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