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eISSN: 2576-4500

Aeronautics and Aerospace Open Access Journal

Research Article Volume 6 Issue 4

Piezoactuator of nanodisplacement for astrophysics

Afonin SM

National Research University of Electronic Technology, MIET, Moscow, Russia

Correspondence: Afonin SM, National Research University of Electronic Technology, MIET, 124498, Moscow, Russia

Received: September 13, 2022 | Published: September 27, 2022

Citation: Afonin SM. Piezoactuator of nanodisplacement for astrophysics. Aeron Aero Open Access J. 2022;6(4):155-158. DOI: 10.15406/aaoaj.2022.06.00155

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Abstract

The structural scheme of a piezoactuator is obtained for astrophysics. The matrix equation is constructed for a piezoactuator. The characteristics of a piezoactuator are received for astrophysics.

Keywords: piezoactuator, structural scheme, nanodisplacement, characteristic, astrophysics

Introduction

For the control system in astrophysics a piezoactuator of the nanodisplacement is applied in very large telescope, interferometer and orbital telescope.1–9 The energy conversion is clearly for the structural scheme of a piezoactuator.10–16 A piezoactuator is used for the nanodisplacement in adaptive optics and telescopes.17–26

Structural scheme and characteristics

The equations27–35 of the piezoeffects have form

( D )=( d )( T )+( ε T )( E ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGebaacaGLOaGaayzkaaGaeyypa0ZaaeWaaeaacaWGKbaacaGLOaGa ayzkaaWaaeWaaeaacaWGubaacaGLOaGaayzkaaGaey4kaSYaaeWaae aacqaH1oqzdaahaaWcbeqaaiaadsfaaaaakiaawIcacaGLPaaadaqa daqaaiaadweaaiaawIcacaGLPaaaaaa@4598@

( S )=( s E )( T )+ ( d ) t ( E ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGtbaacaGLOaGaayzkaaGaeyypa0ZaaeWaaeaacaWGZbWaaWbaaSqa beaacaWGfbaaaaGccaGLOaGaayzkaaWaaeWaaeaacaWGubaacaGLOa GaayzkaaGaey4kaSYaaeWaaeaacaWGKbaacaGLOaGaayzkaaWaaWba aSqabeaacaWG0baaaOWaaeWaaeaacaWGfbaacaGLOaGaayzkaaaaaa@4619@

where ( D ),( d ),( T ),( ε T ),( E ),( S ),( s E ), ( d ) t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGebaacaGLOaGaayzkaaGaaiilamaabmaabaGaamizaaGaayjkaiaa wMcaaiaacYcadaqadaqaaiaadsfaaiaawIcacaGLPaaacaGGSaWaae WaaeaacqaH1oqzdaahaaWcbeqaaiaadsfaaaaakiaawIcacaGLPaaa caGGSaWaaeWaaeaacaWGfbaacaGLOaGaayzkaaGaaiilamaabmaaba Gaam4uaaGaayjkaiaawMcaaiaacYcadaqadaqaaiaadohadaahaaWc beqaaiaadweaaaaakiaawIcacaGLPaaacaGGSaWaaeWaaeaacaWGKb aacaGLOaGaayzkaaWaaWbaaSqabeaacaWG0baaaaaa@51FB@  are matrixes of electric induction, piezomodule, strength mechanical field, dielectric constant, strength electric field, relative displacement, elastic compliance, transposed piezomodule. The matrixes coefficients we have for a PZT piezoactuator.36–52

( d )=( 0 0 0 0 d  15 0 0 0 0 d  15 0 0 d  31 d  31 d  33 0 0 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaam izaaGaayjkaiaawMcaaiabg2da9maabmaabaqbaeqabmqaaaqaauaa beqabyaaaaqaaiaaicdadaWgaaWcbaaabeaaaOqaaiaaicdadaWgaa WcbaaabeaaaOqaaiaaicdadaWgaaWcbaaabeaaaOqaaiaaicdadaWg aaWcbaaabeaaaOqaaiaadsgalmaaBaaabaaeaaaaaaaaa8qacaGGGc WdaiaaigdacaaI1aaabeaaaOqaaiaaicdaaaaabaqbaeqabeGbaaaa baGaaGimamaaBaaaleaaaeqaaaGcbaGaaGimamaaBaaaleaaaeqaaa GcbaGaaGimamaaBaaaleaaaeqaaaGcbaGaamizaSWaaSbaaeaapeGa aiiOa8aacaaIXaGaaGynaaqabaaakeaacaaIWaWaaSbaaSqaaaqaba aakeaacaaIWaaaaaqaauaabeqabyaaaaqaaiaadsgalmaaBaaabaWd biaacckapaGaaG4maiaaigdaaeqaaaGcbaGaamizaSWaaSbaaeaape GaaiiOa8aacaaIZaGaaGymaaqabaaakeaacaWGKbWaaSbaaSqaa8qa caGGGcWdaiaaiodacaaIZaaakeqaaaqaaiaaicdadaWgaaWcbaaabe aaaOqaaiaaicdadaWgaaWcbaaabeaaaOqaaiaaicdadaWgaaWcbaaa beaaaaaaaaGccaGLOaGaayzkaaaaaa@5B79@

( ε T )=( ε  11 T 0 0 0 ε  22 T 0 0 0 ε  33 T ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaeq yTdu2aaWbaaSqabeaacaWGubaaaaGccaGLOaGaayzkaaGaeyypa0Za aeWaaeaafaqabeWadaaabaGaeqyTdu2aa0baaSqaaabaaaaaaaaape GaaiiOa8aacaaIXaGaaGymaaqaaiaadsfaaaaakeaacaaIWaaabaGa aGimaaqaaiaaicdaaeaacqaH1oqzdaqhaaWcbaWdbiaacckapaGaaG OmaiaaikdaaeaacaWGubaaaaGcbaGaaGimaaqaaiaaicdaaeaacaaI WaaabaGaeqyTdu2aa0baaSqaa8qacaGGGcWdaiaaiodacaaIZaaaba GaamivaaaaaaaakiaawIcacaGLPaaaaaa@52B6@

( s E )=( s  11 E s  12 E s  13 E 0 0 0 s  12 E s  11 E s  13 E 0 0 0 s  13 E s  13 E s  33 E 0 0 0 0 0 0 s  55 E 0 0 0 0 0 0 s  55 E 0 0 0 0 0 0 2( s  11 E s  12 E ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaam 4CamaaCaaaleqabaGaamyraaaaaOGaayjkaiaawMcaaiabg2da9maa bmaabaqbaeqabyGbaaaaaeaacaWGZbWcdaqhaaqaaabaaaaaaaaape GaaiiOa8aacaaIXaGaaGymaaqaaiaadweaaaaakeaacaWGZbWcdaqh aaqaa8qacaGGGcWdaiaaigdacaaIYaaabaGaamyraaaaaOqaaiaado halmaaDaaabaWdbiaacckapaGaaGymaiaaiodaaeaacaWGfbaaaaGc baGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaam4CaSWaa0baaeaape GaaiiOa8aacaaIXaGaaGOmaaqaaiaadweaaaaakeaacaWGZbWcdaqh aaqaa8qacaGGGcWdaiaaigdacaaIXaaabaGaamyraaaaaOqaaiaado halmaaDaaabaWdbiaacckapaGaaGymaiaaiodaaeaacaWGfbaaaaGc baGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaam4CaSWaa0baaeaape GaaiiOa8aacaaIXaGaaG4maaqaaiaadweaaaaakeaacaWGZbWcdaqh aaqaa8qacaGGGcWdaiaaigdacaaIZaaabaGaamyraaaaaOqaaiaado halmaaDaaabaWdbiaacckapaGaaG4maiaaiodaaeaacaWGfbaaaaGc baGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaae aacaaIWaaabaGaam4CaSWaa0baaeaapeGaaiiOa8aacaaI1aGaaGyn aaqaaiaadweaaaaakeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaaca aIWaaabaGaaGimaaqaaiaaicdaaeaacaWGZbWcdaqhaaqaa8qacaGG GcWdaiaaiwdacaaI1aaabaGaamyraaaaaOqaaiaaicdaaeaacaaIWa aabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaikda daqadaqaaiaadohalmaaDaaabaWdbiaacckapaGaaGymaiaaigdaae aacaWGfbaaaOGaeyOeI0Iaam4CaSWaa0baaeaapeGaaiiOa8aacaaI XaGaaGOmaaqaaiaadweaaaaakiaawIcacaGLPaaaaaaacaGLOaGaay zkaaaaaa@8F32@

The equation of the mechanical characteristic is written for a piezoactuator

Δl=Δ l max ( 1F/ F max ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam iBaiabg2da9iabfs5aejaadYgalmaaBaaabaGaaeyBaiaabggacaqG 4baabeaakmaabmaabaGaaGymaiabgkHiTmaalyaabaGaamOraaqaai aadAeadaWgaaWcbaGaaeyBaiaabggacaqG4baabeaaaaaakiaawIca caGLPaaaaaa@4692@

where Δ l max = d m i E m l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfs5aejaadY galmaaBaaabaGaaeyBaiaabggacaqG4baabeaakiabg2da9iaadsga lmaaBaaabaGaamyBaabaaaaaaaaapeGaaiiOa8aacaWGPbaabeaaki aadweadaWgaaWcbaGaamyBaaGcbeaacaWGSbaaaa@44A5@  for F=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabg2 da9iaaicdaaaa@3882@  and F max = d m i E m S 0 / s ij E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaWgaa WcbaGaaeyBaiaabggacaqG4baabeaakiabg2da9maalyaabaGaamiz aSWaaSbaaeaacaWGTbaeaaaaaaaaa8qacaGGGcWdaiaadMgaaeqaaO GaamyramaaBaaaleaacaWGTbaakeqaaiaadofalmaaBaaabaGaaGim aaqabaaakeaacaWGZbWcdaqhaaqaaGqaciaa=LgacaWFQbaabaGaam yraaaaaaaaaa@47D5@  for Δl=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam iBaiabg2da9iaaicdaaaa@3A0E@ , l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaaaa@36E8@  is the length, S 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaSWaaS baaeaacaaIWaaabeaaaaa@37B5@  is the area of a piezoactuator.

For the longitudinal piezoactuator the relative displacement8–21 is written

S 3 = d  33 E 3 + s  33 E T 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadofalmaaBa aabaGaaG4maaqabaGccqGH9aqpcaWGKbWaaSbaaSqaaabaaaaaaaaa peGaaiiOa8aacaaIZaGaaG4maaGcbeaacaWGfbWcdaWgaaqaaiaaio daaeqaaOGaey4kaSIaam4CaSWaa0baaeaapeGaaiiOa8aacaaIZaGa aG4maaqaaiaadweaaaGccaWGubWaaSbaaSqaaiaaiodaaOqabaaaaa@46EC@

where d  33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsgalmaaBa aabaaeaaaaaaaaa8qacaGGGcWdaiaaiodacaaIZaaabeaaaaa@3AF0@  is the longitudinal piezomodule.

In the mechanical characteristic of the longitudinal piezoactuator for astrophysics the maximums values of the displacement Δ δ max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeq iTdq2cdaWgaaqaaiaab2gacaqGHbGaaeiEaaqabaaaaa@3BFD@  and the force F max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaqGTbGaaeyyaiaabIhaaeqaaaaa@39BD@  are determined

Δ δ max = d  33 δ E 3 = d  33 U, F max = d  33 S 0 E 3 / s  33 E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfs5aejabes 7aKTWaaSbaaeaacaqGTbGaaeyyaiaabIhaaeqaaOGaeyypa0Jaamiz aSWaaSbaaeaaqaaaaaaaaaWdbiaacckapaGaaG4maiaaiodaaeqaaO GaeqiTdqMaamyramaaBaaaleaacaaIZaaakeqaaiabg2da9iaadsga lmaaBaaabaWdbiaacckapaGaaG4maiaaiodaaeqaaOGaamyvaiaacY cacaWGgbWaaSbaaSqaaiaab2gacaqGHbGaaeiEaaqabaGccqGH9aqp daWcgaqaaiaadsgalmaaBaaabaWdbiaacckapaGaaG4maiaaiodaae qaaOGaam4uaSWaaSbaaeaacaaIWaaabeaakiaadweadaWgaaWcbaGa aG4maaGcbeaaaeaacaWGZbWcdaqhaaqaa8qacaGGGcWdaiaaiodaca aIZaaabaGaamyraaaaaaaaaa@5CD7@

At E 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraSWaaS baaeaacaaIZaaabeaaaaa@37AA@  = 1.5∙105 V/m, d  33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsgalmaaBa aabaaeaaaaaaaaa8qacaGGGcWdaiaaiodacaaIZaaabeaaaaa@3AF0@ = 4∙10-10 m/V, S 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaSWaaS baaeaacaaIWaaabeaaaaa@37B5@ = 1.5∙10-4 m2, δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqgaaa@379C@ = 2.5∙10-3 m, s   33 E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadohaqaaaaa aaaaWdbiaacckal8aadaqhaaqaaiaaiodacaaIZaaabaGaamyraaaa aaa@3BCA@ = 15∙10-12 m2/N for the longitudinal piezoactuator are obtained Δ δ max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeq iTdq2cdaWgaaqaaiaab2gacaqGHbGaaeiEaaqabaaaaa@3BFD@  = 150 nm, F max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaqGTbGaaeyyaiaabIhaaeqaaaaa@39BD@  = 600 N with error 10%.

Therefore, for the mechanical characteristic of the transverse piezoactuator we have its maximums values

Δ h max = d 31 E 3 h= d 31 Uh/δ , F max = d 31 E 3 S 0 / s 11 E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuiLdq KaamiAaOWaaSbaaSqaaKqzGeGaaeyBaiaabggacaqG4baaleqaaKqz GeGaeyypa0JaamizaOWaaSbaaSqaaKqzGeGaaG4maiaaigdaaSqaba qcLbsacaWGfbGcdaWgaaWcbaqcLbsacaaIZaaakeqaaKqzGeGaamiA aiabg2da9OWaaSGbaeaajugibiaadsgakmaaBaaaleaajugibiaaio dacaaIXaaaleqaaKqzGeGaamyvaiaadIgaaOqaaKqzGeGaeqiTdqga aOGaaiilaKqzGeGaamOraOWaaSbaaSqaaKqzGeGaaeyBaiaabggaca qG4baaleqaaKqzGeGaeyypa0JcdaWcgaqaaKqzGeGaamizaOWaaSba aSqaaKqzGeGaaG4maiaaigdaaSqabaqcLbsacaWGfbGcdaWgaaWcba qcLbsacaaIZaaakeqaaKqzGeGaam4uaOWaaSbaaSqaaKqzGeGaaGim aaWcbeaaaOqaaKqzGeGaam4CaOWaa0baaSqaaKqzGeGaaGymaiaaig daaSqaaKqzGeGaamyraaaaaaaaaa@665F@

At E 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaaIZaaabeaaaaa@37AA@  = 2.4×105 V/m, d  31 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsgalmaaBa aabaaeaaaaaaaaa8qacaGGGcWdaiaaiodacaaIXaaabeaaaaa@3AEE@ = 2∙10-10 m/V, h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaaaa@36E4@  = 1∙10-2 m,

δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqgaaa@379C@  =0.5∙10-3 m, S 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaSWaaS baaeaacaaIWaaabeaaaaa@37B5@  = 1∙10-5 m2, s   11 E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadohaqaaaaa aaaaWdbiaacckal8aadaqhaaqaaiaaigdacaaIXaaabaGaamyraaaa aaa@3BC6@ = 12∙10-12 m2/N the parameters are received Δ h max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam iAaSWaaSbaaeaacaqGTbGaaeyyaiaabIhaaeqaaaaa@3B45@  = 480 nm, F max MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaqGTbGaaeyyaiaabIhaaeqaaaaa@39BD@  = 40 N.

The differential equation of a piezoactuator12–52 is written

d 2 Ξ( x,s ) d x 2 γ 2 Ξ( x,s )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam izamaaCaaabeWcbaGaaGOmaaaakiabf65aynaabmaabaGaamiEaiaa cYcacaWGZbaacaGLOaGaayzkaaaabaGaamizaiaadIhalmaaCaaabe qaaiaaikdaaaaaaOGaeyOeI0Iaeq4SdC2aaWbaaSqabeaacaaIYaaa aOGaeuONdG1aaeWaaeaacaWG4bGaaiilaiaadohaaiaawIcacaGLPa aacqGH9aqpcaaIWaaaaa@4C7E@

here Ξ( x,s ),s,x,γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuONdG1aae WaaeaacaWG4bGaaiilaiaadohaaiaawIcacaGLPaaacaGGSaGaam4C aiaacYcacaWG4bGaaiilaiabeo7aNbaa@4155@  are the Laplace transform of the displacement, the parameter, the coordinate and the propagation factor.

The nanodisplacements are obtained for the longitudinal piezoactuator

Ξ( 0,s )= Ξ 1 ( s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuONdG1aae WaaeaacaaIWaGaaiilaiaadohaaiaawIcacaGLPaaacqGH9aqpcqqH EoawlmaaBaaabaGaaGymaaqabaGcdaqadaqaaiaadohaaiaawIcaca GLPaaaaaa@4162@  for  x=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaaicdaaaa@38B4@

Ξ( δ,s )= Ξ 2 ( s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuONdG1aae WaaeaacqaH0oazcaGGSaGaam4CaaGaayjkaiaawMcaaiabg2da9iab f65ayTWaaSbaaeaacaaIYaaabeaakmaabmaabaGaam4CaaGaayjkai aawMcaaaaa@424E@  for  x=δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iabes7aKbaa@399F@

The decision of the differential equation is determined

Ξ( x,s )= { Ξ 1 ( s )s h[ ( δx )γ ]+ Ξ 2 ( s )s h( x γ ) }/ s h( (δ)γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabf65aynaabm aabaGaamiEaiaacYcacaWGZbaacaGLOaGaayzkaaGaeyypa0ZaaSGb aeaadaGadaqaaiabf65aynaaBaaaleaacaaIXaaabeaakmaabmaaba Gaam4CaaGaayjkaiaawMcaaiaabohaqaaaaaaaaaWdbiaacckapaGa aeiAamaadmaabaWaaeWaaeaacqaH0oazcqGHsislcaWG4baacaGLOa GaayzkaaGaeq4SdCgacaGLBbGaayzxaaGaey4kaSIaeuONdG1aaSba aSqaaiaaikdaaeqaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaaGaae 4Ca8qacaGGGcWdaiaabIgadaqadaqaaiaadIhapeGaaiiOa8aacqaH ZoWzaiaawIcacaGLPaaaaiaawUhacaGL9baaaeaacaqGZbWdbiaacc kapaGaaeiAamaabmaabaGaaiikaiabes7aK9qacaGGPaWdaiabeo7a NbGaayjkaiaawMcaaaaaaaa@68C6@

Taking into account the boundary conditions for two faces, we obtain the system of the equations for the structural model of the longitudinal piezoactuator

Ξ 1 ( s )= ( M 1 s 2 ) 1 { F 1 ( s )+ ( χ   33 E ) 1 [ d 33   E 3 ( s )[ γ/ s h( δ γ ) ] ×[ c h( δ γ ) Ξ 1 ( s ) Ξ 2 ( s ) ] ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuONdG LcdaWgaaWcbaqcLbsacaaIXaaaleqaaOWaaeWaaeaajugibiaadoha aOGaayjkaiaawMcaaKqzGeGaeyypa0JcdaqadaqaaKqzGeGaamytaO WaaSbaaSqaaKqzGeGaaGymaaWcbeaajugibiaadohakmaaCaaaleqa baqcLbsacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaajugibi abgkHiTiaaigdaaaGcdaGadaqaaKqzGeGaeyOeI0IaamOraOWaaSba aSqaaKqzGeGaaGymaaWcbeaakmaabmaabaqcLbsacaWGZbaakiaawI cacaGLPaaajugibiabgUcaROWaaeWaaeaajugibiabeE8aJbbaaaaa aaaapeGaaiiOaOWdamaaDaaaleaajugibiaaiodacaaIZaaaleaaju gibiaadweaaaaakiaawIcacaGLPaaadaahaaWcbeqaaKqzGeGaeyOe I0IaaGymaaaakmaadmaajugibqaabeGcbaqcLbsacaWGKbGcdaWgaa WcbaqcLbsacaaIZaGaaG4maaWcbeaajugib8qacaGGGcWdaiaadwea kmaaBaaaleaajugibiaaiodaaSqabaGcdaqadaqaaKqzGeGaam4Caa GccaGLOaGaayzkaaqcLbsacqGHsislkmaadmaabaWaaSGbaeaajugi biabeo7aNbGcbaqcLbsacaqGZbWdbiaacckapaGaaeiAaOWaaeWaae aajugibiabes7aK9qacaGGGcWdaiabeo7aNbGccaGLOaGaayzkaaaa aaGaay5waiaaw2faaKqzGeGaaGjbVdGcbaqcLbsacqGHxdaTkmaadm aabaqcLbsacaqGJbWdbiaacckapaGaaeiAaOWaaeWaaeaajugibiab es7aK9qacaGGGcWdaiabeo7aNbGccaGLOaGaayzkaaqcLbsacqqHEo awkmaaBaaaleaajugibiaaigdaaSqabaGcdaqadaqaaKqzGeGaam4C aaGccaGLOaGaayzkaaqcLbsacqGHsislcqqHEoawkmaaBaaaleaaju gibiaaikdaaSqabaGcdaqadaqaaKqzGeGaam4CaaGccaGLOaGaayzk aaaacaGLBbGaayzxaaaaaiaawUfacaGLDbaaaiaawUhacaGL9baaaa a@9CBA@

Ξ 2 ( s )= ( M 2 s 2 ) 1 { F 2 ( s )+ ( χ   33 E ) 1 [ d 33   E 3 ( s )[ γ/ s h( δ γ ) ] ×[ c h( δ γ ) Ξ 2 ( s ) Ξ 1 ( s ) ] ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuONdG LcdaWgaaWcbaqcLbsacaaIYaaaleqaaOWaaeWaaeaajugibiaadoha aOGaayjkaiaawMcaaKqzGeGaeyypa0JcdaqadaqaaKqzGeGaamytaO WaaSbaaSqaaKqzGeGaaGOmaaWcbeaajugibiaadohakmaaCaaaleqa baqcLbsacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaajugibi abgkHiTiaaigdaaaGcdaGadaqaaKqzGeGaeyOeI0IaamOraOWaaSba aSqaaKqzGeGaaGOmaaWcbeaakmaabmaabaqcLbsacaWGZbaakiaawI cacaGLPaaajugibiabgUcaROWaaeWaaeaajugibiabeE8aJbbaaaaa aaaapeGaaiiOaOWdamaaDaaaleaajugibiaaiodacaaIZaaaleaaju gibiaadweaaaaakiaawIcacaGLPaaadaahaaWcbeqaaKqzGeGaeyOe I0IaaGymaaaakmaadmaajugibqaabeGcbaqcLbsacaWGKbGcdaWgaa WcbaqcLbsacaaIZaGaaG4maaWcbeaajugib8qacaGGGcWdaiaadwea kmaaBaaaleaajugibiaaiodaaSqabaGcdaqadaqaaKqzGeGaam4Caa GccaGLOaGaayzkaaqcLbsacqGHsislkmaadmaabaWaaSGbaeaajugi biabeo7aNbGcbaqcLbsacaqGZbWdbiaacckapaGaaeiAaOWaaeWaae aajugibiabes7aK9qacaGGGcWdaiabeo7aNbGccaGLOaGaayzkaaaa aaGaay5waiaaw2faaaqaaKqzGeGaey41aqRcdaWadaqaaKqzGeGaae 4ya8qacaGGGcWdaiaabIgakmaabmaabaqcLbsacqaH0oazpeGaaiiO a8aacqaHZoWzaOGaayjkaiaawMcaaKqzGeGaeuONdGLcdaWgaaWcba qcLbsacaaIYaaaleqaaOWaaeWaaeaajugibiaadohaaOGaayjkaiaa wMcaaKqzGeGaeyOeI0IaeuONdGLcdaWgaaWcbaqcLbsacaaIXaaale qaaOWaaeWaaeaajugibiaadohaaOGaayjkaiaawMcaaaGaay5waiaa w2faaaaacaGLBbGaayzxaaaacaGL7bGaayzFaaaaaa@9A97@

χ   33 E = s   33 E / S 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4Xdm geaaaaaaaaa8qacaGGGcGcpaWaa0baaSqaaKqzGeGaaG4maiaaioda aSqaaKqzGeGaamyraaaacqGH9aqpkmaalyaabaqcLbsacaWGZbWdbi aacckak8aadaqhaaWcbaqcLbsacaaIZaGaaG4maaWcbaqcLbsacaWG fbaaaaGcbaqcLbsacaWGtbGcdaWgaaWcbaqcLbsacaaIWaaaleqaaa aaaaa@48DA@

where Ξ 1 ( s ), Ξ 2 ( s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuONdG1aaS baaSqaaiaaigdaaeqaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaaGa aiilaiabf65ayTWaaSbaaeaacaaIYaaabeaakmaabmaabaGaam4Caa GaayjkaiaawMcaaaaa@4094@  are the Laplace transforms of the displacements for two faces.

We have the system of the equations for the structural model of the transverse piezoactuator

Ξ 1 ( s )= ( M 1 s 2 ) 1 { F 1 ( s )+ ( χ   11 E ) 1 [ d   31 E 3 ( s )[ γ/ s h( h γ ) ] ×[ c h( h γ ) Ξ 1 ( s ) Ξ 2 ( s ) ] ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuONdG LcdaWgaaWcbaqcLbsacaaIXaaaleqaaOWaaeWaaeaajugibiaadoha aOGaayjkaiaawMcaaKqzGeGaeyypa0JcdaqadaqaaKqzGeGaamytaO WaaSbaaSqaaKqzGeGaaGymaaWcbeaajugibiaadohakmaaCaaaleqa baqcLbsacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaajugibi abgkHiTiaaigdaaaGcdaGadaqaaKqzGeGaeyOeI0IaamOraOWaaSba aSqaaKqzGeGaaGymaaWcbeaakmaabmaabaqcLbsacaWGZbaakiaawI cacaGLPaaajugibiabgUcaROWaaeWaaeaajugibiabeE8aJbbaaaaa aaaapeGaaiiOaOWdamaaDaaaleaajugibiaaigdacaaIXaaaleaaju gibiaadweaaaaakiaawIcacaGLPaaadaahaaWcbeqaaKqzGeGaeyOe I0IaaGymaaaakmaadmaajugibqaabeGcbaqcLbsacaWGKbWdbiaacc kak8aadaWgaaWcbaqcLbsacaaIZaGaaGymaaWcbeaajugibiaadwea kmaaBaaaleaajugibiaaiodaaSqabaGcdaqadaqaaKqzGeGaam4Caa GccaGLOaGaayzkaaqcLbsacqGHsislkmaadmaabaWaaSGbaeaajugi biabeo7aNbGcbaqcLbsacaqGZbWdbiaacckapaGaaeiAaOWaaeWaae aajugibiaadIgapeGaaiiOa8aacqaHZoWzaOGaayjkaiaawMcaaaaa aiaawUfacaGLDbaajugibiaaysW7aOqaaKqzGeGaey41aqRcdaWada qaaKqzGeGaae4ya8qacaGGGcWdaiaabIgakmaabmaabaqcLbsacaWG ObWdbiaacckapaGaeq4SdCgakiaawIcacaGLPaaajugibiabf65ayP WaaSbaaSqaaKqzGeGaaGymaaWcbeaakmaabmaabaqcLbsacaWGZbaa kiaawIcacaGLPaaajugibiabgkHiTiabf65ayPWaaSbaaSqaaKqzGe GaaGOmaaWcbeaakmaabmaabaqcLbsacaWGZbaakiaawIcacaGLPaaa aiaawUfacaGLDbaaaaGaay5waiaaw2faaaGaay5Eaiaaw2haaaaa@9B44@

Ξ 2 ( s )= ( M 2 s 2 ) 1 { F 2 ( s )+ ( χ 11 E ) 1 [ d 31 E 3 ( s )[ γ/ sh( hγ ) ] ×[ ch( hγ ) Ξ 2 ( s ) Ξ 1 ( s ) ] ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuONdG LcdaWgaaWcbaqcLbsacaaIYaaaleqaaOWaaeWaaeaajugibiaadoha aOGaayjkaiaawMcaaKqzGeGaeyypa0JcdaqadaqaaKqzGeGaamytaO WaaSbaaSqaaKqzGeGaaGOmaaWcbeaajugibiaadohakmaaCaaaleqa baqcLbsacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaajugibi abgkHiTiaaigdaaaGcdaGadaqaaKqzGeGaeyOeI0IaamOraOWaaSba aSqaaKqzGeGaaGOmaaWcbeaakmaabmaabaqcLbsacaWGZbaakiaawI cacaGLPaaajugibiabgUcaROWaaeWaaeaajugibiabeE8aJPWaa0ba aSqaaKqzGeGaaGymaiaaigdaaSqaaKqzGeGaamyraaaaaOGaayjkai aawMcaamaaCaaaleqabaqcLbsacqGHsislcaaIXaaaaOWaamWaaKqz GeabaeqakeaajugibiaadsgakmaaBaaaleaajugibiaaiodacaaIXa aaleqaaKqzGeGaamyraOWaaSbaaSqaaKqzGeGaaG4maaWcbeaakmaa bmaabaqcLbsacaWGZbaakiaawIcacaGLPaaajugibiabgkHiTOWaam WaaeaadaWcgaqaaKqzGeGaeq4SdCgakeaajugibiaabohacaqGObGc daqadaqaaKqzGeGaamiAaiabeo7aNbGccaGLOaGaayzkaaaaaaGaay 5waiaaw2faaaqaaKqzGeGaey41aqRcdaWadaqaaKqzGeGaae4yaiaa bIgakmaabmaabaqcLbsacaWGObGaeq4SdCgakiaawIcacaGLPaaaju gibiabf65ayPWaaSbaaSqaaKqzGeGaaGOmaaWcbeaakmaabmaabaqc LbsacaWGZbaakiaawIcacaGLPaaajugibiabgkHiTiabf65ayPWaaS baaSqaaKqzGeGaaGymaaWcbeaakmaabmaabaqcLbsacaWGZbaakiaa wIcacaGLPaaaaiaawUfacaGLDbaaaaGaay5waiaaw2faaaGaay5Eai aaw2haaaaa@917F@

χ   11 E = s 11 E / S 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4Xdm geaaaaaaaaa8qacaGGGcGcpaWaa0baaSqaaKqzGeGaaGymaiaaigda aSqaaKqzGeGaamyraaaacqGH9aqpkmaalyaabaqcLbsacaWGZbGcda qhaaWcbaqcLbsacaaIXaGaaGymaaWcbaqcLbsacaWGfbaaaaGcbaqc LbsacaWGtbGcdaWgaaWcbaqcLbsacaaIWaaaleqaaaaaaaa@478F@

Therefore, we have the system of the equations for the structural model of the shift piezoactuator in the form

Ξ 1 ( s )= ( M 1 s 2 ) 1 { F 1 ( s )+ ( χ 55 E ) 1 [ d   15 E 1 ( s )[ γ/ s h( b γ ) ] ×[ c h( bγ ) Ξ 1 ( s ) Ξ 2 ( s ) ] ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuONdG LcdaWgaaWcbaqcLbsacaaIXaaaleqaaOWaaeWaaeaajugibiaadoha aOGaayjkaiaawMcaaKqzGeGaeyypa0JcdaqadaqaaKqzGeGaamytaO WaaSbaaSqaaKqzGeGaaGymaaWcbeaajugibiaadohakmaaCaaaleqa baqcLbsacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaajugibi abgkHiTiaaigdaaaGcdaGadaqaaKqzGeGaeyOeI0IaamOraOWaaSba aSqaaKqzGeGaaGymaaWcbeaakmaabmaabaqcLbsacaWGZbaakiaawI cacaGLPaaajugibiabgUcaROWaaeWaaeaajugibiabeE8aJPWaa0ba aSqaaKqzGeGaaGynaiaaiwdaaSqaaKqzGeGaamyraaaaaOGaayjkai aawMcaamaaCaaaleqabaqcLbsacqGHsislcaaIXaaaaOWaamWaaKqz GeabaeqakeaajugibiaadsgaqaaaaaaaaaWdbiaacckak8aadaWgaa WcbaqcLbsacaaIXaGaaGynaaWcbeaajugibiaadweakmaaBaaaleaa jugibiaaigdaaSqabaGcdaqadaqaaKqzGeGaam4CaaGccaGLOaGaay zkaaqcLbsacqGHsislkmaadmaabaWaaSGbaeaajugibiabeo7aNbGc baqcLbsacaqGZbWdbiaacckapaGaaeiAaOWaaeWaaeaajugibiaadk gapeGaaiiOa8aacqaHZoWzaOGaayjkaiaawMcaaaaaaiaawUfacaGL DbaajugibiaaysW7aOqaaKqzGeGaey41aqRcdaWadaqaaKqzGeGaae 4ya8qacaGGGcWdaiaabIgakmaabmaabaqcLbsacaWGIbGaeq4SdCga kiaawIcacaGLPaaajugibiabf65ayPWaaSbaaSqaaKqzGeGaaGymaa WcbeaakmaabmaabaqcLbsacaWGZbaakiaawIcacaGLPaaajugibiab gkHiTiabf65ayPWaaSbaaSqaaKqzGeGaaGOmaaWcbeaakmaabmaaba qcLbsacaWGZbaakiaawIcacaGLPaaaaiaawUfacaGLDbaaaaGaay5w aiaaw2faaaGaay5Eaiaaw2haaaaa@98BA@

Ξ 2 ( s )= ( M 2 s 2 ) 1 { F 2 ( s )+ ( χ   11 E ) 1 [ d   31 E 3 ( s )[ γ/ sh( hγ ) ] ×[ c h( h γ ) Ξ 2 ( s ) Ξ 1 ( s ) ] ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuONdG LcdaWgaaWcbaqcLbsacaaIYaaaleqaaOWaaeWaaeaajugibiaadoha aOGaayjkaiaawMcaaKqzGeGaeyypa0JcdaqadaqaaKqzGeGaamytaO WaaSbaaSqaaKqzGeGaaGOmaaWcbeaajugibiaadohakmaaCaaaleqa baqcLbsacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaajugibi abgkHiTiaaigdaaaGcdaGadaqaaKqzGeGaeyOeI0IaamOraOWaaSba aSqaaKqzGeGaaGOmaaWcbeaakmaabmaabaqcLbsacaWGZbaakiaawI cacaGLPaaajugibiabgUcaROWaaeWaaeaajugibiabeE8aJbbaaaaa aaaapeGaaiiOaOWdamaaDaaaleaajugibiaaigdacaaIXaaaleaaju gibiaadweaaaaakiaawIcacaGLPaaadaahaaWcbeqaaKqzGeGaeyOe I0IaaGymaaaakmaadmaajugibqaabeGcbaqcLbsacaWGKbWdbiaacc kak8aadaWgaaWcbaqcLbsacaaIZaGaaGymaaWcbeaajugibiaadwea kmaaBaaaleaajugibiaaiodaaSqabaGcdaqadaqaaKqzGeGaam4Caa GccaGLOaGaayzkaaqcLbsacqGHsislkmaadmaabaWaaSGbaeaajugi biabeo7aNbGcbaqcLbsacaqGZbGaaeiAaOWaaeWaaeaajugibiaadI gacqaHZoWzaOGaayjkaiaawMcaaaaaaiaawUfacaGLDbaaaeaajugi biabgEna0QWaamWaaeaajugibiaabogapeGaaiiOa8aacaqGObGcda qadaqaaKqzGeGaamiAa8qacaGGGcWdaiabeo7aNbGccaGLOaGaayzk aaqcLbsacqqHEoawkmaaBaaaleaajugibiaaikdaaSqabaGcdaqada qaaKqzGeGaam4CaaGccaGLOaGaayzkaaqcLbsacqGHsislcqqHEoaw kmaaBaaaleaajugibiaaigdaaSqabaGcdaqadaqaaKqzGeGaam4Caa GccaGLOaGaayzkaaaacaGLBbGaayzxaaaaaiaawUfacaGLDbaaaiaa wUhacaGL9baaaaa@969B@

χ 55 E = s 55 E / S 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4Xdm McdaqhaaWcbaqcLbsacaaI1aGaaGynaaWcbaqcLbsacaWGfbaaaiab g2da9OWaaSGbaeaajugibiaadohakmaaDaaaleaajugibiaaiwdaca aI1aaaleaajugibiaadweaaaaakeaajugibiaadofakmaaBaaaleaa jugibiaaicdaaSqabaaaaaaa@464C@

The system of the equations for the structural model of a piezoactuator is determined for Figure 1

Ξ 1 ( s )= ( M 1 s 2 ) 1 { F 1 ( s )+ ( χ i j Ψ ) 1 [ ν m i Ψ m ( s )[ γ/ s h( l γ ) ] ×[ c h( l γ ) Ξ 1 ( s ) Ξ 2 ( s ) ] ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabf65aynaaBa aaleaacaaIXaaabeaakmaabmaabaGaam4CaaGaayjkaiaawMcaaiab g2da9maabmaabaGaamytamaaBaaaleaacaaIXaaabeaakiaadohada ahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiab gkHiTiaaigdaaaGcdaGadaqaaiabgkHiTiaadAeadaWgaaWcbaGaaG ymaaqabaGcdaqadaqaaiaadohaaiaawIcacaGLPaaacqGHRaWkdaqa daqaaiabeE8aJnaaDaaaleaacaWGPbaeaaaaaaaaa8qacaGGGcWdai aadQgaaeaacqqHOoqwaaaakiaawIcacaGLPaaadaahaaWcbeqaaiab gkHiTiaaigdaaaGcdaWadaabaeqabaGaeqyVd42aaSbaaSqaaiaad2 gapeGaaiiOa8aacaWGPbaabeaakiabfI6aznaaBaaaleaacaWGTbaa beaakmaabmaabaGaam4CaaGaayjkaiaawMcaaiabgkHiTmaadmaaba WaaSGbaeaacqaHZoWzaeaacaqGZbWdbiaacckapaGaaeiAamaabmaa baGaamiBa8qacaGGGcWdaiabeo7aNbGaayjkaiaawMcaaaaaaiaawU facaGLDbaaaeaacqGHxdaTdaWadaqaaiaabogapeGaaiiOa8aacaqG ObWaaeWaaeaacaWGSbWdbiaacckapaGaeq4SdCgacaGLOaGaayzkaa GaeuONdG1aaSbaaSqaaiaaigdaaeqaaOWaaeWaaeaacaWGZbaacaGL OaGaayzkaaGaeyOeI0IaeuONdG1aaSbaaSqaaiaaikdaaeqaaOWaae WaaeaacaWGZbaacaGLOaGaayzkaaaacaGLBbGaayzxaaaaaiaawUfa caGLDbaaaiaawUhacaGL9baaaaa@8733@

Ξ 2 ( s )= ( M 2 s 2 ) 1 { F 2 ( s )+ ( χ i j Ψ ) 1 [ ν m i Ψ m ( s )[ γ/ s h( l γ ) ] ×[ c h( l γ ) Ξ 2 ( s ) Ξ 1 ( s ) ] ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabf65ayTWaaS baaeaacaaIYaaabeaakmaabmaabaGaam4CaaGaayjkaiaawMcaaiab g2da9maabmaabaGaamytamaaBaaaleaacaaIYaaabeaakiaadohada ahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiab gkHiTiaaigdaaaGcdaGadaqaaiabgkHiTiaadAealmaaBaaabaGaaG OmaaqabaGcdaqadaqaaiaadohaaiaawIcacaGLPaaacqGHRaWkdaqa daqaaiabeE8aJnaaDaaaleaacaWGPbaeaaaaaaaaa8qacaGGGcWdai aadQgaaeaacqqHOoqwaaaakiaawIcacaGLPaaadaahaaWcbeqaaiab gkHiTiaaigdaaaGcdaWadaabaeqabaGaeqyVd42aaSbaaSqaaiaad2 gapeGaaiiOa8aacaWGPbaabeaakiabfI6aznaaBaaaleaacaWGTbaa beaakmaabmaabaGaam4CaaGaayjkaiaawMcaaiabgkHiTmaadmaaba WaaSGbaeaacqaHZoWzaeaacaqGZbWdbiaacckapaGaaeiAamaabmaa baGaamiBa8qacaGGGcWdaiabeo7aNbGaayjkaiaawMcaaaaaaiaawU facaGLDbaaaeaacqGHxdaTdaWadaqaaiaabogapeGaaiiOa8aacaqG ObWaaeWaaeaacaWGSbWdbiaacckapaGaeq4SdCgacaGLOaGaayzkaa GaeuONdG1aaSbaaSqaaiaaikdaaeqaaOWaaeWaaeaacaWGZbaacaGL OaGaayzkaaGaeyOeI0IaeuONdG1aaSbaaSqaaiaaigdaaeqaaOWaae WaaeaacaWGZbaacaGLOaGaayzkaaaacaGLBbGaayzxaaaaaiaawUfa caGLDbaaaiaawUhacaGL9baaaaa@8736@

χ i   j Ψ = s i   j Ψ / S 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeE8aJnaaDa aaleaacaWGPbaeaaaaaaaaa8qacaGGGcWdaiaadQgaaeaacqqHOoqw aaGccqGH9aqpdaWcgaqaaiaadohadaqhaaWcbaGaamyAa8qacaGGGc WdaiaadQgaaeaacqqHOoqwaaaakeaacaWGtbWaaSbaaSqaaiaaicda aeqaaaaaaaa@4673@

where

v m i ={ d  33 ,d   31 ,d   15 g   33 ,g   31 ,g   15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAhadaWgaa WcbaGaamyBaabaaaaaaaaapeGaaiiOa8aacaWGPbaabeaakiabg2da 9maaceaabaqbaeqabiqaaaqaaiaadsgadaWgaaWcbaWdbiaacckapa GaaG4maiaaiodaaeqaaOGaaiilaiaadsgapeGaaiiOa8aadaWgaaWc baGaaG4maiaaigdaaeqaaOGaaiilaiaadsgapeGaaiiOa8aadaWgaa WcbaGaaGymaiaaiwdaaeqaaaGcbaGaam4za8qacaGGGcWdamaaBaaa leaacaaIZaGaaG4maaqabaGccaGGSaGaam4za8qacaGGGcWdamaaBa aaleaacaaIZaGaaGymaaqabaGccaGGSaGaam4za8qacaGGGcWdamaa BaaaleaacaaIXaGaaGynaaqabaaaaaGccaGL7baaaaa@578C@

Ψ m ={ E 3 , E 3 , E 1 D 3 , D 3 , D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiQdK1aaS baaSqaaiaad2gaaeqaaOGaeyypa0ZaaiqaaeaafaqabeGabaaabaGa amyramaaBaaaleaacaaIZaaabeaakiaacYcacaWGfbWaaSbaaSqaai aaiodaaeqaaOGaaiilaiaadweadaWgaaWcbaGaaGymaaqabaaakeaa caWGebWaaSbaaSqaaiaaiodaaeqaaOGaaiilaiaadseadaWgaaWcba GaaG4maaqabaGccaGGSaGaamiramaaBaaaleaacaaIXaaabeaaaaaa kiaawUhaaaaa@4802@

s i j Ψ ={ s   33 E ,s   11 E ,s   55 E s   33 D ,s   11 D ,s   55 D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadohadaqhaa WcbaGaamyAaabaaaaaaaaapeGaaiiOa8aacaWGQbaabaGaeuiQdKfa aOGaeyypa0ZaaiqaaeaafaqabeGabaaabaGaam4Ca8qacaGGGcWdam aaDaaaleaacaaIZaGaaG4maaqaaiaadweaaaGccaGGSaGaam4Ca8qa caGGGcWdamaaDaaaleaacaaIXaGaaGymaaqaaiaadweaaaGccaGGSa Gaam4Ca8qacaGGGcWdamaaDaaaleaacaaI1aGaaGynaaqaaiaadwea aaaakeaacaWGZbWdbiaacckapaWaa0baaSqaaiaaiodacaaIZaaaba GaamiraaaakiaacYcacaWGZbWdbiaacckapaWaa0baaSqaaiaaigda caaIXaaabaGaamiraaaakiaacYcacaWGZbWdbiaacckapaWaa0baaS qaaiaaiwdacaaI1aaabaGaamiraaaaaaaakiaawUhaaaaa@5E2A@

l={ δ, h,b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabg2 da9maaceaabaGaaGjbVlabes7aKjaacYcaaiaawUhaaiaaysW7caWG ObGaaiilaiaaysW7caWGIbaaaa@4288@

γ={ γ E , γ D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaey ypa0ZaaiqaaeaacqaHZoWzdaahaaWcbeqaaiaadweaaaGccaGGSaGa aGjbVlabeo7aNnaaCaaaleqabaGaamiraaaaaOGaay5Eaaaaaa@414A@

c Ψ ={ c E , c D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaCa aaleqabaGaeuiQdKfaaOGaeyypa0ZaaiqaaeaacaaMe8Uaam4yamaa CaaaleqabaGaamyraaaakiaacYcacaaMe8Uaam4yamaaCaaaleqaba GaamiraaaaaOGaay5Eaaaaaa@4260@

The structural scheme on Figure 1 is used for the decision of a piezoactuator in astrophysics. The matrix of the nanodisplacement of a piezoactuator has the form.

( Ξ 1 ( s ) Ξ 2 ( s ) )=( W  11  ( s ) W  12  ( s ) W  13  ( s ) W  21  ( s ) W  22  ( s ) W  23  ( s ) )( Ψ m ( s ) F 1 ( s ) F 2 ( s ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaqbae qabiqaaaqaaiabf65aynaaBaaaleaacaaIXaaabeaakmaabmaabaGa am4CaaGaayjkaiaawMcaaaqaaiabf65aynaaBaaaleaacaaIYaaabe aakmaabmaabaGaam4CaaGaayjkaiaawMcaaaaaaiaawIcacaGLPaaa cqGH9aqpdaqadaqaauaabeqaceaaaeaafaqabeqadaaabaGaam4vam aaBaaaleaaqaaaaaaaaaWdbiaacckapaGaaGymaiaaigdaaeqaaOWd biaacckapaWaaeWaaeaacaWGZbaacaGLOaGaayzkaaaabaGaam4vam aaBaaaleaapeGaaiiOa8aacaaIXaGaaGOmaaqabaGcpeGaaiiOa8aa daqadaqaaiaadohaaiaawIcacaGLPaaaaeaacaWGxbWaaSbaaSqaa8 qacaGGGcWdaiaaigdacaaIZaaabeaak8qacaGGGcWdamaabmaabaGa am4CaaGaayjkaiaawMcaaaaaaeaafaqabeqadaaabaGaam4vamaaBa aaleaapeGaaiiOa8aacaaIYaGaaGymaaqabaGcpeGaaiiOa8aadaqa daqaaiaadohaaiaawIcacaGLPaaaaeaacaWGxbWaaSbaaSqaa8qaca GGGcWdaiaaikdacaaIYaaabeaak8qacaGGGcWdamaabmaabaGaam4C aaGaayjkaiaawMcaaaqaaiaadEfadaWgaaWcbaWdbiaacckapaGaaG OmaiaaiodaaeqaaOWdbiaacckapaWaaeWaaeaacaWGZbaacaGLOaGa ayzkaaaaaaaaaiaawIcacaGLPaaacaaMe8+aaeWaaeaafaqabeWaba aabaGaeuiQdK1aaSbaaSqaaiaad2gaaeqaaOWaaeWaaeaacaWGZbaa caGLOaGaayzkaaaabaGaamOramaaBaaaleaacaaIXaaabeaakmaabm aabaGaam4CaaGaayjkaiaawMcaaaqaaiaadAeadaWgaaWcbaGaaGOm aaqabaGcdaqadaqaaiaadohaaiaawIcacaGLPaaaaaaacaGLOaGaay zkaaaaaa@8396@

Figure 1 Structural scheme of piezoactuator.

The steady-state nanodisplacements are written for two faces of a piezoactuator

ξ 1 = d m i Ψ m  l  M 2 / ( M 1 + M 2 ) ξ 2 = d m i Ψ m  l  M 1 / ( M 1 + M 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeqOVdG 3cdaWgaaqaaiaaigdaaeqaaOGaeyypa0ZaaSGbaeaacaWGKbWcdaWg aaqaaiaad2gaqaaaaaaaaaWdbiaacckapaGaamyAaaqabaGccqqHOo qwdaWgaaWcbaGaamyBaaGcbeaapeGaaiiOa8aacaWGSbWdbiaaccka paGaamytamaaBaaaleaacaaIYaaabeaaaOqaamaabmaabaGaamytam aaBaaaleaacaaIXaaabeaakiabgUcaRiaad2eadaWgaaWcbaGaaGOm aaqabaaakiaawIcacaGLPaaaaaaabaGaeqOVdG3aaSbaaSqaaiaaik daaeqaaOGaeyypa0ZaaSGbaeaacaWGKbWcdaWgaaqaaiaad2gapeGa aiiOa8aacaWGPbaabeaakiabfI6aznaaBaaaleaacaWGTbaakeqaa8 qacaGGGcWdaiaadYgapeGaaiiOa8aacaWGnbWaaSbaaSqaaiaaigda aeqaaaGcbaWaaeWaaeaacaWGnbWaaSbaaSqaaiaaigdaaeqaaOGaey 4kaSIaamytamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaaaa aaaa@6311@

The steady-state nanodisplacements are obtained for two faces of the longitudinal piezoactuator

ξ 1 = d  33  U  M 2 / ( M 1 + M 2 ) ξ 2 = d  33 U  M 1 / ( M 1 + M 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeqOVdG 3cdaWgaaqaaiaaigdaaeqaaOGaeyypa0ZaaSGbaeaacaWGKbWaaSba aSqaaabaaaaaaaaapeGaaiiOa8aacaaIZaGaaG4maaqabaGcpeGaai iOa8aacaWGvbWdbiaacckapaGaamytamaaBaaaleaacaaIYaaabeaa aOqaamaabmaabaGaamytamaaBaaaleaacaaIXaaabeaakiabgUcaRi aad2eadaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaaaaaabaGa eqOVdG3cdaWgaaqaaiaaikdaaeqaaOGaeyypa0ZaaSGbaeaacaWGKb WaaSbaaSqaa8qacaGGGcWdaiaaiodacaaIZaaabeaakiaadwfapeGa aiiOa8aacaWGnbWaaSbaaSqaaiaaigdaaeqaaaGcbaWaaeWaaeaaca WGnbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamytamaaBaaaleaa caaIYaaabeaaaOGaayjkaiaawMcaaaaaaaaa@5B66@

At U = 75 V, M 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIXaaabeaaaaa@37B0@  = 1 kg, M 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIYaaabeaaaaa@37B1@  = 4 kg, d  33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsgalmaaBa aabaaeaaaaaaaaa8qacaGGGcWdaiaaiodacaaIZaaabeaaaaa@3AF0@ = 4×10-10 m/V the steady-state nanodisplacements are determined ξ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3cda Wgaaqaaiaaigdaaeqaaaaa@38A1@  = 24 nm, ξ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3cda Wgaaqaaiaaikdaaeqaaaaa@38A2@  = 6 nm and ξ 1 + ξ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3cda WgaaqaaiaaigdaaeqaaOGaey4kaSIaeqOVdG3cdaWgaaqaaiaaikda aeqaaaaa@3C38@  = 30 nm with error 10%.

The transfer equation of the transverse piezoactuator is determined at one the fixed face and the elastic-inertial load

W( s )= Ξ( s ) U( s ) = k  31 E T t 2 s 2 +2 T t ξ t s+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4vaO WaaeWaaeaajugibiaadohaaOGaayjkaiaawMcaaKqzGeGaeyypa0Jc daWcaaqaaKqzGeGaeuONdGLcdaqadaqaaKqzGeGaam4CaaGccaGLOa GaayzkaaaabaqcLbsacaWGvbGcdaqadaqaaKqzGeGaam4CaaGccaGL OaGaayzkaaaaaKqzGeGaeyypa0JcdaWcaaqaaKqzGeGaam4AaOWaa0 baaSqaaKqzGeaeaaaaaaaaa8qacaGGGcWdaiaaiodacaaIXaaaleaa jugibiaadweaaaaakeaajugibiaaysW7caWGubGcdaqhaaWcbaqcLb sacaWG0baaleaajugibiaaikdaaaGaam4CaOWaaWbaaSqabeaajugi biaaikdaaaGaey4kaSIaaGOmaiaadsfakmaaBaaaleaajugibiaads haaSqabaqcLbsacqaH+oaEkmaaBaaaleaajugibiaadshaaSqabaqc LbsacaWGZbGaey4kaSIaaGymaaaaaaa@6340@

k  31 E = d  31 ( h/δ )/ ( 1+ C l / C  11 E ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgadaqhaa Wcbaaeaaaaaaaaa8qacaGGGcWdaiaaiodacaaIXaaabaGaamyraaaa kiabg2da9maalyaabaGaamizaSWaaSbaaeaapeGaaiiOa8aacaaIZa GaaGymaaqabaGcdaqadaqaamaalyaabaGaamiAaaqaaiabes7aKbaa aiaawIcacaGLPaaaaeaadaqadaqaaiaaigdacqGHRaWkdaWcgaqaai aadoeadaWgaaWcbaGaamiBaaqabaaakeaacaWGdbWaa0baaSqaa8qa caGGGcWdaiaaigdacaaIXaaabaGaamyraaaaaaaakiaawIcacaGLPa aaaaaaaa@4E9E@

T t = M/ ( C l + C  11 E ) , ω t =1/ T t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfadaWgaa WcbaGaamiDaaqabaGccqGH9aqpdaGcaaqaamaalyaabaGaamytaaqa amaabmaabaGaam4qamaaBaaaleaacaWGSbaabeaakiabgUcaRiaado eadaqhaaWcbaaeaaaaaaaaa8qacaGGGcWdaiaaigdacaaIXaaabaGa amyraaaaaOGaayjkaiaawMcaaaaaaSqabaGccaGGSaGaeqyYdC3cda WgaaqaaiaadshaaeqaaOGaeyypa0ZaaSGbaeaacaaIXaaabaGaamiv amaaBaaaleaacaWG0baabeaaaaaaaa@4B96@

where k  31 E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgadaqhaa Wcbaaeaaaaaaaaa8qacaGGGcWdaiaaiodacaaIXaaabaGaamyraaaa aaa@3BC0@  is the transfer coefficient, C l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGSbaabeaaaaa@37DC@ , C  11 E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeadaqhaa Wcbaaeaaaaaaaaa8qacaGGGcWdaiaaigdacaaIXaaabaGaamyraaaa aaa@3B96@  are the stiffness for the load and the transverse piezoactuator, T t , ξ t , ω t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWG0baabeaakiaacYcacqaH+oaElmaaBaaabaGaamiDaaqa baGaaiilaOGaeqyYdC3cdaWgaaqaaiaadshaaeqaaaaa@3F43@  are the time constant, the attenuation coefficient, the conjugate frequency.

At C l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaSWaaS baaeaacaWGSbaabeaaaaa@37DC@  = 0.2×107 N/m, C  11 E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeadaqhaa Wcbaaeaaaaaaaaa8qacaGGGcWdaiaaigdacaaIXaaabaGaamyraaaa aaa@3B96@ = 1.4×107 N/m, M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaaaa@36C9@  = 2 kg the parameters are obtained T t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaSWaaS baaeaacaWG0baabeaaaaa@37F5@  = 0.354×10-3 s, ω t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3cda Wgaaqaaiaadshaaeqaaaaa@38E9@  = 2.8×103 s-1 with error 10%.

The steady-state nanodisplacement of the transverse piezoactuator is written for elastic-inertial load

Δh= d  31 ( h/δ )U 1+ C l / C  11 E = k  31 E U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfs5aejaadI gacqGH9aqpdaWcaaqaaiaadsgalmaaBaaabaaeaaaaaaaaa8qacaGG GcWdaiaaiodacaaIXaaabeaakmaabmaabaWaaSGbaeaacaWGObaaba GaeqiTdqgaaaWccaGLOaGaayzkaaGccaWGvbaabaGaaGymaiabgUca RmaalyaabaGaam4qamaaBaaaleaacaWGSbaabeaaaOqaaiaadoeada qhaaWcbaWdbiaacckapaGaaGymaiaaigdaaeaacaWGfbaaaaaaaaGc cqGH9aqpcaWGRbWaa0baaSqaa8qacaGGGcWdaiaaiodacaaIXaaaba Gaamyraaaakiaadwfaaaa@5231@

At h/δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca WGObaabaGaeqiTdqgaaaaa@389F@  = 20, C l / C  11 E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalyaabaGaam 4qamaaBaaaleaacaWGSbaabeaaaOqaaiaadoeadaqhaaWcbaaeaaaa aaaaa8qacaGGGcWdaiaaigdacaaIXaaabaGaamyraaaaaaaaaa@3D9B@ = 0.14, d 31 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsgalmaaBa aabaGaaG4maiaaigdaaeqaaaaa@399B@ = 2∙10-10 m/V the transfer coefficient of the transverse piezoactuator is received k  31 E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgadaqhaa Wcbaaeaaaaaaaaa8qacaGGGcWdaiaaiodacaaIXaaabaGaamyraaaa aaa@3BC0@  = 3.5 nm/V with error 10%.

Conclusion

The structural scheme of a piezoactuator is constructed for astrophysics. The matrix of the nanodisplacement of a piezoactuator is obtained. The characteristics of a piezoactuator are determined.

Acknowledgments

None.

Conflicts of interest

The Authors declares that there is no Conflict of interest.

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