A Study on the Linear Piezoelectric Motor of Mode Shape

Transcription

1 Ope Joural of Acoustics, 015, 5, Published Olie December 015 i SciRes. A Study o the Liear Piezoelectric Motor of Mode Shape Jwo Mig Jou Departmet of Mechaical Egieerig, Cheg Shiu Uiversity, Taiwa Received 1 October 015; accepted 1 December 015; published 4 December 015 Copyright 015 by author ad Scietific Research Publishig Ic. This work is licesed uder the Creative Commos Attributio Iteratioal Licese (CC BY). Abstract I this paper, we wat to make a ew type liear piezoelectric motor by mode shape coatig or effective electrode surface coatig. The mode shape is derived from the mechaical boudary coditios of the liear piezoelectric motor. We oly have access to the first three modes of formas, the effective electrode surface coatig basis, as well as with the liear piezoelectric motor of ormal shape do compariso. Next, we will ispect their gai or axial velocity through theoretical aalysis, simulatio ad experimet. Accordig to the results of the theoretical aalysis, we have foud that the gai or axial velocity of the liear piezoelectric motors of mode shape is much larger tha the liear piezoelectric motors of ormal shape. However, accordig to the results of simulatio ad experimets, we have foud that the gai or axial velocity of the liear piezoelectric motors of mode shape is much greater tha the liear piezoelectric motors of ormal shape, which is about 1. to 1.4 times. The liear piezoelectric motor of mode shape 3 has the fastest axial velocity, which is about 48 mm/s ad 48 mm/s uder coditios of 180 V p p drivig voltage, 1. khz drivig frequecy (the third vibratio modal), 5 gw loadig ad the positio of loadig or mass at x = 5 mm & 45 mm respectively. Ad its axial velocity is about 1.4 times the liear piezoelectric motor of ormal shape uder the same coditios. Overall, the mode shape coatig helps to ehace the gai or axial velocity of the liear piezoelectric motor. Keywords Liear Piezoelectric Motor (LPM), Mode Shape (), Normal Shape (), Gai, Axial Velocity 1. Itroductio Over the past decade, the liear piezoelectric motors have a very sigificat developmet ad excellet performace, as show i Table 1 [1]-[19]. If we do ot cosider the size ad the drivig coditio of the piezoelectric liear motors, their liear movig fastest is a high-power liear ultrasoic motor usig bedig vibratio How to cite this paper: Jou, J.M. (015) A Study o the Liear Piezoelectric Motor of Mode Shape. Ope Joural of Acoustics, 5,

2 Table 1. The performace compariso of the liear piezoelectric motors [1]-[19]. Year Movemet Speed (mm/s) Thrust or Loadig (N) Drivig Frequecy (khz) Drivig Coditios Refereces ~ V P P [1] V P P [] V rms [3] V rms [4] V rms [5] or V p p or W [6] V p p [7] ~ V rms [8] V P P [9] V p p [10] V rms or W [11] V p p [1] V p p [13] V rms [14] V rms [15] V p p [16] V p p or 17.4 W [17] V p p or 9.3 mw [18] V p p [19] trasducer [15], the ultrasoic liear motor usig travelig surface acoustic wave [6] ad a liear ultrasoic motor usig push-pull type L-B hybrid Lagevi trasducer with sigle foot [3]. Their liear movemet speed of the fastest are 157 mm/s, 1500 mm/s ad 180 mm/s respectively. If we oly pay attetio to thrust or loadig ability, such as the square matrix type ultrasoic motor [7], a -DOF plaar ultrasoic motor usig logitudialbedig hybrid trasducer [5] ad aultrasoic liear motor usig logitudial ad bedig multimode boltclamped Lagevi type trasducer [1] have the best performace. Their maximum thrust or loadig are 14N, 103N ad 100N respectively. If it focuses o eergy efficiecy, they are a liear piezoelectric motor drive by a sigle-phase sigal [1], a multimode ultrasoic motor [10] ad a ultrasoic liear motor usig travelig surface acoustic wave [6]. Their miimum drivig voltages are 16 V p p, 30 V p p ad 80 V p p respectively. Due to the liear piezoelectric motors have superior performace, ad they have a high use value i the precisio mechaical systems. So i additio to the above the liear piezoelectric motors, i ay form or performace of the liear piezoelectric motors are worth tryig to develop. I additio, we, accordig to the rotary piezoelectric motor of mode shape, have the advatage of a high rotatioal speed ad excellet performace [0]. We try to use the same mode shape coatig method to produce a ew liear piezoelectric motor, ad hope the liear piezoelectric motor of mode shape also have a more superior performace with respect to the liear piezoelectric motor usig ormal shape coatig, which is our motivatio ad mai purpose i this paper.. The Fuctio of Mode Shape ad Effective Electrode Surface I this study, the liear piezoelectric motor ca be regarded as a like simple supported beam with pied-pied boudary coditios [0] [1], show i Figure 1. So we ca get the origi fuctio of mode shape φ ( x ) of the liear piezoelectric motor as follows: 154

3 where the dimesioless eigevalues Figure 1. The origi fuctios of mode shape of the liear piezoelectric motors uder pied-pied boudary coditios. φ ( x) si λ x, 0 x 1 λ is defied as: =. (1) λ = π; = 1,, 3,,. () Ad further we ca get aother fuctio of the effective electrode surface FP ( ) follows or show i Figure ad Table : where the FP ( x ) ( ) FP x = C = 1 φ FP ( x ) = λ ( x ) x from Equatio (1) as. (3) C is defied as a costat. The subscript of ad represet ormal shape ad mode shape. 155

4 Figure. The first three fuctios of mode shape ad the effective electrode surface of the liear piezoelectric motors. Table. The first three vibratio modes ad effective electrode surfaces of the mode shape piezoelectric ceramics uder pied-pied boudary coditios. x/l φ φ φ 1 3 FP 1 FP FP The Compositio ad Operatio Priciple Accordig to the fuctios of the effective electrode surface or from Equatio (3), we ca desig a series of liear piezoelectric motors, icludig the liear piezoelectric motors of ormal shape ad mode shape, as show i Figure 3. For the liear piezoelectric motors of mode shape, we just take the first three fuctios of the mode shape ad the effective electrode surface as the subject of research. As for the mai compoets of the liear piezoelectric motor comprise two piezoelectric ceramics ad a alumium base of bech type. Which the polarizatio directios of the two piezoelectric ceramics are i pairs. Whe we operate the liear piezoelectric motor, oly applied drivig voltage (V 3 ) ad resoace frequecy (fr) or proper loadig, we ca make liear motio of the piezoelectric motor, as show i Figure 4. Whe the right membrae force is greater tha the left side of the 156

5 Figure 3. The compositio of the liear piezoelectric motors of ormal shape ad mode shape. Figure 4. The operatio priciple ad motio state of the liear piezoelectric motor of ormal shape or mode shape. 157

6 membrae force, the piezoelectric motor will move to the right. O the cotrary, the piezoelectric motor will the move to the left. 4. The Equatios of Axial ad Trasverse Vibratio I order to distiguish differeces of the liear piezoelectric motor of the ormal shape ad mode shape, we ca simplify the equatios of motio of 3D structure ito the equatio of motio of the two 1D structures. Oe of which is the equatio of motio of the axial vibratio directio, the other is the equatio of motio of the trasverse vibratio directio as follow: ad N 1 M (, ) u xt = ρhl t 1 = ρhl (, ) wxt where the membrae force (N 1 ) ad of bedig momet (M 1 ) uit width are defied as: ad u t N1 = A11 V3 e31 FP x l k = 1 w, (4), (5) ( ) M1 = D11 zv 3e31FP x l k = 1, (6) ( ) k. (7) We ca let Equatio (1)-(3) & (6) (7) are substituted ito Equatio (4) & (5), ad the after fiishig, we ca get the equatios of motio by axial ad trasverse displacemet of the liear piezoelectric motor of the ormal shape ad mode shape as follow: ad u xt (, ) ρhl u( xt, ) = A t 11 ( ) ρhl ( ) ( ) φ ( ) 3 u xt, u xt, Π3 t x = + 3 A11 t Ah 11 PZT λ wxt 4 (, ) ρhl wxt (, ) = 0 D t 4 ( ) ρhl ( ) ( ) φ ( ) k. (8). (9). (10) 4 4 wxt, wxt, Π3 t x + = 4 4. (11) D11 t D11λ Accordig above the Equatios (8)-(11), we ca get the geeral solutios of axial ad trasverse vibratio displacemet ( u ad w) of the liear piezoelectric motors of ormal shape ad mode shape ca be expressed as follow: ad ω ( ) i ( ) ( u1 βi u βi ) j t jωit u xt, = U x e = C si x+ C cos x e jωit (, ) ( ) e si β cos β ( ) u xt = U x = C x + C x + U x u1 i u i p ωi ( ) = ( ) = w x, t W x e Cw sih λix Cw cosh λix Cw si λix Cw cos λix e j t jωit ωi ( ) = ( ) = λ + λ + λ + λ + φ ( ), (1) w x, t W x e C sih x C cosh x C si x C cos x A x e (13), (14) j t jωit w1 i w i w3 i w4 i, (15) 158

7 where the defiitio of costats or symbols i the above Equatios (4)-(15) as follow: A ( hc ) =, (16) k k = 1 A = λ λ w 4 4 w λwi Π D 3 11, (17) D 1 = c z z, (18) 3 3 ( ) ( k k ) k 1 3 k = 1 hk = zk zk 1, (19) ρh = ρ h, (0) k = 1 k k ρhl ωi 4ρhl π fi λ i = =, (1) D D ρhl ω 4ρhl π f λ = =, () D D ρh βi = lωi, (3) A ( ) 11 Π t =Π e = h eve. (4) j ω it jωit 3 3 PZT 31 3 where i above the symbols of A 11, A, c 11, D 11, e 31, f i, f, h, t, U, u, V 3, W, w, x, z, ρ, ω i, ω, λi, λ, βi ad Π 3 are defied as the membrae stiffess costat per uit width, amplitude, Youg s modulus, bedig stiffess costat per uit width, a arbitrary frequecy, the resoace frequecy, total height or thickess of piezoelectric ceramic or alumium base, time, the axial displacemet depedet o the axial coordiate, the axial displacemet depedet o the axial coordiate ad time, the trasverse drivig voltage, the trasverse displacemet depedet o the axial coordiate, the trasverse displacemet depedet o the axial coordiate ad time, axial coordiate, trasverse coordiate, desity, the agle velocity of arbitrary frequecy, the agle velocity of resoace frequecy, the eigevalues of arbitrary frequecy of trasverse directio, the eigevalues of resoace frequecy of trasverse directio, the eigevalues of arbitrary frequecy of axial directio ad the electric momet. Which C wp (p = 1 ~ 4) ad C uq (q = 1 ~ ) represet udetermied coefficiets of trasverse ad axial directio. Where the eigevalues λwi λw ad i R ad I. Where above udetermied coefficiets ca be determied by the followig electro-mechaical boudary coditios: ad ( 0, ) ; ( 1, ) u t = N1L A11 u t = N1R A 11, (5) ( 0, ) ; ( 1, ) u t = N1L A11 u t = N1R A11 (6) M L M w( 0, t) = 0; w ( 0, t) = ; w( 1, t) = 0; w ( 1, t) = D D 1 1 R 11 11, (7) where M L M w( 0, t) = 0; w ( 0, t) = ; w( 1, t) = 0; w ( 1, t) = D D 1 1 R C Π C Π N = N + N = F ; N = N + N = F E M 3 E M 3 1L 1L 1L fl 1R 1R 1R fr hpzt hpzt, (8), (9) 159

8 ad ( x) C Π φ ( x) φ N N N F N N N F E M CΠ3 E M 3 1L = 1L + 1L = ; fl 1R = 1R + 1R = fr hpzt λ h x 0 PZT λ = x= 1 Π Π M = M + M = + N h + F h ; M = M + M = + N h + F h E M 3 E E M 3 E 1L 1L 1L 1L c fl c1 1R 1R 1R 1R c fr c1 λi λi ( x ) φ M = M + M = + N h + F h E M Π3 E 1L 1L 1L 1L c fl c1 λ x = 0 ( x ) φ M M M N h F h E M Π3 E 1R = 1R + 1R = + 1R c + fr c1 λ x = 1 Aother the frictioal force is defied as follow: ;. (30), (31) a 1 1 fl k mass et ; l a F = µ W + W FfR µ k Wmass Wet l = + l, (33) where FfL, FfR, M1L, M1R, N1L ad N 1R represet the frictioal force per uit width of left side, the frictioal force per uit width of right side, the bedig momet per uit width of left side, the bedig momet per uit width of right side, the membrae force per uit width of left side ad the membrae force per uit width of right side. Ad a, Wmass, W et ad µ k represet the itercept, mass or loadig per uit width, et weight per uit width of the piezoelectric motor ad coefficiet of dyamic frictio. Which hc 1 ad h c represet the cetroid of the liear piezoelectric motor ad the itercept of the piezoelectric ceramics. The superscript of E ad M represet the electrical ad mechaical. Because the fuctios of mode shape are equal to zero uder pied-pied boudary coditios as: ( x) φ ( x) φ φ( x) = φ( x) = = = x= 0 x= 1 So the Equatio (30) ad Equatio (3) ca be rewritte as follows: ad where 0 x= 0 x= 1 N = N + N = F ; N = N + N = F E M E M 1L 1L 1L fl 1R 1R 1R fr M = M + M = F h ; M = M + M = F h E M E M 1L 1L 1L fl c1 1R 1R 1R fr c1 E E E E N1L = N1R = 0; M1L = M1R = 0 (3). (34), (35), (36). (37) Let Equatios (5)-(36) are substituted ito Equatios (1)-(15), we ca get the solutio of the axial ad trasverse vibratio displacemet of the first three modes of the liear piezoelectric motors as follow: ad C N N = ; C = N cos β 1L 1R 1L i u1 u A11βi A11βisi βi, (38) N N N cos β Π λ C C U x x 1L 1R 1L i 3 u1 = ; u = ; p( ) = cosλ A11βi A11βi si βi Ah 11 PZT βi M M cosh λ M M cos λ M C = ; C = = C ; C = 1R 1L i 1L 1L i 1R w1 w w4 w3 D11λi sih λi D11λi D11λi si λi (39), (40) 160

9 M M cosh λ M M cos λ M C = ; C = = C ; C = 1R 1L i 1L 1L i 1R w1 w w4 w3 D11λi sih λi λi D11 D11λi si λi. (41) Let Equatios (35)-(41) are substituted ito Equatios (1)-(15), we ca get the particular solutios of the trasverse ad axial vibratio displacemet of the liear piezoelectric motor as follow: ad w ( xt, ) 1 N1R N1L cos β i u( xt, ) = cos βix N1Lsi βix e A11β i si β i jωit, (4) 1 N1R N1L cos β i Π3λ jωit u( xt, ) = cos βix N1Lsi βix + cos λx e A11β i si β i Ah 11 PZT βi 1 sih λix si λix = ( M1R M1L cosh λi) + ( M1L cos λi M1R) + M1L( cosh λix cos λix) e D11λi sih λi si λi ad w ( xt, ) M1R M1L cosh λ i sih λix M1L cos λi M 1R si λix = D11λ + i sih λ i D11λ i si λi ( cosh cos ) M 1L λix λix λ Π 3 jωit + + φ 4 4 ( x ) e. D11λi λ λi D11 Further, we ca make Equatios (4)-(45) simplified as follows: (, ) si ( ω ) jωit (43), (44) (45) u xt = U it+φ, (46) (, ) si ( ω ) u xt = U it+φ (47) (, ) cos( ω ) wxt = W it+φ, (48) (, ) cos( ω ) wxt = W it+φ, (49) where Φ is defied as drivig phase agle. Therefore, we ca oce agai get the equatios of elliptical trajectory of the liear piezoelectric motor of ormal shape ad mode shape at cotact poits o both sides from above Equatios (46)-(49), as follow or show i Figure 4: ad u w + = 1 U W, (50) u w + = 1. (51) U W At this poit we ca see the differece i the liear motors of ormal shape ad mode shape from the solutios of displacemet. The we will be to test their differeces or gais by the case study of theoretical aalysis, computer simulatio ad experimets. 161

10 5. Special Case Study of Theoretical Aalysis, Simulatio Aalysis ad Experimets We have to uderstad the differeces ad gai betwee the liear piezoelectric motors of the ormal shape ad mode shape i this study. We will try to verify the differeces ad gai of the liear piezoelectric motors through theoretical aalysis, simulatio aalysis ad experimets. Where i the associated electrical ad mechaical properties of the liear piezoelectric motors i special case study is show i Table 3. The first is theoretical aalysis i this special case study, which is as follows: (5-1-1) Try to fid the approximate solutio of the trasverse displacemet of the liear piezoelectric motors of the ormal shape ad mode shape i the special case study. We ca take to uderstad their differece ad gai uder coditio of the ormal shape ad differet mode shape electrode coatig. (5-1-) Try to fid the approximate solutio of the axial displacemet of the liear piezoelectric motors of the ormal shape ad mode shape i the special case study. We ca further uderstad the impact of the axial or liear movemet uder coditios of the ormal shape ad differet mode shape electrode coatig or the above modal vibratio. (5-1-3) Try to uderstad the impact of the axial or liear movemet of the liear piezoelectric motors of the ormal shape ad mode shape uder coditios of the differet loadig (or mass) ad positios. Secod is simulatio aalysis by AYS code (its modal aalysis uder differet drivig coditios show i Figures 5-10, its mai aalytical work as follows: (5--1) Try to fid the vibratio modal of the liear piezoelectric motors of the ormal shape ad mode shape i the special case study. (5--) Try to fid the axial (or liear) ad trasverse displacemet from the above vibratio modal uder coditio of et weight. (5--3) Try to fid the axial (or liear) ad trasverse displacemet uder coditios of differet loadig, positio ad steady state. (5-3-1) Fially the experimetal work (its prototype ad structure show i Figure 11, Figure 1 as follows: (5-3-) Try to fid the maximum axial (or liear) speed of liear piezoelectric motors of the ormal shape ad mode shape uder coditio of et weight. (5-3-3) Try to fid the maximum axial (or liear) speed of liear piezoelectric motors of the ormal shape ad mode shape uder coditio of the differet loadig (or mass) ad positio. 6. Results ad Discussio Accordig to the results of theoretical aalysis, simulatio aalysis ad experimets, we foud: 1) Show i Figures 13-15, we have foud that the chage i axial displacemet or velocity of more focus ad smooth of the liear piezoelectric motors of mode shape with respect to the liear piezoelectric motor of ormal shape by theoretical aalysis uder coditios of the first three vibratio modal or drivig frequecy, 180 V p p drivig voltage, et weight, frequecy spacig of 1 Hz ad the first three eigevalues. ) Show i Figure 16, we have foud that the gai i trasverse displacemet or velocity of the liear piezoelectric motors of mode shape is much larger tha the liear piezoelectric motor of ormal shape by theoretical aalysis uder coditios of the first three vibratio modal or drivig frequecy, 180 V p p drivig voltage, et weight, frequecy spacig of 1 Hz ad the first three eigevalues. Table 3. The material properties ad size of the liear piezoelectric motor. Physical Name PZT Al Relative Permittivity (N.A.) ε 11 = ε = 1730, ε 33 = Piezoelectric Stress Costat (V/Nm) e 31 = e 3 = 5.3, e 33 = 15.8, e 4 = e 15 = Youg s Modulus (Pa) c 11 = c = 1.e11, c 1 = c 1 = 7.5e10, c 13 = c 31 = c 3 = c 3 = 7.51e10, c 33 = 1.11e11, c 44 = 3.0e10, c 55 = c 66 =.6e10 c 11 = c = c 33 = 7E10 Desity (kg/m 3 ) Poisso Ratio (N.A.) Size (mm 3 )

11 J. M. Jou Figure 5. The first modal aalysis of the liear piezoelectric motors of mode shape 1 uder 180 Vp p, 17 gw et weight ad pied-pied boudary coditios. Figure 6. The secod modal aalysis of the liear piezoelectric motors of mode shape 1 uder 180 Vp p, 17 gw et weight ad pied-pied boudary coditios. 163

12 J. M. Jou Figure 7. The third modal aalysis of the liear piezoelectric motors of mode shape 1 uder 180 Vp p, 17 gw et weight ad pied-pied boudary coditios. Figure 8. The first modal aalysis of the liear piezoelectric motors of mode shape 1 uder 180 Vp p, 5 gw loadig, the positio of loadig at x = 5 mm ad pied-pied boudary coditios. 164

13 J. M. Jou Figure 9. The secod modal aalysis of the liear piezoelectric motors of mode shape 1 uder 180 Vp p, 5 gw loadig, the positio of loadig at x = 5 mm ad pied-pied boudary coditios. Figure 10. The third modal aalysis of the liear piezoelectric motors of mode shape 1 uder 180 Vp p, 5 gw loadig, the positio of loadig at x = 5 mm ad pied-pied boudary coditios. 165

14 Figure 11. The prototype of the liear piezoelectric motors of ormal shape ad mode shape. Figure 1. The experimetal structure of the liear piezoelectric motors of ormal shape ad mode shape. 166

15 Figure 13. The dimesioless axial displacemet or velocity relative to the dimesioless legth of the liear piezoelectric motors of the ormal shape ad mode shape by theoretical aalysis uder coditios of the first vibratio modal, 180 V p p drivig voltage, 17 gw et weight, frequecy spacig of 1 Hz, eigevalues of λ 1 ad β 1. Figure 14. The dimesioless axial displacemet or velocity relative to the dimesioless legth of the liear piezoelectric motors of the ormal shape ad mode shape by theoretical aalysis uder coditios of the first vibratio modal, 180 V p p drivig voltage, 17 gw et weight, frequecy spacig of 1 Hz, eigevalues of λ ad β. Figure 15. The dimesioless axial displacemet or velocity relative to the dimesioless legth of the liear piezoelectric motors of the ormal shape ad mode shape by theoretical aalysis uder coditios of the first vibratio modal, 180 V p p drivig voltage, 17 gw et weight, frequecy spacig of 1 Hz, eigevalues of λ 3 ad β

16 3) Show i Figure 17, we have foud that the gai i axial displacemet or velocity of the liear piezoelectric motors of mode shape is much larger tha the liear piezoelectric motor of ormal shape by theoretical aalysis uder coditios of the first three vibratio modal or drivig frequecy, 180 V p p drivig voltage, differet positio of loadig or mass, frequecy spacig of 1 Hz ad the first three eigevalues. 4) Show i Figure 18, we have foud that the gai i axial displacemet or velocity of the liear piezoelectric motors of mode shape is much slightly tha the liear piezoelectric motor of ormal shape by simulatio aalysis uder coditios of differet positio of loadig or mass, 180 V p p drivig voltage, 5 gw loadig ad the first three vibratio modal or drivig frequecy. 5) Show i Table 4, the modal resoace frequecy of the liear piezoelectric motors of ormal shape ad mode shape by 1D theoretical aalysis, 3D simulatio aalysis ad experimets, there are.14 khz ~ 3. khz, 8.56 khz ~ 11. khz ad 19.6 khz ~ 1. khz respectively. 6) Show i Figure 19, we have foud that the gai i axial displacemet or velocity of the liear piezoelectric motors of mode shape is much slightly tha the liear piezoelectric motor of ormal shape by experimets Figure 16. The dimesioless trasverse displacemet or velocity relative to the dimesioless legth of the liear piezoelectric motors of the ormal shape ad mode shape by theoretical aalysis uder coditios of the first three vibratio modal, 180 V p p drivig voltage, 17 gw et weight ad frequecy spacig of 1 Hz. Figure 17. The dimesioless axial displacemet or velocity relative to the positio of loadig or mass of the liear piezoelectric motors of the ormal shape ad mode shape by theoretical aalysis uder coditios of the frequecy spacig of 1 Hz, 180 V p p drivig voltage ad the first three vibratio modal. 168

17 Table 4. The modal resoace frequecy of the liear piezoelectric motors of ormal shape ad mode shape by 1D theoretical aalysis, 3D simulatio aalysis ad experimets. Mode Fr(kHz)_1D Theory Aalysis Fr(kHz)_3D Simulatio Aalysis Fr(kHz)_Experimets ~ ~ ~ ~ ~ ~ 1.00 Figure 18. The dimesioless axial displacemet or velocity relative to the positio of loadig or mass of the liear piezoelectric motors of the ormal shape ad mode shape by simulatio aalysis uder coditios of 180 V p p drivig voltage, 5 gw loadig ad the first three vibratio modal. Figure 19. The axial displacemet velocity relative to the positio of loadig or mass of the liear piezoelectric motors of the ormal shape ad mode shape by experimets uder coditios of 180 V p p drivig voltage, 5 gw loadig, ad the drivig frequecy of the first three vibratio modal. uder coditios of differet positio of loadig or mass, 180 V p p drivig voltage, 5 gw loadig ad the first three vibratio modes or drivig frequecy. Wherei the liear piezoelectric motor of mode shape 3 havig the fastest axial velocity, which is about 48 mm/s ad 48 mm/s uder coditios of 180 V p p drivig voltage,

18 khz drivig frequecy (the third vibratio modal), 5 gw loadig ad the positio of loadig or mass at x = 5 mm & 45 mm respectively. Ad its axial velocity is about 1.4 times the liear piezoelectric motor of ormal shape uder the same coditios. 7. Coclusio Accordig to the results of theoretical aalysis, simulatio aalysis ad experimets, we have foud that the chage i axial displacemet or velocity of the liear piezoelectric motors of mode shape is more focus ad smooth tha the liear piezoelectric motor of ormal shape. We also have foud that the gai i axial ad trasverse displacemet or velocity of the liear piezoelectric motors of mode shape is much larger tha the liear piezoelectric motor of ormal shape. We have further foud that the axial velocity of the liear piezoelectric motors of mode shape is much larger tha the liear piezoelectric motor of ormal shape uder coditio of eccetric loadig or mass. We also have foud that the liear piezoelectric motors of ormal shape ad mode shape caot move faster, whe the state of et weight or the loadig (or mass) placed at the ceter positio of the motors. Whe the loadig or mass is farther away from the ceter, it s movig faster. Ad we also have foud that the placemet of the loadig or mass ca chage the directio of axial movemet of the motors. Overall, for all motors, the gai of the axial velocity of the third vibratio modal is much larger tha the first ad secod vibratio modal. From the above results, we have foud that the trasverse mode shape coatig has a positive effect o the axial displacemet. That is, the trasverse mode shape coatig ca make eergy more cocetrated with respect to the ormal shape coatig, so that axial displacemet ca more quickly move. Ackowledgemets This study ca be fiished smoothly, I especially wat to thak MOST of Taiwa of ROC sposor o fudig, (Project No.: MOST104-1-E ). Refereces [1] Shi, S.J., Che, W.S. ad Liu, J.K. (006) A High Speed Ultrasoic Liear Motor Usig Logitudial ad Bedig Multimode Bolt-Clamped Lagevi Type Trasducer. Proceedigs of the 006 IEEE Iteratioal Coferece o Mechatroics ad Automatio, Luoyag, 5-8 Jue 006, [] Jou, J.M. ad Hou, J.L. (007) A Study o the New Type Liear Ultrasoic Motor. 007 IEEE Iteratioal Ultrasoic Symposium, New York, 8-31 October 007, [3] Shi, S.J., Che, W.S., Liu, Y.X., Liu, J.K. ad Xie, T. (008) Desig ad Fabricatio of a Liear Ultrasoic Motor Usig Push-Pull Type L-B Hybrid Lagevi Trasducer with Sigle Foot. 008 IEEE Iteratioal Ultrasoics Symposium, Beijig, -5 November 008, [4] Yao, Z.Y., Yag, D., Wu, X. ad Zhao, C.S. (008) Structure Desig Method of Bar-Structure Liear Ultrasoic Motors. 008 IEEE Iteratioal Ultrasoics Symposiums, Beijig, -5 November 008, [5] Shi, S.J., Liu, J.K., Che, W.S. ad Liu, Y.X. (009) Developmet of a -DOF Plaar Ultrasoic Motor Usig Logitudial-Bedig Hybrid Trasducer. 18th IEEE Iteratioal Symposium o Applicatios of Ferroelectrics, Xia, 3-7 August 009, 1-5. [6] Kurosawa, M.K. (009) Ultrasoic Liear Motor Usig Travelig Surface Acoustic Wave. 009 IEEE Iteratioal Ultrasoics Symposium, Rome, 19-3 September 009, [7] Jou, J.M., Kug, H.-K. ad Huag, B.-W. (009) A Study o the Square Matrix Type Ultrasoic Motor. 009 Iteratioal Symposium o Mechatroic ad Biomedical Egieerig & Applicatios, Taiwa, 5 November 009, [8] Liu, Y.X., Che, W.S., Liu, J.K. ad Shi, S.J. (010) A High-Power Liear Ultrasoic Motor Usig Logitudial Vibratio Trasducers with Sigle Foot. IEEE Trasactios o Ultrasoics, Ferroelectrics, ad Frequecy Cotrol, 57, [9] Jou, J.-M. (011) A Study o the Arch Type Ultrasoic Motor. Iteratioal Coferece o Electric Iformatio ad Cotrol Egieerig, ICEICE 011, IEEE Publicatio, Jiagxi, 6-8 April 011, [10] Takasaki, M., Takao, H., Ji, H. ad Mizuo, T. (011) Modified Trasducer for Multimode Ultrasoic Motor. IEEE/ ASME Iteratioal Coferece o Advaced Itelliget Mechatroics (AIM), Budapest, 3-7 July 011, [11] Liu, Y.X., Liu, J.K., Che, W.S. ad Shi, S.J. (01) A U-Shaped Liear Ultrasoic Motor Usig Logitudial Vibratio Trasducers with Double Feet. IEEE Trasactios o Ultrasoics, Ferroelectrics, ad Frequecy Cotrol, 59,

19 [1] Liu, Y.X., Che, W.S., Liu, J.K. ad Shi, S.J. (013) Aalysis of a Liear Piezoelectric Motor Drive by a Sigle-Phase Sigal. IEEE Iteratioal Ultrasoics Symposium (IUS), Prague, 1-5 July 013, [13] Che, Z.J., Li, X.T., Che, J.G. ad Dog, S.X. (013) A Square-Plate Ultrasoic Liear Motor Operatig i Two Orthogoal First Bedig Modes. IEEE Trasactios o Ultrasoics, Ferroelectrics, ad Frequecy Cotrol, 60, [14] Liu, Y.X., Che, W.S., Liu, J.K. ad Shi, S.J. (013) A Rectagle-Type Liear Ultrasoic Motor Usig Logitudial Vibratio Trasducers with Four Drivig Feet. IEEE Trasactios o Ultrasoics, Ferroelectrics, ad Frequecy Cotrol, 60, [15] Liu, Y.X., Che, W.S., Liu, J.K. ad Yag, X.H. (013) A High-Power Liear Ultrasoic Motor Usig Bedig Vibratio Trasducer. IEEE Trasactios o Idustrial Electroics, 60, [16] Ci, P.H., Che, Z.J., Liu, G.X. ad Dog, S.X. (014) A Square-Plate Piezoelectric Liear Motor Operatig i Two Orthogoal ad Isomorphic Face-Diagoal-Bedig Modes. IEEE Trasactios o Ultrasoics, Ferroelectrics, ad Frequecy Cotrol, 61, [17] Zhou, S.L. ad Yao, Z.Y. (014) Desig ad Optimizatio of a Modal-Idepedet Liear Ultrasoic Motor. IEEE Trasactios o Idustrial Electroics, 61, [18] Guo, M.S., Pa, S., Hu, J.H., Zhao, C.S. ad Dog, S.X. (014) A Small Liear Ultrasoic Motor Utilizig Logitudial ad Bedig Modes of a Piezoelectric Tube. IEEE Trasactios o Ultrasoics, Ferroelectrics, ad Frequecy Cotrol, 61, [19] Li, X.T., Ci, P.H., Liu, G.X. ad Dog, S.X. (015) A Two-Layer Liear Piezoelectric Micromotor. IEEE Trasactios o Idustrial Electroics, 6, [0] Jou, J.-M. (015) A Study o the Mode Shape Piezoelectric Motor. Ope Joural of Acoustics, 5, [1] Jou, J.-M. (014) Theory ad Simulatio Aalysis of the Mode Shape ad Normal Shape Actuators ad Sesors. Ope Joural of Acoustics, 4,