Dividing of the Full Reaction of the Additional Support Contacting With the Plate Into Viscous, Elastic and Inertial Components

Abstract

An original approach for dividing the reaction of a viscoelastic support into inertial, viscous and elastic components is proposed to assess the effect of various characteristics of additional supports on the deformed state of structural elements. The effectiveness of the proposed approach was tested for a mechanical system consisting of a rectangular isotropic plate of medium thickness, hinged-supported along the contour, and an additional concentrated viscoelastic support, taking into account its mass-inertial characteristics. The deformation of the plate is considered within the framework of Timoshenko's hypotheses. Vibrations of the plate are caused by the applying of an external non-stationary loading. The influence of the additional support is modeled by three independent non-stationary concentrated forces. The paper presents the main analytical relations for obtaining a system of three integral Volterra equations, which is solved numerically and analytically. After performing discretization in time, the system of integral equations is transformed into a system of matrix equations. The resulting system of matrix equations is solved using the generalized Cramer algorithm for block matrices and the Tikhonov regularization method. We point out that the material described is applicable to other objects that have additional supports (beams, plates and shells, which can have different supports along the contour and different shapes in plan). The results of a numerical experiment to determine the components (viscous, elastic and inertial) of the full reaction to the plate, arising due to the presence of an additional support, are presented. The reliability of the proposed approach is confirmed by the coincidence of the results of comparing the reactions found by two methods: numerical-analytical for one complete reaction, as in work [1], and numerical for the full reaction (obtained by adding three components).