Recursive eigenvalue search algorithm for transfer matrix method of linear flexible multibody systems

Abstract

The transfer matrix method is a rather unusual strategy of modeling linear multibody systems, however, it is able to elegantly model systems including both discrete and continuous elements and to solve such kind of problems with any precision required. This is achieved by transforming differential to algebraic equations and summarizing all system information in an overall system of linear equations independent of the degrees of freedom. Nontrivial solutions representing vibration modes then require the coefficient matrix to be singular. Thus, the precision of solutions is associated with the ability of finding zeros for the determinants of these coefficient matrices, which may be nonlinear or transcendental, real or complex functions of natural vibration frequencies or complex eigenvalues. The paper reduces the zero search to a minimization problem and suggests two simple, but robust algorithms which are much more efficient than direct enumeration. Further, the problem of noisy determinant computation is addressed and the complex transfer matrix of a rod for damped vibrations is derived. Three basic examples serve for demonstrating the concept and for showing the robustness of the proposed approach. For a rod-damper system, the solution with jumping frequencies for a critical damping value can be proven analytically.