applied sciences
Article Simulation of Motion Interactions of a 2-DOF Linear Piezoelectric Impact Drive Mechanism with a Single Friction Interface
Haojie Xia †, Liling Han *,†, Chengliang Pan, Huakun Jia and Liandong Yu * School of Instrument Science and Opto-electronics Engineering, Hefei University of Technology, Hefei 230009, China; [email protected] (H.X.); [email protected] (C.P.); [email protected] (H.J.) * Correspondence: [email protected] (L.H.); [email protected] (L.Y.); Tel.: +86-551-6290-1780 (L.H.) † Haojie Xia and Liling Han contributed equally to this work.
Received: 20 July 2018; Accepted: 16 August 2018; Published: 19 August 2018
Featured Application: This study provides theoretical and technical references for the practical development of a parallel piezoelectric impact drive mechanism. A controllable complex curve motion can be realized by the method of using coupled motions.
Abstract: A two-degrees-of-freedom (2-DOF) linear piezoelectric impact drive mechanism (PIDM) is actuated by two independent piezoelectric actuators (PAs). The coupled motion interactions of a two orthogonal DOF linear PIDM with a single friction interface are introduced and analyzed. A complete dynamic model of the 2-DOF PIDM is established with the Karnopp friction model considering the distribution of friction in the x-axis and y-axis. The output displacements of the 2-DOF PIDM and two corresponding independent 1-DOF PIDMs are investigated numerically. When the two input exciting signals of a 2-DOF PIDM have the same driving voltage of 100 V with a duty ratio of 98% at 10 Hz and two 1-DOF PIDMs are driving under the same conditions, the step displacements in the two axes of 2-DOF PIDM are improved compared to the corresponding 1-DOF PIDM. When the two input exciting signals of a 2-DOF PIDM have the same driving voltages of 100 V with a duty ratio of 98% but the driving frequency is 10 Hz in the x-axis and 20 Hz in the y-axis, the results show that the displacement of high frequency achieves a slight decrease and displacement of low frequency shows a large increase compared to the two corresponding 1-DOF PIDMs.
Keywords: piezoelectric; coupled motion; simulation; motion interaction
1. Introduction Because piezoelectric impact drive mechanisms (PIDMs) have a compact structure, high positioning accuracy, and a long stroke, they have been widely used in the fields of precision positioning [1,2]. Multiple-degrees-of-freedom (DOF) motions of stages and manipulators with PIDMs can achieve flexible and wide control in actuators. Multi-DOF motions of PIDM can be designed in two structures. The first kind of structure is the complex structure in series [3–5], for which the motions are independent and their corresponding controls are relatively simple. The second kind of structure is the compact structure in parallel [6–9], for which the motions often share a common friction interface [6,8] that leads to a coupled motion when the motions are operated at the same time [8,10]. Recent studies on parallel PIDMs have attained multi-DOF motions by using different modes of vibration under ultrasonic driving [8,11–15], and studies on series PIDMs have achieved multi-DOF motions by avoiding coupled motions [4,5,16]. However, there are few studies on the coupled motion characteristics of the parallel impact PIDM [17,18].
Appl. Sci. 2018, 8, 1400; doi:10.3390/app8081400 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, x FOR PEER REVIEW 2 of 10 achieved multi-DOF motions by avoiding coupled motions [4,5,16]. However, there are few studies Appl.on the Sci. coupled2018, 8, 1400 motion characteristics of the parallel impact PIDM [17,18]. 2 of 10 Establishing the dynamic model of parallel multi-DOF PIDMs is foundational to understanding coupled motion interactions and estimating the influences on motion performances. PIDMs are usuallyEstablishing combined the by dynamicthree basic model elements: of parallel piezoelectric multi-DOF actuator PIDMs (PAs), is foundational a stage, and to understanding a slider. The workingcoupled motionprocess interactions is a stick-slip and motion estimating with fricti the influenceson actuation on motionunder a performances. sawtooth driving PIDMs voltage are usually signal [19].combined In the by dynamic three basic analysis, elements: the PA piezoelectric and stage actuatorare simplified (PAs), as a stage,a spring–mass–damper and a slider. The workinglumped processmodel, and is a the stick-slip slider is motion considered with as friction a mass. actuation The motion under of athe sawtooth slider relies driving on the voltage friction signal between [19]. theIn the stage dynamic and slider analysis, [20]. The the PAfriction and force stage is are nonlinear, simplified so as an a accurate spring–mass–damper friction model lumpedis important model, to studyand the coupled slider is motion considered interactions. as a mass. Many The motionfriction of models, the slider such relies as onthe theCoulomb, friction betweenKarnopp, the LuGre, stage Leuven,and slider and [20 elastoplastic]. The friction friction force models, is nonlinear, have sobeen an widely accurate used friction in the model friction is importantanalysis of to PIDMs study [21–25].coupled The motion Coulomb interactions. model Manyis a basic friction static models, friction such model as the that Coulomb, cannot describe Karnopp, friction LuGre, when Leuven, the relativeand elastoplastic speed is zero. friction The models,Karnopp have model been is an widely improved used static in the friction friction model analysis that ofprovides PIDMs a [simple21–25]. Themanner Coulomb to avoid model the detection is a basic of static zero frictionrelative modelvelocity that by cannotdefining describe a zero velocity friction interval. when the Dynamic relative frictionspeed is models, zero. The such Karnopp as such model as theis LuGre, an improved Leuven static, and elastoplastic friction model models, that provides have high a simple accuracy manner and canto avoid explain the detectionthe presliding, of zero Stribeck, relative and velocity hysteretic by defining effects, a zeroalthough velocity the interval. definitions Dynamic of the frictionmodel parametersmodels, such are as difficult such as [26–30]. the LuGre, The Leuven,Karnopp and friction elastoplastic model is models,used for haveanalysis high when accuracy the precision and can isexplain low. the presliding, Stribeck, and hysteretic effects, although the definitions of the model parameters are difficultIn this [paper,26–30]. a The2-DOF Karnopp parallel friction linear model PIDM is is used considered for analysis as the when research the precision object. A is simplified low. modelIn for this the paper, orthogonal a 2-DOF motions parallel of linear the PIDMPIDM isis considered established as and the a research two-dimensional object. A simplifiedextended Karnoppmodel for thefriction orthogonal model motionsis introduced. of the PIDM Interactio is establishedns of the and orthogonal a two-dimensional motions extended are investigated Karnopp numericallyfriction model by is si introduced.mulations [31]. Interactions of the orthogonal motions are investigated numerically by simulations [31]. 2. Structure Design, Working Process, and Dynamic Model Analysis 2. Structure Design, Working Process, and Dynamic Model Analysis The proposed 2-DOF linear PIDM is shown in Figure 1a and is made up of two orthogonal PAs, The proposed 2-DOF linear PIDM is shown in Figure1a and is made up of two orthogonal PAs, a stage, and a slider. According to present studies, the 2-DOF linear PIDM can be regarded as two a stage, and a slider. According to present studies, the 2-DOF linear PIDM can be regarded as two spring–mass–damper systems. The simplified system of the PIDM is shown in Figure 1b, where mpx spring–mass–damper systems. The simplified system of the PIDM is shown in Figure1b, where mpx is is the equivalent mass of the PA and stage in the x-axis, xp is the distance of the PA in the x-axis, xs is the equivalent mass of the PA and stage in the x-axis, xp is the distance of the PA in the x-axis, xs is the the distance of the slider in the x-axis, kx is the stiffness of the PA and stage in the x-axis, cx is the distance of the slider in the x-axis, kx is the stiffness of the PA and stage in the x-axis, cx is the damper damper of the PA and stage in the x-axis, Fpx(t) is the equivalent driving force of the PA in the x-axis, of the PA and stage in the x-axis, Fpx(t) is the equivalent driving force of the PA in the x-axis, fx is the fx is the friction force in the x-axis, msx is the mass of the slider in the x-axis, mpy is the equivalent mass friction force in the x-axis, msx is the mass of the slider in the x-axis, mpy is the equivalent mass of the of the PA and stage in the y-axis, yp is the distance of the PA in the y-axis, ys is the distance of the PA and stage in the y-axis, yp is the distance of the PA in the y-axis, ys is the distance of the slider in the slider in the y-axis ky is the stiffness of the PA and stage in the y-axis, cy is the damper of the PA and y-axis ky is the stiffness of the PA and stage in the y-axis, cy is the damper of the PA and stage in the stage in the y-axis, Fpy(t) is the equivalent driving force of the PA in the y-axis, fy is the friction force y-axis, Fpy(t) is the equivalent driving force of the PA in the y-axis, fy is the friction force in the y-axis, in the y-axis, and msy is the mass of the slider in the y-axis [32,33]. and msy is the mass of the slider in the y-axis [32,33].
(a) (b)
Figure 1. AA two-degrees-of-freedom two-degrees-of-freedom (2-DOF) (2-DOF) linear linear piezoelectric piezoelectric impact impact drive drive mechanism mechanism (PIDM): (PIDM): (a) (Schematica) Schematic diagram; diagram; (b) (Simplifiedb) Simplified system. system. Appl. Sci. 2018, 8, x FOR PEER REVIEW 3 of 10
Kinematics equations of the 2-DOF orthogonal motions in the parallel 2-DOF PIDM can be extendedAppl. Sci. 2018 from, 8, 1400the 1-DOF PIDM model [34] and expressed as 3 of 10 =−−− mxpx p F px() t cx x p kx x p f x Kinematics equations of the 2-DOF= orthogonal motions in the parallel 2-DOF PIDM can be mxsx s f x , (1) extended from the 1-DOF PIDM model [34=]δ and expressed as Ftpx() x Vt x () .. . mpxxp = Fpx(t) − cxxp − kxxp − fx .. m x ==−−−f , (1) mysxpys p Fx py() t cy y p ky y p f y Fpx(t) = δxVx(t) mysy s f y , (2) .. = δ . mFtpypy()yp = Fpy y Vt( yt ()) − cyyp − kyyp − fy .. msyys = fy , (2) where the subscript x and y represent Fpy (thet)= twoδy Vdirectionsy(t) of motion and δ is the conversion coefficient betweenwhere the Fp(t) subscript and drivingx and voltagey represent V(t). the two directions of motion and δ is the conversion coefficient betweenWhenF pthe(t) andx-axis driving and y voltage-axis motionsV(t). of the 2-DOF PIDM are driven independently, there is no coupledWhen motion the x -axisand the and controly-axis motionsis very simple. of the 2-DOF Taking PIDM the x are-axis driven motion independently, of 2-DOF PIDM there as is an no example,coupled motionthe working and the principle control isis very shown simple. in Figure Taking 2. the Onex-axis working motion cycle of 2-DOF can be PIDM divided as an into example, three parts:the working (1) initial principle state: all is showncomponents in Figure of 2the. One structure working are cycle still can and be they divided stay intoin their three initial parts: positions; (1) initial (2)state: stick all state: components with the slow of the rise structure of the driving are still voltage, and they the stay PA inextends their initial to a length positions; of x1 and (2) stick the stage state: moves along the PA with the distance of x1 in a low speed as a result of the static friction force; (3) with the slow rise of the driving voltage, the PA extends to a length of x1 and the stage moves along the slip state: with the sudden decrease of the driving voltage, the PA contracts back to its original PA with the distance of x1 in a low speed as a result of the static friction force; (3) slip state: with the positionsudden decreasequickly. The of the stage driving is adhered voltage, to the the PA PA, contracts so the stage back moves to its original back to position its original quickly. position The with stage ais high adhered speed. to Because the PA, sothe the slider stage moves moves by back the friction to its original force between position the with contact a high surface speed. of Because the stage the andslider slider, moves the by speed the frictionof slider force is less between than the the sp contacteed of surfacethe stage. of Thus, the stage the andslider slider, cannot the catch speed the of stageslider and is less it moves than theback speed with of a thesmall stage. displacement Thus, the of slider x2. After cannot the catch working the stagecycle, and the itslider moves moves back forward with a stepping displacement of x1–x2. With repeated working cycles, the slider will move with a small displacement of x2. After the working cycle, the slider moves forward with a stepping forward continuously. When the driving voltage rises rapidly and falls down slowly, the slider will displacement of x1–x2. With repeated working cycles, the slider will move forward continuously. moveWhen backward the driving [35]. voltage rises rapidly and falls down slowly, the slider will move backward [35].
Figure 2. Working principle of 1-DOF PIDM. Figure 2. Working principle of 1-DOF PIDM. The working principle of the y-axis is similar to the x-axis motion when the 2-DOF PIDMs are The working principle of the y-axis is similar to the x-axis motion when the 2-DOF PIDMs are driven independently. Because the two motions share a common friction interface, the coupled motion driven independently. Because the two motions share a common friction interface, the coupled will exist when the two PAs are working together. The study of friction distribution and the value in motion will exist when the two PAs are working together. The study of friction distribution and the the x-axis and y-axis of a 2-DOF PIDM is very import. The Karnopp friction model is often used to value in the x-axis and y-axis of a 2-DOF PIDM is very import. The Karnopp friction model is often analyze the friction force in a 1-DOF PIDM, as shown in Figure3. A zero velocity interval is defined used to analyze the friction force in a 1-DOF PIDM, as shown in Figure 3. A zero velocity interval is from −a to a; then, the Karnopp friction model [22] of a 1-DOF motion is described as follows: defined from −a to a; then, the Karnopp friction model [22] of a 1-DOF motion is described as follows: f sgn(v), |v|> a c f = min( fe(t), fs), |v| ≤ a, fe(t) > 0 , (3) max( fe(t), − fs), |v| ≤ a, fe(t) ≤ 0 Appl. Sci. 2018, 8, x FOR PEER REVIEW 4 of 10
fvvasgn() , | |> c =≤> () () fftfvaftmin()es , , , e 0 , (3) max()ft() ,−≤ f , v aft ,() ≤ 0 es e where fc is the Coulomb friction, fs is the maximum static friction, fe(t) is the external force, v is the relative velocity of the friction interface, and a is a nonzero artificial parameter. By extending the Karnopp friction model in a 1-DOF PIDM and considering the friction distribution in the x-axis and y-axis of a 2-DOF PIDM, the value of the friction in the two directions satisfy the following formulations of a 2-DOF Karnopp friction model:
22> while (-)+(-)xpsxyya ps
xx− = ps ffx ()()−+−22 xxpsyy ps yy− = ps (4) ffy ()()−+−22 xpsxyy ps = ffc ,
22≤ while (-)+(-)xpsxyya ps
x = p ffx ()22+ () xypp y = p ffy ()22+ () xypp (5) 22 22 Appl. Sci. 2018, 8, 1400 mx()++≤ () y, mx() () y f 4 of 10 s ppspps f = 22 fm, () x+> () y f ss p p s where fc is the Coulomb friction, fs is the maximum static friction, fe(t) is, the external force, v is the whererelative f is velocity the friction of the between friction interface,the slider andand astage.is a nonzero artificial parameter.
Figure 3. Karnopp friction model.
3. NumericalBy extending Simulation the Karnopp Processes friction model in a 1-DOF PIDM and considering the friction distribution in the x-axis and y-axis of a 2-DOF PIDM, the value of the friction in the two directions satisfyAssume the following that the formulations spring-mass-damper of a 2-DOF system Karnopp of the friction PA and model: stage has a resonant frequency of 1 r kHz and a mechanical. . quality . factor. Q of 50. Through the flexure hinge amplifier based on the 2 2 principlewhile of triangle(xp − x amplification,s) + yp − ys the) > finala output displacement of one PA can be amplified to 60 μm . . xp−xs fx = f r . . 2 . . 2 (xp−xs) + yp−ys . . − yp ys , (4) fy = f r . . . . 2 − 2+ − (xp xs) yp ys f = fc
q . . . . 2 2 while (xp − xs) + (yp − ys) ≤ a
.. xp fx = f r .. .. 2 2+ (xp) yp .. yp fy = f r .. 2 .. 2 + (xp) yp , (5) r r .. 2 .. 2 .. 2 .. 2 ms xp + y , ms xp + y ≤ fs p p f = r .. 2 .. 2 fs, ms xp + yp > fs where f is the friction between the slider and stage.
3. Numerical Simulation Processes Assume that the spring-mass-damper system of the PA and stage has a resonant frequency of 1 kHz and a mechanical quality factor Q of 50. Through the flexure hinge amplifier based on the principle of triangle amplification, the final output displacement of one PA can be amplified to 60 µm with a driving voltage of 100 V. The driving force of the PA can be measured by the theory of spring deformation force; thus, the conversion coefficient δ can be calculated to −0.6 N·V−1. The resonant frequency and damper ratio can be calculated according to the formulas based on free vibration as follows: s 1 k fr = , (6) 2π mp Appl. Sci. 2018, 8, 1400 5 of 10
1 c ξ = = p (7) 2Q 2 kmp where fr is the resonant frequency, k is the stiffness of the PA and stage, mp is the equivalent mass of the PA and stage, c is the damper of the PA and stage, and ξ is the damping ratio. Structural parameters of the PIDM were assumed equal in the x-axis and y-axis. By choosing an appropriate slider, the parameters for numerical simulation were calculated and are listed in Table1.
Table 1. Parameters of the parallel 2-DOF linear PIDM.
Parameter Value Unit −3 mpx (mpy) 2.5 × 10 kg −3 msx (msy) 5.0 × 10 kg 6 −1 kx (ky) 1.0 × 10 N·m −1 cx (cy) 3.2 N·s·m −1 δx (δy) −0.6 N·V fs 1.2 N fc 1.0 N a 1.0 × 10−6 m·s−1
With these simulation parameters, the simulation blocks were established (Figure4) according to the abovementioned kinematics equations [36]. The whole simulation block used a subsystem module, which included a relay module, a saturation module, and a logical operator module. Thus, it could distinguish the friction force when the system was driving under different speeds. The relay module was used to judge whether the slider was sliding on the contact surface between the slider and stage. Because the stage and PA were tightly adhered, the speed of the stage and PA were assumed to be equal. If the relative speed between the slider and PA was beyond the velocity range of a in the Karnopp friction model, the sliding friction was chosen in the kinematics equations. The sliding friction was usually equal to Coulomb friction, and if the relative speed between the slider and PA was in the velocity range of a, the static friction was chosen and used with the kinematics equations according to the logical operator module. The saturation module was used to judge whether the calculated static friction exceeded the maximum static friction. If the calculated static friction was greater than the maximum static friction, the maximum static friction was chosen. If the calculated static friction was less than the maximum static friction, the calculated static friction was chosen and used with the kinematics equations. Appl. Sci. 2018, 8, 1400 6 of 10 Appl. Sci. 2018, 8, x FOR PEER REVIEW 6 of 10
(b)
(a)
(d)
(c)
FigureFigure 4. 4. SimulationSimulation block: block: (a ()a 2-DOF) 2-DOF whole whole block block diagram; diagram; (b) (2-DOFb) 2-DOF subsystem subsystem block block diagram; diagram; ((cc)) 1-DOF 1-DOF whole whole block block diagram; diagram; ( d) 1-DOF subsystem block diagram.
4.4. Simulation Simulation Results Results and and Discussions Discussions TheThe output displacementsdisplacements ofof the the PA PA and and slider slider in in the the direction direction of theof thex-axis x-axis and andy-axis y-axis of the of 2-DOFthe 2- DOFPIDM PIDM were comparedwere compared to the outputto the displacementsoutput displacements of the PA andof the slider PA ofand the twoslider corresponding of the two correspondingindependent 1-DOF independent PIDMs in 1-DOF order toPIDMs analyze in order the coupled to analyze motion the interactions. coupled motion The PAs interactions. were actuated The PAsby sawtooth were actuated driving by voltage sawtooth signals. driving The voltage curves sign of outputals. The displacements curves of output of the displacements PA in the direction of the PAof thein thex-axis direction and y of-axis the inx-axis the and 2-DOF y-axis PIDM in the are 2-DOF shown PIDM as curve-d are shownpx and as curve-d curve-dpxpy and, respectively. curve-dpy, respectively.The output displacement The output curvesdisplacement of the PA curves are the of periodicthe PA sawtoothare the periodic curves (dottedsawtooth curves), curves which (dotted are curves),similar towhich the drivingare similar signals. to the The driving output signals. displacement The output curves displacement of the slider curves in the of the direction slider in of the the directionx-axis and ofy the-axis x-axis in the and 2-DOF y-axis PIDM in the are 2-DOF shown PIDM as curve-d are shownsx and as curve-d curve-dsysx, and and curve-d the outputsy, and curves the outputare usually curves the are stepping usually curves the stepping (solid curves).curves (so Thelid output curves). displacements The output displacements of the PAs and of sliders the PAs in andthe twosliders corresponding in the two correspondin independentg 1-DOF independent PIDMs are1-DOF curve-d PIDMsp1, curve-dare curve-dp2, curve-dp1, curve-ds1, andp2, curve-d curve-ds1s2, , andrespectively. curve-ds2, The respectively. curve-dp1 Theand curve-d curve-dp1s1 andcorrespond curve-ds1 tocorrespond the curves to of the the curves 2-DOF of PIDMthe 2-DOF (curve-d PIDMpx (curve-dand curve-dpx andsx), curve-d which wassx), which driven was by thedrivenx-axis by alone. the x The-axis curve-d alone.p2 Theand curve-d curve-dp2s2 andcorrespond curve-d tos2 correspondthe curves ofto the 2-DOFcurves of PIDM the 2-DOF (curve-d PIDMpy and (curve-d curve-dpy andsy), curve-d which wassy), which driven was by driven the y-axis by the alone. y- axisAccording alone. toAccording the above to principles, the above curve-d principles,px and curve-d curve-dpxp1 andare curve-d the samep1 dottedare the curves,same dotted and curve-d curves,py andand curve-d curve-dpyp2 andare thecurve-d samep2 dottedare thecurves. same dotted curves. InIn orderorder toto verify verify the the correctness correctness of theof frictionthe friction distribution distribution in the in two the orthogonal two orthogonal coupled coupled motions, motions,the 2-DOF the PIDM 2-DOF was PIDM excited was by excited a driving by voltagea driving of 100voltage V at of a frequency 100 V at a of frequency 10 Hz with of a10 duty Hz ratiowith ofa duty98% inratio the of direction 98% in ofthe the directionx-axis and of the a driving x-axis voltageand a driving of 0 V at voltage a frequency of 0 V of at 10 a Hzfrequency with a duty of 10 ratio Hz withof 98% a duty in the ratio direction of 98% of thein they-axis. direction In addition, of the y one-axis. of theIn addition, two corresponding one of the independenttwo corresponding 1-DOF independentPIDMs was excited1-DOF byPIDMs a driving was excited voltage by of a 100driving V at volt a frequencyage of 100 of V 10 at Hz a frequency with a duty of 10 ratio Hz ofwith 98%, a dutyand theratio other of 98%, was and excited the other by a drivingwas excited voltage by a of dr 0iving V. The voltage results of of 0 V. the The mentioned results of eight the mentioned curves are √ eightshown curves in Figure are 5showna. The 2-DOFin Figure PIDM 5a. wasThe driven2-DOF byPIDM a voltage was driven of 100/ by 2a Vvoltage at the frequencyof 100/√2 V of at 10 the Hz frequencywith duty ratioof 10 of Hz 98% with in both duty the ratio direction of 98% of the inx -axisboth andthe y-axis.direction Simultaneously, of the x-axis one and of they-axis. two Simultaneously, one of the two corresponding independent 1-DOF PIDMs was driven by a voltage Appl. Sci. 2018, 8, x FOR PEER REVIEW 7 of 10
of 100 V at a frequency of 10 Hz with duty ratio of 98%, and the other was driven by a voltage of 0 V. The results of the mentioned eight curves are shown in Figure 5b. The results in Figure 5a show that curve-dpx and curve-dp1 are the same sawtooth curves, and the results of curve-dsx and curve-ds1 are the same output stepping displacement curves. All curves in the y direction and the corresponding 1-DOF PIDM are horizontal lines with a magnitude of 0. The results in Figure 5b show that curve-dpx and curve-dpy are the same sawtooth curves (dotted curves), Appl.and Sci.curve-d2018, 8sx, 1400and curve-dsy are also the same sawtooth curves (solid curves). Figure 5b shows7 ofthat 10 Appl.the Sci.maximum 2018, 8, x FORvalue PEER of REVIEWcurve-d p1 is about √2 times than that of curve-dpx (in one cycle), and7 of the 10 maximum value of curve-ds1 is about √2 times than that of curve-dsx (in one cycle). The results indicate ofcorrespondingthat 100 the V attwo-dimensional a frequency independent of 10 extended Hz 1-DOF with PIDMs dutyKarnopp ratio was frictionof driven 98%, modeland by a the voltage in other the of was2-DOF 100 driven V parallel at a by frequency a linearvoltage ofPIDM of 10 0 HzV. is Thewithcorrect. results duty ratio of the of mentioned 98%, and the eight other curves was drivenare shown by a voltagein Figure of 5b. 0 V. The results of the mentioned eight curves are shown in Figure5b. The results in Figure 5a show that curve-dpx and curve-dp1 are the same sawtooth curves, and the results of curve-dsx and curve-ds1 are the same output stepping displacement curves. All curves in the y direction and the corresponding 1-DOF PIDM are horizontal lines with a magnitude of 0. The results in Figure 5b show that curve-dpx and curve-dpy are the same sawtooth curves (dotted curves), and curve-dsx and curve-dsy are also the same sawtooth curves (solid curves). Figure 5b shows that the maximum value of curve-dp1 is about √2 times than that of curve-dpx (in one cycle), and the maximum value of curve-ds1 is about √2 times than that of curve-dsx (in one cycle). The results indicate that the two-dimensional extended Karnopp friction model in the 2-DOF parallel linear PIDM is correct.
(a) (b)
Figure 5. Model comparison: (a) 2-DOF PIDM with single input; (b) The driving voltage of the 1-DOF √ PIDM is √22 times times than than that that of of the the 2-DOF 2-DOF PIDM PIDM in in the the direction direction of of the the xx-axis-axis ( (yy-axis).-axis).
The results2-DOF inPIDM Figure was5a showdriven that at a curve-d frequencypx and of 10 curve-d Hz withp1 are a voltage the same of sawtooth100 V and curves, a duty andratio the of results98% in both of curve-d the x-axissx and and curve-d y-axis directions.s1 are thesame At the output same time, stepping the two displacement corresponding curves. independent All curves 1- inDOF the PIDMsy direction were anddriven the at corresponding the same conditions. 1-DOF Th PIDMe results are of horizontal the mentioned lines witheight acurves magnitude are shown of 0. Thein Figure results 6. inThe Figure output5b of show slider that was curve-d importantpx and and curve-d was analyzed.py are the The same results sawtooth show that curves the average (dotted curves), and curve-d and curve-d are also the same sawtooth curves (solid curves). Figure5b stepping displacementssx of the slidersy for the 2-DOF PIDM√ in the two axes (curve-dsx and curve-d sy) shows that the maximum(a) value of curve-d is about 2 times than that( ofb) curve-d (in one cycle), are both about 19.5 μm, but the average steppingp1√ displacement of the two 1-DOF PIDMspx (curve-ds1 and the maximum value of curve-d is about 2 times than that of curve-d (in one cycle). The results and Figurecurve-d 5. s2Model) are bothcomparison: about 6.0 (a) 2-DOFμs1m. In PIDM the parallel with single 2-DOF input; PIDM, (b) The the driving backlashsx voltage displacement of the 1-DOF of the indicatesliderPIDM was that is decomposed, √ the2 times two-dimensional than thatand ofthe the stepping extended 2-DOF PIDM displaceme Karnopp in the frictiondirectionnt in the model of two the xaxes in-axis the was ( 2-DOFy-axis). improved parallel in linear a circle. PIDM is correct. TheThe 2-DOF 2-DOF PIDM PIDM was was driven driven at at a a frequency frequency of of 10 10 Hz Hz with with a a voltage voltage of of 100 100 V V and and a a duty duty ratio ratio of of 98%98% in in both both the the xx-axis-axis and and y-axisy-axis directions. directions. At At the the same same time, time, the the two two corresponding corresponding independent independent 1- DOF1-DOF PIDMs PIDMs were were driven driven at atthe the same same conditions. conditions. Th Thee results results of of the the mentioned mentioned eight eight curves curves are are shown shown inin Figure Figure 66.. The output of slider was important and was analyzed. The results show that the average steppingstepping displacements ofof thethe sliderslider for for the the 2-DOF 2-DOF PIDM PIDM in in the the two two axes axes (curve-d (curve-dsx andsx and curve-d curve-dsy) aresy) areboth both about about 19.5 19.5µm, μ butm, but the averagethe average stepping stepping displacement displacement of the of twothe two 1-DOF 1-DOF PIDMs PIDMs (curve-d (curve-ds1 ands1 andcurve-d curve-ds2) ares2) bothare both about about 6.0 µ 6.0m. μ Inm. the In parallelthe parallel 2-DOF 2-DOF PIDM, PIDM, the backlash the backlash displacement displacement of the of slider the sliderwas decomposed, was decomposed, and the and stepping the stepping displacement displaceme in thent in two the axes two was axes improved was improved in a circle. in a circle.
Figure 6. Results of two directions in the 2-DOF PIDM and the two 1-DOF PIDMs driven at the same driving signal.
The two directions of the 2-DOF PIDM were driven at the same frequency of 10 Hz with a duty ratio of 98%, but the driving voltage was 100 V in the direction of the x-axis and 50 V in the direction of the y-axis. At the same time, one of the two corresponding independent 1-DOF PIDMs was excited by a voltage of 100 V at a frequency of 10 Hz with a duty ratio of 98%, and the other was excited by a voltage of 50 V at a frequency of 10 Hz with a duty ratio of 98%. The results of the eight curves are shown in Figure 7. The average stepping displacements of the slider for the 2-DOF PIDM in the two FigureFigure 6. 6. ResultsResults of of two two directions directions in in the the 2-DOF 2-DOF PIDM PIDM and and the the two two 1-DOF 1-DOF PIDMs PIDMs driven driven at at the the same same drivingdriving signal. signal.
TheThe two two directions directions of of the the 2-DOF 2-DOF PIDM PIDM were were driven driven at at the the same same frequency frequency of of 10 10 Hz Hz with with a a duty duty ratioratio of of 98%, 98%, but but the the driving driving voltage voltage was was 100 100 V V in in the the direction direction of of the the xx-axis-axis and and 50 50 V V in in the the direction direction ofof the the yy-axis.-axis. At At the the same same time, time, one one of of the the two two corresponding corresponding independent independent 1-DOF 1-DOF PIDMs PIDMs was was excited excited byby a a voltage voltage of of 100 V at a frequency of 1010 HzHz withwith aa dutyduty ratioratio ofof 98%,98%, andand the the other other was was excited excited by by a a voltage of 50 V at a frequency of 10 Hz with a duty ratio of 98%. The results of the eight curves are shown in Figure 7. The average stepping displacements of the slider for the 2-DOF PIDM in the two Appl. Sci. 2018, 8, 1400 8 of 10
voltage of 50 V at a frequency of 10 Hz with a duty ratio of 98%. The results of the eight curves are Appl.Appl. Sci.Sci. 20182018,, 88,, xx FORFOR PEERPEER REVIEWREVIEW 88 ofof 1010 shown in Figure7. The average stepping displacements of the slider for the 2-DOF PIDM in the two sx sy µ axesaxes (curve-d (curve-dsxsx andand curve-d curve-dsysy)) are are about about 10.810.8 andandand 5.85.85.8 μμm,m,m, while whilewhile the thethe average averageaverage stepping steppingstepping displacements displacementsdisplacements of s1 s2 µ ofofthe thethe two twotwo 1-DOF 1-DOF1-DOF PIDMs PIDMsPIDMs (curve-d (curve-d(curve-ds1 s1and andand curve-d curve-dcurve-ds2s2))) are areare about aboutabout 6.0 6.06.0 and andand− −−3.33.3 μμm.m. The The 2-DOF 2-DOF motions motions areare quitequite different different from from the the corresponding corresponding 1-DOF1-DOF motions.motions.
FigureFigure 7.7. ResultsResults ofof twotwo two directionsdirections directions inin thethe 2-DOF2-DOF PIDMPIDM whenwhen drivingdriving underunder differentdifferent voltagesvoltages andand thethe twotwo 1-DOF1-DOF PIDMs PIDMs driving driving at at the the corresponding corresponding conditions. conditions.conditions.
TheThe two two directions directions of of thethe 2-DOF2-DOF PIDMPIDMPIDM werewerewere excited exciexcitedted by byby the thethe same samesamedriving drivingdriving voltage voltagevoltage of ofof 100 100100 V VV with withwith a aaduty dutyduty ratio ratioratio of ofof 98%, 98%,98%, but butbut the thethe driving drivingdriving frequency freqfrequencyuency was waswas 10 1010 Hz HzHz in in thein thethe direction directiondirection of theofof thethex-axis xx-axis-axis and andand 20 Hz2020 HzHz in the inin thethedirection directiondirection of the ofof thethey-axis. yy-axis.-axis. At AtAt the thethe same samesame time, time,time, one oneone of ofof the thethe two twtwoo corresponding correspondingcorresponding independent independentindependent 1-DOF1-DOF PIDMs PIDMs waswas excited excited by by a a driving driving voltage voltage ofof 100100 VV atat a a frequency frequency ofof 10 10 Hz Hz with with a a duty duty ratio ratio of of 98%, 98%, and and the the otherother was was excited excited byby aa voltagevoltagevoltage of ofof 100 100100 V VV at atat a aa frequency frequencyfrequency of ofof 20 2020 Hz HzHz with withwith a a dutya dutyduty ratio ratioratio of ofof 98%. 98%.98%. The TheThe results resultsresults of ofofthe thethe eight eighteight curves curvescurves are areare shown shownshown in Figureinin FigureFigure8. The 8.8. TheThe average averaverage steppingage steppingstepping displacements displacementsdisplacements of the ofof slider thethe sliderslider for the forfor 2-DOF thethe 2-2- PIDM in the two axes (curve-d and curve-d ) are about 23.8 and 43.1 µm, while the average stepping DOFDOF PIDMPIDM inin thethe twotwo axesaxes (curve-d(curve-dsx sxsx andand curve-dcurve-dsy sysy)) areare aboutabout 23.823.8 andand 43.143.1 μμm,m, whilewhile thethe averageaverage displacements of the two 1-DOF PIDMs (curve-d and curve-d ) are 6.0 and 44.3 µm. It is obvious steppingstepping displacementsdisplacements ofof thethe twotwo 1-DOF1-DOF PIDMsPIDMs s1 (curve-d(curve-ds1s1 andands2 curve-dcurve-ds2s2)) areare 6.06.0 andand 44.344.3 μμm.m. ItIt isis obviousobviousthat the that highthat thethe frequency highhigh frequencyfrequency motion achieves motionmotion achievesachieves a slight decrease aa slightslight decrease anddecrease the low andand frequency thethe lowlow frequencyfrequency motion obtains motionmotion a obtainsobtainslarge increase. aa largelarge increase.increase.
FigureFigure 8. 8. ResultsResults ofof twotwo directionsdirections inin thethe 2-DOF 2-DOF PIDMPIDM whenwhen drivendriven underunder differentdifferent frequencies frequencies and and thethe twotwo 1-DOF 1-DOF PIDMs PIDMs driven driven at at the the corresponding corresponding conditions. conditions.conditions.
The results from Figures 6–8 indicate that the coupled motion interactions of the 2-DOF parallel The results from Figures6 6–8–8 indicate indicate thatthat thethe coupledcoupled motionmotion interactionsinteractions ofof thethe 2-DOF2-DOF parallelparallel linear PIDM are complex and they are not a simple superposition of double 1-DOF motions. linear PIDM are complex and they areare notnot aa simplesimple superpositionsuperposition ofof doubledouble 1-DOF1-DOF motions.motions.
5.5. ConclusionsConclusions ThisThis paperpaper presentspresentspresents the thethe coupled coupledcoupled motion motionmotion interactions interactionsinteractions of aofof 2-DOF aa 2-DOF2-DOF parallel parallelparallel linear linearlinear PIDM PIDMPIDM with awithwith single aa singlesinglefriction frictionfriction interface. interface.interface. The complete TheThe completecomplete dynamic dynamicdynamic model with modelmodel the Karnoppwithwith thethe frictionKarnoppKarnopp model frictionfriction was modelmodel established waswas establishedestablishedto analyze the toto analyze distributionanalyze thethe distributiondistribution of friction force ofof frictionfriction in the orthogonalforceforce inin thethe directions. orthogonalorthogonal With directions.directions. different WithWith relationships differentdifferent relationshipsrelationshipsof the two sawtooth ofof thethe twotwo driving sawtoothsawtooth signals, drivingdriving the influences signals,signals, thethe of theinfluencesinfluences coupled ofof motion thethe coupledcoupled interactions motionmotion on interactionsinteractions the 2-DOF ononPIDM’s thethe 2-DOF2-DOF performance PIDM’sPIDM’s and performanceperformance the two independent andand thethe twotwo 1-DOF independentindependent PIDMs are 1-DOF1-DOF contrasted PIDMsPIDMs through areare contrastedcontrasted numerical throughthrough values. numericalThenumerical results values.values. show that TheThe when resultsresults the showshow driving thatthat voltagewhenwhen thethe or drdr drivingivingiving voltagevoltage frequency oror drivingdriving is different frequencyfrequency in the two isis differentdifferent directions inin thethe twotwo directionsdirections ofof thethe parallelparallel 2-DOF2-DOF linearlinear PIDMPIDM,, thethe valuevalue ofof thethe steppisteppingng displacementdisplacement ofof thethe sliderslider willwill change,change, andand eveneven thethe motionmotion directiondirection cancan turnturn overover comparedcompared toto thethe correspondingcorresponding 1-1- DOFDOF PIDM.PIDM. TheThe resultsresults meanmean thatthat byby usingusing thethe coupledcoupled motion,motion, thethe complexcomplex motionmotion curvescurves cancan bebe synthesizedsynthesized byby thethe twotwo directionsdirections ofof thethe 2-DOF2-DOF paparallelrallel PIDM.PIDM. ThisThis studystudy providesprovides theoreticaltheoretical andand Appl. Sci. 2018, 8, 1400 9 of 10 of the parallel 2-DOF linear PIDM, the value of the stepping displacement of the slider will change, and even the motion direction can turn over compared to the corresponding 1-DOF PIDM. The results mean that by using the coupled motion, the complex motion curves can be synthesized by the two directions of the 2-DOF parallel PIDM. This study provides theoretical and technical references for the practical development of a parallel piezoelectric impact drive mechanism. Thus, a controllable complex curve motion can be realized by a parallel piezoelectric impact drive mechanism.
Author Contributions: All authors conceived the structure and analyzed the data; L.H. and C.P. conducted the simulation process and studied the analysis method; H.J. analyzed the datas; all authors wrote the paper; H.X. and L.Y. contributed to revising the paper. Funding: This research was funded by the National Natural Science Foundation of China (No. 51775157), the Natural Science Foundation of Anhui Province (No. 1608085ME108), the Project 111 (No. B12019), and the National Key Scientific Apparatus Development Project (Grant No. 2013YQ220893). Acknowledgments: The authors would like to express their gratitude to the reviewers and editors for their kind help. Conflicts of Interest: The authors declare no conflict of interest.
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