Forced Vibration of the Non-Homogeneously Pre-Stressed System Consisting of the Hollow Cylinder and Surrounding Medium

Abstract

The paper deals with the study of the influence of the non-homogeneous pre-stresses in the system consisting of the hollow cylinder and surrounding elastic medium on the frequency response of this system caused by the time-harmonic ring load acting in the interior of the cylinder. The axisymmetric problem is considered and it is assumed that in the initial state, the system is compressed in the radial direction with homogeneously distributed static forces as a result of which, non-homogeneous pre-stresses (or initial stresses) appear in that. It is also assumed that after these pre-stresses appear in the interior of the cylinder, the point-located (with respect to the axial coordinate directed along the cylinder's central axis) additional time-harmonic ring load acts. Thus, it is required to determine how the pre-stresses influence the frequency response of the system to the noted additional time-harmonic loading. This influence, which is the non-linear effect, is studied within the scope of the so-called three-dimensional linearized equations and relations of the theory of elastic waves in pre-stressed bodies. The solution to the corresponding boundary value problems is found by employing the discrete analytical solution method and numerical results on the influence of the pre-stresses on the frequency response of the interface stresses which appear as a result of the action of the additional time-harmonic forces, are presented and discussed. In particular, it is established that the pre-stresses mentioned above cause to decrease the magnitude of the interface dynamic stresses appearing as a result of the additional time-harmonic loading.