N°: 2008-02 MASTER THESIS - KFUPMfaculty.kfupm.edu.sa/ME/houakad/Hassen's MS Thesis Report (Final... · Master Thesis N°: 2008-02 MASTER THESIS Presented to Tunisia Polytechnic

Republic of Tunisia

Ministry of Higher Education University of the 7th of November at Carthage

Tunisia Polytechnic School Department of Mechanics

Master’s Program in Computational Mechanics

Master Thesis

N°: 2008-02

MASTER THESIS

Presented to

Tunisia Polytechnic School (Department of Mechanics)

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

in

Computational Mechanics

by

Hassen OUAKAD

Nonlinear Dynamics and Control of Microbeam-Based Systems

Defended on January 4th, 2008, in front of the examination committee:

Dr. Sami EL-BORGI – EPT President

Dr. Brahim JEMAI – ITMT Member

Dr. Slim CHOURA – EPT Supervisor

January 2008

République Tunisienne

Ministère de l’Enseignement Supérieur

Université du 7 Novembre à Carthage École Polytechnique de Tunisie

Département de Mécanique

Programme de Mastère en Mécanique Calculatoire (Computational Mechanics)

Mémoire de MASTERE

N° d’ordre: 2008-02

MEMOIRE

présenté à

l’Ecole Polytechnique de Tunisie (Département de Mécanique)

en vue de l’obtention du diplôme de

MASTERE

Dans la discipline Mécanique Calculatoire (Computational Mechanics)

par

Hassen OUAKAD

Dynamique Nonlinéaire et Contrôle des Systèmes à Base de Micropoutre

Soutenu le 4 Janvier 2008, devant le jury:

Dr. Sami EL-BORGI – EPT Président

Dr. Brahim JEMAI – ITMT Membre

Dr. Slim CHOURA – EPT Encadreur

Janvier 2008

i

Nonlinear Dynamics and Control of Microbeam Based Systems

by

Hassen OUAKAD

Tunisia Polytechnic School, Tunis, 2008

- ABSTRACT -

This thesis is concerned with the modeling, nonlinear dynamic analysis and control design of

two types of electrostatically actuated microbeams. The modeling of the first microbeam

accounts for the mid-plane stretching and nonlinear form of the electrostatic force. The

microbeam is fixed at both ends and electrostatically actuated along its span. Such a

microbeam characterizes the principal component of a large class of MEMS devices, such as

microsensors and microresonators. The second microbeam, which typifies another class of

MEMS devices, such as gas microsensors, is fixed at one end and free and coupled to an

electrostatically actuated microplate at the other end. For each microbeam, a reduced-order

model is constructed, using the method of multiple scales (for the first microbeam) and the

Galerkin method (for the second microbeam), to examine its static and dynamics behaviors.

The present work also addresses the control design of the first microbeam for improving its

nonlinear behavior. The main control objective is to make it behave like commonly known

one-degree-of-freedom self-excited oscillators, such as the van der Pol and Rayleigh

oscillators, which depict attractive filtering features. For this, a review of the nonlinear

dynamics of these oscillators is first provided to gain insight into their appealing filtering

characteristics. We then present a novel control design that regulates the pass band of the

fixed-fixed microbeam and derive analytical expressions that approximate the nonlinear

resonance frequencies and amplitudes of the periodic solutions when the microbeam is

subjected to one-point and fully-distributed feedback forces. We also derive closed-form

solutions to the static and eigenvalue problems associated with the second microbeam. The

Galerkin procedure is used to derive a set of nonlinear ordinary-differential equations that

describe the microbeam-microplate dynamics. We then employ a finite-difference method to

compute limit-cycle solution. We apply Floquet theory to ascertain the stability of the limit

cycles.

Keywords: Microbeam, Resonator, van der Pol Oscillator, Rayleigh Oscillator, Method of

Multiple Scales, Finite Difference Method, Galerkin Methods, Feedback, Microplate

ii

Dynamique Nonlinéaire et Contrôle des Systèmes à Base de

Micropoutre

- RESUME-

Cette thèse s'intéresse à la modélisation, l'analyse dynamique nonlinéaire et la mise au point

d’une stratégie de contrôle pour deux types de micro-actionneurs à base de micropoutres.

Pour les deux cas on utilise des modélisations qui prennent en compte l’allongement de la

fibre moyenne et la forme nonlinéaire de la force électrostatique. La première micropoutre est

fixée aux deux extrémités et actionnée par une force électrostatique appliquée le long de la

micropoutre. Une telle micropoutre caractérise la principale composante d'une large classe de

microsystèmes (MEMS), comme les microrésonateurs et microcapteurs. La deuxième

micropoutre, qui caractérise une autre classe de microsystèmes, tels que les microcapteurs à

gaz, est fixée à une extrémité et attachée à l’autre par une microplaque, celle-ci est actionnée

par une force électrostatique. Pour chaque micropoutre, un modèle à ordre réduit est construit,

en utilisant la méthode de perturbation multiéchelles (pour la première micropoutre) et la

méthode de Galerkin (pour la deuxième micropoutre). L’étude permet aussi d'examiner à la

fois la partie statique et dynamique de comportements de chaque micropoutre. Le présent

travail porte également sur la mise au point d’une stratégie de contrôle pour la première

micropoutre afin d'améliorer son comportement. Le principal objectif est de rendre le

comportement de cette micropoutre aussi simple qu’un oscillateur à un seul degré de liberté,

comme l’oscillateur de van der Pol ou de Rayleigh. Ces derniers montrent des performances

de filtrage remarquables. Pour cela, une étude dynamique de ces oscillateurs est effectuée afin

d'avoir un aperçu sur leurs caractéristiques de filtrage. Ensuite, une nouvelle conception du

contrôle, qui régit le passage de la bande fixe de ces micropoutres, est présentée. Des

expressions analytiques pour la détermination des fréquences de résonance nonlinéaire et de

la réponse statique sont aussi présentées. La méthode de Galerkin est utilisée pour obtenir un

ensemble d'équations différentielles qui décrivent la dynamique du système. La méthode des

différences finies est ensuite employée afin de discrétiser les orbites et l’extraction des

solutions périodiques. Ces solutions aux cycles limites sont ensuite testées en utilisant la

théorie de Floquet afin de déterminer leurs stabilités.

Mots-clés: Micropoutre, Microrésonateur, Oscillateur de van der Pol, Oscillateur de

Rayleigh, Méthode des perturbations multiéchelle, Méthode des différences finies, Méthode

de Galerkin, Contrôle, Microplaque

iii

بسم الله الرحمان الرحيم

Dedication

I dedicate this work

To the spirit of my Mother,

To my father and brother Mehdi,

To my fiancée Sara,

To all my family,

To my supervisor,

To my friends at

Tunisia Polytechnic School,

LASMAP and

Especially the Department of Mechanics

iv

Acknowledgements

First of all, I thank Allah, the most Merciful and most Gracious, for this

achievement.

For the many that have assisted and supported this work, I owe a debt of

gratitude. First, I would like to thank my principal supervisor, Dr. Ali NAYFEH,

for his sound advice, assistance and unconditional encouragement, as well as Dr.

Slim CHOURA and Dr. Eihab ABDEL-RAHMAN for their guidance and precious

directions.

Dr. Sami EL-BORGI earns my sincere appreciation and thanks for the continuous

support and encouragement he offered me.

My thanks go to Mr. Fehmi Najar for his assistance, availability, and his

motivation in collaborating with me.

This research work was carried out at the Laboratory of Systems and Applied

Mechanics (LASMAP) of Tunisia Polytechnic School, to which I’d like to address

my special thanks and acknowledgements.

Unlimited appreciation goes to my family and especially my fiancée Sara, without

their support I would have never accomplished my research work.

v

Contents

Chapter 1. Introduction .............................................................................................................1

1.1. Generalities: Applications of MEMS devices .........................................................1

1.2. Literature Survey of the Modeling of MEMS Microbeam ......................................3

1.3. Motivation and Objective .......................................................................................6

1.4. Organization of the Thesis......................................................................................6

Chapter 2. Analysis of One-Degree-of-Freedom Oscillators....................................................8

2.1. The van der Pol Oscillator .....................................................................................8

2.1.1. Free Vibrations ........................................................................................8

2.1.2. Forced Vibrations ....................................................................................9

2.2. The Rayleigh Oscillator........................................................................................10

2.2.1. Free Vibrations ......................................................................................10

2.2.2. Forced Vibrations ..................................................................................11

2.3. Dynamic Analysis Using the Method of Multiple Scales......................................12

2.3.1. The van der Pol oscillator......................................................................12

2.3.2. The Rayleigh oscillator ..........................................................................16

2.4. Parametric Study for Signal filtering ...................................................................17

2.5. Summary ...............................................................................................................19

Chapter 3. Control and Dynamics of an Electrostatically Actuated Clamped-Clamped

Microbeam Resonator .............................................................................................................20

3.1. Modeling of a Clamped-Clamped Microbeam Resonator....................................20

3.2. A One-Point Feedback:.............................................................................22

3.2.1 Analysis ...................................................................................................23

3.2.2 Simulations..............................................................................................27

3.3. A Fully Distributed Feedback...................................................................32

3.3.1 Analysis ...................................................................................................32

3.3.2 Simulations..............................................................................................33

3.4. Implementation of the Feedback Controller.........................................................39

3.5. Summary ...............................................................................................................40

vi

Chapter 4. Dynamics of an Electrostatically Actuated Cantilever Microbeam Resonator.....41

4.1. Mathematical Modeling........................................................................................41

4.1.1. Kinetic Energy .......................................................................................42

4.1.2. Potential Energy ....................................................................................42

4.1.3. Governing equation of motion ...............................................................43

4.2. Static Deflection....................................................................................................46

4.2.1. Simulations.............................................................................................46

4.3. Natural Frequencies and Mode Shapes................................................................47

4.3.1. Eigenvalue Problem...............................................................................48

4.3.2. Natural Frequencies ..............................................................................50

4.3.3. Mode Shapes ..........................................................................................51

4.3.4. Orthogonality Conditions ......................................................................52

4.4. Reduced-Order Model ..........................................................................................53

4.4.1. One-Mode Approximation .....................................................................54

4.4.1.1 Response to combined DC and AC voltages ............................54

4.4.1.2 Phase portraits .........................................................................57

4.4.2. Multi-Mode Approximation ...................................................................60

4.5. Summary ...............................................................................................................61

Chapter 5. Conclusions and Recommendations for Future Research ....................................62

5.1 Conclusions............................................................................................................62

5.2 Recommendations for Future Research.................................................................62

Appendix ..................................................................................................................................64

Bibliography ............................................................................................................................66

vii

List of Figures

Figure 1.1: Some MEMS applications ......................................................................................2

Figure 1.2: MEMS cantilevers used as tip actuators in a MEMS-based probe-storage chip...3

Figure 2.1: (a) Typical phase diagram and (b) time response of the van der Pol oscillator

( ε = 0.1, ω = 0.707 rad/s) .........................................................................................................9

Figure 2.2: (a) Typical phase diagram and (b) time response of the van der Pol oscillator

( ε = 1, ω = 0.707 rad/s) ............................................................................................................9

Figure 2.3 : (a) Typical phase diagram and (b) time response of a harmonically driven van

der Pol oscillator ( γ = 0.01)...................................................................................................10

Figure 2.4 : (a) Typical phase diagram and (b) time response of a harmonically driven van

der Pol oscillator ( γ = 10)......................................................................................................10

Figure 2.5: (a) Typical phase diagram and (b) time response of the Rayleigh oscillator

(ε = 0.1, ω = 0.707rad/s).......................................................................................................11

Figure 2.6: (a) Typical phase diagram and (b) time response of the Rayleigh oscillator

(ε = 1, ω = 0.707rad/s)..........................................................................................................11

Figure 2.7: (a) Typical phase space diagram and (b) time response of a harmonically driven

Rayleigh oscillator ( γ =0.01) ..................................................................................................12

Figure 2.8: (a) Typical phase space diagram and (b) time response of a harmonically driven

Rayleigh oscillator ( γ =10) .....................................................................................................12

Figure 2.9: Comparison of the approximate solution with the numerical solution for an initial

condition of 0x = 0.01, ε =0.1 ................................................................................................15

Figure 2.10: Frequency-response curves for primary resonances of the van der Pol Oscillator

for various forcing amplitudes of γ ( 0ω = 10.3 rad/s) ............................................................15

Figure 2.11: Comparison of the approximate solution with that obtained by integrating

original equation for an initial condition of 0x = 0.01, ε =0.1 ..............................................16

Figure 2.12: Frequency-response curves for primary resonances of the Rayleigh oscillator

for various forcing amplitudes γ ( 0ω = 1 rad/s and ε = 0.01) ..............................................17

Figure 2.13: Time response of the van der Pol oscillator for various values of σ ................18

Figure 2.14: Time response of the Rayleigh oscillator for various values of σ .....................18

Figure 3.1: Schematic of a transversely deflected clamped-clamped microbeam resonator .21

viii

Figure 3.2: Equilibria of an electrostatically actuated microbeam (Bifurcation Diagram) ..28

Figure 3.3: Amplitude versus detuning; VDC = 1 Volt, K1=0, K2=0, and K3=0......................29

Figure 3.4: Amplitude versus detuning; VAC =0.05 Volt, K1=0, K2=0, and K3=0..................29

Figure 3.5: Amplitude versus detuning; VAC =0.05 Volt, K1=0, K2=0, and K3=0..................30

Figure 3.6: Amplitude versus detuning parameter with VAC =0.05 Volt, VDC =1.5 Volt,

K1=0, K2=0, and K3=0 ............................................................................................................30

Figure 3.7: Amplitude versus detuning; VDC =1 Volt, VAC =0.05 Volt, K2=0, and K3=0.......31

Figure 3.8: Amplitude versus detuning; VDC =1 Volt, VAC =0.05 Volt , K1=0, and K3=0......31

Figure 3.9: Amplitude versus detuning;VDC =1.5 Volt,VAC = 0.05Volt ,K1=0.01,K2=1, and

K3=0 ........................................................................................................................................33

Figure 3.10: Amplitude versus detuning; VDC =1 Volt, VAC =0.05 Volt, K2=0, and K3=0.....34


Figure 3.12: Mid-point time response for VDC =1 Volt, VAC =0.05 Volt, K1=0.01, K2=1,

and K3=0 for various values of the detuning parameter σ ....................................................35

Figure 3.13: Amplitude versus detuning; VDC =1 Volt, VAC =0.05 Volt, K1=0, and K2=0....36

Figure 3.14: Amplitude versus detuning; VDC =1 Volt, VAC =0.05 Volt, K1=0, and K2=0....36

Figure 3.15: Amplitude versus detuning; VDC =1 Volt, VAC =0.05 Volt, K1=0.01, and K2=037


Figure 3.17: Amplitude versus detuning; VAC =0.05 Volt, K1=0, K2=0, and K3=3................38

Figure 3.18: Amplitude versus detuning; VDC =1 Volt, VAC =0.01 Volt, K2=0, K3=179.6.....38

Figure 3.19: Analog implementation of feedback controller ..................................................39

Figure 4.1: Schematic of the cantilever microbeam with a plate at the end...........................41

Figure 4.2: Local coordinate system attached to the microplate............................................42

Figure 4.3: Variation of the static deflection with the DC voltage .........................................47

Figure 4.4: Variation of the first natural frequency with VDC.................................................50

Figure 4.5: Variation of the first five natural frequencies with VDC .......................................50

Figure 4.6: The first five mode shapes ....................................................................................51

Figure 4.7: Frequency-response curve of the microbeam for loading case 1 ........................56



Figure 4.10: Phase portrait for loading cases 1 and 2 without damping and forcing............58

Figure 4.11: Phase portrait for loading case 3 without damping and forcing .......................59

Figure 4.12: Phase portrait for loading case 1 and 2 with damping and without forcing s...60

Figure 4.13: Phase portrait for loading case 3 with damping and without forcing ...............60

ix

List of Tables

Table 3.1. The clamped-clamped microbeam parameters ......................................................28

Table 4.1: Geometric and physical parameters of the microbeam-microplate system...........46

Table 4.2: Variation of the first five natural frequencies with VDC.........................................50

Table 4.3: Loading cases.........................................................................................................55

Table 4.4: Location of the unstable fixed point for different numbers of modes.....................61

Table 4.5: Exact location of the unstable fixed point ..............................................................61

Chapter 1. Introduction 1

Chapter 1.

Introduction

There has been a major interest in the modeling of electrostatically actuated microdevices,

which include microactuators, microsensors, microswitches and micromirrors. Most often, the

dynamics of these devices are described by using a lumped mass-spring model, which under

predicts both of the static and dynamic performances [32]. This chapter is primarily devoted

to (i) an overview of the different applications of MicroElectroMechanical Systems (MEMS),

(ii) a literature survey of different models of electrostatically actuated microbeams, and (iii) a

statement of the research objectives adopted in this thesis.

1.1. Generalities: Applications of MEMS Devices

MEMS technology enables the creation of mechanical components on a microscopic scale by

leveraging off the fabrication techniques used in microelectronics. MEMS technology has

opened up a wide variety of potential applications not only in the inertial measurement sector,

but also in areas such as communications (filters, relays, oscillators, LC passives, and optical

switches), biomedicine (point-of-care medical instrumentation, microarrays for DNA

detection and high throughput screening of drug targets, immunoassays, and in-vitro

characterization of molecular interactions), computer peripherals (memory, new I/O

interfaces, and read-write heads for magnetic disks), projection displays, gas detection, and

mass-flow detection.

This study focuses on electromechanical resonators (Figure 1.1 - a), such as quartz-crystal and

ceramic resonators. They are widely used in radio frequency (RF) and intermediate frequency

(IF) applications. Because they are off-chip components, they have to be interfaced with

integrated electronics at the board level, which conflicts with the continuous trend to

miniaturization in modern communication systems. The study focuses also on a new

generation of small, high-performance, low-power RF-MEMS components, such as switches,

phase shifters, tunable capacitors, inductors, and mechanical resonators and filters (Figure 1.1

- b). Miniaturization, while allowing for the integration of transmitters and receivers on the

same chip, puts severe constraints on the circuit power dissipation and electromagnetic

compatibility requirements. This requires very high dynamic range receivers, ultra-clean


transmitters, and careful attention to the overall EMC design of the system. Consequently,

filtering is indispensable for both transmitters and receivers to ensure that they do not

interfere with each other.

RF MEMS switches (Figure 1.1 - c) are a fast-growing area that has gained a great deal of

attention in recent years. RF MEMS switches overcome the limitations of conventional

switches, such as solid-state switches, and present many attractive features, like low-power

consumption, high isolation, and low-insertion loss. Similar to resonators, RF switches rely on

a mechanical element, which is actuated typically by DC electrostatic forces, to close or break

an electric circuit. A major drawback of these devices is the requirement of high driving

voltages and the relatively slow response [45]. It is highly desirable to bring the actuation

voltage to a level compatible or close to that of the circuit devices and to actuate the switch

with a very high speed. However, state-of-the-art RF MEMS switches are far from achieving

these requirements, which forms a barrier towards the development of this technology.

Figure 1.1: Some MEMS applications [24]

In fact, MEMS-based resonators have the potential to offer performance that is significantly

superior to traditional electronic or mechanical resonators. To date, band pass filter designs,

including MEMS versions, have made use of the usual primary resonance in order to attenuate

signals that are outside of a given frequency band. In the present work, we describe the novel

use of parametric resonance in MEMS oscillators for filtering

Another class of MEMS devices is made up of cantilever microbeams [20], which are

commonly used in MEMS-based probe storage chips [22]. As shown in Figure 1.2, two chips,

a- RF MEMS filter c- RF MEMS switch b- MEMS resonator (clamped clamped beam)


produced with post-CMOS micromachining methods, are bonded together to form a MEMS

based non-volatile magnetic mass storage device. The upper chip contains a moveable

magnetic medium, which is addressed by an array of cantilevered probes illustrated on the

bottom chip. Because of fabrication tolerances, a gap is expected between the probe tips and

the medium’s surface after assembly. To read and write small marks, the tips must move to

within a few nanometers of the medium, which in turn will require the cantilevers to move to

within a small distance of the surface. This means that the cantilevers must have controlled

actuation over 90% or so of the initial gap.

Figure 1.2: MEMS cantilevers used as tip actuators in a MEMS-based probe-storage chip [22]

1.2. Literature Survey of the Modeling of MEMS Microbeams

Advances in microfabrication technology have enabled the design and fabrication of MEMS

devices, which promise breakthrough developments in telecommunications, radar systems,

and personal mobiles. Resonant microbeams (resonators) have been widely used as

transducers in mechanical microsensors. As interest grew dramatically in MEMS devices for

wireless communications applications and the demand for high-frequency and quality factor

resonators increased rapidly, MEMS resonators were proposed in the mid nineties as a

feasible alternative to conventional large-size resonators.

Among the numerous actuation methods for MEMS devices is electrostatic actuation, which

is the most well established technique because of its simplicity and high efficiency [24]. In a

microbeam-based resonator, the microbeam is deflected by a DC bias and then driven to

vibrate around its natural frequency by an AC harmonic load. A key issue in the design of

such a device is to tune the electric load away from the pull-in instability, which leads to


collapse of the microbeam and hence the failure of the device [27]. Many studies have

addressed the pull-in phenomenon and presented tools to predict its occurrence to enable

designers to avoid it [8, 9, 20 and 27]. However, these studies investigate the stability of the

static deflection of the microbeam rather than the stability of motions around this deflected

position. Hence they do not account for motions and transients due to the AC loading. This is

particularly important in light of the fact that the behavior of these devices is nonlinear.

Nayfeh et al [26] studied the pull-in instability in MEMS resonators and found that

characteristics of the pull-in phenomenon in the presence of AC loads differ from those

resulting from the use of only DC loads. They analyzed this phenomenon, dubbed dynamic

pull-in, and formulated safety criteria for the design of MEMS resonant sensors and filters

excited near one of their natural frequencies. They also analyzed the dynamics of MEMS

resonators and switches.

Other studies state that increasing the voltage of a class of MEMS resonators yields a decrease

in the gap and the generation of an incremented force [16]. Consequently, the electrostatic

loading has an upper limit beyond which the mechanical force can no longer resist the

opposing electrostatic force, thereby leading to the collapse of the structure. This actuation

instability phenomenon is known as pull-in, and the associated critical voltage is called pull-in

voltage. Several studies, including the pioneering work of Nathanson et al. [34] and Newell et

al [35], investigated this phenomenon under various loading conditions. Such studies

considered a resonant gate transistor modeled by a mass-spring system subjected to an

electrostatic actuation. They predicted and offered a theoretical justification of the so-called

pull-in instability. Since then, numerous investigators have analyzed mathematical models of

electrostatic actuation in attempts to further understand and control the pull-in instability.

Despite more than three decades of work in the area of electrostatically actuated MEMS, the

complete dynamics of the electrostatic-elastic system is relatively unexplored. There are a lot

of aspects to be clarified. Some studies just center their goal in the immediate application of

the sensor, and a simple mass-spring model can approximate the basic dynamics.

Mass-spring-lumped models of MEMS devices do not account for the inherent nonlinearities

of the electrostatic force and beam deflection [9 and 11]. For this reason, distributed-

parameter models, consisting of partial-differential equations and associated boundary

conditions linearized about the working point, were developed [16]. These models lead to

better approximation of the device performance provided that the microbeam deflections

remain small. To account for large-amplitude deflections, researchers [1-6, and 45-50]


developed nonlinear distributed-parameter models that account for midplane stretching and

axial and electrostatic loadings. Nowadays, model accuracy becomes an essential tool for the

design of new generation of high-performance and self-calibrated MEMS devices.

Shaw et al [41] analyzed the dynamics of MEMS oscillators that act like frequency filters,

where parametric resonance was used for frequency selection. Such resonance features nearly

ideal stop band rejection; that is, the response is essentially zero (at the noise floor) outside of

the instability zone, which is taken here to be the filter pass band, thereby offering an

extremely sharp response roll-off in the frequency domain. They also provided a description

of how to use a pair of MEMS oscillators to build a band-pass filter with nearly ideal stop

band rejection. Their design is appropriate for highly tunable microbeams, which offer

minimal packaging constraints, low-power consumption, low damping, ease of parameter

tuning, and relatively simple integration with electronics. Rhoads and co-workers [38-40]

described a filter design based on the nonlinear response of parametrically excited MEMS

oscillators that have significant potential in many communications applications. They

reviewed parametric resonance and discussed its relevance to MEMS and its potential use in

filtering applications. They also modeled a single MEMS oscillator and analyzed its dynamic

response. In addition, they presented a procedure of how to improve oscillator performance,

specifically for filtering applications, and described one possible filter design that utilizes two

tuned MEMS oscillators.

An electrostatically actuated cantilevered microbeam, used as resonator, constitutes the

primary component of another class of MEMS devices. Because of its practical importance,

many studies focused on the vibrations of uniform flexible cantilever microbeams with

different boundary conditions and engineering applications. Yoo and Shin [51] performed a

detailed vibration analysis of rotating cantilever beams with a tip mass. Recently, Kirk and

Wiedemann [18] provided an analytical solution for the natural frequencies and mode shapes

of a flexible beam equipped with a rigid payload at the tip. Gokdag and Kopmaz [14] studied

the coupled flexural-torsional free and forced vibrations of a beam tip with and without span

attachments. More recently, Esmaeili et al. [12] developed the characteristic equation of a

microbeam modeled as a cantilever beam with a tip mass. The microbeam, which is subjected

to a base rotational motion around its longitudinal direction, is considered to vibrate in all

directions. Krylov et al [20] studied the nonlinear dynamics of a microbeam that is electrically

actuated through a microplate coupled at its tip. Using the Galerkin procedure with normal

modes as a basis, they developed a model that accounts for the distributed nonlinear


electrostatic force, nonlinear squeezed film damping, and rotational inertia of the microplate.

They examined the dynamics of the beam near the unstable points. The developed model led

to results that were confirmed experimentally and showed that the voltage that causes

dynamic instability approaches that associated with the static pull-in.

1.3 Motivation and Objective

Motivated by the need to further improve the static and dynamic performances of microbeam

resonators, this thesis considers the modeling, dynamic analysis, and control design of two

types of electrically actuated microbeams. The first one, which accounts for the mid-plane

stretching, is fixed at both ends and electrostatically actuated along the microbeam span. The

second microbeam is fixed at one end and coupled to an electrostatically actuated microplate

at the other end. For each microbeam, a reduced-order model is developed to analyze both of

its static and dynamics behaviors. The present work also addresses the control design for

improving the nonlinear behaviors of these microbeams. The main control objective is to

make these microbeams behave like commonly known one-degree-of-freedom self-excited

oscillators, such as van der Pol and Rayleigh oscillators, which depict attractive filtering

features. For control purposes, a review of the nonlinear dynamics of these oscillators is

provided to gain insight into their appealing filtering characteristics. The design of the

microbeams is based on their reduced-order models, which are deduced using the method of

multiple scales (for the first microbeam) and the Galerkin procedure (for the second

microbeam). Thus, the control design aims at altering the behaviors of the microbeams so that

they act like one of the one-degree-of-freedom self-excited oscillators.

A major contribution of this work is to provide a set of reliable analytical expressions that

describe the system’s behavior. Such expressions make it possible to design feedback

controllers for tuning oscillators and regulating the pass band of microbeam resonators.

1.4 Organization of the Thesis

The graduation project is organized as follows: Chapter 2 reviews the nonlinear behavior of

the van der Pol and Rayleigh oscillators. The method of multiple scales is applied to

approximate the frequency responses of both oscillators. The approximate solution is

compared to the exact solution for different orders of approximation. We also show the

relevance of both oscillators to the problem of microbeam-based filters. In Chapter 3, the


forced vibrations of a nonlinear clamped-clamped microbeam are addressed. We apply a

perturbation technique to the governing integral-partial-differential equation and associated

boundary conditions to approximate the structural response of the microbeam subjected to a

primary-resonance excitation. Based on the perturbation analysis, we derive equations that

describe the nonlinear resonance frequencies and amplitudes of limit-cycle solutions for both

one-point and distributed feedback forces. We study the effect of design parameters on the

nonlinear resonance frequency and the effective nonlinearity of the system. In Chapter 4, the

nonlinear dynamics of an electrostatically actuated cantilever microbeam with an end plate is

addressed. Analytical solutions to the static and linear eigenvalue problems are derived, and

then the Galerkin decomposition is adopted to approximate the dynamic problem. A finite-

difference method is used to compute periodic solutions of the resulting nonlinear ordinary-

differential equations. Finally, Chapter 5 summarizes and concludes this work with some

recommendations for future research.

Chapter 2. Analysis of One-Degree-of-Freedom Oscillators

8

Chapter 2.

Analysis of One-Degree-of-Freedom Oscillators

In recent years, a twofold interest has attracted theoretical, numerical, and experimental

investigations of nonlinear oscillators [17 and 21]. The theoretical investigation reveals their

rich and complex behavior, and experimental investigations of self-excited oscillators

describe the evolution of many biological, chemical, physical, mechanical, and industrial

systems. Recently, the chaotic behavior of these oscillators has been exploited in the field of

communication for coding information. In this chapter, the free and forced vibrations of the

van der Pol and Rayleigh oscillators are reviewed to gain insight into their attractive features.

2.1. The van der Pol Oscillator

2.1.1. Free Vibrations

The van der Pol oscillator was originally investigated by B. van der Pol as a model for the

human heart (van der Pol and van der Mark, 1928 [53]). It also describes a class of oscillatory

vacuum tube and electronic circuits. This oscillator is governed by

2 2- (1- ) 0y y y yε ω′′ ′ + =

(2.1)

where dyydt

′ = , 2

2

d yydt

′′ = , and ε >0 is a measure of damping strength. The term yε ′−

constitutes linear negative damping and the term 2y yε ′+ constitutes nonlinear positive

damping. For 1y < , the yε ′− term dominates and motions grow exponentially. For 1y > , the

term 2y yε ′+ dominates and motions decay with time. In this oscillator, ω controls the level of

voltage injected into the system. Figure 2.1 shows a typical phase diagram and time response

of the oscillator described by Equation (2.1).


9

-2 -1 1 2

-1.5

-1

-0.5

0.5

1

1.5

50 100 150 200

-2

-1

1

2

(a) Phase diagram (b) Time response

Figure 2.1: (a) Typical phase diagram and (b) time response of the van der Pol oscillator ( ε = 0.1, ω = 0.707 rad/s)

Figure 2.2 shows that when increasing the damping coefficient of the van der Pol equation the

nonlinearity becomes more important and significant. This figure illustrates a typical behavior

of a relaxation oscillator for which stress accumulates slowly and is then released rapidly.

Thus, intervals of relatively slow variation alternate with brief intervals of rather rapid

variation, with the ratio between these intervals being governed by a relaxation parameter.

-2 -1 1 2

-2

-1

1

2

50 100 150 200

-2

-1

1

2

(a) Phase diagram (b) Time response

Figure 2.2: (a) Typical phase diagram and (b) time response of the van der Pol oscillator ( ε = 1, ω = 0.707 rad/s)

2.1.2. Forced Vibrations

A harmonically forced van der Pol oscillator is governed by 2 2- (1- ) siny y y y tε ω γ′′ ′ + = Ω

(2.2)


10

Figures 2.3 and 2.4 illustrate the influence of the amplitude γ of the forcing term on the phase

diagram and time response of the oscillator. Figures 2.3 and 2.4 show that, as the forcing

amplitude is increased, the resulting limit cycles develop faster, become larger, and deform

from oval to sharper rectangular shapes.

-2 -1 1 2

-1.5

-1

-0.5

0.5

1

1.5

20 40 60 80 100 120 140

-2

-1

1

2

ε = 0.1, ω = 0.707 rad/s, Ω = 0.707 rad/s

Figure 2.3 : (a) Typical phase diagram and (b) time response of a harmonically driven van der Pol oscillator ( γ = 0.01)

-7.5 -5 -2.5 2.5 5 7.5

-10

-5

5

10

20 40 60 80 100 120

-7.5

-5

-2.5

2.5

5

7.5

ε = 0.1, ω = 0.707 rad/s, , Ω = 0.707 rad/s Figure 2.4: (a) Typical phase diagram and (b) time response of a harmonically driven van der

Pol oscillator ( γ = 10)

2.2. The Rayleigh Oscillator

2.2.1. Free Vibrations

The Rayleigh oscillator is similar to the van der Pol oscillator except for one key difference:

as the voltage is increased, the Rayleigh oscillator is more effective in limiting the size of the

limit cycle than the van der Pol oscillator. The Rayleigh oscillator is described by 2 2- (1- ) 0y y y yε ω′′ ′ ′ + =

(2.3)


11

Again, ω controls the level of voltage injected into the system and ε commands the way

voltage flows through the system. In Figure 2.5, we show a typical phase diagram and time

response of the limit cycle that develops for small ε ( 0 1ε< << ). In Figure 2.6, we show the

limit cycle that develops for largeε . It exhibits a relaxation oscillation in which stress

accumulates slowly and is then released rapidly.

.

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

50 100 150 200

-1.5

-1

-0.5

0.5

1

1.5

(a) The phase diagram (b) The time response

Figure 2.5: (a) Typical phase diagram and (b) time response of the Rayleigh oscillator (ε = 0.1, ω = 0.707rad/s)

-2 -1 1 2

-1

-0.5

0.5

1

50 100 150 200

-2

-1

1

2

(a) The phase diagram (b) The time response

Figure 2.6: (a) Typical phase diagram and (b) time response of the Rayleigh oscillator (ε = 1, ω = 0.707rad/s)

2.2.2. Forced Vibrations

The dynamics of a harmonically forced Rayleigh oscillator is described by

2 2- (1- ) sin( )y y y y tε ω γ′′ ′ ′ + = Ω

(2.4)


12

Figures 2.7 and 2.8 display the influence of the forcing amplitude γ on the phase diagram and

time response of the oscillator.

-1.5 -1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

50 100 150 200

-1.5

-1

-0.5

0.5

1

1.5

ε =0.1, ω =0.707rad/s, Ω =0.707rad/s Figure 2.7: (a) Typical phase space diagram and (b) time response of a harmonically driven

Rayleigh oscillator ( γ =0.01)

-10 -5 5

-4

-2

2

4

50 100 150 200

-10

-5

5

ε =0.1, ω =0.707rad/s, Ω =0.707rad/s

Figure 2.8: (a) Typical phase space diagram and (b) time response of a harmonically driven Rayleigh oscillator ( γ =10)

2.3. Dynamic Analysis using the Method of Multiple Scales Here, we approximate the dynamic responses of the van der Pol and the Rayleigh oscillators

using the method of Multiple Scales. A review of this method was provided in the Graduation

Project [36] and a thorough discussion of it can be found in [28 and 31].

2.3.1. The van der Pol oscillator

We consider the van der Pol oscillator in the case of a primary-resonance excitation; that is,

0( ) , ( )O Oγ ε ω ε= Ω = + . Thus, we consider

2 20 0

² (1 ) cos( ),²

d x dxx x tdt dt

ε ω γε ω εσ− − + = Ω Ω = +

(2.5)


13

We seek an approximate solution of Equation (2.5) using the method of multiple scales. In

general, we consider ( )x t to be a function of multiple (two in this case) independent time

variables or scales. We express x in the form

0 0 1 1 0 1( , ) ( , ) ...x x T T x T Tε= + +

(2.6)

where 0T t= is a fast scale and 1T tε= is a slow scale, characterizing the modulation in the

amplitude and phase caused by the nonlinearity, damping, and resonances. The time

derivatives become

0 1... ...d dT D Ddt t dt T

ε∂ ∂= + + = + +∂ ∂

(2.7)

so that

20 0 1

² 2 ...²

d D D Ddt

ε= + +

(2.8)

where nn

DT∂

=∂

. Substituting Equations (2.6-2.8) into Equation (2.5) and equating the

coefficient of 0ε and ε on both sides, we obtain

2 20 0 0 0

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

0

2 cos( )

D x x

D x x D x x D x D D x T

ω

ω γ

+ =

+ + − + = Ω

(2.9)

(2.10)

The solution of Equation (2.9) is given by

0 0 0 00 1 1( ) e ( ) ei T i Tx A T A Tω ω−= +

(2.11)

Hence, Equation (2.10) becomes 0 0 0 01 32 2 2 3

0 1 0 1 0 0[ 2 e ]e e cci T i Ti TD x x i A A A A i Aω ωσω ω γ ω′+ = − + − + + +

(2.12)

where cc denotes the complex conjugate of the preceding terms. The secular terms can be

eliminated from the solution of 1x if

122 e 0i TA A A A σγ′− + − + =

(2.13)


14

We let 1 e2

iA a β= in Equation (2.13), where a and β are real functions of the slow times

scale T1, separate real and imaginary parts, and obtain

2

0 0

1(1 ) sin( ) cos( )2 4 2 2aa a aγ γλ β λ

ω ω′ ′= − + = −

(2.14, 2.15)

where a' and β' are derivatives of the slow time scale T1 and

11 1

and d dTdT dTλ βλ σ β σ= − = −

(2.16)

Eliminating β from Equations (2.14) and (2.15) yields

0

cos( )2

a a γλ σ λω

′ = +

(2.17)

Therefore to the first approximation, we have

cos( ) ( )x a t Oλ ε= Ω − +

(2.18)

where a and λ are given by Equations (2.14) and (2.18).

For periodic motions, the time variation of the amplitude and phase of the response must

vanish; that is, 0a λ′ ′= = . It follows from Equations (2.14) and (2.17) that

2

0 0

(1 ) sin( ) cos( )2 4 2 2a a aγ γλ σ λ

ω ω− = − = −

(2.19,2.20)

Squaring and adding Equations (2.19) and (2.20) yields the following frequency-response

equation: 2 2

2 20

0

( ) 4 where4 4

aγρ ω ρ σ ρ ρω

− + = =

(2.21)

In Figure 2.9, we compare the approximate solution with that obtained by numerically

integrating Equation (2.5). The agreement is excellent.


15

Figure 2.9: Comparison of the approximate solution with the numerical solution for an initial

condition of 0x = 0.01, ε =0.1

Figure 2.10: Frequency-response curves for primary resonances of the van der Pol Oscillator

for various forcing amplitudes of γ ( 0ω = 10.3 rad/s)

In Figure 2.10, we show frequency-response curves generated using Equation (2.21)

presented in terms of the amplitude2

4aρ = for selected values of the forcing amplitudeγ . For

small γ , the curves consist of two branches: a branch runs close to the σ-axis and the second


16

branch is a closed curve that can be approximated by an ellipse having its center at the ρ-axis.

As γ increases, the ellipses expand, open, and coalesce with the first branch to form a single

branch of solutions. As γ increases further, the response curves are single-valued for all σ .

We note that, in the σ ρ− plane, the frequency-response curves, which are symmetric with

respect to the σ axis, have shapes similar to those of the force-response curves.

2.3.2. The Rayleigh Oscillator

Now, let us consider the Rayleigh oscillator in the case of a primary-resonance excitation

( 0( ) , ( )O Oγ ε ω ε= Ω = + ). The dynamics are described by the following equation:

220

² 1 ( )²

d x dx dx x f tdt dt dt

ε ω⎛ ⎞⎛ ⎞− − + =⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

0( ) cos( ),f t tγε ω εσ= Ω Ω = +

(2.22)

An approach similar to that followed in the preceding section is used to determine the

folowing frequency-response equation of this oscillator: 2 2

2 2 2 2 03(2 ) 4 where4 1/ 2

aρ ωε ρ η ρ σ ρ γ

η⎧ =

− + = ⎨=⎩

(2.23)

In Figure 2.11, we compare the approximate solution with that obtained by numerically

integrating Equation (2.22). The agreement is excellent.

Figure 2.11: Comparison of the approximate solution with that obtained by integrating

original equation for an initial condition of 0x = 0.01, ε =0.1


17

Frequency-response curves generated from Equation (2.23) are presented in Figure 2.12.

Figure 2.12: Frequency-response curves for primary resonances of the Rayleigh oscillator for various forcing amplitudes γ ( 0ω = 1 rad/s and ε = 0.01)

2.4. Parametric Study for Signal Filtering

Here, we examine the effect of varying the detuning σ between the forcing frequency and the

natural frequency of the oscillator on the responses of the two oscillators. Through the use of

feedback control, the detuning parameter plays a key role in shaping the time and frequency-

responses of both oscillators.

Figures 2.13 and 2.14 display the time responses of the van der Pol and Rayleigh oscillators,

respectively, as σ takes different values ( )0.01, 1, 10, 100± ± ± while the remainder of the

system parameters are held constant 0.01ε = , 0 5ω = rad/s, and 1γ = . For both oscillators, it

can be observed that as σ increases from 0.01 to 100, the amplitude of oscillations gets

smaller. Therefore, using a forcing frequency closely matched to the oscillator’s natural

frequency (i.e., 0ω ) using a smallσ yields a more sensitive oscillator signal.


18

10 20 30 40 50

-1

-0.5

0.5

1

ε =0.01, 0ω =5rad/s, γ =1, σ =0.01

10 20 30 40 50

-0.75

-0.5

-0.25

0.25

0.5

0.75

ε =0.01, 0ω =5rad/s, γ =1, σ =±1

10 20 30 40 50

-0.15

-0.1

-0.05

0.05

0.1

0.15

ε =0.01, 0ω =5rad/s, γ =1, σ =±10

10 20 30 40 50

-0.01

-0.005

0.005

0.01

ε =0.01, 0ω =5rad/s, γ =1, σ =±100

Figure 2.13: Time response of the van der Pol oscillator for various values of σ

10 20 30 40 50

-4

-2

2

4

ε =0.01, 0ω =5rad/s, γ =1, σ =0.01

10 20 30 40 50

-2

-1

1

2

ε =0.01, 0ω =5rad/s, γ =1, σ =±1

10 20 30 40 50

-0.2

-0.1

0.1

0.2

ε =0.01, 0ω =5rad/s, γ =1, σ =±10

10 20 30 40 50

-0.015

-0.01

-0.005

0.005

0.01

0.015

ε =0.01, 0ω =5rad/s, γ =1, σ =±100

Figure 2.14: Time response of the Rayleigh oscillator for various values of σ


19

2.5. Summary We reviewed and discussed the nonlinear dynamics of two self-excited oscillators subjected

to external harmonic forces. We used the method of multiple scales to determine two

nonlinear ordinary-differential equations describing the modulation of the amplitudes and

phases of these oscillators. These equations were used to calculate the frequency-response

equations and generate frequency-response curves and compute the resonance frequencies.

The frequency-response equations make enable the design of feedback controllers for tuning

the oscillators. For this purpose, the following chapter synthesizes a feedback controller for a

clamped-clamped microbeam resonator to make it behave like either the van der Pol or the

Rayleigh oscillator, which exhibit attractive features, as shown in this chapter.

Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 20

Chapter 3.

Control and Dynamics of an Electrostatically

Actuated Clamped-Clamped Microbeam Resonator

Most MEMS devices are actuated using electrostatic forces. Among the most commonly used

types of actuators are the parallel or lateral plate actuators. Nevertheless, electrostatic

actuation has some limitations due to its nonlinear nature. This work presents a novel control

design that regulates the response of a microresonator. The feedback is primarily used to

cause the microbeam to behave like either the van der Pol or the Rayleigh oscillator whose

dynamical features are examined in the preceding chapter. In particular, we consider a

resonator formed of an electrostatically actuated clamped-clamped microbeam. We introduce

feedback, as a secondary input to the microbeam, to regulate its behavior and drive it to

mimic the response of the two oscillators discussed in Chapter 2.

3.1. Modeling of a Clamped-Clamped Microbeam Resonator

In this section, a dynamic model of a clamped-clamped electrostatically actuated microbeam

with a feedback input is presented. This model exhibits the main characteristics that can be

found in a large number of MEMS devices, which rely on electrostatic actuation. The analysis

of the different participating terms is presented separately to address each aspect of the system

dynamics.

In MEMS devices, the basic structure is the beam. This mechanical component and its

extension, the plate, constitute the majority of MEMS sensors and actuators. Consequently,

the first step to analyze the behavior of any device is to understand and model the dynamic

characteristics of a beam. For this, we consider the clamped-clamped microbeam shown in

Figure 3.1.


Figure 3.1: Schematic of a transversely deflected clamped-clamped microbeam resonator

The nonlinear dynamics of an electrically actuated microbeam and associated boundary

conditions can be described as follows [3 and 46]:

( )( )( ) ( )

224 2 2DC

4 2 2 20

,2 2 ( )

l V v tw w E A w w bE I A N t dx u x tl xx t x d w

ερ⎡ ⎤ +′∂ ∂ ∂ ∂⎛ ⎞′ + = + + +⎢ ⎥⎜ ⎟∂∂ ∂ ∂ −⎝ ⎠⎢ ⎥⎣ ⎦

∫

(0, ) ( , )(0, ) ( , ) 0, 0w t w l tw t w l tx x

∂ ∂= = = =

∂ ∂

(3.1)

where w is the deflection amplitude, ρ is the beam density, b and h are, respectively, the

width and height of the beam section, l is the beam length, E is Young’s modulus and I is the

second moment of area, ( )N t is an axial force, 21EEν

′ =−

is the modified Young modulus

of elasticity, ν is Poisson’s ratio, and DCV and ( )v t are the DC and AC voltages,

respectively. We note that the microbeam dynamics depend on five elements: the beam

resistance to bending, inertia, beam stiffness due to the externally applied axial load, midplane

stretching, and external force input, including the electrostatic force the term

( )( )2DC

22 ( )V v tb

d wε +

− derived assuming parallel-plate theory and complete overlap of the areas of

the microbeam and the stationary electrode.

For convenience, we introduce the following nondimensional variables:

ˆ , , w x twd l T

ξ τ= = =

(3.2)


where T is a time constant defined by '

4

E ITAlρ

= . Rewriting Equation (3.1) in

nondimensional form, we have

where

xl

ξ =

ˆ wwd

=

'

4

E ItAl

τρ

=2

'ˆ clc

E I Aρ=

2

1 6 dh

α ⎛ ⎞= ⎜ ⎟⎝ ⎠

40

2 3 ' 36 ld E h

εα =4

3 '

lE Id

α =

2

'ˆ lN N

E I=

Next, we consider two types of feedback input: point force and distributed force feedback.

3.2. A One-Point Feedback Let the feedback input be concentrated at the middle point of the microbeam; that is,

In this study, we propose a feedback input that causes the microbeam to behave like the van

der Pol and Rayleigh oscillators. By inspecting the dynamic equations of these oscillators

(Equations (2.2) and (2.4)), we can simply add the linear and nonlinear terms that are missing

using feedback. To this end, we propose to use the following form of ( )u τ :

Therefore, the closed loop dynamics becomes

( )( )

( )222 4 21

DC1 2 32 4 2 2

0

ˆ ˆ ˆ ˆ ˆˆˆ ˆ , ˆ1

V vw w w w wc d N uw

τα ξ α α ξ τ

τ ξτ ξ ξ

⎡ ⎤ ⎡ ⎤+⎛ ⎞∂ ∂ ∂ ∂ ∂ ⎣ ⎦+ + = + + +⎢ ⎥⎜ ⎟∂ ∂∂ ∂ ∂ −⎢ ⎥⎝ ⎠⎣ ⎦∫

[ ]0,1ξ ∈

(3.3)

( ) ( ) ( ) ( )ˆ ˆˆ ˆ0, 1, 0, 0, 1, 0w ww wτ τ τ τξ ξ∂ ∂

= = = =∂ ∂

(3.4)

( ) ( ) 1ˆ ˆ,2

u uξ τ τ δ ξ⎛ ⎞= −⎜ ⎟⎝ ⎠

(3.5)

( ) 3 21 2 3

1 1 1 1ˆ ˆ ˆ ˆ ˆ, , , ,2 2 2 2

u k w k w k w wτ τ τ τ τ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(3.6)


( )( )

222 4 21DC

1 2 22 4 20

3 21 2 3

ˆ ˆ ˆˆ ˆ ˆˆˆ1

1 1 1 1 1ˆ ˆ ˆ ˆ, , , ,2 2 2 2 2

V vw w ww wc d Nw

K w K w K w w

τα ξ α

τ τ ξ ξ ξ

τ τ τ τ δ ξ

⎡ ⎤ ⎡ ⎤+⎛ ⎞∂ ∂ ∂∂ ∂ ⎣ ⎦+ + = + +⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂ ∂ −⎢ ⎥⎝ ⎠⎣ ⎦⎧ ⎫⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − − −⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎩ ⎭

∫

[ ]0,1ξ ∈

(3.7)

( ) ( ) ( ) ( )ˆ ˆˆ ˆ0, 1, 0 0, 1, 0w ww wτ τ τ τξ ξ∂ ∂

= = = =∂ ∂

(3.8)

where 1 3 1K kα= , 2 3 2 K kα= , and 3 3 3 K kα= . At this stage, it should be emphasized that the

proposed control design aims at converting a microbeam filter to either the van der Pol or the

Rayleigh oscillator by adjusting the feedback gains. In other words, the microbeam resonator

can be tuned to meet certain specifications by properly selecting the constants 1 2 3, , and K K K .

3.2.1 Analysis

In order to attain this objective, we apply the method of multiple scales to determine a second-

order approximate solution to Equations (3.7) and (3.8). To this end, we define

20 0 1 1 2 2

0 1 2

T D T D T DT T T

τ ε τ ε τ∂ ∂ ∂= = = = = =

∂ ∂ ∂ (3.9)

2ˆ ˆc cε= ( ) 3AC cos( )v Vτ ε τ= Ω

( )2

1K O ε=

( )2 1K O=

( )3 1K O=

(3.10)

We seek a solution in the following form

( )( ) ( ) ( )2 3

1 0 2 2 0 2 3 0 2

ˆ , , ( ) ( , )

( ) , , , , , , ...s

s

w w u

w u T T u T T u T T

ξ τ ε ξ ξ τ

ξ ε ξ ε ξ ε ξ

= +

= + + + + (3.11)

Let

( ) ( )( ) ( ) ( )1

0

,f g f g dξ ξ ξ ξ ξ′ ′Γ = ∫

Substituting Equations (3.9) and (3.11) into Equations (3.7) and (3.8) and equating like

powers of ε, we obtain


• Order 0ε : (static equation)

( )( )

2'' 2 DC

1 2ˆ , 0

1s s s s s

s

Vw Nw w w w

wα

α′′′′ ′′− − Γ − =−

( ) ( ) ( ) ( )0 1 0, 0 1 0s s s sw w w w′ ′= = = =

(3.12)

• Order 1ε :

( ) ( ) ( )( )

22 2 DC

1 1 1 1 1 0 1 1 1 13

2ˆ , 2 , 01

s s s ss

Vu u Nu w w u D u w u w uw

αα α′′′′ ′′ ′′ ′′= − − Γ + − Γ − =−

L

(3.13)

• Order 2ε :

( ) ( ) ( )( )

2'' '' 22 DC

2 1 1 1 1 1 1 14

3, 2 ,1

s ss

Vu u u w w u u uw

αα α= Γ + Γ +−

L

(3.14)

• Order 3ε :

( ) ( ) ( ) ( )

( )( )

( ) ( )

3 1 1 2 1 2 1 1 1 2

2 DC AC1 1 1 1 0 2 1 0 1 2

2 23 3 22 DC 2 DC

1 2 1 1 0 1 2 0 1 3 1 0 14 5

2 , 2 , 2 ,

2ˆ, 21

6 4 1( ) ( )21 1

s s s

s

s s

u w u u w u u u u w

V Vu u u D D u cD uw

V Vu u u K D u K D u K u D uw w

α α α

αα

α α δ ξ

′′ ′′ ′′= Γ + Γ + Γ

′′+ Γ − − +−

⎡ ⎤+ + + − − −⎣ ⎦− −

L

(3.15)

The solution of Equation (3.13) is assumed to consist of only the directly excited mode.

Accordingly, we express 1u as

( )0 01 0 2 2 2( , , ) ( ) ( )i T i Tu T T A T e A T eω ωξ φ ξ−⎡ ⎤= +⎣ ⎦

(3.16)

where 2( )A T is a complex-valued function, the over bar denotes the complex conjugate, and

ω and ( )φ ξ are the natural frequency and corresponding eigenfunction of the directly excited

mode, respectively. Substituting Equation (3.16) into Equation (3.14), we obtain

0

0

222

2 1 14

222 22

1 14

222 22

1 14

3( ) 2 2 ( , ) ( , )(1 )

3 2 ( , ) ( , )(1 )

3 2 ( , ) ( , )(1 )

DCs s

s

i T DCs s

s

i T DCs s

s

Vu AA w ww

VA e w ww

VA e w ww

ω

ω

α φ α φ φ α φ φ

α φ α φ φ α φ φ

α φ α φ φ α φ φ−

⎛ ⎞′′ ′′= + Γ + Γ⎜ ⎟−⎝ ⎠

⎛ ⎞′′ ′′+ + Γ + Γ⎜ ⎟−⎝ ⎠

⎛ ⎞′′ ′′+ + Γ + Γ⎜ ⎟−⎝ ⎠

L

(3.17)


Equation (3.17) can be rewritten as

( )0 02 22 22( ) 2 ( )i T i Tu AA A e A e hω ω ξ−= + +L

(3.18)

where 2

22 DC1 14

3( ) 2 ( , ) ( , )(1 ) s s

s

Vh w ww

αξ φ α φ φ α φ φ′′ ′′= + Γ + Γ−

(3.19)

The solution of Equation (3.18) can be expressed as follows:

( ) ( ) ( )0 02 22 22 0 2 1 2 2 2 2 1 2( , , ) ( ) 2 ( ) ( ) ( )i T i Tu T T A T e A T A T A T eω ωξ ψ ξ ψ ξ ψ ξ −= + +

(3.20)

where 1ψ and 2ψ are the solutions of the following boundary-value problems:

( ) ( ) 1, 2i iM hψ ωδ ξ=

0 and 0 at 0 and 1, 1, 2j j jψ ψ ξ ξ′= = = = =

(3.21)

δij is the Kronecker delta and the linear differential operator ( , )M ψ ω is defined by

( )( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( )( ) ( )( )( )

( )

21

22 DC

1 3

, ,

22 ,1

s s

s s

s

M w w

Vw ww

ψ ξ ω ψ ξ ω ψ ξ α ξ ξ ψ ξ

αα ξ ψ ξ ξ ψ ξξ

′′′′ ′′= − + Γ

′′− Γ −−

(3.22)

In order to describe the nearness of the excitation frequency Ω to the fundamental natural

frequency ω , we introduce a detuning parameter σ defined by 2ω ε σΩ = +

(3.23)

Substituting Equations (3.16), (3.20), and (3.23) into Equation (3.15) and keeping the terms

that produce secular terms, we obtain

( )( ) ( )

( )0

2

21

33 3 2 2 AC DC

3 2 2

1ˆ 22

21( 3 ) ( )2 (1 )

i T

i T

s

ic A iK A i A A Au e CC NST

V VK K i A A ew

ω

σ

ω ω δ ξ ω φ ξ χ ξ

αω ω φ ξ δ ξ

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

− + − + ⎟⎜ ⎟−⎝ ⎠

L

(3.24)

where A′ denotes the derivative of A with respect to 2T , CC denotes the complex conjugate

of the preceding terms, NST stands for the terms that do not produce secular terms, and ( )χ ξ

is defined by


( ) ( )( ) ( )

( ) ( ) ( )

1 1 1 1 2

1 1 1 2 1 1 1 2

2 2 232 DC 2 DC 2 DC

1 25 4 4

3 ( , ) 2 ( , ) 4 ( , )

2 ( , ) 4 ( , ) 2 ( , ) 4 ( , )

12 6 121 1 1

s s

s s s

s s s

w w

w w w

V V Vw w w

χ ξ α φ φ α ψ α ψ φ

α φ ψ α φ ψ α φ ψ α φ ψ

α α αφ φψ φψ

′′= Γ + Γ + Γ

′′ ′′ ′′+ Γ + Γ + Γ + Γ

+ + +− − −

(3.25)

Multiplying the right-hand side of Equation (3.24) by ( ) 0i Te ωφ ξ − and integrating the result

from 0ξ = to 1ξ = yields the solvability condition

( ) 2

12 2

1

01

2 3 42 3 2 AC DC 2

0

1ˆ 2 ( )2

13 ( ) 2 02 (1 )

i T

s

ic A i A i AK A A d

A Ai K K V V e dw

σ

ω ω ω φ φχ ξ

φω ω φ α ξ

⎛ ⎞′− − + +⎜ ⎟⎝ ⎠

− + + =−

∫

∫

(3.26)

where φ is normalized such that 1

2

0

1dφ ξ =∫ .

Next, we express A in the polar form 12

iA ae β= , where ( )2a a T= and ( )2Tβ β= are real-

valued functions, representing, respectively, the amplitude and phase of the response.

Substituting for A in Equation (3.26) and letting 2Tγ σ β= − , we obtain

( )

( )

2

1 13

2 AC DC 2

0 0

2 3 3 41 3 2

1 1ˆ 22 81

1 1 1 1( ) 3 ( ) 02 2 8 2

i Ti i

s

i i i i

ic e a V V e d e a dw

i K e a K K ie a i e a e a

σβ β

β β β β

φω α ξ φχ ξ

ω φ ω ω φ ω ω β

− + +−

′ ′+ − + − + =

∫ ∫ (3.27)

Separating the real and imaginary parts in Equation (3.27), we obtain the following

modulation equations:

( )1

2 2 4 32 AC DC1 3 22

01 1

32 AC DC2

0 0

21 1 1 1ˆ( ) sin 3 ( )2 2 (1 ) 8 2

2 1cos(1 ) 8

s

s

V Va K c a d K K aw

V Va a d a dw

α φφ γ ξ ω φω

α φγ σ γ ξ χφ ξω ω

⎛ ⎞′ = − + − +⎜ ⎟ −⎝ ⎠

′ = + +−

∫

∫ ∫

(3.28)

(3.29)

Substituting Equations (3.16) and (3.20) into Equation (3.11) and setting ε = 1, we obtain, to

the second approximation, the following microbeam response to the external excitation:

( ) ( ) ( ) ( )21 2

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

u a aξ τ τ γ φ ξ ψ ξ τ γ ψ ξ⎡ ⎤= Ω − + Ω − + +⎣ ⎦ (3.30)


It follows from Equation (3.30) that periodic solutions correspond to constant a and γ ; that

is, the fixed points ( )0 0,a γ of Equations (3.28) and (3.29). Thus, letting 0 and 0aγ ′ ′= = in

Equations (3.28) and (3.29), we obtain

( )2 2 31 0 0 2 3 0

30 0 0

1 1 1ˆ( ) sin 3 02 2 8

1cos 08

FK c a K K a G

F a S a

φ γ ωω

γ σω ω

⎧ ⎛ ⎞− + − + =⎜ ⎟⎪⎪ ⎝ ⎠⎨⎪ + + =⎪⎩

(3.31)

where

1 14

2 AC DC 20 0

12 ( ) 2(1 )s

F V V d G S dwφα ξ φ φχ ξ= = =

−∫ ∫ (3.32)

Eliminating 0γ from Equations (3.31), we obtain the following frequency-response equation:

( )2 222

2 2 2 200 1 2 3 02

1 1 1 1ˆ( ) 38 2 2 2 8a SF a K c K K a Gσ φ ω

ω ω

⎛ ⎞⎛ ⎞ ⎡ ⎤⎜ ⎟= + + − − +⎜ ⎟ ⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠⎝ ⎠ (3.33)

By inspecting Equation (3.33), we note that the feedback gains have been lumped with the

inherent viscous damping in the system into an overall effective damping term. This is a

direct result of the feedback law used, which calls for derivative feedback.

Next, we evaluate numerically the parametersω , φ , 1ψ , 2ψ , and sw associated with Equation

(3.33) using the Differential Quadrature Method (DQM) (see Appendix). Once these

parameters are computed, frequency-response curves can be generated.

3.2.2 Simulations

To describe the dynamic response of the microbeam, we need to determine the natural

frequencyω , the excitation amplitude F, the effective nonlinearity of the system S, and the

viscous damping coefficient c . As a first step, Equation (3.12) is numerically integrated using

DQM to determine the static deflection sw for a given DC voltage. Using the static

solution sw , we solve the boundary-value problem, ( ), 0M φ ω = , using DQM for the

fundamental natural frequency ω and its corresponding eigenfunction φ . Next, we solve the

two boundary-value problems in Equations (3.21) and (3.22) to evaluate the functions 1ψ and

2ψ using DQM. Finally, we evaluate χ , S, F, and G from Equations (3.25) and (3.32).


We consider the microbeam shown in Figure 3.1 with the geometric and physical

characteristics given in Table 3.1. Figure 3.2 shows variation of the mid-point deflection of

Table 3.1. The clamped-clamped microbeam parameters

L b H d N c E'

510 µm 100 µm 1.5 µm 1.18 µm 1.561 10−4 N 0.0239511 166 GPa

the microbeam maxw with the DC voltage. We used DQM to solve the static equation using 11

grid points. The stable (lower) branch and the unstable (upper) branch meet at a saddle-node

bifurcation at the static pull-in instability DCV ≈ 4.8 Volts, resulting in the destruction of both

branches. This static analysis shows that MEMS resonators should be designed to operate

below this value, which serves as an upper bound of the stability limit of the resonator.

Figure 3.2: Equilibria of an electrostatically actuated microbeam (Bifurcation Diagram)

Next, we study the dynamic behavior of the microbeam under an AC harmonic excitation near

its fundamental frequency 1ω = 23.9. We assume a quality factor Q = 1000, which is related

to the damping coefficient by 1cQω

= . Figure 3.3 shows the frequency-response curves

corresponding to 1DCV = Volt and 0.01, 0.02, 0.05, and 0.1 ACV = Volt. The results are

obtained by solving Equation (3.33). It can be seen that, as the AC voltage gets larger, the

frequency-response curve is bent to the right with a noticeable increase in amplitude.


Figure 3.3: Frequency-response curves when VDC = 1 Volt, K1=0, K2=0, and K3=0

Then, we examine the effect of the DC voltage on the frequency response. Figure 3.4 shows

the curves corresponding to ACV = 0.05 Volt and DCV = 0.1, 0.5, 1, and 2 Volts. Similarly, the

frequency-response curve is bent to the right with higher amplitudes as the DC voltage

increases.

Figure 3.4: Frequency-response curves when VAC =0.05 Volt, K1=0, K2=0, and K3=0

Figure 3.5 shows that, for higher values of the DC voltage, the frequency-response curves are

bent to the left, indicating a change in the effective nonlinearity from a hardening type to a

softening. The effective nonlinearity vanishes at 3.27 Volt, and hence the frequency-response

curve, shown in Figure 3.5, is nearly linear.



Now, we examine the influence of the axial force and damping coefficient on the microbeam.

Figure 3.6 displays the frequency-response curves corresponding to ACV = 0.05 Volt and DCV =

1.5 Volts while varying the nondimensional axial load N and damping coefficient c .

Increasing the axial load from negative values (compressive) to positive values (tensile) shifts

the natural frequency 1ω and translates the whole frequency-response curve to the right.

Decreasing the damping allows the frequency-response curve to climb up along the backbone

curve and as a result increases the maximum amplitude and shifts the nonlinear resonance

frequency to the right (for hardening-type nonlinearity).

Figure 3.6: Frequency-response curves when VAC =0.05 Volt, VDC =1.5 Volt, K1=0, K2=0, and K3=0

The effect of varying the feedback gains ( )2

1K O ε= and ( )2 1K O= on the frequency

response is shown in Figures 3.7 and 3.8 with ACV = 0.05 Volt and DCV = 1 Volt. In the

absence of the cubic feedback term 2 0K = , the linear feedback term acts as negative

damping. Increasing 1K has an identical effect to decreasing the value of the damping

coefficient c . It yields higher amplitudes and shifts the nonlinear resonance frequency to the


right. The mathematical foundations of this behavior can be attributed to the structure of the

effective damping term in Equation (3.33). On the other hand, the cubic feedback term acts as

positive damping, in the absence of the linear feedback term 1 0K = . Increasing 2K pushes the

frequency-response curve down on the backbone curve, bringing down the amplitude and

shifting the nonlinear resonance frequency to the left.

Figure 3.7: Frequency-response curves when VDC =1 Volt, VAC =0.05 Volt, K2=0, and K3=0



3.3. A Fully Distributed Feedback

We also consider the control of a microbeam resonator using distributed-force feedback. In

practical situations, the controller is realized by depositing on the microbeam a piezoelectric

patch, which stretches over at least 20% to 30% of the beam length, and thus it cannot be

approximated by a point load. For this, we propose adding the feedback signal to the signal

used to actuate the beam via the electrode underneath the beam. In other words, we let the

feedback be applied over the whole length of the beam. This is similar to the case of

generating a DC and an AC field between the stationary electrode and microbeam. It is also

possible to collocate the actuation force and feedback force at the same electrode.

Therefore, let the distributed feedback input be given by

Hence, the closed-loop dynamics becomes

where 1 3 1 2 3 2 and K k K kα α= = .

3.3.1. Analysis Again, we apply the method of multiple scales to approximate the solutions of Equations

(3.35) and (3.36). By taking advantage of the definitions given in Equations (3.9) and (3.10),

we seek a solution in the following form:

( ) ( ) ( )( ) ( ) ( ) ( )2 3

1 0 2 2 0 2 3 0 2

ˆ , , ,

, , , , , , ...s

s

w w u

w u T T u T T u T T

ξ τ ε ξ ξ τ

ξ ε ξ ε ξ ε ξ

= +

= + + + + (3.37)

Similarly, we adopt the same procedure as for the case of one-point feedback. As a result the

fixed-point problem of the modulation Equations (3.31) and (3.32) is modified as follows:

( ) ( ) ( ) ( ) ( )3 21 2 3ˆ ˆ ˆ ˆ, , , , ,F k w k w k w wξ τ ξ τ ξ τ ξ τ ξ τ= − −

(3.34)

( )( )

22 4 21

12 4 20

2DC 3 2

2 1 2 32

ˆ ˆ ˆˆ ˆ ˆˆ

ˆ ˆ ˆ ˆˆ1

w w ww wc d N

V v tK w K w K w w

w

α ξτ τ ξ ξ ξ

α

⎡ ⎤⎛ ⎞∂ ∂ ∂∂ ∂+ + = +⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

⎡ ⎤+⎣ ⎦+ + − −−

∫

(3.35)

( ) ( ) ( ) ( )ˆ ˆˆ ˆ0, 1, 0, 0, 1, 0w ww wτ τ τ τξ ξ∂ ∂

= = = =∂ ∂

(3.36)


( ) ( )2 31 0 0 2 3 0

30 0 0

1 1ˆ sin 3 02 8

1cos 08

FK c a K K a G

F a S a

γ ωω

γ σω ω

⎧ − + − + =⎪⎪⎨⎪ + + =⎪⎩

(3.38)

where: 1 1 1

42 AC DC 2

0 0 0

2 (1 )s

F V V d G d S dwφα ξ φ ξ φχ ξ= = =

−∫ ∫ ∫ (3.39)

Then eliminating 0γ from the system (3.38), we obtain the following frequency-response

equation:

( )2 222

2 2 200 1 2 3 02

1 1 1ˆ 38 2 2 8a SF a K c K K a Gσ ω

ω ω

⎛ ⎞⎛ ⎞ ⎡ ⎤⎜ ⎟= + + − − +⎜ ⎟ ⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠⎝ ⎠ (3.40)

Next, we present simulations of the controlled frequency response of the microbeam using

distributed-force feedback.

3.3.2. Simulations

A numerical procedure similar to that used in Section 3.1.2.1 is adopted to produce the

results presented here. Figure 3.9 shows the frequency-response curves for DCV = 1.5 Volt and

ACV = 0.05 Volt. The results are obtained by solving Equation (3.33) for the point feedback

and Equation (3.40) for the distributed feedback. As expected, the amplitude becomes lower

when the distributed actuator is used since all points along the microbeam axis are less

compliant than the mid-span point where the point feedback is applied.

Figure 3.9: Frequency-response curves when VDC =1.5 Volt, VAC = 0.05Volt ,

K1=0.01, K2=1, and K3=0


The effects of varying the feedback gains ( )2

1K O ε= and ( )2 1K O= on the frequency

response are shown in Figures 3.10 and 3.11 with VAC = 0.05 Volt and VDC = 1 Volt. As

observed for the case of point feedback, the linear feedback term acts as negative feedback 1K ,

while the cubic feedback term 2K ( 1 0K = ) acts as positive damping.



Figures 3.12 displays the time response of the microbeam midpoint using distributed feedback

for various values of σ ( )0, 0.01, 0.05, 1± ± ± while all other parameters are held constant

at VDC =1 Volt, VAC =0.05 Volt, K1=0.01, and K2=1. It can be observed that, as σ decreases


from 1 to 0, the oscillation increases in size and it is modulated with another frequency within

a small domain (-0.03, 0.03) around zero detuning.

σ =±0.01 σ =0

σ =±1 σ =±0.05

Figure 3.12: Mid-point time response for VDC =1 Volt, VAC =0.05 Volt, K1=0.01, K2=1, and K3=0 for various values of the detuning parameter σ

The purpose of the current work is to design a highly sensitive mass sensor by adjusting the

dynamics of the microbeam resonator. Feedback control is primarily used to make a beam-

based electrostatic resonator behave like the van der Pol oscillator and then drive it to chaos

[17]. The chaotic oscillator is more sensitive to changes in the beam mass than a regular

resonant mass sensor. The van der Pol oscillator is adopted for this purpose since it has more

potential to go chaotic [17]. Therefore, the next set of simulations considers variations of the

feedback gains 1K and 3K with 2K =0.

The effect of varying the feedback gain 3K on the frequency response is shown in Figures

3.13 ( 3 0.1, 0.2, 0.4, and 0.5K = ) and 3.14 ( 3 1, 2, and 3K = ), where ACV = 0.05 Volt, DCV =

1 Volt, 1 0K = and 2 0K = . The behavior of the controlled microbeam resembles that of the

van der Pol oscillator (see Figure 2.9) where, for 3 0.1 and 0.2K = , the frequency-response


curve consists of two branches, one of them is an ellipse and is tilted to the right, indicating a

hardening-type behavior. For 3 0.4and 0.5 and higherK = , the frequency-response curve

becomes a single branch. We show later how these curves can be straightened upward so that

the microbeam acts like the van der Pol oscillator.



Figure 3.15 shows that, as 3K is varied while 1 0.01K = , the amplitude 0a is decreased and

the frequency-response curves (now all with a single branch) are bent more to the right.


Figure 3.15: Frequency-response curves when VDC =1 Volt, VAC =0.05 Volt, K1=0.01, and

K2=0

Varying 1K and keeping 3 1K = yields the frequency-response curves shown in Figure 3.16.

We note that increasing 1K while keeping 3K constant brings the frequency-response curve

down towards the backbone curve, thereby lowering the response amplitude and shifting the

nonlinear resonance frequency to smaller values.


Figure 3.17 shows the effect of varying the DC voltage on the microbeam frequency response

with ACV = 0.05 Volt, 1 0K = , 2 0K = , and 3 3K = . As the DC voltage increases, the number of

branches forming the frequency-response curve reduces from two to one.



For a large value of 3 179.6K = , Figure 3.18 shows the resulting frequency-response curves of

the controlled microbeam. This figure depicts a microbeam behavior that is similar to that of

the van der Pol oscillator (see Figure 2.9). This implies that it is possible to synthesize a set of

feedback gains that makes the response characteristics of a microbeam resemble those of the

van der Pol oscillator.

Figure 3.18: Frequency-response curves when VDC =1 Volt, VAC =0.01 Volt, K2=0, K3=179.6


3.4. Implementation of the Feedback Controller In macro-fabrication, a design issue of significance is to be able to fit additional electronic

components within the limited space of the chip characterizing the microdevice.

Figure 3.19: Analog implementation of the feedback controller. The AD633JNs are voltage

multipliers used to achieve the controller expression given in Equation (3.34).


Taking this issue into consideration, we propose an analog circuit, which is based on the

Analog Device AD633JN voltage multipliers, for building the proposed feedback controller

(see Figure 3.19). This figure depicts that measurements of the microbeam deflection and its

time derivative are needed for the construction of the feedback control, as suggested by the

structure given by Equation (3.34). These variables could be detected by an electronic

interface like the one proposed by Painter [37] and which outs both of these variables.

3.5. Summary We presented a novel control design that regulates the pass band of a microbeam resonator

whose principal component is an electrostatically actuated clamped-clamped microbeam. The

feedback is primarily used to render the microbeam behave like that of either the van der Pol

oscillator or the Rayleigh oscillator whose dynamic features are examined in Chapter 2. Using

the method of multiple scales, we derived two nonlinear ordinary-differential equations that

describe the modulation of the amplitude and the phase of the response with time. These

equations are used to approximate the nonlinear resonance frequencies and amplitudes of

limit-cycle solutions in the presence of either a one-point or a fully distributed feedback force.

In order to broaden the scope of applications of MEMS devices whose principal component is

a microbeam, we address in the next chapter the case of a gas microsensor (Zhou et al [52]).

For this application, we consider a microbeam at one end and coupled to an electrostatically

actuated microplate at the other end.. In this case, the microplate is more likely to experience

pull-in instability for lower DC voltages.

Chapter 4. Dynamics of a Cantilever Microbeam Resonator 41

Chapter 4.

Dynamics of an Electrostatically Actuated

Cantilever Microbeam Resonator There has been a major focus on modeling a cantilever microbeam, a principal component in

a large class of electrostatically actuated MEMS devices [12 and 20], which include gas

microsensors. In this Chapter, we develop a mathematical model for a microbeam resonator

fixed at one end and coupled to an electrostatically actuated microplate at its other end. The

developed model considers the microbeam as a continuous medium, the plate as a rigid body,

and the electrostatic force as a nonlinear function of the displacement and applied DC and AC

voltages. The dynamic behavior of the microbeam is regulated via an electrostatic field

underneath the microplate.

4.1. Mathematical Modeling The cantilevered microbeam shown in Figure 4.1 consists of an elastic beam clamped at one

end and coupled to a microplate at the other end. The microplate is actuated via an electrode

underneath it with a gap width d. The microbeam is modeled as a linear prismatic Euler-

Bernoulli beam of width a , thickness b , length L , Young’s modulus E is, density ρ ,

damping coefficient c, cross section area A ab= , and area moment of inertia 3 /12I ab= .The

microplate is modeled as a rigid body with rotational moment of inertia 213 CJ ML= about its

center of mass, where 2 CL is the length of the plate and M is its mass. The microplate center

of mass is located at C CL x L= − from the tip of the microbeam.

Figure 4.1: Schematic of the cantilever microbeam with a microplate at its end


For convenience, we consider a local coordinate s attached to the rigid microplate (see Figure

4.2).

Figure 4.2: Local coordinate system attached to the microplate

Next, we develop expressions for the kinetic and potential energies of the microbeam-

microplate system.

4.1.1. Kinetic Energy

We let ( , )w x t be the beam displacement at location x and time t and xwwx

∂=∂

and 2

2xxww

x∂

=∂

its slope and curvature at (x, t), respectively. Moreover, the rotation angle of the plate is the

same as the microbeam slope at its tip; that is, ( , )xw L t . Hence, the angular speed of the

microplate is ( , )txw L t . Consequently, the kinetic energy of the microbeam-microplate system

is given

( ) ( ) ( ) ( )( )2 22

0

1 1 1, , ,2 2 2

L

t t c xt xtT A w dx M w L t L w L t J w L tρ ⎡ ⎤= + + +⎣ ⎦∫ (4.1)

where the subscripts x and t denote respectively the partial derivatives with respect to x and t.

4.1.2. Potential Energy The potential energy due to the microbeam elastic deformation is given by

( )2

0

12

L

D xxV EI w dx= ∫ (4.2)

The electrostatic force applied underneath the microplate produces a potential energy that can

be expressed as


( )

( ) [ ]

( )

22

DC AC

0

2 2DC AC 0

2

DC AC

2 ( , ) ( , )

ln ( , ) ( , )2 ( , )

( , ) 2 ( , )ln

2 ( , ) ( , )

c

c

Lp

Ex

Lpx

x

p c x

x

a dsV V Vd w L t w L t s

aV V d w L t w L t s

w L ta d w L t L w L t

V Vw L t d w L t

ε

ε

ε

= − +− −

= + − −

− −⎡ ⎤= + ⎢ ⎥−⎣ ⎦

∫

(4.3)

where ε is the permittivity of free space and DCV and ( )ACV t are, respectively, the DC and

AC voltages applied between the electrode and microplate separated by the gap d w− .

4.1.3. Governing Equation of Motion

We now use Hamilton’s principle to derive the equation of motion and associated boundary

conditions. We have

2

1

0t

tL dtδ =∫ (4.4)

where δ is the variational operator

( ) ( ) ( )( )

( )( ) ( )

22

0

2 2

0

1 1 , ,2 2

1 1,2 2

B E ncL

t t c xt

L

xt x E nc

L T V V W w

A w dx M w L t L w L t

J w L t EI w dx V W w

ρ

= − − −

= + +

+ − − −

∫

∫

(4.5)

and nc tW cw= is the work due to damping. It can be easily shown that the resulting equation of

motion is

0xxxx t ttEIw cw Awρ+ + = (4.6)

subject to the following boundary conditions:

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

2

(0, ) 0(0, ) 0

, , ( ) , ,

, , , ,x

x

xx C tt C xtt E w

xxx tt C xtt E w

w tw t

EIw L t ML w L t ML J w L t V L t

EIw L t Mw L t ML w L t V L t

==

= − − + −

= + +

(4.7)

(4.8)

(4.9)

(4.10)


where

( ) ( ) ( ) ( ) ( ) ( )2

DC1 1,

2 , 2 , ,p

E ACwx c x

aV L t V V

w L t d w L w L t d w L tε ⎡ ⎤

= − + −⎢ ⎥− − −⎢ ⎥⎣ ⎦

( ) ( )( )( )

( )

( )( )

( )

2

DC2

2 ,( , ) 2 ,

,( , )2 , ln

( , ) 2 ,

x

c x

c xpE ACw

x

c x

L w L td w L t L w L ta

V L t V Vd w L tw L t

d w L t L w L t

ε

⎡ ⎤−⎢ ⎥− −⎢ ⎥= − + ⎢ ⎥⎛ ⎞−⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥− −⎝ ⎠⎣ ⎦

(4.11) (4.12)

These boundary conditions can be rewritten as

( ) ( ) ( )

( )( )( )

( )( )

( )( ) ( ) ( )

( ) ( )

2

2DC AC2

2DC AC

(0, ) 0(0, ) 0

, , ( ) ,

2 ,( , ) 2 ,

( )( , )2 , ln

( , ) 2 ,

, , ,

12 , (

x

xx C tt C xtt

c x

c xp

x

c x

xxx tt C xtt

p

x

w tw t

EIw L t ML w L t ML J w L t

L w L td w L t L w L ta

V V td w L tw L t

d w L t L w L t

EIw L t Mw L t ML w L t

aV V

w L t d w

ε

ε

==

= − − +

⎡ ⎤−⎢ ⎥− −⎢ ⎥+ + ⎢ ⎥⎛ ⎞−⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥− −⎝ ⎠⎣ ⎦

= +

− +− ( )

1, ) 2 , ( , )c xL t L w L t d w L t

⎡ ⎤−⎢ ⎥

− −⎢ ⎥⎣ ⎦

(4.13) (4.14)

(4.15)

(4.16)

where the moment of inertia of the microplate about its point of connection with the

microbeam is given by 2 243C CML J ML+ = .

We note that the microbeam dynamics depends on three factors: beam resistance to bending,

inertia due to movement, and electrostatic force.

For convenience, we introduce the following nondimensional variables:

where T is a time constant defined by 4ALT

EIρ

= . In nondimensional forms, Equations

(4.6) and Equations (4.13-4.16) become

ˆˆ ˆ w x tw x td L T

= = = (4.17)

[ ]ˆˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ0 0,1xxxxtt tw cw w x+ + = ∈ (4.18)


where 4

1 32pa L

EIdε

α = ˆ aad

= ˆ2 CLγ =

2

ˆ Lc cEI Aρ

= ˆ MMALρ

= ˆ CC

LLL

=

We decompose the microbeam deflection, under an electric force, into the sum of a static

component due to the DC voltage, denoted by ˆ( )sw x , and a dynamic component due to the

AC voltage, denoted by ˆˆ( , )u x t ; that is,

The static problem can be formulated by setting the time derivatives and AC forcing term in

Equations (4.18-4.19) equal to zero. The result is

( )( )

ˆ

2ˆ ˆˆ ˆ ˆ

2

1 DC AC ˆ2

ˆ ˆˆ

ˆ ˆˆ

ˆˆ (0, ) 0ˆˆ (0, ) 0

4ˆ ˆ ˆ ˆˆ ˆ ˆˆ ˆ ˆ(1, ) (1, ) (1, )3

( ) ˆ ˆˆ ˆ(1, ) 1 (1, )lnˆ ˆ ˆ ˆˆ ˆ ˆ ˆ1 (1, ) (1, ) 1 (1, ) (1, )ˆˆ (1, )ˆˆˆ (1, )

x

xx C Ctt xtt

x

x xx

xxx

w tw t

w t L Mw t L Mw t

V V t w t w tw t w t w t w tw t

w t M

α γγ γ

=

=

= − −

+ ⎡ ⎤⎛ ⎞−+ −⎢ ⎥⎜ ⎟− − − −⎢ ⎥⎝ ⎠⎣ ⎦

=

( )ˆ ˆˆ

2

1 DC AC

ˆ ˆ

ˆ ˆˆ ˆˆ ˆ(1, ) (1, )

( ) 1 1ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ(1, ) 1 (1, ) (1, ) 1 (1, )

Ctt xtt

x x

w t ML w t

V V t

w t w t w t w t

α

γ

+

+ ⎡ ⎤− −⎢ ⎥− − −⎣ ⎦

(4.19)

ˆ ˆˆ ˆ ˆ ˆ( , ) ( ) ( , )sw x t w x u x t= + (4.20)

[ ]ˆ 0 0,1sw x′′′′ = ∈ (4.21)

( )

21

2

21

(0) 0(0) 0

(1) 1 (1)(1) ln

1 (1) (1) 1 (1) (1)(1)

1 1(1)(1) 1 (1) (1) 1 (1)

s

s

DC s ss

s s s ss

DCs

s s s s

ww

V w ww

w w w ww

Vw

w w w w

α γγ γ

αγ

=′ =

⎡ ⎤′ ⎛ ⎞−′′ = −⎢ ⎥⎜ ⎟′ ′− − − −′ ⎢ ⎥⎝ ⎠⎣ ⎦

⎡ ⎤′′′ = − −⎢ ⎥′ ′− − −⎣ ⎦

(4.22)


4.2. Static Deflection

When an electric field is applied, electric charges are introduced and cause the microbeam-

microplate system to deflect. We examine this deflection by developing a closed-form

solution, from which the maximum range of travel and associated voltage are determined. The

general solution of Equation (4.21) can be expressed as

Using the first two boundary conditions in Equation (4.19) yields 0C D= = . The remaining

boundary conditions lead to the following nonlinear algebraic equations:

These equations are solved numerically for A and B .

4.2.1. Simulations The geometric and physical characteristics of the microbeam-microplate system used in the

simulations, shown in Figure 4.1, are given in Table 4.1. For a given ( )0,DC pV V∈ and

Table 4.1: Geometric and physical parameters of the microbeam-microplate system

L a b d E ρ

250 µm 5 µm 1.5 µm 4 µm 160 GPa 2300

pL pa pb ε

50 µm 20 µm 1.5 µm 8.85 10-12

DC pV V≠ , the solution to system (4.22) yields two distinct values for A and B , where pV is

the pull-in voltage.

Figure 4.3 displays variation of the static deflection of the microbeam-microplate system with

the applied DC voltage. It is composed of two branches: a lower branch and an upper branch.

The lower branch corresponds to stable equilibria, whereas the upper branch corresponds to

unstable equilibria. Figure 4.3 also shows that beyond a critical voltage pV , there are no

3 2ˆ ˆ ˆ ˆ( )sw x Ax Bx Cx D= + + + (4.23)

( )

21

2

21

(3 2 ) 16 2 ln1 (3 2 ) 1 (3 2 )3 2

1 16(3 2 ) 1 (3 2 ) 1

DC

DC

V A B A BA BA B A B A B A BA B

VAA B A B A B A B

α γγ γ

αγ

⎧ ⎡ ⎤⎛ ⎞+ − −+ = −⎪ ⎢ ⎥⎜ ⎟− − − + − − − ++ ⎝ ⎠⎪ ⎣ ⎦

⎨⎡ ⎤⎪ = − −⎢ ⎥⎪ + − − − + − −⎣ ⎦⎩

(4.24)


equilibria. This critical point, known as the pull-in point, corresponds to 8.295pV = Volts and

a maximum deflection of 0.2837 .

Figure 4.3: Variation of the static deflection with the DC voltage (Bifurcation Diagram)

One can immediately see that the maximum deflection associated with a cantilevered

microbeam is lower than that associated with a fixed-fixed microbeam (see Figure 3.2).

4.3. Natural Frequencies and Mode Shapes

Substituting Equation (4.20) into Equations (4.18-4.19) and expanding the nonlinear

electrostatic force using Taylor series about 0u = yields the dynamics of the microbeam-

microplate system about its static equilibrium:

ˆ ˆ ˆ ˆ ˆ ˆˆ 0xxxxtt tu cu u+ + = (4.25)

( )

( ) ( ) ( )( ) ( ) ( )

ˆ

2 2ˆ ˆˆ ˆ 1ˆ

2ˆ ˆˆ ˆ ˆ 1ˆ

ˆ0, 0ˆ(0, ) 0

4ˆ ˆ ˆ ˆˆ ˆ ˆ1, 1, 1, 13

ˆ ˆ ˆˆ ˆ ˆ1, 1, 1, 2

x

xx C C DCtt xtt

xxx C DCtt xtt

u t

u t

u t L Mu t L Mu t V EP

u t Mu t ML u t V EP

α

α

=

=

= − − +

= + −

(4.26)


where the terms involving 2

2ACV are dropped since

2 2

2 2AC DCV V and

4.3.1. Eigenvalue Problem

We drop the nonlinear forcing and damping terms in Equations (4.25), (4.26), and (4.27) and

obtain the following linear eigenvalue problem:

where

We solve Equations (4.29) and (4.30) for the mode shapes and corresponding natural

frequencies for a given static deflection ( )ˆsw x . To this end, we let

where ˆ( )xφ is the mode shape and ω is its corresponding nondimensional natural frequency.

Substituting Equation (4.32) into Equations (4.29) and (4.30) yields the following eigenvalue

problem:

( )( )

( )( )( )

( )( ) ( ) ( )

22

ˆ2 3 2

2

ˆ2 2 2

(1) 2 3 (1) 2 lnˆ ˆ1 1, 1,

1

2 1ˆ ˆ2 1, 1,

s s

x

s

sx

w wEP u t u t

w

wEP u t u t

χγ χ γ κγ κχκ κ

χ γ γγχ κ χκ

⎡ ⎤′ ′− + ⎢ ⎥⎣ ⎦= −′

′−= +

(4.27)

( ) ( ) ( )1 1 and 1 1 1s s sw w wχ κ γ ′= − = − − (4.28)

ˆˆˆ ˆ ˆ 0xxxx ttu u+ = (4.29)

( )( )( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( )

ˆ

2 2ˆ ˆˆ ˆ ˆ1 1 2ˆ

2ˆ ˆˆ ˆ ˆ ˆ1 3 1ˆ

ˆ0, 0

ˆ0, 0

4ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ1, 1, 1, 1, 1,3

ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ1, 1, 1, 1, 1,

x

xx C C DC xtt xtt

xxx C DC xtt xtt

u t

u t

u t L Mu t L Mu t V C u t C u t

u t Mu t ML u t V C u t C u t

α

α

=

=

= − − + +

= + − +

(4.30)

( ) ( )( )

( )( )( )( )

22

1 2 32 3 2 22

1 2 3 1 2 ln 2 1

1

s ss

s

w w wC C C

w

χγ χ γ κ χ γγ κ γχκ χ κκ

⎡ ⎤′ ′− − + ⎢ ⎥ ′−⎣ ⎦= = =′

(4.31)

( ) ( ) ˆˆˆ ˆ, i tu x t x e ωφ= (4.32)


The general solution of Equation (4.33) can be expressed as

where the coefficients iλ are functions of the applied voltage and β ω= . Using the first

two boundary conditions in Equation (4.34), we eliminate two of the unknowns, say 3λ and

4λ . This yields two linear algebraic equations in 1λ and 2λ , which can be written in the

following matrix form:

Setting the determinant of the 2 2× matrix M equal to zero leads to the characteristic

equation ( )det 0M = of the microbeam-microplate system.

2 0ivφ ω φ− =

(4.33)

( )( )

( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( )

2 2 2 21 1 2

2 2 21 3 4

0 0

0 04ˆ ˆ ˆ ˆ1 1 1 1 13

ˆ ˆ ˆ1 1 1 1 1

C C DC

C DC

L M L M V C C

M ML V C C

φ

φ

φ ω φ ω φ α φ φ

φ ω φ ω φ α φ φ

=

′ =

′′ ′ ′= + + +

′′′ ′ ′= − − − +

(4.34)

( ) 1 2 3 4cos sin cosh sinhx x x x xφ λ β λ β λ β λ β= + + + (4.35)

[ ] [ ]1 11 12

2 21 22

0 with =m m

M Mm m

λλ⎧ ⎫ ⎡ ⎤

=⎨ ⎬ ⎢ ⎥⎩ ⎭ ⎣ ⎦ (4.36)

( ) ( )( )( )

( ) ( )( )( )

( ) ( )

2 4 211 1 1

2 4 21 2

2 4 212 1 1

2 4 21 2

3 4 221 1

ˆ ˆcos cosh cos cosh

4 ˆ ˆsin sinh3

ˆ ˆsin sinh sin sinh

4 ˆ ˆcos cosh3

ˆsin sinh cos cosh

C DC

C DC

C DC

C DC

DC

m L M V C

L M V C

m L M V C

L M V C

m M V C

β β β β β β α

β β β β α


β β β β α


= + + − + −

⎛ ⎞+ +⎜ ⎟⎝ ⎠

= + + − + +

⎛ ⎞− +⎜ ⎟⎝ ⎠

= − − − − +( )( )( )( ) ( )( )

( )( )

3

4 21 1

3 4 222 1 3

4 21 1

ˆ ˆsin sinh

ˆcos cosh sin sinh

ˆ ˆcos cosh

C DC

DC

C DC

L M V C

m M V C

L M V C

β β β β α


β β β β α

+

+ +

= + − − + −

− +

(4.37)


4.3.2. Natural Frequencies Solving the characteristic equation, we obtain an infinite number of natural frequencies for a

given DC voltage. In Figure 4.4, we show variation of the first natural frequency 1ω with the

applied voltage. In Table 4.2 and Figure 4.5, we show the effect of varying the DC voltage on

the first five natural frequencies.

Figure 4.4: Variation of the first natural frequency with VDC

Table 4.2: Variation of the first five natural frequencies with VDC

DCV 0 2 4 6 8 8.2 8.3 8.5

2ω 13.1911 13.1809 13.1798 13.1773 13.1691 13.166 13.1609 13.1609

3ω 39.5393 39.5327 39.5324 39.5316 39.5291 39.5282 39.5267 39.5267

4ω 80.5697 80.5661 80.5659 80.5655 80.5643 80.5639 80.5632 80.5632

5ω 139.0069 139.0052 139.0052 139.0050 139.0044 139.0043 139.0040 139.0040

ω2

ω3

ω4

ω5

0

50

100

150

0 2 4 6 8 10

V DC

Figure 4.5: Variation of the first five natural frequencies with VDC


It follows from Figure 4.4 that increasing the applied DC voltage leads to a sharp drop in the

first natural frequency followed by the pull-in instability. The first five natural frequencies

(Table 4.2) and higher frequencies obtained by the exact solution are insensitive to the DC

voltage.

4.3.3. Mode Shapes

The natural frequencies iω found in the previous section can be substituted into the matrix

M and the linear system of Equations (4.36) and (4.37) can be solved to determine the mode

shapes. Figure 4.6 displays the first five mode shapes of the microbeam when ˆ 0.8M = :

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8 1φ

x

First mode 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1 2φ

x

Second mode

0.2 0.4 0.6 0.8 1

-0.5

0.5

1 3φ

x

Third mode

0.2 0.4 0.6 0.8 1

-1

-0.5

0.5

14φ

x

Fourth mode

0.2 0.4 0.6 0.8 1

-1

-0.5

0.5

1

0.2 0.4 0.6 0.8 1

-1

-0.5

0.5

1 5φ

x

Fifth mode

Figure 4.6: The first five mode shapes


4.3.4. Orthogonality Conditions

The orthogonality conditions can be derived from the governing equation (4.33) and boundary

conditions (4.34). Pre-multiplying Equation (4.33) by jφ and then integrating the outcome

from 0 to 1, we have

Integrating twice by parts the left term in Equation (4.38), we obtain

or

Because ( ) ( )0 0 and 0 0φ φ′= = , Equation (4.40) reduces to

Interchanging the indices i and j in Equation (4.41) yields

Subtracting Equation (4.42) from Equation (4.41), we obtain

Using the third and fourth boundary conditions in Equation (4.34), we simplify Equation

(4.43) to

1 12

0 0

ivi j i i jdx dxφ φ ω φφ=∫ ∫ (4.38)

1 11 1

2

0 00 0

i j i j i j i i jdx dxφ φ φ φ φ φ ω φ φ⎡ ⎤ ⎡ ⎤′′′ ′′ ′′ ′′′− + =⎣ ⎦ ⎣ ⎦ ∫ ∫ (4.39)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1

2

0 0

1 1 0 0 1 1 0 0i j i j i j i j i j i i jdx dxφ φ φ φ φ φ φ φ φ φ ω φ φ′′′ ′′′ ′′ ′′ ′′ ′′′ ′− − + + =∫ ∫ (4.40)

( ) ( ) ( ) ( )1 1

2

0 0

1 1 1 1i j i j i j i i jdx dxφ φ φ φ φ φ ω φφ′′′ ′′ ′′ ′′′− + =∫ ∫ (4.41)

( ) ( ) ( ) ( )1 1

2

0 0

1 1 1 1j i j i i j j i jdx dxφ φ φ φ φ φ ω φφ′′′ ′′ ′′ ′′′− + =∫ ∫ (4.42)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1

2 2

0

1 1 1 1 1 1 1 1i j i j i j i j j i j idxω ω φφ φ φ φ φ φ φ φ φ′′′ ′′ ′′′ ′′′ ′− = − − +∫ (4.43)


These orthogonality conditions can be used reduce the distributed-parameter problem into a

finite-degree-of-freedom system, which approximates the dynamic response of the

microbeam-microplate system.

4.4. Reduced-Order Model

In reducing the distributed-parameter problem, one can either work with the partial-

differential equations, boundary conditions, and orthogonality conditions, or work with the

Lagrangian. In this thesis, we use the latter approach. To this end, we express the Lagrangian

in nondimensional form and obtain

( ) ( ) ( )

( )( )

( )

1 2 22 2

ˆ ˆ ˆ ˆˆ ˆ

02

DC AC ˆ2ˆˆ 1

ˆ0

ˆ ˆˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ1, 1, 1,2 3

ˆ ˆ( ) ˆ ˆ1 (1, ) 1,ˆ ˆ lnˆ ˆˆ 1 (1, )1,

Ct t xt xt

Lx

xxx

MLL w dx M w t w t w t

V V t w t w tw dx R

w tw t

γ

γ

⎛ ⎞= + + +⎜ ⎟⎝ ⎠

⎛ ⎞+ − −− − ⎜ ⎟

⎜ ⎟−⎝ ⎠

∫

∫ (4.45)

where

4

1 3 3

12 pa LR

Eab dε

=

(4.46)

Using the Galerkin procedure, we approximate the system deflection as

where the ( ) ( )ˆ 1, 2, , i x i nφ = … are the mode shapes defined in Equation (4.35).

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )

1

2

0

1 1

2

0 0

4ˆ ˆ ˆ ˆ ˆ 1 1 1 1 1 1 1 1 03

when

1 1 1 1

when

i j i j C i j j i C i j

i i j i j i j i j

dx M L M L M

i j

dx dx

i j

φφ φ φ φ φ φ φ φ φ

ω φφ φ φ φ φ φ φ

′ ′ ′ ′+ + + + =

≠

′′′ ′′ ′′ ′′′= − +

=

∫

∫ ∫ (4.44)

( ) ( ) ( ) ( )1

ˆ ˆˆ ˆ ˆ ˆ,n

s i ii

w x t w x q t xφ=

= +∑ (4.47)


4.4.1. One-Mode Approximation

In this section, we consider a one-mode approximation; that is, Equation (4.47) reduces to

Substituting Equation (4.48) into Equation (4.45), normalizing the first mode shape such

that ( )1

21

0

ˆ ˆ 1x dxφ =∫ , adding the damping term, and writing down the Euler-Lagrange equations,

we obtain

where

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( )( )

1 2 2 211 1 1 1 1 1 10

1 12 211 1 10 0

1 1 1

10

ˆˆ ˆˆ ˆ 1 2 1 1 2 1 14

ˆ ˆ ˆ ˆ ˆ,

ˆ1 1 ˆ ˆ

C

ij

MM x dx ML

K x dx D c x dx

ccx dx

φ φ φ γφ φ γφ

φ φ

θ φ γφφ

⎡ ⎤ ⎡ ⎤′ ′′= + + + +⎣ ⎦ ⎣ ⎦

′′= =

′= + =

∫

∫ ∫

∫

and χ and κ are defined in Equation (4.28).

4.4.1.1 Response to Combined DC and AC Voltages

In order to examine the periodic response of the microbeam-microplate system to an AC

excitation, we use the Finite Difference Method (FDM) [32] to discretize an orbit whose

period is 2 /T π= Ω , where Ω is the excitation frequency. We discretize the orbit using

1m + points and enforce the periodicity condition 0 mq q= . Such a condition implies that the

first and last points of the orbit (points 0 and m ) are identical. Consequently, the orbit is

time-discretize using m equally-spaced points. At each of these points, we have

( ) ( ) ( ) ( )1 1ˆ ˆˆ ˆ ˆ ˆ, sw x t w x q t xφ= +

(4.48)

( ) ( )

( )( ) ( )( )

( ) ( )( )( ) ( ) ( ) ( )

( )( ) ( ) ( )( )( )

1

11 11 11 10

1 1 121 2

1 1

2 1 11

1 1 1 1 1

ˆ ˆ ˆ

ln / 1

2 1 1

1 1 1

2 1 1 1

j j j s

DC AC

s

sDC AC

s

M q D q K q w x x dx

q qR V V

q w

wR V V

q q w q

φ

κ θ χ φ

φ

χφ φγ

χ φ φ κ θ

′′ ′′+ + = −

⎡ ⎤− −⎣ ⎦+ +′ ′+

′ ′++ +

′ ′− + −

∫

(4.49)


( )( ), , ,

vp p

v vp p p DC AC p

q q

q f q q V V t

⎧ =⎪⎨

=⎪⎩

where 1, 2,...,p m= , ( ) ( ) ( ), , and v vp p p p p pt p t q q t q q t q t= Δ = = =

(4.50)

The FDM can now be applied to system (4.50) to yield a set of nonlinear algebraic equations.

In this case, a two-step explicit central-difference scheme is used to approximate the time

derivatives. Therefore, for an 1m + FDM-discretized orbit, the microbeam dynamics can be

approximated by a set of m nonlinear algebraic equations in m unknown displacements and

velocities. These equations can be solved for the unknowns using the Newton-Raphson

method. The stability of the orbits can then be ascertained by this combining the FDM

discretization with Floquet theory [29, 32]. While the method of characteristic exponents

requires the computation of the eigenvalues of an ( ) ( )3 x 3m n m n− − matrix, Floquet

theory estimates those of an ( ) ( )3 3n n− × − matrix. However, this theory requires the

integration of the associated ( )3n − vectors to determine the monodromy matrix, where n is a

given number, and thus to calculate the Floquet multipliers. We examine the frequency

response of the microactuator-microplate system, described in Table 4.1, and simulate its

response to the loading cases in Table 4.3.

Table 4.3: Loading cases

DCV Unstable fixed point ( )ˆˆ 1,w t 1ω ACV Q

Case 1 3.5 V 0.681662 1.473 0.1 V 300

Case 2 3.5 V 0.681662 1.473 0.5 V 300

Case 3 7.0 V 0.483067 1.238 0.1 V 300

The unstable fixed points ( )ˆˆ 1,w t are obtained by solving Equation (4.50) for sq by setting pq

and pq equal to zero and sq , respectively. The damping coefficient is determined using the

relation 1 /c Qω= , where Q is the quality factor. We simulate the frequency-response curve

associated with the maximum deflection of the microbeam tip maxˆ(1, )w w t= in the

neighborhood of 1ω while fixing the number of FDM time steps per period ( 100m = ) and

5n = . Figures 4.7 and 4.8 display the maximum deflection of the midpoint maxw for loading


cases 1 and 2 as the excitation frequency is varied in the neighborhood of the fundamental

natural frequency 1ωΩ ≈ . We apply Floquet theory to ascertain the stability of the periodic

solutions shown in both figures. We note that the frequency-response curve is composed of

four branches A, B, C and D. The solution is stable on branches A and B and unstable on

branches C and D.

Figure 4.7: Frequency-response curve of the microbeam for loading case 1



Figure 4.9 shows the frequency-response curve of the microbeam obtained for loading case 3.

In this case, the system locally and globally has a softening-type behavior. The frequency-

response curve is characterized by four branches of solutions: stable branches A and C and

unstable branches B and D. It can be observed that there is only one cyclic-fold bifurcation,

instead of three in the preceding case. Moreover, there is a period-doubling bifurcation point

separating branches C and D with one of the Floquet multipliers exiting the unit circle

through -1. The resulting two-period (2T) solution is initially stable but quickly loses stability

as the frequency is reduced, resulting in dynamic pull-in. Note that the grey dashed line in

Figure 4.8 denotes the limit of stability defined by the unstable static deflection (saddle).


4.4.1.2 Phase Portraits Early studies assumed that the fixed points and their basins of attraction are unperturbed by

the presence of an AC voltage. Other studies confirmed that the simulated orbits can cross the

fixed points limit without going to pull-in. Therefore, it is essential to investigate further this

issue by determining the basin of attraction of the sink (stable fixed point) for various values

of the AC voltage. We also examine the influence of the AC voltage on the location of the

fixed points.


In the absence of damping ( 0c = ) and forcing ( 0ACV = ), the fixed points are a center and a

saddle. The orbit that starts at the saddle and returns to the saddle is called a separatrix; it

consists of the union of a stable and unstable manifold of the saddle. The separatrix separates

regions of initial conditions that lead to periodic orbits around the center from initial

conditions that lead to unbounded motions. Using equation (4.49), we can determine

analytically the separatrix equation [32]. This analytical expression is obtained by letting

0c = and 0ACV = in equation (4.49). Using the fact that a stable fixed point is centered, we

integrate the resulting equation once and obtain

where

( ) ( )( ) ( )

( ) ( )( )( ) ( ) ( )( )

( )( ) ( ) ( )( )( )

( ) ( ) ( )

11

2 1 1 1 111 12

1 1 1 1 11 1

1 12 2

1 1 1 1 1

0 0

1 ln1 1 1 1

,1 1 11 1

ˆ 2ˆ ˆ ˆ ˆ ˆ

sDC DC AC

eff ss

seff eff

qq wRF q V V V dq

M q w q qw q

c q x dx q w x x dx qM M

κ θφχ φ γ χφ φ

χ φ φ κ θφ

φ φ

⎡ ⎤⎛ ⎞⎡ ⎤−′⎢ ⎥⎜ ⎟⎢ ⎥ ′ ′− +⎢ ⎥⎜ ⎟⎣ ⎦= + +⎢ ⎥⎜ ⎟′ ′− + −′ ′+⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟′′ ′′ ′′+ + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∫

∫ ∫and the integration constant h is obtained by evaluating Equation (4.51) at the original

unstable fixed points given in Table 4.3. In order to obtain the set of stable and unstable

orbits, we vary the constant h . The resulting curves are shown in Figure 4.10 for both loading

cases 1 and 2 and Figure 4.11 for loading case 3.

Figure 4.10: Phase portrait for loading cases 1 and 2 without damping and forcing

( )21 1

1 ,2 DCq F q V h+ =

(4.51)


Figure 4.11: Phase portrait for loading case 3 without damping and forcing

In these figures, the separatrix, stable orbits, and unstable orbits are, respectively, those curves

in grey, blue and red. The arrows define the stable and unstable manifolds of the saddle

marked `S'.

We now examine the system phase portraits in the presence of damping ( ˆ 0c ≠ ), but no

forcing ( ( )ˆ 0ACV t = ). In this case, the homoclinic orbit (separatrix) is destroyed and the center

becomes a focus `F', a point attractor. The stable manifold of the saddle defines the basin of

attraction of the focus. The locations of the saddle and focus do not change with damping

[32]. Forward and backward integrations in time of Equation (4.51) are used in conjunction

with initial conditions representing points in the vicinity of the saddle along the stable and

unstable manifolds, respectively, of the system linearized about the saddle. Figures 4.12 and

4.13 depict the basins of attraction for loading cases 1, 2 and 3, respectively. In these figures,

we increase damping to observe its effect on the basin of attraction. We note that, in all cases,

the stable and unstable manifolds do not intersect because of the presence of damping.


Figure 4.12: Phase portrait for loading case 1 and 2 with damping and without forcing

Figure 4.13: Phase portrait for loading case 3 with damping and without forcing

4.4.2. Multi-Mode Approximation

In this section, we consider a more accurate approximation using more than one mode. Such

an approximation enables examination of the convergence of the fixed points as the number

of modes n is increased. The simulation results are summarized in Table 4.4. The exact

values of the unstable fixed points, found in Section 4.5, are listed in the Table 4.5.


Table 4.4: Location of the unstable fixed point for different numbers of modes

1n = 2n = 3n = 4n =

Loading cases 1 and 2 0.681662 0.677972 0.677521 0.677494

Loading case 3 0.483067 0.481889 0.481785 0.481770

Table 4.5: Exact location of the unstable fixed point

Loading case 1 and 2 0.677413287

Loading case 3 0.481767898

It follows from Tables 4.4 and 4.5 that the use of 2 modes predicts accurately the location of

the unstable fixed point.

4.5. Summary We developed a mathematical model of an electrostatically actuated cantilever microbeam

coupled to a microplate. We derived closed-form solutions to the static and eigenvalue

problems associated with the microbeam-microplate system. The Galerkin method was used

to derive a set of nonlinear ordinary-differential equations that describes the microsystem

dynamics. We then employed FDM to discretize the orbits and solved the resulting nonlinear

algebraic equations to compute the periodic solutions.

Chapter 5. Conclusions and Recommendations for Future Research 62

Chapter 5.

Conclusions and Recommendations for Future

Research

5.1. Conclusions

The modeling, nonlinear dynamic analysis, and control design of two types of

electrostatically-actuated microbeams were the focal issues of this thesis. The first

microbeam, which characterizes a microresonator, is fixed at both ends and electrostatically

actuated along the microbeam span. The second microbeam, which represents a gas

microsensor, fixed at one end and coupled to an electrostatically actuated microplate at the

other end. In order to examine their static and dynamics behaviors, we developed reduced-

order models for both microbeams using the method of multiple scales and the Galerkin

method. We addressed the control design of the first microbeam for the purpose of enhancing

its nonlinear behavior. We presented a review of the nonlinear dynamics of the van der Pol

and Rayleigh oscillators, which posses attractive filtering features. We then presented a novel

control design that regulates the pass band of the fixed-fixed microbeam and derived

analytical expressions that approximate the nonlinear resonance frequency and amplitude of

the periodic solution subjected to both one-point and fully-distributed feedback forces. We

also derived closed-form solutions to the static and eigenvalue problems associated with the

second microbeam. The Galerkin method was used to derive a set of nonlinear ordinary-

differential equations that describe the microbeam-microplate dynamics. We then employed

the Finite Difference Method for discretizing the orbits to approximate the periodic solutions.

5.2. Recommendations for Future Research

This thesis addressed important issues that impact the modeling and simulation of MEMS

devices. We believe that other key research issues remain to be investigated. These include g:

• Devising a methodology for the optimization of the parameters of both microbeams and those

of the feedback controller to minimize the energy consumption by the electrostatic force and

feedback actuator.

Chapter 5. Conclusions and Recommendations for Future Research 63

• Investigating the microfabrication feasibility of both microbeams to validate their models and

static and dynamic behaviors.

• Exploring the circuitry design of the feedback controller associated with the microbeam

resonator by adding a second electrode within its spatial domain.

Appendix 64

Appendix:

Spatial Discretization - The Differential

Quadrature Method

Due to the complexity of the governing equations of the cantilevered microbeam, it is

necessary to use a numerical method to simulate its response. In this study, as did previously

with the clamped-clamped microbeam, we propose the use of Differential Quadrature Method

(DQM). Various problems in structural mechanics have been solved successfully with the aid

of DQM, and it has been shown that it leads to more accurate results at a lower computational

cost. The basic concept of DQM is to approximate the derivative of a function ( )w ξ with

respect to a space variable ξ at a given sampling point as a weighted linear combination of

the function values at all the sampling points in the domain of ξ [42].

Differential equations will then be transformed to a set of algebraic equations for time-

independent problems and a set of ordinary differential equations in time for initial-value

problems. So, for a dimensionless variable ξ defined in the domain (0,1) and using n

discretization points over the domain, the r th-order derivative of ( )w ξ at iξ ξ= is given by:

1i

nrrij jr

j

w A wξ ξξ= =

∂=

∂ ∑

The off-diagonal terms of the weighting coefficient matrix of the first order derivative turn

out to be:

1,1

1,

( )

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

n

i vv v i

ij n

i j j vv v j

A i j n i j

ξ ξ

ξ ξ ξ ξ

= ≠

= ≠

−

= = ≠

− −

∏

∏

The off-diagonal terms of the weighting coefficient matrix of the higher-order derivative are

obtained through the recurrence relationship:

Appendix 65

11 1 , 1, 2,...,

rijr r

ij ii iji j

AA r A A i j n i j

ξ ξ

−−

⎡ ⎤= − = ≠⎢ ⎥

−⎢ ⎥⎣ ⎦

where 2 1r n≤ ≤ − The diagonal terms are given by:

1,

1, 2,...,n

r rii iv

v v i

A A i n= ≠

= − =∑

where 1 1r n≤ ≤ −

For the accuracy of the numerical results, the following grid point distribution is used:

1 11 cos , 1, 2,...2 1i

i i nn

ξ π⎡ ⎤−⎛ ⎞= − =⎜ ⎟⎢ ⎥−⎝ ⎠⎣ ⎦

Such distribution was found to yield more accurate results and obtain the convergence of the

solution with a smaller number of grid points in comparison with other sampling schemes.

In many cases, before applying the quadrature method, we integrate by parts and use the

boundary conditions given to rewrite the integral terms. For example:

21 1 2

2

0 0

w wdx wdxx x

∂ ∂⎛ ⎞ = −⎜ ⎟∂ ∂⎝ ⎠∫ ∫

This integral is approximated using the Newton-Cotes formula [42] 1

10

n

i i ii

w wdx C w w=

′′ ′′∑∫

where 1

1,0

nv

ii vv v i

x xC dxx x

= ≠

−=

−∏∫

Hassen OUAKAD Bibliography 66

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N°: 2008-02 MASTER THESIS - KFUPMfaculty.kfupm.edu.sa/ME/houakad/Hassen's MS Thesis Report (Final... · Master Thesis N°: 2008-02 MASTER THESIS Presented to Tunisia Polytechnic