Piezoceramic disks with a front layer -
Jan Kocbach, Dept. of Physics, University of Bergen (UoB),
N-5007 Bergen, Norway.
Comparisons of FEM with experimental data
Per Lunde, Christian Michelsen Research AS (CMR),
Fantoftvegen 38, N-5036 Fantoft, Bergen, Norway.
Magne Vestrheim, Dept. of Physics, University
of Bergen (UoB), N-5007 Bergen, Norway.
Erlend Bjørndal, Read Well Services A/S, Gravdalsveien
255, N-5034 Ytre Laksevåg, Bergen, Norway.
A program for finite element modelling (FEM) of axisymmetric piezoceramic
transducers vibrating in a fluid is under development in a cooperation
between UoB and CMR. The implementation and testing of the FEM-code for
a piezoceramic disk in vacuum has been presented earlier 
. In the present paper the addition of
an elastic, isotropic front-layer to the piezoceramic disk is discussed.
Admittance-simulations for a piezoceramic disk with a front layer of varying
thickness are compared with measurements. Three different materials are
used for the front layer, and simulated mode-shapes are shown to get a
better understanding of the effects occuring.
2. Basic Theory
The theory for finite element modelling of a piezoelectric disk is covered
in depth in , and will not be
redone here. Only the steps required to add an elastic front layer in the
simulations will be described briefly.
In the finite element method, the region of analysis is split into elements
(typically houndreds of 8-node isoparametric elements for our problems).
The splitting of the region into elements has to be done in such a way
that every element consists of only one material. Thus one gets piezoelectric
and elastic elements.
For each local element, the mass- and stiffness-matrices 
 are calculated separately. For
the piezoelectric elements, there is no difference in the calculations
compared to the case for a piezoelectric disk, described in .
For the elastic elements, the piezoelectric stiffness matrix 
is equal to zero, because there is no piezoelectric coupling in the material.
This decouples the mechanical variables from the electrical variables,
and the equations for the electric potential in these elements is removed
from the FEM-equations. Afterwards the local matrices are assembled to
global mass- and stiffness-matrices, which gives FEM-equations on the same
form as for the purely piezoelectric case, equation (1) in .
The damping is implemented using complex values for the elastic stiffness
coefficients and for the dielectric constants. The electrical impedance/admittance
and mechanical response is calculated using mode superposition.
Figure 1: Piezoceramic disk with a front layer
Circular piezoceramic disks (T=1.005mm,D=12.87mm) with a front layer
of varying thickness (0-6mm) are studied (see Figure 1). Materials used
as a front layer are titanium, plexiglass and divinycell. These materials
are chosen because they have varying specific acoustic impedance (27 MRayl,
3.2 MRayl and 0.6 MRayl respectively), and because measurements were available.
For the piezoceramic disk the material Pz27 is used in the measurements,
and PZT5A with similar properties in the FEM-simulations. The measurements
for piezoelectric disks with an elastic front layer were carried out in
air, and are described in . In the
simulations, it is for simplicity assumed that the structure is vibrating
in vacuum. The piezoelectric disk is fully covered with electrodes on both
plane faces normal to the thickness direction.
The simulated resonance frequencies for a PZT5A disk with a titanium
layer of 15 different thicknesses (0.3 - 11.5 mm) have been compared with
simulations using the commercial FEM-program Abaqus, and very good agreement
has been found. The maximum relative difference for the lowest 25 resonance
frequencies for these 15 simulations was 5 ppm. Similar results were also
obtained for the two other front-layer materials.
The analysis is started with a titanium front-layer (high specific acoustic
impedance, low loss) for low frequencies (beyond the first radial mode
for the piezoelectric disk), because this gives less complicated resonance
patterns than for the other materials in the frequency range considered
here (up to 250 kHz). Due to lack of space, only one figure is included,
the comparison between measured and simulated electrical input-conductance
for a piezoceramic disc with a titanium front-layer, including some simulated
mode-plots (see Figure 2).
When going from a pure piezoceramic disk to a piezoceramic disc with
a thin titanium-layer, the resonance frequencies increase, and the first
radial mode (which is a symmetric mode at 154.2 kHz for the pure piezoceramic
disk) transforms into a flexural mode (at 236.1 kHz for ) . The lower flexural
modes, which are not excited for a pure piezoceramic disk with the current
electrode configuration due to symmetry, become more pronounced in the
conductance plot as the thickness of the titanium layer increases (see
Figure 2 at 36.0 kHz and 133.5 kHz for the pure piezoceramic disk, and
at 71.0 kHz and 200.1 kHz for the disk with a front-layer of thickness
1.011 mm). These same effects are seen both in the measurements and in
Some of the differences that can be seen between the measured and simulated
results in Figure 2 (with respect to frequency) is probably caused by the
material parameters for titanium used in the simulations, which are book-values.
Other possible reasons for the discrepancy might be the glue-layer between
the piezoceramic and the front layer, and the material parameters for the
For the piezoceramic disk with a divinycell-layer and a plexiglass-layer,
the effects occuring are more complicated in this frequency range, and
there is also poorer agreement between FEM-simulations and measurements.
Simulations with adjusted material parameters for these two cases (not
shown here) gave a far better qualitative agreement.
Average displacement in radial and thickness direction and voltage source
sensitivity has also been calculated, but will not be discussed here.
Figure 2:Comparison of FEM-simulations with measurements for
the conductance of a piezoceramic disk with a front layer of titanium with
varying thickness, including simulated mode-plots of the three lowest modes
for the pure piezoceramic disk and the piezoceramic disk with a titanium
front-layer of thickness .
4. Conclusions and further work
A FEM program for piezoelectric ceramic disks with a front layer has been
implemented and tested against FEM and measurements. A very good agreement
has been found for resonance frequencies with the commercial FEM-program
Abaqus. Qualitative agreement has been found with measurements of electrical
input-conductance, and some of the discrepancies between simulations and
measurements can be attributed to uncertainties in the material parameters
used for the elastic front layer.
This paper is a status report for work in progress. The next steps in
our work will be to simulate a piezoceramic disk with several front layers/backing
layers in vacuum, then a complete piezoceramic axisymmetric transducer
construction in vacuum, and finally the same transducer construction vibrating
in a medium, including radiation. Verification of the results against other
FEM-programs and measurements, and understanding of the effects observed,
will be essential in each step of this work.
The work described here is continued under a Dr. Scient. fellowship
by J. Kocbach granted by the Research Council of Norway for the years 1997-2000.
This work has received support from the Norwegian Supercomputing Committee
through a grant of computing time.
Kocbach J., "Endelig element modellering av piezoelektriske skiver", Cand.
Scient. (M.Sc.) thesis, Dept. of Physics, University of Bergen, Bergen, Norway (1996).
Kocbach J., Lunde P. and Vestrheim M.,"FEM-analysis of piezoelectric disks - method and testing", presented at "Scandinavian Symposium in Physical Acoustics",
Ustaoset Høyfjellshotell, 1997
Bjørndal, E. "Utnytting av resonansmodi i ultralyd transdusarkonstruksjonar",
Scient. (M.Sc.) thesis, Dept. of Physics, University of Bergen, Bergen, Norway (1994).
The slides for the presentation are available as a (large) post-script file.
Fri Feb 27 14:51:36 MET 1998
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