Simulation of Intermediate Viscoelastic Support Under Non-Stationary Vibrations of Timoshenko Beams

Abstract

When modeling mechanical objects and their systems, mathematical models developed for an elastic domain are most often used, which in some cases can lead to significant inaccuracies in calculations. The use of mathematical models that take into account visco-elastic properties or energy dissipation allows to obtain more realistic models, which will make it possible to obtain more accurate calculation results. In the paper non-stationary loading of a mechanical system, consisting of a beam hinged at the edges, and an additional support installed in the span of the beam is considered. We use a beam deformation model, which is based on the hypotheses of S. P. Timoshenko and takes into account the inertia of rotation and shear. The system of partial differential equations describing the deformation of the beam is solved by expanding the desired functions into the corresponding Fourier series and subsequent using the integral Laplace transform. The additional support is assumed to be realistic rather than absolutely rigid, having linear elastic and viscous components. It is assumed that at the point of attachment of the additional support to the beam their displacements coincide. The reaction between the beam and the additional support is replaced by an external unknown concentrated force applied to the beam, which changes with time. The law of change in time of this unknown reaction is determined from the solution of the Volterra integral equation. A method for obtaining an integral equation for an unknown reaction is described. Analytical relations and calculation results for specific numerical parameters are given. The influence of stiffness and viscosity on the determined reaction of the additional support, as well as on the deflections of the beam at various points, is investigated. The results obtained can also be used for damping forced vibrations of mechanical systems.

Modeling a Viscoelastic Support Considering Its Mass-Inertial Characteristics During Non-Stationary Vibrations of the Beam

“…The results of the present paper are based on those for non-stationary deforming of mechanical systems consisting of beams and plates having concentrated viscoelastic supports additional to the main support along its edge. Solutions to direct problems for beams and plates with additional supports are found in [19,20].…”