Surface energy effect on nonlinear free axial vibration and internal resonances of nanoscale rods

Abstract

This study aims to investigate effects of the surface energy parameters on the nonlinear axial vibration and internal resonances of nanoscale rods. The surface energy parameters considered are the surface density and the surface Lamé constants. The nonlinear governing equation of motion is derived using the Hamilton's principle which includes the strain energy and the kinetic energy of the nanorod surface and bulk. The strain and kinetic energies of the nanorod bulk are obtained using the classical theory of elasticity and those of the nanorod surface are obtained using the surface elasticity theory. Then, the nonlinear governing equation of motion is solved using the method of multiple scales and natural frequencies are obtained for fixed-fixed and fixed-free boundary conditions. Effects of the surface energy parameters on the frequencies are examined for various cases. It is observed that the type of effect of the surface energy parameters depends on their sign. It is also seen that the type of effect of the surface energy parameters on nonlinear frequencies may become different at higher frequency numbers. The observations of this study can be a useful reference for more accurate design and fabrication of nano-electro-mechanical systems in which rod-shaped nano-structures, carbon nanotubes, play a significant role.