Research Article Chaotic Dynamics-Based Analysis of Broadband Piezoelectric Vibration Energy Harvesting Enhanced by Using Nonlinearity

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1 Shock an Vibration Volume 16, Article ID 3587, 11 pages Research Article Chaotic Dynamics-Base Analysis of Broaban Piezoelectric Vibration Energy Harvesting Enhance by Using Nonlinearity Zhongsheng Chen, Bin Guo, Congcong Cheng, Hongwu Shi, an Yongmin Yang Science an Technology on Integrate Logistics Support Laboratory, National University of Defense Technology, Changsha 173, China Corresponence shoul be aresse to Zhongsheng Chen; czs Receive 31 August 15; Revise October 15; Accepte 7 October 15 Acaemic Eitor: Carlo Trigona Copyright 16 Zhongsheng Chen et al. This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. Nonlinear magnetic forces are always use to enlarge resonant banwith of vibration energy harvesting systems with piezoelectric cantilever beams. However, how to etermine properly the istance between two magnets is one of the key engineering problems. In this paper, the Melnikov theory is introuce to overcome it. Firstly, the Melnikov state-space moel of the nonlinear piezoelectric vibration energy harvesting (PVEH) system is built. Base on it, chaotic ynamics mechanisms of achieving broaban PVEH by nonlinearity are expose by potential function of the unperturbe nonlinear PVEH system. Then the corresponing Melnikov function of the nonlinear PVEH system is efine, base on which two Melnikov necessary conitions of etermining the istance are obtaine. Finally, numerical simulations are one to testify the theoretic results. The results emonstrate that the istance is closely relate to the excitation amplitue an frequency once geometric an material parameters are fixe. Uner a singlefrequency excitation, the nonlinear PVEH system can generate a perioic vibration aroun a stable point, a large-amplitue vibration aroun two stable points, or a chaotic vibration. The propose metho is very valuable for optimally esigning an utilizing nonlinear broaban PVEH evices in engineering applications. 1. Introuction Nowaays, Internet of Things (IoTs) is a research an application hot spot, which can allow objects to be sense an controlle remotely across existing network infrastructure. A great number of embee or wireless sensor noes (WSNs)areistributeinwieanremoteareas,suchas environmental monitoring, brige health monitoring, an smart housing. However, there is an urgent nee to provie long-term an reliable electrical power for those WSNs. Traitional solutions are to utilize batteries for main sources ofwsns.asweallknowthatbatterieshavetherawbackof limite life span, they nee to be replace regularly. However, it is often costly an ifficult to replace so many batteries, in particular in highly angerous or unreachable areas. Hence, it is essential for WSNs to have autonomous an long-life power sources. Fortunately, avances in low power esign open the possibility of harvesting ambient vibration energy to power WSNs, which is calle vibration energy harvesting incluing electrostatic [1], electromagnetic [], an piezoelectric mechanisms [3]. In particular, piezoelectric mechanisms are of particular interest ue to their high energy ensities, no electromagnetic inference, an integration potential, which is always calle piezoelectric vibration energy harvesting (PVEH). In early stuies, linear PVEH evices compose of a piezoelectric cantilever beam an a mass attache to the free en are wiely investigate. As the piezoelectric cantilever beam oscillates with the host base, the inuce mechanical strain prouces charges across two electroes, which can be captureanstoretogenerateanacvoltage.accoringto existing conclusions [], a linear PVEH system can prouce themaximumpoweronlywhentheexcitationfrequency is equal to its resonant frequency. Otherwise, its harvesting efficiency will greatly ecrease once the excitation frequency eviates a little from the resonant frequency. On the other han, resonant frequency of a PVEH evice is closely relate to its geometries, so its volume often has to be large in orer to reuce the resonant frequency. Consequently, there

2 Shock an Vibration are many limitations of linear PVEH evices in engineering applications, such as self-powere embee wireless sensors. The first reason is that frequency spectrum of ambient vibration is usually broa an time-varying, instea of being sharply peake at some frequency. The secon reason is that frequency spectrums of many ambient vibrations are particularly rich in low-frequency regions, so it is very ifficult to buil small-scale, low-frequency resonant PVEH systems. In orer to overcome the above limitations, nonlinear PVEH systems have been propose for ten years. Gammaitoni et al. [5] firstly iscusse vibration to electricity energy conversion mechanisms by using nonlinear stochastic ynamics. Chen et al. [6] built a general nonlinear PVEH moel by combining nonlinear stiffness, nonlinear amping, an nonlinear piezoelectric coupling coefficient simultaneously. Base on it, the effects of ifferent nonlinear parameters on the PVEH system were investigate. By this way, banwith of useful frequencies has the potential to be extene. Up to now, most works on nonlinear PVEH can be ivie into two classes. The first class is base on Duffing-like equations [5, 7 9]. Uner this case, a Duffing-like moel is buil baseonnonlinearynamicsanthenanalyzebypurely numerical simulations. It can be use to expose the effects of nonlinearity, but the numerical moel has little physical meaning an thus is unable to guie practical esigning. The secon class is base on theoretical analysis an experimental testing [1 1]. For a given PVEH structure, theoretical analysis is one to emonstrate its nonlinearity an experiments areonetovaliateitsbroabancharacteristics.amongall stuies, it is the most easy an common way to construct nonlinear PVEH systems by using magnetic forces [13 17]. Accoring to existing stuies, the istance between two magnets is a key geometric parameter once other geometric an material parameters are fixe. It shoul be etermine carefully to construct an expecte nonlinear PVEH system. However, how to easily etermine the istance between two magnets is still an urgent problem in engineering applications. Otherwise, many trials have to be one, which is much more inconvenient an costly. Thus it is necessary to propose a quantitative criterion of etermining the istance. The Melnikov theory provies a unifie framework for stuying transitions an chaos in a wie class of eterministic an stochastic nonlinear planar ynamic systems with restoring forces. It can be use to fin necessary conitions for homoclinic bifurcations to occur in multiwell systems [18]. This is the same case as nonlinear PVEH systems where restoring forces come from electromechanical coupling force an nonlinear magnetic force. The Melnikov criterion helps in an inirect way in ascertaining the values of ifferent parameters for bifurcations. Stanton et al. [19] numerically stuie a Melnikov-base metho to characterize the ynamics of the bistable piezoelectric inertial generator, but there isnoobviousphysicalmeaning.thegoalofthispaperisto buil necessary conitions of etermining a proper istance between two magnets base on the Melnikov metho. The left contents are organize as follows: state-space moel of one nonlinear PVEH system with nonlinear magnetic force is built in Section. In Section 3, a perturbe state-space representation of the nonlinear PVEH system is presente, P(t) F PZT α z(t) N F { S Magnets Figure 1: Schematic iagram of nonlinear PVEH system with two magnets. base on which chaotic ynamics mechanisms of broaban PVEH enhance by nonlinearity are explaine. In Section, the Melnikov function of the nonlinear PVEH system is efine an calculate. Base on it, two necessary conitions of etermining the istance are propose. Simulations are one in Section 5 to valiate the theoretic results. Finally, conclusions are iscusse in Section 6.. State-Space Moel of One Nonlinear PVEH System with Nonlinear Magnetic Force As shown in Figure 1, a typical nonlinear piezoelectric energy harvesting system is compose of a piezoelectric cantilever beam an a pair of permanent magnets. The first magnet is mounte on the free en of the beam as a proof mass an the secon magnet is oriente with opposite polarity to the first magnet. The istance between the two magnets is enote as. The length, with, an thickness of the piezoelectric beam are enote as l, w, anh, respectively.theproofmassis assume to be a lumpe mass M.Thevibrationisplacement of the mass at time t is enote as z(t). Accoring to previous conclusions [7, 1], electromechanical coupling moel of the nonlinear PVEH system in Figure 1 can be formulate as M eq z (t) +c eq z (t) +K eq z (t) +ηv p (t) =P(t) +F V (t), ηz (t) =C p V p (t) + V p (t), R L where M eq is the equivalent mass of the piezoelectric beam an the proof mass. c eq is the equivalent amper of the piezoelectric beam. K eq is the equivalent stiffness of the piezoelectric beam. η is the piezoelectric coupling coefficient. C p is the equivalent capacitor of the piezoelectric beam. z(t) is the isplacement of M eq. R L is the purely resistive loa. P(t) is the force applie to the mass of the equivalent SDOF moel. is the horizontal istance of two magnets. F V (t) is the vertical component of the repulsive magnetic force (F()). Generally speaking, z(t) is much smaller than,sothatwewillhave F V (t) = F () S N (1) F () z (t) 3 z3 (t). ()

3 Shock an Vibration 3 Furthermore, the repulsive magnetic force, F(), can be approximate as [] F () =WHt 1/3 m B r B () f (), (3) where W, H, ant m correspon to the with, height, an thickness of the magnet, respectively. B r is the resiual flux ensity of the magnet. B() is calculate by [] B () = B r WH π (tan 1 W +H + tan 1 WH ). (l+) W +H +(l+) f() is an experiential moification function, which is calculate by [] () f () = ( e ) 1 6. (5) Accoring to the Melnikov theory, the first step is to rewrite (1) into a state-space form in orer to use the Melnikov metho to analyze nonlinear ynamic systems. For thesakeofsimplicity,p(t) isassumetobeaharmonic excitation; that is, P(t) = P m cos ωt. P m,ωare the excitation magnitue an angular frequency, respectively. Then (1) can be transforme into z (t) +A 1 z (t) +A z (t) = A 3F () A V p (t), V p (t) =A 5 V P (t) +A 6 z (t), z (t) A 3F () 3 z 3 (t) +A 3 P m cos ωt where A 1 =c eq /M eq, A =K eq /M eq, A 3 =1/M eq, A = η/m eq, A 5 = 1/R L C p,an A 6 =η/c p. By enoting z(t) z, V p (t) V p, y(t) = z(t) y, statespace moel of the nonlinear PVEH system can be obtaine as (7) from (6): z=y, y=[ A 3F () A ]z A 3F () 3 z 3 A 1 y +A 3 P m cos ωt A V p, V p =A 5 V p +A 6 y. 3. Chaotic Dynamics Mechanisms of Broaban PVEH Enhance by Nonlinearity In orer to use the Melnikov metho, the PVEH system shoul be represente as a perturbe state-space moel. Here itcanbeeasilyseenthattheperturbationisuetovibration (6) (7) excitation, amping, an electromechanical coupling. Then the perturbe state-space moel can be represente as z=y, y=[ A 3F () A ]z A 3F () 3 z 3 +ε[a 3 P m cos ωt A V p A 1 y], V p =A 5 V p +A 6 y, where ε is a small factor that measures the smallness of the perturbation. TheclassicalMelnikovmethoismainlyusefortwoimensional systems. However, (8) is obviously a threeimensional system, so here it is ifficult to irectly use the Melnikov metho. In orer to overcome this problem, one solution in [19] is referre to for converting the threeimensional system into a two-imensional system. Then, (8) can be represente as a two-imensional form as follows: where z X =( ), y f (X) =( [ A 3F () (8) X = f (X) +εg (X,t), (9) y A ]z A 3F () 3 z 3), g (X,t) =( ). A 3 P m cos ωt A V p A 1 y (1) Obviously, when ε =, the unperturbe system is a nonlinear planar ynamic system. Uner this case, potential function of the unperturbe PVEH system can be calculate as follows: V (z) = 1 [A 3F () A ]z + A 3F () 8 3 z. (11) Itcanbeseenfrom(11)thatpotentialfunctionofthe unperturbe PVEH system is closely relate to the istance (). When the istance () isproperlyselecte,itwillbe similartothecurveinfigure.thatistosay,itwillbea ouble-well ynamic system incluing two potential barriers. Inee, the bistability provies essential physical founation of enhancing broaban PVEH, which has become a research hotspotinrecentyears[1 3]. Accoring to (9), perturbe forces of the nonlinear ynamic system are from excitation, amping, an electromechanical coupling forces when ε =. Depening on the amount of ε, ifferent vibration behaviors of the nonlinear PVEH system will occur. When the perturbe force is much

4 Shock an Vibration V(z) Figure : Potential function of the unperturbe nonlinear planar system. smaller, it is ifficult for the system to cross the potential barrier. Uner this case, the system will perioically be aroun one of stable positions accoring to initial conitions. Whentheperturbeforceistoolarge,thesystemisable to cross the potential barrier. Uner this case, the system will perioically vibrate between two potential wells, leaing to a large-amplitue motion. When the perturbe force is intermeiate, the motion will be irregular. Uner this case, thesystemmayvibratearounoneofstablepositionsor escape stochastically between two stable positions if initial values an excitation frequency satisfy necessary conitions. This kin of transitions is also calle a chaotic motion, which is much expecte for PVEH. The reason is that continuous transitions between two potential wells efinitely provie much more power than chaotic motions, but it may not be guarantee for broabans, while chaotic motions at least can provie intermittent large-amplitue oscillations in a broaban. Thus one can unerstan why nonlinearity can be use to achieve high-efficiency broaban PVEH. In practice, however, the key is how to etermine a proper istance () in orer to make the nonlinear PVEH system generate a chaotic motion uner a given vibration excitation. Fortunately, the Melnikov criterion helps in an inirect way in ascertaining the values of the ifferent parameters for bifurcations. That is to say, when the Melnikov function has simple zeros inepenent of the small perturbation factor, the stable an unstable manifols of the perturbe system will intersect each other transversely. Then it implies chaos-like behaviors may occur in the context of the Smale horseshoe map,whichisalsotheresearchbasisofthispaper.. Melnikov Function-Base Necessary Conitions of Broaban PVEH Using Nonlinearity As state in Section 3, nonlinear magnetic forces can be use to cause the nonlinear PVEH system to be in chaotic conitions. In engineering applications, the istance () between two magnets is the most key parameter once geometric an material properties of the harvesting structure are fixe. However, how to etermine is always a ifficult task. In orer to solve this problem, necessary conitions of eterminingtheistanceareproposebaseonthemelnikov function. z Base on (9), it can be easily seen that unperturbe nonlinear PVEH system is a Hamiltonian system an the Hamiltonian function is efine as follows: H(y,z)= 1 y 1 [A 3F () A ]z + A 3F () 8 3 z =λ, (1) where λ enotes one orbit in the phase plane of the Hamiltonian system, which epens on initial conitions. Next three singular points of the unperturbe nonlinear PVEH system can be calculate as follows by setting y= y= : Q = (, ), Q 1 =( (A 3F () A ),), A 3 F () Q =( (A 3F () A ),). A 3 F () (13) Uner this case, it can be seen from basic efinitions that Q is a sale point an Q 1, Q are two central points locate at two sies of Q.However,itmustbenotethatQ 1 an Q exist only when (1) is satisfie: F () > A A 3. (1) Thus (1) is the first necessary conition of etermining the istance (). Furthermore, the orbit escribe by (1) is a homoclinic orbitwhichwillcrossthesalepoint(q ). Uner this case, λ isequaltozero.thenwewillhave y=±z α βz = z, (15) where α=[a 3 F() A ]/ an β=a 3 F()/ 3.Thesign ± just represents the right an left parts of the orbit in the phase plane. Due to its symmetry, only the sign + is consiere here. Then (15) can be transforme into 1 z α βz z = t. (16) By solving (16), analytic solutions of the homoclinic orbit can be obtaine as z H (t) = α sec (i αt), β y H (t) = αi sec (i αt) tan (i αt), β (17) where the superscript H enotes a homoclinic orbit an i enotes an imaginary number.

5 Shock an Vibration 5 Substituting (17) into (8), the homoclinic orbit (V H p (t))of the output voltage is calculate as V H p (t) = iαa 6 β ea 5t t [ e A 5t sec (i αt ) tan (i αt )t ]. (18) Finally, the Melnikov function of the perturbe nonlinear PVEH system can be efine as follows: + M(t )= f [z H,y H ] g[z H,y H,t+t ]t, (19) where enotes the wege prouct. By substituting (17) an (18) into (19), we will have M(t )= Ω 1 Ω +Ω 3 sin ωt S (ω), () where Ω 1 = A 1 α α/3β, Ω = (A A 6 α /β) + h(t) sec(i αt) tan(i αt)t, Ω 3 = A 3 P m, S(ω) = (πω/ β) sech(πω/ α), an h(t) = e A5t [ t e A 5t sec(i αt ) tan(i αt )t ]. Here, Ω canbecalculatenumerically. Accoring to the Melnikov criteria, homoclinic bifurcation an chaotic motions may occur if M(t ) has simple zeros inepenent of ε. Then the following necessary conition must be satisfie base on (): Ω <M = 1 +Ω <1. (1) Ω 3 S (ω) Ω 1,Ω arecloselyrelateto, so (1) is the secon necessary conition of etermining. Up to now, the propose Melnikov necessary conitions ofeterminingtheistancearecomposeof(1)an(1). At the same time, one can see that the secon necessary conitionisalsocloselyrelatetoexcitationamplitue(p m ) an angular frequency (ω). Then one can expect that the excitation angular frequency (ω) inawiebanmaysatisfy the Melnikov conitions uner a proper istance. By this way, one can unerstan why nonlinearity can be use to enlarge resonant banwith of one PVEH system. 5. Simulation Valiations In orer to valiate the above theoretical results, simulations are carrie out to analyze the effects of key parameters on the Melnikov necessary conitions. Also output response characteristics of the nonlinear PVEH system are investigate uner ifferent vibration excitations. Firstly, geometric an material properties of the nonlinear piezoelectric structure are liste in Table 1. Uner this case, resonant frequency of the piezoelectric cantilever beam is calculate as about 6. Hz without consiering the secon magnet. The excitation frequency is enote as f. Thatistosay,ω=πf.Theloais fixe as R L = kω. Table 1: Geometric an material properties of the nonlinear piezoelectric structure. Structure Properties Value Elastic base Density, ρ s (kg/m 3 ) 85 Young s moulus, E (Gpa) 9 Thickness, h p (mm). Density, ρ s (kg/m 3 ) 75 PZT-5H Young s moulus, E p (Gpa) 66 Piezoelectric constant, 31 (pm/v) 19 Permittivity, ε s 33 /nf/m Length, L (mm) 1 Magnet With, W (mm) 6 Thickness, H (mm) 7 Resiual flux ensity, B r (T) Potential Function of the Unperturbe PVEH System. As state in Section 3, potential function of the unperturbe PVEH system is an important physical basis of achieving broa banwith. Base on (11), potential functions uner ifferent istances are plotte an compare in Figure 3. One can see that the ouble well in the potential function will isappear once the istance is too large or small. The reason may be that when is very small, the magnetic force is so large that the system will be a monostable system. Uner this case, neither a large-amplitue nor a chaotic motion will occur espite the vibration excitation, which is not useful for broaban PVEH. Again, one can unerstan why the istance () shoul be etermine carefully. 5.. The Effects of the Excitation Amplitue on the Melnikov Necessary Conitions. As state in Section, the first Melnikov necessary conition as (1) must be satisfie in orer to cause homoclinic bifurcation an chaotic motion. Base on (1), it can be easily foun that the istance shoul satisfy < 9.5 mm, which is inepenent of vibration excitations. Furthermore, the secon Melnikov necessary conition as (1) is closely relate to the excitation amplitue (P m ), so we will focus on the effects of P m on M The Effects of P m on the Banwith. Here the istance is fixe as =6mm accoring to the first Melnikov necessary conition. Then ifferent excitation amplitues are selecte as P m =.,.,.6,.8. ForeachP m, frequency response of M is calculate numerically as Figure, where the re line enotes the threshol. The following can be seen. (i) For any satisfying (1), there will be a broa frequency ban at which (1) is also satisfie. That is to say, homoclinic bifurcation an chaotic motion may occur in the broaban frequency ban. Atthesametime,thefrequencybanisalwayslocateatlowfrequency regions. (ii) As P m increases, the banwith will be enlarge an the central frequency will also increase. These results testify that broaban PVEH can be achieve by using proper nonlinearity The Effects of P m on the Distance. Here the excitation frequency is fixe as f = Hz. Then ifferent excitation

6 6 Shock an Vibration 6 3 Potential function V(z) Potential function V(z) 1 Potential function V(z) (a) =mm (b) =mm (c) =6mm Potential function V(z) 5 Potential function V(z) Potential function V(z) () =8mm (e) =1mm (f) =mm Figure 3: Potential functions of the unperturbe nonlinear PVEH system uner ifferent istances. amplitues are selecte as P m =.,.,.6,.8. For each P m,thecurvesofm with are calculate numerically as Figure 5, where the re line enotes the threshol. The following can be seen. (i) For any given P m,thereisarangeof so that the two Melnikov necessary conitions are satisfie, which can help to esign proper PVEH evices in practice. (ii) As P m increases, the range of will be extene Melnikov Scale Factor (S(f)). Denoting S(f) S(ω) ω=πf in (), it is obvious that whether the secon Melnikov necessary conition is easily satisfie or not epens on S(f) once an P m are given. Thus here S(f) is calle the Melnikov scale factor. Then the curves of S(f) uner ifferent istances are shown in Figure 6. One can see the following. (i) Uner a given, thereisanoptimal excitation frequency at which S(f) has a maximum value. That is to say, it is the most easy case to satisfy the secon Melnikov necessary conition an generate chaotic motions. (ii) As increase, the optimal excitation frequency will shift to low-frequency regions. 5.. Output Responses of the Nonlinear PVEH System uner a Single-Frequency Excitation. In orer to testify three types of motions of the nonlinear PVEH system uner ifferent excitation frequencies, three kins of single-frequency excitations (f =5Hz, f=1hz, an f=1hz) are use, respectively. Here we choose =6mm an P m =.;thenoutput responses of the nonlinear PVEH system are shown in Figures 7 9, incluing the phase plane, voltage, an isplacement.

7 Shock an Vibration M 5 M Frequency f (Hz) Distance (mm) 9 P m =. P m =. P m =.6 P m =.8 P m =. P m =. P m =.6 P m =.8 Figure : Frequency responses of M uner ifferent excitation amplitues. Figure 5: The curves of M amplitues. with uner ifferent excitation ItcanbeseenfromthephaseplaneinFigure7thatthe nonlinear system vibrates perioically aroun one of the stable positions when f=5hz.uner this case,the stable vibration amplitue is small, leaing to small output voltages. It canbeseenfromthephaseplaneinfigure8thatthenonlinear system vibrates perioically between two stable points an a large-amplitue motion occurs when f=1hz. Uner this case, the stable vibration amplitue is large, leaing to largeoutputvoltages.itcanbeseenfromthephaseplane in Figure 9 that the nonlinear system escapes stochastically between two stable positions when f = 5Hz an a chaotic motion occurs. Uner this case, the stable vibration amplitueisalsolarge. Furthermore, average output powers uner the above three motions are calculate as.5mw, 177.mW, an 87. mw, respectively. It is obvious that average output power uner a large-amplitue or chaotic motion is larger than that uner a small-amplitue motion. Although average output power uner a large-amplitue motion is the largest, it is much more sensitive to the excitation frequency. Thus it is ifficult to keep large-amplitue motions ue to interrupts in engineering applications. A chaotic motion can occur in a broa frequency ban, so it is the most suitable for engineering applications. Next, we choose =mm an f=1hz. Obviously, the first Melnikov necessary conition as (1) is not satisfie uner this istance. At this time, the effect of the nonlinear magnetic force is very small an the nonlinear system egraes to a linear one. Then the phase plane, voltage, an isplacement are also calculate as Figure 1. One can see the nonlinear system vibrates perioically aroun the equilibrium position, instea of a chaotic motion. Uner this case, both the stable vibration amplitue an output voltages are smaller than the counterparts uner a chaotic motion. Its average output power is calculate as.5 mw. By comparing S(f) Excitation frequency f (Hz) =5mm =6mm =7mm =8mm Figure 6: The curves of S(f) uner ifferent istances. Figure 1 with Figure 7, one can see that a nonlinear PVEH system with a too small istance is even worse than a linear PVEH system Output Responses of the Nonlinear PVEH System uner a Broaban Excitation. In orer to valiate broaban characteristics of the nonlinear PVEH system, a time-varying excitation frequency is use here; that is, f=1t, t [, 1]. Two istances, = 6mm an = mm, are chosen for comparisons. Obviously, the former enotes a nonlinear PVEH system an the latter enotes a linear one. Then output voltagesofthetwopvehsystemsarecalculateasfigure11. ItcanbeseenthatthebanwithofanonlinearPVEHsystem is larger than that of a linear one, while the maximum output

8 8 Shock an Vibration.15 z Voltage V o (V) y 1 1 Figure 7: A perioic motion aroun a stable position when f=5hz...1 z Voltage V o (V) y 1. 1 Figure 8: A large-amplitue motion across two stable positions when f=1hz. voltage of a linear system is only achieve near its resonant frequency. 6. Conclusions It has been pointe out that nonlinearity by magnetic force can be use to exten resonant frequency banwith of a PVEH system. However, how to etermine the istance between two magnets is a key problem in engineering applications. In this paper, the Melnikov theory is introuce to solve it. Firstly, the Melnikov state-space moel of the nonlinear PVEH system is built. Chaotic ynamics mechanisms of achieving broaban PVEH are expose by the potential function of the unperturbe nonlinear PVEH system. Then the Melnikov function of the nonlinear PVEH system is efine, base on which two Melnikov necessary conitions of etermining the istance are obtaine. Finally, simulations are one to testify the theoretic results. Main conclusionsincluethefollowing.(i)bothatoolargean a small istance are not helpful to form ouble wells. The istance shoul satisfy the first Melnikov necessary conition. (ii) For a given istance satisfying (1), there will

9 Shock an Vibration 9..1 z Voltage V o (V) y 1. 1 Figure 9: A homoclinic bifurcation an chaotic motion when f=1hz z Voltage V o (V) y Figure 1: A perioic motion aroun the equilibrium position when =mm an f=1hz. beabroafrequencybansatisfyingtheseconmelnikov necessary conition. For a given excitation frequency, there also will be a range of the istance satisfying the two Melnikov necessary conitions. (iii) For any given istance satisfying the first Melnikov necessary conition, there will be an excitation frequency at which the secon Melnikov necessary conition is the most easily satisfie. (iv) Uner a single-frequency excitation, the nonlinear PVEH system can generate a perioic vibration aroun a stable point, a largeamplitue vibration aroun two stable points, or a chaotic vibration. In aition, it must be note that the propose Melnikov conitions are just necessary, not sufficient. The problemneestobeinvestigatefurtherinfutureworks. Despite it, however, the above conclusions will still be much more significant for esigning nonlinear PVEH evices in practice. Conflict of Interests The authors eclare that there is no conflict of interests regaring the publication of this paper. Acknowlegment This work was supporte by the National Natural Science Founation of China (Grants nos an ).

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