On the natural frequencies and mode shapes of a multispan Timoshenko beam carrying a number of various concentrated elements

The purpose of this paper is to utilize the numerical assembly method (NAM) to determine the exact natural frequencies and mode shapes of the multispan Timoshenko beam carrying a number of various concentrated elements including point masses, rotary inertias, linear springs, rotational springs and spring–mass systems. First, the coefficient matrices for an intermediate pinned support, an intermediate concentrated element, left- and right-end support of a Timoshenko beam are derived. Next, the overall coefficient matrix for the whole structural system is obtained using the numerical assembly technique of the finite element method. Finally, the exact natural frequencies and the associated mode shapes of the vibrating system are determined by equating the determinant of the last overall coefficient matrix to zero and substituting the corresponding values of integration constants into the associated eigenfunctions, respectively. The effects of distribution of in-span pinned supports and various concentrated elements on the dynamic characteristics of the Timoshenko beam are also studied.