Design of a Low-Frequency Harmonic Rotary Piezoelectric Actuator

actuators

Article Design of a Low-Frequency Harmonic Rotary Piezoelectric Actuator

Kang Liang 1, Chong Li 1,2,* , Yujian Tong 1, Jiwen Fang 1 and Wei Zhong 1

1 School of Mechanical Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China; [email protected] (K.L.); [email protected] (Y.T.); [email protected] (J.F.); [email protected] (W.Z.) 2 Robotics and Microsystems Center, Scoochow University, Suzhou 215021, China * Correspondence: [email protected]; Tel.: +86-511-8444-5385

Abstract: Piezoelectric actuators usually operate under a high frequency driving signal. Here we report a harmonic rotating piezoelectric actuator by coupling a harmonic wave generator and a friction rotor, in which the actuator can be actuated by a low-frequency sinusoidal signal with positive bias. The harmonic wave is generated by a two-stage magnifying mechanism consisting of a displacement ampliﬁer and a harmonic rod. Applying piezoelectricity theory, the actuator’s output characteristic equations are deduced. What is more, the output characteristics of piezoelectric actuators are tested with the established experimental system. Results show that the generated harmonic displacements can drive the actuator to work normally at a driving voltage of larger than 90 V and the maximum total harmonic displacement of the piezoelectric actuator comes up to 427.6 µm under the driving voltage of 150 V. Meanwhile, the error between the measured and calculated values of the harmonic displacement is less than 7%. Furthermore, the rotational speed of the piezoelectric actuator reaches 5.45 rpm/min at 150 V voltage and 5 Hz driving frequency.

Keywords: low frequency; piezoelectric actuator; harmonic motion; rotational speed

1. Introduction Citation: Liang, K.; Li, C.; Tong, Y.; Piezoelectric actuators use the inverse piezoelectric effect to convert electrical energy Fang, J.; Zhong, W. Design of a into mechanical energy or mechanical motion. Due to the advantages of its simple structure, Low-Frequency Harmonic Rotary light weight and fast response, piezoelectric actuators have been used in Micro Electro Piezoelectric Actuator. Actuators 2021, Mechanical Systems (MEMS), micro-robots, bioengineering and medical sciences, as well 10, 4. https://doi.org/10.3390/ as ﬂuid drive and vibration and noise control ﬁelds. Therefore, the research on piezoelectric act10010004 drive technology has great economic and social value. Numerous achievements in piezoelectric actuating have been obtained. Acosta [1] Received: 8 December 2020 made use of idealized piezoelectric sensors to achieve photoacoustic imaging. Using the Accepted: 24 December 2020 method of matched asymptotic expansions and the basic constitutive relations for piezoelec- Published: 27 December 2020 tricity, the mathematical model for piezoelectric transducers was established. Through the

Publisher’s Note: MDPI stays neu- proposed model, a better imaging effect is obtained. Wang et al. [2] designed a non-resonant tral with regard to jurisdictional claims stepping type bolt-clamped piezoelectric actuator. The proposed piezoelectric actuator was in published maps and institutional operated under a clamping and driving mechanism by using two longitudinal-bending afﬁliations. transducers. Under 400 Vp-p voltages and 1 Hz signal drive, the average step distance was about 10.8 µm. When the load mass is 1.47 kg, the actuator can obtain a velocity of 1.42 µm/s. Applying four rotating traveling wave piezoelectric actuators, Hamamoto et al. [3] designed an aircraft similar to a dragonﬂy. Tomoaki [4] designed and manufac- Copyright: © 2020 by the authors. Li- tured a super-miniature piezoelectric motor with a stator volume of 1 mm3, which became censee MDPI, Basel, Switzerland. This one of the smallest piezoelectric motors in the world. In the motor, the stator consists of article is an open access article distributed a metallic cube with a through-hole and piezoelectric elements adhered to its sides. The under the terms and conditions of the torque and speed were measured experimentally. Ho et al. [5] proposed a spiral driven Creative Commons Attribution (CC BY) piezoelectric motor, which is driven by four piezoelectric stacks and operates in double license (https://creativecommons.org/ shear mode, with a transmission speed of 2.137 mm/s. In addition, Shi et al. [6] proposed a licenses/by/4.0/).

Actuators 2021, 10, 4. https://doi.org/10.3390/act10010004 https://www.mdpi.com/journal/actuators Actuators 2021, 10, 4 2 of 18

cyclic multi-degree-of-freedom piezoelectric ultrasonic motor continuously driven by four legs. The test results showed that the maximum output torque and maximum speed of the motor were 0.118 Nm and 55 rpm/min, respectively. Zeng et al. [7] designed and tested a new type of piezoelectric micromotor driven by asymmetric inertia impact. The resolution of this micromotor is 0.02 µm and the maximum load is 1 kg under the drive of 100 V voltage and 35 Hz. What is more, actuating with ’33’ electromechanical coupling mode, a surface-bondable multilayer piezoelectric actuator [8] was developed. Mazeika et al. [9] designed a multimodal piezoelectric traveling wave actuator with a cylindrical stator. The proposed actuator is clamped at the bottom and driven by a four-phase difference of pi/2 electric signals. Experimental results show the actuator can generate a 115 rpm/min rotation speed under constant initial conditions. Based on magnetostrictive material with simultaneous vibration, a rotating piezoelectric actuator with simultaneous vibration in the longitudinal direction was developed by Titsch et al. [10]. It was shown that the magnetic field and its frequency affect the amplitude of the piezoelectric actuator. Moreover, Ostasevicius et al. [11] used the piezoelectric actuator for separation and purification of biological microparticles. Furthermore, many achievements have been made in the sensor and control of piezoelectric actuators. Liu et al. [12] investigated the monolithic piezoelectric control of soliton microcombs. By monolithically integrating piezoelectric actuators on ultralow-loss pho- tonic circuits, soliton microcombs—a spectrum of sharp lines over a range of optical frequencies—can be modulated at high speeds with megahertz bandwidths. Zhu et al. [13] studied a piezoelectrically actuated fast tool servo (FTS) for the diamond turning of microstructured surfaces. Utilizing the newly developed FTS, two typical microstructured surfaces are ultra-precisely turned. Furthermore, the feasibility of this method is proven. As precision motion of the piezoelectric ultrasonic motor is severely affected by inherent friction, hysteresis nonlinearity and model uncertainties, Feng et al. [14] developed a novel integral terminal sliding-mode-based adaptive integral backstepping control to solve the precision problems. Using piezo response function and global optimization techniques, Sumit et al. [15] investigated the shape control optimization of the piezoelectric bimorph. A total of 28 piezoelectric actuators are used in the piezoelectric bimorph to generate the sinusoidal profile, elliptical profile and arbitrary deformation profile by the external load. In the low-frequency driving field, there are also some piezoelectric applications. Nabavi et al. [16] proposed a piezoelectric MEMS vibration energy harvester that operates at a low resonant frequency of less than 200 Hz. By taking advantage of this efficient AI-based performance estimator, the harvested voltage can increase from 2.5 V to 3.4 V under 0.25 g excitation. To enhance the energy harvesting performance of low-frequency rotational motion, Mei et al. [17] proposed a quad-stable piezoelectric energy harvester. The experiments show that the quad-stable piezoelectric energy harvester with time-varying potential wells exhibits a wider operational frequency range (1–7 Hz) in rotational motion. Using liquid as an energy-capturing medium, Yang et al. [18] developed a piezoelectric vibration energy harvester, which can achieve ultralow frequency, low intensity and multidirectional vibration energy harvesting in a horizontal plane. The proposed harvester can generate 9.8 µW under an ultralow frequency of 2.6 Hz and a low intensity vibration excitation of 0.03 g. Modulated by an electromagnetic field, Xing et al. [19] proposed a low-frequency piezoelectric motor, which combines piezoelectric driving and magnetic modulation. Driven at a frequency of 3 Hz and a modulating voltage of 4.5 V, the output speed and torque are 0.1046 rad/min and 0.405 Nmm, respectively. Applying a piezoelectric torsional vibrator and the giant electrorheological fluid, a bidirectional non-contact rotary motor was developed by Qiu et al. [20]. At an excitation frequency of 118 Hz, the proposed motor generates 1.04 mNm torque when the electric field strength of 2 kV/mm with 30% duty cycle is applied to the giant electrorheological fluid. What is more, Barth [21] developed a harmonic micro motor that combined a harmonic drive gearbox and stack piezoactuators. The motor can realize low-speed motion and large output torque. Actuators 2021, 10, 4 3 of 18

In addition to the above piezoelectric actuators, a variety of new rotary actuators and mechanisms based on different driving principles have appeared one after another. Yang et al. [22] presented a triboelectric micromotor coupled with a micromotor and a triboelectric nanogenerator; the proposed micromotor can be actuated by ultralow- frequency mechanical stimuli. With a sliding range of 50 mm at 0.1 Hz, the micromotor can start to rotate and reach over 1000 rpm/min at 0.8 Hz. The maximum operation efﬁciency of the triboelectric micromotor can reach 41%. For early detection of cardiovascular disease, Pullano et al. [23] proposed a novel passive device based on the triboelectric effect to evaluate pseudo impedance cardiography. The proposed device provides a novel monitoring approach opening a promising alternative for continuous and remote risk assessment. Although many accomplishments have been made in the research of piezoelectric actuators, most of the research, however, has focused on intermediate- and high-frequency driving. The driving principle of the most traditional intermediate- and high-frequency piezoelectric actuators uses the stator resonance to produce the large amplitude, and then produces the motion with the rotor friction. What is more, most of the literature on low-frequency piezoelectric mechanisms are about piezoelectric energy acquisition devices. A few low-frequency piezoelectric actuators use electromagnetic clamping of the rotor or other principles to realize the motion between the stator and rotor. Therefore, the authors present a low-frequency harmonic rotating piezoelectric actuator. Compared with other piezoelectric actuators, the proposed actuator uses the harmonic friction drive between the harmonic rod and the rotor to transmit motion. At the same time, piezoelectric displacement amplifying the mechanism and harmonic rod are used to amplify the output displacement of piezoelectric stacks. Therefore, the end of the harmonic rod can generate a large enough displacement, and then generate a large displacement of the harmonic motion. In turn, it can drive the rotor to produce rotational motion. The proposed motor has the advantages of simple structure, low output speed and low assembly precision. Only two piezoelectric stacks are used as driving sources to generate harmonic motion. Therefore, the actuator is easy to drive and control. At the same time, because the generated harmonics are not affected by the frequency, the proposed actuator can work under the drive of ultralow frequency and obtain ultralow speed rotation motion. In this paper, the operating principle of the low-frequency harmonic rotating piezoelectric actuator is presented. The design process and driving parameters of the proposed piezoelectric actuator are analyzed. In addition, the actuator’s output characteristics are deduced. Using experimental equipment, the output characteristics of piezoelectric actuators were tested. The results lay the theoretical and experimental foundation for the improvement of the prototype of the low-frequency harmonic rotating piezoelectric actuator.

2. Driving Principle and Motion Analysis The proposed low frequency harmonic rotating piezoelectric actuator consists of a displacement amplification mechanism, motion output mechanism and auxiliary assembly mechanism. The displacement amplification mechanism includes piezoelectric stack and flexure hinge amplifying mechanism. The motion output mechanism is composed of a rotor and harmonic rod. Moreover, the auxiliary assembly mechanism includes a top bearing cover, upper bearing, upper cover, body case, lower bearing, threaded rod and preloaded spring, lower cover and lower bearing cover. The driving principle of the proposed piezoelectric actuator is shown in Figure1. The actuator is driven by two sinusoidal signals with 90 degrees of the phase difference. Driven by two signals, the piezoelectric stacks generate different degrees of elongation deformation. The deformation of piezoelectric stacks causes the deformation of the flexure hinge amplification mechanism. At the same time, the amount of deformation drives the harmonic rod to sway from left to right and from front to back. Under the action of a continuous signal, the harmonic rod generates a large displacement harmonic motion. Actuators 2021, 10, x FOR PEER REVIEW 4 of 19

The driving principle of the proposed piezoelectric actuator is shown in Figure 1. The actuator is driven by two sinusoidal signals with 90 degrees of the phase difference. Driven by two signals, the piezoelectric stacks generate different degrees of elongation Actuators 2021, 10, 4 deformation. The deformation of piezoelectric stacks causes the deformation of the4 flex- of 18 ure hinge amplification mechanism. At the same time, the amount of deformation drives the harmonic rod to sway from left to right and from front to back. Under the action of a continuous signal, the harmonic rod generates a large displacement harmonic motion. Furthermore, under the actionaction ofof harmonicharmonic force,force, frictionfriction motionmotion betweenbetween thethe harmonicharmonic rod andand rotorrotor isis generatedgenerated toto drivedrive rotorrotor rotation.rotation.

Figure 1. Working principle of the low-frequency harmonic rotating piezoelectric actuator. ( (aa)) The The position ofof 00 degreesdegrees oror 360360 degrees;degrees; ( b(b)) The The positions positions of of 90 90 degrees; degrees; (c ()c The) The positions positions of of 180 180 degrees; degrees; (d) The positions of 270 degrees. (d) The positions of 270 degrees.

The piezoelectricpiezoelectric actuator actuator presented presented in in this this paper paper is drivenis driven by aby sinusoidal a sinusoidal signal. signal. The piezoelectricThe piezoelectric stack stack is easily is damagedeasily damaged when itwh deformsen it deforms in the negative in the direction,negative therefore,direction, thetherefore, positive the signals positive are appliedsignals are to the applied piezoelectric to the piezoelectric stack as a driving stack source. as a driving The equation source. ofThe the equation driving of signal the driving curve can signal be expressed curve can as:be expressed as:  1  Ut()1=+ U1sin2()π ft  U (t)1 = U pp[−1 +sin(2π f t)] 1 2 2p-p  h π i (1)(1) ( ) = 1 1 ++ π  U2t ()2=++Up-p 1 sin 2ππf t Ut2 Upp− 1sin2 ft 2  22 where Up-p and f are the peak-to-peak value and driving frequency of the driving signal, respectively.where Up-p and f are the peak-to-peak value and driving frequency of the driving signal, respectively.When the piezoelectric stack works, it generates elastic deformation and its stress equationWhen [24 the] is piezoelectric expressed as: stack works, it generates elastic deformation and its stress equation [24] is expressed as:  P 1  ε p1 = s33 P+ 1d33Up-p[1 + sin(2π f t)]  επ=+sdUftS 2h  1sin2+ ()  ppp133p p 33 −  (2) Shpp2 h i  P 1 π  ε p2 = s33 + d33Up-p 1 + sin 2π f t + (2)  Sp P 2hp1 π 2 επ=+sdUft 1sin2++  ppp233 33 −   Shpp22 where P and s33 are the preload and elastic ﬂexibility coefﬁcients of the piezoelectric stack, respectively;where P andd 33s33, Sarep and theh ppreloadare piezoelectric and elastic strain flexibilit constant,y coefficients sectional area of andthe piezoelectric layerstack, thickness respectively; of the d33 piezoelectric, Sp and hp are stack, piezoelectric respectively. strain constant, sectional area and pie- zoelectricSupposing layer thickness the total length of the ofpiezoelectric the piezoelectric stack, stackrespectively. is lnp, and the number of ceramic layers in the piezoelectric stack is n, then lnp = nhp. From the relationship between strain εp, total length lnp and total deformation δnp, the following is obtained:

 P 1  δnp1 = n s33hp + d33Up-p[1+ sin(2π f t)]  Sp 2 h i (3)  P 1 π  δnp2 = n s33hp + d33Up-p 1+ sin 2π f t + Sp 2 2 Actuators 2021, 10, x FOR PEER REVIEW 5 of 19

Supposing the total length of the piezoelectric stack is lnp, and the number of ceramic layers in the piezoelectric stack is n, then lnp = nhp. From the relationship between strain εp, total length lnp and total deformation δnp, the following is obtained:

 P 1 δπ=+1+sin2nsh dU− () ft np133 p 33 p p   S p 2  (3) Actuators 2021, 10, 4 π 5 of 18 δπP 1 + np23333=+1+sin2nsh p dU p− p  ft  S 22  p 

FigureFigure2 2 shows shows the the deformation deformation of of displacement displacement amplification amplification mechanism. mechanism. Here, Here, OO 1 pointpoint isis fixedfixed endend andand OO1 pointpoint isis freefree extensionextension end.end. WhenWhen thethe systemsystem isis working,working, thethe twotwo endsends (points(points AA andand B)B) ofof thethe piezoelectricpiezoelectric stackstack areare stretchedstretched andand deformeddeformed alongalong thethe axis.axis. TheThe flexibleflexible arms,arms,O O11CC andand OO11D,D, ofof thethe displacementdisplacement amplificationamplification mechanismmechanism rotaterotate slightlyslightly aroundaround pointspoints CC andand D.D. Thus,Thus, thethe freefree endendO O11 producesproduces anan overhangingoverhanging displacement.displacement.

O np 1 δd CDld

ABlnp δnp

FigureFigure 2.2. DeformationDeformation ofof displacementdisplacement ampliﬁcationamplification mechanism.mechanism.

The unilateral deformation of piezoelectric stack ∆ can be written as: The unilateral deformation of piezoelectric stack Δnpnp can be written as:   11  ∆np1 == δδnp1  np1122 np  1 (4)(4)  ∆np2 = 1δnp2  =2 δ  np222 np According to the dimension relation in Figure2, the length of the flexible arm can be According to the dimension relation in Figure 2, the length of the flexible arm can written as: be written as:  1  l  2 np1 nhp1  l = 1 = d1 l  cos θnp1 2nh cos θ (5) l =1 2 = p1   d1 lnp2   2cosθθ 2cosnhp2  ld2 = = (5)  cos1 θ 2 cos θ lnp2 nh where θ is the angle between the flexiblel arm= 2 and= thep horizontal.2  d 2 The angle between the flexible arm andcos theθθ horizontal 2cos changes after deformation, and itswhere expression θ is the is: angle between the flexible arm and the horizontal. The angle between the flexible1 arm and the horizontal changes after deformation,  l + ∆ and its expression is: 2 np1 np1 cos θ lnp1 + 2∆np1  cos φ1 = =  l nhp1  1 d1 +Δ (6) 1 lnp11 np θ ()+Δ  coslnp11 2 np  cosφ ==lnp22 + ∆np2 cos θ l + 2∆   21 np2 np2  cos φ2 = lnhdp11=  ld2 nhp2 (6)  1 +Δ lnp22 np cosθ ()l +Δ 2 Hence, the output displacementcosφ ==2 of the displacementnp22 amplification np mechanism can be  2 expressed by:  lnhdp22  " !# nl cos θl + 2∆  p1 np1 np1  δd1 = 2ld1 sin φ1 = sin arccos  cos θ nlp1 " !# (7) nl cos θl + 2∆  p2 np2 np2  δd2 = 2ld2 sin φ2 = sin arccos  cos θ nlp2

The output displacement of the displacement ampliﬁcation mechanism is ampliﬁed by a two-order displacement through a harmonic rod. The relationship between the dimensions is shown in Figure3, where l1 and l2 are two section lengths of the harmonic rod. Actuators 2021, 10, 4 6 of 18

As seen in Figure3, the displacement of the harmonic rod is proportional to the displacement of the displacement ampliﬁcation mechanism. According to the triangle principle, the output displacement of the harmonic rod can be expressed by:

 δd1(l1 + l2)  δh1 = l1 (8) δd2(l1 + l2)  δh2 = l1 Furthermore, the total displacement is the sum of the displacement vectors in both directions, which can be expressed as:

q l + l q = 2 + 2 = 1 2 2 + 2 δh δh1 δh2 δd1 δd2 (9) Actuators 2021, 10, x FOR PEER REVIEW l1 6 of 19

To realize the continuous output of the rotor, the output displacement of the harmonic rod should be larger than its offset. Here, the harmonic offset is 0.2 mm. Hence, the relation between the output displacement of the harmonic rod and its offset is as follows: Hence, the output displacement of the displacement amplification mechanism can  > be expressed by:  δh1 ∆r δh2 > ∆r (10)   θ +Δ nl δ > ∆r cos()lnp11 2 np δφ==2l sinp1 sinh  arccos  dd111cosθ nl Through the above analysis, it is easy to determine the drivingp1 principle  and motion  (7) law of a low-frequency rotating piezoelectric actuator. Theθ () structure+Δ of the low-frequency  nlp2 coslnp22 2 np harmonic rotating piezoelectricδφ==2l sin actuator is sin presented arccos in Figure4, where the relationship dd222cosθ nl between the components is clear. After two-stage displacementp ampliﬁcation,2 the proposed piezoelectric actuator obtained a large harmonic displacement to drive the rotor. To ensure the accuracyThe output of harmonic displacement generation of the anddisplacement assembly amplification precision, a movable mechanism upper is coveramplified was adopted.by a two-order The gap displacement between the through rotor and a harmonic the harmonic rod. lever The canrelationship be adjusted between by adjusting the di- themensions screws is on shown the side. in AtFigure the same3, where time, l1 two and side-by-side l2 are two section bearings lengths are used of tothe ensure harmonic that therod. rotor is not deﬂected.

h h

d d

Figure 3. Displacement calculation diagram of harmonic rod.

As seen in Figure 3, the displacement of the harmonic rod is proportional to the displacement of the displacement amplification mechanism. According to the triangle principle, the output displacement of the harmonic rod can be expressed by:  δ ()ll+ δ = d11 2  h1  l1  (8) δ ()ll+ δ = d 21 2  h2  l1 Furthermore, the total displacement is the sum of the displacement vectors in both directions, which can be expressed as: ll+ δδδ=+=2212 δδ 22 + hhh12 dd 12 (9) l1 To realize the continuous output of the rotor, the output displacement of the harmonic rod should be larger than its offset. Here, the harmonic offset is 0.2 mm. Hence, the relation between the output displacement of the harmonic rod and its offset is as follows:

Actuators 2021, 10, x FOR PEER REVIEW 7 of 19

δ >Δ  hr1 δ >Δ  hr2 (10) δ >Δ  hr Through the above analysis, it is easy to determine the driving principle and motion law of a low-frequency rotating piezoelectric actuator. The structure of the low-frequency harmonic rotating piezoelectric actuator is presented in Figure 4, where the relationship between the components is clear. After two-stage displacement amplification, the proposed piezoelectric actuator obtained a large harmonic displacement to drive the rotor. To ensure the accuracy of harmonic generation and assembly precision, a movable upper cover was adopted. The gap between the rotor and the harmonic lever Actuators 2021, 10, 4 can be adjusted by adjusting the screws on the side. At the same time, two side-by-side7 of 18 bearings are used to ensure that the rotor is not deflected.

(a)

(b)

FigureFigure 4. 4. StructureStructure of of the the low-frequency low-frequency harmon harmonicic rotating rotating piezoelectric piezoelectric actuator. actuator. ( (aa)) The The exploded exploded drawing;drawing; ( (bb)) The The assembly assembly drawing. drawing.

3. Simulation Analysis and Experimental Test 3.1. Output Displacement Analysis of Amplification Mechanism The physical parameters of the piezoelectric stack are shown in Table1. Herein, the piezoelectric stack is provided by a signal generator with offset sinusoidal signals. Meanwhile, the piezoelectric driving power supply produced by Harbin Core Tomorrow Company is used to amplify the voltage of the driving signals. Substituting physical parameters of the piezoelectric stack into Equation (3), the output displacement curve of the piezoelectric displacement amplification mechanism can be obtained as shown in Figure5, where the PZT DAM represents the Piezoelectric Displacement Amplification Mechanism (PZT DAM). From Figure5, the following is obtained: (1) When a sinusoidal driving signal is applied to the piezoelectric amplification mechanism, it produces a sinusoidal displacement change. The displacement phase between the two amplification mechanisms is still 90 degrees. In addition, the maximum output displacement of the two displacement amplification mechanisms is 169.1 µm under a driving voltage of 150 V. Actuators 2021, 10, x FOR PEER REVIEW 8 of 19

3. Simulation Analysis and Experimental Test 3.1. Output Displacement Analysis of Amplification Mechanism The physical parameters of the piezoelectric stack are shown in Table 1. Herein, the piezoelectric stack is provided by a signal generator with offset sinusoidal signals. Meanwhile, the piezoelectric driving power supply produced by Harbin Core Tomor- row Company is used to amplify the voltage of the driving signals. Substituting physical Actuators 2021 10 , , 4parameters of the piezoelectric stack into Equation (3), the output displacement curve of 8 of 18 the piezoelectric displacement amplification mechanism can be obtained as shown in Figure 5, where the PZT DAM represents the Piezoelectric Displacement Amplification Mechanism (PZT(2) DAM).With From the increase Figure of5, the driving following voltage, is obtained: the output displacement of piezoelectric dis- (1) When a sinusoidalplacement driving amplification signal is mechanism applied to increases the piezoelectric in approximate amplification linear proportion. As mechanism, it producestime goesa sinusoidal on, it gives displa risecement to a sinusoidal change. The pattern. displacement phase between the two amplification mechanisms is still 90 degrees. In addition, the maximum At the same time, to verify the correctness of the theoretical calculation model, AN- output displacement of the two displacement amplification mechanisms is 169.1 μm SYS finite element analysis software is used to carry out finite element simulation of the under a driving voltage of 150 V. displacement amplification mechanism, which is presented in Figure6. Here, Figure6a (2) With the increase of driving voltage, the output displacement of piezoelectric shows mesh generation of the displacement amplification mechanism. The grid size is set displacement amplification mechanism increases in approximate linear proportion. As by 0.5 mm. Hence, after mesh generation, the number of nodes and cells is 113,943 and time goes on, it40,692, gives rise respectively. to a sinusoidal pattern.

Table 1. Physical parameters of piezoelectric stack. Table 1. Physical parameters of piezoelectric stack. Parameters Values Parameters Values Piezoelectric strain constant d33 (pm/V) 635 Piezoelectric strain constant d233 (pm/V) 635 Elastic flexibility coefficient s33 (μm /N) 2 18 Elastic ﬂexibility coefﬁcient s33 (µm /N) 18 Sectional area of the piezoelectric stack Sp (mm2) 2 25 Sectional area of the piezoelectric stack Sp (mm ) 25 total lengthtotal of lengththe piezoelectric of the piezoelectric stack l stacknp (mm)lnp (mm) 30 30 3 Density ofDensity the piezoelectric of the piezoelectric stack ρ stackp (kg/mρp (kg/m3) ) 7500 7500 Young's modulusYoung’s modulusof the piezoelectric of the piezoelectric stack E stackp (GPa)Ep (GPa) 76.5 76.5 Poisson’s ratio of the piezoelectric stack νp 0.32 Poisson's ratio of the piezoelectric stack νp 0.32

200 PZT DAM 1 PZT DAM 2

150

100 / μm / d δ

0 0 0.1 0.2 0.3 0.4 0.5 t / s (a) (b) Figure 5. Output displacement curve of piezoelectric displacement amplification mechanism. (a) ActuatorsFigure 2021,5. 10Output, x FOR PEER displacement REVIEW curve of piezoelectric displacement ampliﬁcation mechanism. (a) The displacement in both9 of 19 directions variesThe displacement with time; (b in) The both unidirectional directions varies displacement with time; varies (b) The with unidirectional time and peak–peak displacement voltage. varies with time and peak–peak voltage.

At the same time, to verify the correctness of the theoretical calculation model, ANSYS finite element analysis software is used to carry out finite element simulation of the displacement amplification mechanism, which is presented in Figure 6. Here, Figure 6a shows mesh generation of the displacement amplification mechanism. The grid size is set by 0.5 mm. Hence, after mesh generation, the number of nodes and cells is 113,943 and 40,692, respectively.

(a) (b)

FigureFigure 6. 6. FiniteFinite element element simulation simulation of of displacement displacement amplification ampliﬁcation mechanism. mechanism. (a (a) )Mesh Mesh generation; generation; ((bb)) Deformation Deformation nephogram. nephogram.

TheThe magnifying magnifying mechanism mechanism is is made made of of st steel;eel; Young's Young’s modulus modulus and and Poisson's Poisson’s ratio ratio areare 210 210 GPa GPa and and 0.27, 0.27, respectively. respectively. The The yield yield strength strength of of the the material material is is 355 355 MPa MPa and and the the density is 7850 kg/m3. However, Young's modulus and Poisson's ratio of the piezoelectric material are 76.5 GPa and 0.32, respectively. Meanwhile, the density of the piezoelectric material is 7500 kg/m3. A fixed constraint is adopted at one end of the amplification mechanism, as shown in point O in Figure 2. As the selected piezoelectric stack can output a displacement of 30 μm, a displacement load of 30 μm is applied to both ends of the piezoelectric stack. After finite element simulation, the deformation cloud diagram of the displacement amplification mechanism is shown in Figure 6b. From the figure, we see that the finite element simulation shows that the maximum output displacement of the displacement amplification mechanism is 163 μm. Hence, the error between the simulation value and the theoretical calculation value is 3.6%. The results indicate that the theoretical calculation model is reasonable. Meanwhile, the output displacement experiment of piezoelectric displacement amplification mechanism is carried out to further verify the correctness of the theoretical calculation. The displacement amplification mechanisms and piezoelectric stacks for the experiment were manufactured by the Harbin Tiny Nano Company. Figure 7 presents the experimental test system for the piezoelectric displacement amplification mechanism. Here, the piezoelectric drive power supply adopts the three-channel power supply produced by the Harbin Core Tomorrow Company. The output displacement of the piezoelectric displacement amplifier is measured by LVDT inductance micro displacement measuring instrument. In addition, JUNTEK JDS-2900 signal generator was used to provide sinusoidal drive signals for the piezoelectric stacks. The experimental results of output displacement are shown in Figure 8, where Figure 8a–c represents the measured displacement of piezoelectric displacement amplification mechanisms 1 and 2, as well as the error between the measured value and the theoretical value, respectively. The experimental results indicate that: (1) Compared with the theoretical calculation, the experimental curve of piezoelectric displacement amplification mechanism presents a weak nonlinear variation with the peak–peak voltage change. Driven by a 150 V voltage signal, the maximum output displacements of the two displacement amplification mechanisms are 166 μm and 163 μm, respectively. (2) With the increase of voltage, the experimental error decreases first and then increases, and the error is the smallest when the voltage is 90–100 V. At the same time, the lower the voltage, the greater the error, therefore the lower voltage is not suitable for driving piezoelectric displacement amplification mechanism (3) When the voltage is within the range of 60–150 V, the output displacement error is within 5%. Within this voltage range, the error of displacement amplification mecha-

Actuators 2021, 10, 4 9 of 18

density is 7850 kg/m3. However, Young’s modulus and Poisson’s ratio of the piezoelectric material are 76.5 GPa and 0.32, respectively. Meanwhile, the density of the piezoelectric material is 7500 kg/m3. A fixed constraint is adopted at one end of the amplification mechanism, as shown in point O in Figure2. As the selected piezoelectric stack can output a displacement of 30 µm, a displacement load of 30 µm is applied to both ends of the piezoelectric stack. After finite element simulation, the deformation cloud diagram of the displacement amplification mechanism is shown in Figure6b. From the figure, we see that the finite element simulation shows that the maximum output displacement of the displacement amplification mechanism is 163 µm. Hence, the error between the simulation value and the theoretical calculation value is 3.6%. The results indicate that the theoretical calculation model is reasonable. Meanwhile, the output displacement experiment of piezoelectric displacement amplification mechanism is carried out to further verify the correctness of the theoretical calculation. The displacement amplification mechanisms and piezoelectric stacks for the experiment were manufactured by the Harbin Tiny Nano Company. Figure7 presents the experimental test system for the piezoelectric displacement amplification mechanism. Here, the piezoelectric drive power supply adopts the three-channel power supply produced by the Harbin Core Tomorrow Company. The output displacement of the piezoelectric displacement amplifier is measured by LVDT inductance micro displacement measuring instrument. In addition, JUNTEK JDS-2900 signal generator was used to provide sinusoidal drive signals for the piezoelectric stacks. The experimental results of output displacement are shown in Figure8, where Figure8a–c represents the measured displacement of piezoelectric displacement amplification mechanisms 1 and 2, as well as the error between the measured value and the theoretical value, respectively. The experimental results indicate that: (1) Compared with the theoretical calculation, the experimental curve of piezoelectric displacement amplification mechanism presents a weak nonlinear variation with the peak–peak voltage change. Driven by a 150 V voltage signal, the maximum output displacements of the two displacement amplification mechanisms are 166 µm and 163 µm, respectively. (2) With the increase of voltage, the experimental error decreases first and then increases, and the error is the smallest when the voltage is 90–100 V. At the same time, the lower the voltage, the greater the error, therefore the lower voltage is not suitable for driving Actuators 2021, 10, x FOR PEER REVIEW piezoelectric displacement amplification mechanism. 10 of 19 (3) When the voltage is within the range of 60–150 V, the output displacement error is within 5%. Within this voltage range, the error of displacement amplification mechanism 1 is smaller than that of amplification mechanism 2. At 150 V, the error nism between1 is smaller the than output that displacements of amplification of the mechanism two displacement 2. At 150 amplifiersV, the error is between 1.8%. the output displacements of the two displacement amplifiers is 1.8%.

Figure 7. Experimental test system for piezoelectricpiezoelectric displacementdisplacement ampliﬁcationamplification mechanism.mechanism.

Theoretical datas Experimental datas Errors 180 40 37.0 160 35 140 30 120 25 100 21.4 20 80 15.3 15 60 10.3 40 6.8 10 3.8 20 1.6 1.8 5 0.7 0.1 0.8 0.4 0.1 0.5 1.0 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Peak–peak voltage Up-p / V (a)

(b) Errors / %

(c) Figure 8. Experimental results of output displacement of piezoelectric displacement amplification mechanism. (a) Displacement amplification mechanism 1 test results; (b) Displacement amplification mechanism 2 test results; (c) The error between the experimental result and the theoretical calculation.

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nism 1 is smaller than that of amplification mechanism 2. At 150 V, the error between the output displacements of the two displacement amplifiers is 1.8%.

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Figure 7. Experimental test system for piezoelectric displacement amplification mechanism.

Peak–peak voltage Up-p / V (a)

(b) Errors / %

(c)

Figure 8. FigureExperimental 8. Experimental results of resultsoutput ofdisplacement output displacement of piezoelectric of piezoelectric displacement displacement amplification amplification mechanism.mechanism. (a) Displacement (a) Displacement amplification amplification mechanism mechanism 1 test results; 1 test ( results;b) Displacement (b) Displacement amplifica- amplification tion mechanism 2 test results; (c) The error between the experimental result and the theoretical mechanism 2 test results; (c) The error between the experimental result and the theoretical calculation. calculation. 3.2. Harmonic Displacement Analysis According to Equations (8) and (9), the harmonic displacement curves can be simulated as shown in Figure9, where Figure9a–d represents output harmonic displacement in both directions, total harmonic displacement, harmonic displacements that vary with time and voltage, comparison of total displacement and unidirectional displacement, respectively. In addition, the displacements at different moments in a certain period are also analyzed, which is presented in Figure 10. Here, a few special moments are selected for analysis, and T represents periodic time. From Figures9 and 10, it can be observed that: (1) The harmonic displacements in both directions vary with time by sinusoidal curves; the phase difference is 90 degrees, which is the same as the input signal. The total displacement is approximately sinusoidal with time, and the frequency is the same as the input signal, but the phase is different. Herein, the total displacement is the superposition of the displacements in directions 1 and 2, and the phase difference of the displacement signals in the two directions is 90 degrees. Therefore, there is no case of both being 0, so the total displacement is always greater than 0 in Figure9b. Actuators 2021, 10, x FOR PEER REVIEW 11 of 19

3.2. Harmonic Displacement Analysis According to Equations (8) and (9), the harmonic displacement curves can be simulated as shown in Figure 9, where Figure 9a–d represents output harmonic displacement in both directions, total harmonic displacement, harmonic displacements that vary with time and voltage, comparison of total displacement and unidirectional displacement, respectively. In addition, the displacements at different moments in a certain period are also analyzed, which is presented in Figure 10. Here, a few special moments are selected for analysis, and T represents periodic time. From Figures 9 and 10, it can be observed that: (1) The harmonic displacements in both directions vary with time by sinusoidal curves; the phase difference is 90 degrees, which is the same as the input signal. The total displacement is approximately sinusoidal with time, and the frequency is the same as Actuators 2021, 10, 4the input signal, but the phase is different. Herein, the total displacement is the superpo- 11 of 18 sition of the displacements in directions 1 and 2, and the phase difference of the displacement signals in the two directions is 90 degrees. Therefore, there is no case of both being 0, so the(2) totalUnder displacement the driving is always voltage greater of 150 than V, the0 in maximumFigure 9b. harmonic displacement in one (2) Under the drivingdirection voltage is 355.1 of µ150m, V, whereas the maximum the maximum harmonic total displacement displacement in one is 427.6 µm. The direction is 355.1 μmaximumm, whereas total the maximum displacement total is displacement 20.4% greater is than427.6 the μm. unidirectional The maxi- maximum mum total displacementdisplacement. is 20.4% greater than the unidirectional maximum displacement. (3) As the input peak–peak voltage increases, the total harmonic displacement and its (3) As the inputsingle peak–peak direction voltage displacement increase increases, the total approximately harmonic displacement linearly. and its single direction(4) displacementIn a period, when increase t = 1/8 approximately T, the total harmonic linearly. displacement is the maximum. When (4) In a period,t =when 1/4 T,t = the 1/8 harmonic T, the total displacement harmonic displacement in direction 1 is is the the maximum. maximum, whereas the When t = 1/4 T, thelargest harmonic value displacement of direction in 2 occursdirection at 1 t =is 0the or maximum, T. What is more,whereas the the minimum total largest value of directiondisplacement 2 occurs occurs at t at= 0 t =or 5/8 T. What T and is the more, minimum the minimum value is 72.9 totalµ m.displacement occurs(5) atWhen t = 5/8 t =T 1/2and Tthe and minimum t = 3/4 T, value the system is 72.9 outputsμm. only one-directional displacement. (5) When t =1/2The T singleand t directional= 3/4 T, the displacement system outputs of direction only one-directional 1 occurs at t =displace- 1/2 T, with a value at ment. The single directional176.2 µm. displacement While the single of directionaldirection 1 displacementoccurs at t =1/2 of directionT, with a 2value occurs at t = 3/4 T, at 176.2 μm. While itsthe value single is directional still 176.2 µ m.displacement Meanwhile, of when direction the total 2 occurs displacements at t =3/4 T, take its the maximum value is still 176.2 valueμm. Meanwhile, and the minimum when value,the total the correspondingdisplacements displacements take the maximum in two directions are value and the minimumequal. value, the corresponding displacements in two directions are equal. (6) When the driving voltage is greater than 90 V, the harmonic displacement in all (6) When the drivingdirections voltage is greater is greater than 200 thanµm, 90 whichV, thecan harmonic drive the displacement proposed piezoelectric in all actuator directions is greaterto than work 200 normally. μm, which can drive the proposed piezoelectric actuator to work normally.

400 500 Direction 1 Direction 2 400 300

300 200 / μm / μm h h δ δ 200

100 100

Actuators 2021, 10, x FOR PEER REVIEW 0 0 12 of 19 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 t / s t / s (a) (b)

500 Total displacement Direction 1 displacement 400

300 /μm h δ

200

100 50 75 100 125 150 U / V p-p (c) (d)

Figure 9. CalculationFigure 9. Calculation results of harmonic results of displacement. harmonic displacement. (a) Output ( harmonica) Output displacementharmonic displacement in both directions; in both (b) Total harmonic displacement;directions; (b (c)) Total Harmonic harmonic displacements displacement; that vary(c) Harmonic with time displacements and voltage; (d that) Comparison vary with oftime total and displacement voltage; (d) Comparison of total displacement and unidirectional displacement. and unidirectional displacement. δ δ δ h2 h2 h2 μm μm μm δ h δ h δ h

δ δ δ h1 h1 h1 (a) (b) (c) δ δ h2 h2 μm μm

δ h δ h δ δ h1 h1 (d) (e) δ δ h2 h2 μm μm δ h

δ h

δ δ h1 h1 (f) (g) Figure 10. Harmonic displacements at different periods. (a) t = 0; (b) t = 𝑇; (c) t = 𝑇; (d) t = 𝑇; (e) t = 𝑇; (f) t = 𝑇; (g) t = T.

To investigate the influence of system parameters on harmonic displacement, the parameters’ piezoelectric strain constant d33, the angle between the flexible arm and the

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500 Total displacement Direction 1 displacement 400

300 /μm h δ

200

100 50 75 100 125 150 U / V p-p (c) (d)

Actuators 2021, 10, 4 Figure 9. Calculation results of harmonic displacement. (a) Output harmonic displacement in12 both of 18 directions; (b) Total harmonic displacement; (c) Harmonic displacements that vary with time and voltage; (d) Comparison of total displacement and unidirectional displacement.

δ δ δ h2 h2 h2 μm μm μm δ h δ h δ h

δ δ δ h1 h1 h1 (a) (b) (c) δ δ h2 h2 μm μm

δ h δ h δ δ h1 h1 (d) (e) δ δ h2 h2 μm μm δ h

δ h

δ δ h1 h1 (f) (g) Figure 10. Harmonic displacements at different periods. (a) t = 0; (b) t = 𝑇1; (c) t = 𝑇1; (d) t = 𝑇1; (e) t = 𝑇;5 (f) t = 𝑇;3 (g) Figure 10. Harmonic displacements at different periods. (a) t = 0; (b) t = T;(c) t = T;(d) t = T;(e) t= T;(f) t= T; t = T. 8 4 2 8 4 (g) t = T.

ToTo investigate the influenceinfluence of systemsystem parametersparameters onon harmonicharmonic displacement,displacement, thethe 33 parameters’ piezoelectricpiezoelectric strain constant d33,, the the angle angle between the flexibleflexible arm andand thethe horizontal θ and the length of harmonic rod l1 and l2 are selected as target parameters. The analysis results are shown in Figure 11. From the figure, we see that:

(1) The harmonic displacements change with the variation of d33 and l2 by the same rule. As d33 and l2 increase, the output displacements increase synchronously. (2) The harmonic displacements are inversely proportional to the change of θ and l1. Among them, the angle between the ﬂexible arm and the horizontal of rotation has enormous inﬂuence on the output displacement. Therefore, it is extremely important to set these parameters properly to improve the performance of the low-frequency harmonic rotary piezoelectric actuator. To verify the correctness of the theoretical calculation of harmonic displacement, the harmonic displacement is tested experimentally. By using an LVDT inductance micrometer, the harmonic displacements in two directions are easily measured. Figure 12 shows the test results of harmonic displacement for the proposed piezoelectric actuator. The results indicate that: Actuators 2021, 10, 4 13 of 18 Actuators 2021, 10, x FOR PEER REVIEW 13 of 19

horizontal θ and(1) theThe length measured of harmonic value of rod harmonic l1 and displacementl2 are selected showsas target a nonlinear parameters. trend with peak– The analysis resultspeak are shown voltage in change. Figure This11. From phenomenon the figure, is causedwe see bythat: the hysteresis effect of piezoelec- (1) The harmonictricity displacements of the piezoelectric change material.with the variation of d33 and l2 by the same rule. As d33 and(2) l2 increase,The maximum the output error displace betweenments theoretical increase calculations synchronously. and experimental tests occurs (2) The harmonicat adisplacements driving voltage are of inve 60 Vrsely and proportional is 7.0%. When to the drivingchange of voltage θ and isl1.larger than 80 Among them, the V,angle the errorsbetween are the stable flexible and lessarm than and 3%. the Hence,horizontal the reliabilityof rotation and has accuracy of the enormous influencetheoretical on the output model displacement. of the piezoelectric actuator are further veriﬁed. Therefore, it isTherefore, extremely to important minimize to the se experimentalt these parameters error, aproperly relatively to large improve driving the voltage should performance ofbe the selected low-frequency as far as possibleharmonic when rotary the piezoelectric piezoelectric actuator. actuator is driven.

500 600 d =600pm/V θ=6° 33 d =650pm/V 500 θ=8° 400 33 θ=10° d =700pm/V 33 400 300 / μm / μm h h δ δ 300 200 200

100 100 50 75 100 125 150 50 75 100 125 150 U / V U / V p-p p-p (a) (b)

500 500 l =25mm l =30mm 1 2 l =30mm l =35mm 400 1 400 2 l =35mm l =40mm 1 2 300 300 / μm /μm h h δ δ

200 200

100 100 50 75 100 125 150 50 75 100 125 150 U / V U / V p-p p-p (c) (d)

Figure 11. The influence of parameters on total harmonic displacement. (a) d33 changes; (b) θ Figure 11. The influence of parameters on total harmonic displacement. (a) d33 changes; (b) θ changes; (c) l1 changes; (d) l2 changes. changes; (c) l1 changes; (d) l2 changes.

To verify 3.3.the correctness Output Speed of Test the oftheoretical the Prototype calculation of harmonic displacement, the harmonic displacementIn this is article,tested theexperimentally. average measurement By using an method LVDT isinductance adopted to microm- measure the rotating eter, the harmonicspeed displacements of the prototype. in two Under directions the conditionare easily ofmeasured. stable operation Figure 12 of shows the prototype, the the test resultsaverage of harmonic value ofdisplacement the measured for speed the Nproposedtimes within piezoelectric a certain actuator. time interval The is the average results indicatespeed that: of the system, and the expression can be written as: (1) The measured value of harmonic displacement shows a nonlinear trend with peak–peak voltage change. This phenomenon is caused by1 theN hysteresis effect of piezo- n = n , (11) electricity of the piezoelectric material. av ∑ i N i=1 (2) The maximum error between theoretical calculations and experimental tests occurs at a drivingwhere voltageni is theof 60 rotational V and is velocity 7.0%. When measured the driving at the ivoltage-th time. is larger than 80 V, the errors are stableAn experimentaland less than prototype3%. Hence, of the the reliability low-frequency and accuracy harmonic of the rotating theo- piezoelectric retical model ofactuator the piezoelectri is shownc in actuator Figure 13 are, where further Figure verified. 13a,b represents components and assembly of Therefore,the to machined minimize prototype. the experimental Its experimental error, a testrelatively scheme large is presented driving in voltage Figure 14. Driven by should be selecteda continuous as far as voltagepossible signal, when thethe prototypepiezoelectric can actuator run stably. is driven. Figure 15 shows the states of the piezoelectric actuator as it moves to different positions. Using the average measurement method, the rotational speed test results are shown in Figure 16. The results indicate that:

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(1) When the frequency is constant, the piezoelectric actuator starts to rotate when the driving voltage reaches 90 V. With the increase of the driving voltage, the rotating speed ﬁrst increases and then tends to a stable value. When the driving voltage is 150 V, the maximum rotating speed at 1 Hz, 3 Hz and 5 Hz can reach 1.88 rpm/min, 2.61 rpm/min and 5.45 rpm/min, respectively. (2) When the driving voltage is constant, the output speed presents a nonlinear increase trend with the increase of driving frequency. At the same time, the experimental Actuators 2021, 10, x FOR PEER REVIEW results show that the piezoelectric actuator operates smoothly under different14 of 19 driving

frequencies. m μ / h d Errors / % Displacement

(a)

(b)

(c)

Figure 12. 12. ExperimentalExperimental results results of harmonic of harmonic displaceme displacementnt for the for proposed the proposed piezoelectric piezoelectric actuator. actuator. (a) (a) Direction 1 test results; (b) Direction 2 test results; (c) The error between the experimental result Direction 1 test results; (b) Direction 2 test results; (c) The error between the experimental result and and the theoretical calculation. the theoretical calculation. 3.3. Output Speed Test of the Prototype In this article, the average measurement method is adopted to measure the rotating speed of the prototype. Under the condition of stable operation of the prototype, the average value of the measured speed N times within a certain time interval is the average speed of the system, and the expression can be written as:

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1 N N nn= , = 1 av i (11) nnav i , N i=1 (11) N i=1 where ni is the rotational velocity measured at the i-th time. where ni is the rotationalAn experimental velocity measuredprototype at of the the i-th low-fr time.equency harmonic rotating piezoelectric An experimentalactuator is shownprototype in Figure of the 13, low-fr whereequency Figure harmonic13a,b represents rotating components piezoelectric and assembly actuator is shownof the machinedin Figure 13,prototype. where Figure Its experimental 13a,b represents test scheme components is presented and assembly in Figure 14. Driv- of the machineden by prototype. a continuous Its experimental voltage signal, test the scheme prototype is presented can run in stably. Figure Figure14. Driv- 15 shows the en by a continuousstates of thevoltage piezoelectric signal, the actuator prototype as it canmo vesrun tostably. different Figure positions. 15 shows Using the the average states of themeasurement piezoelectric method,actuator theas it rotational moves to speeddifferent test positions. results are Using shown the in average Figure 16. The re- measurementsults method, indicate the that: rotational speed test results are shown in Figure 16. The results indicate that:(1) When the frequency is constant, the piezoelectric actuator starts to rotate when (1) Whenthe the driving frequency voltage is constant,reaches 90 the V. piez Withoelectric the increase actuator of startsthe driving to rotate voltage, when the rotating the driving speedvoltage first reaches increases 90 V. and With then the tends increase to a stableof the value. driving When voltage, the driving the rotating voltage is 150 V, Actuators 2021, 10, 4 speed first increasesthe maximum and then rotating tends speedto a stable at 1 value.Hz, 3 When Hz and the 5driving Hz can voltage reach is1.88 150 rpm/min,V, 15 2.61 of 18 the maximumrpm/min rotating and speed 5.45 rpm/min, at 1 Hz, respectively.3 Hz and 5 Hz can reach 1.88 rpm/min, 2.61 rpm/min and 5.45(2) rpm/min, When the respectively. driving voltage is constant, the output speed presents a nonlinear in- (2) Whencrease the drivingtrend with voltage the increase is constant, of driving the output frequency. speed presentsAt the same a nonlinear time, the in- experimental The research results verify the feasibility of the proposed low-frequency harmonic crease trendresults with the show increase that theof drivingpiezoelectric frequency. actuator At theoperates same time,smoothly the experimental under different driving piezoelectric actuator. Meanwhile, the investigation in this paper lays a theoretical and results showfrequencies. that the piezoelectric actuator operates smoothly under different driving frequencies.experimental foundation for the improvement of the piezoelectric actuator.

(a) (b) (a) (b) Figure 13. Experimental prototype of the piezoelectric actuator. (a) Prototype components; (b) Figure 13. ExperimentalFigure 13. ExperimentalPrototype prototype assembly. ofprototype the piezoelectric of the piezoelectric actuator. (a )actuator. Prototype (a) components; Prototype components; (b) Prototype (b) assembly. Prototype assembly.

Actuators 2021, 10, x FOR PEER REVIEW 16 of 19

Figure 14. Rotational speed test scheme of th thee proposed piezoelectric actuator. Figure 14. Rotational speed test scheme of the proposed piezoelectric actuator.

FigureFigure 15. PrototypePrototype experimental experimental diagram diagram at atdifferent different rotation rotation positions. positions. (a) Initial (a) Initial position; position; (b) (1/4b) 1/4circle circle position; position; (c) 1/2 (c) 1/2circle circle position; position; (d) 3/4 (d) circle 3/4 circle position. position.

(a) 6 5 4 3 2 1 0 12345 Driving frequency f / Hz

(b) Figure 16. Rotational speed test results of the piezoelectric actuator. (a) The change of rotation speed with driving voltage at different frequencies; (b) The speed varies with frequency.

The research results verify the feasibility of the proposed low-frequency harmonic piezoelectric actuator. Meanwhile, the investigation in this paper lays a theoretical and experimental foundation for the improvement of the piezoelectric actuator. Compared with the harmonic piezoelectric motor developed by Barth [21], the proposed harmonic piezoelectric actuator in this paper just uses two 5 × 5 × 30 mm pie-

Actuators 2021, 10, x FOR PEER REVIEW 16 of 19

Actuators 2021, 10, 4 16 of 18 Figure 15. Prototype experimental diagram at different rotation positions. (a) Initial position; (b) 1/4 circle position; (c) 1/2 circle position; (d) 3/4 circle position.

(a) 6 5 4 3 2 1 0 12345 Driving frequency f / Hz

(b)

Figure 16. Rotational speed testtest resultsresults ofof thethe piezoelectricpiezoelectric actuator. actuator. ( a()a) The The change change of of rotation rotation speed withspeed driving with driving voltage voltage at different at different frequencies; frequencies; (b) The (b speed) The variesspeed withvaries frequency. with frequency.

TheCompared research with results the harmonicverify the piezoelectricfeasibility of motorthe proposed developed low-frequency by Barth [21 ],harmonic the pro- posedpiezoelectric harmonic actuator. piezoelectric Meanwhile, actuator the in investigation this paper just in uses this two paper 5 × lays5 × a30 theoretical mm piezoelec- and tricexperimental stacks as driving foundation elements. for the Barth’s improv harmonicement of piezodrive the piezoelectric motor actuatesactuator. with eight 5 × 5 × 50Compared mm piezoelectric with the stacks. harmonic In addition, piezoelect the harmonicric motor gear developed used for reductionby Barth in[21], Barth’s the motorproposed has harmonic high precision piezoelectric and is difficultactuator toin manufacture.this paper just In uses Ho’s two research 5 × 5 × [305], mm a piezo- pie- electric screw-driven motor operating in two shear vibration modes was developed. The stator structure of the screw-driven motor is formed by obliquely placing four piezoelectric actuators in a four-side hollowed cubic elastic body. In Ho’s motor, four piezoelectric stacks, 2 mm × 3 mm × 9 mm in size, are used as the driving source. The translational velocity of the motor is 1.174 mm/s with a 16 g mechanical load under 16 V peak voltage driving. Since the driving signals of the four piezoelectric stacks are different, it is difficult to drive and control the motors. At the same time, compared with the piezoelectric actuator proposed by the authors, the piezoelectric motor consumes more energy. In Shi’s piezoelectric ultrasonic motor [6], a ring type composite stator with four driving feet uniformly arranged in the inner circumference of the ring stator is used to generate a specific combination of the A(0,5) axial bending modes and R(0,2) radial bending modes. The motor’s structure is relatively complex, and its maximum speed is 55 rpm/min. Since the ultrasonic motor is driven by resonance modes, its speed is not easy to adjust. The low-frequency piezoelectric actuator proposed by the authors not only reaches an ultralow speed, but also easily adjusts the speed by changing the driving frequency. In addition, Xing’s low-frequency piezoelectric motor [19] combines piezoelectric driving and magnetic modulation together and transforms the reciprocating swing of the stator into step running of the rotor via the intermittent magnetic clamping between the rotor and stator. At a driving frequency of 3 Hz, it can output an ultralow speed of Actuators 2021, 10, 4 17 of 18

0.66 rpm/min. Therefore, the motor has high innovation. Although the motor can output ultralow speed, the motor also has disadvantages; it is more complicated to drive and control because of its electromagnetic modulation. However, in the piezoelectric actuator proposed by the authors, only two piezoelectric stacks are used as the driving source. Therefore, its drive and control circuit is simple and easier to use. Furthermore, Qiu’s non-contact rotary piezoelectric motor [20] uses a piezoelectric torsional vibrator and giant electrorheological fluid to generate rotational motion. At an excitation frequency of 118 Hz and electric field strength of 2.5 kV/mm, the motor generates a rotational speed of 6.28 rpm/min. This kind of motor introduces the giant electrorheological fluid into the piezoelectric motor, which has good innovation, but the structure and driving of the motor are too complicated to be of practical use. On the contrary, the piezoelectric actuator presented by the authors is simple in structure and easy to use. Therefore, compared to other piezoelectric actuators, the piezoelectric actuator presented in this paper has the characteristics of reduced energy consumption, a simple structure, low manufacturing cost and is easy to control. As a result, the low-frequency harmonic rotating piezoelectric actuator has high application value.

4. Conclusions In this paper, a low-frequency harmonic rotating piezoelectric actuator based on the harmonic friction principle is proposed. The operating principle of the low-frequency harmonic rotating piezoelectric actuator is presented. Using piezoelectricity and mathematics principle, the actuator’s output characteristic equations are deduced. With experimental equipment, the output characteristics of the piezoelectric actuators were tested. Results show that: (1) The harmonic displacements produced by piezoelectric stacks can drive the proposed piezoelectric actuator to work normally when the driving voltage is larger than 90 V. (2) The maximum total harmonic displacement of the piezoelectric actuator reaches 427.6 µm under the driving voltage of 150 V. (3) The rotational speed of the piezoelectric actuator is 5.45 rpm/min under the driving voltage of 150 V at a frequency of 5 Hz. (4) The error between the measured and calculated values of the harmonic displacement is less than 7%, which veriﬁes the reliability of the theoretical model. These results lay the theoretical and experimental foundation for the improvement of the prototype of the low-frequency harmonic rotating piezoelectric actuator.

Author Contributions: Conceptualization, C.L.; investigation, K.L. and Y.T.; validation, J.F.; writing— original draft, C.L.; writing—review and editing, W.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. 51905228, 51705217], China Postdoctoral Science Foundation [grant no. 2018M640515, 2019M661736], Natural Science Foundation of the Jiangsu Higher Education Institutions [grant no. 18KJB460008] and Science and Technology Innovation Project of Jiangsu University of Science and Technology for Youth [grant no. 1022921801]. Acknowledgments: Thanks for the postdoctoral position and scientific research fund support provided by Suzhou Mingzhi Technology Co., Ltd. Conflicts of Interest: The authors declare no conflict of interest. Actuators 2021, 10, 4 18 of 18

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