In this paper, the Chebyshev polynomial approximation is firstly utilized to analyze the dynamical characteristics of the nonlinear vibration energy harvester with an uncertain parameter. First, the stochastic energy harvester is transformed into a high-dimensional equivalent deterministic system by the Chebyshev polynomial approximation. And the ensemble mean response of the stochastic energy harvester is introduced to discuss the stochastic response. Then, the effectiveness of the approximation method is verified by numerical results. Furthermore, the bifurcation property of the displacement and voltage is analyzed, which is also consistent with the results derived by the top Lyapunov exponent. It is found that random factor can induce the appearance of multi-periodic phenomena and lead to appear the behavior of the periodic bifurcation. The strong random factor induces the fluctuation of the output voltage. In addition, the existence of the random factor greatly influences the property of the sub-harmonics and super-harmonics of the spectrum. Overall, the response mechanism of the nonlinear vibration energy harvester with an uncertain parameter is revealed.
Erturk A and Inman DJ. Piezoelectric energy harvesting. Hoboken, NJ: John Wiley & Sons, 2011.
2.
Priya S and Inman DJ. Energy harvesting technologies. New York: Springer, 2009.
3.
SudevalayamSKulkarniP. Energy harvesting sensor nodes: survey and implications. IEEE Commun Surv Tutorial2011; 13: 443–461.
4.
HarneRLWangKW. A review of the recent research on vibration energy harvesting via bistable systems. Smart Mater Struct2013; 22: 023001.
5.
HuGWangJSuZ, et al.Performance evaluation of twin piezoelectric wind energy harvesters under mutual interference. Appl Phys Lett2019; 115: 073901.
6.
WangJZhouSZhangZ, et al.High-performance piezoelectric wind energy harvester with Y-shaped attachments. Energ Convers Manage2019; 181: 645–652.
7.
MannBPSimsND. Energy harvesting from the nonlinear oscillations of magnetic levitation. J Sound Vib2009; 319: 515–530.
8.
YangWTowfighianS. A hybrid nonlinear vibration energy harvester. Mech Sys Sig Proc2017; 90: 317–333.
9.
LitakGRysakABorowiecM, et al.Vertical beam modal response in a broadband energy harvester. Proc IMechE, Part K: J Multi-body Dynamics2016; 230: 541–552.
10.
HuangDZhouSLitakG. Analytical analysis of the vibrational tristable energy harvester with a RL resonant circuit. Nonlinear Dyn2019; 97: 663–677.
11.
KremerDLiuK. A nonlinear energy sink with an energy harvester: transient responses. J Sound Vib2014; 333: 4859–4880.
12.
FangZZhangYLiX, et al.Integration of a nonlinear energy sink and a giant magnetostrictive energy harvester. J Sound Vib2017; 391: 35–49.
13.
HuangDLiRYangG. On the dynamic response regimes of a viscoelastic isolation system integrated with a nonlinear energy sink. Commun Nonlinear Sci2019; 79: 104916.
14.
BorowiecMLitakGFriswellMI, et al.Energy harvesting in a nonlinear cantilever piezoelastic beam system excited by random vertical vibrations. Int J Struc Stab Dyn2014; 14: 1440018.
15.
BorowiecMSytaALitakG. Energy harvesting optimizing with a magnetostrictive cantilever beam system. Int J Struc Stab Dyn2019; 19: 1941002 .
16.
ChenLJiangW. Internal resonance energy harvesting. J Appl Mech2015; 82: 031004.
17.
ChenLJiangWPanyamM, et al.A broadband internally resonant vibratory energy harvester. J Vib Acoust2016; 138: 061007.
18.
StantonSCMcGeheeCCMannBP. Reversible hysteresis for broadband magnetopiezoelastic energy harvesting. Appl Phys Lett2009; 95: 174103.
19.
ZarepoorMBilgenO. Constrained-energy dynamic cross-well motion of bistable structures subjected to noise disturbance. Int J Struct Stab Dyn2018; 18: 850047.
20.
HuangDZhouSLitakG. Theoretical analysis of multi-stable energy harvesters with high-order stiffness terms. Commun Nonlinear Sci2019; 69: 270–286.
21.
AlevrasPTheodossiadesSRahnejatH. Broadband energy harvesting from parametric vibrations of a class of nonlinear Mathieu systems. Appl Phys Lett2017; 110: 233901.
22.
CottoneFVoccaHGammaitoniL. Nonlinear energy harvesting. Phys Rev Lett2009; 102: 080601.
23.
LitakGFriswellMIAdhikariS. Magnetopiezoelastic energy harvesting driven by random excitations. Appl Phys Lett2010; 96: 214103.
24.
LitakGBorowiecMFriswellMI, et al.Energy harvesting in a magnetopiezoelastic system driven by random excitations with uniform and Gaussian distributions. J Theor Appl Mech-Pol2011; 49: 757–764.
25.
WangXChenCWangN, et al.A frequency and bandwidth tunable piezoelectric vibration energy harvester using multiple nonlinear techniques. Appl Energy2017; 190: 368–375.
26.
AlevrasPTheodossiadesSRahnejatH. On the dynamics of a nonlinear energy harvester with multiple resonant zones. Nonlinear Dyn2018; 92: 1271–1286.
27.
XuYGuRZhangH, et al.Stochastic bifurcations in a bistable Duffing–Van der Pol oscillator with colored noise. Phys Rev E2011; 83: 056215.
28.
SytaALitakGFriswellMI, et al.Multiple solutions and corresponding power output of a nonlinear bistable piezoelectric energy harvester. Eur Phys J B2016; 89: 99.
29.
ZhouSCaoJInman DJ, et al.Impact-induced high-energy orbits of nonlinear energy harvesters. Appl Phys Lett2015; 106: 093901.
30.
Friswell M IBilgenOAli SF, et al.The effect of noise on the response of a vertical cantilever beam energy harvester. ZAMM- Z Angew Math Mech2015; 95: 433–443.
31.
ZhaoSErturkA. On the stochastic excitation of monostable and bistable electroelastic power generators: relative advantages and tradeoffs in a physical system. Appl Phys Lett2013; 102: 103902.
32.
ErturkAHoffmannJInmanDJ. A piezomagnetoelastic structure for broadband vibration energy harvesting. Appl Phys Lett2009; 94: 254102.
33.
ZhouSCaoJErturkA, et al.Enhanced broadband piezoelectric energy harvesting using rotatable magnets. Appl Phys Lett2013; 102: 173901.
34.
HarneRLThotaMWangKW. Concise and high-fidelity predictive criteria for maximizing performance and robustness of bistable energy harvesters. Appl Phys Lett2013; 102: 053903.
35.
PanyamMDaqaqMF. Characterizing the effective bandwidth of tri-stable energy harvesters. J Sound Vib2017; 386: 336–358.
36.
ZhouSZuoL. Nonlinear dynamic analysis of asymmetric tristable energy harvesters for enhanced energy harvesting. Commun Nonlinear Sci2018; 61: 271–284.
37.
ZhouSCaoJLitakG, et al.Numerical analysis and experimental verification of broadband tristable energy harvesters. tm-Tech Mess2018; 85: 521–532.
38.
HuangDLiWYangGD, et al.Analysis of limit cycles and stochastic responses of a real-power vibration isolation system under delayed feedback control. Chaos Solit Fract2018; 112: 125–134.
39.
HuangDLiRZhouS, et al.Theoretical analysis of vibration energy harvesters with nonlinear damping and nonlinear stiffness. Euro Phys J Plus2018; 133: 510.
40.
Nanda A, Karami M and Singla P. Uncertainty quantification of energy harvesting systems using method of quadratures and maximum entropy principle. In: ASME 2015 conference on smart materials, adaptive structures and intelligent systems, Colorado Springs, USA, 21–23 September 2015.
41.
SepahvandKMarburgS. Stochastic dynamic analysis of structures with spatially uncertain material parameters. Int J Struct Stab Dyn2014; 14: 1440029.
42.
PanunzioAMSallesLSchwingshacklCW. Uncertainty propagation for nonlinear vibrations: a non-intrusive approach. J Sound Vib2017; 389: 309–325.
43.
LiYZhouSLitakG. Uncertainty analysis of excitation conditions on performance of nonlinear monostable energy harvesters. Int J Struct Stab Dyn2019; 19: 1950052.
Ghanem RG and Spanos PD. Stochastic finite elements: a spectral approach. New York, NY: Springer-Verlag, 1991.
46.
FangTLengXLSongCQ. Chebyshev polynomial approximation for dynamical response problem of random system. J Sound Vib2003; 266: 198–206.
47.
MaSXuWJinY, et al.Symmetry-breaking bifurcation analysis of stochastic van der pol system via Chebyshev polynomial approximation. Commun Nonlinear Sci2007; 12: 366–378.
48.
WangLXuWLiG, et al.Response of a stochastic Duffing-Van der Pol elastic impact oscillator. Chaos Solit Fract2009; 41: 2075–2080.
49.
WuJZhangYChenL, et al.A Chebyshev interval method for nonlinear dynamic systems under uncertainty. Appl Math Model2013; 37: 4578–4591.
50.
ErturkAHoffmannJInmanDJ. A piezomagnetoelastic structure for broadband vibration energy harvesting. Appl Phys Lett2009; 94: 254102.
51.
ErturkAInmanDJ. Broadband piezoelectric power generation on high-energy orbits of the bistable Duffing oscillator with electromechanical coupling. J Sound Vibrat2011; 330: 2339–2353.
52.
BenettinGGalganiLGiorgilliA, et al.Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory. Meccanica1980; 15: 21–30.
