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Research Papers

New Insights Into Utilizing Bistability for Energy Harvesting Under White Noise

[+] Author and Article Information
Qifan He

Department of Mechanical Engineering,
Clemson University,
Clemson, SC 29634
e-mail: qifanh@g.clemson.edu

Mohammed F. Daqaq

Associate Professor
Nonlinear Vibrations and Energy
Harvesting Laboratory (NOVEHL),
Department of Mechanical Engineering,
Clemson University,
Clemson, SC 29634
e-mail: mdaqaq@clemson.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received October 23, 2014; final manuscript received November 4, 2014; published online December 11, 2014. Assoc. Editor: I. Y. (Steve) Shen.

J. Vib. Acoust 137(2), 021009 (Apr 01, 2015) (10 pages) Paper No: VIB-14-1401; doi: 10.1115/1.4029008 History: Received October 23, 2014; Revised November 04, 2014; Online December 11, 2014

This paper investigates the response of a bistable energy harvester to random excitations that can be approximated by a white noise process. Statistical linearization (SL), direct numerical integration of the stochastic differential equations, and finite element (FE) solution of the Fokker–Plank–Kolmogorov (FPK) equation are utilized to understand how the shape of the potential energy function influences the mean output power of the harvester. It is observed that, both of the FE solution and the direct numerical integration provide close predictions for the mean power regardless of the shape of the potential energy function. SL, on the other hand, yields nonunique and erroneous predictions unless the potential energy function has shallow potential wells. It is shown that the mean power exhibits a maximum value at an optimal potential shape. This optimal shape is not directly related to the shape that maximizes the mean square displacement even when the time constant ratio, i.e., ratio between the time constants of the mechanical and electrical systems is small. Maximizing the mean square displacement yields a potential shape with a global maximum (unstable potential) for any value of the time constant ratio and any noise intensity, whereas maximizing the average power yields a bistable potential which possesses deeper potential wells for larger noise intensities and vise versa. Away from the optimal shape, the average power drops significantly highlighting the importance of characterizing the noise intensity of the vibration source prior to designing a bistable harvester for the purpose of harnessing energy from white noise excitations. Furthermore, it is demonstrated that, the optimal time constant ratio is not necessarily small which challenges previous conceptions that a bistable harvester provides better output power when the time constant ratio is small. While maximum variation of the mean power with the nonlinearity occurs for smaller values of the time constant ratio, this does not necessarily correspond to the optimal performance of the harvester.

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Figures

Grahic Jump Location
Fig. 1

The potential function and associated restoring force of mono- and bi-stable nonlinear energy harvesters

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Fig. 2

A simplified representation of a VEH: (a) piezoelectric and (b) electromagnetic

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Fig. 3

Variation of (a) the potential energy U(x) and (b) the restoring force (dU(x))/dx with the displacement for different values of the nonlinearity coefficient, δ

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Fig. 4

Variation of 〈x〉2,〈x2〉,〈x·2〉, and 〈P〉 with δ obtained for κ = 0.65, ζ = 0.01, α = 0.5, and S0 = 0.01

Grahic Jump Location
Fig. 5

Variation of 〈x〉2 and 〈x2〉 with δ obtained for κ = 0.65, ζ = 0.01, α = 0.5, and S0 = 0.04

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Fig. 6

Cross-sectional views of the resulting stationary PDF for: (a) x1 and x2 when x3 = 0, (b) x1 and x3 when x2 = 0, and (c) x2 and x3 when x1 = 0. Results are obtained for δ = 1.5, α = 0.5, κ = 0.65, ζ = 0.01, and S0 = 0.01.

Grahic Jump Location
Fig. 7

Variation of (a) 〈x〉, (b) 〈x2〉, (c) 〈x·2〉, and (d) 〈P〉 with the nonlinearity δ obtained for α = 0.5, κ = 0.65, ζ = 0.01, and S0 = 0.01. Squares and triangles represent solutions obtained via numerical integration and FEM, respectively.

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Fig. 8

Variation of (a) 〈P〉 and (b) percentage error with the nonlinearity δ obtained for α = 0.5, κ = 0.65, ζ = 0.01, and S0 = 0.01. Triangles represent solutions obtained using FEM while the squares and the crosses represent solutions obtained via numerical integration. The simulated time for squares is ten times that of the crosses.

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Fig. 9

Variation of (a) 〈x2〉 and (b) 〈P〉 with nonlinearity δ obtained for κ = 0.65, ζ = 0.01, and S0 = 0.01 for different α via FE and SL method

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Fig. 10

Variation of the average power 〈P〉 with the nonlinearity δ for different values of α and S0 for (a) S0 = 0.01 and (b) α = 0.5. Results are obtained for κ = 0.65 and ζ = 0.01.

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Fig. 11

Variation of optimal δ with the input variance S0. Results are obtained for κ = 0.65, ζ = 0.01, and α = 0.5.

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Fig. 12

Variation of the optimal average power (a) 〈P〉 and (b) α with the nonlinearity δ. Results are obtained for κ = 0.65 and ζ = 0.01.

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