On the Mathematical Properties of the Characteristic Equation in Multibody System Transfer Matrix Method

The multibody system transfer matrix method (MSTMM) is a powerful tool for analyzing the vibration characteristics of engineering systems. In order to further improve the efficiency of solving the characteristic equation obtained by MSTMM, this paper analyzes the mathematical properties of the characteristic equation, and strictly proves a sufficient condition for the correct use of the recursive eigenvalue search algorithm (RESA). Finally, to ensure the numerical stability of the natural frequency calculation starting from 0, the method for eliminating the poles of the transfer matrix of Euler beam at the natural frequency of 0 is briefly described. It is shown that within the given accuracy range, the bisection search method and RESA can correctly solve the vibration characteristics of the rigid flexible coupling system composed of beams, rigid bodies and elastic hinges, without producing wrong solutions. Moreover, these discussions on the mathematical properties of the characteristic equation in this paper can provide a theoretical basis for further developing the efficient algorithms for solving the characteristic equation.