# A Framework to Determine the Upper Bound on Extractable Power as a Function of Input Vibration Parameters

John D. Heit and Shad Roundy

## Abstract

This paper outlines a mathematical framework to determine the upper bound on extractable power as a function of the forcing vibrations. In addition, the method described provides insight into the dynamic transducer forces required to attain the upper bound. The relationship between vibration parameters and transducer force gives a critical first step in determining the optimal transducer architecture for a given vibration source. The method developed is applied to three specific vibration inputs: a single sinusoid, the sum of two sinusoids, and a single sinusoid with a time-dependent frequency. As expected, for the single sinusoidal case, the optimal transducer force is found to be that produced by a resonant linear spring and a viscous damping force, with matched impedance to the mechanical damper. The resulting transducer force for the input described by a sum of two sinusoids is found to be inherently time dependent. The upper bound on power output is shown to be twice that obtainable from a linear harvester centered at the lower of the two frequencies. Finally, the optimal transducer force for a sinusoidal input with a time-dependent frequency is characterized by a viscous damping term and a linear spring with a time-dependent coefficient.

Funding statement: Funding: Funding for this research was provided by the National Science Foundation under Award Number ECCS 1342070. The authors would also like to gratefully acknowledge the contributions of Dr. Fernando Guevara-Vasquez and Prof. Andrej Cherkaev of the Mathematics Department at the University of Utah.

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Published Online: 2015-3-28
Published in Print: 2016-1-1