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An analytical dynamics approach to the control of mechanical systems

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An analytical dynamics approach to the control of mechanical systems

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Content An Analytical Dynamics Approach to the Control of Mechanical Systems by Harshavardhan Mylapilli A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MECHANICAL ENGINEERING) August 2015 Copyright c 2015 Harshavardhan Mylapilli In loving memory of my father, my hero Acknowledgements It has been nearly eight years since I rst stepped into USC. The years have passed by so quickly that it feels more like a mere instant now. Along this eight year journey, I have had the good fortune of meeting and working with many intelligent minds, all of whom had a profound impact on me - as a teacher, a researcher and a human being. In the following paragraphs, I would like to take this opportunity to thank each and everyone of them who partook in this chapter of my life. Firstly, I would like to convey my sincere thanks and heartfelt gratitude to my advisor Prof. Firdaus E. Udwadia without whose unwavering support, constant guid- ance and persistent motivation over the past six years, this thesis would not have been possible. Prof. Udwadia's incredible understanding of the eld of mechanics, his in- domitable enthusiasm for research, his attention to detail, and the phenomenal clarity of his thinking is what really makes him one of the very best in this eld. One of the rst courses that I took at USC was AME 524 (Advanced Engineering Dynamics) which was taught by Prof. Udwadia. Coming from an education system where most courses are taught in a formulaic and mundane fashion with the professors simply going through the motions in teaching a course, it was extremely refreshing to take a course taught by Prof. Udwadia. There are very few people in this world who can so artfully teach a subject like analytical dynamics in the manner that Prof. Udwadia does. His passion and love for classical mechanics is infectious and it manifested in each and everyone of his lectures. Captivated by his teaching and his deep understanding of mechanics, I iii remember remarking to myself that if I ever do a Ph.D. in Mechanical Engineering, it has to be under a professor like him. And I thank my stars that this indeed came to be true. I am truly indebted to Prof. Udwadia for all the sel ess help and kind generosity that he extended to me over the years. My heartfelt thanks to him for imparting his wisdom to me, for educating me on the ways of doing research, for teaching me how to think, for making me adopt a rigorous work ethic, for patiently going through my work week in and week out, and for bearing with me these past six years. Words are insucient to express how grateful I am to him and I will forever cherish this six-year working relationship that I had with him. I would also like to convey my deepest gratitude and thanks to Prof. Henryk Flash- ner. Prof. Flashner along with Prof. Udwadia helped me through some of the darkest days of my Ph.D. life and I am extremely thankful to them for believing in me and in my potential. If it was not for their faith in me, I probably would not have obtained an admission to the USC Ph.D. program and for that I will always be indebted to both of them. Prof. Flashner has been a pillar of great support and encouragement during my Ph.D. years. I greatly cherish the philosophical conversations I shared with him over the years. Prof. Flashner's friendly, approachable and kind nature is extremely contagious. The three courses I took from him (AME 521, AME 552 and ASTE 585) are some of my most memorable courses at USC and I had so much fun learning from Prof. Flashner. I would like to thank Prof. Flashner from the bottom of my heart for always being there for me, for helping me when I was in need, for supporting me and advicing me over the years. I would also like to thank my dissertation committee members (Prof. Robert Sacker, Prof. Larry Redekopp) and my qualifying exam committee members (Prof. Bingen Yang, Prof. Veronica Eliasson, Prof. Chunming Wang) for very kindly agreeing to be a part of my committee and for their helpful comments and suggestions in improving this dissertation thesis. iv My sincere thanks are also due to Prof. Charles Campbell, Prof. Satwinder Sadhal, Prof. Paul Newton and Prof. Oussama Safadi with whom I have had the pleasurable experience of working multiple times as a teaching assistant for various graduate level courses over the years. They have all had an immeasurable impact in the way I teach and conduct my oce hours and I thank them all very much from the bottom of my heart. Of special mention is Prof. Sadhal who has gone out of his way in procuring me opportunities with the department and believing in my abilities as a teaching assistant and as an instructor. Thank you, Prof. Sadhal. Being a teaching assistant for many graduate level core courses such as AME 525 and AME 526, I have had the thorough pleasure of teaching and mentoring numerous batches of graduate students admitted to the AME department over the past six years. The friendships I made along the way and the time I spent with these students (too many to mention here) will always remain dear to me for the rest of my life. I am also extremely thankful to the many friends that I made over the years who have made this experience all the more enjoyable. I consider myself extremely fortunate to have met some of the most humble and hard-working people (Eric, Sumo, Dayung, Korkut, Shalini, Lan, Chang, Pang, and Kevin) right in my rst year at USC. Eric, Sumo and I were infact so inseparable during our masters' days that professors often referred to us as the \Three Musketeers". The three of us spent many hours solving homeworks, discussing dicult concepts and often on the weekends treating ourselves to the sumptuous varieties of food that Los Angeles oered. Although, Sumo had left us midway after completing his Ph.D., Eric and I have continued our academic journeys together. I am so glad that I had him as a friend to help and support me through the good times and the bad times these past eight years, as otherwise, I cannot imagine myself going through this journey alone. He is the most gentle, kind-hearted, and spiritual human being that I have ever met in my life. His friendship means so much to me and I consider him as one of my best friends. v My special thanks to the Indian cohort of friends: Lavanya, Ketaki, Tripti, Omkar, Pratik, Prakhar, Vikram, Krishna, Sonal, Shaik and Raghu for being such awesome friends over the years and sharing many memorable moments with me. Of special mention is Lavanya, who I met on the rst day at USC and have remained very close friends with her over the years. The comfort level I share with her is next to none. I am extremely grateful to her for her support, words of encouragement and the invaluable advice that she has given me over the years. She is one of my best friends and I cherish her friendship very dearly. My heartfelt thanks to my CET friends, Bo and Elham, with whom I have organized many successful TA training events. Thanks are also due to my table-tennis partners, Aikagra and Moiz, with whom I have spent many hours sharpening my table-tennis skills. I would also like to thank Prasanth for giving me rides in his car every week during the summers to meet Prof. Udwadia at the Arcadia Library. In recent years, I have become good friends with Gauri, Jagan, D.J. and Dejuan, and I thank them all for their benevolent friendship. People have come and gone but Eric, Dayung, Korkut and I have stayed pretty much a constant at USC these past six years and we have become very close friends over time. If not for our weekly fun-lled dinner get-togethers, I would have burnt out a long time back and so thank you all so very much! My special thanks to Korkut for exercising with me every week these past couple of years and keeping me physically t. My thanks are also due to Dayung for being a such a kind and supportive friend. I really cherish the time we spent together these past few years, the conversations we shared on our long walks, and our JPL train rides. Lastly, yet most importantly, I would like to thank my dad, my mom and my sister for all the sacrices that they made in order to send me to the United States so I can pursue my dreams. My father is a man of impeccable character, discipline and honor. He has been my hero my entire life and will always remain so till my last breath. He vi has worked tirelessly, often at the cost of his own health, so that we (his children) could lead a better life. It is one of my greatest regrets that when the time came for me to repay all that he has done for me, it was far too late. The untimely death of my father in 2012 came as a devastating shock to me and my family and it took us quite some time to recuperate. If there is one thing in life that I repent the most, it is that I was not there with my father in the last few months of his life when he needed me most. After my father passed away, I have to applaud my mother's courage in letting me come back to the United States to nish what I have begun here, even though that meant that she had to stay all alone by herself in my hometown. My mother has always been my rock and I thank her very much for the unconditional love and aection that she showers on me everyday. She has always been there for me whenever I needed her and has always lent a patient ear whenever I needed to discuss my troubles. She has unconditionally supported me in all of my endeavours. If it is not for her constant encouragement, sustained support, and unrelenting love, I would not have been able to achieve any of this. I thank her very much for the many sacrices she had to make in her life just so I could pursue my dreams. My thanks are also due to my sister, who despite being the youngest in the family, had to take up all of my responsibilities back in India in my absence. Like my mother, my sister is a strong and independent woman. The courage and resilience she displayed in taking care of my family when my father fell ill is truly commendable and I will forever be grateful for her help. A precious heart, she will always be my best friend, condante and partner-in-crime. I thank her from the bottom of my heart for everything she has done for me. This eight year journey has been a long ride lled with many joyous and heartwrench- ing moments. Above all, it has humbled me and has helped me become a better human being. I am extremely thankful to each and every person who has helped me achieve vii and realize this dream of mine. My apologies (and due thanks) to all the people I have missed acknowledging here. Harshavardhan Mylapilli August, 2015 viii Table of Contents Dedication ii Acknowledgements iii List of Figures xiv List of Tables xxix Abstract xxx Part 1 Preliminaries 1 Chapter 1 : Introduction 2 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Outline of this Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Chapter 2 : Constrained Motion Approach 9 2.1 Unconstrained System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Constrained System and the Fundamental Equation . . . . . . . . . . . 11 2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Part 2 General Nonlinear Lattices 13 Chapter 3 : Energy Control of Inhomogeneous Nonlinear Lattices 14 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Physics of the Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2.1 General Nonlinear Lattice . . . . . . . . . . . . . . . . . . . . . . 18 3.2.2 Inhomogeneous Toda Lattice . . . . . . . . . . . . . . . . . . . . 19 3.3 Unconstrained Equations of Motion . . . . . . . . . . . . . . . . . . . . . 21 3.3.1 General Nonlinear Lattice . . . . . . . . . . . . . . . . . . . . . . 21 3.3.2 Inhomogeneous Toda Lattice . . . . . . . . . . . . . . . . . . . . 23 3.4 Problem Formulation and Constraint Equations . . . . . . . . . . . . . . 24 ix 3.4.1 Formulation of the Constraints . . . . . . . . . . . . . . . . . . . 25 3.4.2 Controlled NDOF Nonlinear Lattice . . . . . . . . . . . . . . . . 27 3.5 Global Asymptotic Convergence . . . . . . . . . . . . . . . . . . . . . . 28 3.5.1 SDOF Toda Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 29 3.5.2 NDOF General Nonlinear Lattice . . . . . . . . . . . . . . . . . . 35 3.6 Results and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.6.1 Toda Lattice Simulations . . . . . . . . . . . . . . . . . . . . . . 39 3.6.1.1 Fixed-Fixed Toda Lattice . . . . . . . . . . . . . . . . . 42 3.6.1.2 Fixed-Free Toda Lattice . . . . . . . . . . . . . . . . . . 49 3.6.2 FPU- Lattice and the General Nonlinear Lattice . . . . . . . . 57 3.6.2.1 Homogeneous FPU- Lattice . . . . . . . . . . . . . . . 58 3.6.2.2 General Nonlinear Inhomogeneous Lattice . . . . . . . . 60 3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Part 3 Multiple Coupled Slave Gyroscopes 67 Chapter 4 : Synchronization of Multiple Coupled Slave Gyroscopes 68 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.1 Master Gyroscope . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.2 Slave Gyroscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2.3 Unconstrained Equations of Motion . . . . . . . . . . . . . . . . 75 4.2.4 Formulation of the Constraints . . . . . . . . . . . . . . . . . . . 75 4.2.5 Constrained Equations of Motion . . . . . . . . . . . . . . . . . . 77 4.2.6 General Coupled Slave Gyroscopes . . . . . . . . . . . . . . . . . 78 4.3 Results and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3.1 Synchronization with the use of a Sleeping Condition . . . . . . . 80 4.3.1.1 Five Gyroscopes using a Sine Coupling . . . . . . . . . 80 4.3.1.2 Six Gyroscopes using an Incidence Matrix . . . . . . . . 85 4.3.2 Synchronization without the use of the Sleeping Condition . . . . 91 4.3.2.1 Five Gyroscopes using Toda Coupling . . . . . . . . . . 91 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Part 4 Incompressible Hyperelastic Beams 98 Chapter 5 : Dynamics of Incompressible Hyperelastic Beams 99 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.2 Absolute Nodal Coordinate Formulation (ANCF) . . . . . . . . . . . . . 104 5.2.1 Classical ANCF Element . . . . . . . . . . . . . . . . . . . . . . 106 x 5.2.2 Lower Order 32 10111 3 Element . . . . . . . . . . . . . . . . . . . 108 5.2.3 Higher Order 3253 Element . . . . . . . . . . . . . . . . . . . . . 109 5.3 Equations of Motion of a Beam Element . . . . . . . . . . . . . . . . . . 110 5.4 Nonlinear Constitutive Material Models . . . . . . . . . . . . . . . . . . 111 5.4.1 Incompressible Neo-Hookean Material Model . . . . . . . . . . . 114 5.4.2 Incompressible Mooney-Rivlin Material Model . . . . . . . . . . 115 5.4.3 Incompressible Yeoh Material Model . . . . . . . . . . . . . . . . 116 5.5 Equations of Motion of the Entire Beam . . . . . . . . . . . . . . . . . . 117 5.6 Cantilever Beam Boundary Conditions . . . . . . . . . . . . . . . . . . . 118 5.6.1 Partially-Clamped Boundary Condition . . . . . . . . . . . . . . 119 5.6.2 Fully-Clamped Cantilever Beam . . . . . . . . . . . . . . . . . . 120 5.6.3 Equations of Motion of a Cantilever Beam . . . . . . . . . . . . . 120 5.7 Integration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.7.1 Volume Integration Using Gaussian Quadrature . . . . . . . . . . 122 5.7.1.1 Full Integration . . . . . . . . . . . . . . . . . . . . . . 123 5.7.1.2 Reduced Integration . . . . . . . . . . . . . . . . . . . . 123 5.7.1.3 Selective Reduced Integration . . . . . . . . . . . . . . 123 5.7.2 St ormer{Verlet Fixed-Time Step Integration Scheme . . . . . . . 125 5.8 Results and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.8.1 Locking eects and their elimination techniques . . . . . . . . . . 128 5.8.1.1 Classical ANCF Finite Element . . . . . . . . . . . . . 128 5.8.1.2 Lower Order 32 10111 3 Element . . . . . . . . . . . . . . 132 5.8.1.3 Higher Order 3253 Element . . . . . . . . . . . . . . . . 137 5.8.1.4 Summary and Discussion of the Results . . . . . . . . . 141 5.8.1.5 Selective Reduced Integration . . . . . . . . . . . . . . 143 5.8.2 Convergence Results: Increasing the Number of Elements . . . . 146 5.8.2.1 Classical ANCF Element . . . . . . . . . . . . . . . . . 146 5.8.2.2 Higher Order 3253 Element . . . . . . . . . . . . . . . . 150 5.8.2.3 Lower Order 32 10111 3 Element . . . . . . . . . . . . . . 152 5.8.3 Analysis of the 150 Element Classical ANCF Beam . . . . . . . . 154 5.8.3.1 Energy Conservation . . . . . . . . . . . . . . . . . . . 154 5.8.3.2 Incompressibility Condition . . . . . . . . . . . . . . . . 158 5.8.3.3 Second Piola-Kircho Stress at the Free End . . . . . . 158 5.8.4 Eect of Clamping on the Deformation of the Beam . . . . . . . 161 5.8.5 Comparison of the Finite Element Models . . . . . . . . . . . . . 163 5.8.6 Evaluation of the St ormer{Verlet Integration Solver . . . . . . . 166 5.8.6.1 Eect of Time Step on the Deformation . . . . . . . . . 166 5.8.6.2 Dierent Solver Comparisons . . . . . . . . . . . . . . . 168 5.8.7 Other Nonlinear Constitutive Models . . . . . . . . . . . . . . . . 170 5.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 xi Chapter 6 : Control of Incompressible Hyperelastic Beams 176 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.2 Unconstrained Equations of Motion of the Cantilever Beam . . . . . . . 179 6.3 Formulation of the Constraints . . . . . . . . . . . . . . . . . . . . . . . 181 6.4 Constrained Equations of Motion of the Cantilever Beam . . . . . . . . 182 6.5 Results and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.5.1 Example 1: Fully-Clamp the Right End of a Cantilever Beam . . 184 6.5.2 Example 2: Zero Displacement of a Material Point in the Fixed- Fixed beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.5.3 Example 3: Circular Tip Motion in the YZ-plane with Constraint Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6.5.4 Example 4: Circular Tip Motion in the YZ-plane without Con- straint Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . 199 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 Part 5 Conclusions 208 Chapter 7 : Conclusions and Future Work 209 7.1 General Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Part 6 Appendices 213 Appendix A : Miscellaneous Proofs and Results 214 A.1 Positive Denite Function . . . . . . . . . . . . . . . . . . . . . . . . . . 214 A.2 Radially Unbounded Function . . . . . . . . . . . . . . . . . . . . . . . . 215 A.3 Radially Increasing and Spherically Increasing . . . . . . . . . . . . . . . 216 A.4 Radially Increasing and Radially Unbounded . . . . . . . . . . . . . . . 218 A.5 Miscellaneous Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 A.6 Requisite Conditions on the Potentials of the Springs . . . . . . . . . . . 222 Appendix B : Energy H is a positive denite function 224 Appendix C : A Closed Form Expression for the Control Force F C 225 Appendix D : Origin O is a single isolated equilibrium point 229 Appendix E : Set is compact 231 Appendix F : Sucient Conditions on the Actuator Placements 234 xii Appendix G : Equations of Motion of a Symmetric Gyroscope 238 Appendix H : Modied St ormer{Verlet Scheme 240 Bibliography 242 xiii List of Figures 3.1 An inhomogeneous nonlinear lattice . . . . . . . . . . . . . . . . . . . . 18 3.2 An inhomogeneous Toda lattice . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Exponential Toda potential a o , b o > 0 . . . . . . . . . . . . . . . . . . . 20 3.4 Toda spring force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.5 A single degree-of-freedom Toda oscillator . . . . . . . . . . . . . . . . . 29 3.6 (a) Phase portrait of the uncontrolled SDOF oscillator with Toda spring stiness (a o = 2; b o = 1), (b) Phase portrait of the controlled SDOF oscillator with Toda spring stiness (a o = 2; b o = 1; o = 1; H = 8). 30 3.7 Pictorial representation of a typical set. . . . . . . . . . . . . . . . . . 33 3.8 Scalar energy error V plotted as a function of X 1 and X 2 (a o = 2; b o = 1; H = 8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.9 Example 1 - Velocity eld of the 101-mass xed-xed homogeneous Toda lattice. The lattice has parameters a i = 2, b i = 1, m i = 1 for all i, H o = 36:27, H = 150, o = 0:1 and the initial displacement of central mass (m 51 ) is 3 units. Actuators are located on masses m 75 and m 76 . . 43 3.10 Example 1 - Velocity eld of the 101-mass xed-xed nonhomogeneous Toda lattice. The nonhomogeneous lattice has parameters a, b and m chosen randomly from a uniformly distributed set of numbers between the limits: 1:5<a i < 2:5, 0:5<b i < 1:5 and 0:5<m i < 1:5,H o = 83:58, H = 150, o = 0:1 and the initial displacement of central mass (m 51 ) is 3 units. Actuators are located on masses m 75 and m 76 . . . . . . . . . . . 44 xiv 3.11 Example 1 - Time history of control forces acting on the two actuator masses m 75 and m 76 of the lattice. Dotted lines denote control forces acting on the homogeneous lattice and the solid lines denote the control forces acting on the nonhomogeneous lattice. Red line denotes the control force acting on the massm 75 and blue line denotes the control force acting on mass m 76 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.12 Example 1 - (Top Subplot) Time history of energy of the lattice fromt = 0 tot = 25 seconds. (Bottom Subplot) Time history of energy error e(t) = H(t)H fromt = 400 seconds tot = 500 seconds. Green line denotes the homogeneous lattice and pink line denotes the nonhomogeneous lattice. 45 3.13 Example 2 - Nonhomogeneous Toda lattice with xed-xed boundary conditions. The lattice has its parameters chosen randomly from a uni- formly distributed set of numbers between the limits: 1:5 < a i < 2:5, 0:5 < b i < 1:5, 0:5 < m i < 1:5, H o = 83:58, H = 150, o = 0:01 and initial displacement of the central mass (m 51 ) is 3 units. . . . . . . . . . 48 3.14 Example 3 - Velocity eld of the 101-mass xed-free homogeneous Toda lattice. The top subplot corresponds to the uncontrolled lattice and the bottom subplot corresponds to the controlled lattice. The lattice has parameters a i = 2, b i = 1, m i = 1 for all i, H o = 149:44, H = 100, o = 0:1 and the initial displacement of central mass (m 51 ) is 4:34 units. Actuators are located on masses m 75 and m 76 . . . . . . . . . . . . . . . 51 3.15 Example 3 - Velocity eld of the 101-mass xed-free nonhomogeneous Toda lattice. The nonhomogeneous lattice has parameters a, b and m chosen randomly from a uniformly distributed set of numbers between the limits: 1:5<a i < 2:5, 0:5<b i < 1:5 and 0:5<m i < 1:5,H o = 149:9, H = 100, o = 0:1 and the initial displacement of central mass (m 51 ) is 3.41 units. Actuators are located on masses m 75 and m 76 . . . . . . . . . 52 xv 3.16 Example 3 - Time history of control forces acting on the two actuator massesm 75 (red line) andm 76 (blue line) of the homogeneous lattice (top subplot) and the nonhomogeneous lattice (bottom subplot). . . . . . . . 53 3.17 Example 3 - (Top Subplot) Time history of energy from t = 0 tot = 250 seconds. (Bottom Subplot) Time history of energy error e(t) = H(t) H from t = 750 seconds to t = 1000 seconds. Green line denotes the homogeneous case and pink denotes the nonhomogeneous case. . . . . . 53 3.18 Example 4 - Velocity eld of the 101-mass xed-free nonhomogeneous Toda lattice subjected to random initial excitation. The nonhomogeneous lattice has parameters a, b and m chosen randomly from a uniformly distributed set of numbers between the limits: 1:5 < a i < 2:5, 0:5 < b i < 1:5 and 0:5 < m i < 1:5, H o = 341:47, H = 250, o = 0:1. The initial displacements and initial velocities are chosen randomly between the limits2 and 2. A single actuator is placed on the last mass of the xed-free lattice (i.e. on m 101 ). . . . . . . . . . . . . . . . . . . . . . . . 55 3.19 Example 4 - Time history of control forces acting on the last mass m 101 of the nonhomogeneous lattice. . . . . . . . . . . . . . . . . . . . . . . . 56 3.20 Example 4 - (Top Subplot) Time history of energy from t = 0 tot = 150 seconds. (Bottom Subplot) Time history of energy errore(t) =H(t)H from t = 500 seconds to t = 750 seconds. . . . . . . . . . . . . . . . . . . 56 3.21 Example 5 - Gradient (q i+1 q i ) of the displacement eld of the homoge- neous lattice. The lattice has parametersa i = 1,b i = 1;m i = 18i;H o = 12;H = 150; o = 0:1 and the initial displacement of the central mass, m 51 , is 2 units. Actuators are located at m 75 and m 76 . . . . . . . . . . . 59 xvi 3.22 Example 6 - Gradient (q i+1 q i ) of the displacement eld of the inhomo- geneous lattice. For each spring in the lattice, rst a nonlinear potential is chosen at random from the set S sp and then its parameters a;b andm are chosen randomly from a uniform distributed set of numbers between the limits 0:5 < a i < 1:5; 0:5 < b i < 1:5; and 0:5 < m i < 1:5, respec- tively. The initial displacement of the central mass, m 51 , is 2 units and H o = 28:4;H = 150; and o = 0:1. Actuators are located atm 75 andm 76 . 61 3.23 Examples 5 & 6 - Time history of control forces, energy convergence and energy errors. (a) Control forces acting on the two actuator masses m 75 (solid line) andm 76 (dash line) of the homogeneous lattice (top) and the inhomogeneous lattice (bottom) for o = 0:1. (b) Energy of the system fromt = 0 tot = 150 seconds (top) and energy error (e(t) =H(t)H ) from t = 100 to t = 150 seconds (bottom). Solid ( o = 0:1) and dotted ( o = 0:005) lines denote the homogeneous lattice whereas dash ( o = 0:1) and dash-dot ( o = 0:005) lines denote the inhomogeneous lattice. . 62 4.1 Symmetrical master gyro (independent) subjected to a vertical harmonic excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2 A system of chain-coupled slave gyroscopes; each slave gyro in the chain is subjected to a vertical harmonic excitation which, when uncoupled from the system, results in regular or chaotic motion . . . . . . . . . . . 72 4.3 A phase plot ( m ; _ m ) of the unconstrained motion of the periodic master gyroscope for 150t 200. . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.4 A phase plot of the unconstrained motion of four slave gyroscopes which are all coupled to one another in a chain fashion (150t 200). . . . 82 4.5 Time history of nutation angle for each individual gyroscope for 0 t 20 prior to synchronization . . . . . . . . . . . . . . . . . . . . . . . 83 xvii 4.6 Time history of nutation angle for each individual gyroscope for 0 t 20 in the constrained system. Slave gyros take less than 10 seconds to synchronize with the motion of the master gyro. . . . . . . . . . . . . 83 4.7 A superimposed phase plot of the constrained motion of ve gyroscopes for 150t 200. The slave gyroscopes have precisely synchronized with the motion of the master gyro. . . . . . . . . . . . . . . . . . . . . . . . 83 4.8 Time history of synchronization errors for 60 t 100. Errors are smaller than the tolerance levels of the integration scheme. . . . . . . . 84 4.9 Time history of Nonlinear Control Torques F c acting on each individual gyroscope for 0t 200 . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.10 General-coupled slave gyroscope system depicting various types of cou- pling/interactions between the slaves. . . . . . . . . . . . . . . . . . . . 86 4.11 A phase plot of the unconstrained motion for each individual gyroscope for 50t 100. The master gyroscope is independent whereas the other remaining ve slave gyroscopes are all coupled using the incidence matrix. 87 4.12 A superimposed phase plot of the constrained motion of ve gyroscopes for 50 t 100 (left) and 150 t 200 (right). Slave gyros have precisely tracked motion of the master gyro. . . . . . . . . . . . . . . . . 87 4.13 Time history of nutation angle for each individual gyroscope for 0 t 20 prior to synchronization . . . . . . . . . . . . . . . . . . . . . . . 88 4.14 Time history of nutation angle for each individual gyroscope for 0 t 20 in the constrained system. . . . . . . . . . . . . . . . . . . . . . . 88 4.15 Time history of synchronization errors for 60 t 200. Errors are smaller than the tolerance levels of the integration scheme. . . . . . . . 89 4.16 Generalized control torques acting on each individual slave gyroscope for 0t 200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 xviii 4.17 Superimposed phase plot depicting the unconstrained motion of the pe- riodic master gyro along with the four chain coupled slave gyros for 50t 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.18 A superimposed phase plot of the constrained motion of ve gyroscopes for 50 t 100. The slave gyros have precisely tracked motion of the master gyro. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.19 Time history of nutation angle for each individual gyroscope for 0 t 20 prior to synchronization . . . . . . . . . . . . . . . . . . . . . . . 93 4.20 Time history of nutation angle for each individual gyroscope for 0 t 20 in the constrained system. . . . . . . . . . . . . . . . . . . . . . . 93 4.21 Time history of synchronization errors for 60 t 200. Errors are smaller than the tolerance levels of the integration scheme. . . . . . . . 93 4.22 Generalized control torques acting on each individual slave gyroscope for 0t 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.1 A typical three-dimensional cantilever beam . . . . . . . . . . . . . . . . 105 5.2 Deformation of a typical nite element containing two nodes, A and B. . 105 5.3 Classical ANCF Element - FULL INTEGRATION - Time history of bending deformation of a ve element partially clamped (shown in blue color) and a fully clamped (shown in red color) beam that is fully inte- grated using a classical ANCF nite element model. The deformation of the partially clamped and fully clamped beams is superposed on top of one other and beams exhibit locking. . . . . . . . . . . . . . . . . . . . . 129 xix 5.4 Classical ANCF Element - REDUCED INTEGRATION - Time history of deformation of a ve element partially clamped (blue color) and a fully clamped (red color) beam using a classical ANCF nite element model. The beam is under integrated in thex-direction and this causes the beam to simply free fall with its left end attached to the xed support in both the fully-clamped and partially-clamped cases. . . . . . . . . . . . . . . 131 5.5 Lower Order 32 10111 3 Element - FULL INTEGRATION - Time history of bending deformation of a 5-element partially clamped (shown in blue color) and a fully clamped (shown in red color) beam that is fully inte- grated using the 32 10111 3 nite element model. Both partially-clamped and fully-clamped beams exhibit locking similar to the classical ANCF nite element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.6 Lower Order 32 10111 3 Element - REDUCED INTEGRATION - Time history of bending deformation of a 5-element partially clamped (shown in blue color) and a fully clamped (shown in red color) beam that is underintegrated in the x-direction using a 32 10111 3 nite element model. The partially clamped 5-element beam shows a non-physical result, but the locking in the deformation of the fully clamped beam appears to have been alleviated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.7 Lower Order 32 10111 3 Element - REDUCED INTEGRATION - Time history of bending deformation of a 30-element partially clamped (shown in blue color) and a fully clamped (shown in red color) beam that is under integrated in thex-direction using a 32 10111 3 nite element model. The bending deformation of the partially-clamped and the fully-clamped beams are superposed on top of each other and the non-physical result of the 5-element partially clamped beam is no longer present in this 30- element beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 xx 5.8 Higher Order 3253 Element - FULL INTEGRATION - Time history of bending deformation of a 5-element partially clamped (shown in blue color) and a fully clamped (shown in red color) beam that is fully inte- grated using a 3253 nite element model. The deformation of the partially clamped and fully clamped beams exhibit locking, although to a lesser extent compared to the Classical ANCF and the 32101113 element. . . 138 5.9 Higher Order 3253 Element - FULL INTEGRATION - Time history of bending deformation of a 30-element partially clamped (shown in blue color) and a fully clamped (shown in red color) beam that is fully inte- grated using a 3253 nite element model. The 30-element beams show a ner de ection prole compared to the 5-element beams but some amount of locking still appears to be present. . . . . . . . . . . . . . . . . . . . 139 5.10 Higher Order 3253 Element - REDUCED INTEGRATION - Time history of deformation of a 5-element partially-clamped (blue color) and a fully- clamped (red color) beam using a 3253 nite element model. The beam is under integrated in thex-direction and this causes the beam to simply free fall with its left end attached to the xed support in both the fully- clamped and partially-clamped cases. . . . . . . . . . . . . . . . . . . . 140 xxi 5.11 SELECTIVE REDUCED INTEGRATION - Time history of deforma- tion of a 30-element beam. The deformation of the fully clamped classi- cal ANCF element is shown in black, partially clamped ANCF element is shown in cyan, fully clamped 32 10111 3 element is shown in magenta, partially clamped 32 10111 3 element is shown in blue, fully clamped 3253 element is shown in green, and partially clamped 3253 element is shown in red. The deformation of the both the fully parameterized nite ele- ments (in the partially clamped and the fully clamped cases) show a close match. The deformation of the lower order element does not match with the fully parameterized elements for a 30-element beam. . . . . . . . . 145 5.12 Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Deformation of a 5 (black), 10 (yellow), 15 (cyan), 30 (magenta), 60 (blue), 90 (green) and 150 (red) element classical ANCF fully-clamped beam at dierent time instances. The 60, 90 and 150 elements show a close match in the deformation throughout the duration of the simulation. 148 5.13 Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Logarithmic root mean square (LRMS) errors of the 5, 10, 15, 30, 60 and 90 element beams averaged over space and time when compared with the deformation of the 150 element beam. The 90-element beam has the least error whereas the 5-element beam has the highest error. . . . . . . . . 149 5.14 Higher Order 3253 Element - SELECTIVE REDUCED INTEGRATION - Deformation of the 5 (black), 10 (cyan), 15 (magenta), 30 (blue), 60 (green) and 90 (red) element beams at t = 0.5 seconds. The deformation of the 60 and 90 element beams closely match each other. . . . . . . . . 151 5.15 Higher Order 3253 Element - SELECTIVE REDUCED INTEGRATION - LRMS errors for the 5, 10, 15, 30 and 60 element beams in comparison to the 90-element beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 xxii 5.16 Lower Order 32 10111 3 Element - SELECTIVE REDUCED INTEGRA- TION - Deformation of the 5 (black), 10 (yellow), 15 (cyan), 30 (ma- genta), 60 (blue), 90 (green) and 150 element (red) beams at t = 0.5 seconds. Unlike the fully parameterized elements, the 32 10111 3 element requires a larger number of elements to reach convergence. . . . . . . . . 153 5.17 Lower Order 32 10111 3 Element - SELECTIVE REDUCED INTEGRA- TION - LRMS errors for the 5, 10, 15, 30, 60 and 90 element beams in comparison to the 150-element beam. . . . . . . . . . . . . . . . . . . . . 153 5.18 Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Deformation of a 150-element fully-clamped classical ANCF beam at dierent time instances. Figures on the left show the response of the beam in the two-dimensional xz-plane whereas the gures on the right show the response of the beam in three-dimensions. . . . . . . . . . . . 156 5.19 Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - (a) Time history of the kinetic energy, potential energy and gravitational potential energy of the 150 element beam from t = 0 to t = 2 seconds. (b) Time history of the total energy of the beam from t = 0 to t = 2 seconds. Energy is conserved throughout the duration of the simulation. 157 5.20 Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Determinant of the deformation tensorJ = det(F ) plotted as function of time fromt = 0 tot = 2 seconds at the midpoints of the 37 th , 75 th , 112 th and the 150 th element of the 150-element beam. The incompressibility condition is satised for all time t. . . . . . . . . . . . . . . . . . . . . . 159 5.21 Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - The normal component of 2 nd Piola-Kircho stress tensor in the x direc- tion calculated at the free end of the beam from t = 0 tot = 0:5 seconds and t = 0 to t = 2 seconds. . . . . . . . . . . . . . . . . . . . . . . . . . 160 xxiii 5.22 Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Time history of the tip end displacement of a 90-element beam for the fully (red) and partially (blue) clamped boundary conditions. . . . . . . 162 5.23 Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Dierences in the deformation of the fully-clamped and partially clamped 90-element beam at the tip end (left) and midpoint (right). . . . . . . . 162 5.24 SELECTIVE REDUCED INTEGRATION - Time history of deformation of a fully clamped 90-element beam using the classical ANCF model (red color), lower order 32 10111 3 element (green color) and the higher order 3253 element (blue color). Selective reduced integration of these beams show that the deformation of the classical ANCF model and the 3253 model match closely, while there exists some deviation in the deformation of the lower order 32 10111 3 model. . . . . . . . . . . . . . . . . . . . . . 164 5.25 SELECTIVE REDUCED INTEGRATION - Dierences in the deforma- tion at the tip end (left) and midpoint (right) between a 90-element fully-clamped classical ANCF model and a 90-element fully clamped 3253 model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.26 SELECTIVE REDUCED INTEGRATION - Time history of deforma- tion of a fully clamped 90-element classical ANCF model (red color), 150-element lower order 32 10111 3 element (green color) and a 90-element higher order 3253 element (blue color). Selective reduced integration of these beams show an extremely close match in the deformation of the beams irrespective of the interpolation polynomial used to model these beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 xxiv 5.27 SELECTIVE REDUCED INTEGRATION - Dierences in the deforma- tion between a 90-element fully-clamped classical ANCF model and a 150-element fully clamped 32 10111 3 model at the tip end (left) and mid- point (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.28 Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Dierences in the deformation of a 60-element fully clamped beam at the tip end and the midpoint between the largest time step (1:6 10 6 ) and the smallest time step (0:8 10 6 ). . . . . . . . . . . . . . . . . . . . . 167 5.29 Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Dierences in the deformation of a 60-element fully clamped beam at the tip end and the midpoint between the time steps (1:1 10 6 ) and the time step (0:8 10 6 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.30 Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Logarithmic mean square (LRMS) errors in the deformation of a 60- element beam where the deformation results of the 1:6 10 6 and the 1:1 10 6 time steps are compared with the deformation results of the 0:8 10 6 time step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.31 Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Dierences in the deformation of an 8-element fully-clamped beam at the tip end and the midpoint between the RK4 scheme and the Verlet scheme. 169 5.32 Classical ANCF Element SELECTIVE REDUCED INTEGRATION Dierences in the deformation of an 8-element fully-clamped beam at the tip end and the midpoint between the ODE113 scheme and the Ver- let scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 xxv 5.33 Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - LRMS errors in the deformation of an 8-element beam, where the defor- mation obtained from the ODE113 solver and the RK4 solver are com- pared with the deformation obtained from the Verlet solver. . . . . . . 169 5.34 Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - MOONEY RIVLIN - Deformation of a 90-element fully-clamped classical ANCF beam at dierent time instances. The elastic forces have been computed using the incompressible Mooney-Rivlin nonlinear constitutive model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.35 Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - YEOH - Deformation of a 90-element fully-clamped classical ANCF beam at dierent time instances. The elastic forces have been computed using the incompressible Yeoh nonlinear constitutive model. . . . . . . . . . . 172 6.1 Control Example 1 - Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Time history of deformation of a controlled 30-element cantilever beam. Control forces are acting at the right end of the can- tilever beam such that its motion is completely restricted i.e., the right end eectively becomes a fully clamped end. . . . . . . . . . . . . . . . 186 6.2 Control Example 1 - Time history of control forces acting on the last node of the beam at the right end. The control forces are nonzero in the x (denoted by F c x ) and the z (denoted by F c z ) directions as shown in the gure. The control forces in the other ten nodal directions are all approximately close to zero. . . . . . . . . . . . . . . . . . . . . . . . . 187 xxvi 6.3 Control Example 1 - Time history of errors in satisfying the 12 con- straints. The errors are of the order of 10 10 , which are considerably small. " 12ne+4 (t) represents error in satisfying the fourth constraint on the last node. Verlet solver with a time step of 2:6 10 6 is used to integrate the equations of motion. . . . . . . . . . . . . . . . . . . . . . 187 6.4 Control Example 2 - Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Time history of deformation of a controlled 30-element cantilever beam with fully-clamped boundary condition at the right end of the beam and a constraint on the displacement of the node located at two-thirds the distance from the left end of the beam (see 21 st node in the gure). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.5 Control Example 2 - Time history of control forces acting on the 21 st (located at two-thirds the distance from the left end of the beam) and 31 st node (last node) of the beam in the x;y and z directions. Only the control forces acting on the 21 st node and the 31 st node in the x and z directions are non-zero as shown in the gure. F c 31x represents the control force acting on the 31 st node in the x-direction. . . . . . . . . . . . . . 191 6.6 Control Example 2 Time history of errors in tracking the 15 constraints. The errors are of the order of 10 8 , which are considerably small. " 31 i (t) represents the error in satisfying thei th constraint of the 31 st node. Verlet solver with a time step of 2:6 10 6 is used to integrate the equations of motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.7 Control Example 3 - Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Time history of the motion of a controlled 30-element highly exible hyperelastic cantilever beam where control forces are ap- plied at the right end of the beam such that it executes circular motion in the yz-plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 xxvii 6.8 Control Example 3 - Time history of displacement of the tip (right) end (31 st node) of the cantilever beam in the x;y and z directions. . . . . . 195 6.9 Control Example 3 - Time history of control forces acting at the tip (right) end (31 st node) of the cantilever beam in the x;y andz directions. 198 6.10 Control Example 3 - Time history of errors in tracking the three con- straints in the x;y and z directions of the tip end of the beam. The errors are of the order of 10 14 , which are all extremely small. 4 th order Runge-Kutta solver with a time step of 2:5 10 6 is used to integrate the equations of motion. . . . . . . . . . . . . . . . . . . . . . . . . . . 198 6.11 Control Example 4 - Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Time history of motion of a controlled 30-element highly exible hyperelastic cantilever beam where control is being applied to the tip end (right end) of the beam such that it executes circular motion in the yz-plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.12 Control Example 4 - Time history of displacement of the tip (right) end (31 st node) of the cantilever beam in the x;y and z directions. . . . . . 203 6.13 Control Example 4 - Time history of control forces acting at the tip (right) end (31 st node) of the cantilever beam in the x;y andz directions. 204 6.14 Control Example 4 - Time history of errors in tracking the three con- straints in the x;y and z directions at the tip end of the beam. The errors are of the order of 10 14 , which are all extremely small. 4 th order Runge-Kutta solver with a time step of 2:5 10 6 is used to integrate the equations of motion. . . . . . . . . . . . . . . . . . . . . . . . . . . 204 A.1 Examples of Positive Denite Functions. . . . . . . . . . . . . . . . . . . 214 A.2 Examples of Radially Unbounded Functions. . . . . . . . . . . . . . . . 216 A.3 Examples of Radially Increasing and Spherically Increasing Functions. . 217 xxviii List of Tables 3.1 Description of the six nonlinear lattice simulations . . . . . . . . . . . . 40 4.1 Parameter and Initial Conditions Sets for the Five Gyroscopes. . . . . . 81 4.2 Parameter and Initial Conditions Sets for the Six Gyroscopes. . . . . . . 90 4.3 Parameter and Initial Conditions Sets for the Five Gyroscopes. . . . . . 95 5.1 Classical ANCF Element Results . . . . . . . . . . . . . . . . . . . . . . 142 5.2 Lower Order 32 10111 3 Element Results . . . . . . . . . . . . . . . . . . . 142 5.3 Higher Order 3253 Element Results . . . . . . . . . . . . . . . . . . . . . 142 xxix Abstract A new and novel approach to the control of nonlinear mechanical systems is presented in this study. The approach is inspired by recent results in an- alytical dynamics that deal with the theory of constrained motion. The control requirements on the dynamical system are viewed from an analytical dynamics perspective and the theory of constrained motion is used to recast these control require- ments as constraints on the dynamical system. Explicit closed form expressions for the generalized nonlinear control forces are obtained by using the fundamental equation of mechanics. The control so obtained is optimal at each instant of time and causes the constraints to be exactly satised. No linearizations and/or approximations of the nonlinear dynamical system are made, and no a priori structure is imposed on the na- ture of nonlinear controller. Three examples dealing with highly nonlinear complex dynamical systems that are chosen from diverse areas of discrete and continuum me- chanics are presented to demonstrate the control approach. The rst example deals with the energy control of underactuated inhomogeneous nonlinear lattices (or chains), the second example deals with the synchronization of the motion of multiple coupled slave gyros with that of a master gyro, and the nal example deals with the control of incompressible hyperelastic rubber-like thin cantilever beams. Numerical simulations accompanying these examples show the ease, simplicity and the ecacy with which the control methodology can be applied and the accuracy with which the desired control objectives can be met. xxx Part 1 Preliminaries Chapter 1 Introduction R ecent advances in the eld of analytical dynamics that stem from a series of papers written by Udwadia and Kalaba [78, 79, 81, 83, 84, 89] provide a new and novel insight into the way the problem of control of general mechanical systems can be approached from an analytical dynamics perspective. The approach relies on recasting the control requirements on the mechanical system as constraints on its dynamics, thereby transforming the control problem into a constrained motion prob- lem. Although not immediately apparent, there exist many parallels between concepts in control theory and concepts in constrained motion theory. For example, what a con- trol theorist might call `uncontrolled system' or `plant', a classical mechanician would call it `unconstrained system'. Similarly, a `control force' and a `constraint force' are dual concepts and so are `controlled' and `constrained' systems. In solving one of the long-standing, fundamental problems of constrained motion theory i.e., the problem of determining explicit closed form expressions for constraint forces in a constrained sys- tem, Udwadia and Kalaba, in essence provided, by making use of this duality between control theory and constrained motion, a simple and unied methodology to compute the control forces in closed form that need to be applied to the uncontrolled system so that the desired trajectory control requirements are realized. This novel approach has blurred the traditional boundaries between analytical dynamics and control, and 2 provided us with a new methodology to approach the control of general nonlinear me- chanical systems. In this dissertation, we will apply this analytical dynamics based control approach to two highly nonlinear discrete dynamical systems and a highly nonlinear continuous mechanical system, and in each of the these cases, we will achieve certain desired control objectives. The rst discrete dynamical system that we consider is an inhomogeneous nonlinear lattice. An inhomogeneous nonlinear lattice is chain of masses that are con- nected together by linear or nonlinear elastic springs, where the qualitative nature of the nonlinear spring elements along the chain can be, in general, dierent from one another. These lattices are known to display intricate structures in their response such as solitons, breathers, phonons etc. and exhibit complex nonlinear wave interactions. Our objective is to control the energy of such an underactuated n-mass inhomogeneous nonlinear lattice. The second discrete dynamical system that we consider is a symmetric gyroscope with linear plus cubic damping that is subjected to a harmonic vertical base excitation. The motion of such a symmetric gyro is known to range from regular periodic motion all the way to chaotic motion depending on the parameters selected. In this study, we consider a system of n such nonidentical general coupled slave gyroscopes and require that each slave gyroscope in the coupled system synchronizes precisely with the motion of a master gyroscope, irrespective of the chaotic or periodic behavior displayed by either the master or the slave system of gyroscopes. The third dynamical system that we study is chosen from the eld of continuum mechanics. A thin, highly exible, rubber-like, hyperelastic, incompressible cantilever beam that exhibits large deformations and large rotations in the nonlinear elastic range is considered. Our aim is to demonstrate accurate (nonlinear, time-varying) tracking control of such a cantilever beam, thereby illustrating that this analytical dynamics based control approach is not just limited to the control of highly nonlinear discrete dynamical systems, but can be readily extended to tackle problems related to the control of highly nonlinear continuous mechanical and structural systems. 3 1.1 Background In their landmark paper published in 1992, Udwadia and Kalaba [81] obtained explicit, general equations of motion for constrained discrete dynamical systems. They provided a simple, explicit, closed form expression for the constraint force that needs to be ex- erted on the unconstrained system, such that the resulting constrained system exactly satises the constraints at each instant of time [81]. The novelty of the formulation is that it treats holonomic and nonholonomic constraints with equal ease, including non- holonomic constraints that are nonlinear in velocities, which most classical formulations nd it dicult to handle [76]. The formulation also allows constraints to be functionally dependent on each other [84]. The formulation is based on Gauss's principle of least constraint and requires no use of Lagrange multipliers [82]. The equations of motion of the constrained system are obtained in the same coordinates as those used to describe the unconstrained system and there is no attempt made to eliminate coordinates as is done in Lagrangian mechanics [77]. The formulation assumes d'Alembert's principle of virtual work to be valid [83]. Later on, the formulation was extended to include systems that violate d'Alembert's principle [32, 75, 86]. The close connections between control theory and analytical dynamics were pointed out by Udwadia [78, 79, 89]. The analytical dynamics based control approach was used for holonomic and nonholonomic trajectory tracking of nonlinear mechanical systems in Ref. [78]. The technique of trajectory stabilization was introduced to handle initial conditions that did not lie on the (constraint) manifold described by the trajectory requirements [78]. Utilizing Gauss's principle of least constraint, insights into Nature's approach as a control engineer were provided [79]. It was shown by Udwadia [89] that the error signal that Nature appears to be using for its feedback control law is related to accelerations, and not to displacements, or velocities, or integrals of the displacement as is commonly done in control theory. It was also illustrated that Nature appears to be solving an optimal control problem at each instant of time, where the control cost it is minimizing is given by the weightedL 2 norm of the control force at each instant of time, where the weighting that it uses is given by the inverse of the mass matrix [79]. Most 4 control engineers, on the other hand, minimize the time integrals of such norms. Explicit closed form expressions for the control force were derived in [79], when one considers more general weighting matrices that are dierent from the inverse of the mass matrix. The interested reader is referred to the following excellent set of references [78, 79, 89] for more on the interrelations and connections between constrained motion of mechanical systems and tracking control of nonlinear mechanical systems. The formulation has been applied to the following problems in the eld of discrete mechanics: rigid body rotational dynamics and control [92, 93], satellite formation keeping problem [13], tumbling control of dumbbell spacecraft systems [62], control of uncertain mechanical systems [97], robot control with redundant degrees of freedom [57], and decentralized control of nonlinear systems [88]. Part of novelty of the present dissertation is the fact that we extend this analytical dynamics based control approach to the eld of continuum mechanics. 1.2 Outline of this Study This dissertation is broadly divided into three parts: 1. Energy Control of Inhomogeneous Nonlinear Lattices: In the rst part of this dissertation, we consider the problem of energy control of an underactuated n- degrees-of-freedom general nonlinear lattice with xed-xed and xed-free boundary conditions. The inhomogeneous lattice consists of dissimilar masses wherein each mass is connected to its nearest neighbour by a nonlinear or linear memoryless spring element. The potential functions of the nonlinear spring elements along the lattice are assumed to be qualitatively dierent. By `qualitatively dierent', we mean that each of the springs along the lattice can have dierent restoring-force characteristics (for example, linear, cubic, quintic, exponential, etc.). Each potential is assumed to be described by a twice continuously dierentiable, strictly convex function, possessing a global minimum at zero displacement, with zero curvature possibly only at zero displacement. The energy control requirement is viewed from an analytical dynamics perspective and is recast as a constraint on the motion of the dynamical system. Given the set of masses at which 5 control is to be applied, explicit closed form expressions for the nonlinear control forces are obtained. Stability analysis of the control so obtained is performed through the use of LaSalle's invariance principle [36]. The principle is used to show global asymp- totic convergence to the desired energy state and to simultaneously obtain sucient conditions on the placement of actuators so that such convergence is guaranteed for the underactuated system. Specically, the control force obtained is shown to provide global asymptotic convergence from any given (nonzero) initial energy state, H o , to any desired (nonzero) nal energy state, H , provided that either (i) the rst mass of the lattice, m 1 , or (ii) the last mass, m n , or alternatively, (iii) any two consecutive masses of the lattice, are included in the set of masses that are actuated. Numerical simula- tions demonstrating energy control of a 101-mass Toda lattice, an FPU- lattice and a general nonlinear lattice illustrate the simplicity and ecacy of the control approach. 2. Synchronization of Multiple Coupled Slave Gyros: In the second part of this dissertation, we consider the problem of synchronization of multiple coupled slave gy- roscopes with that of a master gyroscope. A set of n gyroscopes are coupled together to form a system of `slave' gyroscopes. A simple approach is developed for synchro- nizing the motion of these slave gyroscopes, whose individual motion may be regular or chaotic, with the motion of an independent `master' gyroscope irrespective of the chaotic or regular motion exhibited by the master. The problem of synchronization of these multiple gyroscopes is approached from a constrained motion perspective through the application of the fundamental equation. The approach yields explicit, closed form expressions for the control torques that are required to be applied to each of the coupled slave gyroscopes and achieves `exact' synchronization with the master's motion. The in uence of dierent types of interactions between the slave gyroscopes is investigated through the use of an incidence matrix which describes the coupling between any two of them. The eect of the so-called sleeping condition on the synchronization of the gyroscopes is also explored. To illustrate the ecacy of the methods presented in this 6 paper, we consider three numerical examples involving systems of multiple coupled slave gyroscopes and synchronize them with the motion of a master gyroscope. 3. Dynamics and Control of Incompressible Hyperelastic Beams: The nal part of this dissertation deals with the dynamic analysis and control of three-dimensional rubber-like incompressible hyperelastic beams. The equations of motion of the hyper- elastic beam are derived using the absolute nodal coordinate formulation [64, 102], which is a technique based on the nite element method that accurately describes large- deformation and large-rotation nonlinear motion in structures. Nonlinear constitutive material models based on isotropic hyperelastic theory are utilized to model the elastic deformation of the beam elements and the general continuum mechanics approach is used to compute the elastic forces [48, 54, 61]. Incompressibility of the material is en- forced by adding a volumetric penalty energy function to the strain energy density [9, 43]. Volumetric locking caused by the addition of this function is eliminated through the use of selective reduced integration [54]. Numerical simulations of the dynamics of a three- dimensional, thin cantilever beam of a near-incompressible Neo-Hookean material are performed. Time integration is carried out using the St ormer{Verlet scheme [99]. The eect of partially-clamped and fully-clamped boundary conditions is studied, and results related to convergence as a function of the number of beams elements is illustrated. The performance of the classical ANCF element is also compared and contrasted against a lower order and a higher order element to illustrate the eect of the order of the inter- polation polynomial in capturing the large bending deformations of the beam. Control of such hyperelastic cantilever beams is approached from an analytical dy- namics perspective and the theory of constrained motion is used to recast the control objectives as constraints on the nodes of the beam. The fundamental equation of me- chanics is employed to obtain the explicit generalized nonlinear control forces in closed form that are applied at the nodes of the beam to achieve the desired control objective. No linearizations and/or approximations of the mechanical system are made, and no a priori structure is imposed on the nature of the controller. Four numerical simulations demonstrating accurate tracking control of the tip-end of a thirty-element cantilever 7 beam are presented to show the ecacy of the control methodology in achieving the desired control objectives. 8 Chapter 2 Constrained Motion Approach T he Udwadia-Kalaba equation (also referred to as fundamental equation of mechanics [83]) is used to derive the equations of motion of the constrained (controlled) mechanical system and to obtain the explicit nonlinear con- straint (control) forces that are required to be applied to the uncontrolled system or `plant' to achieve the desired control objectives. The fundamental equation is the sim- plest and most comprehensive equation discovered till date [81] for writing down the equations of motion of a constrained mechanical system. The formulation is known for the relative ease with which the constrained equations of motion of a complex mechan- ical system can be derived in comparison to other classical methods [81, 82, 83]. 2.1 Unconstrained System Consider an unconstrained, discrete dynamic system of n particles [81]. An uncon- strained system is one where the virtual displacements of the system are all independent of one another. The equations of motion of this unconstrained (uncontrolled) system at a certain instant of time t can be written down using Newton's laws or Lagrange's method as M(q;t) q =F (q; _ q;t); q(0) =q o ; _ q(0) = _ q o ; (2.1) 9 where M is the n-by-n symmetric, positive denite mass matrix, q is the n-vector of generalized coordinates of the system, andF is then-vector of generalized `given' forces acting on the unconstrained system. The acceleration a of the unconstrained system is given by a(q; _ q;t) = M(q;t) 1 F (q; _ q;t): (2.2) 2.2 Constraints Consider now that we impose a set ofm constraints on the unconstrained system, all of which may or may not be independent i.e. some of the constraints may be a combination of others [82] i (q;t) = 0; i = 1; 2; ::: ; h; i (q; _ q;t) = 0; i = h + 1; h + 2; ::: ; m; (2.3) The initial conditions stated in equation (2.1) are assumed to satisfy these constraint equations. However, in some cases, it may not be possible to initialize the unconstrained system from points in the phase space where the constraints are satised. Thus, instead of considering the existing set of m constraints described by equation (2.3), we modify the constraint equations as follows [78] i (q; _ q; q;t) = i +c _ i +k i = 0; i = 1; 2; ::: ; h; i (q; _ q; q;t) = _ i + i = 0; i = h + 1; h + 2; ::: ; m; (2.4) where c; k and > 0 are chosen such that the system of equations (2.4) has an equilibrium point described by equation (2.3) and that this equilibrium point is stable. This set ofm modied constraints can now be expressed in the general constraint matrix form as A(q; _ q;t) q =b(q; _ q;t); (2.5) where A is an m-by-n constraint matrix of rank r (i.e., r out of the m constraint equations are independent) while b is a column vector with m entries. 10 2.3 Constrained System and the Fundamental Equation The presence of constraints causes the acceleration of the constrained system to deviate from its unconstrained acceleration at every instant of time t. This deviation in the acceleration of the constrained (controlled) system is brought about by a force, F C , called the constraint (control) force, which is exerted on the system by virtue of the fact that the unconstrained system must now further satisfy an additional set of constraints. The equation of motion of the constrained system can now be written down as M(q;t) q =F (q; _ q;t) +F C (q; _ q;t); (2.6) whereF C is the set of additional forces that arise by virtue of the application of the m constraints. One can also envision F C to be the set of control forces that are required to be applied to the uncontrolled open loop system to obtain the controlled closed loop system. Udwadia and Kalaba [83] proposed the following closed form expression for the constraint force (or the control force) F C (q; _ q;t) = M 1=2 (AM 1=2 ) + (bAa); = A T AM 1 A T + (bAa) (2.7) where (AM 1=2 ) + denotes the Moore-Penrose inverse of the matrix (AM 1=2 ). Equa- tion (2.6) along with (2.7) is referred to as the `fundamental equation of mechanics'. Equation (2.7) provides us with the optimal set of control forces such that the constraints are exactly satised at every instant of timet [79]. The control forces are optimal in the sense that they minimize the control cost given by J(t) = [F C ] T M 1 [F C ] at each in- stant of time. Once the matricesM;F;A; andb are obtained from the description of the unconstrained system and the constraint equations, the constraint (control) force can be readily calculated using equation (2.7). This often greatly reduces the conceptualization eort when compared with other classical methods. 11 2.4 Applications The generality of the formulation makes it applicable in many diverse areas of mechanics. The formulation has been previously applied to the following problems in the eld of discrete mechanics: satellite formation keeping problem [13], tumbling control of dumbbell spacecraft systems [62], control of uncertain mechanical systems [97], rigid body rotational dynamics and control [92, 93], robot control with redundant degrees of freedom [57], and decentralized control of nonlinear systems [88]. The papers in Refs. [13, 57, 62, 88, 92, 93, 97] primarily deal with the use of the fundamental equation of mechanics applied to the full-state control of discrete nonlin- ear systems that have a relatively small number of degrees of freedom. In this study, we use the fundamental equation of mechanics to control mechanical systems that can potentially have a large number of degrees of freedom. In Chapter 3, we use Lyapunov stability theory in conjunction with the fundamental equation of mechanics to inves- tigate highly underactuated, global asymptotic energy control of autonomous systems that can have a very large number of degrees of freedom. Besides considering discrete dynamical systems, we also extend these concepts to control both chaotic and continu- ous mechanical systems. In Chapter 4, we use the fundamental equation of mechanics to investigate synchronization of highly nonlinear nonautonomous chaotic dynamical systems. And in Chapters 5 and 6, we extend these control strategies to the eld of continuum mechanics to control a general continuous structural system that possesses both material and geometric nonlinearities. 12 Part 2 General Nonlinear Lattices Chapter 3 Energy Control of Inhomogeneous Nonlinear Lattices 3.1 Introduction T his chapter deals with the energy control of an inhomogeneous nonlinear one-dimensional lattice. The inhomogeneous nonlinear lattice is made up of a chain of masses connected together by linear or nonlinear elastic springs, where the qualitative nature of the nonlinear spring elements along the chain can be, in general, dierent from one another. The study of energy distribution in nonlinear lat- tices consisting of identical masses and identical spring elements, called as homogeneous lattices, was initiated by Fermi, Pasta, and Ulam (FPU) in 1955 [18]. Contrary to their expectations, the energy in the various modes did not reach a state of equipartition. Instead, the long-term dynamics of the FPU lattice appeared to be periodic with the energy remaining trapped in the small number of modes with which it was initialized. Although, many of the puzzling aspects of the FPU phenomenon are well understood now [7, 60], the fundamental research that ensued following the seminal work of FPU 14 has opened up many new interesting questions which are still being actively pursued to date [5, 19]. The force-displacement curves of the springs studied by FPU included quadratic, cu- bic and quartic spring potentials. Later, other spring potentials were considered such as the potentials of the Toda lattice [72], the so-called 4 lattice [58], and the Klein-Gordon chains [40]. These homogeneous FPU-like lattices are known to display intricate struc- tures in their response such as solitons, phonons, breathers, and nanopterons. Nearly all of the research done in this eld to date has focused on homogeneous lattices. Studies on inhomogeneous lattices in the literature most often either deal with weakly inhomo- geneous lattices [46] or are limited to two degrees-of-freedom systems [41]. However, when dealing with engineering applications, disparities in material stiness in spatially extended mechanical systems is rather the norm than the exception. Hence, it is im- portant to study inhomogeneous lattices from an engineering standpoint. The literature on nonlinear lattices with dissimilar masses wherein the nonlinearity of each of the spring elements in the lattice is qualitatively dierent is, to the best of the author's knowledge, nonexistent. Nevertheless, these type of nonlinear lattices are representative of real life behavior and arise frequently in engineering models. Besides their applications to engineering situations, the theoretical understanding and control of such nonlinear lattices is of fundamental importance. In this study, an n-degrees-of-freedom general nonlinear lattice with xed-xed and xed-free boundary conditions is considered, in which 1. the masses in the lattice can all be chosen to be dissimilar, 2. dierent nonlinear and/or linear memoryless spring elements can be chosen, wherein (a) the qualitative nature of each of the nonlinear spring elements along the lattice can be dierent, and (b) the parameters of the potential functions describing each spring element can also be dierent. 15 The spring force associated with each spring element in the lattice is assumed to be derivable from a potential function which (i) is C 2 , (ii) is strictly convex possessing a global minimum at zero displacement, and (iii), has zero curvature possibly only at zero displacement. Such potential functions lead to a variety of spring forces that include, but are not limited to, the linear spring force, the cubic spring force, the quintic spring force and the Toda spring force. Asymmetries in the potential function would naturally lead to disparities between the magnitudes of the tensile and compressive forces exerted by the springs, as is often the case with many real-life elastic materials. For example, exible cables in suspension bridges are strong under tensile forces but are weak under compressive forces. Thus, one can potentially model these and many such structural subsystems using nonlinear lattices with asymmetrical potential functions. The main focus of the present study is to control the energy of these nonlinear lat- tices and bring them to a desired energy state. The control approach adopted herein distinguishes itself from related work on the subject in ve key ways. First, control of FPU-like lattices is restricted to homogeneous lattices in the literature [55, 59]. In this study, general inhomogeneous lattices are considered. Second, the energy control problem is approached using the theory of constrained motion, wherein the key idea is to recast the energy control requirement on the nonlinear lattice as an energy constraint on the system. The fundamental equation of mechanics [83] is employed to determine explicit closed form expressions for the nonlinear control forces. This approach is in- spired by recent results in analytical dynamics [83]. Third, the methodology developed herein allows us to explicitly determine control forces needed to be applied to any ar- bitrarily chosen subset of masses that are designated to be actuators, and still obtain global asymptotic convergence to any desired nonzero energy state provided that the rst mass, or the last mass, or alternatively, any two consecutive masses of the lattice are included in this subset. Fourth, once the energy of the system is brought to its desired value, the control forces automatically terminate, and the conservative nature of the ensuing Hamiltonian dynamics is utilized to maintain it at the desired energy level for all future time. Finally, inspite of the general nature of the nonlinear lattice considered in this study, the control is obtained in closed form with relative ease without the need 16 to make any approximations or linearizations of the nonlinear dynamical system and without the need to impose any a priori structure on the nature of the controller. When the spring potentials are all exponential in nature, the nonlinear lattice reduces to an inhomogeneous Toda lattice. The Toda lattice [72, 91] was discovered in the 1960s by Morikazu Toda, a Japanese physicist, who was inspired by the FPU paradox. The Toda lattice is an example of a completely integrable Hamiltonian system. But perhaps the most interesting aspect of Toda lattices is the fact that they admit multiple soliton solutions [72]. Besides their obvious importance in theoretical physics, Toda lattices also nd many practical engineering applications primarily due to the disparate nature of the tensile and compressive forces of its elastic spring elements, which arises from an asymmetry in its potential. Throughout this chapter, in addition to providing results on general nonlinear lattices, we will also be drawing parallels of these results to the Toda lattices. The Toda lattice will be used as a tool to elucidate some of the more complicated results that we provide on general nonlinear lattices. This chapter is organized as follows. An introduction to the physics of the nonlinear lattice and the Toda lattice is presented in Section 3.2. The equations of motion of ann-degrees-of-freedom inhomogeneous nonlinear lattice with xed-xed and xed-free boundary conditions are derived in Section 3.3 and a similar derivation is performed for the Toda lattice in Section 3.3.2. In Section 3.4, the energy control problem is formu- lated and closed form expressions for the control forces are derived. In Section 3.5, the invariance principle [36] is used to derive sucient conditions for the placement of the actuators so that the control force obtained in Section 3.4 gives us global asymptotic convergence to any given nonzero desired energy state. The approach is rst illustrated for a single degree-of-freedom Toda oscillator in Section 3.5.1, which is then extended to ann-mass general nonlinear lattice in Section 3.5.2. And nally, in Section 3.6, numeri- cal simulations involving a 101-mass inhomogeneous Toda lattice, an FPU- lattice and a general nonlinear lattice are presented to illustrate the ecacy and simplicity with which the control approach can be eected. Several of the technical details have been placed in the appendices to maintain the ow of thought. 17 3.2 Physics of the Lattice 3.2.1 General Nonlinear Lattice Consider a lattice with n + 1 masses wherein each of the masses is connected to its neighboring mass with the help of a nonlinear memoryless spring element as shown in Figure 3.1. Figure 3.1: An inhomogeneous nonlinear lattice The nonlinear potential of the i th spring element is denoted by u i (x) wherex is the displacement of thei th spring element from the equilibrium position. The nature of the nonlinearity in each of the spring elements can, in general, be dierent provided that each of the spring potentials u i (x);i = 0; 1;:::;n; satises the following three properties (see Appendix A). (1) u i (x) is a C 2 function. (2) u i (x) is strictly convex with a global minimum at x = 0. (3) u 00 i (x) = 0 only at x = 0. By strictly convex, we mean u i (x + (1)y) < u i (x) + (1)u i (y)8 2 (0; 1) and x6=y. Without any loss of generality, let us add a suitable constant to u i (x) such that u i (0) = 0. This along with Property (2) implies that the potentials of the spring elements are strictly positive denite (i.e. u i (0) = 0;u i (x)> 08 x6= 0). Properties (2) and (3) also imply that the potentials are strictly radially increasing (i.e. > 1) u i (x) > u i (x)8 x2<f0g). In one dimension, if a function is strictly radially increasing, then it is also radially unbounded and therefore, in our case, each u i (x) is also radially unbounded (i.e. u i (x)!1 askxk!1). The reader is referred to Appendix A for details regarding these results. 18 The spring force f i (x) of the i th spring element is also nonlinear in general, and is assumed to be derivable from a potential as F spring (x) =F restoring (x) =f i (x) = @u i (x) @x ; where f i (0) = 0, and xf i (x) > 08 x6= 0. Furthermore, it follows from Property (2) that u 00 i (x) = f 0 i (x) 0, with u 00 i (x) = f 0 i (x) = 0 possibly only at x = 0 according to Property (3). Additionally, if the spring potentials are asymmetrical, then the springs exhibit dissimilar tensile and compressive characteristics. 3.2.2 Inhomogeneous Toda Lattice The Toda lattice [72] is a simple model for a nonlinear one-dimensional crystal that describes the motion of a chain of particles with exponential interactions between the nearest neighboring elements (see Figure 3.2). Figure 3.2: An inhomogeneous Toda lattice Consider a single degree-of-freedom (SDOF) spring-mass system with Toda spring stiness that is obtained by setting m i = 08 i = 2; 3;:::;n + 1 (see Figure 3.2). The expression for the nonlinear potential (Figure 2) of the Toda spring is given by the smooth and continuously dierentiable function u o (q 1 ) = a o b o e boq 1 a o q 1 a o b o ; a o > 0;b o > 0 (3.1) where the displacement q 1 is measured from the equilibrium position. A constant (a o =b o ) has been included in the expression of the potential to ensure that u o (0) = 0. The potential has a single stationary point atq 1 = 0 and sinceu 00 o (0) =a o b o > 0,q 1 = 0 is a global minimum of equation (3.1). Further, sinceu 0 o =a o a o e boq 1 1 > 08q 1 > 0 19 Figure 3.3: Exponential Toda potential a o , b o > 0 Figure 3.4: Toda spring force 20 and u 0 o < 08 q 1 < 0, u o is strictly increasing in the interval 0<q 1 <1 and is strictly decreasing in the interval1<q 1 < 0 (see Appendix A). Consequently, the potential function is strictly radially increasing (see Figure 3.3) and hence radially unbounded [16]. It is also strictly positive denite withu o (0) = 0 andu o (q 1 )> 08q 1 6= 0. Further, since u 00 o (q 1 ) = a o b o e boq 1 > 08 q 1 , the potential of the Toda spring (equation 3.1) is a strictly convex function. The reader is once again referred to Appendix A for details regarding these results. The exponential Toda spring force F s (q) for this SDOF spring-mass system is given by F s (q 1 ) =F restoring (q 1 ) = @u o (q 1 ) @q 1 =a o e boq 1 1 (3.2) For suciently small q 1 the spring force is approximately linear. However, the nonlin- earity of the force gains prominence as q 1 increases. Also, a larger force is required to stretch the spring by a unit distance than is required to compress it (see Figure 3.4). Hence, the Toda lattice considered in this study possesses spring elements that are stronger in tension than in compression. Such systems arise frequently in structural sub-systems such as the stringers in suspension bridges. In the present study, although we focus our attention on a Toda spring that is strong in tension and weak under compression (a > 0; b > 0), the theory developed herein is also, in general, applica- ble to a Toda spring that is weak in tension and strong under compression (a< 0; b< 0). 3.3 Unconstrained Equations of Motion 3.3.1 General Nonlinear Lattice Let m i denote the mass of the i th particle in the lattice, where i = 1; 2;:::;n + 1. The coordinate describing the motion of mass m i measured from its equilibrium position in 21 an inertial frame of reference is denoted by q i (see Figure 3.1). The velocity of massm i is denoted by _ q i . The total energy H of the lattice is given by H(q; _ q) =T ( _ q) +U(q) = n+1 X i=1 1 2 m i _ q 2 i + n X i=0 u i (q i+1 q i ); (3.3) where q and _ q denote the vector of displacements and velocities of the lattice, respec- tively. Further, q o 0 because the left end of the lattice is xed for all time t (see Figure 3.1). The energy H is a smooth and continuously dierentiable function. It is also strictly positive denite and radially unbounded (see Appendix E). Using Newton's laws of motion, the equations of motion of the nonlinear lattice can be written down as m i q i =f i (q i+1 q i )f i1 (q i q i1 ); i = 1; 2;:::;n; (3.4) whereq n+1 0 for a xed-xed lattice, andf n 0 for a xed-free lattice (because there is no spring between the masses m n and m n+1 ). When this set of n equations (3.4) is re-written in matrix form, the equation of motion of an n-degrees-of-freedom (NDOF) lattice with xed-xed ends is given byM q =F (q), which when written explicitly yields 2 6 6 6 6 6 6 6 6 6 6 4 m 1 0 ::: ::: 0 0 . . . . . . . . . . . . . . . m i . . . . . . . . . . . . . . . 0 0 ::: ::: 0 m n 3 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 4 q 1 . . . q i . . . q n 3 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 4 f 1 (q 2 q 1 )f o (q 1 ) . . . f i (q i+1 q i )f i1 (q i q i1 ) . . . f n (q n )f n1 (q n q n1 ) 3 7 7 7 7 7 7 7 7 7 7 5 ; (3.5) where the mass matrix M is an n-by-n diagonal matrix and the column vector of generalized `given' forcesF is given by the right hand side of equation (3.5). Further, by setting f n 0 in equation (3.5), we obtain the matrix representation for the equation of motion of an NDOF lattice with xed-free ends. Moreover, we assume that there is no damping in the system since the inclusion of damping adds complexities that go beyond the current scope of this study. 22 3.3.2 Inhomogeneous Toda Lattice Consider an n-degrees-of-freedom (NDOF) nonhomogeneous Toda lattice as shown in Figure 3.2 in which the mass at the i th location is denoted by m i . The kinetic energy of this lattice can be written as T ( _ q) = n+1 X i=1 1 2 m i _ q 2 i ; (3.6) where _ q i denotes the velocity of the i th mass in the lattice. The potential energy of the lattice is composed of exponential interactions between the nearest neighboring elements and is dened by U(q) = n X i=0 u i (q i+1 q i ) = n X i=0 a i b i e b i (q i+1 q i ) a i (q i+1 q i ) a i b i (3.7) where a i ;b i > 0 denote the spring constants of the i th spring element in the lattice and q o 0 because the left end of the lattice is always xed (see Figure 3.2). The total energy H(q; _ q) of the Toda lattice is a smooth and continuously dierentiable function and is given by H(q; _ q) = T ( _ q) +U(q) = n+1 X i=1 1 2 m i _ q 2 i + n X i=0 a i b i e b i (q i+1 q i ) a i (q i+1 q i ) a i b i (3.8) The energy H is a positive denite function with H(0; 0) = 0 and H(q; _ q)> 08 q; _ q6= 0 (see Appendix B). The equations of motion of the n-degrees-of-freedom (NDOF) inhomogeneous Toda lattice can be derived using Newton's laws of motion and are given by m i q i = a i h e b i (q i+1 q i ) 1 i a i1 h e b i1 (q i q i1 ) 1 i ; i = 1; 2;:::;n + 1; (3.9) where q n+1 0 for an n-mass lattice with xed-xed boundary conditions and a n = b n = 0 for an n-mass lattice with xed-free boundary conditions as there is no spring 23 between the masses m n and m n+1 (see Figure 3.2). When equation (3.9) is expressed in matrix form, the equation of motion of an inhomogeneous NDOF Toda lattice with xed-xed boundary conditions is given by M q =F (q), which when written explicitly yields 2 6 6 6 6 6 6 6 6 6 6 4 m 1 0 ::: ::: 0 0 . . . . . . . . . . . . . . . m i . . . . . . . . . . . . . . . 0 0 ::: ::: 0 m n 3 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 4 q 1 . . . q i . . . q n 3 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 4 a 1 e b 1 (q 2 q 1 ) 1 a o e bo(q 1 ) 1 . . . a i e b i (q i+1 q i ) 1 a i1 e b i1 (q i q i1 ) 1 . . . a n e bn(qn) 1 a n1 e b n1 (qn q n1 ) 1 3 7 7 7 7 7 7 7 7 7 7 5 ; (3.10) where the n-by-n mass matrix M is diagonal and the column vector of generalized `given' forces F is given by the right hand side of equation (3.10). Furthermore, by setting a n = b n = 0 in equation (3.10), we obtain the matrix representation for the equation of motion of an inhomogeneous NDOF Toda lattice with xed-free boundary conditions. 3.4 Problem Formulation and Constraint Equations Consider an NDOF nonlinear lattice with appropriate boundary conditions (see equation 3.5). The energy control problem for this unconstrained system is formulated as follows. Given a set of k masses selected from amongst n masses of the n-mass lattice, nd the explicit control forces applicable to this set of k masses such that the total energy of the nonlinear lattice approaches a `given' positive value H as t!1. H(q(t); _ q(t))!H as t!1; H > 0 (3.11) Although we assume at this stage that the locations of these k actuators (where 1 kn) can be arbitrarily selected from amongst then masses in the lattice, we will later show that in order to have global asymptotic convergence, the set of actuator locations 24 need to satisfy certain conditions (see Section 3.5) when the system is underactuated (i.e., when k<n). In this study, the energy control problem (3.11) is approached from a constrained motion perspective which comprises of three vital steps as discussed in Chapter 2. The rst step involves the derivation of the equations of motion of the unconstrained system. For an inhomogeneous NDOF nonlinear lattice, this has already been discussed in Section 3.3. The second step involves the formulation of the constraint equations (see Section 3.4.1) and the last step deals with the use of the fundamental equation of mechanics to obtain the constrained equations of motion of the nonlinear lattice (see Section 3.4.2). In the process, we also nd a closed form expression for the explicit nonlinear control force that is required to be applied to the uncontrolled system to stabilize the energy of the nonlinear lattice at the desired level. 3.4.1 Formulation of the Constraints Consider an unconstrained NDOF nonlinear lattice as described in Section 3.3. The unconstrained acceleration a(q) of this lattice can be computed using equations (2.2) and (3.5). Suppose now that out of these n masses, we apply control inputs to k arbitrarily selected masses, where 1kn. The locations of these k masses where a control input is applied is denoted by the ordered set S C =fi 1 ;i 2 ;i 3 ;:::;i k g where, with no loss of generality, we order these locations along the lattice such that i 1 <i 2 <<i k . Similarly, the set of (nk) masses at which no control is applied is given by the complement of the set S C which we denote by S N =S c C =f1; 2; 3;:::;ngnfi 1 ;i 2 ;:::;i k g =fj 1 ;j 2 ;:::;j nk g where again j 1 < j 2 < < j nk . It is also convenient to represent this information in terms of matrices. The following matrices are dened to simplify the notation. A k-by-n `control selection matrix', C, is dened such that every element of its g th row 25 (1gk) is zero except for thei th g element (wherei g 2S C ), which is unity. Similarly, we dene an (nk)-by-n `no-control selection matrix', N, such that every element of itsh th row (1hnk) is zero except for thej th h element (wherej h 2S N ), which is unity. The mass matrices associated with the set of controlled and uncontrolled masses are represented by M C = diag(m i 1 ;m i 2 ;:::;m i k ) and M N = diag(m j 1 ;m j 2 ;:::;m j nk ); respectively, and the corresponding displacements are represented by the column vectors q C = [ q i 1 q i 2 ::: q i k ] T and q N = q j 1 q j 2 ::: q j nk T , respectively. While dealing with the energy control problem (3.11), we interpret the energy requirement as an energy constraint on the unconstrained NDOF nonlinear lattice. 1. Constraint of `Energy Stabilization'. Using equation (3.3), the energy stabi- lization constraint is given by (q; _ q) =H(q; _ q)H = 1 2 _ q T M _ q +U(q) H = 0 (3.12) where H(q; _ q) is re-written in matrix-vector notation. The constraint (3.12) resembles equation (2.3) and therefore needs to be dierentiated once with respect to time so that it can be expressed in the general form of equation (2.5). Further, we modify the constraint by introducing > 0 as in equation (2.4) so that the nonlinear lattice can be initiated from any arbitrary nonzero initial energy state. The modied energy stabilization constraint is now given by (q; _ q; q) = d dt () + = d dt 1 2 _ q T M _ q +U(q) H + (HH ) = 0 = 1 2 _ q T M +M T d _ q dt + dq dt T @U @q + (HH ) = 0 = _ q T M q _ q T F + (HH ) = 0 (3.13) 2. Constraint of `No Control'. In addition to the energy stabilization constraint, a constraint of `no control' is imposed on all the masses that belong to the setS N that are left unactuated. Since no control is applied to these masses, the prevailing unconstrained 26 motion of these masses (3.4) can themselves be considered as constraints. Thus, this set of (nk) `no control' constraints can be described in matrix form as N(M qF ) = 0 (3.14) When the constraints described by equations (3.13) and (3.14) are expressed in the general constraint matrix form (see equation 2.5), this leads to an (nk + 1)-by-n constraint matrix A and an (nk + 1) sized column vector b given by A = 2 4 _ q T M NM 3 5 = 2 4 _ q T N 3 5 M ; b = 2 4 _ q T F (HH ) NF 3 5 (3.15) 3.4.2 Controlled NDOF Nonlinear Lattice Once the matrices M;F;a;A and b are known for the nonlinear lattice, the explicit nonlinear control force F C can be computed in closed form (see Appendix C for a detailed derivation) as F C (q; _ q) = (H(q; _ q)H ) _ q T C M C _ q C C T CM _ q (3.16) where _ q T C M C _ q C = P k g=1 m ig _ q 2 ig . The control force possesses a singularity when the velocities of the set of masses that are controlled are all simultaneously zero. To avoid this, we choose as (q; _ q) = _ q T C M C _ q C (q; _ q); where (q; _ q)> 0 (3.17) Moreover, for simplicity, we choose (q; _ q) = o , where o is a positive constant that can be suitably altered to control the rate at which the system converges to the desired energy state H . The explicit control force is now given by F C = o (H(q; _ q)H )C T CM _ q =g(q; _ q)C T CM _ q (3.18) 27 Though it might appear that the control force, which depends linearly on the momentum of the controlled masses, resembles a velocity feedback type of control, the nonlinear gain g(q; _ q) changes the nature of the feedback. The equation of motion of the constrained (controlled) NDOF nonlinear lattice (with appropriate boundary conditions) can be written down using equation (2.6) as M q =F +F C =F o (H(q; _ q)H )C T CM _ q (3.19) or alternatively as M q + o (H(q; _ q)H )C T CM _ qF (q) = 0 (3.20) where the `given' force F is obtained from the unconstrained system (3.5) and the constraint force F C is computed using (3.18). Equation (3.20) resembles the familiar form of a self-excited oscillator with nonlinear damping, akin to a Van der Pol-type system. WhenH <H , the damping is negative and the energy of the system is raised. Conversely, when H > H , the damping is positive and the energy of the system is lowered. When H =H is attained, the control force terminates and the conservative nature of the lattice is utilized to remain at H for all future time. 3.5 Global Asymptotic Convergence In this section, our aim is to prove that the control force F c gives us global asymp- totic convergence to any given desired energy state H in< 2n fOg provided that the rst mass, or the last mass, or alternatively, any two consecutive masses of the NDOF lattice are included in the subset of masses that are controlled. To acquire in- sight into the nature of the control, we rst consider a single degree-of freedom (SDOF) spring-mass oscillator with Toda spring stiness and later generalize these results to an inhomogeneous NDOF general nonlinear lattice with xed-xed (and xed-free) bound- ary conditions. To maintain the ow of thought, many of the details of the proofs related to the NDOF system are derived in the Appendices. 28 3.5.1 SDOF Toda Oscillator Consider a SDOF spring-mass system with Toda spring stiness (see Figure 3.5) as discussed in Section 3.2.2. Figure 3.5: A single degree-of-freedom Toda oscillator The uncontrolled (unconstrained) equation of motion of this spring-mass system with unit mass is given by q 1 + a o h e boq 1 1 i = 0; q 1 (0) =q (0) 1 ; _ q 1 (0) = _ q (0) 1 (3.21) where the initial displacement and the initial velocity of the system are specied by q (0) 1 and _ q (0) 1 , respectively. Let us represent this SDOF system (equation 3.21) in an equivalent state-space form: _ X 1 = X 2 ; and _ X 2 = a o h e boX 1 1 i ; (3.22) where X 1 = q 1 and X 2 = _ q 1 . The SDOF system (equation 3.22) has a single isolated equilibrium point at the origin O, which is a center [70]. Hence, the phase space of uncontrolled system (3.22) is composed of concentric closed orbits around the origin with each closed orbit denoting a constant energy level as shown in Figure 3.6(a). 29 (a) Uncontrolled SDOF Oscillator (b) Controlled SDOF Oscillator Figure 3.6: (a) Phase portrait of the uncontrolled SDOF oscillator with Toda spring stiness (a o = 2; b o = 1), (b) Phase portrait of the controlled SDOF oscillator with Toda spring stiness (a o = 2; b o = 1; o = 1; H = 8). 30 Consider now that this uncontrolled system described by equation (3.21) is subjected to an energy stabilization constraint described by equation (3.13). The control force can be computed using equation (3.18) and is given by F c = o H(q 1 ; _ q 1 )H _ q 1 (3.23) where H(q 1 ; _ q 1 ) = 1 2 _ q 2 1 + a o b o e boq 1 a o q 1 a o b o : (3.24) The energy H of the SDOF Toda oscillator is positive denite (see Appendix B) and radially unbounded (see Appendix E). Also, H increases monotonically in every radial direction from the origin. Therefore, any constant energy curve is a closed orbit in phase space. With the control force (equation 3.23) at our disposal, the controlled (constrained) equations of motion of the SDOF Toda oscillator can now be written as q 1 + o H(q 1 ; _ q 1 )H _ q 1 + a o h e boq 1 1 i = 0; (3.25) Equation (3.25) resembles the familiar form of a self-excited nonlinear oscillator with a nonlinear damping term similar to those found in Van der Pol-type systems. When H >H , the damping in the system is positive and the energy of the system is lowered. Conversely, when H < H , the damping is negative and the energy of the system is raised. When H = H is attained, the control force terminates and the conservative nature of the lattice is utilized to maintain its energy at H for all future time. The equivalent pair of rst order equations for the constrained SDOF Toda oscillator system (3.25) are given by _ X 1 = X 2 ; _ X 2 = a o h e boX 1 1 i o H(X 1 ;X 2 )H X 2 ; (3.26) The constrained system (3.26) still possesses a single isolated equilibrium point at the origin, but it is now unstable. The introduction of the control force (3.23) has 31 destroyed the concentric closed orbits of the uncontrolled (unconstrained) system (see Figure 3.6(a)) and has led to the creation of an unstable origin as well as a stable limit cycle described by H(X 1 ;X 2 ) = H in the controlled (constrained) system (see Figure 3.6(b)). As shown in this gure, the controlled system asymptotically tends to the manifold H(X 1 ;X 2 ) = H in the two-dimensional phase space, and to get to this desired manifold, it can take one of many dierent trajectories depending on the initial conditions of the controlled system and the value chosen for the parameter o > 0. We now investigate if we can analytically establish the convergence of all orbits in< 2 fOg to the manifold H(X 1 ;X 2 ) = H for the controlled system. And to do this, we resort to LaSalle's invariance principle [36]. LaSalle's Invariance Principle - The invariance principle in< n is postulated as: Let be a compact set ( D< n ) that is positively invariant. Let V : D!< be a continuously dierentiable function such that _ V (x) 0 in . Let E be the set of all points in where _ V (x) = 0. Let P be the largest invariant set in E. Then, every solution x(t) starting in approaches P as t!1. Consider a continuously dierentiable scalar function V given by V (X 1 ;X 2 ) = 1 2 H(X 1 ;X 2 )H 2 ; H > 0 (3.27) dened on the set described by = (X 1 ;X 2 ) 2 < 2 j H(X 1 ;X 2 ) c (3.28) where 0 < < H < c and H is the energy of the system described by equation (3.24). By choosing > 0, an open region around the origin O (X 1 = X 2 = 0) is excluded from the set . Our basic motive in choosing as in equation (3.28) is to establish that: (i) the origin O is an unstable xed point, and (ii) all trajec- tories in< 2 fOg asymptotically converge to the closed periodic orbitH(X 1 ;X 2 ) =H . 32 Figure 3.7: Pictorial representation of a typical set. Figure 3.8: Scalar energy error V plotted as a function of X 1 and X 2 (a o = 2; b o = 1; H = 8). 33 1. is a compact set: A formal proof of this result is presented in Appendix E. However, given that constant energy curves of unconstrained system (3.10) form closed orbits in< 2 , from Figure 3.7, it is easy to see that the set is indeed closed and bounded and therefore compact. 2. is positively invariant: A setW is said to be positively invariant set ifx(0)2W implies x(t)2W for all t 0 [34]. Figure 3.8 shows V plotted as a function of X 1 and X 2 . We note that the functionV has a positive value everywhere in< 2 (which includes ) except when H(X 1 ;X 2 ) = H , where it is zero. Let us now evaluate _ V along the trajectories of the controlled (constrained) system (3.26) and determine the region in phase space where _ V is guaranteed to be nonpositive. _ V (X 1 ;X 2 ) = (HH ) dH dt = (HH ) d dt 1 2 X 2 2 + a o b o e boX 1 a o X 1 a o b o = (HH ) n X 2 _ X 2 +a o _ X 1 e boX 1 1 o = (HH ) n a 0 e boX 1 1 X 2 o (HH )X 2 2 +a o _ X 1 e boX 1 1 o = o (HH ) 2 X 2 2 8 < 2 (3.29) Thus, we nd that _ V is indeed nonpositive throughout< 2 (which includes the set ). Since, V 0 (by virtue of equation 3.27) and _ V 0 (from equation 3.29) at all points that lie in the set , we deduce that the trajectories that enter the set at t = 0 are conned to it for all future time. This implies that is positively invariant. 3. Set E: The set E is dened as consisting of all points in where _ V = 0. From equation (3.29), we deduce that _ V is zero in the set when E = (X 1 ;X 2 ) 2 < 2 j X 2 0 [ H(X 1 ;X 2 ) H (3.30) 4. Set P : The set P is dened to be the union of all invariant sets within E [65]. The set of all points satisfying H(X 1 ;X 2 ) = H is positively invariant because when 34 H(X 1 ;X 2 ) = H is substituted into the constrained equations of motion (3.26), the control force is zero and we obtain our unconstrained (uncontrolled) system which is conservative and for which the energy remains constant (which in this case is H ) for all time t. On the other hand, substituting X 2 0 (and therefore _ X 2 0 in equation (3.26), we nd thatX 1 0. Thus, the origin O given byX 1 X 2 0 is the only point on the line X 2 = 0 which is invariant and all other trajectories originating on the line X 2 = 0 move away from it. However, since the origin (at which H(0; 0) = 0) is itself excluded from the set , the set P consists of P = (X 1 ;X 2 ) 2 < 2 j H(X 1 ;X 2 ) H (3.31) Then, by the invariance principle, every solution x(t) starting in approaches P as t!1. Thus, we conclude that all trajectories in the set have to eventually converge to the limit cycle (given by equation 3.31) as time t tends to innity. Hence, global asymptotic convergence to the limit cycle has been established in . Now, since c (outer boundary of the set ) can be chosen to be arbitrarily large and (inner boundary of the set encircling the origin) can be chosen to be arbitrarily small, all orbits in< 2 fOg asymptotically tend to P . Furthermore, since the open region around the origin can be made arbitrarily small through a proper choice of , the origin is an unstable xed point. This proves that the control force derived in equation (3.23) for the case of an SDOF Toda oscillator (with initial energy H o > 0) gives us global asymptotic convergence to any given desired energy state H in< 2 fOg. We next proceed to proving the same result for the NDOF general nonlinear lattice system. 3.5.2 NDOF General Nonlinear Lattice In this section, our aim is to show that 1. the control forceF C (equation 3.18) gives us global asymptotic convergence to any nonzero desired energy state H provided that the rst mass, or the last mass, or 35 alternatively, any two consecutive masses of the NDOF lattice are included in the set of masses that are controlled, and 2. the origin O in phase space is an unstable xed point in the controlled system. The controlled NDOF system (described by equation 3.19) possesses a unique and isolated equilibrium point at the origin (see derivation in Appendix D). Our aim is to prove that this xed point at the origin O is unstable. Once again, LaSalle's invariance principle [36] (see Page 32 for the statement) can be used to establish both these results. Similar to the SDOF system, let us consider a continuously dierentiable scalar function V as V (q; _ q) = 1 2 (H(q; _ q)H ) 2 ; H > 0 (3.32) dened on the set described by =f(q; _ q)2< 2n j H(q; _ q)cg; (3.33) where 0 < < H < c. By choosing > 0, an open region around the origin O (prescribed by q _ q 0) is excluded from the set . Our basic motive in choosing as in equation (3.33) is to establish that the origin O is an unstable xed point and all trajectories in< 2n fOg asymptotically converge to the compact and invariant set dened by H(q; _ q) = H . Now, to apply the invariance principle, we need to rst es- tablish that the set is compact and positively invariant in 2n-dimensional phase space. 1. is a compact set: A detailed derivation of this result is presented in Appendix E. 2. is positively invariant: A set W is said to be positively invariant if x(0)2W implies x(t)2W for all t 0 [34]. Let us compute _ V along the trajectories of the controlled NDOF nonlinear lattice (3.19) as shown below. 36 _ V (q; _ q) = (HH ) dH dt = (HH ) d dt 1 2 _ q T M _ q +U(q) = (HH ) _ q T (M q) + _ q T (F ) = (HH ) _ q T F o (HH )C T CM _ q + _ q T (F ) = o (HH ) 2 _ q T C T CM _ q = o (HH ) 2 _ q T C M C _ q C 0 8 < 2n (3.34) Since V 0 (equation 3.32) and _ V 0 (equation 3.34) at all points that lie in the set , we deduce that the set is positively invariant. Note that this result also holds true if were to be given by equation (3.17) instead. 3. Set E: The set E is dened as consisting of all points in the set where _ V = 0. From equation (3.34), we deduce that _ V is zero in the set when E = (q; _ q)2< 2n j _ q C 0 [ H(q; _ q)H (3.35) 4. SetP : The setP is dened to be the union of all invariant sets withinE [36]. The set of all points satisfying H(q; _ q) =H is positively invariant because when H(q; _ q) =H is substituted into the equations of motion of the controlled lattice (equation 3.19), the control force is zero and we obtain our uncontrolled system (3.5), which is conservative and for which the energy remains constant (which in this case is H ) for all time t. Next, we need to ensure that the only invariant set in E( ) is the set dened by H(q; _ q) = H so that all trajectories in are globally attracted to this set. Thus, we would ideally like the invariant set(s) satisfying _ q C 0 to lie outside of . To ensure this, we need to place the actuators appropriately so that _ q C 0 only yields q _ q 0 (origin O) (3.36) 37 which is invariant, and which does not belong to the set . A sucient condition for relation (3.36) to hold in xed-xed (or xed-free) inhomogeneous nonlinear lattices is when the set of locations of the actuators includes atleast one of the following congu- rations (see Appendix F for a detailed derivation). (1) A single actuator is placed on the rst mass m 1 and/or the last mass m n of the lattice. (2) Two actuators are placed on two consecutive masses located anywhere in the lattice. Now, since the origin O is excluded from the set , the largest invariant set in E is P = (q; _ q)2< 2n j H(q; _ q) =H ; H > 0 : (3.37) Then, by the invariance principle, every solution x(t) starting in the set approaches P as t!1. Thus, global asymptotic convergence to the set H(q; _ q) = H has been established in . Now, since c can be chosen arbitrarily large and can be chosen arbitrarily small, all trajectories in< 2n fOg asymptotically tend to P . Moreover, since the open region around the origin can be made arbitrarily small through a proper choice of , the origin is an unstable xed point. Hence, this proves that the control forceF C derived in equation (3.18) for an NDOF nonlinear lattice with xed-xed (or xed-free) ends gives us global asymptotic conver- gence to any given desired energy state H in< 2n fOg provided that the rst mass, or the last mass, or alternatively, any two consecutive masses of the lattice are included in the subset of masses that are controlled. Since actuation at only one mass or any two consecutive masses could guarantee global asymptotic energy control for an n-mass lattice, the control could be highly underactuated. 3.6 Results and Simulations One of the signicant results of the present study is the fact that though the control is highly underactuated, it is still guaranteed to control the energy of the lattice from any 38 nonzero initial state to any other nonzero desired nal state. As shown analytically in Section 3.5.2, to achieve this desired energy state, one could use just a single actuator placed on the rst mass m 1 of the lattice, or on the last mass m n of the lattice, or one can simply actuate two neighboring masses located anywhere in the lattice immaterial of the number of degrees of freedom of the lattice. This is shown to be true for dierent boundary conditions, and whether or not the lattice is homogeneous. The qualitative nature of each of the spring elements in the lattice can also be chosen to be widely dierent (provided the requisite conditions on the potentials are satised). Furthermore, the rate of convergence to the desired energy state can also be controlled, and in addition various sets of actuator locations can be chosen. To illustrate the scope of all these qualitatively dierent results, six numerical simulations are presented in this section as described in Table 3.1. First, we present four simulations involving the Toda lattice in Section 3.6.1, and then, for the sake of comparison, a homogeneous FPU- lattice is juxtaposed with a general nonlinear lattice and the simulations of these lattices are presented in Section 3.6.2. 3.6.1 Toda Lattice Simulations In this subsection, numerical simulations involving a Toda lattice with xed-xed and xed-free boundary conditions are presented to illustrate the ease and ecacy with which the control methods described in this study can be applied. Since n can be any nitely large number, a 101-mass lattice is chosen. Four dierent examples are considered as shown in Table 3.1. They are described below. Boundary Conditions and Homogeneity of the Lattice: Examples 1 and 2 deal with a xed-xed Toda lattice whereas Examples 3 and 4 deal with a xed-free Toda lattice. Examples 1 and 3 consider both a homogeneous lattice (for purposes of comparison) and a nonhomogeneous lattice. Examples 2 and 4 deal exclusively with nonhomogeneous lattices. 39 Example Number Lattice Type Boundary Conditions Homogeneity of Lattice Location of Initial Excitation Energy Raised / Lowered Actuator Locations o 1 Toda Fixed-Fixed Homogeneous & Inhomogeneous m 51 Raised m 75 , m 76 0.1 2 Toda Fixed-Fixed Inhomogeneous m 51 Raised Multiple Sets of Actuator Congurations 0.01 3 Toda Fixed-Free Homogeneous & Inhomogeneous m 51 Lowered m 75 , m 76 0.1 4 Toda Fixed-Free Inhomogeneous Random Initial Excitation (all masses) Lowered m 101 0.1 5 FPU- Fixed-Fixed Homogeneous m 51 Raised m 75 , m 76 0.005 & 0.1 6 General Nonlinear Lattice Fixed-Fixed Inhomogeneous m 51 Raised m 75 , m 76 0.005 & 0.1 Table 3.1: Description of the six nonlinear lattice simulations Specication of the Lattice Parameters: In Example 1, the values of the spring constants (a i ;b i ) and the masses (m i ) of the xed-xed homogeneous lattice (see Figure 3.2) are taken to be a i = 2;b i = 1 for 0 i 101, and m i = 1 for 1 i 101. For the xed-free homogeneous Toda lattice in Example 3, the last spring element (i.e. the spring element with spring constantsa 101 ;b 101 ) is discarded keeping all the other spring element and mass parameter values the same as in Example 1. The values of the spring constants and the masses of the xed-xed nonhomogeneous lattice in Example 1 are selected at random from a uniformly distributed set of numbers between the limits: 1:5 < a i < 2:5, 0:5 < b i < 1:5 for 0 i 101, 0:5 < m i < 1:5 for 1 i 101. The nonhomogeneous lattice in Example 2 has the same parameter values as those in Example 1. The parameter values of the xed-free nonhomogeneous lattices in Examples 3 and 4 are also the same as those in Example 1, except that the last spring element is discarded, as before. Specication of Initial Conditions: In Examples 1, 2 and 3, the lattice is initially excited with all masses having zero initial displacement and zero initial velocity except for the mass located at the center of the lattice, m 51 , which is given an initial displacement. In Example 4, all the masses of the xed-free lattice are excited with random initial displacements and random initial velocities. Specication of Energy Control Requirements: In all of the examples, the aim is to control the energy in these respective lattices and bring them to the desired energy level. In Examples 1 and 2, the energy of the lattice is desired to be increased from its initial state, while in the latter two examples, the energy is desired to be reduced. Actuator Locations: To achieve these desired energy levels, control can be applied to one or more of these 101 masses (provided that the rst mass, or the last mass, or alternatively, any two consecutive masses of the lattice are included in the subset of masses that are controlled). In Examples 1 and 3, control is applied to two consecutive masses, m 75 and m 76 , located at about three quarters the distance from the left end of 41 the lattice (see Figure 3.2). In Example 2, ve dierent sets of actuator congurations are chosen to exhibit the eect of the placement of the actuators on the time it takes for the controlled lattice to get to the desired energy level. In Example 4, only the last mass of the lattice is actuated in order to lower its energy. The constant o (see equation 3.18) which aects the rate at which the controlled lattice converges to the desired energy level is chosen to be 0:1 in all of the examples, except in Example 2 where it is lowered to 0:01 to allow for an easier comparison of the times taken by the dierent sets of actuator congurations to reach the desired energy level. In all of the four examples that are considered in this section, the equations of motion are integrated using ode113 in the MATLAB environment with a relative integration error tolerance of 10 10 and an absolute error tolerance of 10 13 . All quantities are assumed to be in consistent units. 3.6.1.1 Fixed-Fixed Toda Lattice Example 1: In the rst example, we study a 101-mass Toda lattice with xed-xed boundary conditions. Our aim is to raise the energy of both a homogeneous lattice (for comparison) and a nonhomogeneous lattice (both of whose parameters are as described earlier) to a desired level of 150 units in each case. The mass located at the center of the lattice, m 51 , is initially displaced by 3 units, both in the homogeneous and the nonhomogeneous lattice. This causes the initial energy level, H o , of the homogeneous and the nonhomogeneous lattice to be 36:27 units and 83:58 units, respectively. Example 1 - Homogeneous Lattice: Figure 3.9(a) shows the velocity eld for the uncontrolled (unconstrained) homogeneous lattice, where time is plotted on the x-axis and the location of the masses is plotted on they-axis. The velocity of each mass, which is plotted on the z-axis, is instead shown through a color variation (see color scale on the right of the gure). As seen from Figure 3.9(a), the initial displacement of the mass at the center of the lattice gives rise to, what appear to be, multiple soliton structures propagating through the velocity eld. Amongst many small waves, there appear to be 42 (a) Uncontrolled Homogeneous Lattice (b) Controlled Homogeneous Lattice Figure 3.9: Example 1 - Velocity eld of the 101-mass xed-xed homogeneous Toda lattice. The lattice has parameters a i = 2, b i = 1, m i = 1 for all i, H o = 36:27, H = 150, o = 0:1 and the initial displacement of central mass (m 51 ) is 3 units. Actuators are located on masses m 75 and m 76 . 43 (a) Uncontrolled Inhomogeneous Lattice (b) Controlled Inhomogeneous Lattice Figure 3.10: Example 1 - Velocity eld of the 101-mass xed-xed nonhomogeneous Toda lattice. The nonhomogeneous lattice has parameters a, b and m chosen ran- domly from a uniformly distributed set of numbers between the limits: 1:5<a i < 2:5, 0:5 < b i < 1:5 and 0:5 < m i < 1:5, H o = 83:58, H = 150, o = 0:1 and the initial displacement of central mass (m 51 ) is 3 units. Actuators are located on massesm 75 and m 76 . 44 Figure 3.11: Example 1 - Time history of control forces acting on the two actuator masses m 75 and m 76 of the lattice. Dotted lines denote control forces acting on the homogeneous lattice and the solid lines denote the control forces acting on the nonho- mogeneous lattice. Red line denotes the control force acting on the mass m 75 and blue line denotes the control force acting on mass m 76 . Figure 3.12: Example 1 - (Top Subplot) Time history of energy of the lattice fromt = 0 to t = 25 seconds. (Bottom Subplot) Time history of energy error e(t) = H(t)H from t = 400 seconds to t = 500 seconds. Green line denotes the homogeneous lattice and pink line denotes the nonhomogeneous lattice. 45 two large solitons generated at t = 0, one with positive velocity amplitude (shown in dark red) traveling towards the left end of the lattice (towards mass m 1 ) and another with negative velocity amplitude (shown in dark blue) traveling towards the right end of the lattice (towards mass m 101 ). When each of these solitons reaches the xed end, the incident soliton is re ected, and the re ected soliton has its amplitude reversed in sign. The propagation speed of these individual solitons appears to be constant (as seen from the slopes of these lines), with the two large solitons traveling at a faster rate than the smaller waves. The re ections of these two large solitons rst cross each other at around 52 seconds and they continue to propagate undisturbed after they cross. The explicit closed form control given in equation (3.18) is now applied to this homogeneous lattice at masses m 75 and m 76 , so that the controlled lattice achieves the desired energy level of 150 units. A plot of the control forces is shown in Figure 3.11 from t = 0 to t = 25 seconds. The dotted red and blue lines denote the control forces acting on actuator masses m 75 and m 76 , respectively. As seen from the plot, a nite amount of time elapses before the control begins. This is because it takes a nite amount of time for the initial excitation at the center of the lattice to traverse through the lattice and reach the actuator locations at m 75 and m 76 . Since the control forces depend on the velocity of the actuated masses (see equation (3.18)), once the actuator masses are in motion at around 9 seconds, the control begins and the desired energy state of H = 150 (see Figure 3.12, top) is very quickly achieved. Once the desired energy state is achieved, the control forces terminate (see Figure 3.11) and the conservative nature of the lattice is utilized to maintain its energy at the desired level for all future time. Figure 3.9(b) shows a plot of the velocity eld for the controlled homogeneous lattice. From the gure, we observe that coinciding with the application of the control forces, a new set of soliton structures is generated (see black circle in the gure) in addition to those already generated by the initial displacement of the central mass. Some of these newly generated solitons have larger amplitudes and higher propagation speeds when compared to those extant. Figure 3.12 (bottom subplot) shows the energy error, e(t) = H(t) H , in achieving the desired energy level plotted as a function of time from t = 400 seconds to t = 500 46 seconds. As seen from the gure, the magnitude of this error is commensurate with the error tolerances used in integrating the equations of motion of the controlled system. Example 1 - Inhomogeneous Lattice: We next consider the uncontrolled non- homogeneous lattice whose velocity eld is shown in Figure 3.10(a). As seen in the gure, the initial displacement of the mass at the center of the lattice, m 51 , generates waves at t = 0, which traverse through the length of the lattice. The criss-cross pattern shown in the gure is generated by the propagation of these waves and their re ection at the boundaries. The distinct soliton structures that were observed in the velocity eld of the homogeneous lattice are no longer present in the nonhomogeneous lattice and this absence is due to the nonhomogeneity of the lattice. Once again, we apply control (see equation (3.18)) to this nonhomogeneous lattice to raise its energy level from H o = 83:58 units to H = 150 units. The time history of the control forces obtained by using equation (3.18) is shown in Figure 3.11 where the solid red and blue lines denote the control forces acting on the actuator masses m 75 and m 76 , respectively. The control begins at around 11 seconds and again quickly stabilizes the lattice at the desired energy level (see Figure 3.12, top). The control generates its own velocity eld causing waves to emanate at around 11 seconds (see black circle in Figure 3.10(b)) in addition to those already generated by the initial displacement of the central mass. Figure 3.12 (bottom subplot) shows the energy error,e(t) =H(t)H , as before. Example 2: In this example, a 101-mass nonhomogeneous Toda lattice with xed- xed ends is considered with the same parameter values for the spring elements and the masses as in the previous example. An initial displacement of 3 units is given to the mass located at the center of the lattice like in the previous example and this causes the initial energy of the lattice to be H o = 83:58 units. Five dierent sets of actuator congurations are considered (in accordance with the actuator location rules specied in Appendix F) to study their eect on the time it takes for the controlled nonhomogeneous lattice to reach a desired energy level of 150 units. The congurations are (see Figure 3.13(a)): 47 (a) Time history of energy convergence for dierent sets of actuator congurations of the 101-mass xed-xed nonhomogeneous Toda lattice. (b) Zoomed-in plot showing the energy convergence for the ve dierent sets of actuator congurations. Figure 3.13: Example 2 - Nonhomogeneous Toda lattice with xed-xed boundary conditions. The lattice has its parameters chosen randomly from a uniformly distributed set of numbers between the limits: 1:5 < a i < 2:5, 0:5 < b i < 1:5, 0:5 < m i < 1:5, H o = 83:58, H = 150, o = 0:01 and initial displacement of the central mass (m 51 ) is 3 units. 48 (a) only mass m 1 is actuated; (b) only masses m 75 and m 76 are actuated; (c) only masses m 50 and m 51 actuated; (d) every tenth mass along the lattice is actuated starting with mass m 1 ; (e) every fth mass along the lattice is actuated starting with mass m 1 . As described earlier, the parameter o (see equation 3.18) is lowered to 0:01 in the present example. Figure 3.13 shows a plot of the convergence of the lattice's energy to the desired energy level as a function of time for the ve sets of actuator congurations considered. From the gure, we observe that the lattice controlled using two consecutive actuators placed at roughly three quarters the distance from the left end of the lattice at m 75 and m 76 takes the longest time to reach the desired energy state; longer even compared to a single actuator placed on the rst mass (m 1 ) of the lattice. However, if one chooses to place these actuators on two consecutive masses atm 50 andm 51 , closer to the initial excitation at the center of the lattice (i.e. at mass m 51 ), the desired energy state is achieved comparatively faster (in approximately 140 seconds). Furthermore, using 11 and 21 equidistantly placed actuators with the rst of these actuators placed on the rst mass of the lattice helps us in attaining the energy state in approximately 80 seconds and 45 seconds, respectively. We, thus, conclude that the time taken for the controlled lattice to reach the desired energy state depends on the number of actuators, the placement of these actuators, as well as on the nature of the initial excitation of the lattice. Though an interesting problem in itself, we, however, do not delve deeper into it here as it will take us too far aeld from the central focus of this study. 3.6.1.2 Fixed-Free Toda Lattice Example 3: In this example, we study a 101-mass Toda lattice with xed-free boundary conditions. Our aim is to lower the energy of a homogeneous lattice (for comparison) and a nonhomogeneous lattice from an initial energy level of approximately 150 units 49 to a desired level of 100 units. The parameter values of the lattices are as described earlier. To initialize both lattices at an energy level of approximately 150 units, the initial displacement of the mass at the center of the lattice, m 51 , is chosen to be 4:34 units in the homogeneous case (this causes its initial energy level, H o , to be 149:44 units) and 3:41 units in the nonhomogeneous case (initial energy level, H o , is 149:90 units), with all the other masses in the respective lattices given zero initial displacement and zero initial velocity. Example 3 - Homogeneous Lattice: A plot of the velocity eld of the homogeneous lattice is shown in Figure 3.14. Once again, similar to Example 1, we observe that the initial displacement of the central mass gives rise to two large solitons in addition to many small waves in the velocity eld of the uncontrolled homogeneous lattice (see Figure 3.14(a)). Since the left end of the lattice is a xed end, the positive velocity amplitude soliton traveling towards m 1 is re ected and the re ected soliton has its velocity amplitude reversed in sign but its magnitude remains the same as the incident soliton. On the other hand, the large negative amplitude soliton traveling towards the free end of the lattice towards m 101 is imperfectly re ected. The free end has the eect of breaking up the incident soliton into a smaller amplitude (re ected) soliton and many small (re ected) waves that can be seen traversing through the velocity eld (see Figure 3.14(a)). This behavior is consistent with what has been documented in the literature [73]. Next, we apply control to this homogeneous lattice at masses m 75 and m 76 to reduce its energy level to 100 units. From Figure 3.14(b), we observe that the application of the control force breaks the large negative velocity amplitude soliton (generated by the initial displacement of the center mass m 51 ) into two slower moving smaller solitons (see black circle in the gure), one having positive velocity amplitude traveling towards the left end of the lattice and another having negative amplitude traveling towards the right end of the lattice. The time history of control forces acting on the homogeneous lattice are shown in the top subplot of Figure 3.16. Figure 3.17 shows that the desired energy level of 100 units is achieved and the energy error lies 50 (a) Uncontrolled Homogeneous Lattice (b) Controlled Homogeneous Lattice Figure 3.14: Example 3 - Velocity eld of the 101-mass xed-free homogeneous Toda lattice. The top subplot corresponds to the uncontrolled lattice and the bottom subplot corresponds to the controlled lattice. The lattice has parameters a i = 2,b i = 1,m i = 1 for all i, H o = 149:44, H = 100, o = 0:1 and the initial displacement of central mass (m 51 ) is 4:34 units. Actuators are located on masses m 75 and m 76 . 51 (a) Uncontrolled Inhomogeneous Lattice (b) Controlled Inhomogeneous Lattice Figure 3.15: Example 3 - Velocity eld of the 101-mass xed-free nonhomogeneous Toda lattice. The nonhomogeneous lattice has parameters a, b and m chosen randomly from a uniformly distributed set of numbers between the limits: 1:5<a i < 2:5, 0:5<b i < 1:5 and 0:5 < m i < 1:5, H o = 149:9, H = 100, o = 0:1 and the initial displacement of central mass (m 51 ) is 3.41 units. Actuators are located on masses m 75 and m 76 . 52 Figure 3.16: Example 3 - Time history of control forces acting on the two actuator massesm 75 (red line) andm 76 (blue line) of the homogeneous lattice (top subplot) and the nonhomogeneous lattice (bottom subplot). Figure 3.17: Example 3 - (Top Subplot) Time history of energy from t = 0 to t = 250 seconds. (Bottom Subplot) Time history of energy errore(t) =H(t)H fromt = 750 seconds tot = 1000 seconds. Green line denotes the homogeneous case and pink denotes the nonhomogeneous case. 53 close to the tolerance levels specied in the integration algorithm. Example 3 - Inhomogeneous Lattice: Figure 3.15(a) shows a plot of the velocity eld of the uncontrolled nonhomogeneous lattice with xed-free ends, wherein the mass at the center of the lattice, m 51 , is provided an initial displacement of 3:41 units. Once again, like in Example 1, the absence of distinct soliton structures in the velocity eld of the nonhomogeneous lattice is evident. Control is applied to masses m 75 and m 76 to reduce the lattice's energy from 149:9 units to 100 units. From the velocity eld of the controlled lattice (shown in Figure 3.15(b)), we note that the set of masses located beyond massesm 75 andm 76 have amplitudes of motion that are smaller than the other masses in the lattice. The actuators atm 75 andm 76 damp out the motion in the lattice through destructive interference in order to reduce the lattice's energy. Comparing Figure 3.16 to Figure 3.11, we note that the control acts for a longer duration and it takes a longer time for the lattice to reach the desired energy state (both in the homogeneous and nonhomogeneous lattice) when the energy of the lattice is desired to be reduced as compared to when the energy of the lattice is desired to be raised. As before, Figure 3.17 shows that the desired level of 100 units is attained and that the energy errors lie close to the tolerance levels specied by the integration algorithm. Example 4: In this nal example, a xed-free nonhomogeneous Toda lattice subjected to random initial excitation is considered. The initial displacements and initial velocities of each of the 101 masses in the lattice are chosen at random from a uniformly distributed set of numbers between the limits 2 and 2. This random selection of the initial conditions causes the lattice to have an initial energy level H o = 341:47 units, and the aim is to reduce this energy to H = 250 units using only a single actuator located on the last mass (m 101 ) of the xedfree lattice. Figure 3.18(a) shows the velocity eld of the uncontrolled nonhomogeneous lattice subjected to random initial conditions. Control is now applied to the last mass (m 101 ) of the xed-free lattice to reduce its energy. Figure 3.18(b) shows the velocity eld 54 (a) Uncontrolled Inhomogeneous Lattice (b) Controlled Inhomogeneous Lattice Figure 3.18: Example 4 - Velocity eld of the 101-mass xed-free nonhomogeneous Toda lattice subjected to random initial excitation. The nonhomogeneous lattice has parameters a, b and m chosen randomly from a uniformly distributed set of numbers between the limits: 1:5 < a i < 2:5, 0:5 < b i < 1:5 and 0:5 < m i < 1:5, H o = 341:47, H = 250, o = 0:1. The initial displacements and initial velocities are chosen randomly between the limits2 and 2. A single actuator is placed on the last mass of the xed-free lattice (i.e. on m 101 ). 55 Figure 3.19: Example 4 - Time history of control forces acting on the last mass m 101 of the nonhomogeneous lattice. Figure 3.20: Example 4 - (Top Subplot) Time history of energy from t = 0 to t = 150 seconds. (Bottom Subplot) Time history of energy errore(t) =H(t)H fromt = 500 seconds to t = 750 seconds. 56 of the controlled nonhomogeneous lattice, where we observe that the response of the controlled lattice is markedly dierent from the uncontrolled lattice. This is especially evident near right end of the lattice (close to the last mass m 101 ) where the motion of the lattice is being damped out through destructive interference in order to reduce its energy. Figure 3.19 shows a time history of the control forces acting on the last mass of the lattice where we note that the control acts right from the instant t = 0. Figure 3.20 shows a plot of the energy convergence and energy errors in achieving the desired energy state, as before. In all of the examples considered in this section, once the desired energy state is achieved, the control forces automatically become zero (see Figures 3.11, 3.16, 3.19) and the conservative nature of the lattice is thereafter utilized to maintain its energy at the desired level for all future time. Further, in all of the examples, the energy errors are small and lie close to the tolerance levels specied in our integration algorithm (see Figures 3.12, 3.17, 3.20), highlighting the ecacy of the control methodology in achieving the desired energy state. 3.6.2 FPU- Lattice and the General Nonlinear Lattice In this section, numerical simulations involving a nonlinear lattice with xed-xed boundary conditions are presented to illustrate the ease and ecacy with which the control methodology can be applied. Since n can be any nitely large number, a 101- mass lattice is chosen. Given any nonzero initial energy state of the lattice, our aim is to control the energy of the nonlinear lattice and bring it to a desired energy level. To achieve this desired energy level, control can be applied to one or more of these 101 masses (provided ofcourse that the rst mass, or the last mass, or any two consecutive masses, are included in the set of masses that are controlled). In both the examples that we consider in this section (i.e., Examples 5 and 6), the spring elements in the lattice are all taken to be nonlinear and control is applied to two consecutive masses, m 75 and m 76 , located at about a quarter of the lattice's length from the right end (see Figure 3.1). In Example 5, a homogeneous FPU -lattice [7, 18] consisting of 101 unit masses is considered. The nonlinear potential of the FPU -lattice is given by u i (x) = (a i =2)x 2 +(b i =4)x 4 , where the spring constants are chosen to bea i =b i = 18i. 57 Example 6 deals with a 101-mass inhomogeneous nonlinear lattice where the non- linear potential of each spring in the lattice is chosen at random from the following set of potentials, S sp . S sp = a 4 x 4 ; a 2 x 2 + b 4 x 4 ; a 6 x 6 ; a 4 x 4 + b 6 x 6 ; a 8 x 8 ; a 6 x 6 + b 8 x 8 ; a b e bx ax a b (3.38) We note that each potential function in the set S sp is qualitatively dierent, and is characterized here, for illustration purposes, by at most two parameters a and b. The potentials further satisfy all the requisite conditions listed in Section 3.2.1. For each spring in the 101-mass lattice, rst a potential function is chosen at random from the set S sp , and then its parameter values are selected at random from a uniformly distributed set of numbers between the limits: 0:5 < a i < 1:5, and 0:5 < b i < 1:5. Likewise, each mass in the lattice is also chosen at random from a uniformly distributed set of numbers between the limits: 0:5 < m i < 1. One realization of the inhomogeneous lattice from the ensemble of random lattices so produced is used in the example below. In both examples, the lattice is initially excited with all the masses having zero initial displacement and zero initial velocity except for the mass at the center of the lattice, m 51 , which is initially displaced by 2 units. This causes the initial energy level, H o , of the homogeneous and the inhomogeneous lattice to be 12 units and 28:4 units, respectively. The aim is to control the energy in these respective lattices and raise them to a desired energy level of H = 150 units in each case. The equations of motion are integrated usingode113 in the Matlab environment with a relative integration error tolerance of 10 10 and an absolute error tolerance of 10 13 . All quantities are assumed to be in consistent units. Following [15], the gradient of the displacement eld is plotted for Examples 5 and 6 to better envision the dynamics of the lattice. For plots of the velocity eld instead, see Ref. [90]. 3.6.2.1 Homogeneous FPU- Lattice Example 5: Figure 3.21(a) shows a plot of the gradient of the displacement eld (i.e. q i+1 q i ) for the uncontrolled homogeneous FPU -lattice where time is plotted on 58 (a) Uncontrolled Lattice (b) Controlled Lattice Figure 3.21: Example 5 - Gradient (q i+1 q i ) of the displacement eld of the homo- geneous lattice. The lattice has parameters a i = 1, b i = 1;m i = 18 i;H o = 12;H = 150; o = 0:1 and the initial displacement of the central mass,m 51 , is 2 units. Actuators are located at m 75 and m 76 . 59 the x-axis, the location of the masses is plotted on the y-axis and the amplitude of the gradient of the displacement eld is shown through a color variation (see color scale on the right). The initial excitation of the mass at the center of the lattice gives rise to, what appears to be, a breather structure (see Figure 3.21(a)) located at the center of the lattice amidst many small waves propagating through the spatial eld. This breather structure oscillates undisturbed throughout the duration of the simulation. Control is now applied to this homogeneous lattice (as described earlier) to raise its energy level to 150 units. Time histories of the control forces acting on each of the masses that are controlled, namely m 75 (solid line) and m 76 (dash line), are shown in Figure 3.23(a) (top) for o = 0:1 (see equation 3.18). A nite amount of time is seen to elapse before the control sets in. This is because it takes a nite time for the initial excitation (at the center of the lattice) to traverse through the lattice and reach the actuator locations (at m 75 and m 76 ). Since the control forces are proportional to the velocity of the actuated masses (see equation 3.18), once the actuator masses are in motion at around 8:6 seconds, the control begins and the desired energy state of H = 150 is almost immediately achieved (see o = 0:1 case of Figure 3.23(b) top). Further, to illustrate the eect that o has on the rate at which the controlled homogeneous lattice converges to the desired energy state, we show the time histories of energy convergence for o = 0:1 and o = 0:005 in Figure 3.23(b) (top). Figure 3.21(b) shows a plot of the gradient of the displacement eld for the controlled homogeneous FPU-lattice for o = 0:1. From the gure, we observe that in addition to the breather structure generated by the initial excitation of the central mass, multiple soliton structures are generated (see black circle in the gure) coinciding with the application of the control forces. These structures have been further investigated, though for brevity, their analysis is not shown here. Figure 3.21(b) shows the interaction of these structures and their propagation through the spatial eld. 3.6.2.2 General Nonlinear Inhomogeneous Lattice Example 6: Figure 3.22(a) shows a plot of the gradient of the displacement eld for the uncontrolled inhomogeneous nonlinear lattice. The initial excitation of the mass at the 60 (a) Uncontrolled Lattice (b) Controlled Lattice Figure 3.22: Example 6 - Gradient (q i+1 q i ) of the displacement eld of the inhomo- geneous lattice. For each spring in the lattice, rst a nonlinear potential is chosen at random from the set S sp and then its parameters a;b andm are chosen randomly from a uniform distributed set of numbers between the limits 0:5<a i < 1:5; 0:5<b i < 1:5; and 0:5<m i < 1:5, respectively. The initial displacement of the central mass, m 51 , is 2 units andH o = 28:4;H = 150; and o = 0:1. Actuators are located at m 75 andm 76 . 61 (a) (b) Figure 3.23: Examples 5 & 6 - Time history of control forces, energy convergence and energy errors. (a) Control forces acting on the two actuator masses m 75 (solid line) and m 76 (dash line) of the homogeneous lattice (top) and the inhomogeneous lattice (bottom) for o = 0:1. (b) Energy of the system from t = 0 to t = 150 seconds (top) and energy error (e(t) =H(t)H ) from t = 100 to t = 150 seconds (bottom). Solid ( o = 0:1) and dotted ( o = 0:005) lines denote the homogeneous lattice whereas dash ( o = 0:1) and dash-dot ( o = 0:005) lines denote the inhomogeneous lattice. 62 center of the lattice generates waves, which traverse through the length of the lattice as can be inferred from the gure. Until about 40 seconds, the dynamics of the lattice seems conned to only a few masses, following which, it spreads out more rapidly to the other masses in the lattice. The crisscross pattern shown in the gure is generated by the propagation of the waves and their re ection at the boundaries. Once again, we apply control as before to this inhomogeneous lattice to raise its energy level to 150 units. A time history of the control forces acting on the inhomogeneous lattice is shown in Figure 3.23(a) (bottom) for o = 0:1. The control begins at around 59 seconds and stabilizes the lattice at the desired energy level (Figure 3.23(b) top) of 150 units. Like in Example 1, the time histories of energy convergence are plotted for two dierent values of o in Figure 3.23(b) (top) for the controlled lattice. Figure 3.22(b) shows a plot of the gradient of the displacement eld for the controlled system for o = 0:1. The control generates its own displacement eld causing waves to emanate (see black circle in the gure) in addition to those generated by the initial displacement of the central mass. Some of these newly generated waves appear to have larger amplitudes and higher propagation speeds when compared to those extant. The energy error, e(t) =H(t)H , in achieving the desired energy state is plotted as a function of time over the duration of the last 50 seconds of the simulation (see Figure 3.23(b) bottom) for o = 0:1 for both the examples in this section. The gure shows that this error is small and is close to the error tolerance levels specied in our integration algorithm, thus showing the ecacy of the control methodology in achieving the desired energy state. From Figure 3.23(a), we observe that once the desired energy state is achieved, the control forces automatically become zero, and the conservative nature of the lattice is thereafter utilized to maintain its energy at the desired level for all future time. Similar examples can also be generated for a xed-free nonlinear lattice, but we do not present them here for the sake of brevity. 63 3.7 Conclusions The problem of energy control of ann-degrees-of-freedom general nonlinear lattice with xed-xed and xed-free boundary conditions is considered. The nonlinear lattice is composed of a chain of masses wherein each mass is connected to its nearest neighbour by a nonlinear or linear memoryless spring element. The masses in the lattice are assumed to be dierent from one another. The qualitative nature of the nonlinear spring elements along the lattice is also assumed to be dierent as are the parameters of the functions of the potentials describing each of the spring elements. Thus, the nonlinear lattice that is considered in the present study is strongly inhomogeneous. This is in contrast to the current literature on the subject which almost exclusively deals with either homogeneous lattices (in which the masses are taken to be identical as also the nature and parameter values of the spring elements), or weakly inhomogeneous lattices (in which the inhomogeneity in the lattice is created by the presence of small, spatially localized perturbations from uniform values). To the best of the author's knowledge, neither the dynamics nor the control of such highly general nonlinear lattice systems has been hitherto addressed in the literature. The spring force associated with each spring element in the lattice is assumed to be derivable from a potential function that is: (i) twice continuously dierentiable, (ii) strictly convex possessing a global minimum at zero displacement, and (iii), has zero curvature possibly only at zero displacement. These assumptions are in conformity with the structural models used to describe many real-life situations. Additionally, as in many materials, asymmetries in the potential functions, which result in dierences between tensile and compressive behavior, are included. Such general nonlinear lattices exhibit highly nonlinear behavior and can display intricate and disparate structures such as solitons and breathers in their dynamical response, as shown in this study. The control approach adopted in this study is inspired by recent results in analytical dynamics that deal with the theory of constrained motion. Inspite of the general nature of the nonlinear lattice considered in this study, closed form expressions for the explicit 64 nonlinear control forces are obtained with relative ease without the need for any ap- proximations or linearizations of the nonlinear dynamical system, and without the need to impose any a priori structure on the nonlinear controller. The equations of motion of the controlled nonlinear lattice resemble that of a self-excited system akin to a Van der Pol nonlinear oscillator. The control forces, F C , are continuous in time and are optimal; they minimize the control cost given byJ(t) = [F C ] T M 1 [F C ] at each instant of time while causing the energy constraint (3.13) to be exactly satised. The control forces act on the NDOF nonlinear lattice to bring it to the desired energy level. Once this desired value is reached, the control forces terminate and the conservative nature of the lattice is utilized to maintain it at the desired energy level for all future time. The nonlinear lattice is underactuated. Stability analysis of the control is performed through the use of LaSalle's invariance principle. The principle is used to show global asymptotic convergence to the desired energy state and to simultaneously obtain su- cient conditions on the placement of actuators so that such convergence is guaranteed for the underactuated system. Specically, the control force obtained is shown to provide global asymptotic convergence from any given (nonzero) initial energy state, H o , to any desired (nonzero) nal energy state, H , provided that either (i) the rst mass of the lattice, m 1 , or (ii) the last mass, m n , or alternatively, (iii) any two consecutive masses of the lattice, are included in the set of masses that are actuated. To rephrase this, with just one actuator placed at either end of the chain, or with two actuators placed adjacent to each other anywhere in the chain, the energy of the entire lattice can be controlled. The manifoldH(q; _ q) =H forms a globally attracting limit hypersurface in 2n-dimensional phase space and the trajectories of the controlled system asymptotically tend to it. Numerical simulations contrasting the behavior of homogeneous and inhomogeneous Toda lattices, FPU- lattices and general nonlinear lattices containing 101 masses is shown. Complex nonlinear wave interactions are found in both the uncontrolled and the controlled systems. The closed form control obtained analytically is shown to work well for both increasing and decreasing the energy of these highly nonlinear lattices. The generality of the nonlinear lattice that can be considered is illustrated through 65 an example where the value of each mass in the inhomogeneous lattice is chosen at random from a uniformly distributed set of numbers. Each spring potential is randomly chosen from a set of seven qualitatively dierent potential functions (that satisfy the requisite conditions on the potentials described in Section 3.3), and the parameter values dening each potential function are also chosen at random. These simulations demonstrate the ease, simplicity, and accuracy with which the control methodology works. 66 Part 3 Multiple Coupled Slave Gyroscopes Chapter 4 Synchronization of Multiple Coupled Slave Gyroscopes 4.1 Introduction G yroscopes have long been used in navigation due to their ability to mea- sure orientation. From aerospace vehicles to consumer electronic devices, gyroscopes have played an important role in our everyday lives. From an engineering point of view, gyroscopes are highly nonlinear mechanical systems that exhibit a plethora of complicated motions such as periodic behavior, period-doubling behavior, quasi-periodic behavior and chaotic behavior. Studies [22, 23, 74, 98] have shown that when a symmetric gyroscope with linear plus cubic damping is subjected to a harmonic vertical base excitation, the motion thus obtained can range from regular motion to chaotic motion depending on the parameters selected. In analytical mechanics, the problem of synchronization of multiple chaotic systems has been of particular interest in recent times [39, 56]. In particular, the problem of synchronizing a set of slave gyroscopes which may or may not be coupled to each other, to an independent master gyroscope is an important problem in nonlinear dynamics that has received considerable attention [12, 38, 80]. This problem gains prominence in the eld of spacecraft dynamics where attitude control of spacecraft is almost always 68 performed using gyroscopes. When there are multiple gyroscopes onboard a spacecraft, it is often required that they be synchronized to indicate a specic, usable spacecraft pointing. And given that these gyroscopes exhibit a wide range of dynamic behavior, the problem of synchronization becomes even more challenging. The synchronization problem also nds applications in areas of secure communication (e.g. transmission of encrypted messages) and in areas of signal processing [70]. A large number of papers written on this subject only seem to consider the synchronization problem of two iden- tical gyroscopes. Chen [12] considers two identical chaotic gyros with dierent initial conditions and uses numerous classical control laws to show that when the feedback gain (obtained through experimentation) exceeds a particular value, the slave gyro synchro- nizes with the master gyro. Lei and Xu [38] approach the same problem using feedback linearizations wherein the dierence in response between the master and slave gyro is taken as an error signal and a time varying control is chosen to drive the error signal to zero. Aghababa [1] considers an adaptive robust nite-time controller to synchronize two chaotic gyros inspite of unknown uncertainties in the system. These approaches to the synchronization of master/slave gyros appear to work when the number of slaves considered is small in number. However, when a large number of slaves with noniden- tical physical and geometric properties are considered, the need to develop ecient control methods arise. It is also important to note that many papers on this subject assume that the gyroscopes are in the so-called `sleeping position', which in general is an over-simplifying condition, since most gyroscopes do not satisfy it. Recently, Udwadia and Han [80] approached the problem of synchronization of multi- ple chaotic gyroscopes from a new perspective - using the constrained motion approach. In their paper, Udwadia and Han consider a system ofn gyroscopes, each in the so-called `sleeping position', which are all uncoupled from one another having varying physical and geometrical properties, some or all of which exhibiting chaotic behavior, and at- tempt to synchronize these uncoupled slave gyros with a master gyro. The problem of synchronization, which is classically considered as a tracking control problem in control theory, is recast as a problem of constrained motion. The slaves are suitably constrained 69 and the fundamental equation of mechanics [83] is used in arriving at the nonlinear con- trol torques that force exact synchronization of each slave gyro with the master gyro. However, the problem with synchronizing a master gyro with n uncoupled slave gyro- scopes is that it can eectively be simplied inton separate problems of synchronization of the master gyro with n individual slave gyros. In this study, we consider a set ofn coupled gyros such that they form a single system of interacting slave gyroscopes. Next, we attempt to synchronize this nonlinear system of slave gyros with an independent master gyroscope whose motion can be either regular or chaotic. We adopt the methods developed by Udwadia and Han [80] and build on them to derive the constrained equations of motion for the case when the slave gyroscopes are all coupled. In the process, we also obtain explicit and closed form expressions for the generalized nonlinear control torques that are required to be applied to each of the slave gyros to obtain synchronization with the master gyro. We consider both linear and highly nonlinear types of coupling (that include linear plus cubic coupling, sinusoidal coupling and Toda coupling [72]) between the slave gyroscopes. The eect of the `sleeping' condition on the synchronization of the gyros is also studied. Further, utilizing the concept of an incidence matrix, we also examine the case when the slave gyroscopes are coupled in a more general fashion (instead of the usual chain coupling). Finally, to illustrate the ecacy of the methods presented in this study, we provide three numerical simulations. We consider a system of ve gyros with one master and four chain coupled slaves where the slaves are coupled (a) using sinusoidal coupling (b) using Toda coupling. In the Toda coupling case, the eect of the no-sleeping condition on the synchronization of the gyros is studied. The third example consists of a system of six gyroscopes with one master and ve slaves where the slaves are coupled to one another through an incidence matrix with dierent types of strongly nonlinear couplings. 70 4.2 Equations of Motion Consider the system of n + 1 symmetric gyroscopes as shown in Figures 4.1 { 4.2. The master gyro (whose parameters are denoted by the subscript m) is an independent sys- tem separate from the slave gyroscopes. The othern gyroscopes (denoted by subscripts 1 to n) are all coupled together with torsion springs to form the system of slave gyros. In the present study, we use Euler angles to describe the orientation of each gyroscope in the system: (nutation), ' (precession) and (spin). 4.2.1 Master Gyroscope The nonlinear equation of motion of a symmetric gyroscope whose point of support O is subjected to a vertical harmonic excitation of frequency ! m and amplitude ~ d m has been derived by in Appendix G and is given by I m m + (p 'm p m cos m ) (p m p 'm cos m ) I m sin 3 m m m gr m sin m m m r m sin m d m (t) =F d (4.1) where the angular momenta p m = I 3m _ m + _ ' m cos m and p 'm = I m _ ' m sin 2 m + p m cos m are conserved quantities as m ; ' m happen to be cyclic co-ordinates; m m is the mass of the master gyroscope,I m =I 1m +m m r 2 m , whereI 1m =I 2m are the principal equatorial moment of inertia of the master gyroscope and I 3m is the polar moment of inertia, m ; m ; ' m are the Euler angles of rotation associated with the master gyroscope; r m is the distance along the polar axis of the center of mass of the gyro from its point of support (see Figure 4.1) and d m (t) = ~ d m sin(! m t) is the time-varying amplitude of the vertical support motion that has frequency ! m . The nonconservative damping force acting on the master gyroscopeF d =~ c m _ m ~ e m _ 3 m is assumed to be of linear plus cubic type [23]. A stationary symmetric gyro positioned with its axis along the vertical usually falls over. However, when it is given a suciently large spin about the vertical axis, it begins to rotate in a stable fashion, with its axis remaining very close to the vertical. The symmetric gyro showing this type of sleeping top motion is said to be in a `sleeping position' and this usually happens when the angular momenta 71 Figure 4.1: Symmetrical master gyro (independent) subjected to a vertical harmonic excitation Figure 4.2: A system of chain-coupled slave gyroscopes; each slave gyro in the chain is subjected to a vertical harmonic excitation which, when uncoupled from the system, results in regular or chaotic motion 72 p =p ' =p is constant for all time t. The equation of motion of the vertically excited symmetric master gyro with linear plus cubic damping (see equation 4.1) in the `sleeping position' is then given by m + 2 m (1 cos m ) 2 sin 3 m +c m _ m +e m _ 3 m m sin m + m sin m sin(! m t) = 0 (4.2) where P m = ( m = p m I m ; m = m m gr m I m ;c m = ~ c m I m ;e m = ~ e m I m ; m = m m r m ~ d m ! 2 m I m ;! m ;I m ) is the parameter set that describes the physical characteristics of the master gyroscope. 4.2.2 Slave Gyroscopes Consider the system of n symmetric gyroscopes which are coupled in chain fashion as shown in Figure 4.2. The subscripti (wherei = 1; 2;:::;n) refers to thei th gyroscope in the slave system. Like the master gyroscope, each individual slave gyro is subjected to a vertical harmonic base excitation of amplitude ~ d i and frequency ! i and a nonconser- vative damping force which is of linear plus cubic type related only to the -coordinate. The following types of torsion spring couplings are studied: 1. Harmonic coupling : V t i ( i+1 i ) = i [1 cos( i+1 i )] (4.3) 2. Toda torsion coupling [72] : V t i ( i+1 i ) = a i b i e b i ( i+1 i ) a i ( i+1 i ) a i b i (4.4) 3. Linear plus cubic coupling : V t i ( i+1 i ) = f i 2 ( i+1 i ) 2 + g i 4 ( i+1 i ) 4 (4.5) 73 where i ;a i ;b i ;f i ; and g i are the spring constant parameters corresponding to the i th spring element and V t i is the torsional potential energy of the i th spring element in the chain. Consider thei th gyroscope in the slave system. Besides the forcesF gyro i (see equation 4.6) that act on a symmetric gyro as a consequence of it being subjected to nonlinear damping and vertical excitation, the only additional forces that act on it are the coupling forces. Thus, the equation of motion of the i th slave gyroscope in the i direction can be computed using Newton's laws as follows: i + (p ' i p i cos i ) (p i p ' i cos i ) I 2 i sin 3 i + ~ c i I i _ i + ~ e i I i _ 3 i m i gr i I i sin i m i r i I i sin i d i (t) | {z } F gyro i = 1 I i @V t ( i+1 i ) @ i @V t ( i i1 ) @ i | {z } F S i+1;i F S i;i1 (4.6) where p i = I 3 i _ i + _ ' i cos i , p ' i = I i _ ' i sin 2 i +p i cos i , d i (t) = ~ d i sin(! i t) and F sine i+1;i = i sin ( i+1 i ) for a harmonic coupling, F Toda i+1;i = a i e b i ( i+1 i ) 1 for a Toda coupling andF LC i+1;i =f i ( i+1 i )+g i ( i+1 i ) 3 for a linear plus cubic coupling. Because, i and ' i are cyclic coordinates, equation (4.6) alone is sucient to describe the equation of motion of the i th slave gyro. Equation (4.6) can be represented more conveniently as i =F gyro i + 1 I i F S i+1;i F S i;i1 (4.7) where F gyro i are the gyroscopic forces acting on the i th gyro from it being subjected to vertical excitation and nonlinear damping and F S i+1;i is the coupling force exerted by the (i + 1) th gyro on the i th gyro. In the present study, we approach the problem of synchronization of multiple gyroscopes from a constrained motion perspective, which includes three vital steps. The rst step involves the derivation of the equations of motion of the unconstrained system. The second step involves the formulation of the constraint equations and the last step deals with the use of fundamental equation [81, 83, 84] to obtain the constrained equations of motion of synchronized master-slave system. In the process, we also nd a closed form expression for the explicit nonlinear control 74 torques that are required to be applied to each of the slave gyroscopes to synchronize them precisely with the motion of the master gyroscope. 4.2.3 Unconstrained Equations of Motion In the present study, the independent master gyroscope (equation 4.1) along with the system of coupled slave gyroscopes (equation 4.6) forms the unconstrained (uncon- trolled) system [83]. The unconstrained equation of motion of the master-slave system can be written down in matrix form (using equations 4.2 and 4.7) as follows M (n+1)(n+1) 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 m 1 . . . i . . . n 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 |{z} q (n+1)1 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 F gyro m F gyro 1 + F S 2;1 . I 1 . . . F gyro i + F S i+1;i F S i;i1 . I i . . . F gyro n + F S n;n1 . I n 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 | {z } F (n+1)1 (4.8) where M = I (n+1) is an identity matrix of size n + 1. The unconstrained acceleration of the system is can be computed using equation (2.2) and is given by a =M 1 F =F . To this unconstrained system, we impose a set of constraints such that the system of coupled slave gyros precisely follows the master gyroscope in its motion. 4.2.4 Formulation of the Constraints The synchronization problem of the master gyroscope with the coupled slave system of gyroscopes is interpreted in terms of the following set of constraints being imposed on the unconstrained system (equation 4.8). These set of constraints fall into two categories. 1. No Control Force on the Master Gyro : Since the motion of the master gyroscope is considered to be independent of the motion of the slave gyroscopes; the need to apply a control force on the master gyro does not arise in the constrained system. 75 Hence, the unconstrained motion of the master gyroscope can itself be considered as a constraint equation. Thus, we have m =F gyro m (4.9) 2. Synchronization Constraint - Next, our goal is to synchronize the motion of each of the coupled slave gyros with that of the master gyro. Hence, the following n trajectory tracking requirements are imposed as constraints on the unconstrained system i = m i = 0; i = 1; 2;:::;n: (4.10) However, it may not always be possible to initiate the slave gyros from positions in phase space that satisfy the constraints (equation 4.10). This problem is sur- mounted by choosing appropriate trajectory stabilization parameters c;k [78, 79] such that the constraints can be modied as follows i +c _ i +k i = 0; i = 1; 2;:::;n: (4.11) Thus, the unconstrained system (equation 4.8) is subjected to a total ofn+1 cumulative constraints in order to achieve synchronization of each individual coupled slave gyro with the motion of the master gyro. When the constraints (equations 4.9 and 4.11) are expressed in the constraint matrix form, we obtain 2 6 6 6 6 6 6 6 6 6 6 6 4 h 1 i h 0 0 i 2 6 6 6 6 4 1 . . . 1 3 7 7 7 7 5 h I nn i 3 7 7 7 7 7 7 7 7 7 7 7 5 | {z } A (n+1)(n+1) 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 m 1 . . . i . . . n 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 |{z} q (n+1)1 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 F gyro m c _ m _ 1 k ( m 1 ) . . . c _ m _ i k ( m i ) . . . c _ m _ n k ( m n ) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 | {z } b (n+1)1 (4.12) 76 4.2.5 Constrained Equations of Motion With all the requisite matrices (M;F;A;b) at our disposal, the constrained equations of motion can now be calculated using the fundamental equation of mechanics [83, 87]. Since, M =I n+1 and A is a square matrix of size n + 1, the expression for the control force reduces to F c =M 1=2 AM 1=2 + (bAa) =A + (bAa) =A 1 (bAa) (4.13) The constraint matrixA is structured in such a way that it possesses the unique property A 1 =A, and this reduces the explicit expression of the nonlinear control torques to F c =A 1 (bAa) =A 1 bA 1 Aa =Aba = 2 6 6 6 6 6 6 6 6 6 6 6 4 h 1 i h 0 0 i 2 6 6 6 6 4 1 . . . 1 3 7 7 7 7 5 h I nn i 3 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 F gyro m c _ m _ 1 k ( m 1 ) . . . c _ m _ i k ( m i ) . . . c _ m _ n k ( m n ) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 F gyro m F gyro 1 + F S 2;1 I 1 . . . F gyro i + F S i+1;i F S i;i1 I i . . . F gyro n + F S n;n1 In 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 h c _ m _ 1 +k ( m 1 ) i + F gyro m F gyro 1 + F S 2;1 I 1 . . . h c _ m _ i +k ( m i ) i + F gyro m F gyro i + F S i+1;i F S i;i1 I i . . . h c _ m _ n +k ( m n ) i + F gyro m F gyro n + F S n;n1 In 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (4.14) wherec;k are the trajectory stabilization parameters,F gyro are the gyroscopic forces and F S are coupling forces. Thus, equation (4.14) gives us the explicit closed form expression for the nonlinear control torques, devoid of any approximations or simplications, that 77 need to be applied to the unconstrained (uncontrolled) system to obtain the constrained (controlled) system. The constrained (controlled) equations of motion of the master- slave system [83, 84, 87], where the coupled system of slave gyros precisely track the motion of the master gyro, can now be written as M q =F +F c (4.15) whereF is given by the right hand side of equation (4.8) andF c is given by the equation (4.13). Equation (4.15) represents the governing equations of motion of the coupled system of slave gyroscopes that follow the (independent) motion of a given master gyroscope, irrespective of its chaotic or regular motion. 4.2.6 General Coupled Slave Gyroscopes Until now, we have considered the case wherein the slave gyros are all coupled in a chain fashion. But it is indeed possible to consider a more general interaction between the slaves wherein a particular slave gyro can be coupled to any number of other slave gyros without any restrictions. Now, in such a case, it becomes imperative to keep track of the coupling information between any two slave gyros in the system. This can be succinctly done by the use of the incidence matrix. The incidence matrix has the following salient features: The incidence matrix, denoted by , is a symmetric matrix of size n (where n denotes the number of slave gyros). It gives us information about the coupling between any two slave gyroscopes in the system. Each element of the incidence matrix is restricted to have a value of either 0 or 1. A zero value of ij indicates that thei th andj th slaves are uncoupled. Contrarily, a unitary value of ij indicates that the i th slave is coupled to the j th slave. Because a gyroscope cannot be coupled to itself, the diagonal elements of the incidence matrix are all zero (i.e., ii = 0; i = 1; 2; 3;:::;n). 78 Thus, given an incidence matrix that describes the various dierent interactions between a set of coupled slave gyros, the unconstrained equations of motion of the i th slave gyro in the system can be written down as follows i =F gyro i + 1 I i 0 B B @ n X j=1 j6=i ij F Couple ij 1 C C A ; i = 1; 2; 3;:::;n; (4.16) whereF Couple ij =F Couple ji . This set ofn equations along with the unconstrained motion of the independent master gyro (equation 4.2) forms the unconstrained (uncontrolled) system for the master and general coupled system of slave gyros. The reader can now utilize the methods described in this section to re-derive the constrained (controlled) equations of motion of the master and general coupled slave system. One can also compute the analytical expressions for the control torques that are required to be applied to the slave system of gyroscopes such that they synchronize precisely with the motion of the master gyroscope. 4.3 Results and Simulations To better illustrate the ecacy of the methods presented in this chapter, we present three numerical simulations. In the rst example, we consider a system of ve gyros (1 Master + 4 Slaves) with the sleeping condition imposed on all ve gyros. We examine the eect of sinusoidally chain coupling the slave gyros on the synchronization of the gyros. In the second example, we consider a system of six gyros (1 Master + 5 Slaves) and use an incidence matrix to describe the various dierent interactions between the ve coupled slave gyros. The sleeping condition is imposed on all six gyros and dierent types of linear and nonlinear couplings are used between the individual slaves. In the nal example, we consider a ve gyro system (1 Master + 4 Slaves) and explore the eect of the sleeping condition on the synchronization of the gyros. Throughout this section, the integration of the equations of motion (constrained as well as unconstrained) is performed using the ODE45 solver from the software MAT- LAB with a relative error tolerance of 10 9 and an absolute error tolerance of 10 12 . 79 Further, the Lyapunov exponents for the master-slave system of gyros are computed using the methods described in references [94, 95, 96] over a time span of 1000 seconds. To determine these exponents, integration has been performed, once again, using the MATLAB ODE45 solver with a relative error tolerance of 10 9 and an absolute error tolerance of 10 13 . 4.3.1 Synchronization with the use of a Sleeping Condition 4.3.1.1 Five Gyroscopes using a Sine Coupling Consider a system of ve non-identical gyroscopes (1 Master + 4 Slaves) where the set of four slave gyroscopes are all connected in a chain fashion using a sinusoidal coupling (described by equation 4.3) with coecients 12 = 2; 23 = 1:5; 34 = 2:5. All gyros are assumed to be in the sleeping position and so we have p ' i =p i =p i ; i =m; 1; 2; 3; 4: Each individual slave gyro (although coupled to other slaves) is required to precisely track the motion of the master gyro. Table 4.1 gives the parameter sets P i =f i ; i ; c i ; e i ; i ; ! i ; I i g; i =m; 1; 2; 3; 4; that describe the characteristics of each of the gyros of the master-slave system and their initial condition sets IC i = h 0 i ; _ 0 i ; 0 i i ; i =m; 1; 2; 3; 4: Using the parameters in Table 4.1, the Lyapunov exponents for the master gyroscope and the system of slave gyros are computed to be l m ; l s , respectively as shown below. l m =f0:179707; 0:500232; 0g 80 Name Parameter Sets Initial Conditions Master Gyro (Periodic) P m =f10:5; 1; 0:5; 0:02; 38:7; 2:2; 1g IC m = [1; 0:5; 0] 1st Slave Gyro P 1 =f101; 0:5; 0:05; 35:5; 2; 1g IC 1 = [0:5; 1; 0] 2nd Slave Gyro P 2 =f10:5; 1; 0:5; 0:04; 38:5; 2:1; 2g IC 2 = [1;0:5; 0] 3rd Slave Gyro P 3 =f10; 1; 0:5; 0:03; 35:8; 2:05; 1:5g IC 3 = [0:5; 1; 0] 4th Slave Gyro P 4 =f10:5; 1; 0:45; 0:045; 36; 2:05; 1:7g IC 4 = [0:5; 0:5; 0] Table 4.1: Parameter and Initial Conditions Sets for the Five Gyroscopes. Because the values of l m are either negative or zero, the master gyro is said to exhibit regular periodic motion. The slave system on the other hand appears to be chaotic. l s = 8 < : 0:121455; 0:066865; 0:004824; 0:039699; 0:253232; 0:000107; 0:568669; 0:645423; 0:000110; 0:728986; 0:777867; 0 9 = ; The unconstrained system consists of the master gyroscope and the set of four coupled slave gyroscopes whose equations of motion can be written down using equation (4.8). Figure 4.3 shows a phase plot ( m ; _ m ) of the unconstrained motion of the master gyroscope. Clearly, the motion of the master is periodic, as predicted by its Lyapunov exponents. Figure 4.4, on the other hand, shows a phase plot ( i ; _ i );i = 1; 2; 3; 4 of the unconstrained motion of the individual slave gyroscopes. The phase plots (Figures 4.3 - 4.4) are all plotted in the time range 150t 200 seconds. A superimposed image of the time history of the nutation angle of the ve gyros is plotted in Figure 4.5 showing that prior to synchronization, the ve gyros exhibit highly nonlinear behavior and their trajectories vary widely from each other. To this unconstrained system, we impose a set of ve constraints (as described by equation (4.12) with trajectory stabilization constants c = 1 and k = 2) and compute the constrained equations of motion using equation (4.15). A superimposed phase plot of the constrained motion of the master- slave system of gyros is plotted in Figure 4.7 for 150 t 200. As can be seen from the gure, the four slave gyros which are all sinusoidally (chain) coupled to each other synchronize precisely with the periodic motion of the master gyroscope. Figure 4.6 shows the time history of nutation angle post-synchronization; the control forces take 81 Figure 4.3: A phase plot ( m ; _ m ) of the unconstrained motion of the periodic master gyroscope for 150t 200. Figure 4.4: A phase plot of the unconstrained motion of four slave gyroscopes which are all coupled to one another in a chain fashion (150t 200). 82 Figure 4.5: Time history of nutation angle for each individual gyroscope for 0t 20 prior to synchronization Figure 4.6: Time history of nutation angle for each individual gyroscope for 0t 20 in the constrained system. Slave gyros take less than 10 seconds to synchronize with the motion of the master gyro. Figure 4.7: A superimposed phase plot of the constrained motion of ve gyroscopes for 150 t 200. The slave gyroscopes have precisely synchronized with the motion of the master gyro. 83 Figure 4.8: Time history of synchronization errors for 60t 100. Errors are smaller than the tolerance levels of the integration scheme. Figure 4.9: Time history of Nonlinear Control Torques F c acting on each individual gyroscope for 0t 200 84 less than 10 seconds to exponentially reduce the error between the motion of master and the individual slaves to zero. Figure 4.8 shows a time history of synchronization errors for 60 t 200. Note that the error converges exponentially as demanded by equation (4.11). The synchronization errors are approximately of the order of 10e 13 , which is lower than the tolerance levels used in MATLAB ODE45 integration solver, thereby showing the ecacy of the control forces in achieving synchronization. Thus, a system of four non-identical slave gyros which are sinusoidally chain coupled to each other have been shown to precisely track the motion of a independent master gyroscope exhibiting periodic motion. 4.3.1.2 Six Gyroscopes using an Incidence Matrix In the present example, we consider a system of six non-identical gyroscopes (1 Master + 5 Coupled Slaves) where the interaction between the ve slaves is more general and is described by an incidence matrix. The ve coupled gyros are required to precisely track the motion of the independent master gyro (described by the parameter set P m shown in Table 4.2), which in this case exhibits chaotic motion as seen from its Lyapunov exponents l m =f0:206668; 0:896620; 0g: (4.17) The coupling between the slaves (with parameter sets P i ;i = 1; 2; 3; 4; 5 given in Table 4.2) is described by the 5 5 incidence matrix , shown as = 2 6 6 6 6 6 6 6 6 6 6 4 0 1 T 0 1 L 1 S 1 T 0 1 S 0 1 L 0 1 S 0 1 LC 0 1 L 0 1 LC 0 0 1 S 1 L 0 0 0 3 7 7 7 7 7 7 7 7 7 7 5 (4.18) The interaction between thei th andj th slaves is characterized by the value of the (i;j) th element of the matrix . The subscripts L;S;T; and LC of the (i;j) th element of the 85 Figure 4.10: General-coupled slave gyroscope system depicting various types of cou- pling/interactions between the slaves. incidence matrix denotes Linear, Sinusoidal, Toda, and Linear plus cubic couplings, re- spectively. Figure 4.10 depicts a pictorial representation of the incidence matrix showing the various types of couplings applied between the individual slave gyroscopes. The un- constrained equations of motion of the master (equation 4.2) and the ve coupled slave gyroscopes (equation 4.16) which are all assumed to be in the sleeping position can now be written down as m =F gyro m 1 =F gyro 1 + 1 I 1 F Toda 12 +F Linear 14 +F Sine 15 2 =F gyro 2 + 1 I 2 F Toda 21 +F Sine 23 +F Linear 25 3 =F gyro 3 + 1 I 3 F Sine 32 +F Linear + Cubic 34 (4.19) 4 =F gyro 4 + 1 I 4 F Linear 41 +F Linear + Cubic 43 5 =F gyro 5 + 1 I 5 F Sine 51 +F Linear 52 86 Figure 4.11: A phase plot of the unconstrained motion for each individual gyroscope for 50 t 100. The master gyroscope is independent whereas the other remaining ve slave gyroscopes are all coupled using the incidence matrix. Figure 4.12: A superimposed phase plot of the constrained motion of ve gyroscopes for 50 t 100 (left) and 150 t 200 (right). Slave gyros have precisely tracked motion of the master gyro. 87 Figure 4.13: Time history of nutation angle for each individual gyroscope for 0t 20 prior to synchronization Figure 4.14: Time history of nutation angle for each individual gyroscope for 0t 20 in the constrained system. whereF gyro i is dened in equation (4.6);F Toda ij =a ij e b ij ( j i ) 1 ;F Sine ij = ij sin( j i ); F Linear ij = ij ( j i ); F Linear + Cubic ij = f ij ( j i ) +g ij ( j i ) 3 ; and F Couple ij = F Couple ji 8 i < j; i;j = 1; 2; 3; 4; 5. The spring constants are given by the following parameters 8 < : a 12 = 1:5; b 12 = 0:05; 14 = 0:5; 15 = 2; 23 = 1:2; 25 = 0:3; f 34 = 0:8; g 34 = 0:04 9 = ; 88 Figure 4.15: Time history of synchronization errors for 60t 200. Errors are smaller than the tolerance levels of the integration scheme. Figure 4.16: Generalized control torques acting on each individual slave gyroscope for 0t 200. From these parameters, the Lyapunov exponents, l s , for the system of slave gyros are calculated to be l s = 8 < : 0:08267; 0:05103; 0:01575; 0:00752; 0:07520; 0:00019; 0:42477; 0:48301; 0:00040;0:52319;0:64614; 0:000002;0:72396;0:78758; 0 9 = ; Based on these exponents, the slave system appears to be chaotic. Figure 4.11 shows a phase plots ( i ; _ i ); i =m; 1; 2; 3; 4; 5 of the unconstrained system (equation 4.19) for 89 Name Parameter Sets Initial Conditions Master Gyro (Chaotic) P m =f10; 1; 0:5; 0:03; 35:8; 2:05; 1g IC m = [0:5; 1; 0] 1st Slave Gyro P 1 =f10; 1; 0:5; 0:05; 35:5; 2; 2g IC 1 = [0:5; 1; 0] 2nd Slave Gyro P 2 =f10:5; 1; 0:5; 0:04; 38:5; 2:1; 1:3g IC 2 = [1;0:5; 0] 3rd Slave Gyro P 3 =f10:5; 1; 0:5; 0:02; 38:7; 2:2; 0:9g IC 3 = [1; 0:5; 0] 4th Slave Gyro P 4 =f10:5; 1; 0:45; 0:045; 36; 2:05; 1:5g IC 4 = [0:5; 0:5; 0] 5th Slave Gyro P 5 =f10:5; 1; 0:5; 0:05; 38:5; 2; 1:7g IC 5 = [0:5;1; 0] Table 4.2: Parameter and Initial Conditions Sets for the Six Gyroscopes. 50t 100. The master gyroscope exhibits chaotic motion as predicted by l m . From Figure 4.11, we can infer that all six gyros exhibit highly nonlinear behavior and are unsynchronized. The constrained (synchronized) equations of motion for this system of six gyros can once again be computed using equation (4.15) with the application of six constraints (see equation 4.12). The phase plots ( i vs. _ i ); i = m; 1; 2; 3; 4; 5 of the constrained system have been plotted in Figure 4.12 for (a) 50 t 100 and (b) 150 t 200, where we observe that the plot of each gyro has been superimposed on top of another, indicating synchronization of the ve slave gyros with the chaotic motion of the master gyro. Figure 4.14 shows a time history of the nutation angle for 0 t 20. As can be inferred from the gure, it takes less than 10 seconds for the slave gyros to track the motion of the chaotic master gyro. Figure 4.15 shows a time history of the synchronization errors in for each slave gyroscope in the time range of 60t 200. The vertical scale of the gure shows that the error is of the order of 10 13 , signifying that the errors in the simulation are within the tolerance levels specied in the MATLAB ODE45 integration solver. A time history of generalized control torques applied to each individual slave gyroscope to achieve synchronization with the chaotic master gyro is shown in Figure 4.16. Thus, given a set of ve nonidentical general- coupled slave gyros where the coupling between the gyros is prescribed by the incidence matrix, we managed to precisely synchronize the motion of the each of the slave gyros with that of the chaotic master gyro, something quite signicant for chaotic systems since they show extremely sensitive dependence to initial conditions. 90 4.3.2 Synchronization without the use of the Sleeping Condition In this section, we attempt to synchronize a set of nonidentical gyroscopes without using the so-called sleeping condition and show that synchronization can indeed be obtained even without the need to assume that the gyros are in sleeping position. If we remove the simplication of a sleeping condition, then we no longer assume that p ' i =p i =p i and hence p ' i ; p i can be chosen to be two dierent constants. Angular momenta p ' i and p i are associated with cyclic coordinates' i and i , respectively and are therefore constant and cannot change with time. The imposition of the no-sleeping condition brings in an additional two parameters p ' i ; p i into the dynamics of each individual gyroscope and they are appended at the end of the setP to give the new parameter set P i = ( i = 1 I i ; i = m i gr i I i ; c i = ~ c i I i ; e i = ~ e i I i ; i = m i r i ~ d i ! 2 i I i ; ! i ; p ' i ; p i ) 4.3.2.1 Five Gyroscopes using Toda Coupling Consider a system of ve nonidentical gyroscopes (with parameter sets given by Table 4.3) where the set of four slave gyros are coupled in a chain fashion using a Toda torsion spring coupling. Each individual coupled slave gyro is required to exactly track the motion of the master gyro which, in this case, exhibits regular periodic motion as l m =f0:223403; 0:226832; 0g The coecients of the Toda spring elements are given by a 12 = 1:20; a 23 = 1:50; a 34 = 1:25; b 12 = 0:11; b 23 = 0:13; b 34 = 0:105: 91 Figure 4.17: Superimposed phase plot depicting the unconstrained motion of the peri- odic master gyro along with the four chain coupled slave gyros for 50t 100. Figure 4.18: A superimposed phase plot of the constrained motion of ve gyroscopes for 50t 100. The slave gyros have precisely tracked motion of the master gyro. 92 Figure 4.19: Time history of nutation angle for each individual gyroscope for 0t 20 prior to synchronization Figure 4.20: Time history of nutation angle for each individual gyroscope for 0t 20 in the constrained system. Figure 4.21: Time history of synchronization errors for 60t 200. Errors are smaller than the tolerance levels of the integration scheme. 93 Figure 4.22: Generalized control torques acting on each individual slave gyroscope for 0t 100. The system of slave gyros appears to exhibit chaotic behavior as indicated by its Lya- punov exponent set l s = 8 < : 0:107805; 0:015994; 0:000394; 0:121534; 0:634956; 0:000173; 1:390733; 1:850994; 0:000002; 1:881298; 1:976043; 0 9 = ; Because the simplication of the sleeping condition is removed, the unconstrained equa- tions of motion of the master and the individual slave gyros are given by equations (4.1) and (4.6), respectively. A superimposed phase plot of the unconstrained motion of the periodic master gyroscope ( m ; _ m ) along with the unconstrained motion of each individual coupled slave gyro ( i ; _ i ); i =m; 1; 2; 3; 4 is shown in Figure 4.17. The phase plots are all plotted in the time range 50 t 100. The nonidentical gyros exhibit highly nonlinear behavior as expected with their trajectories varying widely from each other. The constrained equations of motion of the master-slave system of gyroscopes are computed using equation (4.15) with the application of ve constraints as described by equation (4.12). A superimposed phase plot of the constrained motion of the master- slave system of gyros is plotted in Figure 4.18 for 50t 100. As can be seen from the gure, the four slave gyros (which are all chain-coupled) synchronize exactly with the periodic motion of the master gyroscope. The phase plot of each gyro has superposed on 94 Name Parameter Sets Initial Conditions Master Gyro P m =f2:5; 1:5; 0:45; 0:045; 35; 3; 1:5; 2:5g IC m = [1;0:5; 0] 1st Slave Gyro P 1 =f1:35; 1:1; 0:44; 0:044; 36:1; 3; 1:9; 2:1g IC 1 = [0:5; 1; 0] 2nd Slave Gyro P 2 =f1; 1:0; 0:44; 0:044; 35; 2:5; 2; 2:2g IC 2 = [1;0:5; 0] 3rd Slave Gyro P 3 =f1:04; 1:0; 0:45; 0:045; 36:0; 2:06; 1:5; 1:8g IC 3 = [1;1; 0] 4th Slave Gyro P 4 =f1; 1:1; 0:45; 0:045; 36:0; 2:05; 2:05; 2:15g IC 4 = [0:5; 0:5; 0] Table 4.3: Parameter and Initial Conditions Sets for the Five Gyroscopes. top of the other, indicating precise synchronization with the master gyro. Figure 4.20 shows the time history of nutation angle for the ve gyros post-synchronization. The control torques take less than 10 seconds to synchronize the motion of individual slaves with that of the periodic master gyroscope. Figure 4.21, on the other hand, shows the time history of synchronization errors for 60t 100. The exponential convergence of the errors to zero is evident. And nally, Figure 4.22 shows a time history of the nonlin- ear control torques acting on the individual slave gyroscopes to obtain synchronization with the master gyro. Thus, we conclude that the removal of the sleeping condition has had little eect on the synchronization of the master-slave system of gyros. 4.4 Conclusions In the present study, we consider the problem of synchronization of a system of n chain-coupled or generally-coupled slave gyroscopes with that of a master gyroscope. However, this classical problem of tracking is approached from a constrained motion perspective. The tracking requirements are recast as constraints and the fundamental equation of mechanics is used to obtain the constrained (synchronized) equations of motion of the master-slave system of gyroscopes. In the process, explicit closed form expressions for the nonlinear control torques are calculated that are used to drive the system of coupled slave gyroscopes to synchronize exactly with the motion of the master gyroscope (irrespective of the chaotic or periodic behavior displayed by the master gyro or the slave system of gyroscopes). 95 1. Previous investigators concern themselves with the synchronization of a set of one or two uncoupled slave gyroscopes with the motion of a master gyroscope. But the theory developed herein allows for a set of n slave gyroscopes that are all coupled to one another to synchronize exactly with the motion of the master gyro. Each slave gyro is a highly nonlinear system. In addition, the slaves are nonlinearly coupled to one another with dierent types of couplings and this leads to a highly nonlinear, complex dynamical system. However, the control torques are found with relative ease showing the power and ecacy of the underlying control methods. 2. The control torques derived in this study are continuous in time and in theory lead to exact synchronization of the master-slave system of gyroscopes. No ap- proximations have been made in the derivation of these nonlinear control torques, or in approximating the highly nonlinear dynamics of either the master or any of the coupled slave gyroscopes. 3. The equations of motion of master-slave system of gyros are derived without ap- plying the simplication of the sleeping condition. Although many authors on this subject assume the gyros to be in sleeping position, in general however, most gyros do not satisfy this condition. In this study, it is shown that removal of the sleeping condition has little eect on the synchronization of the gyroscopes as is evident from numerical simulation. 4. To show the ecacy of the methods presented in this study, numerical simulations involving multiple non-identical gyroscopes, with various types of couplings be- tween the slave gyros, is presented. In all the cases, the control torques provided by the fundamental equation leads to an exact synchronization of the master- slave system of gyros irrespective of the chaotic or regular motions exhibited by the individual gyroscopes. It should be noted that the methods presented in this study are applicable to any number of slave gyroscopes and given any type of gen- eral, nonlinear coupling among them. The speed with which synchronization is 96 achieved can be controlled by appropriately modifying the trajectory stabilization parameters. 97 Part 4 Incompressible Hyperelastic Beams Chapter 5 Dynamics of Incompressible Hyperelastic Beams 5.1 Introduction T his chapter deals with modeling the dynamics of a highly exible three di- mensional rubber-like incompressible hyperelastic beam that exhibits large deformation and large rotation in the nonlinear elastic range. Accurate mod- eling and simulation of highly exible continuous structures involving both geometric and material nonlinearities is of tremendous importance in a wide range of elds such as aerospace, medicine, biomechanics, animation, manufacturing and the rubber and poly- mer industry. Inspite of the signicant geometric and material nonlinearities that arise in these applications, many existing models in the literature use classical Euler-Bernoulli and Timoshenko beam theories. These classical theories have many in-built simplifying assumptions in them making them incompatible for use with structures that exhibit large deformations and/or large rotations [43]. In addition, an approximate description of the elastic forces is used in most such models because of the diculty in incorporating the nonlinear constitutive material characteristics [42]. To counter these issues, in the present study, we employ the absolute nodal coordinate formulation (ANCF) [64, 102] to accurately describe the dynamics of large deformations, and use nonlinear constitutive 99 models along with the continuum mechanics approach to precisely compute the highly nonlinear elastic forces. In the literature, there exist many nite element formulations to handle large defor- mation and large rotation problems. However, the absolute nodal coordinate formula- tion (ANCF) stands apart because of certain key attributes. Unlike the Euler-Bernoulli and Timoshenko beam theories, the ANCF formulation is accurate, allowing for rotation and deformation of the nite element beam cross-section [64]. The ANCF formulation also captures the Poisson modes that result from coupling between the stretch and bend- ing and the deformation of the beam cross section [43]. More importantly, the ANCF formulation uses a global coordinate frame to model bodies (either rigid or exible or a combination of both) and this leads to a symmetric constant mass matrix with no centrifugal or Coriolis forces thereby simplifying the equations of motion, unlike other similar three dimensional formulations that lead to highly nonlinear inertia forces [63]. The elastic forces, on the other hand, are highly nonlinear and a major portion of the computational eort is spent on computing these elastic forces in ANCF. To describe orientation, global slope coordinates are used instead of rotations and no restrictions are imposed on the magnitudes of the rotations [102]. There are also no restrictions imposed on the amount of deformation within the nite element. These attributes make ANCF ideal for application in large deformation and large rotation problems and allow for complex shapes to be represented using a relatively small number of nite elements when compared to other nite element formulations [47]. It is also straightforward to incorporate general linear and nonlinear constitutive models in the framework of ANCF [43], thus allowing for the use of the general continuum mechanics approach to compute the elastic forces. In highly exible materials like rubber and biological tissue, linear elasticity is in- sucient to describe the observed material behavior. These materials exhibit large de- formations in the nonlinear elastic range with negligible internal energy dissipation [53]. They return to their original shape once the loading is removed and exhibit no observ- able hysteretic behavior. Thus, the loading and unloading stress-strain curves of these materials are near-coincident and so the stress in these materials is determined by the 100 current state of deformation and is largely independent of the path or history of defor- mation [30]. Nonlinear elastic materials exhibiting the properties described above are called as hyperelastic materials. Hyperelastic materials (also called as Green elastic materials) are characterized by the existence of a strain energy density function (often called the Helmholtz free energy density function) [30, 53] and the stress-strain relationship in these materials is derived from the strain energy density function. For homogeneous and isotropic hyperelastic materials (e.g. rubber), the strain energy density function is simply a function of the invariants of the strain tensor. Mooney [48] and Rivlin [61] developed nonlinear con- stitutive models to describe the strain energy density of such hyperelastic materials. Materials like rubber are not only homogeneous and isotropic, but also incompressible. To model the incompressibility of these materials, one approach is to add a volumetric energy penalty function to the strain energy density function. By using a high value of the penalty coecient, the incompressibility of the material can be maintained at each instant of time. However, it has to be noted that a high value of this penalty coecient can lead to Poisson (volumetric) locking eects and methods to eliminate these locking eects are discussed in this study. A major portion of the ANCF literature until now has been focused on incorpo- rating linear elastic models within the framework of ANCF theory, and resolving the underlying numerical problems therein [27]. The use of nonlinear constitutive models in the framework of ANCF is relatively new and to the author's knowledge is limited to the following studies: Maqueda & Shabana [43], Jung et al. [31] and Orzechowski et al. [54]. The theoretical foundations for incorporating nonlinear constitutive models within the framework of ANCF were laid down by Maqueda & Shabana [43]. They provided the explicit expressions for the generalized elastic forces of a beam element by dierentiating the strain energy density function with respect to the nodal coordinates. However, they assume the deviatoric part of the strain energy density to be a function of the total strain invariants instead of their deviatoric components. The numerical example they provide { that of a exible pendulum { was perhaps too simplistic to 101 highlight some of the adverse locking eects that can be encountered using the ANCF approach when used for near-incompressible material behavior [54]. Jung et al. [31], on the other hand, carry out a computational-cum-experimental study on a thin, exible, hyperelastic, cantilever beam that is released from a horizontal rest position in a constant gravitational eld. They use the computational approach de- veloped by Maqueda & Shabana [43] to predict the time-dependent deformations of the cantilever beam and compare their numerical results with the data obtained from their experiments. They use ve nite-elements to model the beam and show a remarkably close match between their experimental and computational results. However, we are un- able to replicate the computational results reported in Ref. [31] despite using the same parameter values and constants reported therein. Unlike the results shown in Ref. [31], the ve-element beam is found to exhibit extremely sti bending behavior and appears to be in uenced by signicant locking eects. Ref. [31] contained no description of the boundary conditions of the cantilever beam (i.e., whether the left end of the beam is partially-clamped or fully-clamped), no description of the spatial (volume) integration approach used to integrate the strain energy density function, no description of the time integration method used to integrate the equations of motion of the beam and no mention of the locking eects that other researchers [27] have been reporting when considering thin elastic beams in the ANCF framework. Furthermore, the convergence studies carried out in the present work point out that when dealing with the classical ANCF element [64, 102], the use of just ve nite elements is too small a number to obtain a proper description of the time dependent behavior of such a cantilever. Some of these observations on the use of the ANCF method for simulating the dynamic response of hyperelastic structural systems have, in part, motivated the author to take a closer look at this problem. Although, Orzechowski et al. [54] address some of these issues, a comprehensive treatment of the problem appears to be missing in the literature and it is therefore the objective of this study to provide a detailed solution to the problem. 102 In this study, we consider a three-dimensional rubber-like incompressible hypere- lastic beam and derive the dynamic equations of motion of the beam using the abso- lute nodal coordinate formulation. Three nonlinear constitutive laws for incompress- ible materials (i.e., the incompressible Neo-Hookean material model, the incompressible Mooney-Rivlin material model and the incompressible Yeoh material model) are consid- ered and the correct expressions for the elastic force vectors are derived for each of these cases. The methods described in this study are suitable for large deformation and large rotation applications wherein the material is considered to be incompressible or nearly incompressible. The incompressibility condition is imposed in the dynamic formulation through the use of the penalty method, and the resulting Poisson (volumetric) locking eects due to the high value of the penalty coecient are eliminated through the use of the selective reduced integration [33, 50]. Time integration of the equations of motion is performed using the xed-time step St ormer-Verlet scheme, which is a symplectic scheme that preserves energy throughout the simulation. The example of a thin hy- perelastic cantilever beam considered by Jung et al. [31] is studied. Replicating the set up of Ref. [31], the existence of locking eects is demonstrated, and the elimination of these locking eects using selective reduced integration is shown. Further, the eect of partially clamped and fully clamped boundary conditions on the deformation of the beam is studied. The eect of increasing the number of beam elements on the conver- gence (in the deformation) of the beam is studied. And lastly, the classical ANCF beam element is compared with a lower order 32 10111 3 element [45] and a higher order 3253 element [16] to study the eect of the nature of the interpolation polynomial on the bending deformation in the beam. The chapter is organized as follows. In Section 5.2, a brief review of the absolute nodal coordinate formulation is presented and three dierent types of nite element models are introduced: the classical ANCF element, a lower order 32 10111 3 element and a higher order 3253 element. In Section 5.3, the equations of a motion of a typical beam nite element are derived. Section 5.4 introduces three dierent nonlinear constitutive models: the incompressible Neo-Hookean, the incompressible Mooney-Rivlin and the 103 incompressible Yeoh nonlinear constitutive model to describe the behavior of rubber- like materials. Detailed derivations of the equations of motion of the entire beam are presented in Section 5.5. Section 5.6 discusses two types of xed boundary conditions that are commonly considered in the literature for cantilever beams: the fully-clamped and the partially-clamped boundary conditions. In Section 5.7, the spatial and time integration approaches that are used in this study are discussed. Section 5.8 presents simulation results of a thin hyperelastic cantilever beam. Locking eects as well as their elimination techniques are discussed. Comparative analysis of the dierent beam nite element models, hyperelastic material models and boundary conditions is also performed. And nally, Section 5.9 presents the conclusions of this chapter. 5.2 Absolute Nodal Coordinate Formulation (ANCF) The absolute nodal coordinate formulation (ANCF) is a nite element formulation that was introduced by Shabana [63] in 1996 to handle large deformation and large rotation problems in exible multibody dynamics. In the present section, we brie y review the ANCF approach and introduce three types of beam elements: the classical ANCF ele- ment [64], a lower order 32 10111 3 element [16] and a higher order 3253 beam element [45] that dier in the polynomial used to approximate the displacement eld and the choice of the nodal coordinates. Consider a three-dimensional beam element of length l b , breadth b, and height h (see Figure 5.1). The beam is discretized along its length into n e nite elements each of length l. Figure 5.2 depicts a typical nite element with two nodes (A and B), the beam element (local) coordinate system and global coordinate system that are used in the analysis. The gure also shows an initially undeformed beam element AB, which after rigid body motion and elastic deformation ends up in the deformed state. The global position vector of an arbitrary point P on the beam element in the undeformed state is given by X = [x y z] T ; and in the deformed state is given by r, which is 104 Figure 5.1: A typical three-dimensional cantilever beam Figure 5.2: Deformation of a typical nite element containing two nodes, A and B. 105 assumed to be described by an interpolation polynomial. It is possible to express this position vector r as r =S(x;y;z)e(t) (5.1) where S(x;y;z) is called as the shape function matrix which is only a function of the spatial coordinates of the beam element ande(t) is the vector of nodal coordinates that consist of the global position coordinates and the gradient (or slope) coordinates of the nodes of the beam element. Depending on the choice of the interpolation polynomial used to approximate the displacement eld of the beam element, and the choice of the nodal coordinates, one could theoretically come up with many dierent types of three- dimensional beam elements. In the present study, we restrict our attention to three such elements. 5.2.1 Classical ANCF Element The classical ANCF element, introduced by Yakoub and Shabana [64, 102], is a three- dimensional beam element that is described by the following cubic interpolation poly- nomial. r = 2 6 6 6 6 4 r 1 r 2 r 3 3 7 7 7 7 5 = 2 6 6 6 6 4 a 0 +a 1 x +a 2 y +a 3 z +a 4 xy +a 5 xz +a 6 x 2 +a 7 x 3 b 0 +b 1 x +b 2 y +b 3 z +b 4 xy +b 5 xz +b 6 x 2 +b 7 x 3 c 0 +c 1 x +c 2 y +c 3 z +c 4 xy +c 5 xz +c 6 x 2 +c 7 x 3 3 7 7 7 7 5 31 (5.2) where the coecientsa i ;b i , andc i ;i = 0; 1;:::; 7 are all functions of time. The interpo- lation polynomial is cubic inx, but linear iny andz as the cross-section dimensions are assumed to be small compared to the length of the beam element (i.e., l>>b;l>>h). The inclusion of the constant terms in the displacement eld enable us to capture rigid body translations and the linear terms (x;y and z) help us capture rotary inertia and shear eects of the beam. Each node in the classical ANCF beam element has 12 nodal coordinates given by e j = r j @r j @x T @r j @y T @r j @z T T ; j =A;B; (5.3) 106 wherer j contains 3 components of global position vector of thej th node and @r j @x ; @r j @y ; @r j @z together contain 9 components of the position vector gradients (or slopes) associated with thej th node. The 9 components of position vector gradients account for 3 rotations and 6 deformation modes. The beam element is made up of 2 nodes and hence contains 24 nodal coordinates, which are given by e 241 (t) = h e T A e T B i T (5.4) Using equations 5.3 and 5.4, the displacement eld given by equation 5.2 can be ex- pressed as follows r 31 =S 324 (x;y;z) e 241 (t) = h s 1 I s 2 I s 3 I s 4 I s 5 I s 6 I s 7 I s 8 I i 324 2 4 e A (t) e B (t) 3 5 241 (5.5) where I is the identity matrix of size 3 3, and the shape functions s i ; i = 1; 2;::: 8 are given by s 1 = 1 3 2 + 2 3 s 2 =l( 2 2 + 3 ) s 3 =l(1) s 4 =l(1) s 5 = 3 2 2 3 s 6 =l( 2 + 3 ) s 7 =l s 8 =l where = x=l; = y=l; = z=l and l is the length of the beam element in the unde- formed conguration. 107 5.2.2 Lower Order 32 10111 3 Element The lower order 32 10111 3 element is a linear beam element introduced by Matikainen et al. [45], where the displacement eld given by r = 2 6 6 6 6 4 r 1 r 2 r 3 3 7 7 7 7 5 = 2 6 6 6 6 4 a 0 +a 1 x +a 2 y +a 3 z +a 4 xy +a 5 xz +a 6 yz +a 7 xyz b 0 +b 1 x +b 2 y +b 3 z +b 4 xy +b 5 xz +b 6 yz +b 7 xyz c 0 +c 1 x +c 2 y +c 3 z +c 4 xy +c 5 xz +c 6 yz +c 7 xyz 3 7 7 7 7 5 31 ; (5.6) is linear in x;y and z. Comparing equation 5.6 with equation 5.2, the higher order terms inx (i.e.,x 2 andx 3 terms) are missing in the 32 10111 3 element (and hence in this study, we prex it with the tag `lower order'), but it includes theyz andxyz terms that account for the trapezoidal cross-section mode. Matikainen et al. [45] proposed that the inclusion of this trapezoidal mode helps alleviate some of the Poisson locking eects observed in the classical ANCF element. One of the objectives of our present study is to verify the validity of this statement. The nodal coordinates of the beam element are given bye 241 (t) = h e T A e T B i T where e j = r j @r j @y T @r j @z T @ 2 r j @y@z T T ; j =A;B; (5.7) where now a 2 nd order derivative of the global position vector r is used as a nodal coordinate. Similar to the classical ANCF element (see equation 5.5), we can express r (see equation 5.6) in terms nodal coordinatese(t) (equation 5.7) and the shape function matrix S, where the shape functions s i ; i = 1; 2;::: 8 are given by s 1 = 1 s 2 =l(1) s 3 =l(1) s 4 =l 2 (1) s 5 = s 6 =l s 7 =l s 8 =l 2 where = x=l; = y=l; = z=l and l is the length of the beam element in the unde- formed conguration. 108 5.2.3 Higher Order 3253 Element The higher order 3253 element, introduced by Matikainen et al. [45], is based on the classical ANCF element. In addition to including all the terms in the displacement eld of the classical ANCF element, it also includes the yz and xyz terms that account for the trapezoidal cross-section mode. The interpolation polynomial of the 3253 element is given by r = 2 6 6 6 6 4 r 1 r 2 r 3 3 7 7 7 7 5 = 2 6 6 6 6 4 a 0 +a 1 x +a 2 y +a 3 z +a 4 xy +a 5 xz +a 6 yz +a 7 xyz +a 8 x 2 +a 9 x 3 b 0 +b 1 x +b 2 y +b 3 z +b 4 xy +b 5 xz +b 6 yz +b 7 xyz +b 8 x 2 +b 9 x 3 c 0 +c 1 x +c 2 y +c 3 z +c 4 xy +c 5 xz +c 6 yz +c 7 xyz +c 8 x 2 +c 9 x 3 3 7 7 7 7 5 31 (5.8) where the coecients a i ; b i , and c i ; i = 0; 1;:::; 9 are once again functions of time. Each node of the beam element contains 15 nodal coordinates, given by e j = r j @r j @x T @r j @y T @r j @z T @ 2 r j @y@z T T ; j =A;B; (5.9) and beam element e 301 (t) = h e T A e T B i T contains 30 nodal coordinates. The global position vector r for the 3253 element can be expressed r 31 = h s 1 I s 2 I s 3 I s 4 I s 5 I s 6 I s 7 I s 8 I s 9 I s 10 I i 330 2 4 e A (t) e B (t) 3 5 301 where the shape functions s i ; i = 1; 2;::: 10 are given by s 1 = 1 3 2 + 2 3 s 2 =l( 2 2 + 3 ) s 3 =l(1) s 4 =l(1) s 5 =l 2 (1) s 6 = 3 2 2 3 s 7 =l( 2 + 3 ) s 8 =l s 9 =l s 10 =l 2 (5.10) 109 where = x=l; = y=l; = z=l and l is the length of the beam element in the unde- formed conguration. ANCF elements that include all three rst order gradient vectors are referred to as fully parameterized elements [64] and by this denition both the classical ANCF element and the higher order 3253 elements are fully parameterized beam elements, whereas the lower order 32 10111 3 element does not fall under this denition. 5.3 Equations of Motion of a Beam Element With the global position vector r at our disposal, the velocity _ r and the acceleration r of any point on the beam element can be computed by appropriately dierentiating the vector r with respect to time t as follows r =S(x;y;z)e(t) =) _ r = S _ e =) r = S e (5.11) The equations of motion of the beam element can be written down using the principle of virtual work as W inertia = W elastic + W external (5.12) For the beam element under consideration, the virtual work done by the inertia forces can be written as W inertia = Z V r T rdV = e T Z V S T SdV e = e T M e (5.13) where is the mass density and V is the volume of the nite element. Because the principle of conservation of mass holds, the density and volume of the undeformed conguration can be used in calculating the virtual work of the inertia forces. The virtual work done by the elastic forces are given by W elastic = ( Z V @U @e T dV ) e =Q T elastic e (5.14) 110 where U is the strain energy density function. For isotropic and homogeneous hyper- elastic materials like rubber and biological tissue, the strain energy density function is characterized by the generalized Mooney-Rivlin nonlinear constitutive laws, which will be discussed in Section 5.4. In the present problem, the only external force acting on the beam element is the force of gravity and hence the virtual work done by the gravitational force can be written as W external = Z V g T ext S dV e =Q T external e (5.15) where g T ext = [0 0 g]. Substituting equations (5.13 - 5.15) into the principle of virtual work (equation 5.12) and noting that the nodal coordinates of the beam element are all independent of one another, the equations of motion of the beam element can be derived as M e + Q elastic (e) = Q external (5.16) whereM is the constant symmetric mass matrix,Q elastic (e) is the vector of elastic forces which is a highly nonlinear function of the nodal coordinatese, andQ external is the vector of external forces acting on the beam element. No centrifugal or Coriolis force terms appear in the equation of motion as a global coordinate system is used to describe the beam element. 5.4 Nonlinear Constitutive Material Models In order to compute the elastic force vector of a beam element, a description of the strain energy density function is required. For homogeneous and incompressible hyper- elastic materials, this is given by the generalized Mooney-Rivlin nonlinear constitutive laws. Some additional modications are required to be made to these constitutive laws to account for the incompressibility of the material. But before we introduce these modications, let us brie y dene some terminology. 111 Given the global position vector r of any point on the beam element (see equation 5.1), the matrix of position vector gradients F (also called as the deformation tensor) at the same point can be computed as F = @r @X = 2 6 6 6 6 4 @r 1 @x @r 1 @y @r 1 @z @r 2 @x @r 2 @y @r 2 @z @r 3 @x @r 3 @y @r 3 @z 3 7 7 7 7 5 = 2 6 6 6 6 4 r x;1 r y;1 r z;1 r x;2 r y;2 r z;2 r x;3 r y;3 r z;3 3 7 7 7 7 5 = h r x r y r z i (5.17) The determinant J of the deformation tensor F is given by J = det(F ) = 1 2 3 (5.18) where 1 ; 2 and 3 are the principle stretch ratios. The right Cauchy-Green deforma- tion tensor is dened as C = F T F (5.19) and the invariants of the tensor C are given by I 1 = tr(C) = 1 + 2 + 3 I 2 = 1 2 h ftr(C)g 2 tr(C 2 ) i = 2 1 2 2 + 2 2 2 3 + 2 1 2 3 (5.20) I 3 = det(C) = ( 1 2 3 ) 2 where tr(C) is the trace of the matrixC, and det(C) is the determinant of the matrixC. For homogeneous and isotropic hyperelastic material, the strain energy density function U is simply a function of the strain invariants [30] U = U(I 1 ;I 2 ;I 3 ) = U( 1 ; 2 ; 3 ) (5.21) For incompressible materials, because there is minimal change in the volume of the material, the quantityJ = 1 2 3 = 1 (or equivalentlyI 3 = 1), and hence this condition needs to be maintained throughout the simulation at each material point of the beam 112 element. Consequently, for incompressible materials, the strain energy density function U is simply a function of I 1 and I 2 . The expressionU =U(I 1 ; I 2 ) assumes that the material is perfectly incompressible and so the algebraic constraint J = 1 needs to be introduced into the dynamic formu- lation. There are two ways to achieve this: the rst is the Lagrange multipliers method and the second is the penalty function method [9, 63]. The penalty function method is the easier of the two approaches to implement numerically and in the present study, we employ this approach. In the penalty function method, the strain energy density function is modied to include a volumetric energy penalty function of the form U penalty = 1 2 (J 1) 2 (5.22) where the penalty coecient must be chosen large enough (but not too large as it can create numerical issues). The value of is chosen to be 10 9 in the present study to enforce the incompressibility. The volumetric energy penalty function is added to the strain energy density function U to give us the total strain energy density for the incompressible material. In Maqueda & Shabana [43] and Jung et al. [31], the total strain energy density function for incompressible materials U IC is taken to be U IC = U(I 1 ;I 2 ) + 1 2 (J 1) 2 (5.23) However, after implementing our code, we have realized that using equation (5.23) as the expression for the strain energy density creates serious numerical issues. More- over, it leads to incorrect values of stresses being applied to the beam element and the incompressibility condition is only approximately satised. As discussed in Holzapfel [30] and Bonet & Wood [9], when dealing with an incom- pressible or nearly incompressible material, it is necessary to separate out the distor- tional (shape changing) and the volumetric (volume changing) components of deforma- tion. Such a separation must ensure that the distortional component of the deformation tensor, ^ F , does not imply any change in volume i.e., det( ^ F ) = 1. This can be ensured 113 by choosing ^ F =J 1=3 F . The distortional component of the right Cauchy-Green defor- mation tensor, ^ C, is now ^ C =J 2=3 C. Similarly, with a little algebra, the distortional components of the strain invariants can be derived as ^ I 1 =J 2=3 I 1 and ^ I 2 =J 4=3 I 2 . It is also necessary to separate out the strain energy density function into its distortional and volumetric components as follows U IC = U distortional ( ^ I 1 ; ^ I 2 ) + U volumetric (J) = U( ^ I 1 ; ^ I 2 ) + 1 2 (J 1) 2 (5.24) Comparing equation (5.24) with equation (5.23), it can be observed that the distortional component of the strain energy density function is now a function of the distortional component of the strain invariants, instead of the total invariants as was incorrectly postulated by [31, 43]. This new formulation leads to the correct stresses being applied on the beam element as will be discussed in Section 5.8. To model the deformation of rubber-like materials, dierent nonlinear constitutive models (or strain energy density functions) have been proposed in the literature depend- ing on the amount of strain expected in the material. The generalized Mooney-Rivlin model for incompressible hyperelastic materials is given by [10] U IC = N X i;j=0 ij ^ I 1 3 i ^ I 2 3 j + M X m=1 1 2 m (J 1) 2m (5.25) where 00 = 0, ij are material constants related to the distortional response, which are usually obtained from experiment and m are the material constants related to the volumetric response of the material. 5.4.1 Incompressible Neo-Hookean Material Model The Neo-Hookean hyperelastic model is typically used to model rubber-like materials when the strains in the material are less than 20%. It is the simplest model that can 114 be obtained from the generalized Mooney-Rivlin model by choosing N = M = 1 and 00 = 01 = 11 = 0 in equation (5.25) as follows: U NHIC = 10 ^ I 1 3 + 1 2 (J 1) 2 (5.26) where 10 ==2. Here, and are similar to the shear modulus and bulk modulus of the material. The elastic force vector at any integration point of the beam element for the incompressible Neo-Hookean model can be computed as Q integration point elastic = @U NHIC @e = @ @e 2 ^ I 1 3 + 1 2 (J 1) 2 = @ @e 2 J 2=3 tr(C) 3 + 1 2 (J 1) 2 = 2 J 2=3 @tr(C) @e 2 3 J 1 tr(C) @J @e | {z } distortional + (J 1) @J @e | {z } volumetric (5.27) where the expressions for @J @e and @tr(C) @e are provided in [43]. 5.4.2 Incompressible Mooney-Rivlin Material Model The Mooney-Rivlin hyperelastic material model is suitable for rubber-like materials that exhibit strains upto 100%. By choosing N = M = 1 and 11 = 0 in the generalized Mooney-Rivlin polynomial (equation 5.25), we obtain the incompressible Mooney-Rivlin material model for hyperelastic materials as U MRIC = 10 ^ I 1 3 + 01 ^ I 2 3 + 1 2 (J 1) 2 (5.28) The elastic force vector at any integration point of the beam element for the incom- pressible Mooney-Rivlin model can be derived as Q integration point elastic = @U MRIC @e = @ @e 10 ^ I 1 3 + 01 ^ I 2 3 + 1 2 (J 1) 2 115 = @ @e 8 > < > : 10 J 2=3 tr(C) 3 + 1 2 (J 1) 2 + 01 J 4=3 n 1 2 (tr(C)) 2 tr(C 2 ) o 3 9 > = > ; = 8 > > > > < > > > > : 10 J 2=3 + 01 J 4=3 tr(C) @tr(C) @e 1 2 01 J 4=3 @tr(C 2 ) @e + (J 1) @J @e + n 2 3 10 J 5=3 tr(C) 2 3 01 J 7=3 (tr(C)) 2 tr(C 2 ) o @J @e 9 > > > > = > > > > ; (5.29) where the expressions for @J @e , @tr(C) @e and @tr(C 2 ) @e are provided in [43]. 5.4.3 Incompressible Yeoh Material Model The incompressible Yeoh nonlinear constitutive material model is given by U YIC = 10 ^ I 1 3 + 20 ^ I 1 3 2 + 30 ^ I 1 3 3 + 1 2 (J 1) 2 (5.30) As is evident from equation (5.30), the distortional component of the Yeoh model is a third order polynomial in ^ I 1 , with no dependance on ^ I 2 . This enables it to more accu- rately predict the deformation of the elastomers compared to the Neo-Hookean model, while at the same time it avoids some of the stability issues that may be encountered with the Mooney-Rivlin model [6]. The elastic force vector at any integration point of the beam element for the incompressible Yeoh model can be derived as Q integration point elastic = @U YIC @e = @ @e 10 ^ I 1 3 + 20 ^ I 1 3 2 + 30 ^ I 1 3 3 + 1 2 (J 1) 2 = n 10 + 2 20 ( ^ I 1 3) + 3 30 ( ^ I 1 3) 2 o @ ^ I 1 @e + (J 1) @J @e (5.31) where @ ^ I 1 @e =J 2 3 @tr(C) @e 2 3 J 5 3 tr(C) @J @e . The expressions for @J @e and @tr(C) @e are provided in [43]. 116 5.5 Equations of Motion of the Entire Beam The equations of motion for a single nite element have been derived in Section 5.3. In this section, we stitch up the nite elements in the beam and formulate the equations of motion of the entire beam. Figure 5.1 shows a typical beam with n e nite elements and n n =n e + 1 nodes. The global position vector of any arbitrary point P on the k th beam element is given by r k =S k (x;y;z)e k ; k = 1; 2; ::: n e ; (5.32) where S k (x;y;z) is the element shape function matrix expressed in the beam element coordinate system (x;y andz) ande k = (e k A ) T (e k B ) T T is the column vector of nodal coordinates of the k th nite element. The virtual work for the entire beam can be obtained by summing up the virtual work expression of the individual nite elements (see Section 5.3) and is given by ne X k=1 n W k inertia W k elastic W k external o = 0 =) ne X k=1 M k e k + Q k elastic Q k external e k = 0 (5.33) where M k ;Q k elastic and Q k external are given by M k = Z V k k [S k ] T [S k ]dV k Q k elastic = Z V k @U IC @e k T dV k (5.34) Q k external = Z V k k g T ext S k dV k and k , V k are the mass density and volume of the k th nite element, respectively. As discussed in Section 5.4, the vector of generalized elastic forces depends on the nonlinear 117 constitutive model chosen. The nodal coordinates of the k th nite element, e k , can be mapped to the beam coordinates, e, using the following relation e k = k e =) _ e k = k _ e =) e k = k e (5.35) where k is a Boolean matrix that denes the element connectivity and e = (e 1 ) T (e 2 ) T (e nn ) T T is the stacked vector of nodal coordinates of alln n nodes in the beam. Assuming that the nodal coordinates of the entire beam are independent (and they are when the beam is unconstrained), substituting equation (5.35) in equation (5.33) gives us the governing equations of motion of the entire beam as M e + Q elastic (e) = Q external (5.36) where M = P ne k=1 [ k ] T M k k ;Q elastic (e) = P ne k=1 [ k ] T Q k elastic and Q external = P ne k=1 [ k ] T Q k external . 5.6 Cantilever Beam Boundary Conditions The cantilever beam, considered in this study, is assumed to be xed at the left end and is left free at the right end. There are two types of boundary conditions that are studied in the ANCF literature for cantilever beams: the rst is the partially-clamped boundary condition and the second is the fully-clamped boundary condition [20]. In both cases, conditions are imposed on the motion of the left end of the cantilever beam and no conditions are imposed on the right end as it is a free end. To describe the left end boundary condition, we consider the nite element at the left most end of the beam (labeled as `1' in Figure 5.1). The classical ANCF model is used as an example to illustrate the two dierent boundary conditions. As discussed in Section 5.2, the nite element at the left most end contains two nodes (A andB), with each node containing 12 nodal coordinates given by e 1 A = [e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 e 11 e 12 ] T 118 e 1 B = [e 13 e 14 e 15 e 16 e 17 e 18 e 19 e 20 e 21 e 22 e 23 e 24 ] T ande 1 = (e 1 A ) T (e 1 B ) T T . Recall that the nodal coordinates (e 1 ;e 2 ;e 3 ) = r A describe the global position vector of the nodeA, whereas the coordinatese 4 ; e 5 ; :::; e 12 describe the position vector gradients (or slope coordinates) of the node A as follows Fj A = @r @X A = 2 6 6 6 6 4 @r 1 @x @r 1 @y @r 1 @z @r 2 @x @r 2 @y @r 2 @z @r 3 @x @r 3 @y @r 3 @z 3 7 7 7 7 5 A = 2 6 6 6 6 4 r x;1 r y;1 r z;1 r x;2 r y;2 r z;2 r x;3 r y;3 r z;3 3 7 7 7 7 5 A = 2 6 6 6 6 4 e 4 e 7 e 10 e 5 e 8 e 11 e 6 e 9 e 12 3 7 7 7 7 5 5.6.1 Partially-Clamped Boundary Condition For a partially-clamped beam, only six coordinates out of the twelve nodal coordinates of the xed end (i.e. node A) are restricted. To x the beam at the left end, three coordinates describing the translation (of the node A) are suppressed r(t)j A 0 =) 8 > > > > < > > > > : e 1 (t) 0 e 2 (t) 0 e 3 (t) 0 (5.37) To suppress the rotation of the cross-section (at the node A), the following position vector gradients are restricted. r y;1 (t) =e 7 (t) 0 (captures rotation about z-axis) r y;3 (t) =e 9 (t) 0 (captures rotation about x-axis) (5.38) r z;1 (t) =e 10 (t) 0 (captures rotation about y-axis) The other six nodal coordinates are unrestricted and allowed to change. This enables for the cross-section to stretch and shear at the xed end. Partially-clamped boundary conditions have been used in the literature to model knee-joints [20]. 119 5.6.2 Fully-Clamped Cantilever Beam In the fully-clamped beam, all twelve nodal coordinates of the left end of the beam are restricted. This eliminates all translation, rotation and deformation degrees-of-freedom of the cross-section at the xed end (node A). The twelve conditions are given by r(t)j A 0 =) 8 > > > > < > > > > : e 1 (t) 0 e 2 (t) 0 e 3 (t) 0 (5.39) and 2 6 6 6 6 4 r x;1 r y;1 r z;1 r x;2 r y;2 r z;2 r x;3 r y;3 r z;3 3 7 7 7 7 5 A = 2 6 6 6 6 4 e 4 (t) e 7 (t) e 10 (t) e 5 (t) e 8 (t) e 11 (t) e 6 (t) e 9 (t) e 12 (t) 3 7 7 7 7 5 2 6 6 6 6 4 1 0 0 0 1 0 0 0 1 3 7 7 7 7 5 (5.40) In this study, we investigate the eect of these two types of boundary conditions on the deformation of the cantilever beam. One can similarly derive an equivalent set of these boundary conditions for the lower order 32 10111 3 element and the higher order 3253 element. 5.6.3 Equations of Motion of a Cantilever Beam The equations of motion of the entire beam (5.36) along with the boundary conditions discussed above govern the motion of the cantilever beam. However, recall that the equa- tions of motion (5.36) of the beam have been derived in Section 5.4 under the assumption that the nodal coordinates are all independent (unconstrained). The existence of the cantilever boundary conditions [see equations (5.37 - 5.38) for the partially-clamped beam and equations (5.39 - 5.40) for the fully-clamped beam] make the beam nodal coordinates, e, dependent on each other. 120 By simply discarding the boundary conditions, an independent set of nodal coordi- nates, f, can be arrived at, where the mapping between the beam coordinates e and these new set of coordinates f is given by e =Bf +C =) _ e =B _ f =) e =B f (5.41) where B is a Boolean matrix and C is a Boolean vector that map the independent set of coordinates, f, to the dependent set of coordinates, e. The equations of motion of the beam (equation 5.36) can now be transformed from the dependent coordinates e-space to the independent coordinate f-space as follows: M e + Q elastic (e) = Q external =) M B f + Q elastic (Bf +C) = Q external =) B T MB | {z } M f f + B T Q elastic (Bf +C) | {z } Q elastic;f = B T Q external | {z } Q external;f =) M f f + Q elastic;f (f) = Q external;f (5.42) whereM f =B T MB;Q elastic;f (f) =B T Q elastic (Bf +C) andQ external;f =B T Q external . Equation (5.42) represents the equations of motion of the cantilever beam with the boundary conditions already into account. The equations of motion can be integrated in thef-space to nd the vectorf, and the relation (5.41) can be used to transform the vector f back into the dependent coordinates space (e-space). 5.7 Integration Methods Numerical integration techniques used for integrating in space and time are discussed in the present section. Gaussian quadrature is employed to perform spatial integra- tion (more specically, volume integration) and a xed-time step St ormer{Verlet time marching scheme is used for integrating in time. 121 5.7.1 Volume Integration Using Gaussian Quadrature To write down the equations of motion of the beam (equation 5.36), three quantities (the mass matrix, elastic force vector and external force vector) need to be computed for each nite element in the beam (see equation 5.34) and all three of these quantities involve a volume integral. The volume integral associated with the mass matrix and external force vector are relatively straightforward and closed form explicit expressions for these integrals can be analytically evaluated. The volume integral associated with the elastic force vector, on the other hand, leads to gigantic expressions due to its nonlinear nature. Moreover, attempts to analytically (or symbolically) evaluating this integral should be avoided as the approach is also computationally inecient. Alternately, one could eciently evaluate this volume integral by employing numerical integration techniques such as Gaussian quadrature, where a triple integral is approximated with a triple summation as shown below. Z V F (x;y;z;e)dV = Z l 0 Z b 2 b 2 Z h 2 h 2 F (x;y;z;e) dz dy dx = Z 1 1 Z 1 1 Z 1 1 F l 2 ( + 1); b 2 ; h 2 ;e lbh 8 d d d = nx X i=1 ny X j=1 nz X k=1 w i w j w k F l 2 ( i + 1); b 2 j ; h 2 k ;e lbh 8 (5.43) where w i ;w j and w k are the weights of the Gaussian quadrature and i ; j and k are the base points of the Gaussian quadrature. The values of these weights and base points can be found in any standard Gaussian integration tables. The number of Gauss points (or quadrature points) of integration in each of the x;y and z directions is denoted by n x ;n y andn z respectively. For functionsF that can be approximated by a polynomial, Gaussian quadrature provides extremely accurate results. The number of quadrature points used in the numerical integration denes the accuracy of the integration. There is a number at which the tradeo between accuracy of the result and the computational cost is optimal and this number must be carefully arrived at. Selecting the number of 122 quadrature points greater than the number that gives (a nearly) exact evaluation of the integral is futile as it ends up wasting computational power. 5.7.1.1 Full Integration Full integration implies that the integral given by equation (5.43) is exactly evaluated. However, numerical implementations of the integral such as the one described in (5.43) always include some amount of error. When these errors are negligible enough to ap- proximate the numerically evaluated integral with the exact integral, we term it as full integration. In the present study, for full integration, the number of quadrature points chosen in thex;y andz directions are given byn x = 5;n y = 3, andn k = 3, respectively. The number of quadrature points required is more in the x-direction compared to the y and z directions as the displacement eld is cubic in x and linear in y and z. 5.7.1.2 Reduced Integration When the number of quadrature points chosen are less than what is required for an exact evaluation of the integral (equation 5.43), it is called as reduced integration. In the present study, if the number of quadrature points chosen is n x < 5 in thex-direction or n y < 3 in they-direction orn z < 3 in thez-direction, we denote it as reduced integration. Reduced integration is commonly adopted in nite element analysis to eliminate shear locking eects. This is achieved by discarding some of the high frequency modes that deteriorate element performance but do not have a signicant impact on the accuracy of the solution. However, care has to be taken while using reduced integration as it can sometimes lead to instabilities and absurd, non-physical results. 5.7.1.3 Selective Reduced Integration Selective reduced integration [33, 50] is a technique that is used to prevent locking eects in ANCF elements. When the source of the locking eect in the elastic force expression can be isolated, then selective reduced integration can be used to selectively underintegrate the term that causes locking and fully integrate the rest of the elastic 123 force expression. This approach has been shown to work well [63] in cases where Poisson (or volumetric) locking has been observed. Poisson locking has been documented in the ANCF literature when the continuum mechanics approach has been used along with a linear constitutive material model (to calculate the elastic forces) [25]. The problem has been solved by adopting selective reduced integration. In the present study, we also employ the continuum mechanics approach along with a nonlinear constitutive material model to calculate the elastic forces of the hyperelastic beam. As will be discussed in Section 5.8, Poisson locking is also observed in our case and we eliminate it by adopting selective reduced integration. The method can be best explained with the help of a simple example. Consider the Neo-Hookean material model (described by equation 5.26) for incompressible materials. U(e) = Z V U NHIC dV = Z V 10 ^ I 1 3 dV | {z } NO VOLUMETRIC EFFECTS FULL INTEGRATION + Z V 1 2 (J 1) 2 dV | {z } HAS VOLUMETRIC EFFECTS REDUCED INTEGRATION (5.44) The strain energy density function U NHIC when integrated with respect to the volume of the nite element gives us the strain energyU. Dierentiating the strain energy with respect to the nodal coordinates of the nite element gives us the elastic force vector acting on the element (see equations (5.27) and (5.29) { one can either perform the volume integration rst and then dierentiate with respect to the nodal coordinates OR dierentiate with respect to the nodal coordinates rst and then perform the volume integration and both approaches should lead to the same expression of the elastic forces). Because of the high value of chosen to enforce the incompressibility condition, all three elements discussed in Section 5.2 suer from Poisson locking (some more so than others as discussed in Section 5.8). The source of the Poisson locking can be narrowed down to the term involving in equation (5.44), which is the volumetric component of the strain energy density and therefore has all of the volumetric eects embedded in it. The other term is the distortional component of the strain energy density function and it does not contain any volumetric eects. Thus, the volumetric term, which is the 124 source of Poisson locking, can be underintegrated by choosing to integrate only along the beam axis (n x = 5; n y = 1; n z = 1) of the nite element and the distortional term can be fully integrated (n x = 5; n y = 3; n z = 3) throughout the volume of the nite element. In other words, by using selective reduced integration, one is essentially evaluating the following integral U(e) = Z V U NHIC dV = Z V 10 ^ I 1 3 dV + Z l 1 2 (J 1) 2 dl (5.45) where the integral involving is no longer evaluated on the entire volume, but is only integrated along the centerline of the beam nite element. In equation (5.45), l denotes the length of the beam nite element. 5.7.2 St ormer{Verlet Fixed-Time Step Integration Scheme To integrate the equations of motion of the beam (see equation (5.36), or alternately, equation (5.42)) in time, we employ the xed-time step St ormer{Verlet integration scheme (or simply, Verlet scheme) [99]. The Verlet scheme is used to integrate New- tons equations of motion, M x = F (x;t), where the force F is only a function of the displacement x and time t. For the cantilever beam that we consider in this study, the forces (see equation (5.42)) are only a function of the nodal coordinates (no explicit time or nodal velocities involved in the force expression). Hence, the Verlet scheme can be applied to the beam problem without any issues. The Verlet scheme has been used as far back as in 1909 to predict the orbit of Halley's comet [14] and in molecular dynamics simulations to nd the trajectories of particles [99]. The computation cost of the Verlet scheme is as low as Euler's method, and unlike Euler's method, it has a local error of O(t 4 ) and a global error of O(t 2 ). The integration scheme provides good numerical stability, time-reversibility and is energy preserving (symplectic integrator). And more importantly, it has a single function evaluation per time step, which is the costliest step in the algorithm for the cantilever beam. The method is signicantly faster compared to other higher order methods like the RK4 scheme (Runge-Kutta method), which require four function 125 evaluations per time step (see Section 5.8.6). Because of these salient features, in the present study, we employ the Verlet scheme to integrate the equations of motion of the beam. St ormer{Verlet Integration Algorithm Given the initial displacement vector y o , the initial velocity vectorv o , and the accelera- tion of the system a(x) =M 1 F (x), the displacement Y i , velocity V i , and acceleration A i of the i th time step (where i = 1; 2; :::; N) are calculated as follows: INITIAL STEP: Y 1 =y o ; V 1 =v o ; A 1 =a(y o ) for i = 1 to N 1 STEP 1: Y i+1 =Y i +V i t +A i t 2 2 STEP 2: A i+1 =a (Y i+1 ) STEP 3: V i+1 =V i + t 2 (A i + A i+1 ) end When the acceleration of the system is a function of the displacementsx, velocities _ x and time t, i.e., a = a(x; _ x;t), the St ormer{Verlet scheme described above is inapplicable. In such cases, one can employ the xed time step Runge-Kutta scheme or the modied Verlet scheme (which is discussed in Appendix H). 5.8 Results and Simulations In this section, the deformation of a thin rubber-like hyperelastic cantilever beam under the in uence of gravity is studied. This example is similar to the one considered by Jung et al. [31]. The cantilever beam is xed at its left end and its right end is free. The beam has a length of 0:35 m and a rectangular cross-section of width 7 mm and height 5 mm. The beam, which is made up of a rubber-like material, has a density of 2150 kg/m 3 . In the present study, we use a Neo-Hookean hyperelastic nonlinear constitutive 126 model (see equation 5.25) to model the material behavior. Kircho's modulus of the material is assumed to be 1:906036 MPa and a penalty coecient of 10 9 Pa is chosen to maintain the incompressibility of the material. Acceleration due to gravity g = 9:81 m/s 2 . For the parameters chosen, the equations of motion are written down using equa- tion (5.36), where the mass matrix and the external forces are found exactly through analytical methods, whereas the elastic forces are found numerically using the Gaus- sian quadrature method (described in Section 5.7.1). Time integration of the equations of motion is performed using the St ormer-Verlet scheme (described in Section 5.7.2), where a step size of the order of 10 6 is chosen. In all of the simulations considered in this section, the beam is initially horizontal and is at rest (i.e., all generalized nodal velocities of the beam are initially zero, _ e i (0) = 0 for all i). For an initially horizontal beam, the initial values of the nodal coordinates of the beam are specied by Classical ANCF Element: e 12(i1)+1 = (i 1)l; e 12(i1)+4 = e 12(i1)+8 = e 12(i1)+12 = 1; Lower Order 32 10111 3 Element e 12(i1)+1 = (i 1)l; e 12(i1)+5 = e 12(i1)+9 = 1; Higher Order 3253 Element e 15(i1)+1 = (i 1)l; e 15(i1)+4 = e 15(i1)+8 = e 15(i1)+12 = 1; wherei = 1; 2; ::: ; n e andl is the length of the nite element. The values of all the other nodal coordinates of the beam are taken to be zero [68]. The principle reason behind choosing Jung et al.'s [31] numerical example is to replicate their results. However, given the inconsistences of their paper, there exists considerable doubt regarding the correctness of their results, which we will discuss in this section. We begin by demonstrating in Section 5.8.1 that the methods chosen by Jung et al. [31] lead to locking eects in the hyperelastic beam and that these locking eects can be eliminated by using the selective reduced integration approach. In Section 5.8.2, the eect of dierent number of beam elements on the convergence 127 of the beam (deformation) is studied. In Section 5.8.3, energy conservation, stresses and incompressibility results of a 150-element classical ANCF beam are discussed. The eect of partially-clamped and fully-clamped boundary conditions on the deformation of the beam is studied in Section 5.8.4. The performance of the classical ANCF element is contrasted in Section 5.8.5 against a lower order and a higher order element to illustrate the eect of the order of the interpolation polynomial (and the choice of the gradient coordinates) in capturing the large bending deformation of the beam. And nally, in Section 5.8.6, the performance of the Verlet scheme is compared with other higher order integration schemes. 5.8.1 Locking eects and their elimination techniques 5.8.1.1 Classical ANCF Finite Element Jung et al. [31] use a ve-element (six nodes) beam and a classical ANCF nite element model to describe the deformation of the beam element. There is, however, no mention in their paper of the type of boundary condition of the cantilever beam nor the spatial integration approach used to nd the elastic forces. And therefore, to cover all bases, we consider both partially-clamped and fully-clamped cantilever beams and perform both the full and reduced integration in an attempt to try and replicate their results. Classical ANCF Element - Full Integration The dynamics of a ve-element cantilever beam with the parameters described above is simulated. The elastic forces are fully integrated with n x = 5;n y = 3 and n z = 3 (see Section 5.7.1). Both partially-clamped and fully-clamped boundary conditions of the left end are considered. The classical ANCF element (see Section 5.2) is used to model the deformation of each nite element. The beam is initially horizontal and is released from rest to deform under the in uence of gravity. The four subplots in the Figure 5.3 show the deformation of the cantilever beam at dierent time instances. Deformation of the partially-clamped beam is shown in blue color whereas the deformation of the fully-clamped beam is shown in red. As can be seen from the plots, the deformation of 128 (a) t = 0 s (b) t = 0.05s (c) t = 0.115s (d) t = 0.15s Figure 5.3: Classical ANCF Element - FULL INTEGRATION - Time history of bending deformation of a ve element partially clamped (shown in blue color) and a fully clamped (shown in red color) beam that is fully integrated using a classical ANCF nite element model. The deformation of the partially clamped and fully clamped beams is superposed on top of one other and beams exhibit locking. 129 both beams is exactly similar and is superposed on top of one another. The beams also exhibit signicant locking. The beams start to bend as soon as they are released from rest at t = 0 because of the gravitational forces acting on them. The tip (right) end of the beams achieve a maximum deformation of 0.04 m at t = 0:115 seconds before returning back to their initially horizontal position. The beams continue to oscillate in this fashion for the rest of the duration of the simulation. This overly rigid and sti behavior of the beams in bending is called as locking. This behavior is not representative of the behavior of the beam in nature and only arises during simulation. These results are in stark contrast to the ones shown by Jung et al. [31], where they do not notice any locking eects in the beam and obtain a ne de ection prole by using only 5 nite elements. This discrepancy in the results raises questions regarding the validity of Jung et al.'s results. More so, because ANCF beams (especially, thin ANCF beams like the ones considered in this study) are known to exhibit signicant locking behavior as reported in the literature [26]. The source of this locking behavior can be varied and depending on the source, the phenomenon is classied as membrane locking, or shear locking or Poisson locking. Our aim in this section is to systematically investigate the source of the locking phenomena in the beam and try to eliminate it. Classical ANCF Element - Reduced Integration We now use reduced integration to try and eliminate the locking eect that was ob- served when the elastic forces were fully integrated. Reduced integration is frequently used in nite element analysis to eliminate shear locking in the beam as discussed in Section 5.7.1. In the present case, when the beam is underintegrated in the y-direction orz-direction by choosingn y < 3 orn z < 3, respectively, the locking behavior (that was seen in the full integration case) is once again present (see Figure 5.3). On the other hand, under integrating in the x-direction by choosing n x = 1;n y = 3;n z = 3, we ob- tain a solution that qualitatively does not make any physical sense. Both fully-clamped (denoted by red) and partially-clamped (denoted by blue) cantilever beams simply free fall to the ground with their left ends attached to the xed support. This result is not 130 (a) t = 0 s (b) t = 0.08s (c) t = 0.12s (d) t = 0.15s Figure 5.4: Classical ANCF Element - REDUCED INTEGRATION - Time history of deformation of a ve element partially clamped (blue color) and a fully clamped (red color) beam using a classical ANCF nite element model. The beam is under integrated in thex-direction and this causes the beam to simply free fall with its left end attached to the xed support in both the fully-clamped and partially-clamped cases. 131 surprising as reduced integration is equivalent to discarding mode shapes in a solution, and if one discards too many mode shapes that are important to the solution, instabili- ties set in and absurd results can be obtained very similar to what we have seen in this section. Hence, reduced integration needs to be performed with extreme care. To summarize, the beam appears to be too sti in bending when the elastic forces are fully integrated using the classical ANCF nite element. On the other hand, under integration of the elastic forces yields a qualitatively non-physical result. There are other ways besides reduced integration to deal with the locking phenomena. However, many of these approaches require one to manually write down the individual contributions of the elastic energy of the beam and nd the source of the locking phenomena to suppress it from the elastic energy expression [27]. The problem with using the continuum mechanics approach is that the nonlinear constitutive model (and hence the elastic energy expression) is given to us from which the elastic forces need to be computed. For the continuum mechanics approach, there exist limited options to alleviate the locking eects in the beam. The rst approach is to try an alternate interpolation function and/or higher order gradients for the nodal coordinates in order to capture the bending deformation of the beam better. To this eect, we try a lower order 32 10111 3 element and a higher order 3253 element in the hope that the presence of trapezoidal cross-section modes in them reduces the locking eects currently observed. The second approach is to try the selective reduced volumetric integration that has been shown in the literature to alleviate Poisson locking in the beam. We try this in Section 5.8.1.5. 5.8.1.2 Lower Order 32 10111 3 Element Lower Order 32 10111 3 Element - Full integration Full integration of the elastic forces of a ve-element cantilever beam with lower order 32 10111 3 beam element is considered. Once again, like in the classical ANCF element case, the partially-clamped and the fully-clamped beams exhibit signicant locking with the lower order element as shown in Figure 5.5. The maximum de ection of the tip end of the beam in the lower order 32 10111 3 element is greater than the corresponding 132 (a) t = 0 s (b) t = 0.08s (c) t = 0.17s (d) t = 0.24s Figure 5.5: Lower Order 32 10111 3 Element - FULL INTEGRATION - Time history of bending deformation of a 5-element partially clamped (shown in blue color) and a fully clamped (shown in red color) beam that is fully integrated using the 32 10111 3 nite element model. Both partially-clamped and fully-clamped beams exhibit locking similar to the classical ANCF nite element. 133 de ection in the classical ANCF element case. Hence, the inclusion of the trapezoidal mode did reduce the locking eect to a certain extent but not nearly anything that can be considered as acceptable. The beam still exhibits considerable amount of locking as can be seen in Figure 5.5. Lower Order 32 10111 3 Element - Reduced Integration Like the classical ANCF element case, the beam is underintegrated in the x-direction but fully integrated in the y and z directions using n x = 1;n y = 3;n z = 3. The deformation of the fully clamped (red color) and the partially clamped (blue color) ve- element 32 10111 3 beams are shown in Figure 5.6. The gure shows promising results for the fully clamped beam where the beam deforms as expected and locking in the beam appears to be signicantly alleviated. However, it still remains to be seen whether there remain any left over locking eects in the beam. We have to also make sure that the loss of the higher order terms (such asx 2 andx 3 terms) in the interpolation polynomial has not compromised the dynamics of the beam. These issues will be discussed later on in this section (see Sections 5.8.1.5 and 5.8.5). The partially clamped ve-element beam, on the other hand, shows a non-physical result as can be seen in Figure 5.6. However, repeating the exact same simulation with a 30-element beam (see Figure 5.7) shows a close match between the deformation of the partially-clamped and fully-clamped beams. The non-physical result that was obtained for the ve-element partially-clamped beam is no longer present in this 30- element simulation. The inconsistent result of the partially-clamped ve-element beam can be explained by the fact that reduced integration may sometimes discard mode shapes that are necessary to capture deformation. 134 (a) t = 0 s (b) t = 0.2s (c) t = 0.34s (d) t = 0.5s Figure 5.6: Lower Order 32 10111 3 Element - REDUCED INTEGRATION - Time history of bending deformation of a 5-element partially clamped (shown in blue color) and a fully clamped (shown in red color) beam that is underintegrated in thex-direction using a 32 10111 3 nite element model. The partially clamped 5-element beam shows a non- physical result, but the locking in the deformation of the fully clamped beam appears to have been alleviated. 135 (a) t = 0 s (b) t = 0.2s (c) t = 0.34s (d) t = 0.5s Figure 5.7: Lower Order 32 10111 3 Element - REDUCED INTEGRATION - Time history of bending deformation of a 30-element partially clamped (shown in blue color) and a fully clamped (shown in red color) beam that is under integrated in thex-direction using a 32 10111 3 nite element model. The bending deformation of the partially-clamped and the fully-clamped beams are superposed on top of each other and the non-physical result of the 5-element partially clamped beam is no longer present in this 30-element beam. 136 5.8.1.3 Higher Order 3253 Element Higher Order 3253 Element - Full integration The ve-element fully-clamped and partially-clamped beams are once again considered in this subsection but this time the displacement eld of the beam elements is described by a higher order interpolation polynomial called as the 3253 element (described in Section 5.2). Full integration of this beam yields the results shown in Figure 5.8. The gures show that 3253 element beam, in both the fully-clamped (red color) and the partially clamped (blue color) cases, has larger bending deformation when compared to the full integration cases of the classical ANCF and the lower order 32 10111 3 element. This can be attributed to the existence of the trapezoidal mode in the higher order 3253 element. However, the beam still exhibits considerable amount of locking which needs to be eliminated. Repeating the simulation with a thirty element beam shows a ner de ection prole (Figure 5.9). However, just like the reduced integration case of the 32 10111 3 element, there might be some left over locking eects that are still present in the beam. (see Section 5.8.1.5 for further discussion). Higher Order 3253 Element - Reduced Integration Reduced integration (n x = 1;n y = 3;n z = 3) of a 5-element beam with the higher order 3253 element produces a non-physical result (see Figure 5.10) similar to the classical ANCF element case. The beam simply free falls to the ground with its left end attached to the xed support both in the partially-clamped and fully-clamped cases. 137 (a) t = 0 s (b) t = 0.2s (c) t = 0.35s (d) t = 0.5s Figure 5.8: Higher Order 3253 Element - FULL INTEGRATION - Time history of bending deformation of a 5-element partially clamped (shown in blue color) and a fully clamped (shown in red color) beam that is fully integrated using a 3253 nite element model. The deformation of the partially clamped and fully clamped beams exhibit locking, although to a lesser extent compared to the Classical ANCF and the 32101113 element. 138 (a) t = 0 s (b) t = 0.2s (c) t = 0.333s (d) t = 0.5s Figure 5.9: Higher Order 3253 Element - FULL INTEGRATION - Time history of bending deformation of a 30-element partially clamped (shown in blue color) and a fully clamped (shown in red color) beam that is fully integrated using a 3253 nite element model. The 30-element beams show a ner de ection prole compared to the 5-element beams but some amount of locking still appears to be present. 139 (a) t = 0 s (b) t = 0.075s (c) t = 0.15s (d) t = 0.225s Figure 5.10: Higher Order 3253 Element - REDUCED INTEGRATION - Time history of deformation of a 5-element partially-clamped (blue color) and a fully-clamped (red color) beam using a 3253 nite element model. The beam is under integrated in the x-direction and this causes the beam to simply free fall with its left end attached to the xed support in both the fully-clamped and partially-clamped cases. 140 5.8.1.4 Summary and Discussion of the Results The qualitative results that we obtained until now for the classical ANCF element, the lower order 32 10111 3 element and the higher order 3253 element for dierent types of boundary conditions and spatial integration approaches is summarized in Tables 5.1, 5.2 and 5.3, respectively. There are a few lessons that we have learnt along the way by doing this analysis. The rst is that ve elements are too small a number of elements to generate meaningful conclusions about the behavior of the beam. The results of the 5-element beam are far from converged as we will show later on in Section 5.8.2. Typically, the choice of atleast a 30-element beam (although 60-element or 90-element beams is advisable) is necessary to make meaningful conclusions about the behavior of the beam. This is because the deformation in these 60-element and 90-element beams are very close to the converged results. The second lesson is that utmost care has to be taken when performing reduced integration as it can lead to instabilities and absurd results. In the present case, reduced integration of the two fully parameterized nite elements (i.e. the classical ANCF element and the higher order 3253 element) produces absurd results where the beam simply free falls to the ground whereas reduced integration of the lower order 32 10111 3 element produces a promising result. However, since we are under integrating, there is still some amount of doubt as to whether the exact dynamics of the bending beam are being captured. We discuss this issue in more detail in Section 5.8.5. The third lesson is that the approach of choosing an interpolation polynomial that includes the trapezoidal cross-section mode to describe the displacement eld of the cantilever beam has worked only marginally well. Full integration of both the lower order and higher order elements still appears to leave some residual locking eects in the beams. Moving forward, we will investigate if the results that we have generated until now still have any residual locking eects by comparing the deformation obtained using the full integration and reduced integration approaches with that obtained using the selective reduced integration approach. 141 Table 5.1: Classical ANCF Element Results Classical ANCF Element Partially Clamped Fully Clamped Full Integration Locking Locking Reduced Integration Absurd Result Absurd Result Table 5.2: Lower Order 32 10111 3 Element Results Lower Order Element Partially Clamped Fully Clamped Full Integration Locking Locking Reduced Integration Promising Result (May have residual locking) Promising Result (May have residual locking) Table 5.3: Higher Order 3253 Element Results Higher Order Element Partially Clamped Fully Clamped Full Integration Promising Result (May have residual locking) Promising Result (May have residual locking) Reduced Integration Absurd Result Absurd Result 142 5.8.1.5 Selective Reduced Integration In this section, we consider all three types of nite elements (i.e. the classical ANCF element, the lower order 32 10111 3 element and the higher order 3253 element) and perform selective reduced integration. As discussed in Section 5.7.1, in the selec- tive reduced integration approach, the volumetric energy term is under integrated (n x = 5;n y = 1;n z = 1) along the beam centerline, whereas the distortional energy term is fully integrated (n x = 5;n y = 3;n z = 3). Given the results of the previous sec- tion, we consider a 30-element beam with fully-clamped and partially-clamped boundary conditions. Numerical simulations are run for these six cases (3 types of nite elements 2 types of boundary conditions) for 0.5 seconds duration and a time history of the bending deformation of these six beams at dierent time instances are shown in the subplots of Figure 5.11. From Figure 5.11, we observe that the deformation of the fully parameterized beams (i.e. the classical ANCF element and the higher order 3253 element) is greater than the corresponding full integration cases (see Section 5.8.1.1 and 5.8.1.3) at each time instant. This leads us to conclude that the fully parameterized beams have been experiencing Poisson (volumetric) locking, which appears to have been eliminated through the use of selective reduced integration. Next, we observe that the responses of the two fully parameterized beam elements show a close match in their response at each time instant. And this response is markedly dierent to the lower order 32 10111 3 element, which leads us to believe that the absence of the higher order terms (i.e. x 2 andx 3 terms) has made it dicult for the lower order element to capture the entire dynamics of the cantilever beam using just 30-elements. However, using a larger number of nite elements, we will show in Section 5.8.5 that the results of the lower order element do converge to the results of the fully parameterized elements. Further, Orzechowski et al. [54] have shown that the selective reduced integration result of the 30-element classical ANCF beam matches closely with the result computed using the ANSYS commercial nite element package. The fact that three widely dierent 143 (a) t = 0 s (b) t = 0.2s (c) t = 0.3s (d) t = 0.4s 144 (e) t = 0.5s Figure 5.11: SELECTIVE REDUCED INTEGRATION - Time history of deformation of a 30-element beam. The deformation of the fully clamped classical ANCF element is shown in black, partially clamped ANCF element is shown in cyan, fully clamped 32 10111 3 element is shown in magenta, partially clamped 32 10111 3 element is shown in blue, fully clamped 3253 element is shown in green, and partially clamped 3253 element is shown in red. The deformation of the both the fully parameterized nite elements (in the partially clamped and the fully clamped cases) show a close match. The deformation of the lower order element does not match with the fully parameterized elements for a 30-element beam. 145 nite element models (two with a trapezoidal cross-section mode and one without) show a close match in their response, coupled with the fact that this response is known to match the response computed using commercial nite element packages gives us greater condence in the results. See also Section 5.8.3 where checks are performed on a 150- element fully-clamped classical ANCF beam to make sure that the results make physical sense. Regardless, an experimental verication of the simulation results generated in this section is necessary to conrm that selective reduced integration of fully parameterized nite elements (like the Classical ANCF element and the 3253 element) model real life deformation of hyperelastic cantilever beams. The dierences in the deformation between the partially-clamped and the fully- clamped boundary conditions of the individual nite elements are minimal. The errors in deformation between the two types boundary conditions are discussed in Section 5.8.4. Since, there is no eect of clamping on the deformation of the beam, from here onwards, we will only consider the fully-clamped boundary condition as mathematically this is the more correct boundary condition to model the xed end of a cantilever beam. 5.8.2 Convergence Results: Increasing the Number of Elements In this section, we study the eect of increasing the number of nite elements on the bending deformation of a fully clamped cantilever beam. We consider all three types of nite elements in this section (i.e., the classical ANCF element, the lower order 32 10111 3 and the higher order 3253 element) to perform the analysis. Selective reduced integration is used to nd the elastic forces in all three cases. 5.8.2.1 Classical ANCF Element The thin Neo-Hookean cantilever beam (discussed at the start of this section) is dis- cretized into 5 (black color), 10 (yellow), 15 (cyan), 30 (magenta), 60 (blue), 90 (green) and 150 (red) elements to study the eect of the discretization on the bending deforma- tion of the beam. The cantilever beam is fully clamped at its left end and the classical ANCF element is used to model the displacement eld. Selective reduced integration is 146 (a) t = 0 s (b) t = 0.2s (c) t = 0.4s (d) t = 0.6s 147 (e) t = 0.8s (f) t = 0.9s (g) t = 1s Figure 5.12: Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Deformation of a 5 (black), 10 (yellow), 15 (cyan), 30 (magenta), 60 (blue), 90 (green) and 150 (red) element classical ANCF fully-clamped beam at dierent time instances. The 60, 90 and 150 elements show a close match in the deformation throughout the duration of the simulation. 148 Figure 5.13: Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Logarithmic root mean square (LRMS) errors of the 5, 10, 15, 30, 60 and 90 element beams averaged over space and time when compared with the deformation of the 150 element beam. The 90-element beam has the least error whereas the 5-element beam has the highest error. 149 used to calculate the elastic forces of the beam. Figure 5.12 shows the deformation of the beams at dierent time instances from t = 0 seconds to t = 1 second. The subplots in Figure 5.12 demonstrate that the 60, 90 and 150 elements show a close match in their deformation throughout the duration of the simulation. Thus, the 60, 90 and 150 element beams appear to have converged in their deformation. The 5- element beam is the rst to deviate from the motion of the 150-element beam, followed by the 10, 15 and 30 element beams. The 60-element beam also appears to be marginally deviating from the deformation of the 90 and 150 element beam att = 1 second. Beyond t = 1 second, the motion of the 60-element beam may possibly diverge from the actual solution. Further, one can also observe that as the number of elements is increased, the beam assumes a smoother de ection prole. Assuming that the deformation of the 150-element beam is the converged result, the logarithmic root mean square (LRMS) error for the 5, 10, 15, 30, 60 and 90 element beams is calculated using the following formula: LRMS Error = log 0 @ v u u t 1 N x N t Nx X i=1 Nt X j=1 k i (t j )k 2 1 A (5.46) where i (t j ) =x 150 i (t j )x NE i (t j ), NE = 5; 10; 15; 30; 60; 90, N x is the number of nodal points in the beams where the errors are sampled,N t is the number of time points where the errors are sampled andx i (t j ) is the displacement of the beam at thei th sample point in space and the j th sample point in time. The LRMS error, given by equation (5.46), denote the errors averaged over space and time. A plot of this error in the x and z directions for the dierent number of elements is shown in Figure 5.13. From the gure, we infer that the deformation of the 90-element beam closely matches that of the 150 element beam, whereas the 5-element beam diers the most. 5.8.2.2 Higher Order 3253 Element The eect of increasing the number of nite elements on the deformation of the fully- clamped beam is studied using the higher order 3253 element. Selective reduced inte- gration is used to nd the elastic forces. The deformation of a 5 (black), 10 (cyan), 150 Figure 5.14: Higher Order 3253 Element - SELECTIVE REDUCED INTEGRATION - Deformation of the 5 (black), 10 (cyan), 15 (magenta), 30 (blue), 60 (green) and 90 (red) element beams at t = 0.5 seconds. The deformation of the 60 and 90 element beams closely match each other. Figure 5.15: Higher Order 3253 Element - SELECTIVE REDUCED INTEGRATION - LRMS errors for the 5, 10, 15, 30 and 60 element beams in comparison to the 90-element beam. 151 15 (magenta), 30 (blue), 60 (green) and 90 (red) element beam for 0.5 seconds of sim- ulation time is studied. At the end of the 0.5 seconds, one can clearly observe from Figure 5.14 that the 60 and 90 element beams show a close match in their deforma- tion. This behavior is similar to the classical ANCF element case where the 60, 90 and 150 elements displayed a close match in their deformation. The logarithmic root mean square errors for the 5, 10, 15, 30, and 60 element beams is shown in Figure 5.15 for the 3253 element, where the comparison is done with respect to the deformation of the 90-element beam. Once again, the deformation of the 60-element beam closely matches that of the 90 element beam, whereas the 5-element beam diers the most. 5.8.2.3 Lower Order 32 10111 3 Element The eect of discretization on the deformation of the fully-clamped beam is also studied using the lower order 32 101113 3 element. The deformation of a 5 (black), 10 (yellow), 15 (cyan), 30 (magenta), 60 (blue), 90 (green) and 150 (red) element beam is studied for 0.5 seconds of simulation time. Unlike the fully parameterized beam elements (i.e., the classical ANCF and the 3253 element), the 32 10111 3 element requires a larger number of nite elements to reach convergence. This can be observed in Figure 5.16 which shows the deformation of the dierent beams at the end of 0.5 seconds. There is considerable deviation in the deformation of the 60 and 90 element beams when compared to the deformation of the 150-element beam at the end of the 0.5 seconds. The reason for this is possibly because of the absence of the higher powers (x 2 and x 3 ) in the interpola- tion polynomial of the 32 10111 3 element (see Section 5.2.2 for the exact interpolation polynomial) making it require more number of elements to accurately predict the de- formation of the beam. The 150-element beam (32 10111 3 element), however, appears to have converged in its deformation as it closely matches the deformation predicted by the 90-element fully parameterized beams (see Section 5.8.5 for further discussion). The logarithmic root mean square errors for the 5, 10, 15, 30, 60 and 90 element beams is shown in Figure 5.17 for the 32 10111 3 element, where the comparison is done with respect to the deformation of the 150-element beam. The LRMS errors of the 5, 10, 152 Figure 5.16: Lower Order 32 10111 3 Element - SELECTIVE REDUCED INTEGRATION - Deformation of the 5 (black), 10 (yellow), 15 (cyan), 30 (magenta), 60 (blue), 90 (green) and 150 element (red) beams at t = 0.5 seconds. Unlike the fully parameterized elements, the 32 10111 3 element requires a larger number of elements to reach convergence. Figure 5.17: Lower Order 32 10111 3 Element - SELECTIVE REDUCED INTEGRATION - LRMS errors for the 5, 10, 15, 30, 60 and 90 element beams in comparison to the 150- element beam. 153 15, 30, 60 and 90 element beams are noticeably higher than those of the correspond- ing fully parameterized elements (see Figures 5.15 and 5.13) indicating that there exist considerable deviations in the deformation of the respective beams from the converged result. 5.8.3 Analysis of the 150 Element Classical ANCF Beam A 150-element fully-clamped classical ANCF beam is considered in this section, where the elastic forces of the beam have been computed using selective reduced integration. Figure 5.18 shows the deformation of the 150-element beam fromt = 0 seconds tot = 2 seconds at 0.1 second time intervals. The gures on the left show the deformation of the beam in the xz-plane whereas the gures on the right show the same deformation in three-dimensions. To make sure that the selective reduced integration approach is yielding us the correct results, let us perform some sanity checks and make sure that the results obtained for the 150-element beam make physical sense. To do this, we nd if the energy is being conserved throughout the simulation, if the incompressibility condition is being satised at every time instant and if the stresses at the free end of the beam are zero throughout the duration of the simulation. 5.8.3.1 Energy Conservation The beam considered in the present study is subjected to conservative forces alone. In addition, the Verlet integration scheme is an energy preserving symplectic scheme. Thus, the total energy of the beam should be conserved throughout the duration of the simulation. To do this, we nd the kinetic energy using the mass matrix (equation 5.36) and the nodal velocities of the beam, the elastic potential energy using equation (5.45) and the gravitational potential energy of the entire beam by computing the gravitational potential energy of the individual nite elements and summing them up [8]. A plot of these energy components is shown in Figure 5.19(a). The total energy of the beam is plotted in Figure 5.19(b) and as can be inferred from the gure, the total energy of the beam is indeed conserved throughout the duration of the simulation. 154 (a) Deformation in xz-plane (b) Three-Dimensional View (c) Deformation in xz-plane (d) Three-Dimensional View (e) Deformation in xz-plane (f) Three-Dimensional View 155 (g) Deformation in xz-plane (h) Three-Dimensional View Figure 5.18: Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Deformation of a 150-element fully-clamped classical ANCF beam at dierent time instances. Figures on the left show the response of the beam in the two-dimensional xz-plane whereas the gures on the right show the response of the beam in three- dimensions. 156 (a) Energy Components (b) Total Energy Figure 5.19: Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - (a) Time history of the kinetic energy, potential energy and gravitational potential energy of the 150 element beam from t = 0 to t = 2 seconds. (b) Time history of the total energy of the beam from t = 0 to t = 2 seconds. Energy is conserved throughout the duration of the simulation. 157 5.8.3.2 Incompressibility Condition Since the material model being considered in the present simulation is an incompressible Neo-Hookean material, the incompressibility condition i.e., J = det(F ) = 1 needs to be satised throughout the duration of the simulation at each material point of the 150-element beam. To check if the selective reduced integration approach is indeed maintaining the incompressibility condition, we plot a time history of the variable J (see Figure 5.20) at four equidistantly located points in the 150-element beam (i.e. at the midpoint of the 37 th nite element, the 75 th nite element, the 112 th nite element and the 150 th nite element). As seen from the gure, the incompressibility condition is close to 1 at all four locations of the 150-element beam. 5.8.3.3 Second Piola-Kircho Stress at the Free End The closed form expression for calculating the 2 nd Piola-Kircho stress tensor for a Neo- Hookean material model at a material point, given the nodal coordinates of the beam element, is given by PK2 = 10 J 2=3 I 1 3 (tr(C))C 1 + (J 1)JC 1 (5.47) Since no external forces are being applied at the free (right) end of the beam, the normal component of the 2 nd Piola-Kircho (PK2) stress in the x-direction should be zero for all time t. Figure 5.21 shows a plot of this normal component of the PK2 stress tensor at the free end of the 150-element beam. As can be seen from the gure, the component of this stress is indeed zero until about t = 0:8 seconds of the simulation. But the condition is violated after 0.8 seconds. Incorrect prediction of stresses when using the ANCF framework is a known issue [24]. 158 (a) J at midpoints of 37 th and 75 th nite elements (b) J at midpoints of 112 th and 150 th nite elements Figure 5.20: Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Determinant of the deformation tensor J = det(F ) plotted as function of time from t = 0 to t = 2 seconds at the midpoints of the 37 th , 75 th , 112 th and the 150 th element of the 150-element beam. The incompressibility condition is satised for all time t. 159 (a) t = 0 to t = 0.5s (b) t = 0 to t = 2 s Figure 5.21: Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - The normal component of 2 nd Piola-Kircho stress tensor in the x direction calculated at the free end of the beam from t = 0 to t = 0:5 seconds and t = 0 to t = 2 seconds. 160 5.8.4 Eect of Clamping on the Deformation of the Beam From Section 5.8.1.5, we know that the deformation of the partially-clamped and fully- clamped 30-element classical ANCF cantilever beam show a close match when selective reduced integration is used to calculate the elastic forces. In this section, we perform a quantitative analysis to interpret how closely matched the solutions of the partially- clamped and fully-clamped boundary conditions are for the case of a 90-element classical ANCF beam (instead of the 30-element beam that was considered in Section 5.8.1.5). Figure 5.22 shows a time history of the displacement of the tip end of the beam in the x and the z direction for the fully-clamped and partially-clamped beams for 0.5 seconds simulation time. The gures show an exact match in the tip end displacement both in thex and thez directions. The dierences in the displacement of the fully and partially clamped beams at the tip end and midpoint of the 90-element beam is plotted in Figure 5.23(a) and Figure 5.23(b), respectively. These dierences are of the order of 10 5 indicating an extremely close match between both types of boundary conditions. 161 Figure 5.22: Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Time history of the tip end displacement of a 90-element beam for the fully (red) and partially (blue) clamped boundary conditions. (a) Dierences at Tip end (b) Dierences at Midpoint Figure 5.23: Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Dierences in the deformation of the fully-clamped and partially clamped 90-element beam at the tip end (left) and midpoint (right). 162 5.8.5 Comparison of the Finite Element Models The deformation of a 90-element fully-clamped beam is plotted in Figure 5.24. The classical ANCF element, the higher order 3253 element and the lower order 32 10111 3 element are used to model the deformation of the beam. The elastic forces are computed using the selective reduced integration approach in each case. As discussed in Section 5.8.1.5 and can be seen in the subplots of Figure 5.24, the fully parameterized beam elements i.e., the classical ANCF element (red color) and the higher order 3253 element (blue color) show a close match for the 90-element beam. The dierences in the displacements between the 90-element fully clamped classical ANCF model and the 90-element fully clamped 3253 model at the tip end and midpoint of the beam are plotted in Figure 5.25. These dierences are of the order of 10 3 indicating a close match in the deformation of the beams despite using widely dierent interpolation polynomials. Earlier, in Section 5.8.1.5, the 30-element lower order 32 10111 3 beam did not match the deformation of the fully parameterized beams. The 32 10111 3 element lacks the higher powers ofx in the interpolation polynomial. Hence, it requires more number of elements to accurately predict the deformation of the beam. The 32 10111 3 element (green color) shows a closer match to the deformation of the fully parameterized beam elements when 90 elements are used (see Figure 5.24). However, the deformation of the lower order 90-element beam has still not quite converged yet. When the number of elements is further increased to 150 elements (see Figure 5.26), the dynamic deformation of a 150-element lower order 32 10111 3 model is seen to match closely with the deformation of a 90-element classical ANCF and a 90-element 3253 model. Figure 5.27 shows the dierences in the displacement of the 150-element 32 10111 3 beam and the 90-element classical ANCF beam at the tip end and at the midpoint. These dierences are of the order of 10 3 , respectively, indicating a close match in the deformation at the tip end and at the midpoint of both beams at the end of 0.5 seconds simulation time. It is worth noting that the lower order 32 10111 3 element is signicantly faster to run compared to the fully parameterized elements. 163 (a) t = 0 s (b) t = 0.2s (c) t = 0.4s (d) t = 0.5s Figure 5.24: SELECTIVE REDUCED INTEGRATION - Time history of deformation of a fully clamped 90-element beam using the classical ANCF model (red color), lower order 32 10111 3 element (green color) and the higher order 3253 element (blue color). Selective reduced integration of these beams show that the deformation of the classical ANCF model and the 3253 model match closely, while there exists some deviation in the deformation of the lower order 32 10111 3 model. (a) Dierences at the Tip end (b) Dierences at the Midpoint Figure 5.25: SELECTIVE REDUCED INTEGRATION - Dierences in the deformation at the tip end (left) and midpoint (right) between a 90-element fully-clamped classical ANCF model and a 90-element fully clamped 3253 model. 164 (a) t = 0 s (b) t = 0.2s (c) t = 0.4s (d) t = 0.5s Figure 5.26: SELECTIVE REDUCED INTEGRATION - Time history of deformation of a fully clamped 90-element classical ANCF model (red color), 150-element lower order 32 10111 3 element (green color) and a 90-element higher order 3253 element (blue color). Selective reduced integration of these beams show an extremely close match in the deformation of the beams irrespective of the interpolation polynomial used to model these beams. (a) Dierences at the Tip end (b) Dierences at the Midpoint Figure 5.27: SELECTIVE REDUCED INTEGRATION - Dierences in the deforma- tion between a 90-element fully-clamped classical ANCF model and a 150-element fully clamped 32 10111 3 model at the tip end (left) and midpoint (right). 165 5.8.6 Evaluation of the St ormer{Verlet Integration Solver In this section, we rst study the eect of the (time) step size of the Verlet solver on the accuracy of the results it generates. Second, the Verlet scheme is compared with higher order Runge-Kutta (RK4) and ODE 113 schemes in both speed and accuracy. 5.8.6.1 Eect of Time Step on the Deformation To compare the eect of the size of the time steps on the accuracy of the results, we consider a 60-element fully clamped classical ANCF beam where the elastic forces are calculated using selective reduced integration. The equations of motion of this beam are integrated using three dierent time steps: 1:6 10 6 , 1:1 10 6 and 0:8 10 6 . Assuming that the simulation run with the lowest time step (0:8 10 6 ) gives us the most accurate result, the results of the other two time steps are compared with the results of the smallest time step. Figure 5.28 shows the dierences in the deformation of the beam at the tip end and the midpoint between the largest time step (1:6 10 6 ) and the smallest time step (0:810 6 ). Although the step size has been halved, the dierences in the displacements (at the tip end and the midpoint) are of the order of 10 9 , which is extremely small. Hence, as long as a step size chosen is of the order of 10 6 (which is the order of the step size chosen for all the simulations in this study), the results that we obtain using the Verlet scheme are fairly accurate. Figure 5.29 shows the dierences in the displacements at the tip end and midpoint of the beam between the (time) step sizes of 1:110 6 and 0:810 6 . Figure 5.30 shows the LRMS errors of the 60-element beam when the results of the 1:6 10 6 and the 1:1 10 6 time steps are compared with the 0:8 10 6 . From Figures 5.29 and 5.30, we deduce that as the time step decreases, the results obtained are more accurate. However, the marginal increase in accuracy is not worth the smaller step size as each additional step is computationally expensive. 166 (a) Dierences at the Tip End (b) Dierences at the Mid Point Figure 5.28: Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Dierences in the deformation of a 60-element fully clamped beam at the tip end and the midpoint between the largest time step (1:6 10 6 ) and the smallest time step (0:8 10 6 ). (a) Dierences at the Tip End (b) Dierences at the Midpoint Figure 5.29: Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Dierences in the deformation of a 60-element fully clamped beam at the tip end and the midpoint between the time steps (1:1 10 6 ) and the time step (0:8 10 6 ). Figure 5.30: Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Logarithmic mean square (LRMS) errors in the deformation of a 60-element beam where the deformation results of the 1:6 10 6 and the 1:1 10 6 time steps are compared with the deformation results of the 0:8 10 6 time step. 167 5.8.6.2 Dierent Solver Comparisons To compare the Verlet scheme with a xed-time step RK4 scheme and a variable time step ODE 113 scheme (MATLAB), we choose an 8-element partially clamped classical ANCF beam where the elastic forces are computed using the selective reduced integra- tion approach. A 0.5 second simulation of the 8-element beam is computed using the three solvers. The Verlet solver required a time step of 3:3 10 6 whereas the RK4 solver required a time step of 4 10 6 to give stable results. The ODE 113 solver chooses its own time step as it is a variable time step solver. To complete this 0.5 second simulation, the Verlet solver took 3 hours 10 minutes, the RK4 solver took 16 hours 14 minutes and the ODE 113 solver took 41 hours and 48 minutes. Thus, the Verlet solver is about 5 times faster than the RK4 scheme and about 14 times faster than the ODE 113 scheme (relative error tolerance of 10 3 and global error tolerance of 10 4 ). This result is not surprising as the Verlet method has only a single function evaluation per time step unlike the RK4 and ODE113 schemes which have atleast four function evaluations per time step. The dierences in the displacements of the tip end and the midpoint of the 8- element beam are computed between RK4 scheme and the Verlet scheme in Figure 5.31 and between the ODE113 scheme and the Verlet scheme in Figure 5.32. The errors are of the order of 10 8 at the tip end and about 10 9 at the midpoint. The graphs in Figure 5.31 and Figure 5.32 are almost identical to each other, which leads us to believe that the ODE 113 scheme uses an algorithm similar to the xed-time step RK4 scheme to compute its results. The logarithmic root mean square errors in the deformation of the beam averaged over space and time are calculated between the RK4 and the Verlet scheme, and the ODE113 scheme and the Verlet scheme and shown in Figure 5.33. These errors are of the order of 10 8 , which indicates that the Verlet scheme is fairly accurate when compared to the RK4 and ODE 113 schemes. 168 (a) Dierences at the Tip End (b) Dierences at the Mid Point Figure 5.31: Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Dierences in the deformation of an 8-element fully-clamped beam at the tip end and the midpoint between the RK4 scheme and the Verlet scheme. (a) Dierences at the Tip End (b) Dierences at the Midpoint Figure 5.32: Classical ANCF Element SELECTIVE REDUCED INTEGRATION Dif- ferences in the deformation of an 8-element fully-clamped beam at the tip end and the midpoint between the ODE113 scheme and the Verlet scheme. Figure 5.33: Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - LRMS errors in the deformation of an 8-element beam, where the deformation obtained from the ODE113 solver and the RK4 solver are compared with the deformation ob- tained from the Verlet solver. 169 5.8.7 Other Nonlinear Constitutive Models Until now, we have dealt with the most basic hyperelastic material model i.e., the in- compressible Neo-Hookean model. However, the results generated in this study can be easily extended to the more complicated nonlinear constitutive models such as the incompressible Mooney-Rivlin model (see Section 5.4.2) and the incompressible Yeoh model (see Section 5.4.3) that describe rubber-like behavior more accurately. For the sake of completeness, we present two simple numerical simulations where the elastic forces of the beam are computed using the incompressible Mooney-Rivlin and the in- compressible Yeoh model, respectively. We consider, once again, a cantilever beam with a xed left end and a free right end. The beam is discretized into 90 elements. The parameters of the fully-clamped cantilever beam remain exactly the same as in Section 5.8 except that 10 = 0:8 MPa and 01 = 0:2 MPa for the incompressible Mooney-Rivlin model (see equation 5.28) and 10 = 0:545235 MPa, 20 = 0:0610498 MPa and 30 =0:000802537 MPa for the incompressible Yeoh model (see equation 5.30). A penalty coecient of 10 9 Pa is chosen to maintain the incompressibility of the material in both cases. Acceleration due to gravity g = 9:81 m/s 2 . The classical ANCF element (see Section 5.2) is used to model the deformation of the beam in both cases. The elastic forces are calculated using selective reduced integration. The dynamic deformation of the 90-element fully-clamped cantilever beam is shown in the xz-plane at dierent time instances for the incompressible Mooney-Rivlin model (see Figure 5.34) and the incompressible Yeoh model (see Figure 5.35). As with the Neo-Hookean hyperelastic model (see Section 5.8.2.1), the selective reduced integration approach avoids the volumetric locking phenomena in both the Mooney-Rivlin and Yeoh hyperelastic models. However, an experimental verication of the simulation results needs to be conducted to conrm that the selective reduced integration approach indeed accurately predicts the real life deformation of hyperelastic cantilever beams. This could be an avenue to explore for a future course of study. 170 (a) t = 0s (b) t = 0:1s (c) t = 0:2s (d) t = 0:3s (e) t = 0:4s (f) t = 0:5s Figure 5.34: Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - MOONEY RIVLIN - Deformation of a 90-element fully-clamped classical ANCF beam at dierent time instances. The elastic forces have been computed using the incom- pressible Mooney-Rivlin nonlinear constitutive model. 171 (a) t = 0s (b) t = 0:1s (c) t = 0:2s (d) t = 0:3s (e) t = 0:4s (f) t = 0:5s Figure 5.35: Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - YEOH - Deformation of a 90-element fully-clamped classical ANCF beam at dierent time instances. The elastic forces have been computed using the incompressible Yeoh nonlinear constitutive model. 172 5.9 Conclusions This study deals with the large-deformation and large-rotation problem of three- dimensional, thin, rubber-like, near-incompressible hyperelastic cantilever beams. Ab- solute nodal coordinate formulation is used to describe the dynamic motion of the beam, whereas the continuum mechanics approach along with nonlinear constitutive material models are used to derive the elastic forces. The nonlinear constitutive models stud- ied include the incompressible Neo-Hookean model, the incompressible Mooney-Rivlin model and the incompressible Yeoh model. Incompressibility of the rubber-like mate- rial is maintained by adding a volumetric energy penalty function to the strain energy density of the hyperelastic material. The addition of this term leads to signicant vol- umetric locking eects in the cantilever beam, where the beam exhibits extremely sti behavior in bending. These locking eects are eliminated by using selective reduced integration, where full integration is used to calculate the distortional component of the elastic forces and reduced integration along the beam axis is used to compute the volumetric component. The main contributions and ndings of the study are as follows: 1. Previous investigators incorrectly assume the distortional component of the strain energy density to be a function of the total strain invariants instead of their distortional components, leading to an incorrect description of the elastic forces. We rectify this error in the present study and derive the correct expressions for the elastic forces acting at any material point of a beam element for incompressible Neo-Hookean and incompressible Mooney-Rivlin material models. 2. Locking problem in thin cantilever beams is discussed in considerable detail and techniques to eliminate it are systematically presented. To compute the elastic forces accurately in hyperelastic beams, the use of nonlinear constitutive models and the continuum mechanics approach is a necessity. However, there exist limited options in the case of the continuum mechanics approach to eliminate the source of locking. We try two such options in the present study. The rst approach is to choose an interpolation polynomial that can better describe the displacement eld 173 of the beam element in bending. In addition to the classical ANCF element model (fully parameterized beam element, and has a cubic interpolation polynomial), two new nite elements models: the 32 10111 3 element (lower order linear beam element, and contains the trapezoidal mode) and the 3253 element (fully parameterized, has a cubic interpolation polynomial, and contains the trapezoidal mode), that are better at capturing bending deformation, are tried. However, all three nite element models show either signicant locking eects or absurd nonphysical results when full integration and reduced integration of elastic forces are tried on them. The second approach to eliminate locking is the use of selective reduced integration and this is shown to work well in eliminating the locking eects from all three nite element models. 3. The eect of fully-clamping (all nodal coordinates of the left most node restricted) and partially-clamping (only six nodal coordinates restricted) the left end of the cantilever beam on its bending deformation has been studied. When the beam is discretized using more than 30 nite elements, these dierences have been found to be extremely small (of the order of 10 5 at the tip end for 90 elements). Moreover, there is no computational advantage in choosing one boundary condition over the other. In such a case, it is advisable to choose the correct mathematical representation of the boundary condition, which is the fully-clamped condition. 4. A 150-element classical ANCF beam is simulated using selective reduced inte- gration. The total energy in the beam is shown to be constant throughout the duration of the simulation highlighting the fact that the time marching Verlet scheme is indeed energy preserving. The incompressibility condition is plotted at four dierent material points in the beam and the condition is shown to be main- tained at all points throughout the duration of simulation, reiterating the fact that selective reduced integration is indeed successful in enforcing incompressibility of the material at each time instance. 5. A convergence study on the bending deformation of the cantilever beams has been carried out for the three dierent nite element models. When the beams 174 are discretized using 60 or 90 nite elements using the fully parameterized beam elements (i.e., the classical ANCF element and the higher order 3253 element), the dynamic deformations in these beams match closely with each other (dierences in the tip end deformation are of the order of 10 3 ) at each instant of time. The lower order 32 10111 3 element, on the other hand, because of its linear nature of the interpolation polynomial, requires a larger number of elements (approximately 150 elements) to reach convergence in the deformation. The fact that three widely dierent nite element models yield the same converged response at each instant of time, which also happens to match the response computed using commercial nite element packages gives us greater condence in the results that we have obtained in this study. However, an experimental verication of the results is necessary to be absolutely sure of these computational results. 6. The Verlet scheme has been shown to perform well in integrating the equations of motion of the cantilever beam in comparison to other higher order schemes such as the 4 th order Runge-Kutta (RK4) scheme and the ODE 113 scheme of MATLAB. With a similar time step size, the Verlet scheme is shown to be nearly 5 times faster than the RK4 scheme and 14 times faster than ODE 113 (relative error tolerance of 10 3 and global error tolerance of 10 4 ), with the errors in the deformation at the tip of the order of 10 8 in both cases. This signicant increase in the com- putational speed is possible thanks to a single function evaluation (costliest step in the cantilever beam algorithm) per time step in the Verlet scheme, compared to higher order schemes which require atleast four function evaluations per time step. 175 Chapter 6 Control of Incompressible Hyperelastic Beams 6.1 Introduction T HE rst two parts of this thesis was devoted to solving control problems in discrete mechanics from the perspective of constrained motion. In this chap- ter, we will show that this analytical dynamics based control approach is not just limited to the control of highly nonlinear discrete dynamic systems, but can be readily extended to tackle problems related to the control of highly exible continuous structural systems that contain both geometric and material nonlinearities. To illus- trate the control approach, we use the example of a highly exible, three-dimensional, thin, rubber-like, incompressible, Neo-Hookean, hyperelastic cantilever beam, which was studied in the previous chapter, and demonstrate a few tracking control objectives. The generality of the formulation presented herein allows us to tackle a whole class of other control objectives such as energy control, shape control [4], modal control [52] as well as vibration control [35] of the beams and other such general continuum structures. Control of continuous structures have numerous applications. Underactuated control of exible solar panels is an important problem in the spacecraft industry [17, 21, 69, 100]. Solar panels are almost always made of exible materials and these materials 176 undergo large deformations in the nonlinear elastic range [21, 100]. If the panels are thin and long, a simplied model is to consider them as thin, long cantilever beams. The problem that is currently being actively researched is to control the free end of the cantilever beam, by actuating as much as possible, the left end of the beam alone, which is closer to the axis of the xed support where all the electrical equipment and the motors are stored. In this study, as a rst step, we assume full-state feedback of the cantilever beam and show that the tip end can be controlled by demonstrating a few tracking control objectives that can be time-varying as well. The assumption of full-state feedback is a realistic assumption as currently, it is possible to run wiring through the length of beam and install miniature piezoelectric actuators at the desired locations. However, this is resisted in the industry because of the prohibitive costs and the addition of these devices changes the characteristics of the model. Our future course of work will be devoted to solving the underactuated beam problem, and later on, extend these methods to thin membranes, which model the solar panels more realistically. Other applications related to the control of continuous structures include vibration control of space structures [29], suppression of utter in aircraft panels [2], aeroelastic control of adaptive wings and airfoils [66], and the control of exible biomechanical systems [101]. Accurate prediction of the dynamics of deformable structures is an important rst step in precisely controlling these structures. In the present study, the dynamic de- formation of the cantilever beam is computed using the absolute nodal coordinate for- mulation (ANCF) [63]. ANCF is a nite element approach that is used to accurately describe large-deformation and large-rotation motion in highly exible continuous sys- tems. The approach is more accurate compared to classical beam theories such as the Euler-Bernoulli and Timoshenko beam theories [64, 102]. Control of continuous struc- tures in the framework of ANCF is a relatively new eld, and to the best of the authors knowledge, is restricted to the following references [20, 37, 51, 71]. However, given the fact that ANCF accurately predicts the dynamic motion of large deformable structures, its ease of use in modeling complex multibody exible systems, absence of nonlinear inertia forces and the ability to incorporate the general nonlinear continuum mechanics 177 approach for a precise computation of the elastic forces, make it an ideal candidate to describe the dynamics of controlled structural systems, like we demonstrate in this study. In this study, we consider a three-dimensional rubber-like hyperelastic incompress- ible thin cantilever beam. Absolute nodal coordinate formulation along with a fully parameterized classical ANCF element is used to describe the displacement eld of each nite element and thus the elastic dynamic deformation of the beam. Incompressible Neo-Hookean nonlinear constitutive model [9, 30] is used to describe the highly exible rubber-like material behavior. The general nonlinear continuum mechanics approach along with selective reduced integration technique [54] is used to compute the elastic forces in the beam. Control of such a cantilever beam is approached from an analytical dynamics perspective and the theory of constrained motion is used to recast the control objectives as constraints on the nodes of the beam. The fundamental equation of me- chanics [81, 83, 84] is used to obtain the explicit expressions of the generalized nonlinear control forces in closed form. These control forces are applied at the nodes of the beam to achieve the desired objectives. The constrained motion approach comprises of three vital steps. The rst step involves the derivation of the equations of motion of the unconstrained system. For the fully-clamped cantilever beam that we consider in this study, this is discussed in Section 6.2. The second step involves the formulation of the constraint equations (see Section 6.3) and the last step deals with the use of the fundamental equation of mechanics to obtain the constrained equations of motion of the cantilever beam (see Section 6.4). In the process, we also nd the explicit generalized nonlinear control forces that are applied to the nodes of the beam. Four numerical simulations are presented in Section 6.5 that demonstrate accurate nonlinear tracking control of the tip end of a thirty-element cantilever beam. The simulations demonstrate the ease, simplicity and ecacy with which the control methodology can be applied to achieve the desired control objectives. 178 6.2 Unconstrained Equations of Motion of the Cantilever Beam Consider a three-dimensional Neo-Hookean hyperelastic incompressible beam deforming under gravity (similar to the cantilever beam of Chapter 5). The governing equations of motion of this beam have been derived in detail in Section 5.5 and are reproduced as follows M e + Q elastic (e) = Q external (6.1) where M = P ne k=1 [ k ] T M k k ;Q elastic (e) = P ne k=1 [ k ] T Q k elastic and Q external = P ne k=1 [ k ] T Q k external , M k is the mass matrix associated with the k th nite element, Q k elastic andQ k external are the vector of elastic and external forces acting on the k th nite element, and k is the connectivity matrix that maps the nodal coordinates of the k th nite element to the beam coordinates. The classical ANCF element (discussed in Section 5.2) is used to describe the dis- placement eld of each nite element. An incompressible Neo-Hookean hyperelastic material model is used to describe the material behavior of the beam (see Section 5.4) and the nonlinear continuum mechanics approach along with selective reduced integra- tion is used to calculate the elastic forces (see Sections 5.4.1 and 5.7.1.3 ). Equation (6.1) describes the motion of a beam that is free at both ends. To fully clamp its left end such that a cantilever beam is constructed, we need to impose ad- ditional constraints on the motion of its left end. The equation of motion of such a fully-clamped cantilever beam, that is derived in Section 5.6.3, is given by M f f + Q elastic;f (f) = Q external;f =) M f f = Q f (f) := Q external;f Q elastic;f (f) (6.2) 179 whereM f =B T MB,Q elastic;f (f) =B T Q elastic (Bf +C) andQ external;f =B T Q external , the matrix B and the vector C map the beam coordinates, e, into an independent set of nodal coordinates f using the transformation e 12nn1 = B 12nn12ne f 12ne1 + C 12nn1 (6.3) where n n = n e + 1 is the number of nodes, and n e is the number of nite elements that the beam is discretized into. The independent set of nodal coordinates f are generated by simply discarding the boundary conditions of the xed end from the beam coordinates, e. The cantilever beam is allowed to deform under gravity from an initially horizontal rest position. To characterize zero initial velocity of the beam, all generalized nodal velocities of the beam are initialized to zero (i.e., _ e i (0) = 08 i). To characterize the geometry of an initially horizontal beam, the nodal coordinates of the beam are all initialized to zero except for the following nodal coordinates e 12(i1)+1 = (i 1)l e 12(i1)+4 = 1 e 12(i1)+8 = 1 e 12(i1)+12 = 1 ; i = 1; 2; ::: ; n e : (6.4) where l is the length of the nite element. These initial conditions in the e-space are then transformed into the f-space by using transformation (6.3). In the context of constrained motion theory (see discussion in Chapter 2), equations (6.2 - 6.4) represent the unconstrained (or uncontrolled) equations of motion of the cantilever beam that take into account the fully-clamped boundary conditions of the left end of the beam. The acceleration a f of this unconstrained cantilever beam can be computed as a f (f) = M 1 f Q f (f) (6.5) 180 6.3 Formulation of the Constraints For the unconstrained system described by equation (6.2), in order to achieve the desired control objectives (tracking control or otherwise), a set of constraints are formulated in terms of nodal coordinates. These constraints can either holonomic (described by equation (6.6)) or nonholonomic (described by equation (6.7)) in nature, and are allowed to be functionally dependent on each other. i (e;t) = 0; i = 1; 2; ::: ; h; (6.6) i (e; _ e;t) = 0; i = h + 1; h + 2; ::: ; m; (6.7) The initial conditions of the unconstrained beam (stated in Section 6.2) are assumed to satisfy these constraint equations (6.6) and (6.7). However, there might be situations wherein we would like to initialize the nodal coordinates and nodal velocities of the beam from any arbitrary point in phase space where the constraints are not necessarily satised (see Section 6.5.3). In such a case, instead of considering the existing set of m constraints described by equations (6.6) and (6.7), we appropriately modify these constraint equations as follows [78]. i +c _ i +k i = 0; i = 1; 2; ::: ; h; (6.8) _ i + i = 0; i = h + 1; h + 2; ::: ; m; (6.9) where the parameters c; k; > 0 are chosen such that the system of equations (6.8) and (6.9) have equilibrium points described by (6.6) and (6.7), respectively and that these equilibrium points are asymptotically stable. The parameters c; k and govern the rate and nature of convergence of the controlled dynamical system towards manifold prescribed by equations (6.6) and (6.7). This set of m modied constraints can now be expressed in the general constraint matrix form of equation (2.5) as A(e; _ e;t) e = b(e; _ e;t) (6.10) 181 where A is an m 12n n constraint matrix of rank r (i.e., r out of the m constraint equations are independent) while b is a column vector with m entries. Because the equations of motion of the unconstrained cantilever beam are formulated in the f- space, the constraint equations also need to be transformed into the f-space and this can be done using equation (6.3) as follows A(e; _ e;t) e = b(e; _ e;t) =) A(Bf +C;B _ f;t) B f = b(Bf +C;B _ f;t) =) B T AB | {z } A f f = B T b |{z} b f =) A f (f; _ f;t) f = b f (f; _ f;t) (6.11) where A f is an m 12n e and b f is a column vector with m entries. 6.4 Constrained Equations of Motion of the Cantilever Beam Given the unconstrained equations of motion of the cantilever beam (see equation (6.2)) and the specication of the constraints (see equation (6.11)), we are now in a position to compute closed form explicit expressions of the generalized nonlinear control forces. The fundamental equation (equation 2.7) is used to compute these nonlinear control forces and they are given by Q control;f f; _ f;t = M 1=2 f A f M 1=2 f + b f A f a f = A T f A f M 1 f A T f + b f A f a f (6.12) With the closed form expression for the explicit generalized nonlinear control forces at our disposal, we can now write down the constrained (controlled) equations of motion of the cantilever beam as follows M f f = Q f (f) + Q control;f f; _ f;t 182 = Q external;f Q elastic;f (f) + Q control;f f; _ f;t (6.13) where M f ; Q external;f and Q elastic;f are obtained from the description of the uncon- strained system (equation 6.2) whereas Q control;f is computed using equation (6.12). This completes the discussion of the constrained motion approach in the context of control of cantilever beams (and in general, control of continuum structures). The next section discusses some numerical examples to illustrate the control methodology. 6.5 Results and Simulations The three-dimensional, thin, rubber-like, incompressible, hyperelastic cantilever beam that we studied in the previous chapter (see Section 5.8) is once again considered in this section. The cantilever beam is discretized into 30 nite elements. The beam, which is initially horizontal, is released from rest under the in uence of a constant gravitational eld. The equations of motion of such a beam and its initial conditions are listed in Section 6.2. These equations represent the unconstrained equations of motion of the beam. The parameters of the beam are specied as follows. The beam has an undeformed length of 0.35m, width of 7mm, a height of 5mm and a density of 2150 kg/m 3 . A Neo-Hookean nonlinear constitutive model is used to describe the material behavior where Kircho's modulus of the material is taken to be = 1:906 MPa and a penalty coecient of = 10 9 Pa is chosen to maintain the incompressibility of the material. Acceleration due to gravity g = 9:81 m=s 2 . The dynamic bending deformation of such a 30-element beam is shown in Section 5.8.1.5 (see Figure 5.11). To illustrate the control methods discussed in this chapter, we consider four numeri- cal examples involving this unconstrained cantilever beam, where the examples dier in the control objective that is desired to be achieved. In Example 1, we require the right end of the cantilever beam to be xed such that it eectively becomes a fully-clamped xed-xed beam. In Example 2, in addition to xing the right end of the cantilever 183 beam, we further want to x the displacement of a material point located at about two-thirds the distance from the left end of the beam. The rst two examples deal with constraints that are only functions of the nodal coordinates. For the last two examples, we consider more complicated constraints, where the constraints are functions of nodal coordinates, nodal velocities as well as time. In Example 3, we once again consider an initially horizontal cantilever beam at rest, where we now require the right end of the beam to move in a circle of radius 0.05 m. The zero initial velocity of the beam in this example has the implication that the initial conditions of the beam do not satisfy the constraints and therefore we need to appropriately modify the constraints for the control objective to be achieved (see equation (6.8) and (6.9)) as discussed in Section 6.3. In Example 4, the control objective desired is the same as in Example 3, but now there are no restrictions on the cantilever beam to start from rest. This allows us to approach the problem using an alternate set of the constraints but yet still achieve the same control objective as in Example 3. In each of these numerical examples, the control objective is cast as a set of con- straints in terms of the nodal coordinates and nodal velocities of the beam. With the knowledge of the unconstrained system and the constraints at our disposal, the fun- damental equation of mechanics is used to derive the closed form expressions of the explicit generalized nonlinear control forces. These control forces are required to be applied at the nodes of the beam so that the desired control objectives can be achieved. The constrained equations of motion are written down using these explicit expressions of the control forces, which are then integrated with respect to time to nd the response of the beam. A xed time-step Verlet scheme in used Examples 1 and 2, and a 4 th order Runge-Kutta scheme is used in Examples 3 and 4. 6.5.1 Example 1: Fully-Clamp the Right End of a Cantilever Beam In this rst example, our objective is to fully-clamp the right end of a 30-element unconstrained cantilever beam, such that the controlled system eectively becomes a fully-clamped xed-xed beam. In Section 5.6.3, we looked at one way of imposing a xed end boundary condition by transforming into an independent coordinate space. In 184 the present case, we will look at an alternate way of achieving the same objective. We interpret the boundary conditions as constraints on the system. The generalized control forces that need to be applied to the right end of the beam are then computed such that these constraints are satised for all time. Unlike the rst approach, the need to transform into an independent coordinate space (to maintain the boundary conditions) does not arise in the present case. Instead, we remain in the same coordinate space and add control forces to satisfy the constraints. Unconstrained System: The 30-element, thin, incompressible, hyperelastic cantilever beam forms the unconstrained system, whose equations of motion have been derived in Section 6.2. See equations (6.2 - 6.4) for a description of the unconstrained system. The dynamic bending deformation of this 30-element beam is shown in Figure 5.11. Constraints: To fully clamp the right end of the beam, we interpret the following con- straints to be imposed on the motion of the unconstrained system. e 12ne+1 (t) 0 e 12ne+2 (t) 0 e 12ne+3 (t) 0 and 2 6 6 6 6 4 e 12ne+4 (t) e 12ne+7 (t) e 12ne+10 (t) e 12ne+5 (t) e 12ne+8 (t) e 12ne+11 (t) e 12ne+6 (t) e 12ne+9 (t) e 12ne+12 (t) 3 7 7 7 7 5 2 6 6 6 6 4 1 0 0 0 1 0 0 0 1 3 7 7 7 7 5 (6.14) where n e = 30 in the present case. Thus, a total of 12 constraints given by equation (6.14) are considered to be acting on the unconstrained system. The rst three con- ditions constrain the displacement of the node at the right end (31 st node), whereas the last nine conditions constrain the rotation of the cross-section of the beam at the right end. The initial conditions of the cantilever beam, described in Section (6.2), satisfy the constraints given by equation (6.14) and therefore, the need to modify the constraints does not arise. The constraints can be dierentiated twice with respect to 185 (a) t = 0s (b) t = 0.04s (c) t = 0.08s (d) t = 0.12s (e) t = 0.16s (f) t = 0.2 s Figure 6.1: Control Example 1 - Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Time history of deformation of a controlled 30-element cantilever beam. Control forces are acting at the right end of the cantilever beam such that its motion is completely restricted i.e., the right end eectively becomes a fully clamped end. 186 Figure 6.2: Control Example 1 - Time history of control forces acting on the last node of the beam at the right end. The control forces are nonzero in the x (denoted by F c x ) and the z (denoted by F c z ) directions as shown in the gure. The control forces in the other ten nodal directions are all approximately close to zero. Figure 6.3: Control Example 1 - Time history of errors in satisfying the 12 constraints. The errors are of the order of 10 10 , which are considerably small. " 12ne+4 (t) represents error in satisfying the fourth constraint on the last node. Verlet solver with a time step of 2:6 10 6 is used to integrate the equations of motion. 187 time so that they can be expressed in the general constraint matrix form of equation (6.10), which when transformed into the f-space gives us equation (6.11), where A f = h O 1212(ne1) I 1212 i and b f = h O i 121 (6.15) Control Force: With M f ;Q f ;a f ;A f and b f at our disposal, the nonlinear control force can be readily calculated in closed form using equation (6.12) and is given by Q control;f (f) =M 1=2 f A f M 1=2 f + (b f A f a f (f)) =A T f A f M 1 f A T f + A f a f (f) (6.16) Constrained System: With the control force computed, the constrained (controlled equations of motion) of the fully-clamped xed-xed beam can be written down us- ing equation (6.13) and is given by M f f = Q f (f) + Q control;f (f) = Q f (f) A T f A f M 1 f A T f + A f a f (f) (6.17) where the RHS of equation (6.17) is simply a function of the nodal coordinatesf. Thus, one can use the Verlet scheme described in Section 5.7.2 to integrate these equations of motion. The xed time step size used is 2:6 10 6 . A time history of the dynamic deformation of this controlled xed-xed fully- clamped beam is shown in Figure 6.1 from t = 0 to t = 0:2 seconds. Beyond 0:2 seconds, the beam keeps periodically oscillating in a similar fashion. The gure also shows that the right end of the beam is xed for all time t as desired. A time history of the control forces acting on the rightmost (31 st ) node of the beam are plotted in Figure 6.2. The gure shows that the control forces acting in the x and the z directions are nonzero, with the control forces in the other nodal directions approximately zero. A time history of the errors in tracking the constraints is plotted in Figure 6.3. These errors are of the order of 10 10 , which are considerably small, illustrating the ecacy of the control methodology in achieving the control objectives. For an even stringent 188 bound on the error, one can employ a higher order integration scheme such as the 4 th order Runge-Kutta scheme. 6.5.2 Example 2: Zero Displacement of a Material Point in the Fixed- Fixed beam We once again consider the 30-element classical ANCF cantilever beam with a fully clamped left end as the unconstrained system. The control objective in this example is to constrain the displacement of a point located about two-thirds the distance from the left end of the beam, in addition to fully-clamping the right end of the beam. Unconstrained System: The equations of motion of the unconstrained system along with the initial conditions remain the same as in Example 1. Constraints: In addition to the 12 constraints specied by equation (6.14), which con- strain the motion of the right most node (31 st node) of the beam (and make it a fully- clamped end), the following three constraints need to be additionally imposed on the unconstrained system in this example to further constrain the displacement of the node located at two-thirds the distance from the left end (i.e., 21 st node). e 12( 2 3 ne)+1 = 2 3 L e 12( 2 3 ne)+2 = 0 e 12( 2 3 ne)+3 = 0 (6.18) Thus, the unconstrained system is subjected to a total of 15 constraints (12 constraints given by equation (6.14) and 3 additional constraints given by equation (6.18)). The rst condition of equation (6.18) xes the displacement in the x-direction of the node located at two-thirds the distance from the left end (i.e., 21 st node), whereas the second and the third condition x the displacement of the same node in the y andz-directions, respectively. When these fteen constraints are dierentiated twice with respect to time and transformed into the f-space, the general constraint matrix form in the f-space is 189 (a) t = 0s (b) t = 0.02s (c) t = 0.05s (d) t = 0.07s (e) t = 0.1s (f) t = 0.14s Figure 6.4: Control Example 2 - Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Time history of deformation of a controlled 30-element cantilever beam with fully-clamped boundary condition at the right end of the beam and a con- straint on the displacement of the node located at two-thirds the distance from the left end of the beam (see 21 st node in the gure). 190 Figure 6.5: Control Example 2 - Time history of control forces acting on the 21 st (located at two-thirds the distance from the left end of the beam) and 31 st node (last node) of the beam in the x;y and z directions. Only the control forces acting on the 21 st node and the 31 st node in the x and z directions are non-zero as shown in the gure. F c 31x represents the control force acting on the 31 st node in the x-direction. Figure 6.6: Control Example 2 Time history of errors in tracking the 15 constraints. The errors are of the order of 10 8 , which are considerably small. " 31 i (t) represents the error in satisfying the i th constraint of the 31 st node. Verlet solver with a time step of 2:6 10 6 is used to integrate the equations of motion. 191 given by equation (6.11), where elements of the 15 12n e matrix, A f , are all zeros except the following entries A f i; 12 2 3 n e 1 +i = 1; i = 1; 2; 3; A f (j; 12 (n e 1) +j) = 1; j = 1; 2; ::: ; 12; (6.19) and b f = h O i 151 . Constrained System: The expression for the explicit nonlinear control force is given by equation (6.16), whereA f andb f are now specied by equation (6.19). The constrained equations of motion of the system are, once again, written down using equation (6.17). These equations are integrated using the Verlet scheme with a xed time step size of 2:6 10 6 . A time history of the dynamic deformation of the beam is shown in Figure 6.4 from t = 0 to t = 0:14 seconds. The gures show that in addition to the right end of the beam being xed, the displacement of the node located at two-thirds the distance from the left end (i.e., 21 st node) is also xed. The beam keeps periodically oscillating in this fashion beyond 0.14 seconds. A time history of the control forces acting on the 21 st node and the right end of the beam (31 st node) in the x, y and z directions are shown in Figure 6.5. The control forces acting on the remaining 9 nodal directions of the right end of the beam (31 st node) are approximately zero. A time history of the errors in tracking the desired constraints is shown in Figure 6.6. Once again, these errors are small indicating the accuracy with which the control methodology tracks the desired constraints. For a more stringent error bound, the use of higher order time marching integration schemes are suggested. 6.5.3 Example 3: Circular Tip Motion in the YZ-plane with Con- straint Stabilization Consider a 30-element initially horizontal cantilever beam at rest, similar to the uncon- strained beam considered in Examples 1 and 2. Our objective in this example is to 192 (a) t = 0s (b) t = 0.01s (c) t = 0.03s (d) t = 0.05s (e) t = 0.07s (f) t = 0.08s 193 (g) t = 0.1s (h) t = 0.2s (i) t = 0.3s (j) t = 0.4s (k) t = 0.5s Figure 6.7: Control Example 3 - Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Time history of the motion of a controlled 30-element highly exible hyperelastic cantilever beam where control forces are applied at the right end of the beam such that it executes circular motion in the yz-plane. 194 Figure 6.8: Control Example 3 - Time history of displacement of the tip (right) end (31 st node) of the cantilever beam in the x;y and z directions. control the tip end of the cantilever, such that its x-coordinate is xed at a distance L = 0:35m from the left end of the beam (same as the undeformed length of the beam), while itsy andz coordinates vary such that the tip end executes circular motion (radius of circle R = 0:05m) in the yz-plane. Unconstrained System: The equations of motion of the unconstrained beam and its initial conditions remain the same as in Example 1. Constraints: To achieve the desired motion, we interpret the following constraints to be acting on the unconstrained cantilever beam. e 12ne+1 = L (6.20) e 12ne+2 = R sin(!t) (6.21) e 12ne+3 = R cos(!t) (6.22) 195 where n e = 30 in the present implementation. The rst condition (equation 6.20) restricts x-coordinate of the tip end of the beam to be at L for all time, while equa- tions (6.21) and (6.22) constrain its motion in the yz-plane to lie on the manifold (R sin(!t); R cos(!t)). It should be noted, however, that the initial conditions of tip end of the unconstrained beam do not satisfy the constraints (6.21) and (6.22). And therefore, for the control objective to be achieved, the constraints (6.21) and (6.22) need to be appropriately modied as 2 =e 12ne+2 R sin(!t) = 0 =) 2 + c _ 2 + k 2 = 0 (6.23) 3 =e 12ne+3 R cos(!t) = 0 =) 3 + c _ 3 + k 3 = 0 (6.24) wherec> 0,k> 0 are chosen such that the solutions to the equations (6.23) and (6.24) i.e., 2 and 3 tend to zero asymptotically as t!1. This results in the constraints (6.21) and (6.22) being ultimately satised, as desired. Equations (6.20), (6.23) and (6.24) can be expressed, after appropriately dierentiating with respect to time, in the general constraint matrix form of equation (6.10), which after transforming into the f-space gives us equation (6.11), where the 3 12n e matrixA f and the 3 1 vectorb f are given by A f = h O 312(ne1) I 33 O 39 i and b f = 2 6 6 6 6 4 0 R! 2 sin(!t)c _ f 12(ne1)+2 R!cos(!t) k f 12(ne1)+2 Rsin(!t) R! 2 cos(!t)c _ f 12(ne1)+3 +R!sin(!t) k f 12(ne1)+3 Rcos(!t) 3 7 7 7 7 5 (6.25) 196 where the values of c = 160 and k = 6400 in the present implementation and b f = b f (f; _ f;t). Control Force: The control forces are calculated using the fundamental equation (6.13) as follows Q control;f f; _ f;t = A T f A f M 1 f A T f + b f f; _ f;t A f a f (f) (6.26) Constrained System: The constrained (controlled) equations of motion of the beam can be written down using equation (6.13) as M f f = Q f (f) + Q control;f f; _ f;t = Q f (f) + A T f A f M 1 f A T f + b f f; _ f;t A f a f (f) (6.27) where the RHS of equation (6.27) is now a function of nodal coordinates f, nodal velocities _ f as well as timet. Thus, as discussed in Section 5.7.2, the Verlet integration scheme cannot be used in the present case as the RHS of equation (6.27) is a function of the nodal velocities _ f. We, therefore, use the 4 th order Runge-Kutta scheme to integrate the equations of motion (6.27) in the present example. The xed time step used for the RK4 scheme is 2:510 6 . If a faster run time is of essence, one can employ the modied St ormer{Verlet scheme (see Appendix H) instead to integrate the equations of motion. Figure 6.7 shows a time history of the dynamic motion of the controlled cantilever beam. The beam is initially horizontal as can be seen in Figure 6.7(a), but control acts at the beam tip so that the desired circular manifold is asymptotically reached. The tip end of the beam initially overshoots the desired circular path as can be seen in Figures 6.7(a) - 6.7(f), but eventually returns to the desired manifold. The amount of this overshoot can be controlled by varying the parameters c and k, which control the rate and nature of convergence of the trajectories towards the desired manifold. From Figures 6.7(g) - 6.7(k), we can observe that after the initial overshoot, the required circular trajectory is precisely tracked by the tip end as desired (see also Figure 6.10). A time history of the displacement of the tip end of the beam is plotted in Figure 6.8, 197 Figure 6.9: Control Example 3 - Time history of control forces acting at the tip (right) end (31 st node) of the cantilever beam in the x;y and z directions. (a) t = 0 to t = 0.5s (b) t = 0.4s to t = 0.5s Figure 6.10: Control Example 3 - Time history of errors in tracking the three constraints in the x;y and z directions of the tip end of the beam. The errors are of the order of 10 14 , which are all extremely small. 4 th order Runge-Kutta solver with a time step of 2:5 10 6 is used to integrate the equations of motion. 198 where the overshoot and the return to the desired circular manifold is more apparent. Figure 6.9 shows a plot of the control forces acting at the tip end of the beam in the x;y and z directions for the duration of the simulation. Figure 6.10 depicts the errors in tracking the three constraints, where we observe that error exponentially goes down to zero. The errors at the end of the 0.5s simulation are of the order of 10 14 , which are all extremely small, illustrating the ecacy of the control methodology in accurately tracking the constraints. 6.5.4 Example 4: Circular Tip Motion in the YZ-plane without Con- straint Stabilization In this example, our objective is to, once again, control the tip end of an initially horizontal cantilever beam such that thex-coordinate of the tip end is xed at a distance ofL = 0:35m from the left end of the beam (same as the undeformed length of the beam), while y and z coordinates vary such that the tip end executes the motion of a circle of radius R = 0:05m in the yz-plane. Unlike Example 3, we would like to illustrate the control approach without modifying the constraints and therefore this would mean that the initial conditions that we will be choosing will be such that they satisfy the constraints (see Section 6.3 for details). Unconstrained System: We once again consider the 30-element classical ANCF can- tilever beam with a fully clamped left end as the unconstrained system. The uncon- strained equations of motion of this beam are given by equation (6.2). The initial conditions of the beam in the present example are dierent from those chosen in Ex- amples 1, 2 and 3. Their choice is guided by the need to satisfy the constraints at each instant of time and hence also at the initial time (see discussion below). Constraints: For the beam to achieve the desired motion, the tip end of the uncon- strained system is assumed to be subjected to the following three constraints. e 12ne+1 = L (6.28) e 12ne+2 = R 1 e t sin(!t) (6.29) 199 e 12ne+3 = R 1 e t cos(!t) (6.30) Equation (6.28) xes the x-coordinate of the tip end of the beam at L for all time t. On the other hand, equations (6.29) and (6.30) specify that the y and z coordinates of the tip end of the beam start from the origin (0; 0) in the yz-plane at t = 0, and asymptotically reach the desired circular manifold R sin(!t);R cos(!t) as t!1. The speed with which the circular manifold is reached can be controlled through the parameter, which is chosen to be 40 in the present implementation and the frequency of oscillation, ! is chosen to be 200. These constraints represent an alternate way to achieve the circular motion in the yz-plane of the beam tip end that was obtained in Example 3. Since in the present case, we do not intend to modify the constraints, the initial conditions of the unconstrained beam need to be chosen such that they satisfy constraints (6.28 - 6.30). One set of initial conditions of the tip end that do satisfy the constraints are chosen as follows e 12ne+1 (0) =L; e 12ne+2 (0) = 0; e 12ne+3 (0) = 0; _ e 12ne+1 (0) = 0; _ e 12ne+2 (0) = 0; _ e 12ne+3 (0) =R; (6.31) The initial conditions of all the other nodes in the beam are chosen as specied in Section 6.2. These combined set of initial conditions characterize an initially horizontal cantilever beam, with all nodes having zero nodal velocities except for the node at the tip end, which has an initial positive velocity in the z-direction of magnitude R. The constraints (6.28 - 6.30) are dierentiated twice with respect to time t, and transformed into the f-space. The general constraint matrix form in the f-space is 200 (a) t = 0s (b) t = 0.02s (c) t = 0.04s (d) t = 0.06s (e) t = 0.09s (f) t = 0.16s 201 (g) t = 0.3s (h) t = 0.45s (i) t = 0.5s Figure 6.11: Control Example 4 - Classical ANCF Element - SELECTIVE REDUCED INTEGRATION - Time history of motion of a controlled 30-element highly exible hyperelastic cantilever beam where control is being applied to the tip end (right end) of the beam such that it executes circular motion in the yz-plane. 202 Figure 6.12: Control Example 4 - Time history of displacement of the tip (right) end (31 st node) of the cantilever beam in the x;y and z directions. specied by equation (6.11), where the 3 12n e matrix A f and the 3 1 vector b f are given by A f = h O 312(ne1) I 33 O 39 i and b f = 2 6 6 6 6 4 0 (t) sin(!t) + (t) cos(!t) (t) cos(!t) (t) sin(!t) 3 7 7 7 7 5 31 (6.32) where (t) =R 2 e t + ! 2 1e t and (t) = 2R!e t . Control Force: Once again, with M f ;Q f ;a f ;A f and b f at our disposal, the nonlinear control force is calculated in closed form using equation (6.12) and is given by Q control;f (f;t) =A T f A f M 1 f A T f + b f (t)A f a f (f) (6.33) 203 Figure 6.13: Control Example 4 - Time history of control forces acting at the tip (right) end (31 st node) of the cantilever beam in the x;y and z directions. Figure 6.14: Control Example 4 - Time history of errors in tracking the three constraints in the x;y and z directions at the tip end of the beam. The errors are of the order of 10 14 , which are all extremely small. 4 th order Runge-Kutta solver with a time step of 2:5 10 6 is used to integrate the equations of motion. 204 Constrained System: The constrained (controlled) equations of motion of beam exhibit- ing the desired motion is given by M f f = Q f (f) + Q control;f (f;t) = Q f (f) + A T f A f M 1 f A T f + b f (t)A f a f (f) (6.34) where the RHS of equation (6.34) is a function of both nodal coordinates f and time t. Thus, one can still use the Verlet scheme to integrate these equations of motion. However, to illustrate the eect of the order of the time integration scheme on the accuracy of the solution, we choose to integrate these equations using the 4 th order Runge-Kutta solver with a xed time-step of 2:5 10 6 . Figure 6.11 shows a time history of the dynamic motion of the 30-element cantilever beam with its tip end (right end) executing the motion of a circle in the yz-plane as desired. The beam starts from an initially horizontal position as shown in Figure 6.11(a) and tends to the desired circular manifold asymptotically (see Figures 6.11(e) - 6.11(i)), while precisely satisfying the constraints (equations 6.28 - 6.30) at each instant of time (see Figure 6.14). Figure 6.12 shows a plot of the displacement of the tip end of the beam for the duration of the simulation. From the gure, we observe that for the value chosen in the problem, the tip end of beam reaches the circular manifold at approximatelyt = 0:15 seconds. The control forces acting at the tip end of the cantilever beam in the x;y and z directions are shown in Figure 6.13. The errors in tracking the three constraints are shown in Figure 6.14, where we observe the errors are of the order of the order of 10 14 , which are extremely small highlighting the ecacy of the control methodology in achieving the desired objectives. Also, these errors are smaller than those in Examples 1 and 2 because of the use of a higher order integration scheme (4 th order Runge-Kutta scheme) in the present example to integrate the equations of motion. 205 6.6 Conclusions The main contribution of this study is the extension of the analytical dynamics based control approach, that was previously used to successfully control highly nonlinear dis- crete dynamical systems, to now include the control of general continuous structural systems that possess nonlinearities of both material and geometric nature. The study deals with the control of highly exible, rubber-like, three-dimensional, near-incompressible, hyperelastic cantilever beams. The dynamics associated with the large-deformation and large-rotation of the cantilever beam is modeled using the abso- lute nodal coordinate formulation and the Neo-Hookean nonlinear constitutive model is used to describe its material behavior. Nonlinear continuum mechanics approach along with selective reduced integration is used to compute the elastic forces in the beam. Control of such highly deformable beams is considered in the present study. The control approach is inspired by recent results in analytical dynamics that deal with the theory of constrained motion. To the best of the author's knowledge, analytical results on neither the dynamics nor the control of such highly nonlinear deformable continuum structures that involve both geometric and material nonlinearities exist in the literature. Inspite of the large-deformation and large-rotation motion exhibited by the hyperelastic beam, the control is obtained with relative ease and in closed form without the need to make any approximations or linearizations of the exible nonlinear dynamics associated with the beam and without the need to impose any a priori structure on the nature of controller. The key idea behind the control approach is to recast the control requirements on the continuous system as constraints on its nodes. The fundamental equation of mechanics is then used to determine explicit closed form expressions of the generalized nonlinear control forces that are applicable at these nodes. The control forces so computed are continuous in time. The control has been shown to precisely track the desired constraints (control objectives) at each instant of time. Numerical simulations demonstrating nonlinear time-varying tracking control of a thirty-element cantilever hyperelastic beam is presented. The closed form analytical 206 dynamics based control is shown to perform well demonstrating the ease, simplicity, and accuracy with which the methodology works. The methodologies presented in this study can be readily extended to control other general structural systems in continuum mechanics. 207 Part 5 Conclusions Chapter 7 Conclusions and Future Work 7.1 General Conclusions I n this dissertation, the problem of control of highly nonlinear general mechanical systems is approached from an analytical dynamics perspective. The approach is inspired by recent developments in the eld of analytical dynamics that deal with the theory of constrained motion [78, 79, 81, 83, 84, 89]. Three highly nonlinear control problems from diverse areas of discrete and continuum mechanics are chosen to illustrate the approach. 1. Although the approach was previously applied to control other discrete dynamical systems, in this dissertation, we show for the rst time that it can be applied with equal ease, simplicity and accuracy to control highly exible continuous mechanical systems that possess both geometric and material nonlinearities. 2. Inspite of the general nature of the discrete and continuous mechanical systems that is considered in this study, explicit expressions for the nonlinear control force are obtained in closed form with relative ease, without the need for any approxima- tions or linearizations of either the nonlinear dynamics of the mechanical system being considered, or its control. Further, no a priori structure is imposed on the nature of the nonlinear controller. 209 3. We obtain exact control in each of the three control problems. The constraints are precisely satised at each instant of time, and the errors in satisfying the constraints are within the tolerance levels specied by the integration algorithm. 4. The generalized control forces,F c , are continuous in time. They are also optimal, in the sense that they minimize the control cost given by J(t) = [F c ] T M 1 [F c ] at each instant in time [79]. J(t) is simply the square of the weighted L 2 norm of the control force, where the weighting used is the inverse of the mass matrix, (M 1 ). This is the exact control cost that Nature would also minimize, if Nature were performing the control. 5. Inhomogeneous Nonlinear Lattice: It is interesting to note that although it is extremely dicult to understand the dynamical response of these general nonlinear lattices and obtain anything nearing general closed form analytical solutions for their response - a consequence of which is the virtually nonexistent literature on the subject - its energy control, however, can be done relatively easily and the necessary control forces can be obtained in closed form. This seems to beg the question: Has Nature somehow intentionally made it easier for us to control the energy of nonlinear systems rather than determine their exact nonlinear behavior? 6. Inhomogeneous Nonlinear Lattice: To the best of the author's knowledge, this work represents the rst time that Lasalle's invariance principle has been used to derive sucient conditions on the placement of actuators for an underactuated lattice such that global asymptotic convergence is guaranteed from any nonzero initial energy state to any desired nal (nonzero) energy state. The same ap- proach can be similarly adopted to nd conditions for underactuation in other general nonlinear systems (for example: underactuation in robotic systems is an important problem, where one could apply Lasalle's principle to nd the necessary and/or sucient conditions for underactuating the robot such that asymptotic convergence to the desired constraint manifold can be guaranteed). 7. Numerical simulations accompanying each of the three control problems that we have presented in this dissertation illustrate the ease, simplicity and ecacy with 210 which the control methodology can be implemented in diverse areas of mechanics. Furthermore, they demonstrate the accuracy with which the control objectives are satised in each case. 8. The analytical dynamics based control approach and the fundamental equation are powerful tools to obtain the equations of motion of constrained (controlled) nonlinear mechanical systems. Thus, the ideas and methodologies provided in this dissertation can be readily adopted and expanded to tackle far more complicated problems than those that we have presented herein. 7.2 Future Work Future direction of research could be focused in the following areas: 1. Partitioning of Energy in the Lattice: In the present work, we were able to control the entire lattice at any given nonzero desired energy level. But one could also think of a more general case wherein one part of the lattice is desired to be controlled at a certain level and the rest of the lattice at a dierent level. The practical application of this type of partitioning of energy is that it could be used to localize energy in a certain desired part of the lattice. 2. Extension to 2D and 3D Lattices: In the present study, we have considered one- dimensional lattices (also called as chains). We have seen the existence of periodic solutions (such as solitons and breathers), and highly nonlinear complex wave interactions in these one-dimensional lattices. The two-dimensional and three- dimensional counterparts of the Toda, FPU and Klein-Gordon lattices are known to have an even richer set of dynamics and control of general inhomogeneous two- dimensional and three-dimensional lattices could be an interesting future direction of research. 3. Hyperelastic beam problem: Our work has concentrated on deriving the elastic forces of the beam for the simplest hyperelastic material model i.e., Neo-Hookean 211 material model. However, most studies place the Neo-Hookean model to be accu- rate for strains up to 20% { 30%. For strains larger than these values, one needs to use the Mooney-Rivlin or the Yeoh material models. The methods presented in Chapter 5 and 6 could be easily extended to include these material models. Further, one could also expand this work to include more complicated geometries such as plates, shells and membranes with equal ease. 4. Control of Hyperelastic beams: As discussed in Chapter 6, control of an under- actuated beam is a problem of tremendous practical signicance. A theoreti- cal study could be performed to nd the feasibility of underactuated control in beams, and if it turns out that underactuation is possible, one could derive the necessary/sucient conditions for underactuation (on lines similar to our use of Lasalle's principle in Section 3.5.2) such that the desired control objectives are asymptotically satised. 5. Experimental verication: As discussed in Chapter 5, studies have not yet been performed on experimentally verifying and corroborating the simulation results generated on the dynamic deformation of the hyperelastic beam using the absolute nodal coordinate formulation. This is extremely important to make sure that the absolute nodal coordinate formulation is able to capture the entire nonlinear dynamics of the highly exible hyperelastic beam. Experimental studies could also be performed to verify the simulation results generated on the control methods presented in this study. 212 Part 6 Appendices Appendix A Miscellaneous Proofs and Results A.1 Positive Denite Function Denition: A function u(x) is said to be positive denite if u(0) = 0 and u(x)> 0 for all x6= 0. (a) (b) (c) Figure A.1: Examples of Positive Denite Functions. 214 From the denition, a positive denite function is positive everywhere except at zero where its value is zero. And so, gures A.1(a), A.1(b), A.1(c) and A.2(c) are all examples of positive denite functions in one dimension. However, the example in Figure A.2(b) is not positive denite because u(0)6= 0. A.2 Radially Unbounded Function Denition: A function u(x) is radially unbounded if given any M2< + , there exists an R2< + such that u(x)>M for allkxk>R. (a) For M1 > 0 =) R1 = maxfjr1j;jr2jg, and for M2 > 0 =) R2 = maxfjr3j;jr4jg. (b) For M1 > 0 =) R1 = maxfjr1j;jr2j;jr3j;jr4jg, for M2 > 0 =) R2 = maxfjr5j;jr6j;jr7jg, and for M3 > 0 =)R3 = maxfjr8j;jr9jg. 215 (c) For M1 > 0 =) R1 = maxfjr1j;jr2jg, for M2 > 0 =) R2 = maxfjr3j;jr4j;jr5j;jr6j; jr7j;jr8jg, and for M3 > 0 =)R3 = maxfjr9j;jr10jg. Figure A.2: Examples of Radially Unbounded Functions. Figures A.2(a), A.2(b) and A.2(c) are all examples of radially unbounded functions. To understand the reasoning behind this, consider example A.2(a). According to the denition, given anM 1 > 0, we need to come up with anR 1 > 0 such thatu(x)>M 1 for allkxk>R 1 . For one-dimension, the norm can simply be replaced by its absolute value. Hence, as can be inferred from the picture, for M 1 > 0, one can always come up with an R 1 = maxfjr 1 j;jr 2 jg such that u(x) > M 1 for alljxj > R 1 . Similarly, for example A.2(c), given an M 2 > 0, one can always come up with an R 2 = maxfjr 3 j;jr 4 j;jr 5 j;jr 6 j; jr 7 j;jr 8 jg such that u(x)>M 2 for alljxj>R 2 . A.3 Radially Increasing and Spherically Increasing Radially Increasing Function: Let X be a normed space. A function u :X!< is strictly radially increasing if > 1 implies u(x)>u(x) for all x2X and x6= 0. Spherically Increasing Function: A function u : X!< is strictly spherically increasing ifkxk>kyk implies u(x)>u(y) for all x;y2X. 216 (a) A function that is (strictly) radially increasing but not spherically increas- ing. (b) A function that is both (strictly) radially increasing and (strictly) spheri- cally increasing function Figure A.3: Examples of Radially Increasing and Spherically Increasing Functions. Consider example A.3(a). The function in this gure is strictly radially increasing because for = 2 (needs to be greater than 1), u(215) = u(30) > u(15) and u(2 5) =u(10)>u(5). Continuing this line of reasoning, we can now generalize from the gure that u(x)>u(x)8 x2<f0g and > 1. Hence, the function is strictly 217 radially increasing. However,k 15k >k5k does not imply that u(15) > u(5). In fact, from Figure A.3(a), u(15) < u(5). Hence, the function fails to be spherically increasing. Next, consider example A.3(b). The function is strictly spherically increasing be- causekxk >kyk impliesku(x)k >ku(y)k. Further,kxk >kyk impliesku(x)k > ku(y)k,kxk>kyk impliesku(x)k>ku(y)k andkxk>kyk impliesku(x)k> ku(y)k. Hence, the functionu(x) is indeed strictly spherically increasing. Further, the function is also strictly radially increasing because u(2x) > u(x) for all x2<f0g. Hence, the function in Figure A.3(b) is both strictly spherically increasing and strictly radially increasing. From the examples above, we can now deduce that in one-dimension, a spherically increasing function implies that the function is also radially increasing. However, the converse need not necessarily hold true as illustrated. Spherically increasing functions form a subclass of the more general radially increasing functions. A.4 Radially Increasing and Radially Unbounded Given that a function u(x) in one-dimension is (strictly) radially increasing, it is our aim to show that u is also radially unbounded. Proof: The function u(x); x2< is given to be (strictly) radially increasing. Since u is strictly radially increasing, when we draw a line y = M across the graph of the function, it intersects the graph at precisely two points (see Figure A.3). Thus, for any given M > 0, there will be a maximum of two intersection points r 1 and r 2 such that u(r 1 ) = u(r 2 ) = M, where r 1 < 0 and r 2 > 0. Thus, R = maxfjr 1 j;jr 2 jg + where > 0. 218 Now, for u to be radially unbounded, it suces to show that u(x) > M for all kxk>R. To show this, choosex =R where> 1. Consequently,kxk =kRk = jjkRk =R. As is varied from 1 to1,x correspondingly varies fromR to1. Now, u(R)>u(R)>u(r 2 ) =M =)u(R)>M u(R)>u(R)>u(r 1 ) =M =)u(R)>M asjRj>jRj,R>jr 1 j andR>jr 2 j. This implies thatu(R)>M8kxk>R. Thus, this proves thatu(x) is indeed radially unbounded given that it is already radially increasing. The converse need not necessarily hold true as can be observed from Figures A.2(b) and A.2(c). A.5 Miscellaneous Results Result 1: If u(x) is C 1 and it is positive denite, then u 0 (0) = 0. Proof: Before we begin, let us refresh some of the denitions required to prove these results. The derivative of a function u(x), denoted as u 0 (x), is dened as u 0 (x) = lim h!0 u(x +h)u(x) h If the limit exists, then u is said to be dierentiable at the point x. Next, if u is a C 1 function, then u(x) is a continuous function and additionally, its derivative u 0 (x) exists and is continuous. And nally, a positive denite function has u(0) = 0 and u(x) > 08 x6= 0. Knowing these denitions, we can now detail the proof. Consider u 0 (0) which is given by u 0 (0) = lim h!0 u(0 +h)u(0) h = lim h!0 u(h) h 219 Now, depending on how we approach zero on the real axis i.e., either from the positive side or the negative side, the value of the limit diers and is given by u 0 0 = lim h!0 u(h) h = positive negative = negative u 0 0 + = lim h!0 + u(h) h = positive positive = positive Thus, the derivative u 0 (x) is negative immediately to the left of x = 0, and is positive immediately to the right of x = 0. But since u(x) is a C 1 function, its derivative u 0 (x) must exist and must be continuous everywhere in the domain. Thus, u 0 (x) has to be continuous at x = 0 as well and cannot jump to1. This forces the value of u 0 (0) to be exactly zero. Result 2: If u(x) is a C 2 function that is strictly positive denite and additionally if u 00 (x)> 0 for all x, then u(x) is strictly radially increasing. Proof: Since u(x) is a C 2 function, u itself is a continuous function and the rst and second derivatives ofu exist and are continuous. The second derivative of u is given by u 00 (x) = lim h!0 u 0 (x +h)u 0 (x) h Keeping Result 1 in mind (i.e., if u(x) is C 1 and positive denite, then u 0 (0) = 0), let us evaluate u 00 (x) at x = 0 as u 00 (0) = lim h!0 u 0 (0 +h)u 0 (0) h = lim h!0 u 0 (h) h But since u 00 > 0 for all x, u 00 0 and u 00 0 + are also both greater than zero. Hence, u 00 0 = lim h!0 u 0 (h) h = u 0 (0 ) negative > 0 u 00 0 + = lim h!0 + u 0 (h) h = u 0 (0 + ) positive > 0 220 This means that in a neighborhood of zero, u 0 (h) is given by u 0 (h) = 8 > > < > > : < 0 for h< 0: > 0 for h> 0: (A.1) Thus, u is decreasing immediately to the left of x = 0 and is increasing immediately to the right of x = 0 along with a local minimum at x = 0 (as u 0 (0) = 0). But we know something more about the second derivative i.e., u 00 > 0 for all x2< (and not just the neighborhood of x = 0). And therefore, we have u 00 (x) = lim h!0 u 0 (x +h)u 0 (x) h > 0 8 x For negative h (i.e., approaching x from the left), we have u 0 (x +h)u 0 (x) negative > 0 =) u 0 (x +h)<u 0 (x) 8 x; 8 h< 0 For positive h (i.e., approaching x from the right), we have u 0 (x +h)u 0 (x) positive > 0 =) u 0 (x +h)>u 0 (x) 8 x; 8 h> 0 Summarizing the facts, we have 1. u 0 (0) = 0 2. u 0 (x +h)>u 0 (x) for all h> 0 and for all x2< 3. u 0 (x +h)<u 0 (x) for all h< 0 and for all x2< From the above summary of points, we can conclude that 1. x = 0 is a global minimum. 2. u is strictly increasing in the interval 0<x<1. 3. u is strictly decreasing in the interval1<x< 0. 4. u 0 (x) is strictly increasing for all x2< 221 The above conclusions mean that the function u is a strictly radially increasing. Func- tions with u 00 > 0 having atmost one global minimum are referred to as strictly convex functions (for more on convex functions, see Ref. [11]). An alternate way to prove the same result is detailed as follows. For u(x) to be radially increasing, we need u(x) > u(x) for all > 1 and x2<f0g. Choose = 1 +, where > 0. Thus, we have u(x) u(x) = u ((1 +)x) u(x) = u(x) + u 0 (x) x + u 00 (x) 2 x 2 2 + u(x) = u 0 (x) x + u 00 (x) 2 x 2 2 + H.O.T. = u 0 + u 00 2 2 + H.O.T. If we choose appropriately to be a very small positive number, then x can be ap- proximated by a small number (our intention in choosing to be small is so that we can step a tiny bit to the right of point x). Since is small, the rst two terms in the expansion dominate and the higher order terms can be ignored. When> 0,u 0 > 0 and when < 0, u 0 < 0. Furthermore, 2 > 0 and u 00 > 0. Hence, the right hand side is always positive and sou(x)>u(x). The functionu is thus strictly radially increasing. A.6 Requisite Conditions on the Potentials of the Springs Let us denote the potential of the spring by u(x) and the spring force by f(x). The rst condition on the spring is that it should be a `physical' spring similar to the ones found in nature. By this, we mean that when the displacement is positive (i.e., spring is in tension), the spring force is positive (restoring force is negative) and when the displacement is negative (i.e., spring is under compression), the spring force is negative (restoring force is positive). Furthermore, a zero displacement should produce a zero 222 spring force. This requirement translates to the following two conditions: 1. f(0) = 0 (i.e., when the displacement is zero, the spring force is also zero). 2. xf(x)> 0 for all x6= 0 (positive spring force for positive displacement and negative spring force for negative displacement; this condition restricts the spring force to lie in the rst and third quadrants). There is one additional requirement that is imposed on the spring potential by Ap- pendix F. For the actuator locations proof to go through, we need 3. f 0 (x) =u 00 (x)6= 0 with f 0 (x) =u 0 (x) = 0 only at x = 0. If we desire a `physical' spring force, then f 0 (x) = u 00 (x) 0. Thus, the only remain- ing possibility is to choose f 0 (x) = u 00 (x) 0 with f 0 (x) = u 00 (x) = 0 only at x = 0. Additionally, we require u 00 (x) to be continuous (u(x) is a C 2 function captures this requirement). Discontinuities or jumps inu 00 (x) are not allowed as this correspondingly makes the spring force discontinuous and this is something we want to avoid as it does not capture reality. For completeness, we also want to mention thatu 00 (x) cannot change signs from positive to negative or vice-versa, because in order to change signs, u 00 has to equal zero somewhere and this is not allowed as per the third condition except possibly atx = 0. Even if the change of sign ofu 00 happens atx = 0, it results in a non-physical spring force and hence is disallowed. Thus, these requirements translate to the following three conditions on the spring po- tentials: 1. u(x) is a C 2 function. 2. u(x) is a strictly convex function with a global minimum at x = 0. 3. u 00 (x) = 0 possibly only at x = 0. 223 Appendix B Energy H is a positive denite function The energy function H for the NDOF nonlinear lattice is given by equation (3.3). The kinetic energyT ( _ q) is the sum ofn quadratic terms withT (0) = 0 andT ( _ q)> 08 _ q6= 0. The potential energy U(q) on the other hand, is a sum of (n + 1) functions u i ;i = 0; 1; 2;:::n; for the xed-xed lattice and a sum of n functions u i ;i = 0; 1; 2;:::n 1; for the xed-free lattice (as u n 0). Since each potential is positive denite (see Section 3.2.1 for reasoning), we have u i (0) = 0 and u i ( i ) > 08 i 6= 0 where i = q i+1 q i . Therefore, U(q) = P u i ( i )> 08 q6= 0 (as q6= 0 implies that atleast one of the i 's is not equal to zero). Further, since the left end is xed (q o 0) for both a xed-xed and a xed-free lattice, i = 08 i implies q = 0. Thus, U(0) = P u i (0) = 0. Consequently, we obtainH(0) = 0 andH(x)> 08x6= 0 wherex = (q; _ q)2< 2n . Hence, H is strictly positive denite. 224 Appendix C A Closed Form Expression for the Control Force F C In this appendix, a closed form expression for the explicit nonlinear control force F C is derived using equation (2.7). The constraint matrices A and b are expressed in terms of matrices C and N (see Section 3.4). And therefore, before we compute the control force, let us list some properties of the matrices C and N. (a) C T C +N T N = I n , where I n denotes the identity matrix of size n. (b) CC T = I k , where I k denotes the identity matrix of size k. (c) NN T = I nk , where I nk denotes the identity matrix of size (nk). (d) C T C = C T C for all diagonal matrices . (e) N T N = N T N for all diagonal matrices . (f) NC T = NMC T = [O] (nk)k , where [O] denotes the zero matrix. (g) CN T = CMN T = [O] k(nk) , where [O] denotes the zero matrix. The computation of the control force,F C , involves the evaluation of the Moore-Penrose (MP) inverse of the (nk + 1)-by-n matrix B [83] given by B =AM 1=2 = 2 6 4 _ q T M NM 3 7 5M 1=2 = 2 6 4 _ q T M 1=2 NM 1=2 3 7 5 (C.1) 225 Given any (nk + 1)-by-n matrixB, there exists a uniquen-by-(nk + 1) matrixB + , called the MP inverse of the matrixB, which satises the following four conditions [83]. 1: (BB + ) T =BB + ; 2: (B + B) T =B + B; 3: BB + B =B; 4: B + BB + =B + For a matrix B given by (C.1), we claim that B + is given by B + = M 1=2 C T C _ q _ q T C T CM _ q M 1=2 N T M 1=2 C T C _ q _ q T N T _ q T C T CM _ q (C.2) Assuming thatB + given by equation (C.2) is indeed the correct expression for the MP inverse of B, we show that it satises all four conditions of the MP inverse. (i) BB + = 2 6 4 _ q T M 1=2 NM 1=2 3 7 5 M 1=2 C T C _ q _ q T C T CM _ q M 1=2 N T M 1=2 C T C _ q _ q T N T _ q T C T CM _ q = 2 6 4 _ q T MC T C _ q _ q T C T CM _ q _ q T N T ( _ q T MC T C _ q) _ q T N T _ q T C T CM _ q (NMC T )C _ q _ q T C T CM _ q NN T (NMC T )C _ q _ q T N T _ q T C T CM _ q 3 7 5 =I nk+1 (C.3) By applying Property (d), the (1, 1) block ofBB + is unity and the (1, 2) block simplies to a (nk) sized zero row vector. The (2, 1) block is a (nk) sized column vector which is zero by virtue of Property (f). Similarly, the (2, 2) block is an (nk) sized square matrix which reduces toNN T by applying Property (f), which further simplies toI nk by applying Property (c). This reduces the matrix BB + to an (nk + 1) sized identity matrix. Hence, the rst MP condition is satised. (ii) B + B = M 1=2 C T C _ q _ q T C T CM _ q M 1=2 N T M 1=2 C T C _ q _ q T N T _ q T C T CM _ q 2 6 4 _ q T M 1=2 NM 1=2 3 7 5 = M 1=2 C T C _ q _ q T M 1=2 _ q T C T CM _ q +M 1=2 N T NM 1=2 M 1=2 C T C _ q _ q T N T NM 1=2 _ q T C T CM _ q = M 1=2 C T C _ q _ q T (IN T N)M 1=2 _ q T C T CM _ q +M 1=2 N T NM 1=2 = M 1=2 C T C _ q _ q T C T CM 1=2 _ q T C T CM _ q +N T N (C.4) 226 To arrive at the last equality of (C.4), Properties (a) and (e) have been used. Clearly, the matrix B + B is symmetric and thus the second MP condition is satised. (iii) BB + B =I nk+1 B =B; which directly follows from equation (C.3). (iv) B + BB + = M 1=2 C T C _ q _ q T C T CM 1=2 _ q T C T CM _ q +N T N M 1=2 C T C _ q _ q T C T CM _ q M 1=2 N T M 1=2 C T C _ q _ q T N T _ q T C T CM _ q The B + BB + matrix is a 1-by-2 block matrix, where the (1, 1) block is given by (1; 1) = M 1=2 C T C _ q _ q T C T CMC T C _ q ( _ q T C T CM _ q) 2 + N T (NM 1=2 C T )C _ q _ q T C T CM _ q = M 1=2 C T C _ q _ q T C T (CC T )CM _ q ( _ q T C T CM _ q) 2 = M 1=2 C T C _ q( _ q T C T CM _ q) ( _ q T C T CM _ q) 2 = M 1=2 C T C _ q _ q T C T CM _ q (C.5) In the derivation (C.5) above, the second term of the rst equality drops out by virtue of Property (f) and the rst term is simplied by using Properties (b) and (d). Next, the (1, 2) block of the matrix B + BB + is given by (1; 2) = h M 1=2 C T C _ q _ q T C T (CM 1=2 M 1=2 N T ) _ q T C T CM _ q +N T NM 1=2 N T M 1=2 C T C _ q _ q T C T CM 1=2 M 1=2 C T C _ q _ q T N T ( _ q T C T CM _ q) 2 N T (NM 1=2 C T )C _ q _ q T N T _ q T C T CM _ q i (C.6) The rst and the fourth terms of the (1, 2) block above drop out by virtue of Properties (g) and (f), respectively. When Property (e) is applied to the second term and Property (d) is applied to the third term, the (1, 2) block reduces to (1; 2) = M 1=2 N T (NN T ) M 1=2 C T C _ q _ q T C T (CC T )CM _ q _ q T N T ( _ q T C T CM _ q) 2 = M 1=2 N T M 1=2 C T C _ q( _ q T C T CM _ q) _ q T N T ( _ q T C T CM _ q) 2 = M 1=2 N T M 1=2 C T C _ q _ q T N T _ q T C T CM _ q (C.7) 227 We note that Properties (b) and (c) have been used to simplify the rst equality of (C.7) above. Hence, we obtain B + BB + as B + BB + = M 1=2 C T C _ q _ q T C T CM _ q M 1=2 N T M 1=2 C T C _ q _ q T N T _ q T C T CM _ q =B + (C.8) which satises the fourth MP condition. Since, all four MP conditions are satised, we ascertain that the B + given by (C.2) is indeed the correct expression for the Moore- Penrose inverse of the matrix B. Main Result: The control force can now be calculated as F C (q; _ q;t) =M 1=2 (AM 1=2 ) + (bAa) =M 1=2 B + 2 6 4b 2 6 4 _ q T M NM 3 7 5M 1 F 3 7 5 =M 1=2 B + 2 4 2 4 _ q T F(HH ) NF 3 5 2 4 _ q T F NF 3 5 3 5 =M 1=2 B + 2 6 4 (HH ) [O] (nk)1 3 7 5 =M 1=2 M 1=2 C T C _ q _ q T C T CM _ q M 1=2 N T M 1=2 C T C _ q _ q T N T _ q T C T CM _ q 2 6 4 (HH ) [O] (nk)1 3 7 5 = (HH ) _ q T C T CM _ q MC T C _ q = (H(q; _ q)H ) _ q T C M C _ q C C T CM _ q (C.9) where _ q T C M C _ q C = P k g=1 m ig _ q 2 ig is twice the kinetic energy of the set of controlled masses. From the third equality of (C.9), we note that whenever an energy stabilization constraint is applied to a mechanical system, the term _ q T F always drops out as long as the system under consideration is conservative. This concludes our derivation of the explicit nonlinear control force F C in closed form. 228 Appendix D Origin O is a single isolated equilibrium point Consider a NDOF nonlinear lattice with xed-xed (or xed-free) boundary conditions. The equilibrium points of uncontrolled (unconstrained) and the controlled (constrained) system can be calculated by substituting _ q q 0 in equations (3.5) and (3.19) respectively. In both cases, we obtain F = [O] n1 ; where the i th row of this relation can be written as f i1 q i (t)q i1 (t) = f i q i+1 (t)q i (t) ; i = 1; 2; 3;::: n: (D.1) For the xed-xed lattice, equation (D.1) implies f i q i+1 (t)q i (t) = c(t); i = 0; 1; 2; 3;::: n: (D.2) so thatf 1 i c(t) =q i+1 (t)q i (t); i = 0; 1;:::n. Summing overi on both sides, we have n X i = 0 f 1 i c(t) = n X i = 0 q i+1 (t)q i (t) = q n+1 (t)q o (t) (D.3) Sinceq o (t)q n+1 (t) 0, we have P n i=0 f 1 i c(t) = 0 whose only solution isc(t) = 0 as eachf i is a strictly increasing bijective function with f i (0) = 0; xf i (x)> 08x6= 0 (see 229 Section 3.3). From (D.2) then with c(t) = 0, we have q i+1 (t)q i (t) = 0;i = 0; 1;:::;n; which implies q i (t) = 0;i = 1; 2;:::n; because q o (t) 0. For the xed-free case, when i =n;f n 0 and hence equation (D.1) yields f i q i+1 (t)q i (t) = 0;i = 0; 1;::: (n 1), as in the xed-xed case. And since q o (t) 0; we again obtain q i (t) = 0; i = 1; 2;:::n. Therefore, in 2n-dimensional phase space, the origin O (q _ q 0) is a unique and isolated equilibrium point of the unconstrained (and the constrained) NDOF nonlinear lattice. 230 Appendix E Set is compact In this appendix, our aim is to show that the set (described by equation 3.33) is com- pact. The set can be alternately conceived as =H 1 ([;c]) where 0<<H <c and H 1 denotes the pre-image of the energy function H (described by equation 3.3). To prove that is compact, we use the following result [3, 67]. Let X < 2n and Y < + be Euclidean spaces. A function H : X ! Y is radially unbounded if and only if the pre-image H 1 (K) of every compact setKY is compact in X. To use this result, we need to rst establish that the energy H is radially unbounded. The energy function H is said to be radially unbounded if given any M2< + , there exists an R2< + such that H(x)>M for allkxk>R [44]. We use the innity norm to prove our results. H(x) =H(q; _ q) =T ( _ q) +U(q) = n+1 X i=1 1 2 m i _ q 2 i + n X i=0 u i (q i+1 q i ) = X i h i (E.1) The basic idea behind the approach is to equate each individual term h i of the energy function H to M, and nd the supremum amongst the largest absolute valuesjx i j max of the 2n-coordinates, such that for each of these terms, h i (jx i j max ) equals M. This 231 supremum value gives usR, which is the side length of the hypercube in 2n-dimensional phase space. To nd R, we adopt the following algorithm. STEP 1: Find the supremum R v amongst the largest absolute values of the n ve- locities. To this end, we consider the kinetic energy terms T ( _ q) of the energy function H (equation E.1). When each kinetic energy term is equated to M, out of all the n terms, the term with the inmum mass m k;inf = inffm 1 ;m 2 ;:::;m n g gives us the supre- mum velocity. Therefore, 1 2 m k;inf _ q 2 k = M yields R v =j _ q k j + o = + q 2M m k;inf + o ; where o > 0 is included so that R v is strictly greater than the supremumj _ q k j. STEP 2: Find the supremumR d amongst the largest absolute values of then displace- ments. Consider the potential energy termsU(q) of the energy functionH (see equation E.1). When the rst termu 0 (q 1 ) is equated toM, for any givenM > 0, there exist pre- cisely two real valuesr 1 ;r 2 2< such thatu 0 (r 1 ) =M andu 0 (r 2 ) =M. This is because u 0 (q 1 ) is positive denite and strictly radially increasing (see Section 3.3). Therefore, given anM, a bound on the maximum value ofjq 1 j is given byR 1 = maxfjr 1 j;jr 2 jg+ 1 , where 1 > 0. Next, let us equate the second term of the potential energy,u 1 (q 2 q 1 ), toM. Again, since u 1 is positive denite and strictly radially increasing, for a given M, there exist precisely two real values r 3 ;r 4 such thatu 1 (r 3 ) =M andu 1 (r 4 ) =M. Therefore, given anM, a bound on the maximum value ofjq 2 j is given byR 2 =R 1 +maxfjr 3 j;jr 4 jg+ 2 , where 2 > 0. Continuing this recursive process, for the i th term, given an M > 0, a bound on the maximum value ofjq i j is given by R i = R i1 + maxfjr 2i1 j;jr 2i jg + i , where i > 08 i, R o = 0, and u i1 (r 2i1 ) =u i1 (r 2i ) =M for i = 1; 2;:::n. For a xed-free lattice, there are onlyn potential energy terms in the energy expres- sion and therefore the supremum amongst then displacements is given byR d =R n . On the other hand, for a xed-xed lattice, there are (n + 1) terms and equating the last term of the energy expressionu n (q n ) toM yields yet another estimate for a bound on the maximum value ofjq n j given byR 0 n = maxfjr 2n+1 j;jr 2n+2 jg + n+1 where n+1 > 0 andu n (r 2n+1 ) =u n (r 2n+2 ) =M. Thus, for a xed-xed lattice, the supremum amongst 232 the largest absolute values of the n displacements is given by R d = maxfR n ;R 0 n g. STEP 3: Therefore, given any M, a corresponding bound on the side length R of the hypercube in 2n-dimensional phase space is given by R = maxfR v ;R d g +4 where 4 > 0. Clearly, for allkxk 1 > R, we have H(x) > M. Hence, the energy function H(x) is radially unbounded. Thus, by virtue of the result stated at the beginning of this appendix, since H(x) is a radially unbounded function, it follows that the pre-image H 1 (K) of every compact set K = [;c]2< + is compact in< 2n . But, H 1 ([;c]) is our set and therefore, is compact in< 2n . 233 Appendix F Sucient Conditions on the Actuator Placements In this appendix, we want to show that under certain choices of the actuator placements, the only invariant set belonging to _ q C 0 is the origin (q _ q 0). We will show this to be true provided that one or more of the following actuator congurations are adopted. 1. The rst mass m 1 and/or the last mass m n in the lattice is among the set of controlled masses. 2. A consecutive pair of masses in the lattice is among the set of controlled masses i.e. i x ;i y 2S C such thatji x i y j = 1. To ensure that the set dened by H(q; _ q) = H is the only globally attracting set in , the invariant set(s) satisfying _ q C 0 should lie outside of . But, by actuating an arbitrary set of k masses out of n masses in the lattice, one cannot always guarantee that this holds true as is shown by the following example. Consider a three-mass, homogeneous, nonlinear lattice with xed ends. The con- trolled (constrained) equations of motion of the lattice with a single actuator placed at the second mass of the three-mass lattice is given by 234 m q 1 = f(q 2 q 1 )f(q 1 ); (F.1) m q 2 = f(q 3 q 2 )f(q 2 q 1 ) o (HH )m _ q 2 ; (F.2) m q 3 = f(q 3 )f(q 3 q 2 ); (F.3) where f meets all the criteria of a spring force function as discussed in Section 3.3. Since the second mass alone is controlled S C =f2g, and when _ q C 0, we have _ q 2 0. Consequently, q 2 0. This reduces equation (F.2) to q 3 (t)q 2 (t) =q 2 (t)q 1 (t) (F.4) Dierentiating equation (F.4) with respect to time t and noting that _ q 2 0, we obtain _ q 3 = _ q 1 . Consequently, q 3 = q 1 . Solving this along with equations (F.1) and (F.3) yields q 2 0 and q 1 q 3 . Hence, there exist sets of invariant orbits described by Q =f(q 1 (t); _ q 1 (t); 0; 0;q 1 (t); _ q 1 (t))g that satisfy _ q 2 0 and that lie inside . And therefore, besides H(q; _ q) =H , there are additional invariant sets in E (and hence in ) that satisfy _ q 2 0 and to which the trajectories are conned. Thus, in such a case, one cannot guarantee that the set H(q; _ q) =H is globally attracting in . To ensure that the sets of invariant orbits satisfying _ q C 0 lie outside of , the actuators must be placed appropriately so that _ q C 0 only yields the set q _ q 0 (origin O), which lies outside . Actuator Positions: Consider that the i th mass of the lattice is actuated. The con- strained equation of motion of the i th mass of the lattice is m i q i = f i (q i+1 q i )f i1 (q i q i1 ) o (HH )m i _ q i (F.5) 235 Since thei th mass is controlled,i2S C , and when _ q C 0, we have _ q i 0. Consequently, q i 0. This reduces equation (F.5) to f i (q i+1 q i ) =f i1 (q i q i1 ) (F.6) Dierentiating equation (F.6) with respect to time t, we obtain f 0 i (q i+1 q i ) ( _ q i+1 _ q i ) =f 0 i1 (q i q i1 ) ( _ q i _ q i1 ) (F.7) which simplies to f 0 i (q i+1 q i ) ( _ q i+1 ) =f 0 i1 (q i q i1 ) ( _ q i1 ) (F.8) Now, if we additionally have either _ q i1 0 or _ q i+1 0, then we obtain certain simplifying results. In the present case, we derive our results assuming _ q i1 0 but a similar derivation follows if _ q i+1 0 instead. In equation (F.8), if _ q i1 0, then we obtainf 0 i (q i+1 q i ) ( _ q i+1 ) 0 which implies that either _ q i+1 0 orf 0 i (q i+1 q i ) 0. If f 0 i (q i+1 q i ) 0, then (q i+1 q i ) 0 from Property (3) in Section 3.3, which once again after dierentiation with respect to time t yields _ q i+1 0. Consequently, q i+1 0. Substituting these results into the constrained equation of motion of the (i + 1) th mass of the lattice gives us f i+1 (q i+2 q i+1 ) =f i (q i+1 q i ) (F.9) Dierentiating equation (F.9) with respect to time t gives us f 0 i+1 (q i+2 q i+1 ) ( _ q i+2 _ q i+1 ) =f 0 i (q i+1 q i ) ( _ q i+1 _ q i ) (F.10) Again since _ q i _ q i+1 0, as before equation (F.10) yields _ q i+2 0 (and consequently q i+2 0). Continuing this process of recursive substitution into the constrained equa- tion of motion of the (i + 2) th mass, and then the (i + 3) th mass, and so on and so forth until the n th mass of the lattice, we obtain 236 _ q k q k 0; k =i 1;i;i + 1;i + 2;:::;n: (F.11) On the other hand, since _ q i1 q i1 0, following the steps in equations (F.5 - F.8) for the constrained equation of motion of the (i 1) th mass of the lattice, we obtain _ q i2 0 (and hence q i2 0). Continuing this process of recursive substitution into the constrained equation of motion of the (i 2) th mass, and then the (i 3) th mass, and so on and so forth until the rst mass of the lattice, we obtain _ q k q k 0; k =i;i 1;i 2;i 3;:::; 1: (F.12) Thus, for an NDOF inhomogeneous nonlinear lattice 1. If i x ;i y 2S C andji x i y j = 1, then whenever _ q C 0, we have _ q ix _ q iy 0 with ji x i y j = 1 and for i =i x in equations (F.5 - F.12), we obtain _ q q 0. 2. For a xed-xed lattice, if we actuate the rst mass of the lattice, when _ q C 0 we have _ q 1 0, then i = 1 in equations (F.5 - F.11) along with _ q o 0 yields _ q q 0. On the other hand, if we actuate the last mass of the lattice, when _ q C 0, we have _ q n 0, then i = n in equations (F.5 - F.8, F.12) along with _ q n+1 0 yields _ q q 0. 3. For a xed-free lattice, actuating the rst mass follows a derivation similar to the xed-xed case. On the other hand, if we actuate the last mass of the lattice, when _ q C 0, we have _ q n 0, theni =n in equations (F.5 - F.8, F.12) along with f n 0 yields _ q q 0. From Appendix D, we know that for an NDOF nonlinear lattice, if _ q q 0, then q 0. Thus, for the three cases discussed above, it follows that the origin O (q _ q 0) is the only invariant point satisfying _ q C 0. The set has been constructed in such a way that an open region around the origin O of< 2n is excluded from . Since the origin O lies outside the set , the largest invariant set in E (and hence ) is the set dened by H(q; _ q) =H . 237 Appendix G Equations of Motion of a Symmetric Gyroscope Udwadia and Han [80] derive the following expression for the Lagrangian L of the symmetric gyro: L = 1 2 I m _ 2 m + _ ' 2 m sin 2 m + 1 2 I 3m _ m + _ ' m cos m 2 m m r m _ d m _ m sin m m m gr m cos m (G.1) With the Lagrangian at our disposal, the equations of motion of the symmetric gyro can be derived using Lagrange's method as d dt @L @ _ q @L @q =Q (G.2) where L is given by equation (G.1), Q is the generalized forces not arising from a potential and the generalized coordinates q, in our present case, are ;' and . The equations of motion in the three coordinates ;' and are then respectively given by I m m I m _ ' 2 m sin m cos m +I 3m _ m + _ ' m cos m _ ' m sin m m m r m d m sin m m m gr m sin m =F d (G.3) 238 I m _ ' m sin 2 m +I 3m _ m + _ ' m cos m cos m = const. =p 'm (G.4) I 3m _ m + _ ' m cos m = const. =p m (G.5) In equation (G.3), F d represents the nonconservative damping force acting on the sym- metric gyro in the nutation ( m ) direction. Using equation (G.5), equation (G.4) can be re-written as I m _ ' m sin 2 m +p m cos m =p 'm (G.6) Equations (G.5) and (G.6) can now be substituted into equation (G.3) to simplify it as follows: I m m I m _ ' 2 m sin m cos m +I 3m _ m + _ ' m cos m _ ' m sin m m m r m d m sin m m m gr m sin m =F d =)I m m I m _ ' 2 m sin m cos m +p m _ ' m sin m m m r m d m sin m m m gr m sin m =F d =)I m m + _ ' m sin m I m _ ' m cos m +p m m m r m d m sin m m m gr m sin m =F d =)I m m + p 'm p m cos m I m sin 2 m sin m I m p 'm p m cos m I m sin 2 m cos m +p m m m r m d m sin m m m gr m sin m =F d =)I m m + p 'm p m cos m I m sin m p m sin 2 m p 'm cos m +p m cos 2 m sin 2 m m m r m d m sin m m m gr m sin m =F d which nally reduces to I m m + (p 'm p m cos m ) (p m p 'm cos m ) I m sin 3 m m m r m d m sin m m m gr m sin m =F d (G.7) 239 Appendix H Modied St ormer{Verlet Scheme Given the initial displacement vector y o , the initial velocity vectorv o , and the accelera- tion of the systema(x; _ x;t) =M 1 (x;t)F (x; _ x;t), the displacementY i , velocityV i , and accelerationA i of thei th time step (wherei = 1; 2; :::; N) are calculated as follows: INITIAL STEP: Y 1 =y o ; V 1 =v o ; A 1 =a(t o ;y o ;v o ) for i = 1 to N 1 STEP 1: Y i+1 =Y i +V i t +A i t 2 2 STEP 2: V i+1 =V i +A i t INACCURATE (PREDICTOR STEP) STEP 3: A i+1 =a (t i+1 ;Y i+1 ;V i+1 ) STEP 4: V i+1 =V i + t 2 (A i + A i+1 ) (CORRECTOR STEP) end The St ormer{Verlet scheme discussed in Section 5.7.2 is suitable only for situations where the forceF (and hence the acceleration) in Newton's equation of motionM x =F is a function of the displacement and time alone i.e.,F =F (x;t). 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Abstract (if available)

Abstract A new and novel approach to the control of nonlinear mechanical systems is presented in this study. The approach is inspired by recent results in analytical dynamics that deal with the theory of constrained motion. The control requirements on the dynamical system are viewed from an analytical dynamics perspective and the theory of constrained motion is used to recast these control requirements as constraints on the dynamical system. Explicit closed form expressions for the generalized nonlinear control forces are obtained by using the fundamental equation of mechanics. The control so obtained is optimal at each instant of time and causes the constraints to be exactly satisfied. No linearizations and/or approximations of the nonlinear dynamical system are made, and no a priori structure is imposed on the nature of nonlinear controller. Three examples dealing with highly nonlinear complex dynamical systems that are chosen from diverse areas of discrete and continuum mechanics are presented to demonstrate the control approach. The first example deals with the energy control of underactuated inhomogeneous nonlinear lattices (or chains), the second example deals with the synchronization of the motion of multiple coupled slave gyros with that of a master gyro, and the final example deals with the control of incompressible hyperelastic rubber-like thin cantilever beams. Numerical simulations accompanying these examples show the ease, simplicity and the efficacy with which the control methodology can be applied and the accuracy with which the desired control objectives can be met.

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Asset Metadata

Creator Mylapilli, Harshavardhan (author)

Core Title An analytical dynamics approach to the control of mechanical systems

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Aerospace and Mechanical Engineering (Dynamics and Control)

Publication Date 08/04/2015

Defense Date 01/15/2015

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag absolute nodal coordinate formulation (ANCF),beam control,breathers,cantilever beams,chain-coupling,closed form global asymptotic control,constrained motion,energy control,Fermi–Pasta–Ulam problem,fundamental equation of mechanics,general-coupling,hyperelastic materials,incompressible material,Mooney-Rivlin models,multiple gyroscopes,nodal control,nonlinear control,nonlinear non-homogeneous chains,OAI-PMH Harvest,Poisson volumetric locking,selective reduced integration,solitons,Störmer–Verlet method,synchronization,thin beams,Udwadia-Kalaba equation,under actuation

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Language English

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Advisor Udwadia, Firdaus Erach (committee chair), Redekopp, Larry G. (committee member), Sacker, Robert J. (committee member)

Creator Email harsha.iitb@gmail.com,mylapill@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-625018

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