# Analysis of Dynamic Characteristics of Tristable Piezoelectric Energy Harvester Based on the Modified Model

Authors: Dawei Man; Deheng Xu; Xingchen Kuang; Xiaofang Kang; Qinghu Xu; Yu Zhang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Taking into account the gravitational potential energy of the piezoelectric energy harvester, the size effect, and the rotary inertia of tip magnet, a more accurate distributed parametric electromechanical coupling equation of tristable cantilever piezoelectric energy harvester is established by using the generalized Hamilton variational principle. The effects of magnet spacing, the mass of tip magnet, the thickness ratio of piezoelectric layer and substrate, and the load resistance and piezoelectric material on the performance of piezoelectric energy capture system are studied by using multiscale method. The results show that the potential well depth can be changed by reasonably adjusting the magnet spacing, so as to improve the energy capture efficiency of the system. Increasing the mass of tip magnet can enhance the output power and frequency bandwidth of the interwell motion. When the thickness of the piezoelectric beam remains unchanged, the optimal load impedance of the system increases along with the increase of thickness ratio of piezoelectric layer and substrate. Compared with the traditional model, which neglects the system gravitational potential energy, the eccentricity, and the rotary inertia of the tip magnet, the calculation results of the frequency bandwidth and the peak power of the modified model have significantly increased.


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## Body

## 1. Introduction

The extensive application of wireless sensors in building structural health monitoring, environmental monitoring, and military fields has put forward higher requirements for the cleanliness and sustainability of the power supply. The piezoelectric energy harvester (PEH) can collect vibration energy in the surrounding environment and convert it into sustainable and clean electricity. Due to its simple structure and compact and easy integration, it has great potential to implement wireless sensor nodes and has received extensive attention in recent years [1–4]. The primary structure of the PEH initially is a linear resonator that mainly consists of a piezoelectric cantilever beam with a tip mass. Such linear PEH has a narrow frequency bandwidth that can achieve a peak value only when the external excitation frequency is in agreement with its resonant frequency [5–7]. This makes such linear PEH difficult to meet the requirements of the practical application [8, 9]. To make it more applicable, a variety of complex energy capture systems with active or adaptive techniques have been developed. Among these techniques, nonlinear techniques have been proved to be more suitable for energy harvesting from ambient vibrations in practical applications due to the less sensitivity to variations in the excited frequency of the realistic operational conditions [10,11].The nonlinearity of the PEHs caused by the magnetic force can be generally classified into three main categories, namely, monostable [12, 13], bistable [14–17], and tristable [18–20] PEHs. Bistable piezoelectric energy harvester (BPEH) has two symmetric potential wells and can oscillate between these two potential wells under low-frequency external excitations, which can greatly enhance the power outputs. Stanton et al. [21] established the distribution parameter model of the cantilever BPEH. Based on the harmonic balance method, the analytical expression of the steady-state response of the energy capture system is obtained and its working characteristics are analyzed. Tang et al. [22] investigated the voltage output of a bistable cantilever piezoelectric energy harvesting system under different excitations and analyzed the influence of magnet spacing on the system response. Kim et al. [23] proposed an electromechanical coupling equation for a hysteresis reversible cantilever-type magnetoelastic BPEH. Wang et al. [24] designed an asymmetric bistable piezoelectric energy harvester composed of a piezoelectric cantilever beam and rotatable magnets and further proposed a method for compensating asymmetric potential wells by appropriately changing the magnet's declination angle. These investigations have shown that the working bandwidth and output power of the BPEH can be greatly improved after oscillating into an interwell motion. However, when the external excitation is low, it is difficult for the BPEH to get rid of the confinement of the potential well and move into the large interwell motion.In order to overcome this challenge and to produce high energy output at low excitation intensity, tristable piezoelectric energy harvesters (TPEHs) are proposed. Zhu et al. [25] established a TPEH based on the magnetic bistable piezoelectric cantilever beam model, when the angle and distance between two fixed magnets are appropriate, the potential energy function of the energy capture system presents a three-steady state and the potential barrier decreases. Numerical simulation and experimental results show that the TPEH can generate higher output power than the bistable piezoelectric energy harvester in a wider frequency range under lower excitation intensity. Li et al. [26] analyzed a tristable energy harvester which can obtain a high harvesting efficiency at low-frequency base excitation by tristable coherence resonance. Panyam et al. [27] theoretically and experimentally studied the influence of the electromechanical coupling coefficient on the effective bandwidth of the TPEH. Kim et al. [28] proposed potential energy diagrams and found that the tristability of the TPEH is initiated by a new pitchfork bifurcation or a degenerate pitchfork bifurcation that leads to a pair of saddle-node bifurcations. Leng et al. [29] obtained a more precise magnetic force model for the TPEH using equivalent magnetizing current method and the calculation results of magnetic force are in good agreement with experimental data. Yang et al. [18,19] investigated the energy harvesting of the tristable hybrid vibration energy harvester by using geometric nonlinearity to tune the resonant frequency.In this paper, considering the gravitational potential energy of the system, the eccentricity, and rotary inertia of the tip magnet, a more accurate nonlinear three-steady-state distribution parameter model of the piezoelectric cantilever energy harvester is established based on the generalized Hamilton variational principle, and the analytic solution of the motion equation of the system is obtained by using the multiscale method. In the magnetic dipole model, the influence of eccentricity of the tip magnet is taken into account and the magnetic potential energy expressed by polynomial is obtained. The effects of the distance between the magnets, the mass of the tip magnet, the load resistance, the thickness ratio of the piezoelectric layer to the substrate layer, and the piezoelectric materials on the performance of the energy capture system were studied.


## 2. Theoretical Model of the TPEH

Figure1 shows the configuration of the TPEH with magnetic coupling considered in this paper. It mainly consists of a piezoelectric cantilever beam and three magnets (denoted as A, B, and C). The piezoelectric cantilever beam is fixed at the base and is composed of a metal substrate and a pair of piezoelectric layers (PZTs). Two identical PZTs, poled oppositely in the thickness direction, are tightly bonded on the upper and lower surfaces of the substrate. The two PZTs are electrically connected in series with an equivalent load resistance (R) represented as a low power electronic device. Magnet A (named tip magnet) is attached at the tip of the cantilever beam and two external magnets (B and C) are fixed at the frame. The horizontal gap between the tip magnet and magnets B and C is dh. The vertical gap between magnet B and magnet C is 2dv. Here, l and b denote the length and width of the piezoelectric cantilever beam, respectively; hs and tp denote the thickness of the metal substrate and the PZTs, respectively; e is the eccentricity of the tip magnet.Figure 1
Schematic of the considered TPEH.vbt represents the vibration displacement of the base, and s is the coordinate along the neutral axis of the piezoelectric cantilever beam. vs,t represents the displacement of the beam at s relative to its fixed end. The modeling of the PEH is based on the linear Euler-Bernoulli beam theory and the linear constitutive equations of the piezoelectric beam are assumed as follows:(1)T1s=YsS1sT1p=YpS1p−d31E3D3=d31T1+ε33TE3,where Y is Young’s modulus, subscript/superscript p and s represent the piezoelectric layers and substrate layer, S1 and T1 are the strain and the stress of the piezoelectric cantilever beam, respectively. 1 and 3 indicate x and y directions, D3 is the electric displacement, d31 is the piezoelectric constant, and ε33T is the dielectric constant. E3 is the electric field.The strain generated in the piezoelectric cantilever beam can be expressed as(2)E3=−Vt2tP,(3)S1s=S1p=−yv″.The Lagrange function of the piezoelectric energy harvesting system is as follows:(4)L=Tk+We−Ue−Ug−Um,where Tk is the kinetic energy, Ue is the potential energy, We is the electrical energy, Ug is the gravitational potential energy, and Um is the magnetic potential energy. Tk and We are as follows:(5)Tk=12∫0lmv˙+v˙bt2ds+12Mtv˙l,t+ev˙′l,t+v˙bt2+12Jv˙′l,t2,We=12Ypbd31h+tp2Vt∫0lv″ds+blε33SVt24tp,where m=2ρptpb+ρshsb, in which ρp and ρs are the density of the piezoelectric layers and substrate layer, respectively. Mt and J are mass and the rotary inertia of the tip mass, respectively. ε33s is the permittivity at constant strain.Ue is expressed as(6)Ue=12∫0lYIv″2−Ypbd31h+tp2Vtv″ds,where h=hs/2, YI=2/3Ysbh3+Ypb3h2tp+3htp2+tp3.Ug can be expressed as(7)Ug=mg∫0lv+vbtds+Mtgvl,t+vbt.According to the magnetic dipole model [25] and considering the eccentricity of the tip magnet, the magnetic potential energy can be given by(8)Um=μ04πmBrBA23+mCrCA23−mB⋅rBA3rBArBA25−mC⋅rCA3rCArCA25⋅mA,where μ0=4π×10−7H⋅m−1 is the magnetic permeability constant. As shown in Figure 1, the horizontal displacement can be evaluated by Δx≈e1−cosβ, where β=arctanv′l is the rotation angle of the tip magnet A. rBA=−dh−Δxvl,t+esinβ−dV and rCA=−dh−Δxvl,t+esinβ+dV are the vector directed from the magnetic moment source of magnet B and C to that of magnet C, respectively.mA=MAVAcosβMAVAsinβ, mB=−MBVB0 and mC=−MCVC0 represent the magnetic moment vectors of magnet A, B, and C, respectively, where MA (MB or MC) and VA (VB or VC) are the magnetization intensity and material volume of the magnet A (B or C), respectively.Using the Galerkin approach, the displacementvs,t is assumed as the following form:(9)vs,t=φrsηrt,where φrs and ηrt are the R-order mode shape function and the generalized mode coordinates of the cantilever beam, respectively.The modal shape function satisfies the following orthogonal relations:(10)∫0lφssmφrsds+φslMtφrl+φslMteφr′l+φS′lJ+Mte2φr′l+φS′lMteφrl=δrs,(11)∫0ld2φssds2YId2φrsds2ds=ωr2δrs,where δrs is the Kronecker delta, ωr=λr2YI/ml4 represents the resonance frequency of the r-th mode, and λr is the eigenvalue. The calculation of the oscillation function and eigenvalues is described in the literature [6].Substituting equation (9) into (8) and considering only the 1st order mode, Taylor's expansion of Um at ηt=0 can be written as(12)Um=k0−12k1η12+14k2η14+16k3η16+οη17,where, k0=2κq1/dh2+dv22,(13)k1=4κdh2+dv25/2q12.5q2−17.5q32−5q3q4dh2+dv2−q5k2=8κdh2+dv25/2q1−2.5q6−17.5q3q7+4.38q22−78.75q2q32+144.375q34+q4−2.5q7−17.5q2q3+52.5q33dh2+dv2+q5−2.5q2+17.5q32+5q3q8+q9k3=12κdh2+dv25/2q1−2.5q10+13.125q3q11+8.75q2q6+4.375q72−78.75q32q6+78.25q2q3q7−288.75q33q7+216.55q22q32−6.56q32−938.44q2q34+938.44q26+q41.875q11−17.5q3q6+8.75q2q7−78.75q32q7+39.38q22q3−288.75q2q32+375.375q35+q5−2.5q6−17.5q3q7+4.375q22−78.75q2q32+144.375q34+q8−2.5q7−17.5q2q3+52.5q33+q9−2.5q2+17.5q32+5q3q12+q13.Expressions for the coefficientsκ and qi=1⋯13 are shown in the appendix.Considering only the 1st order mode, equation (2) is substituted into the following Lagrangian variational equation using equation (9).(14)ddt∂L∂η˙−∂L∂η=Ft,ddt∂L∂V˙−∂L∂V=Qt,where F1t=−2ξ1ω1η˙1t is the generalized dissipative force, ω1 denotes the 1st order resonance frequency, ξ1 is the damping ratio, and Qt represents the generalized output charge. Q˙t=Vt/R, where R denotes the load impedance. The electromechanical coupling equations of the piezoelectric beam energy harvesting system can be obtained by using equation (13):(15)η¨1+2ξ1ω1η˙1+ω12η1+g0−k1η1+k2η13+k3η15−θ1Vt=−Γ1v¨bt,(16)CpV˙t+VtR+θ1η˙1=0,where ω12=YI∫0lφ1″ds, g0=mg∫0lφ1sds+Mtgφ1l, Γ1=m∫0lφ1sds+Mtφ1l+eφ1′l, θ1=Ypbd31h+tp/2∫0lφ1″ds, and Cp=blε33s/2tp.Assume that the excitation acceleration isv¨bt=v¯bcosωet, where vv¯b denotes the amplitude of the excitation, ωe denotes the circular frequency, and Cp denotes the capacitance. Introducing the dimensionless parameters shown in equation (17), the electromechanical coupling equations (14) and (16) can be rewritten as equations (18) and (19) in the dimensionless form.(17)x=η1l,V¯=VCplθ1,τ=ω1t,(18)x¨+2ξ1x˙+G0+1−K1x+K2x3+K3x5−ΘV¯=fcosωτ,(19)V¯˙+αV¯+x˙=0,where G0=G0/ω12l, K1=k1/ω12, K2=k2l2/ω12, K3=k3l4/ω12, Θ=θ12/Cpω12, f=−Γ1v¯b/ω12l, and ω=ωe/ω1, α=1/CpRω1.


## 3. Multiscale Method Analysis

The total potential energy function of the system can be expressed as(20)U=G0x+121−K1x2+14K2x4+16K3x6.Figure2 shows the potential energy curve of the system at different magnet spacing. For the traditional model (G0 = 0), when 1−K1>0, K2<0, K3>0 and K22−41−K1K3>0, the two symmetric potential wells associated with the equilibria at xs=∓−K2+K22−41−K1K3/2K3 are separated by the potential well associated with the trivial equilibrium, xs = 0. In this scenario, the potential energy function is symmetrically tristable. With the increase of dv, the middle potential well depth increases while the outer potential well depth decreases when dh remains constant. When the dv is constant, the middle and outer middle potential well depths decrease with the increase of dh. When the modified model (G0 ≠ 0) is used, the potential energy curve will no longer be symmetric, but the variation pattern of potential well depth with magnet spacing is consistent with that of the conventional model.Figure 2
Tristable potential energy curve.Introducing a small perturbation parameterε, a new independent time variable Tn can be expressed as(21)Tn=εnτ.The derivative with respect toτ is(22)ddτ=∑n=0mεnDn.The displacement and output voltage response of the system can be respectively expressed as(23)xτ,ε=∑n=0mεnxnT0,T1,…,Tn,V¯τ,ε=∑n=0mεnV¯nT0,T1,…,Tn.The constant parameters such as nonlinear term coefficients, electromechanical coupling coefficients, and the excitation forcing in equation (18) are scaled so that the viscous damping effect appears at the same order of the perturbation problem. That is, we let(24)ξ1=εξ¯1,K2=εK¯2,K3=εK¯3,Θ=εΘ¯,f=εf¯.Substituting equation (24) into equation (18), we get(25)x¨+ω¯12x+G0+ε2ξ¯1x˙+K¯2x3+K¯3x5−Θ¯V¯=εf¯cosωτ,where ω¯1=1−K1 represents the linearized oscillation frequency within the middle potential well.To express the nearness of the excitation frequency to the linearized resonance frequency within the middle potential well, we let(26)ω2=ω¯12+εσ.Substituting equations (22)–(26) into equation (25) and truncating at order ε and separating the terms of ε and equating them to zero, we obtain(27a)Oε0:D02x0+ω2x0=−G0,(27b)D0V¯0+αV¯0=−D0x0,(28a)Oε1:D02x1+ω2x1=−2D0D1x0−2ξ¯1D0x0−K¯2x03−K¯3x05+σx0+Θ¯V¯0+f¯cosωτ,(28b)D0V¯1+αV¯1=−D0x1−D1x0−D1V¯0.The solutions of equations (27a) and (27b) can be written as(29)x0=AT1expiωT0−G0ω2+cc,(30)V¯0=−ω2+iωαα2+ω2AT1expiωT0+cc,where A(T1) is a complex-valued function, A¯ is the complex conjugate of A, and cc is the complex conjugate of the preceding term. Substituting equations (29) and (30) into equation (28a) and eliminating the secular terms, we get(31)−2iωD1A−2iωξ¯1A−ω2+iωαα2+ω2Θ¯A−3G0ω22K¯2A−5G0ω24K¯3A−3K¯2A2A¯−30G0ω22K¯3A2A¯−10K¯3A3A¯2+σA+f¯2expiσT1=0.A can be expressed in the polar formA=1/2aeiθ, where a represents the steady-state displacement response amplitude and θ is the argument. Introducing γ=σT1−θ, substituting it into equation (29) and then separating the real and imaginary parts, we obtain(32a)D1a=−c1a+c2sinγ,(32b)aD1γ=c3a−c4a3−c5a5+c2cosγ.where c1=ξ¯1+αΘ¯/2ω2+α2, c2=f¯/2ω, c3=σ+σ/2ω−Θ¯ω/2ω2+α2−3K¯2/2ωG0/ω22−5K¯3/2ωG0/ω24c4=3K¯2/8ω+15K¯3/4ωG0/ω22, and c5=5K¯3/16ω.For the steady-state response, let the time derivatives in equations (32a) and (32b) be equal to zero, and square and add the resulting equations to obtain(33)c52a10+2c4c5a8+c42−2c3c5a6−2c3c4a4+c12+c32a2−c22=0where the steady-state displacement response amplitude a can be obtained by equation (33). The steady-state solutions for the displacement and the voltage can then be expressed in the following form:(34a)xτ=acosωτ+ϕ0,(34b)V¯τ=ucosωτ+ϕ0+φ,where u=ω/α2+ω2a is the steady-state voltage response amplitude ϕ0=arcc1/c3−c4a2−c5a4, and φ=arctanα/ω.The steady-state solutions of output power can be written as follows:(35)P=l2θ12u2Cp2R.The stability of the solution can be determined according to the Routh–Hurwitz method.


## 4. Results and Discussion

In this section, we discuss the magnet spacing, the mass of tip magnet, thickness ratio of piezoelectric layer and substrate, load resistance, and piezoelectric material on the performance of TPEH for two different computational models. The geometric and material properties are as follows:l=75mm, b=20mm, hs=0.2mm, Ys=70Gpa, ρs=2700kg/m3, Mt=14.9×10−3kg, MA=MB=MC=1.22×106A/m, VA=VB=VC=1×10−6m3, ξ1=0.01. PZT-5 A: Yp=60.98Gpa, ρs=7750kg/m3, d31=−1.71×10−10C/N, ε33s=1.33×10−8F/m. PZT-5H: Yp=60.61Gpa, ρs=7500kg/m3, d31=−2.74×10−10C/N, and ε33s=2.56×10−8F/m. Figures 3 and 4 depict variations of displacement and output power versus excited frequency for different magnet spacing, respectively, in which Mt = 14.9 g, R = 300 kΩ. Figures 3 and 4 show that the displacement and output power amplitude of the interwell motion increase while the bandwidth decreases as dh increases for a fixed dv. The amplitude of the intrawell motion displacement of the conventional model decreases as dh increases, while the amplitude of the intrawell motion displacement of the modified model is not sensitive to changes in dh. With the dv increasing, the displacement and output power amplitude of the interwell motion decreases, while the bandwidth and intrawell motion displacement amplitude increases in the case of the fixed dh. The trend of interwell power varies with the distance between magnets being the same for the conventional model and the modified model, but the peak power and interwell bandwidth of the two models differ significantly. Taking dv = 7.5 mm, dh = 20 mm and dv = 7.5 mm, dh= 21 mm as examples, when the modified model is adopted, the corresponding peak power of the two cases is increased by 10.4% and 10.8%, respectively, compared with that of the traditional model, and the corresponding interwell bandwidth is increased by 25.8% and 27.3%, respectively, compared with that of the traditional model.Figure 3
Displacement frequency response curve in different distance between three magnets.Figure 4
Power frequency response curve in different distance between three magnets.Figures5 and 6 give the displacement and output power frequency response curves for different tip magnet masses Mt when dv = 7.5 mm, dh = 20 mm, and R = 300 kΩ. Figure 5 shows that increasing the tip magnet mass Mt can significantly increase the peak displacement and interwell bandwidth. When the excited frequency is low, the displacement amplitude of the interwell motion increases with tip magnet mass Mt increasing for the conventional model case while there is no significant change in interwell motion displacement amplitude for the modified model. As can be seen from Figure 6, the interwell motion output power amplitude and peak power for both computational model cases increase significantly with Mt increasing. The peak power calculated by the modified model is 0.13 W and 0.23 W, when Mt = 9.9 g and Mt = 14.9 g, respectively, which is 21.2% and 7.4% higher than the peak power of the conventional model, and the interwell motion bandwidth of the modified model is 38.1% and 25.5% higher than that of the conventional model, respectively.Figure 5
Displacement frequency response curve in different values ofMt.Figure 6
Power frequency response curve in different values ofMt.Figures7 and 8 show the steady-state amplitude response curves of the interwell motion output voltage with the variation of the excited acceleration amplitude for different tip magnet masses Mt when excited frequency ω = 0.5 and ω = 1. Figures 7 and 8 show that there is an excited acceleration threshold, and when the TPEH is excited by an excited amplitude greater than this threshold, it can cross the potential barrier and enter the high energy orbit. The excited acceleration threshold of interwell motion decreases with the increase of Mt when excited frequency ω = 0.5 and ω = 1. Compared to the conventional model, the modified model has a lower excitation acceleration threshold for interwell motion. In addition, the larger the excitation frequency, the larger the excitation acceleration amplitude required for the TPEH to generate interwell motion.Figure 7
Voltage-excitation amplitude response curve in different values ofMt when ω = 0.5.Figure 8
Voltage-excitation amplitude response curve in different values ofMt when ω = 1.The variation of output power amplitude with load resistance for the two calculation models at low excited frequency is studied whendv = 7.5 mm, dh = 20 mm, and Mt = 14.9 g. Figure 9 shows that the output power amplitude at each excitation frequency tends to increase at the beginning and decrease afterwards with the increase of load resistance at each excited frequency. Each excitation frequency corresponds to an optimal load resistance to maximize the amplitude of output power of the TPEH, and the optimal load resistance decreases with the increase of excitation frequency. At lower excited frequencies (in the range of 0.6 to 1.1), the optimal load resistance of the modified model is higher than that of the conventional model, but the corresponding peak power is significantly lower than that of the conventional model.Figure 9
Output power amplitude response of the system with different load resistance: (a) modified model and (b) traditional model.
(a)(b)The effect of load resistanceR on the peak displacement of the cantilever beam for two different piezoelectric materials (PZT-5A, PZT-5H) at dv = 7.5 mm, dh = 20 mm, and Mt= 14.9 g is shown in Figure 10. The results show that the load resistance R has little effect on the peak displacement of the system under the short circuit or open circuit conditions. The peak displacements of the two piezoelectric materials are very similar to each other. The reason is that they have similar elastic modulus and density, the natural frequencies of short circuit and open circuit are close, and the flexibility under short circuit or open circuit resonance excitation is also very close.Figure 10
Peak displacement curves along with resistance change.Figure11 depicts the peak power variation curve with load resistance R for the cantilever beam of two piezoelectric materials (PZT-5A, PZT-5H) when dv = 7.5 mm, dh = 20 mm, Mt = 14.9 g. The results show that the peak power of the system increases sharply with the increase of the load resistance (the first peak power maximum occurs, and the corresponding local optimal load resistance is Ropt1), then decreases slightly and continues to increase (the second peak power maximum occurs, and the corresponding local optimal load resistance is Ropt2), and then decreases gradually. Both local optimum load resistances of the PZT-5A cantilever beam are larger than those of the PZT-5H cantilever beam, but the maximum values of the system peak power with both piezoelectric materials are very close, due to the fact that the magnitude of the peak power depends on the flexibility of the piezoelectric cantilever beam. The peak power maximum values of the modified model for both piezoelectric materials are larger than those of the conventional model, and the corresponding local optimal load resistance Ropt1 is larger than that of the conventional model, while Ropt2 is smaller than that of the conventional model in both cases. Taking the PZT-5A cantilever beam as an example, the Ropt1 = 59K obtained by using the modified mode is 10.1% higher than that of the traditional model, and the Ropt2 = 281K is 21.6% lower than that of the traditional model. The maximum peak power of the modified model is 7% higher than that of the traditional model.Figure 11
Peak power curves along with resistance change.Figure12 shows the variation curve of peak power of PZT-5A and PZT-5H cantilever beams with thickness ratio TP/HS of piezoelectric layer and substrate under the two calculation models when the thickness of piezoelectric beam (2tp + hS) is 0.6 mm. It can be seen from Figure 12 that the peak power of PZT-5A and PZT-5H cantilever beams under both calculation models show a trend of increasing sharply and then leveling off with the increase of tp/hs. Under the traditional model, when tp/hs < 0.5, the peak power of PZT-5H cantilever beam is significantly lower than that of PZT-5A cantilever beam; when tp/hs is between 0.5 and 1.1, the peak power of PZT-5H cantilever beam is slightly larger than that of PZT-5A cantilever beam. After that, with the continuous increase of tp/hs, the peak power of PZT-5H cantilever beam is lower than that of PZT-5A cantilever beam again. Under different piezoelectric materials, when tp/hs is small, the peak power obtained by the two models is close to each other, while when tp/hs exceeds a certain value, the peak power obtained by the modified model is obviously higher than that obtained by the traditional model.Figure 12
Peak power curves along withtp/hs change.The optimal load resistance curves of PZT-5A and PZT-5H cantilever beams with two excitation frequencies (ω = 0.5, ω = 0.5) for dv= 7.5 mm, dh = 20 mm, Mt = 14.9 g, and piezoelectric beam thickness 2tp + hs = 0.6 mm are given in Figure 13. It shows that the optimal load resistance of PZT-5A and PZT-5H cantilever beams under both calculation models decreases with increasing excited frequency and increases with increasing tp/hs, and the growth rate of optimal load resistance slows down with increasing tp/hs. The optimal load resistance of both calculated models is greater for the PZT-5A cantilever beam compared to the PZT-5H cantilever beam for the same excited frequency and tp/hs. Under the same conditions, the optimal load resistance of the modified model is larger than that of the conventional model.Figure 13
Optimal resistance curves along withtp/hs change.


## 5. Conclusions

In this paper, based on the generalized Hamilton variational principle, considering the gravity potential energy of the energy harvester, the size effect, and the rotary inertia of tip magnet, a more accurate electromechanical coupling equation of the nonlinear tristable piezoelectric cantilever energy harvester is obtained, and the analytical solution of the equation is obtained by using the multiscale method. The effects of magnet spacing, the mass of tip magnet, thickness ratio of piezoelectric layer and substrate, load resistance, and piezoelectric material on the performance of piezoelectric energy capture system under the modified and conventional models were investigated and the following conclusions were obtained:(1)
Increasing dh or decreasingdv can improve the interwell displacement and output power amplitude. The interwell bandwidth decreases with the increase of dh and increases with the increase of dv.(2)
IncreasingMt can significantly increase the peak power and bandwidth of the system, but the excited acceleration threshold to generate interwell motion decreases with increasing Mt.(3)
The maximum peak power of two piezoelectric cantilever beams (PZT-5A and PZT-5H) is very close. Whentp/hs is very small or too large, the peak power of PZT-5A cantilever beam is slightly larger than that of PZT-5H cantilever beam, while when tp/hs is moderate, the peak power of PZT-5H cantilever beam is significantly larger than that of PZT-5A cantilever beam. The optimal load resistance of both piezoelectric material cantilever beams increases with tp/hs increasing. Under the same excited frequency and tp/hs, the optimal load resistance of PZT-5A cantilever beam is larger.(4)
The effect on the optimal load resistance of the system is significant after introducing the gravitational potential energy of the TPEH, the eccentricity, and the rotational inertia of the tip magnet into the conventional calculation model. In addition, compared with the traditional model, the peak power and interwell motion bandwidth of the modified model are significantly increased. Therefore, it is of great significance to establish a modified model to improve the accuracy of predicting the energy harvesting efficiency of the TPEH.



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