Indian Journal of Textile ResearchVol. 1, September 1976, pp. 104-107

A Theoretical Study of Yarn Irregularity through theSpectrum

T. NARASIMHAM, K. GANESH & N. R. KOTHARIAhmedaba.d Textile Industry's Research Association, Ahmedabad 380015

Received 7 July 1976; accepted 10 August 1976

Yarn irregularity may be thought of as the sum of three components: the periodic, the totally random and thequasiperiodic variations. A spectral decomposition of these components gives more information about yarn irregu-larity than the variances alone. Methods for such decomposition have been suggested. Examination of the spectraof some yarns has indicated considerations to be taken into account when proposing a model for the arrangement offibres in yarns.

THE factors that go into the making up of thetotal yarn unevenness are both complex andvaried. For example, it is known that yarns

have a minimum unevenness which itself is dependentupon factors such as fibre length and fineness. Inaddition, machine faults that give rise to periodicvariations and drafting waves which give rise to othertypes of variations confound the pattern of fibrearrangement along the length of the yarn and hencethe total yarn irregularity. Martindale! proposeda model for estimating the inherent minimum irre-gularity. Foster", Cavaney and Foster" and others"!studied the irregularity due to drafting waves. Whilethese studies give an idea of the variance in yarncaused by these factors, a study of the entire correlo-gram rather than the variance alone would give moredetailed information about the yarn irregularity.For instance, it is easily seen that periodic variationsof two different wavelengths, but of the same ampli-tude, give rise to equal variance; their effect on fabricappearance, however, may not be the same. Asimilar logic extends to more complex situations. Infact, the correlogram, when studied through thespectrum, gives a split up of the variance into contri-butions due to oscillations in the linear density ofdifferent frequencies. Further, if a reliable methodfor the estimation of the spectrum of the componentscould be developed, valuable information on theprocess of drafting may also be gained. Draftingmay be considered as a process that ·modifies thespectrum of the input material; a comparative studyof the spectra of the components of irregularity ofthe input and output materials would then be a furtherstep towards understanding fibre movement duringdrafting.

The present work attempts to develop reliableestimates of the spectra of the three components,namely periodic variations, totally random varia-tions and quasiperiodic variations and to indicate theusefulness of the methods for analysis.

Components of Yarn IrregularityFor the purpose of this study, yarn irregularity has

been classified as periodic variations, totally randomvariations and quasiperiodic variations.

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Periodic variations - Variations produced byfactors such as eccentric rollers or broken gearscreate a regular pattern of waves along the lengthof the yarn. These may be represented as :

A1 cos W1x + B1 sin W1 x,j = 1, .... , p,where ,.f Al + Bl and W1 are respectively theamplitude and angular frequency of the j th period.

Totally random variations - It is known thatspun yarns have a certain minimum irregularity.Martindale! viewed this as arising out of a totallyrandom arrangement of fibres, i.e. the fibre left-endsforming a Poisson process. Variations arising outof such an arrangement are referred to here as totallyrandom variations. These depend only on the fibrecharacteristics and the mean linear density.

Quasiperiodic variations - All variations whichare neither periodic nor totally random have beentermed quasiperiodic. Generally, a large part ofthis component is due to the drafting wave. How-ever, drafting is known to introduce variations otherthan drafting waves, the nature and causes of whichare not clearly understood today. Even the descrip-tion of the drafting wave as given by Foster" doesnot seem to lend itself to a strict mathematical defi-nition. However, the spectral density", which isused in other fields to describe similar situations,provides an adequate description of the quasiperiods.The graph of the spectral density function is commonlyknown as the spectrogram. In general terms, thearea under the curve between the frequencies wandw . dw is the contribution to variance from the fre-quencies in this range. In the context of textilestrands, the units of frequency are cycles/unit lengthand those of the spectral density, the square of theunits of linear density. In the spectrogram, periodsshow up as sharp peaks and the drafting wave isexpected to show up as a general rise in the densityover a band of frequencies.

The ModelA model that considers these irregularities super-

imposed upon the mean yarn linear density, m, hasbeen proposed.

NARASIMHAN et at. : STUDY OF YARN IRREGULARITY THROUGH THE SPECTRUM

p

hex) = m(x) - m =I [A; cos WJ x + e, sin Wj x]

j=l

+ E(x) + y(x) ... (1)where m(x), the linear density at a distance x measuredfrom an arbitrarily fixed origin is supposed to be astochastic process, stationary in the wide sense;Aj, B, and Wj are the constants defining the periods;and E(x) and y(x) are independent processes withspectral densities go(W) and g2(W) respectivelycorresponding to the totally random and the quasi-periodic variations.

From Eq. (1), it is seen that in the absence ofstrict periods, i.e. when Aj = B, = 0 for all valuesof j, the spectral density gl(W) of hex) is given by

gl(W) = go(W) + g2(W) ... (2)Further, the total yarn variance, V, is given by

p ex> co

V = t I (Aj2 + Bj)2+ Igo(W)dW + J g2(W)dWj=l -co

.. .(3)=V1 + V2 + V3, say.The method of obtaining these components and

the estimation of spectral density are discussed below.

Estimation of the SpectrumThe classical approach to splitting up the variance

into contributions due to different frequencies hasbeen through the Finite Fourier Transform (FFT),known as the periodogram. However, FFT is not aconsistent estimator of the spectral density; itsvariance does not decrease to zero as the sample sizeincreases", A consistent estimator is obtained bysmoothing the periodogram. This smoothing isdone by weighting the covariances such that thoseof a high order receive low weightage. Variousworkers have used different weights in differentsituations.

The weights suggested by Tukey and Hamming"have been used here with a view to keeping the meansquare error low. These weights vanish for covari-ances of an order M much less than N, the numberof observations. Although broad guidelines areavailablet-" for the choice of M, the only reliablemethod, as suggested by Jenkins", seems to be throughstudying the stability of the spectral estimates fordifferent values of M. In the absence of previousdata and past experience, this procedure had neces-sarily to be followed. For medium counts and N=800, a value of M =40 was found suitable.

Defining bandwidth as the distance between half-power points, this choice of M roughly corresponds

to a bandwidth of ~; of the smoothing window.

This means any peak in the spectral density of the

yarn of bandwidth !;, i.e. 0.05 cycles/" em is

resolved.Detection of periods - It was mentioned earlier

that periods show up as peaks in the spectrogram.However, the spectrum is almost never smooth, sothat a peak due to a period becomes one among

many. It is, therefore, necessary to adopt a testto see if any particular peak is really due to a harmoniccomponent.

A test has been suggested by Fisher'? for the nullhypothesis of a uniform spectrum against the alter-native of the presence of a harmonic component.This cannot be directly used in the case of a yarn,since there is always a nonuniform spectral density.In fact, even the spectrum of the ideal yarn is notuniform. Fisher's test uses the statistic

Max IN (AI) 2 "It j. [ N ]~IN(AI) ,AJ= -N-,J=I, .... 2

where IN (A;) are the periodogram ordinates at thepoints AI. A simple modification suggested byWhittle'! is to replace IN(AI) by KN(A;) = IN (A;)/[2 "It f(A;)], where f(A;) is the spectral density at AI.However, this modification is applicable only if weknow f(A) a priori. One can think of replacingf(A)itself by its estimate, but good estimation is possibleonly when there are no peaks in f Further modifica-tions have, therefore, been suggested by Bartlett'",Priestley" and Hannan'! .

Both Hannan's and Priestley's tests correct for theeffect of peak in the spectrum on spectral estimation,take into account the nonuniformity of the spectraldensity and are the most powerful among the testsavailable for the purpose. Hannan's test does notdiffer very much in terms of power from Priestley'sand has been found most suitable for our analysisbecause of ease of application. The test criterion Gof Hannan's test is given by

G = Max KN* (A;)~ KN* (Ai)

where KN*(A;) = -----cI~N--'('--'A;"--)-_..,.-----~2{fN(AI)- ; WN(o)IN(A;) ]

j = 1, .. [ ~ ]

and WN (0) is the value at the origin of the weightfunction used in the estimate (iN (A) of f(A).

It is of interest to note here that the Uster spectro-gram is based on the periodogram and is open toerrors due to its inconsistency. Further, this isplotted in the wavelength domain, thereby distortingthe original spectrum. It, therefore, becomes diffi-cult to distinguish genuine peaks from the spuriousones. The spectral density, when based on thecorrect choice of M, does not suffer from this draw-back. .

Removal of periodic trend - For a complete ana-lysis of the spectrum, it is necessary to remove theperiodic component from hex). This is done by firstestimating by regression the Aj and B; of Eq. (1) foreach of the periods detected. By subtracting them,a new set of observations

hex) = E(x) • y(x)is generated. The spectrum is again estimated and isthe sum of the spectra of the remaining two com-ponents.

Trend removal by this method is known to introduceerrors in the estimation of spectral density. WithN = 800 and M = 40, however, these errors are

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INDIAN J. TEXT. RES., VOL. 1, SEPTEMEER 1976

seen to be negligible.As remarked earlier, the estimate of spectral

density used in the test criterion is itself influenced bythe existence of a period, although to a minimalextent in Hannan's test. Therefore, some new peaksare likely to show up as significant after the removalof the influence of those already detected. The test,therefore, has to be applied again to the new set ofobservations, h. This iterative procedure has beencontinued until no more peaks are detected.

The totally random component - This arises whenthe fibre ends form a Poisson process, as describedby Martindale, and are known to give the spectraldensity

co

) n( 1.06)2 {i2 I (I-cas wi) d (1)go(w = 2 'PI Wo

where Il is the average number of fibres in the cross-section; G, the average fineness of fibre; /, the meanfibre length ; and p (I), the fibre length distribution.

Quasiperiods - The spectral density, g2(W), of quasi-periods has been calculated from the equationg2(W) = gl(W) - go(IV), where gl(w) is the estimatedspectral density of the yarn after removal of theperiodic trend. The mean wavelength of the quasi-periods is calculated from the equations

2TtMean wave length = -- ,

W<II

-co -co

Experimental Measurements and DiscussionSpinning - Yarns were specially prepared under

conditions expected to increase one or more of thethree components discussed earlier. These were :

(A) 30s yarn spun from a 3.6 bank roving madeout of a 120s mixing and fed double to a gooddrafting system;

(8) 30s yarn spun from a normal mixing on agood drafting system from a 1.4 hank singleroving;

(C) 30s yarn from the same roving as in (8), butusing an eccentric front top roller in orderto create a period;

(D) 30s yarn from the same roving as in (8) and(C), but on an old, unused drafting systemwith a large spacing between aprons (draft-ing disturbances here were expected to leadto a large quasiperiodic component); and

(E) a good quality 36s yarn collected from a mill.The Uster charts of these were recorded with

'normal' sensitivity at the highest possible ratio ofchart speed to yarn speed and readings were takenfor analysis at intervals corresponding to a length orof l.16 cm on the yarn.

Spectra of the components - Figs. 1-5 show (i)the complete spectra, g(w), of the yarns and (ii) thespectra, g2(W), of quasiperiodic variations for eachof the five samples.

Examination of these spectra leads to the followingconclusions.

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(i) Periodic componentIn Sample (C), there is a significantly high peak at

a frequency of about 0.14 cycles/or ern (correspondingto a wavelength of about 8.5 ern), This, as can beexpected, arises from the eccentric front top rollerused in this spinning and has an amplitude of 15 %of the mean. After the removal of the periodicity,a small peak continues to exist. This peak, whentested, was found to be statistically not significant.In other words, it arises from the quasiperiodicvariations.

(ii) Quasiperiodic componentIn all the cases, the quasiperiodic component forms

a substantial part of the spectral density of the yarn.

>~ )OOr------------------,...tnZWN_ 200o 'E~ ~;: ~ 100

~ - o!:-··_·· ~:-::----;:'-::::7""~~~~,..J<n 0 0·125 0.250 0.375 0.500

- yarn sp.ctrum---sPKtrum of

quasi-~riods

CYCLES/T emFig. 1.- Spectrum of Sample (A) : 30s yarn from a 120s mixing

N_ 500~----------------~E

.!:!

~ 400_ yarn spec t r u rn-. -spectrum of

quasi-pe r iods~>-- 300enzw 200c~ ,«

10:tI

a: I,l- IU ..-WIL ,V> 0 0125 0·250

CYCLES!. emFig. 2 - Spectrum of Sample (B) : 305 yarn from a normal

mixing3600~----------------------------__,_ yarn spectrum--spKtrum of

quasl-pe rlodl

...>-!:: 2000enzwQ 1600

400

°O~~~~~~,~2~~~=--O~.;2S;O::~=O~.3=7S~~~O~,S-O~OJCYCLES/T em

Fig. 3 - Spectrum of Sample (C) : Yarn spun using a n eccentricroller

NARASIMHAN et al. : STUDY OF YARN IRREGULARITY THROUGH THE SPECTRUM

In fact for the poor drafting system, the density conti-nues to be high for frequencies lower than 0.25cycles/r cm (for wavelengths higher than ab~ut4.5 em). In other words, the spectral density remalI~shigh over a much wider r~nge than expected. ~hlSindicates that a poor drafting system not only raisesthe level of the variance in yarn, but also influencesthe manner in which this rise is brought about.

Components of yarn variance - Table 1 shows thecontribution to variance of the different componentsfor each of the yarns and shows the order of increasein these from the grossly underspun yarn to thatspun on a poor drafting system.

700N_

E 60u-01

3500•..>!:: 400en:z:w 3000-oJoCt 2000::t-UW 1000..en

00

_yarn spectrum---spectrum of

quaSi-periods

0·125 0.250 0·)75 05(){1

CYCLESfT emof Sample (D) : Yarn spun on an old

drafting systemFig. 4 - Spectrum

- Yarn spectrum--- Spectrum of

quasi - periods

a 0·' 0·2 0·3CYCLE S If: em

Fig. 5 - Spectrum of Sample (E) : 36s yarn from a mill

TABLE1 - CoMPONENTSOF VARIANCEYarn unevenness Variance [(gfcm)' x 10-10]

Code due to

U C.V. Variance Periods Totally Quasi-% % (gjcm)" random periods

x 10-10 variations

A 9.2 lZ.2 5.7 0.0 2.8 2.9B 16.0 17.8 12.2 0.0 4.1 8.1C 20.9 25.3 24.9 11.0 4.1 9.8D 19.0 25.2 24.6 0.0 4.1 20.5E 13.8 17.3 7.8 0.0 2.4 5.4

If drafting waves, whose wavelengths are expected tobe bounded in a small range (generally between 5.5and 7.5 em) were the only factor contributing tothis component, one would expect the spectrum toshow high values in this range (0.16-0.21 cycles/,; em) and vanish elsewhere. This range has beenshown by the shaded region in the case of Sample (E)(Fig. 5). However, this component has a nonzerodensity in the entire range, indicating the presenceof other variations. These are likely to be due tofactors not taken into account either by Martindale'smodel or by Foster's description of the range ofwavelengths created by the drafting process.

In the two combed yarns, (A) and (E), thereappears to be no real rise in the spectral density inthe range of frequencies of the drafting wave. Thissuggests that although drafting waves, as understoodtoday, are almost completely eliminated in theseyams (by combing, underspinning, etc.), the othervariations referred to above continue to contributesubstantially to the total variance (Table 1).

In the other cases, (B), (C) and (D), and parti-cularly in the case of the poor drafting system (D),the density is high over a wide range of frequencies.

•1

ConclusionWe now have methods for the detection of periods

and the reliable estimation of the spectral densitiesof yarns, which can be fruitfully used for estimatingquantitatively the contribution of different compo-nents of yarn irregularities. Analysis of the compo-nent irregularities has brought to light three impor-tant points that need to be considered for proposingmodels for yarn irregularity and the drafting process.These three points are as follows:

(a) There exist in yarn, irregularities other thanthose due to drafting waves and totally ran-dom variations. These have a nonzerospectral mass over the whole range of thespectrum.

(b) Poor drafting at the ringframe increases thespectral mass not only over the band of fre-quencies corresponding to the drafting wave,but over a much wider range.

(c) It appears possible to eliminate almost com-pletely drafting waves as they are understoodtoday by combing, underspinning, etc. Thevariations referred to in (a), however, continueto exist.

AcknowledgementThe authors are grateful to Dr B. V. Iyer and

Shri A. R. Garde of ATIRA for their many valuablecriticisms and comments. They thank the Director,ATIRA for permission to publish this paper.

References1. MARTINDALE,J. G., J. Text. Inst., 36 (1945), T 35.2. FOSTER,G. A. R., Manual of cotton spinning, Vol. IV Pt I

(Textile Institute and Butterworths, Manchester, Lo~don),1958.

3. CAVANEY,B., & FOSTER,G. A. R., J. Text. Inst. 46 (1955)T 529. ' ,

4. HANNAH, M., J. Text. Inst., 41 (1950), T 57.5. Cox, D. R., Proc. R. Soc., 197A (1949), 28.6. GRENANDER,U. & ROSENBLATT,M., Statistical analysis of

stationary time series (John Wiley & Sons New York)1~~ "7. PRIESTLEY,M. B., J. R. Slat. Soc., 14C (1965), 33.8. PARZEN, E., Time series analysis papers (Holden-Day

Cambridge), 1967. '9. JENKINS,G. M., J. R. Stat. Soc., 14C (1965), 1.

10. FISHER, R. A., Proc. R. Soc., 12SA (1929), 54.11. WHITTLE,P., Hypothesis testing in time series analysis, Thesis,

Uppsala University, Almquist and Wickswell, Uppsala,Hafner, New York, 1957.

n. Bartlett, M. S., An introduction to stochastic processes(Cambridge University Press), 1956.

13. PRIESTLEY,M. B., J. R. Stat. Soc., 24B (1962), 215.14. HANNAN,E. J., J. R. Stat. Soc., 238 (1961). 394 .

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