Forced vibration analysis of multi-cracked Timoshenko beam with the inclusion of damping by virtue of Green's functions

This paper concentrates on obtaining the closed-formed analytical solutions of the steady-state forced vibration of the multi-cracked Timoshenko beams with damping effects. Varying from the Euler-Bernoulli beam, the mechanical properties of a cracked section of the Timoshenko beam are described as a rotational spring and a transversal spring models. Initially, the Green's function of a one-cracked simple supported Timoshenko beam is derived. With the assistant of the solution of the one-cracked situation and the transfer matrix method, the Green's functions of the multi-cracked Timoshenko beams for various boundary conditions are also obtained. The present Green's functions can be degenerate to the Rayleigh and Euler-Bernoulli cases, and the multiple cracked-models can be reduced to the single cracked cases when the depths of some cracks are specified to be zero. In the numerical section, the FEM results and the solutions in some references are used to validate the present solutions. Some numerical examples in this research are to reveal the influences of the shearing effect on the vibration performances of the one-cracked Timoshenko model as well as the multiple cracks cases. Additionally, the effects of some significant physical parameters, like the crack depth and crack locations, are also discussed in some numerical examples. (C) 2019 Elsevier Ltd. All rights reserved.