# Finite Element Analysis of the Distribution Parameters of a Metal Dot Array in a SAW Gyroscope
## 3 Modeling and Simulation
### Simulation Results and Discussion  
The vibration modes of the SAW in the piezoelectric substrate can be derived by characteristic frequency analysis; when a metal dot array is distributed on a piezoelectric  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Material Constants**} & \multicolumn{5}{c}{**Materials**} \\ \cline{2-5}  & \(\mathbf{128^{\circ}}\)**YX-LiNbO\({}_{3}\)** & \(\mathbf{X112^{\circ}}\)**Y-LiTaO\({}_{3}\)** & \(\mathbf{ST-X}\)**Quartz** & \(\mathbf{Cu}\) \\ \hline SAW type & RSAW & RSAW & RSAW & \\ \hline Euler angle & \((0^{\circ},37.86^{\circ},0^{\circ})\) & \((90^{\circ},90^{\circ},112.2^{\circ})\) & \((0^{\circ},132.75^{\circ},0^{\circ})\) & \\ \hline \(\kappa^{2}\) (\%) & 5.4 & 0.71 & 0.11 & \\ \hline TCF (ppm/\({}^{\circ}\)C) & \(-72\) & \(-18.2\) & 0 & \\ \hline Stiffness constants (\(10^{11}\) N/m\({}^{2}\)) & & & & \\ C\({}_{11}\) & 1.98 & 2.32 & 0.87 & 1.77 \\ C\({}_{12}\) & 0.54 & 0.46 & 0.07 & 1.24 \\ C\({}_{13}\) & 0.65 & 0.83 & 0.12 & \\ C\({}_{14}\) & 0.07 & \(-0.11\) & \(-0.18\) & \\ C\({}_{33}\) & 2.27 & 2.75 & 1.07 & \\ C\({}_{44}\) & 0.59 & 0.95 & 0.58 & 0.82 \\ \hline Piezolectric constants (C/m\({}^{2}\)) & & & & \\ \(e_{x1}\) & & & & \(0.171\) & \\ \(e_{x4}\) & & & & \(-0.0436\) & \\ \(e_{5z}\) & 3.69 & 2.64 & & \\ \(e_{y2}\) & 2.42 & 1.86 & & \\ \(e_{z1}\) & 0.30 & \(-0.22\) & & \\ \(e_{z3}\) & 1.77 & 1.71 & & \\ \(e_{z6}\) & & & & \(0.14\) \\ \hline Dielectric constants (\(10^{-12}\) F/m) & & & & \\ \(\varepsilon_{11}\) & \(45.6\times\varepsilon_{0}\) & \(40.9\times\varepsilon_{0}\) & \(4.5\times\varepsilon_{0}\) & \\ \(\varepsilon_{33}\) & \(26.3\times\varepsilon_{0}\) & \(42.5\times\varepsilon_{0}\) & \(4.6\times\varepsilon_{0}\) & \\ \(\varepsilon_{0}\) & 8.854 & 8.854 & 8.854 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material parameters used in FEM simulations [19; 20; 21].  
Figure 4: A 3D model of the single-period structure.  
substrate, the characteristic frequency will split into two modes: symmetric mode and anti-symmetric mode [18]. According to the model established in Section 3.1, taking the \(128^{\circ}\)YX-LiNbO\({}_{3}\) substrate distributed with a metal dot of size \(1/16\lambda\) as an example, we set the initial operation frequency \(f_{0}\) to be 100 MHz. Since the SAW velocity of the \(128^{\circ}\)YX-LiNbO\({}_{3}\)\(V_{f}=3986\) m/s [40], we set the acoustic wavelength \(\lambda\) to 39.86 \(\upmu\)m, the parameter 'width' is defined to achieve a parametric sweep of the metal dot size, and its value is set from \(1/16\lambda\) to \(15/16\lambda\) with a step interval of \(1/16\lambda\), the grid size is selected to be ultra-refinement, and the number of characteristic frequency searches is set to 30. Finally, a rotate coordinate system is added, and the angular velocity is set to \(2\pi f_{0}\times a\) (\(a\) is a constant used to control the angular velocity of the substrate). The simulation results shown in Figure 5a,b demonstrate the SAW of the symmetric mode and anti-symmetric mode, respectively [41], where the depth of the color indicates the magnitude of the displacement (i.e., amplitude of displacement), and the darker the color, the greater the displacement. In addition, the right side of the figure is the specific value of the displacement [42].  
It can be clearly seen from Figure 5a,b that the Rayleigh waves propagate on the surface of the medium, and the elastic displacement is only in the \(x\)-axis and \(z\)-axis planes; there is no displacement in the \(y\)-axis. Besides, the particle displacement is mainly concentrated in the depth of one wavelength, and is almost invisible in when the depth reaches two wavelengths [43], which is completely consistent with the theory that the energy of the SAW is mainly concentrated in one to two wavelengths [44].