Structural-Parametric Model and Diagram of a Multilayer Electromagnetoelastic Actuator for Nanomechanics

Abstract

In this work, the parametric structural schematic diagrams of a multilayer electromagnetoelastic actuator and a multilayer piezoactuator for nanomechanics were determined in contrast to the electrical equivalent circuits of a piezotransmitter and piezoreceiver, the vibration piezomotor. The decision matrix equation of the equivalent quadripole of the multilayer electromagnetoelastic actuator was used. The structural-parametric model, the parametric structural schematic diagram, and the matrix transfer function of the multilayer electromagnetoelastic actuator for nanomechanics were obtained.

Full text

Actuators 2019, 8, 52; doi:10.3390/act8030052 www.mdpi.com/journal/actuators
Article
Structural-Parametric Model and Diagram of a
Multilayer Electromagnetoelastic Actuator for
Nanomechanics
Sergey M. Afonin
Institute of Microdevices and Control Systems, National Research University of Electronic
Technology (MIET), Moscow 124498, Russia; learner01@mail.ru
Received: 20 May 2019; Accepted: 24 June 2019; Published: 29 June 2019
Abstract: In this work, the parametric structural schematic diagrams of a multilayer
electromagnetoelastic actuator and a multilayer piezoactuator for nanomechanics were determined
in contrast to the electrical equivalent circuits of a piezotransmitter and piezoreceiver, the vibration
piezomotor. The decision matrix equation of the equivalent quadripole of the multilayer
electromagnetoelastic actuator was used. The structural-parametric model, the parametric
structural schematic diagram, and the matrix transfer function of the multilayer
electromagnetoelastic actuator for nanomechanics were obtained.
Keywords: multilayer electromagnetoelastic actuator; multilayer piezoactuator; structural-
parametric model; matrix transfer function; parametric structural schematic diagram; characteristic
1. Introduction
A multilayer actuator provides an increase in the range of movement from a few nanometers to
tens of micrometers in nanomechanical systems for nanotechnology and adaptive optics. In this
work, the parametric structural schematic diagrams of a multilayer electromagnetoelastic actuator
and a multilayer piezoactuator are determined in contrast to the electrical equivalent circuits of a
piezotransmitter and piezoreceiver, the vibration piezomotor [1–12]. An investigation of the static
and dynamic characteristics of the multilayer piezoactuator was necessary for the calculation of
nanomechanical systems in the scanning tunneling microscope and the atomic force microscope used
for nanotechnology [7–29]. In this work, the solution from the matrix equation of the equivalent
quadripole of the multilayer electromagnetoelastic actuator was used compared with the solution
from the wave equation presented in my article [25].
Using the decision matrix equation of the equivalent quadripole of the multilayer
electromagnetoelastic actuator, with an allowance for the corresponding equation for
electromagnetoelasticity and the boundary conditions on its working faces, we constructed the
structural-parametric model of the multilayer electromagnetoelastic actuator [8,9,14,16]. The matrix
transfer function and the parametric structural schematic diagram of the multilayer
electromagnetoelastic actuator were obtained from its structural-parametric model.
2. Parametric Structural Schematic Diagram of the Multilayer Electromagnetoelastic Actuator
In general, the equation for electromagnetoelasticity [11,14,16,20] has the following form:
jijmmiiTsS 

,
(1)

Actuators 2019, 8, 52 2 of 15
where
i
S
,
mi

,
m

,

ij
s
, and
j
T
represent the relative displacement, the electromagnetoelasticity
coefficient (piezomodule or magnetostrictive coefficient
mi
d
), the generalized control parameter
(electric
m
E
, magnetic
m
H
, field strength or electric
m
D
induction), the elastic compliance with
const
, and the mechanical stress, respectively, and i, j, and m are indexes.
The multilayer piezoactuator in Figure 1 consists of piezolayers or piezoplates connected
electrically in parallel and mechanically in series. Let us construct the structural-parametric model of
the multilayer piezoactuator for the multilayer piezoactuator at the longitudinal piezoelectric effect,
where
3
SSi
,
33
d
mi 
,
3
E
m
,
E
ij ss 33


, and
3
TTj
:
3333333TsEdS E

.
(2)
The Laplace transform of the force, which causes the deformation, has the following form:
   
E
s
pESd
pF
33
3033

.
(3)
(a)
(b)
(c)
Figure 1. Kinematic schemes of the multilayer piezoactuator for (a) longitudinal, (b) transverse, and
(c) shift piezoeffects, and the k piezolayer.
We consider the matrix equation for the Laplace transforms of the forces and the displacements
[16] at the input and output ends of the k piezolayer of the multilayer piezoactuator from n
piezolayers. The equivalent T-shaped quadripole of the k piezolayer is shown in Figure 2.
Figure 2. Quadripole for the k piezolayer.

Actuators 2019, 8, 52 3 of 15
The circuit of the multilayer piezoactuator in Figure 3 is compiled from the equivalent T-shaped
quadripole for the k piezolayer and the forces equations, acting on the faces of the piezolayer.
Therefore, we have the Laplace transforms of the corresponding forces on the input and output faces
of the k piezolayer of the multilayer piezoactuator in Figures 2 and 3 in the form of the system of the
equations for the equivalent T-shaped quadripole in the following form:
       
pZpZZpF kk
inp
k1221 

       
pZZpZpF kk
out
k1212 

,
(4)
where
 



ij
s
S
Zth
0
1
,
 


sh
0
2
ij
s
S
Z
represent the resistance of the equivalent quadripole of the k
piezolayer,  is the thickness,

is the coefficient of wave propagation (
 
c
p
, and p is the
Laplace operator,

c
is the speed of sound in the piezoceramics with
const
,

is the
attenuation coefficient),
 
pF inp
k
,
pF out
k
are the Laplace transforms of the forces on the input and
output ends of the k piezolayer, and
 
p
k

,
 
ps
k1

are the Laplace transforms of the
displacements at the input and output ends of the k piezolayer.
Figure 3. Circuit of the multilayer piezoactuator with quadripoles for the k and k + 1 piezolayers.
Accordingly, we have the Laplace transforms for the following system of the equations for the k
piezolayer in Figure 2 in the following form:
     
p
Z
Z
ZpF
Z
Z
pF k
out
k
inp
k1
2
1
1
2
121 

















,
(5)
     
p
Z
Z
pF
Z
pk
out
kk 1
2
1
1
1
1










,
the matrix equation for the k piezolayer:
 
 
 
 
 
















p
pF
M
p
pF
k
out
k
k
inp
k
1
,
(6)
and the matrix
 
M
in the following form:
 





























2
1
2
1
1
1
2
1
2221
1211
1
1
21
Z
Z
Z
Z
Z
Z
Z
Z
mm
mm
M
,
(7)
where
 
 ch1
2
1
2211 Z
Z
mm
,
 








 sh20
1
1
112 Z
Z
Z
Zm
,
 
02
21 sh1
ZZ
m

, and



ij
s
S
Z0
0
.

Actuators 2019, 8, 52 4 of 15
For the multilayer piezoactuator of the Laplace transform, the displacement
 
p
k1

and the
force
 
pF out
k
, acting on the output face of the k piezolayer in Figure 3, correspond to the Laplace
transforms of the displacement and the force, acting on the input face of the k + 1 piezolayer.
The force on the output face for the k piezolayer is equal in amplitude and opposite in direction
to the force on the input face for the k + 1 piezolayer:
   
pFpF inp
k
out
k1

.
(8)
From Equation (7), the matrix equation for the n piezolayers of the multilayer piezoactuator, we
obtain the following form:
 
 
 
 
 
















p
pF
M
p
pF
n
out
n
n
inp
1
1
1
,
(9)
with the matrix multilayer piezoactuator for the longitudinal piezoeffect (see Figure 1a) in the
following form:
 
   
   











n
Z
nnZn
Mnch
sh shch
0
0
.
(10)
The equations of the forces acting on the faces of the multilayer piezoactuator are as follows:
at
0x
,
     
ppMpFSp,Tj1
2
110
0
,
at
lx 
,
     
ppMpFSs,lTj2
2
220 
,
(11)
where
 
p,Tj0
and
 
p,lTj
are the Laplace transforms of the mechanical stresses at two faces of the
multilayer piezoactuator and
0
S
is the cross-sectional area.
We have the Laplace transforms of the displacements and the forces for the first and second faces
of the multilayer piezoactuator in the following form:
at
0x
and
 
p
1

,
 
pF1
,
at
lx 
and
   
pp n12 

,
   
pFpF out
n

2
.
(12)
The structural-parametric model of the multilayer piezoactuator for longitudinal piezoeffect
with
 nl
in Figure 4 is obtained from the result analysis of the equation of the force (3) that causes
deformation, the system of the equations for the equivalent quadripole of the multilayer
piezoactuator (9) and (10), and the equation of the forces (11) on its faces in the following form:
       
     
33 3
11
2
1 33
12
11
sh
ch
E
d E p n
p F p
Mp n p p













    




 


  






,
       
     
33 3
22
2
2 33
21
11
sh
ch
E
d E p n
p F p
Mp n p p













    




 


  






,
(13)
where
03333 SsEE 
.

Actuators 2019, 8, 52 5 of 15
Figure 4. Parametric structural schematic diagram of the multilayer piezoactuator for longitudinal
piezoeffect with voltage control.
For the multilayer piezoactuator at the transverse piezoelectric effect, where
1
SSi
,
31
d
mi 
,
3
E
m
,
E
ij ss 11


, and
1
TTj
, we obtain the following:
1113311TsEdS E

.
(14)
The Laplace transform of the force, which causes the deformation, has the following form:
   
E
s
pESd
pF
11
3031

.
(15)
The matrix multilayer piezoactuator for the transverse piezoeffect (see Figure 1b) has the
following form:
 
   
   











nh
Z
nh nhZnh
Mnch
sh shch
0
0
.
(16)
The structural-parametric model of the multilayer piezoactuator for the transverse piezoeffect
in Figure 5 with length
nhl
is obtained from the result analysis of Equations (15) and (16), and (11)
in the following form:
       
     
31 1
11
2
1 11
12
11
sh
ch
E
d E p nh
p F p
Mp nh p p










   

    




    

  






,
(17)

Actuators 2019, 8, 52 6 of 15
       
     
31 1
22
2
2 11
21
11
sh
ch
E
d E p nh
p F p
Mp nh p p










   

    




    

  






,
where
01111 SsEE 

.
Figure 5. Parametric structural schematic diagram of the multilayer piezoactuator for transverse
piezoeffect with voltage control.
For the multilayer piezoactuator at the shift piezoelectric effect, where
5
SSi
,
15
d
mi 
,
1
E
m
,
E
ij ss 55


, and
5
TTj
, we obtain the following:
5551155TsEdS E

(18)
The Laplace transform of the force, which causes the deformation, has the following form:
   
E
s
pESd
pF
55
1015

.
(19)
The matrix multilayer piezoactuator for the shift piezoeffect (see Figure 1c) has the following
form:
 
   
   











nb
Z
nb nbZnb
Mnch
sh shch
0
0
.
(20)
The structural-parametric model of the multilayer piezoactuator for shift piezoeffect with length
nbl
is obtained from the result analysis of Equations (19) and (20), and (11) in the following form:

Actuators 2019, 8, 52 7 of 15
       
     
15 1
11
2
1 55
12
11
sh
ch
E
d E p nb
p F p
Mp nb p p













    




 


  






,
       
     
15 1
22
2
2 55
21
11
sh
ch
E
d E p nb
p F p
Mp nb p p













    




 


  






,
(21)
where
05555 SsEE 
.
For the multilayer magnetostrictive actuator at the longitudinal magnetostrictive effect, where
3
SSi
,
33
d
mi 
,
3
H
m
,
H
ij ss 33


, and
3
TTj
, we obtain the following:
3333333TsHdS H

.
(22)
The Laplace transform of the force, which causes the deformation, has the following form:
   
H
s
pHSd
pF
33
3033

.
(23)
The matrix multilayer piezoactuator for the longitudinal magnetostrictive effect has the
following form:
 
   
   











n
Z
nnZn
Mnch
sh shch
0
0
.
(24)
The structural-parametric model of the multilayer magnetostrictive actuator for the longitudinal
magnetostrictive effect and the parametric structural schematic diagram in Figure 6 with
 nl
are
obtained from the result analysis of Equations (23) and (24), and (11) in the following form:
       
     
33 3
11
2
1 33
12
11sh
ch
H
d H p n
p F p
Mp n p p













    




 


  






,
       
     
33 3
22
2
2 33
21
11
sh
ch
H
d H p n
p F p
Mp n p p













    




 


  






,
(25)
where
03333 SsHH 
.

Actuators 2019, 8, 52 8 of 15
Figure 6. Parametric structural schematic diagram of multilayer magnetostrictive actuator for
longitudinal magnetostrictive effect with the magnet field strength control.
For the multilayer magnetostrictive actuator at the transverse magnetostrictive effect, where
1
SSi
,
31
d
mi 
,
3
H
m
,
H
ij ss 11


, and
1
TTj
, we obtain the following:
1113311TsHdS H

.
(26)
The Laplace transform of the force, which causes the deformation, has the following form:
   
H
s
pHSd
pF
11
3031

.
(27)
The matrix multilayer magnetostrictive actuator for the transverse magnetostrictive effect has
the following form:
 
   
   











nh
Z
nh nhZnh
Mnch
sh shch
0
0
.
(28)
The structural-parametric model of the multilayer magnetostrictive actuator for the transverse
magnetostrictive effect with
nhl
is obtained from Equations (27) and (28), and (11) in the following
form:
       
     
31 1
11
2
1 11
12
11
sh
ch
H
d H p nh
p F p
Mp nh p p










   

    




    

  






,
       
     
31 1
22
2
2 11
21
11
sh
ch
H
d H p nh
p F p
Mp nh p p










   

    




    

  






,
(29)

Actuators 2019, 8, 52 9 of 15
where
01111 SsHH 
.
For the multilayer magnetostrictive actuator at the shift magnetostrictive effect, where
5
SSi
,
15
d
mi 
,
1
H
m
,
H
ij ss 55


, and
5
TTj
, we obtain the following:
5551155TsHdS H

.
(30)
The Laplace transform of the force, which causes the deformation, has the following form:
   
H
s
pHSd
pF
55
1015

(31)
The matrix multilayer magnetostrictive actuator for the shift magnetostrictive effect has the
following form:
 
   
   











nb
Z
nb nbZnb
Mnch
sh shch
0
0
.
(32)
The structural-parametric model of the multilayer magnetostrictive actuator for the shift
magnetostrictive effect with length
nbl
is obtained from the result analysis of Equations (31) and
(32), and (11) in the following form:
       
     
15 1
11
2
1 55
12
11
sh
ch
H
d H p nb
p F p
Mp nb p p













    




 


  






,
       
     
15 1
22
2
2 55
21
11
sh
ch
H
d H p nb
p F p
Mp nb p p













    




 


  






,
(33)
where
05555 SsHH 
.
Therefore, in general, from (1) for the multilayer electromagnetoelastic actuator, the Laplace
transform of the force that causes deformation has the following form:
   



ij
mmi
s
pS
pF 0
,
(34)
where
0
Ssijij  
and
0
S
is the cross-sectional area of the multilayer actuator.
Accordingly, in general, the matrix for the equivalent quadripole of the multilayer
electromagnetoelastic actuator has the following form:
 
   
   











l
Z
llZl
Mnch
sh shch
0
0
.
(35)
From Equation (35), we have the equivalent quadripole of the multilayer piezoactuator in Figure
1a–с for the longitudinal piezoeffect with length of the multilayer piezoactuator being
 nl
, for the
transverse piezoeffect being
nhl
, and for the shift piezoeffect being
nbl
, where
b,h,
are the
thickness, the height, and the width for the k piezolayer, respectively.
We obtain the equations for the generalized structural-parametric model and the generalized
parametric structural schematic diagram in Figure 7 of the multilayer electromagnetoelastic actuator

Actuators 2019, 8, 52 10 of 15
from the result analysis of the equation of the force (34) that causes deformation, the system of the
equations for the equivalent quadripole (35), and the equation of the forces (11) on its faces in the
following form:
       
     
11
2
1
12
11sh
ch
mi m
ij
pl
p F p
Mp l p p







  






    



 
 

   






,
       
     
22
2
2
21
11sh
ch
mi m
ij
pl
p F p
Mp l p p







  






    



 
 

   






,
(36)
where






nb
nh
n
l
,






153133
153133
153133
d,d,d
g,g,g
d,d,d
vmi
,






13
13
13
H,H
D,D
E,E
m
,







HHH
DDD
EEE
ij
s,s,s
s,s,s
s,s,s
s
551133
551133
551133
,







H
D
E
c
c
c
c
,









H
D
E
.
Figure 7. Generalized parametric structural schematic diagram of the multilayer
electromagnetoelastic actuator.
3. Matrix Transfer Function of the Multilayer Electromagnetoelastic Actuator
From Equation (36), we have in general the matrix transfer function of the multilayer
electromagnetoelastic actuator in the following form:
 
 
 
 
 
 
pPpWp 
,
(37)

Actuators 2019, 8, 52 11 of 15
where
 
 
 
 








 p
p
p
2
1
is the matrix of the displacements,
 
 
 
 
 











pF
pF
p
pP
m
2
1
is the matrix of the
control parameters, and
 
 
     
     






pWpWpW
pWpWpW
pW
232221
131211
is the matrix transfer function, where

       
 
ijijmimAlpMpppW 2th
2
2111  
,

       
 
ij
ijmimAlpMpppW 2th
2
1221  
,

       
 
ij
ijij AlpMpFppW   th
2
21112
,

             
 
ij
ij AlpFppWpFppW  sh
12222113
,

       
 
ij
ijij AlpMpFppW   th
2
12223
,
where
 
   
 
 
   
 
 
2
2
21
3
21
4
2
21 1thth pclMMpncMMpMMA ijijijij  
2
2 
cp
.
For example, for the voltage-controlled multilayer piezoactuator at the longitudinal piezoeffect,
we have the following transfer functions:
       
 
33
2
332333111 2th AnpMdpEppW E
,
       
 
33
2
331333221 2th AnpMdpEppW E
,
       
 
33
2
332331112 th AnpMpFppW EE 
,
             
 
333312222113 sh AnpFppWpFppW E
,
       
 
33
2
331332223 th AnpMpFppW EE 
,
(38)
where
 
   
 
 
   
 
 
2
2
3321
3
3321
4
2
332133 1thth pcnMMpncMMpMMA EEEEE 
2
2 E
cp
.
For the voltage-controlled multilayer piezoactuator, the static displacements of its faces at the
longitudinal piezoeffect and the inertial load at
1
Mm 
,
2
Mm 
and
   
0
21  tFtF
have the
following form:
     
p)U(ppWtm
p
t


 11
0
0
11 limlim
,
   
212331MMMnUdm
,
(39)
     
p)U(ppWtm
p
t


 21
0
0
22 limlim
,
   
211332MMMnUdm
,
(40)
where
m
U
is the amplitude of the voltage,
m
is the mass of the multilayer piezoactuator, and
21 M,M
are the load masses. For the multilayer piezoactuator from the piezoceramics type PZT at
33
d
= 4 × 10−10 m/V,
n
= 8,
m
U
= 100 V,
1
M
= 1.5 kg and
2
M
= 6 kg, we obtain the static
displacements of the faces
 
1
= 256 nm,
 
2
= 64 nm, and
   
 21
= 320 nm.
Let us consider a static characteristic multilayer piezoactuator with one fixed face at the
longitudinal piezoeffect with the voltage control. In Figure 8, we have maximum displacement
m2

for
0
2F
and maximum force
m
F2
for
0
2
in the following form:
mm nUd332
,
(41)

Actuators 2019, 8, 52 12 of 15

Em
ms
SUd
F
33
033
2
.
(42)
For the voltage-controlled multilayer piezoactuator at the longitudinal piezoeffect from the
piezoceramics type PZT with one fixed face at
33
d
= 4 × 10−10 m/V,

= 6 × 10−4 m,
n
= 40,
0
S
= 1.8
× 10−4 m2,
E
s33
= 3 × 10−11 m2/N, and
m
U
= 150 V, we obtain the Figure 8 values of maximum
displacement
m2
ξ
= 2.4 μm and maximum force
m
F2
= 600 N. The measurements were made on a
Universal testing machine UMM-5, Russia in the range of working loads under mechanical stresses
in the multilayer piezoactuator up to 100 MPa. The discrepancy between the experimental data and
the calculation results is 5%.
Figure 8. Static characteristic multilayer piezoactuator for longitudinal piezoeffect.
From Equation (38), we have the transfer function with lumped parameters of the multilayer
piezoactuator for the longitudinal piezoeffect at the voltage control and the elastic-inertial load in the
following form:
   
 
   
121 22
33
33
2



pTpTCC
nd
pU
p
pW
ttt
E
e
,
 
E
et CCMT 332
,
 
 






 E
e
EE
tCCMcCn 33233
23
,
(43)
where
 
p
2

and
 
pU
are the Laplace transforms of the displacement face and the voltage,
t
T
and
t

are the time constant and the damping coefficient, and
 
 nsSC EE 33033
is the rigidity piezoeffect
for
constE
.
Therefore, we obtain the static displacement of the multilayer piezoactuator in the following
form:
E
e
m
mCC
nUd
33
33
21

,
(44)
where
m2

is the steady-state value of the displacement face and
m
U
is the amplitude of the voltage.
From Equation (43), the expression for the transient response of the voltage-controlled
piezoactuator for the elastic-inertial load under the longitudinal piezoeffect is determined in the
following form:
     


















 ttt
tt
mt
Tt
et sin11 2
2
,
ttt T
2
1
,





 tt
t2
1arctg
.
(45)

Actuators 2019, 8, 52 13 of 15
For the voltage-controlled multilayer piezoactuator from the piezoceramics type PZT with one
fixed face for longitudinal piezoeffect and elastic-inertial load at
33
d
= 4 × 10−10 m/V,
n
= 10,
m
U
=
60 V,
M
= 9 kg,
E
C33
= 3 × 107 N/m, and
e
C
= 0.6 × 107 N/m, the steady-state value of the
displacement face
m2
ξ
= 200 nm and the time constant
t
T
= 0.5 × 10−3 s are obtained.
4. Results and Discussion
We obtained the generalized structural-parametric model, the generalized parametric structural
schematic diagram, and the matrix equation of the multilayer electromagnetoelastic actuator from
the equation of the force, that causes deformation, and the matrix equation of the equivalent
quadripole of the multilayer actuator, as well as the equations of the forces on its faces. From the
generalized structural-parametric model of the multilayer electromagnetoelastic actuator, we
obtained the structural-parametric models of the multilayer piezoelectric or magnetostrictive
actuators.
The decision matrix equation of the equivalent quadripole of the multilayer
electromagnetoelastic actuator was used. We derived the generalized matrix for the equivalent
quadripole of the multilayer electromagnetoelastic actuator.
We obtained the structural schematic diagrams of the multilayer electromagnetoelastic actuator
and the multilayer piezoactuator in contrast to the electrical equivalent circuits of the
piezotransmitter and piezoreceiver, the vibration piezomotor.
The structural-parametric model and the parametric structural schematic diagrams of the
multilayer piezoactuator for the longitudinal, transverse, and shift piezoelectric effects were
determined from its structural-parametric models. The static and dynamic characteristics of the
multilayer piezoactuator were constructed for nanomechanics.
5. Conclusions
In this work, we obtained the generalized structural-parametric model of the multilayer
electromagnetoelastic actuator from the equation of the force that causes deformation, the system of
the equations for the equivalent quadripole of the multilayer actuator, and the equations of the forces
on its faces. We obtained the generalized matrix for the equivalent quadripole of the multilayer
electromagnetoelastic actuator and the matrixes for the equivalent quadripole of the multilayer
piezoelectric or magnetostrictive actuators.
We also determined the generalized parametric structural schematic diagram and the
generalized matrix transfer function of the multilayer electromagnetoelastic actuator and, from this,
the generalized structural-parametric model. We derived the structural schematic diagrams of the
multilayer electromagnetoelastic actuator and the multilayer piezoactuator in contrast to the
electrical equivalent circuits of the piezotransducer, the vibration piezomotor.
Finally, we determined the parametric structural schematic diagram and the matrix transfer
function of the multilayer piezoactuator for the transverse, longitudinal, and shift piezoelectric effects
for nanomechanics. From the matrix transfer function of the multilayer piezoactuator, we obtained
the static and dynamic characteristics of the multilayer piezoactuator.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
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