Piezoactuator of nanodisplacement for astrophysics

Abstract

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

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Introduction
For the control system in astrophysics a piezoactuator of the
nanodisplacement is applied in very large telescope, interferometer
and orbital telescope.1–9 The energy conversion is clearly for the
structural scheme of a piezoactuator.10–16 A piezoactuator is used for
the nanodisplacement in adaptive optics and telescopes.17–26
Structural scheme and characteristics
The equations27–35 of the piezoeects have form
( ) ( )( )
( )
( )
ETdD
T
ε+=
( )
( )
( ) ( ) ( )
EdTsS
t
E
+=
where
( )
D
,
( )
d
,
( )
T
,
( )
T
ε
,
( )
E
,
( )
S
,
( )
E
s
,
( )
t
d
are
matrixes of electric induction, piezomodule, strength mechanical eld,
dielectric constant, strength electric eld, relative displacement, elastic
compliance, transposed piezomodule. The matrixes coecients we
have for a PZT piezoactuator.36–52
( )
15
15
3
1 31 33
0000 0
000 00
000
d
dd
ddd


=


( )
11
22
3 3
00
00
00
T
TT
T
ε
εε
ε


=



( )
( )
11 12 13
12 11
13
13 13 33
55
5
11
5
12
00 0
00 0
00 0
000 0 0
0000 0
0 0 0 0 02
EEE
EEE
EEE
E
E
E
EE
sss
sss
sss
ss
s
ss





=




−

The equation of the mechanical characteristic is written for a
piezoactuator
( )
maxmax
1FFll −∆=∆
where max mi m
l d El∆=
for
0=F
and
max 0
E
mi m ij
F d ES s=
for
0=∆l
,
l
is the length, 0
S
is the area of a piezoactuator.
For the longitudinal piezoactuator the relative displacement8–21 is
written
3 33 3 33 3
E
S d E sT= +
where
33
d
is the longitudinal piezomodule.
In the mechanical characteristic of the longitudinal piezoactuator
for astrophysics the maximums values of the displacement
max
δ∆
and the force
max
F
are determined
max 33 3 3 3
d E dU
δδ
∆= =
,
max 3 3 3 30 3
E
F d SE s=
At
3
E
= 1.5∙105 V/m,
3 3
d
= 4∙10-10 m/V,
0
S
= 1.5∙10-4 m2,
δ
=
2.5∙10-3 m,
33
E
s
= 15∙10-12 m2/N for the longitudinal piezoactuator are
obtained
max
δ∆
= 150 nm,
max
F
= 600 N with error 10%.
Therefore, for the mechanical characteristic of the transverse
piezoactuator we have its maximums values
At
3
E
= 2.4∙105 V/m,
1 3
d
= 2∙10-10 m/V,
h
= 1∙10-2 m,
,
0
S
= 1∙10-5 m2,
11
E
s
= 12∙10-12 m2/N the parameters
are received
max
h∆
= 480 nm,
max
F
= 40 N.
The dierential equation of a piezoactuator12–52 is written
( ) ( )
2
2
2
,,0
d xs xs
dx
γ
Ξ−Ξ =
here
( )
s,xΞ
, s,
x
,
γ
are the Laplace transform of the displacement,
the parameter, the coordinate and the propagation factor.
The nanodisplacements are obtained for the longitudinal
piezoactuator
( ) ( )
ss,
1
0Ξ=Ξ
for
0=x
( ) ( )
ss,
2
Ξ=δΞ
for
δ=x
The decision of the dierential equation is determined
Taking into account the boundary conditions for two faces, we
obtain the system of the equations for the structural model of the
longitudinal piezoactuator.
Aeron Aero Open Access J. 2022;6(4):155‒158. 155
©2022 Afonin. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits
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Piezoactuator of nanodisplacement for astrophysics
Volume 6 Issue 4 - 2022
Afonin SM
National Research University of Electronic Technology, MIET,
Moscow, Russia
Correspondence: Afonin SM, National Research University of
Electronic Technology, MIET, 124498, Moscow, Russia,
Email
Received: September 13, 2022 | Published: September 27,
2022
Abstract
The structural scheme of a piezoactuator is obtained for astrophysics. The matrix equation
is constructed for a piezoactuator. The characteristics of a piezoactuator are received for
astrophysics.
Keywords: piezoactuator, structural scheme, nanodisplacement, characteristic,
astrophysics
Aeronautics and Aerospace Open Access Journal
Research Article Open Access

Piezoactuator of nanodisplacement for astrophysics 156
Copyright:
©2022 Afonin
Citation: Afonin SM. Piezoactuator of nanodisplacement for astrophysics. Aeron Aero Open Access J. 2022;6(4):155‒158. DOI: 10.15406/aaoaj.2022.06.00155
33 33 0
EE
sS
χ
=
where
( )
s
1
Ξ
,
( )
s
2
Ξ
are the Laplace transforms of the
displacements for two faces.
We have the system of the equations for the structural model of the
transverse piezoactuator
( )
( )
( )
( )
( ) ( )
( ) ( ) ( )
11
31 3
2
1 1 1 11
12
sh
ch
E
dEs h
s Ms F s
hs s
γγ
χγ
−−


− 
  

Ξ= − +


× Ξ −Ξ 




( )
( )
( )
( )
( ) ( )
( ) ( ) ( )
11
31 3
2
2 2 2 11
21
sh
ch
EdEs h
s Ms F s
hss
γγ
χγ
−−


− 
  

Ξ= − +


× Ξ − Ξ 




11 11 0
EE
sS
χ
=
Therefore, we have the system of the equations for the structural
model of the shift piezoactuator in the form
The system of the equations for the structural model of a
piezoactuator is determined for Figure 1.
Figure 1 Structural scheme of piezoactuator.
( )
( )
( )
( )
( ) ( )
( ) ( ) ( )
1
1
2
111
12
sh
c
h
mi m
ij
sl
s Ms F s
ls s
ν γγ
χγ
−−
Ψ


Ψ − 
  

Ξ= − +


× Ξ −Ξ 




( )
( )
( )
( )
( ) ( )
( ) ( ) ( )
1
1
2
222
21
sh
c
h
mi m
ij
sl
s Ms F s
lss
ν γγ
χγ
−−
Ψ


Ψ − 
  

Ξ= − +


× Ξ − Ξ 




0 ij ij
sS
χ
ΨΨ
=
where
33 31 15
3
3 31 15
, ,
,,
mi
ddd
vggg

=




=Ψ
133
133
D,D,D
E,E,E
m
33 11 55
33 1 55
1
,,
,,
E EE
ij D DD
sss
s
sss
Ψ

=


{
b,h,l δ=
{
DE
,γγ=γ
{
DE
c,cc =
Ψ
The structural scheme on Figure 1 is used for the decision of a
piezoactuator in astrophysics. The matrix of the nanodisplacement of
a piezoactuator has the form
(
)
( )
( ) ( ) ( )
( ) ( ) ( )
( )
( )
( )
1 11 12 13
1
2 21 22 23
2
ms
s WsWsWs Fs
s WsWsW s Fs
Ψ 
  
Ξ
=
  

  
Ξ
  


The steady-state nanodisplacements are written for two faces of a
piezoactuator
( )
( )
1 21 2
2 11 2
mi m
mi m
d lM M M
d lM M M
ξ
ξ
=Ψ+
=Ψ+
The steady-state nanodisplacements are obtained for two faces of
the longitudinal piezoactuator
( )
( )
1 33 2 1 2
2
33 1 1 2
d UM M M
d UM M M
ξ
ξ
= +
= +
At
U
= 75 V,
1
M
= 1 kg,
2
M
= 4 kg,
3 3
d
= 4∙10-10 m/V the
steady-state nanodisplacements are determined
1
ξ
= 24 nm, 2
ξ
= 6
nm and 21
ξ+ξ
= 30 nm with error 10%.
The transfer equation of the transverse piezoactuator is determined
at one the xed face and the elastic-inertial load
( )
( )
31 3 1 11
1
EE
l
k d h CC
δ
= +
( )
1 1
E
tl
T MC C= +
tt T1=ω
,
where 31
E
k is the transfer coecient, l
C
,
E
C11
are the stiness
for the load and the transverse piezoactuator, t
T
, t
ξ
, t
ω
are the time
constant, the attenuation coecient, the conjugate frequency.
At l
C
= 0,2∙107 N/m, 1 1
E
C= 1.4∙107 N/m,
M
= 2 kg the parameters
are obtained
t
T
= 0.354∙10-3 s,
t
ω
= 2.8∙103 s-1 with error 10%.
The steady-state nanodisplacement of the transverse piezoactuator
is written for elastic-inertial load.
( )
31
3
1
1
1
1
E
E
l
dhU
h kU
CC
δ
∆= =
+

Piezoactuator of nanodisplacement for astrophysics 157
Copyright:
©2022 Afonin
Citation: Afonin SM. Piezoactuator of nanodisplacement for astrophysics. Aeron Aero Open Access J. 2022;6(4):155‒158. DOI: 10.15406/aaoaj.2022.06.00155
At
δh
= 20,
11
E
l
CC
= 0.14, 31
d
= 2∙10-10 m/V the transfer
coecient of the transverse piezoactuator is received
1 3
E
k
= 3.5 nm/V
with error 10%.
Conclusions
The structural scheme of a piezoactuator is constructed for
astrophysics. The matrix of the nanodisplacement of a piezoactuator
is obtained. The characteristics of a piezoactuator are determined.
Acknowledgements
None.
Conict of interest
The Authors declares that there is no Conict of interest.
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