Nonlinear free vibration analysis of pre-actuated isotropic piezoelectric cantilever Nano-beams

In this paper, nonlinear free vibration analysis of pre-actuated clamped-free isotropic piezoelectric Euler-Bernoulli nanobeams is discussed. The governing equations of motion are derived on the basis of size-dependent piezoelectricity theory. A more accurate model is developed for the large amplitude vibration analysis of piezoelectric cantliver nanobeams by the consideration of a higher order curvature-displacement relation. In this case, a nonlinear equation of motion is derived. Accordingly, the hardening or softening treatment dependency on the flexoelectric constant and the length scale parameter is examined for the considered nanocantilever. The assumed nano-beam is actuated by a constant voltage. The nonlinear free vibration analysis of pre-actuated nano-beam about the pre-static deformation is examined by Lindstedt-Poincare technique which is applied on the discretized equations of motion. A closed form relation is extracted for the nonlinear natural frequency and the corresponding effective nonlinearity. Some numerical analysis has been performed to peruse the effects of applied voltage, the length scale parameter and flexoelectric coefficient on the static deflection, the nonlinear natural frequencies and the associated effective nonlinearities. The outcomes demonstrate occurrence of very interesting phenomena in the combination of the various magnitudes of the length scale parameter and the flexoelectric coefficient.