# Transfer Matrix Method for Multibody Systems

## Theory and Applications

1. Edition December 2018

768 Pages, Hardcover*Practical Approach Book*

**978-1-118-72480-4**

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.

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