Free and forced vibrations of a Timoshenko beam with any number of translational and rotational springs and lumped masses

Abstract The free vibration and the forced dynamic responses of a uniform cantilever Timoshenko beam carrying any number of concentrated masses, translational and rotational springs and subjected to various external loadings are studied by means of the analytical‐and‐numerical‐combined method (ANCM), where the locations and the magnitudes of the masses, springs and loadings are arbitrary. First of all, the (exact) closed‐form solutions for the natural frequencies and the normal mode shapes of the ‘unconstrained’ Timoshenko beam (without carrying any concentrated elements) are determined; then the eigenvalue equation for free vibration analysis and the equation of motion for forced vibration analysis of the ‘constrained’ Timoshenko beam (carrying the prescribed concentrated elements) are derived analytically by applying the expansion theorem and the mode superposition methodology; finally the approximate natural frequencies and mode shapes of the ‘constrained’ Timoshenko beam and its dynamic responses due to external excitations are calculated numerically. It is found that the ANCM presented in the paper has the advantages of both the pure analytical method (to be able to save computing time) and the pure numerical method (to be able to solve various practical problems).