Quantifying chaotic dynamics of nanobeams with clearance

Abstract

We investigate chaotic dynamics and contact interaction of geometrically nonlinear (according to the Kármán model) beam nanostructures with a small clearance. Contact interaction is described by the Kantor’s model. The constructed mathematical models are based on the first-order (Euler–Bernoulli) and the second-order (Timoshenko) kinematic approximations with the help of the modified couple stress theory. The fundamental governing dynamical equations of two-layer beam nanostructures are yielded by Hamilton’s principle.The paper substantiates the validity of the obtained solutions for the problem of contact interaction of two nanobeams, subject to different kinematic hypotheses. When solving problems by numerical methods, errors accumulate, which are very often mistaken for chaotic oscillations. In this paper, we propose a methodology for identifying “true” chaos for mechanical beam size-dependent structures and obtaining reliable results.The governing PDEs are reduced to ODEs by finite difference method (FDM) of the second-order and the Bubnov–Galerkin method in higher approximations. The obtained ODEs are solved by the Runge–Kutta and Newmark methods. Results of convergence versus the number of partition points along the spatial coordinate in the finite difference method, the number of series members in the Bubnov–Galerkin​ method and time steps are investigated. The revealed chaotic/hyper-chaotic vibrations are studied by nonlinear dynamics methods, wavelet analysis, and the spectrum analysis of the Lyapunov exponents. The numerical methods are validated by estimating the temporal and spatial convergence, while the reliability of the obtained solution is validated by the Lyapunov exponents obtained through the analysis of the results from four different qualitative methods. The numerical solution is obtained by several methods at each simulation step. The influence of control parameters (the transverse harmonic load and the size-dependent parameter) on the two-layer beam vibrations is reported.