basic studies for modeling complex weft knitted fabric ...€¦ · 22.08.2016 · fabric, the...

Textile Research Journal Article

Textile Research Journal Vol 78(3): 198–208 DOI: 10.1177/0040517507082352 www.trj.sagepub.com © 2008 SAGE PublicationsFigures 3–5, 8–9 appear in color online: http://trj.sagepub.com Los Angeles, London, New Delhi and Singapore

Basic Studies for Modeling Complex Weft Knitted Fabric Structures Part I: A Geometrical Model for Widthwise Curlings of Plain Knitted Fabrics

Arif Kurbak1 and Ozgur EkmenDepartment of Textile Engineering, Dokuz Eylül University, Izmir, Turkey

.

Many geometrical models have been created by previousresearchers for plain knitted fabric, including those ofChamberlain [1], Peirce [2], Leaf and Glaskin [3], Leaf [4],Munden [5], Postle [6], Demiroz [7] and Kurbak [8, 9].Some researchers have also tried to modify plain knit mod-els to obtaining models of more complex knitted fabricstructures. Some reasons for creating geometrical modelsinclude the following:

a. to find the effective parameters which cause dimen-sional changes during relaxation;

b. to be able to plan the production of a piece of fabricbefore knitting;

c. for use in technical textile applications;d. to obtain computer simulations of knitted fabrics for

design or fault-assessing purposes;e. to obtain the surface properties of fabrics;f. to prepare a basic work for creating a physical

model, etc.

A group of people at Dokuz Eylül University started alarge-scale project to extend plain knitted fabric models tocomplex weft knitted structures and to meet the aboveaims. In this series of papers, some extensions of plain knit-ted model are given as basic works to obtain the models ofmore complex structures. In this context we deal with thewidthwise curlings in this part I of the series.1

Edge curlings are important problems which occur, inparticular, in plain knitted fabrics owing to the unbalancedyarn bending moment existing in the three-dimensionalnature of the structure. The curlings occur at the upperand lower edges of a piece of fabric towards the front sideand at the left and at the right edges of the fabric towardsthe back side.

A plain knitted fabric structure is shown in Figure 1where the edge curling directions are highlighted by thearrows. The yarn wants to adopt a straight form but it isprevented from doing so by neighboring loops. Thus, curl-ing can start at the edges as there is no neighboring loop onone side to prevent curling.

The edge curlings given above can create some prob-lems during the creation of plain knitted clothing goods.Some suggestions have been made to overcome the prob-lems of edge curlings before clothing operations and theseinclude (a) sticking paper to the back of fabric, (b) layingout a heavier chemical substance on both sides of fabricduring the openwidth processes in finishing and (c) pinningdown at spreading table before cutting, etc.

In addition to the problems discussed above, there arealso positive effects or advantages provided by edge curl-ings, such as the upper side edge curling being used to

Abstract A geometrical model for widthwisecurlings in plain knitted structures is suggestedhere based on the plain knitted fabric model ofKurbak (1998). The main aim is to create modelsof m × n rib fabrics and also to model small diam-eter tubular fabrics. The model obtained is drawnto scale using the 3DS-MAX computer graphicsprogram. With this model, the changes in radius ofcurvature at each part of the loop are shown.

Key words plain knitted fabric, widthwise curl-ing, geometrical model, Kurbak’s model, compu-ter graphics program

1 Corresponding author: e-mail: [email protected]

by Satyadev Rosunee on August 22, 2016trj.sagepub.comDownloaded from

http://trj.sagepub.com/

Part I: A Geometrical Model for Widthwise Curlings of Plain Knitted Fabrics A. Kurbak and O. Ekmen 199 TRJ

form the neck of pullovers, as was the fashion some timeago. In addition to searching for a model of edge curling infabric, the curlings in width direction can be used as basicmodels to obtain the whole m × n rib structures (see Fig-ure 2) and this is the main aim of this study. The other aimis to obtain a model of small diameter tubular fabrics.

To meet the above aims, we create a geometrical modelfor the curlings in the width direction of plain knitted fab-ric to explain how the loop shape changes and also findwhich part of the loop the energy of yarn changes. In thewidthwise edge curlings of fabrics, different amounts ofcurlings probably occur at two arms of a loop. As the aimwas to create a widthwise curling model to obtain the m ×n rib models and tubular fabric model rather than applyingthe model to obtain edge curlings, the same amount ofcurlings at both of the arms of a loop are taken in thispresent model.

Among the above-mentioned plain loop models, Kur-bak’s [9] model is found to be suitable for our purpose.Therefore, our model is based on Kurbak’s [9] plain knitmodel, which is explained further in the following.

Kurbak’s Model

In this model the upper and lower parts of the loop are inthe form of elliptical curves and the arms of the loop havehelical shapes that are wrapped over elliptical cylinders,which are arranged parallel to the wales direction, withvariable helix angles (see Figures 3(a) and (b)).

In Figure 3, we show the elliptical shape representingthe top of the loop, which is situated in a plane that makesan angle α1 with the horizontal y axis directed towards thefabric thickness and extends until a point B making ellipseangles of θ = 90o in both sides in the fabric width (x direc-tion). The major and the minor radii of elliptical cross sec-tions of elliptical cylinders for wrapping the central axis ofloop arms are

(1)

and

(2)

where a is in the y direction (in the thickness direction), b isin the x direction (in the course direction), t1 is the diame-

Figure 1 Edge curlings of plainknitted fabric: (a) front view of theplain knitted fabric; (b) A–A crosssection of the fabric with thearrows showing the edge curlingdirection; (c) B–B cross section ofthe fabric with the arrows showingthe edge curling direction; (d) sche-matic view of the upper and loweredge curlings (towards the front ofthe fabric); (e) schematic view ofthe edge curling in the fabric widthdirection (towards the back of thefabric).

a t1d 2⁄=

b e1d 2⁄=



200 Textile Research Journal 78(3)TRJTRJ

ter of the yarn arm cylinder along the thickness direction,e1 is the diameter of the yarn arm cylinder along the coursedirection and d is the effective yarn diameter.

As an assumption, the equation of the curve betweenthe points B, C and BI in the right arm of the loop wasdenoted by the following formulas:

(3)

(4)

(5)

where θ is the angle measured from the y′′ axis, S1 is thelength measured from the y′′ axis along the perimeter ofthe right section of elliptical cylinder of yarn arm and a′, b′,c′ are constants.

According to Figure 3, the following loop parameterscan be calculated from the curves that represent the yarnaxis of the right arm of the loop:

• Distance between axes z and z′′ = xAC

(6)

where χ = w/d and w is the wale spacing.• Horizontal distance between points A and B = xAB.

Thus, we can write

(7)

If the loops are in contact with each other (wale-jam-ming point) in the direction of courses without any yarncompression in the loop interlacing regions, then e1 = 1.

Equation (7) gives the major radius of the upper ellipti-cal curve of the loop head, thus,

(8)

is obtained and the minor radius of upper elliptical curveof the loop head can be given as

Figure 2 Photographic representation of m × n rib structures [6]. (A) Front view and cross sections of balanced rib fabrics:(a) 1 × 1, (b) 2 × 2, (c) 3 × 3, (d) 4 × 4, (e) 5 × 5 ribs. (B) Cross sections of unbalanced rib fabrics: (a/) 3 × 1, (b/) 3 × 2, (c/) 3 ×3, (d/) 4 × 3, (e/) 5 × 3 ribs (see [6]).

x e1d 2 θsin⁄=

y t1d 2 θcos⁄=

z a′ S1( )2 b′ S1( ) c′+ +=

xAC w 4⁄ χ 4⁄( )d= =

xAB w 4⁄ e1+ d 2⁄ χ 2⁄ e1+( )d 2⁄= =

aχ2--- e1+ d

2---=




(9)

where e is the eccentricity of the ellipse. The ellipticalcurve length between A and B can be found by integratingthe equation between A and B as

(10)

or

(11)

where cos φ = e and E is the complete elliptical integral ofthe second kind.

The estimated radii of curvature at A and B in theupper curve of the loop are

(12)

(13)

S1 is the elliptical length on the cross section of theelliptical cylinder, on which the axis of yarn is wrapped,measured from the y′′ axis; S1 (this is the projection of yarnaxis on the elliptical cross section) can be calculated as

(14)

where is the complete and is theincomplete elliptical integral of the second kind and

Figure 3 Kurbak’s [9] model: (a) schematic drawing of the loop central axis; (b) drawing of the real loop model.

b ea eχ2--- e1+

d

2---= =

dS a 1 1 e2– θ2sin– dθ,=

sAB a 1 1 e2– θ2sin– dθ

θ 0=

π 2⁄

∫=

SAB

χ2--- e1+ d

2--- E π

2--- φ,

=

ρAe1

e---- χ

2e1-------- 1+

d

2---=

ρB e2e1χ

2e1-------- 1+

d

2---=

S1 t1 E1 π 2 φ1,⁄( ) E1 ϕ1 φ1,( )+[ ]d 2⁄=

E1 π 2 φ1,⁄( ) E1 ϕ1 φ1,( )




(15)

and

(16)

can be given.Here θ is measured from y′′ to match with θ of the

upper ellipse angle measured from the point A of the loop.If we denote the derivate of the equation z = aI(S1)

2 +bI(S1) + cI by u as

(17)

then

u = tan α

is obtained, where α is the helix angle of the yarn armmeasured on the horizontal plane

The yarn length in the arm can be calculated as follows:

(18)

whereas, from Equation (17),

(19)

is obtained.Substituting Equations (17) and (19) back into Equa-

tion (18), dS becomes

(20)

and integrating dS from u1 to u2, the yarn length SBCbecomes

(21)

where u2 = 2a′t1[2E1]d/2 + bI = tan α2 and u1= 2a′t1[E1]d/2+ bI = tan α1.

It is assumed that the interlocking points of the loopsare at their widest points, thus, the vertical distancebetween B and C is equal to half of the course spacing

(c = η.d). Since dz/dS1 and dS1 are given in Equations (17)and (19), respectively, we have

(22)

By integrating dz from u1 to u2,

(23)

is obtained.Since zBC is equal to half of the course spacing accord-

ing to the above assumption,

(24)

can be written.The radii of curvature at B and C of the right arm curve

can be estimated by assuming that the yarn is wrappedaround imaginary cylinder at B and C and assuming thatthe curvatures are approximately same as that of ellipticsection at these points. Hence,

(25)

(26)

are obtained. The radius of curvature which is obtainedfrom the upper elliptical curve must be equal to the radiusof curvature which is obtained from the helical armcurve, so that

(27)

and, hence, the equation

(28)

can be written. Then the following equation is obtained:

(29)

If we denote the height of the upper part by z0, itcan be calculated that

φ1cose1

t1----=

ϕ1 θ π 2⁄–=

dzdS1--------- 2a1S1 b′+[ ] u= =

dS dz2 dS12+

dzdS1---------

2

1+ dS1= =

dS1 du 2a1⁄=

dS 12aI-------- 1 u2+ du=

SBC1

4aI-------- u2 1 u2

2+ u1 1 u12+–

u2 1 u22++

u1 1 u12++

------------------------------ln+

=

dz u du

2aI--------=

zBCu2

2 u12–

4aI----------------=

c ηdu2

2 u12–

2aI----------------d= =

ρBI t1

2

e1 α21cos

---------------------- d2---=

ρCe1

2

t1 α22cos

--------------------- d2---=

ρB

ρBI

ρBI ρB=

t12

e1 α21cos

----------------------χ2--- e1+

e2=

u1 α1tane2e1

2

t12

---------χ

2e1-------- 1+

1–= =

AB




(30)

Now we can calculate the constants aI, bI and cI by usingthe equations

(31)

The results of the above analysis are given below as theloop parameters:

(32)

(33)

(34)

(35)

where

(36)

(37)

the complete elliptical integrals of second kind,

and dt is the free yarn diameter in the thickness direction.The radii of curvature at different points of the loop can becalculated and compared with the following formulas:

(38)

For e1 in the equations, the following assumption can bemade:

(39)

where is the stitch length at the wale-jammingpoints and this was found by Kurbak [11] as inhis experiments.

A drawing of Kurbak’s model using the 3DS-MAXgraphics program is given in Figure 4 with the effectiveyarn diameter.

A Model for Widthwise Curlings of Plain Knitted Fabrics

The model of curlings in the widthwise direction of theplain knitted fabric was created by discarding some parts ofKurbak’s model [9] to obtain the head of the next loopmaking an angle with the considered loop seen in Figure 5(the black parts are the discarded parts).

Discarded parts can be calculated as follows. The dis-carded part from the upper side of the loop is shown inFigure 6. By geometrical analysis (not given here) weobtain the following equation from Figure 6:

(40)

The discarded part from the arm of loop could beshown as in Figure 7. The following equation could be alsowritten from Figure 7:

(41)

z0 e χ2--- e1+

u1

1 u12+

------------------- d2---=

aI 1t1E1d-------------- u2 u1–( )

u22 u1

2–2c

----------------= =

bI 2u1 u2–=

cI 1 u12+

t12d

2ee1----------- 2c

u1 u2+-----------------–

u1c2---+=

cd--- η

t1E1

2---------- u2 u1+( )= =

wd--- χ 2e1

t12

e2e12

--------- u12 1+

1–= =

td--- pt

t1

2--- 1

2--- 1 u1

2+t12

ee1-------+ +

=

�

d--- 2e1

χ2e1-------- 1+

Et1E1

u2 u1–( )---------------------+=

u2 1 u22+ u1 1 u1

2+u2 1 u2

2++

u1 1 u12++

------------------------------ln+–

×

E11 1

e12

t12

----–

– ϕ21sin ϕ1d

ϕ1 0=

π 2⁄

∫=

E 1 1 e2– – ϕ2sin ϕd

ϕ 0=

π 2⁄

∫=

pt dt d⁄=

ρ d⁄( )Ae1

e---- χ

2e1-------- 1+

1

2---=

ρ d⁄( )B e2e1χ

2e1-------- 1+

1

2---

t12

e1 α21cos

----------------------12---= =

ρ d⁄( )Ce1

2

t1 α22cos

---------------------12---=

e1l d⁄l0 d⁄( )

---------------=

l0 d⁄( )l0 d⁄( ) 20≅

βtanaD

bD ϕ*tan---------------------=

γtane1D

t1D-------- ϕalttan=




The point B in Figure 6 and the point K′ in Figure 7 are thesame point, thus the curves in Figures 6 and 7 were drawntogether with B and K′ coinciding to the same point. All ofthe defined parameters so far are shown in Figure 8 for acurled loop in the widthwise direction. It should be notedthat the lines LK′ (Figure 7) and BK (Figure 6) are on thesame line.

The following equations could be written for a curledloop in the widthwise direction according to the abovemodel (see also [10]):

1. The loop length could be given as

(42)

where

(43)

(44)

Figure 4 Computer drawing of Kurbak’s model [9] witheffective yarn diameter.

Figure 5 Loop curling in the widthwise direction accord-ing to the present model.

Figure 6 Discarded upper part of the loop (a = aD, b = bD).

Figure 7 Discarded part from the arm of the loop.

l d⁄ 4aD E ϕ* ψ,( )t1DE1D

u2D u1D–----------------------- u2D 1 u2D

2++=

u1D 1 u1D2+

u2D 1 u2D2++

u1D 1 u1D2++

-------------------------------------ln+–

ψtanbD

aD α1Dcos-------------------------=

u1D α1Dtan=




(45)

(46)

(47)

2. The radius of curvature at point B in Figure 6 shouldbe equal to the radius of curvature at point K′ in Fig-ure 7 (ρB = ρK′), thus

(48)

where

(49)

(50)

(51)

(52)

The equality LB = LK′ was enough to equalize ρB = ρK′.3. The torsion at point B in Figure 6 should be equal to

the torsion at point K′ in Figure 7 (τB = τK′), thus

(53)

where

(54)

(55)

Since LB = LK′ was given above, the equality KB =

KK′ is enough to equalize τB = τK′. It should be noted

that the upper part of loop was assumed to be in-plane for simplicity in Kurbak’s model and thus notorsion was taken for the upper part. Here we coulduse the torsion equality at a point that was away at adistance of elementary length, ∆l, above the point Bin Figure 8. This equation was considered to reducethe number of unknown parameters.

4. The course spacing could be calculated by using Kur-bak’s model as

(56)

5. The length between the points M and O′ (the centerof the x–y coordinate system) in Figure 8 wasassumed to be equal to one-quarter of the wale-spac-ing (w/d), thus

(57)

was taken.

In this model, ϕ was taken as a parameter to measurethe amount of curling in the loop while the parameters e1D,

Figure 8 Curled loop model in the widthwise direction(b = bD, a = aD).

u2D α2Dtan=

E ϕ* ψ,( ) 1 1bD

2

aD2 α2

1Dcos--------------------------–

– ϕ2sin ϕd

ϕ 0=

ϕ*

∫=

E1D1 1

e1D2

t1D2

--------–

– ϕ2sin ϕd

ϕ ϕalt=

π 2⁄

∫=

LB

α21Dcos 1 PB

2+----------------------------------------

LK ′

α21Dcos 1 PK ′

2+------------------------------------------=

LBaD

2

bD------ 1 1

bD2

aD2

------–

ϕ2 *sin–

3 2⁄

=

LK ′e1D

2

2t1D---------- 1 1

t1D2

e1D2

--------–

π 2⁄ ϕalt+( )2sin–

3 2⁄

=

PB 2a′LB α1Dcos=

PK ′ 2a′LK′ α1Dcos=

α1Dsin α1Dcos

LB 1 PB2+

------------------------------------ 1 3KBPB

α1Dsin-----------------–

α1Dsin α1Dcos

LK ′ 1 PK′2+

------------------------------------ 1 3

KK ′PK ′

α1Dsin-----------------–=

KBaD

bD------ 1

bD2

aD2

------–

ϕ*sin ϕ*cos=

KK ′e1D

t1D-------- 1

t1D2

e1D2

--------–

π 2⁄ ϕalt+( ) π 2⁄ ϕalt+( )cossin=

ct1DE1D

2----------------- u1D u2D+( )=

w4---

e1D

2--------

ϕaltcosγcos

----------------- βcos+ aD ϕ*sin=

*




t1D, aD, bD, u1D, u2D and ϕalt were calculated according tothe model.

A computer program was written to calculate the curledloop model using normal tightness values as given in theAppendix.

Then the curled plain knitted fabric was drawn to scaleby using the 3DS-MAX computer graphics program, asshown in Figure 9. The model shows loop shapes that aresimilar to the shapes we observed on the real rib fabricsgiven in Figure 2.

The radii of curvature and torsions of loops are given inFigure 10 for Kurbak’s original plain knit model and inFigure 11 for the curled loop model. Detailed informationabout how to calculate the curvature and torsion can befound in [4].

As can be seen from Figures 9 and 10, the torsionreduces with curling while the radius of curvature isincreasing, especially at the mid points of the arms (thepoint C in Figure 3). Since the bending energy is propor-tional to the reverse of the radius of curvature, bendingenergy is also decreasing as that is also the case for tor-sional energy at point C in Figure 3.

Conclusion

A geometrical model for widthwise curlings of plain knit-ted fabrics has been created based on Kurbak’s (1998)plain loop model. The main aims of the creation of thismodel were its application to the m × n rib structures andto obtain a small diameter tubular technical fabric model.

The model was drawn to scale using the 3DS-MAX com-puter graphics program and the loop shapes obtained werethe same shape as those observed experimentally on them × n rib fabrics. Torsional and bending energies werereduced with widthwise curlings, particularly at the midpoints of the loop arms.

Figure 9 The present model of the widthwise curling ofthe plain knitted fabric (ϕ* = 87°, w = 5.8d, c = 4.15d, � =20.19d). Figure 10 Radii of curvatures and torsions of Kurbak’s

plain knit model (d is the effective yarn diameter).

Figure 11 Radii of curvatures and torsions of the presentcurled loop model (d is the effective yarn diameter).




Appendix: Flow Chart of the Loop Widthwise Curling Computer Program




Literature Cited

1. Chamberlain, J., “Hosiery Yarns and Fabrics”, Vol. II, Leices-ter College of Technology and Commerce, Leicester, 1949,p. 107.

2. Peirce, F. T., Geometrical Principles Applicable to the Designof Functional Fabrics, Textil. Res. J., 17, 123–147 (1947).

3. Leaf, G. A. V., and Glaskin, A., Geometry of Plain-KnittedLoop, J. Textil. Inst., 46, T587–T605 (1955).

4. Leaf, G. A. V., Models of Plain Knitted Loop, J. Textil. Inst.,51, T49–T58 (1960).

5. Munden, D. L., The Geometry of a Knitted Fabric in itsRelaxed Condition, Hosiery Times, April, 43 (1961).

6. Postle, R., Structure Shape and Dimensions of Wool KnittedFabrics, J. Appl. Polmer Sci. Appl. Polymer Symp., 18, 1419,1971.

7. Demiroz A, “A Study of Graphical Representation of KnittedStructures”, Ph.D. Thesis, UMIST, Manchester, 1998.

8. Kurbak, A., Some Investigations on the Geometric Propertiesof Plain Knitted Fabrics, Tekstil ve Makine, 2(11), 238 (1988).

9. Kurbak, A., Plain Knitted Fabric Dimensions (Part II), Textil.Asia, April, 36–44 (1998).

10. Ekmen, O., “A Geometrical Model for Widthwise Curlings inPlain Knitted Fabrics”, BSc Thesis, Dokuz Eylül University,Izmir, 2002.

11. Kurbak, A., Plain Knitted Fabric Dimensions (Part I), Textil.Asia, March, 33, 36, 41–44 (1998).



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