Exact solution of free vibration of a uniform tensioned beam combined with both lateral and rotational linear sub-systems

A general exact solution of free vibration of a uniform tensioned beam with any number of both lateral and rotational sub-systems is addressed. This study is completed in complex domain, because of the presence of the non-proportional damping caused by the dampers utilized in sub-systems. The sub-systems constructed herein could degenerate to the conventional supports and sub-systems, and be arbitrarily placed. Accordingly, the solution is also suitable for multi-span beams. The argument principle method (APM), which does not need initial values and iteration, is employed to solve the characteristic equation in complex domain. Nevertheless, the presence of multi-valued square root functions due to the axial loading renders the function non-analytic in complex domain. A variable substitution method, in which the main branch of logarithmic function is chosen, is proposed to solve this problem. The results obtained are compared with those in literature. Furthermore, several typical case studies were completed, and the root loci of the frequency parameters attained are compared with those obtained by the finite element method (FEM). A good agreement is observed, which confirms the validity of the presented methodologies in this paper.