© Strojni{ki vestnik 47 (200 1 )1,28-44 ISSN 0039-2480 UDK 517.4:621.372.6:648.23 Pregledni znanstveni ~lanek (1.02) © Journal of Mechanical Engineering 47 (2001 )1,28-44 ISSN 0039-2480 UDC 517.4:621.372.6:648.23 Review scientific paper (1.02) Spremljanje trenutne frekven~ne vsebine pri zagonu pralnega stroja Monitoring the Instantaneous Frequency Content of a Washing Machine during Startup Igor Simonovski - Miha Bolte`ar Fourierjeva integralska transformacija je zelo uporabno orodje za analizo frekvenčnega stanja ustaljenih procesov. Pogosto pa se srečujemo tudi z neustaljenimi procesi, pri katerih običajne Fourierjeve transformacije ne moremo uporabiti. Uporabiti moramo druge metode za spremljanje frekvenčne vsebine. V tem prispevku sta predstavljeni okenska Fourierjeva transformacija in novejša zvezna valčna transformacija. Valčna transformacija je zaradi uporabe lokalno omejenih osnovnih funkcij primerna za opazovanje neustaljenih procesov. Uporabnosti omenjenih transformacij za spremljanje frekvenčne vsebine neustaljenega procesa smo ugotavljali na primeru zagona pralnega stroja. Izkazalo se je, da je zaznavnost trenutne vrtilne frekvence pri okenski Fourierjevi transformaciji slabša kakor pri zvezni valčni transformaciji. Raztros vrednosti pri okenski Fourierjevi transformaciji namreč ne omogoča zanesljive identifikacije trenutne vrtilne frekvence, ampak samo umestitev v določen frekvenčni interval. Z uporabo zvezne valčne transformacije smo bistveno zozili ta frekvenčni interval. © 2001 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: procesi neustaljeni, transformacije Fourierjeve okenske, transformacije valčne zvezne, valčki Gaborjevi, stroji pralni) The Fourier integral transform is a very useful tool for analyzing the frequency content of steady processes. When dealing with non-stationary processes, however, other methods for determining the frequency content must be applied. This paper deals with the windowed Fourier transform and, the more recent, wavelet transform. The windowed Fourier transform uses basic functions that have an unlimited definition range and requires multiplication of the observed process with a time-limited window function to be able to detect local non-stationarities. The wavelet transform uses basic functions that have a limited definition range for the same purpose. In this paper we compare the ability of the windowed Fourier transform and the continous wavelet transform to monitor the frequency content of a non-stationary process-washing-machine startup. The results show that the windowed Fourier transform is inferior to the continous wavelet transform. The wide spread of windowed Fourier transform values only makes it possible to roughly determine the instantaneous drum-spin frequency band. Using the continous wavelet transform we were able to determine the instantaneous drum-spin frequency more accurately. © 2001 Journal of Mechanical Engineering. All rights reserved. (Keywords: non-stationary processes, windowed Fourier transform, continous wavelet transform, Gabor wave-let, washing machine) 0 UVOD Pri analizi nihanj se že vrsto let uporabljajo spektralne analize, ki temeljijo na Fourierjevi integralski transformaciji. Fourierjeva integralska transformacija je zaradi uporabe sinusnih in kosinusnih osnovnih funkcij primerna predvsem za opazovanje ustaljenih procesov. Če se v opazovanem procesu pojavijo lokalne neustaljenosti, to vpliva na Fourierjevo integralsko transformacijo pri vseh frekvencah. Ta lastnost je pri 0 INTRODUCTION Spectral analyses, based on the Fourier integral transform, have been present in the field of vibration analysis for years. Due to the sine and cosine basic functions, the Fourier integral transform is mostly appropriate for the analysis of stationary processes. If a local non-stationary appears in the observed process its influence is spread over the Fourier integral transform values grin^SfcflMISDSD VBgfFMK stran 28 I. Simonovski - M. Bolte`ar: Spremljanje trenutne - Monitoring the Instantaneous opazovanju neustaljenih procesov nezaželena, saj nas pri teh procesih pogosto zanima trenutek pojava spremembe oziroma neustaljenosti v procesu. Prilagoditev Fourierjeve integralske transformacije za opazovanje neustaljenih procesov je okenska Fourierjeva transformacija [1]. Pri tej opazovani proces pomnožimo s časovno omejeno okensko funkcijo. Zunaj območja definiranosti okenske funkcije je zmnožek procesa in okenske funkcije enak nič. V novejšem času se je pojavila valčna transformacija, pri kateri vrednosti ft) koreliramo s skupino funkcij, katerih definicijsko območje je definirano na končnem območju [a, b]. Te funkcije lahko hkrati premikamo po časovni osi in skaliramo po frekvenčni osi [2]. Imenujemo jih valčki in jih označujemo s y(t). Pomembna lastnost valčne transformacije je časovno-frekvenčno odvisen raztros valčka (sl. 1). Raztros valčka včasovnem območju je premo sorazmeren s skalo valčka, medtem ko je raztros valčka v frekvenčnem območju obratno sorazmeren s skalo valčka. Ta lastnost valčne transformacije omogoča prilagajanje frekvenčne ločljivosti. Zaradi uporabe valčnih funkcij, ki imajo lokalno omejeno definicijsko območje, je valčna transformacija občutljiva za lokalne neustaljenosti. To je zelo pomembno pri analizi neustaljenih procesov. Zato je valčna transformacija primerna za zaznavanje napak v zobniških pogonih [3], [4], strojnih napravah [5], [6] in kompozitnih ploščah [7]. Na področju dinamike se valčna transformacija uveljavlja pri identifikaciji parametrov dinamskih sistemov ([8] do [12]), izračunu časovno odvisne frekvenčne odzivne funkcije [13], linearizaciji nelinearnih sistemov [14], [15] in zaznavanju drdranja pri odrezovalnem procesu [16]. V tem prispevku primerjamo primernost okenske Fourierjeve in zvezne valčne transformacije za spremljanje frekvenčne vsebine neustaljenega procesa. Izbrani neustaljeni proces predstavlja meritev pospeškov pralne grupe pralnega stroja med zagonom. Pri dosedanjih analizah meritev pralnega stroja smo se osredotočili predvsem na spremljanje ustaljenega stanja. at all frequencies and so the time at which the non-stationary appeared cannot be determined. To analyze a non-stationary process the windowed Fourier transform, which multiplies the observed process with a time-limited window function can be used [1]. Outside the window’s definition range, the product of the process and the window func-tion is equal to zero. In recent years the wavelet transform has played an important role in analyzing non-station-ary processes. This transform correlates the ob-served process, f(t), with a family of functions de-fined on a finite interval [a, b]. These functions, called wavelets, y(t), can be simultaneously trans-lated in time and scaled in the frequency domain [2]. One important property of the wavelet transform is its varying time-frequency resolution, Figure 1. In the time domain, the wavelet spread is proportional to the wavelets’ scale, while in the frequency domain the spread is inversely proportional to the wave-lets’ scale. This property makes it possible to vary the frequency or time resolution. Because the wave-let transform uses basic functions with a limited definition area, the wavelet transform is sensitive to the local non-stationarities. This is very important in the analysis of a non-stationary process. Thus far, the wavelet transform has been used for fault de-tection in gears [3], [4], machines [5], [6] and com-posite plates [7]. In the field of dynamics, the wave-let transform is used for parameter identification ([8] to [12]), calculating time-dependent frequency re-sponse functions [13], linearization of non-linear systems [14], [15] and chatter detection during cut-ting [16]. In this paper we compare the ability of the windowed Fourier transform and the continous wavelet transform to monitor the frequency content of a non-stationary process—washing-machine startup. During startup the accelerations of the washing-machine complex have been measured. In w s g K..............................................................................................................I 7 t \ st sw h/S0 0 t Sl. 1. Časovno-frekvenčna ločljivost okenske Fourierjeve (levo) in valčne transformacije (desno) Fig. 1. The time-frequency resolution of the windowed Fourier (left) and the wavelet transform (right) gfin^diJJlMieCSD stran 29 I. Simonovski - M. Bolte`ar: Spremljanje trenutne - Monitoring the Instantaneous Narejene so bile spektralne analize drugega reda [18], pri katerih se je pokazalo, da je največ moči signala katerekoli merjene kinematične spremenljivke zbrane pri frekvenci ožemanja. Bispektralne analize meritev pralnega stroja so pokazale navzočnost kvadratičnih nelinearnosti [19]. Izkazalo se je, da druga, tretja in četrta harmonska niso v celoti samostojna, temveč so delno tudi posledica kvadratičnega sklapljanja faz. Narejena je bila tudi analiza odzivov modela in meritev pralnega stroja v faznem prostoru [18]. Pregled lastnega dosedanjega dela na področju analiziranja in modeliranja dinamike pralnega stroja je podan v [20] in [21]. 1 TEORETIČNE OSNOVE 1.1 Okenska Fourierjeva transformacija Fourierjeva integralska transformacija funkcije f(t) je definirana kot [17]: preliminary analyses of the washing machine dynamics we focused mainly on the stationary. Second-order spectral analyses showed that for all measured signals the drum-spin frequency has by far the high-est power level [18]. Bispectra analyses revealed the presence of quadratic-order non-linearities [19]. Sec-ond, third and fourth harmonics were partially generated by quadratic phase coupling. Analyses in phase space have been applied to responses of both the model and the real system [18]. A review of our previ-ous work relating to the analysis of washing-machine non-linear dynamics can be found in [20] and [21]. 1 THEORETICAL FOUNDATIONS 1.1 The windowed Fourier transform The Fourier integral transform of the func-tion is defined as [17]: +» f(w)=Lf(t ¦dt +» j f ( t )-[cos ( wt ) -i-sin ( wt )]-dt (1), kjer pomeni i=-1, w pa krožno frekvenco v rad/s. Vrednost f( w) je v splošnem kompleksno število. Realni del f( w) opredeljuje amplitudo nihanja kosinusne funkcije pri frekvenci w, imaginarni del f( w) pa amplitudo nihanja sinusne funkcije pri frekvenci w. Kvadrat absolutne vrednosti f( w) poda moč funkcije f(t) pri frekvenci w. Iz izraza (1) je tudi razvidno, da lokalna neustaljenost vpliva na frekvenčno transformiranko f( w) pri vseh frekvencah w. To je posledica dejstva, da je osnovna funkcija e- iwt definirana na območju (-oo, +oo) zato h končni vrednosti integrala (1) prispevajo vsi dogodki, ne glede na čas njihovega pojava. Časovno-frekvenčna ločljivost Fourierjeve integralske transformacije je konstantna (sl. 1) [17]. Pojav okenske Fourierjeve transformacije Sf(u,w) [1] pomeni prvi poskus omejitve vpliva oddaljenih vrednosti f(t) na frekvenčno transformiranko f (w). Tu vrednosti f(t) pomnožimo z okensko funkcijo g (t)= g(t-u), ki je definirana na končnem območju [a, b], zunaj tega območja pa je enaka nič. V skladu s tem se spremenijo integracijske meje, kar omogoči opazovanje neustaljenih pojavov. +» where i is -1 and w is the circular frequency in rad/s. The value of f (w) at a certain frequency w is, in general, a complex number. The real and imaginary parts of f( w) represent the amplitude of the sine and cosine functions, oscillating at frequency w. The absolute value of f( w) squared is the power at the frequency w. It is evident from (1) that the local non-stationary affects the frequency transform of f( w) at all frequencies w, a consequence of the basic function e i¦wt. Since its definition interval is (-oo, +oo), the integration is carried out over this whole interval. The time-frequency resolution of the Fourier integral transform is constant and is presented in Figure 1, [17]. The windowed Fourier transform reduces the influence of time-distant events on the frequency transform, f( w), by multiplying the f(t) with the time-limited window function g (t)= g(t-u). The window function, g (t), is defined on a finite interval [a, b] and is equal to u zero outside this interval. As a consequence, the integral limits change, which makes the windowed Fourier transform useful for observing non-stationary processes. Sf ( u, w ) = jWf ( t )-gu* ( t )-e-t-dt = j f ( t )-gu* ( t )-e-t-dt J-» meje g(t) Ja (2). V gornjem izrazu pomeni znak * kompleksno konjugacijo. Graf gornjega izraza pogosto imenujemo spektrogram. V praksi običajno ne poznamo vrednosti funkcije f(t) pri vseh zahtevanih časih (-oo, +oo), temveč samo v določenem časovnem območju [0, T]. Funkcijo f(t) lahko razširimo na celotno območje (-oo, +oo), tako da predpostavimo, da je funkcija f(t) periodična s periodo T [17]: The symbol * in expression (2) denotes a complex conjugation. The graph of Sf(u,w) is often referred to as a spectrogram. In practice, the function f(t) is usually only known in the time interval [0, T]. In this case f(t) can be extended to the interval (-oo, +oo) with periodization by T [17]: grin^SfcflMISDSD VH^tTPsDDIK stran 30 I. Simonovski - M. Bolte`ar: Spremljanje trenutne - Monitoring the Instantaneous f(t) = f(t±k-T), 00 (4), (5), (6). Izraza (5) in (6) lahko združimo, če uporabimo Expressions (5) and (6) can be combined by complex kompleksni zapis: notation: Xk =ak-i-bk Ker velja: lahko zapišemo naslednji izraz za X: Since: 2-K-k-t] (2-K-k-t e = cos------------ - i ¦ sin TJ V T the following expression can be written: 1 T T r T ¦dt, k>0 (7). (8), (9). V praksi običajno ne poznamo vseh In practice f(t) is usually known only for vrednosti f(t). Opazovani proces vzorčimo s discrete time indexes. The observed process f(t) is frekvenco vzorčenja fs=1/Dt, tako da poznamo samo sampled with the sampling frequency fs=1/Dt, there- vrednosti pri diskretnih časih f=f(k . Dt), fore only the values fk=f(k . Dt), k=0,1,2,...,N-1 are k=0,1,2,...,N-1. Oznaka Dt pomeni časovni razmik known. The symbol N denotes the number of sampled med dvema sosednjima diskretnima točkama, N pa points while Dt stands for the time delay between število točk diskretizacije, sl 2. Integral (9) sampled points, Figure 2. The integral (9) is calcu- izračunamo z numerično integracijo, kjer periodo T lated using numerical integration, where period T is zapišemo kot T=N . Dt: written as T=NDt: 1 j=n-1 -if2--k(j-Dt)] X j=0 1 j=N -1 j=0 k n ^ N (10), (11). 0 1 2 3 N -1 k Sl. 2. Diskretizacija zvezne funkcije Fig. 2. Discretized continuous function isfFIsJBJbJJIMlSiCšD I stran 31 glTMDDC ¦ I. Simonovski - M. Bolte`ar: Spremljanje trenutne - Monitoring the Instantaneous Izraz (11) poznamo pod imenom “diskretna Fourierjeva transformacija”. Če se na začetku signala f(t) pojavi neustaljenost, ki kasneje izgine, ta neustaljenost vpliva na vse člene ak, bk. Ta zakonitost izhaja iz izrazov (5) in (6). Členi ak in bk pomenijo diskretno Fourierjevo transformiranko diskretiziranega signala fk. Začasna neustaljenost v signalu torej vpliva na koeficiente celotne diskretne Fourierjeve transformacije. Ta lastnost otežuje analizo lokalnih lastnosti pojava, zaradi česar je Fourierjeva tansformacija primerna predvsem za analizo ustaljenih pojavov. 1.2 Zvezna valčna transformacija Valček y(t) je normirana funkcija z nično povprečno vrednostjo: y(t) J -CO Expression (11) is known as the “discrete Fourier transform” (DFT). Coefficients ak and bk can therefore be obtained from classical DFT routines. If non-stationary appears at the beginning of the signal and later vanishes, it influences all coefficients ak and bk. This property is the result of integrating over the whole period T ((5) and (6)) and makes it difficult to analyze local events in the signal. Consequently, the Fourier transform is used primarily for analyzing stationary pro-cesses. 1.2 The continuous wavelet transform The wavelet y(t) is a normalized function with an average value of zero: ¦dt = 1 J y(t)-dt = 0 (12), (13). Skupino valčnih funkcij, ki jih uporabimo pri zvezni valčni transformaciji, dobimo s skaliranjem valčka y(t) po frekvenčni osi s koeficientom “s” (skalirni koeficient) in s premikanjem valčka po časovni osi s parametrom premika “u”. Skaliran in premaknjen valček označimo s yu (t): The continuous wavelet transform uses a family of wavelet functions. Scaling a wavelet function y(t) by “s” and translating it by “u” creates this family of wavelet functions. These two coefficients are called the scaling factor and the translation parameter, respectively. The scaled and translated wavelet is denoted as yu,s(t): y A = 1 y t-u) (14). u,s () J~s l s J Tako skaliran valček ohrani lastnost and remains normalized. On the time scale yu (t) is normiranosti. Na časovni osi je središče yu (t) pri “u”, centered at “u”. The frequency resolution is propor- frekvenčna razsežnost pa je sorazmerna s skalo “s”. tional to scale “s”. The frequency transform of the Frekvenčno transformacijo valčka yu ( t) dobimo z uporabo wavelet yu (t) is obtained from Fourier transform rules pravil o tanslaciji in skaliranju Fourierjevih tansformirank: for translation and scaling: kjer y/(w) pomeni Fourierjevo transformacijo nepremaknjenega in neskaliranega valčka y(t). Zvezna valčna transformacija Wf(u,s) je definirana kot: •Vs •ys(s-w) (15), where y( w) represents the Fourier transform of untranslated and unscaled y(t). The continuous wavelet transform Wf(u,s) is defined as: Wf ( u, s ) = \+lf ( t )-yu*s ( t )-dt = \+lf ( t )- fs ¦y t -u dt (16). Graf gornjega izraza pogosto imenujemo skalogram. Valček yu (t) je funkcija, ki je definirana na končnem območju [a, b], zato je valčna transformacija občutljiva na lokalne neustaljenosti. Če se v funkciji f(t) v določenem trenutku pojavi prehodni pojav, se bo pokazal le na valčni transformiranki v okolici tega pojava. Prehodni pojav ne bo vplival na valčno transformiranko pri časovnih vrednostih “u”, ki so oddaljene od trenutka njegovega pojava in trajanja. Valčne funkcije so lahko realne ali kompleksne. Realne valčne funkcije uporabljamo, če želimo zaznati močne prehodne pojave. Kompleksni valčki so primerni za časovno opazovanje frekvenčnih sprememb v prehodnih pojavih. The graph of Wf(u,s) is often referred to as a scalogram. Because the wavelet yu,s(t) is defined on a finite interval [a, b], the continuous wavelet transform is sensitive to local events (non-stationarities). The local transient reflects on the continuous wave-let transform only when the transient appears and only for the duration of the transient. The continous wavelet transform that is distanced in time from a local transient is unaffected by it. Wavelet functions can be real or complex, real wavelets are often used to detect sharp signal transitions. Measuring the time evolution of frequency transients requires the use of a complex wavelet which can separate amplitude and phase components. grin^SfcflMISDSD VH^tTPsDDIK stran 32 I. Simonovski - M. Bolte`ar: Spremljanje trenutne - Monitoring the Instantaneous Inverzna zvezna valčna transformacija je definirana kot: The inverse continous wavelet transform can be defined as: pri čemer je: f ( t ) = 1f+"f+ Wf ( u, s ).1 where: C=\ y( w)-dw<+oo y J0 1 (t - u ) du ds w (17), (18). Dodatno se zahteva, da je valček yu (t) realna funkcija [17]. 1.3 Izračun zvezne valčne transformacije Obravnavajmo diskreten signal z N točkami: Expression (17) requires the wavelet yu,s(t) to be a real function [17]. 1.3 Calculation of the continuous wavelet transform Suppose we have a discrete signal with N fk=f(k-Dt), Signal fk je poznan samo v časovnem območju: points: k = 0, 1, 2,K,N-1 Signal fk is known only in the time interval: t0=0 ^ -e (s2-n) (29). Gaborjev valček spada med analitične The Gabor wavelet belongs to the family of valčke. Za Gaborjev valček lahko dobimo povezavo analytical wavelets. The relationship between the grin^SfcflMISDSD ^BSfiTTMlliC | stran 34 2 3 4 5 I. Simonovski - M. Bolte`ar: Spremljanje trenutne - Monitoring the Instantaneous med skalo in frekvenco iz Fourierjeve integralske transformacije skaliranega in premaknjenega Gaborjevega valčka (22): scale and the frequency can be obtained from the Fourier integral transform of the scaled and trans-lated Gabor wavelet (22): 1 ( h? s2ls2 ^us w, s, h ) = ( 4-n-s2-s2 )-e- iwu -e1w s 2 (30). Velja torej naslednja povezava med skalo in frekvenco: The relationship between the scale and frequency is: w = —, f =-------- s 2-n-s (31). Pri diskretnih signalih mora biti izpolnjen Nyquistov kriterij: f max - iz česar lahko izračunamo vrednost koeficienta h: When using discretized signals, the Nyquist crite-rion must be satisfied: 1 (32), _____ fs 2-Dt 2 and the value of the h coefficient can be calculated: h = pri čemer s i izberemo glede na potrebno časovno oziroma frekvenčno ločljivost (sl. 1). 2 ANALIZA ZAGONA PRALNEGA STROJA Pri zagonu pralnega stroja smo merili pospeŠke pralne grupe v vertikalni in horizontalni smeri (sl. 4). Pralna grupa je del pralnega stroja ter jo sestavljajo kad z rotirajočim bobnom, dodana utež, elektromotor in vibroizolacija V bobnu je na največjem polmeru pritrjena ekscentrična masa. Pri vrtenju bobna povzroči ekscentrična masa neuravnoteženost, s katero lahko simuliramo najslabšo mogočo porazdelitev perila pri pranju. Podrobnosti o meritvah so podane v [18] in [19]. Uporabljena je bila frekvenca vzorčenja fs=1000 Hz. Število diskretnih točk je bilo 30000, kar pomeni časovno 30 s. Izmerjenim 30000 točkam smo na koncu dodali še 2768 diskretnih ničel, tako da smo dobili celotno dolžino 32768=215 diskretnih točk. Pri zagonu smo spremljali časovno spreminjanje frekvenčne slike pospeškov pralne grupe pralnega stroja. Časovno spreminjajočo se frekvenčno sliko smo dobili iz pospeškov z izračunom okenske Fourierjeve (spektrogram) in zvezne valčne Dt (33), Parameter smin is selected in accordance with the de-sired time or frequency resolution, see Figure 1. 2 WASHING-MACHINE STARTUP ANALYSIS Horizontal and vertical accelerations of a washing-machine complex were measured, Figure 4. The washing-machine complex is made up of: the tub in which the drum is rotating; addi-tional weights; the electromotor attached to the tub and the suspension arms holding the tub as vibroisolation. In the drum of the washing-ma-chine complex an excentrical mass was fixed at the maximum radius, to unbalance the rotating parts and simulate the worst possible laundry distribution. Details of the measurements are given in [18] and [19]. A sampling frequency of fs=1000 Hz was used and the length of all measurements was 30000 discrete points, which is equivalent to 30 s. 2768 discrete zeroes were appended to the 30000 measured points, giving a final length of 32768=215 discrete points. During startup, the spectral content of the washing-machine accel- nabojni predojačevalnik charge amplifier osciloskop osciloscope Sl. 4. Shema merilne verige pralnega stroja Fig. 4. The experimental set-up Sin^ObJJPsflDslJSD I stran 35 glTMDDC I. Simonovski - M. Bolte`ar: Spremljanje trenutne - Monitoring the Instantaneous transformacije (skalogram). Spremljanje frekvenčne slike z dvema različnima integralskima transformacijama je omogočilo vpogled v razlike med obema izračunoma. Ker je okenska Fourierjeva transformacija v primerjavi z valčno transformacijo že po svoji naravi manj primerna za opazovanje neustaljenih pojavov, smo okensko Fourierjevo transformacijo izračunali na dva načina: a) Z uporabo 512 vhodnih točk v rutini FFT, brez dodajanja ničel. Frekvenčna ločljivost pri teh izračunih jeAf =1,953 Hz. b) Z uporabo 512+1536=2048=211 vhodnih točk v rutini FFT. 512 točk je predstavljalo točke meritve, preostalih 1536 točk pa je bilo ničel. Ničle smo dodali z namenom izboljšanja frekvenčne ločljivosti. Frekvenčna ločljivost pri teh izračunih je Äf=0,488 Hz. Uporabili smo Gaussovo okensko funkcijo, ki smo jo premikali po signalu meritve za 10 diskretnih točk v smeri naraščanja časovnih indeksov. Zvezno valčno transformacijo smo izračunali z uporabo Gaborjevega valčka, ki nastane iz Gaussove okenske funkcije. Zaradi primerljivosti so spektrogrami in skalogrami izračunani z istima frekvenčnima ločljivostima (1,953 in 0,488 Hz). Podatki o izračunu spektrogramov so podani v preglednici 1, podatki o izračunu skalogramov pa v preglednici 2. erations was monitored by calculating the win-dowed Fourier transform (spectrograms) and the continuous wavelet transform (scalograms). By using two different approaches we were able to compare both methods. The windowed Fourier transforms were calculated with two frequency resolutions: a) Number of points in the FFT procedure 512, no zero padding. Frequency resolution Df=1.953 Hz; b) Number of points in the FFT procedure 512+1536=2048=211, 512 measured points and 1536 zero padded points. Frequency resolution Df=0.488 Hz. The Gaussian window function was used. Window functions were time-shifted by 10 discrete points in the direction of increasing time indexes. Continous wavelet transforms were calculated us-ing the Gabor wavelet, which comes from the Gaussian window function. Spectrograms and scalograms were calculated with two frequency resolutions (1.953 and 0.488 Hz). Details of the spectrogram and scalogram calculations are given in Tables 1 and 2. Preglednica 1. Podatki o parametrih izračuna spektrogramov Tabel 1. Spectrogram calculation data okenska funkcija window function zamik okenske funkcije window function shift širina okenske funkcije brez ničel window function width without zeroes število dodanih ničel added zeroes skupno število točk overall number of points frekvenčna ločljivost frequency resolution Spektrogram Df =1,953 Hz Spectrogram Gaussova Gaussian 10 [diskretne točke] 10 [discrete points] 1,953 Hz Spektrogram Df =0,488 Hz Spectrogram Gaussova Gaussian 10 [diskretne točke] 10 [discrete points] 512 512 0 1536 512 2048 0,488 Hz Preglednica 2. Podatki o parametrih izračuna skalogramov Tabel 2. Scalogram calculation data Skalogram Df =1,953 Hz Scalogram Skalogram Df =0,488 Hz Scalogram valček wavelet gabor gabor s min 2 2 s 1,5 1,5 spodnja meja računanega frekvenčnega območja lower frequency bound 2 Hz 2 Hz zgornja meja računanega frekvenčnega območja higher frequency bound 50 Hz 50 Hz frekvenčna ločljivost frequency resolution 1,953 Hz 0,488 Hz gTifMsfcflMISDSD VH^tTPsDDIK stran 36 I. Simonovski - M. Bolte`ar: Spremljanje trenutne - Monitoring the Instantaneous 2.1 Primerjava spektrogramov in skalogramov Na slikah 5 do 8 so prikazani spektrogrami in skalogrami horizontalnih pospeškov pralne grupe pralnega stroja pri zagonu. Slika 5 prikazuje magnitudo spektrograma. Zaradi preglednosti je prikazano le frekvenčno območje od 0 do 50 Hz. Levi del slike prikazuje skalogram v prostoru, desni del pa tlorisno projekcijo. Frekvenčna ločljivost je 1,953 Hz. Na sliki lahko vidimo greben, katerega magnituda in frekvenca s časom naraščata. Greben je dejansko magnituda koeficientov Fourierjeve transformacije, sestavljajo ga močnostni spektri izračunani pri določenih časovnih indeksih. Zaradi zagona frekvenca s časom narašča. Amplituda nihanja se povečuje, posledično pa se večajo tudi koeficienti izračunane okenske Fourierjeve transformacije. Poleg povečevanja frekvence zaganjanja drugih zaznavnih frekvenčnih komponent ni opaziti. Tu je treba poudariti, da je magnitudna os prikazana na linearni in ne na logaritemski skali. Po 30 s lahko opazimo, da je magnituda spektrograma enaka nič. To je posledica dejstva, da je imela uporabljena meritev le 30000 točk, kar časovno ustreza 30 s. Za 30000 diskretnimi točkami pa smo v signal dodali dodatne ničle, tako da smo dobili skupno dolžino signala 32768 točk. Po 30 s torej ni več amplitude nihanj, magnitudni spektri morajo biti enaki nič pri vseh frekvencah, to pa lahko tudi vidimo na izračunanih spektrogramih. Zaradi uporabe 215=32768 točk smo lahko za izračun skalogramov uporabili postopek hitre Fourierjeve transformacije, kar je pospešilo numerični izračun. Opazimo lahko, da je slika tlorisne projekcije spektrogramov zelo nejasna. Ta nejasnost je posledica slabe frekvenčne ločljivosti in uporabe sinusnih (kosinusnih) funkcij, ki jih uporablja Fourierjeva transformacija za dekompozicijo signala. Raztros moči na sosednje frekvence bi lahko zmanjšali z boljšo izbiro okenske funkcije, vendar bi s tem zmanjšali primerljivost spektrograma in skalograma. Tako pa smo pri okenski Fourierjevi in zvezni valčni Ji, 50.00 */ .32.76 2.1 Comparison of spectrograms and scalograms Figures 5 through 8 show the spectrograms and scalograms of the washing-machine complex’s horizontal accelerations during startup. Figure 5 depicts the spectrogram magnitude. For clarity reasons only the frequency range 0 to 50 Hz is presented. The left part of the picture shows a surface plot presentation while the right part shows a 2D plot of the spectrogram. The frequency resolution is 1.953 Hz. A ridge representing the maximum value of the magnitude of the wavelet function at an appropriate scale (frequency) and translation (time index) can be clearly seen. Due to the startup, the amplitude of vibrations and the revolutions of the drum are increasing. The spectrogram and the ridge represent the magnitude of the coefficients of the Fourier transform at a given time index. The spectrogram magnitude is therefore directly dependent upon the amplitude of the vibrations. The increase in the amplitudes of vibrations manifests itself as the increasing magnitude of the ridge. The frequency of the ridge represents the instantaneous frequency with the highest power. Since this equals the rotating frequency of the drum, the ridge frequency represents the instantaneous rotation frequency of the drum. With the exception of the drum frequency, other frequencies do not have significant power. We must point out, however, that the magnitude is shown on a linear and not on a logarithmic scale. After 30 s the magnitude of the spectrogram vanishes. This is because after 30 s vibrations were no longer recorded and zeros were added into the signal. These zeros state that in the time period after 30 s, the amplitude of the vibrations is zero. The spectrogram therefore must be zero and this can be seen in the figures. Because 215=32768 discrete points were used for calculating the spectrogram we were able to use the fast Fourier transform procedure which significantly reduced the calculation time. The 2D plot of the spectrogram is Čas[s] Time [s] 0.00 6.55 13.10 19.65 26.21 32.76 „ „„ M M 30.00 K K 1 g* Sl. 5. Spektrogram, frekvenčna ločljivost 1,953 Hz Fig. 5. The spectrogram, frequency resolution 1.953 Hz jäfinHJObJJFsflDjSsCsD I stran 37 glTMDDC I. Simonovski - M. Bolte`ar: Spremljanje trenutne - Monitoring the Instantaneous transformaciji uporabili zelo sorodno Gaussovo very unclear, this can be attributed to the low fre- okensko funkcijo in Gaborjevo valčno funkcijo. quency resolution and the Fourier transform basic Frekvenčno ločljivost smo skušali izboljšati functions (sine and cosine). The power leakage tako, da smo širino vsake okenske funkcije povečali could be reduced with the selection of a better win- na 2048 diskretnih točk, in sicer tako, da smo za 512. dow function but that would make it harder to com- diskretno točko dodali ničle. S tem smo izboljšali pare spectrograms with scalograms. In this study a frekvenčno ločljivost iz 1,953 Hz pri 512 diskretnih točkah comparable Gaussian window function and Gabor na 0,488 Hz pri 2048 diskretnih točkah. Frekvenčna wavelet were used which enabled us to compare the ločljivost se je izboljšala za štirikrat. Slika 6 prikazuje calculated spectrograms and scalograms. ustrezno izračunano magnitudo spektrograma. Zaznati Next, we tried to improve the readability of je izboljšanje jasnosti frekvenčne slike. Na the 2D plots of the spectrogram. We increased the tridimenzionalni sliki spektrograma je predvsem opazno frequency resolution by appending zeros to the enakomernejše naraščanje strmine grebena length of each window function. The window length spektrograma. Na tlorisni projekciji se je zmanjšalo was increased to 2048 discrete points. The frequency število kvadratastih struktur, ki jih vidimo na sliki 5 resolution increased by a factor of 4 (from 1.953 to (tlorisna projekcija). Širina grebena se ni bistveno 0.488 Hz). Figure 6 depicts the corresponding spec- spremenila, kar je verjetno posledica nezmožnosti trogram. On the surface plot the slope of the ridge is Gaussove okenske funkcije, da bistveno zmanjša more regular than before. The clarity of the 2D pro- pretakanje moči med sosednjimi frekvencami. Dodatno jection also improved, however, the width of the ridge na širino grebena vpliva neustaljenost signala. Tudi was not changed significally The width of the ridge na tej sliki vidimo, da so po 30 s magnitude is proportional to the power leakage and the prob- spektrograma enake nič, kar je posledica dejstva, da able explanation could be that the Gaussian window smo v tem območju signalu dodali ničle. function does not sufficiently reduce the power leak- Slika 7 prikazuje magnitudo skalograma age. Since after 30 s only zeros can be found in the horizontalnih pospeškov pralne grupe pralnega stroja signal, the magnitudes of the spectrogram in this re- pri zagonu. Skalogram je izračunan s frekvenčno gion are equal to zero. ločljivostjo 1,953 Hz. Levi del slike prikazuje skalogram The scalogram magnitude of the washing- v prostoru, desni del pa tlorisno projekcijo. Na sliki machine complex’s horizontal accelerations during lahko vidimo greben, katerega magnituda in frekvenca startup is shown in Figure 7. The left part of the pic- s časom naraščata. Tlorisna projekcija je zelo nejasna, ture shows a surface plot presentation while the right tako da lahko zaznamo le zelo grob vzorec naraščanja part shows a 2D plot of the scalogram. The scalo- frekvence. Slika 8 prikazuje magnitudo skalograma, gram is calculated with a frequency resolution of 1.953 izračunanega s štirikrat boljšo frekvenčno ločljivostjo Hz. Here again, the ridge with rising magnitude and (0,488 Hz). Tako izračunana frekvenčna slika je frequency can be seen. The 2D projection is very bistveno jasnejša. Magnituda in frekvenca zelo lepo unclear and only a coarse pattern of the frequency- vidnega grebena s časom naraščata. Greben time dependence can be observed. The scalogram predstavlja magnitudo valčnih funkcij določene skale magnitude with improved frequency resolution (0.488 (frekvence) in časovne premaknitve (časovni indeks). Hz) is shown in Figure 8. The readability of the 2D Greben je ozek, kar pomeni, da je odtekanje moči na projection has improved significantly. The magnitude sosednje frekvence majhno. Tudi na tlorisni projekciji and frequency of the ridge increase with time. The skalograma je naraščanje frekvence lepo razvidno. width of the ridge is small, suggesting small power Čas[s] Time [s] Sl. 6. Spektrogram, frekvenčna ločljivost 0,488 Hz Fig. 6. The spectrogram, frequency resolution 0.488 Hz ^BSfiTTMlliC | stran 38 i I. Simonovski - M. Bolte`ar: Spremljanje trenutne - Monitoring the Instantaneous Čas [s] Sl. 7. Skalogram, frekvenčna ločljivost 1,953 Hz Fig. 7. The scalogram, frequency resolution 1.953 Hz Čas [s] Tims [s] Sl. 8. Skalogram, frekvenčna ločljivost 0,488 Hz Fig. 8. The scalogram, frequency resolution 0.488 Hz Prikazano je frekvenčno območje od 2 Hz do 50 Hz. leakage. The frequency range of the scalogram is 2 to Po 30 s ni več nihanja, zato je magnituda skalograma 50 Hz. After 30 s there are no vibrations present, v tem območju enaka nič. therefore the scalogram vanishes. 2.2 Primerjava grebenov spektrogramov in 2.2 Comparison of the ridges of the spectrogram skalogramov and scalogram Na slikah 9 do 14 so prikazani grebeni Figures 9 through 14 show spectrogram and spektrogramov in skalogramov horizontalnih scalogram ridges of the washing-machine complex’s pospeškov pralne grupe pralnega stroja. Na teh horizontal accelerations during startup. The differ- slikah so razlike med posameznimi načini izračuna ences between windowed Fourier and continuous frekvenčne vsebine še dodatno razvidne. Greben wavelet analyses are here emphasized. The ridges dobimo tako, da pri določenem časovnem indeksu are obtained when the maximum magnitude values of prikažemo samo največje vrednosti magnitude the spectrogram (scalogram) at a given time index are spektrograma oziroma skalograma. Iz grebena displayed. It is very difficult to determine the fre- spektrograma s frekvenčno ločljivostjo 1,953 Hz je quency-time dependence from the spectrogram ridge skoraj nemogoče izluščiti vrednost frekvence pri with a frequency resolution of 1.953 Hz. The spectro- posameznem časovnem indeksu. Greben gram ridge with a frequency resolution of 0.488 Hz spektrograma s frekvenčno ločljivostjo 0,488 Hz represents an improvement over the former ridge, but predstavlja izboljšanje, vendar je še vedno neberljiv. it is still dificult to read. In spite of the improved Navkljub izboljšanju frekvenčne ločljivosti se širina frequency resolution, the width of the ridge has not grebena ni bistveno zmanjšala. V nasprotju s prvima changed significantly. In contrast, the scalogram dvema greben skalogramov natančneje prikaže ridges are much more precise. The ridge scalogram, spreminjanje frekvence pri zagonu pralnega stroja. calculated with a frequency resolution of 0.488 Hz is | lgfinHi(s)bJ][M]lfi[j;?n 01-1_____ stran 39 I^BSSIfTMlGC I. Simonovski - M. Bolte`ar: Spremljanje trenutne - Monitoring the Instantaneous Hz 50 45 40 35 30 25 20 15 10 5 0 Ulli -tf— ¦0« 0 5 10 15 čas, time 20 25 s 30 Sl. 9. Greben spektrograma, frekvenčna ločljivost 1,953 Hz Fig. 9. The ridge of the spectrogram, frequency resolution 1.953 Hz Hz 50 45 40 ¦ 35 30 25 20 ¦ 15 10 5 I 0 5 10 15 20 25 30 čas, time Sl. 11. Greben skalograma, frekvenčna ločljivost 1,953 Hz Fig. 11. The ridge of the scalogram, frequency resolution 1.953 Hz Hz 17,2 17,15 17,1 17,05 17 16,95 16,9 16,85 16,8 M wF*^ v^rfit^hid- 0 5 10 15 20 25 30 čas, time s Sl. 13. Greben skalograma, ustaljeno stanje. Frekvenčna ločljivost 0,01 Hz Fig. 13. The ridge of the scalogram, the stationary state. Frequency resolution 0.01 Hz grin^sfcflMISDSD Hz 50-45-40- 35- 30-25-20-15-10^ 5 TflW J^_ 0 i0^ 0 5 10 15 čas, time 20 25 s 30 Sl. 10. Greben spektrograma, frekvenčna ločljivost 0,488 Hz Fig. 10. The ridge of the spectrogram, frequency resolution 0.488 Hz Hz 0 5 10 15 čas, time 20 25 s 30 Sl. 12. Greben skalograma, frekvenčna ločljivost 0,488 Hz Fig. 12. The ridge of the scalogram, frequency resolution 0.488 Hz Hz 17,2- 17,15 — 17,1 jhJIi 17,05-17- 16,95- 16,9- 16,85-16,8^ W^wp^ ™Wflt 35 40 45 50 čas, time 55 60 65 Sl. 14. Greben skalograma, ustaljeno stanje. Frekvenčna ločljivost 0,01 Hz Fig. 14. The ridge of the scalogram, the stationary state. Frequency resolution 0.01 Hz VH^tTPsDDIK stran 40 s I. Simonovski - M. Bolte`ar: Spremljanje trenutne - Monitoring the Instantaneous Natančnejši je skalogram, ki je izračunan pri frekvenčni ločljivosti 0,488 Hz. Frekvence na začetku ni mogoče zaznati iz dveh razlogov: a) Pri izračunu skalogramov smo se omejili na frekvenčno območje 2 do 50 Hz. Spremljanje frekvenčnega območja pod 2 Hz bi zahtevalo časovno bistveno daljše izračune. b) Pri izračunu grebena smo zanemarili vse vrednosti manjše od 1/20 največje vrednosti magnitude skalograma. Amplitude nihanja na začetku zagona so bile tako majhne, da so ustrezne vrednosti magnitude spektrograma padle pod postavljeni kriterij. Frekvence vrtenja bobna, dobljene iz skalograma z boljšo frekvenčno ločljivostjo, so prikazane v preglednici 3. Frekvenčna ločljivost je 0,488 Hz. Po 12 s zagona se vrtilna frekvenca spreminja manj kot 3%. Pri 30 s je vrtilna hitrost 17,304 Hz, kar se zelo dobro ujema z že znanimi rezultati ([18] do [21]), pri katerih je bila izračunana vrtilna frekvenca ožemanja pri ustaljenem stanju 17,58 Hz. 2.3 Spreminjanje frekvence vrtenja bobna v ustaljenem stanju Želeli smo tudi preveriti ustaljenost frekvence vrtenja bobna v ustaljenem stanju. Izračunali smo grebene skalogramov horizontalnih pospeškov pralne skupine v ustaljenem stanju. Pri merjenju pospeškov smo frekvenco vzorčenja povišali na 2000 Hz. Grebene smo izračunali s the most precise of all the calculated ridges. At very small time indexes the ridges do not detect frequency for two reasons: a) Scalograms were calculated only in the frequency region of 2 to 50 Hz. Calculating scalograms at lower frequencies would be significantly more time consuming; b) When calculating ridges, all magnitudes lower then 1/20 of the maximum magnitude were ignored. The amplitude of the vibrations at the begining of the startup fall below this criterion. Drum-spin frequencies, extracted from the scalogram with a frequency resolution of 0.488 Hz, are presented in Table 3. After 12 s the fre-quency changes less than 3%. At 30 s the fre-quency is 17.304 Hz, which matches with previ-ous studies ([18] to [21]). In these studies the dry spin frequency in the steady state was determined to be 17.58 Hz. 2.3 Spin frequency variation in the stationary state The steadiness of the spin frequency in the steady state was checked. Scalogram ridges of the wash-ing-machine complex’s horizontal accelerations in the steady state were calculated. The signal of the horizontal accelerations was sampled with a higher sampling frequency, fs=2000 [Hz]. Ridges were calculated with a Preglednica 3. Spreminjanje trenutne frekvence pri zagonu pralnega stroja. Vrednosti grebena skalograma horizontalnih pospeškov. Frekvenčna ločljivost 0,488 Hz Tabel 3. The instantaneous frequency during a washing-machine startup. Values of the scalogram ridges of the horizontal accelerations. Frequency resolution 0.488 Hz čas v s time in s frekvenca v Hz frequency in Hz 1 2 4 9,496 12,424 14,376 6 8 15,840 16,328 10 16,816 12 17,304 14 17,304 čas v s time in s frekvenca v Hz frequency in Hz 16 18 20 22 24 26 28 30 17,304 17,304 17,304 17,304 17,304 17,304 17,792 17,304 Preglednica 4. Podatki o parametrih izračuna grebenov skalogramov Tabel 4. Ridge scalogram calculation data valček wavelet gabor s min 2 s 1,5 spodnja meja računanega frekvenčnega območja lower frequency bound 16 Hz zgornja meja računanega frekvenčnega območja higher frequency bound 18,18 Hz frekvenčna ločljivost frequency resolution 0,01 Hz gfin^OtJJIMISCSD stran 41 I. Simonovski - M. Bolte`ar: Spremljanje trenutne - Monitoring the Instantaneous frekvenčno ločljivostjo 0,01 Hz. Podatki o izračunu grebenov skalogramov so podani v preglednici 4. Slika 13 prikazuje greben v časovnem območju 0 do 32,7675 s, slika 14 pa greben v časovnem območju 32,760 do 65,5355 s. Opazimo lahko, da frekvenca vrtenja bobna ni konstantna, temveč s časom počasi narašča. Pogonski elektromotor še naprej pospešuje vrtenje bobna. Z večanjem vrtilne frekvence se moment elektromotorja manjša. Posledično se zvečuje vrtilna frekvenca bobna vedno počasneje. Spreminjanja vrtilne frekvence bobna znotraj enega vrtljaja ni mogoče zaznati. To je v nasprotju s pričakovanji, saj se moment pospeševanja bobna znotraj enega vrtljaja spreminja (sl. 15). Mel. frequency resolution of 0.01 Hz. Details of the calcula-tions are given in Table 4. The scalogram ridge for the time period 0 to 32.7675 s is presented in Figure 13, while the scalogram ridge for the time period 32.760 to 65.5355 s is presented in Figure 14. It is evident that the spin frequency is not constant. The driving electromotor slightly accelerates the spinning of the drum and with increasing spin frequency the driving moment of the electromotor decreases. As a consequence, the spin frequency of the drum increases ever more slowly. Within one revolution the change in drum spin frequency cannot be detected. Because the driving moment, act-ing on the drum, depends on the instantaneous angle of the drum, we expected that the spin frequency within one revolution would not be constant. Mel. mut.*g Sl. 15. Vpliv lege uteži na moment pospeševanja bobna pralnega stroja Fig. 15. The influence of the weight position on the washing-machine drum acceleration moment 3 SKLEP V strojništvu pogosto želimo spremljati frekvenčno vsebino določenega procesa. Največkrat uporabljena metoda je izračun magnitude Fourierjeve transformacije. Zaradi svojih lastnosti je Fourierjeva transformacija primerna predvsem za spremljanje ustaljenih procesov. Pogosto pa se srečujemo tudi s neustaljenimi procesi. V teh primerih lahko spremljamo frekvenčno vsebino z uporabo integralskih transformacij, primernih za neustaljene procese. Med te transformacije štejemo okensko Fourierjevo transformacijo, Wigner-Villejevo porazdelitev, modificirane Wigner-Villejeve porazdelitve, valčno transformacijo. V tem prispevku sta prikazani okenska Fourierjeva in zvezna valčna transformacija. Njune zmožnosti spremljanja frekvenčnega vsebine neustaljenega procesa smo ugotavljali pri primeru zagona pralnega stroja. Okenska Fourierjeva transformacija je omogočila grobo spremljanje frekvenčne vsebine. Odtekanje moči na sosednje frekvence je tako veliko, da so izračunani spektrogrami zelo nejasni. Izboljšanje frekvenčne ločljivosti z dodajanjem ničel v signal sicer izboljša jasnost, vendar je kakovost izračunane frekvenčne vsebine slabša kakor pri zvezni valčni transformaciji, izračunani z isto frekvenčno ločljivostjo. To je še posebej razvidno iz izračunanih grebenov, pri katerih spremljamo časovno spreminjanje vrtilne frekvence bobna pralnega stroja. Grebeni magnitude spektrogramov so zelo široki, posledično pa lahko 3 CONCLUSION In mechanical engineering we often want to monitor the frequency content of a certain process. The most widely used method for monitoring frequency content is calculating magnitudes of the Fourier transform. However, due to the type of basic functions, the Fourier transform is only appropriate for stationary processes. Monitoring the frequency content of non-stationary processes requires the use of different methods. The windowed Fourier transform, the Wigner-Ville distribution, modified Wigner-Ville distributions and the wave-let transform are among methods that are appropriate for analyzing non-stationary processes. This paper deals with the windowed Fourier transform and the continous wavelet transform. Their abilities for monitoring the fre-quency content of non-stationary process were stud-ied for the case of a washing-machine startup. Using the windowed Fourier transform we were only able to roughly determine the frequency content. The amount of power leakage resulted in very un-clear spectrograms. Increasing the frequency resolution with zero padding did not significantly improve the readability of the spectrograms. On the other hand, scalograms calculated at the same frequency resolution as spectrograms offer better readability of the frequency content. This can also be seen in the figures of the ridges, which are used for determination of frequency-time dependence. The width of the spectrogram ridges is wide and the drum-spin frequency in the steady state grin^SfcflMISDSD VH^tTPsDDIK stran 42 I. Simonovski - M. Bolte`ar: Spremljanje trenutne - Monitoring the Instantaneous ugotovimo samo, da je vrtilna frekvenca bobna v ustaljenem stanju med 20,5 in 17 Hz. Z uporabo grebenov magnitude skalogramov lahko to napoved bistveno izboljšamo. Ugotovili smo, da je vrtilna frekvenca bobna v ustaljenem stanju 17,304 Hz. Preverili smo tudi samo ustaljeno stanje. Horizontalne pospeške pralne grupe v ustaljenem stanju smo ponovno pomerili, pri čemer smo uporabili višjo frekvenco vzorčenja. Grebene magnitud skalogramov ustaljenega stanja smo izračunali z bistveno boljšo frekvenčno ločljivostjo -0,01 Hz. Ugotovili smo, da se vrtilna frekvenca bobna ne ustali, temveč se s časom počasi zvečuje. Znotraj enega vrtljaja nismo zaznali spreminjanja vrtilne frekvence bobna. can only be placed in the interval 17 to 20.5 Hz. This estimation of the drum spin frequency in the steady state can be improved on by using scalogram ridges, these ridges reveal that the drum spin frequency in the steady state is 17.304 Hz. In the next step the steady state itself was scrutinized. The washing-ma-chine complex’s horizontal accelerations in the steady state were measured using a higher sampling rate. The scalogram ridges were then calculated with a frequency resolution of 0.01 Hz. We determined that the drum spin frequency is not constant, but slowly increases with time. Within one revolution no varia-tions in drum spin frequency could be observed. Zahvala Acknowledgement Avtorji se zahvaljujemo Ministrstvu za The authors acknowledge the financial sup- šolstvo, znanost in šport za finančno podporo, ki jo port of the Slovenian Ministry of Education, Science dobivamo po pogodbi st.. S24-782-007/19910/99. and Sport, contract No. S24-782-007/19910/99. 4 UPORABLJENE OZNAKE 4 USED SYMBOLS koeficient ak coefficient koeficient bk coefficient koeficient Cy coefficient frekvenca v Hz f frequency in Hz funkcija v časovnem območju f(t) function in time domain Fourierjeva integralska transformacija funkcije f f(w) Fourier integral transform of f (t) function f(k-Dt) fk f(k-Dt) največja frekvenca v Hz f max maximum frequency in Hz frekvenca vzorčenja v Hz fs sampling frequency in Hz okenska funkcija gu(t)= g(t-u) window function -1 i -1 koeficient k coefficient število diskretnih točk v signalu N number of discrete points in the signal skala s scale okenska Fourierjeva transformacija Sf (u,w) windowed Fourier transform čas v s t time in s perioda v s T period in s parameter premika v s u time delay in s Gaussova okenska funkcija w Gauss Gaussian window function valčna transformacija Wf (u, s) wavelet transform časovni inkrement v s Dt time increment in s koeficient h coefficient koeficient s coefficient valčna funkcija v časovnem območju y(t) wavelet function in time domain skalirana in translirana valčna funkcija u,s () yt scaled and translated wavelet function Gaborjev valček v časovni domeni y Gabor Gabor wavelet in time domain Fourierjeva integralska transformacija y (t) y>(w) Fourier integral transform of y(t) Fourierjeva integralska transformacija yu,s (t) yu,s ( w) Fourier integral transform of yu, s (t) Fourierjeva integralska transformacija yGabor yoabor Fourier integral transform of yGabor vektor diskretiziranega valčka y k,s discretized wavelet vector frekvenca v rad/s w frequency in rad/s kompleksna konjugacija * complex conjugation konvolucija ® convolution 1 isfiflKiObJJIMIlfiCšD 01 stran 43 MlglTMDDC I. 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Kuhelj (1999) Dynamical behaviour of the planar non-linear mechanical system - Part II: Experiment. Journal of Sound and Vibration, Vol. 226, No. 5, 941-953. Naslov avtorjev: mag. Igor Simonovski doc.dr. Miha Boltežar Fakulteta za strojništvo Univerze v Ljubljani Aškerčeva 6 1000 Ljubljana Authors’ Address: Mag. Igor Simonovski Doc.Dr. Miha Boltežar Faculty of Mechanical Eng. University of Ljubljana Aškerčeva 6 1000 Ljubljana, Slovenia Prejeto: Received: 18.12.2000 Sprejeto: Accepted: 12.4.2001 grin^SfcflMISDSD VH^tTPsDDIK stran 44