Mean square responses of a viscoelastic Timoshenko cantilever beam with different damping mechanisms

Abstract

In this paper, considering the moment of inertia and shear deformation, the partial differential equation sets governing the vibration of viscoelastic Timoshenko beam subjected to random excitations were derived. The excitation form is the concentrated force, and its random characteristic is the ideal white noise in the time domain. Two damping mechanisms, namely, the external viscous damping and the internal viscoelastic damping (Kelvin-Voigt model), are both taken into account simultaneously. Comparing with the existing literature, there are two improvements: on the one hand, considering the interaction of the rotational inertia and shear deformation, that is, there is a fourth derivative term for time in the partial differential equation sets of Timoshenko beam, while the Bresse-Timoshenko truncated model ignores the fourth derivative term of time. On the other hand, by the method of residue integral, the infinite integral in the mean response is transformed into the expression of the modal damping ratio and the natural frequency. The exact solution of the mean square response is obtained in the form of an infinite series finally. Numerical example is supplied, and the numerical results acquired verify the validity of the theoretical analysis.