10Derivation and Computation of Transfer Matrices

10.1 Introduction

Generally speaking, all kinds of dynamics problems of multibody systems (MSs) can be easily solved using the transfer matrix method for multibody systems (MSTMM) once the transfer matrices of elements are obtained. The library of transfer matrices of various kinds of elements related to MSs is listed in Chapter 15. Those transfer matrices which are not included in the transfer matrix library can be derived by using the methods introduced in this book. The transfer matrices of elements can mainly be derived from differential equations of motion, from solving an n‐order differential equation, from n first‐order differential equations and from the stiffness matrix. The transfer matrices of discrete elements such as rigid bodies, lumped masses, springs and rotary springs whose dynamic equations are expressed by ordinary differential equations, can be directly derived from differential equations of motion with simple substitution and rewriting of the dynamic equations of elements. The transfer matrices of continuous elements such as elastic beams, rods, shafts, plates, etc., whose dynamic equations are expressed by partial differential equations, can be derived by solving partial differential equations, which is primarily done by solving the n‐order ordinary differential equations after substitution, or by solving n first‐order differential equations. For continuous elements (including one‐, two‐ and three‐dimensional ...

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