Dynamics of Multibody Systems

Multibody systems are the appropriate models for predicting and evaluating performance of a variety of dynamical systems such as spacecraft, vehicles, mechanisms, robots or biomechanical systems. This book adresses the general problem of analysing the behaviour of such multibody systems by digital simulation. This implies that pre-computer analytical methods for deriving the system equations must be replaced by systematic computer oriented formalisms, which can be translated conveniently into efficient computer codes for - generating the system equations based on simple user data describing the system model - solving those complex equations yielding results ready for design evaluation. Emphasis is on computer based derivation of the system equations thus freeing the user from the time consuming and error-prone task of developing equations of motion for various problems again and again.

I. Introduction.- 1. Multibody Systems.- 1.1 Origins.- 1.1.1 Rise of Rotational Mechanics.- 1.1.2 Rotation in Technology.- 1.1.3 Status: mid-Twentieth Century.- 1.2 Multibody System Dynamics.- 1.2.1 Multibody Renaissance.- 1.2.2 Examples of Applications.- 1.2.3 Goals of Multibody System Investigation.- 1.2.4 Multibody Formalisms and Computer Codes.- 1.3 References.- 2. Mathematical Preliminaries.- 2.1 Terminology and Notation.- 2.1.1 Frames and Coordinates.- 2.1.2 Vectors and Cartesian Tensors.- 2.1.3 Matrices.- 2.1.4 Arrays of Vectors and Dyadics.- 2.2 Vectors, Dyadics and their Matrix Representations.- 2.2.1 Basic Operations.- 2.2.2 Transformations.- 2.2.3 Cartesian Tensors.- 2.3 References.- II. Kinematics of a Rigid Body.- 3. Location and Orientation.- 3.1 Introduction.- 3.1.1 Degrees of Freedom.- 3.1.2 Rotation.- 3.2 Direction Cosine Matrices.- 3.2.1 Representation of General Rotation.- 3.2.2 Infinitesimal Rotation.- 3.3 Angle Representations of Rotation.- 3.3.1 Elementary Rotations.- 3.3.2 Euler Angles.- 3.3.3 Tait-Bryan Angles.- 3.4 Other Representations of Rotation.- 3.4.1 Euler Parameters.- 3.4.2 Euler-Rodrigues Parameters.- 3.4.3 Six-Parameter Methods.- 3.5 References.- 4. Velocity.- 4.1 Foundations.- 4.1.1 Translational Velocity.- 4.1.2 Angular Velocity.- 4.1.3 Velocity Variables.- 4.2 Angular Velocity and Parameter Derivatives.- 4.2.1 Angle Parametrizations.- 4.2.2 Euler and Euler-Rodrigues Parameters.- 4.3 Relative Derivatives.- 4.3.1 General Form.- 4.3.2 Transformations of Velocity and Acceleration.- 5. Kinematical Equations of Motion.- 5.1 Unconstrained Motion.- 5.1.1 Variables.- 5.1.2 Kinematical Differential Equations.- 5.2 Constrained Motion.- 5.2.1 Constraints.- 5.2.2 Modes of Motion.- 5.2.3 State Space Representation of Kinematics.- 5.2.4 Examples.- 5.3 References.- III. Dynamics of a Rigid Body.- 6. Physical Preliminaries.- 6.1 Mass Geometry.- 6.1.1 Mass and Mass Moments.- 6.1.2 Transformations of the Inertia Matrix.- 6.2 Momentum.- 6.2.1 Linear Momentum.- 6.2.2 Angular Momentum.- 6.3 Force and Torque.- 6.3.1 Basics.- 6.3.2 Systems of Forces.- 6.4 References.- 7. Dynamical Equations.- 7.1 Basic Forms.- 7.1.1 Laws of Newton and Euler.- 7.1.2 Center of Mass as a Reference Point.- 7.2 Unconstrained Motion.- 7.2.1 Matrix Representation.- 7.2.2 Accelerated Reference Frames.- 7.2.3 State-Space Equations.- 7.2.4 Generalized Coordinates, Velocities and Forces.- 7.3 Constrained Motion.- 7.3.1 Separation of Forces and Torques by Modes.- 7.3.2 State Space Equations and Constraint Forces and Torques.- 7.3.3 Examples.- 7.3.4 Other Methodologies.- 7.4 References.- IV. Multibody Systems.- 8. Foundations.- 8.1 Basic Multibody Notation.- 8.1.1 Primitive Equations of Motion.- 8.1.2 Gross Motion and Reference Frames.- 8.1.3 Quantities Related to the Interaction of Bodies.- 8.2 Graph Characterization of System Topology.- 8.2.1 Graph Fundamentals.- 8.2.2 Matrices Associated with a Graph.- 8.2.3 Multibody System Graph.- 8.3 References.- 9. Formalisms.- 9.1 Survey on Methodologies.- 9.1.1 Eulerian Approaches.- 9.1.2 Lagrangian Approaches.- 9.2 Summary Description of Present Formalism.- 9.2.1 Reformulation of Primitive Dynamical Equations.- 9.2.2 Selection of Dependent Variables.- 9.2.3 Data to Describe the System.- 9.2.4 Derivation of State Space Equations.- 9.3 References.- 10. Kinematics.- 10.1 Position.- 10.1.1 Formulation of Problems.- 10.1.2 Path Vectors.- 10.1.3 Barycentric Vectors.- 10.1.4 Unification of Results.- 10.1.5 Orientation Matrices.- 10.2 Velocity and Acceleration.- 10.2.1 Velocity.- 10.2.2 Velocity Derivatives.- 10.3 Constraint Equations.- 10.3.1 Classification of Constraints.- 10.3.2 Joint Constraints and Branch Orientation.- 10.3.3 Consistency Conditions in Kinematic Circuits.- 10.3.4 Circuit Connection Vectors.- 10.4 Kinematical Differential Equations.- 10.4.1 Unreduced System Equations.- 10.4.2 Incorporation of Constraints.- 10.5 References.- 11. Dynamics.- 11.1 Unreduced System Equations.- 11.1.1 Translation.- 11.1.2 Rotation.- 11.1.3 Generalized Inertia Dyadics.- 11.1.4 External and Inertial Forces and Torques.- 11.1.5 Matrix Equations.- 11.1.6 Properties of Generalized Inertia Matrix.- 11.1.7 Problem Types.- 11.2 Systems with Tree Structure.- 11.2.1 Separation by Modes.- 11.2.2 State Space Equations.- 11.2.3 Constraint Forces and Torques.- 11.3 Systems with Closed Circuits.- 11.3.1 Constraints across Primary Joints.- 11.3.2 State-Space Equations.- 11.3.3 Choice of Spanning Tree.- 11.4 References.- 12. Special Topics.- 12.1 Forces and Torques.- 12.1.1 Dry Friction.- 12.1.2 Interaction and Feedback Control.- 12.1.3 Gravitational Effects on Multibody Systems.- 12.2 Application to Robotics.- 12.2.1 Multibody Robot Models.- 12.2.2 Recursive Computation of Inverse Dynamics.- 12.3 References.- 13. Linearized Equations.- 13.1 Introduction.- 13.1.1 Collection of Basic Equations.- 13.1.2 Small Variables Representing Relative Motion.- 13.1.3 Absolute and Relative Motion.- 13.2 Kinematics.- 13.2.1 Unreduced Equations of Motion.- 13.2.2 Consistency Conditions.- 13.2.3 Constraint Equations.- 13.2.4 State-Space Equations.- 13.3 Dynamics.- 13.3.1 Introduction.- 13.3.2 Generalized Inertia Matrix.- 13.3.3 Interactions.- 13.3.4 External and Inertial Action.- 13.3.5 Matrices of Consistency Conditions.- 13.3.6 Unreduced System Equations.- 13.3.7 State-Space Representation.- 13.3.8 System Parameters.- 13.4 References.- 14. Computer Simulation.- 14.1 Introduction.- 14.1.1 Requirements on Multibody Software.- 14.1.2 General Orginization of Programs.- 14.2 User-Oriented Input of Data.- 14.2.1 Generalities.- 14.2.2 System Data and Libraries.- 14.2.3 Feasibility of System Configuration.- 14.3 Generation of System Equations.- 14.3.1 Summary of Equations.- 14.3.2 Recursive Computational Schemes.- 14.3.3 Graph Algorithms.- 14.4 Numerical Integration.- 14.4.1 Formulation of Problems.- 14.4.2 Some Solution Techniques.- 14.4.3 Solution of Partially Reduced Equations.- 14.5 References.- Appendix A. Notational Summary.- Appendix B. List Of Symbols.- Appendix C. Bibliography.