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.

References

References
1.
McInnes
,
C. R.
,
Gorman
,
D. G.
, and
Cartmell
,
M. P.
,
2008
, “
Enhanced Vibrational Energy Harvesting Using Nonlinear Stochastic Resonance
,”
J. Sound Vib.
,
318
(
4–5
), pp.
655
662
.10.1016/j.jsv.2008.07.017
2.
Barton
,
D.
,
Burrow
,
S.
, and
Clare
,
L.
,
2010
, “
Energy Harvesting From Vibrations With a Nonlinear Oscillator
,”
ASME J. Vib. Acoust.
,
132
(
2
), p.
021009
.10.1115/1.4000809
3.
Mann
,
B.
, and
Sims
,
N.
,
2008
, “
Energy Harvesting From the Nonlinear Oscillations of Magnetic Levitation
,”
J. Sound Vib.
,
319
(
1–2
), pp.
515
530
.10.1016/j.jsv.2008.06.011
4.
Masana
,
R.
, and
Daqaq
,
M. F.
,
2011
, “
Electromechanical Modeling and Nonlinear Analysis of Axially-Loaded Energy Harvesters
,”
ASME J. Vib. Acoust.
,
133
(
1
), p.
011007
.10.1115/1.4002786
5.
Quinn
,
D.
,
Triplett
,
L.
,
Vakakis
,
D.
, and
Bergman
,
L.
,
2011
, “
Comparing Linear and Essentially Nonlinear Vibration-Based Energy Harvesting
,”
ASME J. Vib. Acoust.
,
133
(
1
), p.
011001
.10.1115/1.4002782
6.
Erturk
,
A.
,
Hoffman
,
J.
, and
Inman
,
D. J.
,
2009
, “
A Piezo-Magneto-Elastic Structure for Broadband Vibration Energy Harvesting
,”
Appl. Phys. Lett.
,
94
(
25
), p.
254102
.10.1063/1.3159815
7.
Cottone
,
F.
,
Vocca
,
H.
, and
Gammaitoni
,
L.
,
2009
, “
Nonlinear Energy Harvesting
,”
Phys. Rev. Lett.
,
102
(8), p.
080601
.10.1103/PhysRevLett.102.080601
8.
Daqaq
,
M. F.
,
Stabler
,
C.
,
Seuaciuc-Osorio
,
T.
, and
Qaroush
,
Y.
,
2010
, “
Investigation of Power Harvesting Via Parametric Excitations
,”
J. Intell. Mater. Syst. Struct.
,
20
(5), pp.
545
557
.10.1177/1045389X08100978
9.
Stanton
,
S. C.
,
McGehee
,
C. C.
, and
Mann
,
B. P.
,
2010
, “
Nonlinear Dynamics for Broadband Energy Harvesting: Investigation of a Bistable Piezoelectric Inertial Generator
,”
Physica D
,
239
(
10
), pp.
640
653
.10.1016/j.physd.2010.01.019
10.
Daqaq
,
M. F.
, and
Bode
,
D.
,
2010
, “
Exploring the Parametric Amplification Phenomenon for Energy Harvesting
,”
J. Syst. Control Eng.
,
225
(4), pp.
456
466
.10.1177/2041304110401145
11.
Abdelkefi
,
A.
,
Nayfeh
,
A. H.
, and
Hajj
,
M.
,
2011
, “
Global Nonlinear Distributed-Parameter Model of Parametrically Excited Piezoelectric Energy Harvesters
,”
Nonlinear Dyn.
,
67
(
2
), pp.
1147
1160
.10.1007/s11071-011-0059-6
12.
Abdelkefi
,
A.
,
Nayfeh
,
A. H.
, and
Hajj
,
M.
,
2011
, “
Effects of Nonlinear Piezoelectric Coupling on Energy Harvesters Under Direct Excitation
,”
Nonlinear Dyn.
,
67
(
2
), pp.
1221
1232
.10.1007/s11071-011-0064-9
13.
Harne
,
R. L.
, and
Wang
,
K. W.
,
2013
, “
A Review of the Recent Research on Vibration Energy Harvesting Via Bistable Systems
,”
Smart Mater. Struct.
,
22
(
2
), p.
023001
.10.1088/0964-1726/22/2/023001
14.
Mann
,
B. P.
,
Barton
,
D.
, and
Owens
,
B.
,
2012
, “
Uncertainty in Performance for Linear and Nonlinear Energy Harvesting Strategies
,”
J. Intell. Mater. Syst. Struct.
,
23
(
13
), pp.
1451
1460
.10.1177/1045389X12439639
15.
Szemplinska-Stupnicka
,
W.
,
2003
,
Chaos, Bifurcations and Fractals Around Us
,
World Scientific
,
Singapore
.
16.
Masana
,
R.
, and
Daqaq
,
M. F.
,
2011
, “
Comparing the Performance of a Nonlinear Energy Harvester in Mono- and Bi-Stable Potentials
,”
ASME
Paper No. DETC2011-47828.10.1115/DETC2011-47828
17.
Gammaitoni
,
L.
,
Neri
,
I.
, and
Vocca
,
H.
,
2009
, “
Nonlinear Oscillators for Vibration Energy Harvesting
,”
Appl. Phys. Lett.
,
94
(
16
), p.
164102
.10.1063/1.3120279
18.
Daqaq
,
M. F.
,
2010
, “
Response of Uni-Modal Duffing Type Harvesters to Random Forced Excitations
,”
J. Sound Vib.
,
329
(
18
), pp.
3621
3631
.10.1016/j.jsv.2010.04.002
19.
Nguyen
,
D. S.
,
Halvorsen
,
E.
,
Jensen
,
G. U.
, and
Vogl
,
A.
,
2010
, “
Fabrication and Characterization of a Wide-band MEMS Energy Harvester Utilizing Nonlinear Springs
,”
J. Micromech. Microeng.
,
20
(
12
), p.
125009
.10.1088/0960-1317/20/12/125009
20.
Halvorsen
,
E.
,
2013
, “
Fundamental Issues in Nonlinear Wide-Band Vibration Energy Harvesting
,”
Phys. Rev. E
,
87
(4), p.
042129
.10.1103/PhysRevE.87.042129
21.
Green
,
P. L.
,
Worden
,
K.
,
Atallah
,
K.
, and
Sims
,
N. D.
,
2012
, “
The Benefits of Duffing-Type Nonlinearities and Electrical Optimisation of a Mono-Stable Energy Harvester Under White Gaussian Excitations
,”
J. Sound Vib.
,
331
(
20
), pp.
4504
4517
.10.1016/j.jsv.2012.04.035
22.
He
,
Q.
, and
Daqaq
,
M. F.
,
2013
, “
Load Optimization of a Nonlinear Mono-Stable Duffing-Type Harvester Operating in a White Noise Environment
,”
ASME
Paper No. DETC2013-13126.10.1115/DETC2013-13126
23.
Litak
,
G.
,
Friswell
,
M. I.
, and
Adhikari
,
S.
,
2010
, “
Magnetopiezoelastic Energy Harvesting Driven by Random Excitations
,”
Appl. Phys. Lett.
,
96
(
21
), p.
214103
.10.1063/1.3436553
24.
Litak
,
G.
,
Borowiec
,
M.
,
Friswell
,
M. I.
, and
Adhikari
,
S.
,
2011
, “
Energy Harvesting in a Magnetopiezoelastic System Driven by Random Excitations With Uniform and Gaussian Distributions
,”
J. Theor. Appl. Mech.
,
49
(3), pp.
757
764
.
25.
Daqaq
,
M. F.
,
2012
, “
On Intentional Introduction of Stiffness Nonlinearities for Energy Harvesting Under White Gaussian Excitations
,”
Nonlinear Dyn.
,
69
(
3
), pp.
1063
1079
.10.1007/s11071-012-0327-0
26.
Ferrari
,
M.
,
Ferrari
,
V.
,
Guizzetti
,
M.
,
Andò
,
B.
,
Baglio
,
S.
, and
Trigona
,
C.
,
2010
, “
Improved Energy Harvesting From Wideband Vibrations by Nonlinear Piezoelectric Converters
,”
Sens. Actuators, A
,
162
(
2
), pp.
425
431
.10.1016/j.sna.2010.05.022
27.
Khovanova
,
N. A.
, and
Khovanov
, I
. A.
,
2011
, “
The Role of Excitations Statistic and Nonlinearity in Energy Harvesting From Random Impulsive Excitations
,”
Appl. Phys. Lett.
,
99
(
14
), p.
144101
.10.1063/1.3647556
28.
Daqaq
,
M. F.
,
2011
, “
Transduction of a Bistable Inductive Generator Driven by White and Exponentially Correlated Gaussian Noise
,”
J. Sound Vib.
,
330
(11), pp.
2554
2564
.10.1016/j.jsv.2010.12.005
29.
Neri
,
I.
,
Travasso
,
F.
,
Vocca
,
H.
, and
Gammaitoni
,
L.
,
2011
, “
Nonlinear Noise Harvesters for Nanosensors
,”
Nano Commun. Networks
,
2
(
4
), pp.
230
234
.10.1016/j.nancom.2011.09.001
30.
Zhao
,
S.
, and
Erturk
,
A.
,
2013
, “
On the Stochastic Excitation of Monostable and Bistable Electroelastic Power Generators: Relative Advantages and Tradeoffs in a Physical System
,”
J. Appl. Phys.
,
102
(10), p.
103902
.10.1063/1.4795296
31.
Masana
,
R.
, and
Daqaq
,
M. F.
,
2013
, “
Response of Duffing-Type Harvesters to Band-Limited Noise
,”
J. Sound Vib.
,
332
(
25
), pp.
6755
6767
.10.1016/j.jsv.2013.07.022
32.
He
,
Q.
, and
Daqaq
,
M. F.
,
2014
, “
Influence of Potential Function Asymmetries on the Performance of Nonlinear Energy Harvesters Under White Noise
,”
J. Sound Vib.
,
33
(
15
), pp.
3479
3489
.10.1016/j.jsv.2014.03.034
33.
Ali
,
S. F.
,
Adhikari
,
S.
,
Friswell
,
M. I.
, and
Narayanan
,
S.
,
2011
, “
The Analysis of Piezomagnetoelastic Energy Harvesters Under Broadband Random Excitations
,”
J. Appl. Phys.
,
109
(
7
), p.
074904
.10.1063/1.3560523
34.
Elvin
,
N.
, and
Erturk
,
A.
,
2013
,
Advances in Energy Harvesting Methods
,
Springer
,
New York
.
35.
Ito
,
K.
,
1944
, “
Stochastic Integral
,”
Proc. Imp. Acad. (Tokyo)
,
20
(
8
), pp.
519
524
.10.3792/pia/1195572786
36.
Jazwinski
,
A. H.
,
1970
,
Stochastic Processes and Filtering Theory
,
Academic
,
New York
.
37.
Langley
,
R. S.
,
1985
, “
A Finite Element Method for the Statistics of Non-Linear Random Vibration
,”
J. Sound Vib.
,
101
(
1
), pp.
41
54
.10.1016/S0022-460X(85)80037-7
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