Download THE MODELLING OF THE ELECTROMECHANICAL MULTILAYER

THE MODELLING OF THE ELECTROMECHANICAL MULTILAYER
PIEZOCERAMIC MICROACTUATORS
Authors : Dr.Eng. Mircea IGNAT Senior Researcher INCDIE CA ,Dep.1/10
Eng. George ZARNESCU Junior Researcher, INCDIE CA ,Dep 1/10
Student Sebastian SOLDAN INCDIE CA ,Dep.1/10
Abstract:
In this paper ,the modelling of a new class of microelectromechanical piezoceramic
actuators (which represent the important application of the inverse piezoelectric effect) ;
multilayer actuators are presented.
the theoretical and experimental relationship between the piezoceramic ratio and
parameters and design elements of piezoceramic micromotors and microactuators
have been developed.
The ultrasonic microactuators advantages include [1] :
- both high torque and high efficiency at low speed;
- the possibility of direct drive;
-ability to maintain its own position;
- good control characteristics at start and stop;
- simple structure and flexibility of shape;
- no restriction through induction (e.g. electromagnetic);
- silent operation.
Essentially a theoretical commentary on the ceramic properties (used in fabrication of
piezoceramic microactuators ) is presented ; dielectric constant ;  31 ,  33 , piezoelectric
parameters ; d 31 , d 33 , compliance, dielectric loss and mechanical coupling ratio.
The most important equation regarding electrical parameters:
D   T  E  d T
(1)
The most important equation regarding mechanical parameters:
u  s T  d  E
(2)
Where u is the displacement, T is the mechanical stress, D is the electric induction and
E is the intensity of electric field.
The theoretical and experimental relationship between the piezoceramic ratio and
parameters and design elements of piezoceramic micromotors and microactuators
have been developed.
The microactuator displacement as a function of force and voltage will be therefore :
 hs E 
 Fpiezo  dU
(3)
u  
 A 
where ; h -is the height of the piezoceramic element, A -the area of the piezoceramic
actuator, s - the elastic compliance, d is the piezoelectric constant, U - the applied
voltage.
The modelling estimation of design microactuators characteristics ; displacement
,force ,torque as a function of input; voltage and piezoceramic properties has been
obtained in a systematic approach from proposed method in this study.
It is presented the case study of microelectromechanical multilayer actuator with the
modelling of the main parameters : microforce and microdisplacent(an example in fig.1).
.Fig.1.D representation of the linear actuator force function by the piezoceramic
ratios; d and s
References
[1] S.Ueha,Y.Tomikawa,M.Kurosawa,N.Nakamura,”Ultrasonic Motors.Theory and
Applications”,Clarendon Press Oxford ,1993.
[2] P.Hagedorn, W.Seemann, “Problems in the Mathematical Modelling of Ultrasonic
Motors “ , in “Modern methods of analytical mechanics and their applications” edited by
V.V. Rumyantsev, A.I.V.Karapetyan ,Springer ,Wien,New York ,1998.
[3]E.C.N.Silva ,S.Nishiwaki, N.Kikuchi,”Topology Design of Flextensional Actuators”,
IEEE Transactions, Ferroelectrics and Frequency Control,vol.47,no.3, may 2000,p.657671.
Proposal topics: Coupled problems. - Poster.
INCDIE CA - National Research Institute of Electrical Engineering –Advanced
Researche.
Splaiul Unirii 313 ,sector 3,Bucharest ROMANIA
Tel.346.72.31, 3467235.
Email : [email protected], [email protected].
1.INTRODUCTION
With the development of new piezoceramic materials a basis for the realization of efficient new
types of piezoelectric micromotors and microactuators was established[1,2]. The functional
principle of these piezoelectric microactuators and micromotors is totally different from that of
conventional electromagnetic actuators and motors.
In electromagnetic conventional motors and actuators the forces and mechanical moments
are generated by magnetic field whereas in piezoelectric microactuators and micromotors ,
piezoceramic elements are used for energy transformation and to obtain high frequency
vibrations for a relative motion between the different parts of the system and different working
principles are possible.
A classification of piezoelectric linear micromotors and microactuators by vibrational modes is
[1]:
-A. Non-self moving :
A1. Traveling –wave using flexural wave (by Sashida).
A2.Driven by layered piezoelectric ceramics.
-B.Self –moving .
B1. Hybrid transducer with longitudinal vibrator .
B2.Hybrid transducer –type motor using flexural vibrator.
B3.  -shaped transducer.
B4.Flexural-flexural double –mode bar.
B5. Mode annular plate actuator.
The ultrasonic micromotors and microactuators advantages include [1] :
- both high torque and high efficiency at low speed;
- the possibility of direct drive;
-ability to maintain its own position;
- good control characteristics at start and stop;
- simple structure and flexibility of shape;
- no restriction through induction (e.g. electromagnetic);
- silent operation.
Its disadvantages include ;
- need for electrical source of high frequency and voltage;
- inevitable wear and tear of the contact surfaces;
- expensive electrical source and piezoelectric elements.
Two important goals in the design of piezoceramic micromotors and microactuators are output
parameters [2,3,4] ;
- Displacement linear or angular,
Force or torque (blocking force or torque).
The main problems of the piezoelectric micromotors and microactuators researche and design
have the following formulations ;
I. The determination of the force or the torque of the piezoelectric motor or actuator from the
given properties of motion (it is adapted the inverse problem of dynamics [8])
II.Given (to see Figure 1);
- the topology of the active piezoelectric micromotor or microactuator ;  m and boundary o ,
- the topology of the domains of mechanical forces or torques ;  t1,  t 2 ,
- the topology of the boundaries where is applied the voltage; t1 , t 2 , s ,
- the properties of the active piezomaterial ; piezoelectric, dielectric ,mechanic and
electromechanic ratios and constants,
to determine the micromotor or actuator torque(F,T) ,angular speed or force and linear speed.
 t 2 , t 2
F
T
0
 t1 , t1 ,
m
F
T
s
Fig.1 Referring to the main problem formulates of piezoelectric
micromotors and microactuators.
The relations between the different aspect of piezoelectric micromotors and microactuators
design field are presented in Figure 2.
A.Piezomaterial constant and
ratios
FORCES
TORQUES
B.Topologies of the micromotors
and microactuators
C.Kinematic elements of the
motion .
LINEAR
MICRODISPLACEMENTS
ANGULAR
MICRODISPLACEMENTS
Fig.2 Relation necessaries for piezoelectric micromotors and actuators problem
formulations.
Because the important dependence of the main micromotors or microactuatos parameters of the
piezoceramic characteristics the last is the essential problem formulation on design.
The force and torque are function of different piezoceramic parameters,by exemple :
F (  ,  ,  , d , g , e, s ),
(1)
T (  ,  ,  , d , g , e, s)
where ;   mass density, Y -Young modulus,  -Poisson modulus, d - piezoelectric distortion
constant, g - voltage output coefficient, s - compliance,  r
piezoceramic material .
- relative permittivity for
The dependence of the piezoelectric properties material is more important comparative with the
conventional electromagnetic machines and actuators.
2.PIEZOELECTRIC COEFFICIENTS [6,11,12,13]
The most important equation regarding electrical parameters:
D   T  E  d T
(2)
The most important equation regarding mechanical parameters:
u  s T  d  E
(3)
Where u is the displacement, T is the mechanical stress, D is the electric induction and E is the
intensity of electric field.
We can consider the case in which beside the electric exterior applied field we apply an
external mechanical stress, it is a cumulative effect for material polarization, we have a
polarization produced by electrical field over which is added the polarization produced by
external mechanical stress, in the second case we have the same cumulative effect for
displacement.
The relation between absolute and relative permittivity for piezoceramic material is, these
are tensors because we have an anisotropic dielectric medium:
 T   0   ij ,
(4)
in which  ij   ji , if cartesian axes (x, y, z) are the same with main polarization axes of crystal,
we have only three relative dielectric constants for each axis  11 ,  22 ,  33 .
In piezoceramic material catalogues generally it is given  11 ,  33 , or the ratio
 1T  3T
, which
,
0 0
represents the relative permittivity, each piezoceramic crystal have a specific type of symmetry
therefore we’ll have less coefficients and some of these coefficients we’ll be equal.
This is the equation for the direct piezoelectric effect in which we encounter the voltage
output coefficient, the other equation of polarization for direct piezoelectric effect is already
integrated in the general equation for electrical parameters.
E  g  T
(5)
Piezoelectric tensors g and d have the same 3 rank.
Most important coefficients for the piezoelectric effect are:
d - piezoelectric distortion constant measured in [C/N] or [m/V], it is the distortion resulting
from the application of an uniform electric field without mechanical stress.
g - voltage output coefficient [V*m/N], referring to the electrical field strength for an uniform
applied stress with no electrical effect for displacement.
s - compliance is the inverse of Young’s modulus Y, it is an elasticity constant measured in
[m2/N].
We can establish a scalar link between these coefficients:
d T g
(6)
Equations above can be written in a matrix form for the principal axes of polarization, P
represents the polarization for all three axes.
 Px   11 0
P    0 
22
 y 
 Pz   0
0
0  Ex 
0    E y 
 33   E z 
(7)
For example Barium Titanate (BaTiO3) has a tetragonal crystalline structure, this category of
materials are most utilized.
We can observe the matrix of piezoelectric distortion coefficient for the direct
piezoelectric effect in BaTiO3 case.
Px   0
  
P y    0
  d
 P z   31
0
0
0
d15
0
0
d15
0
d31
d33
0
0
T 1 
 
T 2 
0  
T3
0    
T 4 
0   
T 5 
 
T 6 
(8)
The general relationship between mechanical stress and deformation deduced from Hook’s law,
from the theory of elasticity:
u  s T
(9)
The matrix of compliances or the compliance tensor for BaTiO3 can be written:
s11
s12
s13
0
0
0
s12
s11
s13
0
0
0
s13
s13
s 33
0
0
0
0
0
0
s 44
0
0
0
0
0
0
s 44
0
0
0
0
0
0
s 66
(10)
3. THEORETICAL ASPECTS
The determination problem of optimum dimension of torque and force of piezoelectric
micromotors and microactuator can be a problem of maximum (extremum) determination of a
multi – valued function [9,10].
Either force function ; F ( x1 , x 2 ,....x p ) and torque function ; T  T ( x1 , x 2 ,....x p ) with
X pm  R p ( X pm is the multitude of the material properties with the following identifications;
x1   , x2  Y , x3   , x4  d , x5  s,...) , the force and torque function have three partial
derivative on X pm and either (a1 , a 2 ,...a p ) , (b1 , b2 ,...b p ) the solution of the equation systems:
F
F
F
 0,
 0,...,
0
x1
x2
x p
T
T
T
 0,
 0,...,
0
x1
x2
x p
(a1 , a 2 ,...a p )
(11)
(b1 , b2 ,...b p )
(12)
a.If the numbers :
 1  A11  0,  2 
where ; Aij 
A11
A12
A21
A22
 0,...,  p 
F (a1 , a 2 ,...a p )
xi x j
or
Aij 
A11
A12
...
A21
A22
... A2 p
...
...
...
A p1
Ap 2
... A pp
T (b1 , b2 ,...b p )
xi x j
A1 p
...
0
,then
(13)
the function F, T have a
minimum in the points ; (a1 , a 2 ,...a p ) , (b1 , b2 ,...b p ) .
b.If the numbers :
*1   A11  0, *2 
A11
A12
A21
A22
 0,..., *p  (1) p
A11
A12
...
A1 p
A21
A22
... A2 p
...
...
...
A p1
Ap 2
... A pp
...
0
(14)
then the functions F, T have a maximum in the points; (a1 , a 2 ,...a p ) , (b1 , b2 ,...b p ) .
An important observation ; in majority the application which involve the piezoelectric
micromotors or microactuators impose a maximum torque or force (b. condition ).
But between the piezoceramic constant and ratios are bond relations (to see relation (3) between
d and s) and the number of variables are reduced.
An other determination properties piezomaterial( constant and ratios) procedure is to explicit
the constant piezomaterial or piezomaterial function the force (or torque) imposed by the design
theme formulation .
Either the force or torque function of a piezoelectric actuator or motor, with the following
representation :
F  K s f cp ( x)U
T  K s f cp ( x)U
(15)
where; K s  ratio of the dimensional parameters of the motor or actuator, U  supply voltage
and
f cp (x) - function of piezomaterial ratio or constants.Then we explicite and analyze this
function ;
f cp ( x) 
F
K sU
T
f cp ( x) 
K sU
(16)
4.RESULTS AND DISCUSSION
1.The displacement and force of a linear microactuator (Fig. 3 ).
Fig. 3 The main element ; displacement and force of a linear actuation
The microactuator displacement as a function of force and voltage will be therefore :
 hs E 
 Fpiezo  dU
(17)
u  
 A 
where ; h -is the height of the piezoceramic element, A -the area of the piezoceramic actuator, s the elastic compliance, d is the piezoelectric constant, U - the applied voltage.
The force as a function of displacement and voltage will be :
F piezo 
A
 u  dU 
hs E
(18)
Fig.4.3D representation of the linear actuator force function by the piezoceramic ratios; d and s
If in relation (18) we introduce relation (3) between the compliance and distorsion coefficient and
explicite the distorsion function by compliance the force relation becomes :
Fpiezo 
AT (u  dU )
h(u  dE )
(19)
and
Fpiezo
AT (Uu  u  dU  UdE)
d
h(u  dE ) 2
Fpiezo
and by
 0 results an extremum value for distorsion coefficient;
d
u (1  U )
d
U (1  E )

(20)
(21)
In Fig.4 is a 3D representation where is evidentied for design maximum force the sensible
subdomain of the compliance ;
distorsion
s  [0...1]  10 11[m 2 / N ] and
; d  [0,5...5]  10 11[m / V ]
with
an
interesting
inflexion
point
placed
on
11
d  [1...2]10 [m / V ] .
2.Bimorph actuators.
Bimorph actuation (Fig. 5) consist of two independent flat piezoelectric, stacked on top of each
other.By driving one element to expand while contracting the other one, the actuator is forced to
bend, creating an out of plane motion.
2
3 L
d 31   U
4 t
2tw d 31
Fpiez 
U
L s11E
u
(22)
(23)
Fig.5 The structure of a piezoceramic bimorph actuator with dimensional representations.
Either the ratio of the dimensional parameters and function of piezomaterial ratio or constants;
Ks 
d
2tw
, f cp  31
L
s11E
(24)
with the explicit function of piezomaterial :
f cp ( x) 
F
K sU
(25)
If the dimensional initial parameters are t  0,5mm, w  5mm, L  20 and the imposed design
parameters are ; ,U  100V , F  100 N we find;
3
d 31  4  10 3 s11
and extremum relations.But for compliance –the imposed distorsion domain
E
maxim ratio is : d 31 / / s11

5  10 10
 50 with the possible d 31  50  s11E  4000s11E ideal
11
1  10
domain. How the force and voltage are imposed parameters result the design condition :
Ks 
F 1 100 1



 0,02 and is necessary an other iterative dimensional calculation.
U f cp 100 50
Fig.6 3D representation of the bimorph actuator force function by the
ratios; d and s .
piezoceramic
Fig.7 3D representation of the bimorph actuator displacement function by the piezoceramic
coefficient d and voltage U.
In Fig.6 is shown a 3D representation of the specific force bimorph actuator on an imposed 2D
domain d  s.
3.The force source of a cantilever beam piezoceramic actuator [7] has the value :
Fpiezo  2d13YwU
(26)
cm
] is the piezoelectric constant , Y - is the Young modulus, w the width of the
V
piezoceramic element (lamellar geometry) , U - the applied voltage .
The displacement, x a , of the microactuator can be estimated imposing conditions of static
where ; d 13 [
equilibrium on the model shown in Fig. 8. Balancing the force at x a yields :
FA  k bd ( x a  x piezo )
(27)
with x a :
 k bd  k pz  F piezo
x a  Fa 
(28)

 k bd k pz  k pz
where k pz -the spring constant of the piezoceramic element and k bd - the micromechanical
connection with :
k pz 
2Ywt
L pz
(29)
where , t  thickness and L pz  length of piezoceramic element.
FA
k bd
xa
F piezo
k pz
x piezo
Fixed rigid support
Fig 8. Electromechanical model of a cantilever microactuator.
The force function in the case of cantilever beam actuator is a linear function without
distinguished commentaries ; this function has not three partial derivative and has not the
maximum or minimum point. A numerical representation is represented in Fig.9.
Fig.9 3D representation of the linear actuator displacement function by the
ratios; d and s .
piezoceramic
5.CONCLUSIONS
The authors propose an new approached procedure to find the optimum relation between the
piezomaterial properties, imposed parameters and dimensional parameters.This procedure
represents an initial segment in the general piezoceramic micromotor or microactuator design
algorithm.
References
[1]
S.Ueha,Y.Tomikawa,M.Kurosawa,N.Nakamura,”Ultrasonic
Motors.Theory
and
Applications”,Clarendon Press Oxford ,1993.
[2] P.Hagedorn, W.Seemann, “Problems in the Mathematical Modelling of Ultrasonic Motors “ ,
in “Modern methods of analytical mechanics and their applications” edited by V.V. Rumyantsev,
A.I.V.Karapetyan ,Springer ,Wien,New York ,1998.
[3]E.C.N.Silva ,S.Nishiwaki, N.Kikuchi,”Topology Design of Flextensional Actuators”,
IEEE Transactions, Ferroelectrics and Frequency Control,vol.47,no.3, may 2000,p.657-671.
[4] K.Ragulskis,R.Bansevicius,R.Barauskas,G.Kulveitis,”Vibromotors for precision microrobots”
,Hemisphere Publishing Corporation,1988.
[5] M.P.Mason “Physical Acoustics,Principles and Methods” vol.I,Part A,New York,Academic
Press,1964.
[6] W.G.Cady
“ Piezoelectricity.An introduction to the theory and applications of
electromechanical phenomena in crystals” vol.I,II, Dover Publications,New York 1964.
[7] W.B.Robbins,D.L.Polla, D.E.Glumac,”High –Displacement Piezoelectric Actuator Utilizing
a Meander-Line Geometry- Part I; Experimental Characterization,Part II- Theory” IEEE
Transactions on Ultrasonics,Ferroelectrics and Frequency Control,” vol.38, no.5,September
1991, pp.454-467.
[8] A.S.Galiulin ,“Inverse problems of dynamics”,MIR Publishers Moscow,1984.
[9] M.Rosculet , “ Analiza matematica” ,Ed.Didactica si pedagogica ,Buc.1967.
[10] M.Nicolescu, N.Dinculeanu, S.Marcus ,”Analiza matematica “vol.I ,II, Ed.Didactica si
pedagogica , Buc.1967.
[11] MURATA (Japonia) ,Piezoceramic catalogue -2005.
[12]FERROPERM(Danemarca), Piezoceramic catalogue , 2003
[13] PI (Germania), Piezoceramic catalogue ,2005.
[14] A.E.Glazounov,S.Wang,Q.M.Zhang,”Piezoelectric Stepper motor with Direct Coupling
Mechanism to Achieve High Efficiency and Precise Control of Motion” IEEE Transactions,
Ferroelectrics and Frequency Control,vol.17,no., July 2000,p.1059-1067.