A continuous-mass TMM for free vibration analysis of a non-uniform beam with various boundary conditions and carrying multiple concentrated elements

This paper presents a modified continuous-mass (model) transfer matrix method (CTMM) to determine the natural frequencies and associated mode shapes of a uniform or non-uniform beam with various classical (or non-classical) boundary conditions (BCs) and carrying multiple sets of concentrated elements with each set consisting of a point mass (with eccentricity and rotary inertia), a translational spring and a rotational spring. To this end, a continuous non-uniform free–free beam is subdivided into several uniform beam segments (each having distributed mass) and any two adjacent beam segments are connected by a node at which various concentrated elements being attached. Next, the transfer matrix for the integration constants of arbitrary two adjacent beam segments joined at an intermediate node is derived, and then the characteristic equation of the entire vibrating system is derived by combining all transfer matrices for all intermediate nodes and considering the BCs of the entire free–free beam. It has been found that, based on the foregoing formulation for a non-uniform free–free beam, one may easily obtain the mathematical model for a uniform or non-uniform beam with various BCs and carrying various concentrated elements by only adjusting the magnitudes of cross-sectional area and length of each beam segment and those of the concentrated elements (such as the lumped mass mi with eccentricity ei and rotary inertia Ji, the translational spring with stiffness kt,i and/or the rotational spring with stiffness kr,i) attached to each node. The reliability of the presented results has been confirmed by comparing them with those of the existing literature or the conventional finite element method (FEM) and good agreement is achieved.