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Research Papers

# Comparing Linear and Essentially Nonlinear Vibration-Based Energy Harvesting

[+] Author and Article Information
D. Dane Quinn

Department of Mechanical Engineering, University of Akron, Akron, OH 44325-3903quinn@uakron.edu

Angela L. Triplett

Department of Mechanical Engineering, University of Akron, Akron, OH 44325-3903alt25@uakron.edu

Lawrence A. Bergman

Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, 104 South Wright Street, Urbana, IL 61801lbergman@illinois.edu

Alexander F. Vakakis

Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, IL 61801avakakis@illinois.edu

The method of averaging generates a uniformly valid approximation only when the evolutions of the amplitudes before averaging (i.e., $ṙ$) are $O(ϵ)$(27).

J. Vib. Acoust 133(1), 011001 (Dec 03, 2010) (8 pages) doi:10.1115/1.4002782 History: Received June 05, 2009; Revised July 30, 2010; Published December 03, 2010; Online December 03, 2010

## Abstract

This work considers the performance of a resonant vibration-based energy harvesting system utilizing a strongly nonlinear attachment. Typical designs serving as the basis for harvesting energy from ambient vibration typically employ a linear oscillator for this purpose, limiting peak harvesting performance to a narrow band of frequencies about the resonant frequency of the oscillator. Herein, in an effort to maximize performance over the broader band of frequency content typically observed in ambient vibration measurements, we employ an essentially nonlinear cubic oscillator in the harvesting device and show that, with proper design, significant performance gains can be realized as compared with a tuned linear attachment. However, we also show that the coexistence of multiple equilibria due to the nonlinearity can degrade system performance, as the system can be attracted to a low amplitude state that provides reduced harvested power. Finally, when multiple equilibria exist in the system, the basins of attraction for the stable states are determined and related to the expected response of the system.

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## Figures

Figure 3

Geometric construction for the equilibrium amplitude of the averaged equations g(x)=αx3(α=1.00,  ω=1.50,  ϵ=0.10,  δ=1.00,  ρ=1.00,  ξ=1.00)

Figure 4

Equilibrium locations of the averaged equations g(x)=αx3(ϵ=0.10,  δ=1.00,  ρ=1.00,  ξ=1.00)

Figure 1

Nonlinear vibration-based energy harvesting model

Figure 2

Geometric construction for the equilibrium amplitude of the averaged equations g(x)=αx3(α=1.00,  ω=1.00,  ϵ=0.10,  δ=1.00,  ρ=1.00,  ξ=1.00)

Figure 5

Phase portraits of the averaged equations g(x)=αx3(α=1.00,  ϵ=0.10,  δ=1.00,  ρ=1.00,  ξ=1.00), cf. Fig. 4

Figure 6

Basin of attraction of the averaged equations with multiple stable equilibrium points g(x)=αx3(α=1.00,  ω=1.50,  ϵ=0.10,  δ=1.00,  ρ=1.00,  ξ=1.00)

Figure 7

Measure of the basins of attraction of the averaged equations g(x)=αx3(α=1.00,  ω=1.50,  ϵ=0.10,  δ=1.00,  ρ=1.00,  ξ=1.00). The equilibrium points are located with the dashed curve while the width of the surrounding region indicates the measure of the basin of attraction. Note that the intermediate branch is unstable and has a basin of attraction of measure zero.

Figure 8

Comparing the original and averaged equations g(x)=αx3(ω=1.00,  ϵ=0.10,  δ=1.00,  ρ=1.00,  ξ=1.00)

Figure 9

Equilibrium locations of the averaged equations (ϵ=0.10,  δ=1.00,  ρ=1.00,  ξ=1.00)

Figure 10

Weighted amplitude R of the equilibrium response, essential nonlinearity (c=1), α=αmax(ϵ=0.10,  δ=1.00,  ρ=1.00,  ξ=1.00)

Figure 11

Weighted value of the stationary power P for the averaged equations with ω=1.00(ϵ=0.10,  δ=1.00,  ρ=1.00,  ξ=1.00). The system exhibits bistability in the shaded regions: the thin solid curves indicate branches of stable equilibria and the thin dashed lines represent unstable equilibria branches.

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