We construct a mathematical model of non-linear vibration of a beam nanostructure with low shear stiffness subjected to uniformly distributed harmonic transversal load. The following hypotheses are employed: the nanobeams made from transversal isotropic and elastic material obey the Hooke law and are governed by the kinematic third-order approximation (Sheremetev–Pelekh–Reddy model). The von Kármán geometric non-linear relation between deformations and displacements is taken into account. In order to describe the size-dependent coefficients, the modified couple stress theory is employed. The Hamilton functional yields the governing partial differential equations, as well as the initial and boundary conditions. A solution to the dynamical problem is found via the finite difference method of the second order of accuracy, and next via the Runge–Kutta method of orders from two to eight, as well as the Newmark method. Investigations of the non-linear nanobeam vibrations are carried out with a help of signals (time histories), phase portraits, as well as through the Fourier and wavelet-based analyses. The strength of the nanobeam chaotic vibrations is quantified through the Lyapunov exponents computed based on the Sano–Sawada, Kantz, Wolf, and Rosenstein methods. The application of a few numerical methods on each stage of the modeling procedure allowed us to achieve reliable results. In particular, we have detected chaotic and hyper-chaotic vibrations of the studied nanobeam, and our results are authentic, reliable, and accurate.
On the chaotic and hyper-chaotic dynamics of nanobeams with low shear stiffness
Note: This paper belongs to the Focus Issue, Recent Advances in Modeling Complex Systems: Theory and Applications.
T. V. Yakovleva, J. Awrejcewicz, V. S. Kruzhilin, V. A. Krysko; On the chaotic and hyper-chaotic dynamics of nanobeams with low shear stiffness. Chaos 1 February 2021; 31 (2): 023107. https://doi.org/10.1063/5.0032069
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