In this work the chaotic dynamics of flexible curvilinear Euler–Bernoulli micro-beams embedded into a stationary temperature field is investigated. The temperature field is modelled based on a the Duhamel–Neumann theory and is free from the restrictions on the temperature field distribution along beam thickness. The von Kármán geometric strain–stress relations are employed. The governing nonlinear PDEs are yielded by the Hamilton principle with an account of the modified couple stress theory. The finite dimension problem is truncated to a finite system of nonlinear ODEs using the finite difference method (FDM) and then the Cauchy problem is solved with a help of the Runge–Kutta method. Action of the 2D thermal field is defined by solution to the heat transfer PDE which is also solved by FDM of the second order of accuracy. The so-called charts of vibration regimes (amplitude-frequency planes) are constructed. In particular, novel features of nonlinear (chaotic) dynamics versus the change of the magnitude of the size (length) dependent parameter are reported. The carried out numerical analysis is supported by the monitoring of frequency power spectra based on the fast Fourier transform (FFT), phase space projections, Poincaré maps and LLEs (largest Lyapunov exponents). We have also analyzed system chaotic dynamics of the classical and size-dependent beam models versus series of values of the two control parameters, i.e. beam curvature and its size-dependent length. Moreover, we have detected and illustrated the novel scenarios of transition form regular to chaotic vibrations of the studied beams, governed by non-linear PDEs.
|Journal||ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik|
|Publication status||Published - Jan 2021|
- Euler–Bernoulli micro-beams
- Lyapunov exponent
ASJC Scopus subject areas
- Computational Mechanics
- Applied Mathematics