Free vibration of axial-loaded multi-step Timoshenko beam carrying arbitrary concentrated elements using continuous-mass transfer matrix method

The continuous-mass transfer matrix method (CTMM) is one of the few practical approaches to yield the “exact” solutions for free vibrations of a non-uniform beam carrying any number of concentrated elements. However, most of the existing CTMM does not consider the effects of shear deformation (SD), rotary inertia (RI), joint action term of SD and RI, and axial load. Thus, the objective of this paper is to present a simple formulation so that one can easily obtain the “exact“ natural frequencies and the associated mode shapes of a multi-step beam carrying arbitrary various concentrated elements (including eccentric lumped masses with rotary inertias, linear springs, rotational springs and spring-mass systems) in various boundary conditions with all the above-mentioned effects considered by using the modified CTMM. In addition to comparing with the existing relevant data, most of the numerical results obtained from the modified CTMM are also compared with those of the conventional finite element method (FEM) and good agreements are achieved.