Nonlinear dynamics of the contact interaction of a three-layer plate-beam nanostructure in a white noise field

Abstract

The mathematical model of the contact interaction of a multilayer nanostructure consisting of two nanoplates and nanobeams between them with small gaps was constructed for the first time. A modified moment theory is used to describe the size-dependent effects of a real nanostructure. The upper and lower layers are nanoplates, obeying Kirchhoff’s kinematic hypothesis, and the middle layer is the Euler-Bernoulli nanobeam. Contact interaction is accounted for by the model B.Ya. Cantor. Nanoplates and nanobeam are isotropic, elastic, and they are connected through boundary conditions. The effect of the gap between the layers and the noise field is studied. To solve and analyze these constructively nonlinear problems, the methods of the qualitative theory of differential equations, wavelet analysis, and methods for analyzing the sign of the largest Lyapunov exponent are used. The differential equations system reduces to the Cauchy problem the Bubnov-Galerkin method in higher approximations and finite difference methods with approximation O(h2) and O(h4) with respect to the spatial coordinate. Next, the Cauchy problem is solved by the Runge-Kutta methods of the 4th, 6th, 8th accuracy order in time. The analysis showed that the gap size essentially depends on the interaction of the elements of the multilayer nanosystem and on the nature of their complex oscillations. Also, the presence of a noise field involves the contact interaction of elements that were at rest with the previous values of the remaining parameters.