Piezoelectric Motor Using In-Plane Orthogonal Resonance Modes of an Octagonal Plate

Home , Inchworm motor, Piezoelectric motor

actuators

Article Piezoelectric Motor Using In-Plane Orthogonal Resonance Modes of an Octagonal Plate

Karl Spanner and Burhanettin Koc * Physik Instrumente GmbH & Co. KG, Auf der Roemerstrasse, 1, 76228 Karlsruhe, Germany; [email protected] * Correspondence: [email protected]; Tel.: +49-721-4846-2416

Received: 26 September 2017; Accepted: 4 December 2017; Published: 6 January 2018 Abstract: Piezoelectric motors use the inverse piezoelectric effect, where microscopically small periodical displacements are transferred to continuous or stepping rotary or linear movements through frictional coupling between a displacement generator (stator) and a moving (slider) element. Although many piezoelectric motor designs have various drive and operating principles, microscopic displacements at the interface of a stator and a slider can have two components: tangential and normal. The displacement in the tangential direction has a corresponding force working against the friction force. The function of the displacement in the normal direction is to increase or decrease friction force between a stator and a slider. Simply, the generated force alters the friction force due to a displacement in the normal direction, and the force creates movement due to a displacement in the tangential direction. In this paper, we ﬁrst describe how the two types of microscopic tangential and normal displacements at the interface are combined in the structures of different piezoelectric motors. We then present a new resonance-drive type piezoelectric motor, where an octagonal plate, with two eyelets in the middle of the two main surfaces, is used as the stator. Metallization electrodes divide top and bottom surfaces into two equal regions orthogonally, and the two driving signals are applied between the surfaces of the top and the bottom electrodes. By controlling the magnitude, frequency and phase shift of the driving signals, microscopic tangential and normal displacements in almost any form can be generated. Independently controlled microscopic tangential and normal displacements at the interface of the stator and the slider make the motor have lower speed–control input (driving voltage) nonlinearity. A test linear motor was built by using an octagonal piezoelectric plate. It has a length of 25.0 mm (the distance between any of two parallel side surfaces) and a thickness of 3.0 mm, which can produce an output force of 20 N.

Keywords: piezoelectric; motor; friction

1. Introduction Piezoelectric motors are the main candidates for use in small mechanical systems [1–16]. The advantages compared to electromagnetic ones with the same size and weight include high power density, insensitivity towards a magnetic ﬁeld, silent, direct drive without a gear mechanism, a quick response without backlash, maintaining a position with no energy consumption and a high dynamic range of velocity with high positioning accuracy [12–16]. According to functional principles, construction, and drive speciﬁcations, piezoelectric motors can be categorized into three groups: piezo-walk-drive, inertia-drive and resonance-drive [6].

1.1. Piezo-Walk-Drive In a piezo-walk-drive mechanism, there are at least two sets of actuator arrangements embedded in the structure and they operate one after the other. The main reason that these motors are called piezo-walk-drive is because the moving sequences in these motors resemble two or four feed walking actions. In these motors, a step motion is realized in two ways.

Actuators 2018, 7, 2; doi:10.3390/act7010002 www.mdpi.com/journal/actuators Actuators 2018, 7, 2 2 of 15

1.1. Piezo-Walk-Drive In a piezo-walk-drive mechanism, there are at least two sets of actuator arrangements embedded Actuatorsin the structure2018, 7, 2 and they operate one after the other. The main reason that these motors are called2 of 14 piezo-walk-drive is because the moving sequences in these motors resemble two or four feed walking actions. In these motors, a step motion is realized in two ways. InIn oneone way,way, therethere are are normal normal and and tangential tangential microscopic microscopic displacements displacements to to the the moving moving direction direction of aof sliding a sliding element. element. Each Each actuator actuator in a in motor a motor that generatesthat generates a microscopic a microscopic displacement displacement has one has single one task,single which task, iswhich either is to either perform to perform a “clamp” a or“clamp” “move” or action. “move” If anaction. actuator If an has actuator the task has of performingthe task of aperforming “clamp” action, a “clamp” the displacement action, the isdisplacement normal to the is slidernormal moving to the direction slider moving and the direction ultimate functionand the ofultimate these actuators function isof tothese increase actuators or decrease is to increase normal or force decrease and thus, normal to hold force friction and thus, force. to Ifhold an actuator friction isforce. required If an to actuator perform is a “move”required action, to perform the displacement a “move” generatedaction, the by displacement the actuator isgenerated tangential by to the movingactuator direction is tangential of the to sliding the moving element. direction Expansion of the and sliding shrinkage element. of this Expansion actuator and creates shrinkage a microscopic of this movementactuator creates of the a slider.microscopic After the movement structure of proposed the slider. by After Brisbane the structure in 1965 [17 proposed], some other by Brisbane structures in operated1965 [17], on some the basisother of structures the piezo-walk-drive operated on principal the basis [18 of– 21the]. Typically,piezo-walk-drive these structures principal consist [18–21]. of threeTypically, actuators, these wherestructures two ofconsist them of are three responsible actuators, for where the clamping two of andthem one are is responsible responsible for the movingclamping action. and one In theseis responsible motors, thefor requiredthe moving displacements action. In these in the motors, normal the and required tangential displacements directions, within the respect normal to aand sliding tangential element, directions, can be generated with re byspect actuators to a sliding that use element, longitudinal, can be shear, generated transverse, by andactuators planar that coupling use longitudinal, of piezoelectric shear, materials. transverse, In the and early planar structures, coupling the of piezoelectric piezoelectric elements materials. used In inthe the early piezo-drive structures, type the motors piezoelect wereric in elements bulk forms, used but in inthe many piezo-drive of the commercializedtype motors were structures, in bulk theforms, actuators but in aremany manufactured of the commercialized in multilayer structures, forms to generatethe actuators sufﬁcient are manufactured displacement atin amultilayer relatively low-drivingforms to generate voltage. sufficient displacement at a relatively low-driving voltage. Assuming thatthat the the activation activation makes makes the the length, length or, the or diameter,the diameter, decrease decrease and release, and release, which causeswhich ancauses actuator an actuator to return to return to its restto its position, rest position, the motion the motion sequence sequence as seen as seen in Figure in Figure1 can 1 becan started be started by activatingby activating one one clamping clamping actuator actuator (A). When(A). When the moving the moving actuator actuator (C) is (C) also is activated, also activated, shrinkage shrinkage of the movingof the moving actuator actuator generates generates a half step. a half At step. this moment,At this moment, the clamping the clamping actuator actuator (A) is released (A) is soreleased that it canso that clamp it can and clamp maintain and the maintain holding the force. holding In the fo followingrce. In the step, following the second step, clamping the second actuator clamping (B) is activatedactuator (B) and is theactivated moving and actuator the moving (C) is released.actuator (C) When is released. the second When clamping the second actuator clamping (B) is released,actuator one(B) is motion released, sequence one motion is ﬁnished. sequence is finished.

Figure 1. Motion sequences of a piezo-walk-drive motor proposed by Brisbane in 1965 [17]. Figure 1. Motion sequences of a piezo-walk-drive motor proposed by Brisbane in 1965 [17]. Later, we can see examples where clamping actuators are attached to the end of feeding actuators,Later, where we can at see least examples two identical where clampingsets are used actuators in each are motor attached structure to the end[22]. of Assuming feeding actuators, that the whereactivation at least makes two the identical length setsof an are actuator used in increase each motor and release, structure making [22]. Assuming an actuator that return the activation to its rest makesposition, the motion length sequence of an actuator of the increase piezo-walk-drive, and release, as makingseen in anFigure actuator 2, can return be started to its by rest activating position, motionclamping sequence actuators of (1C) the piezo-walk-drive, in the first set (step as 1) seen. A half in Figure step can2, can be bemade started when by the activating moving clamping actuators actuatorsin the first (1C) andin inthe thefirst second set (stepset (1F 1) .and A half 2F) stepare activated can be made (step when 2). At the this moving moment, actuators a pushing in the force first is and in the second set (1F and 2F) are activated (step 2). At this moment, a pushing force is generated by the moving actuators (1F) in the first set. When the clamping actuators (2C) in the second set are activated (step 3), all moving and clamping actuators are in an active state. The second half of the step is started by releasing the clamping actuators (1C) in the first set (step 4). After the moving actuators (1F Actuators 2018, 7, 2 3 of 15 generatedActuators 2018 by, 7 ,the 2 moving actuators (1F) in the first set. When the clamping actuators (2C) in3 ofthe 14 second set are activated (step 3), all moving and clamping actuators are in an active state. The second half of the step is started by releasing the clamping actuators (1C) in the first set (step 4). After the movingand 2F) inactuators the first (1F and and in the2F) secondin the firs setst areand released in the second(step 5) sets, the are second released half (step of the 5) step, the is second completed. half ofA newthe step sequence is completed. can be started A new by sequence activating can clamping be started actuators by activating (1C) in clamping the first set,actuators and releasing (1C) in the firstclamping set, and actuators releasing (2C) the in clamping the second actuators set. In order (2C) to in obtain the second smoother set. motion,In order or to for obtain the purpose smoother of motion,better controllability, or for the purpose an overlap of better of activationcontrollability, anddeactivation an overlap of timings activation is possible and deactivation for both clamping timings isand possible moving for actuators both clamping [19,20]. and moving actuators [19,20].

FigureFigure 2. 2. MotionMotion sequences sequences of of a a piezo-walk-drive piezo-walk-drive motor motor proposed proposed by Shamoto in 1993 [22]. [22].

InIn other typestypes ofof structures,structures, two two sets sets of of actuators actuators placed placed next next to eachto each other other make make “clamp-push “clamp-push and andrelease” release” actions actions in sequence in sequence [23,24 [23,24].]. All actuatorsAll actuators in the in structurethe structure are required are required to perform to perform “clamp-push “clamp- pushand release” and release” actions. actions. “Clamp-push “Clamp-push and and release” release” actions actions can can be fulfilledbe fulfilled if motion if motion generated generated by by an anactuator actuator has has an obliquean oblique or ellipticalor elliptical trajectory. trajectory. Oblique Oblique or elliptical or elliptical trajectory trajectory is generated is generated because because each eachactuator actuator is electrically is electrically divided divided into twointo sections two sectio inns the in longitudinal the longitudinal direction direction so driving so driving signals signals cause causethe actuator the actuator to elongate to elongate and deflect and at deflect the same at time.the same Both time. actuator Both sets actuator (a pair) sets perform (a pair) a “clamp-push perform a “clamp-pushand release” actionand release” one after action the other, one after with the a time other, delay with or a a time phase delay shift or in a the phase same shift period in the cycle. same periodMotion cycle. sequence is started by activating one set. This activation makes the moving element clampMotion due to sequence elongation, is andstarted motion by activating is created dueone toset. deflection. This activation At the momentmakes the when moving the other element set is clampinitially due activated, to elongation, the actuator and motion set in clampis created position due to is deflection. then deactivated. At the moment Because when the second the other set hasset isbeen initially activated, activated, the first the actuatoractuator set deactivates.in clamp posi Duringtion is then this time,deactivated. the sliding Because element the doessecond not set move has beenback, activated, but rather the advances first actuator another set step. deactivates. During this time, the sliding element does not move back, but rather advances another step. 1.2. Inertia-Drive 1.2. Inertia-Drive In inertia-drive piezoelectric motors, only the tangential component of the back-and-forth movement,In inertia-drive at the interface piezoelectric between motors, a slider only and ath statore tangential within onecomponent period, generatesof the back-and-forth a movement. movement,In one direction at the ofinterface the tangential between movement, a slider and the a stator stator within element one is period, activated generates slowly. a Duringmovement. this Inactivation one direction time, the of inertiathe tangential force acting movement, on the slider the isstator smaller element than the is frictionactivated force; slowly. the slider During sticks this to activationthe contact time, area the of the inertia stator force and acting moves on with the it.slider In the is smaller opposite than direction the friction of the force; tangential the slider movement, sticks tothe the stator contact is deactivated area of the faster, stator relative and tomoves its initial with position. it. In the During opposite this time,direction the inertiaof the force tangential acting movement,on the slider the is greaterstator is than deactivated the friction faster, force, relative so the sliderto its initial slips and position. stays behindDuring the this contact time, the area inertia of the forcestator acting element. on Atthe the slider end ofis onegreater cycle, than the slidingthe fricti elementon force, makes so the a microscopic slider slips step. and Thestays accumulation behind the contactof these area microscopic of the stator steps element. creates macroscopicAt the end of movement. one cycle, the sliding element makes a microscopic step. DeformationThe accumulation of the of piezoelectric these microsco elementpic steps in various creates modes,macroscopic such asmovement. longitudinal, transverse or shear,Deformation is either directly of the transferredpiezoelectric to element a moving in elementvarious modes, or through such a as coupling longitudinal, element, transverse where theor shear,motion is generated either directly by the transferred piezoelectric to elementa moving is convertedelement or into through a tangential a coupling direction element, at the where interface. the motionDepending generated on the by structure, the piezoelectric while convertingelement is conv a deformationerted into a intotangential tangential direction motion, at the a interface. leverage Dependingmechanism on could the amplifystructure, the while deformation. converting The a amplifieddeformation deformation into tangential could motion, also have a aleverage normal mechanismcomponent [could25,26 ].amplify Nevertheless, the deformation. having a normal The amplified component deformation at the interface could couldalso have make a anormal motor componenthave direction-dependent [25,26]. Nevertheless, performance having parameters, a normal suchcomponent as generated at the force interface and velocity, could make which a shouldmotor be compensated with magnitude and timing of a driving signal.

Actuators 2018, 7, 2 4 of 15 Actuators 2018, 7, 2 4 of 14 have direction-dependent performance parameters, such as generated force and velocity, which should be compensated with magnitude and timing of a driving signal. InIn inertia-drive inertia-drive motors, motors, tangential tangential movement movement can beca generatedn be generated on a slider on ora onslider a stator or on depending a stator ondepending where a piezoelectricon where a piezoelectric element is embedded. element is Ifemb a piezoelectricedded. If a piezoelectric element is embedded element is into embedded the moving into element,the moving the motorelement, can the be motor considered can be a movingconsidered actuator a moving type. actuator If the piezoelectric type. If the elementpiezoelectric is embedded element intois embedded the stator, into the motorthe stator, can bethe considered motor can abe fixed considered actuator a type.fixed actuator type. EvenEven if there there are are some some structures structures that that could could be considered be considered as inertia-drive as inertia-drive piezoelectric piezoelectric motors motors[27,28], [ 27the,28 first], the practical first practical inertia-drive inertia-drive (stick-slip) (stick-slip) structure structure was proposed was proposed by Pohl by in Pohl 1986 in [29], 1986 where [29], wherea piezoelectric a piezoelectric cylinder cylinder is embedded is embedded into a four-bar into a four-bar mechanism mechanism (Figure (Figure3). One3 side). One of sidethe four-bar of the four-barwas attached was attached to a base to and a base a sliding and a mass sliding is attached mass is attachedto the parallel to the bar. parallel When bar. the When actuator the actuatoris driven iswith driven a saw-tooth with a saw-tooth waveform waveform signal, the signal, attached the mass attached (m) moves mass (m)together moves with together the bar. with This the occurs bar. Thisduring occurs the slow during expansion the slow period expansion of the period saw-tooth of the signal saw-tooth (sticking). signal This (sticking). is left behind This during is left behindthe fast duringcontraction the fast period contraction of the signal period (slipping). of the signal Repeating (slipping). the Repeatingcyclic movements the cyclic makes movements the attached makes mass the attachedmove continuously. mass move continuously.When the electric When field the electricmakes the field piezoelectric makes the piezoelectricelement expand element quickly expand and quicklycontract and slowly, contract the slowly,attached the mass attached moves mass in the moves oppo insite the direction. opposite This direction. mechanism This mechanism was applied was to appliedprecise tomulti-degree precise multi-degree motion positioning motion positioning device applic deviceations applications for an atomic for an force atomic microscope. force microscope. Because Becausethe piezoelectric the piezoelectric element element is embedded is embedded into the into stator the statorstructure, structure, this motor this motor could could be considered be considered as a asfixed a fixed actuator actuator type. type.

Figure 3. Inertia-drive piezoelectric motor proposed by Pohl in 1986 [29]. Figure 3. Inertia-drive piezoelectric motor proposed by Pohl in 1986 [29].

In a structure proposed by Higuchi in 1986 [30], a moving mass (m1) and a weight (m2) are attachedIn a structureto both ends proposed of a piezoe by Higuchilectric element. in 1986 The [30], whole a moving structur masse is (m placed1) and on a weighta base plate (m2) areand attachedheld by tofrictional both ends force of aacting piezoelectric between element. the baseThe plate whole and structurethe moving is placedmass (Figure on a base 4). plateWhen and an heldelectric by frictionalfield is applied force acting to the between piezoelectric the base element, plate and rapid the moving expans massion of (Figure the piezoelectric4). When an electricelement ﬁeldcreates is applied an acceleration, to the piezoelectric causing theelement, moving mass rapid to expansion overcome of the the static piezoelectric friction (slip) element so both creates masses an acceleration,move in opposite causing directions. the moving During mass tothe overcome slow contraction the static time friction of (slip)the piezoelectric so both masses element, move the in oppositeacceleration directions. on the moving During themass slow cannot contraction overcome time the of st theatic piezoelectricfriction, so only element, the weight the acceleration moves (stick). on theAt movingthe end massof one cannot motion overcome cycle, a the microscopic static friction, movement so only theis obtained. weight moves Because (stick). the At piezoelectric the end of oneelement motion is embedded cycle, a microscopic into the moving movement element, is obtained. this motor Because could the be piezoelectric considered elementas a moving is embedded actuator intotype. the Many moving of the element, inertia-drive this motor type couldmotors be dev consideredeloped afterwards as a moving operate actuator according type. to Many the above- of the inertia-drivementioned initial type motorsstructures developed [31–45]. We afterwards can also operate find literature according that to proposes the above-mentioned ideal driving signals initial structureswith detailed [31– 45analysis]. We can of motion also ﬁnd at literature the interface that proposes[46,47]. ideal driving signals with detailed analysis of motion at the interface [46,47].

Actuators 2018, 7, 2 5 of 14 Actuators 2018, 7, 2 5 of 15 Actuators 2018, 7, 2 5 of 15

Figure 4. Inertia-drive piezoelectric motor proposed by Higuchi in 1986 [30]. Figure 4. Inertia-drive piezoelectric motor proposed by Higuchi in 1986 [30]. Figure 4. Inertia-drive piezoelectric motor proposed by Higuchi in 1986 [30]. 1.3. Resonance-Drive 1.3. Resonance-Drive 1.3. Resonance-Drive In a resonance-drive piezoelectric motor, there are two types of microscopic motions: oblique In a resonance-drive piezoelectric motor, there are two types of microscopic motions: oblique and and ellipticalIn a resonance-drive (Figure 5). Excitation piezoelectric of a singlemotor, mode there on are a vibratortwo types is enoughof microscopic for generating motions: an obliqueoblique elliptical (Figure5). Excitation of a single mode on a vibrator is enough for generating an oblique andmotion. elliptical Because (Figure oblique 5). Excitation motion can of havea single components mode on a in vibrator the tangential is enough and for normal generating directions, an oblique a net motion. Because oblique motion can have components in the tangential and normal directions, a net motion.microscopic Because motion oblique is obtained motion whencan have a stator components contact point in the touches tangential a rotor and ornormal a slider directions, contact point. a net microscopic motion is obtained when a stator contact point touches a rotor or a slider contact point. microscopicOn the other motion hand, is excitations obtained whenof two a statororthogon contactal modes point touchesare needed a rotor for orgenerating a slider contact an elliptical point. On the other hand, excitations of two orthogonal modes are needed for generating an elliptical motion. Onmotion. the otherIn a multi-mode hand, excitations excitation of piezoelectrictwo orthogon ultrasal modesonic motor,are needed there isfor elli generatingptical movement an elliptical at the In a multi-mode excitation piezoelectric ultrasonic motor, there is elliptical movement at the interface, motion.interface, In which a multi-mode means that excitation there arepiezoelectric movements ultras in twoonic orthogonal motor, there directions is elliptical at movementthe contact atpoint the which means that there are movements in two orthogonal directions at the contact point and phase interface,and phase which shift betweenmeans that these there movements are movements causes inthe two trajectory orthogonal to be directions elliptical at the contactcontact point.point shift between these movements causes the trajectory to be elliptical at the contact point. This elliptical andThis phase elliptical shift trajectory between causes these movementsthe sliding element causes theto perform trajectory microscopic to be elliptical motion at thein every contact cycle. point. trajectoryThis elliptical causes trajectory the sliding causes element the sliding to perform element microscopic to perform motion microscopic in every motion cycle. in every cycle.

Figure 5. In a resonance-drive type piezoelectric motor, there are elliptical or oblique motions at the FigureFigurestator-rotor 5. 5.In In a a(/slider) resonance-drive resonance-drive contact points. type type piezoelectric piezoelectric motor, motor, there there are are elliptical elliptical or or oblique oblique motions motions at at the the stator-rotorstator-rotor (/slider) (/slider) contactcontact points.points. Among the piezoelectric motors, resonance-drive (or ultrasonic) motors were the actuators that wereAmongAmong studied the the the piezoelectric piezoelectric most. The first motors, motors, idea resonance-drive resonance-driveof converting electrical (or (or ultrasonic) ultrasonic) oscillation motors motors into were weremechanical the the actuators actuators movement that that wereweredates studied studiedback to the the1927 most. most. [48], The Theand ﬁrst firstwe idea canidea see of of converting variousconverting attempts electrical electrical to obtain oscillation oscillation longer, into into mechanical mechanical mechanical motion movement movement using datesdatesthe inverse back back to topiezoelectric 1927 1927 [ 48[48],], and and effect we we [49–55]. can can see see We various various have attempts clattemptsassified to toresonance-drive obtain obtain longer, longer, mechanicalpiezoelectric mechanical motion motionmotors using usingbased thetheon inverse variousinverse piezoelectric piezoelectriccriteria such effect effectas, type [ 49[49–55]. –of55 microscopic]. WeWe have have classiﬁed cl motionassified at resonance-drive resonance-drive the interface, and piezoelectric piezoelectric the method motors motors of direction based based ononchange. various various The criteria criteria interested such such as,reader as, type type can of of microscopicrefer microscopic to our previous motion motion at atpaper the the interface, interf[6]. ace, and and the the method method of of direction direction change.change.In Theliterature, The interested interested we can reader reader see piezoelectric can can refer refer to to our ourmotors previous previous operated paper paper at [ resonance6[6].]. [56–60], where the generated modesInIn literature, literature,are not in we weorthogonal can can see see piezoelectric piezoelectric directions. motorsBecause motors operated optheerated generated at at resonance resonance movement [ 56[56–60],–60 at], the where where contact the the generated pointgenerated is in modesmodesthe tangential areare not notin indirection, orthogonal orthogonal these directions. directions. motors should BecauseBecause be thethe considered generatedgenerated as movementmovement inertia-drive atat thethe types. contactcontact There pointpoint are is isalso in in thethepiezo-walk-drive tangentialtangential direction,direction, motors these thesethat are motorsmotors operated should should at resonance be be considered considered [61,62]. as as inertia-drive inertia-drive types. types. ThereThere areare alsoalso piezo-walk-drivepiezo-walk-driveIn the following motors motors section, that that are are we operated operated introduce at at resonance aresonance newly developed [ 61[61,62].,62]. resonance-drive type piezoelectric motorIn structure, the following where section, the excited we introducemicroscopic a newlymovements developed at the resonance-driveinterface are defined type by piezoelectric parameters motorsuch as structure, the frequency, where the magnitude excited microscopic and phase move shiftments of the at thedriving interface signals, are defined and not by parametersmechanical suchdimensions, as the orfrequency, resonance magnitude modes of aand vibrator. phase With shift these of the parameters, driving signals,not only and oblique not ormechanical elliptical, dimensions,but more complex or resonance microscopic modes movements of a vibrator. at the With interface these parameters, are also possible. not only oblique or elliptical, but more complex microscopic movements at the interface are also possible.

Actuators 2018, 7, 2 6 of 14

In the following section, we introduce a newly developed resonance-drive type piezoelectric motor structure, where the excited microscopic movements at the interface are deﬁned by parameters such as the frequency, magnitude and phase shift of the driving signals, and not mechanical dimensions, or resonance modes of a vibrator. With these parameters, not only oblique or elliptical, but more complex microscopic movements at the interface are also possible. Actuators 2018, 7, 2 6 of 15 2. Structure of the Vibrator and Motor Operating Principle 2. Structure of the Vibrator and Motor Operating Principle A resonance-drive type piezoelectric motor is an octagonal piezoelectric plate, which has a thicknessA resonance-drive of 3.0 mm and type a length piezoelectric (the distance motor betweenis an octagonal any of thepiezoelectric two parallel plate, side which surfaces) has ofa thickness25 mm. This of 3.0 was mm introduced and a length and (the is used distance as the between stator of any the motorof the [two63]. parallel Metal electrodes side surfaces) divide of the25 mm.main This surfaces was introduced of the plate and into is twoused equal as the regions. stator of The the electrodesmotor [63]. on Metal one electrodes main surface divide are arrangedthe main surfacesperpendicular of the toplate the electrodesinto two onequal the regions. other main The surface electrodes (Figure on6 a).one Two main alumina surface eyelets, are arranged used as perpendicularthe friction contact to the elements, electrodes are on attached the other symmetrically main surface at (Figure the center 6a). ofTwo the alumina top and eyelets, bottom surfacesused as the(Figure friction6b). contact elements, are attached symmetrically at the center of the top and bottom surfaces (Figure 6b).

FigureFigure 6. 6. ((aa)) Metallization Metallization electrodes electrodes are are oriented oriented orthogon orthogonallyally on on the the main main surfaces surfaces of of the the octagonal octagonal piezoelectricpiezoelectric plate. (b)) TheThe couplingcoupling elements elements are are two two eyelets eyelets that that are are attached attached in thein the middle middle of the of twothe twomain main surfaces surfaces on the on octagonalthe octagonal piezoelectric piezoelectric plate. plat Thee. The side side surfaces surfaces of the of the eyelets eyelets (marked (marked with with red redarrows) arrows) are are in contact in contact with with the slider.the slider.

TheThe driving driving signals signals on on the the stator stator are are applied applied betw betweeneen the the two two surface surface electrodes electrodes on on both both faces. faces. WhenWhen a a signal signal is is applied between between the the two two top top el electrodes,ectrodes, we we can can assume assume that that one one electrode electrode has has A cos(cos (w)1t ) andand the the other other has has −−A cos(cos (w)1 t signals.) signals. Similarly, Similarly, a second a second signal signal is applied is applied between between the twothe twobottom bottom electrodes, electrodes, when when one onebottom bottom electrode electrode has has cos(B cos (w−)2t − ϕ, ),and and the the other other has has −−B cos(cos (w2−)t − ϕ) signals (Figure(Figure6a). 6a). Even Even if the if appliedthe app signalslied signals are between are between the two surfacethe two electrodes, surface electrodes,the resulting the electric resulting ﬁelds electric are between fields are the between top and the the top bottom and the electrodes bottom electrodes in the thickness in the thickness direction. direction.Assuming Assuming that the thickness that the thickness of the octagonal of the octagonal plate is d plate, we canis d, write we can the write electric the ﬁeldelectric for field the four for thequarter four regionsquarter asregions follows. as follows. 1 1 E = cos[A cos −w t cos(− B cos−)(w t − ϕ)] I d 1 2 11 EII = [Acoscos w1t + B cos(cos( w2−)t − ϕ)] d 1 EIII =1 [−A cos w1t + B cos(w2t − ϕ)] d− cos cos( −) 1 E = [−A cos w t − B cos(w t − ϕ)] IV 1d 1 2 − cos − cos( −)

When the magnitudes A, B and thickness d are normalized to 1, the frequencies ( and ) of the signals on the top and on the bottom electrodes are equal and the phase angle () is set to 9 degrees. Because of the “sum to product trigonometric identity”, the electric field equations can be rewritten as:

cos − cos( − 90°) −2sin( − 45°) sin(45°)

cos cos( − 90°) 2cos( − 45°) cos(45°)

− cos cos( − 90°) 2sin( − 45°) sin(45°)

Actuators 2018, 7, 2 7 of 14

When the magnitudes A, B and thickness d are normalized to 1, the frequencies (w1 and w2) of the Actuatorssignals 2018 on the, 7, 2top and on the bottom electrodes are equal and the phase angle (ϕ) is set to 9 degrees.7 of 15 Because of the “sum to product trigonometric identity”, the electric ﬁeld equations can be rewritten as:

◦ ◦ ◦ E−I = cos [cos w −1t cos(− cos (w −1t 90°)− 90 )] −2= − cos(2 sin (w −1t 45°)− 45 cos(45°)) sin (45 ) Since sin(45°) cos(45°) 0.707, the magnitude of the normalized electric fields are 1.414, = [ + ( − ◦)] = ( − ◦) ( ◦) which means that theE IIplatecos is exposedw1t cos to wa 142%t 90 higher electric2 cos w 1field.t 45 Moreover,cos 45 the phase difference ◦ ◦ ◦ of the normalized electricEIII = [field− cos betweenw1t + cos any(w 1oft −the90 tw)]o =adjacent2 sin (w regions1t − 45 is) sinstill( 4590 )degrees (Figure 7). When the phase shift () is 0 degrees, one can easily find that section I and III have zero electric = [− − ( − ◦)] = − ( − ◦) ( ◦) fields, but the magnitudeEIV ofcos thew 1normalizedt cos w2t electric90 field2 in cos sectionw1t II45 andcos IV 45is 2. Similarly, when phaseSince shift sin()( 45is ◦180) = degrees,cos (45◦) section = 0.707 II, theand magnitude IV have zero of theelectric normalized fields and electric the magnitude fields are ± of1.414 the , normalizedwhich means electric that the field plate in issection exposed I and to aIII 42% is zero. higher For electric phase field. shift Moreover,() of other the values, phasedifference the electric of fieldsthe normalized in these regions electric change field between accordingly. any of the two adjacent regions is still 90 degrees (Figure7).

Figure 7. Electric ﬁelds in the four regions of the octagonal piezoelectric plate when the magnitudes Figure 7. Electric fields in the four regions of the octagonal piezoelectric plate when the magnitudes A, B and thickness d are normalized to 1. Frequencies (w1 and w2) for both signals are assumed to be A, B and thickness d are normalized to 1. Frequencies ( and ) for both signals are assumed to be equal and T is the period (phase shift (ϕ) is 90 degrees). equal and T is the period (phase shift () is 90 degrees).

When the phase shift (ϕ) is 0 degrees, one can easily find that section I and III have zero electric When only the bottom electrodes are excited with the signals ( cos(−) and fields, but the magnitude of the normalized electric field in section II and IV is 2. Similarly, when phase − cos(−)), leaving the top electrodes floating, the in-plane modes are excited only in the x- axisshift direction (ϕ) is 180 (note degrees, that sectionfrequencies II and of IVthe have signal zeros between electric the fields top and and the bottom magnitude electrodes of the are normalized assumed toelectric be the field same). in sectionThe corresponding I and III is zero. first Forand phasesecond shift in-plane (ϕ) of mode other shapes, values, which the electric were fieldscalculated in these by usingregions ATILA-GID change accordingly. software (ATILA-GID 2.0.0, Micromechatronics Inc, State College, PA, USA), are B (w t − ϕ) shownWhen in Figure only 8. When the bottom the signals electrodes ( cos( are) and excited− cos( with)) theare applied signals in ( betweencos 1 the two andtop electrodes,−B cos (w1 onlyt − ϕ the)), leavingmodes in the the top y-axis electrodes direction floating, are excited. the in-plane The corresponding modes are first excited and onlysecond in thein- planex-axis mode direction shapes (note in the that y-axis frequencies direction of are the shown signals in betweenFigure 9. theBecause top andthe electrodes bottom electrodes on the main are surfacesassumed are to identical be the same). but orthogonal, The corresponding the excited first in-plane and second modes in-plane are also mode identical shapes, but orthogonal. which were Whencalculated the driving by using signals ATILA-GID are applied software at the (ATILA-GID same time by 2.0.0, setting Micromechatronics the phase shift ( Inc,) to State 90 degrees, College, the PA, resultingUSA), are movement shown in Figureis nothing8. When but a the hula-hoop signals ( trAajectory.cos (w1t )Indeed,and −A thecos microscopic(w1t)) are applied movement in between on the surfacethe two of top the electrodes, center eyelets only can the modesbe controlled in the y full-axisy by direction configuring are excited. the magnitude The corresponding and phase firstshift and of thesecond top and in-plane the bottom mode signals. shapes in the y-axis direction are shown in Figure9. Because the electrodes on the main surfaces are identical but orthogonal, the excited in-plane modes are also identical but orthogonal. When the driving signals are applied at the same time by setting the phase shift (ϕ) to 90 degrees, the resulting movement is nothing but a hula-hoop trajectory. Indeed, the microscopic movement on the surface of the center eyelets can be controlled fully by configuring the magnitude and phase shift of the top and the bottom signals.

Actuators 2018, 7, 2 8 of 14 Actuators 2018, 7, 2 8 of 15

Actuators 2018, 7, 2 8 of 15

Figure 8.8. In-planeIn-plane ﬁrst first and and second second mode mode shapes shapes when wh onlyen the only bottom the surfacebottom electrodes surface electrodes are electrically are Figure 8. In-plane first and second mode shapes when only the bottom surface electrodes are electricallyexcited. Deformations excited. Deformations are in the x-axis are direction.in the x-axis (a) Firstdirection. in-plane (a) modeFirst in-plane at 68 kHz, mode (b) Second at 68 kHz, in-plane (b) electrically excited. Deformations are in the x-axis direction. (a) First in-plane mode at 68 kHz, (b) Second in-plane mode at 154.5 kHz. modeSecondFigure at 154.5 in-plane8. kHz.In-plane mode first at and154.5 second kHz. mode shapes when only the bottom surface electrodes are electrically excited. Deformations are in the x-axis direction. (a) First in-plane mode at 68 kHz, (b) Second in-plane mode at 154.5 kHz.

FigureFigure 9. In-plane 9. In-plane first first and and second second mode mode shapesshapes when when on onlyly the the top top electrodes electrodes are areelectrically electrically excited. excited. FigureDeformations 9. In-plane arefirst in and the ysecond-axis direction. mode shapes (a) First when in-plane on modely the at top 68 electrodeskHz, (b) Second are electrically in-plane mode excited. Deformations are in the y-axis direction. (a) First in-plane mode at 68 kHz, (b) Second in-plane mode DeformationsatFigure 154.5 9. kHz. In-plane are in the first y -axisand second direction. mode (a shapes) First whenin-plane only mode the top at electrodes 68 kHz, (areb) Secondelectrically in-plane excited. mode at 154.5 kHz. at 154.5Deformations kHz. are in the y-axis direction. (a) First in-plane mode at 68 kHz, (b) Second in-plane mode Whenat 154.5 the kHz. magnitudes (A and B) at the top and the bottom signals are set to 1 V, the calculated Whendisplacements the magnitudes are seen ( (inAA andandFigure BB)) at10. at the theThese top top diand andsplacements the the bottom bottom were signals signals calculated are are set set using to to 1 1 V, V,ATILA-GID the calculated displacementssoftware,When on aretheare the seen magnitudes sideseen inof inFigurethe Figureeyelet (A 10and .at These B10.the) at firstThese the displacements in-ptop dilaneandsplacements resonancethe bottom were mode calculatedsignalswere (at arecalculated68.7 set using kHz), to 1 ATILA-GID and V,using the at thecalculated ATILA-GID second software, displacements are seen in Figure 10. These displacements were calculated using ATILA-GID software,on thein-plane side on of theresonance the side eyelet of mode the at theeyelet (at first154.5 at in-plane kHz).the first Magnitude resonancein-plane of resonance the mode displace (at mode 68.7ment kHz), under(at 68.7 and the kHz), same at the and electric second at the fields in-plane second atsoftware, the second on the in-plane side of resonance the eyelet mode at the is first larger in-p thanlane theresonance displacement mode (at at the68.7 first kHz), in-plane and at mode. the second This in-planeresonance resonance mode (at mode 154.5 (a kHz).t 154.5 Magnitude kHz). Magnitude of the displacement of the displace underment theunder same the electric same electric fields atfields the resultin-plane is resonanceexpected because mode (a maximumt 154.5 kHz). displacement, Magnitude of as the seen displace in Figuresment under8 and the9 at same the firstelectric in-plane fields second in-plane resonance mode is larger than the displacement at the first in-plane mode. This result at theresonance,at second the second in-plane is atin-plane the resonanceside resonance surface mode of mode the isoctagonal is larger larger thanthan plate. thethe displacement displacement at atthe the first first in-plane in-plane mode. mode. This This resultis expectedresult is expected is because expected because maximum because maximum maximum displacement, displacement, displacement, as seen inasas Figuresseenseen in in 8Figures Figuresand9 at8 and the8 and first9 at 9 the in-planeat thefirst firstin-plane resonance, in-plane resonance,is at theresonance, side is surfaceat is the at theside of side the surf surf octagonalaceace of of the the plate. octagonal octagonal plate.

Figure 10. Calculated motion trajectories at first (68.7 kHz) and second (154.5 kHz) resonance modes.

Figure 10. Calculated motion trajectories at first (68.7 kHz) and second (154.5 kHz) resonance modes. Figure 10. Calculated motion trajectories at ﬁrst (68.7 kHz) and second (154.5 kHz) resonance modes.

Figure 10. Calculated motion trajectories at first (68.7 kHz) and second (154.5 kHz) resonance modes.

Actuators 2018, 7, 2 9 of 14

Actuators 2018, 7, 2 9 of 15 Movement trajectories on the side of the eyelet at 68.7 kHz for three driving conditions are seen in Figure 11. AtMovement this frequency, trajectories the on displacement the side of the eyelet generated at 68.7 onkHz the for three side driving surface conditions of the coupling are seen elements (eyelets) isin circular. Figure 11. When At this the frequency, magnitude the displaceme of the signalnt generated applied on on the the side top surface and onof the coupling bottom electrodes is the sameelements (A = B (eyelets)= 1 V) is and circular. the phaseWhen the shift magnitude (ϕ) between of the signal the twoapplied signals on the istop 90 and degrees, on the bottom the trajectory is electrodes is the same (A = B = 1 V) and the phase shift () between the two signals is 90 degrees, the a circle. Whentrajectory the is magnitude a circle. When of the the magnitude applied signalof the applied in between signal in the between top electrodes the top electrodes is doubled is (in this case: A =doubled 2 V, B (in= 1this V), case: only A = the2 V, B magnitude = 1 V), only the of magnitude the displacement of the displacement in the iny-axis the y-axis direction direction is doubled. Similarly,is When doubled. the Similarly, magnitude When of the the magnitude applied signalof the applied between signal the between bottom the electrodes bottom electrodes is doubled is (in this case: A =doubled 1 V, B = (in 2 this V), case: only A the = 1 V, magnitude B = 2 V), only of the the magnitude displacement of the displacement in the x-axis in the direction x-axis direction is doubled. is doubled.

Figure 11. Calculated motion trajectories, when the magnitude of the signals applied on the top or on Figure 11.theCalculated bottom electrodes motion is changed. trajectories, when the magnitude of the signals applied on the top or on the bottom electrodes is changed. When the phase angle () between the signals on the top and on the bottom electrodes is changed from 0 to 180 degrees by setting the magnitudes (A and B) to 1 V, the shape of the elliptical motion Whenalso the changes phase (Figure angle 12). (ϕ) Note between that when the signalsthe phas one angle the is top 0 degrees, and on the the mo bottomtion is in electrodes the oblique is changed from 0 todirection. 180 degrees This is by because setting there the are magnitudeselectric fields on (lyA inand the twoB) todiagonal 1 V, the regions shape (II and of IV the as marked elliptical motion also changesin Figure (Figure 6). In 12the). other Note two that diagonal when areas the (regions phase Iangle and III), is the 0 degrees,electric fields the are motion zero. When is in the the oblique phase angle is 180 degrees, the motion is again in the oblique direction. In this case, there are electric direction.fields This only is because in the two there diagonal are regions electric (I fieldsand III), only and the in theother two two diagonaldiagonal regions regions (II and (II andIV) have IV as marked in Figure6zero). In electric the other field. two diagonal areas (regions I and III), the electric fields are zero. When the phase angle is 180 degrees, the motion is again in the oblique direction. In this case, there are electric fields only in the two diagonal regions (I and III), and the other two diagonal regions (II and IV) have zero electricActuators field. 2018, 7, 2 10 of 15

Figure 12. Response of the trajectory to the phase shift between the signals on the top and bottom Figure 12.electrodes.Response of the trajectory to the phase shift between the signals on the top and bottom electrodes. 2.1. Structure of the Linear Motor In the test motor, the piezoelectric vibrating element was placed between two identical plastic holders made from PEEK material and placed together in a steel housing. The steel housing was guided by two linear bearings in the pre-stress direction. The two linear bearings were fixed on a base plate. A slider that has a U-shaped cross-section guided by another larger linear bearing was also fixed on the base plate. Two alumina rods were attached to both ends of the U-shaped slider. The side surfaces of the two eyelets on the vibrator and the two alumina rods on the slider were in contact tangentially (Figure 13). Two or three springs applied pressure against the slider. In the test motor, a position sensor was embedded in the base plate and a reflective surface was attached to the side surface of the U-shaped slider (Figure 14).

Figure 13. Perspective and end views of the vibrator–slider interface. The side surfaces of both eyelets were contacted tangentially to the alumina rods on the U-shaped slider. The distance between any of the two parallel side surfaces of the plate was 25 mm and the thickness was 3.0 mm.

Figure 14. Perspective views (3D CAD drawing and photo) of the test linear motor. The piezoelectric vibrating element was placed between two identical holders.

Actuators 2018, 7, 2 10 of 15 Actuators 2018, 7, 2 10 of 15

ActuatorsFigure2018 12., 7, 2Response of the trajectory to the phase shift between the signals on the top and bottom10 of 14 Figureelectrodes. 12. Response of the trajectory to the phase shift between the signals on the top and bottom electrodes. 2.1. Structure of the Linear Motor 2.1. Structure of the Linear Motor InIn thethe testtest motor,motor, thethe piezoelectricpiezoelectric vibratingvibrating elementelement waswas placedplaced betweenbetween twotwo identicalidentical plasticplastic holdersIn the made test frommotor, PEEKPEEK the piezoelectric materialmaterial andand vibrating placedplaced togethertogether element inin was aa steel steelplaced housing.housing. between The two steel identical housinghousing plastic waswas holdersguided bybymade twotwo from linear linear PEEK bearings bearings material in in the the and pre-stress pre-stress placed direction. togetherdirection. The in The a two steel two linear housing. linear bearings bearings The were steel were fixed housing fixed on a baseonwas a guidedplate.base plate. A by slider twoA slider thatlinear hasthat bearings a has U-shaped a U-shapedin the cross-section pre-stress cross-section di guidedrection. guided byThe another bytwo another linear larger bearingslarger linear linear bearing were bearing fixed was on alsowas a basefixedalso fixedplate. on the onA base sliderthe plate.base that plate. Two has alumina aTwo U-shaped alumina rods cross-section were rods attached were attachedguided to both by to ends anotherboth of theends larger U-shaped of the linear U-shaped slider. bearing The slider. was side alsosurfacesThe fixedside ofsurfaces on the the two base of eyeletsthe plate. two on Twoeyelets the alumina vibrator on the rodsvibrator and thewere twoand attached aluminathe two to alumina rodsboth onends rods the of slider onthe the U-shaped were slider in were contact slider. in Thetangentiallycontact side tangentially surfaces (Figure of the(Figure13). two Two 13).eyelets or Two three on or springs the three vibrator spring applied ands applied pressure the two pressure againstalumina against the rods slider. onthe the Inslider. theslider testIn werethe motor, test in contactamotor, position atangentially position sensor sensor was (Figure embedded was 13). embedded Two inthe or three basein the platespring base andplates applied a and reflective apressure reflective surface against surface was theattached was slider. attached toIn thethe to sidetest the motor,surfaceside surface a of position the of U-shaped the sensor U-shaped sliderwas embeddedslider (Figure (Figure 14 in). the14). base plate and a reflective surface was attached to the side surface of the U-shaped slider (Figure 14).

Figure 13. Perspective and end views of the vibrator–slider interface. The side surfaces of both eyelets Figure 13. Perspective and end views of the vibrator–slider interface. The side surfaces of both eyelets werewere contactedcontacted tangentiallytangentially toto the alumina rods on the U-shaped slider. The distance between any of were contacted tangentially to the alumina rods on the U-shaped slider. The distance between any of thethe twotwo parallelparallel sideside surfacessurfaces ofof thethe plateplate waswas 2525 mmmm andand thethe thicknessthickness waswas 3.03.0 mm.mm. the two parallel side surfaces of the plate was 25 mm and the thickness was 3.0 mm.

Figure 14. Perspective views (3D CAD drawing and photo) of the test linear motor. The piezoelectric Figure 14. Perspective views (3D CAD drawing and photo) of the test linear motor. The piezoelectric Figurevibrating 14. elementPerspective was viewsplaced (3D between CAD drawingtwo identical and photo)holders. of the test linear motor. The piezoelectric vibrating element was placed between two identical holders. vibrating element was placed between two identical holders.

Two phase sinusoidal waveform signals with independent magnitude and phase shift were generated by a “Multifunction Synthesizer” (WF1946B, NF Corp., Yokohama, Japan) and amplified by two power amplifiers (HAS 4011 and HAS4014, NF Corp.). Before applying the two signals to the test motor, two electromagnetic transformers, with their primary and secondary side turn ratio of 1:1, were used to shift the common grounds to a floating state. In this case, when the signal on one of the top electrodes was cos (w1t) , the other signal on the other top electrode would be −A cos(w1t). Magnitudes of these signals control the displacement in the normal direction. The second signal, with a phase shift of 90 degrees from the other power amplifier, was also applied first to an electromagnetic transformer and applied to bottom electrodes. The signals on the bottom electrodes (B cos (w1t − 90) and −(B cos (w1t − 90)) control displacement in the tangential direction. The slider position data were collected by a PC. The position data collected were used to calculate the motor control input and speed, and load characteristics. Actuators 2018, 7, 2 11 of 14

2.2. Speed Characteristics Conventionally, the speed of an ultrasonic motor for one- or two-phase driving type resonance piezoelectric motors is controlled by changing the magnitude of the driving signals, or the phase difference between the two signals, which results in microscopic movements in the normal and tangential directions at the interface, to increase or decrease at the same time. The disadvantage of both driving methods is nonlinearity, such as dead-zone and hysteresis, especially in the low-speed region. In this motor, microscopic motion at the contact points in the tangential and normal directions is generated by the driving signals on the top and on the bottom electrodes, respectively. When the corresponding electrodes that are responsible for generating tangential displacements are excited, displacement is generated only in a tangential direction. Similarly, when the corresponding electrodes that are responsible for generating normal displacements are excited, only normal displacement is generated. Since the magnitude of a normal direction displacement is responsible for the generated force and the magnitude of a tangential displacement is responsible for the slider speed, these two parameters in this motor are controlled independently. In order to clarify the advantages of independent driving, the motor was driven in the following two different cases. Case 1: The control input in this case was the driving signals of which one was applied between the two top and the other was applied between the two bottom electrodes. The signals on the top and the bottom electrodes were changed with equal magnitudes at the same time while keeping the phase shift between them at 90 degrees. As can be seen in Figure 15, with the speed–driving voltage (control input) curve, the threshold voltage was relatively large, and with the increase of pre-stress, the threshold voltage further increased. When the pre-stress was 35 N, the slider started to move at 60 V. When the pre-stress was increased to 70 N, the starting voltage of the slider increased to 100 V. Case 2: The magnitude of the signal that is responsible for making normal displacement was at maximum level (160 V peak-to-peak). According to the actuator orientation, as seen in Figure 14, the signal that was applied between the two top electrodes generated displacement in the normal direction. The other signal that was applied between the two bottom electrodes generated tangential displacement, which was the control input in “Case 2” driving. Speed–control input curves were also obtained for the two different pre-stress conditions at 35 N and 70 N. As can be seen in Figure 15, case 2 driving had smaller dead-zones, or smaller threshold voltages. When the pre-stress force was 35 N, the control input threshold voltage decreased to 5 V from 60 V. Although, the pre-stress was increased to 70 N, the control input threshold voltage decreased to 10 V from 100 V. Actuators 2018, 7, 2 12 of 15

(a) (b)

Figure 15. Speed–driving voltage (control input) characteristics for two cases of driving. (a) Pre-stress Figure 15. Speed–driving voltage (control input) characteristics for two cases of driving. (a) Pre-stress in the normal direction: 35 N, (b) Pre-stress in the normal direction: 70 N. in the normal direction: 35 N, (b) Pre-stress in the normal direction: 70 N.

2.3. Load Characteristics In order to obtain load characteristics, the test motor was driven under different load conditions (1, 5, 10, 20 N) and a series of position data was collected by the embedded encoder. When the magnitude of the signal responsible for generating displacement in the normal direction was 160 V, and the magnitude of the signal responsible for generating displacement in the tangential direction was 140 V (peak to peak), the motor speeds were calculated from the measured position data. A typical load characteristic of the test motor is shown in Figure 16. Even though the main purpose in this study was to reduce speed–control input (driving voltage) nonlinearities, the test motor can produce a maximum force of 20 N and a maximum speed of 70 mm/s.

Figure 16. Load characteristics of the test linear motor. Sinusoidal wave signals generating normal and tangential displacements have magnitudes of 160 and 140 V (peak-to-peak), respectively.

3. Conclusions Although there are piezoelectric motors with various structures, there are not many operating principles which cause them to be categorized as piezo-walk-drive, inertia-drive and resonance-drive types. In the case of the piezo-walk-drive, normal and tangential displacements (thus forces) at the interface are performing “clamp-release-move” or “while clamping push” steps. In the case of the inertia-drive, only a tangential displacement or a direction-dependent tangential force change at the interface is enough to obtain movement. For the resonance-drive, microscopic displacement at the interface has tangential and normal components, where magnitudes and phase difference between these displacements determine the direction of microscopic movements. The resonance-drive type piezoelectric motor introduced in this study covers both single and multi-mode operation types. Oblique motion at the interface is just a special case when the phase difference is either 0 or 180 degrees.

Actuators 2018, 7, 2 12 of 15

(a) (b)

ActuatorsFigure2018 15., 7, Speed–driving 2 voltage (control input) characteristics for two cases of driving. (a) Pre-stress12 of 14 in the normal direction: 35 N, (b) Pre-stress in the normal direction: 70 N.

2.3. Load Characteristics In order to obtain load characteristics, the test motor was driven under different load conditions (1,(1, 5,5, 10,10, 2020 N)N) andand aa seriesseries ofof positionposition datadata waswas collectedcollected byby thethe embeddedembedded encoder.encoder. When the magnitude of the signal responsible for generating displacement displacement in in the the normal normal direction direction was was 160 160 V, V, and the magnitude of the signalsignal responsible for generatinggenerating displacement in the tangential direction was 140140 VV (peak (peak to to peak), peak), the the motor motor speeds speeds were were calculated calculated from from the measured the measured position position data. A data. typical A loadtypical characteristic load characteristic of the testof the motor test motor is shown is sh inown Figure in Figure 16. Even 16. Even though though the main the main purpose purpose in this in studythis study was towas reduce to reduce speed–control speed–control input (driving input (drivi voltage)ng voltage) nonlinearities, nonlinearities, the test motor the test can motor produce can a maximumproduce a maximum force of 20 force N and of a 20 maximum N and a maximum speed of 70 speed mm/s. of 70 mm/s.

Figure 16. Load characteristics of the test linear motor. Sinusoidal wave signals generating normal Figure 16. Load characteristics of the test linear motor. Sinusoidal wave signals generating normal and and tangential displacements have magnitudes of 160 and 140 V (peak-to-peak), respectively. tangential displacements have magnitudes of 160 and 140 V (peak-to-peak), respectively. 3. Conclusions 3. Conclusions Although there are piezoelectric motors with various structures, there are not many operating Although there are piezoelectric motors with various structures, there are not many operating principles which cause them to be categorized as piezo-walk-drive, inertia-drive and resonance-drive principles which cause them to be categorized as piezo-walk-drive, inertia-drive and resonance-drive types. In the case of the piezo-walk-drive, normal and tangential displacements (thus forces) at the types. In the case of the piezo-walk-drive, normal and tangential displacements (thus forces) at the interface are performing “clamp-release-move” or “while clamping push” steps. In the case of the interface are performing “clamp-release-move” or “while clamping push” steps. In the case of the inertia-drive, only a tangential displacement or a direction-dependent tangential force change at the inertia-drive, only a tangential displacement or a direction-dependent tangential force change at the interface is enough to obtain movement. For the resonance-drive, microscopic displacement at the interface is enough to obtain movement. For the resonance-drive, microscopic displacement at the interface has tangential and normal components, where magnitudes and phase difference between interface has tangential and normal components, where magnitudes and phase difference between these displacements determine the direction of microscopic movements. these displacements determine the direction of microscopic movements. The resonance-drive type piezoelectric motor introduced in this study covers both single and The resonance-drive type piezoelectric motor introduced in this study covers both single and multi-mode operation types. Oblique motion at the interface is just a special case when the phase multi-mode operation types. Oblique motion at the interface is just a special case when the phase difference is either 0 or 180 degrees. difference is either 0 or 180 degrees. Microscopic movement at the stator–slider interface of the new resonance piezoelectric motor is fully controllable, with the phase shifts and magnitude of the surface driving signals on the octagonal plate. By driving the top and bottom electrodes with different frequencies (w1 and w2) at the interface, even complex movements, other than oblique or elliptical, are also possible.

Author Contributions: Two authors write this paper together. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Uchino, K.; Giniewica, J.R. Micromechatronics; Marcel Dekker Inc.: New York, NY, USA, 2003. 2. Ervin, E. (Ed.) Vibro-Motors for Precision Micro-Robots; Hemisphere Publishing Co.: Carlsbad, CA, USA, 1988. 3. Ueha, S.; Tomikawa, Y. Ultrasonic Motors—Theory and Applications; Oxford University Press: Oxford, UK, 1993. Actuators 2018, 7, 2 13 of 14

4. Sashida, T.; Kenjo, T. An Introduction to Ultrasonic Motors; Oxford University Press: Oxford, UK, 1993. 5. Zhao, C. Ultrasonic Motors: Technologies and Applications; Science Press: Beijing, China; Springer: Berlin, Germany, 2011. 6. Spanner, K.; Koc, B. Piezoelectric Motors, an Overview. Actuators 2016, 5, 6. [CrossRef] 7. Hunstig, M. Piezoelectric Inertia Motors—A Critical Review of History, Concepts, Design, Applications, and Perspectives. Actuators 2017, 6, 7. [CrossRef] 8. Peng, Y.; Peng, Y.; Gu, X.; Wang, J.; Yu, H. A Review of Long Range Piezoelectric Motors using Frequency Leveraged Method. Sens. Actuators A Phys. 2015, 235, 240–255. [CrossRef] 9. Zhang, Z.M.; An, Q.; Li, J.W.; Zhang, W.J. Piezoelectric Friction–Inertia Actuator—A Critical Review and Future Perspective. Int. J. Adv. Manuf. Technol. 2012, 62, 669–685. [CrossRef] 10. Morita, T. Miniature Piezoelectric Motors. Sens. Actuators A Phys. 2003, 103, 291–300. [CrossRef] 11. Watson, B.; Friend, J.; Yeo, L. Piezoelectric Ultrasonic Micro/Milli Scale Actuators. Sens. Actuators A Phys. 2009, 152, 219–233. [CrossRef] 12. We Evolve Our Ultrasonic Motor. Available online: http://www.shinsei-motor.com/English/index.html (accessed on 10 August 2017). 13. Nanomotion. Available online: http://www.nanomotion.com/ (accessed on 10 August 2017). 14. PI Products. Available online: https://www.physikinstrumente.com/en/products/ (accessed on 10 August 2017). 15. Piezomotor. Available online: http://www.piezomotor.com/ (accessed on 10 August 2017). 16. Newscaletech. Available online: https://www.newscaletech.com/ (accessed on 10 August 2017). 17. Brisbane, A.D. Position Control Device. U.S. Patent 3,377,489, 9 April 1965. 18. Hsu, K.; Biatter, A. Transducer. U.S. Patent 3,292,019, 13 December 1966. 19. Locher, G.L. Micrometric Linear Actuator. U.S. Patent 3,296,467, 3 January 1967. 20. Galutva, G.V.; Ryazantsev, A.I.; Presnyakov, G.S.; Modestov, J.K. Device for Precision Displacement of a Solid Body. U.S. Patent 3,684,904, 15 April 1970. 21. May, W.G. Piezoelectric Electromechanical Translation Apparatus. U.S. Patent 3,902,084, 30 May 1974. 22. Shamoto, E.; Shin, H.; Moriwaki, T. Development of Precision Feed Mechanism by Applying Principle of Walking Drive (1st Report)—Principle and Basic Performance of Walking Drive. J. JSPE 1993, 59, 317–322. (In Japanese) 23. Marth, H.; Gloess, R. Verstelleantrieb aus Bimorphelementen. DE Patent 4,408,618A1, 15 March 1994. 24. Bexell, M.; Johansson, S. Characteristics of a Piezoelectric Miniature Motor. Sens. Actuators A Phys. 1999, 75, 118–130. [CrossRef] 25. Li, J.; Zhou, X.; Zhao, H.; Shao, M.; Hou, P.; Xu, X. Design and Experimental Performances of a Piezoelectric Linear Actuator by Means of Lateral Motion. Smart Mater. Struct. 2015, 24, 065007. [CrossRef] 26. Cheng, T.; He, M.; Li, H.; Lu, X.; Zhao, H.; Gao, H. A Novel Trapezoid-Type Stick-Slip Piezoelectric Linear Actuator Using Right Circular Flexure Hinge Mechanism. IEEE Trans. Ind. Electron. 2017, 64, 5545–5552. [CrossRef] 27. Stibitz, R. Incremental Feed Mechanisms. U.S. Patent 3,138,749, 23 June 1964. 28. Thaxter, J.B. Piezoelectric Wafer Mover. U.S. Patent 4,195,243, 25 May 1980. 29. Pohl, D.W. Dynamic Piezoelectric Transducer Devices. Rev. Sci. Instrum. 1987, 58, 54. [CrossRef] 30. Higuchi, T.; Watanabe, M. Apparatus for Effecting Fine Movement by Impact Force Produced by Piezoelectric or Electrostrictive Element. U.S. Patent 4,894,579, 29 May 1987. 31. Niedermann, P.; Emch, R.; Descouts, P. Simple Piezoelectric Translation Device. Rev. Sci. Instrum. 1988, 59, 368–369. [CrossRef] 32. Yamagata, Y.; Higuchi, T.; Saeki, H.; Ishimaru, H. Ultrahigh Vacuum Precise Positioning Device Utilizing Rapid Deformations of Piezoelectric Elements. J. Vac. Sci. Technol. A 1990, 8, 4098–4100. [CrossRef] 33. Renner, C.H.; Niedermann, P.H.; Kent, A.D.; Fischer, O. A Vertical Piezoelectric Inertial Slider. Rev. Sci. Instrum. 1990, 61, 965–967. [CrossRef] 34. Howald, L.; Rudin, R.; Guntherodt, H.-J. Piezoelectric Inertial Stepping Motor with Spherical Rotor. Rev. Sci. Instrum. 1992, 63, 3909–3912. [CrossRef] 35. Blackford, B.L. A Simple Self-Propelled Two-Dimensional Micropositioner. Rev. Sci. Instrum. 1993, 64, 1360–1361. [CrossRef] 36. Tapson, J.; Greene, J.R. A Simple Dynamic Piezoelectric x-y Translation Stage Suitable for Scanning Probe Microscopes. Rev. Sci. Instrum. 1993, 64, 2387–2388. [CrossRef] Actuators 2018, 7, 2 14 of 14

37. Libioulle, L.; Ronda, A.; Derycke, I.; Vigneron, J.P.; Gilles, J.M. Vertical Two-Dimensional Piezoelectric Inertial Slider for Scanning Tunneling Microscope. Rev. Sci. Instrum. 1993, 64, 1489–1494. [CrossRef] 38. Miyano, M.; Kawasaki, T.; Kuwana, M.; Miyazawa, M.; Ueyama, M.; Tasaka, Y. Lens Moving Apparatus. U.S. Patent 5,587,846, 14 July 1995. 39. Kleindiek, S. Electromechanical Drive Element Comprising a Piezoelectric Element. U.S. Patent 6,741,011, 3 March 2000. 40. Karrai, K.; Heines, M. Inertial Rotation Device. U.S. Patent 6,940,210B2, 22 November 2001. 41. Meyer, C.; Sqalli, O.; Lorenz, H.; Karrai, K. Slip-Stick Step-Scanner for Scanning Probe Microscopy. Rev. Sci. Instrum. 2005, 76, 063706. [CrossRef] 42. Müller, K.D.; Marth, H.; Pertsch, R.; Gloss, R.; Zhao, X. Piezo-Based, Long-Travel Actuators for Special Environmental Conditions. In Proceedings of the 10th International Conference on New Actuators, Bremen, Germany, 14–16 June 2006; pp. 149–153. 43. Thomas, P.; Desailly, R. Optical Adjustment Mounts with Piezoelectric Inertia Driver. U.S. Patent WO 2008087469A2, 18 January 2007. 44. Huebner, R. Piezoelectric rotary Drive for A Shaft. U.S. Patent 9,106,158, 1 August 2012. 45. Kortschack, A.; Rass, C. Method for Controlling A Multi-Actuator Drive. German Patent WO2013128032A1, 4 March 2013. 46. Hunstig, M.; Hemsel, T.; Sextro, W. Stick-Slip and Slip-Slip Operation of Piezoelectric Inertia Drives. Part I: Ideal Excitation. Sens. Actuators A Phys. 2013, 200, 90–100. [CrossRef] 47. Hunstig, M.; Hemsel, T.; Sextro, W. Stick-Slip and Slip-Slip Operation of Piezoelectric Inertia Drives—Part II: Frequency-Limited Excitation. Sens. Actuators A Phys. 2013, 200, 79–89. [CrossRef] 48. Meissner, A. Converting Electrical Oscillations into Mechanical Movement. U.S. Patent 1,804,838, 26 May 1927. 49. Williams, A.L.W.; Brown, W.J. Piezoelectric Motor. U.S. Patent 2,439,499, 13 April 1948. 50. Lavrinenko, V.; Nekrasov, M. Electrical Motor. USSR Patent 217,509, 10 May 1965. 51. Tehon, S.W. Electromechanical Transducer Motors. U.S. Patent 3,211,931, 12 October 1965. 52. Wischnewsky, V.; Kavertsev, V.L.; Kartashev, I.A.; Lavrinenko, V.; Nekrasov, M.; Prez, A.A. Piezoelectric Motor Structures. U.S. Patent 4,019,073, 10 April 1975. 53. Barth, H.V. Ultrasonic Drive Motor. IBM Tech. Discl. Bull. 1973, 16, 2263. 54. Wischnewskiy, V.; Gultajeva, L.; Kartashev, I.; Lavrinenko, V. Piezoelectric Motor. USSR Patent 851,560, 16 February 1976. 55. Bansiavichus, R. Piezoelectric Motor. USSR Patent 693,493, 18 July 1977. 56. Banseviˇcius,B.; Blechertas, V. Ultrasonic Motors for Mass-Consumer Products. ULTRAGARSAS 2006, 4, 1392–2114. 57. Ko, H.P.; Lee, K.J.; Yoo, K.H.; Kang, C.Y.; Kim, S.; Yoon, S.J. Analysis of Tiny Piezoelectric Ultrasonic Linear Motor. Jpn. J. Appl. Phys. 2006, 45, 4782–4786. [CrossRef] 58. Morita, T.; Nishimura, T.; Yoshida, R.; Hosaka, H. Resonant-Type Smooth Impact Drive Mechanism Actuator Operating at Lower Input Voltages. Jpn. J. Appl. Phys. 2013, 52, 07HE05. [CrossRef] 59. Koc, B. Piezoelectric Motor, Operates by Exciting Multiple Harmonics of a Square Plate. In Proceedings of the 12th International Conference on New Actuators, Bremen, Germany, 14–16 June 2010. 60. Tuncdemir, S.; Ural, S.O.; Koc, B.; Uchino, K. Design of Translation Rotary Ultrasonic Motor with Slanted Piezoelectric Ceramics. Jpn. J. Appl. Phys. 2011, 50, 027301. [CrossRef] 61. Galante, T.; Frank, J.; Bernard, J.; Chen, W.; Lesieutre, G.A.; Koopmann, G.H. Design, Modeling, and Performance of a High Force Piezoelectric Inchworm Motor. J. Intell. Mater. Syst. Struct. 1999, 10, 962–972. [CrossRef] 62. Egashira, Y.; Kosaka, K.; Iwabuchi, T.; Kosaka, T.; Baba, T.; Endo, T.; Hashiguchi, H.; Harada, T.; Nagamoto, K.; Watanabe, M.; et al. Sub-Nanometer Resolution Ultrasonic Motor for 300 mm Wafer Lithography Precision Stage. Jpn. J. Appl. Phys. 2002, 41, 5858–5863. [CrossRef] 63. Koc, B. An Ultrasonic Motor Using Side-Push of Centre-Attached Eyelets on an Octagonal Piezoelectric Plate. In Proceedings of the 16th International Conference on New Actuators, Bremen, Germany, 25–27 June 2018.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).