Electromechanical modeling and nonlinear analysis of axially loaded energy harvesters

To maximize the electromechanical transduction of vibratory energy harvesters, the resonance frequency of the harvesting device is usually tuned to the excitation frequency. To achieve this goal, some concepts call for utilizing an axial static preload to soften or stiffen the structure (Leland and Wright, 2006, "Resonance Tuning of Piezoelectric Vibration Energy Scavenging Generators Using Compressive Axial Preload," Smart Mater. Struct., 15, pp. 1413-1420; Morris, 2008, "A Resonant Frequency Tunable, Extensional Mode Piezoelectric Vibration Harvesting Mechanism," Smart Mater. Struct., 17, p. 065021). For the most part, however, models used to describe the effect of the axial preload on the harvester's response are linear lumped-parameter models that can hide some of the essential features of the dynamics and, sometimes, oppose the experimental trends. To resolve this issue, this study aims to develop a comprehensive understanding of energy harvesting using axially loaded beams. Specifically, using nonlinear Euler-Bernoulli beam theory, an electromechanical model of a clamped-clamped energy harvester subjected to transversal excitations and static axial loading is developed and discretized using a Galerkin expansion. Using the method of multiple scales, the general nonlinear physics of the system is investigated by obtaining analytical expressions for the steady-state response amplitude, the voltage drop across a resistive load, and the output power. These theoretical expressions are then validated against experimental data. It is demonstrated that in addition to the ability of tuning the harvester to the excitation frequency via axial load variations, the axial load aids in (i) increasing the electric damping in the system, thereby enhancing the energy transfer from the beam to the electric load, (ii) amplifying the effect of the external excitation on the structure, and (iii) enhancing the effective nonlinearity of the device. These factors combined can increase the steady-state response amplitude, output power, and bandwidth of the harvester.