Flexible multibody dynamics : algorithms based on Kane's method

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
About the Author
Introduction: Background Material on Dynamics and Vibrations
	I.1 Introduction
	I.2 Direction Cosine Matrix
	I.3 Fundamental Theorem on Differentiation of a Vector in Two Frames
	I.4 Quaternions
	I.5 Four Basic Theorems in Kinematics
	I.6 Generalized Coordinates and Generalized Speeds
	I.7 Partial Velocities and Partial Angular Velocities: Key Components in Kane’s Method
	I.8 Definition of Inertia Force and Inertia Torque
	I.9 Vibration of an Elastic Body: Mode Shapes, Frequencies, Modal Effective Mass, and Model Reduction: The Eigenvalue Problem
		I.9.1 Reduction of Degrees of Freedom
	References
Chapter 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 Kane’s Equations of Motion
		1.3.1 Simple Example: Equations for a Double Pendulum
		1.3.2 Complex Example: Equations of Motion for a Spinning Spacecraft with Three Rotors, a Tank with Thruster Fuel Slosh, and a Nutation Damper
		1.3.3 Comparison to Derivation of Equations of Motion by Lagrange’s Method: Application to the Same Complex Example Shown in Figure 1.2
		1.3.4 Boltzmann–Hamel Equations
		1.3.5 Gibbs Equations
			1.3.5.1 Reader’s Exercise
	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
		1.6.1 Summary of Equations of Motion with Undetermined Multipliers for Constraints
		1.6.2 A Simple Application
	Appendix A: Guideline for Choosing Efficient Motion Variables in Kane’s Method
		A.1 Customary Choice of Generalized Speeds
		A.2 Efficient Choice of Generalized Speeds
	Appendix B: Sliding Impact with Friction of a Nose Cap on a Package of Parachute
	Problem Set 1
	References
Chapter 2 Deployment, Station-Keeping, and Retrieval of a Flexible Tether Connecting a Satellite to the Shuttle
	2.1 Equations of Motion for Deployment of a Satellite Tethered to the Space Shuttle
		2.1.1 Kinematical Equations
		2.1.2 Dynamical Equations
		2.1.3 Simulation Results
	2.2 Thruster Augmented Retrieval of a Tethered Satellite to the Orbiting Shuttle
		2.2.1 Dynamical Equations
		2.2.2 Simulation Results
		2.2.3 Conclusion
	2.3 Dynamics and Control of Station-Keeping of the Shuttle-Tethered Satellite
	2.4 Pointing Control with Tethers as Actuators of a Space Station–Supported Platform
		2.4.1 Non-Linear Equations of Motion
		2.4.2 Linearized Equations
		2.4.3 Control Law Design
	Appendix 2.A Formation Flying with Multiple Tethered Satellites
	Appendix 2.B Orbit Boosting of Tethered Satellite Systems by Electrodynamic Forces
	Problem Set 2
	References
Chapter 3 Kane’s Method of Linearization Applied to the Dynamics of a Beam in Large Overall Motion
	3.1 Non-Linear Beam Kinematics with Neutral Axis Stretch, Shear, and Torsion
	3.2 Non-Linear Partial Velocities and Partial Angular Velocities, and Their 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 with Large Rotation in Conventional Analysis
	Problem Set 3
	References
Chapter 4 Dynamics of a Plate in Large Overall Motion
	4.1 Beginning at the End: Simulation Results
	4.2 Application of Kane’s Methodology for Linearization without Deriving the Non-Linear Equations
	4.3 Simulation Algorithm
	4.4 Conclusion
	Appendix 4 Specialized Modal Integrals
	Problem Set 4
	References
Chapter 5 Dynamics of an Arbitrary Flexible Body in Large Overall Motion
	5.1 Dynamical Equations for General Structures with the Use of Vibration Modes
	5.2 Compensating for Premature Linearization by Adding 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
	5.5 Conclusion
	Problem Set 5
	References
Chapter 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 System
	6.3 Kinematics and Generalized Inertia Forces for a Single Body
	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
	6.9 Conclusion
	Problem Set 6
	References
Chapter 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
		7.6.1 Mode Acceleration Method
	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
	7.10 Conclusion
	Problem Set 7
	Appendix 7 Matlab Codes for Structural Dynamics
		7A.1  Results
	References
Chapter 8 Block-Diagonal Mass Matrix Formulation of Equations of Motion for Flexible Multibody Systems
	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
		First Forward Pass
		Backward Pass
		Second Forward Pass
	8.6 Numerical Results Demonstrating Computational Efficiency
	8.7 Modification of the Block-Diagonal Formulation to Handle Motion Constraints
	8.8 Validation of Theory with Ground Test Results
	8.9 Conclusion
	Problem Set 8
	Appendix 8 An Alternative Derivation of Geometric Stiffness due to Inertia Loads
	References
Chapter 9 Efficient Variables, Recursive Formulation, and Multi-Loop Constraints in Flexible Multibody Dynamics
	9.1 Single Flexible Body Equations in Efficient Variables
	9.2 Multibody Hinge Kinematics with Efficient Generalized Speeds
	9.3 Recursive Algorithm for Flexible Multibody Dynamics with Multiple Structural Loops
		9.3.1 Forward Pass
		9.3.2 Forward Pass
	9.4 Explicit Solution of Dynamical Equations Using Motion Constraints
	9.5 Computational Results and Simulation Efficiency for Moving Multi-Loop Structures
		9.5.1 Simulation Results
	Problem Set 9
	Appendix 9 Pseudo-Code for Constrained nb-Body m-Loop Recursive Algorithm in Efficient Variables
		9A.1 Backward Pass
		9A.2 Forward Pass
	Acknowledgement
	References
Chapter 10 An Order-n Formulation for Beams with Undergoing 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 Large Deformation Dynamics Using the Non-Linear Finite Element Method
	10.5 Comparison of the Numerical Performances of the Order-n Formulation and the Finite Element Formulation
	10.6 Conclusion
	Acknowledgment
	Problem Set 10
	References
Chapter 11 Deployment/Retraction of Beams and Cables from Moving Vehicles: Small Deflection Analysis, and Variable-N Order-N Formulations for Large Deflection
	11.1 Small Deflection Analysis of Beam Extrusion/Retraction from a Rotating Base
		11.1.1 Rationale
	11.2 Simulation Results
	11.3 Deployment of a Cable from a Ship to a Maneuvering Underwater Search Vehicle: Use of a Constrained Order-n Formulation
		11.3.1 Cable Discretization and Variable-n Order-n Algorithm for Constrained Systems with Controlled End Body
		11.3.2 Hydrodynamic Forces on the Underwater Cable
		11.3.3 Non-Linear Holonomic Constraint, Control-Constraint Coupling, Constraint Stabilization, and Cable Tension
	11.4 Simulation Results
	11.5 Case of Large Beam Deflection during Deployment/Retraction
		11.5.1 Deployment/Retraction from a Rotating Base
			11.5.1.1 Initialization Step
			11.5.1.2 Forward Pass in an Order-n Formulation
			11.5.1.3 Backward Pass
			11.5.1.4 Forward Pass
			11.5.1.5 Extrusion/Retraction Step
	11.6 Numerical Simulation of Extrusion and Retraction
	11.7 Conclusion
	References
Chapter 12 Flexible Rocket Dynamics, Using Geometric Softness and a Block-Diagonal Mass Matrix
	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 Block-Diagonal Algorithm for a Flexible Rocket with a Swiveling Nozzle
	12.5 Numerical Simulation of Planar Motion of a Flexible Rocket
	12.6 Conclusion
	Acknowledgment
	Problem Set 12
	Appendix 12 Algorithm for Determining Two Gimbal Angle Torques for the Nozzle for the Example Problem
	References
Chapter 13 Large Amplitude Fuel Slosh in Spacecraft in Large Overall Motion
	13.1 Modeling Large Amplitude Sloshing Fuel as Particles Crawling on Walls of Four Tanks in a Spacecraft with Flexible Solar Panels
		13.1.1 Generalized Active Forces
	13.2 Generalized Force due to Normal Forces Enforcing Constrained Motion of Lumped-Mass Particles Crawling on an Elliptical Surface
	13.3 Simulation of Spacecraft Motion with Fuel Slosh for Various Fill-Fractions of the Tank
		13.3.1 Computation of the Total Slosh Force and Torque
	Appendix 13 Linearized Equations of Motion of Fuel Slosh in a Tank, for Spacecraft Control Design
		13A.1 Generalized Active Forces
	References
Appendix A: Modal Integrals for an Arbitrary Flexible Body
Appendix B: Flexible Multibody Dynamics for Small Overall Motion
Appendix C: A FORTRAN Code of the Order-n Algorithm: Application to an Example
Index