# Development of a Novel Gear-like Disk Resonator Applied in Gyroscope
## 2 Design and Analysis
### FEM Simulations  
To verify that the GDR has better performance and immunity to crystal orientation error than the RDR, the FEM was used to compare the two kinds of disk resonators through COMSOL Multiphysics. In the simulation process, the structure parameters of the GDR and RDR were the same. We used the modal analysis and thermal expansion analysis of the software to solve the frequency and quality factor and used Equation (8) to solve the mechanical sensitivity. In order to reduce the computational effort, the constraints could be equated. The principle of equivalence was that there was no effect on the results of the FEM before and after equivalence. The central anchor was not displaced because it was fixed to the substrate, and no temperature gradient due to deformation was generated. Therefore, the constraint that the central anchor was fixed to the substrate could be equated: the central anchor was deleted and the fixed constraint was applied to the contact surface of the central anchor and the spokes, as shown in Figure 2. The material of the resonator was isotropic silicon, and the values of material parameters are shown in Table 2.  
Without considering the frequency split, the resonant frequency, effective stiffness, \(Q_{TED}\), and \(S_{mech}\) of the GDR and RDR were compared, and the simulation results are shown in Figure 3. Compared with the RDR, the resonant frequency of the GDR was reduced by 33.1%, the effective stiffness was reduced by 61.7%, the \(Q_{TED}\) was increased by 19.9%, and the \(S_{mech}\) was increased by 450%. The simulation results demonstrate that the overall performance of the GDR was better, its effective stiffness and resonant frequency were significantly reduced, and its \(Q_{TED}\) and \(S_{mech}\) were significantly improved. \begin{table}
\begin{tabular}{c c} \hline \hline
**Material Parameters** & **Value** \\ \hline Elastic modulus (\(E\)) & 170 GPa \\ Thermal conductivity (\(\kappa\)) & 130 W/(m\(\cdot\)K) \\ Coefficient of thermal expansion (\(\alpha_{T}\)) & 2.6 \(\times\) 10\({}^{-6}\) 1/K \\ Constant pressure heat capacity (\(C_{p}\)) & 700 J/(kg\(\cdot\)K) \\ Density (\(\rho\)) & 2329 kg/m\({}^{3}\) \\ Poisson’s ratio (\(v\)) & 0.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The value of material parameters used in FEM calculation. Figure 2: The equivalent constraint in COMSOL Multiphysics. The elliptical flexural modes of DRGs have a high demand for symmetry. A single-crystal silicon wafer cut in the (111) plane (Si (111)) has isotropic mechanical properties, and it is an attractive material to manufacture DRGs. However, due to the limitations of microfabrication technology, there is an error in the process of fabricating Si (111) wafers, which is manifested by the fact that the crystal plane of actual (111) and ideal (111) is not the same; there is a pinch angle (\(\alpha\)) between them, which can be expressed as the crystal orientation error. This error leads to the stiffness asymmetry of the resonator, which will increase the frequency split and reduce \(S_{mech}\). The finite element model of crystal orientation error can be built in COMSOL Multiphysics by performing tensor transformation [30]. The frequency mismatch between the drive mode and sense mode can be characterized by the relative frequency split:  
\[\eta=\frac{|w_{A}-w_{B}|}{(w_{A}+w_{B})/2} \tag{9}\]  
The value of \(w_{A}\) and \(w_{B}\) affected by the crystal orientation error of the GDR and RDR were calculated, and the effects are shown in Figure 4. It is clear that the increase in \(\eta\) was almost linear with increasing \(\alpha\). Therefore, the value of \(\eta\) can be expressed as follows:  
\[\eta=\lambda\cdot\alpha \tag{10}\]  
where \(\lambda\) is the slope in Figure 4 and represents the sensitivity of the frequency split to a crystal orientation error. The values of \(\lambda\) could be calculated to be 1743.84 ppm/deg and 321.64 ppm/deg for the RDR and GDR, respectively.