Simplified Lumped Parameter MEMS Tuning Fork Gyroscope Model
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## 1 Introduction to Simplified Lumped Parameter MEMS Tuning Fork Gyroscope Model

Figure 1: Generic 2-axis mass-spring-damper model

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 in-plane drive and sense modes

This gyro structure type is symmetric and decoupled having both the drive and sense modes in-plane. 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

 $LaTeX: M_{d}=M_{s}=m_{p}+m_{b}+m_{c}-m_{e}$

where:

$LaTeX: M_{d} / M_{s}$ is the mass of the structure,
$LaTeX: m_{p}$ is the plate mass,
$LaTeX: m_{b}$ is the beam mass,
$LaTeX: m_{c}$ is the comb mass, and
$LaTeX: m_{e}$ is the etch hole mass.
 $LaTeX: K_{d}=\frac{4Et\omega_{b}^{3}}{l_{b}^{3}}$

where:

$LaTeX: K_{d}$ is the in plane stiffness.
 $LaTeX: K_{s}=\frac{4Et^{3}\omega_{b}}{l_{b}^{3}}$

where:

$LaTeX: K_{s}$ is the out of plane stiffness.
 $LaTeX: B_{d}=\frac{\mu A}{\left( 1+2K_{n} \right)g}$ for rarefied flow

where:

$LaTeX: B_{d}$ is the slide film damping.
 $LaTeX: B_{s}=\frac{3\mu \left( L_{p} \right)^4 n_{h}}{8g^3} \left [ 4ln \left( \eta \right) - 3+\frac{4}{\eta^2}-\frac{1}{\eta^4} \right ] + \frac{8\pi \mu t n_{h} \left( L_{p}^{2} - L_{h}^{2} \right)^2}{L_{p}^{4}}$

where:

$LaTeX: B_{s}$ is the squeeze file damping,
$LaTeX: L_{p}$ is the pitch,
$LaTeX: L_{h}$ is the edge length of hole, and
$LaTeX: \eta=\frac{L_{p}}{L_{h}}$.

The in plane motion and the push-pull driving circuit is governed by the equation

 $LaTeX: M_{d}\ddot{Y}+B_{d}\dot{Y}+K_{d}Y=F_{e}$ (1)

where:

$LaTeX: F_{e}$ is teh electrostatic driving force defined by
 $LaTeX: F_{e}=\frac{2.28n\epsilon t V_{dc} V_{ac} \mbox{ sin} \left( \omega_{d} t \right)}{g_{c}}$

When a rotation rate Ωr 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

 $LaTeX: M_{s}\ddot{Z}+B_{s}\dot{Z}+K_{s}Z=F_{c}$ (2)

where the Coriolis force is

 $LaTeX: F_{c}=2M_{s}\Omega_{r}\dot{Y}\mbox{ sin} \left( \omega_{d} t \right)$

The (linearized) change in capacitance is

 $LaTeX: \Delta C_{b}=\frac{\epsilon A_{e}}{g^2}Z$ (3)

Reduced order models can be developed using the governing equations (Eqn. 1-3). The model can be developed to design and predict the performance in the time and frequency domain.

 $LaTeX: \frac{\Delta C_{b}}{\Omega_{r}}=\frac{\left( G_{1} \bar{G_{2}} G_{3} \right)}{\left[ M_{d}M_{s}s^4 + \left( M_{d}B_{s} + M_{s}B_{d} \right)s^3 + \left( M_{d}K_{s} + M_{s}K_{d} + B_{d}B_{s} \right)s^2 + \left( B_{d}K_{s} + B_{s}K_{d} \right)s + K_{d}K_{s} \right]}$ (4)

where:

$LaTeX: G_{1}=2M_{s}$,
$LaTeX: \bar{G_{2}}=\frac{2.28n\epsilon t V_{dc} V_{ac}}{g_{c}}$ and
$LaTeX: G_{3}=\frac{\epsilon A_{e}}{g^2}$
 $LaTeX: \frac{\Delta C_{b}}{\Omega_{r}}=\frac{2YQ_{s}}{\omega_{s}}G_{3}$ (5)

where:

$LaTeX: Y$ is the amplitude of the drive mode and
$LaTeX: Q_{s}=\frac{\sqrt{K_{s}M_{s}}}{B_{s}}$ is the quality factor of the sense mode.

## 4 Resources

### 4.1 Notes

1. Mohite et all, Abstract
2. Mohite et all, pg. 759