Uniform Stress Distribution of Bimorph by Arc Mechanical Stopper for Maximum Piezoelectric Vibration Energy Harvesting

Abstract

To convert as much vibration energy as possible into electrical energy, the design of a high-performance piezoelectric vibration energy harvester (PVEH) has been studied widely in recent years. To overcome the low energy utilization of a traditional piezoelectric cantilever by inhomogeneous strain, a uniform stress distribution of bimorph by an ARC mechanical stopper structure has been designed for maximum piezoelectric vibration energy harvesting. Deflection equations and their simulation at the first-order modal of two classic bimorph cantilever beam models, with transverse tip force and with equal curvature, are derived based on the Euler–Bernoulli beam assumption. Piezoelectric energy from a beam model with equal curvature is four times that of a cantilever beam model with transverse tip force at the theoretical level. The nonlinear frequency response performance of bimorphs by an ARC mechanical stopper and point stopper model could be observed by the numerical simulations of the lumped parameter electromechanical model. PVEH prototypes were manufactured by 3D printing and tested. To verify the high-power generation capacity, PVEH with an ARC stopper has 1.756 times more voltage than that of a PVEH with a point stopper.

1. Introduction

The distributed micro-energy consumption scenario in the Internet of Things era supports the booming development of energy-harvesting technology. Vibration energy harvesting can be widely used in the human body, vehicles, and the industrial environment to power electronic devices such as wireless sensor nodes. A piezoelectric vibration energy harvester (PVEH) is a device that converts the strain mechanical energy of piezoelectric materials into electrical energy through the piezoelectric effect [1]. Due to its compact structure, high electromechanical conversion efficiency, and no electromagnetic interference, PVEH has become a promising vibration power generation device. How to maximize the use of piezoelectric materials for power generation has become an interesting research topic [2]. Generally speaking, PVEH resonates under the excitation of environmental vibration, the stress of piezoelectric materials is large, and PVEH has ideal power output. In previous studies, a piezoelectric bimorph cantilever is usually used for resonant power generation, and the additional tip mass is used to reduce the resonant frequency [3].

A piezoelectric bimorph cantilever beam is composed of upper and lower piezoelectric layers and an intermediate substrate layer. It can transform the transverse vibration into a longitudinal strain of the piezoelectric layer and has the effect of stress amplification. The stress distribution of the piezoelectric layer in vibration deformation determines the effect of power generation. Because the root stress is the largest when the cantilever vibrates, and the stress gradually decreases along the length of the cantilever to the tip, many studies only arrange the piezoelectric sheet at the root of the cantilever [4]. Although the piezoelectric material has high stress utilization at the root of the cantilever [5], it has low stress utilization, large volume, and low power density for the whole device; therefore, the design of some cantilevers with uniform stress distribution along the length appears. For example, instead of using a rectangular cantilever, a trapezoidal cantilever [6], triangular cantilever [7], variable section cantilever, and piezoelectric functional gradient material (PFGM) [8] are designed accordingly.

However, to achieve uniform stress distribution along the length of a rectangular cantilever, it is necessary to produce an arc-shaped mode instead of a cantilever mode in the first-order eigenfrequency [9]. Some curved piezoelectric cantilevers have been proposed [10], but it is difficult to manufacture. It has been studied that an arc deflection limiting device can make the stress uniformly distributed along the axial direction when pressing the piezoelectric cantilever [11]. The arc frame [12] is also used to limit the vibration mode in wind energy harvesting by piezoelectric cantilever and triboelectric nanogenerator [13]; however, the current study focuses on the static characteristics of the arc frame, which is unable to show more complex dynamic features. Meanwhile, there is a lack of research into PZT materials; the bulk of research into PZT materials has ignored the discussion on the arc stopper’s ability to protect the bimorph cantilever.

The design of the mechanical stopper plays an important role in energy-harvesting capacity. On the one hand, the mechanical stopper is needed to protect the piezoelectric cantilever from overload; the deformation is limited by the allowable stress of the piezoelectric material [14]. On the other hand, the nonlinear vibration caused by collision with the stopper [15] widens the frequency bandwidth of energy harvesting [16].

This paper proposed a PVEH with a mechanical arc stopper to achieve high efficiency by maximum uniform stress distribution of the bimorph cantilever. It is confirmed both experimentally and theoretically that PVEH with a mechanical arc stopper can generate more power than PVEH with a traditional tip mechanical stopper at a significant acceleration level. Further, it still provides overload protection and broadband characteristics. We have paid more attention to deriving critical design parameters and studying the dynamic features brought by a mechanical arc stopper. Moreover, 3D-printing technology has been applied in manufacturing the arc frame, which makes the high-efficiency PVEH easier to produce compared to the current design. The novel arc stopper design can be easily applied to most of the existing micro PVEHs with a compact package.

2. Modeling and Theoretical Analysis

Two PVEHs designed in this study are shown in Figure 1; parameters are shown in Table 1. They have the same bimorph cantilever and tip mass but different frames and stopper shapes. Structure A is a traditional design with a parallel frame; the point stopper only works at the 5/6 length point of the cantilever to limit the amplitude during vibration. Structure B is a novel design with a circular arc surface mechanical stopper, which is in contact from the root to the 5/6 length point of the cantilever to limit the amplitude during vibration.

According to the piezoelectric constitutive equation in the d31 mode in Equation (1), the strain $S_1$ is expressed as the thickness of the neutral layer $h_c$ divided by the bending radius r in geometric Equation (2). The maximum stress of the piezoelectric layer in X (1) direction $T_{max}$ can be written as Equation (3).

(1)$\left[\begin{array}{c}{T}_1\\ D_3\end{array}\right]=\left[\begin{array}{cc}{c}_{11}^E& -e_{31}\\ e_{31}& {\epsilon }_{33}^S\end{array}\right]\left[\begin{array}{c}{S}_1\\ E_3\end{array}\right], E_3=\frac{V}{t_p}$

(2)$S_1=\frac{h_c}{r},h_c=t_p+\frac{t_s}{2}$

(3)$T_{max}=\frac{Y_p\left(t_p+\frac{t_s}{2}\right)}{r}-\frac{e_{31}V}{t_p}$

where the $h_c$ is the distance between the piezoelectric layer surface and the neutral layer, $c_{11}^E$ represents the elastic constant in the 11 direction under a constant electric field, and $Y_p$ is the piezoelectric Young’s modulus in bending mode.

The allowable stress of PZT is about 60~80 MPa [17]. The maximum stress of PZT $T_{max}$ is composed of the elastic strain part and electric field part, which shall not exceed the allowable stress of PZT; therefore, the minimum curvature radius $r$ of the piezoelectric bimorph bending can be designed according to Equation (3) in the case of short circuit $V=0$. The stress in the open circuit state should consider the electromechanical coupling effect.

(4)$\frac{1}{r\left(x\right)}=\frac{{\partial }^2\omega \left(x\right)}{\partial x^2}=u_0{\varphi }^{″}\left(x\right)$

Equation (4) gives the differential equation of the deflection curve of the Euler–Bernoulli beam under small deformation. The deflection curve of the cantilever beam $\omega \left(x\right)$ can be separated into tip displacement $u_0$ and displacement distribution shape function $\varphi \left(x\right)$. For the first-order modal dynamic response, tip displacement $u_0$ is a function of time; shape function $\varphi \left(x\right)$ is the first-order mode shape.

Because the shape function of the cantilever beam changes when contacting the stopper, this study adopts the shape function of two models, one is the cantilever beam model with transverse tip force, and the other is the beam model with equal curvature. The displacement shape function of the cantilever beam with transverse tip force and its first-order and second-order differential with respect to x is shown in Equation (5). Here ${\varphi }^{″}\left(x\right)$ can obtain the maximum value ${\varphi }^{″}\left(0\right)=3/l^2$ at $x=0$.

(5)${\varphi }_1\left(x\right)=\frac{x^2\left(3l-x\right)}{2l^3}, {\varphi }_1^{\prime }\left(x\right)=\frac{3x\left(2l-x\right)}{2l^3}, {\varphi }_1^{″}\left(x\right)=\frac{3\left(l-x\right)}{l^3}$

The displacement shape function of a beam with equal curvature and its first-order and second-order differential with respect to x is shown in Equation (6).

(6)${\varphi }_2\left(x\right)=\frac{3x^2}{2l^2}, {\varphi }_2^{\prime }\left(x\right)=\frac{3x}{l^2}, {\varphi }_2^{″}\left(x\right)=\frac{3}{l^2}$

2.1. Stress and Mode Modeling

The minimum radius of curvature r determines the maximum stress distribution of the two models under a short circuit, in which the transverse tip force cantilever model stress distribution is $T_1$, and stress distribution of beam model with equal curvature is $T_2$.

(7)$T_1=\frac{Y_p{h}_c{u}_0}{r}{\varphi }_1^{″}\left(x\right), T_2=\frac{Y_p{h}_c{u}_0}{r}{\varphi }_2^{″}\left(x\right)$

(8)$E_s=\frac{1}{2}{Y}_p\int T^{2}d{V}_p, E_p=k_{31}^2{E}_s$

where $E_s$ is strain energy and $E_p$ is piezoelectric energy; $V_p$ is piezoelectric volume; $k_{31}$ is the electromechanical coupling coefficient of 31 mode.

The minimum radius of curvature r determines the maximum deflection distribution of the two models, in which the transverse tip force cantilever model deflection $D_1=u_0{\varphi }_1\left(x\right)$, and deflection of beam model with equal curvature $D_2=u_0{\varphi }_2\left(x\right)$. PVEH with point stopper adopts the cantilever beam model with transverse tip force. The shape function of the beam model limited by the arc stopper comprehensively uses two model shape functions according to the arc length of the contact limiting as Equation (8). The beam model with equal curvature is used for the contact part with the arc surface, and the sum of the deflection of the cantilever beam model and the deflection and rotation angle of the contact part is used for the non-contact part.

(9)$D_3=\left\{\begin{array}{c}{u}_0{\varphi }_1\left(x-l_s\right)+u_0{\varphi }_2\left(l_s\right)+\left(x-l_s\right)l_s/r,x\ge l_s\\ u_0{\varphi }_2\left(x\right), x\le l_s\end{array}\right.$

2.2. Electromechanical Coupling Modeling

The lumped parameter (LP) model [18] is introduced in the electromechanical coupling analysis of the nonlinear vibration system. The displacement excitation of the base is $y\left(t\right)$. The absolute displacement of mass is $x\left(t\right)$ and relative displacement is $z\left(t\right)=x\left(t\right)-y\left(t\right)$. The electromechanical coupling LP model is given in Equation (10).

(10)$\begin{array}{c}M\ddot{z}+D\dot{z}+Kz+f_s-\theta V={\beta }_{F}m\ddot{y}\\ C_p\dot{V}+\frac{V}{R}+\theta \dot{z}=0\end{array}$

where M, D, and K are the mechanical system parameters of mass, damping, and stiffness, respectively. V, $C_p$, $\theta$ are the electrical system parameters of voltage, capacitance, and electromechanical coupling factor, respectively. R is the load resistance. ${\beta }_F$ is force correction factor. The dot represents the derivative of time.

Compared to the linear system, a nonlinear impact force by the double side stopper is introduced as Equation (11).

(11)$f_s=k_s{\left[z-\left(\left|z+d_s\right|-\left|z-d_s\right|\right)/2\right]}^3$

where $k_s$ is the nonlinear stiffness caused by the stopper.

For the parallel connection of piezoelectric bimorphs, $C_p$ and $\theta$ can be calculated as:

(12)$C_p=\frac{2wl}{t_p}{\epsilon }_{33}^S$

(13)$\theta =2{\varphi }^{\prime }\left(L\right)w\left(t_p+t_s\right)e_{31}={\beta }_{\theta }{\theta }_a, {\theta }_a=2w\left(t_p+t_s\right)e_{31}/L, {\beta }_{\theta }=L{\varphi }^{\prime }\left(L\right)$

where ${\theta }_a$ is the average electromechanical coupling factor, and ${\beta }_{\theta }$ is electromechanical coupling correction factor, which is 1.5 for ${\varphi }_1^{\prime }\left(L\right)$, and 3 for ${\varphi }_2^{\prime }\left(L\right)$.

The electromechanical coupling factor $\theta$ is proportional to the ${\varphi }^{\prime }\left(L\right)$, which is the function of the $l_s$, as shown in Equation (14). When the bimorph contacts the arc stopper, the contact part with length $l_s$ adopts the equal curvature beam model, and the non-contact part adopts the transverse tip force cantilever beam model.

(14)$\begin{array}{c}\theta \left(l_s\right)=2\left(\frac{3}{L}\frac{l_s}{L}+\frac{3}{2L}\frac{\left(L-l_s\right)}{L}\right)w\left(t_p+t_s\right)e_{31}\\ ={\beta }_{\theta }\left(l_s\right){\theta }_a \\ {\beta }_{\theta }\left(l_s\right)=l_s{\varphi }_1^{\prime }\left(L\right)+\left(L-l_s\right){\varphi }_2^{\prime }\left(L\right)\end{array}$

3. Simulation Analysis and Comparison

The performance of two kinds of PVEH was simulated in MATLAB, and the design and simulation parameters are shown in Table 1.

3.1. Stress and Mode Analysis

Derived from Equation (3), and as the simulation results shown in Figure 2a, PZT maximum stress $T_{max}$ under static state, sharply rises with a decrease in curvature radius $r$. Here, the curvature radius of 0.2 m was chosen with a maximum stress of 51 MPa. In Figure 2b, the surface stress distribution of the beam with equal curvature is uniform and the utilization rate of piezoelectric stress is high, while the transverse tip force cantilever has only the maximum stress at the root, and the utilization rate of piezoelectric stress is only half of that of the beam with equal curvature. Since the strain energy is directly proportional to the square of the stress integral, and piezoelectric energy is proportional to strain energy, piezoelectric energy from the beam model with equal curvature is four times that of the cantilever beam model with transverse tip force at the theoretical level.

The deflection simulation curve is shown in Figure 2c. To leave space for the tip mass block, the stopper is set at the beam length point $l_s=50\mathrm{mm}$. The stopper distance of the two PVEHs can be determined according to D₁ and D₂ in Figure 2. D₃ is a deflection simulation curve when $l_s=10 \mathrm{mm}$. For actual machining, the half thickness of the bimorph should be considered in the stopper distance.

3.2. Electromechanical Coupling Analysis

The curve of the electromechanical coupling coefficient with the arc stopper contact length is simulated in Figure 3a. The electromechanical coupling coefficient when the bimorph fully contacts the arc stopper is twice ($\theta$ = −0.005 N/V) that of when the bimorph does not contact the arc stopper ($\theta$ = −0.0025 N/V). It means that an equal curvature beam makes full use of piezoelectric materials. For $l_s$ = 50 mm adopted in this design, the electromechanical coupling coefficient $\theta$ is −0.004591 N/V. A high electromechanical coupling coefficient can result in strong coupling for the system and make it easy to reach the theoretical maximum power in the resonant state [19].

Tip deflection is the independent variable for the vibration system. The simulation curve of the relationship between tip deflection and contact length is shown in Figure 3b. The relationship between tip deflection and electromechanical coupling coefficient is shown in Figure 3c.

The LP model parameters of PVEH are shown in Table 2. For the cubic stiffness induced by the stopper, $K_s$ can be set as 0, 1 × 10⁹, 9 × 10⁶ N/m³ for PVEHs with non-stopper, point stopper, and arc stopper, respectively. Based on the LP model that we present in Equation (10), we apply the ode45 solver from MATLAB to obtain the numerical output of displacement and voltage. Displacement and voltage frequency response simulation curves of PVEHs with non-stopper, point stopper, and arc stopper, respectively, at different acceleration levels, are shown in Figure 4.

From Figure 4d–f, we can indicate that the output voltage of the arc mechanical stopper design is the best among the other designs in this study. The proposed output can be even higher than the linear PVEH when the $acc=1.0 \mathrm{g}$. Figure 4 provides the simulation output of the proposed PVEH in terms of tip displacement and voltage, which indicate the advantages and efficiency of the arc mechanical design in theory. The ability to optimize the distribution of the stress along the beam makes the output data better than that of the point stopper. Meanwhile, from Figure 4a,d, we can indicate that the displacement and voltage of PVEH have the characteristics of a linear system when $acc=0.5 \mathrm{g}$. With an increase in acceleration, the proposed PVEH shows the characteristics of broadband, which is a desirable feature in a stopper.

4. Experimental Validation

As shown in Figure 5, the main equipment for vibration excitation consists of a PC, a vibration test system that generates excitation signals, a power amplifier that enhances the excitation signals, a signal processing unit (NI USB-6009) that collects the output voltage data, a vibration platform (Econ VT9002) that provides excitation and acceleration, and a meter (Econ EV4200) that collect the status of the platform and can be part of the close-loop control, which helps the system generate the stable output signals. The vibration signals generated from the test system could be enhanced by the power amplifier to drive the platform to move frequently. With the excitation applied at the end of the PVEH design, the signal-processing unit could collect the output voltage from the piezoelectric cantilever beam and display it via the PC.

Prototypes of the PVEHs are shown in Figure 6a,b; PVEH geometric design parameters are shown in Table 1. The polylactic acid (PLA) for the connector and stopper were fabricated by 3D printing. The end of the cantilever beam is fixed by ergo 5800 glue. The material of the PZT-5H piezoelectric bimorph is the same as ref. [20]. Two piezoelectric layers of bimorph are connected in parallel, with the same direction of polarization. Meanwhile, the metal layer in the middle shares the electrode.

Experimental voltage frequency response curves at different acceleration levels are shown in Figure 6c. Before touching the stopper, the vibration characteristics of the two PVEHs are the same in the non-resonant region and the resonant region, less than 0.5 g. With the increase in acceleration amplitude, the two PVEHs show different frequency response curves in the resonant region. The point collision of PVEH-1 limits the amplitude, the voltage amplitude at the original resonant frequency point does not increase, and the frequency response curve clipping is obvious. The point collision also increases the system stiffness; the resonant frequency shifts to a high frequency due to the nonlinear frequency extension effect. PVEH-2 has a piezoelectric cantilever contact with the arc stopper gradually from the root and the limitation of the amplitude is a gradual process; therefore, the voltage still increases in the resonant region by more than 1 g. This process gradually increases the strain at the contact point of the cantilever beam to as large as the root strain and releases the complete power generation capacity of the piezoelectric beam. In our experiment, open-circuit voltage $V_{oc}$ of PVEH-2 in 1.5 g acceleration and 20 Hz is 72 V, which is 1.756 times that in PVEH-1 of 41 V. The arc stopper makes the system stiffness increase gradually; the point stopper makes the system stiffness increase sharply. They both have the nonlinear characteristic that the resonant frequency moves to high frequency; PVEH-1 has a wider resonant frequency band than PVEH-2.

The power performance of PVEHs in 1 g acceleration is shown in Figure 7. Theoretical simulation of the curved surface of the RMS power with load resistance and frequency is plotted in MATLAB according to Equation (10) without the $f_s$. The ideal output power is 27.5 mW at 30 kΩ in 19.5 Hz without the stopper; however, stoppers will limit the actual power output. Experimental curves of peak voltage and RMS power with load resistance in the resonate state indicate PVEH-1 only has a maximum RMS power of 12.9 mW at 30 kΩ, while PVEH-2 has the maximum RMS power of 20.7 mW at 20 kΩ. That means PVEH-2 with an arc stopper wastes less power than PVEH-1 with a point stopper.

5. Conclusions

The main goal of the current study was to maximize the utilization of vibration energy by proposing a bimorph harvester with uniform stress distribution by an arc mechanical stopper. The proposed PVEH increases the conversion efficiency by making the maximum stress at the root distribute uniformly along the cantilever. The static model and nonlinearity of the electromechanical vibration model are introduced with a piecewise shape function. The numerical simulation of stress and deflection indicates the strain energy and piezoelectric energy of bimorph in uniform stress distribution is four times that of the gradient distribution. The nonlinearity electromechanical simulation indicates that the high harvesting voltage is caused by the electromechanical coupling coefficient rising with the deflection and contact length. Furthermore, frequency response voltage curvature in simulation and experiments verify the advantages of the proposed design with 1.756 times of maximum voltage. The experimental comparison of the power performance indicated that the PVEH with an arc stopper wastes less power than that with a point stopper. The proposed PVEH with an arc mechanical stopper is beneficial to high voltage output under a compact package and allowable stress requirements.

Author Contributions

Conceptualization, L.W. and Z.W.; methodology, L.W.; software, Y.Z.; validation, G.Q.; formal analysis, Q.W.; investigation, Z.W.; resources, J.S. and P.Y.; data curation, S.L.; writing—original draft preparation, L.W.; writing—review and editing, D.L.; visualization, L.W.; supervision, L.Z.; project administration, Z.J.; funding acquisition, R.M. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Tables

Figure 1. Schematic of PVEH (a) PVEH-1: cantilever configuration with point stopper. (b) PVEH-2: cantilever configuration with arc stopper. (c) Schematic of bimorph thickness. (d) Schematic of bimorph bending cross-section.

Figure 2. (a) Simulation curve of PZT maximum stress with curvature radius r. (b) Simulation surface stress curves of transverse force cantilever model T1 and equal curvature beam model T2 along the x axis. (c) Simulation deflection curves of transverse force cantilever model D1 and equal curvature beam model D2 along the x axis.

Figure 3. (a) Simulation electromechanical coupling coefficient curve along with arc stopper contact length. (b) Simulation curve of the relationship between tip deflection and contact length. (c) Simulation curve of the relationship between tip deflection and electromechanical coupling coefficient.

Figure 4. Simulation frequency response curves of PVEHs with non-stopper, point stopper, and arc stopper, respectively, at different acceleration levels. Displacement response curves at acceleration of (a) 0.5 g, (b) 1.0 g, (c) 1.5 g. Voltage response curves at acceleration of (d) 0.5 g, (e) 1.0 g, (f) 1.5 g.

Figure 5. Schematic of vibration test system and voltage measurement system.

Figure 6. Prototype of PVEHs. (a) PVEH-1: cantilever configuration with point stopper. (b) PVEH-2: cantilever configuration with arc stopper. (c) Experimental voltage frequency response curves at different acceleration levels: PVEH-1 (solid line), PVEH-2 (dotted line).

Figure 7. Power performance of PVEHs in 1 g acceleration. (a) Theoretical simulation curved surface of RMS power with load resistance and frequency. (b) Experimental curves of peak voltage and RMS power with load resistance: PVEH-1 (solid line), PVEH-2 (dotted line).

Table 1

Design and simulation parameters of PVEHs.

Symbol	Parameters	Value
lb	Bimorph length	60 mm
b	Bimorph breadth	20 mm
tp	Piezoelectric layer thickness	0.18 mm
ts	Substrate layer thickness	0.14 mm
lm	Mass length	10 mm
mb	Beam mass	4.6 g
mt	Tip mass	7.8 g
Yp	Piezoelectric layer elasticity	41.2 GPa
Ys	Substrate layer elasticity	110 GPa
rs	arc stopper radius	200 mm
ls	Stopper length	50 mm
ds 1	Stopper distance 1	4.75 mm
ds 2	Stopper distance 2	6.60 mm
e 31	Piezo stress coefficient	−15.65 C/m2
Cp	Piezoelectric capacitance	368 nF

Table 2

The LP model parameters of PVEH.

K	M	D	Cp	θ	βF
123.58 N/m	8.904 g	2.0724 Ns/m	368 nF	−2.5 mN/V	1.072

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