# Modeling and Experiment of a V-Shaped Piezoelectric Energy Harvester

Authors: Yue Zhao; Yi Qin; Lei Guo; Baoping Tang
Journal: Shock and Vibration
 (2018)

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## Abstract

Vibration-based energy harvesting technology is the most promising method to solve the problems of self-powered wireless sensor nodes, but most of the vibration-based energy harvesters have a rather narrow operation bandwidth and the operation frequency band is not convenient to adjust when the ambient frequency changes. Since the ambient vibration may be broadband and changeable, a novel V-shaped vibration energy harvester based on the conventional piezoelectric bimorph cantilevered structure is proposed, which successfully improves the energy harvesting efficiency and provides a way to adjust the operation frequency band of the energy harvester conveniently. The electromechanical coupling equations are established by using Euler-Bernoulli equation and piezoelectric equation, and then the coupled circuit equation is derived based on the series connected piezoelectric cantilevers and Kirchhoff's laws. With the above equations, the output performances of V-shaped structure under different structural parameters and load resistances are simulated and discussed. Finally, by changing the angleθ between two piezoelectric bimorph beams and the load resistance, various comprehensive experiments are carried out to test the performance of this V-shaped energy harvester under the same excitation. The experimental results show that the V-shaped energy harvester can not only improve the frequency response characteristic and the output performance of the electrical energy, but also conveniently tune the operation bandwidth; thus it has great application potential in actual structure health monitoring under variable working condition.


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## Body

## 1. Introduction

Recently, wireless sensor networks (WSNs) have received much attention. WSNs technology has been recognized as an effective technique for the various industrial fields [1], such as environmental monitoring [2], fire detection [3], body area networks [4], and fault diagnosis [5]. But the limited energy associated with WSNs is a great obstacle of WSN technologies. Environmental energy harvesting is the most probable way to overcome this major bottleneck. For example, vibration energy, solar energy, wind energy, thermal energy, and other green energy [6] can be converted into electrical energy which can augment an existing battery supply or replace it entirely given certain conditions.Since vibration widely exists in mechanical equipment, bridge, and human body, the commonly used energy harvesting is based on vibration. The possible vibration-to-electric energy conversion mechanisms include piezoelectric, electromagnetic, and electrostatic transductions, which provide a high power density [7]. Among these three alternatives for vibration-to-electric energy conversion, piezoelectric transduction has received the most attention. Rocha et al. [8] proposed a system integrated in footwear to harvest energy from vibration when people walk. In [9], a piezoelectric energy harvester is developed to harvest energy from the acceleration of tires. P Gambier et al. [10] designed hybrid low-power generator systems with thin-film batteries and storage system using piezoelectric, solar, and thermal sources of energy in order to improve both structural multifunctionality and system-level robustness in energy harvesting.The most commonly used piezoelectric energy harvester configuration is a cantilever with piezoceramic layers located on a vibrating host structure for electrical power generation from bending vibrations under resonance excitation [12–15]. Various piezoelectric transductions are designed to harvest energy in a single direction, whereas they have better energy conversion efficiency only when the excitation frequencies of the vibration sources closely match the resonance frequencies of the devices [16–19]. Unfortunately, when the ambient vibration sources are multidirectional and have a wide spectrum, these piezoelectric harvesters [20] will probably lose their efficacy.To overcome the above problems, various vibration energy harvesters which are suitable for multidirectional vibrations or wide-band vibrations have been explored. For example, Leadenham et al. [21] presented an M-shaped nonlinear bent beam with clamped end conditions and investigated for bandwidth enhancement under base excitation. Harne et al. [22] investigated the versatility of internal resonance-based behaviors for energy harvesting analytically, numerically, and experimentally by exploiting nonlinear, multimodal motions in an L-shaped energy harvester platform. Danzi F et al. [23, 24] investigated to use a topology optimization to design planar resonators and derived a reduced order model to further investigate the dependence of the beams' orientation on the frequencies ratio. These results show an infinite number of topologies that can have integer ratio modal frequencies and in some cases harvest more power than a nominal L-shaped harvester, operating in the linear regime. Kim et al. [25] and Andò et al. [26] designed two-dimensional piezoelectric energy harvester, respectively, which is sensitive to the vibration of two directions. Su et al. [27] studied the three-directional harvester by using spring and magnet. The method to design the multidirectional piezoelectric energy harvester is to optimize the structure so that the harvester could work effectively by inertia when excited at some direction. In order to broaden the effective operation frequency band of the harvester, a multiresonant piezoelectric beam was used by attaching multiple cantilever beams of natural frequencies close to the exciting frequency [28, 29]. Yang et al. [30] proposed a power harvester that consisted of two piezoelectric beams connected with elastic springs, and it could also improve the bandwidth. Another approach is to provide a resonance tuning mechanism. Hu et al. [31] studied the technique to adjust the flexural mode of a piezoelectric bimorph through axial preloads, which is useful for scavenging energy from ambient vibration with varying frequency spectra. Challa et al. [32] presented a vibration energy harvesting device with autonomous resonance frequency tenability by utilizing a magnetic stiffness technique. In addition, there are other methods to broaden the operating frequency bandwidth of the energy harvester, such as monostable and bistable mechanisms [33–36] and piecewise linear mechanisms [37]. However, their tuning mechanisms make the structure of these harvesters more complex, which may hamper their application in engineering. What is more, the monostable and bistable mechanisms will have high conversion efficiency only when the excitation amplitude is large. In many cases, the amplitude of the environmental vibration is small, and it is difficult to sustain the high output of the harvester. From the above, it can be known that most of the energy harvesters seldom pay attention to multidirection and wide operation bandwidth simultaneously.As an alternative to conventional multiresonant piezoelectric beams, Erturk A et al. [38] proposed an L-shaped beam-mass structure as a new piezoelectric energy harvester configuration, which can broaden the operation bandwidth and overcome the shortcomings of the single frequency excitation. However, in their study, the angle of the L-shaped harvester is fixed; thus it is difficult to be applicable to the variable working condition. Su et al. [11] further provided detailed modeling of the V-shaped structure on the basis of Euler-Bernoulli beam theory and compared the voltage and power frequency responses under various angles between the two beams. The V-shaped structure consisted of aluminum substructure and unimorph piezoelectric layer, which can effectively broaden the bandwidth. Nevertheless, the generated voltage and power are rather small, which is difficult to meet the requirements of the actual application.Based on the above considerations, a new V-shaped piezoelectric low frequency vibration energy harvester which can arbitrarily tune the angle between the two piezoelectric bimorph beams is explored in this study. The electromechanical coupling model of the proposed harvester is built up, and then the simulation and experiments are performed under different structural parameters and load resistances. The frequency of the peak voltage is the natural frequency (resonant frequency). When the designed harvester is excited at natural frequency, the most power output can be achieved by resistance matching. Like the previous L-shaped or V-shaped piezoelectric unimorph harvester, the proposed V-shaped harvester also can be used as a broader band energy harvesting system, whereas the comparative results show that the V-shaped harvester can not only improve the output performance of the electrical energy, but also enhance the frequency response characteristic of the operation frequency band in some case. In addition, the proposed V-shaped piezoelectric bimorph harvester can be conveniently tuned by the use of a proper connector; thus when the ambient exciting frequency changes, we can simply tune the angle between beam 1 and beam 2 so as to change the natural frequency and the operation bandwidth. It then follows that the proposed V-shaped harvester has great application potential and value in the health monitoring of complicated mechanical equipment and bridge.This paper is organized as follows. In Section2, an electromechanical coupling model is established by using the Euler-Bernoulli equation and piezoelectric equation. In Section 3, numerical simulations are carried out to test the performance of the V-shaped harvester. In Section 4, several experiments are performed under different intersection angles and load resistances to further validate the performance of the V-shaped harvester. Finally, the conclusions are presented in Section 5.


## 2. V-Shaped Piezoelectric Energy Harvester Configuration and Its Mathematical Model

### 2.1. Configuration of V-Shaped Harvester

Cantilever structure has been widely applied to vibration-to-electric energy transducer. Although the conventional single piezoelectric cantilevered structure can generate a large output voltage when it is excited at its resonance frequency, the output voltage of the converter would decrease sharply when the excitation frequency is not consistent with its natural frequency. Furthermore, the conventional single piezoelectric harvester generates high power just within a narrow frequency range and is not convenient to adjust when the ambient frequency changes. In order to solve these problems, a V-shaped piezoelectric bimorph energy harvester is proposed in this study. Figure2 illustrates the schematic of the V-shaped piezoelectric energy harvester. Obviously, this harvester consists of two cantilevered piezoelectric beams whose intersection angle is defined as θ; and wb(t) is the ambient excitation. The length of beam 1 is L1, and the length of beam 2 is L2; Mt1 and Mt2 are the tip masses of beam 1 and beam 2, respectively. w1(x1,t) is the transverse vibration response of beam 1 and w2(x2,t) is the transverse vibration response of beam 2. Both beam 1 and beam 2 are cantilevered piezoelectric bimorph beams, which are shown in Figure 1. In this figure, the green region represents the piezoelectric layer, and the yellow region represents the metal substrate.Figure 1
Schematic of the V-shaped piezoelectric energy harvester.Figure 2
Schematic of a bimorph cantilever configuration with a tip mass: (a) parallel connection of piezoceramic layers and (b) cross-sectional view of a bimorph cantilever.
(a) (b)


### 2.2. Mechanical Coupling Equation

As shown in Figure2, a bimorph piezoelectric cantilever with a tip mass is presented and the piezoelectric layers are connected in parallel. Based on the hypothesis of Euler-Bernoulli beam, the beam presented here can be recognized as a homogeneous composite beam and satisfies the linear deformation and small vibration conditions. So the influence of shear deformation and moment of inertia can be ignored. The piezoelectric bending structure is generally a thin-walled beam, which makes this assumption reasonable. In addition, the mechanical loss consists of the internal damping (strain) and the external damping (viscous air). Assuming that the piezoelectric layer and the substructure are integrated closely and the electrode layer covering the piezoelectric layer is very thin, so the influence of the electrode layer can be ignored. In order to ensure the integrity of the mathematical model deduction, the procedure details can be acquired in [39].For a homogeneous composite beam, the partial differential equation in the transverse direction can be written as [40](1)∂2Mx,t∂x2+csI∂5wx,t∂x4∂t+ca∂wx,t∂t+m∂2wx,t∂t2=fx,twhere M(x,t) is the moment of the beam at position x and time t, csI is the equivalent damping term in the cross-section of the beam due to the strain rate damping, ca is the viscous air damping coefficient, m is the mass of the beam per unit length, w(x,t) is the absolute transverse motion of the beam at any position x and time t,f(x,t) is the external force at any position x and time t.The absolute transverse motion of the beam at any positionx and time t,  and w(x,t), can be written as(2)wx,t=wbx,t+wrelx,twhere wb(x,t) is the displacement of the base and wrelx,t is the transverse deflection of the beam relative to its base.As shown in Figure2, the motion of the base wbx,t can be written as(3)wbx,t=gt+xhtwhere g(t) is the motion of the base in the y-direction and h(t) is the small rotation of the base.Substituting (3) into (2), we have(4)∂2Mx,t∂x2+csI∂5wrelx,t∂x4∂t+ca∂wrelx,t∂t+m∂2wrelx,t∂t2=-m∂2wbx,t∂t2-ca∂wbx,t∂t-csI∂5wbx,t∂x4∂t+fx,tSince both the viscous air damping and internal strain damping are related to the relative speed of beam motion, and the tip massMt is introduced into the above equation, then (4) can be rewritten as(5)∂2Mx,t∂x2+csI∂5wrelx,t∂x4∂t+ca∂wrelx,t∂t+m∂2wrelx,t∂t2=-m+Mtδx-L∂2wbx,t∂t2where δ(x) is the Dirac delta function. The internal bending moment in (5) can be written as(6)Mx,t=-b∫-hp-hs/2-hs/2TPydy+∫-hs/2hs/2TSydy+∫hs/2hp+hs/2TPydywhere b is the width of the bimorph cantilever beam, hp is the thickness of each piezoceramic layer, and hs is the thickness of the substructure layer, as shown in Figure 1(b). Moreover, TP and TS are the axial stress components in the piezoceramic and substructure layers, respectively, which can be given by(7)TP=c11Eδ1p-e31E3,TS=YSδ1swhere c11E is the elastic stiffness of the piezoceramic layer at constant electric field, e31 is the piezoelectric constant, E3 is the electric field component, YS is Young’s modulus of the substructure layer, and δ1p and δ1s are the axial strain components in the piezoelectric and substructure layers, respectively. Since the axial strain at a certain position y from the neutral axis of the composite beam is simply proportional to the curvature of the beam at position x, δ1p and δ1s can be given by(8)δ1px,y,t=δ1sx,y,t=-y∂2wrelx,t∂x2For the configuration with parallel connection shown in Figure2(a), the top and the bottom piezoceramic layers are assumed to be identical, and e31 has the same sign in top and bottom piezoceramic layers, whereas the instantaneous electric fields are in the opposite directions, i.e., E3=-v(t)/hp in the top layer and E3=v(t)/hp in the bottom layer. Since the piezoelectric coupling term in (6) is only a function of time, before substituting (6) into (5), the electrical term must be multiplied by [H(x)-H(x-L)], where H(x) is the Heaviside function.With the above discussion, the coupled beam equation can be derived as follows:(9)YI∂4wrelx,t∂x4+csI∂5wrelx,t∂x4∂t+ca∂wrelx,t∂t+m∂2wrelx,t∂t2+ϑvtdδxdx-dδx-Ldx=-m+Mtδx-L∂2wbx,t∂t2where ϑ is the backward coupling term, YI is the backward coupling term, and m is the mass per unit length term, which are, respectively, defined as(10)ϑ=e31bhphs24-hp+hs22(11)YI=2b3YShs38+c11Ehp+hs23-hs38(12)m=bρshs+2ρphpwhere ρs and ρp are the mass densities of the substructure and the piezoceramic materials, respectively.According to the above preliminary, we build the schematic of the V-shaped piezoelectric energy harvester and set the corresponding vibration reference coordinate system, as shown in Figure3. In this study, we only consider the first two-order vibration modes and ignore the rotary inertias of two beams, when establishing the model of the V-shaped structure by the segment modeling method.Figure 3
The vibration schematic of the V-shaped structure.As shown in Figure3, beam 1 is not subject to the axial load (Mt1, beam 2 and Mt2 can be regarded as an integrated mass), and the absolute displacement of the base in X1-Y1 coordinate system can be assumed to be zero. Similarly, beam 2 is subject to the axial load Mt2gsin⁡θ, and the absolute displacement of the base in X2-Y2 coordinate system can be assumed to be zero. Hence, by the use of (9), the electromechanical coupling vibration equations of beam 1 and beam 2 can be, respectively, written as(13)Y1I1∂4wrel1x1,t∂x14+cs1I1∂5wrel1x1,t∂x14∂t+ca1∂wrel1x1,t∂t+m1∂2wrel1x1,t∂t2+ϑ1v1tdδx1dx1-dδx1-L1dx1=0(14)Y2I2∂4wrel2x2,t∂x24+cs1I1∂5wrel2x2,t∂x24∂t+ca1∂wrel2x2,t∂t+m2∂2wrel2x2,t∂t2+Mt2gsin⁡θ∂2wrel2x2,t∂x22+ϑ2v2tdδx2dx2-dδx2-L2dx2=0where Y1I1 and Y2I2 are the bending stiffness of beam 1 and beam 2, respectively; m1 and m2 are the mass per unit length of beam 1 and beam 2, respectively; ϑ1 and ϑ2 are the backward coupling coefficient, respectively; cs1I1 and cs2I2 are the internal strain damping of beam 1 and beam 2, respectively; ca1 and ca2 are the viscous air damping of beam 1 and beam 2, respectively; v1(t) and v2(t) are the output voltage of beam 1 and beam 2, respectively; δ(x) is the Dirac function; wrel1(x1,t) is the transverse vibratory displacement of beam 1 relative to the substrate; and wrel2(x2,t) is the transverse vibratory displacement of beam 2 relative to the mass.At the clamped end (x1=0), both the transverse vibratory displacement and angular displacement equal 0; thus the boundary conditions can be written as(15)wrel10,t=0,∂wrel1x1,t∂x1x1=0=0Atx2 = 0(as well as x1 = L1), the transverse vibratory displacement in local coordinate system equals 0, and by using the angular displacement equilibrium relations, we have(16)wrel20,t=0,∂wrel1x1,t∂x1x1=L1=∂wrel2x2,t∂x2x2=0Atx2 = 0 and x2 = L2, the force and moment equilibrium relations are, respectively, given by(17)Y1I1∂3wrel1x1,t∂x13x1=L1=Mt1∂2wrel1x1,t∂t2x1=L1+m2L2cos⁡θ∂2wrel2x2,t∂t2x2=L2/2+Mt2cos⁡θ∂2wrel2x2,t∂t2x2=L2(18)Y1I1∂2wrel1x1,t∂x12x1=L1+Ia1∂3wrel1x1,t∂t2∂x1x1=L1=Y2I2∂2wrel2x2,t∂x22x2=0+m2gL22cos⁡θ+Mt2gL2cos⁡θ(19)Mt2g∂wrel2x2,t∂x2x2=L2+Y2I2∂3wrel2x2,t∂x23x2=L2=Mt2∂2wrel2x2,t∂t2x2=L2(20)Y2I2∂2wrel2x2,t∂x22x2=L2+Ia2∂3wrel2x2,t∂t2∂x2x2=L2=0where Ia1 and Ia2 are the rotary inertias of the tip masses Mt1 and Mt2, respectively.Based on the expansion theorem, the vibratory motion of a beam segment can be represented by an absolutely and uniformly convergent series of the eigenfunctions as(21)wrelkxk,t=∑r=1∞ϕkrxkηrtwhere ϕkrxk is the normalized mass eigenfunction and ηr(t) is the modal response of the rth vibration mode;k denotes the serial number of the beam. With (13) and (14), by ignoring the internal and external damping and separating the variables, the piecewise-defined eigenfunctions can be written as(22)ϕ1rx1=A1rsin⁡αrx1+B1rcos⁡αrx1+C1rsinh⁡αrx1+D1rcosh⁡αrx1(23)ϕ2rx2=A2rsin⁡βrx2+B2rcos⁡βrx2+C2rsinh⁡γrx2+D2rcosh⁡γrx2where(24)αr4=ωr2m1Y1I1,βr2=κ2+κ24+ωr2m2Y2I2(25)γr2=-κ2+κ24+ωr2m2Y2I2,κ=Mt2gY2I2where ωr is the undamped natural frequency of rth vibration mode, and it can be calculated by the proposed method in [41].Substituting (21)–(25) into (15)–(20) results in an 8×8 coefficient matrix, and we can obtain the value of A1r~D2r by solving this matrix. After substituting (21) into (13) and (14), and by applying the orthogonality condition of normalized mass eigenfunctions, we can write the mechanical vibratory equation in modal coordinates as(26)d2ηrtdt2+2ζrωrdηrtdt+ωr2ηrt+χ1rv1t+χ2rv2t=frtwhere v1(t) and v2(t) are the voltages of the crossbeam and the inclined beam, respectively; χkr is the modal electromechanical coupling term; fr(t) is the modal mechanical response function. χkr and fr(t) can be calculated by(27)χkr=ϑkdϕkrxkdxk0Lk(28)frt=-m1d2gtdt2∫0L1ϕ1rx1dx1+d2htdt2∫0L1x1ϕ1rx1dx1-Mt1+Mt2+m2L2ϕ1rL1d2gtdt2+L1d2htdt2


### 2.3. Coupled Circuit Equations

In this study, we only consider the condition that the crossbeam connects with the inclined beam in series, and each piezoelectric layer can be seen as the current source connecting with the capacitor in parallel, as shown in Figure4. The current sources produced by beam 1 and beam 2 can be written as(29)ipkt=∑r=1∞ψkrdηrtdtwhere(30)ψkr=-e31khpckbkdϕkrxkdxk0LkFigure 4
Electrical circuit of the condition that the two beams connect in series.In (30), e31k is the piezoelectric constant, hpck is the distance between the center of the piezoceramic layer and the neutral axis of the cross-section, and bk is the width of piezoceramic layer. The internal capacitor Cpk can be written as(31)Cpk=ε33SkbkLkhpkwhere ε33Sk is the permittivity of the piezoceramic layer at constant strain, Lk is the length of beam, and hpk is the thickness of the piezoceramic layer.Subsequently, as the bimorph piezoelectric plates connect in series, with the Kirchhoff Laws, we can write the coupled circuit equations as(32)Cp1dv1tdt+v1t2R1+v2t2R1=∑r=1∞ψ1rdηrtdtCp2dv2tdt+v2t2R1+v1t2R1=∑r=1∞ψ2rdηrtdtBy solving (26) and (32), v1(t),v2(t), and ηr(t) can be obtained. With v1(t) and v2(t), we can calculate the voltage across the load resistance R1 by(33)vt=v1t+v2t


### 2.4. Coupled Voltage Response for Harmonic Base Excitation

Assuming that the electromechanical system is linear, the modal mechanical response function can be written asfr(t)=Frejωt. Besides, the steady state expressions for the modal response and voltage response of two beams can be represented by harmonic functions: ηr(t)=Hrejωt,v1(t)=V1ejωt, and v2(t)=V2ejωt. Substituting the above equations into (26) and (32), the two equations can be, respectively, written as(34)ωr2-ω2+j2ζrωrωHr+χ1rV1+χ2rV2=Fr(35)12R1+jωCp1V1+V22R1-jω∑r=1∞ψ1rHr=012R1+jωCp2V2+V12R1-jω∑r=1∞ψ2rHr=0Substituting (34) into (33), we have(36)Q11V1+Q12V2=S1Q21V1+Q22V2=S2where(37)Qmk=12R1+jωCpmδmk+∑r=1∞jωψmrχkrωr2-ω2+j2ζrωrω(38)Sm=∑r=1∞jωψmrFrωr2-ω2+j2ζrωrωFrom (36), two maximum voltage amplitudes (V1 and V2) of the closed-loop circuit are obtained. Substituting V1 and V2 into v1(t)=V1ejωt and v2(t)=V2ejωt, respectively, we can obtain the steady state voltage response functions produced by the two beams. Then, the output voltage v(t) of the V-shaped bimorph piezoelectric cantilever is given by(39)vt=v1t+v2t


## 2.1. Configuration of V-Shaped Harvester

Cantilever structure has been widely applied to vibration-to-electric energy transducer. Although the conventional single piezoelectric cantilevered structure can generate a large output voltage when it is excited at its resonance frequency, the output voltage of the converter would decrease sharply when the excitation frequency is not consistent with its natural frequency. Furthermore, the conventional single piezoelectric harvester generates high power just within a narrow frequency range and is not convenient to adjust when the ambient frequency changes. In order to solve these problems, a V-shaped piezoelectric bimorph energy harvester is proposed in this study. Figure2 illustrates the schematic of the V-shaped piezoelectric energy harvester. Obviously, this harvester consists of two cantilevered piezoelectric beams whose intersection angle is defined as θ; and wb(t) is the ambient excitation. The length of beam 1 is L1, and the length of beam 2 is L2; Mt1 and Mt2 are the tip masses of beam 1 and beam 2, respectively. w1(x1,t) is the transverse vibration response of beam 1 and w2(x2,t) is the transverse vibration response of beam 2. Both beam 1 and beam 2 are cantilevered piezoelectric bimorph beams, which are shown in Figure 1. In this figure, the green region represents the piezoelectric layer, and the yellow region represents the metal substrate.Figure 1
Schematic of the V-shaped piezoelectric energy harvester.Figure 2
Schematic of a bimorph cantilever configuration with a tip mass: (a) parallel connection of piezoceramic layers and (b) cross-sectional view of a bimorph cantilever.
(a) (b)


## 2.2. Mechanical Coupling Equation

As shown in Figure2, a bimorph piezoelectric cantilever with a tip mass is presented and the piezoelectric layers are connected in parallel. Based on the hypothesis of Euler-Bernoulli beam, the beam presented here can be recognized as a homogeneous composite beam and satisfies the linear deformation and small vibration conditions. So the influence of shear deformation and moment of inertia can be ignored. The piezoelectric bending structure is generally a thin-walled beam, which makes this assumption reasonable. In addition, the mechanical loss consists of the internal damping (strain) and the external damping (viscous air). Assuming that the piezoelectric layer and the substructure are integrated closely and the electrode layer covering the piezoelectric layer is very thin, so the influence of the electrode layer can be ignored. In order to ensure the integrity of the mathematical model deduction, the procedure details can be acquired in [39].For a homogeneous composite beam, the partial differential equation in the transverse direction can be written as [40](1)∂2Mx,t∂x2+csI∂5wx,t∂x4∂t+ca∂wx,t∂t+m∂2wx,t∂t2=fx,twhere M(x,t) is the moment of the beam at position x and time t, csI is the equivalent damping term in the cross-section of the beam due to the strain rate damping, ca is the viscous air damping coefficient, m is the mass of the beam per unit length, w(x,t) is the absolute transverse motion of the beam at any position x and time t,f(x,t) is the external force at any position x and time t.The absolute transverse motion of the beam at any positionx and time t,  and w(x,t), can be written as(2)wx,t=wbx,t+wrelx,twhere wb(x,t) is the displacement of the base and wrelx,t is the transverse deflection of the beam relative to its base.As shown in Figure2, the motion of the base wbx,t can be written as(3)wbx,t=gt+xhtwhere g(t) is the motion of the base in the y-direction and h(t) is the small rotation of the base.Substituting (3) into (2), we have(4)∂2Mx,t∂x2+csI∂5wrelx,t∂x4∂t+ca∂wrelx,t∂t+m∂2wrelx,t∂t2=-m∂2wbx,t∂t2-ca∂wbx,t∂t-csI∂5wbx,t∂x4∂t+fx,tSince both the viscous air damping and internal strain damping are related to the relative speed of beam motion, and the tip massMt is introduced into the above equation, then (4) can be rewritten as(5)∂2Mx,t∂x2+csI∂5wrelx,t∂x4∂t+ca∂wrelx,t∂t+m∂2wrelx,t∂t2=-m+Mtδx-L∂2wbx,t∂t2where δ(x) is the Dirac delta function. The internal bending moment in (5) can be written as(6)Mx,t=-b∫-hp-hs/2-hs/2TPydy+∫-hs/2hs/2TSydy+∫hs/2hp+hs/2TPydywhere b is the width of the bimorph cantilever beam, hp is the thickness of each piezoceramic layer, and hs is the thickness of the substructure layer, as shown in Figure 1(b). Moreover, TP and TS are the axial stress components in the piezoceramic and substructure layers, respectively, which can be given by(7)TP=c11Eδ1p-e31E3,TS=YSδ1swhere c11E is the elastic stiffness of the piezoceramic layer at constant electric field, e31 is the piezoelectric constant, E3 is the electric field component, YS is Young’s modulus of the substructure layer, and δ1p and δ1s are the axial strain components in the piezoelectric and substructure layers, respectively. Since the axial strain at a certain position y from the neutral axis of the composite beam is simply proportional to the curvature of the beam at position x, δ1p and δ1s can be given by(8)δ1px,y,t=δ1sx,y,t=-y∂2wrelx,t∂x2For the configuration with parallel connection shown in Figure2(a), the top and the bottom piezoceramic layers are assumed to be identical, and e31 has the same sign in top and bottom piezoceramic layers, whereas the instantaneous electric fields are in the opposite directions, i.e., E3=-v(t)/hp in the top layer and E3=v(t)/hp in the bottom layer. Since the piezoelectric coupling term in (6) is only a function of time, before substituting (6) into (5), the electrical term must be multiplied by [H(x)-H(x-L)], where H(x) is the Heaviside function.With the above discussion, the coupled beam equation can be derived as follows:(9)YI∂4wrelx,t∂x4+csI∂5wrelx,t∂x4∂t+ca∂wrelx,t∂t+m∂2wrelx,t∂t2+ϑvtdδxdx-dδx-Ldx=-m+Mtδx-L∂2wbx,t∂t2where ϑ is the backward coupling term, YI is the backward coupling term, and m is the mass per unit length term, which are, respectively, defined as(10)ϑ=e31bhphs24-hp+hs22(11)YI=2b3YShs38+c11Ehp+hs23-hs38(12)m=bρshs+2ρphpwhere ρs and ρp are the mass densities of the substructure and the piezoceramic materials, respectively.According to the above preliminary, we build the schematic of the V-shaped piezoelectric energy harvester and set the corresponding vibration reference coordinate system, as shown in Figure3. In this study, we only consider the first two-order vibration modes and ignore the rotary inertias of two beams, when establishing the model of the V-shaped structure by the segment modeling method.Figure 3
The vibration schematic of the V-shaped structure.As shown in Figure3, beam 1 is not subject to the axial load (Mt1, beam 2 and Mt2 can be regarded as an integrated mass), and the absolute displacement of the base in X1-Y1 coordinate system can be assumed to be zero. Similarly, beam 2 is subject to the axial load Mt2gsin⁡θ, and the absolute displacement of the base in X2-Y2 coordinate system can be assumed to be zero. Hence, by the use of (9), the electromechanical coupling vibration equations of beam 1 and beam 2 can be, respectively, written as(13)Y1I1∂4wrel1x1,t∂x14+cs1I1∂5wrel1x1,t∂x14∂t+ca1∂wrel1x1,t∂t+m1∂2wrel1x1,t∂t2+ϑ1v1tdδx1dx1-dδx1-L1dx1=0(14)Y2I2∂4wrel2x2,t∂x24+cs1I1∂5wrel2x2,t∂x24∂t+ca1∂wrel2x2,t∂t+m2∂2wrel2x2,t∂t2+Mt2gsin⁡θ∂2wrel2x2,t∂x22+ϑ2v2tdδx2dx2-dδx2-L2dx2=0where Y1I1 and Y2I2 are the bending stiffness of beam 1 and beam 2, respectively; m1 and m2 are the mass per unit length of beam 1 and beam 2, respectively; ϑ1 and ϑ2 are the backward coupling coefficient, respectively; cs1I1 and cs2I2 are the internal strain damping of beam 1 and beam 2, respectively; ca1 and ca2 are the viscous air damping of beam 1 and beam 2, respectively; v1(t) and v2(t) are the output voltage of beam 1 and beam 2, respectively; δ(x) is the Dirac function; wrel1(x1,t) is the transverse vibratory displacement of beam 1 relative to the substrate; and wrel2(x2,t) is the transverse vibratory displacement of beam 2 relative to the mass.At the clamped end (x1=0), both the transverse vibratory displacement and angular displacement equal 0; thus the boundary conditions can be written as(15)wrel10,t=0,∂wrel1x1,t∂x1x1=0=0Atx2 = 0(as well as x1 = L1), the transverse vibratory displacement in local coordinate system equals 0, and by using the angular displacement equilibrium relations, we have(16)wrel20,t=0,∂wrel1x1,t∂x1x1=L1=∂wrel2x2,t∂x2x2=0Atx2 = 0 and x2 = L2, the force and moment equilibrium relations are, respectively, given by(17)Y1I1∂3wrel1x1,t∂x13x1=L1=Mt1∂2wrel1x1,t∂t2x1=L1+m2L2cos⁡θ∂2wrel2x2,t∂t2x2=L2/2+Mt2cos⁡θ∂2wrel2x2,t∂t2x2=L2(18)Y1I1∂2wrel1x1,t∂x12x1=L1+Ia1∂3wrel1x1,t∂t2∂x1x1=L1=Y2I2∂2wrel2x2,t∂x22x2=0+m2gL22cos⁡θ+Mt2gL2cos⁡θ(19)Mt2g∂wrel2x2,t∂x2x2=L2+Y2I2∂3wrel2x2,t∂x23x2=L2=Mt2∂2wrel2x2,t∂t2x2=L2(20)Y2I2∂2wrel2x2,t∂x22x2=L2+Ia2∂3wrel2x2,t∂t2∂x2x2=L2=0where Ia1 and Ia2 are the rotary inertias of the tip masses Mt1 and Mt2, respectively.Based on the expansion theorem, the vibratory motion of a beam segment can be represented by an absolutely and uniformly convergent series of the eigenfunctions as(21)wrelkxk,t=∑r=1∞ϕkrxkηrtwhere ϕkrxk is the normalized mass eigenfunction and ηr(t) is the modal response of the rth vibration mode;k denotes the serial number of the beam. With (13) and (14), by ignoring the internal and external damping and separating the variables, the piecewise-defined eigenfunctions can be written as(22)ϕ1rx1=A1rsin⁡αrx1+B1rcos⁡αrx1+C1rsinh⁡αrx1+D1rcosh⁡αrx1(23)ϕ2rx2=A2rsin⁡βrx2+B2rcos⁡βrx2+C2rsinh⁡γrx2+D2rcosh⁡γrx2where(24)αr4=ωr2m1Y1I1,βr2=κ2+κ24+ωr2m2Y2I2(25)γr2=-κ2+κ24+ωr2m2Y2I2,κ=Mt2gY2I2where ωr is the undamped natural frequency of rth vibration mode, and it can be calculated by the proposed method in [41].Substituting (21)–(25) into (15)–(20) results in an 8×8 coefficient matrix, and we can obtain the value of A1r~D2r by solving this matrix. After substituting (21) into (13) and (14), and by applying the orthogonality condition of normalized mass eigenfunctions, we can write the mechanical vibratory equation in modal coordinates as(26)d2ηrtdt2+2ζrωrdηrtdt+ωr2ηrt+χ1rv1t+χ2rv2t=frtwhere v1(t) and v2(t) are the voltages of the crossbeam and the inclined beam, respectively; χkr is the modal electromechanical coupling term; fr(t) is the modal mechanical response function. χkr and fr(t) can be calculated by(27)χkr=ϑkdϕkrxkdxk0Lk(28)frt=-m1d2gtdt2∫0L1ϕ1rx1dx1+d2htdt2∫0L1x1ϕ1rx1dx1-Mt1+Mt2+m2L2ϕ1rL1d2gtdt2+L1d2htdt2


## 2.3. Coupled Circuit Equations

In this study, we only consider the condition that the crossbeam connects with the inclined beam in series, and each piezoelectric layer can be seen as the current source connecting with the capacitor in parallel, as shown in Figure4. The current sources produced by beam 1 and beam 2 can be written as(29)ipkt=∑r=1∞ψkrdηrtdtwhere(30)ψkr=-e31khpckbkdϕkrxkdxk0LkFigure 4
Electrical circuit of the condition that the two beams connect in series.In (30), e31k is the piezoelectric constant, hpck is the distance between the center of the piezoceramic layer and the neutral axis of the cross-section, and bk is the width of piezoceramic layer. The internal capacitor Cpk can be written as(31)Cpk=ε33SkbkLkhpkwhere ε33Sk is the permittivity of the piezoceramic layer at constant strain, Lk is the length of beam, and hpk is the thickness of the piezoceramic layer.Subsequently, as the bimorph piezoelectric plates connect in series, with the Kirchhoff Laws, we can write the coupled circuit equations as(32)Cp1dv1tdt+v1t2R1+v2t2R1=∑r=1∞ψ1rdηrtdtCp2dv2tdt+v2t2R1+v1t2R1=∑r=1∞ψ2rdηrtdtBy solving (26) and (32), v1(t),v2(t), and ηr(t) can be obtained. With v1(t) and v2(t), we can calculate the voltage across the load resistance R1 by(33)vt=v1t+v2t


## 2.4. Coupled Voltage Response for Harmonic Base Excitation

Assuming that the electromechanical system is linear, the modal mechanical response function can be written asfr(t)=Frejωt. Besides, the steady state expressions for the modal response and voltage response of two beams can be represented by harmonic functions: ηr(t)=Hrejωt,v1(t)=V1ejωt, and v2(t)=V2ejωt. Substituting the above equations into (26) and (32), the two equations can be, respectively, written as(34)ωr2-ω2+j2ζrωrωHr+χ1rV1+χ2rV2=Fr(35)12R1+jωCp1V1+V22R1-jω∑r=1∞ψ1rHr=012R1+jωCp2V2+V12R1-jω∑r=1∞ψ2rHr=0Substituting (34) into (33), we have(36)Q11V1+Q12V2=S1Q21V1+Q22V2=S2where(37)Qmk=12R1+jωCpmδmk+∑r=1∞jωψmrχkrωr2-ω2+j2ζrωrω(38)Sm=∑r=1∞jωψmrFrωr2-ω2+j2ζrωrωFrom (36), two maximum voltage amplitudes (V1 and V2) of the closed-loop circuit are obtained. Substituting V1 and V2 into v1(t)=V1ejωt and v2(t)=V2ejωt, respectively, we can obtain the steady state voltage response functions produced by the two beams. Then, the output voltage v(t) of the V-shaped bimorph piezoelectric cantilever is given by(39)vt=v1t+v2t


## 3. Numerical Simulation

In this section, the voltage performance of the V-shaped cantilevered piezoelectric bimorph harvester is simulated by Matlab. Lead zirconate titanate (PZT) is a ceramic perovskite material that has a prominent piezoelectric effect; thus this material has been widely used in various engineering applications. Since the d51 mode can potentially improve the energy harvesting performance of piezoelectric energy harvester [42], and the PZT-51 is commercially available, we use PZT-51 material in the proposed energy harvesting structure. The cross beam and inclined beam have the same geometric and material parameters, which are listed in Table 1. In order to achieve resonant excitation, we can adjust their harmonic frequency by the use of tip masses of cross beam and inclined beam (Mt1 and Mt2). In this study, Mt1 and Mt2 are set as 14g and 14.6g, respectively.Table 1
The material and dimension parameters of piezoelectric bimorph and substructure.
Material and geometric parameters Value/Unit PZT-51 Density,ρp 7.45×103kg/m3 PZT-51 Elastic stiffness constant,c11E 76.9GPa PZT-51 Piezoelectric constant,d31 -186×10-12C/N PZT-51 Permittivity,ε33S 1500ε0F/m Density of copper substructure,ρs 9.0×103kg/m3 Young’s modulus of copper substructure,Y 106GPa PZT-51 Geometry (L×b×hp) 70mm×15mm×0.55mm Geometry of copper substructure(L×b×hs) 70 mm×18mm×0.45mmAssuming that beam 1 and beam 2 intersect at an angle of 90°, the values of load resistance are, respectively, set as 1KΩ, 33 KΩ, 100 KΩ, and 220 KΩ. When the two beams connect in series, the voltage frequency response functions (FRFs) for four different load resistances are, respectively, shown in Figure5. From this figure, we can see that there are two close peak voltages in the output of V-shaped harvester, and the two peak frequencies, respectively, correspond to the first natural frequency and second natural frequency of this harvester, while the single cantilevered piezoelectric beam only has one natural frequency. It immediately follows that the frequency band is successfully broadened. It can be also seen from Figure 5 that the output peak voltage gradually increases with the increase of load resistance value, and the first natural frequency and second natural frequency increase slightly with the increase of load resistance value. Since the first natural frequency and second natural frequency are close to each other, the proposed energy harvester can effectively increase the operation bandwidth and is more applicable to the actual environment compared to the conventional cantilevered harvester.Figure 5
Voltage FRFs for four different load resistances.Obviously, the value ofθ may also affect the natural frequency of this system. Assuming that θ equals to π/4,π/2, and 3π/4, respectively, and the external load resistance is 100KΩ, then the output voltage frequency responses with different θ are, respectively, illustrated in Figure 6. It can be seen from Figure 6 that when θ equals to π/4, the difference between the first natural frequency and the second natural frequency is the largest; thus the effect of bandwidth extension is not satisfactory in such case. Moreover, when θ equals π/2 or 3π/4, their voltage output performance are similar, whereas the difference between the first natural frequency and the second natural frequency by calculation is the smallest when θ equals 3π/4.Figure 6
Voltage FRFs at differentθ.Su et al. [11] proposed a unimorph V-shaped piezoelectric harvester. The first beam is a cantilevered beam with tip mass while the second beam is attached to the end of the first beam with a certain angle. Piezoelectric layers are attached to both beams in unimorph configuration for power generation. Su investigated the influence of θ on the frequency responses and plotted the curves of the frequency response of voltage and power under different θ. The peak voltage and power are important indicators which can reflect the performance of an energy harvester. For comparison, the simulations for both the unimorph piezoelectric harvester and the bimorph V-shaped piezoelectric harvester are performed based on the same material and geometric parameters used in [11], and the corresponding peak voltages and powers at the first resonant frequency are obtained. The comparative results are listed in Table 2.Table 2
Comparison of the newly bimorph V-shaped harvester with the previous unimorph harvester in [11] (load resistance=50 kΩ).
θ π/4 π/2 3π/4 Vlotage (V/g) Present device 9.67 8.78 8.93 Su [11] 3.01 2.96 2.47 Power (mW/g2) Present device 1.87 1.54 1.59 Su [11] 0.18 0.17 0.12It can be seen from Table2 that both the peak voltage and power at the first resonant frequency generated by the previous unimorph V-shaped harvester in [11] are lower than those generated by the proposed bimorph V-shaped harvester in this study under different θ. While the two kinds of energy harvester have the similar structure, the bimorph configuration has a better effect in improving the output performance than the unimorph configuration. In other words, with the same material and geometric parameters, the newly bimorph V-shaped harvester is able to generate more power than the previous unimorph V-shaped harvester and thus is more practical in vibration energy harvesting.


## 4. Experiment

### 4.1. Structure of Energy Harvester

The V-shaped vibration energy harvester consists of two piezoelectric bimorphs, and its schematic is shown in Figure7.Figure 7
The schematic of the proposed vibration energy harvester.The used piezoelectric bimorph is produced by the PANT, as shown in Figure8(a), and its key parameters have been listed in Table 1. The two cantilevered piezoelectric beams constitute the proposed harvester via a connector, which is shown in Figure 8(b). This cylindrical connector is made of aluminum alloy, and it has eight narrow slots which are manufactured by linear cutting in the axial direction. Thus, we can connect the two piezoelectric cantilevers at one given angle. This connector can be also regarded as a tip mass. The tip mass is made of wolfram steel with a high mass density, which is beneficial to tune the natural frequency. The masses of all components in the designed harvester are listed in Table 3.Table 3
The masses of all the components in the proposed harvester.
Component Mass (g) Bimorph 12 Connector 14 Tip mass 14.6Figure 8
The key components of the proposed harvester: (a) piezoelectric bimorph and (b) connector.
(a) (b)


### 4.2. Energy Harvester Test System

The experimental flow chart and setup of the V-shaped energy harvester are, respectively, illustrated in Figures9 and 10. In the process of testing, an actuator (BK4809) which is driven by a power amplifier (BK2706) and a signal generator (Agilent 33521A) is used to generate sinusoidal excitations with different amplitudes and frequencies. Then the output voltage and current under the excitation can be measured by the multimeter (VC9808). The acceleration sensor (LC0103T) is used to measure the vibration acceleration, whose waveform is displayed by the digital oscilloscope 54645D. And the amplitude of the excitation signal is tuned by the power amplifier.Figure 9
The flow chart of the performance measurement for the proposed harvester.Figure 10
Measurement setup for the proposed harvester.


### 4.3. Experimental Results

Firstly, we test the output performance of this V-shaped energy harvester under certain excitation in order to validate the proposed mathematic model. Let the angle ofθ equal 45°, 90°, and 135°, respectively, and the corresponding experimental setups of V-shaped energy harvester are, respectively, shown in Figures 11(a)–11(c).Figure 11
The experimental setups under thez-axis excitation: (a) θ=45°, (b) θ=90°, and (c) θ=135°.
(a) (b) (c)The open circuit voltage versus frequency curves at differentθ are shown in Figure 12. It is easy to note from Figure 12 that whenever the angle of θ is 45° or 90° or 135°, there are two peak values of open circuit voltage at the first natural frequency and the second natural frequency. The frequency response characteristic of the frequency range between the two natural frequencies (i.e., the operation frequency band) can be used to evaluate the performance of energy harvesting. Obviously, when θ is 45°, the operation frequency band is the widest, whereas its frequency response characteristic is not satisfactory; when θ is 135°, compared to the L-shaped structure (i.e., θ is 90°), not only is the operation frequency band wider (see Table 4), but also its frequency response characteristic is better. Furthermore, when θ is 45° and 135°, both the obtained open circuit voltages under the first natural frequency excitation are much larger than that obtained by the L-shaped structure. Therefore, it can be concluded that the V-shaped structure has a better voltage output performance and wider operation bandwidth than the L-shaped structure in these cases.Table 4
The simulation and experimental results under different conditions.
θ Types The first natural frequency (Hz) The second natural frequency (Hz) Open circuit voltage under the first natural frequency excitation (V/g) Open circuit voltage under the second natural frequency excitation (V/g) θ=45° Experimental value 7 54 139 21.6 Simulation value 7 55 127.00 20.48 θ=90° Experimental value 9.2 29 83.2 22.9 Simulation value 10 30 72.87 20.18 θ=135° Experimental value 12.5 34.2 91.6 49.1 Simulation value 14 36 88.70 40.32Figure 12
The measured open circuit voltage versus exciting frequency curves at differentθ.The simulation and experimental results of V-shaped energy harvester under different conditions are compared, which is shown in Figure13, and the natural frequency and the corresponding open circuit voltage are summarized in Table 4. It can be known from Figure 13 and Table 4 that the experimental results are close to the simulation results, which verifies the proposed mathematical model. It should be noted that the experimental results are a little higher than the simulation results. According to the mechanism of the piezoelectricity, there are three modes, d31, d32, d33, which would transform the deformation into energy. In general, the d32 and d33 mode hardly do contribution to the energy generation, so in this paper we only consider the d31 mode; the transverse deformation of the cantilever beam can generate the maximum energy. In order to simplify the model, the d32 and d33 mode are ignored. But the cantilever beams did deform in vertical direction during experiment, so the experimental results are a little higher than the simulation results under some excitations. In addition, there might be some experimental errors. The amplifier is tuned manually, and the amplitude of the actual excitation may be higher than that of the theoretical excitation, which leads to the fact that the experimental values are higher than the theoretical ones.Figure 13
The comparison between the experimental measurements and model predictions at (b)θ=45°, (c) θ=90°, and (d) θ=135°.
(a) (b) (c)To adapt various ambient excitations, we can conveniently tune the angle of V-shaped energy harvester by the connector, which leads to the change of the operation bandwidth. However, for other commonly used harvesters, tuning the mass may be a conventional and effective method. Obviously, the tuning method based on the proposed harvester is more convenient.Then we consider the impedance matching of the proposed energy harvester under different structure parameters and ambient excitations. When the three kinds of V-shaped energy harvesters are excited at the first natural frequency and the second natural frequency under different load resistance, the measured output voltage and power versus load resistance curves are simultaneously plotted in Figure14. It can be seen from this figure that the output voltage will increase with the increase of the load resistance, and the output voltage will be close to the open circuit voltage when the resistance is up to a certain value. However, for the output power, it will firstly increase and then decrease with the increase of load resistance. There is a maximum output power, and the corresponding load resistance is the best matched resistance.Figure 14
The output voltage and output power versus load resistance curves underz-axis excitation: (a) θ=45°, the first natural frequency excitation; (b) θ=45°, the second natural frequency excitation; (c) θ=90°, the first natural frequency excitation; (d) θ=90°, the second natural frequency excitation; (e) θ=135°, the first natural frequency excitation; (f) θ=135°, the second natural frequency excitation.
(a) (b) (c) (d) (e) (f)


## 4.1. Structure of Energy Harvester

The V-shaped vibration energy harvester consists of two piezoelectric bimorphs, and its schematic is shown in Figure7.Figure 7
The schematic of the proposed vibration energy harvester.The used piezoelectric bimorph is produced by the PANT, as shown in Figure8(a), and its key parameters have been listed in Table 1. The two cantilevered piezoelectric beams constitute the proposed harvester via a connector, which is shown in Figure 8(b). This cylindrical connector is made of aluminum alloy, and it has eight narrow slots which are manufactured by linear cutting in the axial direction. Thus, we can connect the two piezoelectric cantilevers at one given angle. This connector can be also regarded as a tip mass. The tip mass is made of wolfram steel with a high mass density, which is beneficial to tune the natural frequency. The masses of all components in the designed harvester are listed in Table 3.Table 3
The masses of all the components in the proposed harvester.
Component Mass (g) Bimorph 12 Connector 14 Tip mass 14.6Figure 8
The key components of the proposed harvester: (a) piezoelectric bimorph and (b) connector.
(a) (b)


## 4.2. Energy Harvester Test System

The experimental flow chart and setup of the V-shaped energy harvester are, respectively, illustrated in Figures9 and 10. In the process of testing, an actuator (BK4809) which is driven by a power amplifier (BK2706) and a signal generator (Agilent 33521A) is used to generate sinusoidal excitations with different amplitudes and frequencies. Then the output voltage and current under the excitation can be measured by the multimeter (VC9808). The acceleration sensor (LC0103T) is used to measure the vibration acceleration, whose waveform is displayed by the digital oscilloscope 54645D. And the amplitude of the excitation signal is tuned by the power amplifier.Figure 9
The flow chart of the performance measurement for the proposed harvester.Figure 10
Measurement setup for the proposed harvester.


## 4.3. Experimental Results

Firstly, we test the output performance of this V-shaped energy harvester under certain excitation in order to validate the proposed mathematic model. Let the angle ofθ equal 45°, 90°, and 135°, respectively, and the corresponding experimental setups of V-shaped energy harvester are, respectively, shown in Figures 11(a)–11(c).Figure 11
The experimental setups under thez-axis excitation: (a) θ=45°, (b) θ=90°, and (c) θ=135°.
(a) (b) (c)The open circuit voltage versus frequency curves at differentθ are shown in Figure 12. It is easy to note from Figure 12 that whenever the angle of θ is 45° or 90° or 135°, there are two peak values of open circuit voltage at the first natural frequency and the second natural frequency. The frequency response characteristic of the frequency range between the two natural frequencies (i.e., the operation frequency band) can be used to evaluate the performance of energy harvesting. Obviously, when θ is 45°, the operation frequency band is the widest, whereas its frequency response characteristic is not satisfactory; when θ is 135°, compared to the L-shaped structure (i.e., θ is 90°), not only is the operation frequency band wider (see Table 4), but also its frequency response characteristic is better. Furthermore, when θ is 45° and 135°, both the obtained open circuit voltages under the first natural frequency excitation are much larger than that obtained by the L-shaped structure. Therefore, it can be concluded that the V-shaped structure has a better voltage output performance and wider operation bandwidth than the L-shaped structure in these cases.Table 4
The simulation and experimental results under different conditions.
θ Types The first natural frequency (Hz) The second natural frequency (Hz) Open circuit voltage under the first natural frequency excitation (V/g) Open circuit voltage under the second natural frequency excitation (V/g) θ=45° Experimental value 7 54 139 21.6 Simulation value 7 55 127.00 20.48 θ=90° Experimental value 9.2 29 83.2 22.9 Simulation value 10 30 72.87 20.18 θ=135° Experimental value 12.5 34.2 91.6 49.1 Simulation value 14 36 88.70 40.32Figure 12
The measured open circuit voltage versus exciting frequency curves at differentθ.The simulation and experimental results of V-shaped energy harvester under different conditions are compared, which is shown in Figure13, and the natural frequency and the corresponding open circuit voltage are summarized in Table 4. It can be known from Figure 13 and Table 4 that the experimental results are close to the simulation results, which verifies the proposed mathematical model. It should be noted that the experimental results are a little higher than the simulation results. According to the mechanism of the piezoelectricity, there are three modes, d31, d32, d33, which would transform the deformation into energy. In general, the d32 and d33 mode hardly do contribution to the energy generation, so in this paper we only consider the d31 mode; the transverse deformation of the cantilever beam can generate the maximum energy. In order to simplify the model, the d32 and d33 mode are ignored. But the cantilever beams did deform in vertical direction during experiment, so the experimental results are a little higher than the simulation results under some excitations. In addition, there might be some experimental errors. The amplifier is tuned manually, and the amplitude of the actual excitation may be higher than that of the theoretical excitation, which leads to the fact that the experimental values are higher than the theoretical ones.Figure 13
The comparison between the experimental measurements and model predictions at (b)θ=45°, (c) θ=90°, and (d) θ=135°.
(a) (b) (c)To adapt various ambient excitations, we can conveniently tune the angle of V-shaped energy harvester by the connector, which leads to the change of the operation bandwidth. However, for other commonly used harvesters, tuning the mass may be a conventional and effective method. Obviously, the tuning method based on the proposed harvester is more convenient.Then we consider the impedance matching of the proposed energy harvester under different structure parameters and ambient excitations. When the three kinds of V-shaped energy harvesters are excited at the first natural frequency and the second natural frequency under different load resistance, the measured output voltage and power versus load resistance curves are simultaneously plotted in Figure14. It can be seen from this figure that the output voltage will increase with the increase of the load resistance, and the output voltage will be close to the open circuit voltage when the resistance is up to a certain value. However, for the output power, it will firstly increase and then decrease with the increase of load resistance. There is a maximum output power, and the corresponding load resistance is the best matched resistance.Figure 14
The output voltage and output power versus load resistance curves underz-axis excitation: (a) θ=45°, the first natural frequency excitation; (b) θ=45°, the second natural frequency excitation; (c) θ=90°, the first natural frequency excitation; (d) θ=90°, the second natural frequency excitation; (e) θ=135°, the first natural frequency excitation; (f) θ=135°, the second natural frequency excitation.
(a) (b) (c) (d) (e) (f)


## 5. Conclusions

Since the ambient vibration is usually broadband and changeable, a novel V-shaped vibration energy harvester is proposed based on the cantilevered piezoelectric bimorph structure. This harvester consists of two cantilevered piezoelectric bimorph beams which intersect at a certain angle. By using the Euler-Bernoulli equation and piezoelectric equation, the mechanical coupling equations of the V-shaped energy harvester are established. And then the coupled circuit equation is derived by Kirchhoff's laws. With the above equations, the output responses of the V-shaped harvester under different structural parameters and load resistances are calculated, and it can be seen from the simulation results that there is a wide operation band between the first natural frequency and the second natural frequency. At last, various experiments are carried out to demonstrate the output performance of the proposed energy harvester, and the obtained results show that(1) the natural frequency and operation bandwidth of the V-shaped structure can be easily tuned by the use of the connector to adapt the ambient vibration; (2) the operation frequency band of V-shaped structure and its corresponding frequency response characteristic can be improved for some θ; (3) the V-shaped structure has a better voltage output performance than the L-shaped structure and the V-shaped unimorph structure in the given examples; (4) the maximum output power can be achieved by resistance matching under the excitation of natural frequency. Therefore, the proposed V-shaped energy harvester can not only increase the output of the electrical energy from the environmental vibration, but also conveniently tune the operation bandwidth, which make it more applicable in various environments. In further study, an adaptive connector (such as motor and spring) can be designed to make the V-shaped energy harvester self-tune with the change of ambient excitation.



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*Source: 7082724-2018-05-27.xml*