Establishment and analysis of coupled dynamic model for dual-mass silicon micro-gyroscope

This paper presents a coupled dynamic model for a dual-mass silicon micro-gyroscope (DMSG). It can quantitatively analyze the influence of left-right stiffness difference on the natural frequencies, modal matrix and modal coupling coefficient of the DMSG. The analytic results are verified by using the finite element method (FEM) simulation. The model shows that with the left-right stiffness difference of 1%, the modal coupling coefficient is 12% in the driving direction and 31% in the sensing direction. It also shows that in order to achieve good separation, the stiffness of base beam should be small enough in both the driving and sensing direction.


Introduction
The Dual-mass Silicon Micro-gyroscopes (DSMG) are employing electrostatic actuation and capacitive detection. Limited by the current MEMS fabrication technology, the stiffness of the support beams for the left and right structures of DSMG will be different with the fabrication error [1]. The left-right stiffness difference (stiffness difference) will change the mechanical properties especially the bias and vibration sensitivity of the gyroscope. Therefore, it is important to analyze the impact of stiffness difference on bias and vibration sensitivity of the DMSG.
The fabrication error impacting on the driving direction of the DSMG had been analyzed in some literatures [2][3][4]. Take into considered of both the drive and sense mode, a non-ideal dynamic model was studied in the literatures [5][6]. However, the stiffness difference of the DSMG in the above mentioned papers was only considered in its driving direction and the modal coupling coefficient was not be proposed.
In this paper, firstly, the coupled dynamic model is proposed for a DMSG. Second, the natural frequencies and modal matrix are calculated. Then the modal coupling coefficient is deduced from the proposed modal matrix. Additionally, the coupled dynamic model can be solved by modal decoupling method. At last, the reliability of the theory is verified by finite element method (FEM) simulation. Fig.1 is the schematic of a dual-mass silicon micro-machined gyroscope. By applying the alternating voltage on the driving comb capacitor, the left and right mass will move in opposite direction along Xaxis (driving direction). When the sensor rotates about Z-axis, the resulting Coriolis force causes the left and right mass to move in opposite direction along Y-axis (sensing direction). The relative motion between the movable detection comb and the fixed detection comb forms the differential capacitance for detection. Ideally, the amount of differential detection capacitance is proportional to the input angular rate.

Structure and operation principle
Herein, msl and msr are the left and right proof masses, mdl and mdr are the left and right drive comb masses, mb and Ib are the proof mass of the base beam, kdl=kd+Δkd/2 and kdr=kd-Δkd/2 are the bending stiffness of the left and right drive beams, ksl=ks+Δks/2 and ksr=ks+Δks/2 are the bending stiffness of the left and right sense beams. kbd and kbs are the bending stiffness of the base beam in its driving and sensing direction. The parameters of the DMSG designed are list in the Tab. 1.
Here r1 and r2 are defined as the mode function. Now, the modal coupling coeffecient which reflects the coupling between the in-phase and anti-phase motion can be express as (7) Finally, the system can be decoupling by applying the modal matrix. As a result the motion equations change to (

8) where [Mp]=[A] T [M] [A] and [Kp]=[A] T [K] [A] are the modal mass and modal stiffness. {q}=[A] T {u}
is the modal freedom. As a consequence, the motion equations can be solved independently.

Drive mode
The natural frequencies and mode functions in its driving direction are shown as:

Sense mode
The natural frequencies and mode functions in its sense direction are shown as: Obviously, the relation in (15) means that the in-phase motion and the anti-phase motion are complete independence. According to (9)-(12), Δkd and Δks are determined by the processing precision. kd and ks determine the drive mode and sense mode. As a result, in order to achieve good separation, the stiffness of base beam should be small enough in both the driving and sensing direction. However, these low-rigidity beams present challenges and often introduce other problems.

Fem simulation
The Finite element software ANSYS was employed to verify the coupling dynamic model of the DSMG. The schematic of a dual mass silicon micro-machined gyroscope is shown in Fig.3. Frist, the modal analysis of the DSMG structure with stiffness difference is done to verify the natural frequencies. The natural frequency variation due to the stiffness difference of 1% is shown in Tab. 2.
The data in Tab. 2 shows that the natural frequency variations due to the stiffness difference of 1% are 0.5Hz in its driving direction and 1.5Hz in its sensing direction. It also shows that the natural frequency variation can be accurately represented by the coupling dynamic model.  Then the harmonic response simulation is done to verify the modal coupling coefficient. Fig. 4 shows the frequency response of the anti-phase drive mode. The frequency response of other modes has the similar results. Fig. 4 shows the in-phase and anti-phase displacements are not independent from each other. Fig. 5 shows the modal coupling coefficient of theory and simulation in each mode. According to Fig. 5, with the stiffness difference of 1%, the modal coupling coefficient is 12% in the driving direction and 31% in the sensing direction. It also shows that the theoretical result and simulation result are in good agreement.

Conclusion
Base on the structural characteristics, a coupling dynamic model is proposed and established for the dual mass silicon micro-gyroscope in this paper. The effect of the stiffness difference on the natural frequencies and mode functions are demonstrated. The modal coupling coefficient is calculated based on the mode functions. The natural frequency variations due to the stiffness difference of 1% are 0.5Hz in its driving direction and 1.5Hz in its sensing direction. The modal coupling coefficient is 12% in the drive direction and 31% in the sense direction due to the stiffness difference of 1%. These theoretical results are verified by the FEM simulation. The good agreement between the theory result and simulation result prove the accuracy of our model and its theory.