Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter March 28, 2015

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

John D. Heit EMAIL logo and Shad Roundy


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.


Bryson, A. E.1975. Applied Optimal Control: Optimization, Estimation and Control. Boca Raton, Florida: CRC Press Inc.Search in Google Scholar

Daqaq, M. F.2010. “Response of Uni-Modal Duffing-Type Harvesters to Random Forced Excitations.” Journal of Sound and Vibration329 (18): 362131. doi:10.1016/j.jsv.2010.04.002Search in Google Scholar

Daqaq, M. F.2011. “Transduction of a Bistable Inductive Generator Driven by White and Exponentially Correlated Gaussian Noise.” Journal of Sound and Vibration330 (11): 255464. doi:10.1016/j.jsv.2010.12.005Search in Google Scholar

Daqaq, M. F.2012. “On Intentional Introduction of Stiffness Nonlinearities for Energy Harvesting under White Gaussian Excitations.” Nonlinear Dynamics69 (3): 106379. doi:10.1007/s11071-012-0327-0Search in Google Scholar

Daqaq, M., R.Masana, A.Erturk, and D. D.Quinn. 2014. “On the Role of Nonlinearities in Vibratory Energy Harvesting: A Critical Review and Discussion.” Applied Mechanics Reviews66: 0408011–040801–23. doi:10.1115/1.4026278Search in Google Scholar

Erturk, A., J.Hoffmann, and D. J.Inman. 2009. “A Piezomagnetoelastic Structure for Broadband Vibration Energy Harvesting.” Applied Physics Letters94 (25): 254102. doi:10.1063/1.3159815Search in Google Scholar

Halvorsen, E.2008. “Energy Harvesters Driven by Broadband Random Vibrations.” Journal of Microelectromechanical Systems17 (5): 106171. doi:10.1109/JMEMS.2008.928709Search in Google Scholar

Halvorsen, E.2013. “Fundamental Issues in Nonlinear Wideband-Vibration Energy Harvesting.” Physical Review E87 (4): 042129. doi:10.1103/PhysRevE.87.042129Search in Google Scholar

Halvorsen, E., C. P.Le, P. D.Mitcheson, and E. M.Yeatman. 2013. “Architecture-Independent Power Bound for Vibration Energy Harvesters.” Journal of Physics: Conference Series476: 012026. doi:10.1088/1742-6596/476/1/012026Search in Google Scholar

Hoffmann, D., B.Folkmer, and Y.Manoli. 2012. Comparative study of concepts for increasing the bandwidth of vibration based energy harvesters. Proceedings of PowerMEMS 2012, 219–222.Search in Google Scholar

Mann, B. P., D. A.Barton, and B. A.Owens. 2012. “Uncertainty in Performance for Linear and Nonlinear Energy Harvesting Strategies.” Journal of Intelligent Material Systems and Structures23 (13): 145160. doi:10.1177/1045389X12439639.Search in Google Scholar

Mitcheson, P. D., T. C.Green, E. M.Yeatman, and A. S.Holmes. 2004. “Architectures for Vibration-Driven Micropower Generators.” Journal of Microelectromechanical Systems13 (3): 112.10.1109/JMEMS.2004.830151Search in Google Scholar

Nguyen, S. D., E.Halvorsen, and I.Paprotny. 2013. “Bistable Springs for Wideband Microelectromechanical Energy Harvesters.” Applied Physics Letters102 (2): 023904. doi:10.1063/1.4775687.Search in Google Scholar

Roundy, S., E. S.Leland, J.Baker, E.Carleton, E.Reilly, E.Lai, B.Otis,, J. M.RabaeyP. K.Wright, and V.Sundararajan. 2005. “Improving Power Output for Vibration-Based Energy Scavengers.” IEEE Pervasive Computing4 (1): 2836. doi:10.1109/MPRV.2005.14Search in Google Scholar

Roundy, S., P. K.Wright, and J.Rabaey. 2003. “A Study of Low Level Vibrations as a Power Source for Wireless Sensor Nodes.” Computer Communications26 (11): 113144. doi:10.1016/S0140-3664(02)00248-7.Search in Google Scholar

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

©2016 by De Gruyter

Downloaded on 30.1.2023 from
Scroll Up Arrow