Quasi-resonance Behaviour of Viscoelastic Rods Subjected to Free and Forced Longitudinal Vibrations

Abstract

A classical viscoelastic rod subjected to longitudinal vibrations is considered. The boundary conditions are so that the left end of the rod is fixed and the right end is free. Moreover it is assumed that the rod is growing and hence, its length is changing in time. The function of growth is assumed to be twice continuously differentiable with respect to time. A particular case of rod's growth proportional to time is of special interest. The change of variables is introduced so that in new variables the rod length becomes constant. The new partial differential equation describing the rod's dynamics is derived in these variables. This equation is simplified using assumptions on slow rate growth constant and small viscoelastic damping factor. A special representation of solution is introduced which uses eigenfunctions of the generating problem, when growth and damping are neglected, and satisfy the boundary conditions. By means of this representation the governing partial differential equation is converted in infinite system of ordinary differential equations. It is shown that solutions of truncated systems converge to solution of the original system of equations. Three major problems of the growing rod vibrations are formulated and solved. In the first problem free undamped vibrations are considered. It is shown that at linear growth of rod length amplitudes of all its modes are also growing linearly in time. The simplified model neglecting the modes cross-coupling is composed for the explanation of this effect. The corresponding differential equation is solved exactly in elementary functions and it is shown that amplitudes of vibration of any modes grow linearly and almost-periods of vibrations have logarithmic dependence on time. In the second problem free damped vibrations of linearly growing rod are considered. It is shown that time behaviour of the rod has two characteristic domains: in the first the vibration amplitudes decay exponentially due to domination of the viscoelastic damping effects; in the second domain these amplitudes start to grow linearly in time due to domination of the effects considered in the first problem. The simplified model describing this effect and neglecting the modal cross-coupling is developed. The exact solution of the corresponding differential equation is obtained in the confluent hypergeometric functions, which qualitatively explain the abovementioned behaviour of the rod. In the third problem the forced damped vibrations of the rod are considered. It is shown that at fixed frequency of excitation the resonant effects are manifested subsequently in all modes of the rod in the process of its growing.