Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness

Rustamkhan Abdikarimov,Dadakhan Khodzhaev,Marco Amabili,Nikolai Ivanovich Vatin

doi:10.3390/app11136029

Abstract

The research object of this work is an orthotropic viscoelastic plate with an arbitrarily varying thickness. The plate was subjected to dynamic periodic load. Within the Kirchhoff–Love hypothesis framework, a mathematical model was built in a geometrically nonlinear formulation, taking into account the tangential forces of inertia. The Bubnov–Galerkin method, based on a polynomial approximation of the deflection and displacement, was used. The problem was reduced to solving systems of nonlinear integrodifferential equations. The solution of the system was obtained for an arbitrarily varying thickness of the plate. With a weakly singular Koltunov–Rzhanitsyn kernel with variable coefficients, the resulting system was solved by a numerical method based on quadrature formulas. The computational algorithm was developed and implemented in the Delphi algorithmic language. The plate’s dynamic stability was investigated depending on the plate’s geometric parameters and viscoelastic and inhomogeneous material properties. It was found that the results of the viscoelastic problem obtained using the exponential relaxation kernel almost coincide with the results of the elastic problem. Using the Koltunov–Rzhanitsyn kernel, the differences between elastic and viscoelastic problems are significant and amount to more than 40%. The proposed method can be used for various viscoelastic thin-walled structures such as plates, panels, and shells of variable thickness.

Highlights

Thin-walled plates and shells of variable thickness are widely used in various technology fields due to the requirements for strength, durability, and thin-walled structural elements’ exterior appearance
An analytical–numerical method was used to study the nonlinear vibrations of an orthotropic rectangular plate of variable thickness under the action of a uniformly distributed vibration load leading to parametric resonance
The solution of the system (1)–(3) is obtained by using the Bubnov–Galerkin method based on a polynomial approximation of the deflection and displacements; discretization is performed in spatial variables at each moment and is reduced to solving systems of nonlinear integrodifferential equations with weakly singular kernels with variable coefficients

Summary

Introduction

Thin-walled plates and shells of variable thickness are widely used in various technology fields due to the requirements for strength, durability, and thin-walled structural elements’ exterior appearance. The study of plates and shells of variable thickness made from homogeneous and inhomogeneous materials is a challenging task, and sometimes it encounters insurmountable difficulties. The difficulties are due to the solution of rather cumbersome equations, which are obtained by the desire to reflect, in mathematical modeling, the real mechanical behavior of the thin-walled structure. Variable thickness plate problems involve computational complexity, i.e., the lack of suitable universal numerical methods for solving the obtained equations, and as a consequence, unified computational algorithms. Studies of parametric vibrations of thin-walled structures have become a separate research area in the mechanics of deformable solids. Thin-walled structures have versatile applications for various mechanical systems, in particular for plates and shells. The solution of dynamic stability problems includes the derivation of the equation of motion, discretization, and determination of areas of dynamic instability of structures

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