Free Vibration Analysis of a Tree Structure by the Transfer influence Coefficient Method (1st Report, Formulation for a Two-Dimensional Tree Structure)

The authors apply the transfer influence coefficient method to a two-dimensional tree structure, and formulate a general algorithm to analyse in-plane longitudinal and flexural coupled free vibration. The tree structure, which is mainly found in the Γ-shaped structure of machine tools, pipeline systems and so on, has some crooked parts and subsystems but has no closed loop in the system. It is modeled as a distributed mass system in the present algorithm. It is theoretically confirmed that some merits of the transfer influence coefficient method are kept also for the two- dimensional tree structure; that is, boundary conditions are easily controlled by the spring constants, and false roots in the use of the bisection method as a solution to the frequency equation can readily be eliminated by using the values obtained in the computational process. The occurrence mechanism of false roots is discussed in detail.