H. Bremer Elastic Multibody Dynamics A Direct Ritz Approach Springer
TABLE OF CONTENTS 1. INTRODUCTION 1 1.1 Background 1 1.2 Contents 5 2. AXIOMS AND PRINCIPLES 7 2.1 Axioms 7 2.2 Principles - the "Differential" Form 7 2.3 Minimal Representation 15 2.3.1 Virtual Displacements and Variations 16 2.3.2 Minimal Coordinates and Minimal Velocities 18 2.3.3 The Transitivity Equation 19 2.4 The Central Equation of Dynamics 20 2.5 Principles - the "Minimal" Form 22 2.6 Rheonomic and Non-holonomic Constraints 25 2.7 Conclusions 26 3. KINEMATICS 29 3.1 Translation and Rotation 29 3.1.1 Rotation Axis and Rotation Angle 30 3.1.2 Transformation Matrices 33 3.1.2.1 Rotation Vector Representation 33 3.1.2.2 Cardan Angle Representation 34 3.1.2.3 Euler Angle Representation 36 3.1.3 Comparison 36 3.2 Velocities 39 3.2.1 Angular Velocity 40 3.2.1.1 General Properties 40 3.2.1.2 Rotation Vector Representation 41 3.2.1.3 Cardan Angle Representation 42 3.2.1.4 Euler Angle Representation 43 v
VI 3.3 3.4 3.5 3.6 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 Accelerations Topology - the Kinematic Chain Discussion 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 44 45 45 46 46 47 49 50 56 59 59 59 61 65 65 66 66 67 68 69 70 71 71 73 75 78 78 78 82 85 85 88 89 91 93 95 100 100 101 102 103 104 107
Vll 4.6 Segmentation: Elastic Body Representation 109 4.6.1 Chain and Thread (Plane Motion) 109 4.6.2 Chain, Thread, and Beam 111 4.7 Conclusion 113 5. ELASTIC MULTIBODY SYSTEMS - THE PARTIAL DIFFERENTIAL EQUATIONS 115 5.1 Elastic Potential 115 5.1.1 Linear Elasticity 116 5.1.2 Inner Constraints, Classification of Elastic Bodies 117 5.1.3 Disk and Plate 119 5.1.4 Beam 121 5.2 Kinetic Energy 123 5.3 Checking Procedures 124 5.3.1 Hamilton's Principle and the Analytical Methods 124 5.3.2 Projection Equation 131 5.4 Single Elastic Body - Small Motion Amplitudes 133 5.4.1 Beams 133 5.4.2 Shells and Plates 153 5.5 Single Body - Gross Motion 159 5.5.1 The Elastic Rotor 159 5.5.2 The Helicopter Blade (1) 162 5.6 Dynamical Stiffening 167 5.6.1 The Cauchy Stress Tensor 167 5.6.2 The Trefftz (or 2nd Piola-Kirchhoff) Stress Tensor 167 5.6.3 Second-Order Beam Displacement Fields 173 5.6.4 Dynamical Stiffening Matrix 177 5.6.5 The Helicopter Blade (2) 183 5.7 Multibody Systems - Gross Motion 190 5.7.1 The Kinematic Chain 190 5.7.2 Minimal Velocities 193 5.7.3 Motion Equations 194 5.7.3.1 Dynamical Stiffening 195 5.7.3.2 Equations of Motion 196 5.7.4 Boundary Conditions 210 5.8 Conclusion 214 6. ELASTIC MULTIBODY SYSTEMS - THE SUBSYSTEM ORDINARY DIFFERENTIAL EQUATIONS 219 6.1 Galerkin Method 219 6.1.1 Direct Galerkin Method 219 6.1.2 Extended Galerkin Method 224 6.2 (Direct) Ritz Method 225 6.3 Rayleigh Quotient 229
Vlll 6.4 Single Elastic Body - Small Motion Amplitudes 235 6.4.1 Plate 235 6.4.1.1 Equations of motion 235 6.4.1.2 Basics 236 6.4.1.3 Shape Functions: Spatial Separation Approach 238 6.4.1.4 Expansion in Terms of Beam Functions 239 6.4.1.5 Convergence and Solution 243 6.4.2 Torsional Shaft 246 6.4.2.1 Eigenfunctions 246 6.4.2.2 Motion Equations 247 6.4.2.3 Shape Functions 249 6.4.3 Change-Over Gear 253 6.5 Single Elastic Body - Gross Motion 255 6.5.1 The Elastic Rotor 255 6.5.1.1 Rheonomic Constraint 257 6.5.1.2 Choice of Shape Functions - Prolate Rotor (Q, - 0) 259 6.5.1.3 Choice of Shape Functions - Oblate Rotor (П = 0) 264 6.5.1.4 Configuration Space and State Space (fl Ф 0) 271 6.5.1.5 The Laval- (or Jeffcott-) Rotor 272 6.5.1.6 Rotor with Fixed Point 277 6.5.1.7 Elastic Rotor Properties 283 6.6 Gross Motion - Dynamical Stiffening (Ritz Approach) 290 6.6.1 Rotating Beam - One-Link Elastic Robot 291 6.6.1.1 Mass Matrix 293 6.6.1.2 Restoring Matrix 293 6.6.1.3 Equations of Motion 295 6.6.2 Translating Beam - Elastic TT-Robot 296 6.6.2.1 Mass Matrix 305 6.6.2.2 Restoring Matrix 305 6.6.2.3 Equations of Motion 306 6.6.2.4 Simplified System 306 6.7 The Mass Matrix Reconsidered (Ritz Approach) 311 6.8 The G-Matrix Reconsidered (Ritz Approach) 315 6.9 Conclusions 321 7. ELASTIC MULTIBODY SYSTEMS - ORDINARY DIFFERENTIAL EQUATIONS 327 7.1 Summary Procedure 327 7.1.1 Rigid Multibody Systems 327 7.1.2 Elastic Multibody Systems 329 7.2 Mixed Rigid-Elastic Multibody Systems 335 7.3 Applications 338 7.3.1 Prismatic Joint - The Telescoping Arm 338 7.3.1.1 On Mass Distribution: Tip Body Influence 339
ix 7.3.1.2 Subsystem Equations 340 7.3.1.3 The Kinematic Chain 345 7.3.2 Revolute Joint 346 7.3.2.1 Subsystem Equations 346 7.3.2.2 The Kinematic Chain 347 7.3.3 Spatial Motion 348 7.3.4 Plane Motion 350 7.4 Plane Motion - Recalculation 352 7.4.1 Minimal Velocities and Projection 352 7.4.2 Subsystem Matrices 354 7.4.3 Dynamical Stiffening 359 7.4.4 The Kinematic Chain 361 7.5 Reduced Number of Shape Functions: Controlled Systems 369 7.6 Remark on Controlled Systems 380 8. A SHORT EXCURSION INTO STABILITY AND CONTROL 383 8.1 Optimality 383 8.1.1 Results from Classical Optimization Theory 387 8.1.2 Riccati- (or LQR-) Control 388 8.1.3 Control Parameter Optimization 390 8.2 Stability 391 8.3 Linear Time-Invariant Systems 395 8.3.1 Fundamental (or Transition) Matrix 395 8.3.2 Theorem of Cayley and Hamilton 399 8.3.3 Stability Theorem for Mechanical Systems 401 8.4 Stabilization of Mechanical Systems 406 8.5 Observers 411 8.5.1 Basic Notation 411 8.5.2 Complete State Observer for Control 413 8.5.3 Disturbance Suppression ("High Gain Observer") 416 8.5.4 Disturbance Observation 419 8.6 Decentralized Control 423 8.7 On Control Input Variables 426 References 431 List of Symbols 437 Index 445