Piezoceramic disks with a front layer - 
Comparisons of FEM with experimental data

Jan Kocbach, Dept. of Physics, University of Bergen (UoB), N-5007 Bergen, Norway. 
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. 

1.  Introduction

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 [1] [2]. 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 [1], 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 [2] [1] are calculated separately. For the piezoelectric elements, there is no difference in the calculations compared to the case for a piezoelectric disk, described in [2]. For the elastic elements, the piezoelectric stiffness matrix [2] 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 [2]. 

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. 

3.  Analysis

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 [3]. 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 the FEM-simulations. 

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 piezoceramic. 

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). (In Norwegian.)
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", Cand. Scient. (M.Sc.) thesis, Dept. of Physics, University of Bergen, Bergen, Norway (1994). (In Norwegian.)

The slides for the presentation are available as a (large) post-script file.  

Jan Kocbach
Fri Feb 27 14:51:36 MET 1998 

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