Nonlinear dynamics of contact interaction of MEMS beam elements accounting the Euler-Bernoulli hypothesis in a temperature field

In this paper is proposed the methodology for determining the true chaos from the position of nonlinear dynamics for distributed mechanical systems in the form of a beam structure of two beams described by the kinematic firstapproximation hypothesis (Euler-Bernoulli). There is a small gap between the beams. The lower beam (beam 2) will be considered as an elastic base for the upper beam (beam 1). The transverse distributed alternating load acts on beam 1. The contact interaction of the beams was taken into account by the Cantor model. This problem has a great nonlinearity due to the account of the geometric nonlinearity of beams according to T. von Karman and the constructive nonlinearity of the structure due to the contact interaction. Differential equations in partial derivatives reduce to the ODE system by the finite differences method (FDM) of the second order accuracy. The resulting system is solved by Runge-Kutta methods of various accuracy orders. In general case, the problem solution essentially depends on the methods of reducing partial differential equations to ODEs and methods for solving the Cauchy problem, boundary and initial conditions. When solving the problem by the method of finite differences with approximation O(h²), the solution will depend on the number of points of the integration interval partition and the time step in solving the Cauchy problem. The analysis of the obtained results is carried out by the nonlinear dynamics methods and the qualitative theory of differential equations. The results are compared for geometrically linear and nonlinear beams, taking into account the contact interaction. The mechanical structure is considered as a system with an infinite number of freedom degrees. A complete coincidence of solutions is achieved, depending on the partitions number in the spatial coordinate in the chaos. The sign of the first Lyapunov exponent is determined by the Kantz, Wolf and Rosenstein methods.