Elastic Multibody Dynamics: A Direct Ritz Approach

1. INTRODUCTION 1.1 Background 1.2 Contents 2. AXIOMS AND PRINCIPLES 2.1 Axioms 2.2 Principles - the 'Differential' Form 2.3 Minimal Representation 2.3.1 Virtual Displacements and Variations 2.3.2 Minimal Coordinates and Minimal Velocities 2.3.3 The Transitivity Equation 2.4 The Central Equation of Dynamics 2.5 Principles - the 'Minimal' Form 2.6 Rheonomic and Non-holonomic Constraints 2.7 Conclusions 3. KINEMATICS 3.1 Translation and Rotation 3.1.1 Rotation Axis and Rotation Angle 3.1.2 Transformation Matrices 3.1.2.1 Rotation Vector Representation 3.1.2.2 Cardan Angle Representation 3.1.2.3 Euler Angle Representation 3.1.3 Comparison 3.2 Velocities 3.2.1 Angular Velocity 3.2.1.1 General Properties 3.2.1.2 Rotation Vector Representation 3.2.1.3 Cardan Angle Representation 3.2.1.4 Euler Angle Representation 3.3 State Space 3.3.1 Kinematic Differential Equations 3.3.1.1 Rotation Vector Representation 3.3.1.2 Cardan Angle Representation 3.3.1.3 Euler Angle Representation 3.3.2 Summary Rotations 3.4 Accelerations 3.5 Topology - the Kinematic Chain 3.6 Discussion 4. RIGID MULTIBODY SYSTEMS 4.1 Modeling aspects 4.1.1 On Mass Point Dynamics 4.1.2 The Rigidity Condition 4.2 Multibody Systems 4.2.1 Kinetic Energy 4.2.2 Potentials 4.2.2.1 Gravitation 4.2.2.2 Springs 4.2.3 Rayleigh's Function 4.2.4 Transitivity Equation 4.2.5 The Projection Equation 4.3 The Triangle of Methods 4.3.1 Analytical Methods 4.3.2 Synthetic Procedure(s) 4.3.3 Analytical vs. Synthetic Method(s) 4.4 Subsystems 4.4.1 Basic Element: The Rigid Body 4.4.1.1 Spatial Motion 4.4.1.2 Plane Motion 4.4.2 Subsystem Assemblage 4.4.2.1 Absolute Velocities 4.4.2.2 Relative Velocities 4.4.2.3 Prismatic Joint/Revolute Joint - Spatial Motion 4.4.3 Synthesis 4.4.3.1 Minimal Representation 4.4.3.2 Recursive Representation 4.5 Constraints 4.5.1 Inner Constraints 4.5.2 Additional Constraints 4.5.2.1 Jacobi Equation 4.5.2.2 Minimal Representation 4.5.2.3 Recursive Representation 4.5.2.4 Constraint Stabilization 4.6 Segmentation: Elastic Body Representation 4.6.1 Chain and Thread (Plane Motion) 4.6.2 Chain, Thread, and Beam 4.7 Conclusion 5. ELASTIC MULTIBODY SYSTEMS - THE PARTIAL DIFFERENTIAL EQUATIONS 5.1 Elastic Potential 5.1.1 Linear Elasticity 5.1.2 Inner Constraints, Classification of Elastic Bodies 5.1.3 Disk and Plate 5.1.4 Bea 5.2 Kinetic Energy 5.3 Checking Procedures 5.3.1 HAMILTON's Principle and the Analytical Methods 5.3.2 Projection Equation 5.4 Single Elastic Body - Small Motion Amplitudes 5.4.1 Beams 5.4.2 Shells and Plates 5.5 Single Body - Gross Motion 5.5.1 The Elastic Rotor 5.5.2 The Helicopter Blade (1) 5.6 Dynamical Stiffening 5.6.1 The CAUCHY Stress Tensor 5.6.2 The TREFFTZ (or 2nd Piola-Kirchhoff) Stress Tensor 5.6.3 Second-Order Beam Displacement Fields 5.6.4 Dynamical Stiffening Matrix 5.6.5 The Helicopter Blade (2) 5.7 Multibody Systems - Gross Motion 5.7.1 The Kinematic Chain 5.7.2 Minimal Velocities 5.7.3 Motion Equations 5.7.3.1 Dynamical Stiffening 5.7.3.2 Equations of Motion 5.7.4 Boundary Conditions 5.8 Conclusion 6. ELASTIC MULTIBODY SYSTEMS - THE SUBSYSTEM ORDINARY DIFFERENTIAL EQUATIONS 6.1 Galerkin Method 6.1.1 Direct Galerkin Method 6.1.2 Extended Galerkin Method 6.2 (Direct) Ritz Method 6.3 Rayleigh Quotient 6.4 Single Elastic Body - Small Motion Amplitudes 6.4.1 Plate 6.4.1.1 Equations of motion 6.4.1.2 Basics 6.4.1.3 Shape Functions: Spatial Separation Approach 6.4.1.4 Expansion in Terms of Beam Functions 6.4.1.5 Convergence and Solution 6.4.2 Torsional Shaft 6.4.2.1 Eigenfunctions 6.4.2.2 Motion Equations 6.4.2.3 Shape Functions 6.4.3 Change-Over Gear 6.5 Single Elastic Body - Gross Motion 6.5.1 The Elastic Rotor 6.5.1.1 Rheonomic Constraint 6.5.1.2 Choice of Shape Functions - Prolate Rotor ( = 0) 6.5.1.3 Choice of Shape Functions - Oblate