## 5 FEM Simulations
### Modal Analysis  
With the objective of determining the drive- and sense-mode resonant frequencies and eigen mode shapes for the proposed gyroscope design, a FEM-based modal analysis is performed in ANSYS APDL module. The effect of electrostatic tuning on the MEMS gyroscope dynamic behavior is investigated by performing two different simulations: The first one is performed without applying any bias tension on tuning electrodes, while the second one is performed by applying an increasing bias tension until compensating the frequency difference between the two modes. Figure 6 shows the first four mode shapes of the MEMS gyroscope obtained without considering the electrostatic tuning effect. The first mode of the structure (Figure 6a) is related to drive mode, in which only the drive mass is moving along the \(x\)-direction and the spring beams, which connect it to the sense frame, are bending in the same direction. This mode implies a purely in-plane deformation of the structure. Similarly, the second mode is described by an in-plane motion and is related to the sense mode (Figure 6b). When the device is subjected to an external angular velocity in the \(z\)-direction, the sense frame moves together with the drive mass in the \(y\)-direction due to the Coriolis force. The third out-of-plane (Figure 6c) and fourth in-plane (Figure 6d) modes are associated with the tilting of the mass and do not affect the operation of the gyroscope. The slight difference between the resonant frequency values related to the two modes is necessary to apply the electrostatic tuning for the compensation of the frequency mismatch due to foundry process uncertainties and environmental device operating variations. Table 3 shows the comparison of modal analysis results obtained through the analytical model and FEM analysis, and show a close agreement, which demonstrates the accuracy of the developed analytical model for the proposed gyroscope design.  
The electrostatic tuning application is considered performing a FEM-based modal analysis in ANSYS by applying an increasing tuning DC voltage on the tuning-comb structures. In this case, the followed first step is to proceed with a static analysis of the MEMS gyroscope with a DC voltage applied to the TRANS126 elements attached to moving tuning-comb fingers, and then perform a pre-stressed modal analysis on the MEMS gyroscope structure. The included pre-stress captures the effects of the applied voltage on the system frequency characteristic [25; 26]. The program outputs are mechanical displacements and eigen frequencies with incorporated electrostatic effects. Figure 7 shows the frequency tuning characteristic of the drive and sense resonant modes of the proposed MEMS gyroscope structure as a function of the tuning voltage applied on the tuning electrodes. As expected, by increasing the DC tuning voltage applied to the fixed electrodes, the drive-mode resonant frequency remains constant at 11014 Hz, while the sense-mode resonance frequency is adjusted due to DC voltage-based tuning of the electrostatic spring constant. The resonant mode frequencies are perfectly matched at the DC tension of 14.903 V.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Mode Shape**} & **Analytical Model** & **FEM Model** & **Error** \\  & **(Hz)** & **(Hz)** & **(\%)** \\ \hline Drive axis & 11,348 & 11,014 & \(-\)2.94 \\ Sense axis & 11,757 & 11,511 & \(-\)2.09 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of natural frequency values for the drive and sense modes obtained through analytical model and FEM model.  
Figure 6: Eigen frequencies and the mode shapes for the proposed dual mass MEMS gyroscope (**a**) 1 st mode (11,014 Hz) (**b**) 2 nd mode (11,511 Hz) and (**c**) 3 rd mode (34,022 Hz) and (**d**) 4 th mode (42,860 Hz).  
Figure 7: FEM-based resonant frequency tuning of the drive and sense resonant modes of MEMS gyroscope with applied tuning tension \(V_{t}\).