Onset of chaos in nano-resonators based on strain gradient theory: Numerical analysis

Abstract

In recent years, experimental and theoretical studies have shown that size effect is non-negligible in mechanical micro- and nano-structures. As a result, classical continuum theory cannot model these structures. To accurately predict the behavior of micro- and nano-structures, non-classical continuum theories should be used. In this paper, the effect of size on the chaotic region of resonators is studied by comparing the results of classical and strain gradient theories. A bifurcation diagram and Lyapunov exponent are used for detecting chaos in nano-resonators.It is shown that the effect of size on the chaotic region in bifurcation diagram varies significantly depending on the bias voltage. Lyapunov exponent is utilized to validate the chaotic region and it is in best agreement with bifurcation diagram. For some DC voltage values, size effect can eliminate chaotic vibrations while in higher DC voltages, it can result in chaos at much lower amplitudes than those predicted by classical theories. Dynamic behaviors of nano-resonators are categorized based on their bias voltages and their chaotic behaviors are analyzed based on classical and non-classical theories in detail. The obtained numerical results are compared with Melnikov and Maximum Velocity methods and it is shown that all methods predict the lower critical actuation amplitude for strain gradient theory.