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Contents
1 Introduction to Simplified Lumped Parameter MEMS Tuning Fork Gyroscope Model
See Also 

The basics of vibratory MEMS gyro are presented in Vibratory MEMS Gyroscopes. The standard modeling technique is presented in MEMS Gyro Modeling.
Typical is described concisely in Suhas et all^{[1]}, quote
Among various MEMS sensors, a rate gyroscope is one of the most complex sensors from the design point of view. The gyro normally consists of a proof mass suspended by an elaborate assembly of beams that allow the system to vibrate in two transverse modes.
In most cases, the FEM analysis becomes prohibitive and one resorts to equivalent electrical circuit simulations... Here, we present a simplified lumped parameter model of the tuning fork gyro and show how easily it can be implemented using a generic tool like Simulink. The results obtained are compared with those obtained from more elaborate and intense simulations... The comparison shows that lumped parameter Simulink model gives equally good results...
The standard proof mass model is shown in Figure 1.
1.1 Main Design Parameters for Vibratory Gyroscopes
The main design parameters are stiffness of the structure, damping coefficients and system response to applied rate. The stiffness is key to mode matching between the resonant axes and the driving frequency. Achieving the desired performance is an iterative modeling and simulation process.
2 Gyroscope Structures
As described in Vibratory MEMS Gyroscopes, mode matching of the resonance is key to maximizing the signal to noise ratio (SNR). The quality (Q) factor is a measure of how large the resonant amplitude will be; when the sense and drive resonances are equal, the output signal is amplified be the Q factor. The larger the Q factor the better the SNR will be. The gyro structure can have a large impact on the Q factor.
2.1 Structure with inplane drive and sense modes
This gyro structure type is symmetric and decoupled having both the drive and sense modes inplane. In this structure the sensing is done with a pair of comb fingers identical to the fingers used for driving the structure. For more details on this structure and design see Suhas et all^{[2]}.
3 Modeling and Simulation of the Tuning Fork Gyroscope
As seen in Figure 1, the suspended proof mass acts as a single degree of freedom (DOF) system for both the drive and sense motions. The lumped parameters of the structure, mass and stiffness, are obtained using energy methods. The relevant boundary conditions are of the suspended beam model are the anchored end and the free end connected to the proof mass.
The lumped parameters are
where:
 is the mass of the structure,
 is the plate mass,
 is the beam mass,
 is the comb mass, and
 is the etch hole mass.
where:
 is the in plane stiffness.
where:
 is the out of plane stiffness.
for rarefied flow 
where:
 is the slide film damping.
where:
 is the squeeze file damping,
 is the pitch,
 is the edge length of hole, and
 .
The in plane motion and the pushpull driving circuit is governed by the equation
(1) 
where:
 is teh electrostatic driving force defined by
When a rotation rate Ω_{r} is applied about the rotational axis. The Coriolis force causes motion along the sense direction (normal to the plane of excitation). The motion caused by the Coriolis force is
(2) 
where the Coriolis force is
The (linearized) change in capacitance is
(3) 
Reduced order models can be developed using the governing equations (Eqn. 13). The model can be developed to design and predict the performance in the time and frequency domain.
(4) 
where:
 ,
 and
(5) 
where:
 is the amplitude of the drive mode and
 is the quality factor of the sense mode.
4 Resources
 Mohite, S., Patil, N., Pratap, R., "Design, modeling and simulation of vibratory micromachined gyroscopes.", Journal of Physics: Conference Series 34 (2006), pp. 757763.
 Thermoelastic Damping and Engineering for High Q MEMS Resonators
 Boa, M. 2004 Micro Mechanical Transducers. 2nd. Elsevier. ISBN 044450558X
 Gaura, E., and Newman, R 2006 Smart MEMS and Sensor Systems. 1st. Imperial College Press. ISBN 1860944930