Equivalent damping and frequency change for linear and nonlinear hybrid vibrational energy harvesting systems M. Amin Karami a,n , Daniel J. Inman b a Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA 24061, United States b Center for Intelligent Material Systems and Structures, Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24061, United States article info Article history: Received 23 June 2010 Received in revised form 3 May 2011 Accepted 27 June 2011 Handling Editor: K. Worden Available online 30 July 2011 abstract A unified approximation method is derived to illustrate the effect of electro-mechanical coupling on vibration-based energy harvesting systems caused by variations in damp- ing ratio and excitation frequency of the mechanical subsystem. Vibrational energy harvesters are electro-mechanical systems that generate power from the ambient oscillations. Typically vibration-based energy harvesters employ a mechanical subsys- tem tuned to resonate with ambient oscillations. The piezoelectric or electromagnetic coupling mechanisms utilized in energy harvesters, transfers some energy from the mechanical subsystem and converts it to an electric energy. Recently the focus of energy harvesting community has shifted toward nonlinear energy harvesters that are less sensitive to the frequency of ambient vibrations. We consider the general class of hybrid energy harvesters that use both piezoelectric and electromagnetic energy harvesting mechanisms. Through using perturbation methods for low amplitude oscillations and numerical integration for large amplitude vibrations we establish a unified approximation method for linear, softly nonlinear, and bi-stable nonlinear energy harvesters. The method quantifies equivalent changes in damping and excitation frequency of the mechanical subsystem that resembles the backward coupling from energy harvesting. We investigate a novel nonlinear hybrid energy harvester as a case study of the proposed method. The approximation method is accurate, provides an intuitive explanation for backward coupling effects and in some cases reduces the computational efforts by an order of magnitude. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction Energy harvesting refers to scavenging small amounts of power from the ambient energy in the environment. This paper focuses on energy harvesting from vibrations. Such ambient energy can come from bridge vibrations, tire motion or the human heart beating. The energy can power up sensor nodes and therefore reduce the wiring complications or eliminate the frequent need of changing batteries. For more information on general energy harvesting the reader may refer to Refs. [1–6]. During the past two years nonlinear energy harvesting has received substantial attention. The main advantage of nonlinear energy harvesters over their linear counterparts is that the nonlinear harvesters scavenge energy over a broader frequency range of vibrations. Vibrational energy harvesters typically have low damping ratios to maximize the harvested power at resonance. The low damping ratio however, makes the harvesting device highly frequency sensitive (Fig. 1a). As depicted in Fig. 1b the presence of nonlinearity makes the peak of frequency response function lean toward higher or lower frequencies (this depends on the type of nonlinearity). The nonlinear energy harvester therefore acts over a wider Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jsvi Journal of Sound and Vibration 0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.06.021 n Corresponding author. E-mail addresses: karami@vt.edu (M.A. Karami), dinman@vt.edu (D.J. Inman). Journal of Sound and Vibration 330 (2011) 5583–5597