About the Book

This book describes how to build mathematical models of multibody systems with elastic components. Examples of such systems are the human body itself, construction cranes, cars with trailers, helicopters, spacecraft deploying antennas, tethered satellites and underwater maneuvering vehicles looking for mines while being connected by a cable to a ship. Equations of motion for these complex systems are derived by using Kane's method. Kane's equations are chosen, for reasons of reduced labor in deriving the equations and simplicity of the final equations. However, the materials the book covers goes beyond a direct application of Kane's equations, to more computationally efficient algorithms like order-n formulations and block-diagonal formulations and differential-algebraic equations for systems with closed structural loops. 

About the Author

Arun Kanti Banerjee worked for 28 year at Lockheed Martin Advanced Technology Center (1982 thru 2010), Palo Alto, California, US

Table of Contents:
Preface 1 Derivation of Equations of Motion 1.1 Available Analytical Methods and the Reason for Choosing Kane's Method 1.2 Kane's Method of Deriving Equations of Motion 1.3 Comparison to Derivation of Equations of Motion by Lagrange's Method 1.4 Kane's Method of Direct Derivation of Linearized Dynamical Equation 1.5 Prematurely Linearized Equations and a Posteriori Correction by ad hoc Addition of Geometric Stiffness due to Inertia Loads 1.6 Kane's Equations with Undetermined Multipliers for Constrained Motion 1.7 Summary of the Equations of Motion with Undetermined Multipliers for Constraints 1.8 A Simple Application 2 Deployment, Station-Keeping and Retrieval of a Flexible Tether Connecting a Satellite to the Shuttle 2.1 Equations of Motion of a Tethered Satellite Deployment from the Space Shuttle 2.2 Thruster-Augmented Retrieval of a Tethered Satellite to the Orbiting Shuttle 2.3 Dynamics and Control of Station-Keeping of the Shuttle-Tethered Satellite 3 Kane's Method of Linearization Applied to the Dynamics of a Beam in Large Overall Motion 3.1 Nonlinear Beam Kinematics with Neutral Axis Stretch, Shear and Torsion 3.2 Nonlinear Partial Velocities and Partial Angular Velocities for Correct Linearization 3.3 Use of Kane's Method for Direct Derivation of Linearized Dynamical Equations 3.4 Simulation Results for a Space-Based Robotic Manipulator 3.5 Erroneous Results Obtained Using Vibration Modes in Conventional Analysis 4 Dynamics of a Plate in Large Overall Motion 4.1 Motivating Results of a Simulation 4.2 Application of Kane's Methodology for Proper Linearization 4.3 Simulation Algorithm 4.4 Conclusion 5 Dynamics of an Arbitrary Flexible Body in Large Overall Motion 5.1 Dynamical Equations with the Use of Vibration Modes 5.2 Compensating for Premature Linearization by Geometric Stiffness due to Inertia Loads 5.2.1 Rigid Body Kinematical Equations 5.3 Summary of the Algorithm 5.4 Crucial Test and Validation of the Theory in Application 6 Flexible Multibody Dynamics: Dense Matrix Formulation 6.1 Flexible Body System in a Tree Topology 6.2 Kinematics of a Joint in a Flexible Multibody Body System 6.3 Kinematics and Generalized Inertia Forces for a Flexible Multibody System 6.4 Kinematical Recurrence Relations Pertaining to a Body and Its Inboard Body 6.5 Generalized Active Forces due to Nominal and Motion-Induced Stiffness 6.6 Treatment of Prescribed Motion and Internal Forces 6.7 "Ruthless Linearization" for Very Slowly Moving Articulating Flexible Structures 6.8 Simulation Results 7 Component Mode Selection and Model Reduction: A Review 7.1 Craig-Bampton Component Modes for Constrained Flexible Bodies 7.2 Component Modes by Guyan Reduction 7.3 Modal Effective Mass 7.4 Component Model Reduction by Frequency Filtering 7.5 Compensation for Errors due to Model Reduction by Modal Truncation Vectors 7.6 Role of Modal Truncation Vectors in Response Analysis 7.7 Component Mode Synthesis to Form System Modes 7.8 Flexible Body Model Reduction by Singular Value Decomposition of Projected System Modes 7.9 Deriving Damping Coefficient of Components from Desired System Damping 8 Block-Diagonal Formulation for a Flexible Multibody System 8.1 Example: Role of Geometric Stiffness due to Interbody Load on a Component 8.2 Multibody System with Rigid and Flexible Components 8.3 Recurrence Relations for Kinematics 8.4 Construction of the Dynamical Equations in a Block-Diagonal Form 8.5 Summary of the Block-Diagonal Algorithm for a Tree Configuration 8.6 Numerical Results Demonstrating Computational Efficiency 8.7 Modification of the Block-Diagonal Formulation to Handle Motion Constraints 8.8 Validation of Formulation with Ground Test Results 8.9 Conclusion 9 Efficient Variables, Recursive Formulation and Multi-Point Constraints in Flexible Multibody Dynamics 9.1 Single Flexible Body Equations in Efficient Variables 9.2 Multibody Hinge Kinematics for Efficient Generalized Speeds 9.3 Recursive Algorithm for Flexible Multibody Dynamics with Multiple Structural Loops 9.4 Explicit Solution of Dynamical Equations Using Motion Constraints 9.5 Computational Results and Simulation Efficiency for Moving Multi-Loop Structures 10 Efficient Modeling of Beams with Large Deflection and Large Base Motion 10.1 Discrete Modeling for Large Deflection of Beams 10.2 Motion and Loads Analysis by the Order-n Formulation 10.3 Numerical Integration by the Newmark Method 10.4 Nonlinear Elastodynamics via the Finite Element Method 10.5 Comparison of the Order-n Formulation with the Finite Element Method 10.6 Conclusion 11 Variable-n Order-n Formulation for Deployment and Retraction of Beams and Cables with Large Deflection 11.1 Beam Discretization 11.2 Deployment / Retraction from a Rotating Base 11.3 Numerical Simulation of Deployment and Retraction 11.4 Deployment of a Cable from a Ship to a Maneuvering Underwater Search Vehicle 11.5 Simulation Results 12 Order-n Equations of Flexible Rocket Dynamics 12.1 Introduction 12.2 Kane's Equation for a Variable Mass Flexible Body 12.3 Matrix Form of the Equations for Variable Mass Flexible Body Dynamics 12.4 Order-n Algorithm for a Flexible Rocket with Commanded Gimbaled Nozzle Motion 12.5 Numerical Simulation of Planar Motion of a Flexible Rocket 12.6 Conclusion Acknowledgment Appendix 12.A Summary Algorithm for Finding Two Gimbal Angle Torques for the Nozzle Problem Set 12 References Appendix A Efficient Generalized Speeds for a Single Free-Flying Flexible Body Appendix B A FORTRAN Code of the Order-n Algorithm: Application to an Example Index