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Physics & Astronomy International Journal

Short Communication Volume 4 Issue 1

Structural-parametric model actuator of adaptive optics for composite telescope and astrophysics equipment

Afonin SM

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

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

Received: January 07, 2020 | Published: January 16, 2020

Citation: Afonin SM. Structural-parametric model actuator of adaptive optics for composite telescope and astrophysics equipment. Phys Astron Int J.2020;4(1):18-21. DOI: 10.15406/paij.2020.04.00198

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Abstract

In this paper, we obtained the structural-parametric model, the matrix transfer function, the static and dynamic characteristics of the multilayer electromagnetoelastic actuator of adaptive optics. It was designed the structural diagram of the multilayer electromagnetoelastic actuator of adaptive optics for composite telescope and astrophysics equipment in contrast to electrical equivalent circuits of the piezotransducer and the vibration piezomotor.

Keywords: multilayer electromagnetoelastic actuator, multilayer piezoactuator, structural diagram, matrix transfer frunction

Introduction

For the adaptive optics of the composite telescope and the astrophysics equipment we used the multilayer electromagnetoelastic actuator nano and micro displacement with the piezoelectric, piezomagnetic, electrostriction, magnetostriction effects with the range of movement from nanometers to hundred of micrometers.1−30 We received the structural-parametric model, the structural diagram of the multilayer electromagnetoelastic actuator in contrast to the electrical equivalent circuits of the piezotransducer and the vibration piezomotor.1–11 The matrix transfer function of the multilayer electromagnetoelastic actuator is calculated for the control system of the composite telescope or the interferometer.14−32 We determined the structural-parametric model and the structural diagram of the multilayer actuator using the equation of the electromagnetoelasticity, the equivalent quadripole and the boundary conditions on the faces of the multilayer electromagnetoelastic actuator.

Structural diagram of multilayer electromagnetoelastic actuator

We received the structural diagram of the multilayer electromagnetoelastic actuator of adaptive optics for composite telescope and astrophysics equipment in difference from Cady's and Mason's electrical equivalent circuits of the piezotransducer and the vibration piezomotor. In this work we used the method of the mathematical physics with Laplace transform for the structural-parametric model and the structural diagram of the multilayer electromagnetoelastic actuator for the adaptive optics of the composite telescope in astronomy.8,14,19,29,30 We have the equation8,9,11,24,29,31 of the electromagnetoelasticity in the form

Si=νmiΨm+sΨijTj

where Si  is the relative displacement, νmi is the coefficient of electromagnetoelasticity in the form dmi  piezomodule or magnetostrictive coefficient, Ψm  is control parameter in variables: electric Em , magnetic Hm  field strengths or electric Dm  induction, sΨij  is the elastic compliance with Ψ=const , Tj  is the mechanical stress, i, j, m are the indexes.

For the multilayer electromagnetoelastic actuator we received the equation of the causes force in the form

F=νmiS0Ψm/sΨij

where S0   is the cross sectional area of the multilayer electromagnetoelastic actuator. The matrix the equivalent quadripole of the multilayer piezoactuator29,31 has the form

[M]n=[ch(lγ)Z0sh(lγ)sh(lγ)Z0ch(lγ)]

where l is the length for longitudinal l=nδ, for transverse l=nh and for shift piezoeffect l=nb,  for the piezolayer δ,h,b are the thickness, the height, the width, γ  is the coefficient propagation.

We obtained the structural-parametric model and the structural diagram of the multilayer electromagnetoelastic actuator of adaptive optics for composite telescope and astrophysics equipment on Figure 1 from the equation of the force that causes deformation, the equivalent quadripole and the boundary conditions with the forces on faces of the actuator in the following form

Figure 1 Structural diagram of multilayer electromagnetoelastic actuator for adaptive optics.

Ξ1(p)=[1/(M1p2)]××{F1(p)+(1/χΨij)[νmiΨm(p)[γ/sh(lγ)][ch(lγ)Ξ1(p)Ξ2(p)]]}

Ξ2(p)=[1/(M2p2)]××{F2(p)+(1/χΨij)[νmiΨm(p)[γ/sh(lγ)][ch(lγ)Ξ2(p)Ξ1(p)]]}

where  vmi={d33,d31,d15g33,g31,g15d33,d31,d15Ψm={E3,E1D3,D1H3,H1sΨij={sE33,sE11,sE55sD33,sD11,sD55sH33,sH11,sH55 ,

l={δhb, 

We have the matrix transfer function of the multilayer electromagnetoelastic actuator of adaptive optics for composite telescope and astrophysics equipment from the generalized structural-parametric model in the form

[ Ξ( p ) ]=[ W( p ) ][ P( p ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaO qaaKqzGeGaeuONdGvcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGa ayzkaaaacaGLBbGaayzxaaqcLbsacqGH9aqpjuaGdaWadaGcbaqcLb sacaWGxbqcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaaa caGLBbGaayzxaaqcLbsacaaMe8Ecfa4aamWaaOqaaKqzGeGaamiuaK qbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaaGaay5waiaa w2faaaaa@5134@

where [ Ξ( p ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaO qaaKqzGeGaeuONdGvcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGa ayzkaaaacaGLBbGaayzxaaaaaa@3E42@ , [ W( p ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaO qaaKqzGeGaam4vaKqbaoaabmaakeaajugibiaadchaaOGaayjkaiaa wMcaaaGaay5waiaaw2faaaaa@3D9A@ , [ P( p ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaMe8 Ecfa4aamWaaOqaaKqzGeGaamiuaKqbaoaabmaakeaajugibiaadcha aOGaayjkaiaawMcaaaGaay5waiaaw2faaaaa@3FAF@  are the matrixes of the displacements the faces, the transfer functions, the control parameters.

In the static we obtained displacements for t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b GaeyOKH4QaeyOhIukaaa@3ADC@  the faces of the voltage-controlled multilayer piezoactuator for the longitudinal piezoeffect and the inertial load at m<< M 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb GaeyipaWJaeyipaWJaamytaKqbaoaaBaaaleaajugWaiaaigdaaSqa baaaaa@3CFF@ , m<< M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb GaeyipaWJaeyipaWJaamytaKqbaoaaBaaaleaajugWaiaaikdaaSqa baaaaa@3D00@ , where m is the mass of the multilayer piezoactuator, M 1 , M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb qcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiaacYcacaaMe8Ua amytaKqbaoaaBaaaleaajugWaiaaikdaaSqabaaaaa@4052@  are the load masses, and the forces on faces F 1 ( t )= F 2 ( t )=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajuaGdaqadaGcbaqcLbsa caWG0baakiaawIcacaGLPaaajugibiabg2da9iaadAeajuaGdaWgaa WcbaqcLbmacaaIYaaaleqaaKqbaoaabmaakeaajugibiaadshaaOGa ayjkaiaawMcaaKqzGeGaeyypa0JaaGimaiaacYcaaaa@4972@  in the following form

ξ 1 ( )= lim p0 p W 11 ( p )( U/δ )/p = d 33 nU M 2 / ( M 1 + M 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqbaoaabmaakeaajugi biabg6HiLcGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWfqaGcba qcLbyacaqGSbGaaeyAaiaab2gaaSqaaKqzadGaamiCaiabgkziUkaa icdaaOqabaqcfa4aaSGbaOqaaKqzGeGaamiCaiaadEfajuaGdaWgaa WcbaqcLbmacaaIXaGaaGymaaGcbeaajuaGdaqadaGcbaqcLbsacaWG WbaakiaawIcacaGLPaaajuaGdaqadaGcbaqcfa4aaSGbaOqaaKqzGe GaamyvaaGcbaqcLbsacqaH0oazaaaakiaawIcacaGLPaaaaeaajugi biaadchaaaGaeyypa0tcfa4aaSGbaOqaaKqzGeGaamizaKqbaoaaBa aaleaajugWaiaaiodacaaIZaaaleqaaKqzGeGaamOBaiaadwfacaWG nbqcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaaOqaaKqbaoaabmaake aajugibiaad2eajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGa ey4kaSIaamytaKqbaoaaBaaaleaajugWaiaaikdaaSqabaaakiaawI cacaGLPaaaaaaaaa@73EA@

ξ 2 ( )= lim p0 p W 21 ( p )( U/δ )/p = d 33 nU M 1 / ( M 1 + M 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqbaoaabmaakeaajugi biabg6HiLcGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaGdaWfqaGcba qcLbyacaqGSbGaaeyAaiaab2gaaSqaaKqzadGaamiCaiabgkziUkaa icdaaOqabaqcfa4aaSGbaOqaaKqzGeGaamiCaiaadEfajuaGdaWgaa WcbaqcLbmacaaIYaGaaGymaaGcbeaajuaGdaqadaGcbaqcLbsacaWG WbaakiaawIcacaGLPaaajuaGdaqadaGcbaqcfa4aaSGbaOqaaKqzGe GaamyvaaGcbaqcLbsacqaH0oazaaaakiaawIcacaGLPaaaaeaajugi biaadchaaaGaeyypa0tcfa4aaSGbaOqaaKqzGeGaamizaKqbaoaaBa aaleaajugWaiaaiodacaaIZaaaleqaaKqzGeGaamOBaiaadwfacaWG nbqcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaaaOqaaKqbaoaabmaake aajugibiaad2eajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGa ey4kaSIaamytaKqbaoaaBaaaleaajugWaiaaikdaaSqabaaakiaawI cacaGLPaaaaaaaaa@73EB@

ξ 1 ( )+ ξ 2 ( )= d 33 nU MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqbaoaabmaakeaajugi biabg6HiLcGccaGLOaGaayzkaaqcLbsacqGHRaWkcqaH+oaEjuaGda WgaaWcbaqcLbmacaaIYaaaleqaaKqbaoaabmaakeaajugibiabg6Hi LcGccaGLOaGaayzkaaqcLbsacqGH9aqpcaWGKbqcfa4aaSbaaSqaaK qzadGaaG4maiaaiodaaSqabaqcLbsacaWGUbGaamyvaaaa@5176@

where U is the voltage.

For the multilayer piezoactuator at d 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGKb qcfa4aaSbaaSqaaKqzadGaaG4maiaaiodaaSqabaaaaa@3ADB@ = 4∙10-10 m/V, n=16, U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGvb aaaa@375F@ =100 V, M 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb qcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaaaaa@3A05@ =1 kg and M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb qcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaaaa@3A06@ =4 kg we obtained the static displacements of the faces the multilayer piezoactuator ξ 1 ( ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqbaoaabmaakeaajugi biabg6HiLcGccaGLOaGaayzkaaaaaa@3F21@ =512 nm, ξ 2 ( ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqbaoaabmaakeaajugi biabg6HiLcGccaGLOaGaayzkaaaaaa@3F22@ =128 nm, ξ 1 ( )+ ξ 2 ( ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqbaoaabmaakeaajugi biabg6HiLcGccaGLOaGaayzkaaqcLbsacqGHRaWkcqaH+oaEjuaGda WgaaWcbaqcLbmacaaIYaaaleqaaKqbaoaabmaakeaajugibiabg6Hi LcGccaGLOaGaayzkaaaaaa@492F@ =640 nm.

We received transfer function of the multilayer piezoactuator at longitudinal piezoeffect with one fixed face and voltage control for the elastic-inertial load at m<< M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb GaeyipaWJaeyipaWJaamytaKqbaoaaBaaaleaajugWaiaaikdaaSqa baaaaa@3D00@  in the following form

W( p )= Ξ 2 ( p ) U( p ) = d 33 n ( 1+ C e / C 33 E ) ( T t 2 p 2 +2 T t ξ t p+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGxb qcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacqqHEoawjuaGdaWgaaWcbaqcLbmaca aIYaaaleqaaKqbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMca aaqaaKqzGeGaamyvaKqbaoaabmaakeaajugibiaadchaaOGaayjkai aawMcaaaaajugibiabg2da9KqbaoaalaaakeaajugibiaadsgajuaG daWgaaWcbaqcLbmacaaIZaGaaG4maaWcbeaajugibiaad6gaaOqaaK qbaoaabmaakeaajugibiaaigdacqGHRaWkjuaGdaWcgaGcbaqcLbsa caWGdbqcfa4aaSbaaSqaaKqzadGaamyzaaWcbeaaaOqaaKqzGeGaam 4qaKqbaoaaDaaaleaajugWaiaaiodacaaIZaaaleaajugWaiaadwea aaaaaaGccaGLOaGaayzkaaqcfa4aa0baaSqaaaqaaaaajuaGdaqada GcbaqcLbsacaWGubqcfa4aa0baaSqaaKqzadGaamiDaaWcbaqcLbma caaIYaaaaKqzGeGaamiCaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaK qzGeGaey4kaSIaaGOmaiaadsfajuaGdaWgaaWcbaqcLbmacaWG0baa leqaaKqzGeGaeqOVdGxcfa4aaSbaaSqaaKqzadGaamiDaaWcbeaaju gibiaadchacqGHRaWkcaaIXaaakiaawIcacaGLPaaaaaaaaa@7F23@

T t = M 2 / ( C e + C 33 E ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamiDaaWcbeaajugibiabg2da9Kqbaoaa kaaakeaajuaGdaWcgaGcbaqcLbsacaWGnbqcfa4aaSbaaSqaaKqzad GaaGOmaaWcbeaaaOqaaKqbaoaabmaakeaajugibiaadoeajuaGdaWg aaWcbaqcLbmacaWGLbaaleqaaKqzGeGaey4kaSIaam4qaKqbaoaaDa aaleaajugWaiaaiodacaaIZaaaleaajugWaiaadweaaaaakiaawIca caGLPaaaaaaabeaaaaa@4F4D@ ξ t = α ( nδ ) 2 C 33 E / ( 3 c E M 2 ( C e + C 33 E ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEjuaGdaWgaaWcbaqcLbmacaWG0baaleqaaKqzGeGaeyypa0tcfa4a aSGbaOqaaKqzGeGaeqySdewcfa4aaeWaaOqaaKqzGeGaamOBaiabes 7aKbGccaGLOaGaayzkaaqcfa4aaWbaaOqabSqaaKqzadGaaGOmaaaa jugibiaadoeajuaGdaqhaaWcbaqcLbmacaaIZaGaaG4maaWcbaqcLb macaWGfbaaaaGcbaqcfa4aaeWaaOqaaKqzGeGaaG4maiaadogajuaG daahaaGcbeWcbaqcLbmacaWGfbaaaKqbaoaakaaakeaajugibiaad2 eajuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqbaoaabmaakeaajugi biaadoeajuaGdaWgaaWcbaqcLbmacaWGLbaaleqaaKqzGeGaey4kaS Iaam4qaKqbaoaaDaaaleaajugWaiaaiodacaaIZaaaleaajugWaiaa dweaaaaakiaawIcacaGLPaaaaeqaaaGaayjkaiaawMcaaaaaaaa@6835@  

where Ξ 2 ( p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHEo awjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqbaoaabmaakeaajugi biaadchaaOGaayjkaiaawMcaaaaa@3E67@ , U( p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGvb qcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaaaaa@3B0E@  are the Laplace transforms the displacement face and the voltage, T t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamiDaaWcbeaaaaa@3A4A@ , ξ t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH+o aEjuaGdaWgaaWcbaqcLbmacaWG0baaleqaaaaa@3B34@  are the time constant and the damping coefficient, C 33 E = S 0 / ( s 33 E nδ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGdb qcfa4aa0baaSqaaKqzadGaaG4maiaaiodaaSqaaKqzadGaamyraaaa jugibiabg2da9KqbaoaalyaakeaajugibiaadofajuaGdaWgaaWcba qcLbmacaaIWaaaleqaaaGcbaqcfa4aaeWaaOqaaKqzGeGaam4CaKqb aoaaDaaaleaajugWaiaaiodacaaIZaaaleaajugWaiaadweaaaqcLb sacaWGUbGaeqiTdqgakiaawIcacaGLPaaaaaaaaa@4F53@  is the rigidity of the multilayer piezoactuator for E=const MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGfb Gaeyypa0Jaae4yaiaab+gacaqGUbGaae4Caiaabshaaaa@3D0B@ .

At the elastic-inertial load for d 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGKb qcfa4aaSbaaSqaaKqzadGaaG4maiaaiodaaSqabaaaaa@3ADB@ =4∙10-10 m/V, n=12, U=200 V, M 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb qcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaaaa@3A06@ =4 kg, C 33 E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGdb qcfa4aa0baaSqaaKqzadGaaG4maiaaiodaaSqaaKqzadGaamyraaaa aaa@3CB3@  = 2∙107 N/m, C e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGdb qcfa4aaSbaaSqaaKqzadGaamyzaaWcbeaaaaa@3A2A@ =0.4∙107 N/m we received the steady-state value of the displacement of the multilayer piezoactuator ξ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOVdmaaBa aaleaacaaIYaaabeaaaaa@3822@ =800 nm and the time constant T t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamiDaaWcbeaaaaa@3A4A@ =0.4∙10-3 s.

Conclusion

The structural-parametric model, structural diagram and the matrix transfer function of the multilayer electromagnetoelastic actuator of adaptive optics for composite telescope and astrophysics equipment are obtained. The static and dynamic characteristics of the multilayer actuator are received with using the matrix transfer function of the multilayer electromagnetoelastic actuator.

Acknowledgments

None.

Conflicts of interest

The author declares there is no conflict of interest.

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