## 5 FEM Simulations
### FEM Analysis of Temperature Variations on Structural Stability  
The performance of the proposed MEMS gyroscope can be affected by operating-temperature variations because they can cause resonant frequency shift [25] of drive and sense modes, which may result in a drastic decrease in the output signal gain. The Young's modulus of the silicon structural material, which was used to design the proposed MEMS gyroscope, changes with temperature and this results in changes in the resonant frequency values. The analytical expression for the effect of operating temperature on [PERSON]'s modulus value of the silicon material is given as [27]:  
\[E(T)=E_{0}-BTexp\left(-\frac{T_{0}}{T}\right) \tag{19}\]  
where \(E_{0}\) is the Young's modulus at reference temperature, \(T\) is the temperature, and \(B\) and \(T_{0}\) are the constants that are calculated as 15.8 MPa and 317 K, respectively. The silicon Young's modulus value varies from 169.03 to 168.93 GPa only in the MEMS gyroscope operational temperature values between \(-40\)\({}^{\circ}\)C to 100 \({}^{\circ}\)C. Accordingly, the effect of temperature variation on Young's modulus and, thus, on stiffness variation is in practice negligible over the desired operating temperature range. However, the device operating temperature from \(-40\)\({}^{\circ}\)C to 100 \({}^{\circ}\)C may lead to thermal deformations, which are expansions and contractions in the microstructure. For the MEMS gyroscope design, these thermal deformations may eventually lead to a change in the air gap and structural planarity of the comb-drive actuators, electrostatic tuning combs and parallel plate-based capacitive sensors. In addition, temperature variations may result in thermal stresses. A FEM-based thermal analysis is performed in the ANSYS APDL module to analyze the resonant frequency changes due to thermal deformations and thermally induced stresses. First, a uniform temperature, varying in the range \(-40\)\({}^{\circ}\)C to 100 \({}^{\circ}\)C, is assigned to all the structural nodes in the FEM model. A static analysis is then performed with pre-stress effects [26] turned on and considering the ambient temperature \(T_{F}\) = 25 \({}^{\circ}\)C as a reference temperature. Figure 10(a),b shows the thermally induced localized stresses resulting from the FEM-based static analysis, at \(-40\)\({}^{\circ}\)C and 100 \({}^{\circ}\)C, respectively. In the region where the sensing spring beams are connected to the structure, there is a large concentration of thermal stress, while nearly no thermal stress is induced in the regions of driving spring beams and in the rest of the structure. Figure 11(a),b shows structural thermal deformation results at \(-40\)\({}^{\circ}\)C and 100 \({}^{\circ}\)C, respectively. The thermal deformation at \(-40\)\({}^{\circ}\)C results in a structural contraction with a maximum value of 0.47 \(\upmu\)m in the sensing microbeams, while at 100 \({}^{\circ}\)C the gyroscope structure expands with a maximum expansion 0.545 \(\upmu\)m in the sensing microbeams. These  
Figure 11: Thermally induced Von Mises stress in the proposed dual mass MEMS gyroscope (**a**) at \(-40\)\({}^{\circ}\)C, (**b**) at 100 \({}^{\circ}\)C, with reference temperature \(T_{F}\) of 25 \({}^{\circ}\)C.  
results show that although the mechanical thermal deformation in the sensing microbeams is high, the mechanical deformation in the drive and sense frame and in the driving microbeams is negligible, with a maximum value of only 0.104 \(\upmu\)m at \(-\)40 \({}^{\circ}\)C and 0.121 \(\upmu\)m at 100 \({}^{\circ}\)C.  
The static analysis is followed by a modal analysis. Prior to performing the modal analysis, in the FEM model the nodal coordinates are updated with the deflections from the previous static analysis and hence the pre-stress effects with the stresses stored from the static analysis are applied to the modal analysis. Figure 13 shows the FEM-based pre-stressed modal analysis results.