Parametric resonances of an electrically actuated piezoelectric nanobeam resonator considering surface effects and intermolecular interactions

This paper investigates the nonlinear dynamics of a parametrically excited doubly clamped piezoelectric nanobeam, actuated by a combined AC and DC loadings. Surface effects, intermolecular van der Waals forces, and fringing effects are incorporated in the nonlinear model. The governing equation of motion is obtained using the extended Hamilton principle. The reduced-order model equation (ROM) is obtained based on the Galerkin method. The multiple-scale method is applied directly to the nonlinear equation of motion and associated boundary conditions to obtain the nanobeam response analytically under small AC voltage loads. The influence of van der Waals forces, piezoelectric voltages, and surface effects is investigated on the natural frequencies, static equilibria, pull-in voltages, and principle parametric resonance (subharmonic resonance of order one-half) of the nanoresonator. It is shown the surface effect profoundly affects the nontrivial parametric responses, trivial stability zones, and bifurcation point’s loci, and it is necessary to consider the surface effects for accurate and exact investigation of the system response. The effect of piezoelectric voltage to control the dynamic instability region is also demonstrated. To validate analytical results, ROM equation is integrated numerically. It is seen that the perturbation results are in accordance with numerical results.