**1. Introduction**

Wireless sensor, wearable devices, and medical implants have shown their significance in modern society [1]. Powering these low-power devices is usually done through conventional batteries, however, these batteries must be regularly recharged or replaced, which can be very costly and cumbersome [2]. Meanwhile, there are environmental issues when disposing of used batteries after operation [3]. Energy harvesting technology holds great potential to achieve the self-powered operation of these devices. Among the various energy harvesting technologies, electromagnetism, electrostatics, and piezoelectricity are the three main methods that generate energy from vibration [4,5]. In particular, vibration-based piezoelectric energy technology can convert kinetic energy from the ambient environment via piezoelectric effect to electric energy, which has received considerable interest for its high energy density, ease of implementation, and miniaturization [6].

At the early stage, research on piezoelectric energy harvesters was mainly based on a linear piezoelectric energy harvester. The linear piezoelectric energy harvester has a high resonance frequency, and when the environmental frequency deviates from its resonance frequency, the power generation performance of the system will drop sharply, resulting in low environmental adaptability [7]. Currently, nonlinear bistable piezoelectric energy harvesters have received great attention. Zhang et al. [8] proposed an arched composite beam magnetically coupled piezoelectric energy harvester. Experiments showed that the

**Citation:** Chen, X.; Zhang, X.; Chen, L.; Guo, Y.; Zhu, F. A Curve-Shaped Beam Bistable Piezoelectric Energy Harvester with Variable Potential Well: Modeling and Numerical Simulation. *Micromachines* **2021**, *12*, 995. https://doi.org/10.3390/ mi12080995

Academic Editors: Kai Tao and Yunjia Li

Received: 24 July 2021 Accepted: 16 August 2021 Published: 21 August 2021

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

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

effective bandwidth of the energy harvester under magnetic coupling was 3.1 times the bandwidth without magnetic force. Rubes et al. [9] conducted research on magnetically coupled bistable piezoelectric energy harvesters, and their research showed that the introduction of nonlinear stiffness can greatly improve the energy harvesting performance of piezoelectric energy harvesters; Erturk and Inman [10] experimentally proved that the nonlinearity of magnetic coupling can cause vibration between the bistable high-energy traps, thereby improving the collection performance of the energy harvester; and Stanton et al. [11] established a complete dynamic model for the output voltage and dynamic behavior of the magnetic coupling bistable piezoelectric energy harvester and proved the availability of the bistable harvester. Under the condition of simple harmonic excitation, Li et al. [12] developed a magnetic-coupled bi-stable flutter-based energy harvester and proved that the proposed system was an effective design approach for enhancing energy harvesting capability in a low air speed range. Singh et al. [13] investigated a bistable piezoelectric energy harvester with SSHI circuit, and their experiments proved that the output power of the bistable piezoelectric energy harvester with the SSHI circuit reached 478 µw, while the corresponding linear structure was only 129 µw.

The above research shows that the bistable piezoelectric energy harvester is effective for improving the performance of an energy harvester. However, the above harvesters all had a fixed barrier height. In practical applications, the excitation level must provide enough energy to overcome the barrier to achieve a large response, otherwise it will not be able to work well, resulting in poor output performance. In order to reduce the barrier height to improve the performance of the bistable piezoelectric energy harvester, many scholars have carried out studies on piezoelectric energy harvesters with variable potential wells. Zhou et al. [14] placed an external magnet in the middle of the fixed beams at both ends and proposed a bistable system with variable potential wells. Experiments proved that the system not only had a low interwell jump threshold, but also produced higher voltage output. Cao et al. [15] proposed a bistable energy harvesting with time varying potential energy to harvest energy from human motion and various motion speed treadmill tests were performed to demonstrate the advantage of time-varying bistable harvesters over linear and monostable. Nguyen et al. [16] proposed a bistable piezoelectric energy harvester with an auxiliary magnet oscillator and their research showed that this design could improve 114–545% bandwidth compared with traditional bistable piezoelectric energy harvesters. Yang et al. [17–19] designed a double-beam piezoelectric energy harvester with variable potential well structure and verified its advantages over traditional bistable piezoelectric energy harvester under random excitation conditions. Lan et al. [20] significantly reduced the barrier height of the traditional bistable piezoelectric energy harvester by adding a small magnet to a traditional bistable energy harvester and compared their design with a three-stable piezoelectric energy harvester, verifying the validity of the proposed device. Shan et al. [21] designed an elastically connected bistable piezoelectric energy harvester based on the straight beam configuration, where the energy harvester had a variable potential barrier during the vibration process. It was experimentally proven that the energy harvesting bandwidth was 60% higher than that of the traditional energy harvester. Li et al. [22] carried out theoretical analysis on the elastically connected straight beam piezoelectric energy harvester, and the results showed that the spring-connected bistable piezoelectric energy harvester had a variable potential function and better energy harvesting performance under low-frequency excitation. Kim et al. [23] designed a multi-degree of freedom (MDOF) vibration energy harvesting system that leverages magnetically coupled bistable and linear harvesters, where the analytical, numerical, and experimental investigations revealed that the novel harvester could facilitate the energetic interwell response for relatively low excitation amplitudes and frequencies by passively and adaptively lowering the potential energy barrier level. Qian et al. [24] developed a broadband piezoelectric energy harvester (PEH) with a mechanically tunable potential function, and the simulations proved that the proposed PEH could harvest vibration energy in a wide frequency range of 0–91 Hz at the excitation level of 0.5 g.

Inspired by the development of variable-potential-energy techniques, this paper proposes a novel bistable energy harvester with a variable potential well. Meanwhile, we used a curve-shaped beam as the energy transducing element to further improve the performance of the piezoelectric energy harvester due to the disadvantages of the straight beam in terms of uneven stress, low conversion efficiency [25,26]. The finite element simulation was performed for the curve-shaped beam and the conventional beam. The results show that the curved beam structure has a special stress distribution and can improve output voltage compared with the straight beam structure. Then, the dynamic model of BPEH-V system is established. Numerical simulation analysis showed that it was easier for the proposed harvester to achieve large-amplitude response in a low-frequency environment compared with the conventional counterpart, and the spring stiffness had an important impact on system performance. The research can provide theoretical guidance for the optimal design and engineering application of the novel piezoelectric energy harvester. proposes a novel bistable energy harvester with a variable potential well. Meanwhile, we used a curve-shaped beam as the energy transducing element to further improve the performance of the piezoelectric energy harvester due to the disadvantages of the straight beam in terms of uneven stress, low conversion efficiency [25,26]. The finite element simulation was performed for the curve-shaped beam and the conventional beam. The results show that the curved beam structure has a special stress distribution and can improve output voltage compared with the straight beam structure. Then, the dynamic model of BPEH-V system is established. Numerical simulation analysis showed that it was easier for the proposed harvester to achieve large-amplitude response in a low-frequency environment compared with the conventional counterpart, and the spring stiffness had an important impact on system performance. The research can provide theoretical guidance for the optimal design and engineering application of the novel piezoelectric energy harvester.

Inspired by the development of variable-potential-energy techniques, this paper

bistable and linear harvesters, where the analytical, numerical, and experimental investigations revealed that the novel harvester could facilitate the energetic interwell response for relatively low excitation amplitudes and frequencies by passively and adaptively lowering the potential energy barrier level. Qian et al. [24] developed a broadband piezoelectric energy harvester (PEH) with a mechanically tunable potential function, and the simulations proved that the proposed PEH could harvest vibration energy in a wide

#### **2. Finite-Element Simulation 2. Finite-Element Simulation**

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frequency range of 0–91 Hz at the excitation level of 0.5 g.

#### *2.1. Stress Analysis 2.1. Stress Analysis*

At present, most of these existing piezoelectric energy harvesters utilize straight beam as the energy transducing elements due to its advantages in terms of simplicity and ease of fabrication, as shown in Figure 1a. At present, most of these existing piezoelectric energy harvesters utilize straight beam as the energy transducing elements due to its advantages in terms of simplicity and ease of fabrication, as shown in Figure 1a.

 **Figure 1.** Finite-element model: (**a**) Straight beam, (**b**) Curve-shaped beam. **Figure 1.** Finite-element model: (**a**) Straight beam, (**b**) Curve-shaped beam.

As we know, the conversion efficiency of piezoelectric materials is closely related to the stress distribution of the base layer. The evenly-distributed stress is helpful for harvesting energy and improving conversion efficiency. According to the theory of material mechanics, the conventional straight cantilever experiences a linear stress distribution on the surface when excited. The base layer considered in this work is schematically shown in Figure 1b, which is built of an arc-shaped and a flat configuration, and experiences different stress distribution from the conventional straight cantilever due to the arc-shaped configuration being introduced to improve the stress condition. The finite element analysis was performed in COMSOL software to analyze the influence of curved beam and traditional straight beam structure on the stress distribution of piezoelectric materials (PVDF). In order to make a fair comparison, both beams had the same rectangular sections; the material parameters used are listed in Table 1. Two identical mass were attached at the free end of both beams to reduce resonance frequency, respectively. Note that the curve-shaped beam had an arch with a central angle of 180 degrees, As we know, the conversion efficiency of piezoelectric materials is closely related to the stress distribution of the base layer. The evenly-distributed stress is helpful for harvesting energy and improving conversion efficiency. According to the theory of material mechanics, the conventional straight cantilever experiences a linear stress distribution on the surface when excited. The base layer considered in this work is schematically shown in Figure 1b, which is built of an arc-shaped and a flat configuration, and experiences different stress distribution from the conventional straight cantilever due to the arc-shaped configuration being introduced to improve the stress condition. The finite element analysis was performed in COMSOL software to analyze the influence of curved beam and traditional straight beam structure on the stress distribution of piezoelectric materials (PVDF). In order to make a fair comparison, both beams had the same rectangular sections; the material parameters used are listed in Table 1. Two identical mass were attached at the free end of both beams to reduce resonance frequency, respectively. Note that the curve-shaped beam had an arch with a central angle of 180 degrees, with a radius of *R* = 10 mm. The PVDF was only adhered to the arc-shaped surface of the curve-shaped beam, with a horizontal length of *Lp* = 31.4 mm. Meanwhile, the identical piezoelectric material (PVDF) was attached on the surface of the straight beam. The same load was applied on both beams, respectively. The stress distribution along the length direction of the piezoelectric materials on the curved beam and straight beam is shown in Figure 2, respectively.


with a radius of *R* = 10 mm. The PVDF was only adhered to the arc-shaped surface of the curve-shaped beam, with a horizontal length of *Lp* = 31.4 mm. Meanwhile, the identical piezoelectric material (PVDF) was attached on the surface of the straight beam. The same load was applied on both beams, respectively. The stress distribution along the length direction of the piezoelectric materials on the curved beam and straight beam is

**Parameter Symbol Value Unit** 

Density *ρ<sup>s</sup>* 8300 kg/m3 Elastic modulus *Es* 128 GPa Arc-shaped radius *Rs* 10 × 10−3 m Horizontal length *Ls* 40 × 10−3 m Height *hs* 2 × 10−4 m Width *bs* 8 × 10−3 m

Density *ρ<sup>p</sup>* 1780 kg/m3

Substrate layer (beryllium bronze)

Piezoelectric layer (PVDF)

**Table 1.** Material parameters for simulation. Elastic modulus *Ep* 3 GPa

shown in Figure 2, respectively.

**Table 1.** Material parameters for simulation.

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**Figure 2.** Comparison of stress distribution of PVDF on the curve-shaped beam and the straight beam. **Figure 2.** Comparison of stress distribution of PVDF on the curve-shaped beam and the straight beam.

*2.2. Generation Performance Comparisons*  It can be seen from Figure 2 that the stress of the piezoelectric material on the straight beam structure decreased linearly from the fixed end. The stress of the piezoelectric material on the curved beam structure was higher than that of the straight beam structure in most areas, and dropped more smoothly than that of the curved beam. The stress distribution was correlated with the bending moment acting on the configuration, the bending moment acting on the straight beam configuration decreased linearly along the fixed end, leading to linearly decreasing stress. However, the bending moment acting on the arc-shaped configuration behaved in a complex manner and decreased nonlinearly along the fixed end according to the theory of material mechanics, thus improving the stress distribution.

#### *2.2. Generation Performance Comparisons*

The piezoelectric coupling analyses are carried out in COMSOL software to compare the power generation performance of the curved beam and the straight beam structure.

Figure 3 shows the voltage comparison diagram of the curved beam and the straight beam structure under two different excitation conditions. At the excitation level of 2 m/s<sup>2</sup> , the resonance voltage of the curved beam was 11 V, and the resonance voltage of the straight beam was 7 V. With an increase in the excitation level to 5 m/s<sup>2</sup> , the resonance voltage of the curved beam was 22 V, and corresponding value of the straight beam was only 15 V in this case. Based on the simulation results, the voltage output of the piezoelectric material on the curved beam structure is always higher than that of the straight beam structure

under two different excitation levels. The relatively large and evenly-distributed stress results in less energy dissipation during charge flowing from the large stress region to low, which contributes to enhance the power output and energy conversion efficiency [27,28]. The special stress distribution of the curved beam configuration is beneficial to improving the output performance of the piezoelectric material. Therefore, the piezoelectric material on the surface of the curved beam produces a higher output voltage than that of the straight beam, and the curved beam has a better performance than the straight beam. At the same time, it can be found that curved beams have a lower resonance frequency than the straight beam, which will also benefit energy harvesting in low-frequency environments. Therefore, the introduction of a curved beam structure to a piezoelectric energy harvester is beneficial to increase the output power and improve the output performance of the conventional energy harvester. evenly-distributed stress results in less energy dissipation during charge flowing from the large stress region to low, which contributes to enhance the power output and energy conversion efficiency [27,28]. The special stress distribution of the curved beam configuration is beneficial to improving the output performance of the piezoelectric material. Therefore, the piezoelectric material on the surface of the curved beam produces a higher output voltage than that of the straight beam, and the curved beam has a better performance than the straight beam. At the same time, it can be found that curved beams have a lower resonance frequency than the straight beam, which will also benefit energy harvesting in low-frequency environments. Therefore, the introduction of a curved beam structure to a piezoelectric energy harvester is beneficial to increase the output power and improve the output performance of the conventional energy harvester.

The piezoelectric coupling analyses are carried out in COMSOL software to compare the power generation performance of the curved beam and the straight beam

Figure 3 shows the voltage comparison diagram of the curved beam and the straight beam structure under two different excitation conditions. At the excitation level of 2 m/s2, the resonance voltage of the curved beam was 11 V, and the resonance voltage of the straight beam was 7 V. With an increase in the excitation level to 5 m/s2, the resonance voltage of the curved beam was 22 V, and corresponding value of the straight beam was only 15 V in this case. Based on the simulation results, the voltage output of the piezoelectric material on the curved beam structure is always higher than that of the straight beam structure under two different excitation levels. The relatively large and

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structure.

**Figure 3.** Output voltage obtained by finite-element simulation at different excitation amplitude: (**a**) *A* = 2 m/s2; (**b**) *A* = 5 m/s2. **Figure 3.** Output voltage obtained by finite-element simulation at different excitation amplitude: (**a**) *A* = 2 m/s<sup>2</sup> ; (**b**) *A* = 5 m/s<sup>2</sup> .

#### **3. BPEH-V Configuration 3. BPEH-V Configuration**

The BPEH-V, shown in Figure 4, is comprised of a curve-shaped beam, magnet A, magnet B (i.e., external magnet), piezoelectric material (PVDF), and base. The piezoelectric material is attached to the surface of the arched part of the curve-shaped beam to realize energy conversion, and the flat part remains free. The external magnet B maintains a magnetic repulsive relationship with magnet A, and imposing bistability on the system. The difference between the proposed system and a conventional bistable piezoelectric harvester is because the external magnet B is connected to the base through a spring. If the BPEH-V is excited by ambient vibrations, the piezoelectric cantilever and magnet A are vibrated with the base, so the oscillation of piezoelectric cantilever would result in the deformation of PVDF, thus the conversion of mechanical energy from ambience into electrical energy via the piezoelectric effect can be achieved. When the end magnet of the cantilever beam moves to the intermediate equilibrium position, the spring is compressed and the potential barrier is lowered. Conversely, if the end magnet moves far away from the intermediate equilibrium position, the spring returns to the zero point The BPEH-V, shown in Figure 4, is comprised of a curve-shaped beam, magnet A, magnet B (i.e., external magnet), piezoelectric material (PVDF), and base. The piezoelectric material is attached to the surface of the arched part of the curve-shaped beam to realize energy conversion, and the flat part remains free. The external magnet B maintains a magnetic repulsive relationship with magnet A, and imposing bistability on the system. The difference between the proposed system and a conventional bistable piezoelectric harvester is because the external magnet B is connected to the base through a spring. If the BPEH-V is excited by ambient vibrations, the piezoelectric cantilever and magnet A are vibrated with the base, so the oscillation of piezoelectric cantilever would result in the deformation of PVDF, thus the conversion of mechanical energy from ambience into electrical energy via the piezoelectric effect can be achieved. When the end magnet of the cantilever beam moves to the intermediate equilibrium position, the spring is compressed and the potential barrier is lowered. Conversely, if the end magnet moves far away from the intermediate equilibrium position, the spring returns to the zero point and the magnetic distance is reduced to maintain the bistable characteristics of the system. Therefore, a bistable piezoelectric energy harvester with variable potential well is formed during the process of the piezoelectric beam vibration.

formed during the process of the piezoelectric beam vibration.

**Figure 4.** Schematic diagram of the BPEH-V. **Figure 4.** Schematic diagram of the BPEH-V.

The BPEH-V not only retains the vibration bistability of the piezoelectric cantilever but could also adjust the potential barrier level, which is beneficial to realizing large-amplitude interwell oscillations under a low excitation level, thus improving the energy harvesting performance. The BPEH-V not only retains the vibration bistability of the piezoelectric cantilever but could also adjust the potential barrier level, which is beneficial to realizing largeamplitude interwell oscillations under a low excitation level, thus improving the energy harvesting performance.

and the magnetic distance is reduced to maintain the bistable characteristics of the system. Therefore, a bistable piezoelectric energy harvester with variable potential well is

#### *3.1. Theoretical Modeling 3.1. Theoretical Modeling*

ynomial, as follows:

#### 3.1.1. Modeling of Nonlinear Restoring Force 3.1.1. Modeling of Nonlinear Restoring Force

Unlike the linear restoring force of the conventional straight beam, the restoring force was nonlinear in the curve-shaped beam due to the existence of the arc-shaped configuration. To model the restoring force, the relationship between deflection and restoring force is extracted by using experimental method. To this end, the curve-shaped beam was fixed on the left end, and the free end of the beam was pushed by the dynamometer to measure the value of the nonlinear restoring force at different displacements. The process was repeated and the measurement results were averaged, then the relationship between the resorting force and transverse displacements were fit to a pol-Unlike the linear restoring force of the conventional straight beam, the restoring force was nonlinear in the curve-shaped beam due to the existence of the arc-shaped configuration. To model the restoring force, the relationship between deflection and restoring force is extracted by using experimental method. To this end, the curve-shaped beam was fixed on the left end, and the free end of the beam was pushed by the dynamometer to measure the value of the nonlinear restoring force at different displacements. The process was repeated and the measurement results were averaged, then the relationship between the resorting force and transverse displacements were fit to a polynomial, as follows:

$$F\_r = k\_1 u^3(L, t) + k\_2 u^2(L, t) + k\_3 u(L, t) \tag{1}$$

 = ଵଷ(, ) + ଶଶ(, ) + ଷ(, ) (1) where ଵ, ଶ, and ଷ are constant coefficients on the third, second, and first-order terms, respectively. Figure 5 shows the measurement results and curve fitting results of the nonlinear restoring force of the curve-shaped beam. It can be observed from Figure 5 that the experimental data and the fitting curve had good agreement, and the restoring force of the curve-shaped beam exhibited a curve due to the existence of the curved configuration. Setting *u* = 0 as the static equilibrium position, it was found that the measurement results were asymmetrical, which is due to the fact that the radius of curvature for the curved configuration is continuously varied in the process of the piezoelectric where *k*1, *k*2, and *k*<sup>3</sup> are constant coefficients on the third, second, and first-order terms, respectively. Figure 5 shows the measurement results and curve fitting results of the nonlinear restoring force of the curve-shaped beam. It can be observed from Figure 5 that the experimental data and the fitting curve had good agreement, and the restoring force of the curve-shaped beam exhibited a curve due to the existence of the curved configuration. Setting *u* = 0 as the static equilibrium position, it was found that the measurement results were asymmetrical, which is due to the fact that the radius of curvature for the curved configuration is continuously varied in the process of the piezoelectric beam vibration, and resulting in asymmetric nonlinear restoring force.

#### beam vibration, and resulting in asymmetric nonlinear restoring force. 3.1.2. Modeling of Magnetic Force

The permanent magnets can be modeled as the point dipoles when calculating the magnetic force between the tip magnet and the external magnet. The schematic diagram of the spatial position of the magnets is shown in Figure 6. Considering the additional degree of freedom (DOF) and rotation of the magnet, the distance vector *rBA* from the center of magnet B to magnet A can be expressed as:

$$r\_{BA} = \begin{bmatrix} -d - q(t), \ u(L, t) \end{bmatrix} \tag{2}$$

where *q*(*t*) is the compression displacement of magnet B, and the magnetic field generated by magnet B on magnet A is obtained as [29]:

$$\begin{aligned} \mathbf{U}\_{MA} &= \frac{\mu\_0}{4\pi} \left[ \frac{m\_B}{||r\_{BA}||\_2^2} - \frac{(m\_B \cdot r\_{BA}) \cdot 3 \cdot r\_{BA}}{||r\_{BA}||\_2^2} \right] m\_A \\\\ \mathbf{0} &= \begin{bmatrix} \frac{\mu\_0}{4\pi} \left[ \frac{(m\_B \cdot r\_{BA}) \cdot 3 \cdot r\_{BA}}{\|r\_{BA}\|\_2^2} \right] m\_A \\\\ \frac{\mu\_0}{4\pi} \left[ \frac{(m\_B \cdot r\_{BA}) \cdot 3 \cdot r\_{BA}}{\|r\_{BA}\|\_2^2} \right] \\\\ \mathbf{0} &= \begin{bmatrix} \frac{\mu\_0}{4\pi} \left[ \frac{(m\_B \cdot r\_{BA}) \cdot 3 \cdot r\_{BA}}{\|r\_{BA}\|\_2^2} \right] \\\\ \frac{\mu\_0}{4\pi} \left[ \frac{(m\_B \cdot r\_{BA}) \cdot 3 \cdot r\_{BA}}{\|r\_{BA}\|\_2^2} \right] \end{bmatrix} \begin{bmatrix} \frac{\mu\_0}{4\pi} \left[ \frac{(m\_B \cdot r\_{BA}) \cdot 3 \cdot r\_{BA}}{\|r\_{BA}\|\_2^2} \right] \\\\ \frac{\mu\_0}{4\pi} \left[ \frac{(m\_B \cdot r\_{BA}) \cdot 3 \cdot r\_{BA}}{\|r\_{BA}\|\_2^2} \right] \end{bmatrix} \end{aligned} (3)$$

**Figure 5.** Displacement-restoring force curve of the curve**-**shaped beam. **Figure 5.** Displacement-restoring force curve of the curve-shaped beam. ‖‖ଶ ‖‖ଶ

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**Figure 6.** Schematic diagram of the spatial position of the magnets. **Figure 6.** Schematic diagram of the spatial position of the magnets.

The magnetic moment vectors and for magnets A and B can be respectively expressed as: The magnetic moment vectors *m<sup>A</sup>* and *m<sup>B</sup>* for magnets A and B can be respectively expressed as:

$$m\_A = [M\_A V\_A \cos \alpha\_\prime \ M\_A V\_A \sin \alpha\_\prime] \tag{4}$$

$$\mathcal{M}\_B = \begin{bmatrix} -M\_B V\_{B'} \ 0 \end{bmatrix} \tag{5}$$

<sup>ହ</sup> (3)

 = [−, 0] (5) where and (*i* = A, B) are the magnetization strength and material volume of where *M<sup>i</sup>* and *V<sup>i</sup>* (*i* = A, B) are the magnetization strength and material volume of magnets A and B, respectively. *α* is the slope of beam at the free end, which is given by:

$$u = \arctan\left(\dot{u}(L, t)\right) \tag{6}$$

= [, ] (4)

= [−, 0] (5)

Substituting Equation (2) and Equation (4) to Equation (6) into Equation (3), the magnetic field *UMA* can be expressed in the following equation:

$$\mathcal{U}\_{MA} = \frac{\mu\_0 M\_A V\_A M\_B V\_B \left(-\mu \left(L, t\right)^2 - 2\left(d + q(t)\right)^2 + 3\left(d + q(t)\right)\mu \left(L, t\right)\dot{\mu}\left(L, t\right)\right)}{4\pi\sqrt{\left(\dot{\mu}\left(L, t\right)\right)^2 + 1}\left(\mu\left(L, t\right)^2 + \left(d + q(t)\right)^2\right)^{5/2}} \tag{7}$$

where and (*i* = A, B) are the magnetization strength and material volume of magnets A and B, respectively. is the slope of beam at the free end, which is given by:

tively expressed as:

#### 3.1.3. Dynamical Model

To predict the response of BPEH-V, considering the Euler–Bernoulli theory and the linear constitutive equations for piezoelectric materials, the coupled governing equations are derived by using the generalized Hamilton principle.

$$\int\_{t1}^{t2} \left[ \delta (T\_k - \mathcal{U}\_r + \mathcal{W}\_\varepsilon) + \delta \mathcal{W}\_{\text{nc}} - \delta \mathcal{U}\_m \right] = 0 \tag{8}$$

where *U<sup>r</sup>* is the elastic potential energy of the piezoelectric beam and *U<sup>m</sup>* is the magnetic potential energy. *W<sup>e</sup>* is the electric potential energy of the piezoelectric layer, and *Wnc* is the external work applied to the system. The whole kinetic energy of the proposed system can be expressed as:

$$T\_k = T\_1 + T\_2 + T\_3 + T\_4 \tag{9}$$

where *T*1, *T*2, *T*3, and *T*<sup>4</sup> represent the kinetic energy of the substrate layer, the piezoelectric layer, the tip magnet A, and the movable magnet B.

$$T\_k = \frac{1}{2} \int\_{V\_S} \rho\_s \dot{u}^2(\mathbf{x}, t)dV\_s + \frac{1}{2} \int\_{V\_T} \rho\_p \dot{u}^2(\mathbf{x}, t)dV\_p + \frac{1}{2} m\_A \dot{u}^2(\mathbf{x}, t)|\_{\mathbf{x}=L} \dots + \frac{1}{2} I\_l \left[ \frac{\partial^2 u(\mathbf{x}, t)}{\partial t \partial \mathbf{x}}|\_{\mathbf{x}=L} \right]^2 + \frac{1}{2} m\_B \dot{q}(t)^2 \tag{10}$$

where *u*(*x*, *t*) is the transverse displacement of the beam; *V<sup>p</sup>* and *V<sup>s</sup>* are the piezoelectric and substrate layer volume, respectively; and *I<sup>t</sup>* is the rotational inertia of the tip magnet with respect to the beam free end. The electric potential energy of the piezoelectric material can be expressed as follows:

$$\mathcal{W}\_{\varepsilon} = \frac{1}{2} \int\_{V\_p} \varepsilon\_{33}^s E\_3^2 V\_p + \frac{1}{2} \int\_{V\_p} e\_{31} E\_3 S\_1 dV\_p \tag{11}$$

where *E*<sup>3</sup> and *S*<sup>1</sup> represent the electrical field and the axial strain, respectively. *ε s* <sup>33</sup> and *e*<sup>31</sup> represent the permittivity component at constant strain and the piezoelectric constant. The external work applied to the BPEH-V system can be written as follows:

$$\delta \mathcal{W}\_{\text{nc}} = -\int\_0^L \delta u(\mathbf{x}, t) m(\mathbf{x}) \ddot{z}(t) d\mathbf{x} - \delta u(L, t) m\_0 \ddot{z}(t) + Q \delta v \tag{12}$$

In this paper, based on the Rayleigh–Ritz principle, it is assumed that a single-mode approximation of the beam deformation is sufficient, and the vibrational displacement of the beam can be expressed as follows:

$$u(\mathbf{x},t) = \sum\_{i=1}^{n} \varphi\_i(\mathbf{x}) r\_i(t) \tag{13}$$

where *ϕi*(*x*) is the *i*th mode shape of the beam and *ri*(*t*) is the time-dependent generalized coordinates. Under the low frequency excitations, the vibration of the beam is mainly concentrated in the first-order mode, so it is sufficient to consider one mode to obtain the reduced-order model. Meanwhile, for the boundary conditions where one end is clamped and the other one is free, the allowable function can be written as [30,31]:

$$\varphi(\mathbf{x}) = 1 - \cos\left(\frac{\pi \mathbf{x}}{2L}\right) \tag{14}$$

Substituting Equations (7) and (10)–(12) into Equation (8), according to Kirchhoff's law, the governing equations of BPEH-V system are obtained:

$$\dot{M}\ddot{r}(t) + \dot{\mathcal{C}}r(t) + k\_1r^3(t) + k\_2r^2(t) + k\_3r(t) - \frac{\partial L\_{MA}}{\partial r(t)} - \theta v = -H\_s\ddot{z}(t) \tag{15}$$

$$
\theta \dot{r}(t) + \mathcal{C}\_p \dot{v}(t) + \frac{v(t)}{R} = 0 \tag{16}
$$

$$m\ddot{q}(t) + kq(t) - F\_q = 0\tag{17}$$

where *M* and *C* refer to the mass coefficient and the damping coefficient, respectively. *θ* is the electromechanical coupling coefficient; *C<sup>p</sup>* is the capacitance of the piezoelectric patch; *R* is the load resistance; and *F<sup>q</sup>* is the horizontal magnetic force component, as follows:

$$M = \int\_{\Omega b} \rho\_b \boldsymbol{\rho}(\mathbf{x})^2 d\_{\Omega b} + \int\_{\Omega p} \rho\_p \boldsymbol{\rho}(\mathbf{x})^2 d\_{\Omega p} + m\_0 \boldsymbol{\rho}(\mathbf{L})^2 + I\_l \boldsymbol{\rho}^{(2)}(\mathbf{L})\tag{18}$$

$$H\_{\rm s} = \rho\_b A\_b \int\_0^L \varphi(\mathbf{x}) d\mathbf{x} + \rho\_p A\_p \int\_0^{L\_p} \varphi(\mathbf{x}) d\mathbf{x} + m\_0 \tag{19}$$

$$\mathcal{C} = \int\_0^L c\varphi(\mathbf{x})^2 d\mathbf{x} \tag{20}$$

$$\theta = \frac{1}{h\_p} \int\_{\Omega p} e\_{\Im 1} z \varphi''(\varkappa) d\_{\Omega p} \tag{21}$$

$$\mathcal{C}\_p = \frac{\varepsilon\_{33}^S b\_p L\_p}{h\_p} \tag{22}$$

$$F\_q = \frac{\partial U\_{MA}}{\partial q(t)}\tag{23}$$

#### **4. Numerical Simulation**

*4.1. Study on the Potential Energy of BPEH-V*

Magnetic potential energy is an important factor that affects the nonlinearity of the system. Different magnetic distances will produce different nonlinear magnetic forces, so the system presents different characteristics. Regarding the BPEH-V system, the magnetic potential energy is continuously varied with vibration due to the external magnet being connected elastically. Figure 7 shows the potential energy curve of the system under the condition of magnetic distance (*d* = 17 mm). In this case, two obvious potential wells are formed, that is, the system becomes bistable. We should notice that magnetic distance *d* is constantly varied during the vibration of the piezoelectric cantilever beam, so the potential energy of the system is different from a traditional bistable piezoelectric energy harvester with a fixed external magnet. The magnetic potential energy is not only affected by the magnetic distance *d*, but also by the compression displacement *q*(*t*) of the spring. As shown in Figure 7, the *x*-axis denotes displacement of the curved-shape beam's tip, the *y*-axis denotes the compression displacement of the spring, and the *z*-axis denotes the potential energy of the system. The height of the barrier between the two wells is pulled down as the compression displacement of the spring gradually increases due to the repulsive force between the tip magnet and the external magnet. In this condition, the system can cross the potential barrier to realize interwell oscillations more easily. The influence of *q*(*t*) in BPEH-V, which is caused by spring compression, equals that of time-varying *d* in the traditional bistable system. When the tip magnet tends to approach its original point (at *u*(*L*, *t*) = 0 in Figure 6), it drives the external magnet away from the equilibrium position due to magnetic repulsion, thus decreasing the potential barrier. Conversely, when the tip magnet moves far away from the original point, the potential barrier gradually becomes high and reaches its maximum. Thus, the design of the BPEH-V provides an adaptive potential using the spring in comparison to the traditional bistable system.

bistable system.

**Figure 7.** Potential curve of the system at different compression displacement of the spring. **Figure 7.** Potential curve of the system at different compression displacement of the spring.

Meanwhile, it can also be seen from Figure 7 that the potential energy curve of the proposed system is inconsistent with the straight beam bistable piezoelectric energy harvester. The potential well is shallower on the left side and deeper on the right side, showing an asymmetrical trend. This is mainly due to the asymmetric restoring force of the curve-shaped beam. Meanwhile, it can also be seen from Figure 7 that the potential energy curve of the proposed system is inconsistent with the straight beam bistable piezoelectric energy harvester. The potential well is shallower on the left side and deeper on the right side, showing an asymmetrical trend. This is mainly due to the asymmetric restoring force of the curve-shaped beam.

down as the compression displacement of the spring gradually increases due to the repulsive force between the tip magnet and the external magnet. In this condition, the system can cross the potential barrier to realize interwell oscillations more easily. The influence of *q(t)* in BPEH-V, which is caused by spring compression, equals that of time-varying *d* in the traditional bistable system. When the tip magnet tends to approach its original point (at (, ) = 0 in Figure 6), it drives the external magnet away from the equilibrium position due to magnetic repulsion, thus decreasing the potential barrier. Conversely, when the tip magnet moves far away from the original point, the potential barrier gradually becomes high and reaches its maximum. Thus, the design of the BPEH-V provides an adaptive potential using the spring in comparison to the traditional

#### *4.2. The Dynamics Analysis of BPEH-V*

*4.2. The Dynamics Analysis of BPEH-V*  According to the potential energy diagram shown in Figure 7, the system becomes bistable and the height of the potential barrier is relatively shallow when the magnet distance is *d* = 17 mm. In this section, the numerical simulations are performed for the separation distance *d* = 17 mm to investigate the influence of the variable potential well According to the potential energy diagram shown in Figure 7, the system becomes bistable and the height of the potential barrier is relatively shallow when the magnet distance is *d* = 17 mm. In this section, the numerical simulations are performed for the separation distance *d* = 17 mm to investigate the influence of the variable potential well on the dynamic characteristics of BPEH-V (the ode45 command of MATLAB was used here).

on the dynamic characteristics of BPEH-V (the ode45 command of MATLAB was used here). The bifurcation diagram of the tip displacement versus the excitation frequency of the BPEH-V and the CBH-C for excitation amplitude *A* = 10 m/s2 is shown in Figure 8. Compared to Figure 8a,b, it can be found that BPEH-V exhibited more complex dynamic behaviors than the CBH-C. At 4 Hz excitation, BPEH-V enters into the chaotic oscillation, which can be concluded from the phase plane portrait (the phase plane portrait is drawn The bifurcation diagram of the tip displacement versus the excitation frequency of the BPEH-V and the CBH-C for excitation amplitude *A* = 10 m/s<sup>2</sup> is shown in Figure 8. Compared to Figure 8a,b, it can be found that BPEH-V exhibited more complex dynamic behaviors than the CBH-C. At 4 Hz excitation, BPEH-V enters into the chaotic oscillation, which can be concluded from the phase plane portrait (the phase plane portrait is drawn by red curves) and Poincaré map (the Poincaré map is drawn by black dots) depicted in Figure 9a. *Micromachines* **2021**, *12*, x FOR PEER REVIEW 11 of 15

**Figure 8.** Bifurcation diagram of the tip displacement versus the excitation frequency for excitation amplitude *A* = 10 m/s2: (**a**) BPEH-V; (**b**) CBH-C. in Figure 8b. **Figure 8.** Bifurcation diagram of the tip displacement versus the excitation frequency for excitation amplitude *A* = 10 m/s<sup>2</sup> : (**a**) BPEH-V; (**b**) CBH-C.

**Figure 9.** Phase plane portrait, Poincaré map, and output voltage histories for excitation frequency *f*  = 4 Hz. (**a**) Phase plane portrait and Poincaré map. (**b**) Voltage histories for BPEH-V, respectively.

However, the CBH-C system only made a small-amplitude intrawell motion at this time, as shown in Figure 9c. Meanwhile, compared with Figure 9b,d, we found that the BPEH-V generated a much higher output voltage than the CBH-C in the low excitation

With the increase in excitation frequency to 5 Hz (Figure 10 shows the simulation results for BPEH-V), the Poincaré map is concentrated in a single point and the phase plane portrait consists of a closed obit, as shown in Figure 10a, which demonstrates that the BPEH-V entered into large-amplitude periodic oscillations. However, the CBH-C system still made a small-amplitude intrawell motion at this time, as can be found from the bifurcation diagram of the tip displacement versus the excitation frequency depicted

(**c**) Phase plane portrait and Poincaré map. (**d**) Voltage histories for CBH-C, respectively.

frequency.

amplitude *A* = 10 m/s2: (**a**) BPEH-V; (**b**) CBH-C.

**Figure 9.** Phase plane portrait, Poincaré map, and output voltage histories for excitation frequency *f*  = 4 Hz. (**a**) Phase plane portrait and Poincaré map. (**b**) Voltage histories for BPEH-V, respectively. (**c**) Phase plane portrait and Poincaré map. (**d**) Voltage histories for CBH-C, respectively. **Figure 9.** Phase plane portrait, Poincaré map, and output voltage histories for excitation frequency *f* = 4 Hz. (**a**) Phase plane portrait and Poincaré map. (**b**) Voltage histories for BPEH-V, respectively. (**c**) Phase plane portrait and Poincaré map. (**d**) Voltage histories for CBH-C, respectively.

**Figure 8.** Bifurcation diagram of the tip displacement versus the excitation frequency for excitation

However, the CBH-C system only made a small-amplitude intrawell motion at this time, as shown in Figure 9c. Meanwhile, compared with Figure 9b,d, we found that the BPEH-V generated a much higher output voltage than the CBH-C in the low excitation frequency. However, the CBH-C system only made a small-amplitude intrawell motion at this time, as shown in Figure 9c. Meanwhile, compared with Figure 9b,d, we found that the BPEH-V generated a much higher output voltage than the CBH-C in the low excitation frequency.

With the increase in excitation frequency to 5 Hz (Figure 10 shows the simulation results for BPEH-V), the Poincaré map is concentrated in a single point and the phase plane portrait consists of a closed obit, as shown in Figure 10a, which demonstrates that the BPEH-V entered into large-amplitude periodic oscillations. However, the CBH-C system still made a small-amplitude intrawell motion at this time, as can be found from the bifurcation diagram of the tip displacement versus the excitation frequency depicted in Figure 8b. With the increase in excitation frequency to 5 Hz (Figure 10 shows the simulation results for BPEH-V), the Poincaré map is concentrated in a single point and the phase plane portrait consists of a closed obit, as shown in Figure 10a, which demonstrates that the BPEH-V entered into large-amplitude periodic oscillations. However, the CBH-C system still made a small-amplitude intrawell motion at this time, as can be found from the bifurcation diagram of the tip displacement versus the excitation frequency depicted in Figure 8b. *Micromachines* **2021**, *12*, x FOR PEER REVIEW 12 of 15

**Figure 10.** Simulation results for BPEH-V under excitation frequency *f* = 5 Hz. (**a**) Phase plane portrait and Poincaré map. (**b**) Output voltage histories. **Figure 10.** Simulation results for BPEH-V under excitation frequency *f* = 5 Hz. (**a**) Phase plane portrait and Poincaré map. (**b**) Output voltage histories.

With the increase in excitation frequency to 7.1 Hz, Figure 11 shows the simulation results for CBH-C, where the Poincaré map is concentrated in a single point and the phase plane portrait consists of a closed obit, which demonstrates that the CBH-C enters into large-amplitude periodic oscillations. Meanwhile, it was observed from Figure 8a that the BPEH-V underwent transient chaotic oscillation at 6.3 Hz excitation, and then returned to large-amplitude interwell oscillations at 7 Hz excitation. With the increase in excitation frequency to 7.1 Hz, Figure 11 shows the simulation results for CBH-C, where the Poincaré map is concentrated in a single point and the phase plane portrait consists of a closed obit, which demonstrates that the CBH-C enters into large-amplitude periodic oscillations. Meanwhile, it was observed from Figure 8a that the BPEH-V underwent transient chaotic oscillation at 6.3 Hz excitation, and then returned to large-amplitude interwell oscillations at 7 Hz excitation.

**Figure 11.** Simulation results for CBH-C under excitation frequency *f* = 7.1 Hz. (**a**) Phase plane

With the still further increase in excitation frequency, the BPEH-V exits large-amplitude interwell oscillations when the excitation frequency exceeds 10.1 Hz. Meanwhile, the CBH-C exits large-amplitude interwell oscillations at a frequency *f* = 10.3

From the above simulations and analyses, we found that the frequency ranges of large-amplitude periodic response of BPEH-V were 5 < *f* < 6.3 Hz and 7 < *f* < 10.1 Hz, and the effective bandwidth was 4.4 Hz. The corresponding frequency range of CBH-C was 7.1 < *f* < 10.3 Hz, and the effective bandwidth was only 3.2 Hz. Accordingly, the effective bandwidth of BPEH-V was 1.37 times that of CBH-C under the same circumstances due to the spring being efficiently introduced to broaden bandwidth, and the BPEH-V was superior to the CBH-C from the aspect of effective bandwidth. The conventional bistable system only made a small-amplitude intrawell motion at low excitation frequency due to the lack of sufficient energy to overcome the potential barrier. However, thanks to the compression adjustment of the spring, it can pull down the potential barrier and form an adaptive potential barrier. The BPEH-V with suitable stiffness can realize large-amplitude interwell motions at the lower excitation frequency, thus improving the

portrait and Poincaré map. (**b**) Output voltage histories.

Hz.

harvesting performance.

returned to large-amplitude interwell oscillations at 7 Hz excitation.

trait and Poincaré map. (**b**) Output voltage histories.

**Figure 11.** Simulation results for CBH-C under excitation frequency *f* = 7.1 Hz. (**a**) Phase plane portrait and Poincaré map. (**b**) Output voltage histories. **Figure 11.** Simulation results for CBH-C under excitation frequency *f* = 7.1 Hz. (**a**) Phase plane portrait and Poincaré map. (**b**) Output voltage histories.

**Figure 10.** Simulation results for BPEH-V under excitation frequency *f* = 5 Hz. (**a**) Phase plane por-

With the increase in excitation frequency to 7.1 Hz, Figure 11 shows the simulation results for CBH-C, where the Poincaré map is concentrated in a single point and the phase plane portrait consists of a closed obit, which demonstrates that the CBH-C enters into large-amplitude periodic oscillations. Meanwhile, it was observed from Figure 8a that the BPEH-V underwent transient chaotic oscillation at 6.3 Hz excitation, and then

With the still further increase in excitation frequency, the BPEH-V exits large-amplitude interwell oscillations when the excitation frequency exceeds 10.1 Hz. Meanwhile, the CBH-C exits large-amplitude interwell oscillations at a frequency *f* = 10.3 With the still further increase in excitation frequency, the BPEH-V exits large-amplitude interwell oscillations when the excitation frequency exceeds 10.1 Hz. Meanwhile, the CBH-C exits large-amplitude interwell oscillations at a frequency *f* = 10.3 Hz.

Hz. From the above simulations and analyses, we found that the frequency ranges of large-amplitude periodic response of BPEH-V were 5 < *f* < 6.3 Hz and 7 < *f* < 10.1 Hz, and the effective bandwidth was 4.4 Hz. The corresponding frequency range of CBH-C was 7.1 < *f* < 10.3 Hz, and the effective bandwidth was only 3.2 Hz. Accordingly, the effective bandwidth of BPEH-V was 1.37 times that of CBH-C under the same circumstances due to the spring being efficiently introduced to broaden bandwidth, and the BPEH-V was superior to the CBH-C from the aspect of effective bandwidth. The conventional bistable system only made a small-amplitude intrawell motion at low excitation frequency due to the lack of sufficient energy to overcome the potential barrier. However, thanks to the compression adjustment of the spring, it can pull down the potential barrier and form an adaptive potential barrier. The BPEH-V with suitable stiffness can realize From the above simulations and analyses, we found that the frequency ranges of large-amplitude periodic response of BPEH-V were 5 < *f* < 6.3 Hz and 7 < *f* < 10.1 Hz, and the effective bandwidth was 4.4 Hz. The corresponding frequency range of CBH-C was 7.1 < *f* < 10.3 Hz, and the effective bandwidth was only 3.2 Hz. Accordingly, the effective bandwidth of BPEH-V was 1.37 times that of CBH-C under the same circumstances due to the spring being efficiently introduced to broaden bandwidth, and the BPEH-V was superior to the CBH-C from the aspect of effective bandwidth. The conventional bistable system only made a small-amplitude intrawell motion at low excitation frequency due to the lack of sufficient energy to overcome the potential barrier. However, thanks to the compression adjustment of the spring, it can pull down the potential barrier and form an adaptive potential barrier. The BPEH-V with suitable stiffness can realize large-amplitude interwell motions at the lower excitation frequency, thus improving the harvesting performance.

#### large-amplitude interwell motions at the lower excitation frequency, thus improving the harvesting performance. *4.3. The Influence of the Spring Stiffness K on Harvesting Performance*

Spring stiffness has a great impact on the system characteristics. In order to investigate the influence of the spring stiffness *K* on energy harvesting performance, the numerical frequency-swept experiments of the BPEH-V system with three distinct spring stiffness were conducted under the excitation amplitude of 5 m/s<sup>2</sup> , as shown in Figure 12. The BPEH-V system with suitable stiffness of *K* = 200 N/m can realize large-amplitude interwell oscillations and have a higher output at a frequency range of *f* = 9.4–11Hz. We decreased the spring stiffness to *K* = 150 N/m. The spring was more easily compressed due to the small spring stiffness, so the system could realize large-amplitude interwell oscillations at lower excitation frequency; the theoretical frequency range of the large-amplitude periodic response was *f* = 8.6–11 Hz; and the effective bandwidth was 2.4 Hz, which was broader than the case of *K* = 200 N/m. Meanwhile, we notice that the system without spring can only realized intrawell oscillations and generated a lower output voltage at the same condition, which was because that the system could not obtain sufficient energy at low excitation level to overcome the potential barrier, thus resulting in poor output performance. Therefore, the BPEH-V with suitable spring stiffness contributed to realize large-amplitude interwell oscillations over a wide range of excitation, especially in low excitation level compared to CBH-C.

pecially in low excitation level compared to CBH-C.

*4.3. The Influence of the Spring Stiffness K on Harvesting Performance* 

Spring stiffness has a great impact on the system characteristics. In order to investigate the influence of the spring stiffness *K* on energy harvesting performance, the numerical frequency-swept experiments of the BPEH-V system with three distinct spring stiffness were conducted under the excitation amplitude of 5 m/s2, as shown in Figure 12. The BPEH-V system with suitable stiffness of *K* = 200 N/m can realize large-amplitude interwell oscillations and have a higher output at a frequency range of *f* = 9.4–11Hz. We decreased the spring stiffness to *K* = 150 N/m. The spring was more easily compressed due to the small spring stiffness, so the system could realize large-amplitude interwell oscillations at lower excitation frequency; the theoretical frequency range of the large-amplitude periodic response was *f* = 8.6–11 Hz; and the effective bandwidth was 2.4 Hz, which was broader than the case of *K* = 200 N/m. Meanwhile, we notice that the system without spring can only realized intrawell oscillations and generated a lower output voltage at the same condition, which was because that the system could not obtain sufficient energy at low excitation level to overcome the potential barrier, thus resulting in poor output performance. Therefore, the BPEH-V with suitable spring stiffness contributed to realize large-amplitude interwell oscillations over a wide range of excitation, es-

**Figure 12.** Frequency-swept voltage response of the system with different spring stiffness under excitation amplitude *A* = 5 m/s2. **Figure 12.** Frequency-swept voltage response of the system with different spring stiffness under excitation amplitude *A* = 5 m/s<sup>2</sup> .

It needs to be mentioned that we should ensure the bistable characteristic of the system when choosing a small stiffness spring as the connection element. Otherwise, the system will be close to a linear one and vibrates around the middle equilibrium point, leading to poor output performance. It needs to be mentioned that we should ensure the bistable characteristic of the system when choosing a small stiffness spring as the connection element. Otherwise, the system will be close to a linear one and vibrates around the middle equilibrium point, leading to poor output performance.

#### **5. Conclusions 5. Conclusions**

This paper proposed a magnetically coupled bistable piezoelectric energy harvester based on an elastically connected external magnet. First, finite-element simulations were performed for the curve-shaped composite and the straight beam to compare the influence of different configurations on the stress distribution and power generation performance. Moreover, the dynamics model of the system was established by using the generalized Hamilton variational principle, and the fourth-order Runge–Kutta algorithm was used to numerically solve the dynamic equations. The dynamic characteristics of the piezoelectric energy harvester were analyzed and compared with the traditional curve-shaped beam bistable harvester. Finally, the influence of the spring stiffness on energy harvesting performance of the system was discussed. The main conclusions are as This paper proposed a magnetically coupled bistable piezoelectric energy harvester based on an elastically connected external magnet. First, finite-element simulations were performed for the curve-shaped composite and the straight beam to compare the influence of different configurations on the stress distribution and power generation performance. Moreover, the dynamics model of the system was established by using the generalized Hamilton variational principle, and the fourth-order Runge–Kutta algorithm was used to numerically solve the dynamic equations. The dynamic characteristics of the piezoelectric energy harvester were analyzed and compared with the traditional curve-shaped beam bistable harvester. Finally, the influence of the spring stiffness on energy harvesting performance of the system was discussed. The main conclusions are as follows:


characteristics and degenerate into a nonlinear monostable system, thus resulting in poor energy harvesting performance.

In addition, experimental investigations will be presented in the future.

**Author Contributions:** Conceptualization, X.Z. and X.C.; Methodology, X.C.; Software, Y.G.; Validation, F.Z., X.C. and L.C.; Writing—original draft preparation, X.C.; Writing—review and editing, L.C.; Visualization, Y.G.; Supervision, X.C.; Project administration, X.Z.; Funding acquisition, X.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (Grant No. 51974228 and No. 51834006), the National Green Manufacturing System Integration Project (Grant No. 2017-327), Shaanxi Innovative Talent Plan Project (Grant No. 2018TD-032), and Key R&D project in Shaanxi (Grant No. 2018ZDCXL-GY-06-04).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** Authors acknowledge the support from the students Lin Wang, Meng Zuo and Xiao She for their assistance in the modeling and experiments.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Dongmei Xu 1,\*, Wenzhong Yang <sup>1</sup> , Xuhui Zhang <sup>1</sup> and Simiao Yu 2,\***


**Abstract:** An ultrasonic motor as a kind of smart material drive actuator has potential in robots, aerocraft, medical operations, etc. The size of the ultrasonic motor and complex circuit limits the further application of ultrasonic motors. In this paper, a single-phase driven ultrasonic motor using Bending-Bending vibrations is proposed, which has advantages in structure miniaturization and circuit simplification. Hybrid bending vibration modes were used, which were excited by only single-phase voltage. The working principle based on an oblique line trajectory is illustrated. The working bending vibration modes and resonance frequencies of the bending vibration modes were calculated by the finite element method to verify the feasibility of the proposed ultrasonic motor. Additionally, the output performance was evaluated by experiment. This paper provides a singlephase driven ultrasonic motor using Bending-Bending vibrations, which has advantages in structure miniaturization and circuit simplification.

**Keywords:** ultrasonic motor; single-phase driven; bending vibration

## **1. Introduction**

The ultrasonic motor is a kind of special motor based on the inverse piezoelectric effect, which has the merits of no electromagnetic interference, no need for lubrication, fast response, and high positioning accuracy [1–3]. Thus, the ultrasonic motor has been used in fields such as camera lens drives, robots, optical fiber connections, biomedical engineering, etc. [4–6].

From the viewpoint of the phase number of excitation power supply, the ultrasonic motor can be divided into single-phase driven ones [7–9], two-phase driven ones [10,11], and multi-phase driven ones [12,13]. As the phase shift of each phase of excitation voltages should be adjustable, the power supplies of the two-phase-driven and multi-phase-driven ultrasonic motors are relatively complex and large. Tian et al. proposed a single-phasedriven piezoelectric actuator, which worked with an eight-shaped trajectory, and the piezo rings were clamped between the flange bolt and the horn [8]. A single-phase-driven piezoelectric actuator using the longitudinal bending coupling mode was proposed by Liu et al.; when one signal voltage with the frequency of the first longitudinal and third bending resonance frequency was applied to the motor and the boundary was unsymmetrical, oblique elliptical movement was generated to push the mover [14]. However, the consistency demand of the frequency of this longitudinal bending hybrid mode is relatively high. Flueckiger et al. proposed a single-phase ultrasonic motor, in which the longitudinal vibration mode was converted to the particular deformation of the resonator [15]. However, to obtain the forward motion, a signal frequency of 84 kHz was utilized, and to achieve the backward motion, a signal frequency of 69 kHz was used; thus, the output mechanical characteristics of the bi-directional motions are not consistent.

**Citation:** Xu, D.; Yang, W.; Zhang, X.; Yu, S. Design and Performance Evaluation of a Single-Phase Driven Ultrasonic Motor Using Bending-Bending Vibrations. *Micromachines* **2021**, *12*, 853. https:// doi.org/10.3390/mi12080853

Academic Editors: Kai Tao and Yunjia Li

Received: 30 June 2021 Accepted: 19 July 2021 Published: 21 July 2021

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

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Based on the structure of the metal base and piezoelectric element, ultrasonic motors can be divided into two types: bonded type [16–18] and sandwich type [19–21]. The sandwich type ultrasonic motor has the advantages of large output force and high velocity. The sandwich type Langevin transducer in the literature [19] had an output mechanical force of 92 N and a no-load velocity of 0.47 m/s. A frog-shaped sandwich type piezoelectric actuator in the literature [20] achieved a maximum speed and a thrust of 287 mm/s and 11.8 N, respectively. However, because of the existence of stud structure, the structure of the sandwich type ultrasonic motor is relatively large. Therefore, in some specific situations, the use of the sandwich type ultrasonic motor is restricted. can be divided into two types: bonded type [16–18] and sandwich type [19–21]. The sandwich type ultrasonic motor has the advantages of large output force and high velocity. The sandwich type Langevin transducer in the literature [19] had an output mechanical force of 92 N and a no-load velocity of 0.47 m/s. A frog-shaped sandwich type piezoelectric actuator in the literature [20] achieved a maximum speed and a thrust of 287 mm/s and 11.8 N, respectively. However, because of the existence of stud structure, the structure of the sandwich type ultrasonic motor is relatively large. Therefore, in some specific situations, the use of the sandwich type ultrasonic motor is restricted. In view of the above situations, a novel single-phase-driven bonded type ultrasonic motor is proposed in this study, which is beneficial to the miniaturization of motor size

and to achieve the backward motion, a signal frequency of 69 kHz was used; thus, the

Based on the structure of the metal base and piezoelectric element, ultrasonic motors

output mechanical characteristics of the bi-directional motions are not consistent.

*Micromachines* **2021**, *12*, x FOR PEER REVIEW 2 of 7

In view of the above situations, a novel single-phase-driven bonded type ultrasonic motor is proposed in this study, which is beneficial to the miniaturization of motor size and drive circuit. Bending-Bending vibrations are utilized to form the desired oblique line driving trajectory. Additionally, there is no need for frequency degeneracy of Bending-Bending vibrations in this study. Section 2 introduces the structure and working principle of the single-phase-driven ultrasonic motor. Finite element analysis of the single-phase-driven ultrasonic motor is illustrated in Section 3. Output performance of this single-phase-driven bonded type ultrasonic motor is evaluated in Section 4. Finally, the conclusion is provided. and drive circuit. Bending–bending vibrations are utilized to form the desired oblique line driving trajectory. Additionally, there is no need for frequency degeneracy of bending– bending vibrations in this study. Section 2 introduces the structure and working principle of the single-phase-driven ultrasonic motor. Finite element analysis of the single-phasedriven ultrasonic motor is illustrated in Section 3. Output performance of this singlephase-driven bonded type ultrasonic motor is evaluated in Section 4. Finally, the conclusion is provided.

#### **2. Structure and Working Principle of the Single-Phase-Driven Ultrasonic Motor 2. Structure and Working Principle of the Single-Phase-Driven Ultrasonic Motor**

The structure of the proposed single-phase-driven ultrasonic motor is shown in Figure 1a, which is composed of one aluminum alloy base and four pieces of PZT ceramic. The integrated base has three functioning parts, which are the base, horn, and driving foot. The horn is designed to magnify the vibration amplitude. In order to demonstrate the two orthogonal bending vibration modes of the ultrasonic motor, the polarization direction of four pieces of PZT ceramic is illustrated in Figure 1b. The bonded type of the proposed ultrasonic motor makes it suitable for miniaturization. The structure of the proposed single-phase-driven ultrasonic motor is shown in Figure 1a, which is composed of one aluminum alloy base and four pieces of PZT ceramic. The integrated base has three functioning parts, which are the base, horn, and driving foot. The horn is designed to magnify the vibration amplitude. In order to demonstrate the two orthogonal bending vibration modes of the ultrasonic motor, the polarization direction of four pieces of PZT ceramic is illustrated in Figure 1b. The bonded type of the proposed ultrasonic motor makes it suitable for miniaturization.

**Figure 1**. Structure and working principle of the proposed single-phase-driven ultrasonic motor: (**a**) structure, (**b**) polarization directions, (**c**) schematic diagram of driving trajectory. **Figure 1.** Structure and working principle of the proposed single-phase-driven ultrasonic motor: (**a**) structure, (**b**) polarization directions, (**c**) schematic diagram of driving trajectory.

In addition, in a traditional case of a hybrid of two orthogonal bending vibration modes, two sinusoidal excitation voltages with a phase shift of 90 degrees are used to form an elliptical driving trajectory [22,23]. In this study, two orthogonal bending vibration modes with a 0-degree phase shift are utilized; thus, displacements in the OX and OY In addition, in a traditional case of a hybrid of two orthogonal bending vibration modes, two sinusoidal excitation voltages with a phase shift of 90 degrees are used to form an elliptical driving trajectory [22,23]. In this study, two orthogonal bending vibration modes with a 0-degree phase shift are utilized; thus, displacements in the OX and OY directions will be generated simultaneously; then the oblique line driving trajectory is formed, as shown in Figure 1c. Under the proposed principle, only single-phase excitation voltage is needed, which is beneficial to reduce the power cost, simplify the circuit, and miniaturize the whole ultrasonic motor.

#### **3. Finite Element Analysis of the Single-Phase-Driven Ultrasonic Motor** 10 N/m 0 0 0 3.3 0 0

*Micromachines* **2021**, *12*, x FOR PEER REVIEW 3 of 7

minum alloy and the PZT ceramics are listed in Table 1.

**Table 1.** The properties of the aluminum alloy and the PZT ceramics.

miniaturize the whole ultrasonic motor.

0 0 0 050

1.6 1.6 3.3 0 0 0 *d* <sup>−</sup> = × − −

15 8.4 6.8 0 0 0 8.4 15 6.8 0 0 0 6.8 6.8 12.9 0 0 0

 

= ×

0 0 0 5 0 0 10 C/N

Piezoelectric matrix <sup>10</sup>

Stiffness matrix 10 2

*E c*

The finite element method is used to calculate the vibration modes and to obtain the resonance frequencies of the Bending-Bending vibration modes. The finite element method (FEM) model of the proposed ultrasonic motor was built in ANSYS, as shown in Figure 2. The numbers of nodes and elements of the FEM model are 58657 and 40698, respectively. The element type of the FEM model is SOLID227. The properties of the aluminum alloy and the PZT ceramics are listed in Table 1. 0 0 0 0 2.8 0 0 0 0 0 0 2.8 Dielectric matrix <sup>9</sup> 8.1 0 0 0 8.1 0 0 10 F/m 0 0 6.7 *T* ε <sup>−</sup> = × Modulus of elasticity *E* =4.72 GPa

**PZT41 Aluminum Alloy** 

Density *ρ* = 2810 kg/m3

Poisson's ratio *μ* = 0.33

directions will be generated simultaneously; then the oblique line driving trajectory is formed, as shown in Figure 1c. Under the proposed principle, only single-phase excitation voltage is needed, which is beneficial to reduce the power cost, simplify the circuit, and

The finite element method is used to calculate the vibration modes and to obtain the resonance frequencies of the bending–bending vibration modes. The finite element method (FEM) model of the proposed ultrasonic motor was built in ANSYS, as shown in Figure 2. The numbers of nodes and elements of the FEM model are 58657 and 40698, respectively. The element type of the FEM model is SOLID227. The properties of the alu-

**3. Finite Element Analysis of the Single-Phase-Driven Ultrasonic Motor** 

**Figure 2.** The FEM model of the proposed ultrasonic motor built in ANSYS. **Figure 2.** The FEM model of the proposed ultrasonic motor built in ANSYS.


The optimized size of the proposed ultrasonic motor is achieved by parameter sensi-**Table 1.** The properties of the aluminum alloy and the PZT ceramics.

The optimized size of the proposed ultrasonic motor is achieved by parameter sensitivity analysis to ensure that the working frequency of the motor is greater than 20 kHz and the amplitude of bending vibration is greater than 1 µm. The total length of the aluminum alloy base is 36 mm, the height of the cross section of the base is 12 mm, the diameter of the driving foot is 3 mm, and the length of the horn is 15 mm. The size of the PZT ceramic is <sup>10</sup> <sup>×</sup> <sup>10</sup> <sup>×</sup> 1 mm<sup>3</sup> , and the position of the PZT ceramic is shown in Figure 3. The detailed dimensions of the single-phase-driven ultrasonic motor using Bending-Bending vibrations are shown in Figure 3.

**Figure 3.** The dimensions of the proposed ultrasonic motor (unit: mm). **Figure 3.** The dimensions of the proposed ultrasonic motor (unit: mm). **Figure 3.** The dimensions of the proposed ultrasonic motor (unit: mm). The calculated bending vibration modes in OX and OY directions are shown in Figure 4, the resonance frequencies are 41,023 Hz and 41,107 Hz, respectively, and the main

*Micromachines* **2021**, *12*, x FOR PEER REVIEW 4 of 7

The calculated bending vibration modes in OX and OY directions are shown in Figure 4, the resonance frequencies are 41,023 Hz and 41,107 Hz, respectively, and the main reason for the frequency deviation is the unsymmetrical mesh of the model. The calculated driving trajectory is shown in Figure 5, which is an oblique line as proposed in Section 2. In addition, a clamping device was designed, which is shown in Section 4; in the FEM model, a displacement constraint was applied to the ultrasonic motor by cylinders to simulate the constraint of the clamping device. The calculated bending vibration modes in OX and OY directions are shown in Figure 4, the resonance frequencies are 41,023 Hz and 41,107 Hz, respectively, and the main reason for the frequency deviation is the unsymmetrical mesh of the model. The calculated driving trajectory is shown in Figure 5, which is an oblique line as proposed in Section 2. In addition, a clamping device was designed, which is shown in Section 4; in the FEM model, a displacement constraint was applied to the ultrasonic motor by cylinders to simulate the constraint of the clamping device. The calculated bending vibration modes in OX and OY directions are shown in Figure 4, the resonance frequencies are 41,023 Hz and 41,107 Hz, respectively, and the main reason for the frequency deviation is the unsymmetrical mesh of the model. The calculated driving trajectory is shown in Figure 5, which is an oblique line as proposed in Section 2. In addition, a clamping device was designed, which is shown in Section 4; in the FEM model, a displacement constraint was applied to the ultrasonic motor by cylinders to simulate the constraint of the clamping device. reason for the frequency deviation is the unsymmetrical mesh of the model. The calculated driving trajectory is shown in Figure 5, which is an oblique line as proposed in Section 2. In addition, a clamping device was designed, which is shown in Section 4; in the FEM model, a displacement constraint was applied to the ultrasonic motor by cylinders to simulate the constraint of the clamping device.

**Figure 4.** Bending vibration modes in OX and OY directions. **Figure 4.** Bending vibration modes in OX and OY directions. **Figure 4.** Bending vibration modes in OX and OY directions. **Figure 4.** Bending vibration modes in OX and OY directions.

**4. Mechanical Characteristics of the Single-Phase-Driven Ultrasonic Motor 4. Mechanical Characteristics of the Single-Phase-Driven Ultrasonic Motor Figure 5.** The calculated driving trajectory. **Figure 5.** The calculated driving trajectory.

#### In order to evaluate the output performance, a prototype was manufactured. The In order to evaluate the output performance, a prototype was manufactured. The **4. Mechanical Characteristics of the Single-Phase-Driven Ultrasonic Motor 4. Mechanical Characteristics of the Single-Phase-Driven Ultrasonic Motor**

stator of the ultrasonic motor was composed of one integrated aluminum alloy base and four pieces of PZT ceramic, the dimensions of which were the same as the optimized simulation results. Additionally, four PZT ceramics were pasted on the stator surface with resin glue at the positions shown in Figure 3, and the curing time was 24 h under the action of preload. The impedance characteristics were tested by an impedance analyzer stator of the ultrasonic motor was composed of one integrated aluminum alloy base and four pieces of PZT ceramic, the dimensions of which were the same as the optimized simulation results. Additionally, four PZT ceramics were pasted on the stator surface with resin glue at the positions shown in Figure 3, and the curing time was 24 h under the action of preload. The impedance characteristics were tested by an impedance analyzer In order to evaluate the output performance, a prototype was manufactured. The stator of the ultrasonic motor was composed of one integrated aluminum alloy base and four pieces of PZT ceramic, the dimensions of which were the same as the optimized simulation results. Additionally, four PZT ceramics were pasted on the stator surface with In order to evaluate the output performance, a prototype was manufactured. The stator of the ultrasonic motor was composed of one integrated aluminum alloy base and four pieces of PZT ceramic, the dimensions of which were the same as the optimized simulation results. Additionally, four PZT ceramics were pasted on the stator surface with resin glue at the positions shown in Figure 3, and the curing time was 24 h under the action

(ZX80A, Zhixin Precision Electronics Co., Ltd., Changzhou, China), as shown in Figure 6.

resin glue at the positions shown in Figure 3, and the curing time was 24 h under the action of preload. The impedance characteristics were tested by an impedance analyzer

(ZX80A, Zhixin Precision Electronics Co., Ltd., Changzhou, China), as shown in Figure 6.

of preload. The impedance characteristics were tested by an impedance analyzer (ZX80A, Zhixin Precision Electronics Co., Ltd., Changzhou, China), as shown in Figure 6. The tested resonance frequency was 41.92 kHz, and the deviation of the simulation resonance frequency and the test one was 855 Hz, which was approximately 2.1% of the simulation resonance frequency. The main reasons for the deviation are the parameters error of the aluminum alloy base and the PZT ceramics, the manufacturing error, and the error caused by the test condition of the impedance analyzer. The tested resonance frequency was 41.92 kHz, and the deviation of the simulation resonance frequency and the test one was 855 Hz, which was approximately 2.1% of the simulation resonance frequency. The main reasons for the deviation are the parameters error of the aluminum alloy base and the PZT ceramics, the manufacturing error, and the error caused by the test condition of the impedance analyzer. *Micromachines* **2021**, *12*, x FOR PEER REVIEW 5 of 7 The tested resonance frequency was 41.92 kHz, and the deviation of the simulation resonance frequency and the test one was 855 Hz, which was approximately 2.1% of the simulation resonance frequency. The main reasons for the deviation are the parameters error

of the aluminum alloy base and the PZT ceramics, the manufacturing error, and the error

**Figure 6.** The impedance characteristics. **Figure 6.** The impedance characteristics. **Figure 6.** The impedance characteristics.

Then, the output performance was tested under the single-phase excitation voltage. The clamping device and the experimental setup are shown in Figure 7. The prototype was clamped and fixed on the foundation support; the driving foot was pressed on the linear guide rail. The excitation voltage was generated by the signal generator, then amplified by s power amplifier (ATA-4051, Agitek, China); the single-phase excitation volt-Then, the output performance was tested under the single-phase excitation voltage. The clamping device and the experimental setup are shown in Figure 7. The prototype was clamped and fixed on the foundation support; the driving foot was pressed on the linear guide rail. The excitation voltage was generated by the signal generator, then amplified by s power amplifier (ATA-4051, Agitek, China); the single-phase excitation voltage, sine signal, was applied to the PZT ceramics of the proposed ultrasonic motor. Then, the output performance was tested under the single-phase excitation voltage. The clamping device and the experimental setup are shown in Figure 7. The prototype was clamped and fixed on the foundation support; the driving foot was pressed on the linear guide rail. The excitation voltage was generated by the signal generator, then amplified by s power amplifier (ATA-4051, Agitek, China); the single-phase excitation voltage, sine signal, was applied to the PZT ceramics of the proposed ultrasonic motor.

**Figure 7.** The clamping device and the experimental setup. **Figure 7.** The clamping device and the experimental setup.

**Figure 7.** The clamping device and the experimental setup. The output velocity versus the input excitation voltage frequency is shown in Figure 8. We can see that changing the frequency is another way to change the output velocity. The maximum output velocity of the mover was achieved at a frequency of 42.1 kHz. As the ultrasonic motor works in a resonance state, when the working frequency is far from the resonance frequency, the output velocity decreases rapidly. The output velocity versus the input excitation voltage frequency is shown in Figure 8. We can see that changing the frequency is another way to change the output velocity. The maximum output velocity of the mover was achieved at a frequency of 42.1 kHz. As the ultrasonic motor works in a resonance state, when the working frequency is far from the resonance frequency, the output velocity decreases rapidly.

The output velocity versus the input excitation voltage frequency is shown in Figure 8. We can see that changing the frequency is another way to change the output velocity. The maximum output velocity of the mover was achieved at a frequency of 42.1 kHz. As the ultrasonic motor works in a resonance state, when the working frequency is far from The output velocity versus the input excitation voltage amplitude is shown in Figure 9, which indicates that we can change the voltage amplitude to increase the output velocity. With the excitation voltage no more than 120 V, the mover cannot be driven. The maximum velocity was approximately 340 mm/s under an excitation voltage of 300 V and 42.1 kHz.

the resonance frequency, the output velocity decreases rapidly.

In addition, the proposed single-phase ultrasonic motor using Bending-Bending vibration modes is feasible, which can also output rotary motion if the linear guide rail is replaced by a ring. The proposed bonded-type single-phase-driven ultrasonic motor not only has the merit of easy miniaturization, but also has a simple and easy miniaturization circuit. This single-phase-driven ultrasonic motor is indeed an impact motor, which has potential to be used in a high-accuracy platform. *Micromachines* **2021**, *12*, x FOR PEER REVIEW 6 of 7

**Figure 8.** The output velocity versus the input excitation voltage frequency. **Figure 8.** The output velocity versus the input excitation voltage frequency. imum velocity was approximately 340 mm/s under an excitation voltage of 300 V and 42.1 kHz.

*Micromachines* **2021**, *12*, x FOR PEER REVIEW 6 of 7

**Figure 9.** The output velocity versus the input excitation voltage amplitude. **Figure 9.** The output velocity versus the input excitation voltage amplitude.

## **5. Conclusions**

**Figure 9.** The output velocity versus the input excitation voltage amplitude. In addition, the proposed single-phase ultrasonic motor using bending–bending vibration modes is feasible, which can also output rotary motion if the linear guide rail is replaced by a ring. The proposed bonded-type single-phase-driven ultrasonic motor not only has the merit of easy miniaturization, but also has a simple and easy miniaturization circuit. This single-phase-driven ultrasonic motor is indeed an impact motor, which has potential to be used in a high-accuracy platform. In addition, the proposed single-phase ultrasonic motor using bending–bending vibration modes is feasible, which can also output rotary motion if the linear guide rail is replaced by a ring. The proposed bonded-type single-phase-driven ultrasonic motor not only has the merit of easy miniaturization, but also has a simple and easy miniaturization circuit. This single-phase-driven ultrasonic motor is indeed an impact motor, which has potential to be used in a high-accuracy platform. **5. Conclusions**  A single-phase-driven ultrasonic motor using bending–bending vibrations was proposed in this paper. The structure of this ultrasonic motor was composed of a metal base and four pieces of PZT ceramic. Additionally, orthogonal bending vibration modes were A single-phase-driven ultrasonic motor using Bending-Bending vibrations was proposed in this paper. The structure of this ultrasonic motor was composed of a metal base and four pieces of PZT ceramic. Additionally, orthogonal bending vibration modes were excited simultaneously by only single-phase voltage, thus an oblique line driving trajectory was formed to drive the mover. The working principle was verified by the finite element method. Additionally, the impedance characteristics of the ultrasonic motor were tested. The output performance was evaluated by experiment. Additionally, the maximum output velocity under 300 Vp-p was 340 mm/s. The practicability of this proposed single-phasedriven ultrasonic motor was verified. This paper provides a single-phase-driven ultrasonic motor, which has merits in the miniaturization of structures and power circuits. In future work, we will focus on the verification of the linear trajectory and its application in a high-accuracy platform.

**5. Conclusions**  A single-phase-driven ultrasonic motor using bending–bending vibrations was proposed in this paper. The structure of this ultrasonic motor was composed of a metal base and four pieces of PZT ceramic. Additionally, orthogonal bending vibration modes were tory was formed to drive the mover. The working principle was verified by the finite element method. Additionally, the impedance characteristics of the ultrasonic motor were tested. The output performance was evaluated by experiment. Additionally, the maximum output velocity under 300 Vp-p was 340 mm/s. The practicability of this proposed **Author Contributions:** Conceptualization, D.X. and S.Y.; methodology, D.X.; software, W.Y.; validation, W.Y. and D.X.; formal analysis, S.Y.; investigation, W.Y.; resources, D.X.; data curation, W.Y.; writing—original draft preparation, S.Y.; writing—review and editing, W.Y.; visualization, D.X.; supervision, D.X.; project administration, X.Z.; funding acquisition, X.Z. All authors have read and agreed to the published version of the manuscript.

excited simultaneously by only single-phase voltage, thus an oblique line driving trajectory was formed to drive the mover. The working principle was verified by the finite ele-

single-phase-driven ultrasonic motor was verified. This paper provides a single-phasedriven ultrasonic motor, which has merits in the miniaturization of structures and power

excited simultaneously by only single-phase voltage, thus an oblique line driving trajec-

tested. The output performance was evaluated by experiment. Additionally, the maximum output velocity under 300 Vp-p was 340 mm/s. The practicability of this proposed single-phase-driven ultrasonic motor was verified. This paper provides a single-phasedriven ultrasonic motor, which has merits in the miniaturization of structures and power circuits. In future work, we will focus on the verification of the linear trajectory and its

**Author Contributions:** Conceptualization, D.X. and S.Y.; methodology, D.X.; software, W.Y.; validation, W.Y. and D.X.; formal analysis, S.Y.; investigation, W.Y.; resources, D.X.; data curation, W.Y.;

**Author Contributions:** Conceptualization, D.X. and S.Y.; methodology, D.X.; software, W.Y.; validation, W.Y. and D.X.; formal analysis, S.Y.; investigation, W.Y.; resources, D.X.; data curation, W.Y.;

application in a high-accuracy platform.

**Funding:** This work was supported in part by the National Natural Science Foundation of China (No. 52005398), in part by the China Postdoctoral Science Foundation (No. 2019M663776), in part by the Shaanxi Natural Science Basic Research Program (No. 2019JQ-805), in part by the Shaanxi Education Department General Special Scientific Research Plan (No. 20JK0774), and in part by the Shaanxi Key Laboratory of Mine Electromechanical Equipment Intelligent Monitoring for the Open Fund (SKL-MEEIM201916).

**Conflicts of Interest:** The authors declare no conflict of interest.
