The ring type piezoelectric actuator generating elliptical movement/Elipsiniam judesiui gauti ziedo formos pjezoelektrinis zadintuvas.
Bauriene, G. ; Mamcenko, J. ; Kulvietis, G. 等
1. Introduction
Piezoelectric actuators have advanced features if compared to
others and are widely used for different commercial applications [1-4].
The demand for new type displacement transducers that can achieve
high resolution and accuracy of the driving object increases nowadays
[5-7].
A lot of design and operating principles are investigated to
transform mechanical vibrations of piezoceramic elements into elliptical
movement of the contact zone of an actuator [8, 9].
Elliptical movement of piezoelectric actuators fall under two
types--rotary and linear. Rotary type actuators are one of the most
popular because of high torque at low speed, high holding torque, quick
response and simple construction. Linear type traveling wave actuators
feature these advantages as well but development of these actuators is a
complex problem [10-12].
Summarizing the following, the following types of piezoelectric
actuators can be specified: traveling wave, standing wave, hybrid
transducer, and multi-mode vibrations actuators. [7, 13, 14].
The ring shaped piezoelectric actuator generating elliptical
movement is presented and analyzed in this paper.
2. The influence of geometric parameters on domination coefficients
Usually, for numerical analysis of piezoactuators the software such
as ANSYS is used. By the algorithm of eigenvalue problem
eigenfrequencies for systems are sorted in the ascending order; thereby
the sequences of eigenforms change. This rule for sorting frequencies is
disadvantageous when numerical analysis of multidimensional
piezoactuators needs to be automated. This problem is also important for
optimization, since calculations are related both to eigenfrequencies
and eigenforms. If the eigenfrequency is chosen incorrectly, the
piezoactuator will not function, so it is very important to numerically
determine eigenforms and place them inside the eigenform matrix of the
construction model [15].
Calculation of eigenfrequencies and forms for a given construction
(Fig. 1) is proposed in this paper.
Then for the nth eigenfrequency the following sum can be formed:
[S.sup.n.sub.k] [r.summation over (i=1)] [([A.sup.n.sub.ik).sup.2],
r = l/k, (1)
where k is the number of degrees of freedom in a node, l is the
number of nodes (degrees of freedom) in the model, r is the size of the
form vector for the kth coordinate, [A.sup.n.sub.ik] is the value of the
eigenform vector for the ith element.
[FIGURE 1 OMITTED]
Then the ratio is formed:
[m.sup.n.sub.jk] = [S.sup.n.sub.j]/[S.sup.n.sub.k], j [not equal
to] k, (2)
where [m.sup.n.sub.jk] is the oscillation domination coefficient.
The sum [S.sup.n.sub.k] corresponds to the oscillation energy of the nth
eigenfrequency in the kth direction, and the ratio [m.sup.n.sub.jk] is
the ratio of oscillation energies of the nth eigenfrequency in the
coordinate directions of j and k.
These coefficients have to be called partial domination
coefficients since they estimate energy only in two coordinate
directions. The domination coefficients discussed above have the
following shortcomings:
Not normalized. Because of this the range of the domination
coefficients calculated varies from 0 to infinity.
In the case of three dimensions, six domination coefficients
result. Such a number of coefficients aggravate analysis.
To solve this problem the following algorithm is proposed: find the
sum of the amplitude squares of piezoactuator oscillations in all
directions of the degrees of freedom for a point, i.e., the full system
energy in all directions [16, 17]:
[S.sup.n.sub.k] = [r.summation over (i=1)]
[([A.sup.n.sub.ik]).sup.2], (3)
where n is the eigenfrequency for a system, k is the number of
degrees of freedom in a node, [A.sup.n.sub.ik] is the value of the
eigenform vector for the ith element.
Then the ratio is calculated [8]:
[m.sup.n.sub.j] = [S.sup.n.sub.j]/ [k.summation over (i=1)]
[S.sup.n.sub.i] (4)
where [m.sup.n.sub.j] is the oscillation domination coefficient
corresponds to the nth eigenform. The index j of domination coefficients
indicates in which direction the energy under investigation is the
largest. j can assume such values: 1 corresponds to the x coordinate, 2
- y, and 3 - z, etc. Having calculated domination coefficients in all
directions of degrees of freedom and having compared them to each other,
we can determine the dominant oscillation type. The domination
coefficients calculated according to formula (4) are normalized, so
their limits vary from 0 to 1. It is very convenient for analyzing the
influence of various parameters on domination coefficients.
To clearly determine the eigenform and its place in the eigenform
matrix of the construction model, it is not enough to calculate only the
oscillation domination coefficients. Domination coefficients only help
to differentiate eigenforms by dominating oscillations, for example,
radial, tangential, axial, etc.
Because of this an additional criterion is introduced into the
process of determining eigenform, individual for each eigenform, i.e.,
the number of nodal points or nodal lines for the form. That depends on
the dimensionality of the eigenform. During calculations the number of
nodal points of beam-like and two-dimensional piezoactuators is
determined by the number of sign changes in oscillation amplitude for
the full length of the piezoactuator in the directions of coordinate
axes.
Summarizing the algorithm for determining eigenforms of
piezoactuator oscillations (Fig. 1), we can note that it is composed of
two integral stages: calculating domination coefficients and determining
the number of nodal points or lines of the eigenform. This algorithm is
not tightly bound to multidimensional piezoactuators, so it can be
successfully applied in analysing oscillations of any constructions.
When solving the problems of piezoactuators dynamics for high precision
microrobots where repeated calculations with higher eigenfrequencies are
involved, it is proposed to modify the general algorithm introducing the
stage of determining eigenforms with the help of domination coefficients
[18].
3. Design and results of numerical modeling
Numerical modeling of piezoelectric actuator was performed to
validate actuator design and operating principle through the modal
analysis.
Modal analysis of piezoelectric actuator was performed to find
proper resonance frequency. Material damping was assumed in the finite
element model [19].
Finite element model software ANSYS 11.0 was employed for
simulation and finite element model was built.
Principle scheme of the analysed piezoelectric actuator is provided
in Fig. 2.
[FIGURE 2 OMITTED]
PZT-8 piezoceramics was used for the ring. The polarization vector
is directed along the width of the ring. The detailed properties of this
material are provided in Table 1.
Geometric parameters of the ring are chosen in such a way that the
eigenfrequency of the 2nd flexional form is as high as possible, since
in this way its rapidity is guaranteed.
The first iteration of calculations of piezoelectric actuator was
performed to find proper resonance frequency and in order to determine
the same eigenform of elliptical movement with diferent inner radius.
During analysis the ring dimensions have been changed. Geometric
parameter's proportions used in the finite element model for modal
analysis (Fig. 3) are provided in Table 2.
[FIGURE 3 OMITTED]
Domination coefficients and eigenfrequencies have been also
calculated, considering when crossply and rotative movement is the
optimal, e.g. geometrical parameters based on dominance coefficients
were optimized. It was examined at what frequency rotation of the ring
is the best and at what frequency it is the most flexible.
A more detailed analysis of domination coefficients (according to
which better flexibility was examined) is provided in Tables 3, 4 and
Figs. 4, 5.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
A more detailed analysis of model eigenfrequencies (by crossply and
rotative movements) is provided in Table 5 and Fig. 6.
FEM represented below reflect the best crossply and rotation
movement.
The 2nd iteration of calculations of piezoelectric actuator was
performed to find proper resonance frequency and in order determine the
same eigenform of elliptical movement with diferent outer radius.
[FIGURE 6 OMITTED]
During analysis in the 2nd iteration of calculations dimensions of
the ring have been changed. Geometric parameter's proportions used
in the finite element model for modal analysis (Fig. 7) are provided in
Table 6.
[FIGURE 7 OMITTED]
During analysis in the 2nd iteration domination coefficients and
eigenfrequencies have been also calculated, considering when crossply
and rotative movement is most optimal, namely were optimized geometrical
parameters based on domination coefficients. It was examined at what
frequency the rotation of the ring is the best and at what frequency it
is the most flexible.
A more detailed analysis of domination coefficients (according to
which better flexibility was examined) is provided in Tables 7, 8 and
Figures 8, 9.
[FIGURE 8 OMITTED]
[FIGURE 9 OMITTED]
A more detailed analysis of model eigenfrequencies (by crossply and
rotative movements) is provided in Table 9 and Fig. 10.
[FIGURE 10 OMITTED]
FEM represented below the best reflect the crossply and rotation
movement.
Having compared the influence of geometric parameters on domination
coefficients and eigenfrequencies in two iterations of the calculation,
it can be claimed that with the help of domination coefficients the
eigenform of elliptical rotation can be partially determined.
Also, during analysis the oscillation amplitude A has to remain
unchanged or change unsignificantly. The resulting construction would
satisfy technical characteristics of the system and be rational from a
technological standpoint.
4. Conclusions
Results of numerical modeling and simulation of piezoelectric
actuator are presented and analyzed in this paper.
Numerical modeling of piezoelectric actuator was performed to
validate design and operating principle of the actuator through its
modal response analysis.
While changing geometrical parameters of piezoelectric actuators
the variation in the modal shape sequence has been observed.
Identification of modal shapes sequence is the necessary step in
order to automate numerical experiments of multicomponent piezoelectric
actuators.
In practical part modal analysis is performed, eigenform determined
and eigenfrequency calculated, the size of inner and outer radius
represented maximum rotation and flexibility.
In two iterations of the calculation, the best result of crossply
movement was obtained in model 3 with coefficient 0.890225, which was
achieved with eigenfrequency of 29191Hz (Table 7).
In two iterations of calculation, the best result of rotation
movement was obtained in model 2 with coefficient 0.835137, which was
achieved with eigenfrequency of 107425 Hz (Table 8).
Experimental studies confirmed that elliptical rotation
oscillations were obtained on the surface of the actuator.
crossref http://dx.doi.org/10.5755/j01.mech.19.6.6017
Acknowledgement
This work has been supported by Research Council of Lithuania,
Project No. MIP-075/2012.
Received February 01, 2013
Accepted December 10, 2013
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G. Bauriene, Kaunas University of Technology, Kestucio 27,
44312Kaunas, Lithuania, E-mail: genaovaite.
[email protected]
J. Mamcenko, Vilnius Gediminas Technical University, Sauletekio al.
11, Vilnius, LT-10223, Lithuania, E-mail:
[email protected]
G. Kulvietis, Vilnius Gediminas Technical University, Sauletekio
al. 11, Vilnius, LT-10223, Lithuania, E-mail: genadijus.
[email protected]
A. Grigoravicius, Vilnius Gediminas Technical University,
Sauletekio al. 11, Vilnius, LT-10223, Lithuania, E-mail:
[email protected]
I. Tumasoniene, Vilnius Gediminas Technical University, Sauletekio
al. 11, Vilnius, LT-10223, Lithuania, E-mail:
[email protected]
Table 1
Properties of the material used for modelling
Material property Piezoceramics PZT-8
Jung modulus, N/[m.sup.2] 8.2764 x [10.sup.10]
Puason coefficient 0.33
Density, kg/[m.sup.3] 7600
Dielectric permittivity [[epsilon].sub.11] = 1.2;
x [10.sup.-3] F/m [[epsilon].sub.22] = 1.2;
[[epsilon].sub.33] = 1.1
Piezoelectric matrix [[epsilon].sub.13] = -13.6;
[[epsilon].sub.23 = -13.6;
x [10.sup.-3]C/[m.sup.2] [[epsilon].sub.33 = 27.1;
[[epsilon].sub.42 = 37.0;
E = 37.0
Table 2
The detailed measurement of geometric parameters of the
piezoceramic ring
Measured Model Model Model Model Model
parameters of 1 2 3 4 5
ring actuator
Outer radius 0.0150 0.0200 0.0150 0.0200 0.0150
R, m
Inner radius 0.0100 0.0080 0.0075 0.0063 0.0050
r, m
Height h, m 0.0020 0.0020 0.0020 0.0020 0.0020
Table 3
The domination coefficients of crossply movement
Model S [tau] (rotative) S[phi] (crossply) Sz (long)
1 0.131347 0.868609 0.000045
2 0.128142 0.871767 0.000091
3 0.139465 0.860425 0.000109
4 0.324006 0.675855 0.000139
5 0.162175 0.837622 0.000203
Table 4
The domination coefficients of rotative movement
Model S [tau] (rotative) S [phi] (crossply) Sz (long)
1 0.772685 0.225999 0.001316
2 0.765639 0.233801 0.000561
3 0.791188 0.207722 0.001090
4 0.821089 0.178561 0.000351
5 0.831918 0.167350 0.000732
Table 5
The eigenfrequencies
(by crossply and rotative movements)
Model 1 2 3 4 5
S [phi] (crossply) 12161 18091 20548 24118 30728
Frequency f,
Hz
S [tau] (rotative) 89599 67579 90757 66160 88491
Frequency f,
Hz
Table 6
The detailed measurement of geometric parameters of the
piezoceramis ring in the 2nd iteration of calculations
Measured parameters of Model 1 Model Model
ring actuator 2 3
Outer radius R, m 0.0100 0.0125 0.0175
Inner radius r, m 0.0050 0.0050 0.0050
Height h, m 0.0020 0.0020 0.0020
Table 7
The domination coefficients of crossply movement in the
2nd iteration of calculations
Model S [tau] (rotative) S [phi] (crossply) Sz (long)
1 0.170595 0.829186 0.000219
2 0.211616 0.788173 0.000211
3 0.109596 0.890225 0.000179
Table 8
The domination coefficients of rotative movement in the
2nd iteration of calculations
Model S [tau] (rotative) S [phi] (crossply) Sz (long)
1 0.718499 0.278451 0.00305
2 0.835137 0.163658 0.001204
3 0.706132 0.293226 0.000642
Table 9
The eigenfrequencies (by crossply and rotative move-
ments) in the 2nd iteration of calculations
Model 1 2 3
S [phi] (crossply) frequency f, Hz 30836 31684 29191
S [tau] (rotative) frequency f, Hz 136067 107425 75342
