On the Reduced-order Modeling of Energy Harvesters

This work addresses the accuracy and convergence of reduced-order models (ROMs) of energy harvesters. Two types of energy harvesters are considered, a magnetostrictive rod in axial vibrations and a piezoelectric cantilever beam in traverse oscillations. Using generalized Hamilton's principle, the partial differential equations (PDEs) and associated boundary conditions governing the motion of these harvesters are obtained. The eigenvalue problem is then solved for the exact eigenvalues and modeshapes. Furthermore, an exact expression for the steady-sate output power is attained by direct solution of the PDEs. Subsequently, the results are compared to a ROM attained following the common Rayleigh-Ritz procedure. It is observed that the eigenvalues and output power near the first resonance frequency are more accurate and has a much faster convergence to the exact solution for the piezoelectric cantilever-type harvester. In addition, it is shown that the convergence is governed by two dimensionless constants, one that is related to the electromechanical coupling and the other to the ratio between the time constant of the mechanical oscillator and the harvesting circuit. Using these results, some interesting conclusions are drawn in regards to the design values for which the common single-mode ROM is accurate. It is also shown that the number of modes necessary for convergence should be obtained at maximum electric loading for the piezoelectric harvester and at minimum electric loading in the magnetostrictive case.