Characterizing harmonic and subharmonic solutions of the bi-stable piezoelectric harvester with a modified Harmonic Balance approach

Abstract

In this study, broadband frequency-response of a bi-stable piezo-magneto-elastic harvester is extended by analytically characterizing many subharmonic-n solutions. The energy principle-based model considers the continuous analytical mode-shapes of a compound beam, rotational inertia of the tip mass, and nonlinear geometry effect. A modified higher-order Harmonic Balance (HB) approach proposed here is effective not only in characterizing the frequency-responses of harmonic-1 and subharmonic-n solutions, but also in estimating the chaotic response zones of any multi-stable harvester. Balancing between the computational cost and solution convergence, intra-well (IW) and cross-well (CW) period-n solutions of general-type upto period-4 and odd-type upto period-7 are characterized. Effectiveness and convergence of the modified HB approach are demonstrated through good qualitative and quantitative agreements observed between the numerical sweep and HB results. Dynamical behaviors of several period-n solutions from different branches are revealed through the self-evident plots of displacement and voltage time histories, frequency spectra, and phase-portraits. Novel observations regarding the role of various sub- and super-harmonics in producing only CW orbits and mixed-orbits; and subsequently, asymmetric solution branches appearing due to the nonlinear geometry effect are reported. Besides the customary voltage and power output analyses, the assessment of maximum and average strains induced in the piezoelectric material is presented. Such an assessment has proved quite useful in identifying the alternate low-strain inducing and next best power-producing solutions to harmonic-1 responses. The characterized subharmonic-n solutions produce 20μW–90 mW of RMS power and span the entire range of 1–30 Hz considered in this investigation.