The application of Transfer matrix method (TMM) ranges from linear/nonlinear vibration, composite structure, and multibody system to calculating static deformation, natural vibration, dynamical response, and damage identification. Generally TMM has two characteristics: (1) the TMM formulae share similarity to the chain mechanics model in terms of topology structure; then TMM often is selected as a powerful tool to analyze the chain system. (2) TMM is adopted to deal with the problems of the discrete system, continuous system, and especial discrete/continuous coupling system with the uniform matrix form. In this investigation, a novel TMM is proposed to analyze the natural vibration of the tree system. In order to make the TMM of the tree system have the two above advantages of the TMM of the chain system, the suitable state vectors and transfer matrices of the typical components of the tree system are constructed. Then the topology comparability between the mechanics model and its corresponding formulae of TMM can be adopted to assembling the transfer matrices and transfer equations of the global tree system. Two examples of natural vibration problems validating the method are given. The formulation of the proposed TMM is mathematically intuitive and can be held and applied by the engineers easily.

Transfer matrix method is a practical technology for vibration analysis of engineering mechanics. In this article, a new version of mixed differential quadrature discrete time transfer matrix method is presented for simulating vibrations. First, ordinary differential equations of the substructure or the element of a mechanical system are determined by classical mechanics rules or the finite element method. These ordinary differential equations can be transformed to a set of algebraic equations at some discrete time points by the application of a differential quadrature method. After extending the state vector of the transfer matrix method, new transfer equations and transfer matrices of the substructures (or elements) of a mechanical system are developed finally. Riccati transformation is used to improve the computational convergence of the method. Several numerical examples show that the proposed method can be regarded as an efficient tool for transient response analysis of vibration systems.