Abstract
Geometrically nonlinear mathematical model of the problem of parametric oscillations of a viscoelastic orthotropic plate of variable thickness is developed using the classical Kirchhoff-Love hypothesis. The technique of the nonlinear problem solution by applying the Bubnov-Galerkin method at polynomial approximation of displacements (and deflection) and a numerical method that uses quadrature formula are proposed. The Koltunov-Rzhanitsyn kernel with three different rheological parameters is chosen as a weakly singular kernel. Parametric oscillations of viscoelastic orthotropic plates of variable thickness under the effect of an external load are investigated. The effect on the domain of dynamic instability of geometric nonlinearity, viscoelastic properties of material, as well as other physical-mechanical and geometric parameters and factors are taken into account. The results obtained are in good agreement with the results and data of other authors.
Highlights
Construction of the dynamic instability domain of thin-walled elastic systems is a very urgent and important task; a multitude of publications are devoted to this issue, of which, first of all, the studies carried out in [1,2,3,4,5,6] should be mentioned
Belyaev [1], parametric oscillations become the subject of numerous studies in application to various mechanical systems with distributed parameters, in particular, to rods, plates and shells
Assume that the plate is subjected to an external dynamic load acting along the edge a, having a periodic character: P(t) P0 P1 cos t
Summary
Construction of the dynamic instability domain of thin-walled elastic systems is a very urgent and important task; a multitude of publications are devoted to this issue, of which, first of all, the studies carried out in [1,2,3,4,5,6] should be mentioned. Beginning with the article by N.M. Belyaev [1], parametric oscillations become the subject of numerous studies in application to various mechanical systems with distributed parameters, in particular, to rods, plates and shells. In the fundamental monograph V.V. Bolotin [2] has analyzed in detail the occurrence of parametric resonance in elastic thin-walled systems. Schmidt [6] contains a detailed review of many aspects concerning these issues
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