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Flexible Multibody Dynamics
Algorithms Based on Kane’s Method



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ISBN 9781032139197
February 28, 2022 Forthcoming by CRC Press
360 Pages 141 B/W Illustrations

 
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Book Description

This book demonstrates how to formulate the equations of mechanical systems. Providing methods of analysis of complex mechanical systems, the book has a clear focus on efficiency, equipping the reader with knowledge of algorithms that provide accurate results in reduced simulation time.

  • The book uses Kane’s method due to its efficiency, and the simple resulting equations it produces in comparison to other methods and extends it with algorithms such as order-n.
  • Kane’s method compensates for the errors of premature linearization, which are often inherent within vibrations modes found in a great deal of public domain software.
  • Describing how to build mathematical models of multibody systems with elastic components, the book applies this to systems such as construction cranes, trailers, helicopters, spacecraft, tethered satellites, and underwater vehicles.
  • It also looks at topics such as vibration, rocket dynamics, simulation of beams, deflection, and matrix formulation.

Flexible Multibody Dynamics will be of interest to students in mechanical engineering, aerospace engineering, applied mechanics and dynamics. It will also be of interest to industry professionals in aerospace engineering, mechanical engineering and construction engineering.

Table of Contents

Chapter I   INTRODUCTORY BACKGROUND MATERIAL ON  DYNAMICS AND VIBRATION I.1 Kinematics of Rotation I.2 The Most Important Theorem in Kinematics I3. Euler Angles I.4 Quaternions I.5 A Rigorous Definition of Angular Velocity 1.6 Four Important Theorems in Kinematics 1.7 Generalized Coordinates and Generalized Speeds 1.8 Partial Velocities and Partial Angular Velocities: key components in Kane’s method 1.9 Vibration of an Elastic Body: Mode Shapes, Frequencies and Reduction of Degrees of Freedom 1  DERIVATION OF EQUATIONS OF MOTION 1.1 Available Analytical Methods and Reasons for Choosing Kane’s Method 1.2 Kane’s Method of Deriving Equations of Motion 1.2.1 Kane’s Dynamical Equations 1.2.2 Simple Example: Equations for a Double Pendulum 1.2.3 Complex Example: Spinning Spacecraft with Three Rotors, Fuel Slosh and Nutation Damper 1.3 Lagrange’s Equation: Comparison in Labor Involved to Derive Equations of Motion for the same Complex Problem 1.3.1 Boltzmann-Hamel Equations and Gibbs Equations 1.4 Kane’s Method of Derivation of Linearized Equations without Deriving Nonlinear Equations 1.5 Prematurely Linearized Equations and a Posteriori Correction by ad hoc Addition of Geometric Stiffness due to Inertia Loads: Simple Example 1.6 Kane’s Equations with Undetermined Multipliers for Constrained Motion, Summary of the Equations of Motion with Undetermined Multipliers
1.6.1 A Simple Application for Constrained Motion Appendix 1.A Guidelines for Choosing Efficient Motion Variables in Kane’s Method 1.B Sliding Impact of a Nose Cap on a Package of Parachute Used for Recovery of a Booster Launching Satellites Problem Set 1 References 2 DEPLOYMENT, STATION-KEEPING, AND RETRIEVAL OF A FLEXIBLE TETHER CONNECTING A SATELLITE TO THE ORBITING SPACE SHUTTLE 2.1 Equations of Motion of a Tethered Satellite Deployment from the Space Shuttle
2.1.1 Simulation Results 2.2 Thruster Augmented Retrieval of a Satellite Tethered to the Orbiting Shuttle 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 Appendix Appendix 2.A: Formation Flying of Multiple Tethered Satellites Appendix 2.B Orbit Boosting of Tethered Satellite Systems by Electrodynamic Forces Problem Set 2 References 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 Needed for 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 Produced by Conventional Analysis Using Vibration Modes Problem Set 3 References 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 Proper Linearization 4.3 Simulation Algorithm Appendix: Specialized Modal Integrals Problem Set 4 References 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 Problem Set 5 References 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 Flexible Multibody System 6.4 Kinematical Recurrence Relations between 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 Problem Set 6 References 7 COMPONENT VIBRATION 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 of Errors due to Model Reduction by Modal Truncation Vectors 7.6 Role of Modal Truncation Vectors in Response Analysis 7.7 Synthesis of Component Modes 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 Appendix  7.A.1 A Matlab Code for Structural Dynamics 7.A.2 Results 7.10 Conclusion References 8 BLOCK-DIAGONAL MASS MATRIX FORMULATION OF EQUATIONS OF MOTION FOR FLEXIBLE MULTIBODY SYSTEMS 8.1 Example Showing the Role of Geometric Stiffness in Stiffness of 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 Theory with Ground Test Results 8.9 Conclusion  Appendix: An Alternative Derivation of Geometric Stiffness due to Inertia Loads Problem Set 8  References 9 FFICIENT 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 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  Appendix: Pseudo-Code for Constrained nb-Body m-Loop Recursive Algorithm Using Efficient Variables Problem Set 9 References 10 AN ORDER-N FORMULATION FOR BEAMS 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 Nonlinear Elastodynamics via the Finite Element Method 10.5 Accuracy and Efficiency of the Order-n Formulation vs. the Finite Element Method 10.6 Conclusion Problem Set 10 References 11 DEPLOYMENT AND RETRACTION OF BEAMS AND CABLES FROM  MOVING VEHICLES: SMALL DEFLECTION ANALYSIS, AND VARIABLE-N ORDER-N FORMULATION FOR LARGE DEFLECTION 11.1 Small Deflection Analysis of Beam Extrusion from a Rotating Base 11.2 Simulation Results 11.3 Deployment of a Cable from a Ship to a Maneuvering Underwater Search Vehicle 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 11.3.2 Nonlinear Holonomic Constraint, Control-Constraint Coupling, Constraint Stabilization and Cable Tension 11.4 Simulation Results 11.5 The Case of Large Deflection in Deployment and Retraction 11.6 Numerical Simulation Problem Set 11 References 12 FLEXIBLE ROCKET DYNAMICS, USING GEOMETRIC  SOFTNESS AND A BLOCK-DIAGONAL MASS MATRIX 12.1 Introduction 12.2 Kane’s Equations for a Variable-Mass Flexible Body 12.3 Matrix Form of the Equations for Variable Mass Flexible Body Dynamics, with Discrete Modal Integrals 12.4 Order-n Algorithm for a Flexible Rocket with Prescribed Nozzle Motion 12.5 Numerical Simulation of Planar Motion of a Flexible Rocket Appendix: Summary Algorithm for Finding Two Gimbal Angle Torques for the Nozzle Problem Set 12 13. Large Amplitude Fuel Slosh in Flexible 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 13.2 Dynamics of Constrained Motion of Particles on an Elliptical Surface, Under Normal and Friction Forces. 13.3 Simulation of Rigid Body Motion and Fuel Slosh for Various Fill-fractions of the Tank Appendix: Linearized Equations of Motion of Fuel Slosh in a Tank, for Spacecraft Control Design APPENDICES  Appendix A: Modal Integrals for an Arbitrary Flexible Body Appendix B: Efficient Generalized Speeds for a Single Free-Flying Flexible Body Appendix C: Flexible Multibody Dynamics for Small Overall Motion Appendix D: A FORTRAN Code of the Order-n Algorithm: An Example INDEX

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Author(s)

Biography

Dr. Banerjee is a leading expert in flexible dynamics. After working in industry for Northrop Services, Martin Marietta Aerospace, Ford Aerospace, and Lockheed-Martin Advanced Technology Research Center, he developed DYNACON, Lockheed-Martin's multi-flexible-body dynamics and control simulation tool. He was awarded 1990 AIAA Engineer of the Year Award in Astronautics, was associate editor for the Journal of Guidance, Control, and Dynamics and is associate fellow of AIAA (American Institute of Aeronautics & Astronautics).