A harmonic balance approach with alternating frequency/time domain progress for piezoelectric mechanical systems

A semi-analytical approach based on harmonic balance (HB) method is proposed to predict steady-state responses of nonlinear piezoelectric mechanical systems. In order to handle complicated non-polynomial forces of the systems, the alternating frequency/time domain (AFT) progress is applied to establish a set of implicit algebraic equations of HB method. The stabilities of periodic responses are analyzed via Floquet theory. The proposed approach is successfully implemented a nonlinear piezoelectric energy harvester with a magnetic oscillator and a Duffing energy harvester with strong nonlinear electric terms. The HB solutions are well supported by the simulation results of the Runge-Kutta method. Besides, an experiment piezoelectric device is developed by a circular laminated plate, and an experimental model is established by system identifications. The model includes both the nonlinear stiffness and nonlinear damping which are characterized by the combined functions of polynomials and hyperbolic tangents. The model is analyzed via the proposed approach. The numerical simulations under different excitations show good agreements with the harmonic solutions. The frequency sweeps experimentally pursue the system has both softening and hardening nonlinearities which are verified by HB results. The proposed approach requires less analysis during the solving processes, and shows high accuracy to the strongly nonlinear piezoelectric mechanical systems, even the system with complicated nonlinearities.