## 5 FEM Simulations
### Compensation of Thermally Induced Frequency Mismatch through Electrostatic Tuning  
The mismatch between the drive- and sense-mode frequency can cause the performance of the MEMS gyroscope to reduce drastically since even a slight mismatch can reduce the amplitude response of the resonant gyroscope significantly. As previously demonstrated in this paper, this error can be mitigated through the electrostatic frequency tuning, which allows tuning the sense-mode frequency to match the drive-mode one, guaranteeing the maximum possible mechanical sensitivity. Because thermal deformations induce air-gap variation between tuning-comb fingers that clearly influences the electrostatic tuning force generated by each fingers pair, a FEM-based modal analysis is performed in ANSYS APDL module to determine the exact value of tuning tension \(V_{t}\) to compensate the mismatch between drive and sense modes for the desired operating temperature range. The exact tuning-voltage value \(V_{t}\) is determined by applying the iterative approach of bisection method. At each iteration, the tuning voltage applied to the fixed electrodes is gradually increased and the drive- and sense-mode resonant frequencies are computed. If the driving frequency value is lower than the sensing one, it is concluded that the applied voltage is below the necessary voltage value to compensate the frequency mismatch. On the other hand, if the driving frequency is higher than the sensing one, it is concluded that the applied voltage is higher than the necessary voltage value. The interval between these two limits is continuously decreased until the voltage interval is smaller than a predetermined accuracy. Figure 15 provides the tuning voltage \(V_{t}\) compensating the difference between the drive- and sense-mode resonant frequency in the gyroscope operational temperature values between \(-40\)\({}^{\circ}\)C and \(100\)\({}^{\circ}\)C. As expected, the temperature increase with respect to the room temperature leads to an increase in the frequency mismatch and, thus, in the necessary tuning-voltage value \(V_{t}\). A mechanical sensitivity analysis is performed to investigate the effect of electrostatic tuning on the proposed MEMS gyroscope performances in the gyroscope operational temperature values between \(-40\)\({}^{\circ}\)C and \(100\)\({}^{\circ}\)C. A FEM-based full harmonic analysis is performed neglecting the effect of temperature variation on the viscous damping coefficients. This means that mechanical sensitivity is only influenced by the frequency mismatch between the modes. Two different harmonic analyses are performed: the first  
Figure 15: Tuning voltage \(V_{t}\) compensating the mismatch between drive and sense modes for a temperature variation in the gyroscope operational temperature values between \(-40\)\({}^{\circ}\)C and \(100\)\({}^{\circ}\)C, with a reference temperature \(T_{F}\) = 25\({}^{\circ}\)C.  
with no tuning voltage applied on the tuning electrodes, and the second with the tuning voltage \(V_{t}\), which compensates the frequency mismatch between the modes. The procedure is first to assign a uniform temperature varying in the range of \(-40\)\({}^{\circ}\)C to \(100\)\({}^{\circ}\)C to all structural nodes in the FEM model. The tuning voltage \(V_{t}\) and the DC actuation voltage are respectively applied to TRANS126 element nodes connected to the tuning and drive combs' moving fingers. A static analysis is performed with the pre-stress effects turned on and considering the ambient temperature \(T_{F}\) of \(25\)\({}^{\circ}\)C. A full harmonic analysis is then carried out by applying an angular velocity \(\Omega_{z}\) = 300 \({}^{\circ}\)/s in the \(z\)-axis and the AC actuation voltage with pre-stress effects due to applied DC voltages and temperature variations. Figure 16 shows the sensitivity percentage change computed with respect to the reference temperature.