3.1. Kinematic Analysis of Type A and B
The spring in
Figure 1 represents the stiffness of the drive beams, the sense beams and the decoupling beams in the design of this paper. The stiffness of these beams is much less than that of the lever, frame and proof mass. On the other hand, the mass of the lever, frame, and proof mass is much larger than that of the beam. Therefore, the lever, frame and proof mass are simplified to inelastic mass and beam to massless spring. It is worth mention here that the lumped-parameter model used in this study is based on linear elastic material behavior, which is very widely used in the research of TFG [
6,
14,
21,
33,
34,
35]. Furthermore, the nonlinear analysis of the overall structure will be discussed in detail later. However, when the stiffness and mass of the beam are relatively large compared to other structures, some errors will occur and will not be adapted to this assumption.
According to
Figure 1a, we can obtain the four-degrees-of-freedom (4-dof) coupling vibration model of the MEMS tuning fork gyroscope with an anchored leverage mechanism, as illustrated in
Figure 2.
In operation, two proof masses and their respective drive mechanisms are electrostatically driven into anti-phase motion with the same amplitude along the drive direction by driving voltages imposed across the differential lateral comb electrodes on the drive mechanism. When an angular rate Ωz is applied, the anti-phase Coriolis acceleration of the proof mass induces linear anti-phase motions that are capacitively detected using differential parallel plate electrodes on the sense mechanism along the y-axis. The input angular rate Ωz can be calculated by the differential output.
Ideally, the structure is entirely symmetrical. Then, the two proof masses systems and damping coefficients are equal. The dynamics in the direction of sense mode are governed by the following:
Proof mass:
where
,
and
are the mass of the proof mass, decoupled frame and sense-mode frame, respectively.
is the damping coefficient of each tine in sense direction.
is the stiffness of the spring connected to
and the drive-mode frame.
represents the stiffness of the spring connected to decoupled frame and lever in the sense direction.
and
represent the force exerted on the lever by the decoupled frame in the left and right tines. Ω
z is the angular rate and the drive velocity
can be defined as:
Sense-mode frame:
where
is the stiffness of the spring connected to the anchor and the sense-mode frame;
represents the stiffness of the spring connected to the sense-mode frame and lever in the sense direction;
is the stiffness in the direction of the spring connected to the sense lever and sense-mode frame; and
and
represent the force exerted on the sense-mode frame by the lever in the left and right tines, respectively.
The leverage mechanism and coordinate relationships are as follows:
where
is the leverage rate (LR),
.
According to the above equations, the kinematic analysis of the dual-mass gyroscope in the sense direction can be expressed as:
Subtracting Equation (7) from Equation (6), we obtain
where
is the Coriolis force at angulate rate Ω
z, which can be given as:
where
is the driving force, and
,
, and
are the resonant frequency, total stiffness, and quality factor, respectively, in the anti-mode.
Adding Equations (6) and (7), we obtain:
Since the vibration output of in- and anti-phase modes needs to be explored, a coordinate transformation is made as follows:
Substituting Equation (11) into Equations (8) and (9), we obtain:
where
,
,
,
, and
, in which
and
are the defined resonant frequencies and
and
are the quality factors of the anti- and in-phase motions, respectively, and
is the total mass in the sense direction.
Equation (12) can be represented as a matrix:
where
,
,
,
, and
.
The natural frequency can be obtained by using the characteristic equation:
where
and
are the first- and second-order resonant frequency, respectively.
The modal superposition technique is used to acquire the steady-state response by solving Equation (13):
where
,
, and
are the magnification factors of amplitude, phase angle, and frequency ratio, respectively.
When
, we can obtain that:
According to Equation (16), the differential detection output of the tuning fork micromechanical gyroscope is:
The mechanical sensitivity of the type A architecture can be obtained from Equation (17):
The 4-dof coupling vibration model of the MEMS gyroscope with equal displacement capacitance detection is shown in
Figure 3.
The differential detection output and mechanical sensitivity of the type B architecture can be acquired by the same technique:
where
,
,
, and
are the quality factor, total stiffness, total mass, and resonant frequency of the anti-phase mode in the drive direction, respectively.
3.2. Optimization Analysis of LR
In order to obtain a more substantial improvement of mechanical sensitivity, it is effective to reduce and increase as much as possible.
Here, the dimensionless parameters
,
, and
are defined in terms of stiffness ratio (SR):
is the stiffness ratio of the power arm (SRPA) and
is the stiffness ratio of the resistance arm (SRRA). The three parameters are given by:
Obviously, tiny LR will lead to an insignificant amplification effect of leverage, which will not be important in truly amplifying the mechanical sensitivity. On the other hand, extremely large LR will increase the detection mode stiffness, thus making it difficult for the tiny Coriolis force to drive the detection mode effectively. Without considering the effect of the quality factor during the design phase, according to Equation (18), we obtain:
When
, we can obtain that:
Equation (23) shows that an optimal solution to this problem exists and different parameters have different effects on the optimal leveraged magnification. Furthermore, the square of is directly proportional to and inversely proportional to and .
3.3. Analysis of IRMS
Type A and type B adopt the same structure and springs, except for the connection between the proof mass and detection frame, and the same fabrication and packaging technique. It is assumed that the damping ratios of the two types of TFG are equal.
The dimensionless parameters
and
are defined by:
where
denotes the improvement rate of mechanical sensitivity (IRMS) and
denotes the mass ratio (MR).
Substituting Equations (18) and (20) into Equation (24), we obtain:
It can be seen that the IRMS is mainly determined by the stiffness ratios
,
, and
and mass ratio
. By solving the partial derivative of function
with respect to these variables, we obtain that:
In general, the values of the stiffness ratios
,
, and
and mass ratio
are greater than 0 and less than 1. From Equations (26)–(29), we can find that:
From Equation (30), the improvement rate of mechanical sensitivity monotonically increases with increasing and , but decreases with increasing and . Within a reasonable range, and are as small as possible and and are as large as possible. The leverage mechanism can improve the displacement of sense frame effectively and obtain a huge improvement in terms of mechanism sensitivity.
The change of and will lead to a corresponding change of the mechanism architecture with no leverage. Furthermore, and are system parameters, not just leverage mechanism parameters, which will be analyzed in later studies. Here, four architectures are designed to explore the impact of and on IRMS. One is based on type B architecture and the other three are based on type A architecture, defined as types A1, A2 and A3. The of type A2 and the of type A3 are slightly larger than those of type A1. This can be achieved by intentionally increasing the spring width of the leverage mechanism in the sense direction.