## 5 FEM Simulations
### Dynamic Analysis  
To study the dynamic frequency response of the MEMS gyroscope structure, a FEM-based harmonic analysis is performed in ANYS APDL module. The drive mass is excited by applying a DC voltage, superimposed on a small-signal AC voltage, to the comb-drive-based actuators, while an electrostatic DC tuning voltage is applied to tuning electrodes for the compensation of the frequency difference between the two modes. For frequency behavior simulation, a pre-stressed static analysis with actuation and electrostatic tuning DC voltage followed by a pre-stressed full harmonic analysis with an AC excitation is carried out. When the device is subjected to an angular velocity in the \(z\)-axis, the sense-mode frequency response can be obtained activating the Coriolis effect in ANSYS. Figure 8 shows a comparison between the FEM-based and analytical frequency-based displacement amplitude of the gyroscope structure in the drive axis, obtained with a drive comb voltage of 50 V DC and 5 V AC. The results show that the FEM-based frequency response is slightly shifted in frequency compared with the one obtained analytically. At the corresponding drive resonant frequency \(\Omega_{d}\), the FEM-based and analytical frequency response amplitude are 46.93 \(\upmu\)m and 44.03 \(\upmu\)m, respectively.  
Figure 9 shows a comparison between the FEM-based frequency response amplitudes in the sense direction, obtained without and with the electrostatic tuning-voltage application and considering an angular velocity of \(\Omega_{z}=300^{\circ}\)/s.  
When no tuning tension is applied to tuning electrodes, the sense response has two distinct peaks due to the frequency mismatch between the modes, which results in a maximum oscillation amplitude of 0.0411 \(\upmu\)m. When the tuning tension \(V_{t}=14.903\) V guaranteeing the perfect match between resonant frequencies is applied, the sense response has only one peak with the maximum oscillation magnitude of 0.13930 \(\upmu\)m. By considering a linear relation between the input angular velocity and the displacement in the sense direction, the mechanical sensitivity for the proposed MEMS gyroscope increases from 1.370 \(\times 10^{-4}\) to 4.643 \(\times 10^{-4}\)\(\upmu\)m/deg/s, meaning that the tuning application allows an increase in the mechanical sensitivity by 238.9%. Figure 10 shows a comparison between FEM-based and analytical frequency response magnitude of the MEMS gyroscope structure in the sense direction obtained without (Figure 10a) and with (Figure 10b) the electrostatic tuning-voltage application. In both cases, the FEM-based and analytical frequency response amplitude are slightly shifted in frequency. The analytical drive and sense frame displacement in the sense direction corresponding to the rotation-induced Coriolis force is 0.0440 \(\upmu\)m and the mechanical sensitivity is 1.467 \(\times 10^{-4}\)\(\upmu\)m/deg/s without the electrostatic tuning application, while in the case of a perfect match between modes the sense oscillation amplitude and mechanical sensitivity become 0.1205 \(\upmu\)m and 4.017 \(\times 10^{-4}\)\(\upmu\)m/deg/s, respectively.  
Figure 8: Comparison between the FEM-based and analytical frequency response amplitudes in the drive direction for the proposed dual mass MEMS gyroscope (logarithmic scale), obtained with 50 V DC and 5 V AC actuation voltage.  
Figure 10: Comparison between the FEM-based and analytical frequency response amplitude in the sense direction for the proposed dual mass MEMS gyroscope (logarithmic scale), obtained with 50 V DC and 5 V AC actuation voltage and \(\Omega_{z}=300^{\circ}/\)s, (**a**) without electrostatic tuning, (**b**) with electrostatic tuning.  
Figure 9: Comparison between the FEM-based frequency response amplitudes in the sense direction for the proposed dual mass MEMS gyroscope (logarithmic scale) obtained with a tuning voltage \(V_{l}=0\) and \(V_{l}=14.903\) V, with 50 V DC and 5 V AC actuation voltage and \(\Omega_{z}=300^{\circ}/\)s.