## 4 Multiphysics FEM Model of the Proposed MEMS Gyroscope  
The performance of the proposed MEMS gyroscope design is verified through FEM based simulations in ANSYSTM. Figure 4 shows the meshed model of the MEMS gyroscope design. The adopted mesh has been designed to optimize as much as possible the required computational resources. The structure is meshed using 2D SHELL181 elements for the drive mass and sense frame, while BEAM188 elements are adopted to model the suspension systems, the comb-drive fingers, the tuning-comb fingers and the sense parallel plates. Since the rotational stiffness around the \(z\)-axis (ROTZ stiffness) of SHELL181 elements is an artificial value, it is inconsistent to connect only one node of a BEAM188 element to a SHELL181 element such that a rotational degree of freedom (DOF) of the beam element corresponds to the rotational around the \(z\)-axis (ROTZ) of the shell element. Accordingly, constraints equations are used to couple the rotational degree of freedom around the \(z\)-axis (ROTZ-DOF) of the single nodes shared by the BEAM188 and SHELL181 elements with the in-plane translations DOFs, namely UX and UY, of adjacent shell element nodes. Fixed supports are added at the end of sense spring beams connected to the external anchors. Figure 3: Sense Quality Factor variation: (**a**) effect of change in temperature in the range of \(-40\)\({}^{\circ}\)C to \(100\)\({}^{\circ}\)C and ambient pressure of 101 kPa (**b**) effect of change in pressure the range of 1 kPa to 101 kPa and ambient temperature of 25 \({}^{\circ}\)C.  
Air viscous damping effects in the drive and sense direction are included in the FEM model based on the analytical estimation of damping coefficients \(c_{x}\) and \(c_{y}\). These coefficients are specified directly as input and the corresponding values as real constants of the special element type having viscous damping characteristics, i.e., COMBIN14 element. Relying on the lumped mass-spring damper model representation shown in Figure 2, at least two COMBIN14 elements should be used: an element oriented according to the \(x\)-axis and connecting the drive mass and sense frame is used for model damping in the drive direction, while sense damping is represented by an element oriented according to the \(y\)-axis and connecting the sense frame with the external anchors. In order to obtain a reliable modeling of damping effects, COMBIN14 and SHELL181 elements are bound together by coupling the corresponding DOFs of the two contact nodes using CP command in ANSYSTM. The comb drives and tuning combs are modeled as a series of capacitors, as each couple of moving fingers forms a pair of parallel plates. The one-dimensional transducer (1-D) TRANS126 elements are used to model the capacitance of comb-drive-based actuators and tuning-comb fingers [24]. As shown in Figure 5, two elements are connected at the end of each moving finger, while the mechanical degrees of freedom of nodes connected to the stationary fingers are fixed. An electromechanical coupling is thus achieved between the distributed mechanical and lumped electrical domains. The TRANS126 element has two DOFs at each node, i.e., the translation in the nodal \(x\), \(y\) or \(z\)-direction and an electric potential. This allows modeling the capacitive response of a mechanical microstructure to a movement in a fixed direction. Since the fringing field along the lateral edges of the comb drive and tuning-comb fingers is neglected, indefinite capacitor theory is adopted. The capacity characteristics of the elements connected to the comb-drive fingers is a function of the \(x\)-displacement and it is set as a real constant, by imposing the coefficients obtained using the following equation:  
\[C(x)=\frac{x_{0d}}{y_{0d}}e_{0}t(\pm)\frac{x}{y_{0d}}e_{0}t \tag{17}\]  
Figure 4: Meshed finite element method (FEM) model of the proposed MEMS gyroscope design. where the sign \(+\) or \(-\) respectively depends on whether the capacity increases or decreases according to the displacement in the \(x\)-direction.