Transfer Matrix Method for Multibody Systems

TRANSFER MATRIX METHOD FOR MULTIBODY SYSTEMS: THEORY AND APPLICATIONS

Xiaoting Rui, Guoping Wang and Jianshu Zhang - Nanjing University of Science and Technology, China

Featuring a new method of multibody system dynamics, this book introduces the transfer matrix method systematically for the first time. First developed by the lead author and his research team, this method has found numerous engineering and technological applications. Readers are first introduced to fundamental concepts like the body dynamics equation, augmented operator and augmented eigenvector before going in depth into precision analysis and computations of eigenvalue problems as well as dynamic responses. The book also covers a combination of mixed methods and practical applications in multiple rocket launch systems, self-propelled artillery as well as launch dynamics of on-ship weaponry.

* Comprehensively introduces a new method of analyzing multibody dynamics for engineers

* Provides a logical development of the transfer matrix method as applied to the dynamics of multibody systems that consist of interconnected bodies

* Features varied applications in weaponry, aeronautics, astronautics, vehicles and robotics

Written by an internationally renowned author and research team with many years' experience in multibody systems Transfer Matrix Method of Multibody System and Its Applications is an advanced level text for researchers and engineers in mechanical system dynamics. It is a comprehensive reference for advanced students and researchers in the related fields of aerospace, vehicle, robotics and weaponry engineering.

Introduction xi
About the Author xiii
Foreword One for the Chinese Edition xv
Foreword Two for the Chinese Edition xvii
Foreword Three for the Chinese Edition xix
Foreword Four for the Chinese Edition xxi
Professor Rui's Method-Discrete Time Transfer Matrix Method for Multibody System Dynamics xxiii
Preface xxv
1 Introduction 1
1. 1 The Status of the Multibody System Dynamics Method 1
1. 2 The Transfer Matrix Method and the Finite Element Method 3
1. 3 The Status of the Transfer Matrix Method for a Multibody System 5
1. 4 Features of the Transfer Matrix Method for Multibody Systems 7
1. 5 Launch Dynamics 12
1. 6 Features of this Book 13
1. 7 Sign Conventions 14
Part I Transfer Matrix Method for Linear Multibody Systems 19
2 Transfer Matrix Method for Linear Multibody Systems 21
2. 1 Introduction 21
2. 2 State Vector, Transfer Equation and Transfer Matrix 22
2. 3 Overall Transfer Equation, Overall Transfer Matrix and Boundary Conditions 31
2. 4 Characteristic Equation 32
2. 5 Computation for State Vector and Vibration Characteristics 36
2. 6 Vibration Characteristics of Multibody Systems 41
2. 7 Eigenvalues of Damped Vibration 56
2. 8 Steady-state Response to Forced Vibration 63
2. 9 Steady-state Response of Forced Damped Vibration 70
3 Augmented Eigenvector and System Response 79
3. 1 Introduction 79
3. 2 Body Dynamics Equation and Parameter Matrices 80
3. 3 Basic Theory of the Orthogonality of Eigenvectors 83
3. 4 Augmented Eigenvectors and their Orthogonality 86
3. 5 Examples of the Orthogonality of Augmented Eigenvectors 96
3. 6 Transient Response of a Multibody System 102
3. 7 Steady-state Response of a Damped Multibody System 111
3. 8 Steady-state Response of a Multibody System 117
3. 9 Static Response of a Multibody System 124
4 Transfer Matrix Method for Nonlinear and Multidimensional Multibody Systems 129
4. 1 Introduction 129
4. 2 Incremental Transfer Matrix Method for Nonlinear Systems 129
4. 3 Finite Element Transfer Matrix Method for Two-dimensional Systems 140
4. 4 Finite Element Riccati Transfer Matrix Method for Two-dimensional Nonlinear Systems 154
4. 5 Fourier Series Transfer Matrix Method for Two-dimensional Systems 162
4. 6 Finite Difference Transfer Matrix Method for Two-dimensional Systems 167
4. 7 Transfer Matrix Method for Two-dimensional Systems 170
Part II Transfer Matrix Method for Multibody Systems 181
5 Transfer Matrix Method for Multi-rigid-body Systems 183
5. 1 Introduction 183
5. 2 State Vectors, Transfer Equations and Transfer Matrices 184
5. 3 Overall Transfer Equation and Overall Transfer Matrix 185
5. 4 Transfer Matrix of a Planar Rigid Body 185
5. 5 Transfer Matrix of a Spatial Rigid Body 187
5. 6 Transfer Matrix of a Planar Hinge 188
5. 7 Transfer Matrix of a Spatial Hinge 189
5. 8 Transfer Matrix of an Acceleration Hinge 192
5. 9 Algorithm of the Transfer Matrix Method for Multibody Systems 193
5. 10 Numerical Examples of Multibody System Dynamics 194
6 Transfer Matrix Method for Multi-flexible-body Systems 199
6. 1 Introduction 199
6. 2 State Vector, Transfer Equation and Transfer Matrix 200
6. 3 Overall Transfer Equation and Overall Transfer Matrix 201
6. 4 Transfer Matrix of a Planar Beam 201
6. 5 Transfer Matrix of a Spatial Beam 205
6. 6 Numerical Examples of Multi-flexible-body System Dynamics 211
Part III Discrete Time Transfer Matrix Method for Multibody Systems 217
7 Discrete Time Transfer Matrix Method for Multibody Systems 219
7. 1 Introduction 219
7. 2 State Vector, Transfer Equation and Transfer Matrix 221
7. 3 Step-by-step Time Integration Method and Linearization 225
7. 4 Transfer Matrix of a Planar Rigid Body 235
7. 5 Transfer Matrices of Spatial Rigid Bodies 242
7. 6 Transfer Matrices of Planar Hinges 251
7. 7 Transfer Matrices of Spatial Hinges 256
7. 8 Algorithm of the Discrete Time Transfer Matrix Method for Multibody Systems 259
7. 9 Numerical Examples of Multibody System Dynamics 259
8 Discrete Time Transfer Matrix Method for Multi-flexible-body Systems 265
8. 1 Introduction 265
8. 2 Dynamics of a Flexible Body with Large Motion 266
8. 3 State Vector, Transfer Equation and Transfer Matrix 276
8. 4 Transfer Matrix of a Beam with Large Planar Motion 277
8. 5 Transfer Matrices of Smooth Hinges Connected to a Beam with Large Planar Motion 282
8. 6 Transfer Matrices of Spring Hinges Connected to a Beam with Large Planar Motion 286
8. 7 Transfer Matrix of a Fixed Hinge Connected to a Beam 292
8. 8 Dynamics Equation of a Spatial Large Motion Beam 296
8. 9 Transfer Matrix of a Spatial Large Motion Beam 300
8. 10 Transfer Matrices of Fixed Hinges Connected to a Beam with Large Spatial Motion 305
8. 11 Transfer Matrices of Smooth Hinges Connected to a Beam with Large Spatial Motion 309
8. 12 Transfer Matrices of Spring Hinges Connected to a Beam with Large Spatial Motion 313
8. 13 Algorithm of the Discrete Time Transfer Matrix Method for Multi-flexible-body Systems 318
8. 14 Planar Multi-flexible-body System Dynamics 318
8. 15 Spatial Multi-flexible-body System Dynamics 322
9 Transfer Matrix Method for Controlled Multibody Systems 327
9. 1 Introduction 327
9. 2 Mixed Transfer Matrix Method for Multibody Systems 328
9. 3 Finite Element Transfer Matrix Method for Multibody Systems 338
9. 4 Finite Segment Transfer Matrix Method for Multibody Systems 341
9. 5 Transfer Matrix Method for Controlled Multibody Systems I 348
9. 6 Transfer Matrix Method for Controlled Multibody Systems II 362
10 Derivation and Computation of Transfer Matrices 377
10. 1 Introduction 377
10. 2 Derivation from Dynamics Equations 378
10. 3 Derivation from an nth-order Differential Equation 388
10. 4 Derivation from n First-order Differential Equations 398
10. 5 Derivation from Stiffness Matrices 401
10. 6 Computational Method of the Transfer Matrix 402
10. 7 Improved Algorithm for Eigenvalue Problems 406
10. 8 Properties of the Inverse Matrix of a Transfer Matrix 408
10. 9 Riccati Transfer Matrix Method for Multibody Systems 417
10. 10 Stability of the Transfer Matrix Method for Multibody Systems 428
11 Theorem to Deduce the Overall Transfer Equation Automatically 433
11. 1 Introduction 433
11. 2 Topology Figure of Multibody Systems 433
11. 3 Automatic Deduction of the Overall Transfer Equation of a Closed-loop System 435
11. 4 Automatic Deduction of the Overall Transfer Equation of a Tree System 435
11. 5 Automatic Deduction of the Overall Transfer Equation of a General System 439
11. 6 Automatic Deduction Theorem of the Overall Transfer Equation 442
11. 7 Numerical Example of Closed-loop System Dynamics 443
11. 8 Numerical Example of Tree System Dynamics 451
11. 9 Numerical Example of Multi-level System Dynamics 470
11. 10 Numerical Example of General System Dynamics 474
Part IV Applications of the Transfer Matrix Method for Multibody Systems 489
12 Dynamics of Multiple Launch Rocket Systems 491
12. 1 Introduction 491
12. 2 Launch Dynamics Model of the System and its Topology 492
12. 3 State Vector, Transfer Equation and Transfer Matrix 496
12. 4 Overall Transfer Equation of the System 502
12. 5 Vibration Characteristics of the System 504
12. 6 Dynamics Response of the System 506
12. 7 Launch Dynamics Equation and Forces Acting on the System 512
12. 8 Dynamics Simulation of the System and its Test Verifying 516
12. 9 Low Rocket Consumption Technique for the System Test 533
12. 10 High Launch Precision Technique for the System 541
13 Dynamics of Self-propelled Launch Systems 545
13. 1 Introduction 545
13. 2 Dynamics Model of the System and its Topology 545
13. 3 State Vector, Transfer Equation and Transfer Matrix 549
13. 4 Overall Transfer Equation of the System 555
13. 5 Vibration Characteristics of the System 555
13. 6 Dynamic Response of the System 557
13. 7 Launch Dynamic Equations and Forces Analysis 563
13. 8 Dynamics Simulation of the System and its Test Verifying 570
14 Dynamics of Shipboard Launch Systems 581
14. 1 Introduction 581
14. 2 Dynamics Model of Shipboard Launch Systems 581
14. 3 State Vector, Transfer Equation and Transfer Matrix 583
14. 4 Overall Transfer Equation of the System 587
14. 5 Launch Dynamics Equation and Forces of the System 589
14. 6 Solution of Shipboard Launch System Motion 598
14. 7 Dynamics Simulation of the System and its Test Verifying 599
15 Transfer Matrix Library for Multibody Systems 607
15. 1 Introdution 607
15. 2 Springs 607
15. 3 Rotary Springs 609
15. 4 Elastic Hinges 610
15. 5 Lumped Mass Vibrating in a Longitudinal Direction 611
15. 6 Vibration of Rigid Bodies 612
15. 7 Beam with Transverse Vibration 615
15. 8 Shaft with Torsional Vibration 620
15. 9 Rod with Longitudinal Vibration 621
15. 10 Euler-Bernoulli Beam 622
15. 11 Rectangular Plate 624
15. 12 Disk 629
15. 13 Strip Element of a Two-dimensional Thin Plate 635
15. 14 Thick-walled Cylinder 638
15. 15 Thin-walled Cylinder 640
15. 16 Coordinate Transformation Matrix 642
15. 17 Linearization and State Vectors 645
15. 18 Spring and Damper Hinges Connected to Rigid Bodies 646
15. 19 Smooth Hinges Connected to Rigid Bodies 648
15. 20 Rigid Bodies Moving in a Plane 649
15. 21 Spatial Rigid Bodies with Large Motion and Various Connections 651
15. 22 Planar Beam with Large Motion 654
15. 23 Spatial Beam with Large Motion 656
15. 24 Fixed Hinges Connected to a Planar Beam with Large Motion 658
15. 25 Fixed Hinges Connected to a Spatial Beam with Large Motion 660
15. 26 Smooth Hinges Connected to a Beam with Large Planar Motion 663
15. 27 Smooth Hinges Connected to a Beam with Large Spatial Motion 666
15. 28 Elastic Hinges Connected to a Beam with Large Planar Motion 668
15. 29 Elastic Hinges Connected to a Beam Moving in Space 672
15. 30 Controlled Elements of a Linear System 675
15. 31 Controlled Elements of a General Time-variable System 676
Appendix I Rotation Formula Around an Axis 681
Appendix II Orientation of a Body-fixed Coordinate System 683
Appendix III List of Symbols 687
Appendix IV International Academic Communion for the Transfer Matrix Method for Multibody Systems 693
References 707
Index 729