Nonlinear dynamics of cantilevered microbeams based on modified couple stress theory

The aim of this paper is to develop a new nonlinear theoretical model for cantilevered microbeams and to explore the nonlinear dynamics based on the modified couple stress theory, taking into account of one single material length scale parameter. The full nonlinear equation of motion, which is valid when the motion is large, is derived using the Hamilton’s principle. The governing partial differential equation is further discretized with the aid of Galerkin’s method. The numerical results, in which the existence of primary resonances of the first mode of the microbeam due to base excitations is demonstrated, are presented in the form of frequency–response curves, phase portraits and time histories. For a cantilevered microbeam subjected to harmonic base excitations, it is found that the frequency–response curve exhibits a clear softening-type behavior. For the same system but with an intermediate linear spring support, it is shown that the linear spring is capable of increasing the resonance frequency and decreasing the resonance amplitudes of the microbeam. Interestingly, it is found that the softening behavior could be changed to a hardening one if an intermediate nonlinear spring is added somewhere along the microbeam’s length.