Submit manuscript...
MOJ
eISSN: 2576-4519

Applied Bionics and Biomechanics

Research Article Volume 6 Issue 1

Piezoengine for nanomedicine and applied bionics 

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: October 28, 2022 | Published: November 11, 2022

Citation: Afonin SM. Piezoengine for nanomedicine and applied bionics. MOJ App Bio Biomech . 2022;6(1):30?33. DOI: 10.15406/mojabb.2022.06.00164

Download PDF

Abstract

The mathematical models of a piezoengine are determined for nanomedicine and applied bionics. The structural scheme of a piezoengine is constructed. The matrix equation is obtained for a piezoengine.

Keywords: piezoengine, structural scheme, nanomedicine and applied bionics

Introduction

A piezoengine is used for nano displacement in tunnel microscopy, for the nano alignment in adaptive optics, microscopy and interferometers in nanomedicine and applied bionics, for the automatic adjustment of the constant optical parameter of the ring quantum generators, for the actively dampen mechanical vibrations in the laser system, for the deform mirrors and operations with penetration in a cells and for the works with a genes.1–15 A piezoengine with a compact design provides positioning of elements of adaptive systems with an accuracy of up to a nanometer in the range of hundreds of nanometers. These precise parameters of a piezoengine are provided by the use of the reverse piezoelectric effect.16–48 To calculate the deformations of nano systems, it is required to build the structural scheme of a piezoengine. A piezoengine is used in adaptive optics systems for phase corrections, for example, in an interferometer to adjust maximum of the interference image. In scanning probe microscopy, an image of a surface is formed using a physical probe to scan an object. For example, a scanning tunneling microscope is used to visualize surfaces at the atomic level. Nano movements of the probe along three coordinates X, Y, Z are carried out using a piezoengines.14–23

Material and methods

A piezoengine works on basis of the reverse piezoelectric effect in the form3–52

S i = s ij E T j + d mi E m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGPbaabeaakiabg2da9iaadohadaqhaaWcbaGaamyAaiaa dQgaaeaacaWGfbaaaOGaamivamaaBaaaleaacaWGQbaabeaakiabgU caRiaadsgadaWgaaWcbaGaamyBaiaadMgaaeqaaOGaamyramaaBaaa leaacaWGTbaabeaaaaa@4495@

where S i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGPbaabeaaaaa@37E8@ , s ij E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa aaleaacaWGPbGaamOAaaqaaiaadweaaaaaaa@39C2@ , T j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGQbaabeaaaaa@37EA@ , d mi MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGTbGaamyAaaqabaaaaa@38EB@ , E m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGTbaabeaaaaa@37DE@  are the relative deformation, elastic compliance, strength mechanical field, piezomodule, strength electric field, ij . m are indexes.

The differential equation is written4–52

d 2 Ξ( x,s ) d x 2 γ 2 Ξ( x,s )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbWaaWbaaeqaleaacaaIYaaaaOGaeuONdG1aaeWaaeaacaWG4bGa aiilaiaadohaaiaawIcacaGLPaaaaeaacaWGKbGaamiEaSWaaWbaae qabaGaaGOmaaaaaaGccqGHsislcqaHZoWzdaahaaWcbeqaaiaaikda aaGccqqHEoawdaqadaqaaiaadIhacaGGSaGaam4CaaGaayjkaiaawM caaiabg2da9iaaicdaaaa@4B66@

Here   Ξ( x,s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuONdG1aae WaaeaacaWG4bGaaiilaiaadohaaiaawIcacaGLPaaaaaa@3BA8@ , s, x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F3@ , γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCgaaa@379D@  are the transform of the deformation, the parameter of the Laplace transform, the coordinate, the propagation factor. For the transverse piezoengine we have at x=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaaicdaaaa@38B3@  the first deformation Ξ( 0,s )= Ξ 1 ( s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuONdG1aae WaaeaacaaIWaGaaiilaiaadohaaiaawIcacaGLPaaacqGH9aqpcqqH EoawlmaaBaaabaGaaGymaaqabaGcdaqadaqaaiaadohaaiaawIcaca GLPaaaaaa@4161@  and at x=h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaadIgaaaa@38E6@  the second deformation Ξ( h,s )= Ξ 2 ( s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuONdG1aae WaaeaacaWGObGaaiilaiaadohaaiaawIcacaGLPaaacqGH9aqpcqqH EoawlmaaBaaabaGaaGOmaaqabaGcdaqadaqaaiaadohaaiaawIcaca GLPaaaaaa@4195@ .

The decision of the differential equation is obtained.

Ξ( x,s )= { Ξ 1 ( s )sh[ ( hx )γ ]+ Ξ 2 ( s )sh( xγ ) }/ sh( hγ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuONdG1aae WaaeaacaWG4bGaaiilaiaadohaaiaawIcacaGLPaaacqGH9aqpdaWc gaqaamaacmaabaGaeuONdG1aaSbaaSqaaiaaigdaaeqaaOWaaeWaae aacaWGZbaacaGLOaGaayzkaaGaae4CaiaabIgadaWadaqaamaabmaa baGaamiAaiabgkHiTiaadIhaaiaawIcacaGLPaaacqaHZoWzaiaawU facaGLDbaacqGHRaWkcqqHEoawdaWgaaWcbaGaaGOmaaqabaGcdaqa daqaaiaadohaaiaawIcacaGLPaaacaqGZbGaaeiAamaabmaabaGaam iEaiabeo7aNbGaayjkaiaawMcaaaGaay5Eaiaaw2haaaqaaiaaboha caqGObWaaeWaaeaacaWGObGaeq4SdCgacaGLOaGaayzkaaaaaaaa@5FAA@

Where Ξ 1 ( s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuONdG1aaS baaSqaaiaaigdaaeqaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaaaa aa@3AEC@ , Ξ 2 ( s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuONdG1cda WgaaqaaiaaikdaaeqaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaaaa aa@3AED@  are the transforms of the deformations.

At x=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaaicdaaaa@38B3@  and x=h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaadIgaaaa@38E6@  we have the system for the transverse piezoengine

T 1 ( 0,s )= 1 s 11 E dΞ( x,s ) dx | x=0 d 31 s 11 E E 3 ( s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaaIXaaabeaakmaabmaabaGaaGimaiaacYcacaWGZbaacaGL OaGaayzkaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaam4CamaaDaaale aacaaIXaGaaGymaaqaaiaadweaaaaaaOWaaqGaaeaadaWcaaqaaiaa dsgacqqHEoawdaqadaqaaiaadIhacaGGSaGaam4CaaGaayjkaiaawM caaaqaaiaadsgacaWG4baaaaGaayjcSdWaaSbaaSqaaiaadIhacqGH 9aqpcaaIWaaabeaakiabgkHiTmaalaaabaGaamizamaaBaaaleaaca aIZaGaaGymaaqabaaakeaacaWGZbWaa0baaSqaaiaaigdacaaIXaaa baGaamyraaaaaaGccaWGfbWaaSbaaSqaaiaaiodaaeqaaOWaaeWaae aacaWGZbaacaGLOaGaayzkaaaaaa@5946@

T 1 ( h,s )= 1 s 11 E dΞ( x,s ) dx | x=h d 31 s 11 E E 3 ( s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaaIXaaabeaakmaabmaabaGaamiAaiaacYcacaWGZbaacaGL OaGaayzkaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaam4CamaaDaaale aacaaIXaGaaGymaaqaaiaadweaaaaaaOWaaqGaaeaadaWcaaqaaiaa dsgacqqHEoawdaqadaqaaiaadIhacaGGSaGaam4CaaGaayjkaiaawM caaaqaaiaadsgacaWG4baaaaGaayjcSdWaaSbaaSqaaiaadIhacqGH 9aqpcaWGObaabeaakiabgkHiTmaalaaabaGaamizamaaBaaaleaaca aIZaGaaGymaaqabaaakeaacaWGZbWaa0baaSqaaiaaigdacaaIXaaa baGaamyraaaaaaGccaWGfbWaaSbaaSqaaiaaiodaaeqaaOWaaeWaae aacaWGZbaacaGLOaGaayzkaaaaaa@59AC@

The mathematical model for the transverse piezoengine has the form.

Ξ 1 ( s )= ( M 1 s 2 ) 1 { F 1 ( s )+ ( χ 11 E ) 1 ×[ d 31 E 3 ( s )[ γ/ sh( hγ ) ] ×[ ch( hγ ) Ξ 1 ( s ) Ξ 2 ( s ) ] ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuONdG1aaS baaSqaaiaaigdaaeqaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaaGa eyypa0ZaaeWaaeaacaWGnbWaaSbaaSqaaiaaigdaaeqaaOGaam4Cam aaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGa eyOeI0IaaGymaaaakmaacmaaeaqabeaacqGHsislcaWGgbWaaSbaaS qaaiaaigdaaeqaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaaGaey4k aSYaaeWaaeaacqaHhpWydaqhaaWcbaGaaGymaiaaigdaaeaacaWGfb aaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqGHsislcaaIXaaaaaGc baGaey41aq7aamWaaqaabeqaaiaadsgadaWgaaWcbaGaaG4maiaaig daaeqaaOGaamyramaaBaaaleaacaaIZaaabeaakmaabmaabaGaam4C aaGaayjkaiaawMcaaiabgkHiTmaadmaabaWaaSGbaeaacqaHZoWzae aacaqGZbGaaeiAamaabmaabaGaamiAaiabeo7aNbGaayjkaiaawMca aaaaaiaawUfacaGLDbaacaaMe8oabaGaey41aq7aamWaaeaacaqGJb GaaeiAamaabmaabaGaamiAaiabeo7aNbGaayjkaiaawMcaaiabf65a ynaaBaaaleaacaaIXaaabeaakmaabmaabaGaam4CaaGaayjkaiaawM caaiabgkHiTiabf65aynaaBaaaleaacaaIYaaabeaakmaabmaabaGa am4CaaGaayjkaiaawMcaaaGaay5waiaaw2faaaaacaGLBbGaayzxaa aaaiaawUhacaGL9baaaaa@7EBF@

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 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuONdG1cda WgaaqaaiaaikdaaeqaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaaGa eyypa0ZaaeWaaeaacaWGnbWaaSbaaSqaaiaaikdaaeqaaOGaam4Cam aaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGa eyOeI0IaaGymaaaakmaacmaaeaqabeaacqGHsislcaWGgbWcdaWgaa qaaiaaikdaaeqaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaaGaey4k aSYaaeWaaeaacqaHhpWydaqhaaWcbaGaaGymaiaaigdaaeaacaWGfb aaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqGHsislcaaIXaaaaaGc baGaey41aq7aamWaaqaabeqaaiaadsgadaWgaaWcbaGaaG4maiaaig daaeqaaOGaamyramaaBaaaleaacaaIZaaabeaakmaabmaabaGaam4C aaGaayjkaiaawMcaaiabgkHiTmaadmaabaWaaSGbaeaacqaHZoWzae aacaqGZbGaaeiAamaabmaabaGaamiAaiabeo7aNbGaayjkaiaawMca aaaaaiaawUfacaGLDbaaaeaacqGHxdaTdaWadaqaaiaabogacaqGOb WaaeWaaeaacaWGObGaeq4SdCgacaGLOaGaayzkaaGaeuONdG1aaSba aSqaaiaaikdaaeqaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaaGaey OeI0IaeuONdG1aaSbaaSqaaiaaigdaaeqaaOWaaeWaaeaacaWGZbaa caGLOaGaayzkaaaacaGLBbGaayzxaaaaaiaawUfacaGLDbaaaaGaay 5Eaiaaw2haaaaa@7D35@

χ 11 E = s 11 E / S 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aa0 baaSqaaiaaigdacaaIXaaabaGaamyraaaakiabg2da9maalyaabaGa am4CamaaDaaaleaacaaIXaGaaGymaaqaaiaadweaaaaakeaacaWGtb WaaSbaaSqaaiaaicdaaeqaaaaaaaa@406D@

At x=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaaicdaaaa@38B3@  and x=l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaadYgaaaa@38EA@  the system in general for a piezoengine is obtained.

T j ( 0,s )= 1 s ij Ψ dΞ( x,s ) dx | x=0 ν mi s ij Ψ Ψ m ( s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGQbaabeaakmaabmaabaGaaGimaiaacYcacaWGZbaacaGL OaGaayzkaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaam4CamaaDaaale aacaWGPbGaamOAaaqaaiabfI6azbaaaaGcdaabcaqaamaalaaabaGa amizaiabf65aynaabmaabaGaamiEaiaacYcacaWGZbaacaGLOaGaay zkaaaabaGaamizaiaadIhaaaaacaGLiWoadaWgaaWcbaGaamiEaiab g2da9iaaicdaaeqaaOGaeyOeI0YaaSaaaeaacqaH9oGBdaWgaaWcba GaamyBaiaadMgaaeqaaaGcbaGaam4CamaaDaaaleaacaWGPbGaamOA aaqaaiabfI6azbaaaaGccqqHOoqwdaWgaaWcbaGaamyBaaqabaGcda qadaqaaiaadohaaiaawIcacaGLPaaaaaa@5E03@

T j ( l,s )= 1 s ij Ψ dΞ( x,s ) dx | x=l ν mi s ij Ψ Ψ m ( s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGQbaabeaakmaabmaabaGaamiBaiaacYcacaWGZbaacaGL OaGaayzkaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaam4CamaaDaaale aacaWGPbGaamOAaaqaaiabfI6azbaaaaGcdaabcaqaamaalaaabaGa amizaiabf65aynaabmaabaGaamiEaiaacYcacaWGZbaacaGLOaGaay zkaaaabaGaamizaiaadIhaaaaacaGLiWoadaWgaaWcbaGaamiEaiab g2da9iaadYgaaeqaaOGaeyOeI0YaaSaaaeaacqaH9oGBdaWgaaWcba GaamyBaiaadMgaaeqaaaGcbaGaam4CamaaDaaaleaacaWGPbGaamOA aaqaaiabfI6azbaaaaGccqqHOoqwdaWgaaWcbaGaamyBaaqabaGcda qadaqaaiaadohaaiaawIcacaGLPaaaaaa@5E71@

Where l={ δ, h,b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabg2 da9maaceaabaGaaGjbVlabes7aKjaacYcaaiaawUhaaiaaysW7caWG ObGaaiilaiaaysW7caWGIbaaaa@4287@  the length for the longitudinal, transverse or shift piezoengine.

Therefore, the mathematical model of a piezoengine is determined on Figure 1.

Ξ 1 ( s )= ( M 1 s 2 ) 1 { F 1 ( s )+ ( χ ij Ψ ) 1 ×[ ν mi Ψ m ( s )[ γ/ sh( lγ ) ] ×[ ch( lγ ) Ξ 1 ( s ) Ξ 2 ( s ) ] ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuONdG1aaS baaSqaaiaaigdaaeqaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaaGa eyypa0ZaaeWaaeaacaWGnbWaaSbaaSqaaiaaigdaaeqaaOGaam4Cam aaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGa eyOeI0IaaGymaaaakmaacmaaeaqabeaacqGHsislcaWGgbWaaSbaaS qaaiaaigdaaeqaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaaGaey4k aSYaaeWaaeaacqaHhpWydaqhaaWcbaGaamyAaiaadQgaaeaacqqHOo qwaaaakiaawIcacaGLPaaadaahaaWcbeqaaiabgkHiTiaaigdaaaaa keaacqGHxdaTdaWadaabaeqabaGaeqyVd42aaSbaaSqaaiaad2gaca WGPbaabeaakiabfI6aznaaBaaaleaacaWGTbaabeaakmaabmaabaGa am4CaaGaayjkaiaawMcaaiabgkHiTmaadmaabaWaaSGbaeaacqaHZo WzaeaacaqGZbGaaeiAamaabmaabaGaamiBaiabeo7aNbGaayjkaiaa wMcaaaaaaiaawUfacaGLDbaaaeaacqGHxdaTdaWadaqaaiaabogaca qGObWaaeWaaeaacaWGSbGaeq4SdCgacaGLOaGaayzkaaGaeuONdG1a aSbaaSqaaiaaigdaaeqaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaa GaeyOeI0IaeuONdG1aaSbaaSqaaiaaikdaaeqaaOWaaeWaaeaacaWG ZbaacaGLOaGaayzkaaaacaGLBbGaayzxaaaaaiaawUfacaGLDbaaaa Gaay5Eaiaaw2haaaaa@8097@

Ξ 2 ( s )= ( M 2 s 2 ) 1 { F 2 ( s )+ ( χ ij Ψ ) 1 ×[ ν mi Ψ m ( s )[ γ/ sh( lγ ) ] ×[ ch( lγ ) Ξ 2 ( s ) Ξ 1 ( s ) ] ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuONdG1cda WgaaqaaiaaikdaaeqaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaaGa eyypa0ZaaeWaaeaacaWGnbWaaSbaaSqaaiaaikdaaeqaaOGaam4Cam aaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGa eyOeI0IaaGymaaaakmaacmaaeaqabeaacqGHsislcaWGgbWcdaWgaa qaaiaaikdaaeqaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaaGaey4k aSYaaeWaaeaacqaHhpWydaqhaaWcbaGaamyAaiaadQgaaeaacqqHOo qwaaaakiaawIcacaGLPaaadaahaaWcbeqaaiabgkHiTiaaigdaaaaa keaacqGHxdaTdaWadaabaeqabaGaeqyVd42aaSbaaSqaaiaad2gaca WGPbaabeaakiabfI6aznaaBaaaleaacaWGTbaabeaakmaabmaabaGa am4CaaGaayjkaiaawMcaaiabgkHiTmaadmaabaWaaSGbaeaacqaHZo WzaeaacaqGZbGaaeiAamaabmaabaGaamiBaiabeo7aNbGaayjkaiaa wMcaaaaaaiaawUfacaGLDbaaaeaacqGHxdaTdaWadaqaaiaabogaca qGObWaaeWaaeaacaWGSbGaeq4SdCgacaGLOaGaayzkaaGaeuONdG1a aSbaaSqaaiaaikdaaeqaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaa GaeyOeI0IaeuONdG1aaSbaaSqaaiaaigdaaeqaaOWaaeWaaeaacaWG ZbaacaGLOaGaayzkaaaacaGLBbGaayzxaaaaaiaawUfacaGLDbaaaa Gaay5Eaiaaw2haaaaa@809A@

χ ij Ψ = s ij Ψ / S 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aa0 baaSqaaiaadMgacaWGQbaabaGaeuiQdKfaaOGaeyypa0ZaaSGbaeaa caWGZbWaa0baaSqaaiaadMgacaWGQbaabaGaeuiQdKfaaaGcbaGaam 4uamaaBaaaleaacaaIWaaabeaaaaaaaa@42C5@

Figure 1 Structural scheme in general of piezoengine.

Where

v mi ={ d 33 , d 31 , d 15 g 33 , g 31 , g 15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBa aaleaacaWGTbGaamyAaaqabaGccqGH9aqpdaGabaqaauaabeqaceaa aeaacaWGKbWaaSbaaSqaaiaaiodacaaIZaaabeaakiaacYcacaWGKb WaaSbaaSqaaiaaiodacaaIXaaabeaakiaacYcacaWGKbWaaSbaaSqa aiaaigdacaaI1aaabeaaaOqaaiaadEgadaWgaaWcbaGaaG4maiaaio daaeqaaOGaaiilaiaadEgadaWgaaWcbaGaaG4maiaaigdaaeqaaOGa aiilaiaadEgadaWgaaWcbaGaaGymaiaaiwdaaeqaaaaaaOGaay5Eaa aaaa@4D8F@

Ψ m ={ E 3 , E 3 , E 1 D 3 , D 3 , D 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiQdK1aaS baaSqaaiaad2gaaeqaaOGaeyypa0ZaaiqaaeaafaqabeGabaaabaGa amyramaaBaaaleaacaaIZaaabeaakiaacYcacaWGfbWaaSbaaSqaai aaiodaaeqaaOGaaiilaiaadweadaWgaaWcbaGaaGymaaqabaaakeaa caWGebWaaSbaaSqaaiaaiodaaeqaaOGaaiilaiaadseadaWgaaWcba GaaG4maaqabaGccaGGSaGaamiramaaBaaaleaacaaIXaaabeaaaaaa kiaawUhaaaaa@4801@

s ij 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 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa aaleaacaWGPbGaamOAaaqaaiabfI6azbaakiabg2da9maaceaabaqb aeqabiqaaaqaaiaadohadaqhaaWcbaGaaG4maiaaiodaaeaacaWGfb aaaOGaaiilaiaadohadaqhaaWcbaGaaGymaiaaigdaaeaacaWGfbaa aOGaaiilaiaadohadaqhaaWcbaGaaGynaiaaiwdaaeaacaWGfbaaaa GcbaGaam4CamaaDaaaleaacaaIZaGaaG4maaqaaiaadseaaaGccaGG SaGaam4CamaaDaaaleaacaaIXaGaaGymaaqaaiaadseaaaGccaGGSa Gaam4CamaaDaaaleaacaaI1aGaaGynaaqaaiaadseaaaaaaaGccaGL 7baaaaa@542D@

γ={ γ E , γ D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaey ypa0ZaaiqaaeaacqaHZoWzdaahaaWcbeqaaiaadweaaaGccaGGSaGa aGjbVlabeo7aNnaaCaaaleqabaGaamiraaaaaOGaay5Eaaaaaa@4149@

c Ψ ={ c E , c D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaCa aaleqabaGaeuiQdKfaaOGaeyypa0ZaaiqaaeaacaaMe8Uaam4yamaa CaaaleqabaGaamyraaaakiaacYcacaaMe8Uaam4yamaaCaaaleqaba GaamiraaaaaOGaay5Eaaaaaa@425F@

The mathematical model and the structural scheme of a piezoengine on Figure 1 are used for the design of a precise control system in nanomedicine and applied bionics.

The matrix of the deformations is written

( 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 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa qabeGabaaabaGaeuONdG1aaSbaaSqaaiaaigdaaeqaaOWaaeWaaeaa caWGZbaacaGLOaGaayzkaaaabaGaeuONdG1aaSbaaSqaaiaaikdaae qaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaaaaaaGaayjkaiaawMca aiabg2da9maabmaabaqbaeqabiqaaaqaauaabeqabmaaaeaacaWGxb WaaSbaaSqaaiaaigdacaaIXaaabeaakmaabmaabaGaam4CaaGaayjk aiaawMcaaaqaaiaadEfadaWgaaWcbaGaaGymaiaaikdaaeqaaOWaae WaaeaacaWGZbaacaGLOaGaayzkaaaabaGaam4vamaaBaaaleaacaaI XaGaaG4maaqabaGcdaqadaqaaiaadohaaiaawIcacaGLPaaaaaaaba qbaeqabeWaaaqaaiaadEfadaWgaaWcbaGaaGOmaiaaigdaaeqaaOWa aeWaaeaacaWGZbaacaGLOaGaayzkaaaabaGaam4vamaaBaaaleaaca aIYaGaaGOmaaqabaGcdaqadaqaaiaadohaaiaawIcacaGLPaaaaeaa caWGxbWaaSbaaSqaaiaaikdacaaIZaaabeaakmaabmaabaGaam4Caa GaayjkaiaawMcaaaaaaaaacaGLOaGaayzkaaGaaGjbVpaabmaabaqb aeqabmqaaaqaaiabfI6aznaaBaaaleaacaWGTbaabeaakmaabmaaba Gaam4CaaGaayjkaiaawMcaaaqaaiaadAeadaWgaaWcbaGaaGymaaqa baGcdaqadaqaaiaadohaaiaawIcacaGLPaaaaeaacaWGgbWaaSbaaS qaaiaaikdaaeqaaOWaaeWaaeaacaWGZbaacaGLOaGaayzkaaaaaaGa ayjkaiaawMcaaaaa@734A@

The settled longitudinal deformations are determined

ξ 1 = d 33 U M 2 / ( M 1 + M 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3cda WgaaqaaiaaigdaaeqaaOGaeyypa0ZaaSGbaeaacaWGKbWaaSbaaSqa aiaaiodacaaIZaaabeaakiaadwfacaWGnbWaaSbaaSqaaiaaikdaae qaaaGcbaWaaeWaaeaacaWGnbWaaSbaaSqaaiaaigdaaeqaaOGaey4k aSIaamytamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaaaaaa a@44EF@

ξ 2 = d 33 U M 1 / ( M 1 + M 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3cda WgaaqaaiaaikdaaeqaaOGaeyypa0ZaaSGbaeaacaWGKbWaaSbaaSqa aiaaiodacaaIZaaabeaakiaadwfacaWGnbWaaSbaaSqaaiaaigdaae qaaaGcbaWaaeWaaeaacaWGnbWaaSbaaSqaaiaaigdaaeqaaOGaey4k aSIaamytamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaaaaaa a@44EF@

For   d  33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamizaO WaaSbaaSqaaKqzGeaeaaaaaaaaa8qacaGGGcWdaiaaiodacaaIZaaa leqaaaaa@3C23@  = 4×10-10 m/V, U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaaaa@36D0@  = 125 V, M 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIXaaabeaaaaa@37AF@  = 1 kg, M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIYaaabeaaaaa@37B0@  = 4 kg we have the settled deformations parameters ξ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3cda Wgaaqaaiaaigdaaeqaaaaa@38A0@  = 40 nm, ξ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3cda Wgaaqaaiaaikdaaeqaaaaa@38A1@  = 10 nm and ξ 1 + ξ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabe67a4TWaaS baaeaacaaIXaaabeaakiabgUcaRiabe67a4TWaaSbaaeaacaaIYaaa beaaaaa@3D4F@  = 50 nm at error 10%.

For the transverse piezoengine at one the fixed face the transfer expression is obtained

W( s )= Ξ( s ) U( s ) = k 31 U T t 2 s 2 +2 T t ξ t s+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaabm aabaGaam4CaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeuONdG1a aeWaaeaacaWGZbaacaGLOaGaayzkaaaabaGaamyvamaabmaabaGaam 4CaaGaayjkaiaawMcaaaaacqGH9aqpdaWcaaqaaiaadUgadaqhaaWc baGaaG4maiaaigdaaeaacaWGvbaaaaGcbaGaaGjbVlaadsfalmaaDa aabaGaamiDaaqaaiaaikdaaaGccaWGZbWcdaahaaqabeaacaaIYaaa aOGaey4kaSIaaGOmaiaadsfalmaaBaaabaGaamiDaaqabaGccqaH+o aElmaaBaaabaGaamiDaaqabaGccaWGZbGaey4kaSIaaGymaaaaaaa@55C2@

k 31 U = d 31 ( h/δ )/ ( 1+ C l / C 11 E ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaDa aaleaacaaIZaGaaGymaaqaaiaadwfaaaGccqGH9aqpdaWcgaqaaiaa dsgalmaaBaaabaGaaG4maiaaigdaaeqaaOWaaeWaaeaadaWcgaqaai aadIgaaeaacqaH0oazaaaacaGLOaGaayzkaaaabaWaaeWaaeaacaaI XaGaey4kaSYaaSGbaeaacaWGdbWaaSbaaSqaaiaadYgaaeqaaaGcba Gaam4qamaaDaaaleaacaaIXaGaaGymaaqaaiaadweaaaaaaaGccaGL OaGaayzkaaaaaaaa@49BD@

T t = M/ ( C l + C 11 E ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWG0baabeaakiabg2da9maakaaabaWaaSGbaeaacaWGnbaa baWaaeWaaeaacaWGdbWaaSbaaSqaaiaadYgaaeqaaOGaey4kaSIaam 4qamaaDaaaleaacaaIXaGaaGymaaqaaiaadweaaaaakiaawIcacaGL Paaaaaaaleqaaaaa@41A0@ ω t =1/ T t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3cda WgaaqaaiaadshaaeqaaOGaeyypa0ZaaSGbaeaacaaIXaaabaGaamiv amaaBaaaleaacaWG0baabeaaaaaaaa@3CC7@

For M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaaaa@36C8@  = 4 kg, C l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaSWaaS baaeaacaWGSbaabeaaaaa@37DB@  = 0.2×107 N/m, C 11 E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaSWaa0 baaeaacaaIXaGaaGymaaqaaiaadweaaaaaaa@392B@  = 1.4×107 N/m we have the parameters T t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaSWaaS baaeaacaWG0baabeaaaaa@37F4@  = 0.5×10-3 s, ω t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3cda Wgaaqaaiaadshaaeqaaaaa@38E8@  = 2×103 s-1 at error 10%.

The settled transverse deformation has the form

Δh= d 31 ( h/δ )U 1+ C l / C 11 E = k 31 U U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam iAaiabg2da9maalaaabaGaamizaSWaaSbaaeaacaaIZaGaaGymaaqa baGcdaqadaqaamaalyaabaGaamiAaaqaaiabes7aKbaaaSGaayjkai aawMcaaOGaamyvaaqaaiaaigdacqGHRaWkdaWcgaqaaiaadoeadaWg aaWcbaGaamiBaaqabaaakeaacaWGdbWaa0baaSqaaiaaigdacaaIXa aabaGaamyraaaaaaaaaOGaeyypa0Jaam4AamaaDaaaleaacaaIZaGa aGymaaqaaiaadwfaaaGccaWGvbaaaa@4D50@

For d 31 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaSWaaS baaeaacaaIZaGaaGymaaqabaaaaa@3883@  = 2∙10-10 m/V, h/δ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca WGObaabaGaeqiTdqgaaaaa@389E@  = 20, C l / C 11 E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca WGdbWaaSbaaSqaaiaadYgaaeqaaaGcbaGaam4qamaaDaaaleaacaaI XaGaaGymaaqaaiaadweaaaaaaaaa@3B30@  = 0.14 the coefficient is determined k 31 U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaDa aaleaacaaIZaGaaGymaaqaaiaadwfaaaaaaa@3965@  = 3.5 nm/V at error 10%.

Conclusion

The mathematical model and the structural scheme of a piezoengine are constructed. The matrix of the deformations of a piezoengine is obtained. The parameters of a piezoengine are determined for the development of a precise control system in nanomedicine and applied bionics.

Acknowledgments

None.

Funding

None.

Conflicts of interest

The authors declare that they have no conflict of interest.

References

  1. Schultz J, Ueda J, Asada H. Cellular Actuators. Butterworth-Heinemann Publisher, Oxford, 2017:382 p.
  2. Afonin SM. Absolute stability conditions for a system controlling the deformation of an elecromagnetoelastic transduser. Doklady Mathematics. 2006;74(3):943–948.
  3. Uchino K. Piezoelectric actuator and ultrasonic motors. Boston, MA: Kluwer Academic Publisher. 1997:350 p.
  4. Afonin SM. Generalized parametric structural model of a compound elecromagnetoelastic transduser. Doklady Physics. 2005;50(2):77–82.
  5. Afonin SM. Structural parametric model of a piezoelectric nanodisplacement transducer. Doklady Physics. 2008;53(3):137–143.
  6. Afonin SM. Solution of the wave equation for the control of an elecromagnetoelastic transduser. Doklady Mathematics. 2006;73(2):307–313.
  7. Cady WG. Piezoelectricity: An introduction to the theory and applications of electromechancial phenomena in crystals. McGraw-Hill Book Company, New York, London. 1946:806 p.
  8. Mason P W (Editor). Physical Acoustics: Principles and Methods. Vol. 1. Part A. Methods and Devices. Academic Press, New York. 1964:515p.
  9. Yang Y, Tang L. Equivalent circuit modeling of piezoelectric energy harvesters. Journal of Intelligent Material Systems and Structures. 2009;20(18):2223–2235.
  10. Zwillinger D. Handbook of Differential Equations. Academic Press, Boston. 1989:673 p.
  11. Afonin SM. A generalized structural-parametric model of an elecromagnetoelastic converter for nano- and micrometric movement control systems: III. Transformation parametric structural circuits of an elecromagnetoelastic converter for nano- and micrometric movement control systems. Journal of Computer and Systems Sciences International. 2006;45(2):317–325.
  12. Afonin SM. Generalized structural-parametric model of an electromagnetoelastic converter for control systems of nano-and micrometric movements: IV. Investigation and calculation of characteristics of step-piezodrive of nano-and micrometric movements. Journal of Computer and Systems Sciences International. 2006;45(6):1006–1013.
  13. Afonin SM. Decision wave equation and block diagram of electromagnetoelastic actuator nano- and microdisplacement for communications systems. International Journal of Information and Communication Sciences. 2016;1(2):22–29.
  14. Afonin SM. Structural-parametric model and transfer functions of electroelastic actuator for nano- and microdisplacement. Chapter 9 in Piezoelectrics and Nanomaterials: Fundamentals, Developments and Applications. Ed. Parinov IA. Nova Science, New York. 2015:225–242.
  15. Afonin SM. A structural-parametric model of electroelastic actuator for nano- and microdisplacement of mechatronic system. Chapter 8 in Advances in Nanotechnology. Volume 19. Eds. Bartul Z, Trenor J, Nova Science, New York. 2017:259–284.
  16. Afonin SM. Electromagnetoelastic nano- and microactuators for mechatronic systems. Russian Engineering Research. 2018:38(12):938–944.
  17. Afonin SM. Nano- and micro-scale piezomotors. Russian Engineering Research. 2012;32(7–8):519–522.
  18. Afonin SM. Elastic compliances and mechanical and adjusting characteristics of composite piezoelectric transducers. Mechanics of Solids. 2007;42(1):43–49.
  19. Afonin SM. Stability of strain control systems of nano-and microdisplacement piezotransducers. Mechanics of Solids. 2014;49(2):196–207.
  20. Afonin SM. Structural-parametric model electromagnetoelastic actuator nanodisplacement for mechatronics. International Journal of Physics. 2017;5(1): 9–15.
  21. Afonin SM. Structural-parametric model multilayer electromagnetoelastic actuator for nanomechatronics. International Journal of Physics. 2019;7(2): 50-57.
  22. Afonin SM. Calculation deformation of an engine for nano biomedical research. International Journal of Biomed Research. 2021;1(5):1–4.
  23. Afonin SM. Precision engine for nanobiomedical research. Biomedical Research and Clinical Reviews. 2021;3(4):1–5.
  24. Afonin SM. Solution wave equation and parametric structural schematic diagrams of electromagnetoelastic actuators nano- and microdisplacement. International Journal of Mathematical Analysis and Applications. 2016;3(4):31–38.
  25. Afonin SM. Structural-parametric model of electromagnetoelastic actuator for nanomechanics. Actuators. 2018;7(1):6.
  26. Afonin SM. Structural-parametric model and diagram of a multilayer electromagnetoelastic actuator for nanomechanics. Actuators. 2019;8(3):52.
  27. Afonin SM. Structural-parametric models and transfer functions of electromagnetoelastic actuators nano- and microdisplacement for mechatronic systems. International Journal of Theoretical and Applied Mathematics. 2016;2(2):52–59.
  28. Afonin SM. Design static and dynamic characteristics of a piezoelectric nanomicrotransducers. Mechanics of Solids. 2010;45(1):123–132.
  29. Afonin SM. Electromagnetoelastic Actuator for Nanomechanics. Global Journal of Research in Engineering: A Mechanical and Mechanics Engineering. 2018;18(2):19–23.
  30. Afonin SM. Multilayer electromagnetoelastic actuator for robotics systems of nanotechnology, Proceedings of the 2018 IEEE Conference EIConRus. Engineering. 2018: 1698–1701.
  31. Afonin SM. A block diagram of electromagnetoelastic actuator nanodisplacement for communications systems. Transactions on Networks and Communications. 2018;6(3):1–9.
  32. Afonin SM. Decision matrix equation and block diagram of multilayer electromagnetoelastic actuator micro and nanodisplacement for communications systems. Transactions on Networks and Communications. 2019;7(3):11–21.
  33. Afonin SM. Condition absolute stability control system of electromagnetoelastic actuator for communication equipment. Transactions on Networks and Communications. 2020;8(1):8–15.
  34. Afonin SM. A Block diagram of electromagnetoelastic actuator for control systems in nanoscience and nanotechnology. Transactions on Machine Learning and Artificial Intelligence. 2020;8(4):23–33.
  35. Afonin SM. Optimal control of a multilayer electroelastic engine with a longitudinal piezoeffect for nanomechatronics systems. Applied System Innovation. 2020;3(4):53.
  36. Afonin SM. Coded сontrol of a sectional electroelastic engine for nanomechatronics systems. Applied System Innovation. 2021;4(3):47.
  37. Afonin SM. Structural scheme actuator for nano research. COJ Reviews and Research. 2020;2(5):1–3.
  38.  Afonin SM. Structural–parametric model electroelastic actuator nano- and microdisplacement of mechatronics systems for nanotechnology and ecology research. MOJ Ecology and Environmental Sciences. 2018;3(5):306‒309.
  39. Afonin SM. Electromagnetoelastic actuator for large telescopes. Aeronautics and Aerospace Open Access Journal. 2018;2(5):270–272.
  40.  Afonin SM. Condition absolute stability of control system with electro elastic actuator for nano bioengineering and microsurgery. Surgery & Case Studies Open Access Journal. 2019;3(3):307–309.
  41. Afonin SM. Piezo actuators for nanomedicine research. MOJ Applied Bionics and Biomechanics. 2019;3(2):56–57.
  42. Afonin SM. Frequency criterion absolute stability of electromagnetoelastic system for nano and micro displacement in biomechanics. MOJ Applied Bionics and Biomechanics. 2019;3(6):137–140.
  43. Afonin SM. Multilayer piezo engine for nanomedicine research. MOJ Applied Bionics and Biomechanics. 2020;4(2):30–31.
  44. Afonin SM. Structural scheme of electromagnetoelastic actuator for nano biomechanics. MOJ Applied Bionics and Biomechanics. 2021;5(2):36–39.
  45. Afonin SM. Multilayer engine for microsurgery and nano biomedicine. Surgery & Case Studies Open Access Journal. 2020;4(4):423–425.
  46. Afonin SM. A structural-parametric model of a multilayer electroelastic actuator for mechatronics and nanotechnology. Chapter 7 in Advances in Nanotechnology. Volume 22. Eds. Bartul Z, Trenor J, Nova Science, New York. 2019:169–186.
  47. Afonin SM. Electroelastic digital-to-analog converter actuator nano and microdisplacement for nanotechnology. Chapter 6 in Advances in Nanotechnology. Volume 24. Eds. Bartul Z, Trenor J, Nova Science, New York. 2020:205–218.
  48. Afonin SM. Characteristics of an electroelastic actuator nano- and microdisplacement for nanotechnology. Chapter 8 in Advances in Nanotechnology. Volume 25. Eds. Bartul Z, Trenor J, Nova Science, New York. 2021:251–266.
  49. Afonin SM. An absolute stability of nanomechatronics system with electroelastic actuator. Chapter 9 in Advances in Nanotechnology. Volume 27. Eds. Bartul Z, Trenor J, Nova Science, New York 2022:183–198.
  50. Afonin SM. Rigidity of a multilayer piezoelectric actuator for the nano and micro range. Russian Engineering Research. 2021;41(4):285–288.
  51. Nalwa HS. Encyclopedia of Nanoscience and Nanotechnology. Los Angeles: American Scientific Publishers. Journal of Nanoscience and Nanotechnology. 2004;10:11–25.
  52. Bhushan B. Springer Handbook of Nanotechnology. New York: Springer. 2004:1222 p.
Creative Commons Attribution License

©2022 Afonin. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.