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Time-frequency analysis of the relationship between system response and excitation characteristics

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Title:
Time-frequency analysis of the relationship between system response and excitation characteristics
Creator:
Jakubowski, Andrzej
Publication Date:
Language:
English
Physical Description:
viii, 180 leaves : illustrations ; 28 cm

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Subjects / Keywords:
Signal processing -- Mathematics ( lcsh )
Wind turbines -- Testing ( lcsh )
Wavelets (Mathematics) ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaves 179-180).
General Note:
Department of Electrical Engineering
Statement of Responsibility:
by Andrzej Jakubowski.

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Source Institution:
|University of Colorado Denver
Holding Location:
|Auraria Library
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
42611989 ( OCLC )
ocm42611989
Classification:
LD1190.E54 1999m .J33 ( lcc )

Full Text
TIME-FREQUENCY ANALYSIS OF THE RELATIONSHIP BETWEEN
SYSTEM RESPONSE AND EXCITATION CHARACTERISTICS
by
Andrzej Jakubowski
M.S., Technical University of Szczecin, 1987
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineeering
1999


This thesis for the Master of Science
degree by
Andrzej Jakubowski
has been approved
by
Jan T. Bialasiewicz
Tamal Bose
Miloje S. Radenkovic
Date


Jakubowski, Andrzej (M.S., Electrical Engineering)
Time-Frequency Analysis of the Relationship Between System Response and
Excitation Characteristics.
Thesis directed by Professor Jan T. Bialasiewicz
ABSTRACT
This study identifies and evaluates feasibility of several time-frequency analysis
tools applied during the investigation of bending loads on the blades of the wind
turbine.
Experimental data were obtained at National Wind Technology Center in Golden,
Colorado. Time series were generated during real time runs on Micon 65 upwind
turbine and Cannon Wind Eagle 300 downwind turbine, simulated data were
generated for Wind Eagle turbine.
The study begins with brief introduction to modem time-frequency analysis tools
like spectrograms, scalograms, wavelets and reassigned spectrograms. Wavelet
based algorithm allowing for fast detection of rapid variations of the bending
loads is presented.
GUI based program was developed, which provides fast view of large data sets,
finds singular features in data sequences, saves selected areas for further analysis
in different formats for Matlab 4 and 5, and generates spectrograms, scalograms,
multiresolution decompositions and reassigned spectrograms of selected signals.
Results of the experiments with different tools are described and compared, and
usefulness of algorithms is accessed.
This abstract accurately represents the content of the candidate's thesis. I
recommend its publication.
Signed
m
Jan T. Bialasiewicz


CONTENTS
Chapter
1. Introduction............................................................1
2. Time Representation and Frequency Representation........................3
2.1 Wavelet Transform......................................................7
2.2 Discrete Wavelet Transform.............................................8
2.3 Localization and Heisenberg-Gabor Principle...........................13
3. Spectrograms and Scalograms............................................15
3.1 Spectrogram...........................................................15
3.2 Scalogram.............................................................17
3.3 Wavelet Power Spectrum................................................18
3.4 The Reassignment Method...............................................20
4. Detection of Singularities.............................................24
5. Experimental Data......................................................30
5.1 Micon 65 Upwind Turbine...............................................34
5.2 Cannon Wind Eagle 300 Downwind Turbine................................36
5.3 Wind Eagle 300 Turbine Adams Simulation...............................38
6. Software Package.......................................................40
6.1 Time Frequency Analysis Software for Matlab ver. 4....................40
6.2 Time-Frequency Analysis Software for Matlab ver. 5...................40
IV


6.3 List of M-scripts and Description of the Task Performed..............42
6.4 New Wavelets WAVE and RAMP.........................................44
7. Conclusion............................................................45
Appendix
A. Micon 65 Upwind Turbine..............................................47
B. Cannon Wind Eagle 300 Downwind Turbine Low Wind Speed..............73
C. Cannon Wind Eagle 300 Downwind Turbine High Wind Speed.............90
D. Cannon Wind Eagle Adams Simulations Low Wind Speed................109
E. Cannon Wind Eagle Adams Simulations High Wind Speed...............130
F. Matlab Scripts......................................................154
References
179


FIGURES
Figures
2.1 Filter Bank...........................................................10
A. 1 Flap wise Bending Load Blade #1 Micon 65............................48
A. 2 Flap wise Bending Load Blade #2 Micon 65............................49
A. 3 Edgewise Bending Load Blade #1- Micon 65..............................50
A. 4 Edgewise Bending Load Blade #2 Micon 65.............................51
A.5 Multiresolution Decomposition Flapwise Bending Load Blade #1............52
A.6 Multiresolution Decomposition Edgewise Bending Load Blade #1..........53
A.7 Detection of Pseudo-Singularities Flapwise Bending Load Blade #1
flpllOl.mat............................................................54
A.8 Wavelet Power Spectrum Flapwise Bending Load Blade #1
flpl 101.mat...........................................................55
A. 9 Detection of Pseudo-Singularities Flapwise Bending Load Blade #1 -
flpl 102.mat...........................................................56
A. 10 Spectrogram flpl 102.mat............................................57
A. 11 Reassigned Spectrogram flpl 102.mat.................................58
A. 12 Detection of Pseudo-Singularities Flapwise Bending Load Blade #1
flpl 104.mat...........................................................59
A. 13 Reassigned Spectrogram flp 1104.mat.................................60
A. 14 Detection of Pseudo-Singularities Edgewise Bending Load Blade #1
edgel lOl.mat..........................................................61
A. 15 Wavelet Power Spectrum edgl 101.mat.................................62
A. 16 Detection of Pseudo-Singularities Edgewise Bending Load Blade #1
edgel 104.mat..........................................................63
A. 17 Reassigned Spectrogram edgl 104.mat..................................64
A. 18 Detection of Pseudo-Singularities Edgewise Bending Load Blade #1
edgel 105.mat..........................................................65
A. 19 1-D Continuous Wavelet Transform edgl 105.mat MHAT..................66
A.20 Reassigned Spectrogram edgl 165.mat.................................67
A.21 1-D Continuous Wavelet Transform edgl 105.mat WAVE..................68
A.22 1-D Continuous Wavelet Transform edgl 105.mat RAMP..................69
A.23 1-D Continuous Wavelet Transform edgl 105.mat Morlet................70
A. 24 RAMP Wavelet..........................................................71
A. 25 WAVE Wavelet..........................................................72
B. 1 Flapwise Bending Load Flex Beam Blade #A CWE 300................... 74
B.2 Multiresolution Decomposition fafb_dl.mat.............................75
VI


B.3 Detection of Pseudo-Singularities fafb_dl .mat.........................76
B.4 Spectrogram fafb_dl .mat...............................................77
B.5 1-D Continuous Wavelet Transform fafb_dl.mat Morlet................78
B.6 Wavelet Power Spectrum fafb_dl.mat.....................................79
B.7 Flapwise Bending Load Blade Shell Blade #A CWE 300.....................80
B.8 Multiresolution Decomposition Blade Shell bafb dl.mat..................81
B.9 Detection of Pseudo-Singularities bafb_dl.mat..........................82
B.10 Spectrogram bafb_dl.mat...............................................83
B. 11 1-D Continuous Wavelet Transform bafb dl .mat Morlet..............84
B.12 Spectrogrambafb_dl.mat................................................85
B. 13 Wavelet Power Spectrum bafb_dl.mat..................................86
B. 14 Edgewise Bending Load Flex Beam Blade #A CWE 300................. 87
B.15 Multiresolution Decomposition faeb_dl.mat.............................88
B. 16 Wavelet Power Spectrum faeb_dl.mat MHAT.............................89
C. 1 Flapwise Bending Load Flex Beam Blade #A.............................91
C.2 Multiresolution Decomposition Blade Shell fafb_dl4.mat.................92
C.3 Spectrogram fafb_dl4.mat...............................................93
C.4 Signal fafb_dl4.mat Selected Detail..................................94
C.5 1-D Continuous Wavelet Transform fafb_dl4.mat Morlet...............95
C.6 1-D Continuous Wavelet Transform fafb_dl4.mat WAVE.................96
C.7 Wavelet Power Spectrum fafb_dl4.mat....................................97
C.8 Flapwise Bending Load Blade Shell Blade #A bafb_dl4.mat................98
C.9 Multiresolution Decomposition bafb_dl4.mat.............................99
C.10 Signal bafb_dl4.mat Selected Detail................................100
C.ll Spectrogrambafb_dl4.mat..............................................101
C.12 1-D Continuous Wavelet Transform bafb_dl4.mat Morlet...............102
C.13 Wavelet Power Spectrum bafb_dl4.mat..................................103
C.14 Edgewise Bending Load Flex Beam Blade #A CWE 300.....................104
C.15 Multiresolution Decomposition faeb_dl4.mat...........................105
C.16 Signal faeb_dl4.mat Selected Detail................................106
C.17 Spectrogram faeb d 14.mat..........................................107
C. 18 1-D Continuous Wavelet Transform faeb_d!4.mat Morlet.............108
D. 1 Root Flapwise Load Blade # A FLAP-A...............................110
D.2 Spectrum View FLAP-A..................................................Ill
D.3 Spectrogram FLAP-A....................................................112
D.4 Root Edgewise Load Blade #A EDGE-A Selected Detail................113
D.5 Spectrum View EDGE-A..................................................114
D.6 Wavelet Power Spectrum EDGE-A Paul Wavelet..........................115
D.7 Root Torsion Load Blade #A TORS-A Selected Detail...................116
D. 8 Spectrum View TORS-A................................................117
D.9 Spectrogram TORS-A....................................................118
vii


D.10 Normal Force Span Station 12 Blade #ANF-12..........................119
D.ll Spectrum ViewNF-12..................................................120
D.12 Spectrogram NF-12...................................................121
D. 13 Wavelet Power Spectrum NF-12......................................122
D. 14 Tangential Force Span Station 12 Blade #A TF-12...................123
D.15 Spectrum View TF-12.................................................124
D. 16 Reassigned Spectrogram TF-12......................................125
D.17 UV Reynolds Stress Component Station 12 Blade #A UV-12..............126
D. 18 Spectrum View UV-12...............................................127
D.19 Spectrogram UV-12...................................................128
D. 20 Wavelet Power Spectrum UV-12 Paul Wavelet.........................129
E. 1 Root Flapwise Load Blade # A FLAP-A..............................131
E.2 Spectrum View FLAP-A.................................................132
E.3 Root Flapwise Load Blade # A FLAP-A Selected Detail..............133
E.4 Spectrogram FLAP-A.................................................134
E.5 Reassigned Spectrogram FLAP-A........................................135
E.6 Root Edgewise Load Blade #A EDGE-A..................................136
E.7 Spectrum View EDGE-A.................................................137
E.8 Root Edgewise Load Blade #A EDGE-A Selected Detail...................138
E.9 Spectrogram EDGE-A.................................................139
E.10 Root Torsion Load Blade #A TORS-A...................................140
E.l 1 Root Torsion Load Blade #A TORS-A Selected Detail................141
E.12 Spectrogram TORS-A..................................................142
E.13 Normal Force Span Station 13 Blade #A NF-13.........................143
E.14 Signal NF-13 Selected Detail........................................144
E.l5 Spectrogram NF-13...................................................145
E.l6 Reassigned Spectrogram NF-13........................................146
E.l7 Tangential Force Span Station 15 Blade #A TF-15.....................147
E.l8 Signal TF-15 Selected Detail........................................148
E.19 Spectrogram TF-15...................................................149
E.20 UV Reynolds Stress Component Blade #A Station 13....................150
E.21 Signal UV-13 Selected Detail........................................151
E.22 Spectrogram UV-13...................................................152
E. 23 Wavelet Power Spectrum UV-13 Morlet Wavelet.......................153
F. 1 Pseudo-Singularities Detector Custom GUI.........................155
F.2 Wavelet Power Spectrum Custom GUI..................................156
viii


1. Introduction
We live in the world filled with signals. A signal can be regarded as a physical carrier
of some information. Sources of signals are diverse, they span over acoustical,
optical, mechanical, electromagnetic and other domains. Usually, the most important
and interesting information is carried by the rapid transients and movements in a
signal, while stationary component often represents noise. Classical signal processing
based upon indisputable supremacy of the Fourier transform directed most of its
efforts towards time-invariant and space-invariant operators, working well with
stationary signals, but failing in the nonstationary domain.
Emergence of the time-frequency analysis has its roots in the inability of the
traditional methods to efficiently describe transient events like bursts, drifts, ruptures
or modulations.
Turbulent air flow caused by differences in the atmospheric pressure, i.e., wind,
excites oscillations in the turning blades of the wind turbine. Bending loads resulting
from those oscillations are good example of the nonstationary signals. Fourier
transform applied to such a signal will reveal frequencies present in the oscillation,
without taking into account changes in the bending moments resulting from wind
speed fluctuation, turbulent air pockets appearing at random in front of the blades or
tower shadow effect. All the rapid variations of the bending loads will be averaged by
Fourier transform, thus potentially harmfull transients will not be noticed affecting
structural integrity of the blade.
In the study we will describe general concepts of time and frequency and the
limitation resulting from Heisenberg-Gabor rule. Short-time Fourier transform,
discrete and continuous wavelet transform, Wigner-Ville distribution will be
described and compared. Those tools will allow us to obtain time-frequency
representations of selected time series of the bending loads measured during the
experiments at National Wind Technology Center.
Measurements of the bending loads generate huge amount of data, which has to be
analyzed in order to identify rapid signal changes. One approach would be to review
the data manually and identify interesting points, then the time-frequency
representations could be produced for selected areas of the signal. Generation of time-
frequency representations like spectrograms, scalograms or reassigned spectrograms
for long records is computationally expensive. Discrete wavelet transform (DWT)
generates multiresolution decomposition even for the long records (10 minutes of
data, 200 Hz sampling frequency = 120,000 samples) relatively fast.
We developed DWT based algorithm allowing for fast finding of samples of the
signal with values changing rapidly in the relation to their immediate neighbors.
l


Algorithm is described in the detail in the chapter 4, where theoretical background
and examples are presented.
In order to speed up the processing of long data vectors, GUI based tool was
developed, chapter 6 contains description of the major components of the developed
software, source code is included in appendix F. Chapter 5 contains description of
experimental data and results of applied time-frequency tools, plots are included in
the appendices A to E.
2


2. Time Representation and Frequency Representation
The time representation is the most natural description of signal, since the majority of
physical signals are obtained from various recording devices, which register variation
with time. The frequency representation, obtained by Fourier transform
oo
X(v)= jx(t)e-J2Th1dt (2.1)
is a very powerful way to describe a signal, since a lot of physical events considered
in different domains (physics, astronomy, biology etc.) can be classified as periodic.
However, closer look at the spectrum X(v) reveals that original signal x(t) was simply
expanded into family of infinite waves exp{j27ivt} completely unlocalized in time.
Computation of one frequency value X(v) requires knowledge of the full record of
the signal ranging from minus infinity to plus infinity.
The Fourier representation is applicable to square integrable functions i.e. function
f(x) belongs to the square-integrable function space:
h
\f2(x)dx< oo (2.2)
a
According to Fourier any function / e lJ\-n n\ can be expressed as infinite sum:
1 "
/(x) = a0 +^ak cos(kx) + bi sin(fcc) (2.3)
2 *=i
Equality 2.3 should be understood in the following sense:
1 CC 2
/(x)-( a0 +^ak cos(kx) + bk sin(fcc)) dx = 0
k= 1
It's possible that function f and its Fourier representation differ in few points, for
example discontinuity points.
Equation 2.3 shows that value ffx) of the signal at one instant x can be understood as
infinite superposition of completely nonlocal waves, mathematical representation
might reveal true properties of a signal in certain cases, but it can easily distort the
3


real nature of the analyzed signal. If signal has transient components vanishing
outside certain time intervals, Fourier representation of the signal will not detect it.
Necessity to describe a signal both in time and frequency brings into light local
quantity that gives the meaning to an "instantaneous" spectral content.
In order to define an "instantaneous frequency" we will introduce first the concept of
analytic signal. For any real valued signal x(t), we assign a complex valued signal
xa(t) defined as:
xa(t)=x{t) + jHT(x(t))
where HT(x) is the Hilbert transform of x. The definition can be interpreted in the
frequency domain since Xa is single-sided Fourier transform where the negative
frequency values have been removed, the strictly positive ones have been doubled
,and the DC component was not changed:
*.(v) = 0 if vcO
Xa(v) = X(0) f v' = 0
Xa(v) = 2X(v) if v > 0
Analytic signal can be obtained from the real signal by forcing to zero its spectrum
for negative frequencies. Instantaneous amplitude and instantaneous frequency can be
defined as follows:
(0 = |x(')|
^^arg^CO
JK) 2n dt
The instantaneous frequency describes a local frequency behavior as a function of
time. Group delay is defined as a local time behavior as function of frequency:
* 2 n dv
Function can be well approximated by finite sum with upper summation limit index
N:
1 A
SN(x) = a0 + ^ak cos(&x) + bk sin( kx)
k= 1
(2.4)
4


Set of functions {sin(k),cos(k), k=l,2,...} together with constant function forms a
basis for function space j}[-n
Important property of the Fourier basis is its orthogonality, which comes immediately
from properties inherent in the sine and cosine functions.
(2.5)
(srn(m),sin(/2)) = J sin(mx)cos(/2x) J m^n
[tt m = n
f m n
= n m = n> 0
2 n o ii ll 5
n
(sin(w*),cos()) = | sm(/nx)cos(Hx)dx- = 0 for m,n>0
Sine and cosine functions can be easily modified so they will yield on orthonormal
basis, i.e. sequence of functions {f,} are orthonormal if the f's are pairwise
orthogonal and ||fi||=l for all i.
If we define gk(x)=7i*1/2sin(kx) and hk(x)=7t1/2cos(kx) with constant function :
M*) =
on xe [-n 7r]
It will make the set of functions orthonormal as well and f(x):
/(*) = (/, K )K (*) + Z £a (*)+ (/. K )K (x)
k =1
For the discrete sequence x(n) Fourier transform will take the form:
where x(n)~
^-\X{eJa,)eJ(md(o
It
(2.6)
5


X(et0) is given by
X(e*)=
(2.7)
If x(n) is single frequency signal i.e. x(n)=e,on then we have :
X(eja>) = 2 ndo((o-Q)0) Q In such a case transform is completely localized at Wo while time domain plot of
e)on is infinite. If the sequence x(n) is generated by sampling a continuous time
function x(t) at instants nT.
It's easy to derive the following:
1 "
(2.9)
n--oo * A*-oo *
For discrete-time sequence, which is periodic with period N i.e. x(n)=x(n+lN), we
have discrete-time Fourier series:
X(*) = f>()IF;* keZ n=0 (2.10)
Z-rw* N k=0 (2.11)
where Wn is the Nth root of unity.
Discrete Fourier transform can be interpreted as the transform of one period of
periodic signal, or a sampling of the DTFT of finite-length signal. Fourier transform
uses as basis functions trigonometric functions, which are nor compactly supported,
so the local transient event is not captured easily.
In order to achieve greater localization, windowed Fourier transform can be defined
as follows:
STFTf(o,T)= \g'{t-T)x(t)e-JaMds (2.12)
which allows greater time-frequency localization. Windowed Fourier transform
can be expressed in discrete version:
6


T (*) = | *(s)g(s ~ nto ye^ds
(2.13)
Equation 2.13 is derived from 2.12 by assigning t=nto and co^mcoo
Function g in both equations should have compact support and be "smooth".
In such situation Short-Time Fourier Transform can be understood as content of
signal x near time t and near frequency co.
2.1 Wavelet Transform
Formulas for wavelet transform are written as follows:
(Tf)(a,b)=\ar (2.14a)
m
C (/) = VI /('MV' -1A )dt (2.14b)
Its assumed that vy (sometimes called mother wavelet) is selected in such a way that:
| From equations 2.12, 2.13 and 2.14a and 2.14b, we can see the fundamental
difference between windowed Fourier transform and the wavelet transform, i.e.,
shapes of the analysing functions.
Functions regardless of the frequency has the same width, while width of
is very narrow for high frequency and broader for lower frequency, because of such
"zooming" capability, wavelet transform can be used to see very fast transients.
Equations 2.14a and 2.14b depict two basic kinds of wavelet transform:
continuous and discrete. Functions can be reconstructed from their continuous
wavelet transform in the following way:
fC? ]]Â¥'*> (2.15)
7


x-b
where y/a*(x) =j a \ 2 y/(--) and < > denotes inner product and
a
A
Cw = 2n J| y/{£, )|2| E, \cU; where -oc
From equation 2.15 we can conclude that function f can be depicted as a
superposition of wavelets y/aJ and coefficients can be calculated from wavelet
transform of function f.
2.2 Discrete Wavelet Transform
-- x-b
Wavelet y/aJ,{x) =j a \ 2 ----) can be discretized in the following way:
a
Vmj,(x) = a02y(X nbm0 ) = a0 2'"x-nb0)
an
(2.15)
For a0= 2 and b0= 1 we have :
Â¥mJ,{x) = 2 2y,(2~mx-n) (2.16)
It can be seen from eq. 2.15 and 2.16 that two parameters m,n provide us with the
access to two distinctive features of the analyzed signal f.
Parameter n gives us "sliding" capability and makes it possible to access different
locations of the signal, while m parameters by dilating the wavelet let us see
components of the signal at different frequencies.
We can achieve by carefully selecting \\i that y/m n constitutes an orthonormal basis.
The oldest example of function iy for which y/mjt gives an orthonormal basis is Haar
function:
1
y/(x) = -
-1
0
0 2
2
otherwise
(2.17)
8


It's easy to see that Haar function forms an orthonormal basis, since support for
(Vmj,)= [2mn,2m(n + \)\ never overlaps at the same scale, which leads to
=S ..
t mji t mJt
Function ip as defined by 2.17 is sometimes called mother wavelet.
From eq. 2.17 we can conclude also that:
oc
f V/mj,(X)^X ~ 0 fr ^
-CO
It can be shown also that any function f e L?(R) can be approximated arbitrary well
by a finite linear combination of the if/mjI 's, such property brings out important
feature of wavelets, i.e., multiresolution analysis.
Let's define a function space F meZ such as:
Vm = \f eL2(R): f is piecewise constant on [nl~m ,{n + \)2~my^ (2.18)
and n e Z
Sequence of spaces Vm represents a ladder of subspaces with increasing resolution
(with the increase of m). Each subspace Vm consists of functions that are piecewise
constant over interval twice the length of those function for Vm_t .
The sequence of subspaces can be characterized by the following properties:
1. ... c= V_2 a V_t e V0 c F, c V2 c...
2 ry.-w
meZ meZ
3- feVm<*f(2.)eVm+i
4. / eV0=> f(*-k) e V0 for all k e Z
5. there exists (p e V0 such that { Property #2 is called downward and upward completeness, property #3 reflects scale
invariance and property #4 reflects shift invariance.
Property #1 and #3 gives us immediate verification that the scaling function
9


Numbers hn are called filter coefficients of function cp, mother wavelet 1\i is generated
by definition:
y/(t) = ^^gn One of the possible approaches to wavelets is through 2 channel filter
bank. Wavelets are determined by filter coefficients.
Two channel filter bank is depicted on Fig. 2.1
H0(z)
A2

________________________!
H,(z)
J t2
k2
^(z) -
Fig 2.1 Filter Bank
H0,H1 analysis filter
F0,Fl synthesis filter
10


If filter bank implements orthogonal transform then f0(n) = hX)(-n) and
fx(n) = /*,(-) Numbers determine properties of cp and vp,
and the problem is to design filters that achieve properties required for
scaling function and basic wavelets. For example: filter coefficients for Haar wavelet
are:

There are two conditions for filter bank to offer perfect reconstruction,
i.e. x(n) = x(n)
I. Alias Cancellation. F0(z)H0(z) + Fl(z)Hl(z) = 0
II. No Distortion F0(z)H0(z) + Fl(z)Hl(z) = 2z-
it can be written in the vector matrix form:
H0(z)
o]
(2.21)
Some properties of the wavelet transform.
1. Linearity
Lets denote operator T such as T\f(t)\ = F[/,w] = {y/m
then:
T[af(t) + *g(()] = aT\fV)\+bT\g{t)]
This property is a immediate result of linearity of the inner product.
2. Shift
If a signal has a scale limited expansion:
neZ m=-oc
then :
11


f{t 2 M'k) <-> F[m,n 2M^k],- 3. Scaling
k
f(2~kt)<-> 22 F[m-k,n\k e Z
4. Parserval's Identity
For orthonormal family we have the following:
mjteZ
5. Time localization.
Let's have wavelet vj/(t) which is compactly supported on the interval [-/2,,/rJ, i.e.
ymja(t) is supported on \-nx2m,n22m J and is supported on
\-nx+n)2m,(th+n)2m\.
If we consider t = tQ at scale m, corresponding coefficients are:
2~mt0-n2 It's easy to see that region of coefficients affected by the value of the function
at t = t0 will be smaller at higher scale.
6. Frequency localization.
We can derive from Parseval's formula:
cc i rn oo
F[m,n)= \)'V(2-a>y2'da> (2.23)
- -00
m
where 2*x¥(2mco)e~J2 nm is the Fourier transform of y/mj,it).
12


Let's assume that a wavelet vj/(t) vanishes in the Fourier domain outside the region
[6Jm.n>&\n>x L therefore component at co0 shall influence the wavelet series at scale m
if:
GO G)
mm < < max
T 2m
from which we can derive :
log2 ( \ l w0 J V w0 J
2.3 Localization and Heisenberg-Gabor Principle
In order to characterize a signal x(t) simultaneously in time and in frequency, one can
consider its mean localization and dispersions in each of these representations.
It can be obtained by analyzing |x(t)|2 and |X(v)|2 as probability distributions, and
looking at their mean values and standard deviations:
T'-ffr-'.yw*
^ X 30
tm =y
x -eo
Bl =Y)(v-vJ\X{vfdv
ym =y ]v\X(vfdv
Where tm average time, vm average frequency, T2 time spreading, B2- frequency
spreading and Ex is the energy of the signal, assumed to be finite.
Signal can be characterized by its mean position (tm, vm) and a domain of main energy
localization whose area is proportional to the time-bandwidth product TxB.
Product TxB is lower bounded such as:
13


TxB> 1
This constraint, known as the Heisenberg-Gabor inequality, can be interpreted that a
signal can not have simultaneously an arbitrary small support in time and in
frequency. The lower bound is reached for gaussian functions:
x(t) = C exp[- a(t -tmf+ jl7tvm (t tm)]
where C and a are real positive numbers. Signals considered in the study are non-
stationary (signal is stationary if it can be written as discrete sum of sinusoid).
A deterministic signal is considered stationary if it can be written as discrete sum of
sinusoids, in the random case, a signal x(t) is said to be wide-sense stationary if its
expectation is independent of time and its autocorrelation function E[x(ti)x (tj]
depends only on the time difference t?- tt It can be shown that the associated
analytic signal has constant instantaneous amplitude and frequency expectations.
Signal will be non-stationary if one of those assumptions is no longer valid.
14


3. Spectrograms and Scalograms
3.1 Spectrogram
Spectral energy density of the locally windowed signal x(u)h*(u-t) is obtained by
squaring modulus of STFT:
2
Sx(^v) = \x(u)h*(u-t)e
(3.1)
Spectrogram is the real-valued and non-negative distribution. Window h of the STFT
is assumed of unit energy and spectrogram satisfies the global energy distribution
property:
Spectrogram can be understood as measure of the energy of the signal contained in
time-frequency domain centered on the point (t,v).
Properties of the spectrogram
Time and frequency covariance
Spectrogram preserves time and frequency shifts:
Time and frequency resolution
Spectrogram is the squared magnitude of the STFT, and its time-frequency
resolution is limited exactly as it is for the STFT. In particular, it exists trade-off
between time resolution and frequency resolution.
(3.2)
y(t) = x{t-tQ) => Sy(t,v) = SJ(t-t0,v)
y{t) = x(t)vx$[j2nv0t] => Sy(t,v) = SJt,v-v0) (3.3)
15


Interference structure
The spectrogram is a quadratic (bilinear) representation and the sum of two
signals is not the sum of the two spectrograms (quadratic superposition
principle):
M0 = x,(/) + x2(/) => ^(/,v) = ^(/,v) + 5,j(r,v) + 2SR{5Vj(/,i/)} (3.4)
where third component of the equation is the cross-spectrogram and 91 denotes
the real part. Cross-spectrogram presents interference terms, given by Sxix2(t,v).
It can be shown that interference terms are restricted to the regions where auto-
spectrograms Sxi and Sx2 overlap. Thus, it can be stated that if two or more signal
components do not overlap significantly, then the interference term will be
almost zero. The property, which can be seen as a practical advantage of the
spectrogram, is a consequence of the spectrogram's poor resolution.
Time-Frequency Toolbox for Matlab from French CNRS (Centre Nationale de la
Recherche Scientifique) was used to obtain spectrograms. Function tffsp
computes the spectrogram distribution of discrete signal x.

^x(u)h*(u t)e j2du\
(3.5)
Synopsis
[tfr,t,f]=tffsp(x,t,N,h,trace)
where:
x analyzed signal (Nx=length(x))
t time instants
N number of frequency bins
h frequency smoothing window, h is being normalized so as to be of unit energy
trace if nonzero, the progression of the algorithm is shown
tfr time-frequency representation
f vector of normalized frequency
Spectrograms were created with Kaiser window for 128 frequency bins. Length of the
Kaiser window 25. Frequency is normalized, i.e. 0.5 in normalized frequency
corresponds to 0.5 of sampling frequency. Sampling frequency for flapwise and
edgewise bending loads is equal 32 Hz, while sampling frequency for wind speed and
16


cross-products is equal 16 Hz. Shorter Kaiser window offers better time resolution,
while loosing some of the accuracy in the frequency domain. Extending the Kaiser
window gives better frequency resolution while details visible in the time domain are
lost. Additionally, STFT is burdened with mentioned before interference structure.
Function tfrscalo from Time-Frequency Toolbox computes the scalogram according
to the formula:
SCx(t,a;h) = \lTx(t,a;h)\2 =
a
Jx(s)/j'(--)ds
(3.6)
Synopsis:
[tff ,t,f wt]=tfrscalo(x,t, wave, finin, fma x,N,trace)
where: x signal to be analyzed. It's analytic version is used (z=hilbert(real(x)))
t time instant!s)
wave half length of the Morlet analyzing wavelet at coarsest scale.
If wave=0 Mexican hat is used.
finin, fmax respectively lower and upper frequency bounds of the analyzed
signal. These parameters fix the equivalent frequency bandwidth.
Both values should be > 0 and <0.5.
N number of analyzed voices.
3.2 Scalogram
Distribution similar to the spectrogram can be defined for the wavelets. It can be
shown that continuous wavelet transform behaves like an orthonormal basis
decomposition and it preserves energy:
j }\T,{t,ayfdti^ = E, (3.7)
L L a
oc cc
where Ex is the energy of x. Scalogram can be defined as the squared modulus of the
continuous wavelet transform. It depicts energy distribution of the signal in the time-
scale plane, associated with the measure dtda/a2. Time and frequency resolutions of
17


the scalogram are related via the Heisenberg-Gabor principle: they depend on
considered frequency. Signals will be "more" localized in time for larger frequencies
and "less" localized for lower frequencies. Frequency resolution is clearly a function
of the frequency, for larger frequencies signals are less localized. Interference terms
of the scalogram will appear in those regions of the time-frequency plane where
corresponding auto-scalograms (signal terms) overlap.
3.3 Wavelet Power Spectrum
The continuous wavelet transform of discrete sequence will be defined as the
convolution of Xn with scaled version of vp0(r|):
N-1

/>=0
(n n)St
(3.8)
(*) denotes complex conjugate.
Picture showing both amplitude of any features of the signal versus the scale and
how this amplitude varies with time can be constructed by varying the scale s and
translation along time index n. Approximation of the continuous wavelet transform
can be obtained by performing the convolution described above N times for each
scale, where N is the number of points in the time series.
Wavelet function at each scale is normalized to have unit energy, so wavelet
transforms are directly comparable to each other and to the transform of other time
series.
Expectation value for |Wn(s)|2 is equal to N times the expectation value for square
modulus of the DFT of the sequence x. For a white-noise time series, expectation
value is equal variance divided by N, i.e. the expectation value for the wavelet
transform is equal variance. Normalization gives a measure of the power relative to
white-noise.
All sequences considered during the study are finite-length time series, it means that
the errors will occur at the beginning and the end of the wavelet power spectrum,
because the Fourier transform assumes the data is cyclic. Time series is padded with
zeros in order to obtain total length of the sequence to the next power of two. Process
of padding with zeros introduces discontinuities at the endpoints and decreases
amplitude near the edges.
The cone of influence is defined as region of the wavelet power spectrum in which
edge effects become important and was defined in the above mentioned paper as the
e-folding time for the autocorrelation of wavelet power at each scale.
Significance levels for Fourier or wavelet spectrum can be determined if appropriate
background spectrum is chosen. For many physical phenomena, background
spectrum is either white noise or red noise (increasing power with decreasing
frequency)
18


Authors of the paper defined the null hypothesis for significance levels as follows:
" It's assumed that the time series has a mean power spectrum, [...]; if a peak in the
wavelet power spectrum is significantly above background spectrum, then it can be
assumed to be a true feature with certain percent confidence. For definitions,
"significance at 5% level" is equivalent to "the 95% confidence level", and implies a
test against certain background level, while "95% confidence interval" refers to the
range of confidence about a given value."
Procedure for applying continuous wavelet transform is as follows:
1) Finding the Fourier transform of the time sequences padding may be required
2) Selection of the wavelet function and set of scales.
3) Construction of the normalized wavelet function, for each scale.
4) Finding the wavelet transform at that scale.
5) Determination of the cone of influence and the Fourier wavelenght at that scale.
6) Steps 3-5 are repeated for all scales, padding is removed and the wavelet power
spectrum is contour plotted.
7) A background Fourier power spectrum is assumed at each scale, and 95%
confidence contour is found by using chi-squared distribution.
Wavelet Analysis Tool developed for Matlab 5 contains option of calculating wavelet
power spectrum (WPS), there are four different wavelet bases available in the tool:
Morlet
Paul
DOG (Derivative of Gaussian) m=2 so called Mexican Hat
DOG m=l so called WAVE
Wavelet bases are described in "A Practical Guide to Wavelet Analysis" by C.
Torrence and G. Compo and published in Bulletin of American Meteorogical Society.
Time-averaged wavelet spectrum over certain period is defined as follows:
H
,%Q =*,
(3.9)
where the new index n is assigned to the midpoint of ni and n2 and na= n2 ni+1 is the
number of points averaged over. Wavelet plot smoothed by a window is created by
repeating equation 3.9. Extreme case of the eq. 3.9, when the average is all over the
local wavelet spectra and leads us to the global wavelet spectrum
19


(3.10)
M n=0
It was shown that the global wavelet power provides an ubiased and consistent
estimation of the true power spectrum of a time series
Scale-averaged wavelet power is defined as the weighted sum of wavelet power
spectrum over scales Si to S2:
a*
n £md
S j~J\
(3.11)
As we can see the scale-averaged wavelet power is simply a time series of the
average variance in a certain band.
3.4 The Reassignment Method
Methods of time-frequency analysis presented in previous sections have their
readability impaired by bilinearity, which means both good concentration of the
signal components and no misleading interference terms are required in order to get
truthful time-frequency representation of the signal. Scalograms obtained for
normalized sequences with statistical significance tests for wavelet spectra which are
developed by deriving theoretical wavelet spectra for white and red noise processes
offer much better time and frequency localization. Idea of reassignment was
introduced in an attempt to improve spectrogram.
Spectrogram can be expressed as 2D-convolution of the Wigner-Ville distribution of
the signal by WVD of the analysis window:
$Wh(t-s,w-$)dsdZ (3.12)
where Wigner-Ville distribution is defined as:
Wx(t,v)= J x(t + r / 2)x* (/ r / 2)e~j2,rvrd z (3.13)
20


Wigner-Ville distribution (WVD) is always real-valued, it preserves time and
frequency shifts and satisfies the marginal properties, it can be interpreted as
probability density.
Distribution from equation 3.12 reduces interference terms of the signal WVD, but it
happens in the expense of opposed time and frequency resolution, and of biased
marginals and first order moments. However, expression Wh(t-s, v-E) delimits a time-
frequency domain at the vinicity of the (t, v), inside which a weighted average of the
signal's WVD values is performed. The fundamental idea of the reassignment method
is that these values have no reason to be symmetrically distributed around (t, v), which
is the geometrical center of this domain. Therefore, their average should not be
assigned at this point, but rather at the center of the gravity of this domain, which
represents much better the local energetic distribution of the signal. Distribution W//t-
s, v-) Wx(s, E) can be considered as a mass distribution, and it is much more accurate
to assign the total mass (i.e. the spectrogram value) to the center of gravity of the
domain rather to its geometrical center.
Reassignment method moves each value of the spectrogram computed at any point
(t, v) to another point (f', v*) which is the center of gravity of the signal energy
distribution around (t, v).
(3.14)
j \sWh{t-stv-zwMZ)*dZ
-oo op____________________
+OC +00
\ \wh(t-s,v-Z)W,(s,Z)dsdZ
J \£Wh{t-s,v-S)W,{s4)dsd!;
vtet,v) = ------------------------
J ]wh{t-s,v-$)Wx{s,S)dsdq
Equation 3.14 leads us to reassigned spectrogram, whose value at any point is the
sum of all the spectrogram values reassigned to this point:
21


(3.15)
S oc oc
Distribution uses the phase information of the short-time Fourier transform, and not
only its squared modulus as in the spectrogram.
It can be seen from the following expressions of the reassignment operators:
(3.16)
t{x-t,v) = -
d v(x;t,v) = v +
dv
d<&At,v,h)
dt
where 0x(t, v;h) is the phase of the STFT of x.
However, those expressions do not lead to an efficient implementation, and have to
be replaced by the following ones:
(3.17)
Fx(t,v;Th)F;(t,v,h)
t(x;t,v) = t-'$l
v(x;/,v) = v- 3


where Th(t)=txh(t) and Dh(t) is the derivative ofh(t). Reassignment spectrograms,
though no longer bilinear, satisfy the time and frequency shifts covariance, the energy
conservation (h(t) has to be of unit energy) and the nonnegative property.
Reassigned spectrogram was computed by employing tfrrsp.m M-file.
Synopsis:
[tff ,rtfr ,hat]=tfrrsp( x,t,N,h,trace)
where: x analyzed signal
t time instants
N number of frequency beans
h frequency smoothing window
trace if nonzero, the progression if the algorithm is shown
22


tfr, rtfr -
hat
time-frequency representation and its reassigned version
complex matrix of reassigned vectors
23


4. Detection of Singularities
Localized signal structures can be detected by applying wavelet transform Long time
series like those obtained from Micon 65 and Wind Eagle 300 can be very difficult to
investigate, because of the extended computing times when tools like reassigned
spectrogram or 1-d continuous wavelet transform are utilized. In order to find areas of
the signal with rapid changes in values, we developed wavelet based algorithm, it will
make possible to identify those areas of concern in one single shot.
Irregular structures and singularities often contain essential information in a signal.
For example, discontinuities in the intensity of an image indicate the presence of
edges in the scene. Radar signals or electrocardiograms contain vital information in
sharp transition.
Bending loads sequences are not different in that aspect, however, singular point
denotes usually the point where derivative does not exist i.e., it approaches infinity.
In real time, it would mean that load changes infinitely fast, which is impossible, of
course.
Therefore, when we refer further in the report to pseudo-singular features, we
understand that those pseudo-singular features denote the area of rapid variation of
bending loads.
Singularities are usually detected by following the wavelet transform local maxima at
fine scales. To measure the local regularity of the signal, it is important to use the
wavelet with narrow frequency support and number of vanishing moments is also
important.
Vanishing moments are defined as follows:
Wavelet function vp has p vanishing moments if
for example Haar wavelet has only one vanishing moment, while Daubechies N
wavelet has N vanishing moments.
It can be shown that wavelet with n vanishing moments can be interpreted as a
multiscale differential operator of order n Singularities are detected by finding the
abscissa where the wavelet modulus maxima converge at fine scales. Term modulus
maxima is used to describe any point (u&so) such that \Wf(uo,so)\ is locally maximum
at u=uo, which implies that:
(4.1)
24


awy du
This local maximum should be strict local maximum either the right or left
neighborhood of uo.
Algorithm developed for the purpose of finding rapid transition of the bending loads
takes into account fact that we deal with pseudo-singularities.
As a first step we decompose the signal using multiresolution decomposition, number
of decomposition levels will depend on sampling frequency of the signal for
example, sampling frequency 32 Hz require at least 5 levels of details, 200 Hz 10
levels of details. Choice of wavelet should be based on the shape of wavelet function
and the signal number of vanishing moments is also crucial in general Haar wavelet
will not work well.
Discrete Laplacian (Matlab function del2) is applied to all detail vectors, it
calculates the difference between each point of the detail and average from four its
neighbors. As result, detail vectors are mapped into vectors containing differences
between each point of the detail and its surroundings.
Third step of the algorithm uses denoising procedure to clean the vectors containing
values of discrete Laplacian, values obtained in such a way are representing points in
which wavelet coefficients are standing out from the "crowd".
Denoising procedure can be selected from the following thresholding options:
Rigrsure adaptive threshold selection using principle of Stein's Unbiased Risk
Estimate.
Heursure heuristic variant of the first option.
Sqtwolog threshold is sqrt(2*log(length(X)))
Minimaxi minimax thresholding.
Few words are necessary about denoising methods. Basic problem in separating
unwanted noise from useful signal is distinguishing one from another. The same
signal in different applications can be regarded once as a noise and another time as
usefUl signal we want to analyze.
Method proposed by D. Donoho and I. Johnston called wavelet shrinkage basically
comes down to the following steps:
-- wavelet transform is applied to the noisy signal
all coefficients below certain size are removed
signal is reconstructed from bigger coefficients i.e. selective wavelet
reconstruction
25


In our algorithm, we apply wavelet transform, then discrete Laplacian, then we
remove all coefficients of the vectors below certain level.
There are two kinds of thresholding:
Hard thresholding:
V*.(0 =
*(0, |*(0| > 8
0, \x(ti Soft thresholding:
(4.3)
Vs(0 =
s/gw(x(/)X|x(0|-£),
0,
|x(0|<£
(4.4)
There are several methods for choosing the threshold. Generally, those methods can
be divided into two categories:
global thresholding
level dependent thresholding
Global thresholding means that single value > is chosen and applied globally to all
levels, while level dependent thresholding allows us to choose different value \ for
each wavelet level j.
Minimax thresholding is a method of global thresholding which applies the optimal
threshold which depends on the sample size n and is divided to minimize constant
term in an upper bound of the risk involved in estimating the function.
For example: n=64 >=1.474
n=128 >=1.669
n=256 >=1.860
n=512 >=2.047
as given by Donoho and Johnston.
Universal thresholding method is another method by Donoho and Johston
which applies global thresholding X = ^2\ogn.
Method offers smoother estimates and is visually more appealing, sometimes, it's
called "VisuShrink" method.
SURE thresholding (Stein's Unbiased Risk Estimate)
26


The method uses the wavelet coefficients at each wavelet level j to choose
a threshold X which shrinks the coefficients at that level. SURE thresholding is base
A. 2
on the finding an estimator f for f that will have small L risk.
R(f,f) = E
1 "
L
n
f ~ ~f

i=i V \n)
\nj)
(4.5)
From equation 4.4 and Parserval's identity we can derive for wavelet coefficients:
/?(/,/) oc£

where QjJc wavelets coefficients
(4.6)
The risk R( f, f) is estimated from the data, minimization of estimated risk is done
by choosing a threshold value for each level. Let's consider Xl,...,Xcl which are
independent observations: Xk ~ N(/uk ,1). Problem is to estimate the mean vector
H = with minimum risk, which comes to estimation of true wavelet
coefficients at any level j, with Xk = yfnco^l d = 2J
We apply soft thresholding functions fik = (Xk)
A
Denoting the vector of observ ations X and letting fj, represent the resulting estimator
of p, Stein method states that the loss can be estimated without a bias for an estimator
of p that can be written /}(X) = X + g(X)
Function g is weakly differentiable:
EJ/iUMI1 =d + Er\g{X)Hl+2Vg(x)j (4.7)
where V g 'Z,^gk(X)
k=i&
&k (i>l
27



(4.8)
t, Xk<-t
so ||g(*)f =22(|^*M)-
d
It should be noted also that V -g = -]T V-*,) so that Stein's estimate of risk
where #S for a set S denotes cardinality of the set.
The threshold level is set so as to minimize the estimate of risk for given data
x\>xd '
The problem with implementing SURE method arises from the fact, that the SURE
method does not perform well in cases where the wavelet representation at any level
is very sparse, i.e. when the vast majority of coefficients are almost zero.
The hybrid scheme, which takes care of the problem was suggested.
Method comes from the heuristic idea to test the coefficients for sparcity at each
level. If the set of coefficients is determined to be sparsely represented, then the
hybrid scheme defaults to universal threshold -y/21ogd otherwise SURE criterion is
applied. Representation at the level is judged to be sparse if:
applied in this situation can be written for any set of observed data
x = (x1,...,xrf)' :
X = argmin/i0 SURE(t,x)
(4.10)
28


If we look at the definition of del2 from MATLAB, we can see that discrete
Laplacian is determined as a difference between each point and its four neighbors.
Laplacian is applied on the detailed results of discrete wavelet transform, thus we can
say, that we are looking at rapid variations of signal appearing at different scales, i.e.,
different frequencies. Effectiveness of the algorithm strongly depends on selected
wavelet.
29


5. Experimental Data
The following time sequences are analyzed:
1) Micon 65 Upwind Turbine sampling frequency 32 Hz, 15 m/s hub mean wind
speed;
File m65_7314.dat was converted to Matlab 4 and 5 formats, signals containing
edgewise and flapwise bending loads were retrieved:
edgl 165.mat edgewise load blade #1
edg2165.mat edgewise load blade #2
edg3165.mat edgewise load blade #3
flpl 165.mat flapwise load blade #1
flp2165.mat flapwise load blade #2
flp3165.mat flapwise load blade #3
2) Cannon Wind Eagle 300 Downwind Turbine, sampling frequency 200 Hz
File df3_dc01.dat @ 6.57 m/s hub mean wind speed; the following signals were
retrieved:
bafb_dl.mat flapwise bending load blade shell blade # A
bbfb_dl.mat flapwise bending load blade shell blade # B
fafbdl.mat flapwise bending load flex beam blade # A
fbfb_dl.mat flapwise bending load flex beam blade # B
fbeb_dl.mat edgewise bending load flex beam blade #B
faeb dl.mat edgewise bending load -flex beam blade # A
3) Cannon Wind Eagle 300 Downwind Turbine, sampling frequency 200 Hz
File df3_dcl4.dat @ 17.40 m/s hub mean wind speed; the following signals were
retrieved:
bafb_dl4.mat flapwise bending load blade shell blade # A
bbfb_dl4.mat flapwise bending load blade shell blade # B
fafb_dl4.mat flapwise bending load flex beam blade # A
fbfb_dl4.mat flapwise bending load flex beam blade # B
fbeb_dl4.mat edgewise bending load flex beam blade #B
faeb_dl4.mat edgewise bending load -flex beam blade # A
4) Adams Wind Eagle Numerical Simulations
File run_dc08.dat @ hub mean wind speeds 9.8 m/s.
30


Retrieved signals are:
rotaz.mat Rotor Azimuth Angle
Integrated Root Loads Blade A
flap_a.mat root flap wise load
rdge_a.mat root edgewise load
tors a. mat root torsion load
Spanwise blade Loads Blade A
spnf_12.mat Normal Force Sta 12
spnf_13.mat Normal Force Sta 13
spnf_14.mat Normal Force Sta 14
spnf_15.mat Normal Force Sta 15
sptf_12.mat Tangential Force Sta
sptf_13.mat Tangential Force Sta
sptf_14.mat Tangential Force Sta
sptf_15.mat Tangential Force Sta
12
13
14
15
Inflow Turbulence At Blade Stations
U'V' Reynolds Stress Components
uv_12.mat Sta 12
uv_13.mat Sta 13
uv_14.mat Sta 14
uv_15.mat Sta 15
UW' Reynolds Stress Components
uw_12.mat Sta 12
uw_13.mat Sta 13
uw_14.mat Sta 14
uw_15.mat Sta 15
VW1 Reynolds Stress Components
vw_12.mat Sta 12
vw_13.mat Sta 13
vw_14.mat Sta 14
vw 15.mat Sta 15
5) Adams Wind Eagle Numerical Simulations
File run_dcl2.dat @ hub mean wind speeds 18.33 m/s.
Retrieved signals are:
31


rotaz_12.mat Rotor Azimuth Angle
Integrated Root Loads Blade A
flap_al2.mat root flapwise load
rdge_al2.mat root edgewise load
tors_al2.mat root torsion load
Spanwise blade Loads Blade A
sptf!212.mat
sptfl312.mat
sptfl412.mat
sptfl512.mat
Normal Force Sta 12
Normal Force Sta 13
Normal Force Sta 14
Normal Force Sta 15
Tangential Force Sta 12
Tangential Force Sta 13
Tangential Force Sta 14
Tangential Force Sta 15
Inflow Turbulence At Blade Stations
U'V' Reynolds Stress Components
uv_1212.mat Sta 12
uv_13I2.mat Sta 13
uv_1412.mat Sta 14
uv_1512.mat Sta 15
LTW Reynolds Stress Components
uw_1212.mat Sta 12
uw_1312.mat Sta 13
uw_1412.mat Sta 14
uw_1512.mat Sta 15
VW Reynolds Stress Components
vw_1212.mat Sta 12
vw_1312.mat Sta 13
vw_1412.mat Sta 14
vw 1512.mat Sta 15
Data sequences will be analyzed and described individually in the following
order:
general description of the signals
examples of multiresolution decomposition
32


application of singularities detector selection of interesting areas
Analysis of selected areas:
spectrograms
scalograms
reassigned spectrograms
wavelet spectral power
continuous 1-D wavelet analysis
33


5.1 Micon 65 Upwind Turbine
Time sequences are quite consistent with each other within the particular family,
i.e., flapwise sequences are similar to each other regardless of the blade, the same
is true for edgewise loads. Therefore, we consider signals for the blade # 1.
(Figures A. 1 to A.4)
Singularities detector was used and several points were selected:
flapwise load blade #1
D1 level points: 250.938 s Figure A.7
250.375 s Figure A.9
D3 level points: 109.406 s Figure A.12
edgewise load blade # 1
D1 level points: 160.250 s Figure A. 14
D2 level points: 3 81.969 s Figure A. 16
105.219 s Figure A.18
Area of +/- 5 seconds for each of those loads was selected and save in .mat format.
Spectrogram applied to flpl 102.mat (250.375 +/- 5.0s) and depicted on Fig. A. 10 also
detects pseudo-singular point in D1 level (16 Hz to 8Hz).
Reassigned spectrograms give even better localization of detected singularities than
spectrograms or 1-D continuous wavelet transform Figures A. 11, A. 13, A. 17 and
A20.
However, wavelet power spectrum does not detect singular points, application of
Mexican Hat wavelet brings nice separation of each part of the sine wave without
providing any minute details, (edgl 101.mat @ 160.25 +/- 5.0s) Figure A. 15
Plots obtained for edgl 105.mat @ 105.219 +/- 5.0 s show us the difference between
Mexican Hat wavelet, WAVE, RAMP and Morlet wavelet figures A. 19, A21, A22
and A23 respectively.
We can conclude that, application of different wavelet transforms can expose
different areas of time-frequency continuum, for example MexHat and WAVE (2nd
and la derivative of Gaussian) showing maximum values of coefficients at lower
scales (higher frequencies), while RAMP and Morlet wavelet transform indicate
maxima at higher scales (lower frequencies) and they are not localized.
Figures A.5 and A.6 depict multiresolution decomposition of the flapwise and
edgewise bending loads, Daubechies 6 wavelet was used.
34


Signal is decomposed into 5 signal containing details (dl, d2, d3, d4, d5) and 1 signal
containing coarse approximation (a5), sum of those signal gives us the original time
series. Features containing the highest frequencies are localized on the first level,
since decomposition is performed by using filter banks (see Fig. 2.1 ) and filter used
are half-band filters dl contains the following frequencies: 8-16 Hz, d2 (4 8 Hz), d3
(2 4 Hz), d4 (1 2 Hz) and d5 (0.5 -1 Hz).
Further decomposition of a5 signal would reveal other features of the analyzed signal
located at lower frequencies.
35


5.2 Cannon Wind Eagle 300 Downwind Turbine
Flapwise bending loads measured at blade shell have richer frequency representation
than flapwise bending loads measured at flex beam. The difference is visible from
discrete wavelet transform decomposition and from rough spectrum plots. Additional
vibration in the case of blade shell spread all over the spectrum. General shape of
approximation A10 is quite similar for both measurements, however, there are
significant differences for details # 7 and # 8, which corresponds to the following
frequencies: D7 1.5625 Hz to 0.7825Hz and D8 from 0.7825 Hz to 0.39125 Hz.
The finest details are showing some kind of unknown artifacts appearing every 10
seconds, in this point, we disregard the phenomenon, assuming that, it was introduced
by data acquisition systems.
Singularities detector was used to detect the following points:
fafb_dl.mat #1 t = 106.110+/-5.0 level D6 saved into fafbOIs.mat
#2 from ti= 100 to t2 = 130 saved into fafb05s.mat
Sequences #2 was based on visual inspection of the time series, selected range
contains rapid changes of the bending load. Wavelet Daubechies 6 was used by the
detector algorithm.
bafb_dl.mat #1 t = 102.170 +/- 5.0 level D7 saved into bafbOls.mat
#2 from ti= 100 to t2 = 130 saved into bafb05s.mat
Wavelet Coiflet 5 was used and sequence #2 was based on visual inspection.
faeb_dl.mat #1 from ti= 100 to t2 = 130 saved into faeb05s.mat
Sequence bbfb05s.mat retrieved from bbfb_dl.mat contains bending loads
starting from 100 second until 130 s.
In addition, other sequences were analyzed, but only those mentioned abovre
included in the study
Singularities detector was set to utilize hard threshold with rigrsure method.
All plots depict versatile methods used for analysis. We can see that for t = 100 s,
when there is a rapid increase of bending load (fafb, bafb, faeb), it brings more
activity in the range of higher frequencies.
For example: fafb01s.mat and bafbOls.mat- spectrograms -Figure B.4 and B.10
Spectrogram for fafb05s.mat shows activities at higher frequency ranges consistent
with those detected by 1-D continuous Morlet wavelet. Figure B. 12 and B. 11
Wavelet power spectrum gives better granularity than spectrogram and shows periods
of oscillation instead of scales like 1-D cont. wavelet.
36


Analysis of the series obtained at 17.40 m/s hub mean wind speed utilized signal
processing tool for preliminary analysis.
The following samples were selected for analysis:
From sequence fafb_dl4.mat two signals were retrieved fafbOlfmat (120 135 s) and
fafb02f.mat (240 255 s). Sequence bafb_dl4.mat gave us bafb03f.mat (120 -135 s)
and bafb04.mat (240 255 s), faeb_dl4.mat brings signal faeb05f.mat (140 -145 s) as
its representation, while bbfb06f.mat (120 -135 s) is a sample ofbbfb_dl4.mat.
Study contains plots starting from 120 s to 135 s.
Criterion for selection of the sample was to obtain representation of the "quiet" areas
next to rapid changes in values. Bursts of energy expanding far into domain of higher
frequencies are visible on spectrograms and wavelet power spectrums. Those bursts
are set against the backdrop of quiet areas, for example fafb04.mat wavelet power
spectrum Figure C. 13
Closer look is required at the figures: C.4, Cl 1 and C. 17. We can notice flame shaped
artifacts appearing when the sine wave reaches minimum, those flames correspond to
moments when passing blade was excited into higher frequency vibration by shadow
of the tower. We will see, later, in the case of simulated time sequences the same
flame shaped artifacts corresponding to the shadow of the tower.
37


5.3 Wind Eagle 300 Turbine Adams Simulations
Time sequences were obtained for two different mean wind speeds from numerical
simulations of the Wind Eagle Turbine. Application of the discrete wavelet transform
and Fast Fourier Transform allows us to analyze the whole series and distinguish by
simple inspection areas for further time-frequency analysis. Discrete wavelet
transform and FFT are numerically inexpensive i.e., computation time of the
relatively long time series is insignificant. By comparing multiresolution
decomposition obtained for fast and slow time series (fast and slow by mean wind
speed) we can conclude that faster wind speed excites higher frequencies in
corresponding integrated root and spanwise loads. Results of FFT confirm also that
conclusion. LTV' and U'W' Reynolds stress components for both time series contain
interesting artifacts easy visible in the FFT plots. Those artifacts can be described as
spikes appearing on the spectrum within 1 Hz distance from each other. Those spikes
are responsible for similar artifacts found in spanwise edgewise and flapwise loads.
They correspond to the effect of tower shadow, however quick comparison with the
real data, (appendix B and C) leads us to the conclusion that existing model for
Adams simulation exaggerates the influence of the tower.
Area from ti = 70.0 s to t2 = 100.0 s was selected for detailed analysis from dc08 run.
The following Matlab files were created:
flapOls.mat
rdge02s.mat
tors03s.mat
spnfl24s.mat
sptfl28s.mat
uv_1212s.mat
uw_1317s.mat
vw 1422s. mat
flapwise load
edgewise load
torsion load
spanwise normal force @ station 12
spanwise tangential force @ station 12
U'V Reynolds stress component @ station 12
U'W' Reynolds stress component @ station 13
VW' Reynolds stress component @ station 14
Area from ti = 50.0 s t2 = 80.0 s was selected for detailed analysis from del2 run.
The following Matlab files were created:
flapOlfmat flapwise load
rdge02f.mat edgewise load
tors03f.mat torsion load
spnfl34f.mat spanwise normal force station 13
sptfl55f.mat spanwise tangential force @ station 15
uv_136f.mat U'V'Reynolds stress component @ station 13
uw_137.mat UW'Reynolds stress component @ station 13
vw_158.mat VW'Reynolds stress component @ station 15
38


For selected sequences spectrograms, reassigned spectrograms, wavelet power spectra
and 1-D Continuous Wavelet Transform were obtained. Again, selected areas are
characterized by the presence of rapid changes in the quiet neighborhood.
Artifacts previously mentioned are also visible on the plots, they appear on the
spectrograms in the form of flame-like features spreading far into the area of higher
frequencies, reassigned spectrograms showing those features in the forms of thin well
defined lines.
39


6. Software Package
Software developed for the analysis consists of two fully independent programs
designed to work under Matlab version 4 and 5. The reason for such a duality is that
Time-Frequency Toolbox is not compatible with version 5 and converting toolbox
would mean the necessity of converting over 100 m-script files, which could be the
formidable project itself.
Software is integrated with GUI, which makes the usage intuitive. It constitutes the
compromise between functionality and robustness, software is provided in the form of
Matlab m-scripts, which are suitable for further modifications.
6.1 Time-Frequency Analysis Software Compatible with Matlab ver. 4
Application can be started by entering: wavapp in the Command Window
Main window menu bar offers the following choices:
File:
Load - selected file is loaded to the workspace and plotted
Clear Last Loaded File it should be used for housekeeping
Exit - it will close the application and clear the workspace
Tools
Wavelet Toolbox - it opens standard Matlab Wavelet Toolbox
Time-Frequency Toolbox it opens custom window allowing for selection of
T-F tools: spectrogram, reassigned spectrogram and
scalogram
Initial values for T-F tools should be entered in the
Command Window when prompted.
Note: frequency displayed on TFT plots is
normalized by Nyquist frequency, axis x displays
samples instead of seconds,
6.2 Time-Frequency Analysis Software Compatible with Matlab ver. 5
Application can be started by entering: wavapp02 in the Command Window
Main window menu bar offers the following choices:
40


File:
Load
Clear Last Loaded File
Exit
- selected file is loaded to the workspace and plotted
- it should be used for housekeeping
- h will close the application and clear the workspace
Tools
Wavelet Toolbox - it opens standard Matlab Wavelet Toolbox
Singularities Detector - it opens custom window, detector can be used for
detecting singular points by applying previously
described algorithms (see chapter 4).
Singular points and their respective neighborhoods
with user selectable span can be saved in .mat format
with additional option allowing for Matlab 4 format.
In addition, user selected part of the signal can be
saved in both formats. User needs to enter starting
and ending points. Selected part of the signal is saved
in the 1 -D vector format, which means that there are
no reference in newly obtained signal to the areas of
the original from which such vector was derived.
Signal Processing Tool it opens standard Signal Processing GUI
Wavelet Power Spectrum it opens custom window, Wavelet Power Spectrum
can be calculated after loading of selected signal.
Choice of the following wavelet basis functions is
possible: Morlet, PAUL, Mexican Hat, WAVE.
Note: Both applications require specific naming convention allowing them to
recognize which signal is being loaded and to display appropriate title.
Micon 65 loads:
For edgewise loads first four letters of the name should be 'edgX' where X can be
1,2, or 3
For flapwise loads: 'flpX' where X can be 1,2, or 3
41


Cannon Wind Eagle 300 loads:
'bafb' flapwise, blade shell, blade A
'bbfb' flapwise, blade shell, blade B
'faeb' edgewise, flex beam, blade A
'fafb' flapwise, flex beam, blade A
'fbeb' edgewise, flex beam blade A
'fbfb' flapwise, flex beam, blade B
Wind Eagle Adams Simulations:
'flap' Root flapwise load
'rdge' Root edgewise load
'rota' Rotor Azimuth Angle
'tors' Root torsion load
'spnf_lX' or 'spnfl X' spanwise blade load normal force, X can be 2,3,4 or 5
'sptf_lX' or 'sptflX' spanwise blade load tangential force, X can be 2,3,4 or 5
'uv_lX' 17V' Reynolds Component, X = 2,3,4, or 5
'uw_lX' 17W' Reynolds Component, X = 2,3,4, or 5
'vw_lX' VW' Reynolds Component. X = 2,3,4, or 5
6.3 List of m-scripts and Description of the Tasks Performed.
Matlab ver. 4
wavapp.m main m-script file, defines main window and its elements
loadfile.m file is called when Load item is selected. User has to select a file for
loading
clearfilm file is called when Clear Last Loaded File is selected.
quit_w.m file is called when Exit item is selected. All windows are closed and
workspace is cleared
tmfrq_pl.m file is called when Time-Frequency Toolbox is selected
ld flbt.m callback from Time-Frequency Toolbox when Load File button is
selected
clr sgn.m callback from TFT when Clear button is selected. Clear button becomes
active after file for analysis was loaded.
spct_rt.m spectrogram routine
respct_rt.m reassigned spectrogram routine
scal_rt.m -scalogram routine
Each of those custom routine prompts user for entry of all necessary parameters and
42


calls internal functions from TFT. Routines have to be started from the Command
Window after loading file for analysis.
Matlab ver. 5
wavapp02.m main function, open main window, defines its menus.
loadfile.m - function is called when Load item is selected. User has to select a file
clearfil.m quitw.m for loading - file is called when Clear Last Loaded File is selected. - file is called when Exit item is selected. All windows are closed and workspace is cleared
sing_det.m ld_file.m clr_file.m cl_sndt.m wv_numb.m proc_fl.m clc_da.m dsp_mlt.m dsc_lap.m thrs_sel.m thrs_prf.m dsplsng.m det_view.m deta_lv.m sele_pt.m span_sel.m spanvue.m sv_matfl.m sav_piec.m - opens main window for singularities detector - load selected file for singularities detector - clear workspace of singularities detector close singularities detector main window, clear workspace - select wavelet for detection - find singularities in loaded files based on selected criteria - calculate approximation and details - display multiresolution decomposition - calculate discrete laplacian - threshold selection for denoising - thresholding - display detected singularities - detailed view of selected singularities - select the level of detected singularities - select detected point - select span of the neighborhood - plot selected point and its neighborhood - save selected point and its neighborhood - save selected part of the signal
wav_spec.m - opens main window for wavelet power spectrum
ld_pws.m load signal into Wavelet Power Spectrum workspace
cls_pws.m clear workspace when Wavelet Power Spectrum is closed
wvjws.m calculate wavelet power spectrum
wavelet.m wavelet transform with optional significance testing
wave_signif.m significance testing for 1-D wavelet transform
chisquare_inv.m inverse of chi_square cumulative distribution function
chisquare_solve.m internal function used by chisquare_inv.m
43


Last three functions were obtained from http://paos.colorado.edu/research/wavelets
6.3 New Wavelets WAVE and RAMP
The complete procedure of adding new wavelet family is described in Wavelet
Toolbox Manual. Matlab function wavemngr allows to add new wavelet families to
the predefined ones. Before wavemngr can be used the following procedure has to be
followed:
1. Choose the full name of the wavelet family (fii)
fit must be a string, for example we have chosen for RAMP wavelet
fn = 'RAMPWavelet'
2. Choose the short name of the wavelet family (fsn)
fsn = 'ramp', fsn has to be a string of four characters or less
3. Determine the wavelet type (wt)
Matlab distinguish four type of wavelets:
Orthogonal wavelets with FIR filter : Haar or Daubechies
Biorthogonal wavelets with FIR fiber: BiorSplines
Orthogonal wavelets without FIR filter but with scale function: Meyer
Wavelets without FIR filter and without scale function: Morlet and Mexican Hat
RAMP and WAVE wavelets are wavelets of fourth kind.
4. Define the orders of wavelets within given family (nums)
This argument is not used for wavelets of type 3 and 4 i.e, nums ';
5. Build MAT-File or M-File
The sysntax of the first line in the M-File must be (type # 4)
Function [psi,t] = file(lb,ub,n)
Which returns values of the wavelet function psi on a regular n-point grid with
intervals of length t and bounded by [lb ub]
6. Define the Effective Support
Definhion is required for for wavelets of type 3 and 4 since they are not
compactly supported.
Two M-files are provided as example how to build the facility of user defined wavelet
bases: addwvOl.m and addwv02.m They were used to add RAMP and WAVE
wavelets. RAMP and WAVE wavelets were defined by S. Collineau and Y. Brunet
Appendix F includes all listings of the scripts and examples of the custom GUI.
44


7. Conclusion
The following tools were used to analyzed a time series:
Multiresolution analysis, discrete wavelet transform with compactly supported
wavelets as a basis, choice of appropriate basis was briefly discussed.
Multiresolution analysis offers preliminary view into time-frequency domains of the
analyzed signals, frequency domain is divided into different scales of detail, scale is
loosely related to the frequency.
Spectrograms, obtained by applying STFT (Short Time Fourier Transform) offer
better frequency resolution, but time localization is lost and plots are fuzzy.
Scalograms, time sequences are analyzed with Morlet, PAUL, WAVE and
RAMP wavelet. Scalograms were depicted in the form of wavelet power spectrum
and 1-D continuous wavelet transform. From plots obtained with Mexican Hat
wavelet and WAVE wavelet we can see relation between those two basis function,
since MHAT is second derivative of Gaussian, while WAVE function is the first one.
Similar correlations can be found after applying Haar wavelet and RAMP, those
similarities can be explained that Haar scaling function phi can seen as first
derivative of RAMP.
Reassigned spectrograms offer high level of granularity in both domains,
bilinearity of classic spectrograms is lost and plots do not contain misleading
interference factors. Localization of singular events in both domains is on the similar
level as for scalograms of normalized sequences, however reassigned spectrogram
gives high quality resolution for very low frequencies.
Original method was developed for finding pseudo-singular points, method is based
on multiresolution decomposition, applying the Laplacian transform and then
denoising the results with different algorithm, preferable SURE.
Method is based in its core on the fact that the wavelet with n vanishing moments
can be written as the n-th order derivative of the basis function.
Experiments were conducted in order to evaluate usefulness of different wavelet
function for detecting spikes in the time-series. Conclusion can be drawn that Haar
wavelet do not perform well in that type of application, which can be explained by
the shape of Haar function which does not fit well into time sequence of U'V'
Reynolds component or time sequences obtained for Micon 65 wind turbine.
45


Wavelets like Daubechies or biorthogonal ones perform much better, examples
showing use of such wavelets are included.
Method for detecting pseudo-singular points finds only points standing out of the
crowd within immediate neighborhood of four points. However, it can be easy
extended on higher number of points, in addition, the method can be calibrated to the
analyzed signal by using different kinds of wavelets, by changing thresholding
method, level, and denoising algorithm.
It would useful also, to derive algorithm for finding clusters of pseudo-singular
points. It could be achieved by calculating mean values and standard deviation after
multiresolution decomposition and then denoising the results in order to find clusters
with rapid changes.
46


Appendix A Micon 65 Upwind Turbine
47


Load [kNm]
Figure A. 1
Flapwise bending load, Blade #1
I l
I I
I l
.5 I__________I__________1_________I____________I___________I__________I
Q 100 200 300 400 500 600
Time [s]
48


Load [kNm]
Figure A. 2
49


Load [kNm]
Figure A. 3
50


Load [kNm]
Figure A. 4
51


Figure A.5 Flapwise Bending Load flpl 165.mat
Sampling Frequency 32 Hz
Sign* ; IS wmemm
Wavelet 1*
Level U_ 3
Analyze
Statistic* Compress
H triogams Deoobe
Display mode: ^
j Fti Oecorapoahon
V ShowSirtheaaedSiQ.
Close
52


Figure A.6 Edgewise Bending Load edgl 165.mat
Sampling Frequency 32 Hz
Sgul Kj*re, 'J&J&X* T;
Wavelet i* .. d6. J
Level Is J
Analyze I
Stabsbc* Compress
Histogram Depose
Display mode:
j M Decompoalion
attewi |s jj
F SHowS vrthescedSig
Close
53


Figure A. 7 Detection of pseudo-singularities. Flapwise Bending Load flpl 165.mat
Settings: Wavelet db6 Point found @ D1 t = 250.938 +/- 5.0 s
Threshold Method: Hard
Threshold Coefficient: 0.7
Threshold Selection: Rigrsure
Flapwise bending load, Blade #1; Wavelet: db 6
54


Period (seconds)
Figure A.8 Wavelet Power Spectrum Flap wise Bending Load flpl 165. mat
Morlet Wavelet: Point found @ D1 t = 250.938 +/- 5.0 s
flpl 101.mat 321 samples
Flapwise bending load, Blade #1
123456789 10
Time (seconds)
55


Figure A.9 Detection of pseudo-singularities. Flapwise Bending Load flpl 165.mat
Settings: Wavelet db6 Point found @ D1 t = 250.375 +/- 5.0 s
Threshold Method: Hard
Threshold Coefficient: 0.7
Threshold Selection: Rigrsure
Flapwise bending load. Blade #1; Wavelet: db 6
56


Figure A. 10 Time-Frequency Toolbox Spectrogram
Signal flpl 165. mat
Point found @ D1 1 = 250.375 +/- 5.0 s
flpl 102.mat Sampling Frequency 32 Hz
Note: Frequency is normalized per Nyquist i.e., 0.5 => 16 Hz
Time axis shows samples
SP, Lh=25, Nf=128, log. scale, Threshold=1e-005%
50 100 150
200 250 300
Time [s]
57


Figure A. 11 Time-Frequency Toolbox Reassigned Spectrogram
Signal flpll65.mat
Point found @ D1 t = 250.375 +/- 5.0 s
flpl 102.mat Sampling Frequency 32 Hz
Note: Frequency is normalized per Nyquist Le., 0.5 => 16 Hz
Time axis shows samples
RSP, Lh=25, Nf=128, log. scale, Thresholdse-005%
50 100 150 200 250 300
Time [s]
58


Figure A. 12 Detection of pseudo-singularities. Flapwise Bending Load flp 1165.mat
Settings: Wavelet db6 Point found @ D3 t = 109.406 +/- 5.0 s
Threshold Method: Hard
Threshold Coefficient: 0.7
Threshold Selection: Rigrsure
Flapwise bending load, Blade #1. Wavelet: db6
59


Figure A. 13 Time-Frequency Toolbox Reassigned Spectrogram
Signal flpll65.mat
Point found @ D3 t = 109.406 +/- 5.0 s
flpl 104.mat Sampling Frequency 32 Hz
Note: Frequency is normalized per Nyquist i.e., 0.5 => 16 Hz
Time axis shows samples
RSP, Lh=25, Nf=128, log. scale, Thresholds eO05%
50 100 150 200 250 300
Time [s]
60


Figure A. 14 Detection of pseudo-singularities.
Edgewise Bending Load edgl 165.mat
Settings: Wavelet db6. Point found @ D1 t = 160.25 +/- 5.0 s
Threshold Method: Hard
Threshold Coefficient: 0.7
Threshold Selection: Rigrsure
Edgewise bending load, Blade #1; Wavelet, db 6
61


Period (seconds)
Figure A. 15 Wavelet Power Spectrum. Edgewise Bending Load edgl 165.mat
Mexican Hat Wavelet: Point found @ D1 1 = 160.25 +/- 5.0 s
edgl 101.mat 321 samples
Edgewise bending load, Blade #1
0.0625
0.125
0.25
123456789 10
Time (seconds)
62


Figure A. 16 Detection of pseudo-singularities.
Edgewise Bending Load edgl 165.mat
Settings: Wavelet db6. Point found @ D21 = 381.969 +/- 5.0 s
Threshold Method: Hard
Threshold Coefficient: 0.7
Threshold Selection: Rigrsure
Edgewise bending load, Blade #1; Wavelet: db 6
63


Figure A. 17 Time-Frequency Toolbox Reassigned Spectrogram
Signal edgl 165.mat
Point found @ D2 t = 381.969 +/- 5.0 s
edgl 104.mat Sampling Frequency 32 Hz
Note: Frequency is normalized per Nyquist Le., 0.5 => 16 Hz
Time axis shows samples
RSP, Lh=25, Nf=128, log. scale. Thresholds e-005%
50 100 150 200 250 300
Time [s]
64


Figure A. 18 Detection of pseudo-singularities.
Edgewise Bending Load edgll65.mat
Settings: Wavelet db6. Point found @ D21 = 105.219 +/- 5.0 s
Threshold Method: Hard
Threshold Coefficient: 0.7
Threshold Selection: Rigrsure
Edgewise bending load, Blade #1; Wavelet: db 6
65


Figure A. 19 1-D Continuous Wavelet Transform. Signal edgl 165.mat
Point found @ D2 t = 105.219 +/- 5.0 s
Edgl 105.mat Sampling Frequency 32 Hz
Mexican Hat Wavelet
Sipui
Wavelet
Scale Values:
1US
[mesh
| Step by Step Mode
**.(> j r
Stop(>0J | 2
] 32
Cotaaton Mode:
[wt+fay scale+ abs
Analyze
Cotamap I*
Nb Cokn -U _1
Cknc 1
66


Figure A.20 Time-Frequency Toolbox Reassigned Spectrogram
Signal edgll65.mat
Point found @ D2 t = 105.219 +/- 5.0 s
edgl 105.mat Sampling Frequency 32 Hz
Note: Frequency is normalized per Nyquist i.e., 0.5 => 16 Hz
Time axis shows samples
RSP, Lh=25, Nf=128, log. scale, Thresholds e 50 100 150 200 250 300
Time [s]
67


Figure A.21 1-D Continuous Wavelet Transform. Signal edgl 165.mat
Point found @ D2 t = 105.219 +/- 5.0 s
Edgl 105.mat Sampling Frequency 32 Hz
WAVE Wavelet 1st Derivative of Gaussian
serf
Wavelet
Scale Values:
(step by Step Mode
Mj i
Step | >0) p 2
Max f<*128J j 5
CdoteUon Mode:
| ini* by scale* ate
Analyze j
Colormap
Mb. Colors
3
jj j zipF
Cbse
68


Figure A.22 1-D Continuous Wavelet Transform Signal edgl 165.mat
Point found @ D2 t = 105.219 +/- 5.0 s
Edgl 105.mat Sampling Frequency 32 Hz
RAMP Wavelet
Sipal
Wavelet
ScMeVeljet:

| Step by Step Mode
Me.(> v | i
S*p(>Q j 2~
CbfcMtion Mode: -
ft* fay icate+ abs
Analyze
Uop lie* j
Nb Cetot J j J" 128
.ffbagt x* y* Xf* &591JMI x| gffNsiH 1 Ckne
$5^ x- Y- xr-
69


Figure A.23 1-D Continuous Wavelet Transform. Signal edgll65.mat
Point found @ D2 t = 105.219 +/- 5.0 s
Edgll05.mat Sampling Frequency 32 Hz
Morlet Wavelet
Wavelet
Scale Values:

| rod
| Step by Step Mode
Mirr f> 0} | f
Step l > 0) | 2~
Max (<>126} | 32
Cofafaten Mode:
| f< +bp scale *abs_____________
Analyze
Cdormap
Nb. Odors
El
.il _l
3
*£
X* Y+ XT*
X- Y- Xf-

Close
70


Figure A. 24
RAMP Wavelet
71


Figure A. 25
WAVE Wavelet

Wavelet |*Mn J
"iv-'*.
> HfllraMnl ' fu 3
.it ---v

r:
_ Dm*v
v ' ... ,v
MonM&ngn
-
**, -r WVFN wavelet

V? i "
*-v\ Wavelets
* 'iT-.v
- £ - '> -'a .

,v.
a." -
rC--'-
'p\ ,vr .

fii&f'JQy-'.'


-i'-7r
i .
ose I
72


Appendix B Cannon Wind Eagle 300 Downwind Turbine Low Wind Speed
73


Figure B.l
FAFB Flapwise Bending Load Flex Beam Blade # A
t i i r
O'-----------1----------1----------1----------1----------1-----------
0 100 200 300 400 600 600
Time [s]
74


Figure B.2 Flapwise Bending Load Flex Beam fafbdl .mat
Discrete Wavelet Transform Daubechies 6
Signal
Wavelet
Level
1*... J
l10 -d
Analyze
S(abides
Compress
Histograms
De-nose
Display mode
j FtJ DecompoaUon
|lQ -}
P Show Sjrtheseed Sjg
Dose
75


Figure B.3 Detection of pseudo-singularities fafbdl.mat
Settings: Wavelet db6 point found @ D6 t = 106.11 +/- 5.0 s
Threshold Method: Hard
Threshold Coefficient: 0.5
Threshold Selection: Rigrsure
FAFB Flapwise Bending Load Flex Beam Blade # A; Wavelet, db 6
76


Figure B.4 Time_Frequency Toolbox fafb_dl.mat
Point found @ D6 t = 106.11 +/- 5.0 s
Sampling Frequency 200 Hz
Note: Frequency is normalized per Nyquist i.e., 0.5 => 100 Hz
Time axis shows samples
SP, l_h=25, Nf=128, log. scale, Thresholds e-005%
0.45
0.4
0.35
£ 0.3
u.
0.25
0.2
0.15
0.1
0.05
0
200 400 600 800 1000 1200 1400 1600 1800 2000
Time [s]
77


Figure B.5 1-D Continuous Wavelet Transform Signal fafb dl.mat
Part of the signal from ti = 100 s to t2 = 130 s
Fafb05s.mat Sampling Frequency 200 Hz
Morlet Wavelet
S^pai

ScMeVMjes:
\ (Step by Step Mode *j
M rtf>01 ] ]
Step(>0) | 16
Mm {<-2048} | 256
Cofcaabon Mode:
Analyze
Cokamap
Mb Colors
1*-.....J
j j ^
: £ >
Scale of^oor$ ftom mih to Ma:
Close
78


Figure B.6 Wavelet Power Spectrum fafb dl.mat
Part of the signal from ti = 100 s to t2 = 130 s
fafb05.mat Sampling Frequency 200 Hz
Morlet Wavelet
FAFB Flapwise Bending Load Flex Beam Blade #A
V)
T5
C
O
L)

A
-o
o
CL
0.0156
0.0312
0.0625
0.125
0.25
5
10 15 20 25 30
Time (seconds)
79


Load [kNm]
Figure B.7
BAFB Flapwise Bending Load Blade Shell Blade # A
0 100 200 300 400 500 600
Time [s]
80


Figure B.8 Flapwise Bending Load Blade Shell bafb_dl.mat
Discrete Wavelet Transform Daubechies 6
sawlet
mm
X* Y+ XY+
X- Y XT-
81


Figure B.9 Detection of pseudo-singularities bafbdl.mat
Settings: Wavelet coiflet 5 point found @ D7 t = 102.17 +/- 5.0 s
Threshold Method: Hard
Threshold Coefficient: 0.7
Threshold Selection: Rigrsure
BAFB Flapwise Bending Load Blade Shell Blade # A; Wavelet: coH5
82


Figure B.10 Time-Frequency Toolbox bafb_dl.mat
Point found @ D7 t = 102.17 +/- 2.0 s
Sampling Frequency 200 Hz
Mote: Frequency is normalized per Nyquist i.e., 0.5 => 100 Hz
Time axis shows samples
SP, Lh=35, Nf=128, log. scale, Threshold^e 83


Figure B.ll 1-D Continuous Wavelet Transform Signal bafb_dl.mat
Part of the signal from ti = 100 s to t2 = 130 s
bafb05.mat Sampling Frequency 200 Hz
Morlet Wavelet
Sipui
Wavelet
Scale Vaim:
]Stapbj> Step Mode 1
MM(>0) p 1
Step( IB
Mat (<-20481 [* 256
Cotaiabon Mode:
|W + by scale ! J
Analyze
Cofomap
Nb Colors
jJ _J J fia"
Qote
84


Figure B. 12 Time-Frequency Toolbox bafb_dl .mat
Part of the signal from ti = 100 s to t2 = 130 s
bafb05.mat Sampling Frequency 200 Hz
Note: Frequency is normalized per Nyquist i.e., 0.5 => 100 Hz
Time axis shows samples
85


Figure B.13 Wavelet Power Spectrum bafb_dl.mat
Part of the signal from ti = 100 s to t2 = 130 s
bafb05.mat Sampling Frequency 200 Hz
Morlet Wavelet
BAFB Flapwise Bending Load Blade Shell Blade # A
(ft
"O
C
o
L>
07
T3
O
" u
03
Q_
0.0156
0.0312
0.0625
0.125
0.25
5 10 15 20 25 30
Time (seconds)
86


Load [kNm]
Figure B.14
87


Figure B.15 Edgewise Bending Load Flex Beam faeb_dl.mat
Discrete Wavelet Transform Daubechies 6
Signal
WmH 1*
Level ln_ 3
Anafcae |
Statnto Compress
HtMograms De-noee
Dapiaj> mode:
j Ful Decomposition 3
atfevd | 10 3
P Show Synthesized Sig
Ckwe
88


Figure B.16 Wavelet Power Spectrum faeb_dl.mat
Part of the signal from ti = 100 s to t2 = 130 s
faeb05s.mat Sampling Frequency 200 Hz
Mexican Hat Wavelet
FAEB Edgewise Bending Load Flex Beam Blade # A
O
c
o
u
OJ
~a
o
o
Q.
0.0156
0.0312
0.0625
0.125
0.25
0.5
5 10 15 20 25 30
Time (seconds)
89


Appendix C Cannon Wind Eagle 300 Downwind Turbine High Wind Speed
90


Load [kNm]
Figure C. 1
FAFB Flapwise Bending Load Flex Beam Blade # A
T--------------------------1----------------------------1--------------------------T
I | i |
I I I I I
.51___________I___________I------------1-----------1-----------1-----------
0 100 200 300 400 500 600
Time [s]
91


Figure C.2 Flapwise Bending Load Flex Beam fafb_dl4.mat
Discrete Wavelet Transform Symlet 5
J5 ~3
L'"d |10 _J
Analyze j
Statistics Compress
Histograms Denoise
Display mode,
| Fti Decompoabon -I
at level j 10 J
r Show Synthesized Sig
dote
92