Productions in the industries of weapons, ships, astronautics, aeronautics, machinery, vehicles including various modern tanks, artilleries, firearms, warships, submarines, planes, helicopters, missiles, launch vehicles, satellites, spaceships, aerospace vehicles, machine tools, trains, cars and electric cars, etc. are multibody systems composed of many bodies.

Multibody system dynamics (MSD) is one of the research focus areas in today's mechanics, and lays a significant foundation for the dynamics performance design as well as experiment design of industrial productions in engineering fields. Various multibody system dynamics methods (MSDMs) developed in the past 50 years have greatly promoted the development of modern science and engineering technology and provided many effective solutions for MSD. It was shown by many engineering practices that there is an urgent need for the theory of computation with high speed, the platform of numerical simulation and design with high efficiency, as well as methods of physical simulation and testing for MSD when the dynamics performance design and experiment design of multibody systems (MSs) are carried out. The following characteristics are the common problems faced by ordinary MSDMs: (i) It is necessary and rather complicated to develop the global dynamics equations of the system. (ii) The order of system matrix is proportional to the number of degrees of freedom (DOFs) of the system, which may become rather high for a complex multibody system. The computational speed decreases dramatically with the increase of the system scale, which has become an urgent problem to be solved. For example, for a multiple launch rocket system (MLRS) with 40 tubes, one needs to find out the optimal firing sequence to achieve the best firing accuracy from 40! (about 8.2 × 10⁴⁷) schemes, which is an amazing number. It can hardly meet the requirement of system dynamics design by using ordinary MSDMs, even when choosing the fastest computer in the world. This is because one has to first compute MSD for each of the schemes, showing how important it is to compute MSD in practical engineering design! In fact, fast computation will be a permanent research direction for complicated MSD. (iii) There was an urgent engineering demand for fast solution of the eigenvalue problems for complex multibody systems. However, one may encounter huge computational efforts and ill‐conditioned problems due to large stiffness gradients by using the finite element method. Also, there was no universal approach for solving the eigenvalue problem of a linear mutli‐rigid‐flexible‐body system. (iv) For the dynamics coupling between rigid and flexible bodies, the eigenvalue problem of a multi‐rigid‐flexible‐body system is usually non‐self‐conjugate, and the orthogonality of eigenvectors of the MS does not exist in an ordinary sense. Thus, the dynamics response of MSs cannot be precisely analyzed using the classical modal method. Consequently, several questions arise. Is it possible to study MSD without the global dynamics equations of the system and simplify the computational procedure of system dynamics? How can the computational scale for MSD be greatly reduced and thus the computational speed increased? How can dynamics performance design with a fast computational dynamics method be realized? How can the vibration characteristics of linear MSs be computed? How can the orthogonality of eigenvectors for linear MSs be constructed and precise analysis of dynamics response realized with the modal method? How can experiments for complex mechanical systems be designed? These are urgent difficult problems noticed by most scientists and engineers around the world. And it is these urgent theoretical and engineering demands that rightly promote the rapid development of the transfer matrix method for multibody systems (MSTMM). According to my long‐term study and engineering practice in launch dynamics and MSD, I have deeply perceived that it is primarily important to establish a new MSD method with high computational speed.

In order to avoid the global dynamics equations when studying MSD and increase computational speed regardless of the number of system DOFs, I first proposed the MSTMM to study MSD in my PhD thesis “Study on Launch Dynamics of Multibody Systems” in 1993, which was published in 1995, where the computational efficiency of MSD was greatly improved. Later on, many scholars and scientists made a lot of important contributions to the development of this method. By continuous improvement and engineering applications as well as extensive international academic exchanges during the past 25 years, the method has been developed into a novel stylized efficient dynamics method for general MSs. It has been applied to the dynamics analysis and design of various large mechanical systems that have received great attention from many scientists and engineers. Further, MSTMM plays an important role in the research, production, and testing process of many complex mechanical systems. Using this method, serious difficulties appearing in some projects of national high‐tech engineering have been solved, and important technology breakthroughs have been achieved. All of these exhibit the strength and application prospect of MSTMM.

We took the lead in developing a strict testing technology using non‐full charge loading test instead of full charge loading test for testing the firing precision of multiple launch rocket systems. The rocket consumption for testing the firing precision of a multiple launch rocket system has been greatly decreased, which used to be a crucial difficult problem for every country with military power. It has been verified by testing that the new technology is much better than that of Russia and other countries, and 12 million Yuan have been saved in each test of the multiple launch rocket system. In fact, a direct economic benefit of over hundred million Yuan has been saved. Another technology for improving the firing precision of weapons has also been developed. The firing precision of multiple launch rocket systems and self‐propelled artilleries within the national high‐tech engineering projects has been improved greatly using this new technology.

According to incomplete statistics to date, there have been six monographs published regarding MSTMM and its applications. Over 300 articles studying the method have been published in more than 100 journals by over 200 authors from over 60 work units. Moreover, over 10 software copyrights, 100 patents, and 4 national technology progress and invention awards related to MSTMM have been achieved, respectively. This method has been successfully applied to a wide range of engineering problems related to about 100 products and over 50 fields, including self‐propelled artillery, shipborne gun, “metal storm” antiaircraft gun, spin tube gun, vehicular MLRS, airborne MLRS, shipborne MLRS, cannon on helicopters, tank, ultra‐precision single‐point diamond fly‐cutting machine tool, spacecraft navigation and its vibration reduction system, launch vehicle, missile, aerospace aircraft, submarines, underwater towed systems, piezoelectric actuator, controlled flexible manipulators, intelligent flexible four‐bar linkage devices, super long stay cable, earthquake resistant civil structures, immersed tunnel, robots, mobile concrete truck boom, vibrating screen, vibration compaction, road roller, wind turbine, wind turbine tower, gas turbine system, low pressure rotor of gas turbine, high pressure compressor of gas turbine, large‐scale rotary machine, feeding platform, parachute‐submissile, rocket projectile, vehicular missile, truck cranes, floating bridge, wing, five‐axis CNC machine tool, heavy duty longmen machine tool, machine tool spindle system, servo turret system, high pressure gas well system, diesel engine, roots blower with double rotor, ship's anti‐vibration mounting system, ship pipeline system, bearing‐rotor, vehicle suspension, etc.

There are mainly three stages in the development of MSTMM: transfer matrix method for linear multibody systems (linear MSTMM) (1993~), discrete time transfer matrix method for multibody systems (MSDTTMM) (1998~), and the new version of transfer matrix method for multibody systems (NV‐MSTMM) (2013~).

In 1997, Rui et al. proposed the concept of “augmented eigenvector” and “augmented operator” for linear MSs, and constructed the orthogonality of augmented eigenvectors of linear multi‐rigid‐flexible‐body systems. The exact analysis of dynamics response of a linear multi‐rigid‐flexible‐body system using a modal method was achieved for the first time. Yun et al. (2006) proposed the transfer matrix method (TMM) for two‐dimensional systems to analyze two‐dimensional system dynamics by using pure TMM. Rui et al. (2006) also proposed the transfer matrix method for linear controlled MSs.

Rui et al. (1998) proposed and gradually improved the discrete time transfer matrix method for multi‐rigid‐body systems. In this method, the high efficiency of MSTMM and various time integration methods are combined. The constant transfer matrices and state vectors described by modal coordinates in linear MSTMM were extended into time‐dependent transfer matrices and state vectors described by physical coordinates. Rui et al. (1999) proposed and gradually improved the discrete time transfer matrix method for multi‐rigid‐flexible‐body systems. The Riccati transformation was then introduced by He et al. (2007) for solving the MSD composed of a huge number of elements. Rong et al. (2010) established the discrete time transfer matrix method for controlled MS. The dynamics of multi‐rigid‐body systems, multi‐rigid‐flexible‐body systems, as well as controlled MSs, especially complex launching systems, were studied using MSTMM, which is suitable for time‐variant, nonlinear systems with large motion, for the first time with fast computational speed and without global dynamics equations.

However, due to linearization of nonlinear functions in MSDTTMM, the time step size should be kept small to guarantee computational precision and stability. Rui et al. (2013) proposed the NV‐MSTMM, where accelerations, angular accelerations, internal forces, and torques are selected as state variables of the state vectors. The local linearization is fully avoided while deducing the transfer equations of elements. In that sense, the transfer equations of elements and overall transfer equation of the system are exact, so MSTMM belongs to accurate analysis methods in theory.

Rui, Zhang, Xue Rui et al. (2011) introduced the topology figure and presented the automatic deduction theorem to deduce the overall transfer equation automatically. Zhou et al. (2016) proposed the deduction method for overall transfer equations of linear controlled MSs. Rui et al. (2006) proposed the mixed method of MSTMM with other MSD methods and the finite element method to take advantages of each of the methods. Rui et al. (2014) developed the visualized simulation and design software MSTMMSim with MSTMM executing as its core. This software has been applied to the dynamics computation of multiple launch rocket systems, self‐propelled artilleries, tanks, and some other large mechanical systems.

This book is based on the “Transfer Matrix Method of Multibody System and its Applications” (Beijing: Science Press, 2008) written by the first author as the chief author. Additionally, over 100 papers published by the authors of the book during the last decade and the latest research results across the world in this field are incorporated into this book. The remarkable research findings of various state basic research development programs of China (973 Programs), exploration projects, pre‐research fund projects, basic scientific research projects, cross‐industry projects, etc. are included in this book. We strive to exhibit a self‐contained system to introduce the basic theories, algorithms, latest developments, and applications of the transfer matrix method for multibody systems. The book is organized in 4 parts with 15 chapters. Chapter 1 presents the characteristics of the transfer matrix method for multibody systems, the finite element method, and the multibody system dynamics methods.

The first part (Chapters 2–4) develops the transfer matrix method for linear multibody systems. It includes the fundamentals of the transfer matrix method for linear multi‐rigid‐flexible‐body systems, the new concepts of the body dynamics equations and augmented eigenvectors, the methods to construct the orthogonality of augmented eigenvectors, to derivate transfer matrices, to evaluate vibration characteristics and dynamics response. Further, transfer matrix method of multi‐dimensional systems and transfer matrix method for nonlinear systems are also introduced.

The second part (Chapters 5–6) introduces the new version of the transfer matrix method for multibody systems, where the accelerations and the internal forces are treated as the state variables of the connecting point. Both the transfer matrix method for multi‐rigid‐body systems and that for multi‐flexible‐body systems are introduced.

The third part (Chapters 7–11) develops the discrete time transfer matrix method for multibody systems. It includes discrete time transfer matrix method for multi‐rigid‐body systems, discrete time transfer matrix method for multi‐rigid‐flexible‐body systems, transfer matrix method for controlled multibody systems, and a hybrid method of these with other dynamics methods. The derivation and computation methods of transfer matrices of elements, as well as the automatic derivation theorem of the overall transfer equation of the system, are also introduced.

The fourth part (Chapters 12–15) shows some practical engineering applications of the transfer matrix method for multibody systems in launch dynamics that are the hotspot in the international armament science at present. The new theories of launch dynamics of multiple launch rocket systems, launch dynamics of self‐propelled artilleries, and launch dynamics of shipboard guns based on the transfer matrix method for multibody systems are developed. Excellent simulation results obtained by the theories lead to a strong improvement in the firing precision of weapons and a great decrease in costs for the testing of multiple launch rocket systems. Finally, a library of transfer matrices of various elements widely used in various multibody systems is provided.

Nearly 100 researchers have participated in the related research work of this book, and dozens of experts and scholars at home and abroad have made great contribution to this book. The four dynamicists, Professor Wenhu Huang, member of the Chinese Academy of Engineering; Professor Kezhi Huang, member of the Chinese Academy of Science; Professor Werner Schiehlen, President of International Union of Theoretical and Applied Mechanics, Professor Jens Wittenburg, Director of the Institute of Engineering Mechanics, Karlsruhe University, offered wonderful forewords for the first edition of the book. The two dynamicists, Professor Dieter Bestle, Cottbus Industrial University, and Professor Haiyan Hu, member of the Chinese Academy of Sciences from Beijing Institute of Technology, went through the whole manuscript of the book and made a recommendation for publication. The personnel who have participated in the work of this book include Associate Professor Fufeng Yang, Associate Professor Hailong Yu, Associate Professor Bao Rong, Associate Professor Bin He, Professor Laifeng Yun, Professor Laith K. Abbas, Lecturer Yan Wang, Lecturer Tao Chen, Dr. Weibo Wei, Dr. Jun Hong, Dr. Junyi He, Dr. Lei Ma, Dr. Zhihuan Zhan, Dr. Hao Xu, Dr. Binbin Feng, Dr. Wenbin Tang, Dr. Xue Rui, Dr. Feifei Liu, Dr. Heng Zhang, Dr. Chao Li, Dr. Junjie Gu, Dr. Haigen Yang, Dr. Minjiao Li, Dr. Qinbo Zhou, Dr. Qicheng Cha, Dr. Gangli Chen, Dr. Jian Gu, Dr. Tianxiong Tu, Dr. Lilin Gu, Dr. Bo Li, Dr. Yu Tao, Dr. Min Wei, Dr. Lu Sun, Dr. Hanjing Lu, Dr. Xun Wang, Professor Bin Hu, and Professor Ming Song. Science Publishing House and deputy editor Baoli Liu strongly supported the publication of this book. Monicka Simon, editor of Wiley publisher, gave the book an elaborate editing and proofreading, and our family also gave us great understanding and support for our work.

I was invited and supported respectively by famous scientists including Professor Werner Schiehlen, President of International Union of Theoretical and Applied Mechanics; Professor Jens Wittenburg, Head of Engineering Mechanics Institute in Karlsruhe University; Professor Erwin Stein, President of Germany Mechanics Society; Professor Peter Eberhard, Head of Institute of Engineering and Computational Mechanics in Stuttgart University; Professor Klaus Thoma, Head of Ernst‐Mach Institute; Professor Dieter Bestle, Head of Institute of Engineering Mechanics and Vehicle Dynamics in Cottbus Technology University; Professor Bodo Heimann, Head of Robot Institute in Hannover University; and Professor Edwin Kreuzer, President of Hamburg‐Harburg Technology University. I was also supported several times by key projects of German Research Council (DFG), as guest professor of Stuttgart University, Karlsruhe University, Hannover University, Ernst‐Mach Institute, Cottbus Technology University, and Hamburg‐Harburg Technology University. Moreover, I was invited by dozens of famous dynamicists, such as, Professor Friedrich Pfeiffer, Chief Editor of Journal Archive of Applied Mechanics; Professor Peter Maisser, Head of Mechanic and Control Institute in Chemnitz Technology University; Professor Karl Popp, Head of Mechanics Institute in Hannover University; Professor Lutz Sperling of Engineering Mechanics Institute in Magdeburg University; Professor Joachim Lückel, Head of Mechanic Control Institute in Paderbor University; Professor Jorg Wauer of Engineering Mechanics Institute in Karlsruhe University and Professor Horst Irretier of Kassel University; Professor Peter Breedveld, Twente University; Professor Ahmed Shabana, University of Illinois at Chicago; and Professor Fen Wu, North Carolina State University. I was also invited by Harbin Engineering University, Xi'an Jiao Tong University, Xi'an Technological University, Suzhou University, Institute of automation of China weaponry and equipment group, Xi'an Modern Chemistry Research Institute, and Chongqing military representative Bureau. I have given over 80 invited academic lectures about the contents of this book in 20 universities and institutes in Europe and America, as well as in various international conferences about mechanics, machinery, and vibration.

I would like to express our great gratitude to the research organizations, institutions, scientists, experts, and our families, which sponsored, supported, and helped our research work, activities of academic exchange, and lectures at home and abroad. Special thanks are given to Professor Dieter Bestle. During his several academic visits to my institute summing up to about one year, he made a systematic study and development of the Rui method and made a lot of creative contributions to the book word by word. Professor Jens Wittenburg commented that “Rui method represents a totally new approach for simulating the dynamics of multibody system and is very promising.” Professor Werner Schiehlen commented that “It is valuable to popularize the method.” Professor John Herbst commented that “Transfer matrix method of multibody system is a totally new and original method for solving multibody system dynamics. It is very valuable to popularize Rui method in the study fields of multibody system dynamics and complicated mechanical engineering.” I would like to dedicate the book to them. I wish that the transfer matrix method for multibody system will play a positive role in benefiting humankind. Any comments from professors and experts in various research fields on the book will be welcomed.