Flexible Multibody Dynamics

Arun Kanti Banerjee, 28 year career at Lockheed Martin Advanced Technology Center (1982 thru 2010), Palo Alto, California, US.Last position — Principal Research Scientist. Main Contribution: Developer of DYNACON, Lockheed's simulation tool for Multi-flexible-body dynamics and Control that has been used for many projects. Dr. Banerjee is one of the foremost experts in the world on the subject of flexible multibody dynamics.

• Preface ix 1 Derivation of Equations of Motion 11.1 Available Analytical Methods and the Reason for Choosing Kane’s Method 11.2 Kane’s Method of Deriving Equations of Motion 21.2.1 Kane’s Equations 41.2.2 Simple Example: Equations for a Double Pendulum 41.2.3 Equations for a Spinning Spacecraft with Three Rotors, Fuel Slosh, and Nutation Damper 61.3 Comparison to Derivation of Equations of Motion by Lagrange’s Method 111.3.1 Lagrange’s Equations in Quasi-Coordinates 14Reader’s Exercise 151.4 Kane’s Method of Direct Derivation of Linearized Dynamical Equation 161.5 Prematurely Linearized Equations and a Posteriori Correction by ad hoc Addition of Geometric Stiffness due to Inertia Loads 191.6 Kane’s Equations with Undetermined Multipliers for Constrained Motion 211.7 Summary of the Equations of Motion with Undetermined Multipliers for Constraints 221.8 A Simple Application 23Appendix 1.A Guidelines for Choosing Efficient Motion Variables in Kane’s Method 25Problem Set 1 27References 282 Deployment, Station-Keeping, and Retrieval of a Flexible Tether Connecting a Satellite to the Shuttle 292.1 Equations of Motion of a Tethered Satellite Deployment from the Space Shuttle 302.1.1 Kinematical Equations 312.1.2 Dynamical Equations 322.1.3 Simulation Results 352.2 Thruster-Augmented Retrieval of a Tethered Satellite to the Orbiting Shuttle 372.2.1 Dynamical Equations 372.2.2 Simulation Results 472.2.3 Conclusion 472.3 Dynamics and Control of Station-Keeping of the Shuttle-Tethered Satellite 47Appendix 2.A Sliding Impact of a Nose Cap with a Package of Parachute Used for Recovery of a Booster Launching Satellites 49Appendix 2.B Formation Flying with Multiple Tethered Satellites 53Appendix 2.C Orbit Boosting of Tethered Satellite Systems by Electrodynamic Forces 55Problem Set 2 60References 603 Kane’s Method of Linearization Applied to the Dynamics of a Beam in Large Overall Motion 633.1 Nonlinear Beam Kinematics with Neutral Axis Stretch, Shear, and Torsion 633.2 Nonlinear Partial Velocities and Partial Angular Velocities for Correct Linearization 693.3 Use of Kane’s Method for Direct Derivation of Linearized Dynamical Equations 703.4 Simulation Results for a Space-Based Robotic Manipulator 763.5 Erroneous Results Obtained Using Vibration Modes in Conventional Analysis 78Problem Set 3 79References 824 Dynamics of a Plate in Large Overall Motion 834.1 Motivating Results of a Simulation 834.2 Application of Kane’s Methodology for Proper Linearization 854.3 Simulation Algorithm 904.4 Conclusion 92Appendix 4.A Specialized Modal Integrals 93Problem Set 4 94References 965 Dynamics of an Arbitrary Flexible Body in Large Overall Motion 975.1 Dynamical Equations with the Use of Vibration Modes 985.2 Compensating for Premature Linearization by Geometric Stiffness due to Inertia Loads 1005.2.1 Rigid Body Kinematical Equations 1045.3 Summary of the Algorithm 1055.4 Crucial Test and Validation of the Theory in Application 106Appendix 5.A Modal Integrals for an Arbitrary Flexible Body 112Problem Set 5 114References 1146 Flexible Multibody Dynamics: Dense Matrix Formulation 1156.1 Flexible Body System in a Tree Topology 1156.2 Kinematics of a Joint in a Flexible Multibody Body System 1156.3 Kinematics and Generalized Inertia Forces for a Flexible Multibody System 1166.4 Kinematical Recurrence Relations Pertaining to a Body and Its Inboard Body 1206.5 Generalized Active Forces due to Nominal and Motion-Induced Stiffness 1216.6 Treatment of Prescribed Motion and Internal Forces 1266.7 “Ruthless Linearization” for Very Slowly Moving Articulating Flexible Structures 1266.8 Simulation Results 127Problem Set 6 129References 1317 Component Mode Selection and Model Reduction: A Review 1337.1 Craig-Bampton Component Modes for Constrained Flexible Bodies 1337.2 Component Modes by Guyan Reduction 1367.3 Modal Effective Mass 1377.4 Component Model Reduction by Frequency Filtering 1387.5 Compensation for Errors due to Model Reduction by Modal Truncation Vectors 1387.6 Role of Modal Truncation Vectors in Response Analysis 1417.7 Component Mode Synthesis to Form System Modes 1437.8 Flexible Body Model Reduction by Singular Value Decomposition of Projected System Modes 1457.9 Deriving Damping Coefficient of Components from Desired System Damping 147Problem Set 7 148Appendix 7.A Matlab Codes for Structural Dynamics 1497.10 Conclusion 159References 1598 Block-Diagonal Formulation for a Flexible Multibody System 1618.1 Example: Role of Geometric Stiffness due to Interbody Load on a Component 1618.2 Multibody System with Rigid and Flexible Components 1648.3 Recurrence Relations for Kinematics 1658.4 Construction of the Dynamical Equations in a Block-Diagonal Form 1688.5 Summary of the Block-Diagonal Algorithm for a Tree Configuration 1748.5.1 First Forward Pass 1748.5.2 Backward Pass 1748.5.3 Second Forward Pass 1758.6 Numerical Results Demonstrating Computational Efficiency 1758.7 Modification of the Block-Diagonal Formulation to Handle Motion Constraints 1768.8 Validation of Formulation with Ground Test Results 1828.9 Conclusion 186Appendix 8.A An Alternative Derivation of Geometric Stiffness due to Inertia Loads 187Problem Set 8 188References 1899 Efficient Variables, Recursive Formulation, and Multi-Point Constraints in Flexible Multibody Dynamics 1919.1 Single Flexible Body Equations in Efficient Variables 1919.2 Multibody Hinge Kinematics for Efficient Generalized Speeds 1969.3 Recursive Algorithm for Flexible Multibody Dynamics with Multiple Structural Loops 2019.3.1 Backward Pass 2019.3.2 Forward Pass 2079.4 Explicit Solution of Dynamical Equations Using Motion Constraints 2099.5 Computational Results and Simulation Efficiency for Moving Multi-Loop Structures 2109.5.1 Simulation Results 210Acknowledgment 215Appendix 9.A Pseudo-Code for Constrained nb-Body m-Loop Recursive Algorithm in Efficient Variables 216Problem Set 9 220References 22010 Efficient Modeling of Beams with Large Deflection and Large Base Motion 22310.1 Discrete Modeling for Large Deflection of Beams 22310.2 Motion and Loads Analysis by the Order-n Formulation 22610.3 Numerical Integration by the Newmark Method 23010.4 Nonlinear Elastodynamics via the Finite Element Method 23110.5 Comparison of the Order-n Formulation with the Finite Element Method 23310.6 Conclusion 237Acknowledgment 238Problem Set 10 238References 23811 Variable-n Order-n Formulation for Deployment and Retraction of Beams and Cables with Large Deflection 23911.1 Beam Discretization 23911.2 Deployment/Retraction from a Rotating Base 24011.2.1 Initialization Step 24011.2.2 Forward Pass 24011.2.3 Backward Pass 24311.2.4 Forward Pass 24411.2.5 Deployment/Retraction Step 24411.3 Numerical Simulation of Deployment and Retraction 24611.4 Deployment of a Cable from a Ship to a Maneuvering Underwater Search Vehicle 24711.4.1 Cable Discretization and Variable-n Order-n Algorithm for Constrained Systems with Controlled End Body 24811.4.2 Hydrodynamic Forces on the Underwater Cable 25411.4.3 Nonlinear Holonomic Constraint, Control-Constraint Coupling, Constraint Stabilization, and Cable Tension 25511.5 Simulation Results 257Problem Set 11 261References 26712 Order-n Equations of Flexible Rocket Dynamics 26912.1 Introduction 26912.2 Kane’s Equation for a Variable Mass Flexible Body 26912.3 Matrix Form of the Equations for Variable Mass Flexible Body Dynamics 27412.4 Order-n Algorithm for a Flexible Rocket with Commanded Gimbaled Nozzle Motion 27512.5 Numerical Simulation of Planar Motion of a Flexible Rocket 27812.6 Conclusion 285Acknowledgment 285Appendix 12.A Summary Algorithm for Finding Two Gimbal Angle Torques for the Nozzle 285Problem Set 12 286References 286Appendix A Efficient Generalized Speeds for a Single Free-Flying Flexible Body 287Appendix B A FORTRAN Code of the Order-n Algorithm: Application to an Example 291Index 301

"The book is intended for readers with backgrounds in rigid body dynamics and structural dynamics. It is well written and may be useful for structural engineers and researchers in applied mechanics." (Zentralblatt MATH, 2016)