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eISSN: 2576-4500

Aeronautics and Aerospace Open Access Journal

Short Communication Volume 2 Issue 5

Electromagnetoelastic actuator for large telescopes

Afonin SM

National Research University of Electronic Technology (MIET), Russia

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

Received: August 17, 2018 | Published: September 18, 2018

Citation: Afonin SM. Electromagnetoelastic actuator for large telescopes. Aeron Aero Open Access J. 2018;2(5):270-272. DOI: 10.15406/aaoaj.2018.02.00060

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Abstract

The structural diagram and the transfer functions, the characteristics of the electromagnetoelastic actuator for the large telescopes and the cosmic telescopes are obtained. The generalized structural diagram, the matrix transfer functions of the electromagnetoelastic actuator are described the characteristics of the actuator with regard to its physical parameters and external load.

Keywords: electromagnetoelastic actuator, nano-and microdisplacement, electromagnetoelasticity, structural diagram, piezoactuator, transfer function

Introduction

The electromagnetoelastic actuator for the nano- and microdisplacement on the piezoelectric, piezomagnetic, electrostriction, magnetostriction effects is used in the electromechanics systems for the large telescopes and for the cosmic telescopes.1−8 The mathematical model, the structural diagram and transfer functions of the electromagnetoelastic actuator are calculated for designing the control system for the large telescopes and the adaptive optics.9−18 The structural diagram and transfer functions the electromagnetoelastic actuator based on the electromagnetoelasticity make it possible to describe the dynamic and static properties of the electromagnetoelastic actuator for the large telescopes, the cosmic telescopes and the adaptive optics with regard to its physical parameters and external load.19−22

Structural diagram

The structural diagram of the electromagnetoelastic actuator for the large telescopes, the cosmic telescopes and the adaptive optics is changed from Cady and Mason electrical equivalent circuits. The method of mathematical physics with Laplace transform is applied for the solution the wave equation and for the determination the structural diagram of the electromagnetoelastic actuator for the nano- and microdisplacement.1−18

The generalized equation of the electromagnetoelasticity8,11,14 has the following form

S i = ν mi Ψ m ( t )+ s ij Ψ T j ( x,t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb qcfa4aaSbaaSqaaKqzadGaamyAaaWcbeaajugibiabg2da9iabe27a ULqbaoaaBaaaleaajugWaiaad2gacaWGPbaakeqaaKqzGeGaeuiQdK vcfa4aaSbaaSqaaKqzadGaamyBaaWcbeaajuaGdaqadaGcbaqcLbsa caWG0baakiaawIcacaGLPaaajugibiabgUcaRiaadohajuaGdaqhaa WcbaqcLbmacaWGPbGaamOAaaWcbaqcLbmacqqHOoqwaaqcLbsacaWG ubqcfa4aaSbaaSqaaKqzadGaamOAaaGcbeaajuaGdaqadaGcbaqcLb sacaWG4bGaaiilaiaadshaaOGaayjkaiaawMcaaaaa@5CB3@  (1)

where S i = ξ( x,t )/ x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb qcfa4aaSbaaSqaaKqzadGaamyAaaWcbeaajugibiabg2da9Kqbaoaa lyaakeaajugibiabgkGi2kabe67a4LqbaoaabmaakeaajugibiaadI hacaGGSaGaamiDaaGccaGLOaGaayzkaaaabaqcLbsacqGHciITcaWG 4baaaaaa@488B@ is the relative displacement along axis i of the cross section of the piezoactuator or the piezoplate, Ψ m ={ E m , D m , H m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHOo qwjuaGdaWgaaWcbaqcLbmacaWGTbaaleqaaKqzGeGaeyypa0tcfa4a aiqaaOqaaKqzGeGaamyraKqbaoaaBaaaleaajugWaiaad2gaaSqaba qcLbsacaGGSaGaamiraKqbaoaaBaaaleaajugWaiaad2gaaSqabaaa kiaawUhaaKqzGeGaaiilaiaadIeajuaGdaWgaaWcbaqcLbmacaWGTb aaleqaaaaa@4C66@ is the control parameter, E m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb qcfa4aaSbaaSqaaKqzadGaamyBaaWcbeaaaaa@3A34@ is the electric field strength for the voltage control along axis m, D m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGeb qcfa4aaSbaaSqaaKqzadGaamyBaaWcbeaaaaa@3A33@ is the electric induction for the current control along axis m, H m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGib qcfa4aaSbaaSqaaKqzadGaamyBaaWcbeaaaaa@3A37@ for magnetic field strength control along axis m, T j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamOAaaWcbeaaaaa@3A40@ is the mechanical stress along axis j, ν mi MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH9o GBjuaGdaWgaaWcbaqcLbmacaWGTbGaamyAaaWcbeaaaaa@3C10@ is the electromagnetoelastic module, for example, the piezoelectric module, s ij Ψ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qcfa4aa0baaSqaaKqzadGaamyAaiaadQgaaSqaaKqzadGaeuiQdKfa aaaa@3E0B@ is the elastic compliance for the control parameter Ψ=const MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHOo qwcqGH9aqpcaqGJbGaae4Baiaab6gacaqGZbGaaeiDaaaa@3DD0@ , and the indexes i=1, 2, … , 6; j=1, 2, … , 6; m=1, 2, 3. The main size of the electromagnetoelastic actuator is determined us the working length l={ δ, h,b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGSb Gaeyypa0tcfa4aaiqaaOqaaKqzGeGaaGjbVlabes7aKjaacYcaaOGa ay5EaaqcLbsacaWGObGaaiilaiaadkgaaaa@41BC@ in form the thickness, the height and the width for the longitudinal, transverse and shift piezoeffect.

For the construction the structural diagrams of the electromagnetoelastic actuator is used the wave equation8,10,14 for the wave propagation in a long line with damping but without distortions. With using Laplace transform is obtained the linear ordinary second-order differential equation with p parameter. The original problem for the partial differential equation of hyperbolic type using the Laplace transform is reduced to the simpler problem8,14 for the linear ordinary differential equation

d 2 Ξ( x,p ) d x 2 γ 2 Ξ( x,p )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGa euONdGvcfa4aaeWaaOqaaKqzGeGaamiEaiaacYcacaWGWbaakiaawI cacaGLPaaaaeaajugibiaadsgacaWG4bqcfa4aaWbaaSqabeaajugW aiaaikdaaaaaaKqzGeGaeyOeI0Iaeq4SdCwcfa4aaWbaaSqabeaaju gWaiaaikdaaaqcLbsacqqHEoawjuaGdaqadaGcbaqcLbsacaWG4bGa aiilaiaadchaaOGaayjkaiaawMcaaKqzGeGaeyypa0JaaGimaaaa@56CA@    (2)

where Ξ( x,p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHEo awjuaGdaqadaGcbaqcLbsacaWG4bGaaiilaiaadchaaOGaayjkaiaa wMcaaaaa@3D65@ is the Laplace transform of the displacement of section of the actuator, γ=p/ c Ψ +α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo WzcqGH9aqpjuaGdaWcgaGcbaqcLbsacaWGWbaakeaajugibiaadoga juaGdaahaaWcbeqaaKqzadGaeuiQdKfaaaaajugibiabgUcaRiabeg 7aHbaa@436D@ is the propagation coefficient, c Ψ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJb qcfa4aaWbaaSqabeaajugWaiabfI6azbaaaaa@3AE5@ is the sound speed for the control parameter Ψ=const MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHOo qwcqGH9aqpcaqGJbGaae4Baiaab6gacaqGZbGaaeiDaaaa@3DD0@ , α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qyaaa@3824@ is the damping coefficient, C and B are constants.

The mathematical model and the generalized structural diagram of the electromagnetoelastic actuator for the nano- and microdisplacement7,14 on Figure 1 are determined, using method of the mathematical physics for the solution of the wave equation, the boundary conditions and the equation of the electromagnetoelasticity, in the following form

Ξ 1 ( p )= [ 1/ ( M 1 p 2 ) ] × ×{ F 1 ( p )+ ( 1/ χ ij Ψ ) [ ν mi Ψ m ( p ) [ γ/ sh( lγ ) ] [ ch( lγ ) Ξ 1 ( p ) Ξ 2 ( p ) ] ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajugibi abf65ayLqbaoaaBaaaleaajugWaiaaigdaaSqabaqcfa4aaeWaaOqa aKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWada Gcbaqcfa4aaSGbaOqaaKqzGeGaaGymaaGcbaqcfa4aaeWaaOqaaKqz GeGaamytaKqbaoaaBaaaleaajugWaiaaigdaaSqabaqcLbsacaWGWb qcfa4aaWbaaSqabeaajugWaiaaikdaaaaakiaawIcacaGLPaaaaaaa caGLBbGaayzxaaqcfa4aaWbaaSqabeaaaaqcLbsacqGHxdaTaOqaaK qzGeGaey41aqBcfa4aaiWaaOqaaKqzGeGaeyOeI0IaamOraKqbaoaa BaaaleaajugWaiaaigdaaSqabaqcfa4aaeWaaOqaaKqzGeGaamiCaa GccaGLOaGaayzkaaqcLbsacqGHRaWkjuaGdaqadaGcbaqcfa4aaSGb aOqaaKqzGeGaaGymaaGcbaqcLbsacqaHhpWyjuaGdaqhaaWcbaqcLb macaWGPbGaamOAaaWcbaqcLbmacqqHOoqwaaaaaaGccaGLOaGaayzk aaqcfa4aa0baaSqaaaqaaaaajuaGdaWadaGcbaqcLbsacqaH9oGBju aGdaWgaaWcbaqcLbmacaWGTbGaamyAaaWcbeaajugibiabfI6azLqb aoaaBaaaleaajugWaiaad2gaaSqabaqcfa4aaeWaaOqaaKqzGeGaam iCaaGccaGLOaGaayzkaaqcLbsacqGHsisljuaGdaWadaGcbaqcfa4a aSGbaOqaaKqzGeGaeq4SdCgakeaajugibiaabohacaqGObqcfa4aae WaaOqaaKqzGeGaamiBaiabeo7aNbGccaGLOaGaayzkaaaaaaGaay5w aiaaw2faaKqbaoaaBaaaleaaaeqaaKqbaoaadmaakeaajugibiaabo gacaqGObqcfa4aaeWaaOqaaKqzGeGaamiBaiabeo7aNbGccaGLOaGa ayzkaaqcLbsacqqHEoawjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaK qbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaKqzGeGaeyOe I0IaeuONdGvcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaajuaGdaqada GcbaqcLbsacaWGWbaakiaawIcacaGLPaaaaiaawUfacaGLDbaaaiaa wUfacaGLDbaaaiaawUhacaGL9baaaaaa@AC6C@    (3)

Ξ 2 ( p )= [ 1/ ( M 2 p 2 ) ] × ×{ F 2 ( p )+ ( 1/ χ ij Ψ ) [ ν mi Ψ m ( p ) [ γ/ sh( lγ ) ] [ ch( lγ ) Ξ 2 ( p ) Ξ 1 ( p ) ] ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajugibi abf65ayLqbaoaaBaaaleaajugWaiaaikdaaSqabaqcfa4aaeWaaOqa aKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWada Gcbaqcfa4aaSGbaOqaaKqzGeGaaGymaaGcbaqcfa4aaeWaaOqaaKqz GeGaamytaKqbaoaaBaaaleaajugWaiaaikdaaSqabaqcLbsacaWGWb qcfa4aaWbaaSqabeaajugWaiaaikdaaaaakiaawIcacaGLPaaaaaaa caGLBbGaayzxaaqcfa4aaWbaaSqabeaaaaqcLbsacqGHxdaTaOqaaK qzGeGaey41aqBcfa4aaiWaaOqaaKqzGeGaeyOeI0IaamOraKqbaoaa BaaaleaajugWaiaaikdaaSqabaqcfa4aaeWaaOqaaKqzGeGaamiCaa GccaGLOaGaayzkaaqcLbsacqGHRaWkjuaGdaqadaGcbaqcfa4aaSGb aOqaaKqzGeGaaGymaaGcbaqcLbsacqaHhpWyjuaGdaqhaaWcbaqcLb macaWGPbGaamOAaaWcbaqcLbmacqqHOoqwaaaaaaGccaGLOaGaayzk aaqcfa4aa0baaSqaaaqaaaaajuaGdaWadaGcbaqcLbsacqaH9oGBju aGdaWgaaWcbaqcLbmacaWGTbGaamyAaaWcbeaajugibiabfI6azLqb aoaaBaaaleaajugWaiaad2gaaSqabaqcfa4aaeWaaOqaaKqzGeGaam iCaaGccaGLOaGaayzkaaqcLbsacqGHsisljuaGdaWadaGcbaqcfa4a aSGbaOqaaKqzGeGaeq4SdCgakeaajugibiaabohacaqGObqcfa4aae WaaOqaaKqzGeGaamiBaiabeo7aNbGccaGLOaGaayzkaaaaaaGaay5w aiaaw2faaKqbaoaaBaaaleaaaeqaaKqbaoaadmaakeaajugibiaabo gacaqGObqcfa4aaeWaaOqaaKqzGeGaamiBaiabeo7aNbGccaGLOaGa ayzkaaqcLbsacqqHEoawjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaK qbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaKqzGeGaeyOe I0IaeuONdGvcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajuaGdaqada GcbaqcLbsacaWGWbaakiaawIcacaGLPaaaaiaawUfacaGLDbaaaiaa wUfacaGLDbaaaiaawUhacaGL9baaaaaa@AC6F@

where v mi ={ d 33 , d 31 , d 15 g 33 , g 31 , g 15 d 33 , d 31 , d 15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaamyBaiaadMgaaSqabaqcLbsacqGH9aqp juaGdaGabaGcbaqcLbsafaqabeWabaaakeaajugibiaadsgajuaGda WgaaWcbaqcLbmacaaIZaGaaG4maaWcbeaajugibiaacYcacaWGKbqc fa4aaSbaaSqaaKqzadGaaG4maiaaigdaaSqabaqcLbsacaGGSaGaam izaKqbaoaaBaaaleaajugWaiaaigdacaaI1aaaleqaaaGcbaqcLbsa caWGNbqcfa4aaSbaaSqaaKqzadGaaG4maiaaiodaaSqabaqcLbsaca GGSaGaam4zaKqbaoaaBaaaleaajugWaiaaiodacaaIXaaaleqaaKqz GeGaaiilaiaadEgajuaGdaWgaaWcbaqcLbmacaaIXaGaaGynaaWcbe aaaOqaaKqzGeGaamizaKqbaoaaBaaaleaajugWaiaaiodacaaIZaaa leqaaKqzGeGaaiilaiaadsgajuaGdaWgaaWcbaqcLbmacaaIZaGaaG ymaaWcbeaajugibiaacYcacaWGKbqcfa4aaSbaaSqaaKqzadGaaGym aiaaiwdaaSqabaaaaaGccaGL7baaaaa@6F90@ , Ψ m ={ E 3 , E 1 D 3 , D 1 H 3 , H 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHOo qwjuaGdaWgaaWcbaqcLbmacaWGTbaaleqaaKqzGeGaeyypa0tcfa4a aiqaaOqaaKqzGeqbaeqabmqaaaGcbaqcLbsacaWGfbqcfa4aaSbaaS qaaKqzadGaaG4maaWcbeaajugibiaacYcacaWGfbqcfa4aaSbaaSqa aKqzadGaaGymaaWcbeaaaOqaaKqzGeGaamiraKqbaoaaBaaaleaaju gWaiaaiodaaSqabaqcLbsacaGGSaGaamiraKqbaoaaBaaaleaajugW aiaaigdaaSqabaaakeaajugibiaadIeajuaGdaWgaaWcbaqcLbmaca aIZaaaleqaaKqzGeGaaiilaiaadIeajuaGdaWgaaWcbaqcLbmacaaI XaaaleqaaaaaaOGaay5Eaaaaaa@594A@ , s ij Ψ ={ s 33 E , s 11 E , s 55 E s 33 D , s 11 D , s 55 D s 33 H , s 11 H , s 55 H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qcfa4aa0baaSqaaKqzadGaamyAaiaadQgaaSqaaKqzadGaeuiQdKfa aKqzGeGaeyypa0tcfa4aaiqaaOqaaKqzGeqbaeqabmqaaaGcbaqcLb sacaWGZbqcfa4aa0baaSqaaKqzadGaaG4maiaaiodaaSqaaKqzadGa amyraaaajugibiaacYcacaWGZbqcfa4aa0baaSqaaKqzadGaaGymai aaigdaaSqaaKqzadGaamyraaaajugibiaacYcacaWGZbqcfa4aa0ba aSqaaKqzadGaaGynaiaaiwdaaSqaaKqzadGaamyraaaaaOqaaKqzGe Gaam4CaKqbaoaaDaaaleaajugWaiaaiodacaaIZaaaleaajugWaiaa dseaaaqcLbsacaGGSaGaam4CaKqbaoaaDaaaleaajugWaiaaigdaca aIXaaaleaajugWaiaadseaaaqcLbsacaGGSaGaam4CaKqbaoaaDaaa leaajugWaiaaiwdacaaI1aaaleaajugWaiaadseaaaaakeaajugibi aadohajuaGdaqhaaWcbaqcLbmacaaIZaGaaG4maaWcbaqcLbmacaWG ibaaaKqzGeGaaiilaiaadohajuaGdaqhaaWcbaqcLbmacaaIXaGaaG ymaaWcbaqcLbmacaWGibaaaKqzGeGaaiilaiaadohajuaGdaqhaaWc baqcLbmacaaI1aGaaGynaaWcbaqcLbmacaWGibaaaaaaaOGaay5Eaa aaaa@8493@ ,

c Ψ ={ c E c D c H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJb qcfa4aaWbaaSqabeaajugWaiabfI6azbaajugibiabg2da9Kqbaoaa ceaakeaajugibuaabeqadeaaaOqaaKqzGeGaam4yaKqbaoaaCaaale qabaqcLbmacaWGfbaaaaGcbaqcLbsacaWGJbqcfa4aaWbaaSqabeaa jugWaiaadseaaaaakeaajugibiaadogajuaGdaahaaWcbeqaaKqzad GaamisaaaaaaaakiaawUhaaaaa@4B72@ , γ={ γ E γ D γ H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo WzcqGH9aqpjuaGdaGabaGcbaqcLbsafaqabeWabaaakeaajugibiab eo7aNLqbaoaaCaaaleqabaqcLbmacaWGfbaaaaGcbaqcLbsacqaHZo WzjuaGdaahaaWcbeqaaKqzadGaamiraaaaaOqaaKqzGeGaeq4SdCwc fa4aaWbaaSqabeaajugWaiaadIeaaaaaaaGccaGL7baaaaa@4A67@ , l={ δ h b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGSb Gaeyypa0tcfa4aaiqaaOqaaKqzGeqbaeqabmqaaaGcbaqcLbsacqaH 0oazaOqaaKqzGeGaamiAaaGcbaqcLbsacaWGIbaaaaGccaGL7baaaa a@401A@ , χ ij Ψ = s ij Ψ / S 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHhp WyjuaGdaqhaaWcbaqcLbmacaWGPbGaamOAaaWcbaqcLbmacqqHOoqw aaqcLbsacqGH9aqpjuaGdaWcgaGcbaqcLbsacaWGZbqcfa4aa0baaS qaaKqzadGaamyAaiaadQgaaSqaaKqzadGaeuiQdKfaaaGcbaqcLbsa caWGtbqcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaaaaaaaa@4D40@ ,

ν mi MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH9o GBjuaGdaWgaaWcbaqcLbmacaWGTbGaamyAaaWcbeaaaaa@3C10@ is the electromagnetoelastic module, Ψ m ={ E m , D m , H m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHOo qwjuaGdaWgaaWcbaqcLbmacaWGTbaaleqaaKqzGeGaeyypa0tcfa4a aiqaaOqaaKqzGeGaamyraKqbaoaaBaaaleaajugWaiaad2gaaSqaba qcLbsacaGGSaGaamiraKqbaoaaBaaaleaajugWaiaad2gaaSqabaaa kiaawUhaaKqzGeGaaiilaiaadIeajuaGdaWgaaWcbaqcLbmacaWGTb aaleqaaaaa@4C66@ is the control parameter, E m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb qcfa4aaSbaaSqaaKqzadGaamyBaaWcbeaaaaa@3A34@ is the electric field strength for the voltage control along axis m, D m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGeb qcfa4aaSbaaSqaaKqzadGaamyBaaWcbeaaaaa@3A33@ is the electric induction for the current control along axis m, H m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGib qcfa4aaSbaaSqaaKqzadGaamyBaaWcbeaaaaa@3A37@ for magnetic field strength control along axis m, s ij Ψ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qcfa4aa0baaSqaaKqzadGaamyAaiaadQgaaSqaaKqzadGaeuiQdKfa aaaa@3E0B@ is the elastic compliance, d mi MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGKb qcfa4aaSbaaSqaaKqzadGaamyBaiaadMgaaOqabaaaaa@3B40@ is the piezomodule at the voltage-controlled piezoactuator or the magnetostrictive coefficient for the magnetostrictive actuator, g mi MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGNb qcfa4aaSbaaSqaaKqzadGaamyBaiaadMgaaSqabaaaaa@3B44@ is the piezomodule at the current-controlled piezoactuator, S 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb qcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaaaaa@3A0A@ is the cross section area, M 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb qcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaaaaa@3A05@ , M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb qcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaaaa@3A06@ are the mass of the load, Ξ 1 ( p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHEo awjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqbaoaabmaakeaajugi biaadchaaOGaayjkaiaawMcaaaaa@3E66@ , Ξ 2 ( p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHEo awjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqbaoaabmaakeaajugi biaadchaaOGaayjkaiaawMcaaaaa@3E67@ and F 1 ( p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajuaGdaqadaGcbaqcLbsa caWGWbaakiaawIcacaGLPaaaaaa@3DAD@ , F 2 ( p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaajuaGdaqadaGcbaqcLbsa caWGWbaakiaawIcacaGLPaaaaaa@3DAE@ are the Laplace transforms of the appropriate displacements and the forces on the faces 1, 2. The structural diagrams of the magnetostrictive actuator, the voltage-controlled or current-controlled piezoactuator are determined from the mathematical model of the actuator.

Matrix transfer function

The matrix transfer function of the electromagnetoelastic actuator8,14,18 for the large telescopes, the cosmic telescopes and the adaptive optics is deduced from its mathematical model (3) in the following form

( Ξ( p ) )=( W( p ) )( P( p ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeuONdGvcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGa ayzkaaaacaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaqadaGcbaqcLb sacaWGxbqcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaaa caGLOaGaayzkaaqcLbsacaaMe8Ecfa4aaeWaaOqaaKqzGeGaamiuaK qbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaaGaayjkaiaa wMcaaaaa@4FF9@    (4)

( Ξ( p ) )=( Ξ 1 ( p ) Ξ 2 ( p ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeuONdGvcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGa ayzkaaaacaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaqadaGcbaqcLb safaqabeGabaaakeaajugibiabf65ayLqbaoaaBaaaleaajugWaiaa igdaaSqabaqcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaa aabaqcLbsacqqHEoawjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqb aoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaaaaaiaawIcaca GLPaaaaaa@5316@ , ( W( p ) )=( W 11 ( p ) W 12 ( p ) W 13 ( p ) W 21 ( p ) W 22 ( p ) W 23 ( p ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaam4vaKqbaoaabmaakeaajugibiaadchaaOGaayjkaiaa wMcaaaGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aaeWaaOqaaKqzGe qbaeqabiqaaaGcbaqcLbsafaqabeqadaaakeaajugibiaadEfajuaG daWgaaWcbaqcLbmacaaIXaGaaGymaaWcbeaajuaGdaqadaGcbaqcLb sacaWGWbaakiaawIcacaGLPaaaaeaajugibiaadEfajuaGdaWgaaWc baqcLbmacaaIXaGaaGOmaaWcbeaajuaGdaqadaGcbaqcLbsacaWGWb aakiaawIcacaGLPaaaaeaajugibiaadEfajuaGdaWgaaWcbaqcLbma caaIXaGaaG4maaWcbeaajuaGdaqadaGcbaqcLbsacaWGWbaakiaawI cacaGLPaaaaaaabaqcLbsafaqabeqadaaakeaajugibiaadEfajuaG daWgaaWcbaqcLbmacaaIYaGaaGymaaWcbeaajuaGdaqadaGcbaqcLb sacaWGWbaakiaawIcacaGLPaaaaeaajugibiaadEfajuaGdaWgaaWc baqcLbmacaaIYaGaaGOmaaWcbeaajuaGdaqadaGcbaqcLbsacaWGWb aakiaawIcacaGLPaaaaeaajugibiaadEfajuaGdaWgaaWcbaqcLbma caaIYaGaaG4maaWcbeaajuaGdaqadaGcbaqcLbsacaWGWbaakiaawI cacaGLPaaaaaaaaaGaayjkaiaawMcaaaaa@75F8@

( P( p ) )=( Ψ m ( p ) F 1 ( p ) F 2 ( p ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamiuaKqbaoaabmaakeaajugibiaadchaaOGaayjkaiaa wMcaaaGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aaeWaaOqaaKqzGe qbaeqabmqaaaGcbaqcLbsacqqHOoqwjuaGdaWgaaWcbaqcLbmacaWG TbaaleqaaKqbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaa qaaKqzGeGaamOraKqbaoaaBaaaleaajugWaiaaigdaaSqabaqcfa4a aeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaaabaqcLbsacaWGgb qcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaajuaGdaqadaGcbaqcLbsa caWGWbaakiaawIcacaGLPaaaaaaacaGLOaGaayzkaaaaaa@59A9@

where ( Ξ( p ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeuONdGvcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGa ayzkaaaacaGLOaGaayzkaaaaaa@3DD9@ is the column-matrix of the Laplace transforms of the displacements for the faces of the electromagnetoelastic actuator, ( W( p ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaam4vaKqbaoaabmaakeaajugibiaadchaaOGaayjkaiaa wMcaaaGaayjkaiaawMcaaaaa@3D31@ is the matrix transfer function, ( P( p ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamiuaKqbaoaabmaakeaajugibiaadchaaOGaayjkaiaa wMcaaaGaayjkaiaawMcaaaaa@3D2A@ the column-matrix of the Laplace transforms of the control parameter and the forces.

Figure 1 Generalized structural diagram of electromagnetoelastic actuator for the nano- and microdisplacement..

Conclusion

The mathematical model, the structural diagram and transfer functions of the electromagnetoelastic actuator for the large telescopes, the cosmic telescopes and the adaptive optics are obtained. The generalized structural diagram, the transfer functions of the electromagnetoelastic actuator for the nano- and microdisplacement are described the characteristics of the electromagnetoelastic actuator with regard to its physical parameters, external load.

Acknowledgements

None.

Conflicts of Interest

The author declares that there is no conflict of interest.

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