# Triangular-Shaped 5-DOF Piezoelectric Robot for Optical Lens Positioning
## 3 Numerical Investigation of Robot
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material Properties** & **Stainless Steel DIN 1.4301** & **PI Ceramics PIC181** & **Aluminum Oxide Ceramic** \\ \hline Density, [kg/m\({}^{3}\)] & 8000 & 7800 & 3980 \\ Young’s modulus, [N/m\({}^{2}\)] & \(193\times 10^{9}\) & 7.6 \(\times\) 10\({}^{10}\) & 41.9 \(\times\) 10\({}^{10}\) \\ Poisson’s coefficient & 0.29 & 0.34 & 0.33 \\ Isotropic structural loss factor & 0.02 & - & 0.2 \(\times\) 10\({}^{-3}\) \\ Relative permittivity & - & \(\epsilon_{11}\)T/\(\epsilon_{0}\) = 1200 & - \\  & & \(\epsilon_{33}\)T/\(\epsilon_{0}\) = 1500 & - \\ Elastic compliance coefficient [10\({}^{-12}\) m\({}^{2}\)/N] & - & S\({}_{11}\)E = 11.80 & - \\ Elastic stiffness coefficient c\({}_{33}\)D, [N/m\({}^{2}\)] & - & 16.6 \(\times\) 10\({}^{10}\) & - \\ Piezoelectric constant d\({}_{33}\) [10\({}^{-12}\) m/V] & - & 265 & - \\ Piezoelectric constant d\({}_{31}\) [10\({}^{-12}\) m/V] & - & \(-\)120 & - \\ Piezoelectric constant d\({}_{15}\) [10\({}^{-12}\) m/V] & - & 475 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Materials properties used in the model. Figure 4: Modal shapes of the piezoelectric robot; (**a**)—the first bending mode of trapezoidal cantilevers at 13.79 kHz; (**b**)—the radial mode of the piezoelectric ring at 95.75 kHz. It can be seen that the resonant frequency of the first bending mode and third radial mode of trapezoidal cantilevers was obtained at a frequency of 13.18 kHz and 95.51 kHz, respectively. There is a minor difference of 1% and 1.5% between modal and resonant  
Figure 5: Impedance and phase-frequency characteristics of the piezoelectric robot in a frequency range of (**a**) 13.78–13.88 kHz and (**b**) 95.0–96.50 kHz. frequencies that occurs because of the different electrical boundary conditions used in the model. Moreover, the effective coupling factor \(k_{\mathit{eff}}\) of the robot is 0.061 and 0.086 when it operates at the frequency of 13.818 kHz and 95.51 kHz, respectively. The next step of the numerical investigation was performed to analyze the displacement amplitudes of the contact points and coupling ratio between vibrations of different segments of the robot when L and R segments of the piezo ceramic ring were excited. The vibrations coupling ratio is a parameter that shows the ratio between displacement amplitudes of active and passive segments of the piezoceramic ring. It allows evaluating how voltage applied on the active segment of the ring affect vibrations of the contacts used to generate planar and angular motion. Firstly, simulations were performed when the L\({}_{\mathrm{ddeg}}\) segment used to generate the planar motion of the robot was affected by a harmonic voltage of 50 V\({}_{\mathrm{p-p}}\). The remaining segments of the piezoceramic ring were set to open-circuit condition. Analyzing the results of the simulation, it can be noticed that the resonant frequency of the first bending mode of the trapezoidal cantilever is obtained at the frequency of 13.83 kHz (Figure 6). Displacement amplitude of the contact located on the particular trapezoidal beam and excited by L\({}_{\mathrm{ddeg}}\) segment reached 12.25 \(\upmu\)m or 245 nm/V\({}_{\mathrm{p-p}}\). It must be noted that vibration amplitudes of contact located on other cantilevers are significantly smaller and do not exceed 3.8 \(\upmu\)m. The coupling ratio between contact vibrations located on the active and passive cantilevers is 3.22. On the other hand, the contacts placed in segments R\({}_{\mathrm{ddeg}}\)-R\({}_{\mathrm{240 deg}}\) vibrate as well when the L\({}_{\mathrm{ddeg}}\) segment is exciting. Displacement amplitudes of these contacts do not exceed 0.52 \(\upmu\)m and the coupling ratio average is 23.56.