Parametric Vibrations of Viscoelastic Rectangular Plates with Concentrated Masses

Abstract

AbstractIn modern technology, construction, and mechanical engineering, thin-walled composite structures such as plates, panels, and shells are widely used. Often, various devices, assemblies, and overlays are attached to such structures. The structures can be subjected to various loads, both static and dynamic. Among such loads, periodic loads take a special place. Therefore, the study of the dynamic behavior of plates, panels, and shells with concentrated mass is of great importance. Parametric vibrations of isotropic viscoelastic rectangular plates with concentrated masses were considered. The equation of motion was obtained on the basis of the Kirchhoff-Love hypothesis, taking into account geometric nonlinearity. The hereditary Boltzmann-Volterra theory was used to describe the viscoelastic properties of the plate material. The effect of concentrated mass was taken into account using the Dirac delta function. The discretization of the system of equations was realized using the Bubnov-Galerkin method and was reduced to a system of ordinary nonlinear integro-differential equations of the Volterra type. In this case, the weakly singular Koltunov-Rzhanitsyn kernel was used as a relaxation kernel. To solve the resulting system, a numerical method based on the use of quadrature formulas was applied. The graphs of the frequency-amplitude response of the oscillations were constructed. The influence of the viscoelastic properties of the plate material, concentrated mass, geometric nonlinearity on the eigen frequencies was investigated.KeywordsViscoelastic rectangular plateConcentrated massesGeometric nonlinearityPeriodic loadParametric oscillationsBubnov-Galerkin method numerical method