A Novel Low-Frequency Piezoelectric Motor Modulated by an Electromagnetic Field

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Article A Novel Low-Frequency Piezoelectric Motor Modulated by an Electromagnetic Field

Jichun Xing * and Yong Qin

School of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China; [email protected] * Correspondence: [email protected]

Received: 28 August 2020; Accepted: 11 September 2020; Published: 13 September 2020

Abstract: For expanding the driving mode of the piezoelectric motor, a novel piezoelectric motor modulated by a magnetic ﬁeld is proposed. This driving system combines piezoelectric driving and magnetic modulation together and can transform the reciprocating swing of the stator into step running of the rotor via the intermittent magnetic clamping between the rotor and stator. For investigating the inherent character of dynamics, the dynamic equations of key parts of the driving system are established. The natural frequencies and mode functions of the driving system are solved. A prototype was fabricated to prove the dynamic analysis and measure the output characteristic. The results show that the nature of the frequency measured from the test is coincident with theoretical analysis. In addition, by applying the driving frequency of 3 Hz, the voltage of the modulating signal of 4.5 V, the phase diﬀerence α between driving signal and modulating signal of 30◦, the ideal outputs are 0.1046 rad/min for velocity and 0.405 Nmm for torque.

Keywords: electromagnetic modulation; piezoelectric motor; modal function; output characteristic

1. Introduction With the continuous development of modern technology, the requirements for driving devices in intelligent manufacturing, optical fiber docking, biomedicine, and focusing systems are increasing. Piezoelectric motors are widely used in these fields because of their high displacement resolution, simple structure, and fast response [1,2]. For example, piezoelectric microrobots can be used in surgery, human digestive system monitoring and drug delivery [3,4]. Although many piezoelectric motors have been designed for various purposes, they can be roughly divided into three categories in terms of the driving principle: ultrasonic motors, inchworm piezoelectric motors and inertia piezoelectric motors [5]. The ultrasonic piezoelectric motor is the earliest and most widely used piezoelectric motor. According to the different driving signal waveforms, it can be divided into traveling wave ultrasonic motors and standing wave ultrasonic motors. Markus K. et al. [6], proposed a new type of traveling wave ultrasonic motor. The double input sliding mode controller, which uses frequency and phase difference as instructions, switches the control domain without interrupting the speed, reduces the control amount of phase difference, and does not produce overshoot. Chen W. et al. [7], proposed a radial bending ring traveling wave ultrasonic motor. The 40 slots on the outer surface of the stator ring are alternately filled with 20 piezoelectric stacks and 20 elastic blocks for generating traveling waves. The maximum speed of the motor can reach 146 r/min and the maximum torque is 1 Nm. Izuhara S. et al. [8], proposed a piezoelectric ultrasonic linear motor with hollow rectangular stator. The motor is a standing wave ultrasonic motor, which can directly drive its internal load through the stator. The structure saves the mechanism space, and has the advantages of fast response speed and high resolution.

Actuators 2020, 9, 85; doi:10.3390/act9030085 www.mdpi.com/journal/actuators Actuators 2020, 9, 85 2 of 17

The inchworm piezoelectric motor is a precision driving device that imitates the movement state of the inchworm and uses piezoelectric actuators. Mohammad T. et al. [9], proposed a two-axis inchworm piezoelectric motor, which consists of two integrated stages. Each stage provides continuous motion on one axis. It can be applied to an integrated precision positioning system with large stroke and multi-axis linkage. Xing J. et al. [10], proposed a new type of rotary inchworm piezoelectric motor. Due to the special structure of the stator, the driving mechanism can produce angular displacement during operation. The inchworm motor has the advantages of realizing adjustable clamping devices, and a simple structure of single parts. Dong H. et al. [11], proposed a linear inchworm piezoelectric motor, which consists of a driving mechanism and two clamping mechanisms. The overall structure of the motor is relatively simple, the power consumption is low, and the maximum operating speed is 0.72 mm/s. An inertial piezoelectric motor is a motor that uses inertial impact force or torque to obtain linear or rotary motion. By varying the applied voltage, the actuators can take any position and therefore achieve the high motion resolution that is typical for piezo ceramics. Andrius C. et al. [12], designed a new type of inertial piezoelectric motor, consisting of a rectangular bronze stator, a clamping frame, a piezoelectric plate, and a conical rotor. The motor has a simple structure, good motion stability, and high space utilization. Wang L. et al. [13], proposed a new type of piezoelectric inertial rotary motor, which can be used to drive underwater micro-robots. The motor is composed of two rotors and a disc stator. It has a simple structure, small size, and good waterproof performance. The emergence of this motor has opened up a new way to develop advanced and flexible multifunctional robots. Mazeika D. et al. [14], designed a 3-degree-of-freedom high-resolution inertial piezoelectric motor. The motor can work in two vibration modes, namely the bending vibration mode of the outer shaft of the disc and the radial vibration mode of the disc. The bending vibration mode is used to rotate the rotor around the x and y-axis. The radial vibration mode is used to rotate the rotor around the z-axis. Therefore, 3 degrees of freedom movement of the motor is realized. In summary, although piezoelectric motors have many advantages compared with traditional electromagnetic motors, the driving pattern of these piezoelectric motors is based on rigid contact between the fixed rotors to transmit power. The long-term effects of friction and preload force will cause damage to the runner or stator and even lead to the failure of the operation. Therefore, controlling the preload to reduce the friction between the contact surfaces is a key issue [15,16]. Introducing electromagnetic force into the piezoelectric motor improves not only the controllability of the clamping force or preload but also the friction and wear between the stator and rotor are effectively reduced. The proposed piezoelectric motor expands the topology structure of piezoelectric motors, and combining the piezoelectric drive and electromagnetic modulation into one motor would enrich the driving principle of the piezoelectric motor. In this paper, we mainly established the dynamics equations of key parts and solved natural frequencies and mode functions of driving system. We also conduct an experiment to measure the output characteristic of a prototype.

2. Motor Structure and Working Principle

2.1. Motor Structure Composition The proposed piezoelectric motor is mainly composed of the driving part, the electromagnetic modulation part and the moving part. The driving part includes two piezoelectric stacks, a ﬂexible amplifying mechanism and an intermediate shaft. The electromagnetic modulation part includes an electromagnet, a disc-shaped slider, and a spring. The rotor shaft and a bearing belong to the moving part of the motor, as shown in Figure1. Actuators 2020, 9, x FOR PEER REVIEW 3 of 17 Actuators 2020, 9, x FOR PEER REVIEW 3 of 17 Actuators 2020, 9, 85 3 of 17 Bearing end cover Actuators 2020, 9, x FOR PEER REVIEW Bearing end cover 3 of 17 Spring Spring Bearing end cover Bearing Bearing Disc-shapedSpring Moving Disc-shaped Upper cover Moving slider Upper cover part slider Bearing part Disc-shaped UpperRotor shaftcover Moving slider Rotor shaft part Set bolt Case Set bolt Case Electromagnet Rotor shaft Electromagnet Lower Electromagnetic Lower Electromagnetic Set bolt Case cover modulation part cover modulation part Electromagnet Lower Electromagnetic Intermediatemodulation part coverIntermediate Pedestal shaft Pedestal shaft Driving Piezoelectric Driving PiezoelectricIntermediate part Pedestal stack part stackshaft Flexible amplification Driving Flexible amplification Piezoelectricmechanism part mechanismstack Flexible amplification mechanism

Figure 1. Motor structure. Figure 1. MotorMotor structure. structure. In one part, two piezoelectric stacksFigure cooperate 1. Motor withstructure. the flexible amplification mechanism for InIn one part, two piezoelectric stacks cooperate with the flexible flexible amplification amplification mechanism for outputting swing motion, and then an intermediate shaft is assembled on the amplification outputting swingswing motion, motion, and and then then an intermediate an intermed shaftiate isshaft assembled is assembled on the amplification on the amplification mechanism mechanismIn one aspart, the two driving piezoelectric part of stacksthe motor. cooperate In another with the part, flexible the electromagnetamplification mechanismmatched with for mechanismas the driving as partthe ofdriving the motor. part Inof anotherthe motor. part, In the another electromagnet part, the matched electromagnet with intermediate matched shaft,with intermediateoutputting swing shaft, spring,motion, andand disc-shaped then an intermed slider isiate combined shaft isas assembledthe electromagnetic on the amplification modulation intermediatespring, and disc-shaped shaft, spring, slider and is combineddisc-shaped as theslider electromagnetic is combined modulationas the electromagnetic part of the motor. modulation Finally, partmechanism of the motor. as the Finally, driving the part drive of part,the motor.the electromagnetic In another part, modulation the electromagnet part, and the matched moving withpart partthe drive of the part, motor. the Finally, electromagnetic the drive modulation part, the electromagnetic part, and the moving modulation part arepart, assembled and the moving through part the areintermediate assembled shaft, through spring, the shell, and disc-shapedend cover, pedestal, slider is etc. combined to form asa complete the electromagnetic motor. modulation are assembled through the shell, end cover, pedestal, etc. to form a complete motor. partshell, of end the cover,motor. pedestal, Finally, etc.the drive to form part, a complete the electromagnetic motor. modulation part, and the moving part 2.2.are Workingassembled Principle through the shell, end cover, pedestal, etc. to form a complete motor. 2.2. Working Working Principle Principle In the working step, the electromagnet attracts the disc-shaped slider and then rotates with the 2.2. WorkingInIn the working Principle step, the electromagnet attracts the the disc-shaped disc-shaped slider slider and and then then rotates with the driving part. Considering that it takes a certain time for the disc-shaped slider to be attracted down driving part. Considering thatthat it it takes takes a a certain certain time time for for the the disc-shaped disc-shaped slider slider to be to attracted be attracted down down after after Inthe the electromagnet working step, is the energized, electromagnet the piezoele attractsctric the stacksdisc-shaped and the slider electromagnetic and then rotates modulation with the afterthe electromagnet the electromagnet is energized, is energized, the piezoelectric the piezoele stacksctric and stacks the electromagnetic and the electromagnetic modulation modulation mechanism mechanismdriving part. are Considering driven by thattwo itsquare takes awave certain signal times withfor the the disc-shaped same frequency slider butto be different attracted phases down mechanismare driven byare two driven square by two wave square signals wave with signal the sames with frequency the same butfrequency different but phases different separately. phases separately.after the electromagnet From Figure is2, energized, Ua is the thepeak piezoele value ctricof the stacks driving and voltagethe electromagnetic for the electromagnetic modulation separately.From Figure From2, U aFigureis the peak2, U valuea is the of peak the driving value voltageof the driving for the electromagneticvoltage for the modulationelectromagnetic part, modulationmechanism part,are driven while byUb istwo the square peak valuewave ofsignal the sdriving with the voltage same forfrequency the piezoelectric but different stacks, phases T is modulationwhile Ub is the part, peak while value Ub of is the the driving peak value voltage of forthe thedriving piezoelectric voltage stacks,for theT piezoelectricis one cycle of stacks, the motor T is oneseparately. cycle of Fromthe motor Figure operation, 2, Ua is andthe Δpeakt is thevalue driving of the time driving difference voltage of thefor twothe drivingelectromagnetic signals. oneoperation, cycle of and the∆ motort is the operation, driving time and di Δfferencet is theof driving the two time driving difference signals. of Thisthe two type driving of piezoelectric signals. Thismodulation type of piezoelectricpart, while U motorb is the will peak present value a of stepping the driving rotary voltage motion, for as the shown piezoelectric in Figure stacks, 3. T is Thismotor type will of present piezoelectric a stepping motor rotary will present motion, a as stepping shownin rotary Figure motion,3. as shown in Figure 3. one cycle of the motor operation, and Δt is the driving time difference of the two driving signals. U This type of piezoelectric motor willU present a stepping rotary motion, as shown in Figure 3. Ub UUb Ua Ua Ub o Uoa Δt T/2 T 3T/2 2T Δt 2T t T/2 T 3T/2 t o Δ Figure 2. Drive signal. t FigureT/2 2. T DriveDrive3T/ signal. signal.2 2T t Disc-shaped Rotor Disc-shaped FigureRotor 2. Drive signal. slider slider Spring Electromagnetic Spring ElectromagneticDisc-shaped Rotor modulation modulationslider θ mechanism Spring θ Electromagneticmechanism modulationFlexible Flexible θ amplificationmechanism amplification θ mechanism θ mechanismFlexible amplification (a) (b)θ (c) mechanism (a) (b) (c) FigureFigure 3. 3. WorkingWorking principle. principle. (a (a) )Attracting Attracting state; state; (b (b) )Clamping Clamping and and rotating; rotating; (c (c) )Restoring Restoring state. state. Figure 3. Working principle. (a) Attracting(a) state; (b) (b)Clamping and rotating; (c) (c ) Restoring state. Figure 3. Working principle. (a) Attracting state; (b) Clamping and rotating; (c) Restoring state. ActuatorsActuators 20202020,, 99,, 85x FOR PEER REVIEW 44 ofof 1717

The details of the working principle are as follows: The details of the working principle are as follows: • A square wave signal with peak value Ua of voltage is supplied to the electromagnetic Amodulation square wave mechanism. signal with peakSo, the value disc-shapedUa of voltage sl isider supplied is attracted to the electromagnetic under the action modulation of the • mechanism.electromagnetic So, theforce disc-shaped and keeps slider in contact is attracted with underthe upper the action surface of theof the electromagnetic electromagnet, force as andshown keeps in Figure in contact 3a. with the upper surface of the electromagnet, as shown in Figure3a. • AfterAfter thethe timetime∆ Δtt,, another another square square wave wave signal signal with with peak peak value valueU aUofa of voltage, voltage, as as shown shown in in Figure Figure2, • is2, appliedis applied to the to piezoelectric the piezoelectric stacks. Twostacks. piezoelectric Two piezoelectric stacks will stacks produce will elongation produce deformation, elongation anddeformation, then drive and the then electromagnetic drive the electromagnetic modulation part modulation and the rotor part shaft and rotation the rotor angle shaftθ throughrotation theangle flexible θ through amplification the flexible mechanism, amplification as shown mechanism, in Figure as3 shownb. in Figure 3b. • WhenWhen thethe highhigh levellevel ofof thethe drivingdriving signalsignal UUaa disappears,disappears, asas shownshown inin FigureFigure2 2,, the the disc-shaped disc-shaped • sliderslider would be be separated separated from from the the electromagne electromagnett under under the theaction action of the of spring the spring force. force.After Afteranother another period period of time of Δt time, the driving∆t, the signal driving Ub signal of the Upiezoelectricb of the piezoelectric stacks is at stackslow level. is at Then, low level.two piezoelectricThen, two piezoelectricstacks restorestacks to their restore original to length their originaland the electromagnetic length and the electromagneticmodulation part modulationreturns to its part original returns position to its original with the position elastic with restoring the elastic force restoring of the forceflexible of theamplifying flexible amplifyingmechanism, mechanism, so the rotor sorotates the rotor an angle rotates θ relative an angle toθ therelative electromagne to the electromagnet,t, as shown in asFigure shown 3c. inThe Figure time3 differentc. The time Δt of di eachfferent cycle∆t of eachtwo driving cycle of signals two driving prevents signals that the prevents rotor shaft that thereverses rotor shaftdue to reverses inertia duewhen to the inertia disc-shaped when the slider disc-shaped is separated slider from is separated the electromagnet. from the electromagnet.Repeating the Repeatingabove steps the can above achieve steps continuous can achieve step continuous rotation of step the rotation motor. of the motor.

3.3. Analysis for Free Vibration ForFor solvingsolving the the natural natural frequency frequency and and modal modal function, function, first, wefirst, need we to need establish to establish the free vibrationthe free dynamicvibration equationdynamic ofequation the proposed of the proposed piezoelectric piezoelectric motor. Themotor. key The part key of part the piezoelectricof the piezoelectric motor mainlymotor mainly includes includes a driving a driving part, anpart, electromagnetic an electromagnetic modulation modulation part, andpart, aand moving a moving part. part. For theFor establishmentthe establishment of the of free the vibration free vibration dynamics dynamics model, model, we consider we consider the system the condition system condition of clamping of theclamping rotor withthe rotor the electromagnetic with the electromagnetic force. Therefore, force. Therefore, the electromagnetic the electrom modulationagnetic modulation part and part the movingand the moving part are part simplified are simplified into one into mass. one Meanwhile, mass. Meanwhile, the flexible the flexible amplification amplification mechanism mechanism also is simplified,also is simplified, as shown as in shown Figure 4in. According Figure 4. toAccording the di fferent to the vibration different and vibration structural and characteristics structural ofcharacteristics each component, of each free component, vibration equationsfree vibration of each equations subsystem of each need subsystem to be established need to be respectively, established andrespectively, then the boundaryand then the condition boundary and condition continuous and condition continuous can becondition utilized can to solvebe utilized the whole to solve system the freewhole vibration system characteristics. free vibration Sincecharacteristics. the structure Sinc ofe the the flexible structure amplification of the flexible mechanism amplification is center symmetrical,mechanism is here, center only symmetrical, one side is here, used only to carry one out side calculations. is used to carry out calculations.

Intermediate Mass shaft E Flexible amplification mechanism Piezoelectric B l3 stack C A D

l2 l1 FigureFigure 4.4. SimpliﬁedSimplified model of piezoelectric system forfor freefree vibration.vibration. 3.1. Free Vibration Modeling of a Piezoelectric Stack 3.1. Free Vibration Modeling of a Piezoelectric Stack The piezoelectric stack elongates or recovers in the axial direction under signal excitation, so it The piezoelectric stack elongates or recovers in the axial direction under signal excitation, so it can be simpliﬁed into a continuous system of axial vibration. The piezoelectric stack dynamics model can be simplified into a continuous system of axial vibration. The piezoelectric stack dynamics is shown in Figure5, in which δ (y, t) is the axial displacement equation of the piezoelectric stack, F is model is shown in Figure 5, in which δ (y, t) is the axial displacement equation of the piezoelectric the axial internal force between the piezoelectric sheets, and f is the axial distribution force. Let the stack, F is the axial internal force between the piezoelectric sheets, and f is the axial distribution force. density of the piezoelectric material be ρp (y), the elastic modulus is Ep (y) = c33, and the cross-sectional Let the density of the piezoelectric material be ρp (y), the elastic modulus is Ep (y) = c33, and the area is Ap. cross-sectional area is Ap. Actuators 2020, 9, x FOR PEER REVIEW 5 of 17 Actuators 2020, 9, 85 5 of 17 Actuators 2020, 9, x FOR PEER REVIEW 5 of 17 y y

lnp

lnp f dy dy f dy dy F y F y Op x Op x Figure 5. Free vibration model of piezoelectric stack. FigureFigure 5. 5. FreeFree vibration vibration model model of of piezoelectric piezoelectric stack. stack. According to the dynamic model, the dynamic differential equation of the piezoelectric stack is establishedAccordingAccording as follows: to to the the dynamic dynamic model, model, the the dynamic dynamic differ differentialential equation equation of of the the piezoelectric piezoelectric stack stack is is establishedestablished as as follows: follows: ∂2∂∂δ22δδ∂2δ ρρpAAcAp 22==c33Ap (1)(1) pp∂∂∂t∂∂2 δδ2233 p∂y2 ρ AcAty= (1) pp∂∂2233 p Let the solution of Equation (1) be: ty Let the solution of Equation (1) be: Let the solution of Equation (1) be: δδφ(y(,),ytt) ==φ(y ()()) yqtq(t) (2) (2) δφ(,)yt= ()() yqt (2) TheThe modal modal function of piezoelectric stack can be obtained as: The modal function of piezoelectric stack can beωω obtained as: φ =+ωyyyωy ()yW12 sinωω W cos (3) φ(yφ) = W=+1 sin aayy+ W2 cos (3) ()yW12 sina W cos a (3) aa = ρ acp 33 / p where a == c33/ρρp,, thethe coecoefficientsfficients WW11,, WW22 and the natural frequency ω cancan be determined by by the the ac33 / p whereboundaryboundary conditions. , the coefficients W1, W2 and the natural frequency ω can be determined by the boundary conditions. 3.2.3.2. Free Free Vibration Vibration Modeling Modeling of of Flexible Flexible Amplification Amplification Mechanism 3.2. Free Vibration Modeling of Flexible Amplification Mechanism HalfHalf of of the flexible flexible amplification amplification mechanism mechanism is is simplified simplified into into two two beams beams ( (ABAB andand BCBC)) and and a a massmassHalf ( (CDCD of).). Twothe Two flexible beams beams amplificationare are considered considered mechanism as as an an elastomer elastomer is simplified and and the the into mass mass two is is beamsconsidered considered (AB and as as rigid rigidBC) andbody. body. a massEachEach ( simplifiedsimplifiedCD). Two unit unitbeams is is connected areconnected considered to to others others as byan flexibleby elastomer flexible hinges. andhinges.The the dynamicThemass dynamic is considered model model of half as of of rigid half the flexiblebody.of the Eachflexibleamplification simplified amplification mechanism unit is mechanism connected is shown to inis Figureothersshown6 by .in The flexibleFigu flexiblere hinges.6. hingeThe isTheflexible simplified dynamic hinge into modelis asimplified combination of half ofinto ofthe aa flexiblecombinationrigid hinge amplification and of a coilrigid spring. mechanismhinge Thisand part,a iscoil shown the spring. beam in This ABFigu ispart,re taken 6. the The as beam a researchflexible AB is subject. hingetaken asis We asimplified designresearch the subject.into length a combinationWeof the design beam the ABof lengthato rigid be greater ofhinge the beamand than a 5ABcoil times to spring. be the greater height This part,than of the 5the times cross-sectional, beam the AB he isight taken so of the theas driving across-sectional, research beam subject.AB sois Wetheconsidered designdriving the asbeam anlength Euler–Bernoulli AB of is the considered beam beam AB asto [ 10anbe]. greaterEuler–Bernoulli The origin than of 5 atimes coordinate beam the [10]. he systemight The of originthe is set cross-sectional, atof point a coordinateA. x-axis so thesystemdirection driving is isset alongbeam at point theAB length isA . consideredx-axis direction direction as of an the is Euler–Bernoulli beamalong ABthe. Whenlength beam the direction beam [10].AB Theof isthe excitedorigin beam of by AB a piezoelectric .coordinate When the systembeamstack, itABis presents setis atexcited point bending byA. x-axispiezoelectric vibration. direction Let stack, the is longitudinalalong it presents the length displacement bending direction vibration. of of the the corresponding beamLet the AB longitudinal. When bending the beamdisplacementvibration AB ofis theexcited of beamthe correspondingby be ypiezoelectric1 (x1, t), the bending longitudinalstack, itvibration presents distribution of bending the forcebeam vibration. on be the y1 beam(x 1Let, t), be thethef 1 , longitudinallongitudinal the length of displacementdistributionbeam AB be forcel 1of, the the on density correspondingthe beam of the be beam f1 , bendingthe be lengthρq, thevibration of elastic beam of modulusAB the be beaml1, betheE bedensity, the y1 cross-sectional(x 1of, tthe), the beam longitudinal areabe ρ beq, theS1 , distributionelasticand the modulus longitudinal force be on E ,the shearthe beamcross-sectional force be be f1,F thes. lengtharea be of S1 ,beam and theAB longitudinalbe l1, the density shear of force the bebeam Fs. be ρq, the elastic modulus be E, the cross-sectional area be S1, and the longitudinal shear force be Fs.

y1 dx1 B y1A ∂ dx Fs 1 x1 B A Fs + dx ∂∂F 1 x F + xs1 1 s ∂ dx1 Fs x1 Fs M dx1 x1 M l1 ∂M l dx1 x1 M + dx f CD1 ∂∂Mx 1 1 M + 1 CD ∂ dx1 f1 x1

FigureFigure 6. 6. FreeFree vibration vibration model model of of driving driving beam. beam. Figure 6. Free vibration model of driving beam. Actuators 2020, 9, x FOR PEER REVIEW 6 of 17 Actuators 2020, 9, 85 6 of 17 According to the dynamic model, the dynamic differential equation of the beam AB is established as follows: According to the dynamic model, the dynamic diﬀerential equation of the beam AB is established ∂∂42yxt(,) yxt (,) as follows: EI11+=ρ S 11 0 ∂4 y (x∂∂, 42t) q 1∂2 y (x , t) (4) 1 1xt1 1 1 EI + ρqS1 = 0 (4) ∂x4 ∂t2 Let the solution of Equation (4) be: 1 Let the solution of Equation (4) be: yxt(,)=φ ()() xqt 11 111 (5) y (x , t) = φ (x )q (t) (5) The modal function of the beam AB 1can1 be obtained1 1 as:1 The modal function ofφββββ the(x ) beam=++CxCxCxCx cosAB can be sin obtained cosh as: + sinh 11 1 1 2 1 3 1 4 1 (6)

4 βρω= φ1(x1) = C1 cos βx1 + C2 sin βx1 + C3 cosh βx1 + C4sinhβx1 (6) qSEI1 / where , the coefficients C1–C4 and natural the frequency ω can be determined by p4 thewhere boundaryβ = ρconditions.qS1ω/EI, the coeﬃcients C1–C4 and natural the frequency ω can be determined by the boundaryAs the conditions. piezoelectric stack forces the beam AB to bending vibrate, the beam BC presents longitudinalAs the piezoelectric vibration and stack slight forces lateral the beamvibration.AB to Here, bending the vibrate, slight lateral the beam vibrationBC presents is negligible. longitudinal We assumevibration the and length slight of lateralthe beam vibration. BC is l2, Here, the cross-sectional the slight lateral area vibration is S2, and is the negligible. density constant We assume of the the beamlength is ofρq the. beam BC is l2, the cross-sectional area is S2, and the density constant of the beam is ρq. PointPoint BB isis set set as origin ofof aa relativerelative coordinatecoordinate system, system, and andx 2x2axis axis is is along along the the length length direction direction of ofbeam beamBC BC. As. As shown shown in in Figure Figure7, F7,2 Fis2 theis the axial axial internal internal force force acting acting on theon the beam beam cross cross section, section,f 2 is f2 the is theaxial axial distribution distribution force, force,lm is the lm lengthis the of length the beam of CDthe. Letbeam the longitudinalCD. Let the vibration longitudinal displacement vibration of displacementbeam BC is u of(x2 beam, t). BC is u (x2, t).

∂u ∂F u+ dx B F + 2dx A ∂x 2 2 ∂ 2 2 x2 l2 C D dx2 f u m 2

F2 x2 lm

Figure 7. Free vibration model of intermediate beam mass. Figure 7. Free vibration model of intermediate beam mass. According to the dynamic model, the dynamic diﬀerential equation of the beam BC is established as follows:According to the dynamic model, the dynamic differential equation of the beam BC is established as follows: ∂2u E ∂2u 2 = 2 (7) ∂t2 ρq x2 ∂∂22∂ 2 uEu22 = (7) Let the solution of Equation (7) be: ∂∂tx22ρ q 2

Let the solution of Equation (7) be: u2(x2, t) = φ2(x2)q2(t) (8) uxt(,)= φ ()() xqt The modal function of beam BC can be22 obtained 222 as: (8) The modal function of beam BC can be obtainedr as: r ρq ρq φ (x ) = B sin ω x + B cos ω x (9) 2 2 1 Eρρ2 2 E 2 φω=+qq ω 22()x BxBx 1 sin 2 2 cos 2 (9) The coeﬃcients B1, B2 and the natural frequencyEEω can be determined by the boundary conditions.

The coefficients B1, B2 and the natural frequency ω can be determined by the boundary conditions. Actuators 2020, 9, 85 7 of 17 Actuators 2020, 9, x FOR PEER REVIEW 7 of 17

3.3.3.3. Free Free Vibration Vibration Modeling Modeling of of the the Intermediate Intermediate Shaft Shaft TheThe intermediate shaftshaft isis utilizedutilized to to connect connect the the electromagnetic electromagnetic modulation modulation mechanism mechanism and and the theflexible flexible amplification amplification mechanism. mechanism. The The subsystem subsystem vibration vibration model model of theof the intermediate intermediate shaft shaft and and the theelectromagnetic electromagnetic modulation modulation mechanism mechanism is shownis shown in Figurein Figure8, the 8, the simplified simplified mass mass is arranged is arranged at anat anend end of theof the shaft. shaft. Considering Considering the intermediatethe intermediate shaft shaftDE to DE be anto idealbe an elastic ideal body,elastic when body, the when motor the is motorrunning, is running, the intermediate the intermediate shaft presents shaft presents torsional torsional vibration. vibration.

z E

l3 dz

z D x

Figure 8. Free vibration model of intermediate shaft. Figure 8. Free vibration model of intermediate shaft.

Assume that the material of the intermediate shaft is uniform, the density constant is ρz, the shear Assume that the material of the intermediate shaft is uniform, the density constant is ρz, the shear modulus is G, and the second moment of the cross section of the shaft is Iz. Point D is set as the modulus is G, and the second moment of the cross section of the shaft is Iz. Point D is set as the origin origin of a relative coordinate system, z axis direction points to simplified mass md, θ is the angular of a relative coordinate system, z axis direction points to simplified mass md, θ is the angular displacement of the shaft torsional vibration, and fT is the torque per unit length in the axis direction. displacementAccording of the to the shaft relationship torsional vibration, between and torque fT is and the angle,torque theper partialunit length differential in the axis equation direction. of the torsionalAccording vibration to the of therelationship intermediate between shaft torque is: and angle, the partial differential equation of the torsional vibration of the intermediate shaft is: 2 2 ∂ θ 2 ∂ θ 22= c (10) ∂t2 2∂z2 22 c (10) tz Let the solution of Equation (10) be: Let the solution of Equation (10) be: ( ) = ( ) ( ) θ z(,)()(), zt tφ3 z zq q3 t t (11) 33 (11) TheThe modal modal function function of of the the shaft shaft DE DE can can be be obtained obtained as as:: ωz ωz φ3(z) = N1 coszz + N2 sin   (12) 3(z ) N 1 cosc  N 2 sin c  (12) cc    p where c = G/ρz, the coefficients N1, N2 and the natural frequency ω can be determined by the cG / z wboundaryhere conditions., the coefficients N1, N2 and the natural frequency ω can be determined by the boundary conditions. 3.4. Boundary and Continuity Conditions 3.4. Boundary and Continuity Conditions After each subsystem vibration mode function of the motor is established, for solving the natural frequenciesAfter each and the subsystem coefficients vibration of the mode mode function, function the of boundary the motor conditions is established, and continues for solving conditions the naturalneed to be frequencies discussed and the the mechanical coefficients relationship of the mode between function, subsystems the boundary determined. conditions and continuesFrom conditions Figures5–8 ,need the following to be discussed relationship and can the be mechanical obtained at rel noationship load: between subsystems determined. 1. The end O of piezoelectric stack is the fixed, so the displacement is zero at x = 0. It can be From Figurep 5 to 8, the following relationship can be obtained at no load: expressed as: 1. The end Op of piezoelectric stack is the fixed,φ(0 ) so = the0 displacement is zero at x = 0. It can(13) be expressed as: (0) 0 (13) Actuators 2020, 9, 85 8 of 17

2. Regarding the beam AB, the displacement at point A of the driving beam AB is zero, and the bending moment is equal to the reverse moment of the coil spring, so: ( φ (0) = 0 1 (14) EIφ 00 (0) = kφ (0) 1 − 10 where k is the bending stiﬀness of the ﬂexible hinge. 3. At the thrust point of the piezoelectric stack, the displacement equals to the elongation of the piezoelectric stack, and the thrust at y = lnp and the shear force at x1 = lFp are equal, so: ( φ(lnp) = φ (lF ) 1 p (15) c33Apφ0(lnp) = EIφ1000 (lFp )

4. At endpoint B of beam AB, since the ﬂexible hinge provides an elastic resilience, the shear force of point B equals the elastic force of the ﬂexible hinge. It can be expressed as:

EIφ 00 (l ) = kφ 0(l ) (16) 1 1 − 1 1 5. Regarding the beam BC, when the ﬂexible ampliﬁcation mechanism is operating, beam BC exist a translational motion, and let the translational displacement be xp. Therefore, the displacement of the endpoint B of the beam AB is equal to the sum of the displacement of the endpoint C of the beam BC and the translational displacement xp. In addition, the shear force at x1=l1 and the thrust at x = 0 are equal, so: 2 ( φ (l ) = φ (0) + x 1 1 2 p (17) EIφ1000 (l1) = ES2φ20(0)

6. At the endpoint C of beam BC, the superposition of the torque generated by the translation of the beam BC and the end moment of the bending beam AB are equal to the reverse moment of the ﬂexible hinge, so: ES φ 0(l )lm + EIφ 00 (l ) = kφ (l ) (18) 2 2 2 1 1 − 2 2 7. For the junction of the intermediate shaft and mass m, the sum of the displacement at point C and the translational displacement xp is equal to the torsional angular displacement times the length lm. The torque at the endpoint E of the intermediate shaft is equal to the reverse torque of the mass md, so the boundary condition of the torsional intermediate shaft is: ( φ2(l2) + xp = lmφ3(0) 2 (19) GIpφ (l ) = Jω φ (l ) 30 3 − 3 3 3

Then, substitute Equations (13)–(19) into the mode functions, respectively, to solve the natural frequency of each order of the system and the component mode coeﬃcients.

4. Result Analysis The flexible amplification mechanism was made of AA 7075 aluminum alloy. The intermediate shaft was made of AISI 1045 steel. The type of the piezoelectric stack was PSt150/5 5/20 (Core × Tomorrow Technology Co., Ltd., Harbin, China). Table1 shows the material properties and geometric parameters of piezoelectric stacks, flexible amplification mechanism and mass. Actuators 2020, 9, 85 9 of 17

Table 1. Free vibration calculation parameters.

3 3 4 2 ρp (g/cm ) ρq (g/cm ) Iq (cm ) md (kg) S1 (mm ) 7.5 2.81 52.08 3.58 10 3 25 × − 2 3 2 S2 (mm ) ρz (g/cm ) Lm (mm) G (Gpa) J (kg/m ) 20 7.85 10.77 80.7 2.11 10 3 × − 2 d33 (C/N) c33 (Gpa) lnp (mm) Ap (mm ) b (mm) 635 10 12 55.6 20 25 5 × − lFp (mm) l1 (mm) l2 (mm) l3 (mm) E (Gpa) 6 40.64 19.73 20 72

Under no-load state, substituting these parameters in Table1 into Equations (13)–(19), the coeﬃcients W1, W2, C1–C4, B1, B2, N1, N2 and the natural frequencies of the piezoelectric stack, beams and intermediate shaft are obtained. The ﬁrst three order natural frequencies are shown in Table2.

Table 2. First three order natural frequency under no-load state.

Order Natural Angular Frequency (rad/s) Natural Frequency (Hz) 1 291 46 2 227,207 36,179 3 503,641 80,198

According to the first three natural frequencies and coefficients W1, W2, C1–C4, B1, B2, N1, N2, the modal function curves of the piezoelectric stack, beams of flexible amplification mechanism and intermediate shaft under no-load state are drawn by MATLAB, as shown in Figures9–11. From Table2 and Figures9–11, the following points can be summarized:

The natural frequencies of the 2nd and 3rd orders are greatly increased comparing with the • 1st order. This is because when the 1st order free vibration is excited, the deflection of the piezoelectric stack, beam AB, and intermediate shaft DE basically present a linear change, and the deformation of the flexible amplification mechanism mainly occurs at the hinge, as shown in Figure9. In addition, the modal sti ffness of the flexure hinge is much smaller than the other beams, so the 1st order natural frequency is much lower than high order natural frequencies. With the increase of the order, the modal functions of each component begin to present bending • deformation. In the 2nd order modal functions curve, the beam AB appears a peak near 0.01 m and a trough near 0.027 m. In the 3rd order modal function curve, the piezoelectric stack appears a peak near 0.008 m, the beam AB appears a trough near 0.023 m and two peaks near 0.012 m and 0.036 m, and the shaft DE appears a trough near 0.01 m. However the beam BC has basically no obvious peaks and troughs. Taking into account that the electromagnetic modulation mechanism is not suitable for working at • high frequency, when the working frequency is selected to operate the motor, it should be less than the 1st order natural frequency. Actuators 2020, 9, x FOR PEER REVIEW 9 of 17

S2 (mm2) ρz (g/cm3) Lm (mm) G (Gpa) J (kg/m2) 20 7.85 10.77 80.7 2.11 × 10−3 d33 (C/N) c33 (Gpa) lnp (mm) Ap (mm2) b (mm) 635 × 10−12 55.6 20 25 5 lFp (mm) l1 (mm) l2 (mm) l3 (mm) E (Gpa) 6 40.64 19.73 20 72

Under no-load state, substituting these parameters in Table 1 into Equations (13)–(19), the coefficients W1, W2, C1–C4, B1, B2, N1, N2 and the natural frequencies of the piezoelectric stack, beams and intermediate shaft are obtained. The first three order natural frequencies are shown in Table 2.

Table 2. First three order natural frequency under no-load state.

Order Natural Angular Frequency (rad/s) Natural Frequency (Hz) 1 291 46 2 227,207 36,179 3 503,641 80,198

According to the first three natural frequencies and coefficients W1, W2, C1–C4, B1, B2, N1, N2, the modal function curves of the piezoelectric stack, beams of flexible amplification mechanism and intermediate shaft under no-load state are drawn by MATLAB, as shown in Figures 9–11. From Table 2 and Figures 9–11, the following points can be summarized: • The natural frequencies of the 2nd and 3rd orders are greatly increased comparing with the 1st order. This is because when the 1st order free vibration is excited, the deflection of the piezoelectric stack, beam AB, and intermediate shaft DE basically present a linear change, and the deformation of the flexible amplification mechanism mainly occurs at the hinge, as shown in Figure 9. In addition, the modal stiffness of the flexure hinge is much smaller than the other beams, so the 1st order natural frequency is much lower than high order natural frequencies. • With the increase of the order, the modal functions of each component begin to present bending deformation. In the 2nd order modal functions curve, the beam AB appears a peak near 0.01 m and a trough near 0.027 m. In the 3rd order modal function curve, the piezoelectric stack appears a peak near 0.008 m, the beam AB appears a trough near 0.023 m and two peaks near 0.012 m and 0.036 m, and the shaft DE appears a trough near 0.01 m. However the beam BC has basically no obvious peaks and troughs. • Taking into account that the electromagnetic modulation mechanism is not suitable for

Actuatorsworking2020, 9, 85at high frequency, when the working frequency is selected to operate the motor,10 of 17it should be less than the 1st order natural frequency.

-3 x 10-3 x 10 9.049 6

)

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9.047 0

0 0.005 0.01 0.015 0.02 0 0.01 0.02 0.03 0.04 (m) Actuators 2020,, 9,, xx FORFOR PEERPEER REVIEWREVIEW x (m) x 10 of 17 (a) (b) -3 -0.99 xx 1010-3 -0.99 6.66.6

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FigureFigure 9.9. TheThe 1st 1st order order modal modal function. function. (a) Piezoelectric (a)) PiezoelectricPiezoelectric stack; ( bstack;stack;) beam ((bAB)) beam;(beamc) beam AB;BC; ((c;()) dbeambeam) intermediate BC;; ((d)) intermediateintermediateshaft DE. shaftshaft DE..

-3 xx 1010-3 88 0.0150.015

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FigureFigure 10. 10. TheThe 2nd 2nd order order modal modal function. function. (a)) (PiezoelectricPiezoelectrica) Piezoelectric stack;stack; stack; ((b)) (beambeamb) beam AB;; AB((c)) ;( beambeamc) beam BC;; BC((d)); intermediateintermediate(d) intermediate shaftshaft shaft DE..DE .

-3 -3 -3 xx 1010-3 xx 1010 66 44

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x 10-3 -0.99 6.6

-1.03 6.2

) ) x x ( ( 3 2 φ -1.07 φ 5.8

-1.11 5.4

0 0.005 0.01 0.015 0.02 0 0.005 0.01 0.015 0.02 x (m) x (m) (c) (d)

Figure 9. The 1st order modal function. (a) Piezoelectric stack; (b) beam AB; (c) beam BC; (d) intermediate shaft DE.

x 10-3 8 0.015

0.01 6

)

) x 0.005 ( x ( 0

φ 4 1 φ 0

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-8 0.2

-9 0 0 0.005 0.01 0.015 0.02 0 0.005 0.01 0.015 0.02 x (m) x (m) (c) (d)

ActuatorsFigure2020 ,10.9, 85The 2nd order modal function. (a) Piezoelectric stack; (b) beam AB; (c) beam BC; (d11) of 17 intermediate shaft DE.

-3 x 10-3 x 10 6 4

4 2

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) x 2 x 0 ( ( 0 1 φ φ 0 -2

Actuators 2020, 9, x FOR-2 PEER REVIEW -4 11 of 17

x-4 10-3 4 0 0.005 0.01 0.015 0.02 0 0.01 0.02 0.03 0.04 x (m) x (m) Actuators 2020, 9, x FOR PEER REVIEW 11 of 17 (a) -0.2 (b)

) 2 x 10-3 x )

( 4 0 x

2 -0.4 ( φ 3 φ -0.2 0

) 2 -0.6 x ) ( x

2 -0.4 ( φ 3 φ -0.8 -2 0 -0.6 -1 0 0.005 0.01 0.015 0.02 -0.8 0 0.005 0.01 0.015 0.02 -2 x (m) x (m) -1 0 0.005(c) 0.01 0.015 0.02 0 0.005 0.01(d) 0.015 0.02 x (m) x (m) Figure 11. The 3rd order modal function. (a) Piezoelectric stack; (b) beam AB; (c) beam BC; (d) (c) (d) intermediate shaft DE. Figure 11. TheThe 3rd 3rd order order modal modal function. function. (a) (Piezoelectrica) Piezoelectric stack; stack; (b) (beamb) beam AB; AB(c);( beamc) beam BC; BC(d); 5. Experiment(intermediated) intermediate and Discussion shaft shaft DE.DE .

5.To Experiment Experiment verify the and and analysis Discussion of free vibration of the piezoelectric motor, the test of the flexible amplificationTo verifyverify mechanism the the analysis analysis under of freeof vibrationfreeexcitation vibration of theof piezoelectricofthe th piezoelectrice piezoelectric motor, thestacks motor, test ofwas the flexibletestconducted. of amplificationthe flexible Then, the outputamplificationmechanism characteristics under mechanism excitation of the prototypeunder of the excitation piezoelectric were measuredof stacksthe piezoelectric was for conducted. speed stacksand Then, torque. was the conducted. output characteristics Then, the outputof the prototype characteristics were of measured the prototype for speed were and measured torque. for speed and torque. 5.1. Test for Free Vibration 5.1. Test for Free Vibration 5.1. Test for Free Vibration In the free vibration experiment, a test system was built. The equipment included a computer, a In the free vibration experiment, a test system was built. The equipment included a computer, In the free vibration experiment, a test system was built. The equipment included a computer, a HPVa piezo HPV piezodrive drive power power (Suzhou (Suzhou Boshi Boshi Robotics Robotics Tec Technologyhnology Co.,Co., Ltd., Ltd., Suzhou, Suzhou, China), China), the drivingthe driving HPV piezo drive power (Suzhou Boshi Robotics Technology Co., Ltd., Suzhou, China), the driving part partof the of themotor, motor, a OptoMET a OptoMET Digital Digital LDV LDV Vector Vector SeriesSerieslaser laser vibrometer vibrometer (OptoMET, (OptoMET, Darmstadt, Darmstadt, part of the motor, a OptoMET Digital LDV Vector Series laser vibrometer (OptoMET, Darmstadt, Hesse,Hesse, Germany), Germany), and and IOtech IOtech 625u 625u data data acquisition acquisition equipment (Beijing (Beijing Perfect Perfect Technology Technology Co., Co., Ltd., Ltd., Hesse, Germany), and IOtech 625u data acquisition equipment (Beijing Perfect Technology Co., Ltd., Beijing,Beijing, China). China). Figure Figure 12 12shows shows th thee flow flow chart chart of of the freefree vibrationvibration test test system. system. Figure Figure 13 shows 13 shows the the Beijing, China). Figure 12 shows the flow chart of the free vibration test system. Figure 13 shows the experimentalexperimental environment environment for for measuring measuring free free vibration. vibration. experimental environment for measuring free vibration.

DataData acquisition acquisition LaserLaser equipmentequipment VibrometerVibrometer

PrototypePrototype ComputerComputer drivedrive part part

PiezoPiezo drivedrive power

Figure 12. Flow chart of the free vibration test system. FigureFigure 12. 12. FlowFlow chart chart of of the the free vibrationvibration test test system. system.

Vibrometer data Vibrometeracquisition card data acquisition card Piezo drive powerPiezo drive power Platform holder Prototype Computer Platform holder Prototype Computer Experiment platform Experiment Laser vibrometer Laser platform direction Laser vibrometer Laser direction Figure 13. The experimental environment for measuring the free vibration.

Figure 13. The experimental environment for measuring the free vibration. Actuators 2020, 9, x FOR PEER REVIEW 11 of 17

x 10-3 4 0

-0.2

) 2 x ) ( x

2 -0.4 ( φ 3 φ 0 -0.6

-0.8 -2

-1 0 0.005 0.01 0.015 0.02 0 0.005 0.01 0.015 0.02 x (m) x (m) (c) (d)

Figure 11. The 3rd order modal function. (a) Piezoelectric stack; (b) beam AB; (c) beam BC; (d) intermediate shaft DE.

5. Experiment and Discussion To verify the analysis of free vibration of the piezoelectric motor, the test of the flexible amplification mechanism under excitation of the piezoelectric stacks was conducted. Then, the output characteristics of the prototype were measured for speed and torque.

5.1. Test for Free Vibration In the free vibration experiment, a test system was built. The equipment included a computer, a HPV piezo drive power (Suzhou Boshi Robotics Technology Co., Ltd., Suzhou, China), the driving part of the motor, a OptoMET Digital LDV Vector Series laser vibrometer (OptoMET, Darmstadt, Hesse, Germany), and IOtech 625u data acquisition equipment (Beijing Perfect Technology Co., Ltd., Beijing, China). Figure 12 shows the flow chart of the free vibration test system. Figure 13 shows the experimental environment for measuring free vibration.

Data acquisition Laser equipment Vibrometer

Prototype Computer drive part

Piezo drive power Actuators 2020, 9, 85 12 of 17 Figure 12. Flow chart of the free vibration test system.

Vibrometer data acquisition card

Piezo drive power

Platform holder Prototype Computer Experiment platform Laser vibrometer Laser direction

Actuators 2020, 9, x FOR PEER REVIEW 12 of 17 FigureFigure 13. 13.The The experimental experimental environmentenvironment fo forr measuring measuring the the free free vibration. vibration. Actuators 2020, 9, x FOR PEER REVIEW 12 of 17 After establishing the test system, 5 test points with equal spacing were selected on the beam After establishing the test system, 5 test points with equal spacing were selected on the beam AB, AB, as Aftershown establishing in Figure 14.the Duetest system, to the limitation5 test points of withexperimental equal spacing conditions, were selected beam ABon theamong beam the as shown in Figure 14. Due to the limitation of experimental conditions, beam AB among the beams beamsAB, asof shownthe flexible in Figure amplification 14. Due to mechanism the limitation was of easierexperimental to measure conditions, than other beam beams, AB among and couldthe of the flexible amplification mechanism was easier to measure than other beams, and could fulfil the fulfilbeams the ofrequirements the flexible amplificationof verifying freemechanism vibration was theoretical easier to measurecalculations, than soother it was beams, used and as couldthe test requirements of verifying free vibration theoretical calculations, so it was used as the test object. On the object.fulfil Onthe requirementsthe beam AB of, the verifying spacing free of vibration the five theoreticalpoints was calculations, chosen by so 8.5 it wasmm. used In addition, as the test the beam AB, the spacing of the five points was chosen by 8.5 mm. In addition, the theoretically calculated theoreticallyobject. On calculatedthe beam AB result, the ofspacing 1st order of the natura five l pointsfrequency was chosenwas about by 8.546.2 mm. Hz, In so addition, the excitation the result of 1st order natural frequency was about 46.2 Hz, so the excitation frequency of 0~100 Hz was frequencytheoretically of 0~100 calculated Hz was result used of for 1st the order frequency natura lsw frequencyeep test. wasIn order about to 46.2 improve Hz, so test the accuracy excitation and used for the frequency sweep test. In order to improve test accuracy and precision, each test frequency precision,frequency each of 0~100test frequency Hz was used was for recorded the frequency and averaged sweep test. after In multiple order to improveexcitations. test accuracy and wasprecision, recorded each and test averaged frequency after was multiple recorded excitations. and averaged after multiple excitations. Laser direction

Laser direction

5 4 3 2 1 5 4 3 2 1

FigureFigure 14.14. TheThe beam beam AB freefree vibrationvibration test. test.

ForFor visualvisual visual analysisanalysis analysis ofof of experimental experimental experimental and andand theoretical theoreticaltical values, values,values, the thethe frequency frequency frequency domain domain domain diagram diagram diagram of of the of the5ththe point5th 5th point point on theon on beamthe the beam beamAB wasAB AB was drawnwas drawndrawn and and usingand using the datathethe datadata of 5 of testof 5 5 test points,test points, points, the the modethe mode mode shape shape shape was was alsowas alsoﬁttedalso fitted by fitted the by by curve, the the curve, curve, as shown as as shown shown in Figure in in Figure Figure 15. 15.15.

-3-3-3 x10 x10 -3-xx10 10 3-x10 x10 3 2 2 1.4 6 6 ExperimentalExperimental ExperimentalExperimental 5 5 TheoreticalTheoretical TheoreticalTheoretical 4 4 1.5 1.5 1 4 1 3 4 3 φ 1(mφ ) 1 A1(m) 1(m) A1(m) A (m) A (m) 1 1 0.6 2 2 1 0.6 2 2 0.5 0.5 0.2 1 0 0.2 1 0 0 0 0 20 40 60 80 100 0 0.0 1 0.0 2 0.0 3 0.0 4 0 20 40 f (H z)60 80 100 0 0.0 1 l1 (m) 0.0 2 0.0 3 0.0 4 f (H z) l1 (m) (a) (b) (a) (b) Figure 15. The 1st order modal experimental results of beam AB. (a) Frequency domain diagram of Figurethe 5th 15. point The 1ston the orderorder beam modalmodal AB; experimental( bexperi) modemental shape. results results of of beam beamAB AB.(a. )(a Frequency) Frequency domain domain diagram diagram of the of the5th 5th point point on theon beamthe beamAB;( ABb); mode(b) mode shape. shape. From Figure 15, the following conclusions can be obtained: From Figure 15, the following conclusions can be obtained: • As shown in Figure 15a, for the 1st order natural frequency, the measured value is 38.5 Hz • Aswhich shown is inslightly Figure lower 15a, than for thethe 1sttheoretical order natu valueral of frequency, 46.3 Hz. For the the measured resonance value displacement is 38.5 Hz whichresponse is slightly amplitude lower of thethan beam the theoreticalAB end, the vameasuredlue of 46.3 value Hz. is For1.35 the mm resonance which also displacement is slightly responselower than amplitude the theoretical of the beam value AB of end, 1.55 themm. measured The errors value are is16.3% 1.35 mmof the which 1st order also isnatural slightly lowerfrequency than theand theoretical12.9% of the value response of displacement.1.55 mm. The errors are 16.3% of the 1st order natural • frequencyAs shown and in 12.9%Figure of 15b, the the response displacement displacement. response of the 5 points marked on the beam AB • Asconforms shown in to Figure the theoretically 15b, the displacement calculated 1st response order freeof the vibration. 5 points The marked resonance on the response beam AB conformsdisplacement to the of 5ththeoretically point is the calculatedlargest, about 1st 1.35 order mm. free The 1stvibration. point is locatedThe resonance at the junction response of displacementthe driving beamof 5th andpoint the is piezoelectricthe largest, about stack, 1.35 and mm.the resonance The 1st point response is located displacement at the junction is the of thesmallest, driving about beam 0.04 and mm. the piezoelectric stack, and the resonance response displacement is the • smallest,For the aboutaforementioned 0.04 mm. errors, two reasons are summarized. The first reason is there was an • Forerror the inaforementioned the simplified model.errors, Thetwo second reasons reason are summarized. is there also willThe befirst errors reason in measurementis there was an and data processing of the experiment. In addition, a wear-resistant gasket was added between error in the simplified model. The second reason is there also will be errors in measurement and data processing of the experiment. In addition, a wear-resistant gasket was added between Actuators 2020, 9, 85 13 of 17

From Figure 15, the following conclusions can be obtained:

As shown in Figure 15a, for the 1st order natural frequency, the measured value is 38.5 Hz which • is slightly lower than the theoretical value of 46.3 Hz. For the resonance displacement response amplitude of the beam AB end, the measured value is 1.35 mm which also is slightly lower than the theoretical value of 1.55 mm. The errors are 16.3% of the 1st order natural frequency and 12.9% of the response displacement. As shown in Figure 15b, the displacement response of the 5 points marked on the beam AB conforms • to the theoretically calculated 1st order free vibration. The resonance response displacement of 5th point is the largest, about 1.35 mm. The 1st point is located at the junction of the driving beam and the piezoelectric stack, and the resonance response displacement is the smallest, about 0.04 mm. For the aforementioned errors, two reasons are summarized. The first reason is there was an • Actuatorserror 2020 in,, 9, the, xx FORFOR simplified PEERPEER REVIEWREVIEW model. The second reason is there also will be errors in measurement13 of 17 and data processing of the experiment. In addition, a wear-resistant gasket was added between the piezoelectric stack and beam AB, which will affect affect the experimental results by changing the contact stistiffness.ffness.

5.2. Test Test of Output Characteristics For the measuring output characteristics of the piezoelectric motor, another test system was established, whichwhich mainly mainly included included the the computer, computer, NI USB-6009 NI USB-6009 data acquisitiondata acquisition card (Shenzhen card (Shenzhen Boruitu BoruituElectronic Electronic Technology Technology Co., Ltd., Co., Shenzhen, Ltd., Shenzhen, China), HPV China), piezo HPV drive piezo power drive (Suzhou power Boshi(Suzhou Robotics Boshi TechnologyRobotics Technology Co., Ltd., Co., Suzhou, Ltd., China),Suzhou, prototype, China), prototype, UNI-T direct UNI-T current direct (DC) current power (DC) supply power (Unitech supply (China)(Unitech Co., (China) Ltd., Dongguan,Co., Ltd., Dongguan, China) and China) HP-5 micro and forceHP-5 gauge micro (Yueqing force gauge Aidebao (Yueqing Instrument Aidebao Co., InstrumentLtd., Yueqing, Co., China Ltd., ). Yueqing, Figure 16 China shows ). the Figure ﬂow chart16 shows of the the output flow characteristic chart of the testoutput system. characteristic Figure 17 testshows system. the experimental Figure 17 shows environment the experimental for measuring environment output characteristics.for measuring output characteristics.

Photoelectric Data coupling relay Data coupling relay DC power acquisition DC power acquisition supply card 1 supply

Computer Micro force gauge Prototype signal Micro force gauge Prototype Data Data Piezo drive power acquisition Piezo drive power card 2 card 2 Figure 16. Flow chart of the output characteristic test system.

Arbitrary signal DC power generator DC power generator supply

Piezo drive power

Prototype Photoelectric Prototype coupling relay

Experiment platform Micro Force Gauge

Figure 17. TheThe experimental experimental environment for me measuringasuring output characteristics.

In the test, the adjustment range of the drive voltage peak of the electromagnetic modulation mechanism was 3.5–5 V, the adjustment range of the drive signals frequency of the electromagnetic modulation mechanism and the piezoelectric stacks were 1–5 Hz, and adjustment range of the phase difference α between the electromagnetic modulation signal and the piezoelectric drive signal was 0–90°. In the speed measurement, the time of the prototype rotating a certain angle was recorded. Multiple measurements for speed were repeated to record the data, and the average speed of the prototype can be calculated. In the prototype torque measurement, a metal rod was mounted horizontally on the rotor shaft, and the thrust of the metal rod was measured with a micro force gauge. The micro force gauge could display the thrust of the metal rod test point, and the output torque could be calculated by the thrust. When the phase difference α was 50°, we investigated the operating status of the motor, as shown in Figure 18. The motor rotated about 9° in 120 s. This kind of motor presents a slow running state, from the measurement results. Actuators 2020, 9, 85 14 of 17

In the test, the adjustment range of the drive voltage peak of the electromagnetic modulation mechanism was 3.5–5 V, the adjustment range of the drive signals frequency of the electromagnetic modulation mechanism and the piezoelectric stacks were 1–5 Hz, and adjustment range of the phase diﬀerence α between the electromagnetic modulation signal and the piezoelectric drive signal was 0–90◦. In the speed measurement, the time of the prototype rotating a certain angle was recorded. Multiple measurements for speed were repeated to record the data, and the average speed of the prototype can be calculated. In the prototype torque measurement, a metal rod was mounted horizontally on the rotor shaft, and the thrust of the metal rod was measured with a micro force gauge. The micro force gauge could display the thrust of the metal rod test point, and the output torque could be calculated by the thrust. When the phase diﬀerence α was 50◦, we investigated the operating status of the motor, as shown in Figure 18. The motor rotated about 9◦ in 120 s. This kind of motor presents a slow running Actuatorsstate, from 2020 the, 9, x measurementFOR PEER REVIEW results. 14 of 17

Time： 0 s Time：40 s Angle：0° Angle：3°

(a) (b) Time：80 s Time：120 s Angle：6° Angle：9°

Figure 18. Experimental eeffectﬀect whenwhen thethe phasephase di differenceﬀerence α αis is 50 50°.◦.( a()a) At At 0 0 s; s; (b ()b at) at 40 40 s; s; (c )(c at) at 80 80 s; (s;d )(d at) 120at 120 s. s.

To investigateinvestigate thethe e ﬀeffectect of of driving driving signal signal parameters parameters on speedon speed and and torque, torque, more more experiments experiments were wereconducted. conducted. The change The relationshipchange relationship curve of thecurve speed of andthe torquespeed withand ditorqueﬀerent drivewith parametersdifferent drive was parametersdrawn, as shown was drawn, in Figure as shown19. in Figure 19.

0.2 0.1

0.1 0.08 (rad/min) (rad/min)

0 0.06 123453.5 4 4.5 5 5.5 f (Hz) U (V) 0.45 0.5

0.4 0.4 (Nmm) (Nmm) M M 0.35 0.3 123453.5 4 4.5 5 5.5 f (Hz) U (V) (a) (b) Actuators 2020, 9, x FOR PEER REVIEW 14 of 17

Time： 0 s Time：40 s Angle：0° Angle：3°

(a) (b) Time：80 s Time：120 s Angle：6° Angle：9°

Figure 18. Experimental effect when the phase difference α is 50°. (a) At 0 s; (b) at 40 s; (c) at 80 s; (d) at 120 s.

To investigate the effect of driving signal parameters on speed and torque, more experiments Actuatorswere conducted.2020, 9, 85 The change relationship curve of the speed and torque with different 15drive of 17 parameters was drawn, as shown in Figure 19.

0.2 0.1

0.1 0.08 (rad/min) (rad/min)

0 0.06 123453.5 4 4.5 5 5.5 f (Hz) U (V) 0.45 0.5

0.4 0.4 (Nmm) (Nmm) M M 0.35 0.3 123453.5 4 4.5 5 5.5 f (Hz) U (V) Actuators 2020, 9, x FOR PEER REVIEW 15 of 17 (a) (b) 0.1

0.05 (rad/min)

0 0 20406080100 () 0.5 (Nmm) M 0 0 20406080100 () (c)

Figure 19.19. TheThe law law of of the the speed speed and and torque torque of the of prototypethe prototype with diwithﬀerent different drive parameters. drive parameters. (a) Change (a) withChange the with driving the frequency;driving frequency; (b) change (b) with change the with electromagnetic the electrom driveagnetic voltage; drive (voltage;c) change (c) with change the phasewith the diﬀ phaseerence differenceα of the two α of drive the two signals. drive signals.

From Figure 1919,, the following conclusionsconclusions cancan bebe drawn:drawn:

• When thethe frequencyfrequency ofof the the driving driving signal signal increases, increases, the the motor motor speed speed gradually gradually increases increases and and the • torquethe torque gradually gradually decreases. decreases. Since Since the electromagnetic the electromagnetic modulation modulation mechanism mechanism sensitivity sensitivity is lower is thanlower piezoelectric than piezoelectric stack, thestack, driving the driving signal frequencysignal frequency is not suitable is not suitable as it is too as high.it is too In orderhigh. In to maintainorder to maintain the good outputthe good performance output performance of the motor, of herethe motor, we take here the drivingwe take signalthe driving frequency signal of 3frequency Hz. This of not 3 onlyHz. ensuresThis not the only normal ensures speed the ofno thermal motor, speed but of alsothe motor, enables but the also motor enables to have the a strongmotor to load have capacity. a strong load capacity. • With thethe increase increase of theof the voltage voltage of the of electromagnetic the electromagnetic driving driving signal, thesignal, motor the speed motor decreases speed • firstdecreases and then first tends and to then be stable. tends Theto overallbe stable. torque The shows overall a trend torque of slowlyshows increasing. a trend of Since slowly the electromagneticincreasing. Since voltage the electromagnetic increases, the disc-shaped voltage increases, slider would the be disc-shaped magnetized, whichslider causeswould itbe to bemagnetized, unable to separatewhich causes from it electromagnet to be unable into time.separate Therefore, from electromagnet the electromagnetic in time. driving Therefore, voltage the cannotelectromagnetic easily exceed driving 4.5 V.voltage cannot easily exceed 4.5 V. • When increasingincreasing the the phase phase di ffdifferenceerence α between α between the two the drive two signals,drive signals, the motor the speed motor increases speed • firstincreases and then first tends and then to be tends stable, to and be thestable, change and law the of change the torque law isof similar the torque to the is speed. similar In to order the tospeed. ensure In theorder normal to ensure operation the normal of the operation motor, the of drive the motor, signal needsthe drive to keep signal a phaseneeds ditoff erence.keep a Inphase this difference. way, it can avoidIn this the way, electromagnet it can avoid and the theelectromagnet disc-shaped and slider the being disc-shaped unable to slider separate being in timeunable during to separate operation. in time It explains during why operation. there is aIt timeexplains difference why ∆theret between is a time the twodifference excitation Δt signalsbetween in the the two driving excitation principle. signals in the driving principle.

6. Conclusions A novel piezoelectric motor modulated by a magnetic field was proposed. The dynamic equations of the piezoelectric system were established for free vibration, the natural frequency and the vibration mode function were solved according to the continuity and boundary conditions, and the modal function curves were drawn and analyzed. In the free vibration test of the prototype, the relative error of the natural frequency between the test and the theoretical value was small, which verified the correctness of the free vibration theoretical analysis. In the output characteristics test, the prototype had a relatively stable output performance when the frequency of the motor drive signal was 3 Hz, the electromagnetic drive voltage was 4.5 V, and the phase difference between the electromagnetic modulation signal and the piezoelectric stack drive signal was 30°. This time, the motor output speed was 0.1046 rad/min, and the output torque was 0.405 Nmm. The novel electromagnetic modulation piezoelectric motor had some deficiencies. For example, due to the large coil volume, the motor was not easy to miniaturize, and the response of the electromagnetic modulation mechanism was slower. However, it had a higher resolution than the electromagnetically modulated piezoelectric motor with a speed of 2.28 rad/min (0.038 rad/s) Actuators 2020, 9, 85 16 of 17

6. Conclusions A novel piezoelectric motor modulated by a magnetic field was proposed. The dynamic equations of the piezoelectric system were established for free vibration, the natural frequency and the vibration mode function were solved according to the continuity and boundary conditions, and the modal function curves were drawn and analyzed. In the free vibration test of the prototype, the relative error of the natural frequency between the test and the theoretical value was small, which verified the correctness of the free vibration theoretical analysis. In the output characteristics test, the prototype had a relatively stable output performance when the frequency of the motor drive signal was 3 Hz, the electromagnetic drive voltage was 4.5 V, and the phase difference between the electromagnetic modulation signal and the piezoelectric stack drive signal was 30◦. This time, the motor output speed was 0.1046 rad/min, and the output torque was 0.405 Nmm. The novel electromagnetic modulation piezoelectric motor had some deficiencies. For example, due to the large coil volume, the motor was not easy to miniaturize, and the response of the electromagnetic modulation mechanism was slower. However, it had a higher resolution than the electromagnetically modulated piezoelectric motor with a speed of 2.28 rad/min (0.038 rad/s) proposed by Xing J. et al. [17]. The proposal of this novel piezoelectric motor provides a new idea for solving the friction problem of piezoelectric motors, and is of great significance for the further development and application of piezoelectric motors and dynamics research.

Author Contributions: Supervision, J.X.; project administration, J.X.; funding acquisition, J.X.; Conceptualization, J.X. and Y.Q.; methodology, J.X.; software, Y.Q.; validation, J.X. and Y.Q.; investigation, Y.Q.; resources, J.X.; data curation, Y.Q.; writing—original draft preparation, J.X. and Y.Q.; writing—review and editing, J.X. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Natural Science Foundation of China, grant number 51605423 and Hebei Natural Science Foundation Project, grant number E2018203116. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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