1.Study of loop.full

STUDY OF LOOP FORMATION PROCESS ON 1 1 V-BED RIB KNITTING MACHINE

PART1: A MATHEMATICAL MODEL FOR LOOP LENGTH

SRINIVASULU K, MONICA SIKKA & J HAYAVADANA

Department of Textile Technology, National Institute of Technology, Jalandhar, India

ABSTRACT

The mathematical model was proposed in order to determine the Loop length of a V-bed rib knitting machine

based on two dimensional coordinates of knitting zone. Those were obtained by stitch cam profile equations. The final

configuration of loop length was divided into 17 segments and each segment is given by an equation. The rib Loop length

depends up on three variables i.e yarn input tension, cam setting, and takedown were given included in the expression. In

this model variables like wrap angles, depth of needles and distance between cast-off loops are considered and these

variables ultimately effect the loop length calculated from the model. A computer program has also been generated in

JAVA to determine the theoretical loop length of V-bed flat knitting machine from the mathematical model proposed.

KEYWORDS: Yarn Input Tension, Cam Setting, Takedown Load, Loop Length, Needle Bed Verges, Wrap Angles,

Depth of Stitch Cam, and Loop Arm Configuration

INTRODUCTION

Loop length plays an important role in knitted fabric production in order to meet buyer specifications and

consumer satisfaction and the investigation of geometrical loop length or theoretical loop length which makes the

production of knitted fabric easy for knitter and it consumes less time to produce the fabric of different specifications.

Ultimately the rate of production will increase if we know the theoretical calculations of knitted fabric parameters.

The loop formation process became a subjective matter of research from past 50 years, and some studies related to

loop formation process are available in literature. The mechanism of single jersey loop formation process as explained by

Knapton and Munden (1966) was based on the concept of robbing back (%). A mathematical model of the single jersey

weft knitted process involving flat bottom stitch cam was formulated by Alsaka. Peat and Spicer developed a geometrical

model of single jersey loop formation process. A mathematical model of single jersey loop formation process involving

non-linear stitch cam and incorporating five different stages of initial geometry of knitting zone was developed by Lau and

Knapton.

In present work an attempt has made to study the impact knitting processing variables like yarn Input tension,

Cam Setting, and Take down Load on loop length and developed a mathematical model of loop formation on 1x1 rib loop

formation process on V-Bed flat knitting machine. The model is developed by considering two-dimensional coordinates of

knitting elements by rotating both front bed and back bed to 450 and considering the tuck point as an origin of coordinate

system. Based on 2D geometry a mathematical model has been developed for rib loop length. The model is validated

experimentally and statistically.

GEOMETRY OF KNITTING ZONE

On V-Bed rib knitting machine the geometry of knitting zone (KZ) was explained by boundary elements, namely

International Journal of Textile and Fashion

Technology (IJTFT)

ISSN 2250-2378

Vol. 3, Issue 2, Jun 2013, 1-14

© TJPRC Pvt. Ltd.

2 Srinivasulu K, Monica Sikka & J Hayavadana

front bed needles (FN) and Back bed needles (BN, the yarn comes in contact with Front bed verge (FBV) and Back bed

verge (BBV). The phase difference between front bed knitting point (FKP) and Back bed knitting point (BKP) on V-Bed

rib knitting machine is adjustable. Hence the geometry of knitting zone will be contour of the stitch cams, cam setting,

front bed gap (FG) and Back bed Gap (BG) between the beds, the phase difference between FKP and BKP, and the

position of the contact points of loop arms with bed verges.

Figure 1: The Knitting Zone Geometry of V-Bed Rib Knitting Zone

The geometry of knitting zone of V-bed machine is three dimensional in nature (As shown in figure 1) and it can

be represented in simplified form on a two-dimensional plane by rotating both the beds through 45o.

STITCH CAM PROFILE OF 1 X 1 V-BED KNITTING MACHINE

Figure 2 Stitch Cam Profile of 1 X 1 V-Bed Knitting Machine

Coordinates of the Needle Inside Knitting Zone and Geometrical Length of Loop

The reference line described by the equation . After crossing clearing point, the needle would follow the

profile of descending side of the stitch cam i.e. OAC′. After crossing knitting point(C), the needle may follow the ascending

Back needle bed

Study of Loop Formation Process on 1 1 V-Bed Rib Knitting Machine 3

Part1: A Mathematical Model for Loop Length

side profile of stitch cam CYZ (shown in figure 2).

The position of any needle inside knitting zone can be determined from equations the stitch cam profiles. The

equations of stitch cam profiles (from figure 2) derived in terms of cam angles, tuck height(Y), half needle spacing (a). So,

if the X-value of any needle known, the X-value of the other needles and Y values of all the needles can be calculated

using the equation of the cam profiles and other machine parameters. If is the position of along X-axis than

, where i =1, 2, 3…….

And = , where a is the half needle spacing, i.e

distance between neighbouring FN and BN.

After initial contact with yarn, as the FN1 is shifted towards the knitting point by an amount Sx, the new positions

of FNs would be given by:

=

And the new positions of BNs given by:

The new coordinates Y values of FNs and BNs corresponding to new X values would then be calculated from

equations of the cam profile.

As the needle take up new positions after a small shift from previous positions the vertical distances of the needles

with respect to the reference line also change. This would results in change in geometric length of loop arms as well as

wrap angles. The new position, the inclination and configuration of loop arms may also change.

Equations of Stitch Cam Profile

Equations of Front Bed Stitch Cam Profile

For descending side (O˂X˂OC).

Where md is magnitude of slope on the descending side of front bed stitch cam.

And ma is magnitude of slope on the ascending side of front bed stitch cam.

For ascending side (OC˂X˂OE)

For ascending side (OE˂X˂OG)

Where G is gap between two beds


Equations of Back Bed Stitch Cam Profile

For descending side (O<X<OD)

Where md’ is magnitude of slope of descending side of back bed stitch cam

For ascending side (OD<X<OI)

Where ma’

is magnitude of slope of ascending side of back bed stitch cam

LOOP ARM CONFIGURATIONS

The configuration of loop length has been changes continuously as both needles will take a new position until

needles reaches to knitting point position. There four different combinations of loop arm configurations were observed.

From figures 3 to 6.

Figure 3: Loop Arm Configuration When Both Fn1 and Bn1 are Between Bed Verges

After needle contact with yarn at feeding point position the front bed needle (FN1) has moved below the reference

line with yarn in hook but yet to reach the level of FBV, and back bed needle has moved below the BBV line. In this case

the leading and trailing arm lengths are not identical.

Figure 4: Loop Arm Configuration with Presence of Cast-Off Loops



From figure 4 it is observed that both FN1 and BN1 are moved below the verge levels, and the depth of needles

also same. The BN2 already takes new position and it reaches to below verge line, the depth of needle is more than that of

both FN1 and BN1.

Figure 5: Loop Arm Configuration When All Three Needles are Moved Away from Bed Verges

In case 5 all three needle were below verge levels but depth of needles are different. As in the last the depth of

FN1 and BN1 is same but here depths of three needles are different. And the leading and trailing arms are not equal.

From figure 6 the crowns of all the three needles (FN1, BN1, and BN2) are positioned below their respective bed

verges. Depth of needles (FN1 and BN1) is same below the verge lines and depth of needle (BN2) is less than that of

remaining two. And the loop configuration is settled in balance position, means needle were reached lower most (knitting

point) positions, and this configuration taken as final loop length diagram.

Figure 6: The Final Loop Arm Configuration When Both the FN1and BN1 Reached their Knitting Point

FORMATION OF WRAP ANGLES AT DIFFERENT STAGES OF LOOP FORMATION ON V-BED

RIB KNITTING MACHINE

Loop formation process was studied at different stages by moving hand lever at very minute speed. It was

observed that the needle come contact with the yarn well before entering into the knitting zone. After making contact with

yarn, the needle passed through different stages while moving downwards, as illustrated in figure 7 To 12.


Figure 7: Initial Yarn Feeding Point Position

Figure 8: Front Needle (Fn1) Needle Contact Position with Yarn

Figure 9: Formation of Wrap Angle around Front Needle Position



Figure 10: Position of Front Needle Below the Reference Line

Figure 11: Position of Front Bed Needle Below Reference Line with Wrap Angle a Around it


Figure 12: Knitting Point Position of Font Bed Needle

DERIVATION FOR WRAP ANGLES (Α, Β)

Figure 13: Initial Yarn Needle Yarn Contact Point

Form figure 13

ED a parallel line CA. DG, AE‖RR′

So

NOW

From ∆CAF,

And AD Perpendicular to CD

Than From ∆CAD

Than

From the same figure



And

FROM ∆BAN

And from ∆BAL

Than

Where

K=

DESCRIPTION OF FINAL LENGTH CONFIGURATION AT KNITTING POINT

The total length of loop was divided into 17 segments (figure 14), and the expressions of length of loop arm

segments are derived segment wise form the values of needles and yarn dimensions, gauge of the machine, different

contact angles, coordinates of needle, angles made by yarn with bed verges, wrap angles, cam angles and depth of needles

from bed verge lines. And the expressions are not similar to expressions derived on Dial and cylinder machine. The contact

angle (φ) between yarns of new loop and old loop has also been considered. The theoretical loop length represents

maximum length of yarn contained in repeating unit (Loop), so the loop length has taken when the FNs reached to its

lowermost position (i.e. knitting position).

Assumptions and Notations

In order to develop a model of the 1x1 loop formation process on v-bed machine the following assumptions were

made.

At their mutual contact points the yarn and needle are circular in cross-section.

The line of contact between yarns in new and cast off loops follows the path of arc of circle.

The line of contact between yarn and needle follows the path of an arc of a circle.

Below the bed verge line the half angle of wrap around any needle is not 90o

The coordinates of a needle corresponding to that of the tip of the crown.

The build up in the yarn tension obeys Amontons’s capatan equation of friction as yarn passes over and under the

knitting elements or across the yarn surface.

The yarn tension develop inside the knitting zone is proportional to the tensile strain in the yarn.

The space between two intersecting points of cast off loops is fixed between two needle separators.

The following notation is used in formulating mathematical relations (Figure 14)


RR′ = A horizontal line where front and back bed needles will coincide.

Y = Vertical distance between the reference line and X-axis.

X = Horizontal distance between the feeding point and Y-axis.

h1 = Depth of Front bed stitch cam below the bed verges at knitting point.

h2 = Depth of back bed stitch cam below the bed verges at knitting point.

a = half needle spacing

H = Feeding point height with respect to reference line.

Ni = The ith

needle.

E = Elastic coefficient of yarn, equalling the load per unit strain (relative rigidity).

= Radius of the yarn.

= Radius of front bed needle.

= Radius of the back bed needle

= Half wrap angle around front bed needle.

= Half wrap angle of trailing arm around back bed needle (Bn1).

= Half wrap angle of leading arm around back bed needle (Bn2).

= Angle made by yarn with cast- off loops.

ϑ = Angle of path of yarn with machine bed verges

= The angle made by yarn with horizontal Bed verge line at front bed knitting point.

= The angle made by yarn with front bed needle position at reference line Bed verge line.

= The angle made by yarn with horizontal Bed verge line at Back bed knitting point.

LOOP LENGTH AT KNITTING POINT



Figure 14 Configuration of Loop Length At Knitting Point

The theoretical loop length (Lt)

=AC+CD+DE+EG+GI+IJ+JK+KL+LM+AN+NO+OP+PQ+QR+RS+ST+TU+UV …… (I)

From figure

Arc AFC

=

From arc CFD

Draw a parallel line to DE through F and CLF is half cast off loop spacing (S/2), DE= FF2

From ∆FF2F′

Loop arm makes an angle φ with cast- off loops

From arc ECLG

From figure II′ is parallel to BBV line and draw a perpendicular from G on II

′

Than from ∆GII′

From arc ICLJ ,

From the figure JK=DE

From arc KBL

From arc LBM

AN=AC

=

ON=CD

=

OP=DE

=


From arc PCLQ

From ∆RQQ′

SR=

ST =

TU =

UV =

Derivation for Expression of Wrap Angle around Front Bed Needle (FN1)

Figure 15: Wrap Angle around Front Bed Needle (FN1)

Consider front bed needle (FN1) is above front bed verge.

The leading arm angle (

Now

From ∆BEF we have

So

From ∆BFK

Than



So

Derivation for Expression of Wrap Angle around Back Bed Needle (BN1)

Figure 16: Wrap Angle around Front Bed Needle (BN1).

The Trailing arm angle (

Now

From ∆FBK

Than

From ∆FBL we have

So

FINAL THEORETICAL LOOP LENGTH (lt) =

CONCLUSIONS

In this model three variables yarn input tension, cam setting, Takedown load has been studied and how these

variables effecting the loop length. In the theoretical expression three variables included as wrap angles (δL), depth needles

from bed verges (h1, h2), space between two cast off loops respectively. The final loop arm configuration divided in to 17

segments each segment is given by an expression based on mathematical geometry. And a JAVA program also generated


in user friendly manner so that the input parameters under studied can be varied according to requirement. After execution

of programme for any combination of input parameters, the required output parameters are noted or printed from result file.

The output parameters were compared with experimental results.

REFERENCES

1. Banerjee, P.K., and Alaiban, T.S (1987)., Mechanism of Loop formation at Extreme cam setting on a sinker top

machine: part II Analysis of limiting conditions, Textile research journal, 57, 568-574.

2. Banerjee, P. k., and Alaiban, T.S (1987)., Mechanism of Loop formation at Extreme cam setting on a sinker top

machine: part I Relation between count, gauge, and tightness factor, Textile research journal, 57, 513-518.

3. Banerjee, P. K., and Ghosh, S (1999)., A model of single-jersey Loop-formation process, Journal of textile

institute, 90,187-208.

4. Carmine Mazza, Paola Zonda (2001), Knitting (A reference book of textile technology), ACIMIT, The Italian

association of textile machinery producers, first edition.

5. C. Prakash and C. V. Koushik (2010)., Effect of loop length on the dimensional properties of silk and model

union knitted fabric, Indian Journal of Science and Technology,7,752-754.

6. David H. Black (1968)., Design And Performance Of Weft-Knitting Machinery, a Ph.D thesis report.

7. Ghosh, S., and Banerjee, P. K (1990)., Mechanics of the single jersey weft knitting process, Textile Research

Journal. 60,203-211.

8. J. J. F. Knapton and T. W.-Y. Lau (1978)., The design and dynamics of non-linear cams for use in high-speed

weft-knitting machines part 1: the theoretical dynamics of non-linear cams., Journal Of Textile Institute ,69,161-

168.

9. Knapton, J.J.F., and Munden, D.L (1966)., A study of mechanism of loop formation on weft knitting machinery

,Textile Research Journal 36, 1072-1090.

10. Knapton, J. J. F (1968)., the dynamics of weft knitting: further theoretical and mechanical analysis, Textile

Research Journal, 38,914-924.

11. Knapton, J.J.F., and Munden, D.L (1966)., A study of mechanism of loop formation on weft knitting machinery,

part II: the effect of yarn friction on yarn tension in knitting and loop formation ,Textile Research Journal 36,

1081-1091.

12. Noboru Aisaka (1971)., Mathematcial considerations of weft knitting process, Journal Of The Textile Machinery

Society Of Japan,24,82-90.

13. Nuiting. T.S., Kinetic Yam Friction and Knitting, Journal of Textile Institute, 51, 190-202(1960).

14. Noboru aisaka, and Tatsuya kawakami (1969)., knitting tension during weft knitting process, Journal of Textile

Machinery Of Japan,22,228-233.

15. Ray, S.C., and Banerjee P.K (2000)., Some preliminary investigations into mechanics of 1x1 rib loop formation

on a dial and cylinder machine, Indian Journal of Fibre & Textile Research. 25, 97-107.

16. Spencer, D.J., knitting technology, third edition, 85(2001), Woodhead Publishing Limited.

17. Sadhan chandra Ray. , and Banerjee P.K (2003)., Mechanics of 1x1 rib loop formation on a dial and cylinder

machine: part I- Modelling of the 1x1 rib loop formation process, Indian Journal of Fibre & Textile Research.

28,185-196.

18. Sadhan chandra Ray., and Banerjee P.K (2003). ., Mechanics of 1x1 rib loop formation on a dial and cylinder

machine: part III-Validation of the Model, Indian Journal of Fibre & Textile Research. 28,246-259.

1.Study of loop.full

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