Mathematical modeling of MEMS elements subjected to external forces, temperature and noise, taking account of coupling of temperature and deformation fields as well as a nonhomogenous material structure

Abstract

In this work, a mathematical model describing the nonlinear dynamics of a plate-beam system is proposed. The model takes account of coupling between temperature and deformation fields as well as external mechanical, temperature, and noise excitation. It considers a system of integro-differential equations of a hyperbolic-parabolic type and of different dimensions. Both the proof of existence and uniqueness of the solution to the problem and the proof of convergence of the Faedo–Galerkin method used for solving the problem are given. The obtained mathematical model governs the work of the members of the micromechanical system. Algorithms aimed at solving the plate-beam structures, constructed based on the Faedo–Galerkin method in higher approximations as well as the 2nd and 4th order finite difference method (FDM), are employed to reduce PDEs to the Cauchy problem. The last is solved by Runge–Kutta methods of different types. In order to define the character of vibrations of the plate-beam structure, the sign of the largest Lyapunov exponent (LLE) is analyzed with the help of Wolf, Kantz, and Rosenstein methods. This complex approach allows one to obtain reliable results and true chaotic orbits. In addition, a few computational examples are provided.