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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 (TMM) has been developed for a long time and has been used widely in engineering mechanics of the linear and nonlinear system. To linear system, Holzer (1921) initially applied TMM to solve the problems of torsion vibrations of rods [

To nonlinear system, Zu and Ji (2002) proposed an improved TMM for steady-state analysis of nonlinear rotor-bearing systems [

Recently Liu (1999) adopted TMM to analyze the plane frame with variable section and branch [

In this investigation the tree structure system is modeled by TMM. A special attention is focused on how the transfer equations and transfer matrices of the global system can be developed conveniently. By defining the state vectors and deducing the transfer matrices of the typical components of the tree system suitably, some interesting phenomena, which are that the topology structure of the mechanics model is almost similar to that of the interrelated formula, are discovered. Then a systemic TMM is proposed that can be used conveniently to deal with the vibration problem of tree structure. This formulation is mathematically and practically convenient. The text is organized as follows. In Section

The tree system is one kind of important nonchain system. It contains many nodes, branches, and hierarchical organizations. Its concrete shape is very similar to the natural biologic tree. In a tree system there is one and only one path leading from one location to any other location in the system.

The tree system includes two kinds of the subsystems: chain subsystem and branched subsystem. Chain subsystem is comprised of some elements with one input end and one output end that are connected with chain form. The branched subsystem at least contains one element with many input ends and one output end, usually named as the branched components. It does not lose generality in this work, the tree system is constituted by the spring, chain lumped mass, and branched lumped mass whose mechanics performance will be introduced in detail in Section

For example, the tree system shown in Figure

Tree-form system and its linearization.

Original tree-form system

Discrete tree-form system

Now that the tree structure system includes the spring and the lumped mass undergoing one dimension motion along the

The transfer equation of the chain subsystem

Model of chain subsystem.

The topology structure of the chain system in Figure

The branched subsystem

Branched component.

According to the definition of the state vector and transfer matrix in the above section, the transfer equations of all subsystems of the tree system shown in Figure

Then natural frequency of the tree system can be resolved like the common transfer matrix method. Since the unknown quantities in the boundary state vector have nonzero solution, the shape function and the natural frequencies can be calculated.

The spring neglecting the mass vibrates by natural circular frequency

The dynamics equation of the lumped mass

Considering that the lumped mass

The aim of this investigation is limited in the multiple-branched system by using TMM. So in order to narrate conveniently, the research object is selected as the relative simple component, such as spring and lumped mass undergoing one-dimensional motion. Furthermore, the transfer matrices of complex components can be found in [

Two applied examples are given to validate the method of this paper.

As for the simple tree system, according to the structure characteristic of the system, the component from one end (root node of the tree system) to the others (leaf node of the tree system) can be denoted as

Simple tree system model.

The model of Figure

Another branched system model.

From the two examples we find that the proposed transfer matrix method can be used to solve the natural frequencies of the vibration system without developing the global system dynamics equation. The only things which must be done are to know the transfer matrix of typical components that have been deduced aforehand, the topology structure, and the boundary condition of the global system. So we can obtain the eigenfrequency equation of the system to calculate the natural frequencies by the proposed transfer matrix method. This method is intuitive, simple, and can be held by the engineers easily.

On the basis of classic transfer matrix method for the chain system, this investigation constructs the transfer matrix method for tree structure natural vibration from the totally novel angle of view. When this method is used, the transfer matrix of the global system can be obtained like assembling the toy block, and the correlative formulations can be verified by comparing the topology structure of the vibration system and corresponding transfer equations. So the error probability can be reduced enormously, and computer programming or manual derivation is very convenient.

For the discrete/continuous coupling system that includes discrete and continuous components, as long as the transfer matrices of these components have been deduced ahead of schedule, the natural frequency of the tree structure system can be calculated by the method. If other methods are used to analyze the natural frequency of this kind of the system, the hybrid ordinary and partial differential equations have to be resolved generally. This requires good mathematical technique and cannot be applied easily by the engineers. An important work in TMM, deducing the transfer matrix of the general elastic body, is complicated and does not obtain easily. The fortunate thing is that FE-TMM can be used to develop the transfer matrix of these components conveniently [

It is noted that the formulation and computation of this paper is finished by symbolic software Mathematica and the computational convergence and stability are not essential problems. And by the method of symbolic computation, if the eigenfrequency equation is obtained, such as (

There are

The work described in this paper was supported by the National Defence Key Foundation of China, NSF of Jiangsu SBK201140044 and research foundation of Nanjing University of Technology 39724001.