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Nonlinear Dynamics of Electrostatically Actuated MEMS Arches
Thesis by
Qais M. Hennawi
In Partial Fulfillment of the Requirements
For the Degree of
Master of Science
King Abdullah University of Science and Technology
Thuwal, Kingdom of Saudi Arabia
May, 2015
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EXAMINATION COMMITTEE APPROVALS FORM
The thesis of Qais M. Hennawi is approved by the examination committee.
Committee Chairperson [Prof.Mohammad I. Younis]
Committee Member [Prof. Sigurdur Thoroddsen]
Committee Member [Prof.Taous Meriem LALEG-KIRATI]
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© 2015
Qais M. Hennawi
All Rights Reserved
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ABSTRACT
Nonlinear Dynamics of Electrostatically Actuated MEMS Arches
Qais M. Hennawi
In this thesis, we present theoretical and experimental investigation into the nonlinear
statics and dynamics of clamped-clamped in-plane MEMS arches when excited by an
electrostatic force. Theoretically, we first solve the equation of motion using a multi-
mode Galarkin Reduced Order Model (ROM). We investigate the static response of the
arch experimentally where we show several jumps due to the snap-through instability.
Experimentally, a case study of in-plane silicon micromachined arch is studied and its
mechanical behavior is measured using optical techniques. We develop an algorithm to
extract various parameters that are needed to model the arch, such as the induced axial
force, the modulus of elasticity, and the initially induced initial rise. After that, we excite
the arch by a DC electrostatic force superimposed to an AC harmonic load. A softening
spring behavior is observed when the excitation is close to the first resonance frequency
due to the quadratic nonlinearity coming from the arch geometry and the electrostatic
force. Also, a hardening spring behavior is observed when the excitation is close to the
third (second symmetric) resonance frequency due to the cubic nonlinearity coming from
mid-plane stretching. Then, we excite the arch by an electric load of two AC frequency
components, where we report a combination resonance of the summed type. Agreement
is reported among the theoretical and experimental work.
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ACKNOWLEDGEMENTS
All the thanks and praise to ALLAH who helped me and made everything easy for me
during the work on this research, without him, I will never do anything.
I would like to gratefully thank and appreciate my advisor, Professor Mohammad I.
Younis for his guidance, patience, and valuable ideas. He inspired me with a lot of ideas,
he never said impossible to me, he has been more than an advisor, he has been a great
teacher and companion. His trustfulness made me very confident in doing any research.
I would also like to deeply thank the committee members Prof.Sigurdur Thoroddsen and
Prof.Taous Meriem LALEG-KIRATI for their valuable advices and for being members in my
thesis defense committee.
I also thank Dr.Abdallah Ramini who taught me about the instruments used during the
experimental work; he helped me a lot in getting the results for this study.
My appreciation also goes to my father, mother, brothers, sisters, and fiancée for their
trustfulness and encouragements.
I also thank all friends and colleagues in the group and the department faculty and staff
for making my time at King Abdullah University of Science and Technology a great
experience.
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TABLE OF CONTENTS
Page
EXAMINATION COMMITTEE APPROVALS FORM ………………………………………………………… 2
COPYRIGHT PAGE ....................................................................................................... 3
ABSTRACT ................................................................................................................... 4
ACKNOWLEDGEMENTS .............................................................................................. 5
TABLE OF CONTENTS.................................................................................................. 6
LIST OF ABBREVIATIONS ............................................................................................ 8
LIST OF FIGURES ......................................................................................................... 9
LIST OF TABLES ........................................................................................................... 12
Chapter 1: Introduction
1.1 Motivation ........................................................................................................ 13
1.2 Literature Review ............................................................................................ 16
1.2.1 Macro and Micro Scale Arches ............................................................ 16
1.2.2 Multifrequency Excitations .................................................................. 22
1.3 Thesis Objectives and Organization ................................................................ 24
Chapter 2: Background
2.1 Electrostatic Sensing and Actuation in MEMS and Parallel Plate Theory ........ 25
2.2 Nonlinearities in MEMS structures .................................................................. 30
2.2.1 Systems with Quadratic Nonlinearity .................................................. 30
2.2.2 Systems with Cubic Nonlinearity ......................................................... 33
2.2.3 Systems with both Quadratic and Cubic Nonlinearities ...................... 35
2.3 Instabilities and Bifurcations in Bistable Structures ......................................... 36
2.3.1 Snap-through Instability ...................................................................... 38
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2.3.2 Pull-in Instability .................................................................................. 39
2.4 Mixed Frequency Excitations and Combination resonances ........................... 40
Chapter 3: Statics and Dynamics of MEMS Arches
3.1 Problem Formulation and Equation of Motion ................................................ 45
3.2 Reduced Order Model Derivation .................................................................... 47
3.3 The Static Response of the Arch ...................................................................... 51
3.3.1 Problem Formulation ........................................................................... 51
3.3.2 Reduced Order Model Convergence Test ........................................... 51
3.4 The Variation of Natural Frequency with Initial Rise ....................................... 54
3.5 The Variation of Natural Frequencies with DC voltage load ............................ 56
3.6 The Variation of Natural Frequencies with Axial Force ................................... 61
3.7 Dynamics of Arches Under Harmonic Electrostatic Excitations ....................... 64
3.7.1 Single Frequency Excitations ............................................................... 64
3.7.2 Mixed Frequency Excitations ............................................................... 68
Chapter 4: Experimental Characterization
4.1 The Fabricated Arches ...................................................................................... 70
4.2 Topography Characterization ........................................................................... 73
4.3 Experimental Static Response .......................................................................... 73
4.4 Resonant Frequencies ...................................................................................... 76
4.5 Experimental Dynamic Response: Single Frequency Sweep ........................... 78
4.6 Experimental Dynamic Response: Mixed Frequency Sweep .......................... 81
4.7 Validation of Theoretical Simulations with Experimental Data ....................... 82
Chapter 5: Conclusions and Future Work
5.1 Summary and Conclusions ............................................................................... 90
5.2 Recommendations for Future Work ................................................................ 91
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LIST OF ABBREVIATIONS
AC Alternative Current
AFM Atomic Force Microscope
cc complex conjugate
DC Direct Current
DRIE Deep Reactive Ion Etching
FFT Fast Fourier Transform
LDV Laser Doppler Vibrometer
MEMS Micro Electro Mechanical Systems
MSA Micro System Analyzer
ODE Ordinary Differential Equation
PDE Partial Differential Equation
PMA Planar Motion Analyzer
RF Radio Frequency
RIE Reactive Ion Etching
ROM Reduced Order Model
SEM Scanning Electron Microscopy
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LIST OF FIGURES
Figure 1.1: MEMS arches applications :(a) A micro valve [2], (b) Muscle actuator [3],(c) Logic memory [4], (d) Switch(Relay) [5] .......................................................14
Figure 2.1: A parallel plate capacitor .............................................................................27
Figure 2.2: The trapezoidal cross section due to the anisotropic etching .....................29
Figure 2.3: Electric field lines in the case of inclined capacitor plates ..........................29
Figure 2.4: Frequency response of a system having quadratic nonlinearity: (a) linear spring, (b) Softening spring ..........................................................................31
Figure 2.5: Frequency response of a system having quadratic nonlinearity: (a) linear spring, (b) Softening spring, (c) Hardening spring .......................................33
Figure 2.6: The frequency response of a system having both quadratic and cubic nonlinearities ...............................................................................................36
Figure 2.7: Bifurcation diagram of an electrostatically actuated arch ..........................37
Figure 2.8: Phase portraits of a bistable system ............................................................37
Figure 2.9: Snap-through motion of an arch ..................................................................38
Figure 2.10: Bifurcation diagram of an electrostatically actuated arch ..........................39
Figure 2.11: Pull in behavior of an arch ...........................................................................40
Figure 3.1: A schematic diagram of the arch under multifrequency electrostatic loading ..........................................................................................................45
Figure 3.2 The first six symmetric mode shapes of a clamped-clamped beam ............49
Figure 3.3: Convergence of the ROM on the static deflection of case study [17] .........52
Figure 3.4: Convergence of the ROM on the static deflection of the arch I ..................53
Figure 3.5: Variation of the first five natural frequencies with initial rise .....................56
Figure 3.6: Variation of the first nondimensional frequency at voltage load VDC = 40 V ......................................................................................................................59
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Figure 3.7: Variation of the first mode shape with the DC voltage load .......................59
Figure 3.8: Variation of the first three symmetric natural frequencies with axial force ......................................................................................................................60
Figure 3.9: Variation of the static deflection with the axial force .................................63
Figure 3.10: Bifurcation diagram of the arch with the axial force ...................................63
Figure 3.11: A single frequency sweep with VDC = 30 V, VAC = 20 V, and ξ = 0.05............65
Figure 3.12: (a) Phase space at 34.5 , (b) Power spectrum at 34.5 , (c) Phase
space at 17.3 , (d) Power spectrum at 17.3 , (e) Phase space at
11.5 , (f) Power spectrum at 11.5 . ................................................66
Figure 3.13: A frequency sweep at a single excitation frequency at a voltage load (VDC =
40 V, VAC = 20 V) ...........................................................................................67
Figure 3.14: A frequency sweep in the presence of a second excitation source at a fixed frequency at a voltage load (VDC = 30 V, VAC1 = 20 V, VAC2 = 20, Ωnon,2 = 10) 69
Figure 4.1: SEM images of an arch fabricated by MEMSCAP © ....................................71
Figure 4.2: A SEM image of an arch of trapezoidal cross section ..................................72
Figure 4.3: A Zygo image of the cross section ................................................................72
Figure 4.4: Topography measurement using Zygo.........................................................73
Figure 4.5: (a) A top view; (b) A 3D view of a sample extracted from Zygo ..................73
Figure 4.6: (a) A schematic top view of the arch with the contact pads colored be the Blue and Red colors; (b) A schematic diagram of the experimental setup used for the measurement ..........................................................................74
Figure 4.7: (a) Mid span cross section without voltage load; (b) Mid span cross section with voltage load ..........................................................................................75
Figure 4.8: Experimental static deflection curves for different arches .........................75
Figure 4.9: (a) Arch prior to snap-through; (b) Arch after snap-through ......................76
Figure 4.10: The first three natural frequencies with the corresponding phase angle obtained experimentally using PMA ............................................................77
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Figure 4.11: Experimental forward sweep at various voltage loads ................................78
Figure 4.12: Experimental snap-though motion for different voltage loads ...................79
Figure 4.13:
Experimental forward sweeps for various voltage loads around the 3rd mode of arch I ..............................................................................................80
Figure 4.14:
Experimental forward sweeps for various voltage loads around the 3rd mode of Arch II .............................................................................................81
Figure 4.15: Forward frequency sweep with mixing at different voltage loads ..............82
Figure 3.16: Algorithm steps for finding the axial force and the pre-etching initial rise .84
Figure 3.17: Results gained from the developed algorithm ............................................85
Figure 4.18: Agreement between the ROM and experimental data ...............................85
Figure 4.19: Potential energy well ...................................................................................86
Figure 4.20: Forward frequency sweep: VDC = 10 V, VAC = 40 V, and ξ ≈ 0.08 ..................87
Figure 4.21: Forward frequency sweep: VDC = 20 V, VAC = 40 V, and ξ ≈ 0.08 ..................87
Figure 4.22: Forward frequency sweep: VDC = 30 V, VAC = 40 V, and ξ ≈ 0.08 ..................88
Figure 4.23: Forward frequency sweep: VDC = 40 V, VAC = 40 V, and ξ ≈ 0.034................89
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LIST OF TABLES
Table 2.1: Possible resonances that may be triggered when having two frequency components .................................................................................................41
Table 2.2: Possible resonances that may be triggered when having six frequency components .................................................................................................44
Table 3.1: The first Six Symmetric Modal Frequencies of a Straight Beam ..................48
Table 3.2: Arch I Parameters .........................................................................................53
Table 4.1: Ring Down Square Wave Signal Parameters ................................................77
Table 4.2: Arch II Parameters ........................................................................................81
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Chapter 1
Introduction
1.1 Motivation
In the recent decades, the advances in micro and nano technologies facilitate the
fabrication of large number of devices having a wide range of applications especially in
the field of sensing and actuation. Nowadays, Micro-Electro-Mechanical-Systems
(MEMS) are used almost everywhere. In electrical systems, they are found in filters,
phase shifters, resonators, and radio-frequency (RF) switches. In biological and chemical
systems, they are found in gas and mass detectors which can detect very small particles
reaching the scale of viruses and bacteria. In mechanical systems, they are found in
accelerometers used in the air bag mechanism in vehicles, pressure sensors, thermal
sensors and actuators, gyroscopes, microvalves, and micropumps [1].
The interesting feature about the fabrication process of MEMS devices is the ability to
fabricate a very large number of devices in one process on a single wafer. Because of
this, the process is called batch process, and it helps reducing their cost. For example,
MEMS accelerometers are sold nowadays in market in almost $ 1. Also, MEMS devices
have small size and light weight which makes them easily-integrable with other systems
[1].
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In most MEMS devices, the core element in the system is a microstructure which may be
a beam, arch, membrane, torsional mirror, tube ...etc. For example, the Atomic Force
Microscope (AFM) used in characterizing microstructures utilizes a micro cantilever
beam with a sharp stylus at its tip.
The micro arch gained lots of interests in the recent few years because of its desirable
characteristics. The most important feature of the arch is its ability to move with large
amplitude (stroke) by snapping through between its potential wells. This makes it
suitable for some applications like micro valves [2], micro muscle actuators [3], logic
memories [4], micro switches or relays [5], and filters [6].
(a) (b)
(c) (d) Figure 1.1: MEMS arches applications: (a) A micro valve [2], (b) Muscle actuator [3], (c)
Logic memory [4], (d) Switch (Relay) [5].
The arch profile can be formed during the fabrication process because of the bimorph
effect. This effect is caused by the differential thermal expansion between the layers
forming the device due to thermal changes in the fabrication environment. In this case,
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the arch is called an imperfect micro beam. Moreover, arches with a deliberate profile
can be fabricated with precise dimensions either in-plane or out-of-plane.
Also when a micro arch is subjected to electrostatic force actuation, some phenomena
appear like the jumping during the snap-though motion, softening and hardening
behaviors, appearance of super and sub harmonic resonances because of the
nonlinearities involved.
Furthermore, in electrostatically-actuated straight micro beams, when the applied
voltage reaches a certain value, the beam collapses toward the other electrode forming
a short circuit which leads to the destruction of the device. This behavior is known as
the pull-in phenomenon. It can be used in some applications like switching applications.
However, the destruction in the device is avoided by insulating the electrodes either by
dimples or a dielectric layer. This adds more steps and effort to the fabrication process.
In micro arches, when the applied voltage reaches a certain value, a sudden jump
happens but it does not lead to the destruction of the device. This behavior is known as
the snap-through phenomenon. Furthermore, the arch after the snap-though can
withstand more voltage loading until reaching its pull-in voltage. In this thesis, we
investigate the linear and nonlinear statics and dynamics of a MEMS arch when
subjected to both single frequency and mixed frequency excitations. Little research has
been done on mixed frequency excitations in MEMS arches and the associated
resonances that might be triggered. Hence, deep understanding is required to figure out
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the behavior of the arch under frequency mixing in order to employ this behavior in
useful applications like sensors and actuators.
1.2 Literature Review
1.2.1 Macro and Micro Scale Arches
Arches or curved beams are called bistable systems. This means the presence of two
stable equilibrium positions where they can oscillate around. This behavior is related to
a phenomenon called buckling which is an unstable behavior happens when a beam is
subjected to a compressive axial load. It was discovered by the Swiss mathematician
Leonard Euler since more than two centuries. Since then, lots of researches were
dedicated to understand this phenomenon. The most important issues about buckling
that attracted scientists and engineers are finding the buckling critical loads, mode
shapes, natural frequencies, as well as their dynamical behavior. In this thesis, we
investigate an arch under an induced axial force.
The research on arches started in the sixties, Humphreys [7] investigated both
analytically and experimentally the dynamic behavior of a shallow circular arch due to
uniform dynamic-pressure loading. He analyzed the problem for both simply supported
and clamped-clamped ends and used many kinds of dynamic loadings like the impulsive,
step, and the rectangular pulse loads.
Lock [8] studied the snap-through motion of a simply supported shallow sinusoidal arch
subjected to a step-wise pressure load for different geometric parameters. He used
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numerical integration schemes to find the critical loading values that affect the snap-
through motion. An infinitesimal stability study was used to find these critical values. His
results showed that there were two mechanisms that control the snap-through action.
These mechanisms are related to the applied load and the involved parameters.
Hsu et al. [9] investigated analytically the dynamic snap-through instability of different
types of shallow arches like simple, sinusoidal, and parabolic types for various loadings
like concentrated and uniformly distributed loads. They revealed some conditions
helping in the determination of stability or instability limits.
Many investigations have been conducted to find the mode shapes and natural
frequencies of arches or buckled beams with different rises. Dawe [10] calculated the
natural frequencies of a shallow arch using the discrete element displacement method
neglecting the longitudinal inertia.
Nayfeh et al. [11] derived the eigenvalue problem of a buckled beam with different
supporting conditions including hinged-hinged, clamped-clamped, and hinged-clamped
conditions. They found the relation between the first six natural frequencies and the
initial rise of the beam. They validated the theoretical results with experiment, and
there was an excellent agreement.
Nayfeh et al. [12] found a closed-form exact analytical solution for the post-buckling
profile of beams in terms of the applied axial load for many boundary conditions like
hinged-hinged, clamped-hinged, and clamped-clamped. They considered the mid-plane
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stretching nonlinearity in their derivations. They also found the critical buckling loads
and the mode shapes, and then studied the stability of the buckling configurations and
found that only the first buckling configuration is the stable equilibrium position.
Cabal et al. [13] investigated the snap-through phenomenon in a thermally actuated
bilayer micro beam. They determined the beam performance in terms of the original
profile and the beam pliability. They concluded that the original profile has a severe
effect for beams having the same material properties and dimensions but different in
the initial rise.
Das et al. [14] studied the snap-through and pull-in instabilities of both parabolic-shaped
and bell-shaped clamped-clamped micro arches and the pull-in instability of a micro
beam under transient parametric electric loading. They used the continuum equations
for the structural part where the finite element method is utilized to get the solution,
and used Maxwell's equations for the electrical part where the boundary element
method is utilized to get the solution. They found that the arch may exhibit a softening
behavior preceding the snap-through.
Zhang et al. [15] derived an analytical expression to predict the snap-through and pull-in
instabilities. Their approach was based on taking one mode shape of the Galerkin
discretization and representing the nonlinear terms using Taylor series with truncating
the higher order terms. They used an error compensation scheme to eliminate the error
resulting from having a single mode and linearization. They solved the problem
numerically using more than one mode shape. They also did some experimental work.
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Finally, they compared the results of the analytical model with numerical and
experimental results. They found that the uncompensated solution loses the accuracy as
the displacement increases and the compensated solution has a good agreement with
the multimodal and experimental results.
Mallona et al. [16] investigated the quasi-static and nonlinear dynamic behavior of a
shallow arch subjected to a dynamic pulse load. They revealed that the critical shock
load can be varied by controlling the shape of the arch. They compared their results
with finite element results. They studied the sensitivity of the static and dynamic
response to some parameters like the damping as well as the arch shape.
Regarding the arches used in MEMS applications, Krylov et al. [17] studied the static
behavior of a shallow arch to a distributed electrostatic force. They also studied the
effect of the arch parameters like the thickness and the initial rise on the relative
location of the snap-through and the pull-in voltages. They came up with a closed form
formula to compute the critical initial rise which guarantees the existence of the snap-
through instability. They found that the critical initial rise in case of electrostatic loading
is higher than the case of deflection-dependent loading.
Younis et al. [18] studied the nonlinear statics and dynamics of a clamped-clamped
MEMS arch subjected to electrostatic force. They developed a multimodal reduced
order model (ROM) up to five symmetric mode shapes of the linear undamped unforced
mode shapes of a straight clamped-clamped beam. They studied the effect of changing
the initial rise on the pull-in and snap-through instabilities. They excited the arch at its
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primary resonant frequency. A softening spring behavior is noticed due to the quadratic
nonlinearity. Also, they found two super harmonics of order two and three because of
the quadratic and cubic nonlinearities, respectively. It is noticed that the quadratic
nonlinearity is dominant over the cubic nonlinearity. They demonstrated the effect of
changing the initial rise on the frequency response keeping the Direct Current (DC) and
Alternating Current (AC) voltage loads the same with the same damping ratio.
Ruzziconi et al. [19] studied theoretically and experimentally the response of an
imperfect micro beam subjected to electrostatic force. They found experimentally the
mode shapes associated with their arch. They performed some frequency sweeps
forward and backward near the first resonance frequency. They developed a two-
degree-of-freedom reduced order model based on the Galerkin approach.
Alkharabsheh et al. [20] investigated the effect of axial forces on the static response of
an electrostatically actuated shallow MEMS arch. They studied the static behavior for
various DC voltage loads. They used a wide range of axial forces from compression loads
beyond the buckling limit to tension loads for various voltage loads. They found that
increasing the applied DC voltage load extends the stable range. They used different DC
voltage loads for different values of the axial force. It is found that changing the axial
force from compression to tension also extends the stable operation range of the arch.
They studied the effect of changing the axial force on the fundamental natural
frequency of the arch for various values of the DC voltage load. They found that applying
a tensile axial force has a softening effect, and hence reducing the natural frequency. On
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the other hand, applying a compressive load has a hardening behavior, and hence
increases the natural frequency. They concluded that the dynamic response is very
sensitive to changes in the axial forces, which makes it possible to adjust or control the
equilibrium positions and the natural frequencies.
Ouakad et al. [21] solved the eigenvalue problem for an arch considering the effect of
the DC voltage load, and then they demonstrated the effect of changing the initial rise
and the DC voltage load on the natural frequencies and mode shapes. They studied the
dynamic behavior of the arch when subjected to an AC voltage load. They used the
method of multiple time scales to obtain an analytical solution for the forced response
of the arch.
The bistability of arches and buckled beams are used to design and fabricate many
devices that have interesting functions. Gerson et al. [22] designed, fabricated, and
characterized an electrostatically actuated micro arch for actuation purposes, they
cascaded several arches with each other to get a large displacement or stroke actuator.
The device is comprised of two membranes working on opposite of each other with
both electrostatic and pneumatic driving. Hwang et al. [23] designed, fabricated, and
tested an in-plane bistable electrostatic actuator to be used as electromagnetic micro
relay permanent memory. Casals-Terre et al. [24] developed a switch that operates
dynamically using the resonance phenomena between the states of a bistable micro
arch. They found that the dynamic actuation saves power by 40 % when compared to
static actuation.
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1.2.2 Mixed Frequency Excitations
Exciting structures with more than one frequency component has been known long time
ago. In linear systems, having more than one driving frequency component yields a
response with all frequencies involved in the forcing term. In nonlinear systems, there
are different behaviors, other resonances appear in the response that are algebraically
related to the exciting frequencies called Combination Resonances. These resonances
depend on the nonlinearities involved in the system.
Yamamoto [25] studied the whirling motion of rotating shafts experimentally. He found
sub harmonic resonances of order 1/2 as well as combination resonances of summed
and difference type. Nayfeh [26] studied the response of multiple degree of freedom
systems involving quadratic nonlinearities and subjected to a harmonic parametric
excitation that involves internal combination resonance of summed type. He used the
method of multiple time scales to develop a first order uniform expansion. Using both
linear and nonlinear theories, he found that there are three zones for the amplitude of
the harmonic excitation where the response decays to zero according to both linear and
nonlinear theories, settle to a steady state value or zero based on the nonlinear theory
and to zero based on the linear theory, or grow up then settle to a steady state value
because of the nonlinearity.
Nayfeh [27] investigated the response of bowed structures to a combination resonance.
He developed a second order expansion using the method of multiple time scales and
found that there is no way to excite the combination resonance of difference types. He
also found that the steady state amplitude of the response is highly affected by the
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amplitude of the second mode excitation. He concluded that there exist certain critical
values of the excitation amplitude of the second mode where the combination
resonances can never be excited, always excited, or may be excited depending on the
initial conditions.
Nayfeh [28] studied the response of a two-degree-of-freedom system involving both
quadratic and cubic nonlinearities and subjected to multifrequency parametric
excitation that involves internal combination resonance of summed type.
Pezeshki et al. [29] examined the dynamics of a magnetically buckled beam subjected to
two-frequency excitation term. They used the Duffing equation as a model with
negative linear stiffness and found that the chaotic behavior of the system can be
controlled by changing the phase angle of the higher frequency component. Deng et al.
[30] developed a method to dynamically test MEMS gyroscope chips using
multifrequency excitation scheme. They found that using the multifrequency sweep
method reduces the testing time severely compared with traditional single frequency
sweep.
1.3 Thesis Objectives and Organization
The objectives of this thesis are:
To investigate theoretically and experimentally the nonlinear static and dynamic
behavior of MEMS arches.
To develop a methodology to extract from the experimental measurements the
parameters needed for an accurate ROM.
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This thesis is organized as follows: In Chapter 1, we present a literature review about
arches in the macro and the micro scale, we also show the applications in which they
are used and the mathematical models used to understand their work. In Chapter 2, we
present a general background study about electrostatic sensing and actuation,
nonlinearities present in micro arches, the associated instabilities, and the combination
resonances that can be found in such systems. In Chapter 3, we show the theoretical
predictions of our ROM in many important problems like the eigenvalue problem used
to study the effect of curvature, axial force, and the DC voltage load on the natural
frequencies of micro arches. Also, we show the static behavior under DC voltage load as
well as the dynamic behavior under harmonic voltage load having one/two frequency
components. In Chapter 4, we present the experimental work that includes the
characterization of the devices and loading them electrostatically. We develop an
algorithm to extract the parameters needed for the model. Also, we compare the
experimental data with the data obtained theoretically. Finally we conclude the thesis in
Chapter 5, and we summarize the work and present future plans for this research.
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Chapter 2
Background
2.1 Electrostatic Actuation and Parallel Plate Theory
The electrostatic phenomenon is based on the interaction between the electric fields of
stationary charged particles or surfaces. The electrostatic force between two charged
objects was first demonstrated experimentally by the French physicist Charles-Augustin
de Coulomb [31]. He found that the electrostatic force is directly proportional to the
surface area of the charged objects and inversely proportional with the square of the
distance between these two objects. Thus, the higher the surface area and the closer
the objects, the higher the electrostatic force. This force may be either attractive or
repulsive depending on the kind of the charge.
In the world of MEMS, the feature of having high surface areas and close surfaces makes
the use of electrostatics very powerful for sensing and actuation applications.
Nowadays, most electrostatic MEMS are fabricated using the surface micromachining
technique because it yields high-aspect-ratio surfaces (large surface areas and close
surfaces). Hence, surface micromachined microstructures are more favorable for
electrostatic applications than other microfabrication techniques.
For MEMS applications, electrostatic sensing and actuation are utilized due to many
advantages over other techniques, such as the high actuation force, fast response,
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controllability, and simplicity [1]. The most common microstructure used for
electrostatic sensing and actuation is the parallel plate capacitor which comes in many
forms like beams, plates, mirrors, arches…etc. The parallel plate capacitor has a movable
electrode that is attracted toward a stationary electrode through the electric field
generated between them. These microstructure are the basis of many important
applications like accelerometers, resonators, micro mirrors, RF switches…etc.
A well-known phenomenon of electrostatic sensing and actuation is the pull-in
phenomenon. It is defined as the collapse of the movable electrode on the stationary
electrode forming a short circuit leading to the destruction of the device. This
phenomenon happens when the microstructure's stiffness cannot withstand further
electrostatic loading. After this limit a sudden jump happens to the movable electrode
toward the stationary electrode. This phenomenon is strongly related to the nonlinear
nature of the electrostatic force and here the system is called mono stable. Many
studies were done to understand this nonlinear phenomenon in order to use it in some
applications like switches and to come up with a method to avoid the failure. One of the
proposed solutions is to use displacement limiters like the dimples or insulating
dielectric layers [17]. Other studies revealed safe and reliable methods of operation to
widen the range of stability without adding fabrication complexities [32], [33].
Consider two parallel plates separated by a gap d , having equal area A , and a uniform
surface charge density as shown in Figure 2.1
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Figure 2.1: A parallel plate capacitor.
Then the generated electric field is found by applying Gauss’s Law using a cylindrical
hypothetical surface as [1]
00
A
QE (2.1)
Where;
0 : The permittivity of the medium between the plates.
Q : The charge accumulated on the plates in Coulomb (C).
The electric field is also known as the potential gradient though the gap between the
plates [1]
d
VE (2.2)
Where V is the potential difference between the plates in Volts.
Equating equation (2.1) and equation (2.2) defines an important parameter related to
the charged plates called the capacitance C which reflects their ability to store electrical
energy and it is given as [1]
d
A
V
QC 0 (2.3)
d
28
Now, based on the definition of the electric field, the attraction force between the
plates is given by [1]
2
2
0
0
2
22 d
VA
A
QQEF
(2.4)
Notice, the formula given in equation (2.4) does not consider the effect of the fringes of
the electric field lines on the boundaries of the plates because the plates are assumed as
infinite sheets. However, some studies the effect of fringes on the electrostatic force
between two parallel plates and develop a modified equation for the electrostatic force
like Mejis-Fokkema [34]. For very small aspect ratio microstructures, the effect of fringes
is found to be neglected as studied in [18] and can be expressed as
2/1
2
4/3
2
2
0 53.0265.012 b
dh
b
d
d
bVF
(2.5)
Where,
b : The width of the microbeam
h : The thickness of the microbeam
Now, the previous discussion was based on the assumption of having parallel plates.
However, some MEMS devices are fabricated using Deep Reactive Ion Etching technique
(DRIE). It is an anisotropic etching technique, where the etching selectivity is dependent
on the etching orientation. This leads to some changes in the cross section and makes
an angle between the plates as shown in Figure 2.2. The trapezoidal profile changes the
electric field lines from straight in the case of parallel plates to curved lines as shown in
Figure 2.3.
29
Figure 2.2: The trapezoidal cross section due to the anisotropic etching.
Figure 2.3: Electric field lines in the case of inclined capacitor plates.
Tay et al. [35] studied the effect of non-parallel plates and the effect of having curved
electric field lines on the capacitance of an accelerometer. They derived an expression
for the electrostatic force given by
cos
)sin2cos(22
2sin2
bdd
AV
eF
(2.6)
For very small angle, equation (2.6) can be approximated as
)2(2
2
bdd
AV
eF
(2.7)
In this thesis and for simplicity, the model developed is based on having parallel plates,
the work on non-parallel and fringed plates is left for future.
30
2.2 Nonlinearities in MEMS Arches and their Effect
The nonlinear behavior of MEMS devices may come from the microstructure itself like
the geometry in case of arches, the sensing and actuating technique like the
electrostatic and electrothermal techniques, the materials used like piezoelectric
materials, or from the damping like the thermo elastic damping [1]. The effect of
nonlinearities on the response of a system can be seen in many forms like spring
hardening, spring softening, hysteresis, jumps, chaos, and appearance of other
resonances like sub harmonics, super harmonics, and combination resonances as well.
Studying and understanding the effect of nonlinearities on MEMS devices can help in
determining the proper use and the reliable range of use of MEMS device, such as
studying the effect of pull in on RF switches. In most MEMS applications, the common
nonlinearities found are either quadratic or cubic types, which will be discussed in the
following sections.
2.2.1 Systems of Quadratic Nonlinearities
A system has a quadratic nonlinearity if the dependent variable or any of its derivatives
has a quadratic term like 222 ,, xxx …etc. This applies also for any quadratic mix between
the dependent variable and its derivatives like xxxx , …etc. For example, a forced
linearly-damped system of a linear angular natural frequency0 that has a quadratic
nonlinearity is modeled as shown [1]
2 2
0( ) ( ) ( ) ( ) cosqx t cx t x t x t F t (2.8)
31
Finding an exact analytical solution for equation (2.8) is difficult. However, an
approximate analytical solution is found using perturbation techniques like the method
of multiple time scales [1] and [36] given by
2 2
2 2
0 02 4
0 0
55( ) cos 1 cos 2 2 3
12 6 6
q q
q
A Ax t A A t t
(2.9)
Where, A and are constants found from the given initial conditions. It is noticed from
equation (2.9) that the new natural frequency of the system is always less than the
natural frequency of the linear system no matter the sign of the quadratic nonlinearity.
This is known as spring softening. Figure 2.4 shows the frequency response curve of a
generic system having a quadratic nonlinearity.
Figure 2.4: Frequency response of a system having quadratic nonlinearity: (a) linear
spring 0q , (b) Softening spring 0q .
When the system is excited at a frequency close to one-half the linear natural frequency
i.e.0
2
1 , a large response is noticed at the linear natural frequency in addition of
the large response at the frequency as the linear case. This is verified by solving
equation (2.8) assuming weak nonlinearity and forcing and using the method of multiple
scales, as shown in equation (2.10)
(a) (b)
32
2 2
0
( ) cos 2 cos
component comes from the nonlinearity
linear component
Fx t A t t
(2.10)
The new component introduced by the nonlinearity is called a super harmonic of order
2, because it appears at twice the excitation frequency. Now, if the system is excited at
a frequency close to twice the linear natural frequency i.e. 02 , a large response is
noticed at the linear natural frequency in addition of the large response at the
frequency as the linear case. Using the method of multiple scales, an approximate
analytical solution is obtained [1]. The new component introduced by the nonlinearity is
called a sub harmonic resonance of order 1/2, because it appears at one-half the
excitation frequency. Using multiple time scales, the response is expressed as
2 2
0
( ) cos cos2 2
component comes from the nonlinearity linear component
Fx t A t t
(2.11)
Quadratic nonlinearities come from many sources like the loading method, material
used, or the structure itself. For example, the electrostatic loading is of quadratic nature
when assuming that the higher order nonlinearities are weaker that the quadratic one.
Also, the initial curvature in arches or buckled beams has the effect of quadratic
nonlinearities.
2.2.2 Systems of Cubic Nonlinearities
A system has a cubic nonlinearity if the dependent variable or any of its derivatives has
cubic like 333 ,, xxx …etc. This applies also for any cubic mix between the dependent
variable and its derivatives like 22,, xxxxxxx …etc. A famous example of systems
33
having cubic nonlinearities is the Duffing oscillator. Equation (2.12) shows a Duffing
oscillator equation of linear angular natural frequency0 with a cubic nonlinearity [1]
2 3
0( ) ( ) ( ) ( ) coscx t cx t x t x t F t (2.12)
Similarly, an approximate analytical solution is found using perturbation techniques [1]
and [36] given by
2
02
0
3( ) cos 1
8
cAx t A t
(2.13)
It is noticed from equation (2.13) that the new natural frequency of the system is either
less or more than the linear natural frequency of the linear system. This depends on the
sign of the cubic nonlinearity. If the nonlinear natural frequency is more than the linear
natural frequency, then the effect of the cubic nonlinearity is said to be spring
hardening. Also, if the nonlinear natural frequency is less than the linear natural
frequency, then the effect of the cubic nonlinearity is said to be spring softening. Figure
2.5 shows frequency response curves of a system having a cubic nonlinearity.
Figure 2.5: Frequency response of a system having quadratic nonlinearity: (a) linear
spring 0c , (b) spring softening 0c , (c) spring hardening 0c .
(a) (b) (c)
34
When the system is excited at a frequency close to one-third the linear natural
frequency i.e. 03
1 , a large response is noticed at the linear natural frequency in
addition to the large response at the frequency . This is verified by solving equation
(2.12) using the method of multiple scales assuming weak nonlinearity and forcing and
[1]
2 2
0
( ) cos 3 cos
component comes from the nonlinearity
linear component
Fx t A t t
(2.14)
The new component introduced by the nonlinearity is called a super harmonic of order
3, because it appears at three times the excitation frequency. Now, if the system is
excited at a frequency close to three times the linear natural frequency i.e. 03 , a
large response is noticed at the linear natural frequency in addition to the large
response at the frequency . Again, using the method of multiple scales, an
approximate analytical solution is obtained [1]
2 2
0
( ) cos cos3 3
component comes from the nonlinearity linear component
Fx t A t t
(2.15)
The new component introduced by the nonlinearity is called a sub harmonic of order
1/3, because it appears at one-third the excitation frequency. An example of a cubic
nonlinearity is the mid-plane stretching in micro beams. This nonlinearity comes from
geometry when the structure undergoes a large deflection compared to its thickness. As
a result, the stiffness of the structure varies according to cubic function.
35
2.2.3 Systems of Quadratic and Cubic Nonlinearities
When a system has both quadratic and cubic nonlinearities, the resultant response
depends on the dominant nonlinearity in the system. If the quadratic nonlinearity is
dominant, then the response is expected to be spring softening. However, if the cubic
nonlinearity is dominant, then the response is expected to be either spring softening or
spring hardening depending on the sign of the cubic nonlinearity coefficient. A generic
system with both nonlinearities is shown as [1].
2 2 3
0( ) ( ) ( ) ( ) ( ) cosq cx t cx t x t x t x t F t (2.16)
An approximate analytical solution is found using perturbation techniques in equation
(2.17) [1], [36]. For a system with both quadratic and cubic nonlinearities, all super
harmonics and sub harmonics may be triggered depending on the exciting frequency.
2 2 2
2 2
0 02 4 2 2 4
0 0 0 0 0
5 53 3( ) cos 1 cos 1 2 3
8 12 6 4 6
q q qc cA
x t A A t A t
(2.17)
An electrostatically-actuated MEMS arch has a quadratic nonlinearity coming from the
electrostatic loading (assuming that is dominant over higher order nonlinearities) and
the initial curvature. It also has a cubic nonlinearity coming from the mid-plane
stretching. Figure 2.6 shows a forward frequency sweep on a MEMS arch where the
frequency shown in the figure is nondimensional. It is noticed that there is a softening
behavior corresponding to the in-well vibration and a hardening behavior corresponding
to the snap-through vibrations. Also, super harmonics of order 2 and 3 are triggered.
36
Figure 2.6: The frequency response of a system having both quadratic and cubic nonlinearities at VDC = 40 V and VAC = 20 V.
2.3 Instabilities and Bifurcations in Bistable Microstructures
A bistable microstructure is a system having two stable solutions or two load deflection
curves, in between these two solutions there is an unstable solution separating them.
These systems are also called two-well systems because they have two potential energy
wells shown in Figure 2.7. The bottom of each well represents a stable equilibrium
position for oscillation. This is also seen from the phase portrait in Figure 2.8. The
oscillation inside the local wells A and B corresponds to the inner orbit due to small
perturbations or initial conditions. The oscillation about the critical point C corresponds
to the so-called separatix orbit, and the oscillation about the global well D corresponds
to the outer orbit outside the separatix orbit due to large perturbations or initial
conditions.
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
7
8
9
10
non
Am
plit
ude(
m)
37
Figure 2.7: Potential energy wells of a bistable system.
Figure 2.8: Phase portraits of a bistable system.
Electrostatically actuated bistable microstructures are known for nonlinear phenomena
like snap-through, pull-in and symmetry breaking instabilities, which will be discussed in
the following sections.
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
u(t)
V(u
)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
u(t)
udot(t
)
D
A
C
B
38
2.3.1 Snap-through Instability
The snap-through instability is an important behavior of bistable structures, it means
the sudden transition of the structure from its current configuration to the opposing
symmetric configuration. In other words, the transition from one potential well into the
other in Figure 2.9 is known as snap-through.
Figure 2.9: Snap-through motion of an arch.
When a bistable microstructure is actuated electrostatically starting from the initial
equilibrium position. The microstructure’s stiffness resists the deflection, this stiffness
has an upper limit. When the electrostatic loading exceeds this limit, the stiffness force
is no longer capable of maintaining the equilibrium and suddenly drops to zero. As a
result, a sudden change in the form of large displacement takes place. This change is
known as a Saddle-node Bifurcation. Bifurcation is a French word used commonly in
nonlinear dynamics community to describe sudden changes on the system that leads to
a qualitative change in the system’s behavior like changing the number of solutions and
their stability [1]. Figure 2.10 shows the bifurcation diagram of the electrostatically
actuated arch studied by Krylov et al. [17]. When a stable solution (blue colored)
coalesces with another unstable solution (red colored), they collapse and destroy each
other. However, the system is attracted toward another stable position. The snap-
through motion corresponds to the transition from position A to position B.
39
Figure 2.10: Bifurcation diagram of an electrostatically actuated arch.
An important feature is noticed about snap-through motion. The microstructure gains
another stable configuration. This means that the microstructure is able to carry further
loading. Also, one can note that if the load is decreased below the snap-through loading,
the microstructure returns back to its original configuration. Many researchers utilized
the snap-through motion in many important applications in sensing and actuation [2-6].
2.3.2 Pull-in Instability
As mentioned in section 2.3.1, the microstructure is capable to withstand further
loading after the snap-through happens because of the increase in stiffness gained in
the other stable position. If the load reaches another limit, another saddle-node
bifurcation occurs, but this time the microstructure’s stiffness force is no longer able to
maintain the equilibrium. As a result, a sudden jump happens toward the actuating
electrode leading to the destruction of the device and pull-in, unless dimples or
0 20 40 60 80 100 1200
2
4
6
8
10
12
14
VDC
(V)
Wm
ax(
m)
Snap-through
Pull-in
A
B C
E
D
Release
40
dielectric materials are added to prevent the direct contact. Figure 2.11 shows the pull-
in behavior. Referring back to Figure 2.10, the pull-in behavior corresponds to the
transition from position C to the lower electrode.
Figure 2.11: Pull in behavior of an arch.
The pull-in instability was explored experimentally by Nathanson et al. [38] when they
studied the behavior of the resonant gate transistor. Since then, many applications have
been proposed utilizing the pull-in instability like RF switches [36].
2.4 Mixed Frequency Excitations and Combination Resonances
When a linear system is excited by a harmonic force having more than one frequency
component, the frequency response is superposed from all frequency components
involved in the forcing term because of the superposition property of linear systems. In
nonlinear systems, new resonances are triggered other than the forcing frequencies,
super harmonics, and sub harmonics of the system. These resonances depend on the
nonlinearities involved in the system. For an arch, there are both quadratic and cubic
nonlinearities. A generic system involves both quadratic and cubic nonlinearities excited
by mixed frequency harmonic force is modeled by
N
n
nnncq tAtutututuctu1
322
0 cos)()()()()( (2.18)
41
Where,
nA : The amplitude of the n-th forcing component.
n : The driving frequency of the n-th forcing component.
n : The phase shift of the n-th forcing component.
In this thesis, we study an arch excited electrostatically by a DC voltage load
superimposed with two AC harmonic voltage loads. In the case of having one AC
harmonic voltage load, the final voltage term after expansion can be written as
tVtVVVVtVVtV ACACDCACDCACDC 2cos2
1cos2
2
1cos)( 22222
(2.19)
This is a special case of two-term excitations studied by Nayfeh [36], where the exciting
frequencies are related to each other such that 1 and 22 . Finding an
approximate analytical solution for equation (2.18) using the straight-forward expansion
method, the resonances that might be triggered are given in Table 2.1
Table 2.1: Possible resonances that may be triggered when having two frequency
components.
Frequency Resonance Type Order of Resonance
22 10 Super harmonic 2
33 10 Super harmonic 3
2
1
2
110 Sub harmonic
1
2
3
1
3
110 Sub harmonic
1
3
3120 Combination Linear of summed type
120 Combination Linear of difference type
42
The presence of a peak at one-third of the primary resonance frequency may be due to
the cubic nonlinearity. In this case, it is called a super harmonic resonance, or due to the
effect of mixing, and here it is called combination resonance of summed type. When
another AC harmonic voltage load is added, the voltage term is expanded as
22
1 1 2 2
2 2 2 2
1 2 1 1 2 2 1 3
2
2 4 1 2 5 1 2 6
( ) cos cos
1 1 12 cos 2 cos cos
2 2 2
1cos 2 cos 2 cos
2
DC AC AC
DC AC AC DC AC DC AC AC
AC AC AC AC AC
V t V V t V t
V V V V V t V V t V t
V t V V t V V t
(2.20)
A six-term excitation is obtained such that
11 22 13 2 24 2 125 126 (2.21)
Next, we use the method of straight-forward expansion to find an approximate
analytical solution for the equation
6
1
3222
0 cos)()()()(2)(n
nncq tAtututututu (2.22)
Where is a bookkeeping scale parameter.
Seeking for an approximate solution in the form
2
2
10)( uuutu (2.23)
Using the following notations for the derivatives
10 DDdt
d (2.24a)
)2(2 20
2
1
2
10
2
02
2
DDDDDDdt
d (2.24b)
Substituting in equation (2.22), we will get equations of order
43
:)( 0O
6
1
0
2
00
2
0 cosn
nn tAuuD (2.25a)
)( 1O : 2
0000101
2
01
2
0 22 uuDuDDuuD q (2.25b)
)( 2O : 3
0
2
11001100200
2
11102
2
02
2
0 22222 uuuuuDuDuDDuDuDDuuD cqq (2.25c)
The solution for equation (2.25a) can be written in the form
cceeTAtun
Ti
n
Ti nn
6
1
)(
10000)()(
(2.26)
Where cc denotes to complex conjugate terms, and
ni
nnn eA 122
0 )(2
1 (2.27)
Substituting back into equation (2.25b)
cce
eeee
eeee
eeAeAeeA
eieAAiuuD
Ti
TniTniTniTni
TiTniTniTni
Tninnn
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n n
n
Ti
n
Ti
n
Ti
n
Ti
q
Ti
n
n
n
Ti
0)65(
0)4(0)3(0)2(0)1(
0)65(0)4(0)3(0)2(
0)1(0000000
000
65
6
5
4
6
4
3
6
3
2
6
2
1
65
6
5
4
6
4
3
6
3
2
6
1
6
2
1
)()(2222
3
1
01
2
01
2
0
2
2222
2222
222
2)(2
(2.28)
In addition to the primary resonance, other resonances may be triggered as shown in
Table 2.2, which are called secular terms of equation (2.28)
44
Table 2.2: Possible resonances triggered when having six frequency components.
Frequency Resonance
Type Order of
Resonance
)(2),(2,4,4,2,2 121221210 Super
harmonic 2
)(2
1),(
2
1,,,
2
1,
2
1121221210 Sub harmonic
1
2
0 2 1 1 2 1 2 1 2 2 1
2 1 2 1 2 1 2 1
,3 ,2 , 2 , 3 ,2 ,
2( ), 3 ,3 ,3 ,
Combination
Linear of summed type
0 2 1 1 2 1 2 2 1 1 2
2 1 2 1 1 2
, ,2 , , 2 ,2 ,
2( ), 3 ,
Combination
Linear of difference
type
Taking higher order terms shows that there are super harmonics of order 1/3 and sub
harmonics of order 3, as well as some combination resonances associated with the cubic
nonlinearity.
45
Chapter 3
Statics and Dynamics of MEMS Arches
In this chapter, we formulate the arch problem and derive the reduced order model
(ROM). Also, we investigate the static response of the arch under DC voltage load. We
solve the eigenvalue problems to study the variation of the arch natural frequency with
curvature, axial force, and the DC voltage load. Finally, we study the dynamic response
of the arch under single frequency and mixed frequency excitations.
3.1 Problem Formulation and Equation of Motion
In this section, the equation of motion governing an initially curved clamped-clamped
micro arch is formulated. We consider an arch of length L , widthb , thickness h , initial
riseob , Young’s modulus E , and mass density . The arch is actuated electrostatically by
a metallic electrode with a separation gap d using a DC voltage load DCV superimposed
to an AC harmonic voltage loads ACV of frequency , respectively as shown in Figure
3.1.
Figure 3.1: A schematic diagram of the arch under electrostatic loading.
46
The equation governing the transverse deflection of an arch assuming Euler-Bernoulli
model and a negligible initial slope 1ow can be written as [1, 18]
224 2 2
0 0
4 2 2 2
0
2
2
0
22
cos( )
2( )
L
DC AC
d w ww w w EA w w wEI A c dx
x t t L x dx x x x
b V V t
d w w
(3.1)
Whereow is the initial profile, and it is assumed to be the first buckling mode as
ˆ2
ˆ ˆ( ) 1 cos2
oo
b xw x
d L
(3.2)
Where E is the effective Young modulus since the width is very large compared to the
thickness hb 5 , we use the cylindrical plate theory assumption
21
E
Eeffective (3.3)
Where : is Poisson's ratio 0.27, and I is the area moment of inertia
3
12
1bhI (3.4)
The boundary conditions are assumed perfectly clamped-clamped, this means the
deflections and the slopes are zeros at both ends all the time
0),0( tw
0),( tLw
(3.5a)
0),0( tw
0),( tLw
(3.5b)
To avoid using very small numbers which make problems in numerical solvers like
MATLAB or Mathematica, and to make the solution procedures easier and more
convenient, the following nondimensional ratios are introduced:
47
d
ww
d
ww 0
0
L
xx
T
tt
n
T
n
1
Where T is a time constant,
EI
ALT
4
Later, the nondimensional governing equation becomes
2124 2 2
0 014 2 2 2
0
2
2
2
0
2
cos( )
2(1 )
DC AC
d w ww w w w w wc dx
x t t x dx x x x
V V t
w w
(3.6)
Also, the nondimensional boundary conditions are
(0, ) 0w t (1, ) 0w t (0, ) 0w t (1, ) 0w t (3.7)
The parameters of equation (3.6) are defined as
2
1 6
h
d
EIT
Lcc
4
3
4
22EId
bL
3.2 Reduced Order Model Derivation
Now, equation (3.6) is a nonlinear integro-differential equation, finding an exact closed
form analytical solution is very difficult. Therefore approximate methods are used. In
this thesis we employ the weighted residual method with the Galerkin approach. This
reduces the system into a discretized system having finite degrees of freedom instead of
the previous continuous system. Hence, it is called Reduced Order Models (ROMs).
Younis et al. [18] developed a ROM for electrostatically actuated MEMS devices, where
they showed a good agreement with experimental data.
48
To derive the ROM, the dynamic deflection is assumed to be a linear combination of the
multiplication of modal time variant functions with spatial functions. These spatial
functions are assumed to be the normalized linear undamped symmetric mode shapes
as shown in Figure 3.2. These functions are called comparison functions because they
satisfy all the boundary conditions.
)()(),(1
tuxtxw i
n
i
i
(3.8)
)(tui: Time varying modal coordinate function.
)(xi : Normalized linear undamped ith mode shape of a clamped-clamped straight beam
given by
, ,
, , , ,
, ,
cos cosh( ) cos cosh sin sinh
sin sinh
non i non i
i non i non i non i non i
non i non i
L Lx x x x x
L L
(3.9)
We assume that the arch is uniform and the initial profile is the first buckling mode,
hence the used mode shapes are the symmetric ones. Also, the anti-symmetric modes
are orthogonal to the electrostatic force, so they do not have any effect in the solution
of the problem. However, for non-uniform arches, the anti-symmetric modes should be
included. Table 3.1 shows the first five symmetric modal frequencies of a Cl-Cl beam
Table 3.1: The first six symmetric modal frequencies.
Mode No. Frequency (nondimensional)
1 22.373290
3 120.90339
5 298.55554
7 555.16525
9 890.73180
11 1305.38
49
Figure 3.2: The first six symmetric mode shapes of a clamped-clamped beam.
To get the dimensional frequency in (rad/s), we use
nonAL
EI
4
Now, multiplying both sides of the equation (3.6) by the denominator of the
electrostatic force 2
0(1 )w w [18] to reduce the cost of computations, we get
4 2
22
0 24 2
21422 0 0
1 0 2 4
0
(1 ) cos( )
(1 ) 2
DC AC
w w wc w w V V t
x t t
d w dww w ww w dx
x dx x x dx
(3.10)
Substituting equation (3.8) into equation (3.6) yields
0 0.5 1-2
0
2
(x/L)
1(x
)0 0.5 1
-2
0
2
(x/L)
3(x
)
0 0.5 1-2
0
2
(x/L)
5(x
)
0 0.5 1-2
0
2
(x/L)
7(x
)0 0.5 1
-2
0
2
(x/L)
9(x
)
0 0.5 1-2
0
2
(x/L)
11(x
)
50
2
(4)
0
1 1 1 1
2
2
2 24
01 0 4
1 1 1
( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( )
cos( )
1 ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) (
n n n n
i i i i i i i i
i i i i
DC AC
n n n
i i i i i i i i
i i i
x u t x u t c x u t w x u t
V V t
d ww x u t x u t x u t x u
dx
1
0
10
)n
i
dwt dx
dx
(3.11)
Multiplying equation (3.11) by the mode shape )(xj , then integrating the outcome
over the normalized domain 1,0x and utilizing the orthogonality of mode shapes
yields
21
(4)
0
1 1 1 10
12
2
0
2 4
01 0 4
1 1
( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( )
( ) cos( )
( ) 1 ( ) ( ) ( ) ( )
n n n n
j i i i i i i i i
i i i i
j DC AC
n n
j i i i i
i i
x x u t x u t c x u t w x u t dx
x V V t dx
d wx w x u t x u t
dx
21 1
0
1 10 0
( ) ( ) 2 ( ) ( )n n
i i i i
i i
dwx u t x u t dx dx
dx
(3.12)
Further reduction gives
21
0
1 10
21
0
1 10
21
(4)
0
1 10
2
( ) ( ) ( ) 1 ( ) ( )
( ) ( ) ( ) 1 ( ) ( )
( ) ( ) ( ) 1 ( ) ( )
n n
i j i i i
i i
n n
i j i i i
i i
n n
i j i i i
i i
DC
u t x x w x u t dx
u t x x w x u t dx
u t x x w x u t dx
V
1
2
0
2 21 14
01 0 4
1 1 10 0
2
1 0
1 1
cos( ) ( )
( ) 1 ( ) ( ) ( ) ( ) ( ) ( )
( ) 1 ( ) ( ) ( ) ( )
AC j
n n n
j i i i i i i
i i i
n n
j i i i i
i i
V t x dx
d wx w x u t x u t x u t dx dx
dx
x w x u t x u t
1 14
0 0
410 0
2 ( ) ( )n
i i
i
d w dwx u t dx dx
dx dx
(3.13)
51
3.3 The Static Response of the Arch
3.3.1 Problem Formulation
The static solution of equation (3.13) shows the variation of the deflection of the arch
with the DC voltage load. To get the static solution, all time derivatives in equation
(3.13) are set to zeros and all time variant parameters are set to constants as given
1
( , ) ( )n
i i
i
w x t C x
(3.14)
So, the final equation is
21
(4)
0
1 10
1
2
2
0
2 21 14
01 0 4
1 1 10 0
( ) ( ) 1 ( )
( )
( ) 1 ( ) ( ) ( )
( )
n n
i j i i i
i i
DC j
n n n
j i i i i i i
i i i
j
C x x w C x dx
V x dx
d wx w C x C x C x dx dx
dx
x
21 14
0 01 0 4
1 1 10 0
1 ( ) ( ) 2 ( )n n n
i i i i i i
i i i
d w dww C x C x C x dx dx
dx dx
(3.15)
3.3.2 Reduced Order Model Convergence Test
The accuracy of the solution depends on the number of modes used in the Galerkin
expansion. We use only symmetric mode shapes because we have a symmetric
deflection at the mid span point, and the anti-symmetric mode shapes are orthogonal to
the electrostatic force. Considering the case of Krylov et al. [17] where the length
mL 1000 , gap md 10 , thickness mh 4.2 , width mb 30 , and the initial rise
mbo 5.3 , we employ equation (3.15) to get static deflection curves shown in Figure
3.3.
52
Figure 3.3: Convergence of the ROM on the static deflection of case study [17].
We notice from Figure 3.3 that five symmetric mode shapes are enough to get accurate
results [18]. The number of modes required for convergence depends on the device
parameters like the length, width, thickness, and initial rise.
Using five modes of the straight beam in the Galerkin expansion may increase the cost
of computation. But, using the mode shapes of an unactuated arch leads to more
accurate results regarding the convergence of the ROM. In shallow arches, the
difference between straight beams and shallow arches is very small and hence using the
mode shapes of the straight beam may be more efficient in terms of computational
cost. For deep arches, there is a significant difference in the mode shapes and natural
frequencies. In the next section, the eigenvalue problem of an unactuated arch is solved
considering the effect of curvature, axial force, and the DC voltage load. Then, the
0 20 40 60 80 100 1200
2
4
6
8
10
12
VDC
(V)
Wm
ax(
m)
1 mode
2 modes
3 modes
4 modes
5 modes
6 modes
53
results are compared with the case of using straight beam mode shapes. We
experimentally investigate the static response of many arches with different
dimensions. Figure 3.4 shows the convergence of the ROM on an arch with the
parameters in Table 3.2
Table 3.2: Arch I parameters.
Parameter Value (µm)
Length (L) 600
Width (b) 27
Thickness (h) 2
Initial rise (bo) 2.7
Gap (d) 8
Figure 3.4: Convergence of the ROM on the static deflection of Arch.
0 20 40 60 80 100 120 140 160 180 2001
2
3
4
5
6
7
8
9
VDC
(V)
Wm
ax (
m)
One mode
Two modes
Three modes
Four modes
Five modes
54
3.4 Variation of Natural Frequency with Initial Rise
In this section, we solve the eigenvalue problem for a linear, undamped, and unforced
system to study the effect of the initial rise on the mode shapes and natural
frequencies. The governing equation in this case is
124 2
0 0
14 2 2
0
2d w dww w w
dxx t dx x dx
(3.16)
The solution of equation (3.16) is in the form
( , ) ( )i t
w x t x e
(3.17)
Substituting equation (3.17) into equation (3.16) yields
1
(4) 2
1 0 0
0
( ) ( ) ( ) 2 ( ) ( )x x w x x w x dx (3.18)
Equation (3.18) is a nonhomogeneous Ordinary Differential Equation (ODE), so there is a
homogeneous solution and a particular solution, the homogeneous solution is
( ) cos( ) sin( ) cosh( ) sinh( )h
x A x B x C x D x (3.19)
The particular solution can be expressed as
( ) cos(2 )p
x F x (3.20)
Next, the complete solution is
( ) ( ) ( )h p
x x x (3.21a)
( ) cos( ) sin( ) cosh( ) sinh( ) cos(2 )x A x B x C x D x F x (3.21b)
Substituting equation (3.21b) into equation (3.18) yields
55
3 2
1 01
2
4 24 21 0 0
2
4
( )sin(2 )4
16h
b
dF x x dxb
d
(3.22)
Introducing the nondimensional ratio (curvature parameter)
2
1 0
2
b
d
(3.23)
Substituting equation (3.23) into equation (3.22), the factor F is
13
4 2
0
4( ) sin(2 )
4 (4 )h
F x x dx
(3.24)
Finally equation (3.21b) is
13
4 2
0
4( ) ( ) ( ) sin(2 ) cos(2 )
4 (4 )h h
x x x x dx x
(3.25)
Now, substituting the boundary conditions of the clamped-clamped case yields an
eigenvalue problem
(0, ) (0, ) 0w t w t (3.26a)
(1, ) (1, ) 0w t w t (3.26b)
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
0
0
0
0
Q Q Q Q A
Q Q Q Q BQ
Q Q Q Q C
Q Q Q Q D
(3.27)
In literature, many studies demonstrated the effect of the initial rise on the mode
shapes and natural frequencies. Nayfeh et al. [11] investigated the mode shapes and
natural frequencies of buckled beams subjected to different boundary conditions. They
noticed that the initial profile affects only symmetric modes if the initial profile is
56
assumed to be the first buckling mode. Also, the natural frequencies of symmetric
modes increase as the initial rise increases. Figure 3.5 shows the variation of the first
five natural frequencies of a clamped-clamped arch with the curvature parameter . It is
seen that the natural frequencies of anti-symmetric modes do not change, whereas the
natural frequencies of symmetric modes change.
Figure 3.5: Variation of the first five natural frequencies with initial rise.
3.5 Variation of Natural Frequencies with the DC voltage load
Next, we solve the eigenvalue problem for a linear, undamped, and unforced system to
study the effect of the DC voltage load on the mode shapes and natural frequencies. We
solve this problem to see the difference between using the exact mode shapes with the
0 20 40 60 80 100 120 140 160 180 2000
50
100
150
200
250
300
350
non
1st mode
2nd mode
3rd mode
4th mode
5th mode
57
DC voltage load or the normal mode shapes without the DC voltage load. Starting from
the normalized equation
21224 2 2
0 02
124 2 2 2
00
21
w dwVw w w w wdx
x t x x x x dxw w
(3.28)
Next, we assume the deflection of the arch to have both a static component and a
dynamic component as
( , ) ( ) ( , )s dw x t w x w x t (3.29)
We expand the electrostatic force term in Taylor series around the static solution )(xws
using first order approximation, and then drop the equilibrium term, which yields
dxwwwwdxwwwwww
ww
Vww ssdsdsd
s
DCdd
1
0
01
1
0
0013
0
2
2)4( 221
2
(3.30)
Then, we assume the dynamic response is separated into two components; a time
variant component and spatial component as
n
i
iid xtutxw1
)()(),( (3.31)
Substituting equation (3.31) into equation (3.30)
12
(4) 2
1 0 03
1 1 1 100
1
1 0
1 0
2( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( )
1
( ) ( ) 2
n n n n
DC
i i i i i i s s i i
i i i is
n
i i s s
i
Vu t x u t x u t x w w w w u t x dx
w w
u t x w w w dx
(3.32)
After that, we multiply both sides of equation (3.32) by the mode shape )(xj , and then
integrate both sides over the domain from 0 to 1 using the orthogonality of mode
shapes.
58
1
0
0
1
0 1
1
1
0
1
0 1
001
1
0 13
0
2
2
1
1
01
1
0
2
2)()()(
)()()(2
)()(1
2)()()()()()()(
dxwwwdxxtux
dxdxxtuwwwwx
dxxtuww
Vxdxxxtudxxxtu
ss
n
i
iij
n
i
iissj
n
i
ii
s
DCj
n
i
iji
n
i
ijii
(3.33)
Finally, we obtain
1
0
0
1
0 1
1
1
0
1
0 1
0
1
0
01
13
0
2
22
,
2
21
2
dxwwwdxu
dxuwwdxwwdxuww
Vuu
ss
n
i
iij
n
i
iissj
n
i
ii
s
DCjjjnonj
(3.34)
To solve for the natural frequencies in equation (3.34), we should solve for the static
solution at each voltage load. After getting the static solution, we plug it into equation
(3.34), and solve for the eigenvalues of the system in equation (3.34).
As case study to check the convergence, we used the case in Krylov et al. [17]. We found
that using three symmetric mode shapes are enough to get accurate results. Figure 3.6
shows the convergence of the first nondimensional natural frequency at a DC voltage
load of 40 V and we notice a quick convergence. Figure 3.7 shows the variation in the
first mode shape for various DC voltage loads. We find that the variation in the mode
shape is slight over a wide range of DC voltage loads. This makes the use of the mode
shapes without considering the effect of the DC voltage load reasonable.
59
Figure 3.6: Variation of the first nondimensional frequency at voltage load VDC = 40 V.
Figure 3.7: Variation of the first mode shape with the DC voltage load.
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 336
37
38
39
40
41
42
# on Symmetric Modes
non1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
x/L
(x
)
20 V
40 V
60 V
80 V
90 V
100 V
# of Symmetric modes
60
Moreover, each mode shape has a formula that includes three mode shapes of the
straight beam. Thus, using more than one mode shape of the modified modes increases
the computational cost. As a result, we conclude that using multimodal study using the
original straight beam modes are more efficient and simpler.
Figure 3.8 shows the variation of the first three symmetric modal frequencies with the
DC voltage load. We notice that there is no substantial change over a wide range of the
voltage loads except near the bifurcation points which are the snap-through point close
to 90 V and pull-in point close to 106 V.
Figure 3.8: Variation of the first three symmetric natural frequencies shapes with the DC voltage load.
0 20 40 60 80 100 1200
50
100
150
200
250
300
350
400
VDC
non
First mode
Third mode
Fifth mode
61
3.6 Variation of the Natural Frequency with the Axial Force
In this section, we solve the eigenvalue problem for a linear, undamped, and unforced
system to study the effect of the axial force on the mode shapes and natural
frequencies. Therefore we solve this problem as part of the parameters extraction, the
governing equation in this case is
1
0
01
2
12
01
2
2
2
2
2
4
4
2 dxx
w
x
w
x
wN
dx
wd
x
w
t
w
x
w (3.35)
We assume the deflection composed of two components, a static component and a
dynamic component, as given by
),()(),( txwxwtxw ds (3.36)
Next, we linearize about the static solution and then substitute in the governing
equation, which yields
dxwwwNwdxwwwwwww ssdsdsdd
1
0
011
1
0
01011
)4( 22 (3.37)
Now, we apply Galerkin procedure as [18]
n
i
iid xtutxw1
)()(),( (3.38)
Substituting equation (3.39) into equation (3.38) yields
1
(4)
1 01 01
1 1 10
1
1 01
1 0
( ) ( ) ( ) ( ) 2 ( ) ( )
( ) ( ) 2
n n n
i i i i s s i i
i i i
n
i i s s
i
u t x u t x w w w w u t x dx
u t x N w w w dx
(3.39)
62
Multiplying both sides of equation (3.40) by the mode shape )(xj , then integrating
both sides over the domain from 0 to 1 using the orthogonality of mode shapes yields
1 1 1 1
2
1 0 0
1 1 10 0 0 0
1 1
1 1 0
10 0
( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( )
( ) ( ) ( ) 2
n n n
i i j i i j i j s s i i
i i i
n
j i i s s
i
u t x x dx u t x x dx x w w w w u t x dx dx
x u t x dx N w w w dx
(3.41)
Finally, we get
1 1 1 1
2
, 1 0 0 1 1 0
1 10 0 0 0
2 2n n
j non j j j s s i i j i i s s
i i
u u w w dx w w u dx u dx N w w w dx
(3.42)
The static solution should be found at each axial force in order to solve for the natural
frequencies from equation (3.41). Then we plug it into equation (3.41). Finally, we get a
system of ODEs. We solve for the eigenvalues of the system in equation (3.41) in order
to get the nondimensional natural frequencies. Now, the system is discretized and the
static solution should converge. Therefore, we test the convergence to find the number
of enough modes used in the Galerkin expansion. We found that three symmetric
modes are enough to get accurate results for the case study [17]. Figure 3.9 shows the
convergence of the solution.
Figure 3.10 shows the full bifurcation diagram where we have stable and unstable
solutions. We notice that the static response curve has an asymmetric pitchfork
bifurcation.
63
Figure 3.9: Variation of the static deflection with the axial force.
Figure 3.10: Bifurcation diagram of the arch with the axial force.
-250 -200 -150 -100 -50 0 50 100 150 200 250-0.5
0
0.5
1
1.5
Nnon
Wm
ax/d
1 mode
2 modes
3 modes
4 modes
-250 -200 -150 -100 -50 0 50 100 150 200 250-0.5
0
0.5
1
1.5
Nnon
Wm
ax(
m)
Stable solution
Unstable solution
64
From Figure 3.10, increasing the tensile axial force decreases the transverse deflection
until reaching an almost straight configuration. In this case, increasing the axial force
does not change or affect the deflection because the beam in this case reaches the
saturation limit. However, increasing the compressive axial force increases the
transverse deflection until reaching certain value where it might reach a pitchfork
bifurcation leading into two stable solutions and one unstable solution. The lower stable
solution in Figure 3.10 represents the original well before bifurcation. The upper branch
or the new born branch represents the other well.
3.7 Dynamics of Arches Under Harmonic Electrostatic Excitations
In this section, we try to explore the response of shallow arches when excited by a single
harmonic forcing and two-harmonic forcing. For demonstration, we consider the case
study of Krylov et al. [17].
3.7.1 Single Frequency Excitations
At first, we study the single frequency excitation, as explained in Chapter 2. When we
expand a single frequency electrodynamic force, we find it has a DC component, an AC
component having the same forcing frequency, and an AC component having twice the
forcing frequency. Hence, the expected resonances are the primary resonance at the
natural frequency of the system and the secondary resonances. Figure 3.11 shows a
simulated single frequency forward sweep.
65
Figure 3.11: A single frequency sweep at VDC = 30 V, VAC = 20 V, and ξ = 0.05.
One can notice that the primary resonance at 34.5 as well as the secondary
resonances. Here, we have a super harmonic resonance of order 2 near 17.3 and a
super harmonic resonance of order 3 near 11.5 . To prove the presence of the
secondary resonances, we draw the phase spaces for each case as shown in Figure 3.12.
(a) (b)
0 5 10 15 20 25 30 35 40 45 500
0.5
1
1.5
2
2.5
3
3.5
non
Am
plit
ude (
m)
1.0 0.5 0.0 0.5 1.0 1.5 2.0
40
20
0
20
40
0 20 40 60 800
2
4
6
8
Vel
oci
ty (
m/s
)
Position (µm) Frequency (nondimensional)
Po
wer
sp
ectr
um
66
(c) (d)
(e) (f)
Figure 3.12: (a) Phase space at 34.5 , (b) Power spectrum at 34.5 , (c) Phase
space at 17.3 , (d) Power spectrum at 17.3 , (e) Phase space at 11.5 , (f)
Power spectrum at 11.5 .
We notice that for the primary resonance, the phase space is only one-loop orbit
representing a periodic motion of period one. For the super harmonic of order 2, the
phase space is two-loop orbit indicating that the period of the motion is doubled due to
the quadratic nonlinearity. For the super harmonic of order 3, the phase space is three-
loop orbit indicating that the period is doubled two times due to the cubic nonlinearity.
0.5 0.0 0.5 1.0
30
20
10
0
10
20
30
0 20 40 60 800
1
2
3
4
5
0.0 0.2 0.4 0.6 0.8
10
5
0
5
0 10 20 30 40 50 600.0
0.5
1.0
1.5
2.0
Vel
oci
ty (
m/s
)
Position (µm)
Po
wer
sp
ectr
um
Frequency (nondimensional)
Vel
oci
ty (
m/s
)
Position (µm) Frequency (nondimensional)
Po
wer
sp
ectr
um
67
Also, we notice from the power spectrum in Figure 3.12 (b) that exciting the first
resonance yields a resonance at the first resonance and twice of the first resonance, this
is because of the electrostatic force effect. When the excitation is close to one-half the
first resonance, resonances appear at twice of both the exciting frequency as seen in
Figure 3.12 (d) due to the quadratic nonlinearity and the electrostatic effect. When the
excitation is close to one-third the first resonance, the resonances appear at three times
and six times of the exciting frequency due to the cubic nonlinearity and the
electrostatic effect. Now, if the voltage load is high enough, then the system has enough
energy to jump between its two potential wells leading to dynamic snap-through. We
will show this experimentally in Chapter 4. Figure 3.13 shows the snap-through motion
of the arch studied by Krylov et al. [17].
Figure 3.13: A frequency sweep at a single excitation frequency at a voltage load (VDC = 40 V, VAC = 20 V).
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
7
8
9
10
non
Am
plit
ude(
m)
68
One can notice from Figure 3.13 that there is a hardening behavior corresponding to the
snap-through motion around the nondimensional frequency 30. The reason of having
such behavior is due to the large motion of the arch compared to its thickness. This
amplifies the stretching effect which changes the arch's stiffness in a cubic regime. Thus,
the cubic nonlinearity dominates in this frequency range, and as a result a hardening
spring behavior occurs. The large stroke of the motion which extends over almost the
gap separating the electrodes from each other helps in casting this motion in some
crucial applications like energy harvesters.
3.7.2 Mixed Frequency Excitations
In this section, we explore the response of shallow arches when excited by a two-
harmonic force. For demonstration, we consider the case study of Krylov et al. [17].
Exciting the microstructure by a two-harmonic electrostatic force yields a six-term
excitation as mentioned in Section 2.4. We solve for the resonances using perturbation
techniques and we get many resonances that are algebraically related to the exciting
frequencies. By sweeping the forcing frequency, we may catch all of these resonances or
some of them. Figure 3.14 shows a frequency sweep of the arch studied by Krylov et al.
[17].
69
Figure 3.14: A frequency sweep in the presence of a second excitation source at a fixed frequency at a voltage load (VDC = 30 V, VAC1 = 20 V, VAC2 = 20, Ωnon,2 = 10).
Figure 3.14 shows the presence of combination resonances of both summed and
difference types. When adding a second frequency component of a nondimensional
frequency 10 and sweeping the first frequency, it is noticed that we have peaks near
frequency 40non . This peak is the primary resonance component because it occurs at
the system's resonance frequency, and another two peaks at 30non and 50non .
These two peaks are combination resonances of summed and difference types,
respectively (adding 30 to 10 or subtracting 10 from 50 gives the natural frequency 40).
We notice that the combination resonances follow a softening behavior like the primary
resonance.
0 10 20 30 40 50 60 700
0.5
1
1.5
2
2.5
3
non
Am
plit
ude(
m)
70
Chapter 4
Experimental Characterization
In this chapter, we show the experimental work and the results we obtained. Then we
validate these results with the simulations done in Chapter 3. Basically, the
measurements were based on the Zygo profilometer and the Micro System Analyzer
(MSA) with the Planar Motion Analyzer (PMA) component.
4.1 The Fabricated Arches
The unwanted curvature in microstructures as pointed out in the introduction of this
thesis is a common problem in MEMS. The variation in the thermal fabrication
environment can lead to this curvature in MEMS devices. Arches or buckled beams with
deliberate curvature are fabricated using very special fabrication processes to move in
either in-plane or out-of-plane directions. Out-of-plane arches have a limitation in the
initial rise when they are fabricated using surface micromachining processes. They
cannot be deep as the gap beneath them is small. In such cases, the arch cannot snap
freely between its potential wells, and hence cannot be characterized well. These arches
have some troubles during the characterization especially when using a point laser for
motion measurements like the Laser Doppler Vibrometer (LDV). The light is diffracted
away from the lens due to curvature. For in-plane arches, it does not matter how deep
the arch is since there are no limitations in the gap. These arches are characterized using
some tools like the PMA.
71
The in-plane MEMS arches under study were fabricated by MEMSCAP©. Basically, the
process starts with normal photolithography procedures (spinning of a photoresist,
Ultra Violet light exposure, and then lift-off). Then the resultant structure is etched
deeply using Reactive Ion Etching (DRIE) to pattern the gaps between the structure and
the electrodes. Figure 4.1 shows SEM (Scanning Electron Microscopy) images of the
tested arches fabricated by MEMSCAP©.
Figure 4.1: SEM images of an inplane arch fabricated by MEMSCAP ©.
However, the cross section of the arch is not exactly rectangular because of the
difference in the selectivity of the etchant material to different crystallographic planes
because this type of etching is anisotropic. Thus, the thickness of the arch varies
throughout the length and the width. Figure 4.2 shows a top schematic view of one of
the arches as well as an SEM image of the cross section.
72
Figure 4.2: (a) A top schematic view of an arch;(b) A SEM image of an arch cross section.
The trapezoidal section is verified using Zygo as shown in Figure 4.3. It is important to
mention that the Zygo image shows the bottom thickness of the arch greater than the
upper thickness. This happens because the light cannot go deeper due to the diffraction.
Figure 4.3: A Zygo image of the cross section.
4.2 Topography Characterization
The topography of the microstructure is very important to get an idea about the exact
dimensions of the microstructure. In this thesis, the Zygo profilometer is used to
investigate the topography. The measurement is done by exposing a white light on the
measured area followed by moving the Zygo’s objective downward then upward. The
change in the light wave length is processed and then related to topography. Figure 4.4
shows a schematic diagram of the measurement steps.
73
Figure 4.4: Topography measurement using Zygo.
We used many samples with different lengths, thicknesses, and initial rises during the
experimental work. Figure 4.5 shows a top view and a 3D view of one of the samples
extracted from Zygo where we can see the curvature obviously.
(a) (b)
Figure 4.5: (a) A top view; (b) A 3D view of a sample extracted from Zygo.
4.3 Experimental Static Response
The static deflection of the arch is investigated using Zygo by sweeping the DC voltage
load in forward and backward schemes. An image is taken at each voltage load showing
the position of the arch with respect to the actuating electrode. Then the image is
compared with the originally undeflected position. In Figure 4.6, we show a schematic
top view of the arch with the contact pads colored by the Blue and Red colors. Also, we
show a schematic diagram of the experimental setup used for the measurement where
74
we used a DC power supply connected to an amplifier. In Figure 4.7, we show a cross
sectional view of an arch at the mid span point before and after applying the voltage
load.
(a) (b)
(c)
Figure 4.6: (a) A schematic top view of the arch (b) A schematic diagram of the experimental setup used for the measurement ;(c) The experimental setup used for the
static measurement.
Zygo DC power
supply
Amplifier Chip
Multimeter
75
(a) (b)
Figure 4.7: (a) Mid span cross section without voltage load; (b) Mid span cross section with voltage load.
We tested many arches, some of them show clear snap-through jump and others do
not. Figure 4.8 shows the static measurement of three samples having the dimensions
given in the legend in microns with the order (length-gap-thickness-rise). Figure 4.9
shows a microscopic image of one of the tested arches just prior and after the snap-
through jump.
Figure 4.8: Experimental static deflection curves for different arches.
0 20 40 60 80 100 120 140 160 180 2000
1
2
3
4
5
6
7
8
9
10
VDC
(V)
Wm
ax (
m)
500-7-2-2
500-7-2-3
600-8-2-3
76
(a) (b)
Figure 4.9: (a) Arch just before snap-through; (b) Arch after snap-through.
4.4 Resonant Frequencies
In this section, we study the resonant frequencies of the arches. The knowledge of
resonant frequencies helps in determining the vital dynamical range of the device. We
measure the arch resonant frequencies using the Polytec© PMA machine as they are
operated in air.
The Polytec© PMA has the ability to excite the microstructure using many waveforms
like the white noise signal which is mostly used especially for out-of-plane
measurements. In this thesis, we use a ring down square wave signal with a wide range
of frequencies for the in-plane device, and then the Fast Fourier Transform (FFT) is
extracted showing the resonance frequencies.
The measurement procedures are as follows; at first a reference frame is added as a
rectangular area focused on the central area of the arch. Then the arch is excited by the
square wave signal which is a built-in signal in the PMA. At each frequency, the steady
state response is measured by taking many frames with a frequency matches the
exciting frequency. In this case the arch looks like stationary, this is called the
77
stroboscopic principle. Afterwards, the deflection is compared with the initial frame
taken in the beginning of the measurement. Table 4.1 shows the parameters of the used
signal for the arch in Table 3.2. Figure 4.10 shows the first three resonance frequencies
with the corresponding phase.
Table 4.1: The ring down signal specification used in the FFT.
Parameter Value
Duty cycle 50 %
Amplitude 120 V
Frequency limit 400 kHz
Frequency increment 100 Hz
Figure 4.10: The first three natural frequencies with the corresponding phase angle obtained experimentally using PMA.
The first resonance frequency is found at 72.03 kHz, the second resonance frequency is
at 130.1 kHz, and the third resonance frequency (second symmetric) is at 260.1 kHz.
0 50 100 150 200 250 300 350 400-200
-180
-160
-140
(kHz)
Magnitude (
dB
)
0 50 100 150 200 250 300 350 400-200
0
200
400
(kHz)
Phase (
deg)
78
Also, we can evaluate the damping ratio for each mode from equation (4.1) [1]. For the
first resonance, we found the damping ratio about 0.034, and for the third resonance
we found it about 0.11.
2 n
(4.1)
Where,
n : The measured resonance peak.
: The frequency band width at -3 dB of the resonance peak.
4.5 Experimental Dynamic Response: Single Frequency Sweep
A single frequency sweep measurement is conducted around the first and third modal
frequencies of the arch given in Table 3.2 to experimentally verify the dynamic behavior
of the arch. Figure 4.11 shows forward frequency sweeps around the first modal
frequency at different DC voltage loads.
Figure 4.11: Experimental forward sweeps for various voltage loads around the 1st mode
of arch I.
50 55 60 65 70 75 800
0.5
1
1.5
2
2.5
3
Frequency (kHz)
Am
plit
ude (
m)
VDC
= 10 V, VAC
= 40 V
VDC
= 20 V, VAC
= 40 V
VDC
= 30 V, VAC
= 40 V
79
From Figure 4.11, we notice that at 10 V DC voltage load, the behavior is almost linear.
Also, increasing the DC voltage load amplifies the effect of the quadratic nonlinearity,
and hence the stiffness of the arch decreases as the frequency increases. This leads to a
softening behavior. In addition, jumps are observed as the voltage load increases. The
dynamic snap-through motion is also observed at high voltage loads. Figure 4.12 shows
the dynamic snap-through behavior for different voltage loads.
Figure 4.12: Experimental snap-though motion at different voltage loads of arch I.
We notice from Figure 4.12 a hardening behavior corresponding to the snap-through
motion and a softening behavior corresponding to the vibration within the snap-through
region of the well. In addition, increasing the voltage load increases the motion’s
amplitude. This amplifies the effect of the cubic nonlinearity and hence leads to more
stretching. As a result, an increase in the snap-through frequency band is observed.
50 55 60 65 70 75 800
0.5
1
1.5
2
2.5
3
3.5
Frequency (kHz)
Am
plit
ude (
m)
VDC
= 40 V, VAC
= 40 V
VDC
= 40 V, VAC
= 50 V
80
Also, the increase in the voltage load increases the shift in the resonance frequency as
noticed.
The dynamic response around the third mode is also investigated. Figure 4.13 shows
several forward frequency sweeps around the third modal frequency at different
voltage loads.
Figure 4.13: Experimental forward sweeps for various voltage loads around the 3rd mode of arch I.
We observe from Figure 4.13 that exciting the third mode at relatively low voltage loads
yields an almost linear behavior. By increasing the voltage load, a hardening behavior is
noticed. For confirmation, we considered another case study for an arch with the
parameters in Table 4.2.
250 252 254 256 258 260 262 264 266 268 2700
0.2
0.4
0.6
0.8
1
Frequency (kHz)
Am
plit
ude (
m)
V
DC= 40 V,V
AC=40 V
VDC
=50 V,VAC
= 50 V
VDC
=50 V,VAC
=60 V
VDC
=60 V,VAC
= 60 V
VDC
=60 V,VAC
=70 V
VDC
=60 V,VAC
=80 V
VDC
=70 V,VAC
=80 V
81
Table 4.2: Arch II parameters.
Parameter Value (µm)
Length (L) 500
Width (b) 27
Thickness (h) 2
Initial rise (bo) 2
Gap (d) 7
Figure 4.14: Experimental forward sweeps for various voltage loads around the 3rd mode of Arch II.
4.6 Experimental Dynamic Response: Mixed Frequency Sweep
After the experimental work on single frequency excitations, we add a new harmonic
voltage load. An external function generator is added in series with the PMA function
generator. Figure 4.15 shows a forward frequency sweep with mixing.
340 345 350 355 360 365 370 375 3800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Frequency(kHz)
Am
plit
ude(
m)
VDC
=60V,VAC
=60V
VDC
=75V,VAC
=75V
VDC
=75V,VAC
=100V
82
Figure 4.15: Forward frequency sweep with mixing at different voltage loads.
We notice the appearance of combination resonances of the additive type without the
subtractive type. However, when we conduct the experiment, the arch starts twisting.
This motion makes a vague resolution on the PMA camera. Thus, random data are taken
because of the asynchronous measurements. For small voltage loads or for small
motion, the data are clear to some extent.
4.7 Validation of Theoretical Simulations with Experimental Data
In this section, we extract the parameters needed for the reduced order model
to match the data obtained from experiment. These unknown parameters are Young’s
modulus E , the axial force N , and the original initial rise before the etching process (or
before the stresses are relieved)01b . The parameters we measured experimentally are
40 50 60 70 80 90 1000
0.5
1
1.5
2
2.5
Frequency(kHz)
Am
plit
ude(
m)
V
DC =20V,V
AC1=40V,V
AC2=20V,f
2=10kHz
VDC
=30V,VAC1
=40V,VAC2
=20V,f2=20kHz
VDC
=30V,VAC1
=40V,VAC2
=20V,f2=10kHz
83
the initial rise after the etching process 02b (static measurement) and the naturel
frequencies 321 ,, fandff (dynamic measurement).
In order to solve the problem, we developed an algorithm that helps in finding the
unknown parameters, starting by assuming a value for the initial rise 01b as an initial
guess, then the natural frequencies (hence the ratio between them) are evaluated from
the following eigenvalue problem over a wide range of axial forces
1
0
011
1
0 1
1
0 1
01
1
0
011
2
, 22 dxwwwNdxudxuwwdxwwuu ss
n
i
iij
n
i
iissjjjnonj (4.2)
Where 02w is the static solution in equation (3.35) around which the linearization is
done. After that, we match the frequency ratio between the first and third frequencies
obtained from equation (4.2) with the value obtained experimentally. The value of the
axial force that matches the frequency ratio between the first and third modes is then
plugged into
dxwwwNwww
1
0
01020210102
)4(
02 2 (4.3)
Next, we evaluate the static deflection02w in equation (4.3) for the axial force obtained
from equation (4.2). If the deflection matches with the initial profile measured
experimentally, then we stop. If not, then another guess is assumed. Figure 4.14
describes the used algorithm.
84
Start
Assume bo
Get R from EQ 4.2
Get N from EQ 4.2
Get Ws for N from EQ 4.3
Ws==Wo
End
Yes
No
Figure 4.16: Algorithm steps for finding the axial force and the pre-etching initial rise.
Considering the arch in Table 3.2 and after iterating, the algorithm converges to a value
for the pre-etching initial rise about 3.5 µm, which corresponds to a nondimensional
axial force of 70.5. The corresponding frequency ratio1 3/ is 0.277. Figure 4.17 shows
the study performed. The left y-axis represents the value of the axial force and the right
y-axis represents the value of the post-etching initial rise.
85
Figure 4.17: Results gained from the developed algorithm.
Now, after getting the unknown parameters, we solve for the static response of the arch
in Table 3.2 using equation (3.15). We get the response shown in Figure 4.18.
Figure 4.18: Agreement between the ROM and experimental data.
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 440
60
80
100
Axia
l fo
rce (
nondim
ensio
nal)
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 42.2
2.3
2.4
2.5
Measure
d initia
l rise (
m)
Original initial rise (m)
Axial force Nnon
Measured bo
0 20 40 60 80 100 120 140 160 1800
1
2
3
4
5
6
7
8
9
10
VDC
(V)
Wm
ax (
m)
Simulation with 5 modes
Experimental
86
As seen from Figure 4.18, there is no obvious jump in the static response as it looks
moving around the local well. Figure 4.19 shows the potential energy well at a DC
voltage load 40 V. The potential energy well gives information about how deep is the
well and the stable equilibrium positions of the arch.
Figure 4.19: Potential energy well.
We notice from Figure 4.19 that there is no clear jumping from one well to another;
instead we have a single well potential that changes shape (width) suddenly.
Now, we study the dynamic response of the arch in Table 3.2. The first frequency sweep
is done using low values of the voltage load as shown in Figure 4.20. One can see that
the response is almost linear. Now increasing the voltage load yields the response in
Figure 4.21. Further increase in the DC voltage load yields the response in Figure 4.22.
1.0 0.5 0.0 0.5 1.0 1.5 2.02000
5000
1 104
2 104
5 104
Position
Po
ten
tial
en
erg
y
87
Figure 4.20: Forward frequency sweep: VDC = 10 V, VAC = 40 V, and ξ ≈ 0.08.
Figure 4.21: Forward frequency sweep: VDC = 20 V, VAC = 40 V, and ξ ≈ 0.08.
50 55 60 65 70 75 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (kHz)
Am
plit
ude (
m)
Experiment
ROM
50 55 60 65 70 75 800
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Frequency (kHz)
Am
plit
ude (
m)
Experiment
ROM
88
Figure 4.22: Forward frequency sweep: VDC = 30 V, VAC = 40 V, and ξ ≈ 0.08.
Until now, there is no dynamic snap-through. Increasing the voltage load gives the
system more energy to jump to larger motion corresponding to the wide region in the
potential well (snap-through). Figure 4.23 shows the dynamic snap-through behavior. It
is worthy to mention that we decrease the damping ratio in order to catch the jump
numerically.
50 55 60 65 70 75 800
0.5
1
1.5
2
2.5
3
Frequency (kHz)
Am
plit
ude (
m)
Experiment
ROM
89
Figure 4.23: Forward frequency sweep: VDC = 40 V, VAC = 40 V, and ξ ≈ 0.034.
From the previous figures, the shift between the peaks obtained experimentally and the
peaks obtained numerically can be obviously noticed. For low voltage loads, the model
managed to catch the resonance obtained from experiments. Increasing the voltage
load, the model starts the deviate from the experimental. One reason may be due to the
assumption of rectangular cross section. Also, assuming constant linear damping might
not be correct.
50 55 60 65 70 75 800
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Frequency (kHz)
Am
plit
ude (
m)
Experiment
ROM
90
Chapter 5
Summary, Conclusions, and Future Work
In this chapter, we summarize the thesis and present concluding remarks and
recommendations for the future work.
5.1 Summary and Conclusions
In this thesis, we presented a theoretical and experimental study to investigate the
static and dynamic behavior of electrostatically actuated in-plane MEMS arches when
excited by a DC voltage load, a single harmonic voltage load, and two-harmonic voltage
loads. The static response of the arches under DC voltage load was studied
experimentally for several arches using the Zygo profilometer based on imaging. The
experimental results were verified by a multimodal ROM. Some arches didn’t show clear
snap-through jump because they have shallow wells, other arches show clear snap-
through jump due to the depth of their wells. These arches were subjected to axial force
coming from the fabrication process. We developed an algorithm to evaluate the value
of the axial force. This algorithm utilizes the experimental ratio between the modal
frequencies which were found through a FFT study using the PMA and the initial rise of
the arch after the etching process. The modulus of elasticity was found by fitting the
static response curve with the experimental data. Then frequency sweeps were done
using the PMA revealing softening behaviors occurred for the in-well vibrations.
Increasing the voltage load gives the system enough energy to snap-through. A
91
hardening behavior occurred for the snap-through vibrations. After that a second
harmonic voltage source is added to study the possible combination resonances that
may be triggered. We managed to catch experimentally the combination resonances of
the summed type. A problem in the measurement took place because when adding
another harmonic component in the excitation, the response will not be
monoharmonic, it has two frequency components! The PMA measures the motion of
single frequency because it is based on the stroboscopic phenomenon, where the
frequency of the measuring light is synchronized with the frequency of the system in
order to capture the motion. However, if the second harmonic voltage load is small, the
motion can be captured. Hence, combination resonances of the summed type only were
observed, when the second harmonic voltage load becomes large. Nothing is
measurable because the imaging becomes vague. Moreover, if the structure is moving
out-of-the-plane, then using the Polytec© MSA captures the motion because it is based
on Doppler effect, and hence the combination resonances can be obtained.
5.2 Recommendations for Future Work
The following is a list of recommendations for future work.
The model used in this thesis deals with arches with rectangular cross sections,
the equation of motion has to be reformulated considering the effect of the
thickness variation through the length and the width of the arch.
The equation of motion used in the model is an approximate formula that works
for shallow arches, a new formula is derived by Nayfeh [41] which is more
accurate than the formula used in this thesis. Based on Nayfeh’s formula, we tried
92
the static response of arches with different initial rises. Also, we solved the
eigenvalue problem to find the variation of natural frequencies with the DC
voltage load. For future, we will solve the dynamic problem with the new
formula.
The initial guess in the developed algorithm was changed manually. For future,
the process will be optimized in order to get the least computational error.
More experimental work will be dedicated on different arches with different
dimensions.
93
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