PREFACE
Multibody systems are used extensively in the investigation of mechanical systems including structural and non-structural applications. It can be argued that among all the areas in solid mechanics the methodologies and applications associated to multibody dynamics are those that provide an ideal framework to aggregate dif-ferent disciplines.
This idea is clearly reflected, e.g., in the multidisciplinary applications in biomechanics that use multibody dynamics to describe the motion of the biological entities, in finite elements where multibody dynamics provides po-werful tools to describe large motion and kinematic restrictions between system components, in system control where the methodologies used in multibody dynamics are the prime form of describing the systems under analysis, or even in many ap- plications that involve fluid-structure interaction or aero elasticity.
The development of industrial products or the development of analysis tools, using multibody dynamics methodologies, requires that the final result of the develop- ments are the best possible within some limitations, i.e., they must be optimal.
Furthermore, the performance of the developed systems must either be relatively insensitive to some of their design parameters or be sensitive in a controlled manner to other variables. Therefore, the sensitivity analysis of such systems is fundamental to support the decision making process.
This book presents a broad range of tools for designing mechanical systems ranging from the kinematic and dynamic analysis of rigid and flexible multibody systems to their advanced optimization. The multibody kinematics and dynamics methodologies are presented from the fundamentals to the point where it is possible to build and analyze complex multibody models of machines, vehicles and biome-
chanics.
The design sensitivity and optimization methods are presented and applied to these systems to solve problems in design optimization, reliability or parameter identification. It is intended that the models developed demonstrate cases of prac- tical importance and that the methods presented are used as tools for advanced design.
The first part of the book, authored by Dr. John Hansen, addresses the basic methodologies based on Cartesian coordinates for kinematics and dynamics of planar and spatial rigid multibody systems. The synthesis of mechanical systems is used to demonstrate the use of optimal approaches to the kinematic design of me- chanisms.
The second part of this book, authored by Prof. Wojciech Blajer, explores the modeling and computational issues in the simulation of multibody systems including differential-geometric aspects of multibody dynamics, dependent and independent variable formulations, accuracy and stability of numerical solutions and a wide number of useful specialized techniques.
TABLE OF CONTENTS:
Preface
1. Planar Multibody Systems 1
1.1. Introduction 1
1.2. Cartesian Coordinates 1
1.3. Kinematic Constraints 4
1.4. Drivers 8
1.5. Solution of the Kinematic Problem 10
1.6. Velocities and Accelerations 13
1.7. Newton’s Equation 15
1.8. Forces 17
1.9. Numerical Integration 20
2. Spatial Multibody Systems 23
2.1. Introduction to Spatial Kinematic Constraints 23
2.2. Rotational Coordinates 23
2.3. Kinematic Constraints 27
2.4. Kinematic Joints 30
2.5. Newton-Euler Equations 33
2.6. Forces 36
2.7. Solution of the Equations of Motion 37
3. Synthesis of Mechanisms 39
3.1. Introduction 39
3.2. The Joint Coordinate Method 42
3.3. Optimization Using Time-Varying Design Variables 47
3.4. Optimization Using Dynamics 52
3.5. Synthesis Allowing for Non-Assembly 54
4. Differential-Geometric Aspects of Constrained System Dynamics 67
4.1. Introduction 67
4.2. Unconstrained System Dynamics 67
4.3. Constraint Equations 74
4.4. Constraint Reactions and Constraint Reaction-Induced Dynamic Equations 77
5. Dependent Variable Formulations 83
5.1. Introduction 83
5.2. Governing Equations in DAE Forms 83
5.3. ODE Forms of the Equations of Motion 89
5.4. Constraint Violation Problem 91
5.5. Aspects of Accuracy of Constraint-Consistent Solutions 97
6. Independent Variable Formulation 107
6.1. Introduction 107
6.2. Joint Coordinate Formulation for Open-Loop Systems 107
6.3. Velocity Partitioning Formulation 112
6.4. General Projective Scheme for Independent Variable Formulations 117
6.5. Treatment of Closed-Loop Multibody Systems 120
7. Other Useful Modeling and Simulation Techniques 131
7.1. Introduction 131
7.2. Augmented Lagrangian Formulation 131
7.3. Augmented Joint Coordinate Method 142
8. Sensitivity Analysis: Linear Static Spring Systems 151
8.1. Introduction 151
8.2. Notation 152
8.3. Static Analysis 154
8.4. Solution Strategy 158
8.5. Finite Element Program 161
8.6. Sensitivity Analysis 172
8.7. Sensitivity Computer Program 182
8.8. Optimization Problems 190
9. Sensitivity Analysis: Nonlinear Static Spring Systems 195
9.1. Nonlinear Linear Static Spring Systems 195
9.2. Newton Raphson Method 197
9.3. Sensitivity Analysis: Nonlinear Elastic Static Spring Systems 206
9.4. Transient Problems 216
10. Sensitivity Analysis: Generalized Coordinate Kinematic Systems 219
10.1. Position Analysis 219
10.2. Velocity and Acceleration Analysis 225
10.3. Inverse Dynamic Analysis 226
10.4. Sensitivity Analysis 231
10.5. Conclusion 235
11. Optimization of Mechanical Systems 237
11.1. Introduction 237
11.2. Optimization Algorithms 240
11.3. An Example from Multibody Dynamics 245
11.4. Concluding Remarks 250
12. Using Augmented Particle Swarm Optimization for Constrained Problems in Engineering 253
12.1. The Basic PSO Algorithm 256
12.2. Augmented LagrangeMultiplier Method 257
12.3. Augmented Lagrange Particle Swarm Optimization 260
12.4. Web-Based Optimization with ALPSO 264
12.5. Engineering Example: Hexapod Robot 265
12.6. Concluding Remarks 269
13. Optimization of Mechatronic Systems Using the Software Package NEWOPT/AIMS 273
13.1. Optimization of Mechatronic Systems 274
13.2. Software Package NEWOPT/AIMS 276
13.3. Example: Hexapod Manipulator 280
13.4. Concluding Remarks 284
14. Topology Optimized Synthesis of Planar Kinematic Rigid Body Mechanisms 287
14.1. Topology Representation of Mechanisms 289
14.2. Genetic Algorithms 291
14.3. Kinematic Analysis and Dimensional Synthesis 292
14.4. Topology Optimization of Mechanisms 296
14.5. Concluding Remarks 299
15. Grid-Based Topology Optimization of Rigid Body Mechanisms 303
15.1. Grid Structures for Topology Optimization 304
15.2. Kinematic Analysis 305
15.3. Mechanism Design Using Grid Structures 307
15.4. Amplifier Mechanism Example 313
15.5. Concluding Remarks 313
16. Lumped Deformations: a Plastic Hinge Approach 317
16.1 Introduction 317
16.2 Flexible Multibody Dynamics by Lumped Deformations 319
16.3 Plastic Hinges Constitutive Relations Implementation 322
16.4 Continuous Contact Force Model 324
16.5 Road Vehicle Multibody Model for Crash Analysis 326
16.6 Application to the Design of Railway Dynamics Crash Tests 336
17. Distributed Deformation: a Finite Element Method 351
17.1 Introduction 351
17.2 Brief Literature Overview 351
17.3 General Deformation of a Flexible Body 354
17.4 Reference Conditions in a Flexible Body: Linear Elastic Deformations 356
17.5 Generalized Elastic Coordinates for Linear Flexible Bodies 358
17.6 Generalized Coordinates for Nonlinear Flexible Bodies 361
17.7 Kinematic Joints Involving Flexible Bodies 362
17.8 Demonstration Examples 368
18. Optimization of Flexible Multibody Systems 375
18.1 Introduction 375
18.2 Road Vehicle Multibody Model 376
18.3 Road Vehicle Simulations for Comfort and Handling 383
18.4 Vehicle Dynamics Optimization for Comfort and Handling 393
18.5 Minimization of the Maximum Deformation Energy 399
18.6 Sensitivity Analysis in Flexible Multibody Dynamics 401
18.7 Demonstrative Example: Flexible Slider-Crank Mechanism 407
18.8 Optimization of the Deployment of a Satellite Antenna 414
18.9 Conclusions 422
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Advanced Design of Mechanical Systems From Analysis to Optimization