Parametric structural schematic diagram of electroelastic actuator in nanotechnology.

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1 Review ricle hp:// Paraeric rucural cheaic diagra of elecroelaic acuaor in nanoechnology. fonin SM* Deparen of Inellecual echnical Sye, Naional Reearch Univeriy of lecronic echnology (MI), Mocow, Ruia brac In hi work he paraeric rucural cheaic diagra of he elecroelaic acuaor or he piezoacuaor i deerined in conra he elecrical equivalen circui ype Cady or Maon for he calculaion of he piezoelecric ranier and receiver, he vibraion piezooor wih he echanical paraeer in for he velociy and he preure. he ehod of aheaical phyic i ued. he paraeric rucural cheaic diagra of elecroelaic acuaor i obained wih he echanical paraeer he diplaceen and he force. he ranfer funcion of he elecroelaic acuaor are deerined. he generalized paraeric rucural cheaic diagra, he generalized arix equaion for he elecroelaic acuaor nano diplaceen are obained. he deforaion of he elecroelaic acuaor for he nanoechnology are decribed by he arix equaion. Keyword: lecroelaic acuaor, Paraeric rucural cheaic diagra, ranfer funcion, Piezoacuaor. cceped on February 5, 8 Inroducion he paraeric rucural cheaic diagra of he elecroelaic acuaor on he piezoelecric, piezoagneic, elecroricion effec, for exaple, he piezoacuaor i deerined in conra elecrical equivalen circui ype Cady or Maon for he calculaion of he piezoranier and piezoreceiver, he vibraion piezooor wih he echanical paraeer in for he velociy and he preure [-8]. he paraeric rucural cheaic diagra of he acuaor i obained wih he echanical paraeer he diplaceen and he force. he elecroelaic acuaor for nanoechnology i ued in he canning unneling icrocope, he canning force icrocope, he aoic force icrocope [8-]. Mehod he ehod of aheaical phyic i applied for he oluion of he wave equaion of he elecroelaic acuaor for he nanoechnology wih uing he Laplace ranfor for he conrucion he paraeric rucural cheaic diagra of elecroelaic acuaor [-6]. he reul of he join oluion of he wave equaion of he acuaor wih he Laplace ranfor, he equaion of he elecroelaiciy, he boundary condiion on he wo loaded working urface of he acuaor, we obain he correponding rucural-paraeric odel and he paraeric rucural cheaic diagra of elecroelaic acuaor. Reul and Dicuion he paraeric rucural cheaic diagra and he arix ranfer funcion of he elecroelaic acuaor for he nanoechnology are obained fro he rucural-paraeric odel of he elecroelaic acuaor wih he echanical 5 paraeer he diplaceen and he force. he rucural diagra of he volage-conrolled or curren-conrolled piezoacuaor are deerined fro he generalized rucuralparaeric odel of he elecroelaic acuaor [7,8]. Generalized paraeric rucural cheaic diagra Le u conider he generalized equaion of he elecroelaiciy [8,,8] in he for: S d + x, () i i ij j where Si ξ( x,) x i he relaive diplaceen along axi i of he cro ecion of he piezoacuaor or he piezoplae, {, D i he conrol paraeer for he volage conrol, D for he curren conrol along axi, j i he echanical re along axi j, d i i he coefficien of elecroelaiciy, for exaple, piezoodule, ij i he elaic copliance for conrol paraeer con, indexe i, j,,, 6;,, 3. he ain ize i deerined u he working lengh l { δ,h, b for he elecroelaic acuaor or he piezoacuaor in for he hickne, he heigh and he widh for he longiudinal, ranvere and hif he piezo effec. For he conrucion he paraeric rucural cheaic diagra of elecroelaic acuaor in nanoechnology i ued he wave equaion [8,,8] for he wave propagaion in a long line wih daping bu wihou diorion. Wih uing Laplace ranfor i obained he linear ordinary econd-order differenial equaion wih he paraeer p. Correpondingly he original proble for he parial differenial equaion of hyperbolic ype uing he Laplace ranfor i reduced o he ipler proble [8,,] for he linear ordinary differenial equaion Maer Sci Nanoechnol 8 Volue Iue

2 Ciaion: fonin SM. Paraeric rucural cheaic diagra of elecroelaic acuaor in nanoechnology. Maer Sci Nanoechnol. 8;():5-9. ( x,p) d Ξ γ Ξ ( x,p) () dx wih i oluion ( x, p) x Ce x Be where ( x, p) γ Ξ + (3) Ξ i he Laplace ranfor of he diplaceen of ecion of he acuaor, γ p c +α i he propagaion coefficien, c i he ound peed for con, α i he daping coefficien, C and B are conan. he generalized rucural-paraeric odel and he generalized paraeric rucural cheaic diagra of he elecroelaic acuaor for nanoechnology on Figure are deerined, uing equaion () and (), he boundary condiion on loaded face and he rain along he axe, in he following for ( Mp) { F ( ) ij i h ( l ) ch ( l ) } Ξ + χ ν γ γ γ Ξ Ξ (4) ( Mp) { F ( ij ) i ( l ) ( l ) h ch } Ξ + χ ν γ γ γ Ξ Ξ where,, ij χ ij, S ij D D D 33,,55 d,d,d ν i, g 33,g 3,g5 3,3, D3,D3,D, D, l { δ,h,b, { c c, c, γ { γ, γ D, v i i he coefficien of he elecroelaiciy, for exaple, d i i he piezoodule for he volage-conrolled piezoacuaor, g i i he piezoodule for he curren-conrolled piezoacuaor, S i he cro ecion area and M, M are he diplaced a on he face of he elecroelaic acuaor, F, F and F (p), F (p) are he Laplace ranfor of he diplaceen and he force on he face of he elecroelaic acuaor, conrol paraeer of he elecroelaic acuaor. i he he arix ae equaion [,,4] for he piezoelecric effec have he for: + ( ε ) D d (5) + S d (6) where he fir equaion decribe he direc piezoeffec, he econd equaion preen he invere piezoeffec, ( D ) i he colun-arix of he elecric inducion along he coordinae axe, (S) i he colun-arix of he relaive deforaion, ( ) i he colun-arix of he echanical ree, ( ) i he colun-arix of he elecric field rengh along he coordinae axe, (d) i he ranpoed arix of he piezoelecric odule, ( ) i he elaic copliance arix, (ε ) i he arix of dielecric conan. he deforaion of he piezoacuaor correpond o i reed ae. If he echanical re are creaed in he piezoeleen, he deforaion S are fored in i [8,]. here are he ix re coponen,, 3, 4, 5, 6. he coponen 3 3 are defined o exenion-copreion ree, he coponen 4 6 are relaed o hear ree. Le u conider he ranvere piezoelecric effec in he piezoacuaor hown in Figure. he equaion of he invere ranvere piezoeffec [8,] S d + x, (7) 3 3 where S ξ( x, ) ) x i he relaive diplaceen of he cro ecion of he piezoacuaor along axi, d 3 i he piezoelecric odule for he ranvere piezoeffec, i he elaic copliance for con along axi, i he re along axi. Figure. Generalized paraeric rucural cheaic diagra of elecroelaic acuaor in nanoechnology. Maer Sci Nanoechnol 8 Volue Iue 6

3 fonin cheaic diagra of he volage-conrolled piezoacuaor for he ranvere piezoelecric effec on Figure 3. Figure. Piezoacuaor for he ranvere piezoelecric effec. he oluion of he linear ordinary econd-order differenial equaion wih he paraeer p () can be wrien a (3) and ubjec o he condiion Ξ (,p) Ξ for x (8) Ξ ( l,p) Ξ for x h herefore he conan C and B for he oluion we obain in he following for: ( h γ hγ ) h B ( Ξ Ξe ) ( hγ) C Ξe Ξ h γ, hen, he oluion (3) can be wrien a: { } h (9) Ξ x,p Ξ p h h x γ +Ξ p h xγ h hγ () he equaion of force acing on he face of he piezoacuaor ha he for: (, p) S F + M p ( h, p) S F M p,p and Ξ for x () Ξ for xh where ( ) are deerined fro he equaion of he invere ranvere piezoeffec. herefore we obain he ye of he equaion for he echanical ree a he face of he piezoacuaor for he ranvere piezoeffec in he for: ( Mp ) { F ( ) 3 3 } d p h h p p Ξ + χ γ h γ ch γ Ξ Ξ (3) ( Mp) { F ( ) 3 3 } d p h h p p h ch Ξ + χ γ γ γ Ξ Ξ χ S, l h where he paraeric rucural cheaic diagra of he volageconrolled or curren-conrolled piezoacuaor for he ranvere, longiudinal, hif piezoelecric effec are deerined fro he generalized rucural-paraeric odel of he elecroelaic acuaor. Marix equaion of diplaceen Fro generalized rucural-paraeric odel of he elecroelaic acuaor, aking ino accoun he generalized equaion of he elecroelaiciy, wave equaion and he equaion of he force on i face, we obain he ranfer funcion of he elecroelaic acuaor. Correpondingly he Laplace ranfor of diplaceen for wo face of he elecroelaic acuaor are dependen fro he Laplace ranfor of he general paraeer of conrol and force on wo face. Marix equaion of he Laplace ranfor of he diplaceen wih he arix ranfer funcion of he elecroelaic acuaor i obained [8,4,8] in he for ( p ) Ξ p W p W p W p 3 F p W p W p W3 p Ξ F (4) Le u conider he diplaceen he face of he volageconrolled he piezoacuaor for he ranvere piezoeffec wih he oupu paraeer diplaceen. 3 d Ξ( x,p ) d (,p) p 3 () 3 dx x d Ξ( x,p ) d (h,p ) p 3 3 dx x h he e of equaion () for echanical ree in piezoacuaor yield he following e of equaion decribing he rucural paraeric odel and paraeric rucural Figure 3. Paraeric rucural cheaic diagra of volageconrolled piezoacuaor for ranvere piezoelecric effec. 7 Maer Sci Nanoechnol 8 Volue Iue

4 Ciaion: fonin SM. Paraeric rucural cheaic diagra of elecroelaic acuaor in nanoechnology. Maer Sci Nanoechnol. 8;():5-9. In arix equaion (4) he ranfer funcion of he volageconrolled he piezoacuaor for he ranvere piezoeffec can be wrien in he following for: Ξ χ +γh( γ ) W p p p d M p h 3 3 χ S { h } ( M M ) h( h ) ( c ) p p c MM χ p + M + M χ c hγ p χ α γ + + α +α Ξ χ +γh( γ ) Ξ χ χ +γ h( γ) W p p 3 p d3 M p h W p p F p M p h Ξ h W p p F p 3 W p Ξ p F p χγ hγ Ξ χ χ +γ h( γ) W3 p p F p M p h he aic diplaceen of he face for he volageconrolled he piezoacuaor for he ranvere piezoeffec are obained fro (4) a < M and < M in he for p α pw p U d hu M ξ ( ) li δp δ M + M á 3 pw p U d hu M ξ ( ) li p äp ä M M 3 ( + ) (5) (6) For he volage-conrolled he piezoacuaor fro PZ under he ranvere piezoeffec a <<M and <<M, d 3. 5 /V, h δ, U 5 V, M kg and M 8 kg he aic diplaceen of he face he piezoacuaor are deerined ξ ( ) n, ξ ( ) 5 n, ξ( ) + ξ( ) 5 n. he ranfer funcion of he volage-conrolled ranvere piezoacuaor i obained fro (4) for elaic-inerial load a M, << M and he approxiaion he hyperbolic coangen by wo er of he power erie in he for: Ξ d3 h δ W (7) U p + C C p + ξ p+ ( e ) ( ) ( e+ ), ξ α hc 3c M ( Ce + C ) M C C where U( p ) i he Laplace ranfor of he volage, i he ie conan and ξ i he daping coefficien of he piezoacuaor. herefore he expreion for he ranien repone of he volage-conrolled ranvere piezoacuaor i deerined in he following for: ξ e (8) ξ ξ in ( ω +φ) ξ d3 ( h δ ) U ξ ξ ξ, ω, φ arcg + Ce C ξ Maer Sci Nanoechnol 8 Volue Iue where ξ i he eady-ae value of diplaceen for he volage-conrolled piezoacuaor, U i he apliude of he volage in he eady-ae. For M, << M. U 5 V, d 3. 5 /V, h δ, M kg, 7 C N/, C e.5. 7 H/ we obain value he eady-ae value of diplaceen and he ie conan of he acuaor ξ 6 3 n,. c. he arix ranfer funcion of he acuaor are deerined for conrol ye wih he elecroelaic acuaor in he nanoechnology. In hi work he generalized paraeric rucural cheaic diagra and generalized rucural-paraeric odel of he elecroelaic acuaor are obained. Fro generalized rucuralparaeric odel of he elecroelaic acuaor afer algebraic ranforaion he ranfer funcion of he elecroelaic acuaor are deerined. he paraeric rucural cheaic diagra, he rucuralparaeric odel of he piezoacuaor for he ranvere, longiudinal, hif piezoelecric effec are deerined fro he generalized rucural-paraeric odel of he elecroelaic acuaor for nanoechnology. Concluion For nanoechnology he generalized paraeric rucural cheaic diagra and he generalized rucural-paraeric odel of he elecroelaic acuaor are conruced wih he echanical paraeer he diplaceen and he force. he paraeric rucural cheaic diagra of he piezoacuaor for he ranvere, longiudinal, hif piezoelecric effec are deerined. he arix ranfer funcion of he elecroelaic acuaor are deerined for conrol ye in he nanoechnology. Reference. Schulz J, Ueda J, ada H. Cellular acuaor. Oxford: Buerworh-Heineann Publiher. 7:38.. fonin SM. bolue abiliy condiion for a ye conrolling he deforaion of an elecroagneoelaic randucer. Doklady aheaic. 6;74(3): Zhou S, Yao Z. Deign and opiizaion of a odalindependen linear ulraonic oor. I ranacion on ulraonic, ferroelecric, and frequency conrol. 4;6(3): Przybylki J. Saic and dynaic analyi of a flexenional randucer wih an axial piezoelecric acuaion. ngineering Srucure. 5;84: Ueda J, Secord, ada HH. Large effecive-rain piezoelecric acuaor uing need cellular archiecure wih exponenial rain aplificaion echani. I/ SM ranacion on echaronic. ;5(5): Karpelon M, Wei GY, Wood RJ. Driving high volage piezoelecric acuaor in icroroboic applicaion. Senor and acuaor : Phyical. ;76:

5 fonin 7. fonin SM. Block diagra of a ulilayer piezoelecric oor for nano- and icro-diplaceen baed on he ranvere piezoeffec. J Copu Sy Sci In. 5;54(3): fonin SM. Srucural paraeric odel of a piezoelecric nanodiplaceen randuer. Doklady Phyic. 8;53(3): fonin SM. Soluion of he wave equaion for he conrol of an elecroagneoelaic randucer. Doklady Maheaic. 6;73(): Cady WG. Piezoelecriciy: n inroducion o he heory and applicaion of elecroechancial phenoena in cryal. New York, London: McGraw-Hill Book Copany. 946;86.. Phyical acouic: Principle and ehod. Par. Mehod and device. d.: Maon W. New York: cadeic Pre. 964;:55.. Zwillinger D. Handbook of differenial equaion. Boon: cadeic Pre. 989; fonin SM. Srucural-paraeric odel and ranfer funcion of elecroelaic acuaor for nano- and icrodiplaceen. Chaper 9 in Piezoelecric and nanoaerial: Fundaenal, developen and applicaion. d. Parinov I. New York: Nova Science. 5; fonin SM. rucural-paraeric odel of elecroelaic acuaor for nano- and icrodiplaceen of echaronic ye. Chaper 8 in dvance in nanoechnology. Volue 9. d. Barul Z, renor J. New York: Nova Science. 7; fonin SM. Nano- and icro-cale piezooor. Ruian engineering reearch. ;3(7-8): fonin SM. laic copliance and echanical and adjuing characeriic of copoie piezoelecric randucer. Mech Solid. 7;4(): fonin SM. Sabiliy of rain conrol ye of nanoand icrodiplaceen piezoranducer. Mech Solid. 4;49(): fonin SM. Srucural-paraeric odel elecroagneoelaic acuaor nanodiplaceen for echaronic. In J Phy. 7;5():9-5. *Correpondence o: fonin SM Deparen of Inellecual echnical Sye Naional Reearch Univeriy of lecronic echnology (MI) Mocow Ruia -ail: learner@ail.ru 9 Maer Sci Nanoechnol 8 Volue Iue