Elastic Multibody Dynamics

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 Rotor ( = 0); 6.5.1.4 Configuration Space and State Space ( 6= 0); 6.5.1.5 The Laval- (or Jeffcott-) Rotor; 6.5.1.6 Rotor with Fixed Point; 6.5.1.7 Elastic Rotor Properties; 6.6 Gross Motion -- Dynamical Stiffening (Ritz Approach); 6.6.1 Rotating Beam -- One-Link Elastic Robot; 6.6.1.1 Mass Matrix; 6.6.1.2 Restoring Matrix; 6.6.1.3 Equations of Motion; 6.6.2 Translating Beam -- Elastic TT-Robot; 6.6.2.1 Mass Matrix; 6.6.2.2 Restoring Matrix; 6.6.2.3 Equations of Motion; 6.6.2.4 Simplified System; 6.7 The Mass Matrix Reconsidered (Ritz Approach); 6.8 The G-Matrix Reconsidered (Ritz Approach); 6.9 Conclusions; 7. ELASTIC MULTIBODY SYSTEMS -- ORDINARY DIFFERENTIAL EQUATIONS; 7.1 Summary Procedure; 7.1.1 Rigid Multibody Systems; 7.1.2 Elastic Multibody Systems; 7.2 Mixed Rigid-Elastic Multibody Systems; 7.3 Applications; 7.3.1 Prismatic Joint -- The Telescoping Arm; 7.3.1.1 On Mass Distribution: Tip Body Influence; 7.3.1.2 Subsystem Equations; 7.3.1.3 The Kinematic Chain; 7.3.2 Revolute Joint; 7.3.2.1 Subsystem Equations; 7.3.2.2 The Kinematic Chain; 7.3.3 Spatial Motion; 7.3.4 Plane Motion;7.4 Plane Motion -- Recalculation; 7.4.1 Minimal Velocities and Projection; 7.4.2 Subsystem Matrices; 7.4.3 Dynamical Stiffening; 7.4.4 The Kinematic Chain; 7.5 Reduced Number of Shape Functions: Controlled Systems; 7.6 Remark on Controlled Systems; 8. A SHORT EXCURSION INTO STABILITY AND CONTROL; 8.1 Optimality; 8.1.1 Results from Classical Optimization Theory; 8.1.2 Riccati- (or LQR-) Control; 8.1.3 Control Parameter Optimization; 8.2 Stability; 8.3 Linear Time-Invariant Systems; 8.3.1 Fundamental (or Transition) Matrix; 8.3.2 Theorem of Cayley and Hamilton; 8.3.3 Stability Theorem for Mechanical Systems; 8.4 Stabilization of Mechanical Systems; 8.5 Observers; 8.5.1 Basic Notation; 8.5.2 Complete State Observer for Control; 8.5.3 Disturbance Suppression ("High Gain Observer"); 8.5.4 Disturbance Observation; 8.6 Decentralized Control; 8.7 On Control Input Variables; References; List of Symbols; Index