“DYNAMIC ANALYSIS OFELECTROSTATICALLY EXITED MICRO CANTILEVER BEAM” Department of Mechanical Engineering

“DYNAMIC ANALYSIS OFELECTROSTATICALLY EXITED MICRO CANTILEVER BEAM”  Department of Mechanical Engineering
“DYNAMIC ANALYSIS OFELECTROSTATICALLY
EXITED MICRO CANTILEVER BEAM”
A Thesis Submitted in partial fulfillment
of the requirements for the award of
Master of Technology
In
Machine Design and Analysis
By
J S N B Vamsee Gowtham A
Roll No: 209ME1179
Department of Mechanical Engineering
National Institute of Technology-Rourkela
Rourkela -769008
2011
“DYNAMIC ANALYSIS OFELECTROSTATICALLY
EXITED MICRO CANTILEVER BEAM”
A Thesis Submitted in partial fulfillment
of the requirements for the award of
Master of Technology
In
Machine Design and Analysis
By
J S N B Vamsee Gowtham A
Roll No: 209ME1179
Under the Guidance of
Dr. J.Srinivas
Department of Mechanical Engineering
National Institute of Technology
Rourkela
2011
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA
CERTIFICATE
This
is
to
certify
that
the
thesis
“DYNAMICANALYSIS
entitled,
OFELECTROSTATICALLY EXITED MICRO CANTILEVER BEAM” by Mr. J S N B
Vamsee Gowtham A in partial fulfillment of the requirements for the award of Master of
Technology Degree in Mechanical Engineering with specialization in “Machine Design &
Analysis” at the National Institute of Technology, Rourkela is an authentic work carried out by
him under my supervision and guidance.
To the best of my knowledge the matter embodied in the thesis has not been submitted to any
other University/ Institute for the award of any Degree or Diploma.
Date:
Dr. J. Srinivas
Dept. of Mechanical Engineering
National Institute of Technology
Rourkela-769008
ACKNOWLEDGEMENT
Successful completion of this work will never be one man’s task. It requires hard work in right
direction. There are many who have helped to make my experience as a student a rewarding one.
In particular, I express my gratitude and deep regards to my thesis guide Prof. J. Srinivas, first
for his valuable guidance, constant encouragement and kind co-operation throughout period of
work which has been instrumental in the success of thesis.
I also express my sincere gratitude to Prof. R. K. Sahoo, Head of the Department, Mechanical
Engineering, for providing valuable departmental facilities.
Finally, I would like to thank my fellow post-graduate students.
J S N B Vamsee Gowtham A
Roll No. 209ME1179
Department of Mechanical Engineering
National Institute of Technology
Rourkela
CONTENTS
Abstract
i
Nomenclature
Chapter 1 -Introduction
1.1
Overview of MEMS
1
1.2
Applications of microbeam Resonators
2
1.3
Types of actuation systems
3
1.3.1 Electrostatic actuation
4
1.3.2 Piezoelectric actuation
5
1.3.3 Magnetic actuation
5
Scope of the present work
6
1.4
Chapter 2 -Literature Review
2.1
Pull-in instability analysis
8
2.2
Transient chaotic dynamics
10
2.3
Pull-in and chaos control
13
Chapter 3 Single degree of freedom model
3.1
Mathematical formulation of equation of motion
18
3.2
Expression for squeeze-film damping
19
3.3
Electrostatic excitation
21
3.4
Non-dimensonalization
22
3.5
Stability analysis
24
3.6
Results and discussions
25
3.6.1
Dynamic pull-in curves
26
3.6.2
Frequency responses with pressure variation
31
3.6.3
Frequency responses with ac voltage variation
33
Chapter 4 Distributed parameter modelling
4.1
Mathematical formulation of equation of motion
36
4.2
Normalization procedure
38
4.3
Reduced order model
40
4.4
Results and discussions
42
4.5
Control of chaotic dynamics
49
Chapter 5 Conclusions
5.1
Summary
50
5.2
Future scope
50
References
52
Appendix
55
ABSTRACT
Microcantilever beams find applications in modern micro and nano devices, such as,
microswitches, microvalves and atomic force microscopes (AFM). These are generally
fabricated from single crystal silicon. Dynamics and pull-in analysis of these cantilever
resonators (Beams) is of vital importance before their fabrication. Microbeams are often actuated
by several actuation techniques such as piezoelectric, electromagnetic, thermal and electrostatic
excitations. In electrostatic actuation a controlled dc voltage is applied together with a harmonic
or suddenly applied ac component. The electrostatic force is in fact a non-linear function of
displacement of the beam. In such beams when electrostatic force exceeds the elastic restoring
force in the beam contact between the beam and supporting substrate occurs. This situation is
referred to as pull-in instability which defines the life of a resonator. The study of pull-in is very
important to design process of microsystems, such as, microgyroscopes to analyze the frequency
response, the dynamic range and sensitivity.
In this line, present work attempts to predict the static and dynamic characteristics of a
cantilever microbeam resonator. A single-DOF model incorporating squeeze-film damping
forces is first considered. Dynamic pull-in voltages are obtained from time-domain analysis and
phase trajectories, for values of ac voltage amplitudes. Parametric studies are carried-out to find
the effect of ac voltage amplitudes and air film pressure on the intermediate stability of system.
For this bifurcation analysis and Poincare maps are employed. The harmonic voltage frequency
effects are also illustrated in the form of frequency-response curves. As stiffness of test structure
reduces due to applied voltage, the spring –softening behavior is observed in frequency-response
curves. Two benchmark cantilever beam geometries available in literature are considered to
illustrate the approach.
For validation of lumped-parameter SDOF model, an approximate solution for the
continuous beam system used in Galerkin’s method is adopted. One mode approximation upto
fifth order terms is formulated to arrive the dynamic pull-in curves and frequency response
diagrams. Results are compared for the two test cases.
Finally the sliding mode-control approach is implemented to convert chaotic motion of
resonator to a desired periodic motion. The non-linear dynamic equations are solved using
Runge-Kutta fourth order time-integration explicit scheme. The integral of cantilever mode
shape functions, phase trajectories and Poincare maps obtained with MATLAB programs. The
organization of thesis is as follows: Chapter-1 presents overall introduction to microresonators
highlighting their importance in various devices. Chapter-2 gives brief description of literature
available in this field. Chapter-3 presents mathematical model and corresponding results with
respect to Single-DOF oscillator model. Chapter-4 deals with Galerkin’s approach and its output
results. Chapter-5 gives overall conclusions of the work.
Nomenclature
L
Length of microbeam (µm)
b
Width of the microbeam (µm)
d, g
Initial gap (µm)
m
Mass of the microbeam (kg)
ρ
Density of microbeam (kg/m3)
y,w
Deflection of microbeam (µm)
E
Young’s modulus (GPa)
υ
Poisson’s ratio
A
Base area of electrode (µm)2
A1
Cross-sectionl areaof micrbeam(µm)2
cv
Viscous damping coefficient (N-s/m)
c(y)
Squeeze film damping coefficient (N-s/m)
k
Structural stiffness of microbeam (N/m)
ε0
Permittivity in free space (F/m)
εr
Relative permittvity of air
Vdc, Vp
dc voltage
VAc, V0
ac voltage
Fe
Electrostatic force developed per unit length
ω1
First natural frequency (rad/s)
ω
Frequency of ac signal (rad/s)
Kn
Knudsen number
σ(y)
Squeeze-film number
µeff
Effective dynamic viscosity (Pa-s)
p0
Working pressure (Pa)
pa
Ambient pressure (Pa)
λ
Mean free path of gas molecule (µm)
C
Capacitance for the microbeam
r
Frequency ratio
φ
Mode shape function
CHAPTER-1
1. INTRODUCTION
This chapter presents a brief introduction to microelectromechanical systems, microresonators
and various types of materials, actuation schemes and objectives of the present work.
1.1 Overview of MEMS
Microelectromechanical System (MEMS) is a new research frontier due to its multiple
physical fields properties. Micro-Electro-Mechanical Systems (MEMS) is the integration of
mechanical elements, sensors, actuators, and electronics on a common silicon substrate through
the utilization of microfabrication technology. While the electronics are fabricated using
integrated circuit (IC) process sequences, the micromechanical components are fabricated using
compatible “micromachining” processes that selectively etch away the parts of the silicon wafer
or add new structural layers to form the mechanical and electromechanical devices.
Micro switches and resonators are simple Micro-Electro-Mechanical System elements, used in
several sensors and actuators. Technology of Micro-Electro-Mechanical systems (MEMS) has
experienced a lot of progress recently. Their light weight, low cost, small dimensions,
low-energy consumption and durability attracted them widely. Compared to the traditional
mechanical systems, the MEMS devices are usually small and their largest size will not exceed 1
cm, sometimes only in micron order. Studying the design parameters of Micro-ElectroMechanical by scientific techniques is of great importance; therefore simulation tools are being
improved.
Micromachining can be done for materials such as silicon, glass, ceramics, polymers, and
compound semiconductors made of group III and V elements and a variety of metals including
titanium
and
tungsten.
Silicon,
however,
remains
the
material
of
choice
for
microelectromechanical systems.
Among the various materials available silicon can be economically manufactured in
single crystal substrates. This crystalline nature provides many significant electrical and
mechanical advantages. Electrical conductivity of the silicon makes use of impurity doping
which is a very core operation of electronic semiconductor devices. Mechanically, silicon is an
elastic and robust material. Over the last few decades the tremendous wealth of information
1
accumulated on silicon and its compounds has made it possible to innovate and explore new
areas of application extending beyond the manufacturing of electronic integrated circuits.
It becomes evident that silicon is a best suitable base material on which electronic,
mechanical, thermal, optical, and even fluid-flow functions can be integrated. Ultrapure,
electronic-grade silicon wafers available for the integrated circuit industry are most common in
the MEMS technology. The relatively low cost of these substrates makes them very attractive for
the fabrication of micromechanical components and systems.
Silicon is a very good thermal conductor with a thermal conductivity greater than that of
many metals available. Its conductivity is approximately 100 times larger than that of glass. In
most of the complex integrated systems, the silicon substrate can be used as an efficient heat
sink. Due to these wide advantages mono crystalline silicon is used as material in MEMS
fabrication both as a substrate and as a structural material for MEMS devices. The elastic
behaviour of silicon depends on the orientation of structure. The possible values of Young’s
modulus ranges from 130 to 188 GPa and those for Poisson ratio range from 0.048 to 0.4.
The three major operations involved in MEMS are:
1) Sensing: It is an important operation that is used to measure the mechanical input by
converting it to an electrical signal e.g a MEMS accelerometer or a pressure sensor. (could
also measure electrical signals as in the case of Current Sensors)
2) Actuation: Using an electrical input signal to cause the displacement (or rotation) of a
mechanical structure e.g A synthetic Jet Actuator.
3) Power Generation: Generates power from a available mechanical input e.g MEMS Energy
Harvesters.
These three operations require some form of transduction schemes such as piezoelectric,
electrostatic, piezoresistive, electrodynamic, Magnetic and Magnetostrictive.
1.2 Applications of Microbeam Resonators
Microbeam resonators find wide applications in the field of medicine, automobiles,
printing. Resonators are the MEMS device components which are well known for their
2
parametric excitation during the application of electrostatic forces. A resonator is a mechanical
structure designed to vibrate at a particular resonant frequency. Resonators can be fabricated
from a range of single crystal materials with micron-sized dimensions using various
micromachining processes such as surface and bulk-micromachining. The resonant frequencies
of such microresonators are extremely stable, enabling them to be used as a time base (the quartz
tuning fork in watches, for example) or as the sensing element of a resonant sensor. Resonant
microbeams (resonators) have been widely used as transducers in mechanical microsensors since
the mid-1980s. As interest grew dramatically in MEMS devices for wireless communications
applications and the demand for high-frequency and quality factor resonators increased rapidly,
MEMS resonators were proposed in the mid-1990s as a feasible alternative to conventional
large-size resonators.
Microresonators find wide application in micro switches, amplifiers,
sensors and accelerometers. Capacitive accelerometers are utilized in automobiles for safety
purpose in seat belts. Fig.1.1 shows one such a vibrating resonator accelerometer.
Fig.1.1 Vibrating inertial accelerometer transducer
Accelerometer sensor MMA means Micro Machined Accelerometer. Micro Machined
technologies have capabilities of sensing the rate of change of velocity or acceleration. In cars,
accelerometer sensor continuously monitors the rate of change of acceleration and accordingly it
gives input to microcontroller where microcontroller scans this value with programmed values. If
acceleration is more than the programmed value it triggers the actuator. MMA1260 senses the
acceleration in only one direction i.e. in Z direction whereas MMA7260 senses the acceleration
in three direction i.e. in X, Y and Z directions. The microcantilevers also find application in
Atomic Force Microscope (AFM). The microcantilever used in AFM plays an important role for
the high resolution scanning enabling the resolution on the order of a nanometer. i.e. more than
1000 times better than optical diffraction limit. Comb drives which consists of series of
3
microbeams can also be used for the actuation purpose in the MEMS devices. Comb-drives are
linear motors that utilise electrostatic forces acting between two metal combs. While comb drives
built at normal human scales (size) are extremely inefficient there is the potential to minimize
them to microscopic or nanoscale devices where more common designs will not function.
Almost all comb-drives are built on the micro or nano scale and are typically manufactured using
silicon. Thus, micro-resonators are an integral part of many MEMS devices (RF filters, mass
sensors and AFM). Electrostatic actuation with a dc voltage superimposed to an ac harmonic
voltage is common way to actuate these resonators.
1.3 Types of actuation systems
MEMS often involve movable parts, requiring some microactuators. These microactuation
requires mechanical energy so as to obtain a vibrating or translating or rotating motion based on
the requirement for our MEMS device. The various types of actuation techniques that are used in
are electrostatic, magnetic, piezoelectric etc. In-spite of all the available actuation techniques we
prefer electrostatic actuation because of its ease in the control and high current density. The
advantages of electrostatic actuation include relatively large displacement, ease of fabrication
and controllable linearity of actuation. The voltage that which we maintain is the control
parameter for the forces developed in the actuation.
1.3.1 Electrostatic actuation
An electric charge is created around the electric field due to potential difference between two
conductors. This electric field applies a force between them. This principle, widely known since
Maxwell's era, has not been so much used during the past decades, but MEMS have a high
interest in using electrostatic actuators. The force is computed by differentiating with respect to
gap g, the energy per unit length stored in the capacitor. That is
Here Cg is capacitive per unit length, V is voltage difference between 2 bodies. Cg comprises of
parallel-plate capacitance and fringing field capacitance.
4
The main problem of electrostatic effect is that it decreases with the square of the distance
between the two charged bodies. In microscopic scale, this is a huge advantage, because most of
the structures have a very low aspect ratio (i.e. width and length are large before thickness and
gap in z direction), so the distance between bodies is really very small. Electrostatics is the most
widely used force in the design of MEMS. In industry, it is used in microresonators, switches,
micromirrors, accelerometers, etc. Almost every kind of microactuator has one or more
electrostatic actuation based version. Electrostatic actuation introduces nonlinear behavior such
as hysteresis, jump and dynamic instabilities. This nonlinear behaviour is used in many devices
suchas ultrasensitive mass-sensors, switches with low actuation voltage etc.
1.3.2 Piezoelectric actuation
Piezoelectric materials, called familiarly piezo, are materials that show a small strain when
they're placed under an electric field. This small strain is not so useful at macroscale, but at a
micro scale, this becomes very interesting source.
The piezoelectric effect is realized as the linear electromechanical interaction between the
mechanical and the electrical state in crystalline materials with no inversion symmetry. The
piezoelectric effect is a reversible process in that materials exhibiting the direct piezoelectric
effect (the internal generation of electrical charge resulting from an applied mechanical force)
also exhibit the reverse piezoelectric effect (the internal generation of a mechanical force
resulting from an applied electrical field). Actuators based on piezoelectric ceramic material
prime movers (or piezo-actuators) are finding broad acceptance in applications where precision
motion and/or high frequency operation is required.
Piezoactuators can produce smooth
continuous motion with resolution levels at the nanometer and subnanometer level.
This
property makes them useful in precision positioning and scanning systems. Piezoelectricity is
found in useful applications such as the production and detection of sound, generation of high
voltages, electronic frequency generation, microbalances, and ultrafine focusing of optical
assemblies. It is also the basis of a number of scientific instrumental techniques with atomic
resolution, the scanning probe microscopes. Atomic force microscope is one among those.
5
1.3.3 Magnetic actuation
A plate is supported by torsional hinge structure of embedded conducting wires, constituting
multi windings positioned at different locations. The conducting wires of these windings are
therefore of different lengths. Two permanent magnets are placed on the side of the plate, such
that the magnetic field lines are parallel to the plane and orthogonal to the torsional hinges.
When current passes through the coils, Lorentz forces will develop and cause rotational torque
on the plate. The direction of the torque depends on the direction of input currents. A MEMS
magnetic Actuator is a device that uses the MEMS process technology to convert an electrical
signal (current) into a mechanical output (displacement) by employing the well known Lorentz
Force Equation or the theory of Magnetism. When a current-carrying Conductor is placed in a
static magnetic field, the field produced around the conductor interacts with the static field to
produce a force. This Force can be used to cause the displacement of a mechanical structure.
1.4 Scope of the present work
Based on available techniques, in the present work, an electrostatically excited resonator is
selected for analysis. Here as dc voltage steadily increased the amplitudes of oscillation of elastic
structure increases up to certain critical value. When the applied voltage increases beyond a
critical value, called pull-in voltage, the elastic force can no longer resist the electrostatic force,
leading thereby to collapse of the structure. That is structure falls on to the substrate. The MEMS
structures are liable to this instability known as the pull-in instability. A key issue in the design
of such a device is to tune the electric load away from the pull-in instability which leads to
collapse of the microbeam and hence the failure of the device. In the present work, initially the
pull-in voltages are predicted for microcantilever resonator subjected to both dc polarization
voltage and harmonic ac voltage. By calculating the pull-in values for a particular system
considered, the operating region of microcantilever beam resonator is predicted. After predicting
pull-in voltage it is necessary to know the periodic and non periodic operating regions of
resonator under various operating variables. Time domain and phase plane diagrams are used to
determine the region where the system can be operated safely without any instability.
Single DOF and one-mode-continuous system approximations are employed to know the
dynamic response and stability regions of electrostatically actuated micro cantilever beam
6
resonator. Squeeze-film damping forces of air between substrate and beam are considered with
two dimensional Reynold’s equation. An expression for damping coefficient is adopted in terms
of squeeze number, absolute viscosity, air pressure and Knudsen number. The entire work is
classified into three broad headings:
i.
Prediction of dynamic pull-in voltages from time-domain responses, study the variation
of natural frequency with applied voltages.
ii.
Analyze the stability of resonator as a function of ac harmonic voltage and air pressure in
the gap.
iii.
Analyze the sliding-mode control strategy is convert any aperiodic states into desired
periodic output. Fig.1.2 shows the block diagram of the proposed scheme.
Use Galerkin’s decomposition scheme to
validate SDOF model by predicting dynamic
Pull-in curves, from time-domain analysis
and Frequency response
Predict a non-periodic motion for
different values of ac amplitudes
within the pull-in value by mean
of phasediagram and Poincare
Apply the SMC law to convert
the motion to periodic
Fig.1.2 Block diagram of the various stages
Computer programs are implemented for each task in MATLAB environment. Results are
validated with two benchmark examples from literature.
7
CHAPTER-2
2. LITERATURE REVIEW
This chapter is intended for reporting list of several available works on microbeam resonator
studies, starting from pull-in analysis to chaotic motion control. Many researchers have
investigated the nonlinear dynamical behaviors of micro-cantilever based instrument in MEMS
under various loading conditions. Recent works highlighted the importance of resonator beam
dynamics. All these works are classified into three broad headings:
i.
Pull-in analysis
ii.
Transient analysis
iii.
Control of resonators
2.1 Pull-in Instability studies.
This section classifies the work under pull-in instability analysis, squeeze-film damping models
and reduced-order modeling approaches.
Ouakad and Younis [1] presented an investigation of the nonlinear dynamics of carbon
nanotubes when actuated by both ac and dc load. Dynamic analysis is conducted to explore the
nonlinear oscillation of carbon nanotube near its fundamental natural frequency. The carbon
nanotube is described by an Euler–Bernoulli beam model that accounts for the geometric
nonlinearity and the nonlinear electrostatic force. A reduced-order model based on the Galerkin
method is developed and utilized to simulate the static and dynamic responses of the carbon
nanotube. The nonlinear analysis is carried out using a shooting technique to capture periodic
orbits combined with the Floquet theory to analyze their stability. Subharmonic-resonances are
found to be activated over a wide range of frequencies, which is a unique property of CNTs. The
complex nonlinear dynamics phenomena, such as hysteresis, dynamic pull-in, hardening and
softening behaviors and frequency bands with an inevitable escape from a potential well are
presented.
Lee et al. [2] proposed an improved theoretical approach to predict the dynamic behavior
of long, slender and flexible microcantilevers affected by squeeze-film damping at low ambient
pressures. Theoretical frequency response functions are derived for a flexible microcantilever
beam excited both inertially and via external forcing. They investigated the relative importance
of theoretical assumptions made in the Reynolds-equation-based approach for flexible
8
microelectromechanical systems. The uncertainties in damping ratio prediction introduced due to
assumptions on the gas rarefaction effect, gap height and pressure boundary conditions are
studied. They attempted to calculate squeeze-film damping ratios of higher order bending modes
of flexible microcantilevers in high Knudsen number regimes by theoretical method.
Nayfeh and Younis [3] studied the pull-in instability in microelectromechanical(MEMS)
resonators and found that the characteristics of the pull-in phenomenon in the presence of AC
loads differ from those under purely DC loads. They analyzed this phenomenon, dubbed
dynamic pull-in, and formulated safety criteria for the design of MEMS resonant sensors and
filters excited near one of their natural frequencies. This phenomenon is utilized to design a lowvoltage MEMS RF switch actuated with a combined DC and AC loading. The frequency or the
amplitude of the AC loading can be adjusted to reduce the driving voltage and switching time.
The new actuation method has the potential of solving the problem of high driving voltages of
RF MEMS switches. The reduced-order model with minimum of three modes are employed used
to predict the transient and steady-state responses.
Yagubizade et al. [4] presented the squeeze-film damping effects on the dynamic
response of electrostatically-actuated clamped-clamped microbeams. They utilized the coupled
nonlinear Euler-Bernoulli beam equation (1D) and the nonlinear Reynolds equation (2D). A
Galerkin-based reduced-order model and finite difference method have been used for beam and
Reynolds equations respectively. They also considered the nonlinearity effect of squeeze-fluid
near the pull-in voltage. So, from the nonlinearity behavior in large deflection, single equivalent
value consideration for damping ratio or quality factor is incorrect and cannot predict its
behavior exactly and so it must be solved the beam or plate equation coupled with Reynolds
equation. In their study made theoretically calculated pull-in voltages and pull-in times are in
good agreement with available experimental data.
Huang et al. [5] investigated theoretically the squeeze-film damping and undergoing
normal vibrations. Their model attempts to isolate viscous damping clearly from squeeze-film
damping.
Bicak and Rao [6] presented an analytical method for the solution of squeeze film
damping based on Green’s function to the nonlinear Reynold’s equation as a forcing term.
9
Approximate mode shapes of the rectangular plate are used to calculate the frequency shift and
damping ratio.
Pasquale and Soma [7] studied the effects of electro-mechanical coupling on dynamic
characterization of MEMS experimentally and analytically. The common experimental problems
relating to dynamic characterization of electrostatically actuated Microsystems are described.
Alsaleem et al. [8] presented experimental and theoretical investigations of dynamic pullin of an electrostatically actuated resonators. The regimes of ac forcing amplitude versus ac
frequency are predicted where a resonator is forced to pull-in if operated within these regimes.
Effects of initial conditions, ac excitation amplitude, ac frequency, excitation type and sweeping
type are investigated.
Krylov [9] investigated the dynamic pull-in instability of double clamped microscale
beams actuated by a suddenly applied distributed electrostatic force and subjected to nonlinear
squeeze-film damping. Galerkin’s decomposition model with undamped linear modes as base
functions was adopted. Largest Lyapunov exponent used to evaluate the stability.
Gutschmidt [10] employed a nonlinear continuum model to describe a doubly clamped
microbeam subjected to two cases of electromechanical actuation. He illustrated difference
between various reduced-order models.
2.2 Transient Chaotic dynamics
Chaos is a state of system instability in which there is a strong attractor and system rarely returns
to original state. In micro resonators, the beam undergoes sometimes chaotic motion during
different conditions of ac voltage and gap pressures. This section presents related works
available in literature.
Younis and Nayfeh [11] studied the nonlinear dynamic response of a microbeam that is
actuated by a general electric load subject to an applied axial load, accounting for mid-plane
stretching. Various vibration tools like perturbation method, the method of multiple scales, is
used to obtain two first order nonlinear ordinary-differential equations that describe the
modulation of the amplitude and phase of the response and its stability. The dc electrostatic load
is found to affect the qualitative and quantitative nature of the frequency response curves,
10
resulting in either a softening or a hardening behavior. The results obtained show that an
inaccurate representation of the system nonlinearities may lead to an erroneous prediction of the
frequency response. Also this work presents that the non linear model is capable of simulating
the mechanical behavior of microbeams for general operating conditions and for a wider range of
applied electric loads.
Chaterjee and Pohit [12] formulated a comprehensive model of an electrostatically
actuated microcantilever with larger gap between the electrodes. The model accounts the
nonlinearities of the system arising out of electric forces, geometry of the deflected beam and the
inertial terms. Static analysis is carried out to know the deflections and dynamic analysis with
five modes is carried out to known frequencies. They have observed that the reduced order
model exhibits good convergence when five or more number of modes is considered for the
analysis.
Younis and Nayfeh [13] presented a method for simulating squeeze-film damping of
microplates actuated by large electrostatic loads. The method uses nonlinear Euler’s beam
equation, Von-Karmann’s plate equation and compressible Reynold’s equation. Effect of
pressure and electrostatic force on mode shapes, natural frequencies and quality factor were
shown.
Zhang and Meng [14] presented the nonlinear responses and dynamics of the
electrostatically actuated MEMS resonant sensors under two-frequency parametric and external
excitations. For the combination resonance, response and dynamics of MEMS resonator are
studied. The responses of the system at steady-state conditions and their stability are predicted
using the method of multiple scales. The effect of varying the dc bias, the squeeze-film damping,
cubic stiffness and ac excitation amplitude on the frequency response curves, resonant
frequencies and nonlinear dynamic characteristics are examined. Nonlinear dynamic
characteristics of microbeam based resonant sensors in MEMS are investigated.
Towfighian et al. [15] presented the closed –loop dynamics of a chaotic electrostatic
microbeam actuator. Bifurcation diagrams are derived by sweeping ac voltage amplitudes and
frequencies. Period doubling, one well and two well chaos, super harmonic resonances and onoff chaotic oscillations were found in frequency sweep.
11
Zhang and Meng [16] presented a simplified model for the purpose of studying the
resonant responses nonlinear dynamics of idealized electrostatically actuated micro-cantilever
based devices in micro-electro-mechanical systems (MEMS). The underlying linearized
dynamics of a periodic or quasi-periodic system described by a modified nonlinear Mathieu
equation. The harmonic balance method is applied to simulate the resonant amplitude frequency
responses of the system under the combined parametric and forcing excitations. The resonance
responses and nonlinearities of the system are studied under different parametric resonant
conditions, applied voltages and various gaps between the capacitor plates. The possible effects
of cubic nonlinear spring stiffness and nonlinear response resulting from the gas squeeze film
damping on the system can be ignored are discussed in detail. The nonlinear dynamical
behaviors are characterized using phase portrait. The investigation provided the nonlinear and
chaotic characteristics of micro-cantilever based device in MEMS.
Ostaseviciusa et al. [17] reported the results of numerical analysis of squeeze-film
damping effects on the vibrations of cantilever microstructure effects under free and forced
vibrations. The operation of microelectromechanical systems (MEMS) with movable parts is
strongly affected by a fluid–structure interaction. Microelectromechanical devices often operate
in ambient pressure, therefore air functions as an important working fluid. The influence of air in
MEMS devices manifests as viscous air damping, which can be divided into two categories:
slide-film damping and squeeze-film damping. The usage of particular form of Reynolds
equation for modelling air-damping effects in case of developed electrostatic microswitch is
suggested.
Zhang and Zhao [18] presented the one-mode analysis method on the pull-in instability of
micro-structure under electrostatic loading. The one-mode analysis is the combination of
Galerkin method and Cardan solution of cubic equation. This one-mode analysis offers a direct
computation method on the pull-in voltage and displacement. Multi-mode analysis on predicting
the pull-in voltages for three different structures (cantilever, clamped–clamped beams and the
plate with four edges simply-supported) studied here. For numerical multi-mode analysis, they
have shown that, using the structural symmetry to select the symmetric mode can greatly reduce
both the computation effort and the numerical fluctuation.
12
Younis et al. [19] proposed a novel approach to generate reduced order models
(macromodels) for electrically actuated microbeam based MEMS and used them to study the
static and dynamic behavior of these devices.
Linear and nonlinear elastic restoring forces
and the nonlinear electric forces generated by the capacitors are accounted. Here the macromodel
uses few linear-undamped mode shapes of a straight microbeam as basis functions in a Galerkin
procedure which is obtained by discretizing the distributed parameter system in order to reduce
the complexity of the simulation and reduces significantly the computational time.
Chaterjee and Pohit [20] developed a semi analytical model of an electrostatic actuated
microcantilever beam taking into account the dependency of the effective viscosity on variable
gap spacing through changing Knudsen number. Quality factors of the system are obtained
numerically by coupled-field FE analysis.
Batra et al. [21] analyzed the vibrations of a fixed-fixed narrow microbeam excited by
electrostatic force. Electrostatic fringe-field and residual stress effects were considered. The
results of approximate one degree-of-freedom model loaded with distributed force were
compared with those of three-dimensional FE simulations.
Zamanian and Khadem [22] studied the stability of a microbeam under an electric
actuation. The stability of microresonator is studied using phase-plane diagram and Poincare
mapping. The prediction of possible chaotic behavior is studied using Melnikov theorem. It was
observerd that the pull-in instability occurs before going into the chaotic behavior for their
dimensions. That is the system does not realize any chaotic behavior.
Ibrahim et al.[23] presented the effects of common MEMS nonlinearities on their shock
response-spectrum. A case study of capacitive accelerometer is selected to show theoretically
and experimentally the effects of nonlinearities due to squeeze-film damping and electrostatic
actuation.
2.3 Pull-in and Chaos control
Pull-in avoidance by varying ac voltages as well as conversion of dangerous unstable nonperiodic states of resonator to periodic motion has been illustrated in several papers. This deals
with this classification.
13
Haghighi and Markazi [24] investigated the chaotic dynamics of micro mechanical
resonator with electrostatic forces on both sides. They obtained Phase diagram, Poincare map
and bifurcation diagram from numerical analysis which are necessary to predict chaos.
Bifurcation adopted in this study corresponds to a transient chaotic behavior of the system and
with some further increasing of the AC voltage amplitude can lead to the persistent chaotic
motion. Suppression of pull-in instability in MEMS using a high frequency actuation is
presented. They have Utilized the Melnikov function, an analytical criterion for homoclinic
chaos in the form of an inequality is written in terms of the system parameters. A robust adaptive
fuzzy control algorithm for controlling the chaotic motion is applied to eliminate chaotic motion.
Lakrad and Belhaq [25] studied the effect of a high-frequency ac tension on the pull-in
instability induced by a dc tension in a microelectromechanical system. Analysis of steady states
of slow dynamic enables us to depict the effect of the ac voltage on the pull-in. They have shown
that when the microstructure is in the pull-in instability situation, adding a high-frequency ac
actuation can create a new attracting solution which can avoid the failure of the device. This
result indicates that by implementing a HFV, pull-in instability can be prevented in capacitive
MEMS operating in regimes with relatively large amplitude. Taking advantage of the results
obtained, current efforts are directed analytically and numerically to control and suppress
undesirable nonlinear characteristics in MEMS as dynamic pull-in and hysteresis.
Yau et al. [26] analyzed the chaotic behavior of a micromechanical resonator with
electrostatic forces on both sides and investigated the control of chaos. Phase diagram and
maxmum Lyapunov exponent are used to identify chaotic motion. Byusing Fuzzy-sliding mode
control methods the chaotic motion was turned to periodic.
Alsaleem and Younis [27] investigated the stability and integrity of parallel-plate
microelectromechanical systems resonators using a delayed feedback controller. Two case
studies are investigated: a capacitive sensor made of cantilever beams with a proof mass at their
tip and a clamped-clamped microbeam. Dover-cliff integrity curves and basin-of-attraction
analysis are used for the stability assessment of the frequency response of the resonators for
several scenarios of positive and negative gain in the controller. It is found that in the case of a
positive gain, a velocity or a displacement feedback controller can be used to effectively enhance
the stability of the resonators.
14
CHAPTER-3
3. SINGLE DEGREE OF FREEDOM MODEL
Electrostatic devices have received wide popularity in MEMS field because of the fact that they
can be operated at high frequency, they give high energy density and almost infinite life.
Cantilever beam (Fig.3.1) as movable electrode is the most common example. The
Microcantilever beam that we consider is studied with two kinds mathematical models. First is
the single degree-of-freedom spring-mass model with consideration of squeeze-film damping
and electrostatic nonlinearity terms.
Microcantilever beam
L
Y
t
b
Z
d
X
Electrostatic force
Fixed substrate
Fig.3.1 Physical setup of cantilever beam
The system comprises of two components. One is fixed and other movable. The fixed component
is known as substrate. The substrate is the fixed supporting structure for the microbeam. Both
movable microbeam and fixed substrate, both have electrodes. The movable component we
considered here is of cantilever type which is made of silicon. A dc voltage (Vp) superimposed
by harmonic ac voltage (V0cosωt) is applied between the movable microbeam and the supporting
substrate. Then the electrostatic force is produced between them make the microcantilever beam
to oscillate with a particular amplitude relative to the applied voltage. Here V= Vp+ V0cosωt.
Microcantilever beam that we considered is modeled as a spring-mass system as shown in
Fig.3.2 by considering damping into account.
16
k
cv
c(y)
M
Fe
Fig.3.2 Pictorial representation of spring-mass model (m=ρAL, k=ω1m)
Electrostatic actuation is provided for the spring-mass system and so the developed electrostatic
force ‘Fe’ is responsible for the mass to come into motion. The spring-mass model internally has
the structural resistance which acts in the direction opposite to the motion of the mass. Not only
the structural resistance i.e. stiffness of the microbeam, but also the damping [c(y)+cv] that
opposes the motion of the mass. Damping in microstructures is also of special interest in the
dynamic analysis of microbeams. The system which we consider posses both squeeze-film c(y)
as well as viscous damping (cv). Squeeze film damping is an important source of extrinsic
damping. One of the important issues in MEMS modeling is simulation and characterization of
the interaction between the vibrational structure with squeezed fluid between the structure and
the substrate. When the system is undergoing the vibration, squeeze film damping starts acting. It
occurs due to resistance of the viscous fluid present in between the beam and substrate. Squeeze
film damping is dependent on the air-pressure (p0) between substrate and beam surface.
Corresponding to the particular pressure in the system the beam amplitude and behavior of
system as well varies. This interaction of Squeeze-Film damping in microstructures strongly
affects their characteristics especially their dynamic behavior. Thus, the system under study
possesses these damping forces apart from the non-linear electrostatic force. Following section
deals with the mathematical equations of motion and their dimensionless form.
The microcantilever beam that we considered is modeled as a spring-mass system as
shown in Fig.3.2 by considering damping into account. Electrostatic actuation is provided for the
17
spring-mass system and so the developed electrostatic force ‘Fe’ is responsible for the mass to
come into motion. The spring-mass model internally has the structural resistance which acts in
the direction opposite to the motion of the mass. Not only the structural resistance i.e. stiffness of
the microbeam, but also the damping [c(y)+cv] that opposes the motion of the mass. Damping in
microstructures is also of special interest in the dynamic analysis of microbeams. The system
which we consider posses both squeeze-film c(y) as well as viscous damping (cv). Squeeze film
damping is an important source of extrinsic damping. One of the important issues in MEMS
modeling is simulation and characterization of the interaction between the vibrational structure
with squeezed fluid between the structure and the substrate. When the system is undergoing the
vibration, squeeze film damping starts acting. It occurs due to resistance of the viscous fluid
present in between the beam and substrate. Squeeze film damping is dependent on the
air-pressure between substrate and beam surface. Corresponding to the particular pressure in the
system the beam amplitude and behavior of system as well varies. This interaction of SqueezeFilm damping in microstructures strongly affects their characteristics especially their dynamic
behavior. Thus the system under study posses these damping forces apart from the non-linear
electrostatic force.
3.1 Mathematical formulation of Equation of motion
The governing equation for a microcantilever beam resonator system subjected to electrostatic
force under viscous damping is given by:
!"#$
&
% If the squeeze-film damping affect is also considered then the above equation of motion is
modified as follows:
' ( !"#$
&
% The right hand term in the eqn.2.2 represents the electrostatic force required for the actuation.
This electrostatic force is denoted by ‘FE’. Here is permittivity of free space=8.8541878)
*+,F/m, is relative permittivity=1 for air, A=b)L is cross section, d is initial gap.
18
- .
01
!" #$ !"#$/
% 2 !"#$ !"#$3
% &&
3.2 Expression for squeeze-film damping
Damping in the system is of important concern and its effect is studied for designing a system.
Gas damping is the source of most MEMS and it is a strong function of viscosity. The absolute
viscosity ‘µ’ is the ratio of the shear stress ‘τ0’ between flow layers to the velocity gradient,
through the fluid channel. The nature of viscosity of a liquid and the gas are different. Viscosity
in liquid results from cohesion between adjacent molecules, but viscosity in a gas results from
intermolecular collisions within the gas. As the molecular collisions rate depends on temperature
in a gas, viscosity increases with temperature. In most of the macroscale engineering problems
viscosity is unaffected by the pressure variations. However, when the distance between the micro
beam and fixed substrate become comparable to the mean path (d ≈ λ), the viscosity becomes
sensitive to pressure variation. In this situation, molecular collisions rarely occur within the gas
layer and the momentum is directly transferred between the gas molecules and MEMS surface,
yielding a pressure-dependent viscosity.
Consider nonlinear Reynolds eq.
5% 6
482 5%&9
4 1
$
7
where 48 is pressure gradient, d is film thickness, 7 is absolute viscosity, 5 is gas density.
At low ambient pressure or in very thin films, molecular interactions with surfaces need to be
taken into account. The theory of rarefied gas flow developed by Knudsen (early 1900) is applied
here. Veijola has given a function that approximates the pressure and film width dependency of
viscosity in narrow gap. Based on different derivation considerations, one can get a different
viscosity definition. The raynolds equation is applicable only in the continuum flow regime; the
relationship between Kn and flow regimes is shown in Table 3.1
19
Table 3.1 Knudsen number and corresponding flow regimes
Kn
Flow regime
< 0.01
Continuum
0.01<Kn<0.1
Slip
0.1<Kn<10
Transitional
Kn>10
Molecular
:;<<
=:
*>[email protected] A
*@ A
7BB
=7 A *>[email protected]
***C A
*A
7BB
=7 A 7BB
=7 A *@
7BB
=7 A ***C
There are many types of damping which influence the dynamic properties of the system. Viscous
damping, squeeze-film damping and slide-film damping are the few well known types of
damping. By considering the viscous damping and squeeze-film damping for our system under
consideration, the dynamic properties are analyzed. In order to increase the efficiency of
actuation and improve the sensitivity of detection, the distance between the capacitor beams is
minimized and the area of the electrode is maximized. Under such conditions, the so-called
squeeze-film damping is pronounced. This phenomenon occurs as a result of the massive
movement of the fluid underneath the beam, which is resisted by the viscosity of the fluid. This
type of damping occurs frequently in MEMS devices. Blench model analytically solves the
linearized Reynold’s equation with trivial boundary condition. It is the squeeze-film damping
coefficient is expressed by considering the first term in Blench series as follows:
EF @9G8H K
M
G
9 L
I
I J %
20
where,
[email protected] &?
G
I J % 6 Q
Q
Q
Q
P9 R S T
I
#OBB
&@
8H % Q
O
&C
>@&NUV,,WX
G OBB
UV Y Y 8H
&N
8 %
%
y is vertical deflection of resonating beam having effective mass m, A is area of electrode, A1
is the crosssection area of the microbeam, d is initial gap width, c(y) is squeeze film damping,
b is viscous damping coefficient, k is effective linear mechanical stiffness of the system, UV is
knudsen number, Y is mean free path of gas molecule,8 is working pressure, 8H is ambient
pressure (1.013bar), σ(y) is squeeze number, O is nominal dynamic viscosity of air,OBB is the
effective dynamic viscosity, is permittivity in free space, Vp is dc voltage, V0 is ac voltage,
ω is the frequency of ac signal, Y @?Z.
3.3 Principle of Electrostatic actuation
Various physical properties of different materials and their interactions are used to achieve the
desired sensing and actuation in the Micro domain. For example, different actuation methods
include thermo-actuation, use of shape memory alloy, piezo actuation, magnetostatic actuation
and electrostatic actuation.
Among the different actuation principles, the electrostatic actuation is predominantly
employed because of short response time, high energy densities, low power consumption and
being compatible with integrated circuit processes. So, many microelectromechanical systems
make use of electrostatic forces to actuate their micro-devices such as microcantilever beam.
The electrostatic force is obtained by applying dc polarization voltage superimposed by
harmonic or sudden ac voltage between the microcantilever beam and the fixed substrate. In
microswitch system dc polarization voltage causes beam to deflect whereas the both dc
polarization voltage and harmonic ac voltage are considered. In a microresonator system in
21
which dc part is used to apply a constant deflection to microbeam and ac part is used to excite
harmonic modes of microbeam about the constant position.
Due to the nonlinear nature of electrostatic forces, the electromechanical response of
microcantilever beam considered is often nonlinear. This nonlinearity may cause loss of stability
in the electromechanical response and may limit the range of stable states. These operating states
are to be predicted inorder to operate the MEMS device which makes use of microcantilver beam
resonator.
Because of the application of voltage between the movable microcantilever beam and the
fixed substrate electric field in induced which is a cause for the electrostatic actuation. The
microcantilever beam tends to move back and forth from the fixed substrate.
As in the case of parallel plate capacitor the capacitance for the movable cantilever
microbeam is given by ‘C’.
&>
% The energy ‘E’ of a capacitor with a voltage ‘V’ across the movable microcantilver beam and
fixed substrate is given by
[
&*
% The electrostatic force between the microcantilever beam electrode and substrate can be
determined by differentiating the energy function, E with respect to the coordinate in the
direction of the force. In essence, this is the potential energy term of Lagrange’s equation.
[
- &
% 3.4 Non-dimensionalization of equation of motion
As it is more convenient to use non-dimensional form of eq. of motion, we introduce:
Let
\&
22
[email protected] 8H , \&&
IJ
#OBB S &9
& R
8H I Considering the following dimensionless variables:
] # $\ ^ `
e
#
\ _ &?
%
#
S
\b % 6 # Ra a c
c
\d &@
Ra S
Ra S
[g
\ # &[email protected]
&C
5hL
#
The governing equation 3.2 with respect to non-dimensional variable τ reduces to the
following form:
_ ii , % _
`
b
d
i
i
0 L
3
_
e_
_
!"^j
!"^j
_ _
_
# 9% _L (3.18)
Rewriting the dynamic system of equations in state-space form using
We get
_, _\
_ _ i klm_6 ^j&>
_,i _ &*
_i _, , % _, `
0 L
3 _ e_ L
_, # 9% _, b
d
!"_6 !"_6 &
_, _, i
_6
^&
23
The a non-dimensional differential equations of motion in state space (3.20-3.22) are solved
using the Runge-Kutta fourth order numerical technique with non-dimensional time ranging
from 0 to 1000 in substeps of approximately 0.01. An interactive computer program is
implemented in MATLAB so as to obtain the time histories and phase plane diagram
corresponding to the input voltages.
3.5 Stability analysis
The purpose of the stability analysis is to check whether the movable electrode will remain
within the stable equilibrium range or not corresponding to the voltages applied. If the net force
approaches an unstable point, the electrodes would have the possibility to hit the fixed structures
and/or brake away. The voltage at that point is known as pull-in voltage.
Interaction of electrostatic force with linear elastic force results in a phenomenon called
pull-in instability, which limits the functionality. Initially pull-in voltages are to be predicted to
find the operating zones for the MEMS cantilever beam resonator. In dynamic stability analysis,
it is always ensured that the resonator maintains periodic motion. In order to know the
periodicity status, one often ascertained with time-histories, phase trajectories (diagrams) and
frequency response curves. Graphs that plot one state variable against another during a dynamic
event, with time as a parameter, are very useful. They are called phase-plane plots and are
routinely used to capture the behavior of systems with several state variables.
There is a class of nonlinear systems that have an even more complex behavior, called
chaotic behavior. A characteristic of chaotic behavior is that the state of a system after an
interval of time is highly sensitive to the exact initial state. That is, a chaotic system, if started
from two initial states very close to each other, after time can be in states that are arbitrarily far
away in the accessible phase space of the system. Chaotic systems can also undergo complex
jumps between regions of phase space. So the intermediate choatic and quasi-perodic states are
analyzed for different values of ac harmonic amplitudes and frequencies as well as squeeze-film
pressure between the beam and substrate surfaces. In addition to the pull-in analysis, the
vibration tools such as Poincare maps and bifurcation diagrams are helpful to distinguish
between periodic and non-periodic motion, between different kinds of periodic motion and
24
between chaotic and truly random motion. Poincare maps are plotted in order to distinguish the
periodic and non-periodic states easily when the beam is under motion.
Poincare map is a dotted phase diagram in which each dot represents the state of system
at the end of one period. These dots or points may either be finite or forming a curve or
irregularly spread-over the diagram. Poincare map with finite number of points is said to be
stable with a particular periodicity depending on the number of points that appear in the Poincare
map. And if more points are spread in the Poincare map leading to irregular pattern, it is an
indication of chaos / non-periodicity. Bifurcation diagram is the mark for the qualitative changes
in system behavior that may occur when the parameters of the system is varied. For numerical
computation of bifurcation diagram, the variable (parameter) is increased in a constant step and
state variables at the end of the integration are used as the initial values for the next value of the
parameter. Our variable parameter in the study of this bifurcation diagrams is the ac voltage.
3.6 Results and discussions
The parametric study is carried out by varying one parameter at a time, while keeping the
remaining parameters constant. The variables for this study are ac voltage (V0), gap-pressure (p0)
and the frequency ratio (r). Dynamic analysis is carried out for the two set of microbeams having
the dimensions as given below in the Table 3.2.
Table.3.2 Dimensions of Microbeam under consideration
Parameter
Case-1
Case-2
Width of beam (b) (µm)
80
50
Thickness of beam (t) (µm)
4.5
3
Length of beam (L) (µm)
200
350
3
1
Young’s modulus E (GPa)
166
169
Poisson’s ratio (ν)
0.06
0.06
Density (ρ) (Kg/m3)
2331
2330
8.854×10-12
8.854×10-12
Initial air gap (d) (µm)
Permittivity of free space (ε0) (F/m)
25
Geometrical dimensions of microbeam greatly influence the dynamic analysis results. Dynamic
pull-in voltages vary drastically with the change in length, width, thickness and gap. The other
input parameters are choosen randomly. Computer program in MATLAB is developed for
solving the non-dimensional equation of motion. The function file providing the second state
variable is presented as follows:
%%%%%%%%FUNCTION FILE CONTAINING STATE VARIABLE X2%%%%%%%%%%%%%%%%
function y=g2(x1,x2,t)
b=80e-6;L=200e-6;th=4.5e-6;d=3e-6;
A=b*th;
I=b*th^3/12;
E=166e9/(1-0.06^2);rh=2331;
ep=8.8541878e-12;
Vp=86.1;
%DC VOLTAGE to be varied for pull-in prediction
V0=0.5;
%AC VOLTAGE to be varied for intermediate chaos prediction
zi=1e-6;
% VISCOUS DAMPING RATIO
p0=0.2*133.32; % 1 TORR=133.32 Pa, 1 atm=101.325e3 Pa=760 torr
mu=18.3e-6;
%Dynamic viscosity of air
pa=1.01325e5; %Ambient pressure
L0=65e-9;
%mean free path of air particles
r=0.75;
% non-dimensional parameter
m=rh*A*L;
k=(m)*3.516^2*E*I/(rh*A*L^4);
om0=sqrt(k/m); %NATURAL FREQUENCY
kn=pa*L0/(p0*d*(1-x1));%Knudsen number
mueff=mu/(1+9.638*kn^1.159); %effective visocity
om=r*om0;
%external frequency of AC voltage
A1=b*L;
% AREA
c0=e*A1;
c1=768*mueff*A1^2/pi^6;
c2=(12*A1*om*mueff/(pa*pi^2))^2;
al=c0*(Vp^2+0.5*V0^2)/(2*m*d^3*om0^2);
be=2*c0*V0*Vp/(2*m*d^3*om0^2);
de=0.5*c0*V0^2/(2*m*d^3*om0^2);
term1=2*zi;
%VISCOUS
term2=2*c1*d*(1-x1)/(m*om0*(4*d^4*(1-x1)^4+c2));%SQUEEZE-FILM
term3=term1+term2;
y=-x1-term3*x2+al*(1/(1-x1)^2)+(be/(1-x1)^2)*cos(r*t)
+(de/(1-x1)^2)*cos(2*r*t);%-ga*x1^3;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3.6.1 Dynamic pull-in curves
A plot drawn between dc voltage and non-dimensional amplitude at three conditions of ac
voltage by setting pressure, frequency ratio as constant. The dynamic pull-in value is identified
from the time-history amplitudes at different dc voltages. As this voltage is reached, the
amplitude shoots-up and non-dimensional displacement exceed unity, which physically means
26
that the amplitude exceeds the gap space. The results of the dynamic analysis for two cases
considered in Table-3.2 are shown below:
Case1: Here, we varied the amplitude of dc voltage and obtained in each case the time-history
and phase trajectory and noted down the amplitude of oscillation. In one of the cases where
V0=10 volts and Vdc=70 volts, the following time-history and phase diagram are observed (see
Fig.3.3). The time-step maintained in the solver is h=2π/r/1000=0.0126. It is seen that initially it
oscillates with stable center and slowly it tries to escape as seen from the oval shape of phase
trajectory.
(a) Time history
0.6
x1
0.4
0.2
0
-0.2
0
100
200
300
400
500
600
700
τ
(b)Phase portrait
0.2
x2
0.1
0
-0.1
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
x1
0.15
0.2
0.25
0.3
0.35
Fig.3.3 Time history and phase plane diagrams for spring mass model
at Vac=10v, r=0.5, Pa=800mtorr (1Torr = 133.32Pa).
By calculating the minimum and maximum deflection at particular voltage, in the time history
and phase plane diagrams obtained in the Fig.3.3, the amplitude of microbeam can be obtained.
The obtained amplitude at various dc voltages are plotted to obtain the dynamic pull-in curves.
This dc pull-in voltage predicted is very much important in the design of MEMS devices that
27
makes use of the microcantilever beams. Only within this limit of dc pull-in voltage, MEMS
devices can be operated without failure. So that safety can be ensured at the operating zone of the
device in which this microcantilever beams are used. Corresponding to the ac voltages 10v, 20v,
and 30v considered in our analysis the dynamic pull-in voltages (dc) are predicted as shown in
the Fig.3.4. It is observed that the dynamic pull-in voltage falls with the rise of harmonic ac
voltage. That is voltage greatly influences the amplitude of microcantilever beam resonator.
DC voltage vs Non-dimensional deflection
0.8
0.7
Va.c=10v
0.6
X=y/d
0.5
Va.c=20v
0.4
Va.c=30v
0.3
0.2
0.1
0
0
10
20
30
40
50
60
70
80
90
DC Voltage (Vdc)
Fig.3.4 dc voltage to amplitude is plotted at different ac voltages with pressure=800mtorr,
r=0.5. Pull-in voltages for the three curves are 83.5v, 79.95v, 69.7v
Static pull-in voltage and variation of resonant frequency of microbeam resonator are sometimes
of importance. Especially, the resonant frequency of MEMS is a very important parameter as it
determinates the amount of force the structure can exert. The resonant frequency indirectly
affects design parameters such as quality factor and effects of external noise. The resonant
frequency of a MEM structure depends on both the electrostatic force and its deformation and
may accordingly demonstrate “spring-softening” or “spring-hardening” effect of the electrostatic
force. Fig.3.5 shows the variation of frequency ratio as a function of dc voltage. Here the zero
frequency voltage corresponds to each curve is an approximate static pull-in voltage value.
28
Frequency ratio variation with dc voltage: keff~=k-eps*A*V2/d3
1
d=2µm
d=3µm
d=4µm
0.9
0.8
0.7
keff/k
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
100
V P (volts)
120
140
160
180
Fig.3.5 Variation of frequency ratio at different gap heights (V0=0)
It is seen that the resonant frequency increases with the increase of the gap at the same dc bias
voltage. The resonant frequency is a monotonically decreasing function of increasing dc bias
voltage, when the ac voltage is ignored. More accuracy in frequency results can be obtained from
frequency response curves, which includes higher order terms.
Case2: Here again the same dynamic analysis is carried out and the pull-in curves are obtained
from the amplitudes of time-histories. Fig.3.6 shows the time history and phase diagram at
V0=0.5 volts and VP=2.5 volts. It is seen that there is some initial chaos in the system and there
after system become periodic. Such a set of time histories and phase plane diagrams are utilized
for the calculation of amplitude of microbeam. The difference of maximum to minimum
deflection of microbeam under electrostatic actuation gives the amplitude at that particular
voltage. Here the minimum and maximum of x1 gives the amplitude at the voltage that we
considered. Fig.3.7 shows the Poincare map at this configuration, where 54 points have been
converged into only 9 points showing overall periodicity of the system.
29
Fig.3.6 Time history and phase plane diagrams for spring mass model
at Vac=0.5v, r=0.5, Pa=800mtorr (1Torr = 133.32Pa)
Fig.3.7 Poincare map at V0=0.5 v and VP=2.5 volts
Fig.3.8 shows the pull-in curves obtained for three different ac voltage amplitudes.
30
DC voltage vs Non-dimensional deflection
0.4
0.35
0.3
Va.c=1
X=y/d
0.25
Vac=0.5
0.2
0.15
Vac=0.02
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
DC voltage (Vdc)
Fig.3.8 dc voltage to amplitude is plotted at different ac voltages with pressure=800mtorr,
r=0.5. Pull-in voltages for the three curves are 3.25v, 3.23v, 3.17v.
With the increase of ac voltage the corresponding dynamic pull-in voltage falls. And hence we
can notice that high ac voltage minimize the region of stability. For a device to be operated, there
are many parameters to be considered. Based on the requirements and the application at which
the device is used these data obtained from pull-in curves are utilized in order to evaluate
whether the system can be operated effectively without any failure or not.
3.6.2 Frequency response with pressure variation
Case1: The squeeze-film pressure effect in terms of gap pressure po is studied. As shown in
Fig.3.9, at different operating pressures the response curve amplitudes are changing. The system
behavior is examined with the help of frequency response plots that are plotted between the
frequency ratio and the amplitudes of non-dimensional deflection. These curves are also obtained
from the time-history plots.
31
Frequency ratio vs Non-dimensional deflection
1
0.9
0.8
Pa=600mtorr
0.7
X=y/d
0.6
Pa=800mtorr
0.5
Pa=1torr
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
Frequency ratio(r)
Fig.3.9 Frequency ratio vs Non-dimensional deflection for
spring-mass model at Vac=20v, Vdc=50v.
With the rise of pressure it is found that the amplitude falls due to the fact that the damping
becomes prominent at high pressures. Hence the peak of frequency response curve falls as
pressure increases. These plot is for the first case of dimension considered.
Case2: For the second case also, a similar analysis is carried out and the effect of frequency ratio
on the amplitudes of response is predicted. Fig. 3.10 shows the corresponding frequency
response curve. Based on the dimensions of microbeam the operating pressure varies. Hence
before design the dynamic analysis is carried out for finding the operating parameter of the
device. Different microbeams have different operating regions due to the variation in
dimensions. Our study of these frequency response plots corresponding to system parameters
such as pressure indicate the stable operation region of device to be operated at a particular
environment. The frequency response is also obtained directly from FFT analysis and one can
analyze the curves to known the dynamic stability of the system at corresponding operating
pressures.
32
DC voltage vs Non-dimensional deflection
0.6
0.5
Pa=150mtorr
X=y/d
0.4
Pa=200mtorr
0.3
Pa=250mtorr
0.2
0.1
0
0
0.5
1
1.5
2
2.5
Frequency ratio (r)
Fig.3.10 Frequency ratio vs Non-dimensional deflection
for spring-mass model at Vac=1v, Vdc=2v.
3.6.3 Frequency response with ac voltage variation
Case 1: Frequency response curves are then obtained by varying the ac voltage amplitudes at
constant pressure condition of p0=800 milli-torr. Fig.3.11 shows these frequency domain plots.
It is observed that with the increase of ac voltage the amplitude of microbeam increases. And
another observation is that the peak of frequency response curve rises with the rise in ac voltage
that is applied between the microbeam and the fixed substrate.
Case 2:
Fig 3.12 shows the system behavior at different ac voltages. With the increase of ac voltage the
peak of frequency response curve rises. High amplitude is obtained corresponding to high input
ac voltage that is given to the microcantilever beam system.
33
Frequency ratio vs Non-dimensional deflection
0.7
0.6
VA.c=5v
X=y/d
0.5
Va.c=10v
0.4
0.3
Va.c=20v
0.2
0.1
0
0
0.5
1
Frequency ratio (r)
1.5
2
2.5
Fig.3.11 Frequency ratio vs Non-dimensional deflection
for spring-mass model at Vdc=50v, pressure p0=800mtorr.
Frequency ratio vs Non-dimensional deflection
0.35
0.3
Va.c=0.5
0.25
X=y/d
Va.c=0.75
0.2
Va.c=1
0.15
0.1
0.05
0
0
0.5
1
Frequency ratio (r)
1.5
2
Fig.3.12 Frequency ratio vs Non-dimensional deflection for spring-mass
model at Vdc=2v, pressure p0=800mtorr.
34
2.5
CHAPTER-4
35
4. DISTRIBUTED PARAMETER MODELS
The dynamics of MEMS microcantilever beams are represented by partial-differential equations
(PDEs) and associated boundary conditions. The most widely adopted method to treat these
distributed-parameter problems is to reduce them to ordinary-differential equations (ODEs) in
time, using the Galerkin reduced-order technique. Hence the reduced-order modeling of MEMS
is gaining attention as a way to balance the need for enough fidelity in the model against the
numerical efficiency necessary to make the model of practical use in MEMS design.
The formulated reduced order equations are then solved numerically using the fourthorder Runge-Kutta numerical technique in the MATLAB. This reduced-order model is utilized to
investigate the non-linear dynamics of microcantilever beam.
4.1 Mathematical formulation of equation of motion
A precise prediction of the pull-in plays an important role in the device operations. The
prediction of pull-in started herein by establishing the governing distributed equations of motion
for the deformable micricantilever beam as shown in Fig.4.1. The formulated equation is given
by:
[i g
L
5n
o
- p 9
_ L
$ $
_ Where,
[i [
q Here [ i is the effective Young’s modulus, A=bt and g rs t
,
Beam
Electrode
Fig.4.1. Microbeam oscillator
36
The axial force N=Gn can be taken as zero for the cantilever beam as there is no residual stress
in it. The electrostatic force per unit length developed by the electrostatic actuation is given by:
- u 9
% FE represents the equivalent pressure on the deformed plate due to the applied electrostatic force
developed per unit beam length.
Here,
' !"#$(.
In order to model squeeze-film damping force FA, following are considered:
The two dimensional Reynolds equation for the fluid flow is given by:
5H 6 8
5H 5H 6 8
1
2
1
2
9&
7BB $
_ 7BB _
where µ, p,5H and g represents the effective air viscosity, pressure, air density, and air-film
thickness respectively.
The effective air viscosity ‘7BB ’ is obtained similar to that in a SDOF model, using Veijola’s
theory. That is:
7BB 7
>@&NUV,,WX 99
Intermolecular collisions play a prominent role and the flow properties will affect the Knudsen
number ‘UV ’, which is a dimensionless measure of the relative magnitudes of the gas mean free
path and flow characteristic length. One of the most widely used applications for the Knudsen
number is in microfluidics and MEMS device design. We adopted
Y
UV 9?
where UV is Knudsen number, λ=0.064µm and g = (d-w).
Assuming the gap is much less than the beam length, i.e. g<<L, one can obtain the equivalent
pressure force per unit beam length due to the squeeze-film effect as follows:[see Krylov and
37
Maimon, ‘Pull-in dynamics of elastic beam actuated by continuously distributed electrostatic
force’ J. Vib. Acoust, Vol.126, pp.332-342, 2004]
p Uv
where,
w w
[email protected]
$
&7x y
\ Uv \ x 9C
6
w
%
%
In common practice, since B >> g, Uv could be large enough such that p is too large to neglect
in the governing equation (4.1) as compared to electrostatic force term - .
For the convenience of ensuring analysis, the system
equation (4.1) is further
non-dimensionalized to be of the form
|
|
zL {
| z {
v /
.~
U
`x- 9N
]
z}w L zj
Where
|
_
$
\ _w \ ] \ x 9>
h
€
%
%
5hL
hL
€f
\ ~ \ x- 9*
[g
[g€
|
`
hL
7~ x 6
hL
v \
7~
7\
U
9
[g€
|6
[g%6
In equation (4.8), the beam width x affects the squeeze-film effect through the term associated
v . With a complete system equation in hand, model decomposition is performed to derive
with U
the pull-in voltages.
4.2 Normalization procedure
The method of Galerkin decomposition is employed herein to approximate the system
equation (4.8). So the equation (4.8) can be reduced to an ordinary differential equation in terms
38
of the first-order approximations of the model shape function. The deflection of the beam can be
approximated as:
„
|_w\ ]  ‚V _wƒV ] 9
V…,
where ‚V _w is nth cantilever beam mode shape, no and ƒV ] is nth generalized coordinate. M is
number of modes considered. Thus it results-in M ordinary 2nd order differential equations in
terms of qn, n=1, 2, 3. . . M.
‚_w Where,
o '"†Z bh‡ "†Zn bh‡ o, !" bh‡ !"n bh‡(ˆ‰ŠEkl‹ŒŽŽŠ9&
o
"†Z bh "†Zn bh
99
!" bh !"n bh
o "†Z bh "†Zn bh o, !" bh !"n bh9?
where bh NC? for first mode of cantilever beam. In present case one term (mode)
approximate is considered. i.e. | ‚, _wƒ, ] ‚, ‡ƒ, ]
‚‡ Is chosen for normalization such that,
,
 ‚ %_ [email protected]
By using the Taylor series of expansion, the equation (4.8) is expanded up to four terms. The
expanded equation is simplified as:
L
| |
|
‘~ 7~ x 6 &
| @
| *
| 6 ?
| L ’
L
_w
]
]
` | &
| 9
| 6 ?
| L 9C
We know that,
| ‚_wƒ]9N
So,
39
“
“
L
|
L‚
ƒ
ƒbhL ‚ ƒ#, ‚
_ L
_ L
L
|
ƒ#, ‚9>
_ L
Also
|
ƒ ′′ ‚9*
]
|
ƒ ′ ‚9
]
4.3 Reduced order model
By substituting the equations (4.19-4.21) in the equation (4.17) we obtain:
ƒ#, ‚ ƒ ′′ ‚ ‘ 7~ x 6 &‚ƒ @‚ ƒ *‚ 6 ƒ 6 ?‚ L ƒ L ’ƒ ′ ‚
` ‚ƒ &‚ ƒ 9‚ 6 ƒ 6 ?‚ L ƒ L 9
Integrating both sides with respect to ‘_w’ after multiplying by ‘‚’ hence the equation is as
follows:
,
 ”ƒ#, ‚ ƒ ii ‚ ‘ 7~ x 6 &‚ƒ @‚ ƒ *‚ 6 ƒ 6 ?‚ L ƒ L ’ƒ i ‚ • %_
,
 '` ‚ ‚ ƒ &‚ 6 ƒ 9‚ L ƒ 6 ?‚ W ƒ L ( %_w9&
on simplification the equation (4.23) with equations(4.19-4.21), we get:
x6
x6
,
x6 ,
x6
,
x6
,
ƒ –. 7~ / &7~ ƒ  ‚ %_ @7~ ƒ  ‚ %_ *7~  ‚ %_ ?7~  ‚ J %_— ƒ i
ii
6
40
L
W
,
,
,
,
ƒ#, ` – ‚%_ ƒ & ƒ  ‚ 6 %_ 9ƒ 6  ‚ L %_ ?ƒ L  ‚ W %_—99
Using the state variables x1, x2 the equation 4.24 can be written as:
“ _ ‘ 7~ x6 &_, $ @_, $, *_, 6 $ ?_, L $ ’_ 'bhL ` (_,
` $L &_, $ 9_, 6 $, ?_, L $ 9?
Here,
ƒ _, [email protected]
ƒ ′ _,′ _ 9C
ƒ ′′ _,′′ _ 9N
also
$ ˜ ‚ J %_ \ $ ˜ ‚ W %_ \ $, ˜ ‚ L %_ \ $ ˜ ‚ 6 %_ \ $6 ˜ ‚ %_ 9>
,
,
,
,
,
The obtained results can be utilized to study the behavior of present electrostatic cantilever
resonators and capacitive resonators.
Based on above analysis, programs are developed in MATLAB to predict the integrals of the
mode shape function and define the state variable functions in order to solve the equation (4.25).
Following function program shows the integration procedure during calculation of static and
dynamic pull-in states.
%%%%%%%%%%%%%%%%%%%%%%%%%%
syms x phi;
mu=(sin(1.875104)+sinh(1.875104))/(cos(1.875104)+cosh(1.875104));
y=0:0.01:1;
bl=1.875104;
n2=sin(1.875104)-sinh(1.875104)-mu*(cos(1.875104)-cosh(1.875104));
t5=((sin(1.875104.*y)-sinh(1.875104.*y)-mu.*(cos(1.875104.*y)cosh(1.875104.*y))))/n2;;
ma=max(t5);
phi=(sin(1.875104*x)-sinh(1.875104*x)-mu*(cos(1.875104*x)cosh(1.875104*x)))/n2;
41
t3=eval(int(phi^2,0,1));
t2=eval(int(phi^3,0,1))/t3;
t4=eval(int(phi,0,1))/t3
t1=eval(int(phi^4,0,1))/t3;
t3=1;
eps=8.854e-12;
b=8e-6;t=2e-6;
L=100e-6;
E=166e9;%/(1-0.06^2);
I=b*t^3/12;
d=0.5e-6;
alpha=eps*b*L^4/(2*E*I*d^3);
q=0:0.01:1;
vdc=sqrt((1.875104)^4*(t1.*q.^3-2*t2.*q.^2+t3.*q)/(alpha*t4));
vp=max(vdc);
figure(1);
plot(vdc,q);
st1=3*E*I/L^3;
%FIRST PULL-IN VALUE IS ESTIMATED FROM ONE-MODE GALERKIN APPROACH AND USING
THIS STIFFNESS IS COMPUTED WITH ASSUMPTION THAT PULL-IN OCCURS AT 1/3RD OF
INITIAL GAP
st=27*vp^2*eps*b*t/(8*d^3);
vdc1=sqrt(2*d^3.*(1-y).^2.*y*st/(eps*b*t));% IDEAL CURVE
figure(2)
plot(vdc,y,vdc1,y);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
4.4 Results and discussions
Case1: Static pull-in behaviour can be studied by avoiding first two terms in eq.(4.24). Fig.4.2
shows the pull-in curve obtained from one-mode approximation using Galerkin’s method in
comparison with the theoretical pull-in curve. It is seen that the non-dimensional gap height w/d
at which pull-in occurs is at round 0.46 in comparison to 0.33 for the theoretical approach. This
agrees well with the literature available. The dynamic equations (4.24) are solved again using RK solver with following sub-function:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function y=ji(x1,x2,t,Vp)
to=0.4876; t1=0.5872;t2=0.7389;t3=1;t4=1.566;too=0.4171;
eps=8.854e-12;
l=350e-6;b=50e-6;th=3e-6;d=1e-6;
ro=2330;E=169e9/(1-0.06^2);
bm=b/d;
A=b*th; I=b*th^3/12;
Vo=0.02;
%AC VOLTAGE
alpha=eps*b*l^4/(2*E*I*d^3);
r=1;
% DAMPING COEFFICIENT FROM NON-DIMENSIONAL PARAMETER
42
mu=18.6*10^-6;
pa=0.8*133.32;
kn=0.0064/(pa*d);
mueff=mu/(1+9.638*kn^1.159);
wo=(1.875)^2*sqrt((E*I)/(ro*A*l^4));
MU=(mueff*l^4*wo)/(E*I);
c=0.73; % non dimensional damping quotient;
B=1.875104;
V=Vp+Vo*cos(r*t);
Z=MU*bm^3;
y=-(c+Z*(1+3*x1*t2+6*x1^2*t1+10*x1^3*to+15*x1^4*too))*x2-(B^42*alpha*V^2)*x1+alpha*V^2*(t4+3*x1^2*t2+4*x1^3*t1+5*x1^4*to);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Fig. 4.2 Static pull-in for case-1
By varying the dc voltage VP, we can obtain again a non-dimensional amplitude plots
representing the dynamic pull-in curves. Fig.4.3 shows the time-history and phase diagram at
VP=79.25 volts and V0=10 volts, where system is at the verge of pull-in state.
43
Time history for Vac= 10 v,V Dc =79.25 v
0.8
x1
0.6
0.4
0.2
0
0
100
200
300
400
500
600
700
800
900
1000
τ
Phase plane diagram for Vac=10v , Vdc=79.25v
1
X2
0.5
0
-0.5
-1
0
0.1
0.2
0.3
0.4
X1
0.5
0.6
0.7
0.8
Fig.4.3 Time history and phase plane diagrams for reduced–order Galerkin model at
Vac=10v, r=0.5, Pa=800mtorr (1Torr = 133.32Pa)
By calculating the minimum and maximum deflection at particular voltage, in the time history
and phase plane diagrams, the amplitude of microbeam can be obtained. The amplitude at
different dc voltages is calculated and a plot is drawn between the dc voltage and the nondimensional deflection as shown in Fig.4.4.
Non-dimensional deflection vs DC voltage
1.2
1
Va.c=10v
X=y/d
0.8
0.6
Va.c=20v
0.4
Va.c=30v
0.2
0
0
10
20
30
40
50
60
70
80
90
DC Voltage (Vdc)
Fig.4.4 dc voltage to amplitude is plotted at different ac voltages with pressure=800mtorr,
r=0.5. Pull-in voltages for the three curves are 79.25v, 70.55v, 61.92v.
These curves are at three different ac voltages. It is shown that with the rise of ac voltage applied
the dynamic pull-in voltage falls.
44
Case2: A similar approach is adopted for the second set of geometry and it is found that the
dynamic pull-in voltage for V0=0.5 volts occurs at 2.75 volts. By calculating the minimum and
maximum deflection at particular voltage, in the time history and phase plane diagrams as
obtained in the Fig.4.5, the pull-in curve can be plotted as shown in Fig.4.4.
Time history at Vac=0.5v , Vdc=2.75v
0.8
x1
0.6
0.4
0.2
0
0
100
200
300
400
500
600
700
800
900
1000
τ
Phase plane diagram at Vac=0.5v , Vdc=2.75v
1
x2
0.5
0
-0.5
-1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x1
Fig.4.5 Time history and phase plane diagrams for reduced–order Galerkin model at
Vac=0.5v and Vdc=2.75v, r=0.5, Pa=800mtorr (1Torr = 133.32Pa)
Vdc vs Non-dimensional deflection
0.8
0.7
Vac=0.02v
Vac=0.5v
Vac=1v
0.6
X=y/d
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
DC voltage (Vdc)
2.5
3
3.5
Fig.4.6 Dynamic pull-in curves at three different ac voltages with pressure=800mtorr,
r=0.5. Pull-in voltages for the three curves are 3.25v, 3.23v, 3.17v
45
Dynamic pull-in curves as shown in the Fig.4.6 indicate the pull-in voltages at three different ac
voltages. It is shown that with the rise of ac voltage applied the dynamic pull-in voltage falls.
The pull-in voltage is the primary and essential analysis to be carrired out for the prediction of
stability for the MEMS microbeams. The zone of stability predicted can be made use for the
design of these MEMS devices with makes use of this microbeams. Fig.4.6 shows the frequency
response diagram for case-1 at a pressure p0=600 mtorr.
Frequency ratio vs non-dimensional deflection
0.7
0.6
X=y/d
0.5
Pa=600milltorr
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
Frequency ratio (r)
1.2
1.4
1.6
1.8
Fig.4.7 Frequency ratio vs Non-dimensional deflection for
reduced-order Galerkin model at Vac=20v , Vdc=50v
Frequency response curves are drawn with the variation of system input parameters such as ac
voltage and the pressure. As indicated in the Fig.4.7, with the increase of ambient pressure the
peak of the frequency response falls due to the increased damping effect which opposes the
deflection of the microbeam. With the variation of the pressures, it is observed that the curves
deviated very slightly. Fig.4.8 shows the frequency response curve for the second case under
consideration at a gap pressure p0=150 mtorr. It is seen from the figure that pull-in amplitude
occurs at this pressure due to very small damping effect before it reaches primary resonant zone
r=1. The figure shows the softening nonlinearity in the system with bent of backbone curve
backwards. Fig.4.9 shows the effect of variation of ac voltage amplitude on frequency response
for case-1. It is seen that with increase of ac volage the peak values are increasing.
46
Frequency ratio vs Non-dimensional deflection
1
X=y/d
0.8
0.6
Pa=150milltorr
0.4
0.2
0
0
0.5
1
1.5
Frequency ratio (r)
2
Fig.4.8 Frequency ratio vs Non-dimensional deflection for
reduced-order Galerkin model at Vac=1v , Vdc=2v.
Frequency ratio vs non-dimensional deflection
X=y/d
0.7
0.6
Vac=5v
0.5
Vac=10
v
Vac=20
v
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Frequency ratio (r)
Fig.4.9 Frequency ratio vs Non-dimensional deflection for
reduced-order Galerkin model at Vdc=50v, pressure =800mtorr.
A similar frequency response plot is drawn for case-2 as shown in Fig.4.10.
47
1.8
Frequency ratio vs Non-dimensional deflection
X=y/d
1
0.8
Vac=0.5v
0.6
Vac=0.75v
0.4
Vac=1v
0.2
0
0
0.5
1
1.5
2
Frequency ratio (r)
Fig.4.10 Frequency ratio vs Non-dimensional deflection
for reduced-order Galerkin model at Vdc=2v, pressure =800mtorr.
At three different ac voltages the frequency response curves are drawn. Here also with rise of ac
voltage the amplitude of microbeam as well the peak of frequency response plot rises. For higher
ac voltages, more nonlinear spring softening behaviour is observed from this figure. Table 4.1
shows the comparison of dynamic pull-in results obtained for two cases using the spring-mass
model and Galerkin model. It is seen that the discrepancy occurs due several factors such as
ignoring higher order terms in mode shape expansion and difference in damping expressions etc.
Tabel.4.1 Comparision of present analytic results for model 1 and model 2
micro-beams with Case1 at V0=10v and Case 2 at V0=1volt.
Pull-in voltage
%
Case
Error
Model 1
Model 2
1
83.5
79.25
5.08
2
3.25
3.05
6.15
48
4.5 Control of chaotic dynamics
A better performance of the resonator can be achieved when the oscillations are made periodic
with large amplitudes. Also the energy enhancement and subsequent increase in performance is
achieved by making the system to be periodic. In present work an adaptive control approach is
employed for stabilization of the system in high amplitude oscillation state. The control law can
be stated as:
&x& =f (x, x& ,t) +u
where u is appended control input and is given as:
u= -f + &x& d + kv ( x& d − x& )
Here xd is desired trajectory chosen as periodic function and kv is a positive constant. The ac
voltage amplitude is set at 2.76 volts and desired trajectory is selected as 0.3sin0.5τ. The
working pressure p0=200 mtorr and r=0.5 are maintained. Effectiveness of the controller is
illustrated in Fig.4.11. It is seen that the controller is activated at time τ=25. Here kv=2.
(a) Time history
(b) Phase-diagram
Fig.4.11 Outputs of the controller
Motion maintains a periodic behaviour after this time.
49
CHAPTER-5
49
5. CONCLUSIONS
5.1 Summary
In this work, the results of dynamic analysis of electrostatically excited single-row cantilever
micro beam resonator have been presented. Even though a lot of work has been already done in
literature, present work attempted the problem in a new direction. Using both spring-mass model
and the reduced-order Galerkin techniques the dynamic pull-in voltages for the two cases of
microcantilever beam considered were predicted. A criterion for prediction of dynamic pull-in
has been shown with reference to time-histories and phase-trajectories. Nonlinear squeeze-film
model of Blench was used to model the damping along with linear viscous damping. The
dependency of gap pressure on the dynamic performance of resonator has been shown. It is seen
that with pressure increase, the corresponding amplitudes in frequency response reduced. Also,
as other parameter, ac voltage amplitude was varied and the effects on pull-in voltage, system
intermediate chaos have been predicted. The accuracy of solution from one-mode Galerkin
approximation method was verified with that of single-degree of spring mass model. In
summary, following things were performed:
(1) Considering a dc voltage less than that of the pull-in voltage the frequency response
curves are drawn at different ac voltages. And so the effect of ac voltage on the system is
investigated.
(2) The dynamic analysis results obtained from the spring –mass model are validated with
that of the results obtained from reduced-order model.
Due to difference of squeeze-film damping models considered for the spring-mass model and the
reduced-order Galerkin model, a variation in the dynamic pull-in voltages and the frequency
responses was observed. Poincare maps were simultaneous obtained for all the cases to know
the status of intermediate non-periodicity in the system is predicted.
5.2 Future work
The electrostatic forces act along the gap between micro beam and the substrate. Such an effect
is known as fringing field effect. The effect of fringing field on the dynamics of microcantilever
beam resonator can be included in the model as a future scope to get more accuracy. The
obtained results for single-degree-of-freedom spring-mass system and reduced-order Galerkin
50
model may be compared with that of the multi-degree-of-freedom modeling of cantilever
resonator by including damping effects. Finite element approach with mutli-physics options has
to be adopted to get more insights into the resonator dynamics in micro level. Experimental
characterization has to be done to predict the frequency response curves of the electrostatically
excited cantilever beam model so as to know the real time damping effects on overall system
dynamics. Even some of the above things were attempted; the results have not been completely
obtained due to lapse of time.
51
REFERENCES
[1] H.M.Ouakad and M.I.Younis, ‘Nonlinear Dynamics of Electrically Actuated Carbon
Nanotube Resonators’, journal of computational and nonlinear dynamics, vol.5, pp.0110099-1,
2010.
[2] Jin Woo Lee and Ryan Tung, Arvind Raman, Hartona Sumali and John P Sullivan,‘Squeezefilm damping of flexible Microcantilevers at low ambient pressures: theory and experiment’,
journal of micromechanics and microengineering, vol.19,14pp, 2009.
[3] A. H.Nayfeh, M.I. Younis and E.M.Abdel-Rahman, ‘Dynamic pull-in phenomenon in
MEMS resonators’, Nonlinear Dyn, vol. 48, pp.153-163, 2007.
[4] H.Yagubizade, M.Fathalilou, G.Rezazadeh and S.Talebian, ‘Squeeze-Film Damping effect
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APPENDIX
RUNGE-KUTTA SOLUTION FOR TIME INTEGRATION
Consider set of n-simultaneous ordinary diff. eqs in canonical form:
%,
™, $\ , \ \ š V %$
%
™ $\ , \ \ š V %$
%V
™V $\ , \ \ š V %$
Expanding using 4th order R-K formulas we get,
›œ,\ › .,  &6 9L / *nW ž Ÿ \\&\ š Z
@
, n™ $› \ ›, \ › \ š ›V ,,
,
,V
n
\ › \ š ›V  n™ $› \ ›, ,
V
n
\ › \ š ›V 6 n™ $› \ ›, 6
6V
6,
\ › \ š ›V L n™ $› n\ ›, This method is programmable using nested loops.
In MATLAB, the values of k, y can be put into vectors to easily evaluate in matrix form.
Following pseudo code is adopted in this work:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
x01=0;x02=0;
r=0.5;pe=1000;
h=2*pi/r/pe;
tma=700;
i=1;
for t0=0:h:tma
X1(i)=x01;X2(i)=x02;
k1=h*g1(x01,x02,t0); l1=h*g2(x01,x02,t0);
k2=h*g1(x01+0.5*k1,x02+0.5*l1,t0+h/2);
l2=h*g2(x01+0.5*k1,x02+0.5*l1,t0+h/2);
k3=h*g1(x01+0.5*k2,x02+0.5*l2,t0+h/2);
l3=h*g2(x01+0.5*k2,x02+0.5*l2,t0+h/2);
k4=h*g1(x01+k3,x02+l3,t0+h); l4=h*g2(x01+k3,x02+l3,t0+h);
x1n=x01+(k1+2*k2+2*k3+k4)/6; x2n=x02+(l1+2*l2+2*l3+l4)/6;
x01=x1n;
x02=x2n;
i=i+1;
55
end
% Generation of x1(nt) and x2(nt)
T=[0:h:tma];
for j=2*pe:pe:length(T)
n=(j-pe)/pe;
X3(n)=X1(n*pe);
X4(n)=X2(n*pe);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
56
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