the effects of noise on parametrically excited systems
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The Effects of Noise on Parametrically Excited Systems with Nonlinear Damping
by
Donghao Li
Bachelor of ScienceDepartment of Mechanical And Civil Engineering
Florida Institute of Technology2020
A thesissubmitted to the College of Engineering and Science
of Florida Institute of Technologyin partial fulfillment of the requirements
for the degree of
Master of Sciencein
Mechanical Engineering
Melbourne, FloridaJuly, 2021
© Copyright 2021 Donghao Li
All Rights Reserved
The author grants permission to make single copies.
We the undersigned committeehereby approve the attached thesis
The Effects of Noise on Parametrically Excited Systems with Nonlinear Damping
by Donghao Li
Steven W. Shaw, Ph.D.ProfessorMechanical and Civil EngineeringMajor Advisor
Brian A. Lail, Ph.D.ProfessorComputer Engineering and SciencesCommittee Member
Hector Gutierrez, Ph.D.ProfessorMechanical and Civil EngineeringCommittee Member
Ashok Pandit, Ph.D.Professor and Department HeadMechanical and Civil Engineering
ABSTRACT
Title:
The Effects of Noise on Parametrically Excited Systems with Nonlinear Damping
Author:
Donghao Li
Major Advisor:
Steven W. Shaw, Ph.D.
Parametric oscillators are a class of resonating systems in which a parameter, such as
stiffness in a mechanical system or capacitance in an electrical system, is periodically
modulated in order to alter the system response in a desired manner. A resonance ef-
fect occurs when the pump frequency is near twice of resonant frequency. Such systems
are said to be “parametrically pumped,” and this pump can, above a certain amplitude
threshold, destabilize the system in the absence of nonlinearities. Parametric resonance
is widely observed in nature and has been employed in a large variety of engineered
systems, most notably in micro-electro-mechanical systems (MEMS). Considering both
open and closed loop operations of a parametric oscillator, this work expands on previ-
ous studies by embracing nonlinear damping and multiplicative noise in the modeling
and analysis and investigates their effects. Fluctuations due to noise, signal-to-noise
ratio (SNR), and power spectral density (PSD) for an open loop system are computed
and are compared with stochastic simulations. Phase diffusion for a phase-locked loop
(PLL) is also analyzed, which plays a pivotal role in time-keeping devices. The main
conclusions are relevant to SNR aspects of sensors and to the frequency stability of
time-keeping systems. It is shown that multiplicative noise serves as an ultimate lim-
iting factor in the resolution of sensors and precision of clocks.
iii
Table of Contents
Abstract iii
List of Figures vi
Acknowledgments xiv
1 Introduction 1
1.1 Parametric Resonance and Its Applications . . . . . . . . . . . . . . . . 3
1.2 Sources of Nonlinearity and Noise in N/MEMS . . . . . . . . . . . . . . 5
1.3 Noise Squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Deterministic Nonlinear Behavior 10
2.1 Open Loop Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Phase-Locked Loop Model . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Open Loop Operation with Noise 22
3.1 Theory Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Stochastic Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4 Phase-Locked Loop Operation with Noise 42
4.1 Theory and Simulation Results . . . . . . . . . . . . . . . . . . . . . . 43
iv
4.2 Precision of Time Keeping . . . . . . . . . . . . . . . . . . . . . . . . . 45
5 Conclusions and Future Work 48
5.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.3 Future Work: Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 50
References 52
A Background in Nonlinear Stiffness 61
B Background in Nonlinear Damping 69
C Nonlinear Analysis 71
v
List of Figures
2.1 Steady-state response, with ω0 = 1, α = 0.05, β = 0.05, and γ =
0.019245. (a) Response in the amplitude-frequency space. The three
curves correspond to λ = 0.25 (light blue), λ = λcr = 0.4 (blue), and
λ = 0.65 (dark blue), respectively. The solid curves indicate stable re-
sponse and the dashed curve corresponds to unstable response. Trivial
response exists across the entire frequency domain. (b) Two parame-
ter bifurcation diagram. The blue curve shows the pitchfork bifurca-
tion condition, or Arnold tongue (AT), and the orange curve shows the
saddle-node bifurcation condition (SN). The critical condition is where
the two curves meet at the bottom . . . . . . . . . . . . . . . . . . . . 17
2.2 Steady-state response, with ω0 = 1, α = 0.01, β = 0.01, and γ =
2/225. The three curves correspond to λ = 0.0425 (light blue), λ =
λcr = 0.05 (blue), and λ = 0.625 (dark blue), respectively. (a) Steady-
state amplitude versus imposed phase shift. The peak amplitude always
occurs at ∆Φ = −π/2. (b) Operating frequency versus imposed phase
shift. (c) Amplitude-frequency space. The shapes of the curves coincide
with those of an open loop system; however, the closed loop response is
always stable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
vi
3.1 Stochastic simulation with ω0 = 1, α = 0.05, β = 0.05, λ = 0.16,
κξ = 0.0001, and κη = 0.0001. (a) A single time domain realization
showing A. (b) A single time domain realization showing B . . . . . . 33
3.2 Realization distribution showing in the A and B plane, with the same
device parameters as Fig. 3.1. (a) Parametric pump is off, serving as a
reference. (b) Parametric pump level at 80% threshold (using the data
shown in Fig. 3.1), illustrating the noise squeezing effect in A . . . . . 34
3.3 Comparison for the probability density of A and B, using the data shown
in Fig. 3.1. The orange curves indicate theory, and the blue curves
illustrate the simulation results. (a) Probability density for A. (b)
Probability density for B . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Realization distribution at twice the threshold (λ = 0.4), with the same
device parameters as Fig. 3.1 (a) In the A and B plane, the determin-
istic steady state is not localized to a fixed point, and the trajectory of
(Adet, Bdet) is shown by the green curve. (b) In the δA and δB plane,
where the distribution only comes from the noise . . . . . . . . . . . . 36
3.5 Comparison for the probability density of A and B, using the data shown
in Fig. 3.4. The orange curves indicate theory, and the blue curves
illustrate the simulation results. (a) Probability density for A. (b)
Probability density for B . . . . . . . . . . . . . . . . . . . . . . . . . . 36
vii
3.6 Comparison between the theory (solid curves) and simulations (dots)
for the means and variances of A (dark colors) and B (light colors),
with the same device parameters as Fig. 3.1 and with a parametric
sweep of λ. The orange curves represent theory, and the blue dots
show the simulation results. The theoretical results are given by Eqs.
(3.54)–(3.55) below threshold and Eqs. (3.56)–(3.57) above threshold.
The threshold is at λ = 0.2, as clearly suggested by the figures. (a)
Mean values of A and B. (b) Variances of δA and δB (when below
threshold, δA = A and δB = B) . . . . . . . . . . . . . . . . . . . . . . 37
3.7 Comparison between the theory (solid curves) and simulations (dots)
for the variance of δx and signal-to-noise ratio, with the same device
parameters as Fig. 3.1 and with a parametric sweep of λ. The orange
curves represent theory, and the blue dots show the simulation results.
The threshold is at λ = 0.2, as clearly suggested by the figures. (a)
variance of δx versus parametric pump (δx = x when below threshold).
(b) SNR versus parametric pump . . . . . . . . . . . . . . . . . . . . . 39
3.8 Power spectral density of x, with the same device parameters as Fig. 3.1.
The black curves show the PSD of a parametrically pumped systems,
and the gray curves show the PSP of an unpumped system, serving as
a reference. (a) λ = 0.16. Spectral narrowing can be observed, which is
the same phenomenon shown in [1]. (b) λ = 0.4. This shows the PSD
of a system with multiplicative noise and a pump level above the AT
threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
viii
4.1 Stochastic simulation with ω0 = 1, α = 0.01, β = 0.01, λ = 0.05,
κξ = 0.00004, and κη = 0.00004. (a) Ten realizations showing R. (b)
Comparison for the probability density of R . . . . . . . . . . . . . . . 44
4.2 Characterizing the strength of additive and multiplicative noise by vary-
ing the amplitude. The orange solid curve indicates the theory, the green
dashed curve represents the asymptotic line based on theory, and the the
blue dots shows the stochastic simulation results . . . . . . . . . . . . . 47
C.1 Nonlinearities in MEMS devices. (a) MEMS clamped-clamped beam
with electrodes on both sides. (b) Amplitude-dependent natural fre-
quency illustrating: (i) the system can exhibit hardening (mechanical
nonlinearity) to softening (electrostatic nonlinearity) effect and, (ii) the-
ory using first-principle modeling (blue curve) agrees well with the COM-
SOL time-domain simulation for the ringdown response (red dots). (c)
MEMS vibrating ring gyroscope with a mode shape of φ = cos 2θ. (d)
Amplitude-dependent natural frequency illustrating: (i) the ring does
not have mechanical linearity (indicated by the green curve) due to its
boundary conditions and, (ii) the static deflection characterization (blue
and green curves) shows excellent agreement with the dynamic ringdown
characterization (red dots) . . . . . . . . . . . . . . . . . . . . . . . . . 75
ix
C.2 Phase portrait of a typical MEMS resonator and dynamic pull-in effect.
The gray dot in the middle is the center equilibrium, the arrows indi-
cate the direction of strength of the vector field, and the brown curves
represent the invariant manifolds of the two saddle points (shown as two
crossing points). Between the saddle points is a non-homoclinic orbit,
only within which can the system operate as desired. These two saddle
points indicate the dynamic pull-in amplitude . . . . . . . . . . . . . . 76
C.3 Free decay of x+ 2 (α + βx2) x+ x = 0, with α = 0.01. The blue curve
shows the linear system, with β = 0, while the orange curve shows the
nonlinear system, with β = 0.01 . . . . . . . . . . . . . . . . . . . . . . 77
x
List of Abbreviations and Symbols
AFM Atomic force microscopyAT Arnold tongueEOM Equation of motionFFT Fast Fourier transformJPA Josephson parametric amplifierMEMS Micro-electro-mechanical systemNEMS Nano-electro-mechanical systemPCB Printed circuit boardQFT Quantum field theoryRMS Root mean squareSDE Stochastic differential equationSN Saddle-node bifurcationSNR Signal-to-noise ratioSQUID Superconducting quantum interference deviceTIA Transimpedance amplifier! Factorial⟨·⟩ Average value∑
Summation∫Integral
∇ Gradient1 Indicator functioncr Critical SN conditiondet DeterministicE Expected valueK CovarianceN Normal distributionO OrderR Correlationsto StochasticVar Variance
xi
A Quadrature 1; cross-sectional areaB Quadrature 2Cd Drive mode capacitanceCs Sense mode capacitanced Electrode gap distance
D(n)i Kramers-Moyal coefficient
D(1)i Drift coefficient
D(2)i Diffusion coefficient
e Euler’s numberE Young’s modulusfAC Direct drive amplitudeF Averaged time derivative of A or R
F Unaveraged time derivative of A or RFAC Direct drive amplitude*Fes Electrostatic force per unit lengthG Averaged time derivative of B or Φ
G Unaveraged time derivative of B or Φh Height of the beamI Second moment of areaJ Jacobianl Longitudinal position along the beamL Length of the beamp Externally applied loadP Externally applied load*Q CoordinateR Amplitudes Deformed length of the beamS Power spectral densityt TimeT Kinetic energyT Time*u Nodal displacementVAC AC voltageVd Drive mode voltageVDC DC bias voltageV Mechanical potential energyVs Sense mode voltageW Probability density; workx DisplacementX Displacement*α Linear damping ratioαi Damping coefficient
xii
β Cubic nonlinear damping coefficientγ Duffing coefficientγi Stiffness coefficientΓi Stiffness coefficient*δ Variation∆ω Frequency detuning∆φ Imposed phase shift∆Φ Long time phase shiftε Indicator for small variableε0 Vacuum permittivityεr Relative permittivityη Multiplicative noiseκ Noise strengthλ Parametric pump amplitudeξ Additive noiseρ Mass density per unit lengthτ Correlation time; axial tensionτ0 Axial pretensionφ Mode shapeφ0 Phase constantΦ Phaseω Drive frequencyω0 Eigenfrequencyωb Backbone (amplitude-dependent natural frequency)Ω Frequency in PSD; drive frequency*Ωeigen Eigenfrequency*
The asterisk (*) indicates the variable is before nondimensionalization.
xiii
Acknowledgements
First and foremost, I would like to express my sincere gratitude to my academic advisor,
Professor StevenW. Shaw, who has been guiding, supporting, and encouraging me since
we first met more than four years ago, at which time he enthusiastically introduced
to me his advancements in research. Without him, my academic programs here would
not have been so exhilarating, fulfilling, and meaningful. I really enjoy the times with
him, whether in his office or online, as I can learn not only nonlinear dynamics, but
also far beyond that. I am really grateful to Prof. Shaw for the precious opportunities
he has given me, including a summer visit at Michigan State University, conference
talks at the IDETC-CIE 2019 in Anaheim and at an NSF workshop in New Orleans,
and a summer visit at Stanford University for experiments. He has also introduced
me to many researchers and invited me to join their discussions, which have greatly
expanded my view and sparked my passion for research. All the times that I spend
with Prof. Shaw will become a priceless and everlasting treasure for me.
While I am working on my thesis, many people have offered help to me and con-
tributed to this work. I would like to thank Prof. Thomas W. Kenny and Dr. James
M. L. Miller at Stanford University for offering me the precious opportunity to perform
experiments which are relevant to the thesis. Prof. Kenny has been very nice, and his
labs are equipped with sophisticated and delicate instruments. James possesses ex-
cellent skills, and he has been very helpful and considerate during my training there.
xiv
I wish to thank Prof. Mark I. Dykman at Michigan State University for helping me
during a summer visit there. I am impressed with his warm welcome, and several
meetings with him have guided me about the learning of stochastic theories. I would
also like to thank Prof. Oriel Shoshani at Ben-Gurion University for helping with the
more sophisticated stochastic analyses pertinent to this thesis.
Collaborators in the nonlinear dynamics and MEMS community have also enriched
my research experience. I would like to thank Prof. Philip Feng, Dr. Jaesung Lee,
and Tahmid Kaisar at the University of Florida, who have shared exciting experimental
results and invited me to co-author a paper. I also wish to thank Dr. Pavel M. Polunin
at SiTime, who has helped co-author a paper and provided insightful suggestions.
During my studies, there have been many researchers whose online meetings, in-
person talks, or email correspondences have benefited me. I would like to show appreci-
ation to Prof. Richard H. Rand at Cornell University, Prof. Brian F. Feeny at Michigan
State University, Dr. David A. Czaplewski, Dr. Daniel Lopez, and Dr. Changyao Chen
at Argonne National Laboratory, Prof. Kimberly L. Foster at Tulane University, Prof.
Ho Bun Chan at HKUST, Prof. Alexander Eichler at ETH Zurich, Dr. B. Scott Stra-
chan at SiTime, Prof. Slava Krylov at Tel Aviv University, Gabrielle D. Vukasin and
Nicholas E. Bousse at Stanford University, and Prof. Michael L. Roukes at Caltech.
I would like to thank committee members Prof. Hector Gutierrez and Prof. Brian
A. Lail at Florida Tech, who have provided helpful suggestions for this thesis. Prof.
Gutierrez’s intellectually stimulating courses on mechatronics and instrumentation
have always been a joyful experience, and they have prepared me well for my on-
going experiments at Stanford. Prof. Lail’s elegant course on guided waves has helped
me gain a better understanding of many devices.
This work has been supported in part by NSF grants CMMI-1561829 and CMMI-
1662619, BSF grant 2018041, and Florida Institute of Technology.
xv
Chapter 1
Introduction
Parametric oscillators have been receiving ever-increasing attention from both academia
and industry, in particular the nonlinear dynamics and micro/nano-device communi-
ties. This recent interest is owing to the fact that parametric resonance and parametric
amplification have appealing merits, for instance, in the amplification of dynamic re-
sponses for signal and sensitivity enhancement [2, 3, 4], narrowing thermomechanical
noise spectra [5], improving effective quality factors (up towards the limit of infinity)
[6], and the capability to be tuned to exhibit desirable nonlinear behavior [7]. These
virtues have far-reaching implications in numerous fields of engineering, and many of
the enticing advantages are yet to be exploited to their full potential. For example,
by embracing parametric resonance with a felicitous phase, one can increase the rate
sensitivity of vibratory rate sensing gyroscopes by amplifying the sense mode response
[4, 8] or enhance the phase sensitivity of a resonator using parametric suppression
[9]. An improvement in signal-to-noise ratio (SNR) can also be accomplished thanks
to the thermomechanical noise squeezing effect associated with parametric pumping
[5, 1, 10]. Moreover, a higher effective quality factor can entail a positive consequential
impact on frequency selectivity [1], frequency stability [6], and dissipation engineering
[11, 12]. In addition, the excellent tunability of parametrically pumped resonators sug-
gests promising applications, especially in nano- and micro-electro-mechanical systems
1
(N/MEMS) [13].
One aspect of particular interest is the noisy behavior exhibited in electrical and
mechanical systems used for sensing and time-keeping applications [13, 14, 15]. The
performance of these systems are oftentimes limited by electric noise and/or thermo-
mechanical noise, which leads to amplitude and frequency fluctuations, hindering the
SNR [16]. Typically, thermomechanical noise results in a type of Brownian motion
of the mechanical structures, contributing as additive noise to amplitude fluctuations
and multiplicative noise to frequency fluctuations. In addition, amplitude-to-frequency
noise conversion, which stems from nonlinear stiffness in a resonator, also gives rise to
frequency noise [17]. In addition to the resonator element, many of these systems em-
ploy a phase-locked loop (PLL) with feedback in order to maintain oscillation. This
form of operation has important applications in time-keeping and sensing. In a PLL
there are copious sources of noise, which can originate from phase shifters, amplifiers,
filters, and frequency multipliers. The combined effects of all the noises in system with
a resonator and a PLL ultimately turn into phase noise, which directly affects the the
precision of time-keeping operations.
One primary advantage of using parametric pumping in these applications is that
the drive used to sustain oscillation, that is, the parametric pump, is at twice the
frequency of the operating frequency, and therefore many noise sources, such as those
associated with the PLL operation, are not aligned with the operating frequency [18].
This frequency separation is being explored as a means of improving the SNR in such
systems [19]. Another potential advantage of using a parametric pump is that one can
alter the noise amplitudes in different quadratures of the response (that is, the in-phase
and out-of-phase components), offering a means of squeezing the noise in a beneficial
manner [5]. Therefore, there are significant motivations to examine the impact of
additive and multiplicative of noise in parametric oscillators.
2
All previous studies of noise in parametric oscillators have considered resonators
with linear damping that can be modeled using a simple quality factor Q. However,
in recent years it has been found that many N/MEMS systems exhibit an amplitude-
dependent Q, that is, they have nonlinear damping. Of course, these effects are more
pronounced as signal strength is increased, and thus there is growing interest and
expanding knowledge of nonlinear damping [20, 21, 22], including its significance for
parametric resonance [23, 24].
In this work we demonstrate how noise affects the behavior on both open loop and
PLL operations of parametric oscillators.
1.1 Parametric Resonance and Its Applications
Observed and utilized in a vast variety of physical systems, parametric resonance is
a ubiquitous phenomenon enjoying characteristics that are of great interest to re-
searchers. Principal parametric resonance is achieved by periodically varying the stiff-
ness of the system near twice the resonant frequency, which can either be artificially
produced or can naturally arise due to external effects.
To form an intuitive view, consider a pendulum being operated using a small time-
periodic displacement in two ways: if one moves the pivot horizontally at its natural
frequency, then the pendulum is directly excited at resonance; however, if one varies
the length of the pendulum at twice the natural frequency, then it is parametrically
pumped. In electric and/or mechanical systems, parametric resonance can be realized
in several ways. One method is to use a periodically modified electrostatic force stem-
ming from capacitive effects [3]. Another approach is to base-excite a mechanical beam
in the longitudinal direction [25]. Additionally, parametric resonance can be achieved
indirectly, for instance, through intermodal coupling [4, 8].
3
In terms of modeling, a usual external drive is modeled by a time-harmonic term
such as f cosωt, where f is the drive amplitude, whereas parametric pumping is mod-
eled by a state-dependent time-harmonic term such as px (t) cosωpt, where p is referred
to as the pump amplitude. If a resonator has eigenfrequency ω0, primary resonance
occurs for ω ∼ ω0 whereas principal parametric resonance occurs for ωp ∼ 2ω0. In both
cases, the resulting response is dominated by a harmonic near ω0.
The applications in which parametric pumping is most commonly used are circuits
and MEMS, and our focus is on the latter. To begin with, parametric pumping is able
to improve the transmission characteristics of a bandpass filter in terms of exhibiting
a very sharp response rolloff and nearly ideal stopband rejection [26]. It is also used
for mass sensing, where the change in mass can be detected from the boundary of
frequency shift, and parametric pumping provides a sharper rolloff from resonance [3].
Self-induced parametric amplification has been employed in MEMS vibratory gyro-
scopes, where the sense mode is parametrically pumped via dispersive coupling to the
drive mode, which leads to a significant improvement in the rate sensitivity [4, 8, 27].
Moreover, spin waves, induced by parametric resonance, can be generated in nanoscale
magnetic tunnel junctions, which can have applications related to spintronics [28].
Parametric resonance can also be used as repulsive force actuators for MEMS mir-
ror, which enables large travel ranges by avoiding pull-in instability [29]. In addition,
three-mode parametric resonance has been realized in phononic frequency comb with
applications in resonant frequency tracking [30].
It is also worth mentioning that parametric resonance is widely observed in quantum
physics, cosmology, opto-mechanics, and cell biology. For instance, researchers have
studied two-particle scattering and memory effects in the context of quantum field
theory (QFT) [31], collective neutrinos exhibiting self-induced parametric resonance
[32], and the entanglement evolution in the dynamical Casimir effect [33]. Parametric
4
resonance has also been used to study the exponential amplification of cosmological
fluctuations during the initial reheating stages [34]. It is reported that with the use
of opto-thermal drive, 14 mechanical modes can be brought into parametric resonance
in graphene membranes [35]. Furthermore, ion parametric resonance model has been
developed to study the magnetic field interactions on cell membranes [36, 37].
1.2 Sources of Nonlinearity and Noise in N/MEMS
Virtually all small resonators currently in use operate in their linear range. This
provides well behaved response that is conveniently modeled using linear systems tech-
niques. However, the demands for improved SNR have motivated many investigations
into the nonlinear behavior of N/MEMS resonators. Reviews of these efforts include
[13, 38, 39].
Nonlinearity in resonators comes in two basic forms, conservative and non-conservative,
which can be roughly associated with stiffness and dissipation, respectively. Sources
of these effects are sometimes understood from first principles, such as mechanics and
electrostatics, while in other cases they are not well understood but are still evident.
This is described further in the appendix.
Noise is ubiquitous in resonator systems, and represents unmodeled dynamics that
arise from coupling of the vibration mode of interest to its environment, that is, to
other modes, including electronics, thermal vibrations, etc. The sources of these effects
are sometimes known from first principles, such as microscopic models, but are more
often not easily modeled from basic physics [40, 41, 42]. Of particular interest is noise
squeezing, which attenuate fluctuations in one quadrature of a dynamic system.
5
1.3 Noise Squeezing
Thermal noise is omnipresent and its microscopic sources are well understood from
statistical mechanics. Specifically, it is observed in field-effect transistors [43], Joseph-
son junctions [44], V-shaped cantilevers of atomic force microscopy (AFM) [45], and
many more. However, attention has been raised from a comprehensive literature re-
view that, frequency stability are orders of magnitude larger in all NEMS compared to
the limit imposed by thermomechanical noise [16]. Since the exact sources and mecha-
nisms of noise remain largely unknown, one can simply model it as additive noise [46]
and multiplicative noise [47, 48], assuming that there is sufficient understanding about
the system and the ability to characterize it and the noise. The strengths of these two
types of noise, manifested by the amount of diffusion they cause in the system response,
can be characterized by varying the response amplitude, as shown subsequently in this
work.
In past decades, there have been several important research on noise squeezing using
parametric resonance. In the 1980s, there had been numerous ongoing research related
to Josephson parametric amplifiers (JPA), achieved using superconducting quantum
interference device (SQUID), where squeezed states were experimentally observed [49].
Then, it was first shown by Ruger and Grutter that thermomechanical noise can be
squeezed, that is, the response distributions associated with the quadratures can be
manipulated, using a mechanical parametric amplifier [5]. Various related topics have
been studied since that time, and there has been a resurgence in the topic with the
development and application of N/MEMS resonators. In recent years, it is found that
a continuous weak measurement, such as a quantum measurement, can enhance the
squeezing effect for a mechanical oscillator if the parametric pump is optimally tuned
[50]. Parametric phase noise filtering technique is also demonstrated using a solid-
6
state parametric amplifier, which allows for significant reduction in phase noise [51].
In addition, spectral narrowing has been recently demonstrated and analyzed in a
parametrically pumped resonator, where the spectrum of the thermomechanical noise
is shown to exhibit characteristics of a system with a much higher Q value [1]. These
experimental and analytical works have shed important light on the feasibility and
opportunities for the noise squeezing effect and its utilization in multifarious situations.
1.4 Overview
There have been several recent theoretical works on the noisy behavior of parametric
resonance. Ishihara has investigated the effects of white noise on parametric resonance
using a scalar field theory called the λϕ4 theory [52]. Bobryk and Chrzeszczyk, on
the other hand, have studied the influence of colored noise on parametric oscillators
[53]. The SNR of parametric resonance and parametric amplification have been ana-
lyzed by Batista and Moreira for a resonator in a general context using the Green’s
function method [54]. Small quantum fluctuations in tunable superconducting cavi-
ties, whose nonlinearities are introduced by SQUID, have been investigated by Wust-
mann and Shumeiko [55]. Lin, Nakamura, and Dykman have researched the fluctua-
tions of a quantum parametric oscillator, including the interstate switching near the
period-two bifurcation [56]. Miller et al. have studied the spectral narrowing for a
thermomechanical-noise-driven oscillator under the action of parametric pumping, and
the predicted power spectral density (PSD) is verified by experiments [1].
This thesis expands on these previous works by including nonlinear damping and
multiplicative noise in the resonator model. Embracing nonlinear damping gives rise to
two major advantages. First, it is commonly observed in a range of N/MEMS systems
[57, 58, 59, 60, 61, 62, 63, 64, 65], and there are a few theoretical works [20, 66, 67]; these
7
are elucidated in Appendix B. In addition, nonlinear damping provides a mechanism,
alternate to nonlinear stiffness, whereby parametric oscillators can operate above the
parametric instability threshold (see Section 2.1), in which case the dynamic responses
are bounded by nonlinear damping. Similarly, the inclusion of multiplicative noise also
enjoys two benefits: it is intrinsic in various systems such as feedback oscillators, and it
exhibits phenomena that are otherwise not described by additive noise. Of fundamental
interest is the fact that such noise provides an upper limit on the SNR improvement
one can achieve by increasing the signal amplitude. This was recently demonstrated
for systems with usual resonant drive [17], and confirmed for parametric resonance in
this work.
The mathematical model considered in this thesis describes lightly damped single
mode vibration of an electro-mechanical system, typically of small size (N/MEMS).
While it may appear to be highly idealized, it is in fact quite generic since it describes
a normal form for weakly nonlinear vibrations with the two most important types of
noise, namely additive and multiplicative, both of which arise from coupling to the
environment. Roughly speaking, additive noise results in a stochastic effect that adds
to the underlying deterministic response, and multiplicative noise represents a random
variation in the system natural frequency. Random variations in damping, say, Q, have
negligible effect on the response since dissipation is inherently small in N/MEMS. The
nonlinear effects considered include nonlinear stiffness, which provides an amplitude-
dependent frequency, and nonlinear damping, which results in non-exponential decay
during free response. These nonlinear and noise terms provide a generic model that
describes a wide range of vibratory M/NEMS.
This work is organized as follows. In the appendix, background related to the rel-
evant nonlinear phenomena is introduced. In Chapter 2, the deterministic models are
provided for open loop and PLL operations, and the nonlinear behavior is described.
8
The analysis of the noisy response of the systems of interest builds on these determinis-
tic results. In Chapter 3, the open loop operation with noise is expatiated, wherein the
method is formulated and the results are interpreted. Chapter 4 explicates the closed
loop PLL operation with noise and the results are discussed. Finally, some conclusions
are drawn and future work are explained in Chapter 5.
9
Chapter 2
Deterministic Nonlinear Behavior
Prior to the discussions related to the noisy behaviors of a parametric oscillator, it is
first important to examine the dynamic response using a deterministic model. This
chapter introduces to the audience the open loop and closed loop operations of the
system, helps the understanding pertaining to the nonlinear behavior in the transient
and steady-state responses, and offers a glimpse at the significance of how averaging
simplifies the analysis of a periodically time-varying system.
The open loop case is typically used for system characterization while the closed
loop case is relevant to frequency generation and time-keeping.
2.1 Open Loop Model
For the open loop operation, we consider a single degree-of-freedom parametric oscil-
lator with nonlinear stiffness and nonlinear damping. The effects of damping, nonlin-
earities, and parametric pumping are assumed to be small relative to the inertial and
linear stiffness effects. The notation ε is a parameter introduced and used to track
the small parameters, in other words, 0 < O (ε) ≪ 1. The equation of motion for a
10
noise-free open loop parametric oscillator is given by
x+ 2ε(α + βx2
)ω0x+ (1 + ελ cos 2ωt)ω2
0x+ εγx3 = 0, (2.1)
where α and β represent the linear and nonlinear damping coefficients (both of which
are assumed to be positive), ω20 and γ denote the linear and nonlinear stiffness co-
efficients, λ describes the amplitude of the parametric pump, and 2ω (the factor of
2 is selected so that parametric resonance occurs for ω near ω0) specifies the pump
frequency. Since only the principal parametric resonance is of interest, ω is assumed
to be close to ω0.
Under the stated assumptions, the amplitude and phase of x (t) are slowly-varying
functions of time and perturbation methods can be applied. Using the method of
averaging [68], the time-invariant equations that govern the system’s dynamic response
are obtained. To begin, we apply the van der Pol transformation
x (t) = R (t) cos [ωt+ Φ(t)], (2.2)
x (t) = −ωR (t) sin [ωt+ Φ(t)], (2.3)
where R (t) and Φ (t) are the slowly-varying polar coordinates representing the ampli-
tude and phase of x (t). This transformation allows us to obtain expressions for R (t)
and Φ (t), which are composed of slow drift with small oscillating terms. To remove
the oscillations and obtain the desired equations, we average R (t) and Φ (t) over one
period, 2π/ω, with the assumption that R and Φ remain constant during this time.
The notation ⟨·⟩ is used to denote the mean value of a variable. We obtain the following
11
averaged equations
F = ⟨R⟩ = −εαω0R− ε
4βω0R
3 +ελω2
0R sin (2Φ)
4ω, (2.4)
G = ⟨Φ⟩ = ω20 − ω2
2ω+
3εγR2
8ω+
ελω20 cos (2Φ)
4ω, (2.5)
which provide an approximate model for the drift dynamics of the amplitude and phase.
It is clear that all terms on the right hand side of Eqs. (2.4)–(2.5) are small so that
R (t) and Φ (t) vary slowly in time, consistent with the physical assumptions. These
equations are particularly useful since they do not depend explicitly on time.
The steady-state condition is obtained by solving R = 0 and Φ = 0 simultaneously
for Eqs. (2.4)–(2.5), which provides solutions for the steady-state amplitude R and
phase Φ, representing a fixed point of the autonomous dynamical system. For this
parametric drive, there always exists a trivial solution R = 0 for which the phase Φ is
undefined. The nontrivial solutions can also be solved for explicitly, but are of little
importance for the discussions below, and are thus omitted here.
The stability of a fixed point can be determined by the eigenvalues of its Jacobian
matrix. With only the leading order terms kept, the Jacobian evaluated at the steady-
state equilibrium is given by
J = εω0
⎡⎢⎣−α− 34βR
2+ 1
4λ sin (2Φ) 1
2λR cos
(2Φ
)3
4ω20γR −1
2λ sin
(2Φ
)⎤⎥⎦ . (2.6)
Specifically, the stability can be determined from either the Hartman-Grobman theo-
rem or from the linearized system using the Poincare diagram. Correspondingly, if the
real parts of all eigenvalues are strictly negative, or equivalently, if the trace of the Ja-
cobian is negative and the determinant is positive, then the fixed point is stable, in fact,
12
asymptotically stable. On the other hand, if at least one real part of an eigenvalue is
strictly positive, or alternatively, if the trace is positive or the determinant is negative,
then the fixed point is unstable. In the case of a nonlinear parametric oscillator, the
onset of the pitchfork bifurcation, which corresponds to the appearance of nontrivial
solutions, corresponds to an eigenvalue’s real part, evaluated from the trivial solution’s
Jacobian, becoming zero.
To facilitate further analytical calculations, we focus on the steady-state response
in the amplitude-frequency space. In addition to the trivial solution that exists across
the entire frequency domain, we turn our attention to the nontrivial response of the
system. The frequency detuning parameter is defined as
ε∆ω = ω2 − ω20/(2ω0) ∼= ω − ω0, (2.7)
which appears in the right hand side of Eq. (2.5). By eliminating the phase Φ, the
steady-state condition for the amplitude can be obtained, rewritten as an explicit
expression for detuning as a function of steady-state amplitude R
∆ω =3γR
2
8ω0
± ω0
4
√λ2 −
(4α + βR
2)2
. (2.8)
Several observations can be made regarding to Eq. (2.8). First, the plus-minus
sign indicates that there are two response branches spaced equally on either side of the
backbone curve (defined immediately below)
∆ωb =3
8ω0
γR2. (2.9)
The backbone curve manifests the amplitude-dependent frequency of vibration for the
unforced system and is represented by a path in the amplitude-frequency space along
13
which an uncoupled and lightly damped system freely decays. Typically, it is also
the locus of the amplitude extremum, in this case, a global maximum, or the peak
amplitude, of the pumped system. By examining Eq. (2.8) further, it follows that the
peak amplitude can be acquired when the radicand becomes zero. As a consequence,
the peak amplitude is
Rpeak =
√λ− 4α
β. (2.10)
An important result is the condition for which nontrivial solutions can exist. Again,
a necessary condition lies in the radicand of Eq. (2.8), in which case the radicand and
R have to be simultaneously positive: λ > 4α. Clearly, this expression is consistent
with Eq. (2.10), but is not general. The general result is obtained by using R = 0 into
Eq. (2.8) and rearranging the terms, which gives
λAT = 4
√α2 +
(∆ω
ω0
)2
. (2.11)
This parametric instability threshold is a V-shaped curve in the pump amplitude versus
drive frequency parameter plane known as an Arnold tongue (AT). For λ > λAT, the
linear model response becomes unbounded. In the presence of nonlinear stiffness or
damping the response for λ > λAT is nontrivial and finite, limited by nonlinear effects.
When the system operates below the threshold, only the stable trivial response is
possible. It is worth noticing that the expression of Eq. (2.11) is independent of
nonlinear parameters, which is rooted in the fact that this instability occurs at zero
amplitude at which nonlinear effects do not come into play.
At zero detuning, as the pump level crosses the instability threshold a supercritical
pitchfork bifurcation occurs for which the stable trivial equilibrium becomes unstable
and from which two stable equilibria with equal amplitudes and distinct phases emerge.
Then, as the parametric pumping strength is further increased, these responses continue
14
to grow in amplitude and coexist with the unstable trivial equilibrium.
It is interesting to consider a frequency sweep for λ > λAT. The nature of the
response during such a sweep depends on the sign of the nonlinear stiffness and the
pump amplitude. Here we take γ > 0 and note that the situation is reversed in a
straightforward way for γ < 0. At low and high frequencies, outside of the AT, the zero
response is stable, and inside the AT it is unstable. As the frequency passes through
the AT from the left, a supercritical pitchfork bifurcation occurs, resulting in a pair of
stable, finite amplitude responses with period twice that of the parametric pump; this
is a period doubling bifurcation. These two responses have equal amplitudes and a
phase difference of π and are therefore physically indistinguishable. Without nonlinear
damping, the model predicts that this response never terminates as the frequency
continues to increase, and keeps growing in amplitude while following the backbone
curve. With nonlinear damping, the response terminates at a finite frequency in one
of two ways, depending on the nonlinear stiffness (Duffing) parameter γ and the pump
amplitude λ. For a sufficiently large value of γ and/or large λ, this termination occurs
via a saddle-node bifurcation at a critical value of the frequency that is above the
right side of the AT. Here this stable response merges with a finite amplitude unstable
response (described below), leaving only the re-stabilized zero response. In such cases,
the unstable response is born, as the frequency increases, at the right branch of the AT,
in a subcritical pitchfork bifurcation that restablizes the zero solution. These unstable
responses are also a π phase shifted pair. Note that this scenario results in range of
frequencies to the right of the AT where both zero and the nontrivial responses co-exist,
separated by the unstable responses. This implies hysteresis as the frequency is swept
up and down. For smaller (including zero) values of Duffing parameter γ and/or lower
pump amplitude λ, the stable nontrivial responses simply re-collapse onto the zero
response at the right branch of the AT, restabilizing the zero response. In this case,
15
bistability and hysteresis do not occur and the frequency response is quite simple—the
trivial response is observed outside of the AT and the nontrivial pair of responses are
observed inside the AT. Fig. 2.1 shows a case with both of these scenarios, and is
described in more detail below.
The steady-state amplitude at the above-mentioned saddle-node bifurcation condi-
tion can be obtained from Eq. (2.8) and by setting ∂∆ω/∂R = 0 with R > 0, which
yields
RSN (λ) =
√3 |γ|λ
β√4β2ω4
0 + 9γ2− 4α
β, (2.12)
RSN (∆ω) = 2
√6γ∆ωω0 − 4αβω2
0
4β2ω40 + 9γ2
. (2.13)
The saddle-node bifurcation in the λ − σ parameter plane is of great interest, which
can be solved directly from Eqs. (2.12)–(2.13), given by
λSN =12α |γ|+ 8βω2
0 |∆ω|√4β2ω4
0 + 9γ2. (2.14)
This expression indicates that the saddle-node bifurcation condition is a straight line
in the λ− σ parameter plane, as indicated by the SN line in Fig. 2.1.
The SN critical condition is important for determining whether there exists multi-
stability, or hysteresis, which stems from the saddle-node bifurcation, in the frequency
response. When the parametric pump level is above the AT threshold but below the
SN critical condition, there are only two physically indistinguishable (equal amplitudes
despite distinct phases) stable responses. By examining the locus of the saddle-node
bifurcation from Eqs. (2.12)–(2.13), it can be known that the critical condition occurs
at zero amplitude: Rcr = 0. Thus, by imposing this on Eqs. (2.12)–(2.13), the SN
16
-0.2 -0.1 0 0.1 0.20
1
2
3
Δω
R
AT AT
SN
-0.2 -0.1 0 0.1 0.20
0.2
0.4
0.6
0.8
1
Δω
λ
(a) (b)
Figure 2.1: Steady-state response, with ω0 = 1, α = 0.05, β = 0.05, and γ = 0.019245.(a) Response in the amplitude-frequency space. The three curves correspond toλ = 0.25 (light blue), λ = λcr = 0.4 (blue), and λ = 0.65 (dark blue), respectively.The solid curves indicate stable response and the dashed curve corresponds to unstableresponse. Trivial response exists across the entire frequency domain. (b) Two param-eter bifurcation diagram. The blue curve shows the pitchfork bifurcation condition, orArnold tongue (AT), and the orange curve shows the saddle-node bifurcation condition(SN). The critical condition is where the two curves meet at the bottom
critical condition is obtained
λcr =4α
√4β2ω4
0 + 9γ2
3 |γ|, (2.15)
∆ωcr =2αβω3
0
3γ, (2.16)
below which the system does not experience hysteresis.
Fig. 2.1 illustrates the steady-state response of an open loop nonlinear parametric
oscillator with both nonlinear stiffness and damping. When 0 < λ < 4α, the system
only has a trivial response that is stable across the entire frequency domain. When
4α < λ < λcr (shown in light blue), the system has nontrivial responses between the
17
two supercritical pitchfork bifurcation points, and that the nontrivial responses are
stable. When λ > λcr (shown in dark blue), the system’s stable nontrivial responses
are bounded by the supercritical pitchfork and saddle-node bifurcations, while the
unstable nontrivial responses are bounded by the subcritical pitchfork and saddle-node
bifurcations. Notice that each nontrivial response always has another corresponding
nontrivial response with an equal amplitude, a phase difference of π, and the same
stability characteristic. The nontrivial response, on the other hand, is unstable between
the pitchfork bifurcations, that is, inside the AT, and stable elsewhere.
Note that if the system does not have nonlinear stiffness (γ = 0), the saddle-node
bifurcation does not occur and the AT dictates the entire response.
2.2 Phase-Locked Loop Model
A parametric phase-locked loop (PLL) is a feedback oscillator that uses a parametric
pump at twice the resonant frequency, wherein the response of the resonating element
is measured, amplified, phase shifted, doubled in frequency, and then applied back as
a parametric pump to the resonator [18]. The parametric PLL can have important
applications in time-keeping, since it does not require an external time reference to
generate the signal that excites the system. The equation of motion for a noise-free
parametric PLL is given by
x+ 2ε(α + βx2
)ω0x+ [1 + ελ cos (2ω0t+ 2Φ (t) + ∆φ)]ω2
0x+ εγx3 = 0, (2.17)
where ∆φ is the phase shift, which is selected by the user and remains constant during
operation.
Similar to the previous section, with the method of averaging, the displacement is
transformed into the rotating frame and averaged, in this case over 2π/ω0, resulting in
18
an autonomous equation governing the slow dynamics. Using the slowly-varying polar
coordinates R and Φ, The time-invariant closed-loop dynamics are governed by the
following differential equations
F = ⟨R⟩ = −εαω0R− ε
4βω0R
3 − ε
4λω0R sin∆φ, (2.18)
G = ⟨Φ⟩ = 3εγR2
8ω0
+ε
4λω0 cos∆φ. (2.19)
The steady state for a parametric PLL is slightly different from the open loop
model since there is no external time reference. In this case, the steady-state response
has a fixed amplitude and the frequency is set by that amplitude and the feedback
loop parameters, as described next. The steady-state amplitude is obtained by simply
setting R = 0, which yields
R =
√−λ sin∆φ− 4α
β. (2.20)
This expression sheds light on the two important requirements in order to operate a
parametric PLL: (i) the parametric pump must be driven above the instability thresh-
old, namely, λ > 4α/(− sin∆φ), and (ii) there must exist nonlinear damping in the
system (β > 0). Moreover, another attribute of a parametric PLL is that the response
is always stable, thanks to the fact that the stability only depends on R given by Eq.
(2.18), as opposed to the entire Jacobian of an open loop parametric oscillator.
The time derivative of the phase, Φ, together with the eigenfrequency ω0, depicts
the frequency at which the parametric PLL operates. Hence, the operating frequency
of the parametric oscillator is ω = ω0 + Φ, whose full expression for a deterministic
19
system is
ω = ω0 +3εγR
2
8ω0
+ε
4λω0 cos∆φ. (2.21)
As expected, the phase shift, the pump strength, and all device parameters will affect
the operating frequency of the system.
Fig. 2.2 demonstrates the steady-state response of a parametric PLL. The three
curves in each figure indicate the cases for 4α < λ < λcr (shown in light blue), λ = λcr
(shown in blue), and λ > λcr (shown in dark blue). As can be seen in Fig. 2.2(a),
the maximum amplitude always occurs at ∆Φ = −π/2, as expected. Fig. 2.2(b)
displays how the phase shift can be used to tune the operating frequency of the system.
Finally, Fig. 2.2(c) illustrates the response in the amplitude-frequency space, obtained
by varying ∆φ; while the shapes of the curves are identical with those of an open loop
system, the PLL response is always stable. Note that Fig. 2.2(c) can be used to select
a pump strength and phase shift to achieve a desired amplitude and frequency of the
system.
20
-150 °-120 ° -90 ° -60 ° -30 °0
0.5
1
1.5
Δφ
R
-150 °-120 ° -90 ° -60 ° -30 °
0.99
1
1.01
Δφ
ω
(a) (b)
0.99 1 1.010
0.5
1
1.5
ω
R
(c)
Figure 2.2: Steady-state response, with ω0 = 1, α = 0.01, β = 0.01, and γ = 2/225.The three curves correspond to λ = 0.0425 (light blue), λ = λcr = 0.05 (blue), andλ = 0.625 (dark blue), respectively. (a) Steady-state amplitude versus imposed phaseshift. The peak amplitude always occurs at ∆Φ = −π/2. (b) Operating frequencyversus imposed phase shift. (c) Amplitude-frequency space. The shapes of the curvescoincide with those of an open loop system; however, the closed loop response is alwaysstable
21
Chapter 3
Open Loop Operation with Noise
The equation of motion of an open loop nonlinear parametric oscillator with additive
noise ξ (t) and multiplicative noise η (t) is given by
x+ 2ε(α + βx2
)ω0x+ [1 + ελ cos (2ωt+ φ0) + εη (t)]ω2
0x+ εγx3 = εω20ξ (t) , (3.1)
where φ0 is a phase constant used to align the quadratures with the eigen-directions
of the linearized system when convenient. It is assumed that the two types of noise
are band-limited white noise, their correlation times τcorr are extremely small, and that
they are not cross-correlated, that is, they are statistically independent. The strengths
of the additive and multiplicative noises are characterized by constants as
∫ ∞
−∞⟨ξξτ ⟩ cosωτdτ = κξ (ω) = κξ, (3.2)
∫ ∞
−∞⟨ηητ ⟩ cosωτdτ = κη (ω) = κη, (3.3)
which are independent of frequency; this is consistent with the stated assumptions. In
fact, one needs only to know the strength of these noises in the linewidth of certain
frequencies for the following results to be valid.
22
3.1 Theory Formulation
The quadratures are used to transform the coordinates of this system
x (t) = A (t) cosωt+B (t) sinωt, (3.4)
x (t) = −ωA (t) sinωt+ ωB (t) cosωt. (3.5)
It follows that the time derivative of the transformed, slowly-varying coordinates A (t)
and B (t) can be expressed in the form of
A = F det (A,B, t) + F sto (A,B, t) , (3.6)
B = Gdet (A,B, t) + Gsto (A,B, t) , (3.7)
where F and G indicate the exact transformations into A and B coordinates, separated
using the contributions from the deterministic and stochastic components. Here the
subscript “det” implies the deterministic terms, while the subscript “sto” suggests the
stochastic terms. The oscillating terms in Eqs. (3.6)–(3.7) can be averaged over one
period, where the first-order approximation is written in the form of
F = Fdet (A,B) + Fsto (A,B, t) , (3.8)
G = Gdet (A,B) +Gsto (A,B, t) . (3.9)
23
Note that the averaged deterministic parts do not depend explicitly on time, just as
considered in the previous chapter, but the stochastic terms include noise-related time-
dependent terms that have been transformed by the averaging process. Here we apply
the method of stochastic averaging to analyze the noisy response. For the purpose of
this work, the first-order approximation described in [69] is considered sufficient, whose
results are verified with stochastic simulations.
The nonoscillatory deterministic terms are obtained in similar fashion as the pre-
vious chapter with an accuracy up to O (ε), given by
Fdet =εω0
4
[−4αA− βA
(A2 +B2
)− λ (A sinφ0 +B cosφ0)
−4∆ωB/ω0 + 3γB(A2 +B2
)/(2ω2
0
)],
(3.10)
Gdet =εω0
4
[−4αB − βB
(A2 +B2
)− λ (A cosφ0 −B sinφ0)
+4∆ωA/ω0 − 3γ A(A2 +B2
)/(2ω2
0
)].
(3.11)
The steady-state solution acquired from the deterministic dynamical system is denoted
as(A, B
), and the Jacobian matrix evaluated at the fixed point, which will be used
later, is obtained from
J =
⎡⎢⎣ ∂Fdet
∂A∂Fdet
∂B
∂Gdet
∂A∂Gdet
∂B
⎤⎥⎦(A, B
). (3.12)
To analyze the drift effect of the stochastic system response, we introduce the
Fokker-Planck equation [70]
∂W
∂t= −
2∑i=1
∂
∂Qi
(D
(1)i W
)+
2∑i=1
2∑j=1
∂2
∂Qi∂Qj
(D
(2)ij W
), (3.13)
where W (Q, t) is the time evolving probability density, Q1 = A, and Q2 = B. The
24
D(n)i ’s are the Kramers-Moyal coefficients, where the D
(n)i ’s function as the drift coeffi-
cients for n = 1 and as the diffusion coefficients for n = 2, and higher-order coefficients
are zero for the given Gaussian noise [70]. The drift coefficients are computed using
[69]
D(1)i =
⟨Qi
⟩+
2∑j=1
∫ 0
−∞K
[∂⟨Qi⟩∂Qj
, ⟨Qj,τ ⟩
]dτ, (3.14)
where K is the covariance (in this case, the cross-covariance of two random variables).
Therefore, it is clear that the noise only has an effect at O (ε2) that results in a change
of the diffusion coefficients, which suggests that
D(1)1 = Fdet +O
(ε2), (3.15)
and
D(1)2 = Gdet +O
(ε2). (3.16)
As a consequence, the parts of the drift coefficients due to the noise are not of concern
in this work.
As for the diffusion of the response, we start with the following unaveraged expres-
sions
F sto =εω2
0
2ω
[−2ξ (t) sinωt+ η (t)
(2B sin2 ωt+ A sin 2ωt
)], (3.17)
Gsto =εω2
0
2ω
[2ξ (t) cosωt− η (t)
(2A cos2 ωt+B sin 2ωt
)]. (3.18)
The diffusion coefficients are calculated as follows [69]
D(2)11 =
∫ 0
−∞K[F sto, F sto,τ
]dτ =
ε2ω20
16
[4κξ +
(A
2+ 3B
2)κη
], (3.19)
25
D(2)12 =
∫ 0
−∞K[F sto, Gsto,τ
]dτ =
ε2ω20
16
AB
κη, (3.20)
D(2)21 =
∫ 0
−∞K[Gsto, F sto,τ
]dτ =
ε2ω20
16
AB
κη, (3.21)
D(2)22 =
∫ 0
−∞K[Gsto, Gsto,τ
]dτ =
ε2ω20
16
[4κξ +
(3A
2+ B
2)κη
]. (3.22)
The above yields
Fsto =εω0
4
[2ξA (t) +
√A
2+ 3B
2ηA (t)
], (3.23)
Gsto =εω0
4
[2ξB (t) +
√3A
2+ B
2ηB (t)
], (3.24)
where ∫ ∞
−∞⟨ξAξA,τ ⟩ cosωτdτ =
∫ ∞
−∞⟨ξBξB,τ ⟩ cosωτdτ =
κξ
2, (3.25)
∫ ∞
−∞⟨ηAηA,τ ⟩ cosωτdτ =
∫ ∞
−∞⟨ηBηB,τ ⟩ cosωτdτ =
κη
2, (3.26)
where ξA (t) and ξB (t), ηA (t) and ηB (t), are each identically distributed and mutu-
ally independent random variables. These are the additive and multiplicative noises
projected onto A and B coordinates.
The correlations of the system response are of great interest because they are able
to provide useful information regarding to the stochastic properties, for instance, the
probability distributions of the quadratures and mean square of the response. This
is carried out in the rotating plane using the averaged equations. The associated
quantities for the time response x(t) can then be computed. First, we define the
26
fluctuations about the deterministic model’s steady-state trajectories
δA = A− Adet, (3.27)
δB = B −Bdet. (3.28)
Typically, with weak damping and weak parametric pump, we can assume that Adet and
Bdet are highly localized about the constants A and B (the fixed point of the averaged
system) respectively, that is,Adet − A
≪ |δA| and
Bdet − B
≪ |δB|. However, if
this assumption does not hold true, i.e., the noisy effect does not dominate over the
higher harmonics (in this case, cubic harmonics), then the noisy behavior will have to
be distilled. This point will be elaborated later in this chapter.
With the elements of the Jacobian, the time derivatives of the variations δA and
δB can be written in the form [69]
δA = J11δA+ J12δB + F sto, (3.29)
δB = J21δA+ J22δB + Gsto. (3.30)
The solution of Eq. (3.29) can be expressed as
δA =
∫ t
−∞eJ11(t−t′)
[F sto (t
′) + J12δB (t′)]dt′, (3.31)
27
where the transient term is omitted [70]. The auto-correlation of δA then becomes
⟨δAδAτ ⟩ =∫ t
t′=−∞
∫ t+τ
t′′=−∞eJ11(2t+τ−t′−t′′)
⟨[F sto (t
′) + J12δB (t′)] [
F sto (t′′) + J12δB (t′′)
]⟩dt′dt′′.
(3.32)
With the change of variables t′ − t′′ = σ − τ and t′ + t′′ = 2µ [69], the process for
simplifying the stochastic terms into nonoscillatory terms is shown as follows
⟨sinωt′ sinωt′′⟩ → 1
2cosω (σ − τ), (3.33)
⟨sin2 ωt′ sin2 ωt′′
⟩→ 1
4cos2 ω (σ − τ) +
1
8cos 2ω (σ − τ), (3.34)
⟨sin 2ωt′ sin 2ωt′′⟩ → 1
2cos 2ω (σ − τ). (3.35)
The omitted oscillatory terms will not have an effect on the final expressions because
they integrate to zero.
Here, we make two assumptions to simplify the analysis for the effects of the noise:
(i) there is no frequency detuning (∆ω = 0), and (ii) there is no Duffing nonlinearity
(γ = 0). The first of these assumptions restricts the drive frequency to be at reso-
nance, which still provides interesting results. The second restricts devices to those
for which nonlinear effects are from damping. With these stated assumptions, we can
select the phase constant φ0 = π/2 to align the quadratures with the eigen-directions.
Accordingly, the Jacobian elements J12 and J21 become zero, and the cross-correlation
between δA and δB also vanishes. The auto-correlation of δA, after the change of
28
variables, eventually simplifies to
⟨δAδAτ ⟩ =ε2ω2
0
16J11
[4κξ +
(A
2+ 3B
2)κη
]eJ11|τ |. (3.36)
Similarly, for the auto-correlation of δB,
⟨δBδBτ ⟩ =ε2ω2
0
16J22
[4κξ +
(3A
2+ B
2)κη
]eJ22|τ |. (3.37)
When the system operates below the threshold (λ < 4α for zero detuning), there is
a monostable trivial solution with A = 0 and B = 0. The Jacobian of the underlying
deterministic system is given by
J = ε
⎡⎢⎣−14(4α + λ)ω0 0
0 −14(4α− λ)ω0
⎤⎥⎦ . (3.38)
As can be seen, the Jacobian is a diagonal matrix, and the diagonal elements corre-
spond to the eigenvalues in A and B directions. The auto-correlations given by Eqs.
(3.36)–(3.37) simplifies to
RδAδA (τ) = ⟨δAδAτ ⟩ =εω0κξ
4α + λe−
ε4(4α+λ)ω0|τ |, (3.39)
RδBδB (τ) = ⟨δBδBτ ⟩ =εω0κξ
4α− λe−
ε4(4α−λ)ω0|τ |. (3.40)
When the system operates above the threshold (λ > 4α), there are two stable
nontrivial solutions with equal amplitudes and phase difference of π, and the trivial
solution is unstable. This steady-state solution can be obtained by letting Fdet = 0
and Gdet = 0, which yields A = 0 and B = ±√
λ−4αβ
. The Jacobian for this response
29
is given by
J = ε
⎡⎢⎣−12λω0 0
0 −12(λ− 4α)ω0
⎤⎥⎦ . (3.41)
Again, it can be seen that φ0 = π/2 diagonalizes the Jacobian matrix, The auto-
correlations for δA and δB are given by
RδAδA (τ) = ⟨δAδAτ ⟩ =εω0
(4κξ + 3B
2κη
)8λ
e−ε4λω0|τ |, (3.42)
RδBδB (τ) = ⟨δBδBτ ⟩ =εω0
(4κξ + B
2κη
)8 (λ− 4α)
e−ε2(λ−4α)ω0|τ |. (3.43)
These results are instructive but need to be generalized to account for stiffness nonlin-
earity and off-resonance drive; this is left for future work.
3.2 Stochastic Simulations
Stochastic simulation is an important tool to verify that the theory provides good
predictions. The stochastic differential equations (SDE) used for simulations are given
by
dA = F detdt+ Fξ
√2dtdξA + Fη
√2dtdηA, (3.44)
dB = Gdetdt+Gξ
√2dtdξB +Gη
√2dtdηB, (3.45)
which are computed in the style of a Riemann-Stieltjes integral. In the above SDE,
the coefficients Fξ, Fη, Gξ, and Gη come from the following expressions in comparison
30
to Eqs. (3.17)–(3.18)
F sto = Fξ (t) ξA (t) + Fη (A,B, t) ηA (t) , (3.46)
Gsto = Gξ (t) ξB (t) +Gη (A,B, t) ηB (t) . (3.47)
The increments of the Wiener processes displayed in Eqs. (3.44)–(3.45) are
dξA =√
κξ/2N (0, 1) , (3.48)
dηA =√
κη/2N (0, 1) , (3.49)
dξB =√
κξ/2N (0, 1) , (3.50)
dηB =√
κη/2N (0, 1) , (3.51)
where N (0, 1) indicates a random variable satisfying the normal distribution with zero
mean and a variance of unity. These four normally distributed random variables are
independent from each other.
Because the system preserves ergodicity, only one realization, assuming the time is
adequately long, is needed to simulate the stochastic process.
31
3.3 Results
If the system is relatively linear near the fixed point, i.e., away from the bifurcation
conditions, and in this case, the AT threshold, then the probability distribution in
the eigen-coordinates can be assumed to be Gaussian. This property suggests that
the mean and variance are sufficient to describe the probability density for A and
B. Their means are simply A and B, up to the first-order approximation. And with
the assumption that the deterministic steady state (Adet, Bdet) is highly localized to a
stable fixed point(A, B
), the variances of A and B, on the other hand, can be derived
from the auto-covariance K at τ = 0, which is the same as the auto-covariance of its
deviation (δA or δB), which then becomes the deviation’s auto-correlation (due to zero
mean), that is
Var (A) = KAA (τ = 0) = KδAδA (τ = 0) = RδAδA (τ = 0)− E2 [δA] = RδAδA (τ = 0) ,
(3.52)
Var (B) = KBB (τ = 0) = KδBδB (τ = 0) = RδBδB (τ = 0)− E2 [δB] = RδBδB (τ = 0) .
(3.53)
For the trivial solution, with Eqs. (3.39)–(3.40), the variance of A and B are given
by
Var (A) =εω0κξ
4α + λ, (3.54)
Var (B) =εω0κξ
4α− λ. (3.55)
It is seen that at zero pump level the two variances are equal, and as the pump level
is increased from zero the variance of A decreases and can reach a minimum value
32
-0.2
-0.1
0.0
0.1
0.2
t
A
-0.2
-0.1
0.0
0.1
0.2
t
B
(a) (b)
Figure 3.1: Stochastic simulation with ω0 = 1, α = 0.05, β = 0.05, λ = 0.16, κξ =0.0001, and κη = 0.0001. (a) A single time domain realization showing A. (b) A singletime domain realization showing B
of εω0κξ/ (8α) whereas the variance of B increases without bound as the threshold
is approached. Of course, this situation is the result of the zero eigenvalue point
(bifurcation) being approached, where B spreads out and nonlinear effects come into
play near the threshold. It is also seen that the multiplicative noise does not affect the
variances for the trivial solution, at least to leading order in the analysis. This can be
understood by noting that this noise is multiplied by the response and is thus higher
order.
Fig. 3.1 shows a single, long time realization of the system, which describes how
A and B fluctuates as time evolves. Fig. 3.2 depicts the realization distribution in
the A and B plane with parametric pump off and with a pump level close to the
threshold. By examining Eq. (3.54), it is suggested that at threshold the noise can be
squeezed to one half compared to the case where the parametric pump is off. This noise
squeezing phenomenon is also exhibited by the comparison of the two figures. Fig. 3.3
demonstrates the distinction in the variance of the two eigen-coordinates, where the
noise for A is squeezed and the opposite for B. Furthermore, this shows that the theory
33
(a) (b)
Figure 3.2: Realization distribution showing in the A and B plane, with the samedevice parameters as Fig. 3.1. (a) Parametric pump is off, serving as a reference. (b)Parametric pump level at 80% threshold (using the data shown in Fig. 3.1), illustratingthe noise squeezing effect in A
Theory
Simulation
-0.2 -0.1 0.0 0.1 0.20
5
10
15
20
A
ProbabilityDensity
Theory
Simulation
-0.2 -0.1 0.0 0.1 0.20
2
4
6
8
B
ProbabilityDensity
(a) (b)
Figure 3.3: Comparison for the probability density of A and B, using the data shownin Fig. 3.1. The orange curves indicate theory, and the blue curves illustrate thesimulation results. (a) Probability density for A. (b) Probability density for B
agrees well with the stochastic simulation.
For the nontrivial solution, using Eqs. (3.42)–(3.43) and Eqs. (3.52)–(3.53), the
34
variance of A and B are given by
Var (A) =εω0
(4κξ + 3B
2κη
)8λ
=εω0
(4κξ
B2 + 3κη
)8β λ
λ−4α
, (3.56)
Var (B) =εω0
(4κξ + B
2κη
)8 (λ− 4α)
=εω0
(4κξ
B2 + κη
)8β
, (3.57)
where B2= (λ− 4α) /β. Note that as the pump level increases, the averaged amplitude
B grows like√λ, and therefore at large pump levels the effects of additive noise are
reduced but the effects of multiplicative noise saturate to a finite value for A and
remain a constant for B. This follows since that noise is multiplied by the steady-state
response. Therefore, one can improve the SNR only to a limit set by the multiplicative
noise.
It is important to point out that we have made an important assumption—the
deterministic steady state is localized to a fixed point. This assumption, however,
may not necessarily hold true if the effect of damping or parametric pump is strong,
which induces cubic harmonic content to the response, in which case the steady state
(Adet, Bdet) has a periodic component. Hence, if the assumption fails, then Var (δA)
and Var (δB) are used in place of Var (A) and Var (B).
Fig. 3.4 demonstrates an example of a steady state as viewed in the rotating
plane. The trajectory of the deterministic model (Adet, Bdet) is indicated in green,
with a period of 1/ (2ω). Therefore, in order to distill the noisy behavior, a direct
integration is carried out to obtain the varying Adet and Bdet. This method eliminates
the undesired periodic variation from cubic harmonics and helps to extract the data
whose randomness is pertinent to the noise only. A Gaussian-like distribution, thereby,
is acquired, as shown in Fig. 3.4(b).
35
(a) (b)
Figure 3.4: Realization distribution at twice the threshold (λ = 0.4), with the samedevice parameters as Fig. 3.1 (a) In the A and B plane, the deterministic steady stateis not localized to a fixed point, and the trajectory of (Adet, Bdet) is shown by the greencurve. (b) In the δA and δB plane, where the distribution only comes from the noise
Theory
Simulation
-0.2 -0.1 0.0 0.1 0.20
5
10
15
δA
ProbabilityDensity
Theory
Simulation
-0.2 -0.1 0.0 0.1 0.20
5
10
15
δB
ProbabilityDensity
(a) (b)
Figure 3.5: Comparison for the probability density of A and B, using the data shownin Fig. 3.4. The orange curves indicate theory, and the blue curves illustrate thesimulation results. (a) Probability density for A. (b) Probability density for B
Fig. 3.5 demonstrates the probability density for δA and δB. This provides a
intuitive illustration for the distilled noisy effect. The fact that the theory has an
36
B
A
0 0.1 0.2 0.3 0.4 0.5 0.60
1
2
λ
E[A],E[B]
B
A
0 0.1 0.2 0.3 0.4 0.5 0.60
1E-3
2E-3
λ
Var(δA),Var(δB)
(a) (b)
Figure 3.6: Comparison between the theory (solid curves) and simulations (dots) forthe means and variances of A (dark colors) and B (light colors), with the same deviceparameters as Fig. 3.1 and with a parametric sweep of λ. The orange curves representtheory, and the blue dots show the simulation results. The theoretical results are givenby Eqs. (3.54)–(3.55) below threshold and Eqs. (3.56)–(3.57) above threshold. Thethreshold is at λ = 0.2, as clearly suggested by the figures. (a) Mean values of A andB. (b) Variances of δA and δB (when below threshold, δA = A and δB = B)
excellent agreement with the stochastic simulation verifies the validity of the theory
for the nontrivial case.
Fig. 3.6 demonstrates the means and variances of A (dark colors) and B (light
colors) with a parametric sweep of λ, where the comparisons between the theory (solid
curves) and simulations (dots) are shown. It can be seen in Fig. 3.6(a) that the
pitchfork bifurcation of (A,B) = (0, 0) occurs at the threshold λ = 0.2. In Fig. 3.6(b),
the variance of A reaches the minimum, which is one half of its value at λ = 0.
Meanwhile, the variance of δB becomes extremely large near the bifurcation condition
since the dynamics of B is on a slow manifold (center manifold for λ = 0.2) that has
near-zero eigenvalues that leads to large excursions. The quantification of Var (δB) near
bifurcation, however, cannot be predicted using the simple linearization method since
the response here is dominated by nonlinear effects. As λ is further increased above
37
the threshold, Var (δB) and Var (δA) show that the effects of additive noise continually
diminish and the variances converge to constant values that are proportional to the
level of multiplicative noise; see Eqs. (3.56)–(3.57).
Now, we turn our attention from the quadratures A and B back to x in order to
analyze its noisy behavior. This allows us to examine the root mean square (RMS)
of x, which can be of particular interest to experimentalists. This is because with the
knowledge of RMS, one can also obtain the information pertinent to the signal-to-noise
ratio.
First of all, the fluctuations of x due to noise is
δx = x− xdet
= A cosωt+B sinωt− Adet cosωt−Bdet sinωt
= δA cosωt+ δB sinωt,
(3.58)
where xdet is the steady-state response of x. When the system operates above the
threshold, xdet consists of both the averaged response A cosωt+ B sinωt and periodic
variation due to higher harmonics. For the trivial response, it is evident that δx = x.
The variance of δx is computed using
Var (δx) = ⟨(δA cosωt+ δB sinωt)2⟩
= ⟨(δA)2 cos2 ωt⟩+ ⟨(δB)2 sin2 ωt⟩+ ⟨δAδB sin 2ωt⟩
=1
2⟨(δA)2⟩+ 1
2⟨(δB)2⟩
=1
2[Var (δA) + Var (δB)] .
(3.59)
For the trivial response,
Var (x) =4εαω0κξ
16α2 − λ2, (3.60)
38
0 0.1 0.2 0.3 0.4 0.5 0.60
1E-3
2E-3
λ
Var(δx)
0 0.1 0.2 0.3 0.4 0.5 0.60
50
100
150
λ
Signal-to-NoiseRatio
(a) (b)
Figure 3.7: Comparison between the theory (solid curves) and simulations (dots) forthe variance of δx and signal-to-noise ratio, with the same device parameters as Fig.3.1 and with a parametric sweep of λ. The orange curves represent theory, and the bluedots show the simulation results. The threshold is at λ = 0.2, as clearly suggested bythe figures. (a) variance of δx versus parametric pump (δx = x when below threshold).(b) SNR versus parametric pump
which shows that the variance increases without bound, and cannot be modeled using
simple linearization, as the parametric pump approaches the AT threshold. For the
nontrivial response,
Var (δx) =εω0
[2 (λ− 2α)κξ + (λ− 3α) B
2κη
]4λ (λ− 4α)
=εω0
4
[2 (λ− 2α)κξ
λ (λ− 4α)+
(λ− 3α)κη
βλ
],
(3.61)
where B2= (λ− 4α) /β. Similar as before, this expression suggests that as paramet-
ric pump continue to increase, the variance will eventually converges to a constant
determined by multiplicative noise, since the effect of additive noise vanishes.
In experiments, the RMS of δx, which is a signal with zero-mean, corresponds to
the square root of its variance Var (δx). Therefore, the signal to noise ratio is defined
to be the ratio of the amplitude to the RMS of the noise.
39
Fig. 3.7(a) shows the variance in δx, wherein the theory is shown in orange curves
and the simulations are indicated in blue dots. From the figure, it can be seen that the
variance extends to very large values near the AT bifurcation condition due to near-zero
eigenvalue, and converges to a constant as the parametric pump level becomes large.
Fig. 3.7(b) illustrates the SNR of of the system above threshold, which continuously
increases above the threshold.
The power spectral density (PSD) is also of great interest for experimentalists, as
a means of characterizing the effects of noise, and the present model allows one to
compare theoretical and experimental results. The PSD of x can be obtained from the
auto-correlations of A and B, expressed in Eqs. (3.39)–(3.40) and Eqs. (3.42)–(3.43),
using the Fourier transform, which is given by [1]
Sxx (Ω) =
∫ ∞
−∞e−iΩτ 1
2[RAA (τ) + RBB (τ)] cosωτdτ (3.62)
for systems under the stated assumptions.
Fig. 3.8 demonstrates the PSD of x below and above the threshold. When the
parametric pump level is slightly below the AT threshold, spectral narrowing can be
observed, which is consistent with the results given in [1]. When the parametric pump
level is above the threshold, the effects of nonlinear damping and multiplicative noise
manifest. For the case shown above threshold the noise peak is flattened, but this
will depend in a nontrivial manner on the system and pump parameters. It will be
interesting to see how the theoretical results compare with simulation and experimental
results when available, and how nonlinear damping and multiplicative noise in devices
affect the spectrum.
We now turn to the closed loop case, which is more relevant to applications.
40
0 0.5 1 1.5 20
0.05
0.1
0.15
Ω
PSD
(a)
0 0.5 1 1.5 20
0.01
0.02
Ω
PSD
(b)
Figure 3.8: Power spectral density of x, with the same device parameters as Fig. 3.1.The black curves show the PSD of a parametrically pumped systems, and the graycurves show the PSP of an unpumped system, serving as a reference. (a) λ = 0.16.Spectral narrowing can be observed, which is the same phenomenon shown in [1]. (b)λ = 0.4. This shows the PSD of a system with multiplicative noise and a pump levelabove the AT threshold
41
Chapter 4
Phase-Locked Loop Operation with
Noise
The parametric oscillator can operate as a phase-locked loop. This can have some very
important applications such as frequency generation in time-keeping [13, 14, 15]. Here
the system self-oscillates without any external periodic drive, and the frequency is set
by loop and device parameters. The dynamics are quite different here since there is no
absolute time reference.
The equation of motion for PLL with additive and multiplicative noises is given by
x+2ε(α + βx2
)ω0x+[1 + ελ cos (2ω0t+ 2Φ (t) + ∆φ) + εη (t)]ω2
0x+εγx3 = εω20ξ (t) .
(4.1)
The deterministic response has been studied in Section 2.2 and here we consider fluc-
tuations about that response.
42
4.1 Theory and Simulation Results
Similar to the results from the previous chapter, in order to simplify the analysis,
we make two assumptions: (i) there is no Duffing nonlinearity (γ = 0), and (ii) the
phase shift is selected to be ∆φ = −π/2, a value typically selected for self-oscillation
[18]. With these assumptions, the steady-state amplitude for the deterministic PLL is
R =√
λ−4αβ
, and the frequency at which the system operates is the resonator natural
frequency ω = ω0.
Using a similar approach as the previous section, the stochastic parts of the un-
averaged time derivatives of R and Φ are given by
F sto =εω0
2[−2ξ (t) sin (ω0t+ Φ) + η (t)R sin (2ω0t+ 2Φ)] , (4.2)
Gsto =εω0
R
[−ξ (t) cos (ω0t+ Φ) + η (t)R cos2 (ω0t+ Φ)
]. (4.3)
The Jacobian matrix for the deterministic part of the dynamical system is given by
J = ε
⎡⎢⎣−12(λ− 4α)ω0 0
0 0
⎤⎥⎦ , (4.4)
which possesses the expected form. The zero eigenvalue for the phase is consistent with
the fact that there is no external time reference, so that it will not fluctuate around a
fixed value but will rather experience continual phase diffusion. This diffusion is a key
measure of the quality of a time-keeping device [17]. The amplitude will fluctuate about
its deterministic steady value, and the auto-correlation for the amplitude deviation is
given by
RδRδR (τ) = ⟨δRδRτ ⟩ =εω0
(4κξ + R
2κη
)8 (λ− 4α)
e−ε2(λ−4α)ω0|τ |. (4.5)
43
0.8
0.9
1.0
1.1
1.2
t
R
Theory
Simulation
0.8 0.9 1.0 1.1 1.20
2
4
6
8
R
ProbabilityDensity
(a) (b)
Figure 4.1: Stochastic simulation with ω0 = 1, α = 0.01, β = 0.01, λ = 0.05, κξ =0.00004, and κη = 0.00004. (a) Ten realizations showing R. (b) Comparison for theprobability density of R
Notice that this auto-correlation shares the same form as Eq. (3.43), which is as
expected. Additionally, as elaborated in Section 3.3, the variance of R is obtained
from the auto-correlation at τ = 0, is given by
Var (R) =εω0
(4κξ + R
2κη
)8 (λ− 4α)
=εω0
(4κξ
R2 + κη
)8β
. (4.6)
Fig. 4.1 illustrates the realization and the probability density of R. The selected
damping coefficients are smaller compared to the previous chapter, which allows for
less periodic variation of the steady-state response, a characteristic that is of vital
importance for a PLL. This is because PLLs are oftentimes used for time-keeping, and
minimizing the fluctuations, even those that are periodic and come from deterministic
effects, is beneficial to the operation. In practice, one uses a notch filter to get rid of
such oscillations, and only the signal and noise in the notch are of interest.
44
The details for the variance of the phase over a relatively long period of time is
elucidated in the following section.
4.2 Precision of Time Keeping
The behavior of the phase in this case has a completely different character as compared
to the case with harmonic excitation. The fact that there is no external time reference
results in a zero eigenvalue in the phase direction, and the phase will diffuse and
eventually spread out over the entire circle. The speed at which this occurs is an
important measure of the stability of frequency generator, that is, a clock. Of particular
interest is the frequency stability, which is a consequence stemming from the phase
noise. It characterizes the precision of a time-keeping system, for instance, how a clock
performs, since it describes the diffusion of the variance of the times of individual cycles
over many cycles (a cycle being a phase of 2π). With this purpose, the phase shift of
the PLL over a relatively long time T (T ≫ τcorr) is defined as
∆Φ = δΦT − δΦ. (4.7)
45
Using Eq. (4.3) and Eq. (4.7), and with Φ acting as a random variable uniformly
distributed among [0, 2π) [69], it follows that
⟨(∆Φ)2
⟩=
∫ T
t′=0
∫ T
t′′=0
⟨Φsto (t
′) Φsto (t′′)⟩dt′dt′′
=ε2ω2
0
R2
∫ T
t′=0
∫ T
t′′=0
⟨ξ (t′) sin (ω0t′ + Φ)ξ (t′′) sin (ω0t
′′ + Φ)⟩ dt′dt′′
+ ε2ω20
∫ T
t′=0
∫ T
t′′=0
⟨η (t′) cos2 (ω0t
′ + Φ)η (t′′) cos2 (ω0t′′ + Φ)
⟩dt′dt′′
=ε2ω2
0
R2
∫ T
t′=0
∫ T
t′′=0
⟨ξ (t′) ξ (t′′)⟩[1
2cosω0 (t
′ − t′′)
]dt′dt′′
+ ε2ω20
∫ T
t′=0
∫ T
t′′=0
⟨η (t′) η (t′′)⟩[1
8+
1
8+
1
8cos 2ω0 (t
′ − t′′)
]dt′dt′′
=ε2ω2
0
R2
∫ T
−T
⟨ξξτ ⟩(1
2cosω0τ
)(T − |τ |) dτ
+ ε2ω20
∫ T
−T
⟨ηητ ⟩(1
4+
1
8cos 2ω0τ
)(T − |τ |) dτ
∼=ε2ω2
0
R2
κξ
2T + ε2ω2
0
3κη
8T.
(4.8)
Therefore, the variance of the phase shift becomes
Var (∆Φ) =⟨(∆Φ)2
⟩= ε2ω2
0
(κξ
2R2 +
3κη
8
)T. (4.9)
This variance is proportional to the time interval T , and depends on both additive
noise and multiplicative noise.
From Eq. (4.9), it can be seen that the strength of additive and multiplicative noise
can be characterized by varying the operating amplitude of the system. The theory
and stochastic simulation results are displayed in Fig. 4.2. The variance in phase shift
reduces as the amplitude is increased; this is because the effect of additive noise is
diluted by amplitude. As the amplitude is further increased, the phase shift variance
approaches a non-zero minimum, which is a limit imposed by the multiplicative noise.
46
0 1 2 30
2E-5
4E-5
6E-5
0.04 0.05 0.06 0.07
R2
Var(Δ
Φ)/T
λ
Figure 4.2: Characterizing the strength of additive and multiplicative noise by varyingthe amplitude. The orange solid curve indicates the theory, the green dashed curve rep-resents the asymptotic line based on theory, and the the blue dots shows the stochasticsimulation results
This is a common feature of time-keeping systems, that increasing the amplitude helps
improve SNR. However, stiffness nonlinearity, which has been ignored here, sets a limit
on this effect, since it results in amplitude-to-phase noise conversion, resulting in an
operating amplitude at which the phase noise is minimized [17].
47
Chapter 5
Conclusions and Future Work
The thesis extends previous results in that it examines the effects of additive and multi-
plicative noise on nonlinearly damped parametric oscillators, with both open loop and
PLL operations considered. Most notably this work is new in that its modeling and
analysis: (i) includes nonlinear damping, which is widely observed and plays a signifi-
cant role in parametrically excited systems, especially those at micro and nano scales,
and (ii) considers multiplicative noise, which has a far-reaching impact on frequency
fluctuations.
5.1 Main Results
In this work, predictive results based on theoretical analysis and stochastic simula-
tions are developed, and their results are compared with each other. The implica-
tions of the results have also been extensively discussed. For open loop operation, the
auto-correlation and probability density functions for the uncorrelated quadratures
(eigen-coordinates) are computed in the absence of frequency detuning and stiffness
nonlinearity. The impact of noise on SNR and PSD are also demonstrated. For PLL
operation, the variance of phase, which stems from frequency fluctuations, is obtained
in the absence of stiffness nonlinearity and for the resonant value of the phase shift. All
48
the main results of theory under the stated assumptions are verified using stochastic
simulations.
5.2 Future Work
There are a list of extensions to the analysis that can be carried out based upon the
results shown in the thesis. First and foremost, the analytical results can be expanded
for systems with cubic and quintic stiffness nonlinearities, frequency detuning, and
parametric amplification; possible challenges may arise from the calculation of the
cross-correlations and higher order polynomials, which necessarily complicates the cal-
culations. Second, higher-order approximations can be used for drift coefficients to
improve accuracy; likewise, alternative stochastic simulation methods can be consid-
ered. Third, while this work only considers white noise, it would interesting to examine
the effects of colored noise. Eqs. (3.33)–(3.35), (4.8) may shed light on these issues.
It would be of great interest if this work can motivate future research in following
directions: One is to consider the amplitude-to-frequency noise conversion and examine
the limit of noise in regards to operation at a zero-dispersion point [17]. The other is
to analyze the noise behavior in the context of a multi-mode system and investigate
how nonlinear mode coupling enhances or attenuates noise squeezing and suppression
[71].
Another line of related work involves experimental measurements to confirm the
analytical results. Such experiments have been initiated, as described next.
49
5.3 Future Work: Experiments
At the moment of the submission of this thesis, experiments are being carried out by
the author and collaborators at the Mechanical Engineering Research Lab (MERL) at
Stanford University. The device, based on the one described in [18], had been fabricated
and encapsulated using the EpiSeal process. The main structure of the resonator is
an in-plane cantilever, whose base consists of an engine beam and a sense beam. The
device can be actuated using either capacitive drive or thermal drive. Both capacitive
and piezoresistive sensing are available for the measurements of the response.
A temperature chamber is used to maintain a set desired temperature during opera-
tion; it also allows for varying the temperature in order to examine how noisy behavior
is affected by thermal effects. Inside the chamber, a breakout board, resting on a sponge
pad, securely attaches the device and cables. Two power supplies are employed, where
each can generate up to 60V of bias voltage, adding up to a combined total of 120V,
which facilitates excellent transduction for capacitive sensing. A Zurich Instruments
(ZI) lock-in amplifier is implemented for signal generation and measurement. A tran-
simpedance amplifier (TIA) is connected to the electrode used for capacitive sensing,
which also gives rise to electrical noise during measurement. A vector signal generator
is utilized to visualize the noise spectrum and determine the thermomechanical noise
contribution.
The encapsulated chip, containing multiple MEMS resonators, is wire bonded on
a printed circuit board (PCB). Each MEMS device has six electrodes, which are con-
nected to six bond pads on the PCB. Some electrodes are used for capacitive drive and
sensing, while others are related to the thermal drive and piezoresistive sensing. Be-
tween the chip and the copper substrate is the photoresist, which is to be subsequently
removed by acetone so as to float the chip. Inside the fume hood, the photoresist is
50
applied on the center copper substrate of a PCB, the chip is carefully affixed to the
photoresist, and the entire PCB is placed on a hot plate for heating. After that, the
chip, along with the PCB, is safely removed from the fume hood and arranged for
wire bonding at a suitable height and orientation under a microscope. Threading is
oftentimes needed, which requires a meticulous manipulation of tweezers.
The software tools that are used for the experiment include National Instruments
(NI) Measurement & Automation Explorer (MAX), ziControl, and MATLAB. Specif-
ically, NI MAX is used to link the devices and interfaces together, which utilizes a
PCI-GPIB to connect the hardware components, including the temperature chamber,
two power supplies, a spectrum analyzer, and a multimeter. ziControl is employed to
control the demodulators and signal amplitudes of the direct drive and the parametric
pump; it is also able to act as an oscilloscope, carry out frequency sweeps in both
directions, and compute fast Fourier transforms (FFT). MATLAB is used to extract
experimental data from the device, process the data, and generate pertinent plots.
During the experiments, the bias voltage should be slowly increased so as to avoid
pull-in, an instability that results in a short circuit. A pull-in occurrence typically dam-
ages both the device and the TIA; the latter has to be replaced at a soldering station,
where a soldering iron, hot air pencil, and airbath are used. Currently, preliminary
experimental data have been obtained and analyzed, and more systematic experiments
are underway.
51
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60
Appendix A
Background in Nonlinear Stiffness
Although linear models enjoy the advantages in simplicity and the ability to be solved
analytically, they are inadequate for describing many types of behaviors observed in
a variety of systems such as MEMS. In fact, nonlinear models can play a pivotal
role to account for some important and practical properties of the response of MEMS
resonators. As a consequence, it is oftentimes desirable to include pertinent nonlinear
terms in the modeling and analysis.
In general, both phenomenological modeling and first-principle modeling can be em-
ployed to derive a governing equation of motion. Phenomenological models are usually
more commonly used due to their expediency: the nonlinear terms can be obtained
directly from the theories of Taylor expansion and normal forms. This approach is
useful when the source of nonlinearity is unknown or difficult to model. Of course, in
this approach an understanding of the physical system can still be helpful to ensure a
sensible modeling process and to understand the effects of device parameters. Typi-
cally, finite element analysis and/or experiments are then carried out to determine the
system parameters of the single-mode model and to verify the validity of the model.
First-principle modeling, on the other hand, can be more involved and may be unnec-
essary in many instances; however, whenever possible, it is always beneficial to derive
the system’s equation of motion using such approach in order to gain more complete
61
insight about the effects of parameters on system response. Furthermore, such models
can be used to benchmark computational and/or experimental results, which is a very
favorable capability.
Since our focus is on analytical work done in support of experiments, we consider
models of nonlinear oscillations governed by a single mode of vibration, that is, one
degree of freedom. Even for such a simple class of models, it is vital to include in
them the important terms that capture the characteristics of interest. These can be
introduced in an ad hoc manner, but it is more desirable to derive them from first
principles in order to possess an adequate understanding of their physical sources.
There are many sources of nonlinear behavior, and in this appendix we describe two of
the most common ones, namely, finite deformation mechanics and electrostatic fields,
which lead to nonlinear stiffness effects. Nonlinear damping is more challenging to
describe from first principles (as is linear damping, for that matter), but it is widely
observed in in large spectrum of systems and is described phenomenologically in the
next appendix.
For nonlinear stiffness, this work provides an example of a typical mechanical
structure under electrostatic forcing, demonstrating the physics behind the nonlin-
ear phenomenon in such systems. Because such a system is highly representative in
electromechanical devices (especially in micro- and nano-scale, where mechanical and
electrostatic forces are comparable), it allows one to understand the underlying sources
of nonlinearities in a variety of similar systems.
A clamped-clamped beam subjected to electrostatic forces is considered, where this
system can be, in general, nonuniform along the longitudinal direction. For mechanical
nonlinearity, the majority of structures with such type of boundary conditions, such as
strings, disks, and membranes, typically share similar properties in their nonlinear phe-
nomenon accordingly. In contrary, cantilevers and rings, which are not fully clamped
62
at their boundaries, thus have minimal mechanical stiffness nonlinearity due to their
negligible midline stretching.
To acquire the dynamics of the mechanical structure, the extended Hamilton’s
principle is first used, written in the form [72].
∫ T2
T1(δT − δVm + δWes) dT = 0, (A.1)
where δT represents the variation in kinetic energy, δVm denotes the variation in me-
chanical potential energy, and δWes indicates the variation in the work done by the
electrostatic force respectively. The kinetic energy of a mechanical beam is given by
T =1
2
∫ L
0
ρA
(∂u
∂T
)2
dl, (A.2)
where L denotes the total length of the beam, ρ is the mass density per unit length,
A (l) represents the cross-sectional area, and u (l, T ) represents the transverse nodal
displacement at the longitudinal position l and time T . In the common case of thin
structures, the rotary inertia and the shear deformation effect can be neglected [73]
With the use of Taylor expansion, the potential energy due to the mechanical force
can be derived from
Vm =1
2
∫ L
0
EI
(∂2u
∂l2
)2
dl +
∫ L
0
τ (ds− dl)
=1
2
∫ L
0
[EI
(∂2u
∂l2
)2
+ τ
(∂u
∂l
)2]dl +O
[(∂u
∂l
)3],
(A.3)
where E is the Young’s modulus, I (l) represents the second moment of area, ds de-
scribes the length of a differential element dl in displaced position, and τ denotes the
axial tension. Only the leading-order term in the axial tension is kept because higher
order terms are generally not needed. The axial tension in the expression can be written
63
as
τ = τ0 +E
L
∫ L
0
A (ds− dl)
= τ0 +E
2L
∫ L
0
A
(∂u
∂l
)2
dl +O
[(∂u
∂l
)3],
(A.4)
where τ0 represents the axial pretension and the second term arises from the axial
extension of the beam, which is a nonlinear effect. Again, only the leading-order term
is kept.
For a clamped-clamped beam, the two ends of the beam satisfy the following bound-
ary conditions
u (0, T ) = u (L, T ) = 0, (A.5)
∂u
∂l(0, T ) =
∂u
∂l(L, T ) = 0. (A.6)
It is important to point out that not all structures share these boundary conditions,
and that others can result in distinct nonlinear properties. For instance, cantilever
beams are typically considered to be more linear-like systems due to their minimal
midline stretching (other nonlinear effects exist but are much less pronounced).
For a clamped-clamped beam, The variation in the kinetic energy is calculated using
Fubini’s theorem and integration by parts with the boundary conditions invoked, which
is given by ∫ T2
T1δTdT = −
∫ T2
T1
∫ L
0
ρA∂2u
∂T 2δudldT . (A.7)
The variation in the mechanical potential energy is calculated using integration by
parts, which is given by
δVm =
∫ L
0
[E
∂2
∂l2
(I∂2u
∂l2
)− τ
∂2u
∂l2
]δudl. (A.8)
64
The variation in the work done by the electrostatic force is given by
δWes =
∫ L
0
Fesδudl, (A.9)
where Fes is the electrostatic force per unit length.
At this point, the equation of extended Hamilton’s principle becomes
∫ T2
T =T1
∫ L
l=0
[−ρA
∂2u
∂T 2− E
∂2
∂l2
(I∂2u
∂l2
)+ τ
∂2u
∂l2+ Fes
]δudldT = 0. (A.10)
Due to the arbitrariness of the virtual displacement δu, it follows that
ρA∂2u
∂T 2+ E
∂2
∂l2
(I∂2u
∂l2
)−
[τ0 +
E
2L
∫ L
0
A
(∂u
∂l
)2
dl
]∂2u
∂l2= Fes. (A.11)
This is a fourth-order partial differential equation that governs the beam vibration.
For the electrostatic force, this paper considers parallel-plate capacitances, where
the capacitances per unit length for the drive electrode and sense electrode are given
by
dCd
dl=
ε0εrh
dd − uand
dCs
dl=
ε0εrh
ds + u, (A.12)
respectively, where ε0 is the vacuum permittivity, εr is the relative permittivity, dd (l)
denotes the drive electrode gap distance, ds (l) denotes the sense electrode gap distance,
and h is the height of the beam. Ignoring the fringing effect, the total electrostatic
force per unit length can be expressed by
Fes =1
2V 2d ∇
dCd
dl+
1
2V 2s ∇
dCs
dl=
ε0εrh
2
[V 2d
(dd − u)2− V 2
s
(ds + u)2
]1[l1,l2], (A.13)
where Vd (l) represents the bias voltage of the drive electrode, Vs (l) represents the bias
voltage of the sense electrode, and 1[l1,l2] (l) denotes the indicator function with the
65
interval [l1, l2] indicating the longitudinal position range of the electrodes. The bias
voltages for the drive and sense electrode are defined to be
Vd = VDC,d + VAC cosΩT and Vs = VDC,s, (A.14)
respectively, where VDC,d (l) is the DC bias voltage of the drive electrode, VDC,s (l) is the
DC bias voltage of the sense electrode, VAC (l) is the AC bias voltage, and Ω indicates
the drive frequency. It then follows that
V 2d = V 2
DC,d +1
2V 2AC + 2VDC,dVAC cosΩT +
1
2V 2AC cos 2ΩT . (A.15)
Using Taylor expansion and averaging, the total electrostatic force per unit length can
be approximated by a polynomial in u, which is given by
Fes∼= ε0εrh
∑i=1,3,···
i+ 1
2
[(V 2DC,d +
12V 2AC
)ui
di+2d
+V 2DC,su
i
di+2d
]+
VDC,dVAC
d2dcosΩT
1[l1,l2].
(A.16)
The solution to the above PDE, Eq. (A.11), is assumed to be in the following form
u (l, T ) =∑n
φn (l)Xn (T ). (A.17)
where φn (l) is the nth modal displacement and Xn (T ) is the physical displacement of
the nth mode. With the assumption that the modes are orthogonal, this displacement
can be obtained using the operator∫ L
0dl and by projecting φn onto
∑m
[ρAφmXm + E (Iφ′′
m)′′Xm − τ0φ
′′mXm − E
2L
∫ L
0
A (φ′m)
2X3
mdlφ′′mXm
]= Fes.
(A.18)
66
This allows to obtain a second-order ordinary differential equation in Xn (T )
ρ
∫ L
0
Aφ2ndlXn + E
∫ L
0
φn (Iφ′′n)
′′Xn − τ0
∫ L
0
φnφ′′ndlXn
− E
2L
∫ L
0
A (φ′n)
2dl
∫ L
0
φ′′nφndlX
3n =
∫ L
0
φnFesdl.
(A.19)
The integrals can be simplified using the boundary conditions Eqs. (A.5)–(A.6)
∫ L
0
φn (Iφ′′n)
′′dl = φn (Iφ
′′n)
′ |L0 − φ′nIφ
′′n|L0 +
∫ L
0
I (φ′′n)
2dl =
∫ L
0
I (φ′′n)
2dl, (A.20)
∫ L
0
φnφ′′ndl = φnφ
′n|L0 −
∫ L
0
(φ′n)
2dl = −
∫ L
0
(φ′n)
2dl. (A.21)
The equation of motion for the nth mode can now be express as
Xn+E∫ L
0I (φ′′
n)2 dl + τ0
∫ L
0(φ′
n)2 dl
ρ∫ L
0Aφ2
ndlXn+
E∫ L
0A (φ′
n)2 dl
∫ L
0(φ′
n)2 dl
2ρL∫ L
0Aφ2
ndlX3
n =
∫ L
0Fesφndl
ρ∫ L
0Aφ2
ndl.
(A.22)
The complete equation of motion for the nth mode can be rewritten in the form
Xn + Γ1,nXn + Γ3,nX3n + Γ5,nX
5n + · · · = FAC,n cosΩT , (A.23)
with the stiffness coefficients and forcing expressed as
Γ1,n =E∫ L
0I (φ′′
n)2 dl
ρ∫ L
0Aφ2
ndl+τ0∫ L
0(φ′
n)2 dl
ρ∫ L
0Aφ2
ndl−ε0εrh
∫ l2l1
[(V 2DC,d +
12V 2AC
)d−3d + V 2
DC,sd−3s
]φ2ndl
ρ∫ L
0Aφ2
ndl,
(A.24)
Γ3,n =E∫ L
0A (φ′
n)2 dl
∫ L
0(φ′
n)2 dl
2ρL∫ L
0Aφ2
ndl−
2ε0εrh∫ l2l1
[(V 2DC,d +
12V 2AC
)d−5d + V 2
DC,sd−5s
]φ4ndl
ρ∫ L
0Aφ2
ndl,
(A.25)
67
Γi=5,7,··· ,n = −(i+ 1) ε0εrh
∫ l2l1
[(V 2DC,d +
12V 2AC
)d−i−2d + V 2
DC,sd−i−2s
]φi+1n dl
2ρ∫ L
0Aφ2
ndl, (A.26)
FAC,n =ε0εrh
∫ l2l1VDC,dVACd
−2d φndl
ρ∫ L
0Aφ2
ndl. (A.27)
68
Appendix B
Background in Nonlinear Damping
In this present work, nonlinear damping, also known as nonlinear dissipation or nonlin-
ear friction, is taken into account, in addition to stiffness nonlinearity. This is of practi-
cal interest because nonlinear damping is frequently observed in a huge variety of small
structures. For example, it has oftentimes been observed in NEMS resonators based
on carbon nanotubes [57], graphene [57, 58, 59], and diamond [60]. Likewise, nonlinear
damping has also been commonly observed in micro-structures such as non-contacting
atomic force microscope (AFM) microbeams [61] and MEMS clamped-clamped beams
[21, 22]. In addition, is has been observed in macroscopic mechanical systems, for in-
stance, in large-amplitude ship rolling motions [62], concrete structures [63], stainless
steel rectangular plates, stainless steel circular cylindrical panels, and zirconium alloy
hollow rods [64]. Nonlinear damping of a given mode can also result from mode interac-
tions including induced two-phonon processes [65] and internal resonances [11, 12, 59].
In addition to the experimental observations, various theoretical works have also been
published in this area, covering the topics of the relaxation of nonlinear oscillators in-
teracting with a medium [20], estimation using Melnikov theory [66], estimation using
analytic wavelet transform [67], dynamic response to harmonic drive [21], and charac-
terization using the ringdown response [22].
Nonlinear damping has great significance for systems with parametric resonance due
69
to a phenomenon known as Arnold tongue, above which the systems in the absence of
nonlinearities become unstable, and the response branches without nonlinear damping
are not closed from simple perturbation theory. This will be elucidated in Section 3.1
in detail.
In contrary to the first-principle modeling demonstrated for nonlinear stiffness,
nonlinear damping, on the other hand, is immensely more challenging to model from
first principles in light of its physical natural; therefore, it is common to simply include
it in the model when needed. Consequently, here we model in a phenomenological way
using polynomials, whose coefficients can be easily characterized using the ringdown
response.
For nonlinear damping, the damping of the system for the nth mode is modeled
using
2√Γ1,n
(α1,n + α3,nX
2/X20 + α5,nX
4/X40 + · · ·
)X, (B.1)
where α1,n is the linear damping coefficient, and αi>1,n’s are the nonlinear damping
coefficients; all the damping coefficients are dimensionless. The coefficient “2” at the
front is convenient for the exponential term that describes the decay of the system.
Here we note that a model that uses terms such as xm, m = 1, 3, · · · for damping are
dynamically equivalent to the model presented here.
70
Appendix C
Nonlinear Analysis
Assume that we operate the system near the eigenfrequency of th nth mode,denoted
as Ωeigen,n =√
Γ1,n. This frequency will subsequently be used to define the time scale.
The nondimensionalized equation of motion for the nth mode is given by
x+2(α1,n + α3,nx
2 + α5,nx4 + · · ·
)x+xn+γ3,nx
3n+γ5,nx
5n+ · · · = fAC,n cosωt, (C.1)
where xn = Xn/X0, t = Ωeigen,nT , γi,n = Γi,nXi−10 /Γ1,n, fAC,n = FAC,n/
(Ω2
eigen,nX0
),
and ω = Ω/Ωeigen,n. Therefore, it is also implied that γ1,n = 1. Here, X0 is the
characteristic displacement, and ω is close to unity because only the near resonance
frequency for the mode of interest is of concern. The effects of nonlinearities and
drive are expected to be small. In other words, γi>1,n ≪ 1 and fn ≪ 1. When
these conditions are satisfied, the equation of motion can be analyzed using standard
perturbation techniques for weakly nonlinear oscillators.
The method of averaging is employed here to analyze the system dynamics [68].
It provides an autonomous pair of equations that govern the amplitude and phase of
the response, which vary slowly under some conditions that are met by most MEMS
resonators. To put the equation of motion in the required form for averaging we use
71
the van der Pol transformation, which is given by
xn = Rn (t) cos [ωt+ Φn (t)], (C.2)
xn (t) = −ωRn (t) sin [ωt+ Φn (t]), (C.3)
where Rn (t) and Φn (t) are the dimensionless slowly varying amplitude and phase
coordinates of the response of the nth mode in the rotating frame. To obtain the
averaged equations, their time derivatives are directly integrated with the assumption
that Rn and Φn remain constant over one period, and the oscillatory terms can be
pushed out to higher order by averaging, which yield the averaged equations
Fn = −∑
i=1,3,···
(i− 1)!
2i−1(i+12!) (
i−12!)αi,nR
in −
fAC,n sinΦn
2ω, (C.4)
Gn =1− ω2
2ω+
∑i=3,5,···
(i+ 1)!γi,nRi−1n
2i+1(i+12!)2
ω− fAC,n cosΦn
2ωRn
, (C.5)
where usually only a few terms in the expansion are retained. In MEMS, fifth order
is typically sufficient to describe the dynamics of interest [22].Here, the frequency
detuning parameter is defined as
∆ωn =ω2 − 1
2∼= ω − 1. (C.6)
Steady conditions of the averaged system are obtained by simultaneously solving
Fn = 0 and Gn = 0, which results in equations that can be solved for the steady-state
amplitude R and phase Φ. Steady-state responses are the stable solutions, and the
stability is determined in the usual way by linearization [68].
72
The backbone curve is of particular interest; it is the frequency detuning in the
absence of damping and drive (αi,n = 0 and fn = 0), expressed by
∆ωn =∑
i=3,5,···
(i+ 1)!
2i+1(i+12!)2γi,nRi−1
n . (C.7)
It represents the amplitude dependence of the natural frequency. It also describes
the manner in which frequency response curves bend as a function of amplitude. If a
frequency sweep is applied, the frequency response may exhibit nonlinear effects, which
can be purely hardening, purely softening, hardening-to-softening, or even mixed, and
these effects are captured by the backbone curve [23].
To analyze a system model, it is important to characterize the coefficients in the
model. The above development describes how one can predict values for the coefficients
from first principles, but that is not possible except for very simple systems. Two other
common characterization methods are introduced below. The first is the dynamic
ringdown response, which is convenient in experiments. It uses the free decay of a
vibration mode, during which the damping and mode couplings are assumed to be
sufficiently small so that the system tracks the modal backbone curve during decay.
The ringdown frequency as a function of amplitude is given by
ω2 =∑
i=1,3,···
(i+ 1)!
2i(i+12!)2γi,nRi−1
n , or equivalently, Ω2 =∑
i=1,3,···
(i+ 1)!
2i(i+12!)2Γi,n|Xn|i−1,
(C.8)
where ω and Ω are the nondimensioanal and physical ringdown frequencies of the nth
mode, and |Xn| indicates the physical amplitude of the nth mode. Curve fitting can
be used to acquire the nonlinear coefficients of the system [22, 27].
For finite element analysis, taking the advantage of finite element software, an ex-
peditious approach can be used to obtain the linear and nonlinear stiffness parameters,
73
which is to apply a (virtual) external load onto the system. The static deflection of
the system implies the balance between the externally applied load and the restoring
force of the system. Curve fitting is then used to match the following expression:
p =∑
i=1,3,···
γi,nxin, or equivalently, P =
∑i=1,3,···
Γi,nXin, (C.9)
where p is the nondimensional modal load with modal mass normalized to unity, and
P is the physical modal load applied onto the system.
Figure C.1 demonstrates two examples of stiffness nonlinearities exhibited in MEMS
devices. For the clamped-clamped beam, a hardening-to-softening effect can be seen,
which stems from mechanical nonlinearity dominance (γ3 > 0) at small amplitudes and
electrostatic nonlinearity dominance (γi=3,5,··· < 0) at larger amplitudes. Moreover,
it also exemplifies how first-principle modeling can provide confidence for COMSOL
analysis. For the MEMS vibratory gyroscope, due to its boundary condition (sup-
port springs), the ring does not undergo midline stretching, and thus the mechanical
nonlinearity is negligible and only electrostatic softening is observed. In addition, the
comparison between the two characterization methods are also shown here, the ex-
cellent agreement of which verifies the validity of both approaches. It should also be
noted that the linear stiffness, which has both mechanical and electrostatic sources, is
also well captured by these characterization methods.
In MEMS devices, in the absence of electrostatic forces, the maximum nonlinear
dynamic range is mechanically limited by the allowed gap distance. When the DC bias
voltage is added, the maximum nonlinear dynamic range is then limited by the dynamic
pull-in amplitude, which stems from the attractive nature of the electrostatic forces.
This occurs at the amplitude of unstable equilibrium points, represented by saddle
points in the model, and which corresponds to the local maximum of the combined
74
(a) (b)
(c) (d)
Figure C.1: Nonlinearities in MEMS devices. (a) MEMS clamped-clamped beam withelectrodes on both sides. (b) Amplitude-dependent natural frequency illustrating: (i)the system can exhibit hardening (mechanical nonlinearity) to softening (electrostaticnonlinearity) effect and, (ii) theory using first-principle modeling (blue curve) agreeswell with the COMSOL time-domain simulation for the ringdown response (red dots).(c) MEMS vibrating ring gyroscope with a mode shape of φ = cos 2θ. (d) Amplitude-dependent natural frequency illustrating: (i) the ring does not have mechanical linearity(indicated by the green curve) due to its boundary conditions and, (ii) the staticdeflection characterization (blue and green curves) shows excellent agreement with thedynamic ringdown characterization (red dots)
potential. As the amplitude approaches the dynamic pull-in value, the spring softening
effect becomes infinitely large, and the vibration frequency approaches zero. Therefore,
the dynamic pull-in amplitude can be obtained by solving the equation
∑i=1,3,···
γi,nxin = 0, or equivalently,
∑i=1,3,···
Γi,nXin = 0, (C.10)
75
-d 0 d
0
Displacement
Velocity
0.2 0.4 0.6 0.8 1.0
Figure C.2: Phase portrait of a typical MEMS resonator and dynamic pull-in effect.The gray dot in the middle is the center equilibrium, the arrows indicate the directionof strength of the vector field, and the brown curves represent the invariant manifoldsof the two saddle points (shown as two crossing points). Between the saddle points isa non-homoclinic orbit, only within which can the system operate as desired. Thesetwo saddle points indicate the dynamic pull-in amplitude
where the smallest real solution corresponds to the dynamic pull-in amplitude. Figure
C.2 illustrates the condition for dynamic pull-in to occur. The brown curves show the
stable and unstable manifolds of the two saddle points, and the gray dot in the middle
indicates the center equilibrium, which is Lyapunov stable but not asymptotically
stable. Also, electrostatic buckling of the system can occur, corresponding to the
destabilization of the central equilibrium. This happens when the pull-in amplitude
reaches zero, which occurs at a critical value of the DC bias voltage [14, 15].
For a unforced system with linear and nonlinear damping, the effects of damping
76
Linear decay
Nonlinear decay
0 50 100 150 2000.2
0.4
0.6
0.8
1
Time
Amplitude
Figure C.3: Free decay of x + 2 (α + βx2) x + x = 0, with α = 0.01. The blue curveshows the linear system, with β = 0, while the orange curve shows the nonlinearsystem, with β = 0.01
during the free decay (ringdown response) is given by
Fn = −∑
i=1,3,···
(i− 1)!
2i−1(i+12!) (
i−12!)αi,nR
in, or equivalently,
d
dt|Xn| = −
√Γ1,n
∑i=1,3,···
(i− 1)!
2i−1(i+12!) (
i−12!)αi,n|Xn|i.
(C.11)
This shows the free decay that deviates from the exponential decay of a linear system
during the ringdown response of the system. Figure C.3 demonstrates this deviation,
where the system with nonlinear damping initially decays faster than the linear system,
but eventually transitions to the linear decay rate.
77
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