Electroelastic Actuators for Nano- and Microdisplacement

Transcription

1 SCIRA Journal of Physics hp:// Augus 3, 08 Volue 3, Issue, April 08 lecroelasic Acuaors for Nano- and Microdisplaceen Sergey M. Afonin Deparen of Inellecual echnical Syses, Naional Research Universiy of lecronic echnology (MI), Moscow, Russia, 4498 ail: Absrac In he general for for he equaion of he elecroelasiciy he generalized srucuralparaeric odel and he generalized arix ransfer funcion of he elecroelasic acuaor wih he oupu paraeers displaceens are deerined fro he soluions of he wave equaion wih using he Laplace ransfor. he paraeric srucural scheaic diagra and he ransfer funcions of he elecroelasic acuaor are obained. he srucural-paraeric odel of he piezoacuaor for he ransverse, longiudinal, shif piezoelecric effecs are consruced. he dynaic and saic characerisics of he piezoacuaor wih he oupu paraeer displaceen are obained. Keywords:lecroelasic acuaor, piezoacuaor, srucural-paraeric odel, paraeric srucural scheaic diagra, deforaion, ransfer funcions 8

2 . Inroducion and saeen of proble he elecroelasic acuaor on he piezoeffec and he elecrosricion effec is used for precise alignen in he adapive opics and he nanoechnology [ 6]. he piezoacuaor uses he inverse piezoeffec and serves for he acuaion of echaniss or he anageen and convers he elecrical signals ino he displaceen and he force [,, 3, 6]. he piezoacuaor is applied for he drives of he scanning unneling icroscopes and he aoic force icroscopes [4, 5, 6]. Le us consider he generalized srucural-paraeric odel and he generalized paraeric srucural scheaic diagra of he elecroelasic acuaor are consruced by solving he wave equaion wih he Laplace ransfor for he equaion of he elecroagneolasiciy, he boundary condiions on loaded working surfaces of he acuaor, he srains along he coordinae axes. he ransfer funcions and he paraeric srucural scheaic diagras of he piezoacuaor are obained fro he generalized srucuralparaeric odel. In [6, 7, 8] was deerined he soluion of he wave equaion of he piezoacuaor. he srucural-paraeric odel of he elecroelasic acuaor was deerined in conras elecrical equivalen circui ypes Cady and Mason for calculaion of piezoelecric ransier and receiver [9, 0, ]. In he paper [] presens he classic analyical wo-por luped-eleen odel (LM) ypes Cady and Mason of he piezoelecric coposie circular plae wih he oupu pressure. In he paper [3] considers he developen of various luped-eleen odels as pracical ools o design and anufacure he acuaors wih he oupu velociy. In he [4, 6, ] were obained he srucural-paraeric odels, he scheaic diagras for siple piezoacuaor and were ransfored o he srucuralparaeric odel of he elecroelasic acuaor. In [8, 8] was used he ransfer funcions of he piezoacuaor for he decision proble absolue sabiliy condiions of syse conrolling he deforaion of he elecroelasic acuaor. he elasic copliances and he echanical and adjusing characerisics of he piezoacuaor were found in [9] for calculaion is ransfer funcions and he srucural-paraeric odels. he srucural-paraeric odel of he ulilayer and copound piezoacuaor was deerined in [9] wih he oupu displaceen. In his paper is solving he proble of building he generalized srucural paraeric odel and he generalized paraeric srucural scheaic diagra of he elecroelasic acuaor for he equaion of he elecroelasiciy. he difference of his work fro he papers [0,, 3, 6] is ha he consrucion of he srucure-paraeric odel of he elecroelasic acuaor is produced iediaely in he general for, and no by he ehod of aheaical inducion fro he individual exaples for he odels of he piezoacuaors. 8

3 . Srucural-paraeric odel and arix ransfer funcions of elecroelasic acuaor Le us consider he general srucural-paraeric odel and he paraeric srucural scheaic diagra of he elecroelasic acuaor wih he oupu paraeer displaceen. For he elecroelasic acuaor are presened six sress coponens,, 3, 4, 5, 6, where he coponens - 3 are relaed o exension-copression sresses, 4-6 o shear sresses. In he elecroelasic acuaor is deforaion corresponds o sressed sae. For polarized piezoceraics PZ he arix sae equaions [, 4] wih he elecric and elasic variables have he for wo equaions, where he firs equaion represens he direc piezoeffec, he second equaion describes he inverse piezoeffec D d ε () S s d () where D is he colun arix of elecric inducion; S is he colun arix of relaive deforaions; is he colun arix of echanical sresses; is he colun arix of elecric field srengh; s is he elasic copliance arix for cons ; ε is he arix of dielecric consans for cons ; d is he ransposed arix of he piezoelecric odules. Le us consider he elecroagneoelasic acuaor. In general he equaion of elecroagneoelasiciy [6, ] has following for S i d s x, (3) i j where x, x S i is he relaive displaceen along axis i of he cross secion of he acuaor,, D conrol along axis, is he conrol paraeer for he volage conrol, D for he curren j is he echanical sress along axis j, di is he coefficien of elecroelasiciy, for exaple, piezoodule, cons. s is he elasic copliance for conrol paraeer he piezoacuaor (piezoplae) has he following properies: is he hickness, h is he heigh, b is he widh, respecively l,h, b he lengh of he piezoacuaor for he longiudinal, ransverse and shif piezoeffec. he direcion of he polarizaion axis Р, i.e., he direcion along which polarizaion was perfored, is usually aken as he direcion of axis 3. he equaion of he inverse piezoeffec for conrolling volage [6, ] has he for 83

4 where S i x, x, U Si is he relaive displaceen of he cross secion of he piezoacuaor along axis i, x, is he displaceen of he secion of he piezoacuaor, di is he piezoodule, is he elecric field srengh along axis, U is he volage beween he elecrodes of acuaor, s is he elasic copliance for cons, indexes i, j =,,, 6; =,, 3. he ain size l,h, b for he piezoacuaor, respecively, he hickness, he heigh, he widh for he longiudinal, ransverse, shif piezoeffec. For calculaion of he elecroelasic acuaor is used he wave equaion [6, 7,, 4] for he wave propagaion in a long line wih daping bu wihou disorions. Afer Laplace ransfor is obained he linear ordinary second-order differenial equaion wih he paraeer p, where he original proble for he parial differenial equaion of hyperbolic ype using he Laplace ransfor is reduced o he sipler proble [6, 4, ] for he linear ordinary differenial equaion d x, p dx x, p 0 (4) wih is soluion x x x, p Ce Be (5) where x, p is he Laplace ransfor of he displaceen of he secion of he elecroelasic acuaor, p c is he propagaion coefficien, c is he sound speed for cons, is he daping coefficien. he consans C and B of he soluion he linear ordinary second-order differenial equaion [7] are deerined fro he boundary condiions for he elecroelasic acuaor 0 p p for x 0 (6),, p p for x l l whence we obain he consans in he for l e C sh, l e B sh l l herefore, he soluion (5) can be wrien in following for 84

5 x, p psh l x psh x (7) shl he equaions for he forces on he faces of he elecroelasic acuaor 0, ps 0 F p M p p for x 0, (8) j l, p)s F p M p p for l j 0 x, where j 0, p and l, p j are he echanical sress. Fro equaions of forces acing on he faces of he elecroelasic acuaor we obain he boundary condiions on loaded surfaces j d x, p di 0, p p (9) s dx x0 s j l, p s d x, p dx xh d s i p where S0 is he cross secion area and M, M are he displaced ass on he faces of he elecroelasic acuaor. Fro (4), he boundary condiions on loaded surfaces (5), he srains along he axes he syse of equaions for he generalized srucural-paraeric odel and he generalized paraeric srucural scheaic diagra are deerined for Figure of he acuaor wih he oupu paraeers he Laplace ransfor for he displaceens he elecroelasic acuaor in he for where s S 0 D,, i p p M p M p d33,d3,d5 d i, g33,g3,g5 F F p p di sh p l ch p, p for he faces of l p p di sh p l ch l p p 3,3, D3,D3,D, s s s D 33,s,s D,s,s 55 D 55 (0) c c, c, 33 D, l,h, b, d is he coefficien of he elecroelasiciy (piezoodule), F p, p are he Laplace ransfor of he forces on he faces. Figure shows he generalized paraeric F 85

6 srucural scheaic diagra of he elecroelasic acuaor corresponding o he se of equaions (0) for he Laplace ransfor of he displaceens of he faces. he generalized ransfer funcions of he of he elecroelasic acuaor are he raio of he Laplace ransfor of he displaceen of he face acuaor and he Laplace ransfor of he corresponding conrol paraeer or he force a zero iniial condiions. Fro (0) he generalized arix equaion of he ransfer funcions of elecroelasic acuaor has he for p p W W p Wp W3p p W p W p 3 F F p p p () Figure. Generalized paraeric srucural scheaic diagra of he of he elecroelasic acuaor Le us consider he saic displaceens of he faces for he elecroelasic acuaor wih he oupu paraeer displaceen. For M and M he saic displaceens of he faces of he piezoacuaor for he ransverse piezoeffec are obained fro () in he for 86

7 0 li pw p U p0 p ξ α0 pw p U li p0 δp 0 0 d3hu 0M M M d3hu 0M δ M M () (3) For he piezoacuaor fro PZ under he ransverse piezoeffec a M and M, d /V, h, U 50 V, M kg and M 4 kg, he saic displaceens of he faces are deerined 600 n, 50 n, ξ 750 n. ξ ξ ξ Le us consider he dynaic displaceen of he piezoacuaor under he ransverse piezoeffec wih one fixed face. Using equaion () for unloaded piezoacuaor under he ransverse piezoelecric effec, we wrie he expression for he ransfer funcion a M 0 in he for W M and p p 3p d3 chh. (4) We wrie he resonance condiion for he piezoacuaor under he ransverse piezoeffec wih one fixed face cg c 0 h (5) his eans ha he piezoacuaor is he quarer-wave vibraor wih he resonance frequency f c h. (6) r 4 Le us consider he resonance condiion of he piezoacuaor fro PZ under he ransverse 3 piezoeffec a с 30 /s and h. 80, he resonance frequency is deerined fr 60 khz. For he approxiaion of he hyperbolic coangen by wo ers of he power series in ransfer funcion () he following expressions of he ransfer funcion of he piezoacuaor wih one fixed face is obained for he elasic-inerial load a M, M under he ransverse piezoeffec and conrol volage (Figure ) for he resisance R 0 source in he for W p U p p C C p p e d 3 h of he volage (7) M C C, C 3c M C C e h e 87

8 where Up is he Laplace ransfor of he volage, is he ie consan and is he daping coefficien of he piezoacuaor. Figure. Paraeric srucural scheaic diagra of he volage-conrolled piezoacuaor wih one fixed face under ransverse piezoeffec for elasic-inerial load Le us consider he ransien response of he elecroelasic acuaor wih he oupu paraeer displaceen. he expression for he ransien response of he volage-conrolled piezoacuaor for he elasic-inerial load under he ransverse piezoeffec is deerined e sin (8) ξ d3 h δ U, C C e, arcg where is he seady-sae value of displaceen of he piezoacuaor, U is he apliude of he volage. For he volage-conrolled piezoacuaor fro he piezoceraics PZ under he ransverse piezoelecric effec for he elasic-inerial load M, M and inpu volage 0 wih apliude U 5 V a d. 5 /V, h, 4 7 M kg, C 0 N/, 7 Ce 0.50 N/ are obained values ξ n, c. For calculaions conrol syses wih he elecroelasic acuaor he paraeric srucural scheaic diagras and he ransfer funcions of he elecroelasic acuaor are obained 3. Resuls and Discussions he elecroelasic acuaor solves proble of precise aching, copensaion of eperaure and graviaional deforaions. he srucural-paraeric odel and paraeric srucural scheaic diagras of he volage-conrolled piezoacuaor for he longiudinal, ransverse and shif piezoelecric effecs are deerined fro he generalized srucural-paraeric odel of he elecroelasic acuaor wih he replaceen of he generalized paraeers on he paraeers of he piezoacuaor. 88

9 he ransfer funcions in arix for are describe deforaions of he elecroelasic acuaor. he generalized srucural-paraeric odel, he generalized paraeric srucural scheaic diagra and he arix equaion of he elecroelasic acuaor wih he oupu paraeers displaceens are obained fro he soluions of he wave equaion wih he Laplace ransfor and fro is deforaions along he coordinae axes. Fro he generalized arix equaion for he ransfer funcions of he elecroelasic acuaor afer algebraic ransforaions are consruced he arix equaions of he piezoacuaor for he longiudinal, ransverse and shif piezoelecric effecs. 4. Conclusions he generalized srucural-paraeric odel, he generalized paraeric srucural scheaic diagra, he arix equaion of he elecroelasic acuaor wih he oupu paraeers displaceens are obained in he general for. he srucural-paraeric odel, he arix equaion and he paraeric srucural scheaic diagra of he piezoacuaor for he ransverse, longiudinal, shif piezoelecric effecs are deerined fro he generalized srucural-paraeric odel of he elecroelasic acuaor. Fro he soluion of he wave equaion, fro he equaion of he elecroelasiciy and he deforaions along he coordinae axes he generalized srucural-paraeric odel and he generalized paraeric srucural scheaic diagra of he elecroelasic acuaor wih he oupu paraeers displaceens are consruced. he deforaions of he elecroelasic acuaor are described by he arix equaion for he ransfer funcions of he acuaor. References [] Schulz, J.; Ueda, J.; Asada, H. Cellular Acuaors. Buerworh-Heineann Publisher: Oxford, 07, 38 p. [] Uchino, K. Piezoelecric acuaor and ulrasonic oors. Kluwer Acadeic Publisher : Boson, MA, 997, 347 p. [3] Przybylski, J. Saic and dynaic analysis of a flexensional ransducer wih an axial piezoelecric acuaion. ngineering srucures, 05, 84, 40 5, doi:0.06/j.engsruc [4] Ueda, J.; Secord,.; Asada, H. H. Large effecive-srain piezoelecric acuaors using nesed cellular archiecure wih exponenial srain aplificaion echaniss. I/ASM ransacions on Mecharonics, 00, 5, 5, , doi:0.09/mch

10 [5] Karpelson, M.; Wei, G.-Y.; Wood, R.J. Driving high volage piezoelecric acuaors in icroroboic applicaions. Sensors and Acuaors A: Physical, 0, 76, 78-89, doi:0.06/j.sna [6] Afonin, S.M. Soluion of he wave equaion for he conrol of an elecroagneoelasic ransduser. Doklady aheaics, 006, 73,, , doi:0.34/s [7] Afonin, S.M. Srucural paraeric odel of a piezoelecric nanodisplaceen ransduser. Doklady physics, 008, 53, 3, 37-43, doi:0.34/s [8] Afonin, S.M. Sabiliy of srain conrol syses of nano-and icrodisplaceen piezoransducers. Mechanics of solids, 04, 49, Journal 96-07, doi:0.303/s [9] alakokula, V. ; Bhalla, S.; Ball, R.J.; Bowen, C.R.; Pesce, G.L.; Kurchania, R.; Bhaacharjee, B.; Gupa, A.; Paine, K. Diagnosis of carbonaion induced corrosion iniiaion and progressionin reinforced concree srucures using piezo-ipedance ransducers. Sensors and Acuaors A: Physical, 06, 4, 79-9, doi:0.06/j.sna [0] Cady, W.G. Piezoelecriciy: An inroducion o he heory and applicaions of elecroechancial phenoena in crysals. McGraw-Hill Book Copany : New York, London, 946, 806 p. [] Physical Acousics: Principles and Mehods. Vol.. Par A. Mehods and Devices. d.: Mason, W. New York: Acadeic Press. 964, 55 p. [] Prasad, S.; Gallas, Q.; Horowiz, S.; Hoeer, B. Analyical elecroacousic odel of a piezoelecric coposie circular plae. AIAA Journal, 006, 4, 0, 3 38, doi:0.54/ [3] Chiao, M.; Capuano, F.; Coppola, G.; de Luca, L. LM characerizaion of synheic je acuaors driven by piezoelecric eleen: A Review. Sensors, 07, vol. 7, no. 6, 6-46, doi:0.3390/s7066. [4] Afonin, S.M. Srucural-paraeric odel and ransfer funcions of elecroelasic acuaor for nano- and icrodisplaceen. Chaper 9 in Piezoelecrics and Nanoaerials: Fundaenals, Developens and Applicaions. d. Parinov, I.A. Nova Science: New York, 05, pp [5] Afonin, S.M. Srucural-paraeric odel elecroagneoelasic acuaor nano- and icrodisplaceen for precision engineering. Precision ngineering. ngineering and echnology, 06, 3, 6,

11 [6] Afonin, S.M. Srucural-paraeric odels and ransfer funcions of elecroagneoelasic acuaors nano- and icrodisplaceen for echaronic syses. Inernaional Journal of heoreical and Applied Maheaics, 06,,, 5-59, doi:0.648/j.a [7] Afonin, S.M. Block diagras of a ulilayer piezoelecric oor for nano- and icrodisplaceens based on he ransverse piezoeffec. Journal of copuer and syses sciences inernaional, 05, 54, 3, , doi:0.34/s [8] Afonin, S.M. Absolue sabiliy condiions for a syse conrolling he deforaion of an elecroagneoelasic ransduser. Doklady aheaics, 006, 74, 3, , doi:0.34/s [9] Afonin, S.M. lasic copliances and echanical and adjusing characerisics of coposie piezoelecric ransducers. Mechanics of solids, 007, 4,, 43-49, doi:0.303/s [0] Afonin, S.M. Srucural-paraeric odel elecroagneoelasic acuaor nanodisplaceen for echaronics. Inernaional Journal of Physics, 07, 5,, 9-5, doi: 0.69/p [] Afonin, S.M. A srucural-paraeric odel of elecroelasic acuaor for nano- and icrodisplaceen of echaronic syse. Chaper 8 in Advances in Nanoechnology. Volue 9. ds. Barul, Z., renor, J. Nova Science: New York, 07, pp [] Springer Handbook of Nanoechnology. d. by Bhushan B. Springer: Berlin, New York, 004, p. [3] Afonin, S.M. Srucural-paraeric odel of piezoacuaor nano- and icrodisplaceen for nanoscience. AASCI Journal of Nanoscience, 07, 3, 3, -8. [4] Afonin, S.M. Wave equaion and paraeric srucural scheaic diagras of elecroagneoelasic acuaors nano- and icrodisplaceen. Inernaional Journal of Maheaical Analysis and Applicaions, 06, 3, 4, [5] Afonin, S.M. Srucural-paraeric odel elecroagneoelasic acuaor nano and icrodisplaceen for precision engineering. ngineering and echnology, 06, 3, 6, 0-. [6] Afonin, S.M. Decision Wave quaion and Block Diagra of lecro Magneo lasic Acuaor Nano - and Micro Displaceen for Counicaions Syses. Inernaional Journal of Inforaion and Counicaion Sciences, 06,,, -9. doi:0.648/j.ics