Noisy contact interactions of multi-layer mechanical structures coupled by boundary conditions

Abstract

In this work mathematical models of temporal part of chaos at chosen spatial locations of a plate locally reinforced by ribs taking into account an interplay of their interactions are derived and studied numerically for the most relevant dynamical parameters. In addition, an influence of the additive external noise on chaotic vibrations of multi-layer beam–plate structures coupled only by boundary conditions is investigated. We illustrate and discuss novel nonlinear phenomena of the temporal regular and chaotic contact/no-contact dynamics with the help of Morlet wavelets and Fourier analysis. We show how the additive white noise cancels deterministic chaos close to the boundary of chaotic region in the space of parameters, and we present windows of on/off switching of the frequencies during the contact dynamics between structural members.In order to solve the mentioned design type nonlinear problem we apply methods of qualitative theory of differential equations, the Bubnov–Galerkin method in higher approximations, the Runge–Kutta methods of 4th, 6th and 8th order, as well as the computation and analysis of the largest Lyapunov exponent (Benettin׳s and Wolf׳s algorithms are used). The agreement of outcomes of all applied qualitatively different numerical approaches validate our simulation results. In particular, we have illustrated that the Fourier analysis of the studied mechanical structures may yield erroneous results, and hence the wavelet-based analysis is used to investigate chaotic dynamics in the system parameter space.