Nonlinear dynamic response of beam and its application in nanomechanical resonator

Abstract

Nonlinear dynamic response of nanomechanical resonator is of very important characteristics in its application. Two categories of the tension-dominant and curvaturedominant nonlinearities are analyzed. The dynamic nonlinearity of four beam structures of nanomechanical resonator is quantitatively studied via a dimensional analysis approach. The dimensional analysis shows that for the nanomechanical resonator of tension-dominant nonlinearity, its dynamic nonlinearity decreases monotonically with increasing axial loading and increases monotonically with the increasing aspect ratio of length to thickness; the dynamic nonlinearity can only result in the hardening effects. However, for the nanomechanical resonator of the curvature-dominant nonlinearity, its dynamic nonlinearity is only dependent on axial loading. Compared with the tension-dominant nonlinearity, the curvature-dominant nonlinearity increases monotonically with increasing axial loading; its dynamic nonlinearity The project was supported by the National Natural Science Foundation of China (10721202 and 11023001) can result in both hardening and softening effects. The analysis on the dynamic nonlinearity can be very helpful to the tuning application of the nanomechanical resonator.

Tunable Micro- and Nanomechanical Resonators

“…The governing equation can be given by [ 20 , 181 ]: where the actual axial loading and consists of the following two parts: where is the axial loading depending on the built-in strain [ 15 ], fabrication process [ 200 ], residual stress [ 201 ], temperature [ 202 ], and surface stress [ 203 ]. The second part is the tension due to nonlinear mid-plane stretching [ 15 , 204 ]. The resonant frequency can be obtained by employing Rayleigh’s energy method as [ 183 ]: where is a mode-dependent coefficient and satisfies , in which represents an approximate shape function for a particular mode .…”

Determining the effects of surface elasticity and surface stress by measuring the shifts of resonant frequencies

“…The finite deformation effects can be important in some cases [26]. The higher-order terms of strain owing to finite deformation will result in a nonlinear governing equation such as the Duffing equation [47]. [49].…”

“…The finite deformation effects can be important in some cases [26]. The higher-order terms of strain owing to finite deformation will result in a nonlinear governing equation such as the Duffing equation [47]. [48] mode shape of 4(ξ 2 /2 − ξ 3 /3 + ξ 4 /12) is also presented for comparison (the factor of 4 is to normalize the shape).…”