Dynamic instability of Euler-Bernoulli nanobeams subject to parametric excitation

Little research on the dynamic instability of nanobeams caused by parametric resonance has been reported in the literature. This paper presents an accurate and analytical method for investigating the dynamic instability of nanobeams based on the nonlocal continuum mechanics. The governing equation of transverse vibration of nanobeams subject to axial dynamic load is derived using the Hamilton's principle and the nonlocal theory to establish the Mathieu-Hill equation of dynamic stability, based on which the equations of critical excitation frequencies are derived by the Bolotin's theory to determine the regions of dynamic instability. Especially, the matrix singularity problem encountered in solving the equations of critical frequencies is overcome by matrix transformation. For verifying the accuracy of obtained regions of dynamic stability, the dynamic responses of nanobeams are computed by the fourth-order Runge-Kutta approach. By comprehensively exploring the size dependence of dynamic stability of nanobeams, it is found that the size scale parameter influences the regions of dynamic instability mainly through the nonlocal natural frequency and the nonlocal Euler buckling load. As the size scale parameter increases, the nonlocal natural frequency and the nonlocal buckling load decrease, which leads to the reduction of the value and bandwidth of critical excitation frequencies. Moreover, the size effects are found to decrease with an increase of length of nanobeam.