## 6 Discussion and Verification
### Apc-855
#### 6.1.3 Influence on Self-Heating in Resonance  
For the out-of-plane modes, the piezoceramics was operated at lower frequencies and excited on almost one-hundred voltages. The piezoelectric material remained at almost the same temperature, which was measured by thermography at room temperature. However, we speculate that the in-plane parameters in Ref [27] are more accurate than the material constants obtained using RM in this study because APC-855 is a soft piezoelectric ceramic that is more prone to self-heating at high in-plane resonant frequencies. Thus, we used new specimens to conduct the in-plane vibration measurements. A comparison of Figures 5 and 6 show that the main difference lies in the duration of excitation. In Figure 6, the in-plane vibrations in the experiment were controlled so as to prevent self-heating caused by over-excitation [28; 29]. The lower impedance occurs at the resonant frequency. The experimental measurement of in-plane vibrations requires higher excitation voltage, since ESPI  
Figure 5: In-plane vibration modes for APC-855, after self heating in resonance.  
measurement is sub-micrometer resolution. Under high-voltage, high-frequency, and low-impedance excitation conditions, high temperatures are often generated in piezoelectric components, and the material properties of piezoelectric components change at high temperatures, which may alter the capacitance and material rigidity of piezoelectric ceramic. The results in Tables 5 and 6 show that after self-heating, the errors between the resonant frequency measurements of APC-855 and the results of FEM calculations using the material constants obtained with RM become significant. When the voltage and excitation time are controlled under only several seconds to confirm slight temperature rising, the resonant frequencies from the experimental ESPI measurements and impedance analysis compared to FEM are almost identical. The RM results also become accurate, with errors less than 5.314%. This verifies that the parameters measured using RM can also be used to calculate in-plane vibration values with accuracy. The errors of the Ref [27] parameters input into FEM instead become higher than they were before the self-heating of the specimen, ranging from 7% to 11%. From Equations (1) through (4), the electro-thermal-elastic model illustrated that the physical parameters and material constants were influenced by a thermal effect. In the experimental results, it is concluded that material constants and dynamic characteristics of the piezoelectric material depend on temperature. [PERSON] et al. [22] also show the influence on dielectric loss, dielectric constant, shifting resonant frequency of piezoelectric material through the thermal effect in their results. Hence, the RM used in the measurement of piezoelectric material constants has to be considerate of the temperature dependency in the actual application. This study therefore demonstrated that the material constants obtained using RM can be used to accurately calculate the out-of-plane and in-plane vibration characteristics of piezoelectric materials. However, the impact of self-heating must be taken into account.  
Figure 6: In-plane vibration mode for APC-855 before self heating in resonance.