Internal resonances of nanorods in presence of surface energy effect: Nonlinear torsional vibration

Abstract

In this paper, effects of the surface energy on the nonlinear torsional vibrations and internal resonances of nanorods are investigated. To this end, the second-order term with respect to angle of rotation is considered for the displacement field. In addition, the geometrically nonlinear strain–displacement relations are obtained based on the von-Kármán theory. Then, Hamilton’s principle is implemented to derive the governing equation of motion for torsional vibration of nanorod. In the governing equation of motion, the surface energy parameters are included by the surface elasticity theory. The multiple-scale method is employed to solve the governing equation of motion for fixed-free and fixed-fixed end conditions. The nanorod considered is made of aluminum and silicon because of different values of their surface parameters. The effect of surface energy parameters on the torsional frequencies of nanorods is investigated for different values of length, radius, frequency number, and amplitude of the nonlinear vibrations. In addition the cases of occurring, the internal resonances are reported. The results obtained in this research may be useful for better design of nanoelectromechanical devices such as nanobearings and rotary servomotors.