A lumped-mass TMM for free vibration analysis of a multi-step Timoshenko beam carrying eccentric lumped masses with rotary inertias

In the conventional lumped-mass (model) transfer matrix method (LTMM), one must use different frequency equation and the associated initial parameters to determine the natural frequencies and the corresponding mode shapes of a beam with different boundary (supporting) conditions as one may see from the Appendix at the end of this paper. Besides, the eccentricity of each attached lumped mass is also neglected in the existing LTMM. The purpose of this paper is to present a modified LTMM so that one may easily determine the natural frequencies and the corresponding mode shapes of a multi-step Timoshenko beam with various boundary (supporting) conditions and carrying various concentrated elements with eccentricity of each lumped mass considered, by using the same formulation developed from a beam with “free–free” boundary conditions.To this end, by considering the effect of rotary inertia of each beam segment and the eccentricity of each attached lumped mass, the transfer matrix for a typical station carrying three kinds of concentrated elements is derived. Meanwhile, by considering the shear deformation of each beam segment, the transfer matrix for a typical field is also deduced. It has been found that, by changing the magnitude (from zero to infinity) of each concentrated element, one may easily model many problems studied in the exiting literature concerned. The reliability of the presented approach has been confirmed by the good agreement between the numerical results of this paper and those of the existing literature or the conventional finite element method (FEM).