EFFICIENT SIMULATION OF FLEXIBLE BODY SYSTEMS WITH FRICTIONAL CONTACT/IMPACT By Naresh N. Khude A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mechanical Engineering) at the UNIVERSITY OF WISCONSIN-MADISON 2015 Date of final oral Examination: 11/24/2014 The dissertation is approved by the following members of the Final Oral Committee: Dan Negrut, Associate Professor, Mechanical Engineering Darryl Thelen, Professor, Mechanical Engineering Michael Zinn, Associate Professor, Mechanical Engineering Robert Rowlands, Professor, Mechanical Engineering Peter Qian, Associate Professor, Department of Statistics © Copyright by Naresh Khude 2015 All Rights Reserved i To Nileshdada ii ACKNOWLEDGMENTS I would like to thank my advisor, Professor Dan Negrut, for his support and guidance. I would also like to thank the committee members, Professor Darryl Thelen, Professor Michael Zinn, Professor Robert Rowlands, and Professor Peter Qian for sharing their expertise and for their time. I owe a large debt of gratitude to my friends and colleagues, especially Arman Pazouki and Praveen Yadav, who have been generous with their technical expertise and friendship during the past six years. Finally I would like to thank my parents, my brothers, and my wife for believing in me and keeping me motivated. This thesis would not have been possible without their unconditional love and support. iii TABLE OF CONTENTS LIST OF TABLES........................................................................................................................ vii LIST OF FIGURES ..................................................................................................................... viii ABSTRACT.................................................................................................................................. xii 1 Introduction............................................................................................................................. 1 1.1 Motivation........................................................................................................................ 1 1.2 Thesis Overview............................................................................................................... 3 1.3 Specific Contributions...................................................................................................... 5 2 Literature Review.................................................................................................................... 8 2.1 Multibody Dynamics........................................................................................................ 8 2.2 Numerical Solution Methods for DAEs......................................................................... 10 2.2.1 Stabilization Methods ............................................................................................. 11 2.2.2 Projection methods ................................................................................................. 11 2.2.3 State Space Methods............................................................................................... 13 2.2.4 Other Direct Methods ............................................................................................. 14 2.3 Flexible Multibody Dynamics........................................................................................ 14 2.3.1 Floating Frame of Reference Formulation.............................................................. 15 2.3.2 Nonlinear Finite Element Approach ....................................................................... 17 3 Low Order Implicit Integration Methods.............................................................................. 27 3.1 Low Order Integration Algorithms ................................................................................ 28 3.1.1 Implementation Details........................................................................................... 35 iv 3.2 Models Considered......................................................................................................... 36 3.3 Numerical Experiments and Results .............................................................................. 39 3.3.1 Global Convergence Analysis ................................................................................ 39 3.3.2 Energy Preservation................................................................................................ 42 3.3.3 Kinematic Constraint Drift ..................................................................................... 45 3.3.4 Runtime Comparison .............................................................................................. 48 3.4 Concluding Remarks...................................................................................................... 50 4 Absolute Nodal Coordinate Formulation.............................................................................. 52 4.1 Gradient Deficient ANCF beam..................................................................................... 53 4.1.1 A Curved GD ANCF Beam.................................................................................... 56 4.2 GD ANCF with Bilateral Constraints ............................................................................ 58 4.3 Numerical Experiments and Results .............................................................................. 62 4.3.1 Validation of the Gradient Deficient ANCF Elements........................................... 63 4.3.2 Comparison with Original ANCF Beam ................................................................ 67 4.3.3 Convergence Analysis ............................................................................................ 67 4.3.4 Benchmark Tests..................................................................................................... 68 4.4 Concluding Remarks...................................................................................................... 73 5 Frictional Contact/Impact Modeling – Penalty Approach.................................................... 74 5.1 Introduction and Theoretical Background ..................................................................... 74 5.2 Continuous Contact Force Model (CCFM).................................................................... 76 5.3 Spherical Decomposition and Contact Force Computation ........................................... 79 5.4 Implementation of CCFM with ANCF beams............................................................... 82 v 5.5 Numerical Experiments and Results .............................................................................. 83 5.5.1 Convergence analysis of ANCF+CCFM ................................................................ 85 5.5.2 Validation of ANCF+CCFM .................................................................................. 88 5.5.3 Energy analysis of ANCF+CCFM.......................................................................... 94 5.6 Concluding Remarks...................................................................................................... 96 6 Frictional Contact/Impact Modeling – Variational Approach.............................................. 97 6.1 Theoretical Background of DVI..................................................................................... 97 6.2 Differential Variational Inequality (DVI) ...................................................................... 99 6.2.1 Extension of DVI for Flexible Beams .................................................................... 99 6.2.2 Generalized Contact Coordinates ........................................................................... 99 6.2.3 The Complementarity Condition .......................................................................... 101 6.2.4 DVI Formulation for Frictionless Contact............................................................ 103 6.2.5 DVI Formulation for Frictionless Impact ............................................................. 104 6.2.6 DVI Formulation for Frictional Contact............................................................... 105 6.2.7 Solution of the Cone Complementarity Problem.................................................. 109 6.3 DVI Implementation .................................................................................................... 113 6.4 Numerical Experiments and Results ............................................................................ 114 6.4.1 Energy Analysis.................................................................................................... 115 6.4.2 Validation of DVI approach ................................................................................. 119 6.5 Concluding Remarks.................................................................................................... 123 7 Demonstration of Technology ............................................................................................ 124 vi 7.1 ANCF Implementation in Adams ................................................................................ 124 7.2 ANCF Model Applications .......................................................................................... 127 7.2.1 A Satellite with Self Deployable Solar Panel ....................................................... 127 7.2.2 An Elevator Cable Model ..................................................................................... 130 8 Conclusions........................................................................................................................