Citation |

- Permanent Link:
- http://digital.auraria.edu/AA00003682/00001
## Material Information- Title:
- Time-frequency analysis of the relationship between system response and excitation characteristics
- Creator:
- Jakubowski, Andrzej
- Publication Date:
- 1999
- Language:
- English
- Physical Description:
- viii, 180 leaves : illustrations ; 28 cm
## Subjects- 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.
## Record Information- 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 )
## Auraria Membership |

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) [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 Â£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 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/{Â£, )|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 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 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
Numbers hn are called filter coefficients of function cp, mother wavelet 1\i is generated
One of the possible approaches to wavelets is through 2 channel filter |