We examine multiple responses of a vibrational energy harvester composed of a vertical
          beam and a tip mass. The beam is excited horizontally by a harmonic inertial force while
          mechanical vibrational energy is converted to electrical power through a piezoelectric
          patch. The mechanical resonator can be described by single or double well potentials
          depending on the gravity force from the tip mass. By changing the tip mass we examine the
          appearance of various solutions and their basins of attraction. Identification of
          particular solutions of the energy harvester is important as each solution may provide a
          different level of power output

A. Syta

Arkadiusz Syta

E. Esmailzadeh

Grzegorz Litak

J. Aguirre

J. Cao

J. Twiefel

L. Zavodney

M.F. Daqaq

M.I. Friswell

Michael I. Friswell

P.D. Mitcheson

R.L. Harne

S.P. Pellegrini

Sondipon Adhikari

T.Y. Li

Springer - Publisher Connector

Multiple solutions and corresponding power output of a nonlinear bistable piezoelectric energy harvester

Crossref

The European Physical Journal B

Michael Friswell

Cronfa at Swansea University

We examine multiple responses of a vibrational energy harvester composed of a vertical beam and a tip mass. The beam is excited horizontally by a harmonic inertial force while mechanical vibrational energy is converted to electrical power through a piezoelectric patch. The mechanical resonator can be described by single or double well potentials depending on the gravity force from the tip mass. By changing the tip mass we examine the appearance of various solutions and their basins of attraction. Identification of particular solutions of the energy harvester is important as each solution may provide a different level of power output

Syta, Arkadiusz

Litak, Grzegorz

Friswell, Michael I.

Adhikari, Sondipon

Enlighten

Eur. Phys. J. B (2016) 89: 99DOI: 10.1140/epjb/e2016-60699-0Regular ArticleTHE EUROPEANPHYSICAL JOURNAL BMultiple solutions and corresponding power output of a nonlinearbistable piezoelectric energy harvesterArkadiusz Syta1, Grzegorz Litak1,a, Michael I. Friswell2, and Sondipon Adhikari21 Faculty of Mechanical Engineering, Lublin University of Technology, Nadbystrzycka 36, 20-618 Lublin, Poland2 College of Engineering, Swansea University Bay Campus, Fabian Way, Crymlyn Burrows, Swansea SA1 8EN, UKReceived 22 February 2016 / Received in final form 24 February 2016Published online 13 April 2016c© The Author(s) 2016. This article is published with open access at Springerlink.comAbstract. We examine multiple responses of a vibrational energy harvester composed of a vertical beamand a tip mass. The beam is excited horizontally by a harmonic inertial force while mechanical vibrationalenergy is converted to electrical power through a piezoelectric patch. The mechanical resonator can bedescribed by single or double well potentials depending on the gravity force from the tip mass. By changingthe tip mass we examine the appearance of various solutions and their basins of attraction. Identificationof particular solutions of the energy harvester is important as each solution may provide a different levelof power output.1 IntroductionThe fast development in the miniaturisation of electron-ics and new trends in monitoring of systems and struc-tures has motivated the development of small scale ofsmall energy harvesting to power these small electronicdevices and extend their lifetime and/or remove the needto charge batteries [1]. Ambient vibration energy harvest-ing is one potential solution. In this field, linear devicesare only efficient at resonance, i.e. they are only sensi-tive to a single frequency are not efficient for variableexcitation conditions. On the other hand nonlinear vi-bration energy harvesters showing broadband frequencytransduction are based on nonlinear phenomena. Conse-quently these material and/or geometrical nonlinearitiesimprove the efficiency of energy harvesting from variableambient sources via synchronization and nonlinear reso-nance. However, the nonlinearity also allows the appear-ance of multiple solutions.A range of vibration energy harvesting devices havebeen proposed [2–5]. Recently, in the context of broad-band energy harvesting, bifurcations and chaotic vibra-tions have been studied in several papers. Cao et al. [6,7]studied chaos in the fractionally damped broadband piezo-electric energy generator in a system with additionalmagnets. Syta et al. [8] analysed the dynamic responseof piezoelectric material attached to a bistable laminateplate. In this paper we study the response of a vibra-tional energy harvester composed of a vertical beam anda e-mail: g.litak@pollub.pla tip mass where the gravitational acceleration introducesa double well potential. This system has shown a num-ber of possible solutions with different power outputs [9].The appearance of periodic or chaotic solutions, or so-lutions with single and cross potential well motions, de-pend on the initial conditions. In this paper we identifythese different motions and investigate their dynamics us-ing basins of attraction and Fourier spectrum tools.2 The modelThe proposed nonlinear harvester model is a vertical flex-ible beam of length L with piezoelectric layers and a tipmass, shown schematically in Figure 1. The base is ex-cited in the transverse direction with harmonic base exci-tation z(t) = z0 cosωt. The beam carries a concentratedtip mass, Mt, with moment of inertia, It, at the end of thebeam. The horizontal and vertical elastic displacements atthe tip mass are v and u respectively, and s represents thedistance along the neutral axis of the beam. The model ofthis system was described and verified experimentally inour previous paper [9].The beam is assumed to have uniform inertia and stiff-ness properties, although a non-uniform beam is easilymodeled by including the mechanical beam properties inthe energy integrals. The beam has cross sectional area A,mass density ρ, equivalent Young’s modulus E, and secondmoment of area I.Assuming a single mode approximation the equationof motion of the beam-mass system is derived in terms ofPage 2 of 7 Eur. Phys. J. B (2016) 89: 99Table 1. Parameter values used in the simulation.Beam and tip mass Energy harvesterρ 7850 kg/m3 Lc 28 mmE 210 GN/m2 bc 14 mmb 16 mm hc 300 μmh 0.254 mm γc −4.00 × 10−5 Nm/VL 300 mm Cp 51.4 nFIt/Mt 40.87 mm2 Rl 100 kΩPz t z t0uvvpupMtsFig. 1. Schematic representation of the vertical flexible beamwith piezoelectric layer and a tip mass. Here, Mt denotes thetip mass, v and u are is the horizontal and vertical displace-ments, P is an arbitrary point on the beam, and z(t) is theharmonic base excitation, while s represents the distance alongthe neutral axis of the beam.the displacement of the tip mass using Lagrange’s equa-tions [9–11] as:[N25 It +Mt + ρAN1 +(ρAN3 +MtN24 +N45 It)v2]v̈+[ρAN3+MtN24 +N45 It]vv̇2+[EIN6−N9ρAg−N4Mtg+2EIN7v2]v −Θ1U −Θ2v2U = − [ρAN2 +Mt] z̈ (1)where U is output voltage. The derivation of this equa-tion and the definitions of the constants are outlined inAppendix. The electrical equation of motion is:CpU̇ +URl+Θ1v̇ + Θ2v2v̇ = 0 (2)where Cp is the capacitance of the piezoelectric patch andRl is the load resistance.The system is base excited by the harmonic horizontaldisplacementz = z0 cos(ωt), (3)where z0 and ω are amplitude and frequency of excitation.The equilibrium positions with no forcing are obtainedby setting the velocity and acceleration terms to zero inequation (1) to give[EIN6 −N9ρAg −N4Mtg + 2EIN7v2]v = 0. (4)This equation has either one or three solutions, and v = 0is always a solution. Since N4 > 0, there are three solu-tions ifMt >EIN6 −N9ρAgN4g= Mtb, (5)where Mtb is the tip mass so that the beam is about tobuckle.We use the same set of system parameters as Friswellet al. [9], as given in Table 1. The post buckled responsehas two stable equilibrium positions above the critical tipmass, Mtb = 10g.3 Numerical solutionsThe energy harvesting system given by equations (1)and (2) is simulated using the Runge-Kutta fourth or-der method with a fixed time step δt = 0.01 s. Figure 2shows the six different solutions obtained with the systemparameters from Table 1 and a tip mass of Mt = 10.19 g.Note that our choice of Mt implies the beam is buckledwith two equilibrium points. The average harvested power,P , for a given solution is calculated from the voltage timeseries, Ui, by:P =1NRlN∑i=1U2i (6)where N denotes number of samples.The different solutions in Figure 2 represent differ-ent attractors, obtained from different initial conditions.The excitation amplitude was defined by z0 = 16 mmand the driving frequency was fixed to 0.5 Hz. Attrac-tors (a) and (b) are nonresonant period one single wellsolutions, and the resulting energy harvested is very low.Attractors (c) and (d) are period three cross well solutionsthat produce increased harvested power. In contrast solu-tion (e) is chaotic and (f) is a period one resonant solution,both of which give increased power output.To fully understand the different solutions, Figure 3shows the corresponding time series, and Figure 4 showsthe Fourier spectra, for both displacement and voltagetime series. The low energy solutions (a) and (b), andthe medium energy solutions (c) and (d), have trajecto-ries that represent the reflections in corresponding equiv-alent pairs. This is due to the symmetry of the doubleEur. Phys. J. B (2016) 89: 99 Page 3 of 7-0.6-0.3 0 0.3 0.6-0.2 -0.1  0  0.1  0.2velocity (m/s)displacement (m)(a)-0.6-0.3 0 0.3 0.6-0.2 -0.1  0  0.1  0.2velocity (m/s)displacement (m)(b)-0.6-0.3 0 0.3 0.6-0.2 -0.1  0  0.1  0.2velocity (m/s)displacement (m)(c)-0.6-0.3 0 0.3 0.6-0.2 -0.1  0  0.1  0.2velocity (m/s)displacement (m)(d)-0.6-0.3 0 0.3 0.6-0.2 -0.1  0  0.1  0.2velocity (m/s)displacement (m)(e)-0.6-0.3 0 0.3 0.6-0.2 -0.1  0  0.1  0.2velocity (m/s)displacement (m)(f)Fig. 2. The response of the system with a tip mass Mt = 10.19 g, with kinematic forcing of the fixed amplitude, z0 = 16 mm,and frequency, f = ω/2π = 0.5 Hz. The lines give the phase portraits, and the points give the corresponding Poincaré sections.The initial conditions and the average harvested power for each case are: (a) (x0, y0) = (−0.2990 m, −0.1805 m/s), P = 1.7 μW,(b) (−0.2430 m, −0.2145 m/s), 1.7 μW, (c) (−0.2105 m, −0.2135 m/s), 4.9 μW, (d) (−0.2949 m, −0.1930 m/s), 4.9 μW,(e) (−0.2765 m, −0.1775 m/s), 21 μW, (f) (−0.2985 m, 0.2399 m/s), 150 μW.well potential. More power output can be harvested fromthe larger trajectories (e) and (f). The Fourier transformsconfirm that trajectories (a) to (d) and (f) are regular.The single peaks for solutions (a) and (b) correspond tothe excitation frequency. The three discrete peaks for so-lutions (c) and (d) correspond to the excitation frequencyand two subharmonics at 1/3 and 2/3. Interestingly, pe-riod three implies chaos [12], and hence it is not surprisingthat the next solution (e) is chaotic. The non-periodicityis clearly visible in the time series (Fig. 3e) and also in theFourier transform (Fig. 4e) where discrete peaks changeto a continuous spectrum.A similar mechanical system (without the coupling tothe piezoceramic) was investigated by Litak et al. [13];this system showed Melnikov chaos with Wada patterns ofcomplex mixtures of three basins of attraction. The Wadapattern is a generalization of two mixed areas separatedby a fractal border [14]. To investigate this possibility westudy the corresponding basins of attraction.4 Influence of initial conditionsTo investigate the global dynamical properties of the en-ergy harvester model one can use basins of attraction.In this approach, the stability of particular solutions canbe measured by the volume of their basins of attraction,and the borders between them. Consider first the solu-tions for the harvester with tip mass Mt = 10.19 g,Page 4 of 7 Eur. Phys. J. B (2016) 89: 990 10 20 30 40 50−0.06−0.04−0.020time t (s)dispacement x (m)0 10 20 30 40 50−0.200.2time t (s)voltage U (mV)(a)0 10 20 30 40 5000.020.040.06time t (s)dispacement x (m)0 10 20 30 40 50−0.200.2time t (s)voltage U (mV)(b)0 10 20 30 40 50−0.100.1time t (s)dispacement x (m)0 10 20 30 40 50−0.500.5time t (s)voltage U (mV)(c)0 10 20 30 40 50−0.100.1time t (s)dispacement x (m)0 10 20 30 40 50−0.500.5time t (s)voltage U (mV)(d)0 10 20 30 40 50−0.200.2time t (s)dispacement x (m)0 10 20 30 40 50−101time t (s)voltage U (mV)(e)0 10 20 30 40 50−0.200.2time t (s)dispacement x (m)0 10 20 30 40 50−202time t (s)voltage U (mV)(f)Fig. 3. Displacement and voltage time series corresponding to the solutions presented in Figure 2.corresponding to the solutions in Figures 2 to 4. Fig-ure 5 shows the initial conditions that lead to the at-tractors, which have very different levels of power out-put. The grid of initial conditions considered in Figure 5is (x0, y0) ∈ [−0.3, 0.3] m×[−0.3, 0.3] m/s and V = 0.The percentage area for the different attractors is as fol-lows: Attractor (f), 61%; attractor (e), 32%; attractors (a)and (b), 6.99%; attractors (c) and (d), 0.01%. Obviously,the resonant regular attractor (f) is the most common.This attractor has regions of the basin with intensive mix-ing with other attractors, but also has has a compactregion. For small ranges of tip mass displacement andvelocities, the single well solutions (a) and (b), are mixedwith the regular cross-well solutions (c) and (d), andalso with the chaotic cross-well solution (e). Interestingly,large amplitude solutions (f) are also present in these re-gions. In a larger scale, the compact region of solution(f) has a definite border to the mixed solution regions.Additionally, single well solutions (a) and (b) are mostprobable in the center of studied area (represented by thetwo red islands), while (c) and (a) can be classified as rareattractors [15]. Finally, one can distinguish the Wada pat-terns with the mixture of all attractors (a) to (f) embeddedin the compact area of the resonant regular solution (f).By increasing the tip mass Mt to 10.24 g (Fig. 6) weobserve the quantitative change of the basins of attractionplotted in Figure 5 with small changes to the probabilityof their appearance.However for Mt = 10.39 g the change is more signif-icant (Fig. 7). The cross well non-resonant solutions (a)Eur. Phys. J. B (2016) 89: 99 Page 5 of 70 0.2 0.4 0.6 0.8 1 1.200.020.04frequency f (Hz)displacemnt x (m)0 0.2 0.4 0.6 0.8 1 1.200.10.2frequency f (Hz)voltage U (mV)(a)0 0.2 0.4 0.6 0.8 1 1.200.020.04frequency f (Hz)displacemnt x (m)0 0.2 0.4 0.6 0.8 1 1.200.10.2frequency f (Hz)voltage U (mV)(b)0 0.2 0.4 0.6 0.8 1 1.200.050.1frequency f (Hz)displacemnt x (m)0 0.2 0.4 0.6 0.8 1 1.200.10.2frequency f (Hz)voltage U (mV)(c)0 0.2 0.4 0.6 0.8 1 1.200.050.1frequency f (Hz)displacemnt x (m)0 0.2 0.4 0.6 0.8 1 1.200.10.2frequency f (Hz)voltage U (mV)(d)0 0.2 0.4 0.6 0.8 1 1.200.020.04frequency f (Hz)displacemnt x (m)0 0.2 0.4 0.6 0.8 1 1.200.10.2frequency f (Hz)voltage U (mV)(e)0 0.2 0.4 0.6 0.8 1 1.200.10.20.3frequency f (Hz)displacemnt x (m)0 0.2 0.4 0.6 0.8 1 1.2012frequency f (Hz)voltage U (mV)(f)Fig. 4. Displacement and voltage Fourier spectra corresponding to the solutions presented in Figure 2.to (d) disappear. Consequently, the percentage appear-ance of the attractors is now: attractor (f), 57%; and at-tractor (e), 43%. Note that the Wada pattern is changedto a more simple Melnikov chaos [16] with a fractal borderbetween the two basins of attraction.5 ConclusionsThe examined energy harvester exhibited six coexisting vi-bration responses with regular and chaotic solutions. Thepower output increased once the system changed from asingle well solution to a cross well solution. Further in-creases in power were related to increases in the size ofthe trajectory of the tip mass. The corresponding basinsof attraction penetrated each other creating Wada pat-terns with complex mixtures of three or more basins ofattraction. We observe these transitions in the basins ofattraction by changing the tip mass. The system was ex-cited by an harmonic inertial force at a constant frequency,and the harvester responded at this frequency and alsowith additional subharmonics dependent on the value ofthe tip mass.It is clear that solutions of topologically different at-tractors would lead to different power outputs in the en-ergy harvesting process. The given basins of attractionenable the stability of the solution to be determined. Bycombining the distributions of the probability of particularPage 6 of 7 Eur. Phys. J. B (2016) 89: 99Fig. 5. Basins of attraction calculated for the tip mass Mt =10.19 g: attractors (a) and (b), red; attractors (c) and (d),black; attractor (e), green; attractor (f), blue.Fig. 6. Basins of attraction calculated for the tip mass Mt =10.24 g: attractors (a) and (b), red; attractors (c) and (d),black; attractor (e), green; attractor (f), blue.Fig. 7. Basins of attraction calculated for the tip mass Mt =10.39 g: attractor (e), green; attractor (f), blue.solutions occurring, and the associated power output, onecan estimate the effective power output of the nonlinearharvester.GL and AS gratefully acknowledge the support of the Pol-ish National Science Center under grant No. 2012/05/B/ST8/00080, and No. DEC-2013/11/D/ST8/ 03308, respectively.The authors acknowledge the support of the Engineering andPhysical Sciences Research Council through grant numberEP/K003836.AppendixWe assume the mode shape of the beam (Fig. 1) ψ(s) isψ(s) = 1 − cos(πs2l1). (A.1)With this assumed mode shape, the displacements v and u(Fig. 1) are not independent, and u can be expressed interms of v. Using this single mode approximation [9–11],the kinetic and potential energies of the system in termsof the transverse displacement of the tip mass, v, areT =12ρA∫ L0[(v̇ψ(s) + ż)2 +(vv̇∫ s0(ψ′(ξ))2dξ)2]ds+12Mt⎡⎣(v̇ + ż)2 +(vv̇∫ Lt0(ψ′(s))2ds)2⎤⎦+12It[v̇ψ′(Lt) +12v2v̇(ψ′(Lt))3]2(A.2)=12ρA[N1v̇2 + 2N2v̇ż + ż2L+N3 (vv̇)2]+12Mt[(v̇ + ż)2 +N24 (vv̇)2]+12It[N5v̇ +12N35 v2v̇]2(A.3)andΠ =12EI∫ L0[vψ(s)′′ +12v3(ψ′(s))2ψ′′(s)]2ds− 12ρAgv2∫ L0[∫ s0(ψ′(ξ))2dξ]ds− 12Mtgv2∫ Lt0(ψ′(s))2ds (A.4)=12EI[N6v2 +N7v4 +14N8v6]− 12N9ρAgv2− 12N4Mtgv2 (A.5)Eur. Phys. J. B (2016) 89: 99 Page 7 of 7where the beam is assumed to have uniform inertia andstiffness properties. A non-uniform beam is easily mod-eled by including the mechanical beam properties in theenergy integrals. The constants N1 to N9, are based onthe particular integrals of the assumed mode shape ψ(s),and are given by:N1 =∫ L0(ψ(s))2ds =(3π − 82π)L,N2 =∫ L0ψ(s)ds =(π − 2π)L,N3 =∫ L0(∫ s0(ψ′(ξ))2dξ)2ds =π2(2π2 − 9)3841L,N4 =∫ L0(ψ′(s))2ds =π281L,N5 = ψ′(L) =π21L,N6 =∫ L0(ψ′′(s))2ds =π4321L3N7 =∫ L0(ψ′(s)ψ′′(s))2ds =π6291L5N8 =∫ L0(ψ′(s))4(ψ′′(s))2ds =π840961L7N9 =∫ L0[∫ s0(ψ′(ξ))2dξ]ds = −14+116π2 (A.6)where s is a variable measured along the beam (Fig. 1).Similarly, the electromechanical coupling constants (inEq. (1)) areΘ1 = γcψ′(Lc) and Θ2 =12γc(ψ′(Lc))3(A.7)where Lc is the active length of the piezoelectric material,and γc depends on the properties and arrangement of thepiezoelectric material [9].The Lagrange equations of the second type lead toequations (1) and (2).References1. P.D. Mitcheson, E.M. Yeatman, G.K. Rao, A.S. Holmes,T.C. Green, Proc. IEEE 96, 1457 (2008)2. M.F. Daqaq, R. Masana, A. Erturk, D.D. Quinn, Appl.Mech. Rev. 66, 040801 (2014)3. R.L. Harne, K.W. Wang, Smart Mat. Struct. 22, 023001(2013)4. S.P. Pellegrini, N. Tolou, M. Schenk, J.L. Herder, J. Intell.Mater. Syst. Struct. 24, 1303 (2013)5. J. Twiefel, H. Westermann, J. Intell. Mater. Syst. Struct.24, 1291 (2013)6. J. Cao, S. Zhou, D.J. Inman, Y. Chen, Nonlinear Dyn. 80,1705 (2014)7. J. Cao, A. Syta, G. Litak, S. Zhou, D.J. Inman, Y. Chen,Eur. Phys. J. Plus 130, 103 (2015)8. A. Syta, C.R. Bowen, H.A. Kim, A. Rysak, G. Litak,Meccanica 50, 1961 (2015)9. M.I. Friswell, S.F. Ali, O. Bilgen, S. Adhikari, A.W. Lees,G. Litak, J. Intell. Mater. Syst. Struct. 23, 1505 (2012)10. L. Zavodney, A. Nayfeh, Int. J. Non-Linear Mechanics 24,105 (1989)11. E. Esmailzadeh, G. Nakhaie-Jazar, Int. J. Non-LinearMechanics 33, 567 (1998)12. T.Y. Li, J.A. Yorke, Amer. Math. Monthly 82, 985 (1975)13. G. Litak, M. Coccolo, M.I. Friswell, S.F. Ali, S. Adhikari,A.W. Lees, O. Bilgen, in Proceedings of NSC 2012 - 4thIEEE International Conference on Nonlinear Science andComplexity, August 6-11, 2012, Budapest, Hungary, pp.113–116.14. J. Aguirre, J.C. Vallejo, M.A.F. Sanjuan, Phys. Rev. E 64,066208 (2001)15. M.V. Zakrzhevsky, in Proceedings of the ASME –IDETC/EIE, August 30-September 2, 2009, San Diego,California, USA, DETC2009-86657, pp. 1411–1418.16. J. Guckenheimer, P. Holmes, Nonlinear Oscillations,Dynamical Systems, and Bifurcations of Vector Fields(Springer, 1983)Open Access This is an open access article distributedunder the terms of the Creative Commons AttributionLicense (http://creativecommons.org/licenses/by/4.0), whichpermits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

https://link.springer.com/content/pdf/10.1140%2Fepjb%2Fe2016-60699-0.pdf

Multiple solutions and corresponding power output of a nonlinear bistable piezoelectric energy harvester

Abstract

Similar works

Full text

Available Versions

Springer - Publisher Connector

Springer - Publisher Connector

Crossref

Cronfa at Swansea University

Enlighten