Copyright 1999 IEEE. To appear in: 1999 IEEE International Ultrasonics Symposium Proceedings See also interactive demonstration of vibrational modes in a PZT5A disk with a cork front layer of varying thickness and other publications by Jan Kocbach. FE Simulations of Piezoceramic Disks with a Front Layer of Varying Thickness Jan Kocbach1, Per Lunde2 and Magne Vestrheim1 1University of Bergen, Department of Physics, Allégt. 55, N-5007 Bergen, Norway 2Christian Michelsen Research AS, P.O. Box 6301, N-5892 Bergen, Norway Abstract: Piezoceramic disks with a front layer of varying thickness have been investigated using the FE method. Emphasis is laid on the frequency bands around the fundamental thickness extensional mode (TE1) and the first radial mode (R1) of the disks. The influence of varying the density and the compressional and shear velocity of the front layer material has been investigated using electrical input conductance, average displacement and mode shapes. For the frequency band around the TE1 mode, agreement with 1D transmission line model simulations has been found, and further interpretation has been possible. For the frequency band around the R1 mode, similar effects to the ones for the frequency band around the TE1 mode are seen for a front layer material with low characteristic impedance. For higher-impedance front layer materials, the situation changes significantly.

1. Introduction

Many experimental and theoretical studies of the frequency spectra of resonant vibration in disks of isotropic and piezoceramic materials with varying D/T (Diameter/ Thickness) ratio have been reported in the literature, e.g. [1] [2] [3] [4], including some systematic investigations of piezoceramic disks using the Finite Element (FE) method, e.g. [5] [6]. In the present work the more complex case of a piezoceramic disk with a front layer (see Fig. 1) is investigated using the FE method. Many authors have investigated piezoceramic disks with a coupling layer operating around the TE1 mode of the piezoceramic disk using 1D transmission-line equivalent models (see e.g. [7]), but good analytical models are not available for the study of piezoceramic disks with a front layer in the frequency band around the R1 mode. The case of a long rectangular PZT5H bar with a matching layer of varying thickness operating in water has been studied using the FE method [8], but systematic studies of piezoceramic disks with a front layer of varying thickness using the FE method are felt to be needed.

In the present work FE simulations are compared to simulations obtained using the Mason model for the frequency band around the TE1 mode for several different front layer materials. Also, the effect of changing the thickness and material parameters of the front layer for the frequency band around the R1 mode is investigated using the FE method, and compared to the effects seen for the frequency band around the TE1 mode. The motivation for these investigations is the optimization of piezoceramic air and gas transducers, and especially the analysis of the effect of the thickness and material properties of the coupling layer for such transducers. Although this work is preliminary, it is considered to represent an important step in understanding the vibration and response of piezoceramic transducers.

  

Figure 1:

A piezoceramic disk of thickness Tpiezo and diameter D with a front layer of thickness Tfront.

2. Method

The results presented are simulated using FEMP, Finite Element Modeling of Piezoelectric structures [9], a FE code developed at the University of Bergen and Christian Michelsen Research AS. For all simulations, 5 elements per smallest wavelength (shear) are used in both the piezoceramic disk and in the front layer. Systematic convergence tests have shown that the relative error in calculated resonance frequencies is less than 0.2%, and that the error in calculated electrical and mechanical response functions is less than 2% of their maximum value for each front layer thickness. The large errors in response functions are due to the movement of resonance frequencies to lower frequencies when the element division is improved, leading to a corresponding movement of the peaks of the electrical and mechanical response functions occurring close to the resonance frequencies.

The losses in the piezoceramic and elastic media are described using complex material constants, where a single value is used for all elastic Q-factors (Q_m=75) and a single value is used for the dielectric Q-factors (Q_e=50) for the piezoceramic material, and a single value is used for both elastic Q-factors (Q=25) in the isotropic elastic front layer. Electrical and mechanical response functions are calculated using the mode superposition method, where the eigenmodes are calculated using a complex eigensolver. The FE model is axisymmetric, and the effect of a fluid medium surrounding the structure has been neglected in these simulations.

The FE code has been validated against the FE codes ABAQUS, ANSYS and CAPA. For all cases considered, the relative deviation for calculated resonance frequencies compared to these FE codes has been less than 0.0008% [9]. Validations have been performed for piezoceramic disks with and without a front layer. Additionally several tests have been performed against literature results.

3. Results

It is studied how the resonance frequencies, the electrical input conductance and the average surface normal displacement of a structure, consisting of a piezoceramic disk with a front layer, varies when the thickness of the front layer varies from zero (pure piezoceramic disk) to well above the quarterwave matching thickness for the disk. As an example, two different PZT5A disks with thickness $T_{piezo}=1.0 \, mm$ and D/T_piezo ratios of 10 and 3 are chosen for the study.

PZT5A disk without a front layer

As a basis for the investigations, some results for piezoceramic disks without a front layer are given [9]. It is well known that the resonance frequencies and response functions of piezoceramic disks vary largely with the D/T_piezo ratio of the disks [5]. In the frequency spectrum of PZT5A [10] disks given in Fig. 2a), the disk with D/T_piezo=10 is identified using a vertical line. The electrical input conductance and the average normal displacement per volt over the front surface of the disk (the transfer function U_av,z/V) are shown as functions of frequency in b) and c), respectively. It is seen that there are several modes contributing to these electrical and mechanical response functions around the TE1 mode (which is predicted by the 1D Mason model [11] at $fT_{piezo} = 1935 \, kHz \cdot mm$), but there is nevertheless one major peak in the electrical input conductance approximately at this frequency. The electrical input conductance for the disk with a front layer of varying thickness is shown as a contour plot (all contour plots in the present work are given on dB scale) in Fig. 3a)-c). At the left edge of these figures, the electrical input conductance for the disk without a front layer (corresponding to Fig. 2b)) is seen. The peaks in the electrical input conductance shown in Fig. 2b) are represented by dark areas along the left edge of Fig. 3a)-c).

  

Figure 2:

In a) the frequency spectrum of PZT5A disks with D/Tpiezo ratio between 0.1 and 20 is shown, displaying the resonance frequency-thickness product as a function of the D/Tpiezo ratio of the disks. For each calculated resonance frequency, a circle with radius proportional to the electrical input conductance is displayed ( $T_{piezo}=1.0 \, mm$ for all disks). Note that most circles are too small to be seen on the figure. In b) and c) the electrical input conductance and the average normal displacement per volt over the front surface of the disk, the transfer function Uav,z/V, are shown as a function of frequency.

PZT5A disk with a front layer: TE1 mode region

The PZT5A disk with D/T_piezo=10 and a front layer of varying thickness T_front is studied. Three different front layer materials with characteristic impedances varying from 2.64 Mrayl (epoxy) to 0.15 Mrayl (cork) are used (see Table 1). These materials are chosen because the density is approximately equal for epoxy and RTV, whereas the compressional velocity for epoxy is twice the value for RTV. Furthermore, the compressional velocity is approximately equal for RTV and cork, whereas the density of cork is 1/9 of the density of RTV. That is, using these three materials for the front layer, it is possible to analyze the effect of independently changing the compressional velocity and density of the front layer. Poisson's ratio $\sigma$ is set to 0.25 for all three front layer materials considered in this section to keep this variable unchanged in the simulations.

 

Table 1: Material parameters for the three different front layer materials used [12]. The epoxy is DER332 (50phr V140, rt cure), and the RTV is RTV-511. Poisson's ratio is given as 0.40 for the epoxy, whereas no values are given for cork and RTV. In the simulations the value 0.25 is used for all three front layer materials where nothing else is stated.

	Epoxy	RTV	Cork
Density $\rho$ [kg/m3]	1130	1180	130
Compressional velocity cl [m/s]	2340	1110	1150
Characteristic impedance Z0 [Mrayl]	2.64	1.31	0.15
Poisson's ratio $\sigma$	0.25/0.40	0.25	0.25

In previous studies using 1D transmission-line equivalent models, the occurrence of two peaks with positions varying with the thickness of the front layer has been reported [7]. This is shown in Fig. 3 d)-f), where the electrical input conductance for the piezoceramic disk with three different front layer materials of varying thickness simulated using the Mason model [11], are displayed. The peaks are symmetric with equal height at the quarterwave matching thickness of the TE1 mode, $T_{\lambda/4,TE1} = c_l/4f_{TE1}$ (where c_l is the compressional velocity of the front layer and f_TE1 is the resonance frequency of the TE1 mode predicted by the 1D Mason model), and the spacing between the two peaks becomes smaller when the characteristic impedance of the front layer is decreased.

In Fig. 3 a)-c) corresponding figures simulated using the FE method are shown. As expected from the electrical input conductance for a piezoceramic disk without a front layer presented in Fig. 2b), this case is far more complex, with several minor peaks in the electrical input conductance, in addition to the two major peaks predicted by the Mason model. However, the overall pattern is very similar, and the observations made for the 1D simulations above are also seen for the FE simulation. Some of the peaks, e.g. the two major peaks, are not due to a single vibrational mode, but due to the superposition of several different vibrational modes which are close to each other in frequency.

When a thin front layer with low characteristic impedance (e.g. cork, see Fig. 3c)) is attached to the piezoceramic disk, the resonance frequencies of the structure are virtually unchanged up to a certain thickness of the front layer. This front layer thickness depends on the frequency and on the shear velocity of the front layer material, and is found to be around $\lambda _s/4$, where $\lambda_s$ is the wavelength of shear waves in the front layer. This thickness is therefore not predicted by the Mason model, which neglects the effect of shear waves in the structure. At this front layer thickness many additional modes are introduced into the spectrum, leading to a sudden drop in the resonance frequencies of the structure. By comparing the spectrum in Fig. 3c) with the frequency spectrum of cork disks with the bottom of the disk clamped (not shown here), it is found that these "new" modes are associated with vibrational modes in the front layer. This effect is is discussed in detail for the frequency region around the R1 mode in the next section. The dense band of modes around the front layer thickness $\lambda _l/4$, where $\lambda_l$ is the wavelength of compressional waves in the front layer, is also associated with vibration in the front layer.

For a front layer with higher characteristic impedance (e.g. epoxy, see Fig. 3a)), the coupling between vibration in the piezoceramic disk and in the front layer is stronger, and therefore the drop in resonance frequencies is not that sudden. Changing the shear velocity of the front layer does not affect the electrical input conductance as a function of front layer thickness much, but the front layer thickness $\lambda _s/4$, where new modes are introduced into the spectrum, is changed (not shown here).

  

Figure 3:

In a)-c) frequency spectra of PZT5A disks with D/Tpiezo=10 and a front layer (epoxy, RTV and cork respectively) of varying thickness Tfront, simulated using the FE method, are shown. The resonance frequency-thickness product $f \cdot T_{piezo}$ is shown as a function of normalized front layer thickness (normalized to the quarterwave matching thickness of the front layer at the TE1 mode) using gray lines. Dashed lines show the front layer thicknesses $\lambda _l/4$(rightmost dashed line) and $\lambda _s/4$ (leftmost dashed line). In addition the value of the electrical input conductance is shown as a contour plot, where a dark color indicates a peak in the electrical input conductance. In d)-f) corresponding contour plots of the electrical input conductance simulated using the Mason TE mode model are shown.

PZT5A disk with a front layer: R1 mode region

For the frequency region around the R1 mode, results for a disk of PZT5A with D/T_piezo=3 are shown as an example, with either a cork or an epoxy front layer of varying thickness. In Fig. 4 the average normal displacement per volt over the front surface of the structure (the transfer function U_av,z/V) is shown as a function of $f \cdot T_{piezo}$ and the normalized front layer thickness, along with resonance frequencies and selected mode plots, for the case with a cork front layer.

From the figure it is evident that close to the quarterwave matching thickness for the R1 mode, $T_{\lambda/4,R1} = c_l/4f_{R1}$ (where f_R1 is the resonance frequency of the R1 mode found by FE simulations), there is a splitting into two major peaks in U_av,z/Valong the frequency axis. In addition there is a maximum for U_av,z/V along the thickness axis close to this front layer thickness. This maximum occurs approximately at the frequency for the R1 mode of the disk without a front layer. This is analogous to the result found for the TE1 mode, which is however not shown in the present work. The electrical input conductance of the structure (Fig. 5) is far less affected by the cork front layer than the average displacement shown in Fig. 4: the dark areas indicating peaks in the electrical input conductance are nearly horizontal in Fig. 5. However, a splitting into two major peaks which are very close to each other in frequency is evident around the quarterwave matching thickness. That is, the mechanical and electrical response of the piezoceramic disk with a cork front layer in the frequency region around the R1 mode is analogous to the case for the frequency region around the TE1 mode. Our studies have shown that this is also the case for disks with other D/T_piezoratios with a cork front layer in the frequency region around the R1 mode (not shown here).

However, when other front layer materials are attached to the piezoceramic disk, this is not always the case. It may be impossible to identify two major peaks in both electrical input conductance and average displacement. Also, the maximum of the average displacement over the front of the disk may be shifted to another front layer thickness, and in some cases no distinct maximum may be found at all. Furthermore, in contrast to the case for the frequency region around the TE1 mode, the shear velocity has a major impact on both the electrical and mechanical response of the structure. As an example, consider a piezoceramic disk with an epoxy front layer (see Table 1 for material parameters). If Poisson's ratio is taken to be 0.25, a splitting of the electrical input conductance and the average displacement into two major peaks is seen around the front layer thickness $T_{\lambda/4,R1}$ (see Fig. 6a) for average displacement), similar to the case for the cork front layer. If, however, Poisson's ratio is taken to be 0.40, the situation is changed (see Fig. 6b)). For this front layer material, there are several peaks in both the electrical input conductance and in the average displacement around the quarterwave matching thickness for the R1 mode.

Analogous to the case for the frequency band around the TE1 mode, the resonance frequencies of the structure are virtually unchanged up to the front layer thickness $\lambda _s/4$ for the case with a cork front layer shown in Fig. 4. From the two leftmost mode plots it is also evident that the cork front layer does not affect the vibration of the piezoceramic much until the front layer thickness approaches $\lambda _s/4$. At this front layer thickness, the mode shape and actual deformation of the structure changes significantly: There is very little deformation in the piezoceramic part of the structure, whereas most of the vibration occurs in the front layer (see modes at 186 kHz and 193 kHz in Fig. 4). The deformation seen in the cork front layer is very similar to the eigenmode of a cork disk with thickness T_front where the bottom of the disk is clamped (see eigenmode of cork disk with frequency 191 kHz in Fig. 4). Also, the resonance frequency of the cork disk where the bottom of the disk is clamped, nearly coincides with the resonance frequency of the structure (191 kHz versus 186 kHz and 193 kHz). Actually, all lines in the spectrum shown in Fig. 4 and Fig. 5 which seem to be crossing the nearly horizontal lines emerging from the left side of the spectrum (eigenmodes in PZT5A disks without a front layer), are associated with eigenmodes in cork disks with thickness T_front where the bottom of the disk is clamped. This is shown in Fig. 5, where the eigenfrequencies of cork disks with the bottom of the disk clamped are superimposed onto the spectrum using dotted lines. Furthermore, in regions close to the dotted lines, most of the deformation in the two-layered structure is in the cork front layer, whereas there is very little deformation in the piezoceramic. In regions which are further away from the dotted lines, there is more vibration in the piezoceramic disk, and the deformation of the piezoceramic disk is similar to the mode shape of the piezoceramic disk without a front layer.

  

Figure 4:

Frequency spectrum of PZT5A disk with D/Tpiezo=3 with a cork front layer of varying thickness. The average normal displacement per volt over the front surface of the structure is shown as a function of $f \cdot T_{piezo}$ and the normalized front layer thickness for 80 different front layer thicknesses. Dashed lines show the front layer thicknesses $\lambda _l/4$(rightmost dashed line) and $\lambda _s/4$ (leftmost dashed line). In addition, selected "mode plots" are presented for various points in the spectrum (shown by markers). The left half of these "mode plots" is the actual eigenmode, whereas the right part is the actual deformation at this eigenfrequency calculated by mode superposition. For each "mode plot" the resonance frequency and the average displacement over the front surface of the disk for an input voltage of 1V is given. Also shown is the first eigenmode of a pure cork disk with thickness $0.56 T_{\lambda /4,R1}$ where the bottom of the disk is clamped.

  

Figure 5:

Frequency spectrum of a PZT5A disk with D/Tpiezo=3, with a cork front layer of varying thickness (solid lines). A contour plot gives the value of the electrical input conductance as a function of $f \cdot T_{piezo}$ and the normalized front layer thickness. Dotted lines show the resonance frequencies of cork disks where the bottom of the disk is clamped.

  

Figure 6:

In a) the frequency spectrum of a PZT5A disk with D/Tpiezo=3 is given for an epoxy front layer of varying thickness with $\sigma =0.25$. The transfer function Uav,z/Vis given as a contour plot. A corresponding figure where $\sigma =0.4$ for the epoxy is shown in b). In both figures the front layer thicknesses $\lambda _s/4$ and $\lambda _l/4$ are shown by dashed lines.

4. Conclusions

A program for FE analysis of piezoceramic disks has been developed. The FE program has been validated against several other FE codes, and the accuracy of the calculated results has been established through convergence tests. The frequency bands around the TE1 mode and the R1 mode for piezoceramic disks with a front layer of varying thickness have been studied.

For the frequency band around the TE1 mode, agreement with 1D transmission line model simulations has been found. In addition the effect of changing the characteristic impedance of the front layer has been demonstrated, and it has been shown that the shear velocity does not affect the electrical input conductance in the TE1 mode region significantly.

For the frequency band around the R1 mode, similar effects to the ones for the frequency band around the TE1 mode are seen for a cork front layer. However, for other front layer materials, the situation changes significantly. Additionally, the shear velocity in the front layer is important for the response of the structure, and therefore a description of shear vibrations in the front layer is necessary for accurate modeling of R1 mode transducers. Furthermore, it has been shown that for the front layer materials and frequency regions considered, the resonance frequencies and response of the piezoceramic disk is not much affected by a front layer thinner than $\lambda _s/4$.

For a low-impedance front layer material (e.g. cork), all lines in the spectrum which seem to be crossing the nearly horizontal lines emerging from the left side of the spectrum (eigenmodes in a piezoceramic disk without a front layer), are associated with eigenmodes in the front layer. For a higher-impededance front layer material, the coupling between the vibration in the piezoceramic disk and in the front layer is stronger. Therefore, the vibration in the piezoceramic disk is more affected by the front layer, and the lines emerging from the left side of the spectrum are not that horizontal.

The present work extends previous systematic analyses on piezoceramic disks by accounting for a front layer in the FE analysis. Although the work is preliminary, it is considered to represent an important step in understanding the vibration and response of ultrasonic air and gas transducers.

Bibliography

1

E.A.G. Shaw,
``On the resonant vibrations of thick Barium Titanate disks,''
J. Acoust. Soc. Am., vol. 28, pp. 38-50, 1956.

2

S. Ikegami, L. Ueda, and S. Kobayashi,
``Frequency spectra of resonant vibration in disk plates of PbTiO₃ piezoelectric ceramics,''
J. Acoust. Soc. Am., vol. 55, pp. 339-344, 1974.

3

S. Ueha, S. Sakuma, and E. Mori,
``Measurement of vibration velocity distributions and mode analysis in thick disks of Pb(ZrTi)O₃,''
J. Acoust. Soc. Am., vol. 73, pp. 1842-1847, 1983.

4

D.C. Gazis and R.D. Mindlin,
``Extensional vibrations and waves in a circular disk and a semi-infinite plate,''
J. Appl. Mech., vol. 27, pp. 541-547, 1960.

5

H.A. Kunkel, S. Locke, and B. Pikeroen,
``Finite-element analysis of vibrational modes in piezoelectric ceramic disks,''
IEEE Trans. Ultrason. Ferr. Freq. Contr., vol. 37, pp. 316-328, 1990.

6

N. Guo and P. Cawley,
``Measurement and prediction of the frequency spectrum of piezoelectric disks by modal analysis,''
J. Acoust. Soc. Am., vol. 6, pp. 3379-3387, 1992.

7

N.G. Pace and P.D. Thorne,
``Efficiency of face-plated underwater acoustic transducers,''
Ultrasonics, vol. 21, pp. 65-69, 1983.

8

G.L. Wojcik, C. DeSilets, L. Nikodym, D. Vaughan, N. Abboud, and J. Mould Jr.,
``Computer modeling of diced matching layers,''
in Proc. of 1999 IEEE Ultrason. Symp., 1996.

9

J. Kocbach, P. Lunde, and M. Vestrheim,
``FEMP, Finite Element Modeling of Piezoelectric structures. Theory and verification for piezoceramic disks,''
Scientific/Technical Rep. no. 1999-07 , Department of Physics, University of Bergen, Norway, 1999. Available HTTP: http://www.fi.uib.no Directory: $\sim$jankoc/ File: pub.html.

10

Five piezoelectric ceramics, Bulletin 66011/F,
Vernitron Ltd., England, 1976 (now Morgan Matroc Ltd.).

11

J.H. Goll,
``The design of broad-band fluid-loaded ultrasonic transducers,''
IEEE Trans. Son. Ultrason., vol. SU-26, pp. 385-393, 1979.

12

A.R. Selfridge,
``Approximate material properties of isotropic materials,''
IEEE Trans. Son. Ultrason., vol. 32, pp. 381-394, 1985. Updated tables available at HTTP: http://www.ultrasonic.com Directory: tables/ File: index.htm.

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FE Simulations of Piezoceramic Disks with a
Front Layer of Varying Thickness

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