Analysis of 2:1 Internal Resonance in MEMS

Applications

by

Navid Noori

B.Sc., University of Tehran, 2014

Thesis Submitted in Partial Fulfillment of the

Requirements for the Degree of

Master of Applied Science

in the

School of Mechatronic Systems Engineering

Faculty of Applied Sciences

Navid Noori 2018

SIMON FRASER UNIVERSITY

Spring 2018

ii

Approval

Name: Navid Noori

Degree: Master of Applied Science

Title: Analysis of 2:1 Internal Resonance in MEMS Applications

Examining Committee: Chair: Mohammad Narimani Lecturer

Behraad Bahreyni Senior Supervisor Associate Professor

Farid Golnaraghi Supervisor Professor

Ash M. Parameswaran Internal Examiner Professor School of Engineering Science

Date Defended/Approved:

Jan 12th, 2018

iii

Abstract

Micromachined resonators are typically used within their linear range of operation.

Recently, there has been an increasing interest in understanding nonlinearities and

potentially employing them to improve the performance of resonance-based devices. The

focus of this thesis is to study the nonlinear mode coupling at 2:1 internal resonance both

experimentally and analytically. It is shown that quadratic nonlinearities can couple two

vibrational modes of a micro-resonator with a 2:1 ratio between two of its mode

frequencies. This nonlinear coupling of modes can lead to the transfer of energy between

these two modes through internal resonance. To study the phenomenon, a modified T-

beam structure is proposed, and a simplified mathematical model of operation including

the nonlinearities is developed for this system. Perturbation solutions of the mathematical

model, along with finite element and reduced-order method analysis are used to describe

the nonlinear behaviour of the system. Experiments are performed on a modified micro T-

beam structure with 2:1 ratio between its resonance frequencies. Nonlinear modal

interactions between vibrational modes, jump and saturation phenomena, and bandwidth

enhancement are also observed both in experiments and numerical simulations. The

effect of damping on the behaviour of the system is also studied. Some of the potential

applications of internal resonance in sensing are also proposed and discussed throughout

the thesis.

Keywords: Microresonator; Nonlinear vibrations; Internal resonance; Saturation; Modal interactions

iv

Dedication

To my beloved parents, Soheila and Fereidoon, and my

siblings, Niloofar and Pedram for their support and

dedications.

v

Acknowledgements

I would like to thank Dr. Behraad Bahreyni for his continuous support, patience,

and motivation. He has always been very generous with his time and knowledge, and he

was always available whenever I had a concern or question about the research. His

multidisciplinary expertise in MEMS area has always been a great inspiration to me. This

thesis would not have been completed without his support and advice.

My special thanks should also be extended to Dr. Farid Golnaraghi for his

comments and technical recommendations through my research. His invaluable guidance

through my study helped me to successfully reach this exciting destination and learn many

lessons along the path.

I would also like to acknowledge members of Integrated Multi-Transducer Systems

(IMuTS) Laboratory for their helps and shared memories during my research. My special

thanks go to my friends Atabak Sarrafan, Behdad Bahrani, Jade Zhang, Hooman Rahimi,

and Oldooz Pouyanfar for their supports.

Finally, I would like to thank my parents, Soheila and Fereidoon, and my siblings,

Niloofar and Pedram, for their support, unconditional love and encouragements through

my life and during my graduate study.

vi

Table of Contents

Approval .......................................................................................................................... ii Abstract .......................................................................................................................... iii Dedication ...................................................................................................................... iv Acknowledgements ......................................................................................................... v Table of Contents ........................................................................................................... vi List of Tables ................................................................................................................. vii List of Figures................................................................................................................ viii List of Acronyms ............................................................................................................. xi

Chapter 1. Introduction ............................................................................................. 1 1.1. Motivation ............................................................................................................... 1 1.2. Outline of the Thesis ............................................................................................... 3

Chapter 2. Background and Literature Review ........................................................ 5 2.1. Micromachined Resonators .................................................................................... 5 2.2. Nonlinearities in MEMS .......................................................................................... 7 2.3. Internal Resonance ................................................................................................ 9 2.4. Coriolis Vibratory Gyroscopes .............................................................................. 10

Chapter 3. Modified T-beam Structure with 2:1 Internal Resonance .................... 18 3.1. Description of Modified the T-beam Structure ....................................................... 18 3.2. Mathematical Lumped Element Modeling of the modified T-shape structure ........ 20

3.2.1. Non-dimensional Equation of Motion ....................................................... 23 3.2.2. Scaling of the Equation of the Motion ...................................................... 25 3.2.3. Perturbation Solution ............................................................................... 26

Modified T-beam as resonator ............................................................................... 35 3.3. Numerical Simulations .......................................................................................... 40

Chapter 4. Fabrication Process .............................................................................. 47 4.1. SOIMUMPS Fabrication Process .......................................................................... 47

Chapter 5. Experimental Results ............................................................................ 52 5.1. Structure Characterization .................................................................................... 54

Effect of DC voltage on resonance frequencies .................................................... 55 5.2. Frequency Sweep Response of a 2:1 Modified T-beam Structure ........................ 57 5.3. Effect of Pressure in 2:1 Modified T-beam with Internal Resonance ..................... 61

Chapter 6. Conclusion and Future Work ................................................................ 64

References ................................................................................................................ 68

vii

List of Tables

Table 3-1: Numerical values of nondimensionalized parameters ................................... 36

Table 4-1: Thickness of layers in SOIMUMPs Process .................................................. 48

Table 4-2: Comparison between dimensions of designed structure in CoventorWare and fabricated structure with SOIMUMPS process ......... 50

viii

List of Figures

Figure 1-1. a) Response of a mode matched gyroscope b) Desired response from a gyroscope with a wide output bandwidth ....................................... 2

Figure 2-1: Quality factor dependence on operating pressure ......................................... 7

Figure 2-2: Common sources of nonlinearities in MEMS [14] .......................................... 8

Figure 2-3 A simple spring-pendulum model with 2:1 internal resonance ........................ 9

Figure 2-4. Coriolis effect on an object with translational velocity of in x-

direction, and angular velocity of in z-direction ................................. 11

Figure 2-5. Lumped element model of a simple 2-D single mass Coriolis Vibratory Gyroscope (CVS) .................................................................... 12

Figure 3-1: Schematic of modified T-beam design. The anchors and electrodes are shown in gray colour. The motion of the beams is assumed to be in-plane. ............................................................................................ 19

Figure 3-2. FEM modal simulation using ANSYS showing first two in-plane vibrational modes, drive mode (right) and sense mode (left) .................. 20

Figure 3-3: Lumped element model of a 2DOF T-shaped resonator .............................. 21

Figure 3-4: Nonlinear frequency curve from perturbation solution in the absence of internal resonance .............................................................................. 37

Figure 3-5: Nonlinear frequency curve from perturbation solution in the presence

of internal resonance ( ) ............................................................... 37

Figure 3-6: Nonlinear frequency curve of modified T-beam microresonator from

perturbation solution ( 2 0.01 ) ............................................................ 38

Figure 3-7 Saturation curve system ( , Vdc=50 V). Energy starts to flow

from the drive mode into the sense mode after a certain threshold......... 39

Figure 3-8 Nonlinear frequency sweep response of sense mode for different excitation voltage amplitude (vac). The structure is perfectly tuned

to a 2:1 frequency ratio ( ) ............................................................ 39

Figure 3-9 Nonlinear frequency sweep response of drive mode for different excitation voltage amplitude (vac). The structure is perfectly tuned

to a 2:1 frequency ratio ( ) ............................................................ 40

Figure 3-10: Meshed structure in ANSYS used in modal analysis to determine the natural frequencies and mode shapes of the structure ..................... 41

Figure 3-11 Time domain response of the system in ANSYS FEM simulations showing transfer of energy from drive mode to the sense mode ............. 41

Figure 3-12: Schematic view of modified T-beam structure in CoventorWare® .............. 42

Figure 3-13: A 3D view of the designed structure in CoventorWare ............................... 42

v

z

2 0

1 2 0

2 0

2 0

ix

Figure 3-14 Resonance frequencies of drive and sense modes for Vdc=40v .................. 43

Figure 3-15: Effect of DC voltage on the resonant frequency of the drive beam ............ 44

Figure 3-16: Time response of the system showing internal resonance and transfer of energy between drive and sense modes ............................... 44

Figure 3-17: Comparison of nonlinear frequency sweep curves of sense beam for different Vac amplitudes .......................................................................... 45

Figure 3-18: Comparison of nonlinear frequency sweep curves of drive beam for different Vac amplitudes .......................................................................... 45

Figure 3-19. Saturation curve for VDC=40 v and frequency ratio of 2.0007 and quality factor of Qs=2000 , Qd=1500 ....................................................... 46

Figure 4-1: Process flow of SOIMUMPS process [42] ................................................... 49

Figure 4-2: (a) Final layout of modified T-beam designed in Coventorware®; structural dimensions are presented in the ............................................. 50

Figure 4-3: Final packaged device ................................................................................. 51

Figure 5-1: Packaged MEMS device inside the vacuum chamber ................................. 52

Figure 5-2: Virtual capacitors formed between drive/sense electrodes and the structure ................................................................................................. 53

Figure 5-3. Experimental test setup used for under-vacuum experimental tests for structural resonance frequency measurements ...................................... 54

Figure 5-4. Resonance frequencies of drive and sense mode of modified micro T-beam structure for Vdc=40 V ................................................................... 55

Figure 5-5: Effect of DC voltage on the resonance frequency of sense mode ............... 56

Figure 5-6: Effect of DC voltage on the resonance frequency of drive mode ................. 56

Figure 5-7: Experimental test setup used for frequency sweep response of the system ................................................................................................... 57

Figure 5-8: Backward frequency sweep response of the system ................................... 58

Figure 5-9: Forward frequency sweep response of the system ...................................... 59

Figure 5-10: Overlap of forward and backward frequency sweep response of the system ................................................................................................... 59

Figure 5-11: Sense beam's response to different excitation amplitudes. It shows mode coupling and internal resonance occur after excitation amplitude reaches a threshold ............................................................... 60

Figure 5-12: Quality factor of sense mode in linear modified T-beam microresonator in a range of operating pressures .................................. 61

Figure 5-13: Effect of pressure on the bandwidth in backward frequency sweep response ................................................................................................ 62

Figure 5-14: Effect of pressure on the response bandwidth in forward frequency sweep .................................................................................................... 62

x

Figure 5-15: Effect of pressure on the "full width at half-maximum (FWHM)" and "output signal amplitude" in a forward frequency sweep ......................... 63

Figure 5-16: Effect of pressure on the "full width at half-maximum (FWHM)" and "output signal amplitude" in a backward frequency sweep ..................... 63

Figure 6-1: Response of the system to a ramp angular velocity .................................... 66

Figure 6-2: Sensitivity curve showing the sense-beam amplitude for different input angular velocities ........................................................................... 66

xi

List of Acronyms

CVG Coriolis Vibratory Gyroscope

FEM Finite Element Method

DOF Degree of Freedom

MEMS Micro-Electro-Mechanical Systems

MUMPS Multi-User MEMS Processes

1

Chapter 1. Introduction

1.1. Motivation

Microgyroscopes are widely used microsensors for measuring the angular velocity.

Applications of micro gyroscopes include inertial navigation, robotics, and military

applications. Almost all available micro gyroscopes in the market are using vibratory

microresonators to operate, and they work on the principle of using Coriolis acceleration

to measure the rotational velocity [1]. Micro gyroscopes usually consist of a proof mass

suspended by flexural beams that are anchored to the substrate and is free to oscillate in

usually two main modes: drive mode and sense mode. The Coriolis force (acceleration)

causes transfer of energy between the driving mode and the sensing mode. Most of the

conventional linear micromachined gyroscopes are designed to operate in the mode-

matched condition where natural frequencies of drive and sense modes are chosen to be

matched. By matching the drive and sense resonance frequencies, the device will operate

in its resonance state which enhances the sensitivity drastically. Systems operating in the

resonance state usually use mechanical structures with high quality factors to achieve

large response amplitudes and high sensitivity. However, there is always a trade-off

between the sensitivity and the bandwidth in which the device is operating. In high quality

factors, the sensitivity is high, but the bandwidth is narrow and the resonant peak is sharp.

Current fabrication processes are not ideal and can produce fabrication parameter

fluctuations and errors like asymmetries and dimension changes causing natural

frequency shifts. Even a small gap between excitation frequency (drive mode’s frequency)

and the sense mode’s frequency can lead to a significant drop in amplitude and loss of

sensitivity [2]. It is desired for a gyroscope to have a wide response bandwidth to minimize

the effect of these parameter fluctuations on the response. The response of a mode-

matched gyroscope and an ideal gyroscope with a wide output bandwidth is shown in

Figure 1-1. There are some solutions and methods to overcome the frequency ratio

2

detuning issue like using a DC bias voltage [3], using a mode-matching controller [4], [5],

or using a multi DOF system [6]. However, many of these approaches need adding more

complexity to the system, more degrees of freedom, and higher excitation amplitudes.

Figure 1-1. a) Response of a mode matched gyroscope b) Desired response from a gyroscope with a wide output bandwidth

The first step in improving the performance of a micro gyroscope is to enhance the

performance of the microresonator that is used in the structure of the micro gyroscope. In

microresonators with high quality factors, nonlinearities play a significant role in their

performance. These nonlinearities can be used in favour of performance enhancements.

However, neglecting to consider nonlinearities in a system can cause errors in predicting

the behaviour of the system. The focus of this thesis is to study the nonlinear mode

coupling caused by 2:1 internal resonance to enhance the bandwidth of a microresonator.

The microresonator with wide bandwidth can then be used as a gyroscope. Nonlinear

modal interactions can occur in the vibrational modes of nonlinear systems causing

transfer of energy from a directly excited vibrational mode (drive mode) to an indirectly

excited mode (sense mode) through internal resonance [7], [8]. Using 2:1 internal

resonance to autoparametrically excite the sense mode will increase the sense mode’s

bandwidth and makes a nearly flat region on top of the bandwidth and makes the system

less sensitive to the noise and changes in environmental conditions. Using this idea in

micro gyroscopes eliminates the need for operation in the mode-matched condition. This

idea can also be extended to be used in other sensing applications such as pressure

sensors, and mass sensors. Marzouk [9] used nonlinear mode coupling to increase the

bandwidth of operation in a macro-scaled T-shaped structure with 2:1 frequency ratio.

3

Current thesis basically tries to extend this work to the micro-scale to show bandwidth

enhancement and mode coupling in a MEMS device both through simulations and

experiments.

It is shown throughout the thesis that in a MEMS resonator with a natural frequency

ratio of 2:1 between the drive and sense modes, exciting the system in the higher mode

will excite the lower mode autoparametrically and energy can transfer between these

modes. A modified T-beam structure is proposed, and a mathematical model is developed

for this structure. Perturbation methods are used to solve the equation of motion of the

structure and transfer of energy between modes and improvement in the sense mode’s

bandwidth is observed in the perturbation solutions. Reduced-order simulations in

CoventorWare® and Finite Element analysis in ANSYS® are also done for designing and

optimizing the mechanical structure of the device before fabrication. The MEMS device is

then fabricated by SOIMUMPS process. Experimental tests of the fabricated device verify

the simulation results qualitatively. Effect of damping on the nonlinear behaviour of this

system is also studied both in simulations and experimental tests.

1.2. Outline of the Thesis

In Chapter 2, an introduction to MEMS gyroscopes and resonators, their dynamics,

and principles of operation are presented. Nonlinear internal resonance for performance

improvement of microresonators is also discussed in this chapter.

In Chapter 3, a modified T-beam structure is proposed. The mathematical model

of the structure is derived and then solved using perturbation methods. This structure is

also simulated using ANSYS® and CoventorWare© to show the transfer of energy between

vibrational modes and bandwidth enhancement through nonlinear modal interactions.

In Chapter 4, the fabrication process used to fabricate the designed microresonator

is reviewed, and a comparison is made between dimensions of designed and fabricated

devices to verify the desired 2:1 frequency ratio.

4

In Chapter 5, the experimental test setup and test results of the fabricated device

are presented. These results show the nonlinear mode coupling and confirm the

simulations results qualitatively. The effect of excitation amplitude and operating pressure

on the response of the system in both forward and backward frequency sweeps are also

studied.

Finally, Chapter 6, concludes the thesis and summarizes the contributions and

findings of this thesis. There is also a discussion about ideas for future research and

development on this subject.

5

Chapter 2. Background and Literature Review

MEMS devices have been attracted immense interest during the recent years.

These devices integrate electronic circuit components and movable micro-scaled

mechanical parts like beams and plates and can be used in sensing or actuation

applications [10]. The idea of microsystems first presented by Richard Feynman in 1950s

with a paper entitled “There is Plenty of Room at The Bottom” [11]. However, this idea

stayed dormant for many years due to lack of fabrication mechanisms for very small

systems. The first notable progress in the fabrication of microdevices was made by

Nathanson [12] with the driving force for manufacturing of integrated circuits (ICs). In 1982,

Kurt Peterson [13] introduced the possibility of using IC manufacturing techniques to

fabricate small mechanical systems with the paper “Silicon as a Mechanical Material.” It

was the first step toward the development of what we call today “Micro Electro-Mechanical

Systems (MEMS).” MEMS devices have many advantages in comparison to macro

systems and opened new gates for implementing devices in places where macro devices

do not usually fit [14], for example inside of a human body’s organ. MEMS devices are

more robust, reliable, cost-effective and consume less power compared to the macro-

scaled systems. They can be batch fabricated in large quantities at a very low cost. MEMS

devices have a wide range of applications in micro pressure sensors [15], gas sensors

[16], RF filters [17], biological sensors [18], and inertial micro sensors consisting of micro

gyroscopes [1], [19], [20] and micro accelerometers [1], [19]

2.1. Micromachined Resonators

Microresonators are MEMS devices which use resonantly excited structures to

operate. In microresonator sensors, a change in a physical variable (i.e. temperature,

pressure) leads to a change in the resonant frequency in a specific vibrational mode, and

therefore, the change in the resonant frequency can be measured and then calibrated into

the desired physical variable. Microresonators can also be used in Inertial micro sensors

including accelerometers and gyroscopes. These sensors convert the inertial forces acting

on the mechanical structure like linear or angular velocity into the physical changes in the

structure like deflection of a beam. The physical change caused by inertial forces can then

6

be captured by a transducer and transformed into an electrical signal for further signal

processing. Micromachined resonators can also be used as a mass sensing device [21],

where a shift in resonant frequency occurs due to the addition of mass.

Damping of a microresonator can be represented by the quality factor of the

system (Q-factor) or equivalent damping of the system (ξ) [10]. Quality factor is one of the

most important parameters of microresonators which directly relates to the resonance

amplitude of the microresonator [22]. Quality factor of a system can be defined as the ratio

of stored energy in the system to the dissipated energy of the system:

2Energy Stored

QEnergy Dissipated per Cycle

( 2-1 )

In linear resonators, the quality factor is inversely proportional to the damping of

the system (1/2Q). Most of the microresonators are designed to operate in their linear

regime, and there is always a trade-off between the quality factor and the bandwidth in

which the microresonator is operating. High quality factor resonators provide higher

resonance amplitudes, but they are more sensitive to the frequency variations because in

these resonators the resonance peak is sharp, and the bandwidth is narrow. Nonlinearities

can play a major role in MEMS devices especially in those with high quality factors, and

therefore, it is essential to identify the sources of energy dissipation in these devices.

There are various energy dissipation mechanisms like anchor loss, viscous damping,

thermos-elastic damping, intrinsic damping that contributes to the total damping of the

system. Each of these mechanisms has their quality factor, and the total quality factor of

the system can be expressed as:

intrinsic

1 1 1 1 1

tot viscous anchor loss other mechanismsQ Q Q Q Q ( 2-2 )

Therefore, the lowest quality factor determines the total quality factor of the

system. Among the mentioned damping mechanisms, the major damping mechanisms in

MEMS devices are viscous damping and anchor loss. In anchor damping, the energy of

7

vibrating structure is transferred to the substrate. In viscous damping, energy can dissipate

to the surrounding air or gas when mechanical parts of MEMS device move. Therefore,

many of the MEMS sensors like micro vibratory gyroscopes are designed to operate in

near vacuum condition for achieving higher quality factors [14]. Pressure fluctuations can

have a large impact on the quality factor microresonators and their response. Figure 2-1

shows the quality factor of a MEMS device in different pressure regimes. In the first region,

A, the pressure is very low, near vacuum condition, and the effect of pressure on quality

factor is negligible. In this region, structural damping is the dominant damping mechanism.

In the second region, B, the pressure is higher, and viscous damping is the dominant

damping mechanism. In this region, the quality factor is predicted to be inversely

proportional to pressure ( 1Q

P ). In the third region, C, the damping is independent of

pressure [14].

Figure 2-1: Quality factor dependence on operating pressure

2.2. Nonlinearities in MEMS

Study of nonlinearities in MEMS has gained momentum during the recent years.

Nonlinearities are present in almost all micro and macro systems. However, in MEMS,

due to the small dimensions and high-quality factors (low damping), nonlinearities can be

dominant and have larger impact on the behaviour of the system. Neglecting to consider

8

the nonlinearities can sometimes cause erroneous predictions of system’s response [14].

Research in this area has been active for almost a decade which allowed the designers

to beneficially exploit nonlinearities in the course of MEMS device designs for actuation,

sensing, signal processing, switching, and timing applications [23]. Younis and Nayfeh

[24] investigated the effect of nonlinearities in the response of a resonant microbeam. Hu

and Raman [25] studied the chaotic response of microcantilever in atomic force

microscopy (AFM) and the resulting deformations in the AFM images. Tilmans an

Legtenberg [26] developed a model to approximate the nonlinear resonance frequency of

a microresonator, and they concluded that the impact of nonlinearities on resonance

frequency is more severe in microresonators with high quality factors. Nonlinearities in the

MEMS devices may have many sources including forcing (i.e., electrostatic force),

damping (i.e., squeeze-film [14]), geometric nonlinearities [7], and material nonlinearities

[27]. Figure 2-2 shows a schematic illustrating these nonlinear sources.

Figure 2-2: Common sources of nonlinearities in MEMS [14]

Nonlinear mode coupling between vibrational modes is one of the outcomes of the

presence of nonlinearities in microsystems. Nonlinear mode coupling is basically a

mechanism of energy leakage from an intentionally excited mode to lower modes of

vibration. There are two main mechanisms for nonlinear mode coupling [28]. The first

mechanism is internal resonance which is the focus of this thesis and will be discussed in

detail in the next section. The second mechanism for nonlinear mode coupling is mode

veering [28]– [30]. In this mechanism, resonance peaks are close to each other, and they

deviate and move away from each other after some change in a parameter.

Nonlinearities in MEMS

Forcing

Actuation: electrostatic

Damping

Squeez film

Stiffness

Geometric nonlinearities: large

deformation

Material nonlinearities, piezoelectric

9

2.3. Internal Resonance

Nayfeh and Mook [7] studied the nonlinear vibrations of systems with quadratic

and cubic nonlinearities. Quadratic and cubic nonlinearities can lead to an interesting

phenomenon called internal resonance. Internal resonance, also called autoparametric

resonance, refers to the transfer of energy from a vibrational mode to another vibrational

mode. Internal resonance can occur when natural frequencies of a system are

commensurable or nearly commensurable [7]. Internal resonance can happen between

vibrational modes that are in-plane or out-of-plane. Depending on the natural frequency

ratio, internal resonance can be one-to-one (1:1), two-to-one (2:1), or three-to-one (3:1).

In a system with 2:1 internal resonance where there is a 2:1 natural frequency

ratio, two vibrational modes are coupled through quadratic nonlinearities and energy can

be exchanged among these modes [31]. Internal resonance can be used in many

applications such as mass sensing, energy harvesting [32], and noise suppression [28]. A

very simple example of a system with 2:1 internal resonance is a 2-DOF spring-pendulum

system [7], [8] as depicted in Figure 2-3. In this system with 2:1 ratio between natural

frequencies of spring mode and pendulum mode, energy can transfer from the directly

excited higher order mode with lower amplitude to the non-excited lower mode of vibration.

This may cause the oscillation amplitude of indirectly excited mode to be an order of

magnitude higher than the directly excited mode’s amplitude [14].

Figure 2-3 A simple spring-pendulum model with 2:1 internal resonance

10

In a system with 2:1 internal resonance, by exciting the system near its primary

mode’s (drive mode) natural frequency, the amplitude of primary mode increases linearly

with the excitation amplitude until the excitation amplitude reaches a threshold value. From

this point, the system saturates to the forcing and any additional increase in the excitation

amplitude does not change the primary mode’s amplitude. After the system saturates,

additional energy added to the system by increasing the excitation amplitude will be

channelled to the secondary mode of the system (sense mode) [33], and the amplitude of

this mode starts to grow. This phenomenon is known as saturation phenomenon and can

be used in many applications. There are several reported vibrational absorbers that exploit

saturation phenomena to suppress nonlinear vibrations of a system and improve its

stability [34]– [36].

Nonlinear coupling caused by internal resonance can also significantly increase

the bandwidth of the lower mode’s response. A 2:1 microresonator with a wide bandwidth

caused by internal resonance can then be used to replace mode-matched gyroscopes.

The wide response bandwidth caused by internal resonance can reduces the vulnerability

of the gyroscope to the fabrication errors. In the next section, dynamics of a mode-mode

match gyroscope and its limitations are studied.

2.4. Coriolis Vibratory Gyroscopes

Gyroscopes are sensors that measure and detect the angular motion of an object.

Advances in the micro-machining technologies made it possible for micro-machined

gyroscopes to be an alternative for bulky, expensive macro gyroscopes. Micro gyroscopes

can be used in combination with micro accelerometers in Inertial Measurement Units

(IMUs) for navigation and tracking applications. Almost all of the MEMS gyroscopes in the

market use vibrational mechanical elements and work on the same principle of using

Coriolis acceleration to measure the rotational displacement or velocity [1]. Coriolis

acceleration, named after the scientist Gaspard-Gustave de Coriolis (1792-1843), is an

inertial acceleration resulted in a rotating frame, as illustrated in Figure 2-4, and is

proportional to the translational velocity of an object and the rate of rotation of the rotating

frame.

11

Vibratory gyroscopes can be classified into two main categories based on the type

of their measured variable: angle gyroscopes, and rate gyroscopes. As it is obvious from

their name, the angle gyroscope measures the rotational angle, while rate gyroscopes

measure the rate of angular rotation. Although most of the micromachined vibratory

gyroscopes in the market are rate gyroscopes, implementing the angle gyroscopes on the

micro scale is not impossible, and there are opportunities for implementing them in micro

scale [20].

Figure 2-4. Coriolis effect on an object with translational velocity of in x-

direction, and angular velocity of in z-direction

A basic 2-DOF Coriolis Vibratory Gyroscope (CVG) is shown in Figure 2-5.

Lumped element model of a simple 2-D single mass Coriolis Vibratory Gyroscope (CVS)

is composed of a proof mass supported by two sets of suspensions. These suspensions

give the proof mass the freedom needed to move in two orthogonal directions. The proof

mass is excited to vibrate in the drive direction (Y-axis) by using a sinusoidal force. The

forced vibration of the proof mass in the drive direction in the presence of an angular

velocity along the z-axis can induce Coriolis force along the sense direction (X-axis). The

amplitude of this Coriolis force is proportional to the input angular velocity ( )z , and

therefore, vibrations of the proof mass in the sense direction can be measured and then

calibrated to determine the input angular velocity. To simplify the discussion, it is assumed

that springs used in the suspensions are in their linear region and they produce linear

elastic forces of (k x,k )suspensions x yF y . Although a simple in-plane 2-DOF vibratory

v

z

12

gyroscope is discussed in this section, the discussion is generic and can be applied to

other device architectures with torsional and out-of-plane modes.

Figure 2-5. Lumped element model of a simple 2-D single mass Coriolis Vibratory Gyroscope (CVS)

A similar analysis to investigate the equation of motion of a macro gyroscope is

done in [9], [37]. Let consider {S} a fixed inertial reference frame, in which non-inertial

reference frame {T} is moving with an angular velocity of z along z-axis. An inertial frame

of reference is a coordinate frame where Newton’s laws of motion are valid. The position

of the proof mass in the fixed inertial reference frame {S} and non-inertial rotating

reference frame are given by vectors (X,Y, Z)S

mr and ( , , )T

mr x y z , respectively.

Therefore, using vector addition, proof mass position in respect to the reference frames

{T} and {S} is given as:

ˆ ˆT

m

S T S

m m T

r x i y j

r r r

( 2-3 )

Where, i , j , k and are unite vectors along x, y, and z-axes of reference frame {T},

respectively. Velocity vector of the proof mass can be derived by taking time derivative of

13

the proof mass position vector, and according to rules of differentiation in a rotating

reference frame, velocity vector is given by

ˆT

S Tmm z m

drr k r

dt ( 2-4 )

Combining the equations ( 2-3 ) and ( 2-4 ), the velocity of the proof mass can be

rewritten as

ˆ ˆ( ) (y x)S

m z zr x y i j ( 2-5 )

Acceleration vector of the proof mass is then derived by differentiation of velocity

from equation ( 2-5 ) :

2 2 ˆˆ( 2 ) (y 2 y) jS

S mm z z z z z z

d rr x y y x i x x

dt ( 2-6 )

From Newton’s second law the forces acting upon the proof mass can be written

as:

2 2 ˆˆ[ ( 2 ) (y 2 y) j ]S

m z z z z z zF m x y y x i x x ( 2-7 )

The force vector in ( 2-7 ) is the reaction force of the system to the angular rotation

and is made from three force types of Coriolis, Euler, and Centrifugal forces [9]. In the

case of forced excitation, which is common among rate gyroscopes [37], a sinusoidal force

sin( )excitationF f t is applied to the proof mass along the drive direction (Y-axis). By

considering the stiffness and damping effects, the equation of motion of the proof mass

can be written as:

14

2

2

m(x y 2 ) 0

m(y 2 y) sin( )

z z z x x

z z z y y

y x k x c x

x x k y c y f t

( 2-8 )

Considering a constant rotational rate ( 0z z ), the terms z x and z y

can be neglected. The term 2 z x can also be neglected due to the small effect of sense

direction vibrations on the vibration of drive direction [26]. Also, the input angular velocity

is very smaller in comparison to the natural frequencies in both sense and drive directions

and therefore 2 2 2

x z x and 2 2 2

y z y . By using undamped natural frequencies

,x y , and the damping factors ,x y which are the ratio of damping coefficient to critical

damping 2cC km , the equation of motion can be rewritten as:

2

2

x 2 x 2

y 2 y sin( )

x x x z

y y y

x y

fy t

m

( 2-9 )

Where:

,

,2 2

yxx y

y yx xx y

c cx y

kk

m m

c cc c

c cmk mk

( 2-10 )

The system with the equation of motion described in equation ( 2-9 ) has a

harmonic force in the drive mode. Therefore the solution of this equation is expected to

have a form of ( ) ( ) ( )h py t y t y t , where ( )hy t is the homogeneous solution, and ( )py t

15

is the harmonic response of the system to the forced vibration which has the form of

( ) sin( )y t Y t . The amplitude of the forced vibration response is given by:

2 2

2 2

1

2

1(1 ( )

1

tan

( ) 1

y

y y y

y y

y

f

mY

Q

Q

( 2-11 )

Substituting ( 2-11 ) into equation ( 2-9 ), the equation of motion for sense mode

will be:

2x 2 x 2 cos( )x x x zx Y t ( 2-12 )

Similarly, the solution of equation( 2-12 ) is in the form of ( ) sin( )x t X t and

therefore, it can be written as:

2 2

2 2

1

2

2

1(1 ( )

1

tan

( ) 1

z

x

x x x

x xx

x

YX

Q

Q

( 2-13 )

Where Qy and Qx are quality factors of drive-mode and sense-mode, respectively

and can be defined as:

16

1

2

1

2

y

y

y y

xx

x x

mQ

c

mQ

c

( 2-14 )

To Amplify the response of the sense-mode the structure can be designed in a

way that natural frequency of sense mode is closed to drive mode’s natural frequency.

According to equation ( 2-9 ), frequency and phase of drive oscillation directly determine

the frequency and phase of Coriolis force ( 2 z y ) and therefore by matching the natural

frequencies of drive-mode and sense-mode, the system will be excited to the resonance

in both sense and drive modes. The mode-matching approach allows the amplitude of

sense mode to be amplified by its quality factor, which means that the sensitivity of the

system to the input angular velocity is improved by orders of magnitude. In case of mode-

matching (i.e. y x ) the amplitudes of drive-mode and sense-mode in resonance

can be simplified to:

2

2 2

resonance y

y

z zresonance x x y

x x x

fY Q

m

Y fX Q Q Q

k

( 2-15 )

Looking at resonance amplitude of the drive-mode (Yresonance) and sense-mode

(Xresonance) from ( 2-15 ), the sensitivity of the gyroscope to the input angular velocity z

can be improved by Increasing the quality factor (i.e. vacuum packaging in MEMS),

increasing the excitation amplitude of drive mode, and maximizing the effective mass that

produces the Coriolis force [22].

However, matching the drive and sense mode’s natural frequency is not always

feasible especially in MEMS devices due to inevitable fabrications errors or changes in

the environmental conditions. In high quality factor devices like micro gyroscopes, even a

17

small deviation of operating frequency from natural frequency can cause a significant drop

in sensitivity. Although there are some methods to tune the resonance frequency, most of

these methods add more complexity to the system. Therefore, there is a need for a

solution which can increase the sense mode’s bandwidth and makes the system less

vulnerable to parameter fluctuations.

18

Chapter 3. Modified T-beam Structure with 2:1 Internal Resonance

In this chapter, utilizing internal resonance for transfer of energy between

vibrational modes of a modified T-beam microresonator is discussed. The modified T-

beam structure is designed to have a frequency ratio of 2:1 between its first an second in-

plane mode shapes. A 2-DOF lumped element mathematical model is derived and then

solved using perturbation methods to investigate the vibrational modal interactions in the

system. It has been shown that by exciting the structure with an amplitude above a certain

threshold, transfer of energy between vibrational modes occurs. Also, jump phenomena

and saturation phenomena can be observed in the response of the system. The effect of

external angular velocity the response is also discussed.

3.1. Description of Modified the T-beam Structure

A modified micro T-beam resonator is designed to operate on the principle of

nonlinear modal interactions between its vibrational modes due to the 2:1 internal

resonance. This modified T-beam is designed to have a frequency ratio of 2:1 between

the resonance frequencies of sense mode and drive mode. Exciting the drive beam at its

resonance frequency excites the sense beam autoparametrically through structural

quadratic nonlinearities. This mode coupling can flow the energy from the drive mode into

the sense mode. The modified T-beam design is motivated by the ordinary T-beam design

in [9], [10], [39]. A schematic top view of the proposed modified T-beam microresonator is

shown in Figure 3-1. This design is symmetric with respect to the y-axis and consists of

three beams: (a) the drive beam (bottom beam) which is used for actuation and is

anchored to the substrate from its both ends; (b) the narrowed beam which is connected

to the center of the drive beam and is perpendicular to it; and (c) a sense beam which is

connected to the narrowed sense beam. The narrowed beam is designed to have a lower

width, and therefore lower stiffness- in comparison to the sense beam and drive beam.

Lower stiffness of the narrowed beam enables it to have larger deformation amplitudes,

and due to this large deformation, the effects of nonlinearities are considerable. The sense

beam has a significantly larger mass and stiffness compared to the narrowed beam.

19

Therefore, the sense beam can basically be considered as an added rigid mass to the

narrowed beam which can increase the structure’s sensitivity to the angular velocity.

Figure 3-1: Schematic of modified T-beam design. The anchors and electrodes are shown in gray colour. The motion of the beams is assumed to be in-plane.

For the simplicity of the analysis, only the first two in-plane mode shapes of the

structure are taken into account. These mode shapes are extracted using modal analysis

in ANSYS and are shown in Figure 3-2. Dimensions of the structure are carefully chosen

to ensure the 2:1 frequency ratio between the first and second modes. However, having

a perfectly 2:1 frequency ratio is not always practically feasible, and even small changes

in structural dimensions can deviate the nominal 2:1 ratio. Active tuning and feedback

control can be used to tune the structure to a 2:1 frequency ratio. One of the simplest

techniques for tuning the detuned structures is using a bias DC voltage between the

structure and electrodes. The electrostatic force caused by this bias DC voltage will deflect

the structure (i.e., drive beam) and changes its mechanical stiffness and consequently its

natural frequency. However, tuning by bias DC voltage can only be used for devices that

have a small deviation from frequency ratio of 2:1, and other methods should be used if

the frequency ratio is far from 2:1.

20

Figure 3-2. FEM modal simulation using ANSYS showing first two in-plane vibrational modes, drive mode (right) and sense mode (left)

3.2. Mathematical Lumped Element Modeling of the modified T-shape structure

In general, micromechanical systems can be modelled in discrete elements by

using rigid-body dynamics. A mechanical structure with n degrees of freedom can be

modeled by using n generalized coordinates, q1, q2,…, qn, and time, t. One can use

Lagrange’s equation to derive the equation of motion [40]:

,( ) 1,2,...,nc i

i i

d L LQ i n

dt q q

( 3-1 )

Where L T U is Lagrangian, T is the total kinetic energy of the system,

equalling the sum of kinetic energies of every particle in the system, and U is potential

energy of the system arising from the conservative forces. The term ,nc iQ represents non-

conservative forces like dissipation forces. If viscous damping is the only source of energy

dissipation in the system, then the Lagrange’s equation can be written as [41]:

,( ) 1,2,..., next i

i i i

d L L FQ i

dt q q q

( 3-2 )

Where, F is the Rayleigh dissipation force, and ,ext iQ is the generalized external

force associated with the coordinate qi.

21

In this section, this method is implemented to derive the equation of motions

describing lumped element model of a 2-DOF microresonator which works on the principle

of 2:1 internal resonance. This is a general mathematical model and can be extended to

be used for any 2 DOF system with two orthogonal mode shapes. A schematic view of the

lumped element model of the structure is shown in Figure 3-3. This structure consists a

drive beam and a sense beam. In this model, M1, M2 represent the effective mass of drive

beam and sense beams, respectively and Ci, Ki are the effective damping coefficients and

spring constants of these beams, respectively. A similar study on a macro T-shape

structure has been done in [9]. In addition, Golnaraghi [8] studied a 2-DOF robot arm which

can also be extended to a 2-DOF structure with two orthogonal mode shapes such as T-

beams. Therefore, the study that has been done in this section is similar in many aspects

to [8], [9].

Figure 3-3: Lumped element model of a 2DOF T-shaped resonator

The displacement vector for each of the masses is given by:

1

2

1

1 2

ˆm

m

r r i

r r r

( 3-3 )

And then the velocities of the masses can be written as:

22

1

2

1 1 1

1 2 1 2 2 1 1 2 1 2 2

ˆ ˆ

ˆ ˆ[ ( )sin( )] [ ( )cos( )]

m

m

v ri r j

v r r i r r j

( 3-4 )

Where r1 represents the displacement of the center of mass of the drive beam and

2 2, are angular deflection and velocity of the sense beam. 1 1 1, , are the input angular

displacement, velocity, and acceleration of the structure, respectively. It should also be

mentioned that equilibrium position of this structure is at 1 1 2 20, 0, 0, 0r r . The

kinetic energy T is given by:

1 1 2 21 2

2 2 2 2 2 2 2 2

1 1 1 1 2 1 1 1 2 1 2

1 2 1 2 2 1 2 1 1 2 2

1 1. .

2 2

1 1[r r ] [r r ( )

2 2

2 ( )sin( ) 2 ( ) cos( )]

m m m mT m r r m r r

m m r

r r r r

( 3-5 )

And the potential energy of the system, U, can also be expressed as:

2 2

1 1 2 2

1 1U

2 2k r k ( 3-6 )

According to Lagrange formula nonlinear equation of motion of the system is

derived as:

23

2

1 2 1 1 1 1 1 2 2 1 2 1 2 1 2 2

2 2

2 2 2 2 2 2 1 2 1 1

2

2 2 2 2 2 2 2 2 2 1 2 2 1 2 1 2

2 1 2 1 2 1

( ) r (( sin( ) cos( ) 2 cos( ))

( sin( ) cos( )) (m m ) r ( )

sin( ) 2 cos( )

( cos( )

drive

m m c r k r m r

m r F t

m r c k m r r m r r

m r r

2 2

2 2 2 1sin( )) m r

( 3-7 )

In this equation, 1r , 2 are generalized coordinates and 2r , 1 are given while

cos( )drive dF A t is the harmonic excitation force and 1 2 cos( )zF t is the external

rotational velocity. The equation of motion consists of two coupled nonlinear equations

and can be written in the form of matrixes as:

1 1 1 1

2 22 2

r r r FM C K

F

( 3-8 )

Such that:

1 2 2 2 2 1 2 2 1 2

2

2 2 2 2 2 2 2 1 2 2

2 222 2 2 1 211 1 2 1

222 2 1 2 1 2

sin( ) 2 cos( )

sin( ) 2 cos( )

( ) cos( )(m m ) 0

0 sin( )

drive

m m m r c m rM C

m r m r m r c

m r FFkK

Fk m r r

3.2.1. Non-dimensional Equation of Motion

To generalize the equation of motion ( 3-7 ) to any T-shaped structure, the

equations need be non-dimensionalized. The non-dimensionalized equation of motion can

be used for any macro-scale or micro-scale (MEMS) T-beam structure. The final non-

dimensionalized equation is in the form of:

24

2 2

1 1 1 1 1 1 2 1 2 1 2 2

2 2 2

2 2 2 2 1 1 1 1 1

2

2 2 2 2 2 1 2 1 1 2 1 1 2

2

1 2 1

( sin( ) cos( ) 2 cos( ))

( sin( ) cos( )) cos( )

sin( ) 2 cos( ) ( cos( )

sin( ))

m

m F

( 3-9 )

In the equation ( 3-9 ), is the non-dimensional time and is defined as:

t ( 3-10 )

Also, dots represent the differentiation with respect to the non-dimensional time

variable . In addition, is a frequency used for non-dimensionalizing the equations.

Here is assumed to be equal to natural frequency of sense mode. Other non-

dimensional parameters used in ( 3-9 ) can be defined as:

1 21

2 1 2

1 2

2

1 2 2 2 1 21 2 1 2 2

1 2 2 2( )

dr mm

r m m

k k

m m m r c c

m m m r

( 3-11 )

Where represents the first non-dimensional generalized variable (the second

variable is 2 ) and m is the non-dimensional mass. Also, 1 and 2 are non-dimensional

natural frequencies and 1 , 2 are non-dimensional damping coefficients.

In the process of non-dimensioning the equation of motion ( 3-7 ), first derivatives

of non-dimensional generalized variables are written as:

25

1 1 1 12 2

2 2 2 2

(r )( ) r

( )( )

r r d

t dt

d

t dt

( 3-12 )

Also, second derivatives of non-dimensional generalized variables are written as:

2 221 1 1 1

2 22 2

2 222 2 2 2

2 2

( )(r ) r

( )( )

r dr d

t t dt dt

d d

t t dt dt

( 3-13 )

With the same method, first and second derivatives of input angular velocity 1 are

written as:

1 1 1

2 221 1 1 1

2 2( )( )

d

t dt

d d

t t dt dt

( 3-14 )

3.2.2. Scaling of the Equation of the Motion

After non-dimensionalizing the equations of motion of the system, next step is to

scale these equations. To solve these nonlinear coupled equations and study the effect of

nonlinearities in the equations, a small dimensionless 1 is chosen to perturb the

linear equations. This shows the order of nonlinearity in the equations [8]. Thus, some

variable changes are considered as below:

1 1 2 1 1, , ,n j kF f ( 3-15 )

26

Next step is to substitute the equation ( 3-15 ) into ( 3-9 ) and use of Taylor series

to expand the equations in particular sin( ) ... and cos( ) 1 ... , for small

values of . Then the equations become:

2 1 2 2

1 1

1 2 2 2 2 2 2

1 1 1

2 1

2 2

2 1 2

2 ) cos( )

2

n n n j j

j j n k

n n j

n j n j j

m m

m m m f

( 3-16 )

According to [8], the order of linear terms in ( 3-16 ) is required to be lower than

nonlinear terms and therefore, n j 1 and 1k . Excitation magnitude is determined by

the value of k . For our study, we choose 1k . Also, damping coefficients 1 and 2 are

scaled to 1 and 2 , respectively. This leads to:

2 2 2 2

1 1

2 2

1 1 1

2 3 2

2 2

( 2 ) ( )

cos( )

( 2 )

m m

m f

( 3-17 )

Where:

2 2sin( )f ( 3-18 )

3.2.3. Perturbation Solution

In this section, perturbation method is used to study the behaviour of the system

near internal resonance ( 1 22 ) and in the absence of internal resonance ( 1 22 ).

27

These studies give us a better understanding of dynamics of the system which is an

essential step in improving the design and performance of the system. Using two-variable

perturbation method, these two new variables are used [8]:

, ( 3-19 )

The underlying idea of the two-variable method is that independent variables,

and , are functions of two independent variables and where is a fast time scale

and is a slow one. By using the chain rule, the derivatives with respect to t become

expansions in terms of partial derivatives with respect to and :

2 2 2 22

2 2 22

d

d

d

d

( 3-20 )

Also, and are expanded as:

2

0 1

2

0 1

( )

( )

O

O

( 3-21 )

We consider the excitation frequency ( 1 ) to be close to the natural frequency of

drive mode ( 1 ). By introducing a detuning parameter ( 1 ), the excitation frequency can

be written as:

1 1 1 ( 3-22 )

Therefore, one can write:

28

1 1 1 1 1 1 1

1 1 1 1 1 1 1

cos( ) cos( ) cos( )cos( ) sin( )sin( )

sin( ) sin( ) sin( )cos( ) cos( )sin( )

( 3-23 )

Substituting ( 3-19 ) - ( 3-21 ) into the ( 3-17 ) and equating the coefficients of like powers

of , leads to the following sets of equations.

For order :

2

0 1 0

2 2

0 2 0 2 2 2

0

cos( )f

( 3-24 )

For order 2 :

2 2

1 1 1 0 0 0 0 1 0

2

2 2 0 2 2 2 0 2

2 2 2 2

2 2 2 1 1 1

2

1 1 1 0 2 0 0 0 2 2 0 2

2

2 2 0 2

2 ( )

2 cos( ) sin( )

cos ( ) f cos( )

2 2 cos( )

sin( )

m m

m f f m

f m

f

f

( 3-25 )

The solution of ( 3-24 ) can be expressed in the form of:

0 1 1 2 1

0 3 2 4 2 1 2

( )sin( ) ( )cos( )

( )sin( ) ( )cos( ) sin( )

B B

B B P

( 3-26 )

Where

29

2

2 21 2 2

2 2

fP

( 3-27 )

Note that for simplicity of the mathematical calculations, only 0 0, components

of the solutions are considered and higher order components like 1 1, are neglected.

For increasing the accuracy of the perturbation method these neglected components can

be taken into consideration.

The case 1 2ω 2ω (absence of internal resonance)

In this case, two-to-one internal resonance is not present in the system. Next step

is to substitute ( 3-26 ) into ( 3-25 ) and eliminate the secular terms. It is convenient to

introduce these new polar transformations:

1 1 1 2 1 1

3 2 2 4 2 2

sin( ) , cos( )

sin( ) , cos( )

B a B a

B a B a

( 3-28 )

Where ia and i are real functions of . Using these transformations, secular-

term equations can be achieved:

' 1 1 11 1

' 22 2

1 11

1

sin( )2 2

2

' cos( )2

fa a

a a

f

a

( 3-29 )

Where

30

1 1 2, arbitary constant ( 3-30 )

Note that all the derivatives used in the equations are with respect to . The

steady-state response corresponds to ' ' '

1 2 0 ( )a a and therefore, the solution

of solvability equations ( 3-29 ) will be:

1 1 11 2

212 1

1

, 0 arctan2

24

fa a

( 3-31 )

Thus, the steady-state solution ( ) of the system can be written by

substituting the modal amplitudes achieved in ( 3-31 ) into the ( 3-26 ):

1 11

22 11

2

2 222 2

2 2

cos( ) O( )

24

sin( ) ( )

f

fO

( 3-32 )

This solution corresponds to the linear response of the system in the absence of

internal resonance. In the next section, the response of the system in the presence of 2:1

internal resonance is studied.

The case of Internal Resonance ( 1 22 )

In this case, 1 is close to 22 . A detuning parameter 2 is introduced:

31

1 2 22 ( 3-33 )

Using ( 3-33 ), one can write the following transformations:

1 2 2 2 2

2 2 2 2

1 2 2 2 2

2 2 2 2

cos( ) cos(2 ) cos(2 )

cos(2 )cos( ) sin(2 )cos( )

sin( ) sin(2 ) sin(2 )

sin(2 )cos( ) cos(2 )sin( )

( 3-34 )

Similar to the previous case, by using the polar transformation ( 3-28 ), the

solvability equations can be written as:

2' 22 1 1 11 2 2 1 12

1

2' 1 22 1 2 2 2

2

2' 22 1 11 2 2 1

1 1 1

2' 12 1 2

2

sin( ) sin( )2 2

sin( )4 2

cos( ) cos( )2 2

cos( )4

m fa a a

a a a a

m fa

a a

a

( 3-35 )

Where

1 1 1 2 2 2 1, 2 ( 3-36 )

By eliminating 1 and 2 from equations ( 3-35 ), following set of equations is obtained:

32

2' 22 1 1 11 2 2 1 1

1

2' 1 22 1 2 2 2

2

2' 22 1 11 2 2 1 1

1 1 1

2 2' 21 2 1 12 1 2 2 2 1 2

2 1 1 1

sin( ) sin( )2 2 2

sin( )4 2

cos( ) cos( )2 2

cos( ) cos( ) cos( )2 2 2

m fa a a

a a a a

m fa

a a

m fa a

a a

( 3-37 )

Steady state solution corresponds to ' ' '

1 2 0 ( )a a . Therefore, there

are two possibilities. The first is:

1 11 2

22 11

11 2

1

, 0

24

arctan ,2

fa a

arbitrary

( 3-38 )

Substituting ( 3-38 ) into ( 3-26 ), the steady-state response can be written as:

1 11 1

22 11

2

2 222 2

2 2

cos( ) O( )

24

sin( ) ( )

f

fO

( 3-39 )

Similar to the previous case, this is the solution of the system in its linear range.

The second possibility for solving the equation set ( 3-37 ) is:

33

2 221 1 2 22

1

2 6 2

1 1 1 2

2

2 1

2 3 2 3

2 2 2 1 1 1

1 3 2 3 2

2 2 1 2 1 1 1

22

1 2

2( )

1

1

2arctan( )

arctan

a

fa

m

a a

m a a

( 3-40 )

Where

1 2 1 1 2 1 2 2 2 1 1 2 1 24 ( ) 2 , 2 ( ) 4 ( 3-41 )

Therefore, by substituting ( 3-40 ) into ( 3-26 ), one can write:

0 1 1 2 1

1 1 1

2 221 2 2 1 12

1

( ) ( )sin( ) ( ) cos( ) O( )

cos( ) ( )

2( ) cos( ) ( )

O B B

a O

O

( 3-42 )

And

0 3 2 4 2 1 2

2 2 2 1 2

2 6 2 21 1 1 2 2 2

2 2 22 2

2 1 2 2

( ) ( )sin( ) ( ) cos( ) P sin( ) O( )

cos( ) P sin( ) ( )

1cos( ) sin( ) ( )

O B B

a O

f fO

m

( 3-43 )

34

By using ( 3-22 ), ( 3-33 ), and ( 3-36 ) following expansions can be written:

1 1 1 1 1 1

1 1 1 1

1 2 2 1 22 2

1 1 1 2

1 1 2

cos( ) cos(( ) ( ))

cos( ) cos( )

cos( ) cos( )2 2

cos( )2 2

cos( )2 2

( 3-44 )

Therefore, the steady state response of the system can be rewritten as:

2 221 2 2 1 12

1

2 6 2

1 1 1 2 1 21

2 1

2

2 222 2

2 2

2( ) cos( ) O( )

1 1cos( )

2 2

sin( ) O( )

f

m

f

( 3-45 )

As it can be seen in ( 3-45 ), when internal resonance exists in the system, an extra

term appears in the solution. This extra term is a harmonic term with a frequency equal

to the half of the excitation frequency 1 . In addition, the steady state perturbation

solution for (drive-mode amplitude) is independent of forcing amplitude 1F and

saturates to a constant value.

35

Modified T-beam as resonator

To model the same T-beam structure as a resonator, the input angular velocity

should be set to zero. Basically, modelling the structure as a resonator is a special case

of a gyroscope with no input angular velocity. Therefore, scaled non-dimensionalized

equations of motion can be written as:

2 2 2

1 1 1 1 1

2

2 2

( ) cos( )

0

m f

( 3-46 )

By following the steps to achieve the perturbation solution similar to what has been done

before, the perturbation solution of ( 3-46 ) can be written as:

2 221 1 2 22

1

2 6 2

1 1 1 2

2

2 1

2 3 2 3

2 2 2 1 1 1

1 3 2 3 2

2 2 1 2 1 1 1

22

1 2

2( )

1

1

2arctan( )

arctan

a

fa

m

a a

m a a

( 3-47 )

It should be noted that 1 2,a a are dimensionless modal amplitudes. Therefore, one

can write the final solution as:

36

2 221 2 2 1 12

1

2 6 2

1 1 1 2 1 21

2 1

2( ) cos( ) O( )

1 1cos( ) O( )

2 2

f

m

( 3-48 )

Using the numerical values for the parameters, an estimate of system’s response

can be obtained. Numerical values of the nondimensionalized parameters, obtained from

experimental tests are presented in Table 3-1. It should be also noted that electrostatic

force is used to excite the structure. A DC voltage of 50V and AV voltage of 6V is used for

electrostatic excitation.

Table 3-1: Numerical values of nondimensionalized parameters

Nondimensionalized parameter Symbol Value

Drive mode frequency ω1 2

Sense mode frequency ω2 1

Perturbation parameter ε 0.01

Detuning frequency (1 1 1 ) σ1 -0.1 to 0.1

Detuning frequency (1 2 22 ) σ2 0

Drive beam damping µ1 0.0288

Sense beam damping µ2 0.0167

For a system with a ratio frequency far from 2:1, there will be no internal resonance

as shown in the perturbation solution. Figure 3-4 depicts the linear response of the system

in the absence of internal resonance. As it can be seen, there is no displacement in the

sense beam and drive beam reaches to a resonance.

37

Figure 3-4: Nonlinear frequency curve from perturbation solution in the absence of internal resonance

Figure 3-5 shows the nonlinear frequency curve of a 2:1 tuned modified T-beam

microresonator. This figure shows the non-dimensional modal amplitudes versus the

excitation frequency. As it can be seen in this figure, energy transfer from the drive mode

to the sense modes causes the drive mode’s amplitude to drop and sense mode’s

amplitude to rise.

Figure 3-5: Nonlinear frequency curve from perturbation solution in the presence of

internal resonance ( )

2 0

38

Figure 3-6 shows the frequency sweep response of a detuned system with

detuning coefficient of 2 0.01 . As it can be seen the response of a detuned system is

not symmetric anymore.

Figure 3-6: Nonlinear frequency curve of modified T-beam microresonator from

perturbation solution ( 2 0.01 )

As it was discussed earlier, in a structure with 2:1 natural frequency ratio, before

reaching a threshold in excitation amplitude, the system operates in its linear regime and

amplitude increases linearly with excitation amplitude. However, after reaching a certain

threshold in excitation amplitude, mode coupling occurs, and energy will start to channel

from the drive mode into the sense mode. After this point, the drive mode’s amplitude

saturates, and the excessive energy caused by the increase in excitation amplitude will

transfer to the sense mode. Saturation curve for a perfectly 2:1 tuned ( 2 0 ) modified

micro T-beam is depicted in Figure 3-7.

39

Figure 3-7 Saturation curve system ( , Vdc=50 V). Energy starts to flow

from the drive mode into the sense mode after a certain threshold.

It is also expected that by increasing the excitation amplitude, the bandwidth of

response becomes wider, and the mode coupling between modes becomes stronger as it

is depicted in Figure 3-8 and Figure 3-9. It is also assumed that increasing the excitation

amplitude does not change the resonance frequencies of sense or drive modes.

Figure 3-8 Nonlinear frequency sweep response of sense mode for different excitation voltage amplitude (vac). The structure is perfectly tuned to

a 2:1 frequency ratio ( )

1 2 0

2 0

40

Figure 3-9 Nonlinear frequency sweep response of drive mode for different excitation voltage amplitude (vac). The structure is perfectly tuned to

a 2:1 frequency ratio ( )

3.3. Numerical Simulations

In this section, numerical simulations are used to design the mechanical structure

and simulate the dynamical behaviour of the proposed structure. These simulations are

necessary to ensure achieving a 2:1 frequency ratio between the drive and sense modes

and to investigate the presence of internal resonance in it. We used FEM analysis of

ANSYS in the structural design stage and then we switched to CoventorWare reduced-

order simulations. These simulations are used to optimize the design of the structure and

visualize the behaviour of the structure before it is fabricated which helps to save a lot of

time and resources. The designed structure is first meshed in the ANSYS as it can be

seen in Figure 3-10 and then a modal analysis is performed to find the mode shapes and

natural frequencies of the structure. These mode shapes were presented previously in

Figure 3-2.

2 0

41

Figure 3-10: Meshed structure in ANSYS used in modal analysis to determine the natural frequencies and mode shapes of the structure

In the next step, a finite element transient analysis is done in the ANSYS to show

the time domain response of the system and to observe the internal resonance

phenomenon and transfer of energy between sense and drive modes. As it can be seen

in Figure 3-11 when the mode coupling starts to occur, energy is channelled from drive

mode into the sense mode.

Figure 3-11 Time domain response of the system in ANSYS FEM simulations showing transfer of energy from drive mode to the sense mode

Using finite element method simulations is very time consuming and needs a huge

amount of processing power. Therefore, Coventorware software was chosen to perform

reduced-order simulations which are quicker and need less processing power. The

schematic of the system in the Coventorware is shown in Figure 3-12. In these simulations,

it is assumed that the structure has only in-plane vibrations for simplicity. In addition, in

the first step simulations are done in the absence of angular velocity and gyroscopic test

42

are done at the end to show the effect of input angular velocity on the behaviour of the

system.

Figure 3-12: Schematic view of modified T-beam structure in CoventorWare®

A 3D view of the structure used in CoventorWare can be seen in Figure 3-13. As

it is shown, an electrode is used for electrostatic excitation of the drive beam.

Figure 3-13: A 3D view of the designed structure in CoventorWare

43

Device dimensions are chosen in a way to ensure that there is a 2:1 frequency

ratio between the drive and sense mode. A small signal AC analysis is done to find the

resonance frequencies of the structure. Figure 3-14 shows the resonance frequencies of

drive and sense mode. Here the structure dimensions are chosen to have a frequency

ratio of 2.0007 which is very close to the ideal 2:1 ratio.

Figure 3-14 Resonance frequencies of drive and sense modes for Vdc=40v

It should be noted that changing the DC voltage in the excitation electrodes will

change resonance frequencies of the structure which will then change the frequency ratio.

Therefore, it is important to consider the effect of DC voltage on the resonance frequency.

Increasing the DC voltage will cause an increase in the electrostatic force applied to the

beam and will change the effective stiffness of the beam and therefore will change the

resonance frequency of the structure. Figure 3-15 shows the drive frequency variations

for different values of DC voltage. As it was mentioned before changing the DC voltage is

one of the methods for tuning a detuned structure.

360560

721390

44

Figure 3-15: Effect of DC voltage on the resonant frequency of the drive beam

A transient analysis is then performed to show the nonlinear mode coupling

between the drive and sense modes. To carry out the simulation, a 40v DC voltage is

applied to the whole structure, and drive beam is excited to its resonance frequency with

an AC voltage. The response shows that before the mode coupling occurs, there is no

displacement in the sense beam. However, when the mode coupling begins, the energy

starts to channel from drive mode into the sense mode. This transfer of energy causes the

drive amplitude to drop and the sense amplitude to rise until steady state is reached.

Figure 3-16: Time response of the system showing internal resonance and transfer of energy between drive and sense modes

45

To show the bandwidth enhancement caused by internal resonance, frequency

sweep curve for three different AC voltage amplitude is depicted in Figure 3-17. This figure

shows that increasing the excitation amplitude increases the response amplitude and

bandwidth.

Figure 3-17: Comparison of nonlinear frequency sweep curves of sense beam for different Vac amplitudes

Figure 3-18 shows the amplitude of drive beam in different excitation amplitudes. It shows

that after the energy starts to flow from drive mode to the sense mode, the drive’s

amplitude does not change by changing the excitation amplitude.

Figure 3-18: Comparison of nonlinear frequency sweep curves of drive beam for different Vac amplitudes

46

As it was mentioned earlier, in systems with 2:1 internal resonance saturation

phenomenon occurs after increasing the excitation amplitude beyond a certain threshold.

Before reaching this threshold point, drive’s amplitude increases linearly with respect to

the excitation amplitude. After this point, the drive’s amplitude remains constant, and the

energy starts to flow into the sense mode and rises the sense mode’s amplitude. Figure

3-19 shows the saturation curve of the structure showing the transfer of energy between

vibrational modes after excitation amplitude reaches a threshold.

Figure 3-19. Saturation curve for VDC=40 v and frequency ratio of 2.0007 and quality factor of Qs=2000 , Qd=1500

47

Chapter 4. Fabrication Process

4.1. SOIMUMPS Fabrication Process

One of the challenging and limiting factors in MEMS is the fabrication. There are

many limitations in the fabrication process which a MEMS designer should consider during

the design process. Each fabrication process has a specific design rule which is

conservatively chosen and is aimed to guarantee the highest yield possible. These design

rules constrain minimum gap, minimum and maximum feature sizes, maximum etched

area, the thickness of deposited layer for different mask layers, etc. The modified T-beam

structure used in this work is designed to be fabricated by SOIMUMPS process which is

a process for microfabrication using Multi-User MEMS Processes (MUMPS). SOIMUMPS

is a general purpose microfabrication process introduced by MEMSCAP for

micromachining of devices with highly planar surfaces in a Silicon-on-Insulator (SOI)

framework. This process is a simple 4-mask level SOI patterning and etching and is a

great option for proof of concept fabrication. This process has a minimum feature size of

2µm and the minimum gap between any two silicon parts is also 2µm. A brief review of

this process can be seen in Figure 4-1. More detailed information on SOIMUMPS process

can be found in SOIMUMPS design handbook [42]. This fabrication method has following

steps:

Starting substrate: a silicon-on-insulator wafer is used as the starting substrate. An oxide

layer is initially present on the bottom side of the starting substrate. This process provides

two options for silicon thickness: 1) 10µm silicon thickness on 400µm handle wafer 2)

25µm silicon thickness on 400µm handle wafer.

Step1: The wafer is doped by photosilicate glass later (PSG) and annealed at 1050°C for

an hour. This layer is later removed using wet etching.

Step2: Pad metal is patterned through liftoff.

Step3: Silicon is patterned using deep reactive ion etching (DRIE) to etch the silicon down

to the oxide layer.

48

Step4: A frontside protection layer is applied to the top surface of the silicon layer.

Reactive ion etching (RIE) is used to remove the bottom side oxide layer. A DRIE silicon

etch is then used to etch through the substrate, stopping on the oxide layer.

Step5: Frontside protection material is stripped by a dry etch process. The remaining

oxide layer on the top surface is removed using a vapour HF etching process.

Step6: A separately fabricated shadow mask is aligned and temporarily bonded to the

wafer. The blanket metal layer, consisting of 50nm Cr+600nm Au is deposited through the

shadow mask.

Step7: By removing the shadow mask, a patterned metal layer on SOI wafer is left. The

wafers are ready to be diced and packaged.

The thickness of layers used in this SOIMUMPs process are also listed in Table 4-1 below:

Table 4-1: Thickness of layers in SOIMUMPs Process

Material Layer Thickness (µm)

Pad metal 0.52 (500nm Au+20nm Cr)

Silicon 10±1

Oxide 1.0±0.05

Substrate 400±5

Blanket Metal 0.65 (600nm Au+ 50nm Cr)

49

Figure 4-1: Process flow of SOIMUMPS process [42]

After reviewing all the design rules and process details, the modified T-beam structure is

designed, and mask layouts are prepared by CoventorWare®, as it can be seen in Figure

4-2 (a). As it can be noticed in this figure, there is a safe gap (5µm) between sense

electrodes and feedback electrodes to avoid any interference between their output signals.

It should be noted that even if two identically designed devices are fabricated in different

50

areas of a wafer, they may have not the same structural dimensions and resonant

frequencies due to inevitable fabrication nonuniformities and imperfections. Therefore, it

is important to consider multiple designs throughout the wafer to achieve the desired 2:1

resonance frequency ratio. After multiple tries and errors in fabrication, a modified T-beam

design with a nearly perfect 2:1 frequency ratio could be fabricated. This fabricated device

is shown in Figure 4-2 (b).

Figure 4-2: (a) Final layout of modified T-beam designed in Coventorware®; structural dimensions are presented in the Table 4-2. (b) Fabricated modified T-beam with SOIMUMPS process

Table 4-2: Comparison between dimensions of designed structure in CoventorWare and fabricated structure with SOIMUMPS process

Parameter Designed in CoventorWare

Fabricated

Drive Beam length 11 µm 10.95 µm

width 445 µm 445.3 µm

Narrowed beam length 32 µm 32.21 µm

width 3 µm 3.05 µm

Sense beam length 80 µm 79.56 µm

width 12 µm 12.2 µm

51

As it can be seen in Table 4-2, the error between dimensions of the fabricated

device and the designed device is within an acceptable range of error for MEMS

fabrication. However, even a fairly small error can cause a large shift in natural

frequencies, and therefore, only experimental tests can determine whether this device has

the goal frequency ratio of 2:1. The fabricated device is then packaged and wire-bonded.

The final packaged device can be seen in Figure 4-3. The packaged device will be used

for experimental tests.

Figure 4-3: Final packaged device

52

Chapter 5. Experimental Results

Fabricated devices are employed for experiments to show the concept of internal

resonance and transfer of energy between vibrational modes of the modified T-beam

microresonator. The experimental test results verify the perturbation solution and

numerical simulations qualitatively and show bandwidth enhancement in the response.

Fabricated modified T-beam microresonator (previously presented in Figure 4-2b)

consists of three beams: a drive beam which is anchored to the substrate at its both ends;

a narrowed beam that is attached to the middle of the drive beam; and a sense beam

attached to the end of the narrowed beam. With the help of ANSYS and CoventorWare

simulations, structural dimensions are chosen to achieve a design with 2:1 frequency ratio

between the drive and sense modes. Experimental tests will reveal the frequency ratio of

the fabricated device.

To enhance the sensitivity and increase the quality factor of the structure, the

packaged device is placed under high vacuum inside a vacuum chamber as depicted in

Figure 5-1. It will minimize the energy dissipation caused by damping effect of the air

around the structure. Connections from the device to other instruments are made through

built-in feedthroughs on the chamber wall.

Figure 5-1: Packaged MEMS device inside the vacuum chamber

53

Drive electrode is used to electrostatically excite the structure. Sense electrodes

that are placed on both sides of the sense beam are also used for capacitive sensing of

sense beam’s displacements. Movements of sense beam change the gap between sense

beam and sense electrodes which leads to a change in the capacitors formed between

the sense beam and sense electrodes. These virtual capacitors are shown in Figure 5-2.

Figure 5-2: Virtual capacitors formed between drive/sense electrodes and the structure

Four sets of experiments are conducted to show mode coupling and to show the

response bandwidth enhancement. It was shown in numerical simulations of chapter 3

that there is a coupling between vibrational modes of the structure when there is a 2:1

ratio between the resonance frequencies of drive and sense modes. The first set of

experiments is to measure the resonance frequency of first and second modes to obtain

the frequency ratio. The effect of DC voltage on the resonance frequencies is also

investigated in this experiment set. In the second set of experiments, forward and

backward frequency sweeps are done to show the modal interactions and bandwidth

increase in the sense beam’s response. The third set of experiment investigates the

response of the system in different excitation amplitudes. Test results show that there is

no modal interaction before the excitation amplitude reaches a certain threshold. And

finally, in the fourth set of experiments, results show the effect of pressure on the response

of the system.

54

5.1. Structure Characterization

The first step is to perform a resonance frequency characterization for drive and

sense modes. The experimental setup used to conduct this experimental test consists of

(1) The fabricated modified micro T-beam (2) Vacuum chamber (3) a DC voltage source

(4) a signal amplifier (5) network analyzer to both excite the system and to monitor the

output signal. This experimental setup is depicted in Figure 5-3.

Figure 5-3. Experimental test setup used for under-vacuum experimental tests for structural resonance frequency measurements

In this experiment, a network analyzer is used to both excite the drive beam and

to monitor the amplified output signal from the sense beam. A 40V DC voltage is applied

to the structure and, the drive AC signal from network analyzer is applied to the drive

electrode to excite the structure to its resonances. Network analyzer performs a frequency

sweep to find the resonance frequency of the structure. These resonance frequencies are

shown in Figure 5-4. The resonance frequencies of drive mode and sense mode are

722.590 kHz and 361.135 kHz, respectively. These resonance frequencies lead to a

frequency ratio of 2.0009 which is very close to the ideal ratio of 2:1.

55

Figure 5-4. Resonance frequencies of drive and sense mode of modified micro T-beam structure for Vdc=40 V

Effect of DC voltage on resonance frequencies

Resonance frequencies of vibrational mode are dependent on the amplitude of the

excitation force. Increasing the DC voltage increases the excitation force. Figure 5-5 and

Figure 5-6 show the impact of DC voltage on the resonance frequency of vibrational

modes. As it can be seen in these figures, increasing the DC voltage shifts the resonance

frequencies of sense and drive modes to lower frequencies due to the electrostatic spring

softening in the mechanical structure. This shift in resonance frequency can be used for

56

tuning the devices that have a small frequency ratio detuning. However, there is always a

limit for increasing the DC voltage. Increasing the DC voltage beyond the pull-in voltage

makes the negative spring constant larger than the restoring mechanical spring constant,

and the structure collapses.

Figure 5-5: Effect of DC voltage on the resonance frequency of sense mode

Figure 5-6: Effect of DC voltage on the resonance frequency of drive mode

57

5.2. Frequency Sweep Response of a 2:1 Modified T-beam Structure

From numerical simulations, it is expected that that in a modified micro T-beam

microresonator with 2:1 frequency ratio, first and second vibrational modes can couple,

and energy can transfer from one mode to the other one. To experimentally observe the

mode coupling between modes and response bandwidth enhancements, frequency

sweep tests are conducted. The experimental setup used to conduct the frequency sweep

tests consists of (1) The fabricated modified micro T-beam (2) Vacuum chamber (3) a DC

voltage source (4) a function generator for excitation (5) a signal amplifier (6) spectrum

analyzer to monitor the output signal. The block diagram of the experimental setup is

depicted in Figure 5-7. This test setup is very similar to the setup that was used for

resonance frequency characterization, and only the excitation and output monitoring

devices are changed.

Figure 5-7: Experimental test setup used for frequency sweep response of the system

58

In this experiment set, a constant DC voltage of 40 V is applied to the structure,

and an AC signal from a function generator is applied to the drive electrode. To observe

the frequency sweep response of the system, the amplitude of excitation signal (Vac) is

kept constant, and its frequency is being changed from 714kHz to 724kHz for the forward

sweep and from 724kHz to 714kHz for the backward sweep. These frequency sweeps are

performed in a 180s time frame. The sense output signal is then fed through an amplifier

with the gain of 100k to the spectrum analyzer for monitoring. Figure 5-8 and Figure 5-9

show the backward and forward frequency sweeps of the system for different excitation

amplitudes (Vac). As it can be seen in these figures, increasing the excitation amplitude

widens the bandwidth of response.

Figure 5-8: Backward frequency sweep response of the system

59

Figure 5-9: Forward frequency sweep response of the system

It can also be observed in the results that there is an overlapping region between

forward and backward frequency sweeps. This region can be seen in Figure 5-10. This

region relates to the frequency range that response of the system is not dependent on the

direction of the sweep. This frequency range is a suitable for gyroscopic applications. It is

also observed in the experimental results that the bandwidth of this overlapping region

also increases by increasing the excitation amplitude.

Figure 5-10: Overlap of forward and backward frequency sweep response of the system

60

As it was discussed in previous chapters, in a microresonator tuned to 2:1

resonance mode coupling only occurs if the excitation amplitude is greater than a certain

threshold. In next round of experiment, the same test setup as frequency sweep test setup

is used. To perform the test, the excitation frequency is held constant at drive beam’s

resonance frequency, and the excitation amplitude is being increased. As it can be seen

in Figure 5-11, there is no mode coupling before the excitation amplitude reaches a

threshold (2.3V). After this threshold, drive mode and sense mode couple, and the sense

beam’s amplitude starts to grow. From this point the amplitude of drive beam saturates to

the excitation amplitude and any extra energy added to the system by increasing the

excitation amplitude will channel to the sense mode.

Figure 5-11: Sense beam's response to different excitation amplitudes. It shows mode coupling and internal resonance occur after excitation amplitude reaches a threshold

61

5.3. Effect of Pressure in 2:1 Modified T-beam with Internal Resonance

In a linear microresonator, a linear relation is expected between quality factor(Q)

and amplitude at resonance. The quality factor of the sense mode’s resonance in different

operating pressures can be seen in Figure 5-12.

Figure 5-12: Quality factor of sense mode in linear modified T-beam microresonator in a range of operating pressures

A set of experiment is conducted to investigate the effect of operating pressure on

the response of the microresonator in the presence of 2:1 internal resonance. To do this

test, vacuum chamber’s pressure is being changed slowly from near vacuum condition

(here 10mTorr) to certain pressures. After fixing the pressure, frequency sweeps are

performed as before to show the internal resonance phenomenon. Figure 5-13 and Figure

5-14 show the effect of pressure variations on the backward and forward frequency sweep

response of the system. As it can be seen in both figures, the bandwidth of response

increases by decreasing the pressure. However, the amplitude of the response does not

change dramatically with changes in the operating pressure. Also, after decreasing the

pressure to near vacuum (<10 mTorr), the system reaches a state in which changes in

pressure does not have a significant effect on the response. In this state, air damping is

not the dominant damping mechanism, and other mechanisms are responsible for the

damping of the system.

62

Figure 5-13: Effect of pressure on the bandwidth in backward frequency sweep response

Figure 5-14: Effect of pressure on the response bandwidth in forward frequency sweep

To better understand the effect of the pressure on the response, full bandwidth at half-

maximum (FBHM) and amplitude of output signal at 360.95 kHz for both backward and

forward frequency sweeps are shown in Figure 5-15 and Figure 5-16.

63

Figure 5-15: Effect of pressure on the "full width at half-maximum (FWHM)" and "output signal amplitude" in a forward frequency sweep

Figure 5-16: Effect of pressure on the "full width at half-maximum (FWHM)" and "output signal amplitude" in a backward frequency sweep

These figures show that unlike linear case, amplitude of response in the presence

of 2:1 internal resonance does not change significantly for pressures below 1200 mTorr.

This is particularly desirable for applications like micro gyroscopes because the system

has almost the same response over a wide range of pressures and the designer should

not be worried about designing a structure that operates in a special pressure. This

eliminates the need for high vacuum packaging of the device which can reduce the costs

of mass production of device.

64

Chapter 6. Conclusion and Future Work

This thesis studied the utilization of nonlinear 2:1 internal resonance in

microresonators to increase the response bandwidth through nonlinear modal

interactions. The main motivation of this work was to use nonlinear mode coupling to

overcome the sensitivity loss in the mode-matched gyroscopes caused by fabrication

errors.

A novel 2:1 modified T-beam design was proposed, and a simple 2-DOF

mathematical model including the nonlinearities was derived for this system. Perturbation

methods were used to solve the equations of motion of the system. Perturbation solution

along with numerical simulations in ANSYS and CoventorWare were used to show the

concept of utilizing internal resonance in the proposed modified micro T-beam structure.

Finite Element analysis in ANSYS and reduced order simulations in CoventorWare were

used for the structural design of the proposed modified micro T-beam. It is shown in these

numerical simulations that quadratic nonlinearities can couple two vibrational modes of a

system with 2:1 ratio between two of its modal frequencies. The modal coupling in the

system can act as a bridge to transfer energy from a directly excited mode into an indirectly

excited mode. In addition, it is shown through simulations that transfer of energy from the

directly excited mode to the indirectly excited mode can cause bandwidth enhancement

in sense mode’s response. The response of the system for different excitation amplitudes,

minimum excitation amplitude needed for modal coupling, and effect of input angular

velocity on the response are also studied in the simulations.

The designed structure was then fabricated with SOIMUMPS process. A

comparison between structural dimensions of the fabricated device and the designed

structure showed a very small error between these dimensions. The experimental results

also showed an almost perfect 2:1 frequency ratio between the drive and sense modal

frequencies. These results also verified the simulation results qualitatively. Saturation

phenomenon and a significant response bandwidth enhancement in forward and

backward frequency sweeps are also observed in the test results. Similar to the simulation

65

results, response of the system to different excitation amplitudes, effect of damping, and

minimum excitation amplitude were studied in the experiments. Also, it was shown that

increasing the excitation amplitude boosts the modal coupling and widens the response

bandwidth. Additional enhancements in the bandwidth can be achieved in future by

feedbacking the nonlinear terms using the two feedback electrodes that are placed beside

the drive beam.

The overlapping region between forward and backward sweeps relates to a

frequency range in which the response of the system is independent of the frequency

sweep direction. This frequency range can be used for gyroscopic applications. In

addition, study the effect of pressure on the response of the system showed that for

operating pressure from vacuum condition to almost 1200mTorr, the amplitude of

response does not change dramatically with the change in operating pressure. This effect

eliminates the necessity of near vacuum operation of micro gyroscopes which can lead to

a massive cut in fabrication and packaging costs.

At the time of writing this thesis, our group was working on gyroscopic tests of this

structure and similar structures. To better understand the response of the microstructure,

gyroscopic simulations are performed in CoventorWare. To perform the gyroscopic

simulation, a ramp angular velocity is applied to the system. This angular velocity is

applied to the system after internal resonance occurs and the system reaches to steady

state. Figure 6-1 shows the response of the system to a ramp angular velocity profile. The

response of the system shows an increase in both sense and drive amplitudes. Figure 6-2

depicts the sensitivity of the system to different input angular velocities. This figure shows

that the amplitude of sense beam increases almost linearly with increasing the input

angular velocity. It can also be noticed that, the direction of input angular velocity does not

affect the response of the system. This is an undesirable issue for a gyroscope and needs

to be addressed properly in future experimental tests.

66

Figure 6-1: Response of the system to a ramp angular velocity

Figure 6-2: Sensitivity curve showing the sense-beam amplitude for different input angular velocities

The concept of using internal resonance for increasing the bandwidth can also be

used for developing sensors other than micro gyroscopes. For instance, adding a small

mass to a structure with 2:1 frequency ratio will change the resonant frequencies of the

structure and consequently deviates the frequency ratio from ideal 2:1 ratio. This ratio

deviation will cause a change in response bandwidth and amplitude which can then be

67

detected and then calibrated to the added mass. Furthermore, since the autoparametric

excitation of lower frequency mode is a threshold phenomenon, one can exploit the

transitions across this threshold as a switch. In addition, exciting autoparametric excitation

of lower frequency mode leads to an output signal frequency at half of the excitation

frequency which can be used as a frequency divider.

68

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