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 3.11: Amplitude versus detuning; VDC =1 Volt, VAC =0.05 Volt, K1=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 3.16: Amplitude versus detuning; VDC =1 Volt, VAC =0.05 Volt, K2=0, and K3=1.....37
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 4.8: Frequency-response curve of the microbeam for loading case 2 ........................56
Figure 4.9: Frequency-response curve of the microbeam for loading case 3 ........................57
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
Chapter 1. Introduction 2
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)
Chapter 1. Introduction 3
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
Chapter 1. Introduction 4
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]
Chapter 1. Introduction 5
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
Chapter 1. Introduction 6
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
Chapter 1. Introduction 7
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).
Chapter 2. Analysis of One-Degree-of-Freedom Oscillators
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)
Chapter 2. Analysis of One-Degree-of-Freedom Oscillators
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)
Chapter 2. Analysis of One-Degree-of-Freedom Oscillators
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)
Chapter 2. Analysis of One-Degree-of-Freedom Oscillators
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)
Chapter 2. Analysis of One-Degree-of-Freedom Oscillators
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)
Chapter 2. Analysis of One-Degree-of-Freedom Oscillators
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.
Chapter 2. Analysis of One-Degree-of-Freedom Oscillators
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
Chapter 2. Analysis of One-Degree-of-Freedom Oscillators
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
Chapter 2. Analysis of One-Degree-of-Freedom Oscillators
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.
Chapter 2. Analysis of One-Degree-of-Freedom Oscillators
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
Chapter 2. Analysis of One-Degree-of-Freedom Oscillators
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.
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 21
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)
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 22
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)
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 23
( )( )
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
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 24
• 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)
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 25
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
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 26
( ) ( )( ) ( )
( ) ( ) ( )
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)
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 27
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).
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 28
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.
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 29
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.
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 30
Figure 3.5: Frequency-response curves when VAC =0.05 Volt, K1=0, K2=0, and K3=0
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
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 31
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
Figure 3.8: Frequency-response curves when VDC =1 Volt, VAC =0.05 Volt, K1=0, and K3=0
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 32
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)
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 33
( ) ( )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
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 34
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.
Figure 3.10: Frequency-response curves when VDC =1 Volt, VAC =0.05 Volt, K2=0, and K3=0
Figure 3.11: Frequency-response curves when VDC =1 Volt, VAC =0.05 Volt, K1=0, and K3=0
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
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 35
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
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 36
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 3.13: Frequency-response curves when VDC =1 Volt, VAC =0.05 Volt, K1=0, and K2=0
Figure 3.14: Frequency-response curves when VDC =1 Volt, VAC =0.05 Volt, K1=0, and K2=0
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.
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 37
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 3.16: Frequency-response curves when VDC =1 Volt, VAC =0.05 Volt, K2=0, and K3=1
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.
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 38
Figure 3.17: Frequency-response curves when VAC =0.05 Volt, K1=0, K2=0, and K3=3
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
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 39
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).
Chapter 3. Control and Dynamics of a Clamped-Clamped Microbeam Resonator 40
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
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 42
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
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 43
( )
( ) [ ]
( )
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)
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 44
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)
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 45
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)
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 46
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)
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 47
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)
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 48
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)
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 49
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 β β α
β β β β α
β β 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 β β α
β β β β α
+
+ +
= + − − + −
− +
(4.37)
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 50
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
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 51
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
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 52
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)
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 53
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)
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 54
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)
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 55
( )( ), , ,
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
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 56
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 4.8: Frequency-response curve of the microbeam for loading case 2
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 57
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).
Figure 4.9: Frequency-response curve of the microbeam for loading case 3
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.
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 58
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)
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 59
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.
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 60
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.
Chapter 4. Dynamics of a Cantilever Microbeam Resonator 61
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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