This article was downloaded by: [Northeastern University]On: 20 October 2014, At: 06:57Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

The Journal of The TextileInstitutePublication details, including instructions for authorsand subscription information:http://www.tandfonline.com/loi/tjti20

A Modified Analysis of ThreadContraction in Woven FabricsYoung Jin Jeong a & Joon Seok Lee ba Department of Textile and Fashion Engineering , KumohNational University of Technology , Kumi, KyungBuk,South Koreab School of Textiles, University of Yeungnam , Kyungsan,KyungBuk, South KoreaPublished online: 30 Mar 2009.

To cite this article: Young Jin Jeong & Joon Seok Lee (2001) A Modified Analysis of ThreadContraction in Woven Fabrics, The Journal of The Textile Institute, 92:1, 103-112, DOI:10.1080/00405000108659560

To link to this article: http://dx.doi.org/10.1080/00405000108659560

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information(the “Content”) contained in the publications on our platform. However, Taylor& Francis, our agents, and our licensors make no representations or warrantieswhatsoever as to the accuracy, completeness, or suitability for any purposeof the Content. Any opinions and views expressed in this publication are theopinions and views of the authors, and are not the views of or endorsed byTaylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor andFrancis shall not be liable for any losses, actions, claims, proceedings, demands,costs, expenses, damages, and other liabilities whatsoever or howsoever causedarising directly or indirectly in connection with, in relation to or arising out of theuse of the Content.

This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly

forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

06:

57 2

0 O

ctob

er 2

014

A Modified Analysis of Thread Contractionin Woven FabricsYoung Jin Jeong '* and Joon Seok Lee^

'Department of Textile and Fashion Engineering, Kumoh NationalUniversity of Technology, Kumi, KyungBuk, South Korea^School of Textiles, University of Yeungnam, Kyungsan, KyungBuk,South KoreaCorresponding author

Received 29.8.2000 Received in revised version 21.2.2001 Accepted for publication 2.10.2001

An earlier theoretical analysis of yam contraction is modified by introducing the geometricalmodel of Love's racetrack shape. The modified theory reasonahly predicts the amount ofthread contraction and geometrical parameters of woven fabric for any weave using loomsetting conditions.

Notation

Cj Local cr imp in the interlace zone (%)Cj Thread contraction (%)dj Thread diameter (mm)D Sum of warp and weft diameters (mm)hj Crimp amplitude (mm)1/ Number of intersections hy j yams in the weave repeatKj Cover factorIjf Local length between; yams at points of no interlacing (mm)Iji Local length between j yams at points of interlacing (mm)Ljf Total value of Ijf in the weave repeatLji Total value of /y, in the weave repeatLjK Thread length per weave repeat (mm)LRJ Spacing per weave repeat (mm)Nr. Reed number (number of dents/10 cm)Pjf Local spacing between j yams at points of no interlacing (mm)Pji Local spacing between j yams at points of interlacing (mm)Rj Number of y yams in the weave repeatSj Thread density (number of yams/10 cm)Tj Thread linear density (tex)Z Ends per dent of reed0, Weave angle6 Density of thread (mg/mm^)

The subscript,/ has the values of 1 and 2, to represent warp and weft, respectively.

1. INTRODUCTION

One of the things that a fabric designer would like to predict before weaving is theperformance of woven fabric. This has motivated many researchers to be attracted to thesubject of mathematical modeling of woven fabric. Peirce (1937) was the first to

J Text, Inst,. 2001. 92 Pan I. No. I © Textile Institute 103

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

06:

57 2

0 O

ctob

er 2

014

Jeong and l^ee

introduce elaborate mathematical modeling to study fabric geometry. He defined a wovenfabric structure through seven non-linear equations involving eleven variables. To solvePeirce's equations, prior specifications of four parameters are required. A fabric madefrom a given set of yams with a given set (thread spacing) gives three of the fourparameters to be predetermined. This makes it difficult to predict the geometricalproperty of the woven fabric in the stage of loom setting.

In this study, we suggest a method of predicting geometrical parameters from loomsetting conditions. Also the yam contraction will be determined based on the geometricalparameters. This work has been done by extending the theory of Yukhin and Yukhina(1996).

2. REVIEW AND THE PROBLEMS

According to the work developed by Yukhin and Yukhina (1996), fabric geometry isdescribed as shown in Fig. 1. The thread in the cloth is divided into the parts that form the

Fabric centre line

Fig. 1 Seciion of ihe cloth in the plane of the axis of a warp thread: l\, - the warp length In a single weftfloat; /[, - the warp length in a single intersection; p2i - the weft spacing in the imersccUon (YuWiinand Yukhina, 1996).

floats of the cloth (/|/) and the parts that are situated between the adjacent crossingthreads at points of interlacing (/|,). The authors defined the yam contraction as thepercentage excess of the length of the yam over the cloth length. Thus, the warpcontraction is calculated as follows:

c, - (1)

To obtain the warp contraction, Yukhin and Yukhina derived the geometricalrelationship between the loom setting parameters and the geometrical parameters of clothunit.

104 J. Text. ln,\t.. 2001. 92 Pan I. No. I © Textile Institute

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

06:

57 2

0 O

ctob

er 2

014

A Modified Analysis of Thread Contraction in Woven Fabrics

The total length of thread in the weave repeat is

UB = Lxi+Uf (2)

and the warp length in the floats is determined from the following equation:

Uf=p2,{R2-ix) (3)

L|, is determined by multiplying l\, and (|, i.e.

Ui = hi X /i (4)

To determine IM, Yukhin and Yukhina developed the following relationship based ontheir experience:

Ll={U2-Uff+{hyiyf (5)

The above equation is equivalent to the following one:

ll=pl + h\ (6)

Lff2 is determined from the following relationship:n

LR2=-=r = Pii X i\ + Pif[R2 - i\) (7)•52

By substituting Equations (2)-(4), (6), and (7) in Equation (1), the equation for C\ isobtained. However, two parameters have to be previously determined to obtain thecontraction, i.e. the local spacing between yams at points of interlacing or no interlacing(P2i or p2f) and crimp amplitude (/i| or /i?). Yukhin and Yukhina used the relationshipapplied to Equation (6) to obtain /i? as:

hl=pl-ll (8)

To obtain /12 from the above equation, the following equation was used:

h = ~^-PviRi-h) • (9)

However, Equation (9) needs the value of p]/ to calculate /2,. Thus, Yukhin andYukhina used the cover factor to obtain p\f as:

_ Warp density _ 5) (10)

Maximum warp density lOOR]/{R]d\

(11)

Using the value of pi/ obtained from the above equation. /12 was calculated fromEquations (8), (9), and (12). /?]Was also obtained using /i2 from Peirce's equation, i.e.

p

ffi =~ = Pu X i2+Pif{Ri -i2) (12)

At this stage, we can see two errors in deriving the equations. The relationship appliedto Equations (6) and (8) is true only when that part of the thread (/i, and l2i) is a straight

J. Text. Inst., 2001. 92 Part I. No. I © Textile Institute 105

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

06:

57 2

0 O

ctob

er 2

014

Jeong and Lee

Interlaced zone Fabric centre line

Fig. 2 Section of the cloth in the plane of a warp thread by geometry model of Love's racetrack shape{Love. 1954).

line. Ytikhin and Yukhina showed this section to be a curve (see Fig. 1). Another mistakeappears in deriving Equation (11) to calculate/?!/ (orpi,). Equation (11) does not give thethread spacing of no interlacing because K\ and K2 represent overall cover factorsincluding the spacing of interlacing and no interlacing. Thus, Equation ( i i ) cannot beused to obtain the length of the float thread

3. MODIFIED ANALYSIS

To develop more rigorously the theory for the prediction of yam contraction and clothgeometry, we introduce the cloth geometry of Love's racetrack shape as shown in Figs 2and 3 (Love, 1954).

Interlaced zone Fabric centre line

. 3 Section of the cloth in the plane of a weft thread by geometry model of Love's racetrack shape(Love, 1954).

106 ./- rf.tr. Inst,. 2001. 92 Part I. No. I © Textile Institute

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

06:

57 2

0 O

ctob

er 2

014

A Modified Analysis of Thread Contraction in Woven Fabrics

To obtain the yam contraction, we have to calculate the parameters of Love's clothgeometry. First, we can obtain L^x and Ln2 using Equation (7). We can also derive theequation for thread lengths of no interlacing and interlacing for any weave using Love'sgeometry model and mathematical deduction as follows:

pu = ^^^^ (14)'2

To calculate the other geometrical parameters in the region of interlacing, we usedPeirce's model whose geometry is depicted in the interlaced zone of Figs 2 and 3. Fromthese, Peirce derived the following geometric relationships:

c,={lulP2i)-\ (15)

C2 = [hilpM)-\ (16)

PI,. = {/2, - 002) cos 82+ZJ sin 62 (17)

P2i = {hi -OeOcose , +Dsinei (18)

hi = (/i ,-Z)ei)sinei +D(1 -cosG,) (19)

h2 = {hi - DB2) sin 02 + D(l - cos62) (20)

D=^h\+h2=d\+d2 (21)

To solve the above system of equations, some parameters have to be determined. Theparameter In is obtained as follows;

^ ? ^ ^ (22)'2

Z,2« can be obtained using the value of Lfn [which can be determined from Equation(12)1 and C2 as follows:

L2 = {\ + C2)xP\ (23)

and weft contraction can be obtained from the conditions of loom setting asfollows:

^ n,-NrZ (24)n.

In the same way, Lnz, Ly, and P2i can be obtained. Using pu and In, 82 can bedetermined from Equation (17), which is non-linear. We obtained 02 by solving theequation numerically with the Newton-Raphson method (William et ai., 1988). Then, thevalues of h\ and /12 are determined from Equations (20) and (21). There are still twoparameters to be determined, i.e. hi and 6|. To obtain these values. Equations (18) and(19) must be solved simultaneously. The Newton-Raphson method (William et ai, 1988)was used again to solve the equations numerically. Instead of using a numerical method.

/ Text. Inst.. 2001. 92 Part I. No. I (?) Textile Institute 107

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

06:

57 2

0 O

ctob

er 2

014

Jeong and Lee

approximate solutions for Peirce's equations can be used. Equations (25) and (26) showthe approximate solutions.

(25)

^ I (26)p]i J

From the obtained lu. the rest of the geometry parameters are detemiined asfollows:

^ ^ (27)P2i

L,, = luxi,+L2f (28)

Now, we can determine the contraction of warp yam using Equation (1).

4. PRACTICAL EXAMPLES AND DISCUSSIONS

In this section, we give a practical example to demonstrate the use of the newlyintroduced thread contraction and the calculation of geometrical parameters from loomsetting conditions. Also, we compare our results with those of Yukhin and Yukhina(1996). The data used in this comparison are from Yukhin and Yukhina's work.

A cotton ciolh Art.3080 has the following specifications: T\ and T2 = 42 tex. 5| = 298ends/10 cm. 52 = 200 picks/10 cm. weave - 2/2 twill. A', = 71 (per 10 cm), Z ^ 4. Theyam diameter was calculated from the following equation:

d] — d2 — O.\ CvO.! /"(mm) (29)

and

(30)where 6 = yam density in mg/mm^ .

Yukhin and Yukhina calculated Cas 1.25 for the yams, and the diameter of the warp andweft yams as 0.256 mm. For 2/2 twill weave, Ri — R2 — 4, (t — '2 = 2. Using the initialvalues and derived equations, the yam contraction is calculated in the following way:

. 0.0470 or 4.7%

/?i/5| X 100 = 4/298 x 100 = 1.342

= R2/S2 X 100 - 4200 X 100 = 2.0

Yukhin and Yukhina's solutions for Lij- and Lif were 0.448 and 0.666, respectively.

.1. Text, hist., 2001. 92 Part I. No. I © Textile Institute

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

06:

57 2

0 O

ctob

er 2

014

A Modified Analysis of Thread Contraction in Woven Fabrics

The local spacing between yams at points of interlacing are:

l A ^ ^ 2 . 0 - 0 . ^ ^ ^ ^ ^(i 2

The length of the weft yam per weave repeat, the length of the weft yam and the localcrimp between yams at points of interlacing are:

LzM = (I + C2) XLRI =(1+0.047) X 1.342- 1.408

^ ^ g ^ - t , 1.408-0.40212

^ ^ , ^ ^ 0 - 5 0 3 0 . 4 7 0

pu 0.470

Using the values of Izi, d\, d2 andpi,, the value of 02 can be obtained by solvingEquation (17) numerically or using the approximate formula of Equation (25).

The numerical solution of 02 is 24.94°, and the approximate solution of 02 is:

02 = lOe^/cI - 106 X yo.0704 - 28.12°

From Equations (20) and (21), A2 and hi can be obtained. The solutions using thenumerical value of 02 are:

cos02) = 0.166

/,, ^ D - ; i 2 - 0 . 3 4 7

The solutions using the approximate value of 62 are:

sin 62 + 0(1 - c o s e 2 ) ^ 0.179

hi =D-h2 =0.333

Yukhin and Yukhina's solutions for h\ and /12 were 0.327 and 0.175, respectively.For given values of D, p^ and h[, the values of 0| and/i; can be obtained by

solving Equations (18) and (19) numerically or by using Equations (25) and (26)directly.

The numerical solutions of 0| and lu are 27.17" and 0.878, respectively; hence C[ isgiven as:

^ ° ^ ^ ^ - ° ^ ^ ^ ^0.0901 or 9.901%

The approximate solutions of C], 0| and /i, are:

J. Text. Inst., 2001. 92 Part I, No. I © Textile Institute 10')

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

06:

57 2

0 O

ctob

er 2

014

Jeong and Lee

-- O ro •*m r i o —I \D in— ro Tt — o OCd d d d f i 1 !

—• rJ Tf oc <O d o

O O O nooooooo-OOOO

m

\^ O 00 00T <ri 00 — r^ooTi-in roooo — r ) * * C — 00 —o i-i d d d d rf r.-! ri

ooooooovooio— — (*^O^coo

in u-) — ov — — 00 r j

d: d d> d d '00

_ _ qd d d d

— — 00

0

v c v ^ O — - o c ooooooo-

ddddoom d o o o

ro 00 00 ^^ p <ij fo ^fiTirl — r J . — MNrooo

OTf ddddrirj ddddo —I

c; d d c --

m mooooooo\ — 00o

"^ -o OC ^ -o ^ o

dddddco ddo'd 0000- - ro O\m

(O OV \O —in in (N Ov i n • ^— fi -It O O- - ^d d d d d ro

— rj " ^ O

dddd

ro iqm ro Tt —

11.11sa

in u-i— o r ^ r o -a%0'Or^oc'C ' f l - r - 'Oro^ Oooooooood d d d o o f i u d d d d

•g

a

.2 o —

111•O .H 1-

6u-j i n ^ 00 00 00 00

E O O O O 00 ro

I

Ic0

o

ract

ion

ti iT3U

u

c0

oore

a0

S u e

"'III

4J 1> c3 U

no 7. rf.i:r. Inst.. 2001, 92 Pan 1, No. I © Te.xtile Institute

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

06:

57 2

0 O

ctob

er 2

014

A Modified Analysis of Thread Contraction in Woven Fabrics

h.^Pii X (1-Kci) - 0.799 X (1+0.0977) =0.877

Hence, the numerical and approximate solutions of warp contraction, respectively, are

UR - hi X ix +L2f = 0.878 x2 + 0.4023 = 2.158

^ UR-LR2 2 .158-2.0C l ^ L ^ = 0.0733 or 7.33%L{R 2.158

and

- /|, X /, -hZ^/ = 0.877 X 2 + 0.4023 = 2.156

Z.1/? - LR2 2.156-2.02.156

= 0.0725 or 7.25%

Yukhin and Yukhina obtained the contractions of warp and weft yams as 7.4 and4.67%. respectively. Despite the contradictions in their model, the results fit well withour calculations. However, looking into the geometrical parameters, we can find aconsiderable difference in piy, p2/ (more than 10%) and a small difference in h] and hj.Also, the value of p2! (0.333 mm) calculated by Yukhin and Yukhina is greater thanthe yam diameter (0.256 mm). This means that the weft yams in the fabric are apartfrom each other. This cannot happen in a woven fabric. This error is due to the lack ofaccuracy of their model as they stated in their paper. The reason for their model to givea good result, in spite of low accuracy of the geometrical parameters, is explained asfollows: 'the errors of pi/ and pi/ are compensated by increasing or decreasing thevalues of h\ and hj'. This is realized by constraining the yam length per weave repeatto be unchanged through Equation (6). Thus, the resultant values of yam contractionare close to our results.

Yukhin and Yukhina designed eight cotton fabrics to verify their theory as shown inTable 1. We have also calculated the contractions using our approach to compare withYukhin and Yukhina's results. Considering the inherent variations in textile assemblythat are very different from general engineering materia, the differences in yamcontractions between the predicted results of Yukhin and Yukhina and our approach do notlook significant.

5. CONCLUSIONS

The geometry used in Yukhin and Yukhina's original analysis contains some limitingassumptions. We modified the theory by introducing the geometry of Love's racetrackmodel. We solved the theoretical model with both numerical and approximate methodsand found good agreement between the methods. The modified theory enables areasonable prediction to be made of parameters such as the amount of thread neededfor the given dimensions in the final cloth and the geometrical parameters for anyweave using loom setting conditions.

ACKNOWLEDGMENTS

This work was performed under a grant from the KumOh National University ofTechnology. We would also like to express our gratitude to Dr. D.G. Phillips of theTextile and Fibre Technology of CSIRO for help during the course of this work.

J. Te,xt. Inst.. 2001. 92 Pan I, No. I © Textile Institute 111

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

06:

57 2

0 O

ctob

er 2

014

Jeong and Lee

REFERENCESLove, L.. 1954. Graphical Relationships in Cloth Geometry for Plain. Twill and Sateen Weave.s. Text. Res. J..24, 1073-1083.Peirce. F.T.. 1937. Tlie Geometry of Cloth Stnicture. / Text. Inst.. 28, T45-T112.William. H.P. Brian, P.F. Saul. A.T . and William T.V.. 1988. Numerical Recipes in C. Cambridge Univosityftess. New York. USA.Yukhin. S.S., and Yukhina, Ye. A., 1996. A Theoretical Consideration of the Warp and Weft Contractions inWoven Fabrics. / Text, Inst.. 87. 532-541.

112 / Text, Inst., 2001. 02 Pun I, No. I © Textile Instiuite

Dow

nloa

ded

by [

Nor

thea

ster

n U

nive

rsity

] at

06:

57 2

0 O

ctob

er 2

014