VM175

VM175
Natural Frequency of a Piezoelectric Transducer

Overview

Reference: D. Boucher, M. Lagier, C. Maerfeld, "Computation of the Vibration Modes for Piezoelectric Array Transducers Using a Mixed Finite Element Perturbation Method", IEEE Trans. Sonics and Ultrasonics, Vol. SU-28 No. 5, 1981, pg. 322, table 1.
Analysis Type(s): Mode-frequency Analysis (ANTYPE = 2)
Element Type(s):
3D Coupled-Field Solid Elements (SOLID5)
3D 20-Node Coupled-Field Solid (SOLID226)
Input Listing: vm175.dat

Test Case

A piezoelectric transducer consists of a cube of PZT4 material with its polarization direction aligned along the Z axis. Electrodes are placed on the two surfaces orthogonal to the polarization axis. Determine the first two coupled-mode (breathing-type deformation) natural frequencies for the short circuit (resonance) case and the open circuit (anti-resonance) case.

Figure 254: Piezoelectric Transducer Problem Sketch

Piezoelectric Transducer Problem Sketch

Material PropertiesGeometric Properties
ρ = 7500 kg/m3
See "Constitutive Matrices"
= .02 m

Constitutive Matrices

PZT4 Dielectric Matrix [εr]

PZT4 Piezoelectric Matrix [e] C/m2

PZT4 "Stiffness" Matrix [c] x 1010 N/m2

Analysis Assumptions and Modeling Notes

The electroded regions represent equipotential surfaces and are not modeled explicitly. For the short-circuit case the top and bottom electrodes are grounded (voltages are set equal to zero). For the open-circuit case only the bottom electrode is grounded. The short-circuit case represents excitation by potential while the open-circuit case represents excitation by charge.

A one-quarter symmetry sector is modeled with symmetry boundary conditions applied. The mesh density selected for analysis along the axes (X, Y, Z) are (2,2,4) elements respectively. All non-specified voltage degrees of freedom are condensed out during matrix reduction to allow for electro-elastic coupling.

The KEYOPT(1) that is used does not have TEMP or MAG degrees of freedom.

The modes that produce a breathing-type deformation pattern indicate the desired results. POST1 is used to display the mode shapes for determination of the desired natural frequencies.

Results Comparison

 Target[1]Mechanical APDLRatio
SOLID5
Short Circuitf1, kHz 66560 664470.998
f2, kHz 88010 907091.031
Open Circuitf1, kHz 81590 842611.033
f2, kHz 93410 969881.038
SOLID226
Short Circuit f1, kHz 66560 651220.978
f2, kHz 88010 835110.949
Open Circuit f1, kHz 81590 79922 0.980
f2, kHz 93410 938111.004

  1. Experimentally measured values (f1,f2) represent breathing mode frequencies.

Figure 255: Short Circuit Case, Plot 3: First Breathing Mode using SOLID5 Elements

Short Circuit Case, Plot 3: First Breathing Mode using SOLID5 Elements

Figure 256: Short Circuit Case, Plot 6: Second Breathing Mode using SOLID5 Elements

Short Circuit Case, Plot 6: Second Breathing Mode using SOLID5 Elements

Figure 257: Open Circuit Case, Plot 15: First Breathing Mode using SOLID5 Elements

Open Circuit Case, Plot 15: First Breathing Mode using SOLID5 Elements

Figure 258: Open Circuit Case, Plot 19: Second Breathing Mode using SOLID5 Elements

Open Circuit Case, Plot 19: Second Breathing Mode using SOLID5 Elements

Figure 259: Open Circuit Case, Plot 23: Second Breathing Mode using SOLID226 Elements

Open Circuit Case, Plot 23: Second Breathing Mode using SOLID226 Elements

Figure 260: Open Circuit Case, Plot 26: Second Breathing Mode using SOLID226 Elements

Open Circuit Case, Plot 26: Second Breathing Mode using SOLID226 Elements

Figure 261: Open Circuit Case, Plot 35: Second Breathing Mode using SOLID226 Elements

Open Circuit Case, Plot 35: Second Breathing Mode using SOLID226 Elements

Figure 262: Open Circuit Case, Plot 39: Second Breathing Mode using SOLID226 Elements

Open Circuit Case, Plot 39: Second Breathing Mode using SOLID226 Elements