This project report by Nemish Kanwar focuses on the simulation and sensitivity analysis of MEMS vibratory gyroscopes, emphasizing their use in consumer products due to their compact size and low cost. The document includes detailed analyses of the gyroscope's design, including mathematical models, damping calculations, and simulation results using MATLAB. Acknowledgments highlight the guidance from professors and peers throughout the project, concluding with a comprehensive overview of the system's dynamics and operating principles.

1. A Project Report On
MEMS Vibratory Gyroscope
Simulation and Sensitivity
Analysis
SUBMITTED TO:
ASST PROF SACHIN U. BELGAMWAR
MECHANICAL ENGINEERING
BITS PILANI, PILANI CAMPUS
BY
NEMISH KANWAR
2012A4PS305P

2. A REPORT
ON
MEMS Vibratory Gyroscope Simulation and Sensitivity Analysis
By
Nemish Kanwar
2012A4PS305P
Prepared in the partial fulfilment of
Study Oriented Project
(ME F266)
Under the guidance of
Asst Prof SACHIN U. BELGAMWAR
(MECHANICAL ENGINEERING)
BITS PILANI, PILANI CAMPUS
June, 2015
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE
PILANI (RAJASTHAN)
1

3. CERTIFICATE
This is to certify that the work embodied in this report has been undertaken by: BITS ID:
2012A4PS305P, Nemish Kanwar, under my supervision, for the course ME F266 titled Study
Project at BITS Pilani, Pilani Campus references made to the authors of the original work.
Asst Prof Sachin U. Belgamwar
Mechanical Engineering
BITS Pilani, Pilani Campus
2

4. ACKNOWLEDGEMENTS
I want to sincerely thank Mr. SACHIN U. BELGAMWAR, Professor, Mechanical Engineering
Department, BITS Pilani, Pilani Campus for giving us an opportunity to make this esteemed
project on the topic: “MEMS Vibratory Gyroscope Simulation and Sensitivity Analysis”
under his guidance and support.
I would also like to extend my sincere thanks to Dr. Ankush Jain, scientist at CEERI Pilani for
finding out time from his busy schedule to assist and guide me in this project. This project
wouldn’t have been possible without him.
At last but not the least I would like to thank my other two group members, Varun Sharma and
Akershit Agarwal, with whom I used to discuss all the doubts and most of the times got my
doubts cleared.
Nemish Kanwar
2012A4PS305P
3

5. ABSTRACT
MEMS based gyroscopes have gained in popularity for use as rotation rate sensors in
commercial products like cars and game consoles because of their cheap cost and small sized
compared to traditional gyroscopes. The MEMS gyroscope consists of the basic mechanical
structure an electronic transducer to excite the system as well as an electronic sensor to detect the
change in the mechanical structures modal shape. Among various MEMS sensors, a rate
gyroscope is one of the most complex sensors from design point of view. The gyroscope consists
of a proof mass suspended by an assembly of cantilever beams, allowing system to vibrate in 2
transverse modes. In this paper, simplified lumped parameter model of tuning fork gyroscope is
modeled, and simulated using MATLAB. The structure of Gyroscope was analyzed for drive
frequency to gain maximum amplitude. The device was then simulated for various angular rates
for derived frequency to obtain the sensitivity of device.
4

6. TABLE OF CONTENTS
CERTIFICATE............................................................................................................................... 2
ACKNOWLEDGEMENTS............................................................................................................ 3
ABSTRACT.................................................................................................................................... 4
TABLE OF CONTENTS................................................................................................................ 5
TABLE OF FIGURE...................................................................................................................... 6
LIST OF TABLE............................................................................................................................ 6
1.GYROSCOPE.............................................................................................................................. 7
2.HISTORY .................................................................................................................................... 7
3.PRINCIPLE ................................................................................................................................. 7
4.DIMENSIONS OF GYROSCOPE.............................................................................................. 8
5.MASS CALCULATIONS........................................................................................................... 9
6.SPRING CALCULATIONS........................................................................................................ 9
7.DAMPING CALCULATIONS ................................................................................................... 9
7.1.FOR SLIDE FILM DAMPING........................................................................................... 10
7.2.FOR SQUEEZE FILM DAMPING.................................................................................... 10
8.DRIVING FORCE..................................................................................................................... 12
9.NATURAL FREQUENCY ....................................................................................................... 13
10.DAMPING RATIO.................................................................................................................. 13
11.EQUATION OF MOTION...................................................................................................... 15
11.1DRIVE AXIS (Z-AXIS)..................................................................................................... 15
11.2SENSE AXIS (Z-AXIS)..................................................................................................... 15
11.3.OUTPUT (CURRENT)..................................................................................................... 16
12.RESULTS ................................................................................................................................ 17
13.BIBLIOGRAPHY.................................................................................................................... 20
14.APPENDIX.............................................................................................................................. 21
14.1.CODE................................................................................................................................ 21
5

7. TABLE OF FIGURE
Figure 1: Gyroscope........................................................................................................................ 8
Figure 2: beta convergence plot.................................................................................................... 11
Figure 3: Comb drive.................................................................................................................... 12
Figure 4: Code for BETA convergence ........................................................................................ 21
Figure 5: Code for MEMS device initialization ........................................................................... 22
Figure 6: Code for MEMS device Vibration analysis .................................................................. 22
Figure 7: Drive frequency v/s Amplitude response in sense direction......................................... 17
Figure 8: Comparison of Sense Amplitude for same and different natural frequencies in drive and
sense directions............................................................................................................................. 18
Figure 9: Sense Response v/s Angular rate................................................................................... 18
Figure 10: Sense Current v/s Angular rate.................................................................................... 19
LIST OF TABLE
Table 1: List of Constants............................................................................................................. 13
6

8. GYROSCOPE
Gyroscopes are Physical sensors used to measure Angular rate of an object relative to inertial
frame of reference[1]. The term gyroscope is attributed to French physicist Leon Foucault.
Gyroscopes can track an objects angular rate and orientation and enable Heading Reference
System (AHRS).
Combining 3 gyroscopes and 3 accelerometers in a 6-axis Inertial Measurement Unit (IMU)
enables Inertial Navigation Systems (INS) for navigation and guidance.
Gyroscopes are divided into
1. Rate Gyroscopes which measures the angular rate of object
2. Angle gyroscopes which measures angular position
Essentially all existing Micro-Electro-Mechanical-Systems (MEMS) gyroscopes are of the rate
measuring type and are typically employed for motion detection. When rate gyroscope is used to
track orientation, its output signal is integrated over time to give orientation angle
Micromachined inertial sensors, consisting of accelerometers and gyroscopes, are one of the
most important types of silicon- based sensors.[2] Microaccelerometers alone have the second
largest sales volume after pressure sensors. It is believed that gyroscopes will soon be mass-
produced at similar volumes once manufacturers are able to meet a $10 price target
HISTORY
As early as the 1700s, spinning devices[3] were being used for sea navigation in foggy
conditions. The more traditional spinning gyroscope was invented in the early 1800s, and the
French scientist Jean Bernard Leon Foucault coined the term gyroscope in 1852. In the late
1800s and early 1900ís gyroscopes were patented for use on ships. Around 1916, the gyroscope
found use in aircraft where it is still commonly used today. Throughout the 20th century
improvements were made on the spinning gyroscope. In the 1960s, optical gyroscopes using
lasers were first introduced and soon found commercial success in aeronautics and military
applications. In the last ten to fifteen years, MEMS gyroscopes have been introduced and
advancements have been made to create mass-produced successful products with several
advantages over traditional macro-scale devices.
PRINCIPLE
The underlying physical principle is that a vibrating object tends to continue vibrating in the
same plane as its support rotates. In the engineering literature, this type of device is also known
as a Coriolis vibratory gyro because as the plane of oscillation is rotated, the response detected
by the transducer results from the Coriolis term in its equations of motion.
𝐹𝐹 = 2𝑚𝑚Ω × 𝑣𝑣 [1]
7

9. Where Ω and 𝑣𝑣 are the angular velocity and velocity of the device. It is observed that Coriolis
force is perpendicular to drive velocity. So, motion in the force direction can be used to calculate
Angular rate of the device
DIMENSIONS OF GYROSCOPE
FIGURE 1: GYROSCOPE
Proof Mass Dimensions, 𝑎𝑎 × 𝑏𝑏 × ℎ = 1000𝜇𝜇𝜇𝜇 × 1000𝜇𝜇𝜇𝜇 × 5𝜇𝜇𝜇𝜇
Clearance from base plate, ℎ0 = 5𝜇𝜇𝜇𝜇
Clearance between comb drives, 𝑔𝑔 = 2𝜇𝜇𝜇𝜇
Number of combs, 𝑛𝑛 = 10
Beam dimensions, 𝐿𝐿 × 𝑤𝑤 × 𝑡𝑡 = 300𝜇𝜇𝜇𝜇 × 15𝜇𝜇𝜇𝜇 × 5𝜇𝜇𝜇𝜇
Substrate Properties:
Silicon Single Crystal: Density, 𝜌𝜌 = 2330𝑘𝑘𝑘𝑘/𝑚𝑚3
Young’s Modulus, 𝐸𝐸 = 1.79 × 1011
𝑃𝑃𝑃𝑃
Surrounding properties:
Nitrogen: Pressure, 𝑝𝑝 = 760𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
Viscosity, 𝜇𝜇 = 1.791 × 10−5
𝑘𝑘𝑘𝑘/𝑚𝑚𝑚𝑚
8

10. Dielectric constant, 𝜖𝜖𝑟𝑟 =1.0053
MASS CALCULATIONS
Mass equals density times volume
𝑚𝑚 = 𝜌𝜌 × 𝑎𝑎 × 𝑏𝑏 × ℎ = 1.1650 × 10−8
𝑘𝑘𝑘𝑘
SPRING CALCULATIONS
According to Hooke’s Law, Force required to extend a spring by a distance,𝑥𝑥 Force, 𝐹𝐹required is
directly proportional to stiffness. This law is named after 17th
century British physicist Robert
Hooke.
𝐹𝐹 = −𝑘𝑘𝑘𝑘
Each beam behaves as a cantilever beam in both drive and sense axis, spring constant for a
cantilever beam can be found out [4]
Spring Constant for a
cantilever beam
𝑘𝑘 =
6𝐸𝐸𝐸𝐸
𝐿𝐿3 [2]
𝐼𝐼𝑥𝑥 =
1
12
𝑤𝑤3
𝑡𝑡 =
1
12
∗ (15𝑒𝑒 − 6)3
∗ (5𝑒𝑒 − 6) = 1.4063 × 10−21
𝑚𝑚4
𝑘𝑘 =
6𝐸𝐸𝐼𝐼𝑥𝑥
𝐿𝐿3
For single beam 𝑘𝑘𝑥𝑥 = 55.9375𝑁𝑁/𝑚𝑚, since, there are 4 beams, 𝐤𝐤 𝐱𝐱𝐱𝐱𝐱𝐱𝐱𝐱 = 𝟐𝟐𝟐𝟐𝟐𝟐. 𝟕𝟕𝟕𝟕𝟕𝟕/𝐦𝐦
Similarly,𝐼𝐼𝑧𝑧 =
1
12
𝑡𝑡3
𝑤𝑤 = 1.5625 × 10−22
𝑚𝑚4
𝐤𝐤𝐳𝐳𝐳𝐳𝐳𝐳𝐳𝐳 = 𝟐𝟐𝟐𝟐. 𝟖𝟖𝟖𝟖𝟖𝟖𝟖𝟖𝟖𝟖/𝐦𝐦
DAMPING CALCULATIONS
The air enclosing the proof mass will damp the system decreasing its potential amplitude.
In kinetic theory the mean free path of a particle, such as a molecule, is the average distance the
particle travels between collisions with other moving particles.
Mean free path,𝜆𝜆 = 𝑝𝑝 × 5.1 × 10−5
(p in torr)
= 0.0388𝑚𝑚
[3]
The Knudsen number 𝐾𝐾𝐾𝐾 is a dimensionless number defined as the ratio of the molecular mean
free path length to a representative physical length scale. This length scale is the clearance
between base plate and proof mass. The number is named after Danish physicist Martin
Knudsen.
9

11. Knudsen number,𝐾𝐾𝐾𝐾 =
𝜆𝜆
ℎ0
= 7752 [4]
Slide and squeeze film damping models play important role in the dynamics of the micro-electro-
mechanical systems (MEMS) devices as they are important design parameters, which
significantly affect the frequency-domain behavior and the quality factor (Q-factor) of the
vibrating MEMS devices for both lateral and vertical motions.
For slide film Damping
In our model the flow of fluid below will be a case of couette flow. In fluid dynamics, Couette
flow is the laminar flow of a viscous fluid in the space between two parallel plates, one of which
is moving relative to the other. The flow is driven by virtue of viscous drag force acting on the
fluid and the applied pressure gradient parallel to the plates. This type of flow is named in honor
of Maurice Marie Alfred Couette.
The effective coefficient of viscosity can be determined[5]
𝜇𝜇𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 =
𝜇𝜇
1 + 2𝐾𝐾𝐾𝐾 + 0.2𝐾𝐾𝑛𝑛0.788 𝑒𝑒−
𝐾𝐾𝐾𝐾
10
[5]
= 1.1551 × 10−9
𝑘𝑘𝑘𝑘/𝑚𝑚𝑚𝑚
To determine Damping Coefficient[6]
Area of Plate, 𝐴𝐴 = 1000𝜇𝜇𝑚𝑚2
Damping coefficient, 𝐜𝐜𝐱𝐱 = 𝜇𝜇𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ×
𝐴𝐴
ℎ0
= 𝟐𝟐. 𝟑𝟑𝟑𝟑𝟑𝟑𝟑𝟑 × 𝟏𝟏𝟏𝟏−𝟏𝟏𝟏𝟏
𝐍𝐍 − 𝐬𝐬/𝐦𝐦
For squeeze film damping
Squeeze film effects naturally occur in dynamic MEMS structures because most of these
structures employ parallel plates or beams that trap a very thin film of air or some other gas
between the structure and the fixed substrate. An accurate estimate of the effect of squeeze film is
important for predicting the dynamic performance of such devices.
To determine damping coefficent[7][8]
𝜇𝜇𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 =
𝜇𝜇
1 + 9.638𝐾𝐾𝑛𝑛1.159
[6]
𝜇𝜇𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 5.7713 × 10−11
𝑘𝑘𝑘𝑘/𝑚𝑚𝑚𝑚
10

12. 𝒄𝒄𝒛𝒛 = 𝜇𝜇𝑧𝑧 × 𝑎𝑎 ×
𝑏𝑏3
ℎ0
3 𝛽𝛽(1)
[7]
Where 𝛽𝛽(1) = �1 −
192
𝜋𝜋5
∑
1
𝑛𝑛5 tanh �
𝑛𝑛𝑛𝑛
2
�∞
𝑛𝑛=1,3,5 � = 𝟎𝟎. 𝟒𝟒𝟒𝟒
FIGURE 2: BETA CONVERGENCE PLOT
Squeeze film damping coefficient was calculated = 𝟏𝟏. 𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗 × 𝟏𝟏𝟏𝟏−𝟕𝟕
𝐍𝐍 − 𝐬𝐬/𝐦𝐦
11

13. DRIVING FORCE
The force required to drive the proof mass in constant motion is provided by comb drive as
shown below. The fingers on the comb drive and the side of the proof mass increase the surface
area of the capacitor, which increases the capacitance
FIGURE 3: COMB DRIVE
Capacitance of one finger of the proof mass is given by
𝐶𝐶 =
2𝜖𝜖𝑟𝑟 𝜖𝜖0 𝑙𝑙ℎ
𝑔𝑔
[8]
Where 𝑙𝑙 is initial overlap between fingers of proof mass is, ℎ is the height of gyroscope and 𝑔𝑔 is
the gap between the fingers. 𝜖𝜖0 and 𝜖𝜖𝑟𝑟 are absolute permittivity and dielectric constant
Electrostatic Force [9] on proof mass is given by
Energy stored in Capacitor 𝐸𝐸 =
1
2
𝐶𝐶𝑉𝑉2
𝐹𝐹 =
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
=
𝑛𝑛𝜖𝜖𝑟𝑟 𝜖𝜖0 𝑙𝑙ℎ
𝑔𝑔
Δ𝑉𝑉2
𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐, 𝜖𝜖0 = 8.854 × 10−12
Fm−1
DC Voltage applied to the anchor of the proof mass, while an AC current is supplied to drive
electrode[9]
𝐹𝐹 =
𝑛𝑛𝜖𝜖𝑟𝑟 𝜖𝜖0 𝑙𝑙ℎ
𝑔𝑔
(𝑉𝑉𝐷𝐷𝐷𝐷 + 𝑉𝑉𝐴𝐴𝐴𝐴)2
𝑉𝑉𝐴𝐴𝐴𝐴 = 𝑉𝑉0 sin(𝜔𝜔𝜔𝜔)
12

14. 𝐹𝐹 =
2𝑛𝑛𝜖𝜖𝑟𝑟 𝜖𝜖0ℎ
𝑔𝑔
𝑉𝑉𝐷𝐷𝐷𝐷 𝑉𝑉0 sin(𝜔𝜔𝜔𝜔) [9]
𝐅𝐅 = 𝟒𝟒. 𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒 × 𝟏𝟏𝟏𝟏−𝟖𝟖
𝐍𝐍
NATURAL FREQUENCY
The frequency with which undamped system vibrates
𝝎𝝎𝒙𝒙 = ��
𝑘𝑘𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥
𝑚𝑚
� = 1.3859 × 105
𝑟𝑟𝑟𝑟𝑟𝑟
𝑠𝑠
= 𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐
𝝎𝝎𝒛𝒛 = ��
𝑘𝑘𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧
𝑚𝑚
� = 4.6195 × 104
𝑟𝑟𝑟𝑟𝑟𝑟/𝑠𝑠 = 𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕
DAMPING RATIO
The damping ratio is a dimensionless measure describing how oscillations in a system decay
after a disturbance. The damping ratio provides a mathematical means of expressing the level of
damping in a system relative to critical damping. For a damped harmonic oscillator with mass m,
damping coefficient c, and spring constant k, it can be defined as the ratio of the damping
coefficient in the system's differential equation to the critical damping coefficient
𝜁𝜁 =
𝑐𝑐
𝑐𝑐𝑐𝑐
, 𝑐𝑐𝑐𝑐 = 2√𝑘𝑘𝑘𝑘
𝜻𝜻𝒙𝒙 =
𝑐𝑐𝑥𝑥
2�𝑘𝑘𝑥𝑥 𝑚𝑚
= 𝟏𝟏. 𝟖𝟖𝟖𝟖𝟖𝟖 × 𝟏𝟏𝟏𝟏−𝟏𝟏𝟏𝟏
𝜻𝜻𝒛𝒛 =
𝑐𝑐𝑧𝑧
2�𝑘𝑘𝑧𝑧 𝑚𝑚
= 𝟓𝟓. 𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐 × 𝟏𝟏𝟏𝟏−𝟏𝟏𝟏𝟏
TABLE 1: LIST OF CONSTANTS
S.no Property x-direction z-direction
1 Mass (𝑘𝑘𝑘𝑘) 1.1650 × 10−8
1.1650 × 10−8
2 Spring constant (𝑁𝑁/𝑚𝑚) 223.75 24.8611
3 Damping Coefficient(𝑁𝑁 − 𝑠𝑠/𝑚𝑚) 2.3102 × 10−10
1.9392 × 10−7
4 Damping ratio 1.865 × 10−13
5.2181 × 10−11
5 Natural frequency (𝐻𝐻𝐻𝐻) 22067 7355
13

15. The system can be assumed to be a simple lumped mass model, with the system
properties equivalent to as mentioned above.
FIGURE 4: LUMPED MASS MODEL
Using Newton’s Law F = ma, mathematical model can be made in the form of
differential equations and solution can be obtained.
14

16. EQUATION OF MOTION
Drive Axis (z-axis)
The equation for damped system-mass system
𝑚𝑚
𝑑𝑑2
𝑥𝑥
𝑑𝑑𝑡𝑡2
+ 𝑐𝑐𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
+ 𝑘𝑘𝑥𝑥 𝑥𝑥 = 𝐹𝐹
𝑚𝑚
𝑑𝑑2
𝑥𝑥
𝑑𝑑𝑡𝑡2
+ 𝑐𝑐𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
+ 𝑘𝑘𝑥𝑥 𝑥𝑥 =
2𝑛𝑛𝜖𝜖0ℎ
𝑔𝑔
𝑉𝑉𝑑𝑑𝑑𝑑 𝑣𝑣0 sin(𝜔𝜔𝑑𝑑 𝑡𝑡)
𝑚𝑚
𝑑𝑑2
𝑥𝑥
𝑑𝑑𝑡𝑡2
+ 𝑐𝑐𝑥𝑥
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
+ 𝑘𝑘𝑥𝑥 𝑥𝑥 = 𝐴𝐴 sin(𝜔𝜔𝑑𝑑 𝑡𝑡)
Using the natural frequency of the simple harmonic oscillator and damping ratio, we can rewrite
this equation as
𝑑𝑑2
𝑥𝑥
𝑑𝑑𝑡𝑡2
+ 2𝜁𝜁𝑥𝑥 𝜔𝜔𝑛𝑛𝑛𝑛
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
+ 𝜔𝜔𝑛𝑛𝑛𝑛
2
𝑥𝑥 = 𝐴𝐴 sin 𝜔𝜔𝑑𝑑 𝑡𝑡 [10]
Where,
𝑨𝑨 =
𝟐𝟐𝟐𝟐𝝐𝝐𝟎𝟎 𝝐𝝐𝒓𝒓 𝒉𝒉
𝒎𝒎𝒎𝒎
𝑽𝑽𝒅𝒅𝒅𝒅 𝒗𝒗𝟎𝟎 [11]
Displacement in x-direction,
As General solution tends to 0 for over damped case, therefore, neglecting it. Taking particular
solution of
𝑥𝑥 = 𝐴𝐴 sin 𝜔𝜔𝑑𝑑 + 𝐵𝐵 cos 𝜔𝜔𝑑𝑑 which can also be written as x = X cos(ωdt − ϕ)[10]
Where
𝑋𝑋 =
𝐴𝐴
�(𝜔𝜔𝑛𝑛𝑛𝑛
2 − 𝜔𝜔𝑑𝑑
2)2 + (2𝜁𝜁𝑥𝑥 𝜔𝜔𝑛𝑛𝑛𝑛 𝜔𝜔𝑑𝑑)2
; tan 𝜙𝜙 =
2𝜁𝜁𝜔𝜔𝑛𝑛𝑛𝑛 𝜔𝜔𝑑𝑑
𝜔𝜔𝑛𝑛𝑛𝑛
2 − 𝜔𝜔𝑑𝑑
2 [12]
Sense Axis (z-axis)
Due to angular rate of device in y-axis, a Coriolis force is induced in z-axis. The force can be
given as 𝐹𝐹 = 2𝑚𝑚Ω ×
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
Where Ω is the angular velocity of device in y-axis
The equation of motion is similar to Drive equation
𝑚𝑚
𝑑𝑑2
𝑧𝑧
𝑑𝑑𝑡𝑡2
+ 𝑐𝑐𝑦𝑦
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
+ 𝑘𝑘𝑘𝑘 = 2𝑚𝑚Ω
dx
dt
This can be rewritten as
15

17. 𝑑𝑑2
𝑧𝑧
𝑑𝑑𝑡𝑡2
+ 2𝜁𝜁𝑧𝑧 𝜔𝜔𝑛𝑛𝑛𝑛
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
+ 𝜔𝜔𝑛𝑛𝑛𝑛
2
𝑧𝑧 = 𝐵𝐵 cos(𝜔𝜔𝑑𝑑 𝑡𝑡) , 𝐁𝐁 =
𝟐𝟐𝛚𝛚𝐝𝐝 𝛀𝛀𝛀𝛀
��𝛚𝛚𝐧𝐧𝐧𝐧
𝟐𝟐 − 𝛚𝛚𝐝𝐝
𝟐𝟐
�
𝟐𝟐
+ (𝟐𝟐𝛇𝛇𝐱𝐱 𝛚𝛚𝐧𝐧𝐧𝐧 𝛚𝛚𝐝𝐝)𝟐𝟐
Where
𝐙𝐙 =
𝐁𝐁
��𝛚𝛚𝐧𝐧𝐧𝐧
𝟐𝟐 − 𝛚𝛚𝐝𝐝
𝟐𝟐
�
𝟐𝟐
+ (𝟐𝟐𝛇𝛇𝐳𝐳 𝛚𝛚𝐝𝐝 𝛚𝛚𝐧𝐧𝐧𝐧)𝟐𝟐
; 𝐭𝐭𝐭𝐭 𝐭𝐭 𝛅𝛅 =
𝟐𝟐𝟐𝟐𝛚𝛚𝐧𝐧𝐧𝐧 𝛚𝛚𝐝𝐝
𝛚𝛚𝐧𝐧𝐧𝐧
𝟐𝟐 − 𝛚𝛚𝐝𝐝
𝟐𝟐
[13]
Output (Current)
By measuring the current 𝑖𝑖 from sense mode transducer[9], the displacement of proof mass in
sense direction can be found out by following relation
Since, Q, electric charge can be expressed as 𝐶𝐶𝑉𝑉𝑑𝑑𝑑𝑑, Current can be calculated by taking time
derivative. Electric current can be written as
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
=
𝑑𝑑
𝑑𝑑𝑑𝑑
(𝐶𝐶𝑉𝑉𝑑𝑑𝑑𝑑)
Since, 𝑉𝑉𝑑𝑑𝑑𝑑is constant
𝑖𝑖 = 𝑉𝑉𝑑𝑑𝑑𝑑
𝑑𝑑
𝑑𝑑𝑑𝑑
𝐶𝐶
[14]
Where
𝐶𝐶 =
2𝑛𝑛𝜖𝜖0 𝜖𝜖𝑟𝑟ℎ
𝑔𝑔
(𝑙𝑙0 + 𝑥𝑥) [15]
Therefore
𝑖𝑖 =
𝐶𝐶0
ℎ0
𝑉𝑉𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
i =
C0
h0
Vdcωdzamp [16]
16

18. RESULTS
Amplitude v/s frequency response was plotted; Maximum Amplitude was obtained at lower
natural frequency of the x and z directions. (𝜔𝜔𝑧𝑧 =7355𝐻𝐻𝐻𝐻)
Device was simulated at 𝜔𝜔𝑑𝑑 = 7355𝐻𝐻𝐻𝐻
Response of amplitude v/s Angular rate was compared between square beam and rectangular
beam with dimensions 5𝜇𝜇𝜇𝜇 and 5𝜇𝜇𝜇𝜇 𝑎𝑎𝑎𝑎𝑎𝑎 15𝜇𝜇𝜇𝜇 respectively.
FIGURE 4: DRIVE FREQUENCY V/S AMPLITUDE RESPONSE IN
SENSE DIRECTION
Omega=7355
17

19. FIGURE 5: COMPARISON OF SENSE AMPLITUDE FOR SAME AND DIFFERENT NATURAL
FREQUENCIES IN DRIVE AND SENSE DIRECTIONS
If 𝑡𝑡 = 𝑤𝑤, therefore, natural frequency for sense and drive mode will be same
Having same natural frequency for drive and sense mode increased amplitude of device
manifold, which is unsafe for device
So, design was changed to a rectangular cantilever beam from a square beam.
Dimensions of the beam were set to 5𝜇𝜇𝜇𝜇 × 15𝜇𝜇𝜇𝜇
FIGURE 6: SENSE RESPONSE V/S ANGULAR RATE
18

20. Transducers are used in sense axis which uses current output to measure the sense displacement
By [17] Current v/s Angular velocity was plotted to get the sensitivity of the device.
Sensitivity is defined as 𝑆𝑆 =
𝑐𝑐ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖 𝑖𝑖 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
𝑐𝑐ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖 𝑖𝑖 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣
which is equal to slope the curve.
Sensitivity of this device was found out to be 𝟐𝟐. 𝟐𝟐 × 𝟏𝟏𝟏𝟏−𝟖𝟖
𝐀𝐀/ �
𝐫𝐫𝐫𝐫𝐫𝐫
𝐬𝐬
�
−𝟏𝟏
FIGURE 7: SENSE CURRENT V/S ANGULAR RATE
19

21. BIBLIOGRAPHY
[1] A. A. Trusov, “Overview of MEMS Gyroscopes : History , Principles of Operations , Types of
Measurements,” Irvine,CA,USA, 2011.
[2] S. Nasiri and S. Clara, “A Critical Review of MEMS Gyroscopes Technology and Commercialization
Status,” Avaliable online
http//invensense.com/mems/gyro/documents/whitepapers/MEMSGyroComp. pdf (accessed 25
May 2012), p. 8, 2009.
[3] A. Burg, A. Meruani, A. Sandheinrich, and M. Wickmann, “MEMS Gyroscopes and their
Applications,” Introd. to Microelectromechanical Syst., 2011.
[4] J. E. Shigley, C. R. Mischke, R. G. Budynas, and K. J. Nisbett, Mechanical Engineering Design, 8th
ed. New York: McGraw-Hill, 2010.
[5] H. K. Sharma, A. Jain, and R. Gopal, “Estimation of slide film damping in laterally moving
microstructures using FEM simulations,” in National Conference on Innovations in
Microelectronics, Signal and Communication Technologies, 2014.
[6] J. A. Fay, Introduction to Fluid Mechanics. New Delhi: PHI Learning Private Limited, 2012.
[7] H. K. Sharma, A. Jain, and R. Gopal, “FEM simulations for estimating squeeze film damping in
microstructures,” in International Conference on Advance Trends in Engineering and Technology,
2013, pp. 112–114.
[8] M. Bao and H. Yang, “Squeeze film air damping in MEMS,” Sensors Actuators, A Phys., vol. 136,
no. 1, pp. 3–27, 2007.
[9] A. Bristow, T. Barton, and S. Nary, “MEMS Tuning-Fork Gyroscope Final Report.”
[10] H. Dong and X. Xiong, “Design and Analysis of a MEMS Comb Vibratory Gyroscope,” in UB - NE
ASEE 2009 Conference, 2009, p. 12.
20

22. APPENDIX
Code
𝛽𝛽 Equation was solved in MATLAB file beta1.m
rat=1;
c=0;
for i=1:100
g=2*i-1;
beta=1-(192/3.14^5)*c/rat;
b1 = (1/g^5)*tanh(g*3.14*rat/2);
y(i)=beta;
x(i)=i;
c=b1+c;
end
FIGURE 8: CODE FOR BETA CONVERGENCE
21

23. Code for MEMS Gyroscope Vibration simulation was solved in MATLAB file Memsgyro.m
clear
clc
rho=2330;%kg/m^3
E=179e9;%Pa
p=760;%torr
lambda=p*5.1e-5;%mean free path
mu=1.791e-5;%kg/ms
%dim of proof mass
a=1000e-6;
b=1000e-6;
h=5e-6;
h0=5e-6;
A=a*b;
m=rho*A*h;
%spring coefficient x
t=5e-6;
w=5e-6;
L=300e-6;
Ix=1/12 *w^3* t;
kx=6*E*Ix/L^3 ;
ksx=4*kx;%4 springs
Iz=1/12*w*t^3;
kz=6*E*Iz/L^3;
ksz=4*kz;
%Damping
Kn=lambda/h0;
mu_x=mu/(1+2*Kn+0.2*(Kn^0.788)*exp(-Kn/10));
mu_z=mu/(1+9.638*Kn^1.159);
cx=mu x*A/h0;
FIGURE 9: CODE FOR MEMS DEVICE INITIALIZATION
22

24. zx=cx/2*sqrt(ksx*m);
zz=cz/2*sqrt(ksz*m);
%natural freq
nfx=sqrt(ksx/m);
nfz=sqrt(ksz/m);
%force
n=10;
ep=8.854e-12*1.0053;
Vdc=10;
V0=10;
g=2e-6; %gap bw combs
F=2*n*ep*h*Vdc*V0/g;
ohm=2;
for p=1:108000
f(p)=p/(2*3.14);
Ax(p)=abs(F/(m*sqrt((nfx^2-p^2)^2+(2*zx*p)^2)));
Az(p)=abs(2*ohm*p*Ax(p)/sqrt((nfz^2-p^2)+(2*zz*p)^2));
end
plot(f,Az)
anfx=nfx/(2*3.14);
anfz=nfz/(2*3.14);
aaaz=max(Az);
aaanf=find(Az==aaaz)/(2*3.14);
%Capacitance
C0=ep*a*b/h0;
clear ohm p
for p=1:101
ohm(p)=p-1;
fd=aaanf*2*3.14;
Axd(p)=abs(F/(m*sqrt((nfx^2-fd^2)^2+(2*zx*fd)^2)));
Azd(p)=abs(2*ohm(p)*fd*Axd(p)/sqrt((nfz^2-fd^2)+(2*zz*fd)^2));
Current(p)=(C0/h0)*fd*Azd(p)*Vdc;
end
plot(ohm,Azd)
phi=2*zx*nfx*fd/(nfx^2-fd^2);
del=2*zz*nfz*fd/(nfz^2-fd^2);
tphi=atand(phi);
tdel=atand(del);
FIGURE 10: CODE FOR MEMS DEVICE VIBRATION ANALYSIS
23