altering the

entire structure.

A unique experimental facility, the University of Floridas low speed wind tunnel [9],

is used to experimentally investigate and determine the static and dynamic stability and

control characteristics of a membrane-wing MAV. This dynamic characterization of a

membrane wing is a first of its kind.

The effect of wing battens on the flight dynamics is studied. Battens are approximated

using stepwise changes in torsional stiffness across the wing. Characterizing the

direct effect of battens on the flight dynamics is a unique contribution to the body of

knowledge.

After understanding the effects of wing stiffness on the flight dynamics, optimal

control is applied to the flexible aircraft. The impact of changing the wing stiffness on

the closed-loop aircraft performance and control actuation is studied. A model-following

control approach is used in a novel attempt to virtually change the stiffness of the wing.

Different types of gusts at varying frequencies are applied to the flexible aircraft

models. The aeroelastic effect of the wing stiffness on the gust sensitivity of the aircraft

is analyzed. A regulation controller is used to analyze the effects of wing stiffness on

the gust rejection properties. The results are compared to traditional metrics for gust

sensitivity to determine their usefulness for a flexible aircraft.

The primary numerical tool used is ASWING [45]. Its focus on rapid conceptual

analysis of aeroelastic vehicles is well-suited to this research. The primary vehicle used

for this study is the generic MAV (GenMAV), developed at the Munitions Directorate

of the Air Force Research Laboratory at Eglin Air Force Base, Florida [163, 164]. The

GenMAV was specifically designed to provide a versatile platform for MAV research and

development.

36

1.3 Document Organization

Chapter 1 introduces the document by addressing the relevant scientific methods

and surveying the current body of knowledge. The research goals are introduced and

discussed in light of the current state of the art.

Chapter 2 recalls the theory necessary to proceed with the research. First, the

classic linearized rigid body equations of motion are developed. Second, fundamental

structural dynamics theory is introduced by way of a mass-spring-damper system. The

topic of aeroelasticity is introduced, including static and dynamic aeroelasticity with an

overview of unsteady aerodynamics. Finally, the necessary control theory is reviewed.

Chapter 3 introduces ASWING, the primary modeling and simulation tool used for

the necessary portions of the research. A review of its theoretical basis is undertaken

with an emphasis on application to the current research. In addition, the primary vehicle

of interest, the GenMAV, is introduced.

Chapter 4 reports on the methodology and results of an experimental investigation

to determine the static and dynamic stability and control characteristics of a membrane-wing

MAV.

Chapter 5 gives a detailed account of the investigation into the relationship

between wing stiffness and the flight dynamics of the vehicle. The emphasis is on a

uniform stiffness distribution across the wing. Chapter 6 relates the results of a similar

investigation with selected non-uniform distributions of wing stiffness.

Chapter 7 introduces the concept of wing battens and reports on the methodology

and results for an investigation into the effect of wing battens on the flight dynamics of

an MAV.

Chapter 8 applies the results of Chapter 5 and examines their interaction with a

typical optimal flight control scheme. A model-following control scheme is also analyzed,

which attempts to virtually change the stiffness of the wing.

37

Chapter 9 takes the aircraft developed in Chapter 5 and analyzes their gust

sensitivity and gust rejection properties in light of the aeroelastic effects.

Chapter 10 concludes the dissertation and presents recommendations for future

research opportunities.

38

CHAPTER 2THEORY

2.1 Flight Equations of Motion

2.1.1 Reference Frames

The first step in analyzing the kinematics or kinetics of a system is to choose a

reference frame. A reference frame is the perspective from which motion is observed

and consists of at least three noncolinear points that move in three-dimensional

Euclidean space (R3). The distance between points in a reference frame does not

change as the frame moves.

2.1.1.1 Earth reference frame

The Earth is assumed to be flat and stationary for the purposes of local atmospheric

flight. A reference frame FE is attached to the earth and is considered an inertial frame

in which Newtons laws of motion are valid.

A coordinate system, called the Earth coordinate system, is created in the Earth

frame using a right-handed set of basis vectors with their origin arbitrarily located on

the surface of the Earth. The vertical unit vector in FE (denoted zE) points toward the

center of the earth. The unit vectors xE and yE are chosen to point North and East,

respectively. The coordinate system is denoted by FE(OE , xE , yE, zE) and is illustrated

in Fig. 2-1, where the origin of the coordinate system in FE is denoted OE .

2.1.1.2 Body reference frame

The body reference frame, denoted FB, is fixed to the aircraft. A coordinate system,

called the body coordinate system, is defined in the body reference frame with its origin

at the center of gravity. Note that gravity is assumed to be uniform and thus the aircraft

center of gravity (CG) is coincident with the aircraft center of mass (CM). The xB axis is

defined to run from the CG out the nose and the yB axis runs parallel to the right wing.

The zB axis results from the cross product xB yB. The coordinate system is denoted

by FB(OB, xB, yB, zB) and is illustrated in Fig. 2-2.

39

OE xE

yE

zE

Figure 2-1. Earth coordinate system

The body coordinate system is of primary concern since it is the system in which

the bodys inertia is most easily defined. The equations of motion will be developed in

this coordinate system.

xB

yB

zB

OB

Figure 2-2. Body coordinate system

A second coordinate system, called the stability coordinate system, is defined in

the body frame. An axis xS is aligned with the projection of relative wind on the xB zB

plane and is found by rotating the body coordinate system around yB through an angle

40

, called the angle of attack. This coordinate system is shown in Fig. 2-3 and is denoted

by FB(OS, xS, yS, zS). This system is useful for determining the aerodynamic forces and

moments since they are dependent on the orientation of the aircraft with respect to the

oncoming flow.

xB

xS

yB, yS

zS zB

OB,OS

Figure 2-3. Stability coordinate system

2.1.2 Coordinate Transformations

The angular relations of the Earth, body, and stability coordinate systems are shown

in Fig. 2-4.

Earth Body Stability, ,

Figure 2-4. Overview of coordinate transformations

A vector may be transformed from the Earth frame into the body frame with a 3-2-1

Euler rotation sequence through the heading angle , pitch angle , and roll angle .

These angles are commonly known as the Euler angles. A vector expressed in the Earth

frame (denoted {a}E) can thus be transformed to the body frame through the series of

rotations shown in Eq. (21), where a rotation about the x-axis through the angle is

41

denoted R1().

{a}B = R1()R2()R3() {a}E (21)

A rotation about the x-axis through the angle is accomplished by:

R1() =

1 0 0

0 cos sin

0 sin cos

(22)

A rotation about the y-axis through the angle is accomplished by:

R2() =

cos 0 sin

0 1 0

sin 0 cos

(23)

A rotation about the z-axis through the angle is accomplished by:

R3() =

cos sin 0

sin cos 0

0 0 1

(24)

The resulting rotation matrix from the Earth coordinate system to the body coordinate

system (RBE) is shown in Eq. (26).

RBE = R1()R2()R3() (25)

=

cos cos cos sin sin

sin sin cos cos sin cos cos + sin sin sin sin cos

sin sin + cos sin cos cos sin sin sin cos cos cos

(26)

A vector in the stability coordinate system may be transformed into the body

coordinate system through the transformation shown in Eq. (27), where RBS is defined

42

according to Eq. (29).

{a}B = RBS {a}S (27)

RBS = R2() (28)

=

cos 0 sin

0 1 0

sin 0 cos

(29)

For example, the aerodynamic forces of drag and lift are commonly expressed in the

stability coordinate system but must be transformed to the body coordinate system. This

transformation can be accomplished by applying Eq. (27) as shown in Eq. (210).

Fx

Fy

Fz

B

=

cos 0 sin

0 1 0

sin 0 cos

D

Fy

L

S

(210)

Because a rotation matrix is an orthogonal matrix, it has the special property that

R1 = RT . Thus, the reverse transformations can be obtained with either the inverse or

transposed rotation matrices. For example, a vector expressed in the body frame can be

transformed into the earth frame via Eq. (211) or Eq. (212).

{a}E = RTBE {a}B (211)

= R1BE {a}B (212)

2.1.3 Nonlinear Equations of Motion

Twelve quantities completely describe the aircraft motion over time: position (x, y, z),

translational velocity (u, v, w), orientation (, , ), and angular velocity (p, q, r). Unless

otherwise stated, all quantities are expressed in the body coordinate system. Twelve

equations are needed to obtain a solution for these quantities.

43

2.1.3.1 Translational dynamics

Equations for the translational motion of the center of mass are derived from

Newtons second law which states that the time rate of change of momentum is equal to

the forces acting on the body.

E d

dt(mV) = F (213)

Assuming mass is constant and knowing thatE ddtV = Ea, Eq. (213) can be written as

Eq. (214). The notation E{a}B denotes the acceleration vector expressed in the body

coordinate system as viewed by an observer in the inertial reference frame and, since all

quantities are expressed in the body coordinate system unless otherwise noted, can be

written as Ea.

mEa = F (214)

Let the position of the bodys center of mass as measured with respect to the body

frame be denoted r = Br.

r =

x

y

z

(215)

The velocity as observed in the body frame, given in Eq. (216), is simply the time

derivative.

BV =B d

dtr =

x

y

z

(216)

The velocity as observed in the Earth frame is denoted BV and shown in Eq. (217).

EV =E d

dtBV = BV + EB r (217)

44

Note that the transport theorem must be employed since the time rate of change of

the vector is being observed in a different reference frame (the transport theorem is

introduced in Appendix A.2). The angular velocity vector EB is the angular velocity of

reference frame FB as viewed by an observer in reference frame FE and consists of the

individual rates p, q, r. The result is defined in Eq. (218).

EV =

u

v

w

(218)

Now the acceleration of the body in the Earth frame may be formulated. Note that the

transport theorem must be employed again. The result, shown in Eq. (221), represents

the acceleration of the aircraft as viewed by an observer in the Earth frame, expressed

in the body coordinate system.

Ea =E d

dtEV =

B d

dtEV + EB EV (219)

=

u

v

w

+

0 r q

r 0 p

q p 0

u

v

w

(220)

Ea =

u+ qw rv

v + ru pw

w + pv qu

(221)

Now the right-hand side of Eq. (214) is developed. The forces that will be included

are gravitational (Fg), aerodynamic (Fa), and propulsive (FT ), as shown in Eq. (222).

Each of these forces must be expressed in the body coordinate system to maintain

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consistency with Eq. (221).

{F}B =

Fgx + FAx + FTx

Fgy + FAy + FTy

Fgz + FAz + FTz

(222)

First, the gravitation force is easily expressed in the Earth coordinate system as shown

in Eq. (223).

{Fg}E =

0

0

mg

(223)

The gravitational force may then be expressed in the body coordinate system by

applying the rotation described in Eq. (26), resulting in Eq. (224).

{Fg}B =

mg sin

mg sin cos

mg cos cos

(224)

The aerodynamic forces of lift, drag, and side force are natively expressed in the stability

coordinate system as shown in Eq. (225).

{Fa}S =

D

Fay

L

(225)

These forces can be expressed in the body coordinate system by applying Eq. (29),

resulting in Eq. (226).

{Fa}B =

D cos + L sin

Fay

D sin L cos

(226)

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Finally, the thrust forces act in the body reference frame with T defined as the

angle between the thrust vector and xB. The thrust forces can be rotated into the

body coordinate system through the standard rotation RT2 (T ). The result is given in

Eq. (227).

{FT}B =

T cosT

0

T sin T

(227)

The final set of body forces is grouped as shown in Eq. (228).Fx

Fy

Fz

=

mg sin D cos + L sin+ T cosT

mg sin cos + Fay

mg cos cos D sin L cos T sin T

(228)

The results of Eqs. (221), (224), (226) and (227) can be assembled into the

form of Eq. (214), as shown in Eq. (229).

m

u+ qw rv

v + ru pw

w + pv qu

=

mg sin D cos + L sin + T cosT

mg sin cos + Fay

mg cos cos D sin L cos T sinT

(229)

Equation (229) represents the translational motion of the aircrafts center of mass in

response to the forces acting on the aircraft.

2.1.3.2 Rotational dynamics

Expressions for the rotational equations of motion of the aircraft will now be

obtained. These expressions will be formulated using Eulers second law of motion,

shown in Eq. (230), which states that the time rate of change of the angular momentum

of a rigid body relative to point O in the inertial reference frame FE is equal to the

moment of the body about the same point in the same inertial reference frame [137].

The angular momentum relative to point O as viewed by an observer in FE is denoted

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