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Journal of Pharmacy
and Pharmaceutics ISSN: 2377-1313
OPEN ACCESS
Structural-Parametric Model of Electro Elastic
Actuator for Nanotechnology and Biotechnology
Afonin SM*
Abstract:
The generalized parametric structural schematic diagram, the generalized matrix equation for the electro elastic ac-
tuator for the nanotechnology and the biotechnology are obtained. The deformations of the electro elastic actuator are
described by the matrix equation. The structural-parametric model and the parametric structural sсhematic diagram of
the electro elastic actuator or the piezoactuator are determined in contrast the electrical equivalent circuit types Cady
or Mason for the calculation of the piezoelectric transmitter and receiver, the vibration piezomotor with the mechanical
parameters in form the velocity and the pressure. The parametric structural schematic diagram of electro elastic actuator
is obtained with the mechanical parameters the displacement and the force. The transfer functions of the electro elastic
actuator are determined. The method of mathematical physics is used.
Keywords: Electro elastic actuator; Piezoactuator; Structural-parametric model; Parametric structural
schematic diagram; Transfer function.
Department of Intellectual Technical Systems, 124498, Moscow, Russia
*Corresponding author: Afonin SM, National Research University of Electronic Technology (MIET), Department of Intellectual Technical
Systems, 124498, Moscow, Russia, E-mail: learner01@mail.ru
Citation: Afonin, S.M. Structural-Parametric Model of Electro
Elastic Actuator for Nanotechnology and Biotechnology. (2018)
J Pharm Pharmaceutics 5(1): 8-12.
Received date: May 4, 2018
Accepted date: May 23, 2018
Publish date: May 28, 2018
Introduction
The electro elastic actuator for the nanotechnology
and the biotechnology is used in the scanning tunneling micro-
scopes, the scanning force microscopes, the atomic force micro-
scopes[1-19].
The nano- and micromanipulators with the electro elas-
tic actuators are part of the equipment of the precision engineer-
ing, the adaptive optics and the medical surgical manipulators.
The piezoactuators are used in the medical technology for the
accurate instrument delivery, in the laser equipment for the con-
trolling radiation power and the beam divergence in the optical
beam scanning systems[1-8,14].
The piezoactuators are used for the nanodisplacement
in the spectrometers, the tunnel microscopes, the interferome-
ters, where the high positioning accuracy and the parallelism of
the elements displacement are required simultaneously with the
small ranges of the displacements.
As the result of the joint solution of the wave equation
of the electro elastic actuator with the Laplace transform, the
equation of the electro elasticity and the boundary conditions
on the two loaded working surfaces of the electro elastic actu-
ator, we obtain the corresponding structural-parametric model
and the parametric structural schematic diagram of the electro
elastic actuator
The parametric structural sсhematic diagram of the
electro elastic actuator on the piezoelectric, piezomagnetic, elec-
trostriction effects, for example, the piezoactuator is determined
in contrast electrical equivalent circuit types Cady or Mason for
the calculation of the piezotransmitter and piezoreceiver, the
Short commentary DOI: 10.15436/2377-1313.18.1881
page no:8
Copyright: © 2018 Afonin, S.M. This is an Open access article
distributed under the terms of Creative Commons Attribution 4.0
International License.
vibration piezomotor with the mechanical parametres in form
the velocity and the pressure[1-8]. The parametric structural sche-
matic diagram of the actuator is obtained with the mechanical
parameters the displacement and the force.
The method of mathematical physics is applied for the
solution of the wave equation of the electro elastic actuator for
the nanotechnology and biotechnology with using the Laplace
transform for the construction the parametric structural schemat-
ic diagram of electro elastic actuator.
The parametric structural schematic diagram and the
matrix transfer functions of the electro elastic actuator for the
nanotechnology and biotechnology are obtained from the struc-
tural-parametric model of the electro elastic actuator with the
mechanical parameters the displacement and the force. The
parametric structural schematic diagrams of the voltage-con-
trolled or current-controlled piezoactuator are determined from
the generalized structural-parametric model of the actuator.
page no: 9
Short title:
Electro Elastic Actuator for Nanotechnology and Biotechnology
Afonin, S.M.
Parametric structural schematic diagram
For the determination of the structural-parametric mod-
el and the parametric structural schematic diagram of the elec-
troelastic actuator let us consider the generalized equation of the
electroelasticity[8,11,18] in the form
( ) ( )
,
i mi m ij j
S d t s T xt
Y
=Y+
(1)
where
( )
,/
i
S xt x
x
=∂∂
is the relative displacement along axis i of the
cross section of the piezoactuator or the piezoplate,
{,
m mm
EDY=
is the control parameter E for the voltage control, D for the cur-
rent control along axis m, Tj is the mechanical stress along axis j,
dmi is the coefcient of electro elasticity, for example, the piezo-
module,
Y
ij
s
is the elastic compliance for the control parameter
Y = const, the indexes i, j = 1, 2, … , 6; m = 1, 2, 3.
The main size is determined us the working length for
the electro elastic actuator or the piezoactuator in form the thick-
ness, the height and the width for the longitudinal, transverse
and shift piezo effect.
For the construction the parametric structural schemat-
ic diagram of electro elastic actuator in nanotechnology is used
the wave equation[8,11,18] for the wave propagation in a long line
with damping but without distortions.
With using Laplace transform is obtained the linear or-
dinary second-order differential equation with the parameter p.
correspondingly the original problem for the partial differential
equation of hyperbolic type using the Laplace transform is re-
duced to the simpler problem[8,11,12] for the linear ordinary differ-
ential equation
(2)
with its solution
( )
,
xx
x p Ce Be
gg
−
X=+
(3)
where X(x,p) is the Laplace transform of the displacement of
section of the actuator, g = p/cY + a is the propagation coef-
cient, cY is the sound speed for the control parameter Y = const,
a is the damping coefcient, C and B are constants.
The generalized structural-parametric model and the
generalized parametric structural schematic diagram of the elec-
tro elastic actuator the nanotechnology and the biotechnology
on Figure 1 are determined, using equation of the electro elastic
actuator (1) and the linear ordinary differential equation (2), the
boundary conditions on loaded faces and the strains along the
axes, in the following form
(4)
where
vmi is the coefcient of the electro elasticity, for example, dmi is
the piezomodule for the voltage-controlled piezoactuator, gmi is
the piezomodule for the current-controlled piezoactuator, S0 is
the cross section area and M1, M2 are the displaced mass on the
faces of the electro elastic actuator, X1(p), X2(p) and F1(p), F2(p)
are the Laplace transform of the displacements and the forces on
the faces of the electro elastic actuator, Ym is the control param-
eter of the electro elastic actuator.
Figure 1: Generalized parametric structural schematic diagram of elec-
tro elastic actuator in nanotechnology.
The matrix state equations[2,11,14] for the piezoelectric effect have
the form
(5)
(S) = (sE) (T) + (d)t (E) (6)
where the rst equation describes the direct piezoeffect, the
second equation presents the inverse piezoeffect, (D) is the col-
umn - matrix of the electric induction along the coordinate axes,
(S) is the column-matrix of the relative deformations, (T) is the
column - matrix of the mechanical stresses, (E) is the column -
matrix of the electric eld strength along the coordinate axes,
(d)t is the transposed matrix of the piezoelectric modules, (sE)
is the elastic compliance matrix, is the matrix of dielectric
constants.
The deformation of the piezoactuator corresponds to its
stressed state. If the mechanical stress T are created in the piezo-
actuator, the deformations S are formed in the piezoactuator[8,11].
There are the six stress components T1, T2, T3, T4 , T5,
T6. The components T1-T3 are dened to extension-compression
stresses, the components T4-T6 are related to shear stresses.
Let us consider the transverse piezoelectric effect in the
piezoactuator. The equation of the inverse transverse piezoef-
fect[8,11] in the piezoactuator can be written in the following form
( ) ( )
1 31 3 11 1 ,
E
S d E t sT xt= +
(7)
where
( )
1
,xt
Sx
x
∂
=∂
is the relative displacement of the cross sec-
tion of the piezoactuator along axis 1, d31 is the piezoelectric
module for the transverse piezoeffect, is the elastic com-
pliance for E = const along axis 1, T1 is the stress along axis 1.
The solution of the linear ordinary second-order differ-
ential equation with the parameter p (2) can be written as (3) and
subject to the conditions.
page no: 10
Citation: Afonin, S.M. Structural-Parametric Model of Electro Elastic Actuator for Nanotechnology and Biotechnology. (2018) J Pharm Pharma-
ceutics 5(1): 8-12.
www.ommegaonline.org
X (0,p) = X1(p) For x = 0, (8)
X (h,p) = X2(p) For x = h
Therefore the constants C and B for the solution we obtain in the
following form
(9)
Then, the solution of the linear ordinary second-order differen-
tial equation (2) in form (3) can be written as
(10)
The equations of forces acting on the faces of the piezoactuator
has the form
T1(0,p) S0 = F1(p) + M1p2 X1(p) For x = 0 (11)
T1(h,p) S0 = -F2(p) - M2p2 X2(p) For x = h
where T1(0,p) and T1(h,p) are determined from the equation of
the inverse transverse piezoeffect. Therefore we obtain the sys-
tem of the equations for the mechanical stresses at the faces of
the actuator for the transverse piezoeffect in the form
(12)
The set of equations (12) for mechanical stresses in pie-
zoactuator yields the following set of equations describing the
structural parametric model and parametric structural schematic
diagram of the voltage-controlled piezoactuator for the trans-
verse piezoelectric effect on Figure 2.
(13)
where
11
11
0
E
E
s
s
χ
=
, l=h.
Figure 2: Parametric structural schematic diagram of voltage-con-
trolled piezoactuator for transverse piezoelectric effect.
The parametric structural schematic diagrams of the
voltage-controlled or current-controlled piezoactuator for the
transverse, longitudinal, shift piezoelectric effects are deter-
mined from the generalized structural-parametric model of the
electro elastic actuator.
Transfer functions
From generalized structural-parametric model of the
electro elastic actuator, taking into account the generalized equa-
tion of the electro elasticity, wave equation and the equation of
the forces on its faces, we obtain the transfer functions of the
electro elastic actuator. Correspondingly the Laplace transforms
of displacements for two faces of the actuator are dependent
from the Laplace transforms of the general parameter of control
and forces on two faces
X1(p) = W11(p) Ym (p) + W12(p)F1(p) + W13(p) F2(p) (14)
X2(p) = W21(p) Ym (p) + W22(p)F1(p) + W23(p) F2(p)
Matrix equation of the Laplace transforms of the dis-
placements with the matrix transfer functions of the electro elas-
tic actuator is obtained[8,14,18] in the form
11 12 13
1
1
21 22 23
2
2
()
() () ()
() ()
() () ()
() ()
m
p
WpWpWp
pFp
WpWpWp
pFp
Y
X
=
X
(15)
Let us consider the displacements the faces of the volt-
age-controlled the piezo actuator for the transverse piezoeffect
Ym = E3 with the output parameter displacement. The transfer
functions of the voltage-controlled the piezoactuator for the
transverse piezoeffect can be written in the form
The static displacement of the faces the electro elastic actuator
1
()
x
∞
and
2
()
x
∞
can be written in the form
( )
( )
( )
02
11
12
01
22
12
1 2 12 0
2
( ) lim ( )
2
( ) lim ( )
( ) ( ) lim () ()
mi m
t
mi m
t
mi m
t
vl M m
tMMm
vl M m
tMMm
t t vl
xx
xx
x x xx
→∞
→∞
→∞
Y+
∞= = ++
Y+
∞= = ++
∞+ ∞= + = Y
(16)
(17)
(18)
where m is the mass actuator, M1, M2 are the load masses.
Let us consider example at m<<M1 and m<<M2 for the
voltage-controlled the piezoactuator from PZT under the longi-
tudinal piezoeffect d33 = 4·10-10 m/V, U = 150 V, M1 = 1 kg and
M2 = 4 kg we obtain the static displacements of the faces of the
piezo actuator
1
()
x
∞
= 48 nm,
2
()
x
∞
= 12 nm,
1
()
x
∞
+
2
()
x
∞
= 60 nm.
page no: 11
Short title:
Electro Elastic Actuator for Nanotechnology and Biotechnology
Afonin, S.M.
The static displacements of the faces for the volt-
age-controlled the piezoactuator for the transverse piezoeffect
are obtained from (15) at m<<M1 and m<<M2 in the form
( )
( )
11 0 31 0 2
1012
0
21 0 31 0 1
2012
0
()
( ) lim
()
( ) lim
p
p
pW p U d hU M
p MM
pW p U d hU M
p MM
a
a
xdd
xdd
→
→
→
→
∞= = +
∞= = +
(19)
(20)
For the voltage-controlled the piezoactuator from PZT
under the transverse piezoeffect at m << M1 and m << M2, d31
= 2·10-10 m/V, h/d = 20, U = 100 V, M1 = 2 kg and M2 = 8 kg
the static displacements of the faces the piezoactuator are deter-
mined
1
()
x
∞
= 320 nm,
2
()
x
∞
= 80 nm,
1
()
x
∞
+
2
()
x
∞
= 400 nm.
The transfer function of the voltage-controlled trans-
verse piezoactuator is obtained from (15) for elastic-inertial load
at M1→ ∞ , m << M2 and the approximation the hyperbolic co-
tangent by two terms of the power series in the form
( )( )
( ) ( )
()
31
2
22
11
2
2 11 11 2 11
()
() () 1 21
,3
E
e t tt
E EE E
t et e
dh
p
Wp Up C C Tp T p
T M CC hC cMCC
d
x
xa
X
= = + ++
= += +
(21)
where the Laplace is transform of the voltage, Tt is the time
constant and xt is the damping coefcient of the piezoactuator.
Therefore the expression for the transient response of the volt-
age-controlled transverse piezoactuator is determined in the fol-
lowing form
(22)
where xm is the steady-state value of displacement for the volt-
age-controlled piezoactuator, Um is the amplitude of the voltage
in the steady-state. For
1
M→∞
, m<<M2· Um = 50 V, d31= 2·10-10
m/V, h/d = 20, M2 = 4 kg,
7
11
2.4 10
E
C= ⋅
H/m,
7
0.1 10
e
C= ⋅
H/m we ob-
tain values the steady-state value of displacement and the time
constant of the actuator xm = 192 nm, Tt = 0.4·10-3 c.
The matrix transfer functions of the actuator are deter-
mined for control systems with the electro elastic actuator for the
nanotechnology and the biotechnology.
In this work the generalized parametric structural sche-
matic diagram and generalized structural-parametric model of
the electro elastic actuator are obtained. From generalized struc-
tural-parametric model of the electro elastic actuator after alge-
braic transformations the transfer functions of the electro elastic
actuator are determined. The parametric structural schematic
diagrams, the structural-parametric models of the piezoactuator
for the transverse, longitudinal, shift piezoelectric effects are
determined from the generalized structural-parametric model of
the electro elastic actuator for the nanotechnology and the bio-
technology.
Conclusions
For the nanotechnology and the biotechnology the gen-
eralized parametric structural schematic diagram and the gener-
alized structural-parametric model of the electro elastic actuator
are constructed with the mechanical parameters the displace-
ment and the force.
The parametric structural schematic diagrams of the
piezoactuator for the transverse, longitudinal, shift piezoelectric
effects are determined.
The matrix transfer functions of the electro elastic actu-
ator are determined in the control systems for the nanotechnolo-
gy and the biotechnology.
page no: 12
Citation: Afonin, S.M. Structural-Parametric Model of Electro Elastic Actuator for Nanotechnology and Biotechnology. (2018) J Pharm Pharma-
ceutics 5(1): 8-12.
www.ommegaonline.org
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