Elastic Multibody Dynamics

Transcription

1 H. Bremer Elastic Multibody Dynamics A Direct Ritz Approach Springer

2 TABLE OF CONTENTS 1. INTRODUCTION Background Contents 5 2. AXIOMS AND PRINCIPLES Axioms Principles - the "Differential" Form Minimal Representation Virtual Displacements and Variations Minimal Coordinates and Minimal Velocities The Transitivity Equation The Central Equation of Dynamics Principles - the "Minimal" Form Rheonomic and Non-holonomic Constraints Conclusions KINEMATICS Translation and Rotation Rotation Axis and Rotation Angle Transformation Matrices Rotation Vector Representation Cardan Angle Representation Euler Angle Representation Comparison Velocities Angular Velocity General Properties Rotation Vector Representation Cardan Angle Representation Euler Angle Representation 43 v

3 VI State Space Kinematic Differential Equations Rotation Vector Representation Cardan Angle Representation Euler Angle Representation Summary Rotations Accelerations Topology - the Kinematic Chain Discussion RIGID MULTIBODY SYSTEMS 4.1 Modeling Aspects On Mass Point Dynamics The Rigidity Condition 4.2 Multibody Systems Kinetic Energy Potentials Gravitation Springs Rayleigh's Function Transitivity Equation The Projection Equation 4.3 The Triangle of Methods Analytical Methods Synthetic Procedure(s) Analytical vs. Synthetic Method(s) 4.4 Subsystems Basic Element: The Rigid Body Spatial Motion Plane Motion Subsystem Assemblage Absolute Velocities Relative Velocities Prismatic Joint/Revolute Joint - Spatial Motion Synthesis Minimal Representation Recursive Representation 4.5 Constraints Inner Constraints Additional Constraints Jacobi Equation Minimal Representation Recursive Representation Constraint Stabilization

4 Vll 4.6 Segmentation: Elastic Body Representation Chain and Thread (Plane Motion) Chain, Thread, and Beam Conclusion ELASTIC MULTIBODY SYSTEMS - THE PARTIAL DIFFERENTIAL EQUATIONS Elastic Potential Linear Elasticity Inner Constraints, Classification of Elastic Bodies Disk and Plate Beam Kinetic Energy Checking Procedures Hamilton's Principle and the Analytical Methods Projection Equation Single Elastic Body - Small Motion Amplitudes Beams Shells and Plates Single Body - Gross Motion The Elastic Rotor The Helicopter Blade (1) Dynamical Stiffening The Cauchy Stress Tensor The Trefftz (or 2nd Piola-Kirchhoff) Stress Tensor Second-Order Beam Displacement Fields Dynamical Stiffening Matrix The Helicopter Blade (2) Multibody Systems - Gross Motion The Kinematic Chain Minimal Velocities Motion Equations Dynamical Stiffening Equations of Motion Boundary Conditions Conclusion ELASTIC MULTIBODY SYSTEMS - THE SUBSYSTEM ORDINARY DIFFERENTIAL EQUATIONS Galerkin Method Direct Galerkin Method Extended Galerkin Method (Direct) Ritz Method Rayleigh Quotient 229

5 Vlll 6.4 Single Elastic Body - Small Motion Amplitudes Plate Equations of motion Basics Shape Functions: Spatial Separation Approach Expansion in Terms of Beam Functions Convergence and Solution Torsional Shaft Eigenfunctions Motion Equations Shape Functions Change-Over Gear Single Elastic Body - Gross Motion The Elastic Rotor Rheonomic Constraint Choice of Shape Functions - Prolate Rotor (Q, - 0) Choice of Shape Functions - Oblate Rotor (П = 0) Configuration Space and State Space (fl Ф 0) The Laval- (or Jeffcott-) Rotor Rotor with Fixed Point Elastic Rotor Properties Gross Motion - Dynamical Stiffening (Ritz Approach) Rotating Beam - One-Link Elastic Robot Mass Matrix Restoring Matrix Equations of Motion Translating Beam - Elastic TT-Robot Mass Matrix Restoring Matrix Equations of Motion Simplified System The Mass Matrix Reconsidered (Ritz Approach) The G-Matrix Reconsidered (Ritz Approach) Conclusions ELASTIC MULTIBODY SYSTEMS - ORDINARY DIFFERENTIAL EQUATIONS Summary Procedure Rigid Multibody Systems Elastic Multibody Systems Mixed Rigid-Elastic Multibody Systems Applications Prismatic Joint - The Telescoping Arm On Mass Distribution: Tip Body Influence 339

6 ix Subsystem Equations The Kinematic Chain Revolute Joint Subsystem Equations The Kinematic Chain Spatial Motion Plane Motion Plane Motion - Recalculation Minimal Velocities and Projection Subsystem Matrices Dynamical Stiffening The Kinematic Chain Reduced Number of Shape Functions: Controlled Systems Remark on Controlled Systems A SHORT EXCURSION INTO STABILITY AND CONTROL Optimality Results from Classical Optimization Theory Riccati- (or LQR-) Control Control Parameter Optimization Stability Linear Time-Invariant Systems Fundamental (or Transition) Matrix Theorem of Cayley and Hamilton Stability Theorem for Mechanical Systems Stabilization of Mechanical Systems Observers Basic Notation Complete State Observer for Control Disturbance Suppression ("High Gain Observer") Disturbance Observation Decentralized Control On Control Input Variables 426 References 431 List of Symbols 437 Index 445