Parametric resonances of nonlinear piezoelectric beams exploiting in-plane actuation

A nonlinear Euler–Bernoulli model of slender piezoelectric beams is employed to investigate parametric resonance motions driven by a pulsating voltage with a DC component. The beam model is based on 3D electric charge conservation and 1D reduction of the momentum balance laws assuming as space coordinate the arclength along the beam base line as space coordinate. The 3D constitutive relationships for a piezoelectric material are specialized according to the Euler–Bernoulli ansatz of transverse unshearability. The nonlinear piezoelastic problem of the parametrically excited beam is directly attacked by the method of multiple scales up to the fifth nonlinear order overcoming modal projection drawbacks and severe restrictions on the oscillation amplitude range. The transition curves, separating regions of stable and unstable trivial solutions in the voltage frequency–amplitude plane, are obtained for various modes of a PVDF beam. The onset of parametric resonances and the post-critical large motion are investigated upon variations of meaningful parameters such as the prestress DC voltage amplitude. The analytical outcomes are compared with those provided by the finite element code ABAQUS which yields the numerical solution of the full 3D nonlinear piezoelectric problem. Moreover, the frequency–response curves to within fifth order are computed applying the pseudo-arclength method to the algebraic frequency–response equation. The frequency– response curves of the lowest few modes reveal new aspects of the parametric resonance motions which can be exploited in applications such as dynamic morphing of thin surfaces.