## 3 Numerical Investigation of the Actuator
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material Properties** & **Beryllium Bronze C17200** & **PI Ceramics** & **Aluminum Oxide** \\  & & **PIC181** & **Ceramic** \\ \hline Density, [kg/m\({}^{3}\)] & 8360 & 7800 & 3980 \\ Young’s modulus, [N/m\({}^{2}\)] & \(10\times 10^{9}\) & 7.6 \(\times\) 10\({}^{10}\) & 41.9 \(\times\) 10\({}^{10}\) \\ Poisson’s coefficient & 0.34 & - & 0.33 \\ Isotropic structural loss factor & 0.02 & - & 0.2 \(\times\) 10\({}^{-3}\) \\ Relative permittivity & - & \(\epsilon_{33}\)T/\(\epsilon_{0}\) = 1200 & - \\  & & \(\epsilon_{33}\)T/\(\epsilon_{0}\) = 1500 & - \\ Elastic compliance coefficient [10\({}^{-12}\) m\({}^{2}\)/N] & - & \(\text{S}_{11}\)E = 15.00 & - \\ Elastic stiffness coefficient \(\text{c}_{33}\)D, [N/m\({}^{2}\)] & - & 1.6 \(\times\) 10\({}^{10}\) & - \\ Piezoelectric constant \(\text{d}_{33}\) [10\({}^{-12}\) m/V] & - & 225 & - \\ Piezoelectric constant \(\text{d}_{31}\) [10\({}^{-12}\) m/V] & - & \(-\)97 & - \\ Piezoelectric constant \(\text{d}_{15}\) [10\({}^{-12}\) m/V] & - & 330 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Material properties. 100 V\({}_{\text{p-p}}\) while remaining electrodes were set to open-circuit conditions. Results are given in Figure 5. It can be seen that resonant frequencies of the first radial and the third radial modes are obtained at 18.01 kHz and 46.172 kHz, respectively. Minor differences between natural and resonant frequencies occur due to slight differences in electrical boundary conditions. It can be seen that phase change at the resonant frequency is 12.1 degrees and 1.9 degrees (Figure 5). This means the corresponding vibration modes have relatively low efficiency. Figure 5: Impedance and phase characteristics in the frequency range of 17.76–18.29 kHz (**a**) and 45.96–46.46 kHz (**b**). Harmonic response analysis also was performed to analyze amplitudes of contacting points vibration. The calculations were made when the voltages of 50 V\({}_{\text{P-P}}\) and 200 V\({}_{\text{P-P}}\) were applied for both modes. A single electrode was excited. Coupling between vibrations of different contact points was analyzed as well. Figure 6a shows that the highest displacement amplitude was obtained at a frequency of 18.017 kHz. The frequency value is in good agreement with the results of impedance frequency analysis. Also, it can be seen that the contact point located at the support under the excited electrode (Planar\({}_{1}\)) has the highest displacement of 3.91 \(\upmu\)m. The ratio between displacement amplitude and applied voltage is 78.2 nm/V\({}_{\text{P-P}}\). On the other hand, it can be observed that contact points located on the passive segments (Planar\({}_{2}\) and Planar\({}_{3}\)) generate notably lower displacement amplitudes, i.e., 380 nm and 340 nm. In addition, contact points located on segments for angular motion, Rotary\({}_{1}\)-Rotary\({}_{3}\), also vibrate with amplitudes of 183 nm, 151 nm, and 136 nm, respectively. The coupling ratio between vibration amplitudes of excited and passive planar segments does not exceed 11.5 times and is not more than 28.75 times between excited segment and passive segments of angular motion. Figure 6b represents the displacement-frequency characteristics of the actuator while excitation signal amplitude was set to 200 V\({}_{\text{P-P}}\) and was applied on electrode Planar\({}_{1}\). The highest displacement amplitude of the active segment reached 15.61 \(\upmu\)m or 78.05 nm/V\({}_{\text{P-P}}\). The contact points located on Planar\({}_{2}\) and Planar\({}_{3}\) segments have amplitudes of 3.81 \(\upmu\)m and 3.45 \(\upmu\)m, respectively. Therefore, the ratio between active and passive planar segments is 4.52 times. Analyzing vibration amplitudes of contact points located on Rotary\({}_{1}\)-Rotary\({}_{3}\) segments, it can be seen that it reaches 751 nm, 648 nm, and 557 nm, respectively. The coupling ratio of vibrations amplitudes is around 21 times in this case. Therefore, coupling between vibrations increases when excitation voltage increases as well.