Single-degree-of-freedom energy harvesters by stochastic...
100
Single-degree-of-freedom energy harvesters by stochastic excitation by Han Kyul Joo Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2014 © Massachusetts Institute of Technology 2014. All rights reserved. Author .............................................................. Department of Mechanical Engineering May 9, 2014 Certified by .......................................................... Themistoklis P. Sapsis ABS Career Development Assistant Professor Thesis Supervisor Accepted by ......................................................... David E. Hardt Ralph E. & Eloise F. Cross Professor of Mechanical Engineering Chairman, Committee on Graduate Students
Transcript of Single-degree-of-freedom energy harvesters by stochastic...
Single-degree-of-freedom energy harvesters by
stochastic excitation
by
Han Kyul Joo
Submitted to the Department of Mechanical Engineeringin partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
© Massachusetts Institute of Technology 2014. All rights reserved.
Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Department of Mechanical Engineering
May 9, 2014
Certified by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Themistoklis P. Sapsis
ABS Career Development Assistant ProfessorThesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .David E. Hardt
Ralph E. & Eloise F. Cross Professor of Mechanical EngineeringChairman, Committee on Graduate Students
2
Single-degree-of-freedom energy harvesters by stochastic
excitation
by
Han Kyul Joo
Submitted to the Department of Mechanical Engineeringon May 9, 2014, in partial fulfillment of the
requirements for the degree ofMaster of Science in Mechanical Engineering
Abstract
In this thesis, the performance criteria for the objective comparison of different classesof single-degree-of-freedom oscillators under stochastic excitation are developed. Foreach family of oscillators, these objective criteria take into account the maximumpossible energy harvested for a given response level, which is a quantity that is directlyconnected to the size of the harvesting configuration. We prove that the derivedcriteria are invariant with respect to magnitude or temporal rescaling of the inputspectrum and they depend only on the relative distribution of energy across differentharmonics of the excitation. We then compare three different classes of linear andnonlinear oscillators and using stochastic analysis tools we illustrate that in all cases ofexcitation spectra (monochromatic, broadband, white-noise) the optimal performanceof all designs cannot exceed the performance of the linear design. Subsequently, westudy the robustness of this optimal performance to small perturbations of the inputspectrum and illustrate the advantages of nonlinear designs relative to linear ones.
Thesis Supervisor: Themistoklis P. SapsisTitle: ABS Career Development Assistant Professor
3
4
Acknowledgments
I would like to acknowledge my thesis advisor, Prof. Themistoklis Sapsis, for his
support and academic advice. It is my honor to work with him. It should be men-
tioned that this work is supported from Kwanjeong Educational Foundation as well
as a Startup Grant at MIT. I would also like to acknowledge lab mates, visitors, and
UROP (Undergraduate Research Opportunities Program) at SANDLAB for sharing
time for precious discussions. GAME (Graduate Association of Mechanical Engineer-
ing) is one of the most exciting and amazing society I’ve ever belonged to. Thank
you very much for your support. Furthermore, I would also like to thank every mem-
ber in KGSA (Korean Graduate Student Association) as well as KGSAME (Korean
Graduate Student Association of Mechanical Engineering) for mentoring. I am also
very much grateful for my undergraduate advisor, Prof. Takashi Maekawa, for his
kind support and cares. Last but not least, I would like to sincerely thank my parents
and my younger sister for their enduring love and support. This thesis is dedicated
to them.
5
6
Contents
1 Introduction 13
1.1 Ocean Wave Energy Harvesting . . . . . . . . . . . . . . . . . . . . . 13
1.2 Vibration Energy Harvesting . . . . . . . . . . . . . . . . . . . . . . . 15
2 An Overview of Probability and Stochastic Processes 19
2.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Elements of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 Probability Distribution . . . . . . . . . . . . . . . . . . . . . 21
2.2.2 Noncentral and Central Moments . . . . . . . . . . . . . . . . 25
2.2.3 Characteristic Function . . . . . . . . . . . . . . . . . . . . . . 28
2.2.4 Correlation and Covariance . . . . . . . . . . . . . . . . . . . 30
2.2.5 Gaussian Distribution Function . . . . . . . . . . . . . . . . . 32
2.3 Stationarity and Ergodicity for Stochastic Processes . . . . . . . . . . 34
2.3.1 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.2 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.3 Examples of Stationary and Ergodic Processes . . . . . . . . . 35
2.4 Power Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Energy Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Probabilstic description of water waves 43
3.1 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Ocean Waves with a Gaussian Probability Distribution . . . . . . . . 44
3.3 Sea Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7
3.4 Expression of Ocean Wave Elevation . . . . . . . . . . . . . . . . . . 48
4 Statistical Steady State Response of SDOF Oscillators 51
4.1 Analytical Steady State Response of Linear Systems . . . . . . . . . . 51
4.1.1 System Properties . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1.2 Response Power Spectral Density of Linear Systems . . . . . . 53
4.2 Analytical Steady State Response of Nonlinear Systems Excited by
Gaussian White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.1 Fokker - Planck and Kolmogorov Equation . . . . . . . . . . . 57
4.3 Numerical Simulation of Nonlinear Systems Excited by Colored Noise 59
4.4 Gaussian Closure for Nonlinear SDOF Oscillators Excited by Colored
Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5 Quantification of Power Harvesting Performance 65
5.1 Absolute and Normalized Harvested Power Ph . . . . . . . . . . . . . 66
5.2 Size of the Energy Harvester B . . . . . . . . . . . . . . . . . . . . . 67
5.3 Harvested Power Density ρe . . . . . . . . . . . . . . . . . . . . . . . 67
5.4 Quantification of Performance for SDOF Harvesters . . . . . . . . . . 71
5.5 Results of Performance Quantification . . . . . . . . . . . . . . . . . 74
5.5.1 SDOF Harvester under Monochromatic Excitation . . . . . . . 74
5.5.2 SDOF Harvester under White Noise Excitation . . . . . . . . 79
5.5.3 SDOF Harvester under Colored Noise Excitation . . . . . . . 81
5.6 Results of the Moment Equation Method . . . . . . . . . . . . . . . . 85
6 Performance Robustness 89
7 Conclusions and Future Work 95
8
List of Figures
2-1 A random variable is a function which maps elements in the sample
space to values on the real line. . . . . . . . . . . . . . . . . . . . . . 19
2-2 An ensemble of random signals with five different realizations. . . . . 20
2-3 (a) Gaussian probability distribution function. (b) Gaussian probabil-
ity density function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2-4 (a) Double sided power spectral density. (b) Single sided power spectral
density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4-1 A system with an impulse response of h(t). . . . . . . . . . . . . . . . 53
4-2 Input and output relation in terms of stationarity, ergodicity and gaus-
sian process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4-3 (a) Autocorrelation function of the periodic ocean wave elevation sig-
nal. (b) Autocorrelation function of the aperiodic ocean wave elevation
signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5-1 Various spectral curves obtained by magnitude and temporal rescaling
of the Pierson-Moskowitz spectrum. Amplification and stretching of
the input spectrum will leave the effective damping and the harvested
power density invariant. . . . . . . . . . . . . . . . . . . . . . . . . . 68
5-2 The shapes of potential function U(x) = 12k1x
2+ 14k3x
4. (a) The monos-
table potential function with k1 > 0 and k3 > 0. (b) The bistable
potential function with k1 < 0 and k3 > 0. . . . . . . . . . . . . . . . 72
9
5-3 Linear and nonlinear SDOF systems: (a) Linear SDOF system, (b)
Nonlinear SDOF system only with a cubic spring, and (c) Nonlinear
SDOF system with the combination of a negative linear and a cubic
spring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5-4 Response level B. and power harvested for the case of monochromatic
spectrum excitation over different system parameters. The response
level B is also presented as a contour plot in the power harvested plots.
All three cases of systems are shown: linear (top row), cubic (second
row), and negative stiffness with ν = 1. . . . . . . . . . . . . . . . . . 76
5-5 (a) Maximum harvested power, and (b) Power density for linear and
nonlinear SDOF systems under monochromatic excitation. . . . . . . 77
5-6 A nonlinear system with the combination of a negative linear (ν = 1)
and a cubic spring. Blue solid line corresponds to a local minimum
of the performance in Fig. 5-4: k3 = 0.1 and λ = 0.2. Red dashed
line corresponds to a local maximum of the performance in Fig. 5-4:
k3 = 0.25 and λ = 0.2. (a) Response in terms of displacement. (b)
Fourier transform modulus |q (ω)|. . . . . . . . . . . . . . . . . . . . . 78
5-7 (a) Maximum harvested power, and (b) Power denstity for linear and
nonlinear SDOF systems under white noise excitation. . . . . . . . . 81
5-8 Response level B and power harvested for the case of excitation with
Pierson-Moskowitz spectrum over different system parameters. The
response level B is also presented as a contour plot in the power har-
vested plots. All three cases of systems are shown: linear (top row),
cubic (second row), and negative stiffness with ν = 1. . . . . . . . . . 83
5-9 (a) Maximum harvested power, and (b) Power density for linear and
nonlinear SDOF systems under Pierson-Moskowitz spectrum. . . . . . 84
5-10 Harvested power density ρe for the three different types of excitation
spectra. The linear design is used in all cases since this is the optimal. 84
10
5-11 Results of the moment equation method for the cubic system under
the colored noise excitation. (a) The size of the device with respect to
system parameters. (b) The harvested power with respect to system
parameters. (c) Maximum Harvested Power. (d) Harvested Power
Density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5-12 Results of the moment equation method for the negative linear system
under the colored excitation. (a) The size of the device with respect to
system parameters. (b) The harvested power with respect to system
parameters. (c) Maximum Harvested Power. (d) Harvested Power
Density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6-1 Robustness of (a) the response level, (b) the power harvested, and (c)
the harvested power density for the monochromatic excitation under
three regimes of operation: B = 0.5, B = 1, and B = 8. . . . . . . . . 91
6-2 Robustness of (a) the response level, (b) the power harvested, and
(c) the harvested power density for the PM spectrum excitation under
three regimes of operation: B = 0.5, B = 1, and B = 8. . . . . . . . . 93
11
12
Chapter 1
Introduction
Energy is everywhere; waiting to be harvested. This powerful truth opens up an am-
ple opportunities for energy harvesting devices to bring benefits to this world: from
structural health monitoring and autonomous marine sensors to forest fire preven-
tion and in-situ medical care devices. All of these are based on energy harvesting
techniques to power small devices by means of targeted energy transfer from a given
source, such as mechanical vibrations and ocean water waves.
1.1 Ocean Wave Energy Harvesting
The main purpose of the research is harvesting energy from large phenomena, such as
ocean water waves, by means of targeted energy transfer techniques. Sea waves are
generated by the turbulent interaction of the wind with the ocean surface. Due to
gravity forces, water waves can propagate over the ocean for large distances making
the process energetically dense over a wide range of frequencies. In terms of analysis,
extensive work has been done on the physics of water wave evolution and energy
spectrum propagation. However, the problem of energy harvesting from ocean waves
is still treated in a rudimentary way, through linear techniques which have the major
disadvantage of energy absorption through a narrow and pre-tuned frequency band.
13
There are a wide variety of water wave-powered mechanisms for the conversion of
wave energy to mechanical energy. These mechanisms can be categorized depending
on their location of operation: shoreline, near-shore, and off-shore structures. An
important representative from the first class is the oscillating column of water (see
e.g. [1–3]), where the water surface elevation is used to create airflow through an
air turbine generator. The simple principle of operation as well as the conventional
technology required for its construction are the main advantages for this class of wave
energy harvesters. Moreover, since it is constructed on the shoreline, it has low main-
tenance cost, although the wave potential is much higher offshore than in shallow
water. For this reason it requires very specific location characteristics and, hence,
is not suitable for all coastal regions. Another drawback is the difficulty of building
and anchoring the main structure so that it is able to withstand the roughest sea
conditions and yet generate a reasonable amount of power from small waves.
Near-shore mechanisms usually operate within 3-5 miles of shore and are the most
widely used devices for energy extraction. Typically, these devices have the form of
a buoy that oscillates on the ocean surface (see e.g. [4,5]), converting the mechanical
energy due the vertical component of the oscillation to electrical energy. This is done
by means of a permanent magnet that moves inside a fixed coil inducing electric cur-
rent. Since, these are smallscale devices, they are generally deployed in large numbers
forming a grid. Important benefits of this concept are lower construction costs and
small environmental impact. On the other hand, while the offshore site of operation is
characterized by intense wave power,there can be high costs associated with electricity
transmission to land; and since power generation is based on the vertical oscillation,
the energy conversion mechanism must be pre-tuned (optimized) for a specific range
of wave frequencies, which is usually much narrower than the spectrum of the ocean
waves.
The third class includes off-shore configurations which are generally much large
in scale and which operate in the open sea, away from the coast. The usual principle
14
of operation for these devices is to follow the water surface and act as wave attenua-
tors. An example of a surface following device is the Pelamis Wave Energy Converter.
The sections of the device articulate with the movement of the waves, each resisting
motion between it and the next section, creating pressurized oil to drive a hydraulic
ram which further drives a hydraulic motor. As with the near-shore configurations,
these devices operate in an environment with very strong energy potential; however,
the cost to transfer the energy to the land may be prohibitive, especially for the case
where the point of operation is a significant distance from shore.
We choose the second class of energy-harvesters as the focus of our consideration,
where the main challenge we address is to significantly increase the efficiency and
robustness of the energy-capture mechanism. To this end, we will apply techniques
of targeted energy transfer through essentially nonlinear (i.e., nonlinearizable) local
resonators which act, in essence, as nonlinear energy harvesters. Due to their non-
linearizable character, these nonlinear resonators have no preferential frequency and
are, therefore, able to harvest energy over an extremely broad range of the energy
spectrum. This concept has been applied successfully to a wide range of applications
involving energy absorption and dissipation, including seismic mitigation in structural
systems and flutter suppression in aeroelastic systems. We thus expect that this will
lead to the design of efficient and robust energy harvesting mechanisms and strategies
that will operate effectively over a wide range of the wave energy spectrum.
1.2 Vibration Energy Harvesting
Energy harvesting is the process of targeted energy transfer from a given source (e.g.
ambient mechanical vibrations, water waves, etc) to specific dynamical modes with
the aim of transforming this energy to useful forms (e.g. electricity). In general, a
source of mechanical energy can be described in terms of the displacement, velocity or
acceleration spectrum. Moreover, in most cases the existence of the energy harvesting
device does not alter the properties of the energy source i.e. the device is essentially
15
driven by the energy source in a one-way interaction.
Typical energy sources are usually characterized by non-monochromatic energy
content, i.e. the energy is spread over a finite band of frequencies. This feature has
led to the development of various techniques in order to achieve efficient energy har-
vesting. Many of these approaches employ single-degree-of-freedom oscillators with
non-quadratic potentials, i.e. with a restoring force that is nonlinear see e.g. [6–18].
In all of these approaches, a common characteristic is the employment of intensional
nonlinearity in the harvester dynamics with an ultimate scope of increasing perfor-
mance and robustness of the device without changing its size, mass or the amount of
its kinetic energy. Even though for linear systems the response of the harvester can be
fully characterized (and therefore optimized) in terms of the energy-source spectrum
(see e.g. [9, 19]), this is not the case for nonlinear systems which are simultaneously
excited by multiple harmonics - in this case there are no analytical methods to ex-
press the stochastic response in terms of the source spectrum. While in many cases
(e.g. in [8, 11, 13, 18]) the authors observe clear indications that the energy harvest-
ing capacity is increased in the presence of nonlinearity, in numerous other studies
(e.g. [6,7,10,12]) these benefits could not be observed. To this end it is not obvious if
and when a class (i.e. a family) of nonlinear energy harvesters can perform “better”
relative to another class (of linear or nonlinear systems) of energy harvesters when
these are excited by a given source spectrum.
Here we seek to define objective criteria that will allow us to choose an optimal
and robust energy harvester design for a given energy source spectrum. An efficient
energy harvester (EH) can be informally defined as the configuration that is able to
harvest the largest possible amount of energy for a given size and mass. This is a
particularly challenging question since the performance of any given design depends
strongly on the chosen system parameters (e.g. damping, stiffness, etc.) and in order
to compare different classes of systems (e.g. linear versus nonlinear) the developed
measures should not depend on the specific system parameters but rather on the form
16
of the design, its size or mass as well as the energy source spectrum. Similar chal-
lenges araise when one tries to quantify the robustness of a given design to variations
of the source spectrum for which it has been optimized.
To pursue this goal we first develop measures that quantify the performance of
general nonlinear systems from broadband spectra, i.e. simultaneous excitation from
a broad range of harmonics. These criteria demonstrate for each class of systems the
maximum possible power that can be harvested from a fixed energy source using a
given volume. We prove that the developed measures are invariant to linear trans-
formations of the source spectrum (i.e. rescaling in time and size of the excitation)
and they essentially depend only on its shape, i.e. the relative distribution of energy
among different harmonics. For the sake of simplicity, we will present our measures
for one dimensional systems although they can be generalized to higher dimensional
cases in a straightforward manner.
Using the derived criteria we examine the relative advantages of different classes of
single-degree-of-freedom (SDOF) harvesters. We examine various extreme scenarios
of source spectra ranging from monochromatic excitations to white-noise cases (also
including the intermediate case of the Pierson-Moskowitz (PM) spectrum). We prove
that there are fundamental limitations on the maximum possible harvested power that
can be achieved (using SDOF harvesters) and these are independent from the linear
or nonlinear nature of the design. Moreover, we examine the robustness properties
of various SDOF harvester designs when the source characteristics are perturbed and
we illustrate the dynamical regimes where non-linear designs are preferable compared
with the linear harvesters.
17
18
Chapter 2
An Overview of Probability and
Stochastic Processes
Probability theory and stochastic processes are one of the essential backgrounds of
this thesis. Specifically, the harvested power of the SDOF oscillators under the colored
noise are investigated in terms of stochastic processes and power spectral density. In
this chapter, We first offer a brief overview of the basic probability and stochastic
processes.
2.1 Random Variables
Ω3
Ω4
Ω2
Ω1
Re
Ω
x1
x2
x3
x4
Figure 2-1: A random variable is a function which maps elements in the sample spaceto values on the real line.
A random variable is a function which maps elements in the sample space to values on
the real line. In many physical problems, including base oscillating energy harvesters,
19
their dynamics can be described in terms of probabilistic approach. This is the
concept of stochastic processes.
Then, the stochastic process should be defined. A stochastic process, or equiva-
lently random process, is the family of time dependent random variables which obey
a specific probabilistic law. In other words, it represents a time evolution of a ran-
dom variable. For example, as will be fully explored in following chapters, heights
and amplitudes of ocean waves are representative examples of stochastic processes.
In this case, we can consider that the outcomes of mapping from the entire sample
space to real numbers are connected in terms of ocean wave heights and amplitudes.
Mathematically, a stochastic process is expresses by x(t,Ω), where t represents a
parameter for time and Ω indicates a parameter for probability. This mathematical
expression can be interpreted in two ways in terms of time and probability parameters,
respectively. For a fixed time t, it becomes a function of probability parameter Ω,
and this is called “random variable”. On the other hand, for a fixed probability
parameter Ω, it becomes a function of time t, which is called “realization”. In general,
the collection of time history data, realizations, is denoted as “ensemble”. All these
notions are clearly illustrated in the Figure (2-2).
x(t, Ω=1)
x(t, Ω=2)
x(t, Ω=3)
x(t, Ω=4)
x(t, Ω=5)
Realization
Random Variable
t=t0
Figure 2-2: An ensemble of random signals with five different realizations.
20
2.2 Elements of Probability
Random variables as well as stochastic processes should be accompanied by their
probability distributions in order to fully describe their behaviors. Probability dis-
tribution represents how probabilities are distributed over the values of random vari-
ables. Thus the review of the probability distribution function and the probability
density function are offered in the following sections. For the brevity, counter part
definitions for the discrete random variable are omitted. Interested readers can refer
to elementary probability textbooks [20].
2.2.1 Probability Distribution
According to how the random variables are distributed (i.e. discrete or continuous),
there are two ways to express the distribution of probability of a random variable.
If the random variable is discrete, the discrete random variable has Probability Mass
Function and Probability Distribution Function, while the continuous random variable
has Probability Density Function and Probability Distribution Function. Either of
those two functions will fully describe the probability of a random variable. Here
we will introduce the probability density function and the probability distribution
function for continuous random variables.
• Probability Distribution Function (PDF)
The probability distribution function of random variable X is defined as the proba-
bility that the random variable X is less than or equal to an element x. It is clear
that this probability depends on the assigned element x.
FX(x) = PX ≤ x. (2.1)
Here, the upper case subscript X denotes a random variable and the lower case
x denotes its arbitrary element. This probability distribution function has several
important properties. First, it is obvious that the function takes a value only between
0 and 1. And, it is a non-negative and non-decreasing function with respect to x.
21
Thus we can expect that the function takes 0 at negative infinity and takes 1 at
positive infinity. These properties are summarized as follows:
FX(−∞) =0, (2.2)
FX(+∞) =1, (2.3)
P (a < X ≤ b) =FX(b)− FX(a), (2.4)
P (X ≤ b) =P (X ≤ b) + P (a < X ≤ b), (2.5)
where a and b are two real numbers such that a < b.
• Probability Density Function (pdf)
The probability density function for a continuous random variable X is defined as the
derivative of the probability distribution function with respect to its element x.
fX(x) =d
dxFX(x). (2.6)
Since the probability distribution function is a continuous function with respect to
the element x, the probability density function exists for all values x. Similarly, sev-
eral important properties are introduced. It is obvious that the probability density
function is a non-negative function. Also, it is important that the probability den-
sity function does not give the probability itself, but the underneath area gives the
probability. Please note that this is different from the fact that the probability mass
function for a discrete random variable gives the probability.
fX(x) ≥0, (2.7)
FX(x) =
∫ x
−∞fX(u)du, (2.8)
FX(∞) =
∫ ∞−∞
fX(u)du = 1, (2.9)
P (a < X ≤ b) =
∫ a
b
fX(u)du. (2.10)
22
−5 0 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
FX(x)
−5 0 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
x
f X(x
)
(a) (b)
Figure 2-3: (a) Gaussian probability distribution function. (b) Gaussian probabilitydensity function.
• Joint Probability Distribution Function (JPDF)
In many cases, we may encounter situations with more than one random variable.
Then it becomes our concern that how those random variables behave jointly. The
joint probability distribution function and the joint probability density function fully
describe two random variables. The joint probability distribution function for two
continuous random variables X and Y is defined as the probability that the random
variable X takes an element less than or equal to x and the random variable Y takes
an element less than or equal to y. This can be mathematically expressed as follows:
FXY (x, y) = PX ≤ x ∩ Y ≤ y. (2.11)
It is also clear that FXY (x, y) is a non-negative and non-decreasing function with
respect to x and y.
FXY (−∞,−∞) = 0, (2.12)
FXY (∞,∞) = 1, (2.13)
FXY (x,−∞) = 0, (2.14)
FXY (−∞, y) = 0. (2.15)
23
Marginal distribution function for X and Y can be obtained by replacing each element
with positive infinity as follows:
FX(x) = FXY (x,∞), (2.16)
FY (y) = FXY (∞, y). (2.17)
• Joint Probability Density Function (jpdf)
The partial derivative of the joint probability distribution function with respect to x
and y is defined as the joint probability density function of two continuous random
variables X and Y .
fXY (x, y) =∂2
∂x∂yFXY (x, y). (2.18)
Since the joint probability distribution function is a non-negative and non-decreasing
function, the second partial derivative is also a non-negative function having following
properties.
FXY (x, y) =
∫ y
−∞
∫ x
−∞fXY (u, v)dudv, (2.19)
FXY (∞,∞) =
∫ ∞−∞
∫ ∞−∞
fXY (u, v)dudv = 1. (2.20)
Marginal density functions can be obtained by integrating with respect to each ele-
ment.
fX(x) =
∫ ∞−∞
fXY (u, v)dv, (2.21)
fY (y) =
∫ ∞−∞
fXY (u, v)du. (2.22)
For more than two continuous random variables, the joint probability distribution
function can be defined as
FX(X) = PX1 ≤ x1 ∩X2 ≤ x2 · · · ∩Xn ≤ xn, (2.23)
24
and the joint probability density function can also be defined as follows:
fX(X) =∂n
∂xFX(X). (2.24)
2.2.2 Noncentral and Central Moments
Even though the probability distribution function and the probability density func-
tion fully describe a random variable, it is sometimes necessary to evaluate simple
numbers containing its probabilistic features. Those simple numbers are non-central
and central moments which can be expressed as the expectation of various orders of
a random variable. In this thesis, only the continuous random variables are treated
and introduced. For more information on the discrete random variables are available
on [20,21].
• Expectation
For a real valued function of a random variable, the expectation, or the mean value, is
defined using the probability density function. The symbol E reads the expectation.
By definition, the expectation of a arbitrary function of a random variable X, g(X),
is given as follows:
Eg(X) =
∫ ∞−∞
g(x)fX(x)dx, (2.25)
where fX(x) is the probability density function of the random variable X. Above
expectation is defined only if the absolute integral∫∞−∞ |g(x)|fX(x)dx <∞ converges.
Then it is clear that the expectation of a random variable X can be expressed as
follows:
EX =
∫ ∞−∞
xfX(x)dx. (2.26)
Some important properties of expectation operator are introduced.
25
Ec =c, (2.27)
Ecg(X) =cEg(X), (2.28)
Ecg(X) + dh(Y ) =cEg(X)+ dEh(Y ), (2.29)
where c and d are constants and g(X) and h(Y ) are functions of random variables
X and Y . The linearity of the expectation holds regardless of the independence of
random variables.
• Non-central moments
Non-central moments are the expectation of several orders of a function of a random
variable, and the nth order of non-central moments of a random variable is often
denoted as αn.
αn = EXn =
∫ ∞−∞
xnfX(x)dx. (2.30)
Obviously, the expectation is the first order non-central moment of a random variable.
• Central moments
Central moments are defined as the expectation of several order of a function of a
random variable with respect to its mean value. nth order central moments of a
random variable are represented as βn.
βn = E(X − µ)n =
∫ ∞−∞
(x− µ)nfX(x)dx, (2.31)
where µ represents its mean value.
• Variance
The second order central moment of a continuous random variable is denoted as
variance. Variance indicates how the random variable is distributed with respect to
26
its mean. Large variance represents a large spread of a random variable from its
mean, while small variance indicates a small dispersion of the random variable. The
definition of the variance follows.
σ2X = V arX = E(X − µ)2 =
∫ ∞−∞
(x− µ)2fX(x)dx, (2.32)
where µ is the expectation of a random variable X. Variance has several important
properties and some of those are introduced as follows:
σ2 = α2 − µ2, (2.33)
V ar(X + c) = V ar(X), (2.34)
V ar(cX) = c2V ar(X). (2.35)
• Standard Deviation
The positive square root of variance is defined as standard deviation.
σX =√V arX =
√E(X − µ)2. (2.36)
Standard deviation has the same unit as the expectation, thus it can be easily com-
pared with the mean on the same scale to obtain the degree of spread.
• Non-dimensional coefficients
There are dimensionless numbers that represent several features of a random vari-
able. A dimensionless number denoted as the coefficients of variation represents the
dispersion relative to the mean value. A large value indicates a wide spread while a
small value indicates a narrow spread.
νX =σXµ. (2.37)
Coefficients of skewness is a dimensionless number which gives the measure of sym-
metry of a distribution. When the distribution is symmetrical about its mean, the
27
value becomes zero. It is positive if the distribution has a dominant tail on the right,
and negative if it has a dominant tail on the left.
γ1 =β3σ3X
. (2.38)
Coefficients of excess is a dimensionless number which gives the degree of a distribu-
tion around its mean. The value is positive if the distribution has a slim and sharp
peak, while the value is negative if the distribution has a flattened peak. It becomes
zero when the distribution is Gaussian.
γ2 =β4σ4X
− 3. (2.39)
For more than one continuous random variable, joint non-central moments and joint
central moments can also be defined in the similar way. Joint non-central moments
are defined as
αnm = EXnY m =
∫ ∞−∞
∫ ∞−∞
xnymfXY (x, y)dxdy, (2.40)
and the joint central moments are defined as follows:
βnm = E(X − µX)n(Y − µY )m =
∫ ∞−∞
∫ ∞−∞
(x− µX)n(y − µY )nfXY (x, y)dxdy.
(2.41)
2.2.3 Characteristic Function
The characteristic function of a continuous random variable X is defined as the ex-
pectation EejtX. It is the expectation of a complex function and therefore it is
generally complex valued.
φX(t) =EejtX =
∫ ∞−∞
ejtXfX(x)dx, (2.42)
fX(x) =1
2π
∫ ∞−∞
e−jtXφX(t)dt, (2.43)
28
where t is a real valued parameter. Further, the characteristic functions have following
properties.
φX(t = 0) = 1, (2.44)
φX(−t) = φ∗X(t), (2.45)
|φX(t)| ≤ 1, (2.46)
where ∗ represents the complex conjugate. The characteristic functions provide useful
tools to investigate stochastic processes. One of important properties of the character-
istic function is the moments generating function. This is the process of determining
moments of a random variable. Taylor’s expansion with respect to t = 0 (equivalently
the MacLaurin series) gives
φ(t) = φ(0) + φ′(0)t+1
2φ′′(0)t2 +
1
6φ′′′(0)t3 + · · · = 1 +
∞∑n=1
(jt)n
n!αn. (2.47)
From the above relation, we can deduce that
αn = (j)nφ(n)(0). (2.48)
Therefore the knowledge of the characteristic function provides the moments of all
order of a random variable. Another important property is that the characteristic
function can also be extended to the cumulant generating function as follows:
log φX(t) =∞∑n=1
(jt)nλnn!
, (2.49)
where the coefficient λn is obtained from
λn = (j)−ndn
dtnlog φX(t)|t=0. (2.50)
29
There is a simple relation between the coefficients λn and the moments αn.
λ1 =α1, (2.51)
λ2 =α2 − α21, (2.52)
λ3 =α3 − 3α1α2 + 2α31. (2.53)
Here we can observe that λ1 is the mean, λ2 is the variance, and λ3 is the third central
moment. The coefficients λn are denoted as the cumulants of a continuous random
variable.
2.2.4 Correlation and Covariance
In the case that there are more than one continuous random variable, the interde-
pendence of those random variables become also important. The central expectation
of two continuous random variables X and Y with respect to two different time is
denoted as the covariance function. If those two random variables are the same, it is
denoted as autocovariance, or simply covariance. If those two random variables are
different, it is denoted as crosscovariance. The autocovariance can be expressed with
two different time t and s, or equivalently with the time difference τ as follows:
CXX(t, s) =E(X(t)− µX)(X(s)− µX)
=E(X(t)− µX)(X(t+ τ)− µX) = CXX(τ). (2.54)
The crosscovariance of two random variables X and Y can be written as
CXY (t, s) =E(X(t)− µX)(Y (s)− µY )
=E(X(t)− µX)(Y (t+ τ)− µY ) = CXY (τ). (2.55)
The non-central expectation of two continuous random variablesX and Y with respect
to two different time is defined as correlation function. Similarly, if those two randoms
variables are the same, it is called autocorrelation, however, on the other hand, if
30
those two random variables are different, it is called crosscorrelation. Autocorrelation
function of a random variable X is
RXX(t, s) =EX(t)X(s)
=EX(t)X(t+ τ) = RXX(τ). (2.56)
The crosscorrelation of two different random variables X and Y is as follows:
RXY (t, s) =EX(t)Y (s)
=EX(t)Y (t+ τ) = RXY (τ). (2.57)
There are several important properties and relations between the covariance function
and the correlation function. In general, the covariance function can be expressed
with correlation function as
CXX(τ) = RXX(τ)− µ2X , (2.58)
CXY (τ) = RXY (τ)− µXµY . (2.59)
In the case that the expectation of each random variable is zero, it reduces to
CXX(τ) = RXX(τ), (2.60)
CXY (τ) = RXY (τ). (2.61)
Furthermore, the covariance function and the correlation function are even functions.
CXX(−τ) = CXX(τ), (2.62)
RXX(−τ) = RXX(τ). (2.63)
The physical meaning behind the covariance function and the correlation function
is very important. Positive covariance indicates positive correlation while negative
covariance represents negative correlation. Zero covariance is called uncorrelated.
31
It is important that if two random variables are independent, it is uncorrelated.
However, uncorrelation does not necessarily indicate independence of two random
variables. Moreover, positive correlation indicates that as one variable increases the
other variable also increases. Similarly, negative correlation represents that as one
random variable increase the other variable decreases.
2.2.5 Gaussian Distribution Function
One of the most important examples of the probability distribution is the gaussian
distribution. A continuous random variable is denoted as the Gaussian process if its
probability distribution function and probability density function have the following
expressions.
fX(x) =1√2πσ
exp −(x− µ)2
2σ2, (2.64)
FX(x) =
∫ x
−∞fX(u)du =
1√2πσ
∫ x
−∞exp −(u− µ)2
2σ2du. (2.65)
Graphical illustration can be found in the Figure (2-3). A random variable becomes
the standard Gaussian random variable if its probability distribution has the Gaussian
distribution with zero mean and unit variance. In general, a random variable with
Gaussian probability distribution has its expectation and variance as follows:
EX = µ, (2.66)
V arX = σ2. (2.67)
Furthermore, the characteristic function of Gaussian distribution is given as
φX(t) = exp jµt− 1
2σ2t2. (2.68)
As illustrated in the previous section, we can derive several properties of central
moments by using the characteristic function. In the case that a random variable
follows the Gaussian probability distribution, the expression for the central moments,
32
βn, becomes much simpler.
βn =
0 n = odd
1 · 3 · 5 · · · (n− 1) · σn n = even
(2.69)
33
2.3 Stationarity and Ergodicity for Stochastic Pro-
cesses
2.3.1 Stationarity
For many physical phenomena the associated stochastic processes are characterized
by interesting properties such as stationarity or ergodicity. Here we define these
properties in detail.
A strictly stationary process is defined such that the joint probability distribution of
a stochastic process does not change with respect to the time shift. However, this
definition is sometimes too strong for the engineering sense. Thus a rather relaxed
definition is introduced. A stochastic process X(t) is called “weakly stationary” if the
mean, EX(t), and the autocorrelation, EX(t)X(t+ τ), are both independent of
time t. These conditions can be written as
µX =EX(t) = constant, (2.70)
RXX(τ) =EX(t)X(t+ τ) = function of τ . (2.71)
For a zero mean stochastic process, above conditions reduce to
µX =EX(t) = constant, (2.72)
CXX(τ) =RXX(τ)−m2X = function of τ . (2.73)
Thus, a stationary random process has constant mean and variance for all time t.
34
2.3.2 Ergodicity
Another important process is ergodicity. A stochastic process X(t) is called “ergodic”
if the ensemble mean can be replaced by a temporal average over a single realization.
µX =EX(t) = limT→∞
1
2T
∫ T
−Tx(s)ds, (2.74)
RXX(τ) =EX(t)X(t+ τ) = limT→∞
1
2T
∫ T
−Tx(s)x(s+ τ)ds. (2.75)
If we assume a stochastic process to be an ergodic process, then we automatically
assume the property of stationarity. However, an important point is that a stationary
process does not necessarily guarantee the process to be an ergodic process. Therefore,
in order to assume a stochastic process to be ergodic, it is required to be a stationary
process.
2.3.3 Examples of Stationary and Ergodic Processes
Several commonly adopted stochastic processes are discussed and investigated their
properties in terms of stationarity and ergodicity.
Example 1: X(t) = A cos(ωt+ ϕ)
Let’s consider one of the simplest stochastic processes. The amplitude A and fre-
quency ω are constants, and the phase ϕ is the only random variable with uniform
distribution among 0 to 2π. As illustrated in the previous section, the stationarity of
the above process has been investigated.
EX(t) =EA cos(ωt+ ϕ) =A
2π
∫ ∞−∞
cos(ωt+ ϕ)dϕ = 0, (2.76)
EX(t)X(t+ τ) =EA2 cos(ωt+ ϕ) cos(ω(t+ τ) + ϕ) = · · · = 1
2A2 cos(ωτ).
(2.77)
It is clear that this stochastic process X(t) is weakly stationary since its mean and
35
autocorrelation do not depend on time t. Also, the process has µX = 0 and σ2X = 1
2A2.
Now, let’s look into the ergodicity property.
EX(t) = limT→∞
1
2T
∫ T
−TA cos(ωt+ ϕ)dt = 0, (2.78)
EX(t)X(t+ τ) = limT→∞
1
2T
∫ T
−TA2 cos(ωt+ ϕ) cos(ω(t+ τ) + ϕ)dt =
1
2A2 cos(ωτ).
(2.79)
It is also clear that the ensemble averages of mean and autocorrelation exactly match
with the temporal averages. Thus, it guarantees ergodicity for the given stochastic
process.
36
Example 2: X(t) =∑
N An cos(ωnt+ ϕn)
The next example is the superposition of several cosines with different amplitudes,
frequencies and phases. As the previous example, amplitudes An and frequencies ωn
are constants, and phases ϕn are random variables with uniform distribution among
0 to 2π. Following the same approach, the stationarity and the ergodicity will be
investigated. Followings are about the stationarity.
EX(t) =E∑N
An cos(ωnt+ ϕn) = · · · = 0, (2.80)
EX(t)X(t+ τ) =E∑N
A2n cos(ωnt+ ϕn) cos(ωn(t+ τ) + ϕn)
= · · · = 1
2
∑N
A2n cos(ωnτ). (2.81)
It is obvious that the random process X(t) is weakly stationary, and has µX = 0 and
σ2X = 1
2
∑N A
2n. For the ergodicity, we will have
EX(t) = limT→∞
1
2T
∫ T
−T
∑N
An cos(ωnt+ ϕn)dt = 0, (2.82)
EX(t)X(t+ τ) = limT→∞
1
2T
∫ T
−T
∑N
A2n cos(ωnt+ ϕn) cos(ωn(t+ τ) + ϕn)dt
=1
2
∑N
A2n cos(ωnτ). (2.83)
The ensemble averages perfectly match to the temporal averages, which guarantees
the ergodicity property for the given stochastic process.
37
Example 3: X(t) = A cos(ωt) +B sin(ωt)
Third example has a constant frequency ω, but two amplitudes A and B are random
variables. In this case, we have two random variables, which differ from previous
examples. Following the same steps for the stationarity, we have
EX(t) =EA cosωt+ EB sinωt =√EA2 + EB2 cos(ωt− θ),
(2.84)
EX(t)X(t+ τ) = · · · = 1
2EA2 +B2 cosωτ
+
√1
4EA2 + EB2+ EAB2 cos(2ωt+ ωτ − φ), (2.85)
where θ and φ are corresponding phases. In order for the stochastic process to be
weakly stationary, it should meet following conditions.
EA = EB = 0, (2.86)
EAB = 0, (2.87)
EA2 = EB2. (2.88)
Thus, the random process becomes weakly stationary process if and only if EA =
EB = EAB = 0 and EA2 = EB2. Then the process will have µX = 0
and σ2X = σ2
A = σ2B. One can easily follow the same steps to derive that the above
stochastic process cannot be ergodic in terms of the autocorrelation.
2.4 Power Spectral Density
It is well known that Fourier analysis is used to decompose the time history data
into the sums of sines and cosines over frequency domain. A periodic time history
data can be decomposed as discrete components of frequencies, while an aperiodic
time history can be decomposed as continuous components of frequencies. However,
this Fourier analysis can only be applied if the time history data diminishes as time
38
grows. In the case of the stochastic process, realizations are not generally periodic
and do not diminish with respect to time. Therefore, this difficulty is overcome by
introducing the power spectral density. For a stationary stochastic process, the power
spectral density is defined to be the Fourier transform of the covariance function.
SX(ω) =
∫ ∞−∞
CXX(τ)e−jωτdτ, (2.89)
CXX(τ) =1
2π
∫ ∞−∞
SX(ω)ejωτdω. (2.90)
Specifically, for the case of zero mean stochastic process, above relations can be re-
written as
SX(ω) =
∫ ∞−∞
RXX(τ)e−jωτdτ, (2.91)
RXX(τ) =1
2π
∫ ∞−∞
SX(ω)ejωτdω. (2.92)
The above Fourier transform pairs are called Wiener-Khintchine relation. It is often
the case that the input power spectral density is a given information. By plugging
τ = 0 into the above relation, we will obtain
σ2X = RXX(0) =
1
2π
∫ ∞−∞
SX(ω)dω, (2.93)
which represents that the power spectral density can be viewed as a distribution of
variance over the frequency domain. By using the properties of Fourier transform, it
is shown that the covariance function and the correlation function are even and real
functions. Then it is also obvious that the power spectral density is an even and real
function. Considering that negative frequency components do not have any physical
meaning, we now introduce the one sided power spectral density defined as follows.
We denote the one sided power spectral density with a + superscript.
S+X(ω) =
2SX(ω) ω ≥ 0
0 ω < 0
(2.94)
39
It should be noted that many practically developed power spectral density is defined
only for positive frequency. However, the above Wiener-Khintchine relation holds for
the original power spectral density, thus one is required to convert one sided spectrum
into double sided spectrum before applying the Wiener-Khintchine relation. Note that
the expression for the variance changes to
σ2X =
1
2π
∫ ∞0
S+X(ω)dω. (2.95)
The unit of power spectral density can be evaluated from above relation. Since
the unit of variance of a random variable X is square of its unit, the unit of the
power spectral density is the square of the unit of X(t) divided by radian per second.
Throughout this thesis, the stochastic process X(t) represents the elevation of ocean
waves, and therefore, the power spectral density has the unit of [m2s/rad].
−15 −10 −5 0 5 10 150
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
ω [rad/s]
Po
we
r S
pe
ctr
um
[m
2s/r
ad
]
0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ω [rad/s]
Po
we
r S
pe
ctr
um
[m
2s/r
ad
]
(a) (b)
Figure 2-4: (a) Double sided power spectral density. (b) Single sided power spectraldensity.
40
2.5 Energy Spectral Density
Let x (t) be a stationary and ergodic signal for which we assume that it has finite
power, i.e.
limT→∞
1
2T
T∫−T
|x (t)|2 dt <∞. (2.96)
Also recall that the correlation function is given as
RX (τ) = limT→∞
1
2T
T∫−T
x (t)x (t+ τ) dt = x (t)x (t+ τ), (2.97)
where the bar denotes ensemble average and the last equality follows from the as-
sumption of ergodicity. Note that we always have the property
|RX (τ)| ≤ RX (0) . (2.98)
Based on the correlation function, we can compute the power spectral density as
introduced before
SX (ω) = F [RX (τ)] = limT→∞
1
2T
∣∣∣∣∣∣T∫
−T
x (t) e−iωtdt
∣∣∣∣∣∣2
, (2.99)
where the Fourier transform is given by
F [RX (τ)] =
∞∫−∞
RX (τ) e−iωτdτ. (2.100)
The power spectral density describes how the energy of a signal x (t) is distributed
among harmonics in an averaged sense. Therefore, the averaged energy of a signal
41
can be expressed using the power spectral density as
EX = |x (t)|2 = RX (0) =1
2π
∞∫−∞
SX (ω) dω. (2.101)
In contrast, it should be noted that the usual energy spectrum is defined by the square
of the magnitude of the Fourier transform of a signal.
Se (ω) = |F [h (t)]|2 . (2.102)
Furthermore, it is important that the power spectral density can also be defined for
a signal for which the energy∫ T−T |h(t)|2dt is not finite. Therefore we should see the
power spectral density as a time or ensemble average of the energy distributed over
different harmonics.
42
Chapter 3
Probabilstic description of water
waves
In this chapter, ocean waves are introduced as an example of stochastic processes.
General reviews for the ocean energy harvesting can be found in [22–24]. Through-
out this thesis, we are interested in harvesting energy from external vibrations using
SDOF oscillators, and ocean waves along with winds and earthquakes are representa-
tive sources of energy. In the offshore industry, it is common that the state of ocean
remains constant for a short period of time and small range of area. Within these
conditions, it is possible to model the ocean waves as stationary random processes,
and therefore let us have several important tools to asses the response of offshore
structures. In the following section, it is shown that the ocean wave elevation is a
stationary and ergodic stochastic process which obeys Gaussian probability distribu-
tion. As a basis, the central limit theorem is briefly introduced first.
3.1 Central Limit Theorem
The central limit theorem gives a very general class of random phenomena whose
distribution can be approximated by the normal distribution. When the randomness
in a physical phenomena is the result of many small additive random effects, it tends
to be a normal distribution, irrespective of the individual distribution. For example,
43
let’s denote Xn to be a sequence of mutually independent and identically distributed
random variable with mean µ and variance σ2. It can be written as
Y =N∑j=1
Xj, (3.1)
and, by normalizing the random variable, we have a new normalized random variable
with zero mean and unit variance.
Z =Y −N · µ√
Nσ=
∑Nj=1Xj −N · µ√
Nσ. (3.2)
The probability distribution of a new normalized random variable Z will converge to
the standard Gaussian distribution as the number of sequence N goes to infinity.
3.2 Ocean Waves with a Gaussian Probability Dis-
tribution
The ocean wave processes are assumed to be Gaussian stochastic process, which gives
a reasonably good approximation to reality. This can be proved based on the central
limit theorem. Let’s consider the structure of random ocean wave as a superposition
of many waves with different frequencies, different phases, and different amplitudes.
By constraining the ocean wave in one directional wave, we can then express the wave
elevation at fixed time t, η(t), as follows:
η = X1 +X2 + · · ·+Xn, (3.3)
where Xi are statistically independent random variables, obeying the same but un-
known probability density function. This indicates that the mean and the variance
44
can be written as
EXi =0, (3.4)
V arXi =σ2, (3.5)
then it is obvious to express as follows:
Eη =EX1 +X2 + · · ·+Xn = 0, (3.6)
V arη =V arX1 +X2 + · · ·+Xn = n · σ2. (3.7)
As illustrated in the previous section, the random variable η is normalized and a new
normalized random variable Z is introduced.
Z =η − µηση
=η√nσ
=
∑nX
2i√
nσ. (3.8)
The new random variable Z has zero mean and unit variance. Next, the characteristic
functions φX and φZ are illustrated. There are two important properties for the
characteristic function.
Y = aX + b → φY (t) = ejbtφX(at), (3.9)
K = X + Y → φK(t) = φX(t) · φY (t), (3.10)
where X, Y , and K are random variables while a and b are constants. Then, the
normalized random variable Z can be written in terms of characteristic functions.
φZ(t) = φX(t√nσ
)n. (3.11)
This characteristic function can then be expanded as
φX(t) = 1− 1
2t2EX2+ o(t2), (3.12)
45
where o indicates little o notation which implies more rapidly approaching function
of t toward zero. Therefore, applying this expansion, the characteristic function of Z
can be simplified.
φX(t√nσ
) = 1− t2
2n+ o(
t2
n), (3.13)
φZ(t) = 1− t2
2n+ o(
t2
n)n. (3.14)
Recalling limn→∞(1 + xn)n = ex, it can be easily observed that as the number n
increases, the characteristic function for a new random variable Z approaches to
φZ(t) = e−12t2 . (3.15)
Therefore the normalized random variable Z obeys the standard Gaussian distribution
with zero mean and unit variance. Hence, it is proved that the ocean wave elevation
η has zero mean and variance n · σ2, and it is a Gaussian random variable.
µη =0, (3.16)
σ2η =n · σ2. (3.17)
46
3.3 Sea Spectra
Examples of widely adopted power spectral densities are introduced [25–28]. Those
are developed from empirical data for the ocean wave elevation at a specific location
in North Sea. Following power spectral density functions can be used for prelimi-
nary design of energy harvesting devices since it provides with a sufficiently realistic
modeling of spectral properties.
• Pierson-Moskowitz Spectrum
S+X(ω) = 0.0081
g2
ω5exp (−0.74(
g/U
ω)4). (3.18)
Here U indicates the wind-speed at 19.5m above the sea surface and g is the gravita-
tional acceleration.
• Bretschneider Spectrum (Modified Pierson-Moskowitz Spectrum)
S+X(ω) =
1.25
4
ω4p
ω5H2s exp (−1.25(
ωpω
)4). (3.19)
Here Hs is the significant wave height and ωp is the peak frequency as which S+X(ω)
is a maximum.
• JONSWAP (Joint North Sea Wave Project) Spectrum
S+X(ω) =
αg2
ω5exp (−1.25(
ωpω
)4)γr, (3.20)
r =− (ω − ωp)2
2σ2ω2p
, (3.21)
α =0.076(U2
Fg)0.22, (3.22)
ωp =22(g2
UF)1/3. (3.23)
Here F is the fetch, the distance from a lee shore, U is the wind speed, and γ is the
sharpness parameter. The JONSWAP spectrum is developed for the limited fetch
and is used extensively in the offshore industry.
47
• Ochi-Hubble Spectrum
S+X(ω) =
1
4
(4λ+14ω4p)λ
Γ(λ)
ζ2
ω4λ+1exp −(
4λ+ 1
4)(ωpω
)4, (3.24)
where λ is the width of the spectrum and Γ is the gamma function. Ochi-Hubble
spectrum is an extension of the Bretschneider spectrum, allowing to make it wider
for developing seas, or narrower for the swell.
All the power spectral density functions developed for the ocean waves are assumed
to be uni-directional. This can be modified by introducing a direction function D(θ).
S+X(ω, θ) = S+
X(ω)D(θ), (3.25)
where the directional function describes the spread of energy over different directions.
Thus the following should be satisfied.
∫ π
−πD(θ)dθ = 1. (3.26)
An example of directional function is provide.
D(θ) =2
πcos2(θ). (3.27)
48
3.4 Expression of Ocean Wave Elevation
Based on the proof in the previous section, we can safely state that the wave process
is assumed to be a Gaussian stochastic process. However, even though the station-
ary and ergodic stochastic process with Gaussian probability distribution is widely
adopted as an ocean wave description, it should be mentioned that this is not ac-
ceptable in many cases. Depending on the situation we consider, other assumptions
may be taken into consideration. However, in the context of harvesting energy which
is the main interest of this thesis, a Gaussian ocean wave elevation is an enough
approximation.
Let’s consider a fully developed stationary ocean wave whose wave amplitude is
much smaller than the wavelength. Under this condition, we can assume that the
ocean waves are deep water waves which have a dispersion relation of
ω2 = gk, (3.28)
where ω is the frequency of the ocean wave in rad/s, k indicates the wave number,
and g is the gravitational acceleration. Then the ocean wave elevation η(x, y, t) in
terms of space coordination x and y can be represented as follows:
η(x, y, t) =N∑i=1
M∑j=1
√S+X(ωi, θj)∆ωi∆θjAij cos(ωit− kx cosωj − ky sinωj)
+Bij sin(ωit− kx cosωj − ky sinωj), (3.29)
where the coefficients Aij and Bij are mutually independent Gaussian random variable
with zero mean and unit variance. By applying the dispersion relation for the deep
water, the equation reduces to
η(x, y, t) =N∑i=1
M∑j=1
√S+X(ωi, θj)∆ωi∆θjAij cos(ωit−
ω2i
g(x cosωj + y sinωj))
+Bij sin(ωit−ω2i
g(x cosωj + y sinωj)). (3.30)
49
Please note that the ocean wave elevation does not depend on the spatial coordinate.
Thus by choosing the coordinate at the origin (0, 0), above general equation reduces
to
η(t) =N∑i=1
M∑j=1
√S+X(ωi, θj)∆ωi∆θjAij cos(ωit) +Bij sin(ωit). (3.31)
And by further combining cosine and sine terms with same frequency components, it
can be written as
η(t) =N∑i=1
M∑j=1
√S+X(ωi, θj)∆ωi∆θjRij cos(ωit+ ψij), (3.32)
where R2ij = A2
ij +B2ij and Rij is Rayleigh distributed random variables. Also ψij are
random variables uniformly distributed among 0 to 2π which represent phases. Then
this can be simplified into
η(t) =N∑i=1
M∑j=1
√2S+
X(ωi, θj)∆ωi∆θj cos(ωit+ ψij). (3.33)
The above simplification does not guarantee the ocean wave elevation to be a Gaussian
random process. However, as illustrated in the previous section, the stochastic process
η(t) becomes a Gaussian random process due to the central limit theorem. If we ignore
the directionality and reduce to uni-directional expression, it becomes as follows:
η(t) =N∑i=1
√2S+
X(ωi)∆ωi cos(ωit+ ψi). (3.34)
This expression is a reasonable approximation of uni-directional ocean wave elevation
and thus it is widely adopted in the offshore industry. There are several technical
details which must be paid attention to before conducting simulations. These details
will be discussed in the following chapter.
50
Chapter 4
Statistical Steady State Response
of SDOF Oscillators
4.1 Analytical Steady State Response of Linear
Systems
In this section, the statistical steady state response of SDOF linear oscillators is ana-
lytical illustrated. For a stationary process, the power spectral density of the response
can be easily deduced from the knowledge of the power spectral density of the input
excitation. Three different types of excitations which will be treated throughout this
thesis are the monochromatic excitation, the Gaussian white noise excitation, and the
colored noise excitation. These excitations can be viewed and differentiated in terms
of the shape of the power spectral density function. The monochromatic excitation
has a delta function while the Gaussian white noise excitation has a constant value
in the power spectral density function (i.e. no characteristic frequency). The colored
noise excitation can be described with a band limited power spectral density function.
In the following section, several important properties of the system are discussed.
51
4.1.1 System Properties
In the previous chapter, several properties of power spectral density of an excitation,
i.e. ocean waves, are explored. Now we want to find out how the response of a system
can be characterized in terms of the power spectral density of the response. Thus,
several properties which define a system will be discussed first [29].
Dynamic/Static System
A system is called dynamic if the system depends not only on the present time, but
also on the past or future time. On the contrary, a system is called static if the system
depends only on the present time.
Linear System
A system is called linear if the superposition holds.
Time Invariant System
A system is called time invariant if the time shift in the input corresponds to the time
shift in the output.
Causal System
A system is called causal if the output of the system does not depend on the future
input.
Stable System
A system is called stable if a bounded input gives a bounded output.
Throughout this section, we assume that the transfer function of a system can be
modeled as a linear and time invariant system. Consider a system with an impulse
response of h(t) where the input is x(t) and the output is y(t). Then the input and
52
the output can be related in terms of convolution as follows:
y(t) = h(t) ∗ x(t) =
∫ ∞−∞
x(τ)h(t− τ)dτ. (4.1)
If the system is causal, above relation reduces to
y(t) =
∫ ∞0
x(τ)h(t− τ)dτ. (4.2)
Here, the equation can be interpreted in the way that x(t) is one realization of the
ocean wave elevation and h(t) is the impulse response of the proposed structure, and
y(t) is one realization of the response.
Systemh(t)INPUT OUTPUT
x(t) y(t)
Figure 4-1: A system with an impulse response of h(t).
4.1.2 Response Power Spectral Density of Linear Systems
Consider that there are N realizations of ocean wave elevations x(t) and it is denoted
as X(t). Also, the realizations of response of the structure y(t) are denoted as Y (t).
Then the expectation of response can be expressed as
EY (t) = E∫ ∞0
X(τ)h(t− τ)dτ =
∫ ∞0
EX(τ)h(t− τ)dτ. (4.3)
Therefore, the above relation simplifies to
mX = mY
∫ ∞0
h(t− τ)dτ = H(0)mY , (4.4)
where mX and mY are mean values of X(t) and Y (t), respectively, and H(0) is the
frequency response of the system evaluated at zero frequency. Thus we can conclude
that the mean value of the response of a linear time invariant system is the mean
53
value of the input multiplied by the system response of zero frequency. As the case
for the ocean wave elevation, if the excitation has zero mean, then the response of
the system also has zero mean.
Now assume that the input is a stationary stochastic process with a zero mean. As we
have shown before, it is obvious that the output also has a zero mean. Let’s consider
the autocovariance function.
EY (t)Y (t+ τ) =E∫ ∞0
h(τ1)X(t− τ1)dτ1 ·∫ ∞0
h(τ2)X(t+ τ − τ2)dτ2
=E∫ ∞0
∫ ∞0
X(t− τ1)X(t+ τ − τ2)h(τ1)h(τ2)dτ1dτ2
=
∫ ∞0
∫ ∞0
EX(t− τ1)X(t+ τ − τ2)h(τ1)h(τ2)dτ1dτ2
=
∫ ∞0
∫ ∞0
CX(τ + τ1 − τ2)h(τ1)h(τ2)dτ1dτ2. (4.5)
Therefore, the above equation follows that
CY (τ) =
∫ ∞0
∫ ∞0
CX(τ + τ1 − τ2)h(τ1)h(τ2)dτ1dτ2. (4.6)
Thus it can be generalized that under a linear time invariant system a stationary
input produces a stationary output. Furthermore, this can be easily extended that
under a linear time invariant system a stationary ergodic input with Gaussian prob-
ability distribution produces a stationary ergodic output with Gaussian probability
distribution.
The mean value and the variance play an important role on describing the response
statistics. In the previous part, we have shown that the mean value of the response
can be easily deduced from the knowledge of the mean value of the input in addition
to the frequency response. It is also shown that the autocovariance function of the
response can be related by the autocovariance function of the input. Now we will
investigate the relation between the variance of the input and the output by taking
54
Fourier transform of the autocovariance functions. As described in the previous part,
the Fourier transform of the input and the output covariance functions are given as
SY (ω) =
∫ ∞0
h(τ1)
∫ ∞0
h(τ2)
∫ ∞−∞
CX(τ + τ1 − τ2)e−jωτdτdτ2dτ1
=
∫ ∞0
h(τ1)ejωτ1dτ1
∫ ∞0
h(τ2)e−jωτ2dτ2 · SX(ω)
=H(−ω)H(ω)SX(ω)
=H(ω)H∗(ω)SX(ω)
=|H(ω)|2SX(ω). (4.7)
Thus in terms of the one sided power spectral density, there is an explicit relation
between the input and the output power spectral density as follows:
S+Y (ω) = |H(ω)|2S+
X(ω). (4.8)
Then the variance of the output can be expressed as follows:
σ2Y =
∫ ∞−∞
S+Y (ω)dω =
∫ ∞−∞|H(ω)|2S+
X(ω)dω. (4.9)
LTI
Staonary
Ergodic
Gaussian
Staonary
Ergodic
Gaussian
INPUT OUTPUT
Figure 4-2: Input and output relation in terms of stationarity, ergodicity and gaussianprocess.
Knowledge of the transfer function of a linear system in addition to the knowledge
of the form of the power spectral density function of the input excitation will let
us derive the response power spectral density function analytically. For example,
consider a dynamical system whose governing equation can be written in terms of a
55
second order differential equation as follows:
my + λy + ky = −x, (4.10)
where m is the mass, λ is the dissipation coefficient, k is the stiffness coefficient,
and x(t) is the displacement of the external excitation. It is obvious that above
representative system is linear (because there is no higher order terms), and time
invariant. Furthermore, the transfer function of the system can be easily obtained by
taking Fourier transform of the governing equation.
m(jω)2 + λ(jω) + kY (ω) = −(jω)2X(ω). (4.11)
Then, by combining relevant terms in both sides, the transfer function H(ω) can be
obtained as follows:
H(ω) =Y (ω)
X(ω)=
ω2
−mω2 + λ(jω) + k. (4.12)
Please note that the response power spectral density is related with the input power
spectral density with the multiplication with the square of the absolute value of the
transfer function.
|H(ω)|2 = H(ω)H∗(ω), (4.13)
where ∗ indicates complex conjugate.
56
4.2 Analytical Steady State Response of Nonlinear
Systems Excited by Gaussian White Noise
In this section, the statistical steady state response of nonlinear systems is investi-
gated. In contrast to the linear system, we do not have an explicit expression of the
transfer function for the nonlinear systems. However, under a specific situation, the
Gaussian white noise, it is possible to obtain a statistical steady state solution for the
probability density function of the response. The statistical steady state probabil-
ity density function of the response will fully describe the nonlinear system, and this
knowledge will be fully adopted to investigate the harvested power of SDOF nonlinear
oscillators. Thus, the procedure of obtaining steady state probability density function
under the Gaussian white noise excitation is first given in the following section.
4.2.1 Fokker - Planck and Kolmogorov Equation
Fokker - Planck Equation, or equivalently Kolmogorov forward equation, describes
the time evolution of probability distribution of a stochastic process. The derivation
of the Fokker- Planck equation is not provided in this thesis. Interested readers can
find the detailed derivation in [21]. Let’s consider a 2D stochastic differential equation
such as
dX(t) = µ(X(t), t)dt+ σ(X(t), t)dW (t), (4.14)
whereW (t) represents Wiener processes. Also µ denotes a drift vector and σ denotes
a diffusion tensor. The probability density function f(X(t), t) for the random vector
X(t) satisfies the following Fokker-Planck equation
∂
∂tf(X, t) =− ∂
∂x1[µ1(X)f(X, t)]− ∂
∂x2[µ2(X)f(X, t)] +
∂2
∂x21[D11(X)f(X, t)]
+∂2
∂x1∂x2[D12(X)f(X, t)] +
∂2
∂x1∂x2[D21(X)f(X, t)] +
∂2
∂x22[D22(X)f(X, t)],
(4.15)
57
where x1 and x2 are elements of two dimensional vector X(t), µ1 and µ2 are elements
of two dimensional drift vector µ, and σ11, σ12, σ21 and σ22 are elements of 2× 2 di-
mensional diffusion tensor σ. Specifically, we will consider an example of a dynamical
system under the Gaussian white noise excitation whose governing equation can be
expressed as a second order differential equation. For the generality, the normalized
function F (x) which represents the stiffness of the system with any order is used here.
x+ λx+ F (x) = ζ(t), (4.16)
where ζ(t) is the Gaussian white noise excitation with zero mean and intensity of 2D.
By replacing x1 = x and x2 = x, the governing equation can be rewritten as follows:
dx1dt
=x2, (4.17)
dx2dt
=− λx2 − F (x1) + ζ(t). (4.18)
Equivalently, the can be written as dx1
dx2
=
x2
−λx2 − F (x1)
dt+
0 0
0 2D
dW1
dW2
. (4.19)
Thus, by plugging these into the Fokker-Planck equation, we will obtain
∂
∂tf(X, t) = − ∂
∂x1(x2f) +
∂
∂x2(λx2 + F (x1)) +D
∂2
∂x22f. (4.20)
For the steady state, the left hand side of the equation vanished and we will have
D∂2
∂x22fst −
∂
∂x1(x2fst) +
∂
∂x2[(λx2 + F (x1))fst] = 0. (4.21)
By rearranging the above equation, we will obtain
∂
∂x2[F (x1)fst +
D
λ
∂fst∂x1
] + (λ∂
∂x2− ∂
∂x1)[x2fst +
D
λ
∂fst∂x2
]. (4.22)
58
The solution should satisfy
F (x1)fst +D
λ
∂fst∂x1
= 0, (4.23)
x2fst +D
λ
∂fst∂x2
= 0. (4.24)
By integrating, we will finally have
fst(x1, x2) = C exp− λD
[x222
+
∫ x1
0
F (x)dx], (4.25)
where x1 = x, x2 = x, and C is an integration constant. This steady state probability
density function will be used for investigating the performance of SDOF oscillators
under the Gaussian white noise in the following chapter.
4.3 Numerical Simulation of Nonlinear Systems
Excited by Colored Noise
We have shown that the response of SDOF linear oscillators is analytically described
by the power spectral density of excitation and the transfer function of the system.
Also, under the Gaussian white noise excitation, the response of the nonlinear SDOF
oscillators also can be analytically expressed. In this section, the response of nonlinear
SDOF oscillators under the colored noise excitation is investigated. Since nonlinear
SDOF oscillators do not have explicit expressions for its transfer functions, we should
evaluate the response of the system numerically.
A colored noise excitation can be viewed as an excitation with a narrow banded power
spectral density. Ocean wave spectra such as Pierson-Moskowitz spectrum are repre-
sentative examples. In the previous chapter, the expression of ocean wave elevation
with those power spectral density is fully explored. In this section, several simula-
tional details will be discussed.
59
Recall that the expression of the ocean wave elevation is given as
η(t) =∑N
√2S+
X(ωn)∆ω cos (ωnt+ ϕn), (4.26)
where S+X(ωn) is a given one sided power spectral density, ωn are frequencies, and ϕn
are random phases with uniform distribution between 0 and 2π. An important point
which should be discussed is the aperiodic property of the ocean wave elevation. It is
obvious that ocean waves do not repeat as time passes, thus the generated time history
data of ocean wave elevation should be an aperiodic function. However, depending
on the choice of frequency components ωn, the signal can be either of a periodic or an
aperiodic signal. Thus it is critical to make sure whether the generated ocean wave
elevation is an aperiodic stationary Gaussian random process. There are two ways
of selecting corresponding frequency components. One way is adopting uniformly
distributed frequency components as follows:
ωn = ω0 + (n− 1)δω, (4.27)
where ω0 is initial frequency and δω is the frequency span between two adjacent
frequency components. Depending on the selection of ω0 and δω, there is a possibility
to generate a periodic ocean wave elevation. Thus, as an alternative approach, we
will introduce an additional term into the previous equation as follows:
ωn = ω0 + (n− 1)δω +1
2ψnδω, (4.28)
where ψn is a random number between −1 and +1. By introducing a small pertur-
bation into the frequency components, it is now guaranteed that the generated ocean
elevation signal is an aperiodic signal. This can be observed in Figure (4-3) that the
autocorrelation function of the first case has multiple peaks while the autocorrelation
function of the second case has no peak as time increases.
60
0 500 1000 1500 2000 2500 3000−0.3
−0.2
−0.1
0
0.1
0.2
0.3
t
Rxx(τ
)
0 500 1000 1500 2000 2500 3000−0.3
−0.2
−0.1
0
0.1
0.2
0.3
t
Rxx(τ
)
(a) (b)
Figure 4-3: (a) Autocorrelation function of the periodic ocean wave elevation signal.(b) Autocorrelation function of the aperiodic ocean wave elevation signal.
4.4 Gaussian Closure for Nonlinear SDOF Oscil-
lators Excited by Colored Noise
There are many different methods to tackle down the nonlinear random vibration
problems including statistical linearization method [30–33], moment closure method
[34], perturbation method [35, 36], Monte Carlo simulation method [37, 38], and so
on. Among those techniques, Monte Carlo simulation method is primarily adopted
for SDOF nonlinear oscillators under the colored noise excitation for the previous sec-
tions. Obviously, the accuracy of the results will increase as we increase the number
of simulation records. However, this will lead the computational cost of simulation
significant. Therefore, with an aim of increasing accuracy while keeping the compu-
tational cost low, an innovative moment equation technique is illustrated.
Let’s consider a SDOF nonlinear oscillator whose normalized governing equation is
in the form of
x+ λx+ k1x+ k3x3 = y, (4.29)
where λ is the dissipation coefficient, k1 is the linear stiffness coefficient, and k2 is
61
the cubic stiffness coefficient. Here, we assume that excitation, y(t), is stationary
random process with zero mean, y = 0, and the power spectral density function has
a band limited frequency components. Also, for the vibrational system it is natural
to assume that mean value of the response is zero, x = 0.
In order to derive the moment equations, we first multiply the governing equation
(4.29) with y(s) and take the mean value operator. Here, y(s) is a function of new
time parameter s.
x(t)y(s) + λx(t)y(s) + k1x(t)y(s) + k3x(t)3y(s) = y(t)y(s). (4.30)
By taking the partial derivatives out, above equation can be rewritten as
∂2
∂t2x(t)y(s) +
∂
∂tλx(t)y(s) + k1x(t)y(s) + k3x(t)3y(s) =
∂2
∂t2y(t)y(s). (4.31)
Recall that the expectation of the multiplication of two random variables are the
correlation function. In this specific case, we have zero mean for both of x(t) and
y(t), the correlation function equals with the covariance function. Now, each term in
the above equation will be replaced by C which denotes either of the autocovariance
function or crosscovariance function depending on the parameters indicated in the
subscripts.
∂2
∂t2Ctsxy + λ
∂
∂tCtsxy + k1C
tsxy + k3x(t)3y(s) =
∂2
∂t2Ctsyy. (4.32)
where the super scripts denote time parameters. Similarly, if we multiply the govern-
ing equation (4.29) with x(s) and take the mean value operator, we will have
x(t)x(s) + λx(t)x(s) + k1x(t)x(s) + k3x(t)3x(s) = y(t)x(s), (4.33)
Then, it can also be expressed in terms of the covariance function as
∂2
∂t2Ctsxx + λ
∂
∂tCtsxx + k1C
tsxx + k3x(t)3x(s) =
∂2
∂t2Ctsxy. (4.34)
62
Now, we have obtained two moment equations from the governing equation, but the
problem is that there is higher order terms in the moment equations. We thus need a
closing technique in order to solve the moment equations. Considering the excitation
is a stationery and ergodic Gaussian random process, we can assume that the response
is also a stationary and ergodic Gaussian random process. This enables us to apply
the Gaussian closure assumption in order to express forth-order moments in terms of
second order moments, in accordance to Isserlis’ Theorem as follows:
x3(t)y(s) = 3CttxxC
tsxy = 3σ2
XCtsxy, (4.35)
x3(t)x(s) = 3CttxxC
tsxx = 3σ2
XCtsxx. (4.36)
For the convenience, the variance of x(t), Cttxx is replaced by σ2
X . This will lead the
two moment equations into simpler forms.
∂2
∂t2Ctsxy + λ
∂
∂tCtsxy + (k1 + 3k3σ
2X)Cts
xy =∂2
∂t2Ctsyy, (4.37)
∂2
∂t2Ctsxx + λ
∂
∂tCtsxx + (k1 + 3k3σ
2X)Cts
xx =∂2
∂t2Ctsxy. (4.38)
With the help of Wiener-Khintchine relation, we will have the power spectral density
equations by taking Fourier Transform of above two moment equations.
(jω)2 + λ(jω) + k1 + 3k3σ2XSxy(ω) = (jω)2Syy(ω), (4.39)
(jω)2 + λ(jω) + k1 + 3k3σ2XSxx(ω) = (jω)2Sxx(ω). (4.40)
Thus, we will have a relation between the input excitation spectrum and the output
response spectrum as follows:
Sxx(ω) =ω4
(k1 − ω2 + 3k3σ2X)2 + λ2ω2
Syy(ω). (4.41)
If the excitation is given as a colored noise excitation with a power spectral density
63
in the form of Pierson-Moskowtiz spectrum, such as
S+(ω) =1
ω5exp (− 1
ω4). (4.42)
The equation (4.44) reduces to
Sxx(ω) =ω4
(k1 − ω2 + 3k3σ2X)2 + λ2ω2
1
ω5exp (− 1
ω4), (4.43)
and if we take integration from 0 to ∞,
σ2X =
∫ ∞0
Sxx(ω)dω =
∫ ∞0
ω4
(k1 − ω2 + 3k3σ2X)2 + λ2ω2
1
ω5exp (− 1
ω4)dω. (4.44)
This will give us a nonlinear equation for σ2X as follow.
σ2X =
∫ ∞0
ω4
(k1 − ω2 + 3k3σ2X)2 + λ2ω2
1
ω5exp (− 1
ω4)dω. (4.45)
The above nonlinear equation can be solved numerically by estimating the square of
the absolute difference as follows:
ε = |σ2X −
∫ ∞0
ω4
(k1 − ω2 + 3k3σ2X)2 + λ2ω2
1
ω5exp (− 1
ω4)dω|2. (4.46)
Please note that the above nonlinear equation can have more than one solutions. By
solving above equation, we can fully describe the response of SDOF nonlinear oscil-
lators under the colored noise excitation. Please note that the stationary and ergodic
Gaussian random process is assumed for both of input and output. Computational
results will be presented in the following chapter.
64
Chapter 5
Quantification of Power Harvesting
Performance
We study the energy harvesting properties of a SDOF oscillator subjected to ran-
dom excitation. In the energy harvesting setting, randomness is usually introduced
through the excitation signal which although is characterized by a given spectrum,
i.e. a given amplitude for each harmonic, the relative phase between harmonics is
unknown and to this end is modeled as a uniformly distributed random variable. We
consider the following system consisting of an oscillator lying on a basis whose dis-
placement h (t) is a random function of time with given spectrum. The equation of
motion for this simple system has the form
mx+ λ(x− h
)+ F (x− h) = 0, (5.1)
where m is the mass of the system, λ is a dissipation coefficient expressing only
the harvesting of energy (we ignore in this simple setting any mechanical loses),
and F is the spring force that has a given form but free parameters, i.e. F (x) =
F (x; k1, ..., kn) . One could think of F as a polynomial: F (x; kp) =∑
p=1,...,N
kpxp.
We assume that the excitation process is stationary and ergodic having a given
spectrum Shh (ω). We also assume that after sufficient time the system converges to a
statistical steady state where the response can be characterized by the power spectrum
65
Sqq (ω) . We expect a stationary response given that we have only one structural
mode involved and thus we do not expect to have non-stationary phenomena due
to nonlinear energy transfers. For this system the harvested power per unit mass is
given by
Ph =λ
m
(x− h
)2, (5.2)
where the bar denotes ensemble or temporal average in the statistical steady state
regime of the dynamics. For convenience we apply the transformation x − h = q to
obtain the system
q + λq + F (q) = −h, (5.3)
where λ = λm
and F = Fm.
Through this formulation we note that the mass can be regarded as a parameter
that does not need to be taken into account in the optimization procedure. This
is because for any optimal set of parameters λ and F , the energy harvested will
increase linearly with the mass of the oscillator employed (given that λ and F remain
constant).
5.1 Absolute and Normalized Harvested Power Ph
In the present work, we are interested to compare the maximum possible performance
between different classes of oscillators and to this end we ignore mechanical losses
and assume that the damping coefficient λ describes entirely the energy harvested.
In terms of the spectral properties of the response, the absolute harvested power Ph
can then be expressed as
Ph = λq2 = λ
∞∫−∞
ω2Sqq (ω) dω. (5.4)
This quantifies the amount of energy harvested per unit mass.
66
5.2 Size of the Energy Harvester B
An objective comparison between two harvesters should involve not only the same
mass but also the same size. We chose to quantify the characteristic size of the
harvesting device using the mean square displacement of the center of mass of the
system. For the SDOF setting, this is simply the typical deviation of the stochastic
process q (t) given by
d =
√q2 =
√√√√√ ∞∫−∞
Sqq (ω) dω. (5.5)
Our goal is to quantify the maximum performance of a harvesting configuration for
a given typical size d and for a given form of input spectrum. To achieve invariance
with respect to the source-spectrum magnitude, we will use the non-dimensional ratio
B =q2
h2, (5.6)
which is the square of the relative magnitude of the device compared with the typical
size of the excitation motion√h2. The above quantity also expresses the amount of
energy that the device carries relative to the energy of the excitation and to this end
we will refer to it as the response level of the harvester. It will be used to parametrize
the performance measures developed in the next section with respect to the typical
size of the device.
5.3 Harvested Power Density ρe
For each response level B, we define the harvested power density ρe as the maximum
possible harvested power maxλ,ki | B
Ph (for a given excitation spectrum and under the
constraint of a given response level B) suitably normalized with respect to the response
67
Figure 5-1: Various spectral curves obtained by magnitude and temporal rescaling ofthe Pierson-Moskowitz spectrum. Amplification and stretching of the input spectrumwill leave the effective damping and the harvested power density invariant.
size q2 and the mean frequency of the input spectrum
ρe(B) =
maxλ,ki | B
Ph
ω3hq
2=
maxλ,ki | B
(λq2)
ω3hq
2, (5.7)
where the mean frequency of the input spectrum is defined as
ωh =1
h2
∞∫0
ωShh (ω) dω. (5.8)
This measure should be viewed as a function of the response level of the device B. As
we show below it satisfies an invariance property under linear transformations of the
excitation spectrum, i.e. rescaling of the spectrum in time and magnitude (Figure 1).
More specifically we have the following theorem.
Theorem 1 The harvested power density ρe is invariant with respect to linear trans-
formations of the input energy spectrum Shh (ω) (uniform amplification and stretch-
ing). In particular, under the modified excitation g (t) = a√bh (bt) or equivalently the
input spectrum Sgg (ω) = a2Shh(ωb
), where a > 0 and b > 0 , the curve ρe (B) remain
invariant.
68
Proof. Let λ0 and ki,0 be the optimal parameters for which the quantity Ph at-
tains its maximum value for the input spectrum Shh (ω) under the constraint of a
given response level B0 = q2
h2. For convenience, we will use the notation F0 (q) =
F(q; k1,0, ..., kn,0
). For these optimal parameters we will also have the optimum re-
sponse q0 (t) that satisfies the equation
q0 + λ0q0 + F0 (q0) = −h. (5.9)
We will prove that under the rescaled spectrum Sgg (ω) = a2Shh(ωb
)the harvested
power density curve ρe (B) remains invariant. By direct computation, it can be verified
that the modified spectrum Sgg (ω) corresponds to an excitation of the form
g (t) = a√bh (bt) . (5.10)
Moreover, by direct calculation we can verify that
g2 = a2bh2 and ωg = bωh. (5.11)
We pick a response level B0 for the system excited by h (t) and we will prove that
ρe,g (B0) = ρe,h (B0) . Under the new excitation the system equation will be
q + λq + F (q) = −a√bd2h (bt)
dt2. (5.12)
We apply the temporal transformation bt = τ. In the new timescale, we will have
(differentiation is now denoted with ′)
b2q′′ + λbq′ + F (q) = −ab52h′′. (5.13)
For q2
g2= B0, we want to find the set of parameters λ and ki that will maximize
Pg = λq2 given the dynamical constraint (5.12). This optimized quantity can also be
written as
Pg = λq2 = b2λq′2, (5.14)
69
where q′ is described by the rescaled equation (5.13). However, the optimization
problem in equations (5.13) and (5.14) is identical with the original one given by
equation (5.9) and it has an optimal solution when λ = bλ0 and F (q) = ab52 F0
(q
a√b
).
For this set of parameters, equation (5.13) coincides with equation (5.9) and the
solution to (5.13) will be q (t) = a√bq0 (bt). Note that for this solution we also have
q2
g2=a2bq20a2bh2
=q20h2
= B0, (5.15)
and therefore the optimized solution that we found corresponds to the correct response
level. The last step is to compute the harvested power density for the new solution.
These will be given by
ρe,g(B0) =
maxλ,ki | B0
(λq2)
ω3gq
2=
maxλ,ki | B0
(b2λq′2
)(b3ω3
h)(a2bq20
) =
(bλ0
)(b3a2q′20
)(b3ω3
h)(a2bq20
) =λ0q′20
ω3hq
20
= ρe,h(B0).
(5.16)
This completes the proof.
We emphasize that the above property can be generalized for multi-dimensional
systems; a detailed study for this case will be presented elsewhere. Through this
result we have illustrated that both uniform amplification and stretching of the input
spectrum (see e.g. Figure 5-1 various amplified, and stretched versions of the Pierson-
Moskowitz) will leave the harvested power density unchanged, and therefore the shape
of spectrum is the only factor (i.e. relative distribution of energy between harmonics)
that modifies the harvested power density.
Another important property of the developed measure is its independence of the
specific values of the system parameters since it always refers to the optimal config-
uration for each design. Thus, it is a tool that characterizes a whole class of systems
rather than specific members of this class. To this end it is suitable for the compar-
ison of systems having different forms e.g. having different function F(q; k1, ..., kn
)since it is only the form of the system that is taken into account and not the specific
parameters λ and k1, ..., kn.
70
These two properties give an objective character to the derived measure as it
depends only on the form of the employed configuration and the form of the input
spectrum. For this reason, it can be used to perform systematic comparisons and
optimizations among different classes of system configurations, e.g. linear versus
nonlinear harvesters. In addition to the above properties, the curve ρe (B) reveals the
optimal response level q2 so that the harvested power over the response magnitude is
maximum, achieving in this way optimal utilization of the device size.
We note that for a multi-dimensional energy harvester it may also be useful to
quantify the harvester performance using the effective harvesting coefficient λe which
is defined as the maximum possible harvested power maxλ,ki | B
Ph (for a given excitation
spectrum and under the constraint of a given response level B) normalized by the total
kinetic energy of the device EK :
λe(B) =
maxλ,ki | B
Ph
ωhEK, (5.17)
where we have also non-dimensionalized with the mean frequency of the input spec-
trum so that the ratio satisfies similar invariant properties under linear transforma-
tions of the input spectrum. Although for MDOF systems the above measure can
provide useful information about the efficient utilization of kinetic energy, for SDOF
systems of the form (5.1) we always have λe(B) = λ and to this end we will not study
this measure further in this work.
5.4 Quantification of Performance for SDOF Har-
vesters
We now apply the derived criteria in order to compare three different classes of non-
linear SDOF energy harvesters excited by three qualitatively different source spectra.
In particular, we compare the performance of linear SDOF harvesters with two classes
of nonlinear oscillators: an essentially nonlinear with cubic nonlinearity (mono-stable
71
system) and one that has also cubic nonlinearity but negative linear stiffness (double
well potential system or bistable) as illustrated in the Figure (5-2). The first family
of systems has been studied in various contexts with main focus the improvement
of the energy harvesting performance from wide-band sources. The second family of
nonlinear oscillators is well known for its property to maintain constant vibration am-
plitudes even for very small excitation levels, and it has also been applied to enhance
the energy harvesting capabilities of nonlinear energy harvesters. More specifically
we consider the following three classes of systems (Figure 5-3):
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
5
10
15
20
25
30
x
U(x)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2
0
2
4
6
8
10
x
U(x)
x
U(x)
U(x)
x
(a) (b)
Figure 5-2: The shapes of potential function U(x) = 12 k1x
2 + 14 k3x
4. (a) The monos-
table potential function with k1 > 0 and k3 > 0. (b) The bistable potential functionwith k1 < 0 and k3 > 0.
q + λq + k1q = −h, (linear system) (5.18)
q + λq + k3q3 = −h, (cubic system) (5.19)
q + λq − νq + k3q3 = −h. (negative stiffness) (5.20)
Our comparisons are presented for three cases of excitation spectra, namely: the
monochromatic excitation, the white noise excitation, and an intermediate one char-
acterized by colored noise excitation with Gaussian, stationary probabilistic structure
72
Figure 5-3: Linear and nonlinear SDOF systems: (a) Linear SDOF system, (b) Non-linear SDOF system only with a cubic spring, and (c) Nonlinear SDOF system withthe combination of a negative linear and a cubic spring.
and a power spectrum having the Pierson-Moskowitz form
Shh =1
ω5exp
(−ω−4
). (5.21)
The monochromatic and the white noise excitations are characterized by diametri-
cally opposed spectral properties: the first case is the extreme form of a narrow-band
excitation, while the second represents the most extreme case of a wide-band ex-
citation. Our goal is to understand and objectively compare various designs that
have been employed in the past to achieve better performance from sources which
are either monochromatic or broad-band. We are also interested to use these two
prototype forms of excitation in order to interpret the behavior of SDOF harvesters
for intermediate cases of excitation such as the PM spectrum.
We first present the monochromatic and the white noise cases where many of
the results can be derived analytically. We analyze the critical differences in terms
of the harvester performance and subsequently, we numerically perform stochastic
optimization of the nonlinear designs for the intermediate PM spectrum. For the PM
excitation, we employ a discrete approximation of the excitation h in spectral space,
with harmonics that have given amplitude but relative phase differences modeled
as uniformly distributed random variables. The responses of the dynamical systems
(5.18) and (5.19) are then characterized by averaging (after sufficient time so that
transient effects do not contribute) over a large ensemble of realizations, i.e. averaging
73
over a large number of excitations h generated with a given spectrum but randomly
generated phases.
5.5 Results of Performance Quantification
5.5.1 SDOF Harvester under Monochromatic Excitation
Linear system. We calculate the harvested power density ρe for the the linear oscil-
lator under monochromatic excitation, i.e. the one-sided power spectrum is given by
Shh (ω) = α2δ (ω − ω0). For this case the computation can be carried out analytically.
In particular for the linear oscillator we will have the power spectrum for the response
given by
Sqq (ω) =ω4(
k1 − ω2)2
+ λ2ω2
Shh (ω) . (5.22)
Thus, the response level can be computed as
B =q2
h2=
ω40(
k1 − ω20
)2+ λ2ω2
0
, (5.23)
where h2 is simply α2. Moreover, the average rate of energy harvested per unit mass
will be given by
Ph = λq2 = λα2 ω60(
k1 − ω20
)2+ λ2ω2
0
. (5.24)
Then we will have from equation (5.23)
(k1 − ω2
0
)2+ λ2ω2
0 =ω40
B. (5.25)
Thus, for a given B, the mean rate of energy harvested will be given by
Ph = λq2 = λq2ω20. (5.26)
74
Therefore the mean rate of energy harvested will become maximum when λ is max-
imum. For fixed B, equation (5.25) shows that the maximum legitimate value of λ
will be given by λ = ω0√B and this can be achieved when k1 = ω2
0. Therefore we will
have
Ph =ω30q
2
√B
= ω30h
2√B = α2ω3
0
√B, (5.27)
ρe =
maxλ,ki |B
Ph
ω3hq
2=
1√B. (5.28)
Hence, for a linear SDOF system under monochromatic excitation, the harvested
power density is proportional to the magnitude of the square root of B while the
harvested power is proportional to the square root of the response level.
Cubic and negative stiffness harvesters. For a nonlinear system the response
under monochromatic excitation cannot be obtained analytically and to this end the
computation will be carried out numerically. In figure 5-4, we present the response
level B for all three systems (linear, cubic, and the one with negative stiffness with
ν = 1) for various system parameters. We also present the total harvested power
superimposed with contours of the response level B.
For both the linear and the cubic oscillator, we can observe the 1:1 resonance
regime (see plots for the response level B). For these two cases, we also observe
a similar decay of the response level with respect to the damping coefficient. This
behavior changes drastically in the negative stiffness oscillator where the response
level is maintained with respect to changes of the damping coefficient. This is expected
if one considers the double well form of the corresponding potential that controls the
amplitude of the nonlinear oscillation. Despite the robust amplitude of the response,
the performance (i.e. the amount of power being harvested) drops similarly with the
other two oscillators (especially the cubic one) as the damping coefficient increases.
Thus constant response level does not guarantee the robust performance level with
respect to system parameters. To quantify the performance, we present in figure 5-5
the maximum harvested power and the harvested power density for the three different
75
(a) (b)
Lin
ea
r S
yste
mC
ub
ic S
yste
mN
eg
ative
Stiffn
ess
λλ
λ
BB
B
λ k1 k
1
λ k3 k
3
λ k3 k
3
Figure 5-4: Response level B. and power harvested for the case of monochromaticspectrum excitation over different system parameters. The response level B is alsopresented as a contour plot in the power harvested plots. All three cases of systems areshown: linear (top row), cubic (second row), and negative stiffness with ν = 1.
76
Figure 5-5: (a) Maximum harvested power, and (b) Power density for linear andnonlinear SDOF systems under monochromatic excitation.
oscillators. We observe that in all cases the linear design has superior performance
compared with the nonlinear configurations. In addition, we note that the cubic
and the negative stiffness oscillators have strongly variable performance which are
non-monotonic functions with respect to the response level B.
To better understand the nature of this variability, we pick two characteristic
values of B (one close to a local minimum i.e. B =8.5 and one at a local maximum,
i.e. B =8.1) for the negative stiffness oscillator (Figure 5-5). From these points, we
can observe that the strong performance for the nonlinear oscillator is associated with
signatures of 1:3 resonance in the response spectrum. We also note that the small
amplitude of the higher harmonic is not sufficiently large to justify the difference
in the performance. On the other hand, the significant amplitude difference on the
primary harmonic, which can be considered as an indirect effect of the 1:3 resonance,
justifies the strong variability between the two cases.
Independently of the super-harmonic resonance occurring in the nonlinear designs
for certain response levels, it is clear that the best performance for SDOF systems
under monochromatic excitation can be achieved within the class of linear harvesters.
To understand this result, we consider the general equation (5.3) multiplying with q
and applying the mean value operator. This will give us the following energy equation
1
2
d
dt
(q2)
+ λq2 + F (q) q = −hq. (5.29)
77
0 1 2 3 4 5 6 7 8 9 1010
−4
10−3
10−2
10−1
100
101
ω
|q(ω
)|
k = 0.1 and λ = 0.2
k = 0.25 and λ = 0.2
200 210 220 230 240 250 260 270 280 290 300−5
−4
−3
−2
−1
0
1
2
3
4
5
t
q
k = 0.1 and λ = 0.2
k = 0.25 and λ = 0.2
|q(ω)|
t ω
q
(a) (b)3
3
3
3
Figure 5-6: A nonlinear system with the combination of a negative linear (ν = 1) anda cubic spring. Blue solid line corresponds to a local minimum of the performance inFig. 5-4: k3 = 0.1 and λ = 0.2. Red dashed line corresponds to a local maximum of theperformance in Fig. 5-4: k3 = 0.25 and λ = 0.2. (a) Response in terms of displacement.(b) Fourier transform modulus |q (ω)|.
In a statistical steady state, we will have the first term vanishing. This is also the
case for the third term, which represents the overall energy contribution from the
conservative spring force. Moreover, the harvested power is equal to the second term
and thus we have
Ph = λq2 = −hq. (5.30)
For the monochromatic case, we have h (t) = −αω20 cosω0t. We represent the arbitrary
statistical steady state response as
q =∑i
qi cos (ωit+ φi) , (5.31)
with qi > 0, and φi are phases determined from the system. From this representation,
we obtain
Ph =∑i
qiαω20ωi lim
T→∞
1
T
T∫0
cosω0t sin (ωit+ φi) dt. (5.32)
The quantity inside the integral will be nonzero only when i = 0. Thus,
78
Ph = q0αω30
ω0
2π
2πω0∫0
cosω0t sin (ω0t+ φ0) dt =1
2q0αω
30 sinφ0. (5.33)
Note that from the representation for q, we obtain
q2 =∑i,j
qiqjcos (ωit+ φi) cos (ωjt+ φj)
=∑i
q2i cos2 (ωit+ φi) =1
2
∑i
q2i . (5.34)
It is straightforward to conclude that for constant response level q2 the harvested
power will become maximum when q0 is maximum, and this is the case only when
all the energy of the response is concentrated in the harmonic ω0, a property that
is guaranteed to occur for the linear systems. Thus, for SDOF harvesters, excited
by monochromatic sources, the optimal linear system can be considered as an upper
bound of the performance among the class of both linear and nonlinear oscillators.
5.5.2 SDOF Harvester under White Noise Excitation
We investigated the monochromatic excitation case of both linear and nonlinear sys-
tems as an extreme case of a narrow-band excitation. The opposite extreme, the one
that corresponds to a broadband excitation, is the Gaussian white noise. We con-
sider a dynamical system governed by a second order differential equation under the
standard Gaussian white noise excitation W (t) with zero mean and intensity equal
to one (i.e. W 2 = 1).
q + λq + F (q) = αW (t). (5.35)
For this SDOF system, the probability density function is fully described by the
Fokker-Planck-Kolmogorov equation which for the statistical steady state can be
solved analytically providing us with the exact statistical response of system (5.35)
79
in terms of the steady state probability density function (see e.g. [39])
pst (q, q) = C exp
(− λ
α2
[q2
2+
∫ q
0
F (x)dx
]), (5.36)
where C is the normalization constant so that
∫∫pst (q, q) dqdq = 1.
In order to use previously developed measures, we define h2 = α2 (the typical
amplitude of the excitation is equal to the intensity of the noise). Moreover, since
there is no characteristic frequency we can choose without loss of generality ω2h = 1.
Using expression (5.36), we can compute an exact expression for the harvested power
PW = λq2 = α2. (5.37)
which is an independent quantity of the system parameters - the above result can be
generalized in MDOF system as shown in [40]. We observe that in this extreme form
of broadband excitation the harvested power is independent on the system parameters
and depends only on the excitation energy level α. In addition, the harvested power
density ρe will be given by
ρe(B) =
maxλ,ki | B
Ph
ω3hq
2=α2
q2=
1q2
h2
=1
B. (5.38)
Similarly with the harvested power, we observe that the harvested power density is
also independent of the employed system design (Figure 5-7). Moreover, when we
compare with the monochromatic excitation case (where we illustrated that the best
possible performance can be achieved with linear systems), we see that the harvested
power density drops faster with respect to the device size B when the energy is spread
(in the spectral sense) compared with the case where energy is localized in a single
input frequency.
80
Figure 5-7: (a) Maximum harvested power, and (b) Power denstity for linear andnonlinear SDOF systems under white noise excitation.
5.5.3 SDOF Harvester under Colored Noise Excitation
The third case of our analysis involves a colored noise excitation, the Pierson-Moskowitz
form (equation 5.21), which can be considered as an intermediate case between the
two extremes presented previously. For a general excitation spectrum, the compu-
tation of the performance measures for the nonlinear systems has to be carried out
numerically. However for the linear system the computation of the mean square
amplitude and the mean rate of energy harvested per unit mass can be computed
analytically [39]
q2(k1, λ
)=
∞∫0
ω4(k1 − ω2
)2+ λ2ω2
1
ω5exp
(−ω−4
)dω, (5.39)
Ph
(k1, λ
)= λ
∞∫0
ω6(k1 − ω2
)2+ λ2ω2
1
ω5exp
(−ω−4
)dω. (5.40)
For the nonlinear systems, we employ a Monte-Carlo method since the computational
cost for simulating the SDOF harvester is very small. In particular, we generate ran-
dom realizations which are consistent with the PM spectrum using a frequency domain
method [41]. The results are presented in Figure 5-8. We can still observe similar
features with the monochromatic excitation even though the variations of response
81
level and performance are now much smoother (compared with the monochromatic
case). For the linear system, we do not have the sharp resonance peak that we had
in the monochromatic case while the two nonlinear designs behave very similarly in
terms of their performance maps. However, the characteristic difference of the neg-
ative stiffness design, related to the persistence of the response level even for large
values of damping, is preserved in this non-monochromatic excitation case. Note that
similarly to the monochromatic case this robustness in the response level does not
necessarily imply strong harvesting power.
A comparison of the linear system and the nonlinear systems under the Pierson-
Moskowitz spectrum excitation is shown in Figure 5-9. As it can be seen from Figure
5-9b, the linear oscillator has the best performance compared to two nonlinear designs
(note that for the negative stiffness oscillator a wide range of values ν was employed
and in all cases the results for the power density were qualitatively the same - to
this end only the case ν = 1 is presented). This is expected for any colored noise
excitation, given that for the monochromatic extreme we have shown rigorously that
the optimal performance of any nonlinear oscillator cannot exceed the optimal linear
design, while for the white noise excitation all designs have identical performance.
An important qualitative difference between the response under the Pierson-
Moskowitz spectrum and the monochromatic excitation is the behavior of the har-
vested power for larger values of B. While for the monochromatic case the harvested
power scales with√B, this is not the case for the colored noise excitation where the
harvested power seems to converge to a finite value (a behavior that is consistent
with the white noise excitation). Therefore, we can conclude that for small values
of response level B the optimal performance under colored noise excitation behaves
similarly with the monochromatic excitation while for larger values of B the optimal
performance seems to be closer to the white-noise response. The above conclusions
are also verified from Figure 5-10 where the three optimal harvested power density
curves (corresponding to the three forms of excitation) are presented together.
82
(a) (b)
Lin
ea
r S
yste
mC
ub
ic S
yste
mN
eg
ative
Stiffn
ess
λλ
λ
BB
B
λ k1 k
1
λ k3 k
3
λ k3 k
3
Figure 5-8: Response level B and power harvested for the case of excitation withPierson-Moskowitz spectrum over different system parameters. The response level B isalso presented as a contour plot in the power harvested plots. All three cases of systemsare shown: linear (top row), cubic (second row), and negative stiffness with ν = 1.
83
Figure 5-9: (a) Maximum harvested power, and (b) Power density for linear andnonlinear SDOF systems under Pierson-Moskowitz spectrum.
0 1 2 3 4 5 6 7 8 9 1010
−1
100
101
102
103
B
ρe
Monochromatic
Colored
White
B
ρe
Figure 5-10: Harvested power density ρe for the three different types of excitationspectra. The linear design is used in all cases since this is the optimal.
84
5.6 Results of the Moment Equation Method
In this section, the results of the moment equation technique are illustrated and com-
pared with the results obtained by Monte Carlo method.
As stated in the previous chapters, we can expect a stationary and ergodic Gaussian
random process for the response only if the SDOF oscillator is linear. In the case
that the SDOF oscillator is nonlinear, such as including the cubic stiffness, there
is no guarantee of having a stationary and ergodic Gaussian random process for the
response. However, throughout this section, the stationary and ergodic Gaussian ran-
dom process for the response is assumed and the Gaussian closure approximation is
applied correspondingly. There are other closing techniques such as cumulant closure
approximation and the results for these methods will be considered for future work.
In the Figure (5-11), the results of the moment equation method for the SDOF nonlin-
ear oscillator with the cubic stiffness under the colored noise excitation are illustrated.
If we compare the moment equation method with the Monte Carlo method, the 3D
surface maps of the size of the device for both methods are very close. However, the
harvested power for the combination of different parameters of the stiffness and the
damping gives a difference that the moment equation method overestimates the har-
vesting power and gives a broader range of high harvesting power range. This can be
obviously observed in the maximum harvested power curve with respect to the size of
the device. Since the moment equation method in this section assumes a stationary
and ergodic Gaussian random process for the response, it overestimates the perfor-
mance of the oscillator. This overestimation is due to the energy transfer to higher
harmonics (because of the nonlinear terms). More accurate closing approximation
methods may enable us to take into account this inherently nonlinear behavior that
cannot be captured in the context of the Gaussian closure.
The results of the moment equation method for the SDOF nonlinear oscillator with
85
0 1 2 3 4 5 6 7 8 9 1010
−1
100
101
102
B
ρe
Linear
Cubic (ODE45)
Cubic (NEW)
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
B
Ph
Linear
Cubic (ODE45)
Cubic (New)
B
λ k3 k
3
λ
Ph
B B
ρe
(a) (b)
(c) (d)
Figure 5-11: Results of the moment equation method for the cubic system under thecolored noise excitation. (a) The size of the device with respect to system parameters.(b) The harvested power with respect to system parameters. (c) Maximum HarvestedPower. (d) Harvested Power Density.
86
the cubic and negative linear stiffness are presented in the Figure (5-12). On the
contrary to the results of the cubic oscillator, it can be observed that the 3D surface
map of the size of the device for the moment equation method dramatically differs
from the Monte Carlo method at small stiffness coefficients. This is also the case
where the response of the system is not a stationary and ergodic Gaussian random
process. Overestimation in the harvested power can be observed as well. The maxi-
mum harvested power curve gives even more deviated result and this indicates that
more accurate closing method is required.
0 1 2 3 4 5 6 7 8 9 1010
−1
100
101
102
B
ρe
Linear
Nega (ODE45)
Nega (NEW)
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
B
Ph
Linear
Nega (ODE45)
Nega (New)
B
λ k3 k
3
λ
Ph
B B
ρe
(a) (b)
(c) (d)
Figure 5-12: Results of the moment equation method for the negative linear systemunder the colored excitation. (a) The size of the device with respect to system pa-rameters. (b) The harvested power with respect to system parameters. (c) MaximumHarvested Power. (d) Harvested Power Density.
87
88
Chapter 6
Performance Robustness
We have examined the optimal performance for different designs of SDOF harvesters
under various forms of random excitations. Even though the linear design has the
optimal performance for fixed response level B, the robustness of this performance
under perturbations of the input spectrum characteristics (and with fixed optimal
system parameters) has not been considered. This is the scope of this chapter where
we investigate how linear and nonlinear systems with optimal system parameters
behave when the excitation spectrum is perturbed.
More specifically, we are interested to investigate robustness properties with re-
spect to frequency shifts of the excitation spectrum. Clearly, the harvested power and
the response level (that characterizes the size of the device) will be affected by the
spectrum shift. To quantify these variations we consider the following three ratios
δ =Bshifted
B0
, τ =(Ph)shifted
(Ph)0, σ =
(ρe)shifted(ρe)0
, (6.1)
where δ quantifies the variation of the response level B0 which essentially expresses
the size of the device, τ quantifies exclusively the changes in performance while σ
shows the changes in harvested power density, i.e. it also takes into account the
variations of the response level B.
89
Monochromatic excitation. For the monochromatic excitation, perturbation
in terms of spectrum shift can be expressed as
Shh(ω − ε) = δ(ω − ω0 − ε), (6.2)
where ω0 = 1. In Figure 6-1, we present the ratios describing the variation of the
response level δ, the harvested power τ and the harvested power density σ in terms
of perturbation ε for various levels of the unperturbed response level B. For small
response levels, i.e. when the system response is smaller than the excitation (B =0.5)
we observe that the negative stiffness oscillator has more robustness to maintaining its
response level when it is excited by lower frequencies (ε < 0). For the same case, the
harvested power decays in a similar fashion with the other two oscillators. Therefore,
for ε < 0 and B =0.5 the nonlinear oscillator with negative stiffness has the most
robust performance. For faster excitations (ε > 0) we observe that all oscillators drop
their response level in smaller values than the design response level B0 with the linear
system having the most robust behavior in terms of the total harvested power. We
emphasize that as long as δ < 1 robustness is essentially defined by the largest value
of τ among different types of oscillators.
For B =1, we can observe that for all values of ε the negative stiffness oscillator
has the most robust behavior in terms of the excitation level while the behavior of
the harvested power is also better compared with the other two classes of oscillators.
For larger values of the response level (B =8), we note that the response level ratio δ
is maintained in levels below 1; therefore the size of the device will not be exceeded
due to input spectrum shifts. On the other hand when we consider the variations
of the harvested power, we observe that all in all the linear oscillators has the most
robust behavior, while the two linear oscillators drop suddenly their performance to
very small levels for larger, positive values of ε.
90
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
ǫ
Ratio
δ
Linear
Only Cubic
Negative Stiffness
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.2
0.4
0.6
0.8
1
1.2
1.4
ǫ
Ratio
γ
Linear
Only Cubic
Negative Stiffness
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
3
3.5
4
ǫ
Ratio
σ
Linear
Only Cubic
Negative Stiffness
ε
(a) (b) (c)
ε ε
Ra
tio
δ
Ra
tio
τ
Ra
tio
σ
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
ǫ
Ratio
δ
Linear
Only Cubic
Negative Stiffness
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
3
3.5
4
ǫ
Ratio
σ
Linear
Only Cubic
Negative Stiffness
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
ǫ
Ratio
γ
Linear
Only Cubic
Negative Stiffness
ε
(a) (b) (c)
ε ε
Ra
tio
δ
Ra
tio
τ
Ra
tio
σ
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.2
0.4
0.6
0.8
1
1.2
1.4
ǫ
Ratio
δ
Linear
Only Cubic
Negative Stiffness
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.2
0.4
0.6
0.8
1
1.2
1.4
ǫ
Ratio
γ
Linear
Only Cubic
Negative Stiffness
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
3
3.5
4
ǫ
Ratio
σ
Linear
Only Cubic
Negative Stiffness
ε ε ε
Ra
tio
δ
Ra
tio
τ
Ra
tio
σ
B =
0.5
B =
1B
= 8
(a) (b) (c)
Figure 6-1: Robustness of (a) the response level, (b) the power harvested, and (c)the harvested power density for the monochromatic excitation under three regimes ofoperation: B = 0.5, B = 1, and B = 8.
91
Colored noise excitation. Similarly with the monochromatic case, we consider
a small perturbation ε for the colored noise excitation spectrum:
Shh(ω − ε) =1
(ω − ε)5exp
(−(ω − ε)−4
). (6.3)
The results are presented in Figure 6-2 for three different cases of unperturbed exci-
tation levels B0. In contrast to the monochromatic case, the ratios δ, τ, and σ have
much smoother dependence on the perturbation ε. Moreover, their variation is very
similar for all three response levels B0. More specifically, we can clearly see that the
two classes of nonlinear oscillators can better maintain their response level over all
values of ε. On the other hand, the linear oscillator obtains a larger response level
B when the spectrum is shifted to the right (ε > 0) without substantially increasing
the harvested power compared with the other two nonlinear oscillators. For ε < 0,
all three families of oscillators harvest the same amount of energy. Thus, for colored
noise excitation, the two families of nonlinear oscillators achieve the most robust per-
formance. Hence, as long as the nonlinear design is chosen so that it has comparable
optimal performance with the family of linear oscillators, it is the preferable choice
since it has the best robustness properties.
92
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
3
3.5
ǫ
Ratio
δ
Linear
Only Cubic
Negative Stiffness
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
3
3.5
ǫ
Ratio
γ
Linear
Only Cubic
Negative Stiffness
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
1
2
3
4
5
6
ǫ
Ratio
σ
Linear
Only Cubic
Negative Stiffness
ε
(a) (b) (c)
ε ε
Ra
tio
δ
Ra
tio
τ
Ra
tio
σ
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
3
3.5
ǫ
Ratio
δ
Linear
Only Cubic
Negative Stiffness
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
3
3.5
ǫ
Ratio
γ
Linear
Only Cubic
Negative Stiffness
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
1
2
3
4
5
6
ǫRatio
σ
Linear
Only Cubic
Negative Stiffness
ε
(a) (b) (c)
ε ε
Ra
tio
δ
Ra
tio
τ
Ra
tio
σ
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
3
3.5
ǫ
Ratio
δ
Linear
Only Cubic
Negative Stiffness
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
3
3.5
ǫ
Ratio
γ
Linear
Only Cubic
Negative Stiffness
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
ǫ
Ratio
σ
Linear
Only Cubic
Negative Stiffness
ε ε ε
Ra
tio
δ
Ra
tio
τ
Ra
tio
σ
B =
0.5
B =
1B
= 8
(a) (b) (c)
Figure 6-2: Robustness of (a) the response level, (b) the power harvested, and (c)the harvested power density for the PM spectrum excitation under three regimes ofoperation: B = 0.5, B = 1, and B = 8.
93
94
Chapter 7
Conclusions and Future Work
We have considered the problem of energy harvesting using SDOF oscillators. We
first developed objective measures that quantify the performance of general nonlinear
systems from broadband spectra, i.e. simultaneous excitation from a broad range
of harmonics. These measures explicitly take into account the required size of the
device in order to achieve this performance. We demonstrated that these measures
do not depend on the magnitude or the temporal scale of the input spectrum but
only the relative distribution of energy among different harmonics. In addition they
are suitable to compare whole classes of oscillators since they always pick the most
effective parameter configuration.
Using analytical and numerical tools, we applied the developed measures to quan-
tify the performance of three different families of oscillators (linear, essentially cubic,
and negative stiffness or bistable) for three different types of excitation spectra: an
extreme form of a narrow band excitation (monochromatic excitation), an extreme
form of a wide-band excitation (white-noise), and an intermediate case involving col-
ored noise (Pierson-Moskowitz spectrum). For all three cases, we presented numerical
and analytical arguments that the nonlinear oscillators can achieve in the best case
equal performance with the optimal linear oscillator, given that the size of the device
does not change. We also considered the robustness of each design to input spectrum
shifts concluding that the nonlinear oscillator has the best behavior for the colored
noise excitation. To this end, we concluded that, under a situation of designing a har-
95
vester with specific power, a nonlinear oscillator designed to achieve a performance
that is close to the optimal performance of a linear oscillator is the best choice since
it also has robustness against small perturbations.
Future work involves the generalization of the presented criteria to MDOF os-
cillators and the study of the benefits due to nonlinear energy transfers between
modes [42–45]. Preliminary results indicate that the application of nonlinear energy
transfer ideas can have a significant impact on achieving higher harvested power den-
sity by distributing energy to more than one modes achieving in this way smaller
required device size without reducing its performance level.
96
Bibliography
[1] D. Evans, “Wave-power absorption by systems of oscillating surface pressuredistributions,” Journal of Fluid Mechanics, vol. 114, 1982.
[2] A. J. Sarmento and A. d. O. Falcao, “Wave generation by an oscillating surface-pressure and its application in wave-energy extraction,” Journal of Fluid Me-chanics, vol. 150, 1985.
[3] J. Falnes, Ocean waves and oscillating systems. 2002.
[4] R. Clare, D. Evans, and T. Shaw, “Harnessing sea wave energy by a submergedcylinder device.,” vol. 73, 1982.
[5] D. Pizer, “Maximum wave-power absorption of point absorbers under motionconstraints,” Applied Ocean Research, vol. 15, no. 4, 1993.
[6] M. F. Daqaq, “Response of uni-modal duffing-type harvesters to random forcedexcitations,” Journal of Sound and Vibration, vol. 329, p. 3621, 2010.
[7] M. F. Daqaq, “Transduction of a bistable inductive generator driven by whiteand exponentially correlated gaussian noise,” Journal of Sound and Vibration,vol. 330, no. 11, pp. 2554–2564, 2011.
[8] P. Green, K. Worden, K. Atallah, and N. Sims, “The benefits of duffing-typenonlinearities and electrical optimisation of a mono-stable energy harvester underwhite gaussian excitations,” Journal of Sound and Vibration, vol. 331, no. 20,pp. 4504–4517, 2012.
[9] N. Stephen, “On energy harvesting from ambient vibration,” Journal of Soundand Vibration, vol. 293, no. 1, pp. 409–425, 2006.
[10] E. Halvorsen, “Fundamental issues in nonlinear wideband-vibration energy har-vesting,” Physical Review E, vol. 87, no. 4, p. 042129, 2013.
[11] D. A. Barton, S. G. Burrow, and L. R. Clare, “Energy harvesting from vibrationswith a nonlinear oscillator,” Transactions of the ASME-L-Journal of Vibrationand Acoustics, vol. 132, no. 2, p. 021009, 2010.
[12] P. L. Green, E. Papatheou, and N. D. Sims, “Energy harvesting from human mo-tion and bridge vibrations: An evaluation of current nonlinear energy harvestingsolutions,” Journal of Intelligent Material Systems and Structures, 2013.
97
[13] R. Harne and K. Wang, “A review of the recent research on vibration energyharvesting via bistable systems,” Smart Materials and Structures, vol. 22, no. 2,p. 023001, 2013.
[14] V. Mendez, D. Campos, and W. Horsthemke, “Stationary energy probabilitydensity of oscillators driven by a random external force,” Physical Review E,vol. 87, p. 062132, 2013.
[15] M. Brennan, B. Tang, G. P. Melo, and V. Lopes Jr, “An investigation into thesimultaneous use of a resonator as an energy harvester and a vibration absorber,”Journal of Sound and Vibration, vol. 333, no. 5, 2014.
[16] D. Watt and M. Cartmell, “An externally loaded parametric oscillator,” Journalof Sound and Vibration, vol. 170, no. 3, 1994.
[17] C. McInnes, D. Gorman, and M. Cartmell, “Enhanced vibrational energy har-vesting using nonlinear stochastic resonance,” Journal of Sound and Vibration,vol. 318, no. 4, 2008.
[18] B. Mann and N. Sims, “Energy harvesting from the nonlinear oscillations ofmagnetic levitation,” Journal of Sound and Vibration, vol. 319, no. 1, 2009.
[19] V. Mendez, D. Campos, and W. Horsthemke, “Efficiency of harvesting energyfrom colored noise by linear oscillators,” Physical Review E, vol. 88, no. 2,p. 022124, 2013.
[20] D. P. Bertsekas, Introduction to Probability: Dimitri P. Bertsekas and John N.Tsitsiklis. 2002.
[21] T. T. Soong and M. Grigoriu, “Random vibration of mechanical and structuralsystems,” NASA STI/Recon Technical Report A, vol. 93, 1993.
[22] A. F. d. O. Falcao, “Wave energy utilization: A review of the technologies,”Renewable and sustainable energy reviews, vol. 14, no. 3, pp. 899–918, 2010.
[23] A. Clement, P. McCullen, A. Falcao, A. Fiorentino, F. Gardner, K. Hammar-lund, G. Lemonis, T. Lewis, K. Nielsen, S. Petroncini, et al., “Wave energyin europe: current status and perspectives,” Renewable and sustainable energyreviews, vol. 6, no. 5, pp. 405–431, 2002.
[24] J. Falnes, “A review of wave-energy extraction,” Marine Structures, vol. 20, no. 4,pp. 185–201, 2007.
[25] W. J. Pierson and L. Moskowitz, “A proposed spectral form for fully developedwind seas based on the similarity theory of sa kitaigorodskii,” Journal of geo-physical research, vol. 69, no. 24, pp. 5181–5190, 1964.
[26] M. K. Ochi and E. N. Hubble, “Six-parameter wave spectra,” Coastal Engineer-ing Proceedings, vol. 1, no. 15, 1976.
98
[27] K. Hasselmann, T. Barnett, E. Bouws, H. Carlson, D. Cartwright, K. Enke,J. Ewing, H. Gienapp, D. Hasselmann, P. Kruseman, et al., “Measurementsof wind-wave growth and swell decay during the joint north sea wave project(jonswap),” 1973.
[28] K. B. Dysthe, K. Trulsen, H. E. Krogstad, and H. Socquet-Juglard, “Evolutionof a narrow-band spectrum of random surface gravity waves,” Journal of FluidMechanics, vol. 478, pp. 1–10, 2003.
[29] A. V. Oppenheim, R. W. Schafer, J. R. Buck, et al., Discrete-time signal pro-cessing, vol. 2. Prentice-hall Englewood Cliffs, 1989.
[30] R. C. Booton, M. V. Mathews, and W. W. Seifert, Nonlinear Servomechanismswith Random Inputs. MIT Dynamic Analysis and Control Laboratory, 1953.
[31] R. C. Booton, “Nonlinear control systems with random inputs,” Circuit Theory,IRE Transactions on, vol. 1, no. 1, pp. 9–18, 1954.
[32] I. Kazakov, “Approximate method for the statistical analysis of nonlinear sys-tems,” Rep. of Zhukowskii Institute, no. 394, p. 56, 1954.
[33] I. Kazakov, “Approximate probability analysis of the operational precision of es-sentially nonlinear feedback control systems,” Automation and Remote Control,vol. 17, p. 423, 1956.
[34] R. L. Stratonovich, Topics in the theory of random noise, vol. 2. CRC Press,1967.
[35] J. J. Stoker, Nonlinear vibrations. 1966.
[36] S. H. Crandall, “Perturbation techniques for random vibration of nonlinearsystems,” The Journal of the Acoustical Society of America, vol. 35, no. 11,pp. 1700–1705, 2005.
[37] M. Shinozuka, “Monte carlo solution of structural dynamics,” Computers &Structures, vol. 2, no. 5, pp. 855–874, 1972.
[38] R. Y. Rubinstein and D. P. Kroese, Simulation and the Monte Carlo method,vol. 707. John Wiley & Sons, 2011.
[39] K. Sobczyk, Stochastic Differential Equations. Dordrecht, The Netherlands:Kluwer Academic Publishers, 1991.
[40] R. Langley, “A general mass law for broadband energy harvesting,” Journal ofSound and Vibration, In Press.
[41] D. B. Percival, “Simulating gaussian random processes with specified spectra,”Computing Science and Statistics, vol. 24, pp. 534–538, 1992.
99
[42] T. P. Sapsis, A. F. Vakakis, O. V. Gendelman, L. A. Bergman, G. Kerschen,and D. D. Quinn, “Efficiency of targeted energy transfer in coupled oscillatorsassociated with 1:1 resonance captures: Part II, analytical study,” Journal ofSound and Vibration, vol. 325, pp. 297–320, 2009.
[43] T. P. Sapsis, A. F. Vakakis, and L. A. Bergman, “Effect of stochasticity on tar-geted energy transfer from a linear medium to a strongly nonlinear attachment,”Probabilistic Engineering Mechanics, vol. 26, pp. 119–133, 2011.
[44] T. P. Sapsis, D. D. Quinn, A. F. Vakakis, and L. A. Bergman, “Effective stiff-ening and damping enhancement of structures with strongly nonlinear local at-tachments,” ASME J. Vibration and Acoustics, vol. 134, p. 011016, 2012.
[45] A. F. Vakakis, O. V. Gendelman, L. A. Bergman, D. M. McFarland, G. Kerschen,and Y. S. Lee, Nonlinear Targeted Energy Transfer in Mechanical and StructuralSystems. Springer-Verlag, 2008.
100
