Flexible Multibody Dynamics Efficient Formulations and Applications

This book provides methods of analysis of complex mechanical systems that can be simulated in less computer time than other methods. It equips the reader with knowledge of algorithms that provide accurate results in reduced simulation time.

Arun Kanti Banerjee, 28 year career at Lockheed Martin Advanced Technology Center (1982 thru 2010), Palo Alto, California, US.
Last position — Principal Research Scientist. Main Contribution: Developer of DYNACON, Lockheed's simulation tool for Multi-flexible-body dynamics and Control that has been used for many projects. Dr. Banerjee is one of the foremost experts in the world on the subject of flexible multibody dynamics.

1 Derivation of Equations of Motion 1

1.1 Available Analytical Methods and the Reason for Choosing Kane’s Method 1

1.2 Kane’s Method of Deriving Equations of Motion 2

1.2.1 Kane’s Equations 4

1.2.2 Simple Example: Equations for a Double Pendulum 4

1.2.3 Equations for a Spinning Spacecraft with Three Rotors, Fuel Slosh, and Nutation Damper 6

1.3 Comparison to Derivation of Equations of Motion by Lagrange’s Method 11

1.3.1 Lagrange’s Equations in Quasi-Coordinates 14

Reader’s Exercise 15

1.4 Kane’s Method of Direct Derivation of Linearized Dynamical Equation 16

1.5 Prematurely Linearized Equations and a Posteriori Correction by ad hoc Addition of Geometric Stiffness due to Inertia Loads 19

1.6 Kane’s Equations with Undetermined Multipliers for Constrained Motion 21

1.7 Summary of the Equations of Motion with Undetermined Multipliers for Constraints 22

1.8 A Simple Application 23

Appendix 1.A Guidelines for Choosing Efficient Motion Variables in Kane’s Method 25

Problem Set 1 27

References 28

2 Deployment, Station-Keeping, and Retrieval of a Flexible Tether Connecting a Satellite to the Shuttle 29

2.1 Equations of Motion of a Tethered Satellite Deployment from the Space Shuttle 30

2.1.1 Kinematical Equations 31

2.1.2 Dynamical Equations 32

2.1.3 Simulation Results 35

2.2 Thruster-Augmented Retrieval of a Tethered Satellite to the Orbiting Shuttle 37

2.2.1 Dynamical Equations 37

2.2.2 Simulation Results 47

2.2.3 Conclusion 47

2.3 Dynamics and Control of Station-Keeping of the Shuttle-Tethered Satellite 47

Appendix 2.A Sliding Impact of a Nose Cap with a Package of Parachute Used for Recovery of a Booster Launching Satellites 49

Appendix 2.B Formation Flying with Multiple Tethered Satellites 53

Appendix 2.C Orbit Boosting of Tethered Satellite Systems by Electrodynamic Forces 55

Problem Set 2 60

References 60

3 Kane’s Method of Linearization Applied to the Dynamics of a Beam in Large Overall Motion 63

3.1 Nonlinear Beam Kinematics with Neutral Axis Stretch, Shear, and Torsion 63

3.2 Nonlinear Partial Velocities and Partial Angular Velocities for Correct Linearization 69

3.3 Use of Kane’s Method for Direct Derivation of Linearized Dynamical Equations 70

3.4 Simulation Results for a Space-Based Robotic Manipulator 76

3.5 Erroneous Results Obtained Using Vibration Modes in Conventional Analysis 78

Problem Set 3 79

References 82

4 Dynamics of a Plate in Large Overall Motion 83

4.1 Motivating Results of a Simulation 83

4.2 Application of Kane’s Methodology for Proper Linearization 85

4.3 Simulation Algorithm 90

4.4 Conclusion 92

Appendix 4.A Specialized Modal Integrals 93

Problem Set 4 94

References 96

5 Dynamics of an Arbitrary Flexible Body in Large Overall Motion 97

5.1 Dynamical Equations with the Use of Vibration Modes 98

5.2 Compensating for Premature Linearization by Geometric Stiffness due to Inertia Loads 100

5.2.1 Rigid Body Kinematical Equations 104

5.3 Summary of the Algorithm 105

5.4 Crucial Test and Validation of the Theory in Application 106

Appendix 5.A Modal Integrals for an Arbitrary Flexible Body 112

Problem Set 5 114

References 114

6 Flexible Multibody Dynamics: Dense Matrix Formulation 115

6.1 Flexible Body System in a Tree Topology 115

6.2 Kinematics of a Joint in a Flexible Multibody Body System 115

6.3 Kinematics and Generalized Inertia Forces for a Flexible Multibody System 116

6.4 Kinematical Recurrence Relations Pertaining to a Body and Its Inboard Body 120

6.5 Generalized Active Forces due to Nominal and Motion-Induced Stiffness 121

6.6 Treatment of Prescribed Motion and Internal Forces 126

6.7 “Ruthless Linearization” for Very Slowly Moving Articulating Flexible Structures 126

6.8 Simulation Results 127

Problem Set 6 129

References 131

7 Component Mode Selection and Model Reduction: A Review 133

7.1 Craig-Bampton Component Modes for Constrained Flexible Bodies 133

7.2 Component Modes by Guyan Reduction 136

7.3 Modal Effective Mass 137

7.4 Component Model Reduction by Frequency Filtering 138

7.5 Compensation for Errors due to Model Reduction by Modal Truncation Vectors 138

7.6 Role of Modal Truncation Vectors in Response Analysis 141

7.7 Component Mode Synthesis to Form System Modes 143

7.8 Flexible Body Model Reduction by Singular Value Decomposition of Projected System Modes 145

7.9 Deriving Damping Coefficient of Components from Desired System Damping 147

Problem Set 7 148

Appendix 7.A Matlab Codes for Structural Dynamics 149

7.10 Conclusion 159

References 159

8 Block-Diagonal Formulation for a Flexible Multibody System 161

8.1 Example: Role of Geometric Stiffness due to Interbody Load on a Component 161

8.2 Multibody System with Rigid and Flexible Components 164

8.3 Recurrence Relations for Kinematics 165

8.4 Construction of the Dynamical Equations in a Block-Diagonal Form 168

8.5 Summary of the Block-Diagonal Algorithm for a Tree Configuration 174

8.5.1 First Forward Pass 174

8.5.2 Backward Pass 174

8.5.3 Second Forward Pass 175

8.6 Numerical Results Demonstrating Computational Efficiency 175

8.7 Modification of the Block-Diagonal Formulation to Handle Motion Constraints 176

8.8 Validation of Formulation with Ground Test Results 182

8.9 Conclusion 186

Appendix 8.A An Alternative Derivation of Geometric Stiffness due to Inertia Loads 187

Problem Set 8 188

References 189

9 Efficient Variables, Recursive Formulation, and Multi-Point Constraints in Flexible Multibody Dynamics 191

9.1 Single Flexible Body Equations in Efficient Variables 191

9.2 Multibody Hinge Kinematics for Efficient Generalized Speeds 196

9.3 Recursive Algorithm for Flexible Multibody Dynamics with Multiple Structural Loops 201

9.3.1 Backward Pass 201

9.3.2 Forward Pass 207

9.4 Explicit Solution of Dynamical Equations Using Motion Constraints 209

9.5 Computational Results and Simulation Efficiency for Moving Multi-Loop Structures 210

9.5.1 Simulation Results 210

Acknowledgment 215

Appendix 9.A Pseudo-Code for Constrained nb-Body m-Loop Recursive Algorithm in Efficient Variables 216

Problem Set 9 220

References 220

10 Efficient Modeling of Beams with Large Deflection and Large Base Motion 223

10.1 Discrete Modeling for Large Deflection of Beams 223

10.2 Motion and Loads Analysis by the Order-n Formulation 226

10.3 Numerical Integration by the Newmark Method 230

10.4 Nonlinear Elastodynamics via the Finite Element Method 231

10.5 Comparison of the Order-n Formulation with the Finite Element Method 233

10.6 Conclusion 237

Acknowledgment 238

Problem Set 10 238

References 238

11 Variable-n Order-n Formulation for Deployment and Retraction of Beams and Cables with Large Deflection 239

11.1 Beam Discretization 239

11.2 Deployment/Retraction from a Rotating Base 240

11.2.1 Initialization Step 240

11.2.2 Forward Pass 240

11.2.3 Backward Pass 243

11.2.4 Forward Pass 244

11.2.5 Deployment/Retraction Step 244

11.3 Numerical Simulation of Deployment and Retraction 246

11.4 Deployment of a Cable from a Ship to a Maneuvering Underwater Search Vehicle 247

11.4.1 Cable Discretization and Variable-n Order-n Algorithm for Constrained Systems with Controlled End Body 248

11.4.2 Hydrodynamic Forces on the Underwater Cable 254

11.4.3 Nonlinear Holonomic Constraint, Control-Constraint Coupling, Constraint Stabilization, and Cable Tension 255

11.5 Simulation Results 257

Problem Set 11 261

References 267

12 Order-n Equations of Flexible Rocket Dynamics 269

12.1 Introduction 269

12.2 Kane’s Equation for a Variable Mass Flexible Body 269

12.3 Matrix Form of the Equations for Variable Mass Flexible Body Dynamics 274

12.4 Order-n Algorithm for a Flexible Rocket with Commanded Gimbaled Nozzle Motion 275

12.5 Numerical Simulation of Planar Motion of a Flexible Rocket 278

12.6 Conclusion 285

Acknowledgment 285

Appendix 12.A Summary Algorithm for Finding Two Gimbal Angle Torques for the Nozzle 285

Problem Set 12 286

References 286

Appendix A Efficient Generalized Speeds for a Single Free-Flying Flexible Body 287

Appendix B A FORTRAN Code of the Order-n Algorithm: Application to an Example 291

Index 301

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