Modeling and Vibration Analysis of a Rocking–mass

Gyroscope System

by

Masoud Ansari

A Thesis Submitted in Partial Fulfillment

of the Requirements for the Degree of

Master of Applied Science Mechanical Engineering

in

The Faculty of Engineering and Applied Science

Mechanical Engineering

University of Ontario Institute of Technology

April 2008

©Masoud Ansari, 2008

iii

ABSTRACT

Gyroscopes are one of the most widely used devices for measuring the

angle of rotation or the rate of change of angular rotation. In the last few

years, the advent of micromachining technology has made it possible to

fabricate high performance Micromachined Electro-Mechanical Systems

(MEMS) gyroscopes at a relatively low cost. Of the various types of MEMS

gyroscopes, vibrating beam type, are the most commonly used, but they have

a main drawback (cross-axis effect) which significantly affects their

measurement and results in much error. In view of this, the present work

focuses on a gyroscope, called rocking-mass gyroscope, which does not have

that drawback at all and can give a very precise measurement.

The rocking-mass gyroscope consists of an assembly of four cantilever

beams with a rigid mass attached to them in the middle subjected to base

rotations. Due to the gyroscopic effect, the beams undergo coupled flexural-

torsional vibrations. The main goal of the research is to develop an accurate

model of such a system and along this line a detailed mathematical modeling

of the gyroscope is developed for a macro-scaled system. The equations of

motion clearly show the presence of the gyroscopic couplings in all cantilever

beams. In order to analyze the effectiveness of this type of gyroscope a

computer simulation model in its most general form has been developed.

iv

Before deriving the governing equations of motion for a rocking-mass

gyroscope, a detailed mathematical model of a single beam gyroscope was

considered to investigate the cross-axis effect in this type of gyroscopes.

The characteristic equation of a rocking-mass gyroscope, using an

exact method, is derived, and the fundamental frequency of the vibration and

the corresponding mode shape are obtained. Finally, the time history diagram

of this system is presented to illustrate the dynamic response of the system.

Simulation results show that the bending vibrations induced in the second and

the forth beams are directly proportional to the magnitude of the base rotation.

Therefore, the gyroscope can be widely used as a device for measuring the

base rotation velocity.

Keywords

Rocking-mass gyroscope; Vibrating beam gyroscope; Extended

Hamilton’s principle; Frequency analysis; Mathematical modeling.

v

Dedication

To my father Jamshid and my mother Shahnaz who influenced my life

the most.

vi

Acknowledgments

I would like to sincerely and wholeheartedly thank Prof. E. Esmailzadeh for

his guidance and kindness throughout this work. His patience as an advisor,

boundless energy while teaching, promptness while reviewing all my writing,

and passion for doing research are to be commended and worth emulating.

This thesis would not have been possible without his support.

I would also like to thank my advisory committee members Dr. Dan Zhang

and Dr. Greg Lewis for giving their valuable time and for serving on my

thesis committee.

I am grateful to Dr. Nader Jalili of Clemson University for his constant

guidance and motivation during this research work.

A special thanks to Ms. Jane Dubis and Ms. Kathryn Hanson in the university

library for all their cooperation and help during my course of study at UOIT.

Most importantly, I would like to thank my parents, Jamshid and Shahnaz,

and my sisters Maryam and Marzieh, for their unconditional support, love and

affection. Their encouragement and never-ending kindness made everything

easier to achieve.

vii

Table of Contents

Title Page……………………………………………………………...……….i

Abstract………………………………………………….…………………...iii

Dedication………………………………………………..……………….…...v

Acknowledgements……………………………………..……………….…...vi

Table of Contents……………….…………………………………….……..vii

List of Tables……………………………………………………...………….xi

List of Figures……………………………………………………..………...xii

List of Appendices…………………………………………….….......…….xiii

CHAPTERS

1. INTRODUCTION……………………………………………………….....1

Research Background….………………………………………………1

Thesis Contributions…….……………………………………………..4

Thesis Outline………….………………………………………………5

2. OVERVIEW OF GYROSCOPE TECHNOLOGY………………...………7

Introduction……………………………………………………………7

viii

Micromachined Vibratory Rate Gyroscopes and Their Operational

Principles……………………………………………………………..11

Specifications of MEMS Gyroscopes ……………………..…………13

Applications of MEMS Gyroscopes……………….…………………14

Different Types of MEMS Vibratory Gyroscopes …..………………16

Control Strategies of MEMS Gyroscopes ……..….…………………23

Summary………………………………………………………...……26

3. OVERVIEW OF SINGLE BEAM GYROSCOPE…………………….…27

Introduction……………………………………………..……………27

Governing Equations of Motion…………...…………………………31

Beam Kinematics……………………………………………………..32

Translational Motion……………………………………………..…..33

Rotational Motion……………………………………………….……37

Piezoelectric Modeling……………………………………………….40

Constitutive Equations...…………………………………………...…41

Piezoelectric Patch Actuator....………………...………………….….42

Effects of Piezoelectric Actuator………………………...…………...43

Equations of Motion…………………..……………………………...45

ix

Time Response………………...………………..……………………48

Cross-axis Effects..…………………………………………………...52

Summary………………………………………………………...……54

4. MODELING OF THE ROCKING-MASS GYROSCOPE……………….55

Introduction………………………………………………………..…55

Principle of Operation...…………………………………………….. 56

Governing Equations of Motion...……………………………………59

Translational Motion….……………………………………………...60

Rotational Motion………………………………………………….…61

Equations of motion………...………………………………………...63

Summary………………….…………………………………………..70

5. ROCKING-MASS GYROSCOPE ANALYSIS AND NUMERICAL

SIMULATIONS……………………………………………………………..71

Development of the Frequency Equation…….………………………72

Validation of the Utilized Method for Solving EVP………..………..83

Frequency Analysis of a Timoshenko Beam…………………………83

Time Response of the System………………..………………………89

Summary………………………………………..….…………………93

x

6. CONCLUSIONS AND FUTURE WORK………..………………………94

Recommendations for Future Work………………………………….95

APPENDICES………………………………………….…………………………96

A: Detailed Derivation of Equations of Motion for a Single Beam

Gyroscope…………...………………………………………………..97

B: Frequency Equation…….……………………………………..…101

C: Maple Code to Develop the Frequency Equation………………..104

REFERENCES.………………………………………………….…………110

xi

List of Tables

2.1 Performance requirements of different gyroscopes ..................................14

3.1 Physical parameters of the system ............................................................51

5.1 Physical parameters of the system………………………………...……..81

5.2 The first three natural frequencies of the fixed-free Timoshenko beam...88

xii

List of Figures

1.1 A conventional gyroscope…………………………………………….…..2

2.1 Spinning wheel gyroscope ......................................................................... 8

2.2 (a) Optical gyroscopes (Ring laser gyroscope)...........................................9

2.2 (b) Optical gyroscopes (Fiber optic gyroscope) ........................................9

2.3 The Coriolis effect.....................................................................................10

2.4 Principle of operation of MEMS gyroscopes............................................12

2.5 Applications of MEMS gyroscopes……………...………………………15

2.6 Tuning fork gyroscope ..............................................................................16

2.7 Vibrating beam gyroscope ........................................................................17

2.8 Cylinder gyroscope operating principle.....................................................18

2.9 (a) Schematic of vibrating ring gyroscope ................................................19

2.9 (b) Flexural modes of vibrating ring gyroscope .......................................19

2.10 SEM image of the MEMS tuning fork gyroscope ..................................21

2.11 SEM image of a comb driven surface micromachined vibratory

gyroscope.........................................................................................................22

2.12 Dual axis rate gyroscope .........................................................................22

2.13 SEM image of ring gyroscope.................................................................23

xiii

3.1 Schematic of a single beam gyroscope…………………………………..27

3.2 Schematic of a flexural-torsional beam gyroscope………………...…….28

3.3 Cantilever beam kinematics……………………………………………...33

3.4 End mass kinematics…………………………………………………….35

3.5 Euler angle rotations……………………………………………….…….37

3.6 Piezoelectric patch actuator ……………………………………………..43

3.7 Schematic of a piezoelectric actuator attached on the beam…………….44

3.8 System response (torsional deflection θ(L,t)) to secondary base rotation;

(a) Ω1=0 rad/s , (b) Ω1 = 0.05 rad/s , (c) Ω1 = 0.1 rad/s , and (d) Ω1 = 0.5

rad/s……………………………………………………………………….…53

4.1 Schematic of a rocking-mass gyroscope………………………………....56

4.2 Primary rocking motion of the mass……………………………………..57

4.3 Secondary rocking motion of the mass…………………………………..58

4.4 Top view of the rocking-mass gyroscope with the coordinate systems…60

5.1 First mode shape of the system…………………………………………..82

5.2 Mode shapes of a cantilever Timoshenko beam (a) First mode (b) Second

mode (c) Third mode ……………………………………………………..…88

5.3 System response (a) bending deflection w1 and bending deflection w2 to

different base rotation rates: (b) Ω3=0, (c) Ω3=25 rad/s and (d) Ω3=50

rad/s………………………………………………………………………….91

xiv

List of Appendices

A Detailed Derivation of Equations of Motion for a Single Beam

Gyroscope……...…………………………………………………………….97

B Frequency Equation…………………………………………………...…101

C Maple Code to Develop the Frequency Equation…….………………….104

1

Chapter 1

INTRODUCTION

Research Background

A gyroscope is a device for maintaining orientation or measuring the

angle of rotation or the rate of change of angular rotation. In spite of the

conventional gyroscopes that use the conservation of angular momentum of a

spinning wheel for sensing the angular rate, a vibrating structure gyroscope,

which is a type of gyroscope that functions much like the halteres* of insects,

and can be used as a relatively inexpensive type of Attitude indicator†, uses

transfer of energy between two vibrating modes of a structure caused by

Coriolis acceleration, for measurement. The Coriolis acceleration is an

acceleration which arises as a result of motion of a particle relative to a

rotating reference. Only the components of motion in a plane parallel to the

equatorial plane are influenced. The effect is named after Gaspard-Gustave

Coriolis, a French scientist who described it in 1835, though the mathematics

appeared in the tidal equations of Pierre-Simon Laplace in 1778.

* Halteres, also known as balancers or poisers, are small knobbed structures found as a pair in some two-winged insects; they are flapped rapidly to maintain stability when flying. † An attitude indicator (AI), also known as gyro horizon or artificial horizon, is an instrument used in an aircraft to inform the pilot of the orientation of the airplane relative to earth.

2

Figure 1 shows a sample of conventional gyroscopes, and its

components. Since the conventional gyroscopes have a few major

disadvantages, namely their large size, high cost and limited life, they were

soon replaced by vibratory gyroscopes.

Figure 1. A conventional Gyroscope.

As mentioned before, the MEMS gyroscopes can be batch fabricated at

a very low cost (around US$30 per part in quantity as of late 2007). Since

there is no rotating part in this kind of gyroscope, they are commonly used in

different applications, to name a few, automotive active suspension, air bag

activation, consumer electronics and guided missiles. The vibrating element in

this type of gyroscope can have different shapes, but the most commonly used

elements are tuning forks, rings and beams. Among these, beams are being

used more commonly.

3

In vibrating beam gyroscope, the beam with a mass attached to its tip

(vibrating resonator), when rotated, is subjected to Corioli's effect that causes

secondary vibration orthogonal to the original vibrating direction. By sensing

the secondary vibration, the rate of turn can be detected. For vibration exert

and detection the piezoelectric effect is often used, therefore vibrating gyros

are often called "piezo", "ceramic", or "quartz" gyro, although in fact

vibration and detection do not necessary use the piezo effect.

Although these gyroscopes offer many advantages, they are extremely

sensitive to variations in system parameters such as length of the beam and

magnitude of primary excitation. They are prone to cross-axis effects and

quadrature errors that require advanced control strategies for their elimination.

But usually even with very good control strategies, it is very hard to eliminate

the cross-axis effect, in single beam gyroscopes. Therefore, a rocking-mass

gyroscope will be studied in this research. This gyroscope consists of four

beams and never encounter with cross-axis effect, and will not face the errors

discussed before. So it would be one of the most precise gyroscopes, if not the

most precise one. The main challenge is the complexity of the modeling of

this gyroscope. In spite of this fact, a detailed mathematical analysis of a

rocking-mass gyroscope will be performed in this work.

4

Thesis Contributions

As mentioned before, due to the advantages of the vibrating mass

gyroscopes, they are being used in many applications, and therefore, most of

the researches have conducted different studies on them. But the main

problem is that not many researchers have considered the components of the

gyroscope as elastic and continuous parts, although this is not a good

assumption, and causes error in calculations. In one of the latest works [1]

which deals with the beam gyroscope as a continuous system comprising of

the vibrating mass attached to the rotating base through flexible beam, it has

been concluded that there is a significant error in this kind of gyroscope,

called cross-axis effect, caused by the secondary base rotations in the system.

The work outlined in this thesis deals with a special kind of vibrating

mass gyroscope called rocking-mass gyroscope. In the past, few work has

been done on the rocking-mass gyroscopes. They are mostly focused on the

manufacturing aspects of this kind of gyroscope, and no researcher has

worked on the detail mathematical modeling and analysis of such a system.

In the present work, we offer a detail mathematical modeling and

analysis of a rocking-mass gyroscope. The gyroscope consists of four beams,

undergoing flexural-torsional vibrations, attached to a rocking mass in the

middle. The research primarily focuses on developing an accurate

5

mathematical model of this type of gyroscope. Governing equations of the

system are derived and simulated to analyze the effectiveness of this type of

gyroscope. The fundamental frequency of the system and its corresponding

mode shape will be found as well. Finally the time response of the system will

be presented.

Thesis Outline

The present work has five other chapters, as follows:

In chapter 2, we describe the gyroscopic systems and their basic

principles; however the main focus is on the vibratory gyroscopes and their

operational principles. Different types of the Micro-machined Electro-

Mechanical Systems (MEMS) gyroscopes and their principle of operation will

be pointed out. Finally the application of the discussed gyroscopes as well as

the control strategies will be briefly discussed.

In order to study the main drawback of the single-beam gyroscopes, we

will study a single beam gyroscope in chapter 3. Detailed governing equations

of motion will be derived for a macro-scaled cantilever beam gyroscope

undergoing coupled bending and torsional vibrations. After modeling the

system in full detail, we will perform a time-domain analysis to investigate

the cross-axis effect in this type of gyroscope.

6

Chapter 4 covers the detailed mathematical modeling of the rocking-

mass gyroscope consisting of four flexible beams, undergoing coupled

bending and torsional vibrations, and a rocking-mass attached to them in the

middle. Equations of motion and the boundary conditions are presented, for a

macro-scaled system.

Chapter 5 deals with the development of the frequency equation and

simulation of the system equations. Eigenvalue problem is solved using an

exact method. Simulations are carried out to study the gyroscopic coupling

present in the system as well as the effects of base rotation on the magnitude

of the gyroscopic effect. The time-domain analysis will be carried out to

check the effectiveness of the rocking-mass gyroscope.

The conclusions and suggestions for the future work are presented in

Chapter 6.

7

Chapter 2

OVERVIEW OF GYROSCOPE TECHNOLOGY

Introduction

Gyroscopes or gyro rate sensors are widely used for navigation,

stabilization, general rate control, pointing, autopilot systems, missile

guidance control, etc. A typical example is the application of yaw rate sensors

to automobiles to provide input to the control systems for suspension, braking

and steering. During recent years there have been attempts to develop low

cost gyroscopes suitable for mass production. A promising concept of such a

device is the vibratory gyroscope which can be fabricated using surface-

micromachining technology.

Conservation of momentum is used in gyroscopes to measure the

angular velocity or acceleration. In fact they use the principle of precession

which is actually Newton’s third law of motion. This principle expresses if an

unbalanced force is applied to a stationary object, the object will resist motion

in that direction. We can measure this force, to state the angular velocity or

acceleration of that object.

8

Gyroscopes can be categorized in different ways. Based on the

principle of operation, they fall into three groups: (a) Spinning mass

gyroscopes, (b) Optical gyroscopes and (c) Vibrating mass gyroscopes.

The spinning mass gyroscope is based on the principle that the spin

axis of a spinning mass will remain in a fixed direction in space unless acted

upon by an external influence. It has a mass spinning steadily with free

movable axis, called gimbals. In order to measure the angle of rotation, the

precession principle is used. When the gyroscope is tilted, gyroscopic

effect causes precession (motion orthogonal to the direction tilt sense) on

the rotating mass axis, hence gives the angle moved by the mass. A typical

spinning mass gyroscope is illustrated in Figure 2.1.

Figure 2.1: Spinning wheel gyroscope.

In optical gyroscopes, laser ray reflects round around many times

within the enclosure. If the enclosure rotates, the duration between the

moments of laser emittance to eventual reception will be different, so it is

9

based on the Sagnac effect [2]. In an RLG (Ring Laser Gyro), the laser

go-around is done by mirrors inside the enclosure, and in a FOG (Fiber Optic

Gyro) the laser go-around is done by a coil of optical-fiber. Laser emitter

deteriorates with time, and the fiber has its life fragile. Figure 2.2 shows the

two types of optic gyroscopes and their components.

(a) (b)

Figure 2.2: Optical gyroscopes (a) Ring laser [3] and (b) Fiber optic [4].

The next type of gyroscopes is vibrating gyroscopes. Among all the

mentioned types, this type is the most commonly used. A vibrating element

(vibrating resonator), when rotated, is subjected to Coriolis effect that causes

secondary vibration orthogonal to the original vibrating direction. By sensing

the secondary vibration, the rate of turn can be detected.

Almost all reported micromachined gyroscopes use vibrating

mechanical elements to sense rotation. They have no rotating parts that

require bearings, and hence they can be easily miniaturized and batch

fabricated using micromachining techniques. All vibratory gyroscopes are

10

based on the transfer of energy between two vibration modes of a structure

caused by Coriolis acceleration. To understand the Coriolis effect, imagine a

particle traveling in space with a velocity vector V. An observer sitting on the

x-axis of the xyz coordinate system, shown in Figure 2.3, is watching this

particle. If the coordinate system along with the observer starts rotating

around the z-axis with an angular velocity Ω, the observer thinks that the

particle is changing its trajectory toward the x-axis with an acceleration equal

to 2V×Ω. Although no real force has been exerted on the particle, to an

observer, attached to the rotating reference frame an apparent force has

resulted that is directly proportional to the rate of rotation. This effect is the

basic operating principle underlying all vibratory structure gyroscopes [5].

Figure 2.3: The Coriolis effect [5].

11

Micromachined Vibratory Rate Gyroscopes and Their

Operational Principles

Even though an extensive variety of micromachined gyroscope designs

and operation principles exists, almost all of the reported micromachined

gyroscopes use vibrating mechanical elements to sense angular rate. The

concept of utilizing vibrating elements to induce and detect Coriolis force

involves no rotating parts that require bearings, and have been proven to be

effectively implemented and batch fabricated in different micromachining

processes [6]. They are based on the transfer of energy between two modes of

vibration of a structure [7]. Various elements such as tuning forks, beams,

shells, rings, discs and cylinders are used as the proof mass in MEMS

gyroscope.

The operation principle of the vast majority of all existing

micromachined vibratory gyroscopes relies on the generation of a sinusoidal

Coriolis force due to the combination of vibration of a proof-mass and an

orthogonal angular-rate input. The proof mass is generally suspended above

the substrate by a suspension system consisting of flexible beams. Figure 2.4

illustrates an EMS gyroscope consists of a vibrating proof mass suspended

over a substrate via elastic beams. A force is applied to the proof mass to

vibrate along the x-axis (drive mode). When the gyroscope is subjected to an

12

angular rotation, a sinusoidal Coriolis force is induced in the direction

orthogonal to the drive-mode oscillation at the driving frequency. The

magnitude of this force is given as

Ω×= VmFc 2 (2.1)

where m is the vibrating mass, V is the velocity in the primary direction and Ω

is the applied rotation rate. As mentioned before, due to the Coriolis force, the

mass starts vibrating in the secondary direction (sense mode), and the

magnitude of the sense mode vibration, which is proportional to the rate of

rotation, can be measured to determine the rate of rotation. Different methods

can be utilized for actuating and sensing the vibrations, such as electrostatic,

piezoelectric or electromagnetic.

Figure 2.4: Principle of operation of MEMS gyroscopes [8].

Ideally, it is desired to utilize resonance in both the drive and the sense

modes in order to attain the maximum possible response gain, and hence

13

sensitivity. This is typically achieved by designing and electrostatically tuning

the drive and sense resonant frequencies to match. Alternatively, the

sense-mode is designed to be slightly shifted from the drive-mode to improve

robustness and thermal stability, while intentionally sacrificing gain and

sensitivity. However, the limitations of the photolithography-based

micromachining technologies define the upper-bound on the performance and

robustness of micromachined gyroscopes [6].

Specifications of MEMS Gyroscopes

The performance of a gyroscope is determined through three important

parameters of resolution, drift, zero-rate output (ZRO), and scale factor. In the

absence of rotation, the output signal of a gyroscope is a random function that

is the sum of white noise and a slowly varying function [9]. The white noise

defines the resolution of the sensor and is expressed in terms of the standard

deviation of equivalent rotation rate per square root of bandwidth of detection

[(o/s)/√Hz or (o/h)/√Hz]. The so-called “angle random walk” in o/√h may be

used instead. The short- or long-term drift of the gyroscope corresponds to the

peak-to-peak value of the slowly varying function [9]. The last very important

factor or the gyroscope is the Zero Rate Output (ZRO), which represents the

output of the device in the absence of a rotation rate [5]. In another way of

classification, gyroscopes fall into three categories: inertial-grade, tactical-

14

grade, and rate grade devices. Table 2.1 summarizes the requirements for each

of these categories ([10], [11]).

Table 2.1: Performance requirements of different gyroscopes [5]

Parameter Rate Grade Tactical Grade Inertial Grade

Angle Random Walk, (o/ h ) >0.5 0.5-0.05 <0.001

Bias Drift, (o/h) 10-1000 0.1-10 <0.01 Scale Factor Accuracy, (%) 0.1-1 0.01-0.1 <0.001 Full Scale Range, (o/sec) 50-1000 >500 >400 Max. Shock in 1 msec, (g’s) 103 103-104 103 Bandwidth, (Hz) >70 ~100 ~100

Applications of MEMS Gyroscopes

Applications for MEMS gyroscopes are very broad. Some example for

these applications are; automotive; vehicle stability control, rollover detection,

navigation, load leveling/suspension control, event recording, collision

avoidance; consumers, computer input devices, handheld computing devices,

game controllers, virtual reality gear, sports equipment, camcorders, robots;

industrial., navigation of autonomous (robotic) guided vehicles, motion

control of hydraulic equipment or robots, platform stabilization of heavy

machinery, human transporters, yaw rate control of wind-power plants;

aerospace/military; platform stabilization of avionics, stabilization of pointing

systems for antennas, unmanned air vehicles, or land vehicles, inertial

measurement units for inertial navigation, and many more. Different

15

application areas of MEMS gyroscopes based on different accuracy and range

requirements, are presented in Figure 2.5.

Anti-rollover mechanisms, GPS navigation and electronic stability

control can be achieved in the automotive sector, using gyroscopes. A lot of

automobile manufacturers and researchers are conducting research on new

applications (like active suspension, skid control, and …) of MEMS gyros in

cars. Moreover MEMS gyroscopes are being used by various consumer

electronics. To name a few, they are being used in camcorder stabilization,

game controllers, handheld computing devices, virtual reality gear and robots.

Furthermore, MEMS gyroscopes are also used in inertial navigation of

autonomous guided vehicles, platform stabilization of heavy machinery, yaw

rate control of wind power plants and inertial navigation for military

applications [12].

Figure 2.5: Applications of MEMS gyroscopes [13].

16

Different Types of MEMS Vibratory Gyroscopes

A number of vibratory gyroscopes have been demonstrated, including

tuning forks [14–17], vibrating beams [18], and vibrating shells [19]. Tuning

forks are a classical example of vibratory gyroscopes. The tuning fork, as

illustrated in Figure 2.6, consists of two tines that are connected to a junction

bar. In operation, the tines are differentially resonated to fixed amplitude, and

when rotated, Coriolis force causes a differential sinusoidal force to develop

on the individual tines, orthogonal to the main vibration. This force is detected

either as differential bending of the tuning fork tines or as a torsional vibration

of the tuning fork stem. The actuation mechanisms used for driving the

vibrating structure into resonance are primarily electrostatic, electromagnetic,

or piezoelectric. To sense the Coriolis-induced vibrations in the second mode,

capacitive, piezoresistive, or piezoelectric detection mechanisms can be used.

Optical detection is also feasible, but it is too expensive to implement [5].

Figure 2.6: Tuning fork gyroscope [20].

17

In a vibratory gyroscope, especially vibrating beam gyroscope which

consists of a metallic beam, the drive mode is a flexural vibration which has

been induced through a piezoelectric actuator, placed on the beam. A

secondary vibration (sense-mode), normal to the drive mode vibrations, is

induced in the beam due to the Coriolis force arises in the presence of the

rotation about the longitudinal axis of the beam. The secondary vibration can

be sensed by sensors placed on the beam as shown in Figure 2.7. From this

secondary vibration, the rate of rotation can be determined.

Figure 2.7: Vibrating beam gyroscope [1].

The main concept in a vibratory cylinder gyroscope is: the nodes on the

circumference of a vibrating cylinder (ring) do not stay fixed with respect to

the cylinder itself when it is rotated around its central axis, but they move by a

18

quantity proportional to the turn, as sketched in Figure 2.8. This is due to the

Coriolis coupling between the two vibration modes that change as cos2θ

(primary) and sen2θ (secondary) around the circumference. A certain number

of electrodes are located around the circumference with the aim to excite and

detect both the primary and secondary vibration modes [3].

Figure 2.8: Cylinder gyroscope operating principle [3].

Figure 2.9 (a) illustrates a vibrating ring gyroscope which comprises a

ring, eight semicircular support springs, and drive, sense and control

electrodes [19]. Symmetry considerations require at least eight springs to

result in a balanced device with two identical elliptically-shaped flexural

modes that have equal natural frequencies and are 45o apart from each other

[19].

The antinodes of the second flexural mode are located at the nodes of

the first flexural mode (see Figure 2.9 (b)). The ring is electrostatically

19

vibrated into the primary flexural mode with fixed amplitude. When device is

subjected to rotation around its normal axis, Coriolis force causes energy to be

transferred from the primary mode to the secondary flexural mode, which is

located 45o apart from the primary mode, causing amplitude to build up

proportionally in the latter mode; this build-up is capacitively monitored. The

amplitude of the second mode (sense mode) is proportional to the rotation rate

and can be measured to determine the rate of rotation [21].

(a)

(b)

Figure 2.9: Vibrating ring gyroscope; (a) Schematic of the gyroscope [21] and (b) Flexural

modes of a vibrating ring gyroscope [21].

20

Micromechanical gyros are usually designed as an electronically driven

resonator, often fabricated out of a single piece of quartz or silicon. Such

gyros operate in accordance with the dynamic theory that when an angular

rate is applied to a translating body, a Coriolis force is generated. When this

angular rate is applied to the axis of a resonating tuning fork, its tines

experience a Coriolis force, which then produces torsional forces about the

sensor’s axis. These forces, which are proportional to the applied angular rate,

cause displacements that can be measured capacitively in a silicon instrument

or piezoelectrically in a quartz instrument. The output is then demodulated,

amplified and digitized to form the device output [7].

In spite of having high quality factors, quartz vibratory gyroscopes do

not have compatibility of being processed with integrated circuit fabrication

technology. Based on this fact, they were replaced by silicon, when the batch

fabrication technology was introduced.

The Charles Draper Laboratory is one of the pioneers in manufacturing

the micromachined silicon rate gyroscopes. The first MEMS silicon gyro, they

built, had a double gimbals vibratory gyroscope supported by torsional

flexures, with the vibrating mechanical element made from p++ silicon [22].

After that, they built an improved silicon-on-glass tuning fork gyroscope [23].

It was fabricated through the dissolved wafer process [24]. In order to achieve

large amplitude of motion (10 μm), a set of interdigitated comb drives are

21

used to vibrate this gyroscope electrostatically in its plane [25]. The entire

structure will undergo an out-of-plane rocking motion, if it rotates about a

direction perpendicular to the drive mode. This motion can be capacitively

measured, to determine the rate of rotation. A Scanning Electron Microscope

(SEM) image of such device is illustrated in Figure 2.10.

Figure 2.10: SEM image of MEMS tuning fork gyroscope [26].

Different mechanisms are being used for sensing and actuation in

Micromachined gyros. Electromagnetic actuation has been utilized in some

tuning forks to achieve large amplitude of motion [27-29]). Reference [30]

discusses the use of piezoresistive detection in some of the MEMS

gyroscopes. Some researchers at University of California, Berkeley and

Samsung electronics, developed single and dual-axis polysilicon surface-

micromachined gyroscopes. The vibratory gyroscope developed by Samsung

is illustrated in Figure 2.11. It consists of a 7-μ m-thick polysilicon resonating

mass supported by four fishhook-shaped springs [31-35].

22

Figure 2.11: SEM image of comb driven surface micromachined vibratory gyroscope [31].

Figure 2.12 illustrates a surface-micromachined dual-axis gyroscope

which is based on rotational resonance of a polysilicon rotor disk [36]. The

sensor can sense rotation equally about these two axes, since the disk is

symmetric in two orthogonal axes. A bulk-micromachined, precision silicon

MEMS vibratory gyroscope for space applications was fabricated buy the Jet

Propulsion Laboratory (JPL), in collaboration with the University of

California, Los Angeles [37], [38]. The vibrating ring gyroscope which was

built by Researchers at General Motors and the University of Michigan,

consists of a ring, semicircular support springs, and drive, sense, and balance

electrodes, which are located around the structure [39].

Figure 2.12: Dual axis rate gyroscope [36].

23

An SEM image of a 1.7×1.7 mm2 PRG is illustrated in Figure 2.13.

Different important features of high-performance gyroscopes, like small ring-

to-electrode gap spacing for increasing the sense capacitance, large structural

height for increasing the radius and sense capacitance and reducing the

resonant frequency, and good structural material (polysilicon) with an

orientation-independent Young’s modulus are available in this device [5].

Figure 2.13: SEM image of ring gyroscope ([40], [41]).

Control Strategies of MEMS Gyroscopes

In MEMS vibrating mass gyroscopes, one of the modes of the mass is

actuated into a known oscillatory motion. Due to the Coriolis acceleration a

secondary vibration is induced in the other mode of the mass (sense-mode).

By measuring the response of the sense mode, the angular rate can be

determined.

24

The conventional mode of operation is classified into the open-loop

mode and the closed-loop mode. The main difference between the closed-loop

and open-loop mode of operation lies in that in the former the displacement of

the sense axis is controlled to zero, while in the latter it is measured. Most

MEMS gyroscopes are currently operated in the open-loop mode. The main

advantage of the open-loop mode of operation is that circuitry used for the

operation of gyroscope in this mode is simpler than in the other modes, since

there is no control action in the sense axis. Thus, this mode can be

implemented relatively easily and cheaply. However, under an open-loop

mode of operation, the gyroscope's angular rate scale factor is very sensitive

and not constant over any appreciable bandwidth, to fabrication defects and

environment variations. Therefore, the application areas for the open-loop

mode are limited to those which require low-cost and low-performance

gyroscopes. In contrast to the open-loop mode of operation, in the closed-loop

mode of operation, the sense amplitude of oscillation is continuously

monitored and driven to zero. As a consequence, the bandwidth and dynamic

range of the gyroscope can be greatly increased beyond what can be achieved

with the open-loop mode of operation [42]. However, under conventional

closed-loop mode of operation, it is difficult to ensure a constant noise

performance, in the face of environment variations such as temperature

changes, unless an on-line mode tuning scheme is included. Moreover, there

are practical difficulties in designing a feedback controller which closed-loop

25

system is stable and sufficiently robust, for gyroscopes with high Q (quality

factor) systems. Therefore, the application areas for conventional closed-loop

mode of operation are those which requires medium-cost and medium-

performance (large bandwidth but limited resolution) gyroscopes [43].

The control system of the MEMS gyroscope has to perform four main

tasks: (i) initiate drive axis vibrations at resonant frequency, (ii) maintain the

amplitude of primary vibration at a desired level, (iii) eliminate the cross

coupling errors in the gyroscope and finally (iv) determine the input angular

velocity [43].

Over the years, various controllers were developed for performing

these basic tasks for MEMS gyroscopes. To drive the input frequency of the

drive axis to resonance, phase locked loop technique is used by some

researchers [44]. In phase locked loop, the input frequency is adjusted until

the output of the drive axis is out of phase with the input by -90o, indicating

resonance [45].

In reference [46], an adaptive controller has been developed in which

the system parameters are adjusted using a feedback loop such that resonance

is achieved at a given input frequency. Different methods may be utilized to

adjust the amplitude of the primary vibrations, such as using an automatic

gain control loop [47] or using adaptive controller as described in [48].

26

Several different strategies were used for canceling the quadrature errors and

estimating the angular velocity as follows:

• For open-loop mode of operation: Feedforward scheme [49] and

Feedback control scheme [50];

• For closed-loop mode of operation, a Kalman filter based approach

[51] and force-balancing feedback scheme [52].

an observer-based adaptive controller that is self-calibrating,

compensates for fabrication errors and estimates the angular velocity is

offered in [43]. In reference [53] the Control scheme for a z -axis MEMS

vibrational gyroscope is developed using basic linear system techniques.

Summary

Different types of gyroscope were introduced and they were

categorized in different ways, based on their shapes, applications, and ….

Basic principles of conventional gyroscopes as well as the operational

principles of different types of gyroscope were investigated. Moreover,

several applications of MEMS gyroscopes were briefly presented. Finally, the

main tasks of control systems in gyroscopes and different control strategies of

MEMS gyroscopes were discussed.

27

Chapter 3

OVERVIEW OF SINGLE BEAM GYROSCOPE‡

Introduction

As shown in Figure 3.1, a vibrating beam gyroscope comprises a

cantilever beam with a tip mass, attached to a moving base. According to the

discussions of chapter 2, the gyroscope is based on the principle of Coriolis

acceleration. A lateral vibratory motion is induced in the beam using a

suitable actuation mechanism. In the presence of the angular rotation of the

beam along its longitudinal axis, secondary lateral vibrations are induced in

the beam in the direction orthogonal to the primary oscillations (Flexural-

Flexural). By measuring these secondary vibrations, rate of angular rotation

can be determined.

Figure 3.1: Schematic of a single beam gyroscope.

‡ This chapter is based on the work done in Ref. [1]. Since most of the concepts used in this chapter will be used in the next chapter, we will treat it in detail.

28

This chapter deals with a single beam gyroscope. First we derive the

equation of motion for such a system. Some sample simulations will be

presented, and finally we will investigate the cross-axis effect in this kind of

gyroscope.

A second type of vibrating beam gyroscope will be studied while a

cantilever beam with a tip mass is subjected to a combination of flexural-

torsional vibrations. Similar to the case of flexural-flexural vibrating beam

gyroscope, a flexural vibration is induced in the beam using piezoelectric

actuator (drive mode) and due to the Coriolis force, in presence of the rotation

about the z-axis, a secondary torsional vibration is induced in the beam (sense

mode). A schematic of such a system is illustrated in Figure 3.2.

Figure 3.2: Schematic of a flexural-torsional beam gyroscope [1].

29

The secondary torsional vibrations, which are proportional to the rate

of rotation of the beam, can be measured using an accelerometer, laser sensors

or piezoelectric sensor (as shown), in order to determine the rate of rotation.

The effect of the end mass is to improve the performance of the gyroscope by

increasing the gyroscopic effect.

The measurement of the rate of angular rotation will be accurate, if the

beam only rotates around the Z -axis. In practice, however, the base of the

gyroscope is subjected to secondary rotations as well (e.g. rotation about the

longitudinal axis), which can produce significant errors in the measurement of

the primary angular velocity. In fact, one of the major sources of error in the

vibrating beam gyroscope is the presence of these secondary base rotations

[54].

Many researchers have worked on the problem of coupled bending-

torsion vibrations of cantilever beams, because of its practical importance in

various applications. The theory of coupled flexural-torsional vibrations of

thin walled beams was first developed by Timoshenko and Young [55]. In

their research, they obtained the exact modal solutions for such systems. The

coupled free vibration frequencies of a cantilever beam, was calculated by

Dokumaci [56]. Bercin and Tanaka [57] included warping, shear deformation

and rotary inertia effects into the previous studies. Banerjee [58] developed a

dynamic stiffness matrix analysis method to obtain the natural frequencies and

30

mode shapes of the coupled Euler-Bernoulli beam. The effect of tip mass was

not considered in the mentioned works. Modeling of the tip mass, using the

Euler-Bernoulli beam theory is discussed in [59] and [60]; also Bhat and

Wagner [61] tried to find the frequency equations of a cantilever beam with

tip mass.

More recently, Kirk and Wiedemann [62] performed a study on the

free vibration of a flexible beam with rigid payloads at the tip. They used the

Euler-Benoulli theory but the effect of torsion was not considered.

Oguamanam [63] carried out research on a cantilever beam with a rigid tip

mass, whose center of gravity was not coincident with the attachment point.

This work was extended by Gokdag and Kopmaz [64] through analyzing the

coupled flexural-torsional free and forced vibrations of a beam with tip and in

span attachments. H. Salarieh and M. Ghorashi [65] performed the analyses

on the same system but having Timoshenko beam. Most of these works

analyze the cantilever beam when the base is stationary.

Esmaeili et al. in a series of publications studied the flexural-flexural

vibrations of a cantilever beam with tip mass and subjected to general support

motion [66- 68]. Bhadbhade et al [1] carried out a study on a piezoelectrically

actuated flexural/torsional vibrating beam gyroscope and investigated the

effects of secondary rotation (cross-axis effect) on the precision of the

gyroscope.

31

Governing Equations of Motion

Since we are to the study the coupled flexural-torsional vibrations of

the system, we will derive the two linear partial differential equations of

motion of the system. Several methods can be applied to derive the equations

of motion, such as Newton’s second law of motion, Lagrange’s equation and

extended Hamilton’s principle. In this work, we will use the extended

Hamilton’s Principle to derive the equations of motion.

Since the system is studied in the macro scaled, a few assumptions are

made accordingly. The beam is assumed to follow the Euler-Bernoulli theory

and i.e. the effects of warping and shear deformation are neglected. The beam

is considered to be a slender beam (with small thickness to length ratio). The

Poisson effect is also neglected.

In order to implement the extended Hamilton’s principle, we need to

know the total kinetic energy and potential energy of the system as well as the

total non-conservative work done on it. Beam kinematics should be studied to

define different motions in the systems and their relations, so that we can

define the total kinetic and potential energy of the system.

32

Beam Kinematics

Let’s consider the system shown in Figure 3.3. A rigid tip mass M of

finite dimensions (with length l) is attached to the end of a uniform and

straight metallic cantilever beam with length L and mass per unit length bρ .

We will define two different coordinate sytems: Inertial and rotating

coordinate. The inertial coordinate system is denoted by ( 1 2 3, ,A A A ). The

moving (rotating) coordinate system is denoted by (X,Y,Z) with orthogonal

unit vectors ( , ,X Y Za a a ). We also define a local curvilinear coordinate system

at point p, which is denoted by (x,y,z) and has orthogonal unit vectors

( , ,x y za a a ) but is not shown in the figure. As mentioned before, the primary

bending vibrations w(x,t) in the beam, using piezoelectric actuator attached on

the beam. The base is subjected to two angular rotations; (i) the primary

rotation 3Ω which is to be measured by the gyroscope and (ii) the

secondary rotation 1Ω which causes errors in the measurement of primary

rotation. In presence of the primary rotation 3Ω (about the Z -axis), due to the

Coriolis effect, the secondary torsional vibrations ( , )x tθ are induced in the

beam. After defining the basic requirements, now we can study different types

of motion in the system.

33

Figure 3.3: Cantilever beam kinematics [1].

Translational Motion

The base motion causes each point on the neutral axis to undergo an

elastic deformation and a rotation. Three Cartesian variables ui, i=1, 2, and 3

(where u1=u, u2=v, u3=w) measured in moving coordinate system (X,Y,Z), are

used in order to describe the translational motion of the beam (as shown in

Figure 3.3). As a result of this, the point P on the neutral axis of beam will be

moved to point P*. The position and velocity of point P* in reference frame

Ai can be defined as [54]:

* *

* *

*

*

( ) ( )( )

( ) ( ) ( )

pp p

p pbase p p

p ii j j X i i

r r u

d r d ur u

dt dtd r d u a a sa u a

dt dt

= +

= +Ω × +

= +Ω × +

(3.1)

34

where jΩ is the rotation of the base relative to the reference frame ( 1 2 3, ,A A A )

and s is the position of point P in the moving coordinate system. In this case,

we assume that, the beam has no axial and lateral vibrations, hence u=0 and

v=0. If we simplify the vector products used in Eq. (3.1), different velocities

of point P* can be expressed as:

*( )P

X Y Z

d rfa ga ha

dt= + + (3.2)

where

1 3

0fg w s

wht

== − Ω + Ω∂

=∂

(3.3)

Therefore, kinetic energy of the beam due to the translational motion is

given as

2 2 21 0

1 ( )2

L

bT f g h dxρ= + +∫ (3.4)

The same approach and procedure can be used, in order to calculate the

translational kinetic energy of the tip mass, since the mass is attached to the

end of the beam, with the center of mass collinear with the beam centroid.

35

According to Figure 3.4, position and velocity of the center of gravity

of the mass is

*

*

*

( )( ) ( ) ( )

Mass q mq

qMass mbase q mq

r r u r

d ud r d r r u rdt dt dt

= + +

= + +Ω × + +(3.5)

where q is the point where end mass attaches to the beam, q* is the deformed

position of point q and rm is the position vector of center of gravity of the end

mass from point q* , in the deformed position given as

1 1cos sin2 2m L X L Zr a aψ ψ= + (3.6)

where Lx L

wx

ψ=

∂=∂

.

Figure 3.4: End mass kinematics [1].

36

Consequently, we will have

( ) ( ) sin cos2 2

cos sin2 2

Mass i L Li L X L Z

j j i i X L X L Z

d r d u l la a adt dt t t

l la u a La a a

ψ ψψ ψ

ψ ψ

∂ ∂⎛ ⎞= + − +⎜ ⎟∂ ∂⎝ ⎠⎛ ⎞+Ω × + + +⎜ ⎟⎝ ⎠

(3.7)

By simplifying Eq. (3.7), we can get

( )MassM X M Y M Z

d r f a g a h adt

= + + (3.8)

where

3 1 3 1

sin2

cos sin2 2

cos2

( , ) .

LM L

M L L L

L LM L

L x L

lft

l lg L w

w lht t

w w x t

ψψ

ψ ψ

ψψ

=

∂= −

∂

= Ω −Ω +Ω −Ω

∂ ∂= +

∂ ∂=

(3.9)

Since we are working on a macro-scaled system, we can assume small

deflections and ignore the nonlinear terms. This yield

3 1 3 1

0

2 2

2

M

M L L

L LM

fl lg L w

w lht t

ψ

ψ

=

= Ω −Ω +Ω −Ω

∂ ∂= +

∂ ∂

(3.10)

37

Therefore, the translational kinetic energy of the end mass can be given

as:

( )2 2 21

12M M M MT M f g h= + + (3.11)

Rotational Motion

In order to describe the deformation of the system from its original

configuration, we use the Euler angle rotations. The rotating coordinate

system with orthogonal unit vectors ( , ,X Y Za a a ) is denoted by (X,Y,Z) (See

Figure 3.5). The X-axis coincides with the longitudinal/centroidal axis of the

beam before deformation. The local curvilinear coordinate system at arclength

s in the deformed position is denoted by (x,y,z), with orthogonal unit vectors

( , ,x y za a a ) [69].

Figure 3.5: Euler angle rotations [69].

38

Each cross section of the beam will have an elastic displacement of its

neutral axis and a rotation. We can find the displacement components

according to the equations given in the previous section. we use successive

Euler angle (counterclockwise) rotations with the angle of rotations denoted

by ψ(x,t) and θ(x,t), in order to describe the rotation of the neutral axis, from

the undeformed to the deformed position, as shown in Figure 3.5, where

( , )( , ) w x tx tx

ψ ∂=

∂.

(X,Y,Z) coordinate system will be taken to (x',y'=Y,z') by the first

rotation ψ, about Y. The second rotation θ about x' takes (x',y',z') to the final

orientation (x=x',y,z). The relationship of the three unit vector triads is given

in this form [69]

[ ] [ ]'

'

'

x x X

y y Y

Tz z Z

a a aa T a T T aa a a

θ θ ψ

⎧ ⎫ ⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎡ ⎤= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎣ ⎦⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭ ⎩ ⎭

(3.12)

where

[ ]

[ ]

1 0 0 cos 0 sin0 cos sin , 0 1 00 sin cos sin 0 cos

cos 0 sinsin sin cos sin coscos sin sin cos cos

T T

T

θ ψ

ψ ψθ θθ θ ψ ψ

ψ ψθ ψ θ θ ψθ ψ θ θ ψ

−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎡ ⎤= =⎣ ⎦⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

−⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥−⎣ ⎦

(3.13)

39

Thus, the angular velocity of the beam is given as

1 3X Z Y xa a a at tψ θω ∂ ∂

= Ω +Ω + +∂ ∂

(3.14)

Since the transformation matrices [Tθ], [Tψ] and [T] are orthogonal

matrices, they posses the property[ ] [ ]1 TQ Q− = . Using Eqs. (3.13) and (3.14)

and this property, we can obtain the absolute angular velocity ω(x,t) in this

form

1 3

1 3

1 3

cos sin

cos sin cos sin cos

sin cos sin cos cos

x

y

z

at

at

at

θω ψ ψ

ψ θ θ ψ θ ψ

ψ θ θ ψ θ ψ

∂⎛ ⎞= +Ω −Ω⎜ ⎟∂⎝ ⎠∂⎛ ⎞+ +Ω +Ω⎜ ⎟∂⎝ ⎠∂⎛ ⎞+ − +Ω +Ω⎜ ⎟∂⎝ ⎠

(3.15)

According to Eq. (3.15), the components of the absolute angular

velocity of the beam can be expressed as

1 3

1 3

1 3

cos sin

cos sin sin sin cos

sin cos sin cos cos

x

y

z

t

t

t

θω ψ ψ

ψω θ θ ψ θ ψ

ψω θ θ ψ θ ψ

∂⎛ ⎞= +Ω −Ω⎜ ⎟∂⎝ ⎠∂⎛ ⎞= +Ω +Ω⎜ ⎟∂⎝ ⎠∂⎛ ⎞= − +Ω +Ω⎜ ⎟∂⎝ ⎠

(3.16)

Using the assumption of small angles of bending and torsion, we get

40

( )

1 3

1 3

1 3

x

y

z

t

t

θω ψ

ψω θψ θ

ω ψ

∂⎛ ⎞= +Ω −Ω⎜ ⎟∂⎝ ⎠∂⎛ ⎞= +Ω +Ω⎜ ⎟∂⎝ ⎠

= Ω +Ω

(3.17)

The kinetic energy of the beam and end mass due to the rotational

motion is

( )2 2 22 0

12

L

xb x yb y zb zT I I I dxω ω ω= + +∫ (3.18)

( )2 2 22

12M xM x yM y zM zT I I Iω ω ω= + + (3.19)

where Ixb,Iyb,Izb and IxM,IyM,IzM are the mass moments of inertia of the beam

and end mass, about the X,Y and Z axes, respectively.

Piezoelectric Modeling

Piezoelectricity is the ability of some materials (notably crystals and certain

ceramics) to generate an electric potential in response to applied mechanical stress.

This may take the form of a separation of electric charge across the crystal lattice. If

the material is not short-circuited, the applied charge induces a voltage across the

material. The word is derived from the Greek piezein, which means to squeeze or

press. The piezoelectric effect is reversible in that materials exhibiting the direct

piezoelectric effect (the production of electricity when stress is applied) also exhibit

the converse piezoelectric effect (the production of stress and/or strain when an

41

electric field is applied). For example, lead zirconate titanate (PZT) crystals will

exhibit a maximum shape change of about 0.1% of the original dimension.Due to

their precise operation, piezoelectric actuators are becoming increasingly important

in micro-positioning technology. The direction of expansion with respect to the

direction of the electrical field depends on the constitutive equations of the

piezoelectric material [70].

Constitutive Equations

The fundamental relations for the piezoelectric materials are [70, 71]

E PS s T dE= + (3.20)

T PD dT Eε= + (3.21)

where d is the piezoelectric constant, S is the strain, EP is the electric field, T

is the mechanical stress, εT is the permittivity matrix under constant stress,

and sE refers to the compliance of material when the electric field is constant.

In Eq. (3.21), d relates the electric charge per unit area D (the dielectric

displacement) to the stress T under a zero electric field. We can also rewrite

Equs. (3-20) and (3-21) in the following form

E PT c S eE= − (3.22)

s PD eS Eε= + (3.23)

42

where cE = 1/sE is the Young’s modulus matrix under constant electric field,

e=d/sE is a constant matrix relating the charge per unit area to the strain, and

εS is the permittivity matrix under constant strain [70, 72].

Piezoelectric Patch Actuator

As shown in Figure 3.6, piezoelectric patch actuator comprises of a

thin piezoelectric film bonded on the structure. The geometrical arrangement

is such that d31 (the piezoelectric coefficient that relates the electrical field in

thickness direction to the strain in longitudinal direction) dominates the design

and the useful direction of expansion is normal to that of the electrical field.

Using standard engineering notations and one-dimensional deformation

assumption, Eq. (3.22) within the piezoelectric layer for patch actuator

reduces to [1, 70]

31( )

x P x PP

v tE E dt

σ ε= − (3.24)

where EP is the Young’s modulus of elasticity of the piezoelectric actuator

and v(t) is the applied voltage to the actuator.

43

Figure 3.6: Piezoelectric patch actuator [72].

Effect of Piezoelectric Actuator

In order to find the effect of the piezoelectric actuator, we first write

the equilibrium equation of a classical Euler-Bernoulli beam as follows

2 2

2 2

( , ) ( , )w x t M x tAt x

ρ ∂ ∂= −

∂ ∂ (3.25)

where M(x,t) is the cross-sectional bending moment acting at distance x from

the clamped end of the beam. Since the thickness of piezoelectric layer is very

much less than the thickness of the beam, we can assume that the neutral axis

of the beam does not change in the beam. Therefore, the bending moment is

expressed in this form [72]

2

312

( , ) 1( , ) ( ) ( )( ) ( )2 P b P

w x tM x t EI x bE d v t t t S xx

∂= + +

∂(3.26)

44

where

1 2( ) ( ) and ( ) ( ) ( )b PEI x EI S x EI S x H x l H x l= + = − − − (3.27)

In these equations, b and tb are the width and thickness of the beam,

respectively; tP, EP and d31 are the thicknesses, elastic modulus and

piezoelectric constant of the actuator, respectively; v(t) is the voltage applied

to the actuator, EIb and EIP are the flexural rigidities of the beam and actuator,

respectively. As shown in Figure 3.7, the piezoelectric layer is just bound on a

finite part of the beam, so H(x), the Heaviside function, is used to show that it

is located from l1 to l2. Introducing (3.26) into (3.25) yields

2 2 2 2

312 2 2 2

( , ) ( , ) 1( ) ( )( ) ( )2

Pb P

w x t w x tA EI x bE d v t t t S xt x t x

ρ⎛ ⎞∂ ∂ ∂ ∂ ⎛ ⎞+ = − +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ⎝ ⎠⎝ ⎠

(3.28)

Figure 3.7: Schematic of a piezoelectric actuator attached on the beam.

If we compare the typical Euler-Bernoulli beam theory with Eq. (3.28)

we can conclude that the distributed actuator is equivalent to adding

45

concentrated moments MP at the boundaries of the actuator [73], where MP is

expressed as

31 01 ( ) ( ) ( ) ( ) ( )2P P b P PM bE d t t v t S x M v t S x= − + = (3.29)

Equations of Motion

As mentioned in previous chapters, the extended Hamilton’s principle

is used to derive the two linear partial differential equations (PDEs) governing

the flexural-torsional vibrations of the beam. The extended Hamilton’s

principle for a dynamic system is expressed as

2

1

0t

nctT V W dtδ δ δ− + =∫ (3.30)

where T is the total kinetic energy of the system, V is the total potential energy

and Wnc is the total non-conservative work done on the system. The total

kinetic energy of the beam can be defined as the summation of the kinetic

energy due to the translational motion and the rotational motion. Ignoring the

rotary inertia terms, the total kinetic energy of the beam can be obtained from

Eqs. (3.4) and (3.18) as follows

( )2 2 2 2

0

1 ( ) ( )2

L

b x xT x f g h I x dxρ ω⎡ ⎤= + + +⎣ ⎦∫ (3.31)

46

Combining Eq. (3.31) with Eqs. (3.11) and (3.19), we can write the

total kinetic energy of the system as

( )

( ) ( )

2 2 2 2

0

2 2 2 2 2 2

1 ( ) ( )21 12 2

L

x x

M M M xM x yM y zM z

T x f g h I x dx

M f g h I I I

ρ ω

ω ω ω

⎡ ⎤= + + +⎣ ⎦

+ + + + + +

∫ (3.32)

where

( )( )

1 2

( ) ( )

( ) ( )( ) ( ) ( )

b P

x xb xP

x S x

I x I S x IS x H x l H x l

ρ ρ ρ= +

= +

= − − −

(3.33)

and ρp and Ixp are the mass per unit length and mass moment of inertia of the

piezoelectric actuator, respectively.

The total potential energy of the system can be written as:

2 22

20

1 ( ) ( )2

L wV EI x GJ x dxx x

θ⎡ ⎤⎛ ⎞∂ ∂⎛ ⎞⎢ ⎥= +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦∫ (3.34)

where E and G are elastic and shear moduli of the beam, respectively. EI(x)

and GJ(x) are given as

( )( )

( ) ( )

( ) ( )b P

b P

EI x EI S x EI

GJ x GJ S x GJ

= +

= + (3.35)

47

where EIb and EIP are the flexural rigidities of the beam and actuator,

respectively, and GJb and GJP are the torsional rigidities of the beam and

actuator, respectively.

In the Hamiltonian approach, the piezoelectric actuator control moment

MP and the damping effects are collected in the following virtual work

expression [74]

2

20 0 0

12

L L LPnc B T

M wW wdx C wdx C dxx t t

θδ δ δ δθ∂ ∂ ∂= + +

∂ ∂ ∂∫ ∫ ∫ (3.36)

where CB and CT are the damping coefficients in bending and torsion,

respectively. The actuator control moment is given as

31 01 ( ) ( ) ( ) ( ) ( )2P P b P PM bE d t t v t S x M v t S x= − + = (3.37)

Finally, we should take the variations, from the defined equations, in

order to achieve the equations of motion and boundary conditions which are

expressed as Eqs. (3.38) through (3.43) (the detailed derivation of equations

of motion and boundary conditions is given in Appendix A). It should be

noted that, in these equations damping has been ignored in both bending and

torsion.

2 2 221 1 3 3 32 2

22 2

2 2 2

( ) ( )

( )

x

P

w wx w s I xt t x x

MwEI xx x x

θρ⎛ ⎞ ⎛ ⎞∂ ∂ ∂

− Ω + Ω Ω − Ω +Ω⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎛ ⎞⎛ ⎞ ∂∂ ∂

+ =⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠

(3.38)

48

2 2

32( ) ( ) 0xwI x GJ x

t t x x xθ θ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞+Ω − =⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠⎝ ⎠

(3.39)

0 00

0 , 0 , 0x x

x

wwx

θ= =

=

∂= = =

∂ (3.40)

1 3 3 3 1 1

2 3 3

2 2 3

( ) ( ) ( )2 2

( ) 02

xx L x L

x L x L

w l l wI x M L wt x x

w l w wM EI xt t x x

θ

= =

= =

∂ ∂ ∂⎛ ⎞ ⎛ ⎞+Ω −Ω Ω + −Ω + +Ω + Ω⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞∂ ∂ ∂− + + =⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

(3.41)

2 3

1 1 3 32 2

3 2

1 3 1 3 12

2

1 3 1 2

2 2 2

( )

( ) 0

xMx Lx L

yM yM L

x L x L

zMx L x L

l l w l w wM It t x t x

w w wI It x t t t x x

w wI EI xx x

θ

θψ θ θ θ θ

==

= =

= =

⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞−Ω − + + +Ω −Ω Ω⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠⎝ ⎠

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂− +Ω +Ω + +Ω +Ω Ω⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞∂ ∂⎛ ⎞− Ω +Ω Ω − =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

(3.42)

2 2 2

3 1 3 12

2

1 3 3 ( ) 0

xM yMx Lx L x L

yMx Lx L

w w w wI It t x t x x x

w wI GJ xt x x x

θ θ θ

θθ θ

== =

==

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂− −Ω + +Ω +Ω Ω⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞∂ ∂ ∂⎛ ⎞− +Ω +Ω Ω − =⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

(3.43)

Ω3, which is the primary base rotation velocity, is supposed to be

measured in these equations. As seen from Eqs. (3.38) and (3.39), the system

governing equations are coupled through the base rotation velocity Ω3. In the

absence of the base rotation, the governing equations become decoupled. In

order to analyze the error caused in the output of the gyroscope due to the

presence of the secondary base rotation, Ω1 has been considered.

49

Time Response

In order to perform the time-domain analysis on the system, we use the

Assumed Mode Model (AMM) expansion to truncate the original partial

differential governing equations of motion to ordinary differential equations

[54, 75, 76]. In this method, the lateral displacement w(x,t) and torsional

displacement θ(x,t) are assumed to be linear functions of assumed modes and

generalized coordinates, in this form

1

1

( , ) ( ) ( )

( , ) ( ) ( )

n

j jj

n

j jj

w x t x p t

x t x q t

φ

θ ψ

=

=

=

=

∑

∑ (3.44)

where φj(x) and ψj(x) are the mode shapes of a cantilever beam (with

no base rotation and rigid mass) under bending and torsion, respectively; and

pj(t) and qj(t) are the generalized coordinates for bending and torsion,

respectively. The mode shapes used for this case are

( ) ( )( ) sin sinh cos coshn n n n nx x x x xφ β β α β β= − − − (3.45)( )2 1

( ) sin2

n xx

Lπ

ψ⎡ ⎤+

= ⎢ ⎥⎣ ⎦

(3.46)

where nβ is the nth natural frequency of a cantilever beam and

sin sinhcos cosh

n nn

n n

L LL L

β βαβ β

−=

+ (3.47)

50

Although the cantilever beam is subjected to base rotation and it has a

rigid mass attached to its end, we have used the mode shapes for a regular

cantilever beam (with no base rotation and rigid mass). The mode shapes

given by Eqs. (3.45) and (3.46) satisfy the geometrical boundary conditions

for the beam (Eq. (3.40)) but they do not satisfy the natural boundary

conditions (Eqs. (3.41-3.43)). Hence, these mode shapes can be called as

admissible functions [77].

For two reasons, using these admissible functions does not

significantly affect the results: (i) the base rotations do not considerably

change the natural frequencies [1]. Hence, the mode shapes for the beam

without any base rotations can be safely used, and (ii) the mode shapes for the

cantilever beam with a rigid mass attached to its end are very complicated.

The main purpose of this chapter is to study the cross-axis effect in the beam,

hence for simplifying the analysis we have not considered the end mass. This

is a valid simplification since ( )xφ and ψ(x) are the admissible functions.

Using these admissible functions does not alter the results as the generalized

coordinates pj(t) and qj(t) in Eq. (3.44) change accordingly to give correct

response for w(x,t) and θ(x,t) .

The system governing equations can now be obtained by substituting

Eq. (3.44) into Eqs. (3.38) and (3.39) and ignoring damping in the system as

follow [1]:

51

( )2 21 1 3 1 1 3 1 1 1 1 3 1( ) ( ) ( )M p t C q t K D E p t G F+ Ω + + Ω − Ω + Ω Ω = (3.48)

2 2 3 2( ) ( ) ( ) 0M q t C p t K q t+ Ω + = (3.49)

where

[ ]

1 10 0

1 10 0

1 10 0

1 0 2 1

2

( ) ( ) ( ) , ( ) ( ) ( ) ,

( ) ( ) ( ) , ( ) ( ) ( ) ,

( ) ( ) ( ) , ( )

( ) ( ) ( ) , , 1, 2,...,

(

L L

ij i j ij x i j

L L

ij x i j ij x i j

L L

ij i j ij

ij P i i

ij x

M x x x dx C I x x x dx

K EI x x x dx D I x x x dx

E x x x dx G x xdx

F M v t l l i j n

M I

ρ φ φ φ ψ

φ φ φ φ

ρ φ φ ρ

φ φ

′= =

′′ ′′ ′′= =

= =

′ ′= − =

=

∫ ∫∫ ∫∫ ∫

20 0

2 0

1 2 1 2

) ( ) ( ) , ( ) ( ) ( ) ,

( ) ( ) ( ) , , 1, 2,...,

, ,..., , , ,...,

L L

i j ij x i j

L

ij i j

T Tn n

x x x dx C I x x x dx

K GJ x x x dx i j n

p p p p q q q q

ψ ψ ψ φ

ψ ψ

′=

′ ′= =

= =

∫ ∫∫

(3.50)

Equations of motion given by Eqs. (3.48) and (3.49) can be solved by

MATLAB for two modes and system parameters given in Table 3.1. In this

work two modes have been considered, however using more numbers of the

modes will lead to more accurate results.

Table 3.1: Physical parameters of the system [1].

Parameter Notation Value Beam length (m) L 0.15 Beam thickness (m) tb 0.8×10-3 Beam width (m) b 1.5×10-2 Mass per unit length (kg/m) ρb 3960×bt Beam elastic modulus (Gpa) E 70 Beam shear modulus (Gpa) G 30 End mass length (m) l 0.01 End mass width (m) bM 0.02 End mass height (m) hM 0.02

52

Cross-Axis Effects

The vibrating beam gyroscope is basically used in order to measure the

rotational rate around one of the axes. In practice, however, there are always

some secondary rotations present in the system. These secondary base

rotations can produce significant errors in measurement of the gyroscope

output. These errors are sometimes referred as ‘cross-axis’ effects. In this

section, we discuss the effects of these secondary vibrations on the output of

the gyroscope.

Figure 3.8 illustrates the output of the gyroscope when it is subjected to

primary (Ω3) as well as secondary (Ω1) base rotations of constant angular

velocity. To analyze the ‘cross-axis’ effects, the magnitude of the secondary

base rotation (Ω1) is varied from 0 to 0.5 rad/s while keeping a constant

piezoelectric excitation voltage (V=300 volts) and primary base rotation

(Ω3=20 rad/s). Figure 3.8(a) shows the desired gyroscopic output which is to

be measured. Figures 3.8 (b), (c) and (d) show the gyroscopic output from the

system when the base has secondary rotations of very small magnitude (0.05 –

0.5 rad/s). It can be seen that the gyroscopic output is increased significantly

(almost 40 times more) even for such a small magnitudes of secondary

rotation. This increased output could be interpreted as a gyroscopic output due

to primary base rotation. Such interpretation can produce errors in the

measurement of base velocity.

53

Figure 4.8: System response (torsional deflection θ(L,t)) to secondary base rotation; (a) Ω1 = 0 rad/s , (b) Ω1 = 0.05 rad/s , (c) Ω1 = 0.1 rad/s , and (d) Ω1 = 0.5 rad/s [1].

This is the most important drawback of the single beam gyroscopes,

and is an important factor to take into account in the design of the vibrating

beam gyroscope and effective control strategies have to be developed to

54

eliminate this secondary output (‘cross axis’ effects). The rocking-mass

gyroscope, which will be explained in the next chapter, will eliminate this

error and give a very precise measurement.

Summary

A detailed mathematical modeling of a vibrating beam gyroscope

undergoing flexural-torsional vibrations was presented, in this chapter. The

extended Hamilton’s principle was used in order to derive the governing

equations and boundary conditions. Two base rotations were considered for

the beam: (i) primary base rotation about z-axis, Ω3, and secondary base

rotation about x-axis Ω1. Moreover, the adverse effect of secondary base

rotations (cross-axis effects) on the gyroscopic output signals was discussed.

We concluded that the main drawback of the single beam gyroscopes is the

significant error caused by cross-axis effects.

55

Chapter 4

MODELING OF THE ROCKING-MASS

GYROSCOPE

Introduction

In the previous chapters, we discussed a single beam gyroscope which

comprises a cantilever beam with a tip mass undergoing coupled flexural-

torsional vibrations. As mentioned before, the main drawback of this type of

gyroscope is the difficulty in measuring the secondary torsional vibrations

induced due to the Coriolis force. In this chapter we will work on a rocking-

mass gyroscope, which can be considered as an extension of the single beam

gyroscope, and can overcome the limitations of the single beam gyroscope.

The governing equations of motion as well as the boundary conditions for

such a system, which comprises a set of four cantilever beams and a rocking

mass attached to them in the middle, will be derived in full detail. A

schematic of a rocking-mass gyroscope is shown in Figure 4.1.

56

Figure 4.1: Schematic of a rocking-mass gyroscope.

Principle of Operation

As shown in Figure 4.1, the rocking-mass gyroscope consists of four

beams attached to a rocking mass in the middle. In order to induce and sense

the vibrations in the beams, piezoelectric actuators are attached to beam 1 and

beam 3, and piezoelectric sensors are attached to beams 2 and 4. The primary

bending vibration is induced in the beams 1 and 3, by supplying a sinusoidal

voltage to the piezoelectric patches on them. Due to the bending of the beams

1 and 3, the rocking mass will rotate and produces a torsional vibration in

beams 2 and 4 as schematically shown in Figure 4.2. In presence of the base

angular rotation about the vertical axis, due to the Coriolis force a secondary

rocking motion of the mass is induced. As shown in Figure 4.3, bending is

induced in beams 2 and 4, as a result of the secondary rocking motion of the

mass. The amplitude of this bending vibration is proportional to the angular

57

velocity of the base. This secondary bending vibration, which can be

measured by the piezoelectric sensors placed on beams 2 and 4, gives the

angular velocity of the base.

Similar to the single beam gyroscope discussed in the previous chapter,

the rocking-mass gyroscope, uses the secondary induced vibrations to

determine the rate of rotation. It is usually difficult to measure the secondary

torsional vibrations for the single beam gyroscope, as their amplitude is

relatively small. This drawback is overcome by the rocking-mass gyroscope.

In the rocking-mass gyroscope, the torsional vibrations produced in two

beams are transferred to other two beams as bending vibrations which can be

easily sensed by placing sensors on the beams.

Figure 4.2: Primary rocking motion of the mass.

58

Figure 4.3: Secondary rocking motion of the mass.

Not many researchers have focused on the modeling and performance

evaluation of the ‘rocking-mass’ gyroscope. Tang and Gutierrez [78] dealt

with the fabrication and design of a rocking-mass gyroscope, but the operating

principle of the device was not discussed. Royle and Fox [79] presented an

analysis of the mechanics of an oscillatory rate gyroscope that is actuated and

sensed using thin piezoelectric actuators and sensors.

In this chapter, we derive the equations of motion and boundary

conditions governing the motion of a macro-scaled ‘rocking-mass’ gyroscope.

A method similar to the one developed for the single beam gyroscope in

chapter 3, is used here to derive the equations of motion, since the gyroscope

consists of four beams undergoing coupled flexural-torsional vibrations.

59

Governing Equations of Motion

The eight linear partial differential equations and their corresponding

boundary conditions governing the flexural-torsional motion of the four

beams of the gyroscope will be developed in this section, using the extended

Hamilton Principle. All beams are assumed to follow the Euler-Bernoulli

theory and accordingly the effects of warping and shear deformation are

neglected. The beams are considered to be slender. The poison effect is also

neglected. These assumptions usually hold for macro-scaled system not

micro-scaled ones.

We will follow the same methodology that discussed in chapter 3 to

develop the equations of motion and boundary conditions, since the rocking-

mass gyroscope similar to a single beam gyroscope. Figure 4.4 illustrates the

four coordinate systems that we will use. All the beams are identical with

mass per unit length ρb and thickness tb.

For this case, we consider a system with identical beams of equal

lengths denoted as Li (i=1,2,3,4). The length of the rocking-mass is denoted

by l. Bending and torsional deformations of the beams are denoted by wi

(i=1,2,3,4) and θi (i=1,2,3,4) respectively. In derivation of the two coupled

governing equations (one for bending and one for torsion) for each of the four

beams, we consider the rocking-mass to be attached to the first beam.

60

Figure 4.4: Top view of the rocking-mass gyroscope with the coordinate systems.

Similar to the single beam gyroscope, we need to find the total

translational kinetic energy, total rotational kinetic energy, total potential

energy and the non-conservative work done on the system, sequentially.

Translational Motion

The total translational kinetic energy of the four beams can be

determined by Extending Eq. (3.4) in the following form

2 2 2 2 2 21 1 1 1 1 2 2 2 20 0

2 2 2 2 2 23 3 3 3 4 4 4 40 0

1 1( ) ( )2 21 1( ) ( )2 2

L L

b b b

L L

b b

T f g h dx f g h dx

f g h dx f g h dx

ρ ρ

ρ ρ

= + + + + +

+ + + + + +

∫ ∫

∫ ∫(4.1)

where

61

1 2 3 4 4 2

1 1 1 1 3 2 2 3 3 3 1 3 3 4 4 3

31 2 41 2 2 2 3 4 4 2

0, 0, 0, ,, , , ,

, , , .

f f f f wg w s g s g w s g s

ww w wh h s h h st t t t

= = = = − Ω= − Ω + Ω = Ω = Ω + Ω = Ω

∂∂ ∂ ∂= = − Ω = = + Ω

∂ ∂ ∂ ∂

(4.2)

Also we can extend Eqs. (3.10) and (3.11), to define the translational

kinetic energy of the rocking mass, according to the following equation

( )2 2 21

12M M M MT M f g h= + + (4.3)

where

3 1 1 1 3 1 1

1 1

0,

,2 2

2

M

M L L

L LM

fl lg L w

w lht t

ψ

ψ

=

= Ω −Ω +Ω −Ω

∂ ∂= +

∂ ∂

(4.4)

and

1 1

1 1

11 1 1 1

1

( , ) , .L Lx Lx L

ww w x tx

ψ=

=

∂= =

∂ (4.5)

Rotational Motion

In order to describe the deformation of the beams and the rocking mass

from their original configuration, we use the Euler angle rotations. Again like

what we did for a single beam gyroscope, the two successive angles of

62

rotation for each beam are considered and denoted as ψi (i =1,2,3,4) and θi

(i=1,2,3,4). Similar to Eq. (3.17), if we assuming small angles of bending and

torsion and ignore nonlinear terms (which is fine for a macro-scaled system),

the angular velocity components of four beams and rocking mass are

expressed as

( )

( )

( )

1 11 1 3 1 1 1 1 1 3 1 1 1 1 3

2 22 3 2 2 2 3 2 2 2 2 3

3 33 1 3 3 3 1 3 3 3 3 3 1 3 3

44 3 4

, ,

, ,

, ,

x y z

x y z

x y z

x

t t

t t

t t

t

θ ψω ψ ω θψ θ ω ψ

θ ψω ψ ω θ ω θ

θ ψω ψ ω θψ θ ω ψ

θω ψ

∂ ∂⎛ ⎞ ⎛ ⎞= +Ω −Ω = +Ω +Ω = Ω +Ω⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠∂ ∂⎛ ⎞ ⎛ ⎞= −Ω = +Ω +Ω = Ω +Ω⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠∂ ∂⎛ ⎞ ⎛ ⎞= +Ω −Ω = +Ω +Ω = −Ω +Ω⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠∂⎛= −Ω⎜ ∂⎝

( )44 2 3 4 4 2 4 3, ,y zt

ψω θ ω θ∂⎞ ⎛ ⎞= +Ω +Ω = Ω +Ω⎟ ⎜ ⎟∂⎠ ⎝ ⎠

(4.6)

where , 1, 2,3,4ii

i

w ix

ψ ∂= =∂

.

In this case again, we ignore the rotary inertia terms for the beams, i.e.,

Iybωy and Izbωz, therefore, the rotational kinetic energy of four beams can be

written as

1 2

3 4

2 22 1 1 2 20 0

2 23 3 4 40 0

1 12 21 12 2

L L

b xb x xb x

L L

xb x xb x

T I dx I dx

I dx I dx

ω ω

ω ω

⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞+ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∫ ∫

∫ ∫ (4.7)

And the rotational kinetic energy of the rocking mass is

63

( )2 2 22 1 1 1

12M xM x yM y zM zT I I Iω ω ω= + + (4.8)

where Ixb, Iyb, Izb and IxM, IyM, IzM are the mass moments of inertia of the beams

and rocking mass about the X, Y and Z axes, respectively.

Equations of Motion

The eight linear partial differential equations and boundary conditions

for the rocking-mass gyroscope are developed in this section. As mentioned in

Chapter 3, the extended Hamilton Principle for a dynamic system is expressed

as

2

1

0t

nctT V W dtδ δ δ− + =∫ (4.9)

where T is the total kinetic energy of the system, V is the total potential

energy and Wnc is the total non-conservative work done on the system.

Combining Eqs. (4.1), (4.3), (4.7) and (4.8), we can find the total kinetic

energy of the system as:

( )

1 2

3 4

2 2 2 2 2 2 2 21 1 1 1 1 1 1 2 2 2 2 2 2 20 0

2 2 2 2 2 2 2 23 3 3 3 3 3 3 4 4 4 4 4 4 40 0

2 2 2 2 21 1

1 1( ) ( ) ( ) ( )2 21 1( ) ( ) ( ) ( )2 21 12 2

L L

x x x x

L L

x x x x

M M M xM x yM y

T x f g h I x dx x f g h I x dx

x f g h I x dx x f g h I x dx

M f g h I I I

ρ ω ρ ω

ρ ω ρ ω

ω ω

⎡ ⎤ ⎡ ⎤= + + + + + + +⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤+ + + + + + + +⎣ ⎦ ⎣ ⎦

+ + + + + +

∫ ∫

∫ ∫

( )21zM zω

(4.10)

where

64

( )( )( )( )

1 2

3 4

1 2

3 4

( ) ( ) , ( )

( ) ( ) , ( )

( ) ( ) , ( )

( ) ( ) , ( )

b p b

b p b

x xb xp x xb

x xb xp x xb

x S x x

x S x x

I x I S x I I x I

I x I S x I I x I

ρ ρ ρ ρ ρ

ρ ρ ρ ρ ρ

= + =

= + =

= + =

= + =

(4.11)

In Eq. (4.11) ρp and Ixp are the mass per unit length and mass moment

of inertia of the piezoelectric actuator, respectively, and S(x) is used to define

the finite length of the piezoelectric and is given as

1 2( ) ( ) ( )S x H x l H x l= − − − (4.12)

where l1 and l2 are the starting and end position, respectively, of piezoelectric

actuator on beam 1.

There is no vertical motion for the rocking mass, therefore, the total

potential energy of the system consists of only the four beams which can be

stated as:

1 2

3

2 2 2 22 21 1 2 2

1 1 1 2 2 22 20 01 1 2 2

2 22 23 3 4

3 3 3 4203 3 4

1 1( ) ( ) ( ) ( )2 2

1 1( ) ( ) ( )2 2

L L

L

w wV EI x GJ x dx EI x GJ x dxx x x x

w wEI x GJ x dx EI xx x x

θ θ

θ

⎧ ⎫ ⎧ ⎫⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎪ ⎪ ⎪ ⎪= + + +⎨ ⎬ ⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭⎧ ⎫⎛ ⎞ ⎛ ⎞∂ ∂ ∂⎪ ⎪+ + +⎨ ⎬⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

∫ ∫

∫4

2 2

44 420

4

( )L

GJ x dxxθ⎧ ⎫⎛ ⎞ ⎛ ⎞∂⎪ ⎪+⎨ ⎬⎜ ⎟ ⎜ ⎟∂⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

∫

(4.13)

Where EIb and EIp are the flexural rigidities of the beam and actuator, and GJb

and GJp are the torsional rigidities of the beam and actuator respectively. And

also

65

( )( )( )( )

1 2

3 4

1 2

3 4

( ) ( ) , ( )

( ) ( ) , ( )

( ) ( ) , ( )

( ) ( ) , ( )

b p b

b p b

x xb xp x xb

x xb xp x xb

x S x x

x S x x

I x I S x I I x I

I x I S x I I x I

ρ ρ ρ ρ ρ

ρ ρ ρ ρ ρ

= + =

= + =

= + =

= + =

(4.14)

The piezoelectricity and the effect of piezoelectric, as well as its

controlling moment were comprehensively discussed in chapter 3. Based on

the obtained equations, the piezoelectric actuator control moments Mp of

beams 1 and 3 (ignoring damping) are collected in the following virtual work

expression

1 32 2

1 1 3 32 20 01 3

L Lp pnc

M MW w dx w dx

x xδ δ δ

∂ ∂= +

∂ ∂∫ ∫ (4.15)

where

( )31 01 ( ) ( ) ( ) ( )2p p b p pM bE d t t v t S x M v t S x= − + = (4.16)

In these equations the parameters are defined similar to those defined

in chapter 3, but for beams 1 and 3. In other words, b is the width of the

beams 1 and 3; Ep and d31 are the elastic modulus and piezoelectric constant

of the actuators placed on the two beams, respectively; v(t) is the voltage

applied to the actuators and H(x) is the Heaviside function.

66

Finally, we need to take the variations of these equations in order to

derive the governing equations of motion and corresponding boundary

conditions for the rocking-mass gyroscope. Thus

2 2 221 1 1

1 1 1 1 1 3 1 3 32 21 1

2 221

12 2 21 1 1

( ) ( )

( )

x

P

w wx w s I xt t x x

w MEI xx x x

θρ⎛ ⎞⎛ ⎞∂ ∂ ∂

− Ω + Ω Ω − Ω +Ω⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎛ ⎞⎛ ⎞∂ ∂∂

+ =⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠

(4.17)

2 21 1 1

1 3 121 1 1

( ) ( ) 0xwI x GJ x

t t x x xθ θ⎛ ⎞⎛ ⎞ ⎛ ⎞∂ ∂ ∂∂

+Ω − =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠ (4.18)

2 2 2 2222 2 2 2

2 2 2 2 3 3 22 2 2 22 2 2 2

( ) ( ) ( ) 0xw w wx w I x EI xt t x x x x

θρ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂∂

− Ω − Ω +Ω + =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ (4.19)

2 22 2 2

2 3 222 2 2

( ) ( ) 0xwI x GJ x

t t x x xθ θ⎛ ⎞⎛ ⎞ ⎛ ⎞∂ ∂ ∂∂

+Ω − =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠ (4.20)

2 2 223 3 3

3 3 1 3 1 3 3 3 32 23 3

2 223

32 2 23 3 3

( ) ( )

( )

x

P

w wx w s I xt t x x

w MEI xx x x

θρ⎛ ⎞⎛ ⎞∂ ∂ ∂

− Ω + Ω Ω − Ω +Ω⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎛ ⎞⎛ ⎞∂ ∂∂

+ =⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠

(4.21)

2 23 3 3

3 3 323 3 3

( ) ( ) 0xwI x GJ x

t t x x xθ θ⎛ ⎞⎛ ⎞ ⎛ ⎞∂ ∂ ∂∂

+Ω − =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠ (4.22)

2 2 2 2224 4 4 4

4 4 2 4 3 3 42 2 2 24 4 4 4

( ) ( ) ( ) 0xw w wx w I x EI xt t x x x x

θρ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂∂

− Ω − Ω +Ω + =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ (4.23)

2 24 4 4

4 3 424 4 4

( ) ( ) 0xwI x GJ x

t t x x xθ θ⎛ ⎞⎛ ⎞ ⎛ ⎞∂ ∂ ∂∂

+Ω − =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠ (4.24)

and the boundary conditions:

67

1 1

1

11 10 0

1 0

0 , 0 , 0 ,x x

x

wwx

θ= =

=

∂= = =

∂ (4.25)

2 2

2

22 20 0

2 0

0 , 0 , 0,x x

x

wwx

θ= =

=

∂= = =

∂ (4.26)

3 3

3

33 30 0

3 0

0 , 0 , 0 ,x x

x

wwx

θ= =

=

∂= = =

∂ (4.27)

4 4

4

44 40 0

4 0

0 , 0 , 0,x x

x

wwx

θ= =

=

∂= = =

∂ (4.28)

1 1 1 1

1 1 1 1

1 1 11 3 1 3 1 3 1 1 1

1 1

2 3 31 1 1

12 2 31 1

( )2 2

( ) 02

x

x L x L

x L x L

w wl lI x M L wt x x

w w wlM EI xt t x x

θ

= =

= =

⎛ ⎞⎛ ⎞ ⎛ ⎞∂ ∂ ∂⎛ ⎞Ω +Ω −Ω + Ω −Ω + +Ω +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞ ⎛ ⎞∂ ∂ ∂− + + =⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

(4.29)

( )1 1 1 1

1 11 1

1 1

2 31 1 1 1

1 3 1 32 21 1

3 21 11 1 1 1

1 3 1 1 1 1 3 121 1 1

11 1 3

1

2 2 2 xM

x L x L

yM yM L

x Lx L

zM

x L

w w wl l lM It t x t x

w w wI It x t t t x x

wI Ex

θ

θψ θ θ θ θ

= =

==

=

⎛ ⎞⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎜ ⎟−Ω − + + Ω +Ω −Ω⎜ ⎟ ⎜ ⎟⎜ ∂ ∂ ∂ ⎟ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞∂ ⎛ ⎞∂ ∂ ∂ ∂− +Ω +Ω + Ω +Ω +Ω⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

⎛ ⎞∂− Ω Ω +Ω −⎜ ⎟∂⎝ ⎠

1 1

21

1 21

( ) 0x L

wI xx

=

⎛ ⎞∂=⎜ ⎟∂⎝ ⎠

(4.30)

1 11 1 1 1

1 1 1 1

2 2 21 1 1 1 1

3 1 1 3 1 121 1 1 1

21 1 1

3 1 1 3 1 11 1 1

( ) 0

zM yMx Lx L x L

yM

x L x L

w w w wI It t x t x x x

w w wI GJ xt x x x

θ θ θ

θ θ

== =

= =

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂− −Ω + +Ω +Ω Ω⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞∂ ∂ ∂− Ω +Ω +Ω − =⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

(4.31)

If we look at the boundary conditions (4-29) through (4-31), we can

see that they are all written in terms of the first beam, however in reality the

mass is attached to all the four beams, not jut the first beam. The first

derivative and torsional displacement of beams 2, 3 and 4, are related to those

68

of the first beam. These relationships can be defined according to the

following 9 continuity equations.

( ) ( ) ( )2 1 1 2 1 2 1, , ,2L L L L L L Llw wδ δ δ ψ δ ψ δθ δθ δ ψ= + = = (4.32)

( ) ( ) ( )3 1 1 3 1 3 1, , ,L L L L L L Lw w lδ δ δ ψ δ ψ δ ψ δθ δθ= + = − = − (4.33)

( ) ( ) ( )4 1 1 4 1 4 1, , .2L L L L L L Llw wδ δ δ ψ δ ψ δθ δθ δ ψ= + = − = − (4.34)

Where

1 1 2 2

3 3 4 4

1 1 1 2 2 2

3 3 3 4 4 4

( , ) , ( , ) ,

( , ) , ( , ) .L Lx L x L

L Lx L x L

w w x t w w x t

w w x t w w x t= =

= =

= =

= = (4.35)

1 1 2 2

3 3 4 4

1 1 1 2 2 2

3 3 3 4 4 4

( , ) , ( , ) ,

( , ) , ( , ) .L Lx L x L

L Lx L x L

x t x t

x t x t

θ θ θ θ

θ θ θ θ= =

= =

= =

= = (4.36)

1 1 2 2 4 43 3

31 2 41 2 3 4

1 2 3 4

, , , .L L L Lx L x L x Lx L

ww w wx x x x

ψ ψ ψ ψ= = ==

∂∂ ∂ ∂= = = =∂ ∂ ∂ ∂

(4.37)

and δ( ) represents variation of corresponding terms.

Governing equations of motion, i.e. Eqs. (4.17-4.24), shows that the

two equations of each beam are coupled with each other through the base

rotation velocity Ω3 and each of the beams experiences a coupled flexural-

torsional vibration due to the gyroscopic terms such as 2

3i

xi

It xθ∂

Ω∂ ∂

and

69

2

3i

xi

wIt x∂

Ω∂ ∂

present in the equations of motion. Consequently, in absence of the

base rotation velocity, the two equations of each beam will be decoupled. This

is not the only coupling in the system. The second type of coupling is

expressed through the continuity equations (Eqs. (4.32–4.34)). The continuity

equations represent the relationship between the flexural and torsional

vibrations of beams 2, 3 and 4 in terms of the corresponding vibrations of

beam 1. As discussed before, bending of beams 1 and 3, induces a rocking

motion in the mass. In presence of the base rotation, this induced motion

produces a flexural vibrations in beams 2 and 4 and torsional vibrations in

beam 1 and 3. Since all the beams are connected to each other through the

rocking mass, the flexural and torsional deflections of each beam are also

dependent on each other.

To validate the mathematical modelling, if the beams 2, 3 and 4 are

neglected in the obtained equations, we will get to the governing equations of

a single beam gyroscope, which shows the accuracy of the attained equations.

It is noteworthy to mention, since in contrast with the single beam

gyroscopes, the piezoelectric sensors are placed on top of the beams 2 and 4,

not the sides, the secondary base rotation does not affect the measurement of

the primary base rotation, because the PZT sensors sense the bending of the

70

beams 2 and 4, not torsion. Therefore, the rocking-mass gyroscope is not

subject to the cross-axis effect.

Summary

In this chapter, we first introduced the rocking-mass gyroscope, which

consists of four beams with a finite mass attached to them in the middle, to

overcome the main drawback of the single beam gyroscopes. The principle of

operation of this kind of gyroscope was expressed and then compared with

that of the single beam gyroscope. Finally the eight linear partial differential

equations of motion as well as the boundary conditions were derived for the

71

Chapter 5

ROCKING-MASS GYROSCOPE ANALYSIS AND

NUMERICAL SIMULATIONS

In chapter 4, we developed the mathematical model of the system. We

also discussed different types of coupling available in the system, through the

governing equations of motion and the continuity equations. In this chapter

we mainly focus on the development of the frequency equations and

simulations of the gyroscopic system.

No work has been done on frequency analysis of the rocking-mass

gyroscope, in the past, however some work has been done on developing the

frequency equations of the cantilever beam undergoing coupled flexural-

torsional vibrations. Meirovitch [80-92] established some fundamentals on the

analysis of gyroscopic systems. Oguamanam [63] and Gokdag and Kopmaz

[64] developed the frequency equations of a cantilever beam with bending-

torsion vibrations. However, the effects of base rotation were ignored in their

work. Esmaeili et al. have worked on the problem of a cantilever beam

gyroscope with coupled flexural-flexural vibrations [66-68]. Frequency

equations of the system were developed as well as the gyroscopic effects

induced in the beam due to base rotations were demonstrated by simulating

the system governing equations. Vikrant et al [1] worked on a

72

piezoelectrically actuated flexural/torsional single beam gyroscope, and

performed some analysis on the system, in frequency and time domain.

This chapter essentially deals with the development of the frequency

equation and simulation of the equations of motion of the rocking-mass

gyroscope. The frequency equation is developed and the fundamental natural

frequency of the system and the corresponding mode shape will be found.

Finally a time-domain analysis will be performed to evaluate the effectiveness

of this kind of gyroscope.

Development of the Frequency Equation

An exact method is utilized to develop the frequency equation.

Assuming harmonic motion with frequency ω, the solutions of the equations

of motions can be represented in the following form

( , ) ( )1,2,3,4

( , ) ( )

i tn n

i tn n

w x t P x en

x t Q x e

ω

ωθ⎧ =

=⎨=⎩

(5.1)

where Pn(x) and Qn(x) are the amplitudes of the sinusoidally varying bending

and torsional displacements, respectively.

Substituting Eq. (5.1) into Eqs. (4.17–4.34) yields the following set of

equations of motion

73

0)()()()( 11111231113111

21 =−′′Ω+′Ω+ xPEIxPIxQiIxP IV

xx ωωρ (5.2)

0)()()( 1111131112

1 =″+′Ω− xQGJxPiIxQI xx ωω (5.3)

0)()()()()( 22222232223222

22

22 =−′′Ω+′Ω+Ω+ xPEIxPIxQiIxP IV

xx ωωρ (5.4)

0)()()( 2222232222

2 =″+′Ω− xQGJxPiIxQI xx ωω (5.5)

0)()()()( 33333233333333

23 =−′′Ω+′Ω+ xPEIxPIxQiIxP IV

xx ωωρ (5.6)

0)()()( 3333333332

3 =″+′Ω− xQGJxPiIxQI xx ωω (5.7)

0)()()()()( 44444234443444

22

24 =−′′Ω+′Ω+Ω+ xPEIxPIxQiIxP IV

xx ωωρ (5.8)

0)()()( 4444434442

4 =″+′Ω− xQGJxPiIxQI xx ωω (5.9)

and boundary conditions

,0)0(,0)0(,0)0( 111 ==′= QPP (5.10)

,0)0(,0)0(,0)0( 222 ==′= QPP (5.11)

,0)0(,0)0(,0)0( 333 ==′= QPP (5.12)

,0)0(,0)0(,0)0( 444 ==′= QPP (5.13)

2 2 21 1 1 3 1 1 1 1 1 1 3 1 1

1( ) ( ) ( ) ( ) 02 x xM P L Ml I P L EI P L iI Q Lω ω ω⎛ ⎞ ′ ′′′+ − Ω + + Ω =⎜ ⎟

⎝ ⎠ (5.14)

( )( )( )

2 2 2 2 21 1 3 1 1 1 1 1

3 1 1

2 ( ) 4 4 ( ) 4 ( )

4 ( ) 0

xM yM

xM yM

Ml P L Ml I I P L EI P L

i I I Q L

ω ω ω

ω

′ ′′+ − Ω + −

⎡ ⎤+ Ω − =⎣ ⎦

(5.15)

( ) ( )( )[ ] 0)()()( 1131111123

2 =′−Ω+′−Ω− LPIIiLQGJLQII yMxMyMxM ωω (5.16)

74

2 2 1 1 1 1( ) ( ) ( )2lP L P L P L′= + (5.17)

)()( 1122 LQLP =′ (5.18)

)()( 1122 LPLQ ′= (5.19)

3 3 1 1 1 1( ) ( ) ( )P L P L lP L′= + (5.20)

)()( 1133 LPLP ′−=′ (5.21)

)()( 1133 LQLQ −= (5.22)

4 4 1 1 1 1( ) ( ) ( )2lP L P L P L′= + (5.23)

)()( 1144 LQLP −=′ (5.24)

)()( 1144 LPLQ ′−= (5.25)

where 1−=i and ( )´ represents the derivative with respect to xi.

To solve Eqs. (5.2 – 5.9) an exact approach is utilized. We assume the

solutions of Eqs. (5.2) and (5.3) are in the following forms

1)( 11sxAexP = (5.26)

1)( 11sxeBxQ = (5.27)

where, “s” is an expression in terms of the system parameters and ω, and can

be found through Eq. (5-32).

Substituting (5.26) and (5.27) into (5.2) and (5.3), yields

041

223131

21 =−Ω+Ω+ AsEIAsIsBiIA xx ωωρ (5.28)

02131

21 =+Ω− BsGJsAiIBI xx ωω (5.29)

75

From (5.28) and (5.29) we have

41

2231

21

31

sEIsIsiI

BA

x

x

−Ω+Ω−

=ωρ

ω (5.30)

siIsGJI

BA

x

x

31

21

21

Ω+

=ω

ω (5.31)

Comparing (5.30) and (5.31), we can conclude

( )( ) ( )6,...,2,1,

0231

21

21

41

2231

21

=⇒=Ω++−Ω+

nssiIsGJIsEIsI

n

xxx ωωωρ(5.32)

Therefore, the solutions of (5.2) and (5.3) can be written in the

following form

∑=

=6

111

1)(n

xsn

neAxP (5.33)

∑=

=6

111

1)(n

xsnn

neAxQ α (5.34)

where

6,...,2,1,21

21

31 =+Ω

= nsGJI

siI

nx

nxn ω

ωα (5.35)

We can assume the solutions of Eqs. (5.4) and (5.5) are in the

following forms

76

2)( 22sxAexP = (5.36)

2)( 22sxeBxQ = (5.37)

Substituting (5.36) and (5.37) into (5.4) and (5.5), yields

0)( 42

223232

22

22 =−Ω+Ω+Ω+ AsEIAsIsBiIA xx ωωρ (5.38)

02232

22 =+Ω− BsGJsAiIBI xx ωω (5.39)

From (5.38) and (5.39) we have

42

2232

22

22

32

)( sEIsIsiI

BA

x

x

−Ω+Ω+Ω−

=ωρ

ω (5.40)

siIsGJI

BA

x

x

32

22

22

Ω+

=ω

ω (5.41)

Comparing (5.40) and (5.41), we can conclude

[ ][ ] ( )12,...,8,7,

0)( 232

22

22

42

2232

22

22

=⇒=Ω++−Ω+Ω+

nssiIsGJIsEIsI

n

xxx ωωωρ (5.42)

Therefore, the solutions of (5.4) and (5.5) can be written in the

following form

∑=

=12

722

2)(n

xsn

neAxP (5.43)

∑=

=12

722

2)(n

xsnn

neAxQ α (5.44)

77

where

12,...,8,7,22

22

32 =+Ω

= nsGJI

siI

x

xn ω

ωα (5.45)

The solutions of Eqs. (5.6) and (5.7) can be assumed in the following

forms

33 3( ) sxP x Ae= (5.46)

3)( 33sxeBxQ = (5.47)

Substituting (5.46) and (5.47) into (5.6) and (5.7), yields

043

223333

23 =−Ω+Ω+ AsEIAsIsBiIA xx ωωρ (5.48)

02333

23 =+Ω− BsGJsAiIBI xx ωω (5.49)

From (5.48) and (5.49) we have

43

2233

23

33

sEIsIsiI

BA

x

x

−Ω+Ω−

=ωρ

ω (5.50)

siIsGJI

BA

x

x

33

23

23

Ω+

=ω

ω (5.51)

Comparing (5.50) and (5.51), we can conclude

( )( ) ( )18,...,14,13,

0233

23

23

43

2233

23

=⇒=Ω++−Ω+

nssiIsGJIsEIsI

n

xxx ωωωρ(5.52)

78

Therefore, the solutions of (5.6) and (5.7) can be written in the

following form

∑=

=18

1333

3)(n

xsn

neAxP (5.53)

∑=

=18

1333

3)(n

xsnn

neAxQ α (5.54)

where

18,...,14,13,23

23

33 =+Ω

= nsGJI

siI

nx

nxn ω

ωα (5.55)

Finally, we take the solutions of Eqs. (5.8) and (5.9) in the following

forms

4)( 44sxAexP = (5.56)

4)( 44sxeBxQ = (5.57)

Substituting (5.56) and (5.57) into (5.8) and (5.9), yields

0)( 44

223434

22

24 =−Ω+Ω+Ω+ AsEIAsIsBiIA xx ωωρ (5.58)

02434

24 =+Ω− BsGJsAiIBI xx ωω (5.59)

From (5.58) and (5.59) we have

79

44

2234

22

24

34

)( sEIsIsiI

BA

x

x

−Ω+Ω+Ω−

=ωρ

ω (5.60)

siIsGJI

BA

x

x

34

24

24

Ω+

=ω

ω (5.61)

Comparing (5.60) and (5.61), we can conclude

[ ][ ] ( )24,...,20,19,

0)( 234

24

24

44

2234

22

24

=⇒=Ω++−Ω+Ω+

nssiIsGJIsEIsI

n

xxx ωωωρ (5.62)

Therefore, the solutions of (5.8) and (5.9) can be written in the

following form

∑=

=24

1944

4)(n

xsn

neAxP (5.63)

∑=

=24

1944

4)(n

xsnn

neAxQ α (5.64)

where

24,...,20,19,24

24

34 =+Ω

= nsGJI

siI

x

xn ω

ωα (5.65)

In summary the solutions of Eqs. (5.2)-(5.9) can be written as

∑∑==

==6

111

6

111

11 )(,)(n

xsnn

n

xsn

nn eAxQeAxP α (5.66)

80

∑∑==

==12

722

12

722

22 )(,)(n

xsnn

n

xsn

nn eAxQeAxP α (5.67)

∑∑==

==18

1333

18

1333

33 )(,)(n

xsnn

n

xsn

nn eAxQeAxP α (5.68)

∑∑==

==24

1944

24

1944

44 )(,)(n

xsnn

n

xsn

nn eAxQeAxP α (5.69)

Since each of these si’s is a very long expression in terms of the other

parameters, it is best to present the Maple code in Appendix C to obtain the

roots.

Substituting Eqs. (5.66)-(5.69) into Eqs. (5.10)-(5.25), we will have the

following system of equations. (Details of this system of equations can be

found in Appendix B.)

[ ] 01242424 =× ×× AC (5.70)

Therefore, the frequency equation is

[ ]( ) ( ) 0,,,det 232424 =ΩΩ=× ωGeometryfC (5.71)

The complete Maple code to get the characteristic equation of the

system is available in Appendix C. It is noteworthy that the usage of

traditional commands to calculate the determinant of the matrix “C” will not

work here; since “C” is a 24×24 matrix and each of its elements is a very long

expression in terms of the system parameters. So in the code provided, we

81

used the command “LUdecomp” to change the matrix to a lower-triangular

matrix and therefore, the product of the elements, located on the diagonal of

the new matrix will give the determinant.

Since the final determinant is a very long and complicated problem,

solving that equation and finding its roots needs very high computational cost

(The final expression is 5000 pages long).

Table 5.1: Physical parameters of the system. Parameter Notation Value Beam length (m)

1L 0.15 Beam length (m)

2L 0.15 Beam length (m)

3L 0.15 Beam length (m)

4L 0.15 Beam thickness (m)

1bt 0.8×10-3 Beam thickness (m)

2bt 0.8×10-3 Beam thickness (m)

3bt 0.8×10-3 Beam thickness (m)

4bt 0.8×10-3 Beam width (m)

1b 1.5×10-2 Beam width (m)

2b 1.5×10-2 Beam width (m)

3b 1.5×10-2 Beam width (m)

4b 1.5×10-2 Mass per unit length (kg/m)

1ρ 3960

Mass per unit length (kg/m) 2ρ 3960

Mass per unit length (kg/m) 3ρ 3960

Mass per unit length (kg/m) 4ρ 3960

Beam elastic modulus (Gpa) E 70 Beam shear modulus (Gpa) G 30 End mass length (m) l 0.01 End mass width (m) bM 0.02 End mass height (m) hM 0.02 Base rotation (rad/s) 3Ω 10 Secondary rotation (rad/s) 2Ω 5

82

For a system with specifications according to Table 5.1, the frequency

equation is solved and the fundamental frequency is found to be 131 rad/sec.

Finding the higher natural frequencies, which are of less importance in

compression to the fundamental one, needs a very strong computer with high

computational power.

Substituting s1 through s24 into Eqs. (5.66)-(5-69) for the fundamental

frequency, we will find the first mode shape of the system which corresponds

to its fundamental natural frequency. The first mode is illustrated in Figure

5.1. It can be seen that the first mode deals with the bounce of the rocking

mass.

Figure 5.1: First mode shape of the system.

83

Validation of the Utilized Method for Solving EVP

In order to validate the applied method for solving the Eigenvalue

problem (EVP), an analysis will be performed on a Timoshenko beam to

solve its EVP, since the governing equations of a Timoshenko beam are very

similar to the ones available in the rocking-mass gyroscope.

Frequency Analysis of a Timoshenko Beam

The Timoshenko beam is an extension of the Euler-Bernoulli beam in

which the effect of shear deformation and rotary inertia are included in the

governing equations. The equations governing the motion of a uniform

Timoshenko beam are [93]:

2 2

2 2

w wGA Ax x t

ψκ ρ⎛ ⎞∂ ∂ ∂

− =⎜ ⎟∂ ∂ ∂⎝ ⎠ (5-72)

2 2

2 2

wEI GA Ix x tψ ψκ ψ ρ∂ ∂ ∂⎛ ⎞+ − =⎜ ⎟∂ ∂ ∂⎝ ⎠

(5-73)

Nondimensional variables are introduced according to

* * * *, , ,w x tw x tL L T

ψ ψ= = = = (5-74)

Substitution of Equ. (5-74) into Equs. (5-71) and (5-72) and dropping

the *s sign from nondimensional variables leads to

84

2 2

2 2

w wx x t

ψ∂ ∂ ∂− =

∂ ∂ ∂ (5-75)

2 2

1 22 2

wx x tψ ψη ψ η∂ ∂ ∂

+ − =∂ ∂ ∂

(5-76)

where T has been chosen for convenience as

T LGρκ

= (5-77)

and

1 22 2,EI IGL A AL

η ηκ

= = (5-78)

A set of solutions can be considered for Equations (5-75) and (5-76) in

the following form

( , ) ( ). i tw x t P x e ω= (5-79)

( , ) ( ). i tx t Q x e ωψ = (5-80)

Substitution of Equs. (5-79) and (5-80) into Equs. (5-75) and (5-76)

leads to

2 0P Q Pω′′ ′− − = (5-81)

21 2(1 ) 0P Q Qη η ω′ ′′+ − + = (5-82)

85

To solve EVP, a set of solutions can be considered for Equations

(5-81) and (5-82) in the following form

sxP Ae= (5-83)

sxQ Be= (5-84)

Substitution of Equs. (5-83) and (5-84) into Equs. (5-81) and (5-82),

yields

2 2 0As Bs Aω− + = (5-85)

2 21 2 0As B s B Bη η ω+ − + = (5-86)

Therefore,

( )4 2 2 4 21 1 2 2 0

, 1, 2,3, 4n

s s

s n

η η η ω η ω ω⎡ ⎤+ + − =⎣ ⎦⇒ =

(5-87)

Finally, the exact solutions of the Equs. (5-81) and (5-82) can be

expressed as

4

1

ns xn

nP A e

=

= ∑ (5-88)

4

1

ns xn n

n

Q A eα=

=∑ (5-89)

where

86

2 2n

nn

ssωα +

= (5-90)

For a fixed-free beam (cantilever beam) the four boundary conditions

are

(0, ) 0 , (0, ) 0 , (1, ) (1, ) 0 , (1, ) 0ww t t t t tx x

ψψ ψ∂ ∂= = − = =

∂ ∂ (5-91)

Introducing Equs. (5-88) and (5-98) into Equs. (5-91), will give the

frequency equation, as follow

( ) ( ) ( ) ( )

[ ]

31 2 4

31 2 4

1

1 2 3 4 2

1 1 2 2 3 3 4 4 3

1 1 2 2 3 3 4 4 4

1 1 1 1

0ss s s

ss s s

C

AA

s e s e s e s e As e s e s e s e A

α α α αα α α αα α α α

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ =⎢ ⎥ ⎢ ⎥− − − −⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦

(5-92)

Hence, the frequency equation (characteristic equation) is

[ ]( )det 0C = (5-93)

Solving Equ. (5-93), we will find the natural frequencies of the system.

It is noteworthy to mention that in the latest work done before this

research [93] the solutions of the Equs. (5-81) and (5-82) were chosen as

1 2 3 4cosh( ) sinh( ) cos( ) sin( )P A x A x A x A xμ μ υ υ= + + + (5-94)

87

2 2 2 2

1 2

2 2 2 2

3 4

sinh( ) cosh( )

sin( ) cos( )

Q A x A x

A x A x

ω μ ω μμ μμ μ

ω υ ω υυ υυ υ

⎛ ⎞ ⎛ ⎞+ += +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞− −

+ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(5-95)

where

( ) ( )1/2

22 4 21 2 1 2 1

1

1 42

μ η η ω η η ω η ωη

⎡ ⎤= − + + − +⎢ ⎥⎣ ⎦ (5-96)

( ) ( )1/2

22 4 21 2 1 2 1

1

1 42

υ η η ω η η ω η ωη

⎡ ⎤= + + − +⎢ ⎥⎣ ⎦ (5-97)

It should be noted, although the solutions given by Equs. (5-94) and

(5-95) are the most precise ones offered by that time, but substitution of these

equations into Equ. (5-81), yields

4 0Aε = (5-98)

where ε is a small number which is not equal to zero. However if we

substitute Equs. (5-88) and (5-89) into (5-81), we will get

0 0= (5-99)

This shows the precision of the applied method in this work.

88

The first three natural frequencies of the fixed-free Timoshenko beam

are presented in Table 5.2 for the utilized method and Reference [93] when

31 1.6 10η −= × and 4

2 6.25 10η −= × .

Table 5.2: The first three natural frequencies of the fixed-free Timoshenko beam

nω Mode number (n) Ref [93] Proposed Method

Percentage of error

1 0.1383 0.139920099 1.17 2 0.8417 0.8512000000 1.13 3 2.2586 0.2826000000 1.06

It can be seen that, the application of the offered method, yields to a

very exact calculation.

The first three mode shapes of a cantilever Timoshenko beam, found

through the exact method, is illustrated in Figure V-1.

(a) (b) (c)

Figure 5.2: Mode shapes of a cantilever Timoshenko beam (a) First mode (b) Second mode (c) Third mode

89

Time Response of the System

The time response of the system can be found, using the mode

superposition principle [94]. For this, the bending and torsional motion of

each beam will be assumed as:

1 1 1 1 1 1 1 1 1 11 1

( , ) ( ) ( ) , ( , ) ( ) ( )n n n nn n

w x t P x t x t Q x tψ θ ϕ∞ ∞

= =

= =∑ ∑ (5.100)

2 2 2 2 2 2 2 2 2 21 1

( , ) ( ) ( ) , ( , ) ( ) ( )n n n nn n

w x t P x t x t Q x tψ θ ϕ∞ ∞

= =

= =∑ ∑ (5.101)

3 3 3 3 3 3 3 3 3 31 1

( , ) ( ) ( ) , ( , ) ( ) ( )n n n nn n

w x t P x t x t Q x tψ θ ϕ∞ ∞

= =

= =∑ ∑ (5.102)

4 4 4 4 4 4 4 4 4 41 1

( , ) ( ) ( ) , ( , ) ( ) ( )n n n nn n

w x t P x t x t Q x tψ θ ϕ∞ ∞

= =

= =∑ ∑ (5.103)

Where win (i=1,2,3,4) is the nth normal mode of each beam.

Substituting Eqs. (5.100)-(5.103) into equations of motion of the system, i.e.

Eqs. (4.17)-(4.24) and integrating over the intervals [0,Li], we will have a set

of eight ordinary differential equations in terms of ψi and φi (i=1,2,3,4) which

will be solved by MATLAB© to find the time response of the system.

Since defining the exact frequency equation and the exact mode shape

is hard to do, usually assumed-modes (comparison or admissible functions

are used instead of the normal modes in determining the time response and

therefore in that case it is need to consider enough number of modes [77] to

90

find the exact response. But in the present work, since we have found the

exact equation for the mode shape of the system, using the fundamental mode

shape will give us a very good and acceptable time response, however using

more numbers of the modes will lead to more accurate results.

Figure 5.3 (a) shows the bending of the beam 1 (drive direction), for

different cases of Ω3=0, Ω3=25 rad/s and Ω3=50 rad/s. The bending of the

second beam (sense direction) for these three cases are shown in Figures 5.3

(b), 5.3 (c) and 5.3 (d) respectively. This figure shows that, as the magnitude

of the base rotation increases, due to the corresponding increase in the

gyroscopic coupling, vibrations of the second beam also increase

proportionally. Hence, it can be concluded that the amplitudes of the

vibrations of the second and fourth beams are directly proportional to the

magnitude of the base rotation. This is an important conclusion as it shows the

effectiveness of this type of gyroscope as a device for measuring base angular

velocity.

91

(a)

(b)

92

(c)

(d)

Figure 5.2: System response (a) bending deflection w1 and bending deflection w2 to different base rotation rates: (b) Ω3=0, (c) Ω3=25 rad/s and (d) Ω3=50 rad/s

93

Summary

In this chapter, the exact frequency equation of the system was

developed in full detail, using an exact approach. The fundamental frequency

of the system and the corresponding mode shape was found as well.

Furthermore, by simulating the system, through mode superposition, the

presence of the gyroscopic coupling present in the system was validated. It

was shown that the gyroscopic output from the system is directly proportional

to the base rotation rate. So by sensing the output we would be able to

measure the rate of the base rotation.

94

Chapter 6

CONCLUSIONS AND FUTURE WORK

The primary goal of this research was to develop and analyze a new

type of vibrating gyroscope, called rocking-mass gyroscope. The operating

principle of the gyroscope, which consists of a set of four cantilever beams

with a rigid mass attached to them in the middle while subjected to base

rotations, was presented. First by considering a single beam gyroscope and the

cross-axis effect in them, the main drawback of this type of gyroscopes was

pointed out. For the rocking-mass gyroscope, a thorough analysis was carried

out in order to obtain the governing equations of motion of the system in their

most general form. The analysis was further extended to obtain the frequency

equation. The system response was also obtained for different conditions. The

effectiveness of the rocking-mass gyroscope was analyzed by simulating the

equations using mode superposition method. The results demonstrated that

this type of gyroscope can be used for sensing the rotational motions

accurately. An experimental setup can be used to validate the accuracy of the

simulations.

95

Recommendations for Future Work

A thorough design optimization can be performed on the system, using

powerful computing facilities, to find the optimum parameters such that the

best system performance is obtained. It is also a good idea to build an

experimental setup to study the system in practice, and validate the theoretical

studies in another way too.

The ultimate goal of this research could be to develop a new type of

MEMS vibrating gyroscope. As described in this thesis, MEMS gyroscopes

have tremendous potential for being used in many applications. To this end,

the simulations have to be extended to a micro-scaled gyroscope. In the

present work, the focus was mainly on a macro-scaled gyroscope. Extending

these simulations to a micro-scaled gyroscope is an important step in

determining the effectiveness of a similar MEMS gyroscope.

96

APPENDICES

97

APPENDIX A Detailed Derivation of Equations of Motion for a

Single Beam Gyroscope

The extended Hamilton’s Principle is given as

(A.1)

Using the expressions for kinetic energy, potential energy and virtual

work (Eqs. (3.32), (3.34) and (3.36)), and ignoring damping, we can express

the different components of Eq. (A-1) as

Kinetic energy

(A.2)

Substituting values of f, g, h, ωx, ωy and ωz from Eqs. (3.3) and (3.16),

we can simplify the above expression as follows

(A.2)

98

(A.3)

Integrating by parts, we get

(A.4)

Simplifying and combining similar terms we get

(A.5)

99

Total kinetic energy of the end mass is given as

(A.6)

Taking the variation of the above expression yields,

(A.7)

Substituting values of f, g, h, ωx, ωy and ωz from Eqs. (3.10) and (3.16),

we can simplify the above expression as follows

(A.8)

Potential energy

(A.9)

100

Simplifying and combining similar terms we get,

(A.10)

Virtual work is represented as

(A.11)

where [69]

(A.12)

Using Eqs. (A-1 to A-11) and taking into account the fact that

, , Lw wδ δθ δ and Lwxδ∂

∂could have any arbitrary values; the coefficients of

these terms in Hamilton’s equation must vanish. Hence, after substituting

values of f, g, h, ωx, ωy and ωz and ignoring rotary inertia for the beam, the

equations of motion and boundary conditions can be obtained as given in Eqs.

(3.38-3.43) [91].

101

APPENDIX B Frequency Equation

102

1 2 3 4 5 6

1 2 3 4 5 6

7 8 9 10 11 12

7 8 9 10 11 12

1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 00 0

s s s s s s

s s s s s s

α α α α α α

α α α α α α

13 14 15 16 17 18

13 14 15 16 17 18

19 20 21 22 23 24

19 20 21 22 23 24

131 132 133 134 135 136

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

s s s s s s

s s s s s s

C C C C C C

α α α α α α

α α α α α α

141 142 143 144 145 146

151 152 153 154 155 156

161 162 163 164 165 166 167 168 169 1610 1611 1612

171 172 173 174 175 176 177 178 179 1710 171

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

C C C C C CC C C C C CC C C C C C D D D D D DC C C C C C D D D D D 1 1712

181 182 183 184 185 186 187 188 189 1810 1811 1812

191 192 193 194 195 196 1913 1914 1915 1916 1917 1918

201 202 203 204 205 206 2013 2014 2015 2016 20

0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0

DC C C C C C D D D D D DC C C C C C E E E E E EC C C C C C E E E E E 17 2018

211 212 213 214 215 216 2113 2114 2115 2116 2117 2118

221 222 223 224 225 226 2219 2220 2221 2222 2223 2224

231 232 233 234 235 236 2319 2320 2321 232

0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0

EC C C C C C E E E E E EC C C C C C F F F F F FC C C C C C F F F F

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

1

2 2323 2324

241 242 243 244 245 246 2419 2420 2421 2422 2423 24240 0 0 0 0 0 0 0 0 0 0 0

AAAAAAAAAAAAAAAAAAA

F FC C C C C C F F F F F F

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

9

20

21

22

23

24

13

000000000000000000000000

n

AAAAA

C Mω

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

= [ ] [ ] [ ]1 1 1 1

1 1 1

2 2 2 31 3 1 1 3 16 19 22

2 2 2 2 2 214 3 1 3 17 20 23

(0.5. ) ; 1 0.5 ; 1 ; 1 0.5

2 ( 4 4 ) 4 4 ( ) ; ; ;

n n n n

n n n

s L s L s L s Lx n n x n n n n n n n

s L s L s Ln xM yM n n n xM yM n n n n

Ml I s EI s iI e C ls e C ls e C ls e

C Ml Ml I I s EI s i I I e C e C s e C

ω ω α

ω ω ω α ω α

⎡ ⎤+ − Ω + + Ω = + = + = +⎣ ⎦⎡ ⎤= + − Ω + − + Ω − = =⎣ ⎦

1

1 1 1 1

2 2 2

3 3

2 215 3 1 3 18 21 24

16 17 18

19 20 21

1,2,...,6

( ) ( ) ; ; ;

; ; , 7,8,...,12

; ;

n

n n n n

n n n

n n

s Ln n

s L s L s L s Ln xM yM n n n xM yM n n n n n n n

s L s L s Ln n n n n

s L s Ln n n

e n

C I I GJ s i I I s e C s e C e C s e

D e D s e D e n

E e E s e E

α

ω α α ω α

α

⎫⎪⎪= =⎬⎪

⎡ ⎤= − Ω − + Ω − = = = ⎪⎣ ⎦ ⎭= − = − = − =

= − = 3

4 4 422 23 24

, 13,14,...,18

; ; , 19,20,..., 24

n

n n n

s Ln n

s L s L s Ln n n n n

e n

F e F s e F e n

α

α

= =

= − = = =

103

1 2 3 4 5 6

1 2 3 4 5 6

7 8 9 10 11 12

7 8 9 10 11 12

1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 00 0

s s s s s s

s s s s s s

α α α α α α

α α α α α α

13 14 15 16 17 18

13 14 15 16 17 18

19 20 21 22 23 24

19 20 21 22 23 24

131 132 133 134 135 136

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

s s s s s s

s s s s s s

C C C C C C

α α α α α α

α α α α α α

141 142 143 144 145 146

151 152 153 154 155 156

161 162 163 164 165 166 167 168 169 1610 1611 1612

171 172 173 174 175 176 177 178 179 1710 171

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

C C C C C CC C C C C CC C C C C C D D D D D DC C C C C C D D D D D 1 1712

181 182 183 184 185 186 187 188 189 1810 1811 1812

191 192 193 194 195 196 1913 1914 1915 1916 1917 1918

201 202 203 204 205 206 2013 2014 2015 2016 20

0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0

DC C C C C C D D D D D DC C C C C C E E E E E EC C C C C C E E E E E 17 2018

211 212 213 214 215 216 2113 2114 2115 2116 2117 2118

221 222 223 224 225 226 2219 2220 2221 2222 2223 2224

231 232 233 234 235 236 2319 2320 2321 232

0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0

EC C C C C C E E E E E EC C C C C C F F F F F FC C C C C C F F F F 2 2323 2324

241 242 243 244 245 246 2419 2420 2421 2422 2423 2424

0

0 0 0 0 0 0 0 0 0 0 0 0F F

C C C C C C F F F F F F

=

104

APPENDIX C Maple Code to Develop Frequency Equation

105

106

107

108

109

110

REFERENCES

[1] Bhadbhade V., Jalili N. and Mahmoodi N, “A novel piezoelectrically

actuated flexural/torsional vibrating beam gyroscope”, Journal of Sound and

Vibration, 311, pp. 1305-1324, 2008.

[2] Post E. J., “Sagnac effect,” Review of Modern Physics, 39, pp. 475-493,

1967.

[3] Armenise M. N., Ciminelli C., De Leonardis F., Diana R., Passaro F.,

Peluso F., “Gyroscope technologies for space applications,” 4th Round Table

on Micro/Nano Technologies for Space, ESTEC, Noordwijk, Netherlands,

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