Multibody Dynamics (eBook)

Preface 
6 
Contents 
8 
A Flexible Multibody Pantograph Model for the Analysis of the Catenary–Pantograph Contact 10
1 Introduction 10
2 Flexible Multibody Systems 13
2.1 Flexible Body Equations of Motion 13
2.2 Kinematic Joints with Virtual Bodies 16
3 Co-simulation of Multibody and Finite Element Codes 18
3.1 Integration of the Finite Elements Equations of Motion 19
3.2 Integration of the Multibody Equations of Motion 20
3.3 Co-simulation Using Different Codes 20
4 Analysis of the Pantograph–Catenary Contact Problem 23
4.1 Pantograph Multibody Models 23
4.2 Catenary Finite Element Model 30
4.3 Simulation Scenario and Results 31
5 Conclusions 33
References 34
Maneuvering Multibody Dynamics: New Developments for Models with Fast Solution Scales and Pilot-in-the-Loop Effects 37
1 Introduction 37
2 Coupled Pilot-Vehicle Model 41
3 Formulation of Maneuvers as Optimal Control Problems 42
4 Direct Solution of Maneuver Optimal Control Problems 44
4.1 Direct Transcription 44
4.2 Direct Multiple and Hybrid Single–Multiple Shooting 45
5 Applications and Results 47
5.1 Lateral Reposition MTE 47
5.2 Category-A Fly-Away 50
6 Concluding Remarks 53
References 54
Optimization of Multibody Systems and Their Structural Components 57
1 Introduction 57
2 Optimization of Flexible Multibody Systems 60
2.1 Equations of Motion 60
2.2 Formulation of the Optimization Problem 61
2.3 Time Integration Method 61
2.4 Evaluation of the Objective Function and of the Design Constraints 62
2.5 Sensitivity Analysis 63
2.6 Optimization Algorithms 65
3 Topology Optimization Techniques 67
3.1 Application to the Design of Static Trusses 68
3.2 Topology Optimization of Multibody Systems 69
4 Example 71
4.1 Problem Description 71
4.2 Optimization 72
5 Conclusions 74
References 74
Real-Time Aeroservoelastic Analysis of Wind-Turbines by Free Multibody Software 77
1 Introduction 77
2 Approach 78
3 Wind-Turbine Description 80
4 Baseline Controller 81
5 Multibody Model 83
5.1 Unconstrained Dynamics 83
5.2 Constrained Dynamics 84
5.3 Structural Flexibility 85
5.4 Numerical Integration 85
5.5 CART Wind Turbine Multibody Model 86
6 Real-Time Simulation 88
7 Conclusions 91
8 Additional Material 92
References 92
Comparison of Planar Structural Elements for Multibody Systems with Large Deformations 95
1 Introduction 96
2 Floating Frame of Reference Formulation 97
3 Absolute Nodal Coordinate Formulation 103
4 Numerical Examples 106
4.1 Static Example Problem 106
4.2 Dynamic Example Problems 108
5 Conclusions 112
References 112
Modeling and Analysis of Rigid Multibody Systems with Translational Clearance Joints Based on the Nonsmooth Dynamics Approach 114
1 Introduction 114
2 Basic Set-Valued Elements 116
3 Set-Valued Force Laws for Frictional Unilateral Contacts 118
4 Dynamics of Nonsmooth Rigid Multibody Systems 120
5 Moreau's Time-Stepping Method 123
4 Demonstrative Application to a Slider-Crank Mechanism 128
5 Conclusions 135
References 136
Application of General Multibody Methods to Robotics 138
1 Introduction 138
2 Absolute Coordinates Approach to Robot Kinematic Analysis and Singular Configuration Detection 139
2.1 Theoretical Background 140
2.2 Robotic Example 143
3 Simulation Study of Stewart Platform with Model-Based Control 145
3.1 Methods 145
3.1.1 Manipulator Kinematics 146
3.1.2 Manipulator Dynamics 148
3.1.3 Friction in Actuators 149
3.1.4 DC Motor 149
3.1.5 Control System 150
3.2 Results 151
4 Dynamic Analysis of a Flexible Power Transmission Mechanism 152
4.1 An Outline of Dynamic Analysis of Flexible MBS 153
4.2 Power Transmission Mechanism of POLYCRANK Robot 154
4.3 Dynamic Analysis of Power Transmission Mechanism 156
5 Conclusions 158
References 159
Energy Considerations for the Stabilization of Constrained Mechanical Systems with Velocity Projection 160
1 Introduction 160
2 Constrained Dynamics Formulation 161
3 Coordinate Projection 162
4 Total Energy Balance 164
5 Projection Energy Balance 167
5.1 Some Preliminary Results 167
5.2 Conditions for Energy Dissipation 168
6 Numerical Experiments 169
6.1 Two Particle System 169
6.2 Five-Bar Pendulum 173
7 Conclusions 176
References 177
A General Purpose Algorithm for Optimal Trajectory Planning of Closed Loop Multibody Systems 179
1 Introduction 179
2 Optimal Trajectory Planning Problem of Multibody Systems 181
2.1 Problem Formulation 1: Minimal Form of Equation of Motion 181
2.2 Problem Formulation 2: Augmented Equations 182
2.3 Solution Procedure for Optimal Trajectory Planning Problems 183
3 Optimal Trajectory Planning Algorithm Using Coordinate Partitioning and Embedding Techniques 185
4 Optimal Trajectory Planning Algorithm Based on an Augmented Formulation 188
5 Numerical Examples 191
5.1 Non-redundant Actuation System 191
5.2 Redundant Actuation System 194
6 Conclusions 197
References 198
Real-Time Simulation of Extended Vehicle Drivetrain Dynamics 200
1 Introduction 200
1.1 Notations 204
2 Powertrain System and Vehicle Dynamics 204
2.1 Multibody System 205
2.2 Control Units 207
2.3 Electrical System 207
2.4 Coupled System 208
2.5 Analysis Tasks 208
3 Time Integration 210
3.1 Office Simulation 210
3.2 Realtime Office 210
3.3 Realtime 211
4 Application 212
4.1 Model 213
4.2 Simulation Results 213
4.2.1 Realtime Calculations 214
4.2.2 Comparison Implicit and Explicit Solution 215
4.2.3 Comparison Realtime and Office 215
4.2.4 Comparison with Respect to System Evaluation 216
5 Conclusion 217
References 217
Assessment of Antagonistic Muscle Forces During Forearm Flexion/Extension 220
1 Introduction 221
1.1 Redundancy Problem Formulation 221
1.2 Classification of Solving Methods 222
1.3 Objective of this Study 223
2 Material and Methods 224
2.1 Principle 224
2.1.1 Protocol Step 1: Force Calibration 224
2.1.2 Protocol Step 2: Force Quantification 224
2.2 Experimental Set-up 225
2.3 Model and Hypotheses 226
2.4 Process of Muscle Force Quantification 228
2.4.1 Muscle Force Calibration 228
2.4.2 Muscle Force Quantification 232
3 Results 235
3.1 Joint Kinematics and Dynamics 235
3.2 Muscle Forces 236
3.2.1 Muscle Force Assessment 236
3.2.2 Statistical Validation 237
4 Discussion 238
4.1 Joint Kinematics and Dynamics 238
4.2 Muscle Force Quantification 239
4.2.1 EMG Processing and Parameter Choice 239
4.2.2 Comparison to Existing Methods 239
4.3 Prospects 240
References 241
Computing Time Reduction Possibilities in Multibody Dynamics 244
1 Introduction 244
2 Determination of a Suitable Initial Value 246
2.1 Kinematics 246
2.2 Kinetics 247
2.2.1 Geometry of the Excitation 247
2.2.2 Relationship of Tangential and Radial Belt Velocity 248
2.2.3 Tangential Balance of Forces 249
2.2.4 Radial Balance of Forces 250
2.2.5 Axial Balance of Forces 250
2.2.6 Change of the Running Radius 251
2.2.7 Summary of the Stationary Belt Model and Environment Interaction 251
2.2.8 Reduction of the Final Equations 253
3 Computational Effort During Integration 254
3.1 Stabilising Equations of Motion 255
3.1.1 Coordinate Settings 255
3.1.2 Equations of Motion 258
3.1.3 Analysis of Instability 259
3.2 Parallel Computing Architectures 261
4 Conclusion 262
References 263
Optimization-Based Design of Minimum Phase Underactuated Multibody Systems 265
1 Introduction 265
2 Trajectory Tracking Control 266
2.1 Input–Output Normal-Form 267
2.2 Analysis of the Internal Dynamics 268
2.3 Feedback Linearization 269
2.4 Feed-forward Control Design 270
3 Design of Stable Zero-Dynamics 272
3.1 Identification of Possible Design Parameters 272
3.2 Optimization Criteria 274
3.3 Particle Swarm Optimization 276
4 Application Examples 277
4.1 Manipulator with One Passive Joint 277
4.1.1 System Without Disturbances and Uncertainties 279
4.1.2 System Under Disturbances and Uncertainties 281
4.2 Manipulator with Two Passive Joints 281
5 Conclusions 285
References 286
GPU-Based Parallel Computing for the Simulation of Complex Multibody Systems with Unilateral and Bilateral Constraints: An Overview 287
1 Introduction 287
2 Review of Computing on the Graphics Processing Unit 289
3 Large Scale Multibody Dynamics on the GPU 296
3.1 The Formulation of the Equations of Motion 296
3.2 The Time Stepping Solver 299
3.3 The GPU Formulation of the CCP Solver 302
4 Numerical Experiments 306
5 Conclusions and Directions of Future Work 309
References 310
Investigation of Gears Using an Elastic Multibody Model with Contact 312
1 Introduction 313
2 Classical Models 313
2.1 Finite Element Model 314
2.2 Rigid Body Model 314
2.3 Comparison 315
3 Elastic Multibody Model 317
3.1 Contact Algorithm 318
3.1.1 Coarse Collision Detection 318
3.1.2 Fine Collision Detection 319
3.2 Integration 320
4 Simulation Results 322
4.1 Spur Gears 322
4.2 Numerical Efficiency 323
4.3 Helical Gears 324
5 Experimental Results 326
6 Conclusions 328
References 329