Surface effects on the quasi-periodical free vibration of the nanobeam: semi-analytical solution based on the residue harmonic balance method

Abstract

The study of surface effects on quasi-periodical vibration is significant because of its promising applications in sensors of nano/micro-electro-mechanical systems (N/MEMS). In this study, the surface effects parameters are adopted from the atomistic calculations. The governing equation of the nanobeam considering the nonlinear curvature and surface effects is established. Considering the first two modes, the Garlerkin method is used to translate the partial differential equation into ordinary differential equations. The multi-level residue harmonic balance method (RHBM) with two time variables is developed to solve these ODEs. The influences of surface and aspect ratio on the two frequencies and corresponding amplitudes are analyzed. Both of the frequency and amplitude discrepancies of the nanobeam to the counterparts of the classical beam become greater when the height of the beam shrinks at nanoscale. For the beam with the same height, both of the first- and second-frequencies decrease with the increase of the aspect ratio, and for the beams with the same aspect ratio, they decrease with the increase of the height. The solution of RHBM fits well with the molecular dynamics and Runge–Kutta numerical simulations. The study provides insight into the mechanism of the nonlinear dynamics of nanostructures, and shed light on quantitative design of the elements in N/MEMS, sensors, actuators, and resonators.