A STUDY OF MECHANICS OF SPIRALITY
IN PLAIN KNITTED FABRICS
A THESIS
Presented to
The Faculty of the Graduate Division
by
Dhairya Prakash
In Partial Fulfillment
of the Requirements for the Degree
Master of Science in Textile Engineering
Georgia Institute of Technology
January 1976 % A STUDY OF MECHANICS OF SPIRALITY
IN PLAIN KNITTED FABRICS
Approved:
i*X-- L. Hpward 0IsO|i, Chairman
"Amad Tay^i
David R. Gentry^
Date approved by Chairman: Jan. 29, 1976 Dedicated to my loving and understanding wife,
Madhuri, and respected mother and father with out whose cooperation this work could not have
been possible.
# iii
ACKNOWLEDGMENTS
At the conclusion of this thesis I wish to express my sincere appreciation and gratitude to the following persons :
Dr. L. Howard Olson, thesis advisor, and Dr. Amad Tayebi, the member of the reading committee who initially conceived the thesis topic
Their continued interest, encouragement and advice throughout the study is greatly appreciated.
Dr. David R. Gentry, for his services as a member of the reading committee.
Dr. M. V. Konopasek, without whose help solutions of the energy expressions would have been most difficult.
Dr. W. Denny Freeston, Jr., Director, School of Textile Engineer ing for granting partial financial assistance during the major part of the study.
Dr. Stayanadhan Atluri of Engineering Science and Mechanics for his advice and guidance. iv
TABLE OF CONTENTS
Page ACKNOWLEDGMENTS iii
LIST OF TABLES vi
LIST OF ILLUSTRATIONS vii
SUMMARY viii
NOMENCLATURE ix Chapter I. INTRODUCTION 1
1.1 Statement of the Problem 1.2 Literature Survey 1.3 Purpose of the Research 1.4 Method of Research
II. MATERIALS, INSTRUMENTATION AND EQUIPMENT 6
2.1 Materials 2.2 Instrumentation and Equipment
III. PROCEDURE 8
3.1 Rubber Rod Structure 3.2 Nylon Structure 3.3 Polyester Structure 3.4 Dry Relaxation
IV. DEVELOPMENT OF GEOMERIC MODEL AND ENERGY EQUATIONS 14
4.1 Physical Observations • 4.1.1 Torque in a Twisted Yarn Loop 4.2 Geometrical Model 4.3 Energy Equations 4.3.1 Equations of Curvature and Torsion Along CD
V. RESULTS AND DISCUSSION 28
5.1 Experimental Results 5.2 Theoretical Results ^ TABLE OF CONTENTS (Concluded)
Chapter Page VI. CONCLUSIONS AND RECOMMENDATIONS 38
Conclusions Recommendations
Appendices A. GEOMETRY OF THE HELIX Al
B. CHANGE OF TWIST DUE TO BUCKLING 45
C. LAWSON-HEMPHILL F. A.K 48
D. QUESTION ANSWERING SYSTEM 50
BIBLIOGRAPHY 52 vi
LIST OF TABLES
Table Page 1. Lawson-Hemphill F.A.K. Settings and Amounts 13
2. Actual and Nominal Twists per Inch 28
3. Spirality in Degrees 30
4. Minimum Loop Energy 31 VI1
LIST OF ILLUSTRATIONS
Figure Page 1. Rubber Structure 9
2. Tracing of Nylon Structure Through Enlarger 11
3. Untwisted Yarn Loop 15
4. Forces in a Twisted Yarn Loop 15
5. Energy Dissipation by Rotation of the Loop 15
6. Energy Dissipation by Buckling of the Loop 15
7. Forces at the Points of Interlocking of the Loops 17
8. Geometrical Loop Model 19
9. Curvature and Torsion Along CD 23
10. Angles Q , Q™* ^ ^^d Course Spacing p for Loop Length L = 0.28 Inch 33
11. Angles Qg, Qg, P and Course Spacing p for Loop Length L = 0.35 Inch 34
12. Angles Q , Q2, 3 and Course Spacing p for Loop Length L = 0.42 Inch 35
13. Comparison of Angles Qg, Q^, P for Loop Lengths 0.28, 0.35 and 0.42 Inch 36 14. Geometry of the Helix 42
15. Change in Twist Due to Buckling 46 r 16. Helix 46 Vlll
SUMMARY
The cause of spirality in knitted fabrics and hosiery remained obscure for a long time. It was known to be influenced by the feed yarn twist level and the construction of the fabric. Twist liveliness and not the twist itself was responsible for spirality. The fabrics knitted with twisted yarns which were set prior to knitting practically showed no spirality. This thesis attempted to develop mathematical relations in volving the physical properties of the yarn and knitted loop parameters, to predict the spirality of a knitted structure with known parameters.
Knit structures made of solid rubber rods and nylon monofilament fishing line were studied for loop shape. A geometrical model of a knitted loop was developed on the assumption that in a relaxed fabric, the loop would acquire a minimum energy configuration. Mathematical ex pressions were developed to express the energy of the loop. The energy expression was minimized to determine equilibrium. Empirical data were obtained on spirality of fabrics knitted with polyester yarns twisted to various levels of twist and using different loop lengths.
The spirality increased with loop length and turns per inch in the yarn. There was a definite range of combination of loop length and turns per inch which produced stable fabrics. Below that range, fabric production was not possible. Above that range, the fabrics distorted into rope like structures. Results of the energy expressions suggested that friction at the interlocking points could not be neglected and a more in tense study of the problem was desired. IX
NOMENCLATURE
S3nnbol for Symbo1 Term Q.A.S.
helix angle of S helix ANGS
helix angle of Z helix ANGZ
'VR radius of S helix s 0 'VR radius of Z helix 0 z *L length of yarn along S helix LTHS s length of yarn along Z helix LTHZ z 'VL length of yarn along CD = AB + EF LTHC c .vp angle of inclination of the axis of helices BETA
.Vp course spacing CSPG
loop length along ABCDEF LLTH
.VA length of the helix axis 0
W wale spacing 0
change of twist per unit length in S helix -^s due to buckling
change of twist per unit length in Z helix due to buckling 0
initial turns per inch in the yarn TWTY
twists per unit length in S helix TWTS
°^z twists per unit length in Z helix TWTZ K curvature of S helix CUVS s K curvature of Z helix CUVZ z UBS energy due to bending in S helix UBS
T^t^ X
Symbol for Symbo1 Term Q.A.S,
UBZ energy due to bending in Z helix UBZ
UBC energy due to bending in AB + CD + EF UBC
UTS energy due to torsion in S helix UTS
UTZ energy due to torsion in Z helix UTZ
UTC energy due to torsion in AB + CD + EF UTC
UTOT total energy of the loop = UBS + UBZ + UBC + UTS + UTZ + UTC UTOT
E elastic modulus MOOT
6 shear modulus MODS
I moment of inertia MMTI
I polar moment of inertia PMTI P EI bending rigidity BER
GI torsional rigidity TOR
Note: 0 indicates these terms were not used in Q.A.S. (Question Answering System).
* indicates these symbols are used with reference to Figure 8. CHAPTER I
INTRODUCTION
1.1 Statement of the Problem
In a plain weft knitted fabric, the lengthwise columns of stitches are called wales and the crosswise rows of stitches are called courses.
A distortion in plain tubular knitted fabrics in which the wales follow a helical path around the axis o£ the tube is known as spirality. The wales are inclined to the edge of the tube. If a single feed is used in knitting the tube, the wales will be perpendicular to courses. Another form of distortion quite often confused with spirality is due to round drop of the fabric knitted on a multifeed circular machine. Although the wales are not perpendicular to the courses, they are still parallel to the edge of the fabric.
1.2 Literature Survey
Though spirality has been known to exist in knitted fabrics for a long time, not much work has been done in this direction. Davis and
Edwards (1,2) carried out extensive studies on woolen and cotten spun yarns and showed that spirality was directly related to the twist in the yarn. Fabrics knitted with Z twisted yarn showed a bias to the right and those knitted with S twisted yarn showed a bias to the left. Increasing the loop length (slack structure) increased spirality. The direction of rotation of the machine had no effect on spirality. They also noted that setting of yarn prior to knitting reduced spirality. Nutting (3) stated that it was not the presence of the twist but the twist liveliness in the yarn which was responsible for spirality. In a fabric knitted with a well set twisted yarn, spirality was practically absent. Spirality of the fabrics reduced when the fabrics were relaxed (4) proving thereby that it was the unrelieved torque in the yarn which caused spirality in the knitted structure.
Doyle (5) and Munden (6) recognized that the controlling factor in a knitted fabric was the loop length. It was suggested by Munden that the loop shape was a geometrical property of the loop and was independent of the physical properties of the yarn or the amount of yarn knitted into the loop. He also noted that a knitted loop was a three dimensional unit.
In order to knit a structure, the yarn had to bend in the plane of the fabric and also in a plane perpendicular to the plane of the fabric.
In a stable relaxed state the whole structure tended toward a state of minimum energy. He showed that for a relaxed plain knit fabric the following relations must hold:
C X W = K /j^^
C = K /j^
W = K /i w where C and W are courses and wales per inch, i is the stitch length in inches and K , K and K are constants. He did not specify numerical sew tr J values of the constants which depended on the actual shapes of the loop.
Experimentally their values were found to be: K 21.6 ^ s ,w K 5.3 , Wet Relaxed Fabrics c ,w K w,w 4.J and
II
K , = 19.0 s ,d K = 5.0 \^ Dry Relaxed Fabrics
K - = 3.8 w,d
A number of attempts has been made by researchers to develop geo metrical and mathematical models of the knit loop to fit experimental re sults. The first loop model was given by Chamberlain (7). This was a two dimensional model. It was based on the assumption that knit loop was made up of circular arcs joined by straight lines. Shinn (8) examined this model in great detail, but it was discarded because it was two di mensional and actual knitted loop is three dimensional. Pierce (9) im proved upon Chamberlain's model and constructed a third dimension by laying the two dimensional model on the surface of a circular cylinder whose generators were parallel to the lines of courses. A serious defect of Pierce's model was pointed out by Leaf and Glaskin (10) who showed that there were discontinuities of torsion along the loop and consequently a stable fabric could not be made from the loop of the shape proposed by
Pierce.
Leaf (11) proposed two models which were based on elastica. He assumed that the knitted loop takes the shape of an elastica. The first
'W-' model was based on the elastica lying on the surface of a circular cylin
der whose generators were parallel to the courses, thus giving the third
dimension to the loop structure. This model fitted well to the result of wet relaxed fabrics but was not suitable for dry relaxed fabrics. On
closer examination he found that in the third dimension, the loop followed
a sine wave instead of a circular path. He then developed his second model based on the elastica lying on a cylinder whose cross section was
of sine wave instead of circular. The second model was much more compli
cated than the first.
All the above models were developed on the assumption of some
geometric shape for the knitted loop and were not derived from equilibrium
considerations of the forces and couples applied to a loop by its neigh bors. Postel and Munden (12) developed first a two dimensional model as
suming a system of forces and couples acting on the loops at the inter
locking points. The forces and couples were derived from physical con
siderations of equilibrium and loop S3nTimetry. The two dimensional model was later extended to a three dimensional structure. Empirical methods were still used to fit these models to experimental results. Shanahan
and Postel (13) studied the relaxed plain knitted fabric and derived the
loop configuration based on the reaction forces and couples acting within
the structure. The magnitude of the reaction forces and couples was de
termined by the yarn displacement necessary for loop interlocking. They
showed that a minimum energy structure existed and was independent of
fabric tightness and yarn properties. All above models did not take into
consideration the twist in the yarn and both legs of the loops were con- sidered to be symmetrical.
1.3 Purpose of the Research
The purpose of this research was to develop mathematical relations involving knitted fabric parameters (loop length) and physical properties of yarns, such as bending rigidity, torsional rigidity, elastic modulus, and turns per indh, to predict the spirality of the relaxed knit struc ture with known parameters.
1.4 Method of Research
Knit structures made from twisted solid rubber rods and monofila ment nylon fishing line were studied for loop shape. A geometric model was developed to conform to the observed loop shape as closely as possible.
A minimum energy configuration was assumed for the knit loop in the fully relaxed state. Based on the geometric model and minimum energy configura tion, mathematical relations were developed to express the total energy of the loop as a function of fabric parameters and yarn properties. The total energy of the loop was minimized using the Q.A.S. (Question Answer ing System) developed by Dr. M. S. Konapasek with respect to getting the angle of spirality.
Samples with different loop lengths were also knitted on a Laws-
Hemphill F A K machine using polyester yarn twisted to various levels of twists. The angle of spirality for these samples was measured in a dry relaxed state. CHAPTER II
MATERIALS, INSTRUMENTATION AND EQUIPMENT
2.1 Materials
Solid rubber rods, Du Pont nylon fishing line and multifilament polyester yarn were used to make knit structures to study loop shapes and spirality.
Rubber rods were made of Buna-n-Nitrile manufactured by the Manta- line Corporation. The nominal diameter of the rod was 3/32 inch. The rod was hand knitted on a rectangular (l/8 inch diameter steel wire) as sembled to support the fabric structure. The diameter of the nylon fish ing line was 0.009 inch. The polyester yarn used was 150 denier, 34 filament. This yarn was twisted to different twist levels and machine knitted to study spirality.
2.2 Instrumentation and Equipment
Existing Equipment
The following equipment was used for the fabrication of the samples.
1. The Saco-Lowell slubbing frame
2. The Saco-Lowell spinomatic spinning frame
3. The S.K.F. spinster 82
4. The F.A.K.(Fabric Analysis Knitter) by Lawson-Hemphill. Physical Testing and Observation Instruments
To study the structures and perform the physical tests, the follow ing laboratory instruments were used:
1. Low power microscope
2. Photographic enlarger
3. Large magnifying glass with lamp
4. Alfred Suter Twist Tester
5. A large protractor was made to measure the spirality of the
polyester knit samples. CHAPTER III
PROCEDURE
3.1 Rubber Rod Structure
Two metallic rods bent at right angles and connected by rubber bands to form a rectangle were used in making the rubber structure.
Wooden clips were slid over the two opposite sides of the rectangle.
Using 28 diameters as the loop length, for six loops, the required length of the rubber rod was 15.75 inches. Rods were cut to 18 inch lengths and two points were marked 16 inches apart. Small pieces of wires, cut from paper clips were glued at these points using Permabond instant glue, such that the wires were horizontal when the untwisted rod laid flat on the table. Marks were also made at 15.75 inches from one end to give the length of six loops.
One end of each rod was clamped in wooden clips, sliding over one side of the rectangle, at the point where the wire was glued. The other end was turned four times to give Z twist of 0.25 turn per inch. After twisting, another piece of wire was glued horizontally at the 15.75 inch mark and clamped between another wooden clip on the other side of the rectangle. The horizontal wires glued to the rods prevented slippage of rods between the wooden clips. Then using wire clips, these rubber rods were interknitted to give the rubber knit structure shown in Figure 1.
It was observed that the loops were of unequal sizes. The unequal size of the loops was due to friction between the rubber rods and the method (U »^ 3 -u o 3 U 4-1
CO u (U Xi Xi i
(U u p bO •H Fm 10
of making the loops. In spite of heavy lubrication, using a spray lubri cant, balanced loop size could not be obtained. Nonetheless, the loops did show a regular feature in all loops. One leg of the loop was raised, while the other leg of the loop was depressed from the plane of the struc ture. One leg of the loop straightened out while the other leg buckled.
The three dimensional geometry of the loop was clearly visible.
3.2 Nylon Structure
Nylon fishing line with 0.009 inch diameter, made by Du Pont, was used to prepare the nylon knit structure. It was twisted to 0.5 turn per inch Z twist on a Saco-Lowell slubbing frame. No tension was applied to the fishing line during twisting. The twisted filament was knitted on a
Lawson-Hemphill Fabric Analysis Knitter to give 25 diameters of nylon line per loop. The knitted fabric was taped onto a glass slide and ob served under a low power microscope (magnification 25). It was observed that the left leg of the loop straightened out and was almost perpendicu lar to the course line, while the right leg of the loop buckled. The sinker loop was narrower than the needle loop. The difference in the needle and sinker loops could be due to change in the physical properties of nylon as a result of high rate of distortion during knitting, thereby causing a permanent plastic type of deformation in the nylon. A projected tracing of the structure is shown in Figure 2.
3.3 Polyester Structure
Du Pont 150 denier, 34 filament polyester yarn was used to get empirical data for spirality. The yarn was twisted to six levels of twist on a Saco-Lowell spinomatic spinning frame and S.K.F. spinster 82. 11
< Sinker Loop
Needle Loop
Figure 2. Tracing of Nylon Structure Through En larger 12
Yarn was fed directly through the front rollers, under no tension. Two different machines were used for twisting because all the five levels of twist could not be obtained on one machine. The twisted yarn was condi tioned for over 72 hours in standard atmosphere at 75°F and 65 percent relative humidity before knitting on the Lawson-Hemphill F.A.K. Samples were knitted with 24 inches per course to 44 inches per course with an increment of 4 inches per course. Table 1 shows the F.A.K. settings for various structures. The two hundred needle, 3 1/2 inch diameter cylinder was used for knitting the fabrics.
3.4 Dry Relaxation
For dry relaxation the knit fabrics were placed flat on the surface of a table for 72 hours in a conditioned atmosphere. The angles of spir- ality of the relaxed fabrics were measured using a magnifying glass lamp and protractor. The courses and wales per inch were also measured. 13
Table 1. Lawson-Hemphill F.A.K. Settings arid Amounts
Length of yarn per 24 28 32 36 40 44 course (inches)
Gear ratio 4:1 8:1 8:1 8:1 8:1 8:1
Positive feed 3.5 4.0 4.5 5.0 5.5 circumference (inches)
Number of needles 200 200 200 200 200 200 in the cylinder
Loop length (inches) 0.12 0.14 0.16 0.18 0.20 0.22 14
CHAPTER IV
DEVELOPMENT OF GEOMERIC MODEL AND ENERGY EQUATIONS
4.1 Physical Observations
Throughout the following discussion it is assumed that
1. the yarn (or rod) had a Z twist,
2. the material was linearly elastic, and
3. there was no friction between the yarns (rods).
4.1.1 Torque in a Twisted Yarn Loop
An untwisted rod bent into a loop as shown in Figure 3 was mechani cally balanced in that configuration and stayed in the plane of the paper.
It was sjnnmetrical about the central line and the angle Q^, which the left part of the loop made with the vertical, was equal to the angle Q , which the right part of the loop made with the vertical as shown. When the rod was twisted and bent into a loop, it no longer remained in the plane of the paper. There was an unbalanced torque in the two legs of the loop. The right leg of the loop had a tendency to come out of the plane of the paper (designated by a dot inside a circle), while the left leg of the loop had a tendency to go into the plane of the paper (designated by a cross inside a circle), as shown in Figure 4. The loop could relieve itself of the torque by rotating around its two legs as shown in Figure 5.
In knitted fabrics which are formed by the interlocking of such loops to adjacent loops, the dissipation of torque by rotation around the legs was not feasible. The loop thus restricted from rotation by adjacent loops. 15
Figure 3. Untwisted Yarn Loop
Figure 4. Forces Figure 5. Energy- Figure 6. Energy in a Twisted Yam Dissipation by Dissipation by Buck Loop Rotation of the ling of the Loop Loop 16
could dissipate the torque by buckling into two helices. The left leg buckled into an S helix while the right leg buckled into a Z helix as shown in Figure 6,
A Z twisted yarn gained Z twist while buckling into an S helix and gained S twist while buckling into a Z helix (Appendix B). The net result of the buckling, therefore, was that the left leg of the loop, called the S helix, gained Z twist, so that the resultant twist in the yarn of the S helix was higher than that of the original yarn. In the right leg of the loop, called the Z helix, the yarn gained S twist, so that the resultant twist in the yarn of the Z helix was less than that of the original yarn. Due to this redistribution of twist as a result of buckling, along a wale line, one leg of the loop had higher torsional energy than the other leg.
In a knitted structure, yarn in the loops did not remain in one plane. It was bent from the plane of the fabric to accommodate the pass ing over and under of the adjacent loops during knitting. At the places where loops interlocked each other, the tendency of both intersecting loops was either to come out of the plane of the fabric or to go into the plane of the fabric as shown in Figure 7. These tendencies were in the opposite directions. If one loop coming out of the plane of the fabric tended to rotate to the left, the other loop coming out of the plane of the fabric tended to rotate to the right thus interlocking the other loop.
The same thing was observed for loops going into the plane of the fabric.
Both tended to go into the plane of the fabric, but their rotational tendencies were in the opposite directions. 17
Figure 7. Forces at the Points of Interlocking of the Loops 18
Thus in a knitted fabric, a loop was under the action of forces of bending, buckling and differential twist levels in the two legs of the loop. When allowed to relax, the loop came to an equilibrium configura tion when the total sum of energy in the loop due to bending, twist and buckling was minimum.
Furthermore, due to the symmetry of the structure, the forces of interaction of loops at A were exactly the same as those at A (Figure 7),
Similarly, the forces of interaction at B and B were the same. Due to the definite dimensions of the yarn, there was a definite region of con tact of the interlocking loops. But for theoretical purposes, a point contact between interlocking loops at A, A , B and B was assumed. If a parallel set of axes was drawn at points A and A , the coordinates of B with respect to A would be the same as the coordinates of B with respect to A . In other words, if points A and A and B and B were joined by straight lines, AA would be parallel and equal to BB .
4.2 Geometrical Model
Based on the above observations, a geometrical model of a relaxed knitted fabric loop in minimum energy configuration was developed as shown in Figure 8. ABCDEF represented the central line of the loop.
B,C,D and E were the points of contact with the adjacent loops. The fol lowing assumptions were made to develop the mathematical relations for energy of the loop in an equilibrium (minimum energy) configuration.
1, The yarn formed one complete S helix between the points of contact B and C,
2. The yarn formed one complete Z helix between the points of 19
L (L) / %^V s z
2TTRs (Rz )
Figure 8. Geometrical Loop Model 20
contact D and E.
3. BC and DE formed the axis of the helices.
4. The torsion and curvature changed uniformly and linearly between points C,D; A,B; and EF. L - (L +L ) r^ S Z 5. Due to symmetry CD = L = x where
L = length of the arc CD
L = length of the loop along ABCDEF
L = length of yarn BC along the S helix
L = length of the yarn DE along Z helix also
Q = helix angle of S helix s ^ Q = helix angle of Z helix
R = radius of S helix s R = radius of Z helix z P = angle of inclination of the helices
p = course spacing
A = length of the helix axis.
4.3 Energy Equations
From the geometry of the helix,
A = 2TT R cot Q . (1)
A = 2TT R cot Q (2)
L =^ A sec Q (3) s s ^ -^
See Appendix A. 21
\= A sec Q^ (4)
P/A = cos 3 ' (5)
A = p sec 3 (6)
^s=^-¥-^ (7) s . 2 ^ sm Q
If T is the twist per inch (Z) in the original yarn, then change of twist per inch in S and Z helices is given by
, , sinQ
s
•kickj-ju. sinQ^z
Z
Therefore final angles of twist per unit length in the S and Z helices are
a - 2TTT + T (11) s s ^ ^ a = 2TTT - T (12) z z ^ "^
4.3.1 Equations of Curvature and Torsion Along CD
It was assumed that curvature and torsion varied uniformly and linearly along the length of the curve CD. Length of the curve CD was 0 at C and L at D. c
See Appendix A. On buckling into S helix, yarn gains Z twist and hence + sign. On buckling into Z helix, yarn gains S twist and hence - sign. See Appendix B. 22
Figure 9 shows the graph of curvature (torsion) along the curve CD. The curvature equation was given by
K = iTU^ + c (13) where
K = curvature at any length i along CD
i = length of the curve CD at any point (0 < i < L )
m = slope of the curvature line K K ^ s z c = intercept on the curvature (torsion) axis or K - K K = -~ Je + K . (14) JL S
Check If J^ = 0 K = K s £ = L K = K c z
Similarly, the torsion equation along CD was given by
a - oi a = -A: ^ Ji + a (15) JLi S
From equations (1) and (2)
R = —- tan Q (16) s 2n s ^
R = ~ tan Q (17) Z ZTT Z
Substituting A from (6) into 3, 4, 16 and 17,
L = p .sec Q sec p (18) s s 23
K s (Ois )
c o •1^ en U O H
(U U
•M nJ K (a ) > z z 5-1 U
Figure 9. Curvature and Torsion Along CD 24
L = p sec Q sec P (19)
p tan Q sec P R = :r (20) s 2TT p tan Q sec p \- i (^^>
Energy due to bending (14) along a curve is given by
U = j'^dixi (22) i 2EI 0
2EI where
M = EI/R
E = elastic modulus
I = moment of inertia
Jl = length along the curve
R = radius of curvature FT Substituting M == -r- in (22)
U^gi (23)
1 2 = ^ EI K Jl where
EI = bending rigidity
K = curvature 25
Using equation (23), energies of bending of S and Z helices are
UBS = ^ EI K ^ L (24) 2 s s
UBZ = ^ EI K ^ L (25) 2 z z
Due to the sjnranetry of the structure,
Energy of bending of AB + CD + EF = 2 of'cF or L c UBC = 2 ^ [J K^ dx]
Substituting K from (14)
^c ,K -K .2 UBC = EI [IJ „, (26) '"' 0 i-^'^h)' '^c '\i
EI r 2 2 T UBC = -^ L [K + K + K K ] or 3 c s z s z
Energy due to torsion (14) was given by
U = i i GI a^ dx (27) 2 P h^ 0 = GI A P where
G = shear modulus
I = polar moment of inertia P
a = twist per unit length (radians)
Jl = length of the curve 26
Using equation (27) energies of torsion of S and Z helices were given
by
UTS = i GI a ^ L (28) 2 p s s
12 UTZ = TT GI 0^ L (29) 2 p z z ^
As before, due to the sjnnmetry of the curve,
Energy of torsion in AB + CD + EF = 2 of CD
GI ^^c UTC = 2 r 2 ^ '0 a dx.
Substituting a from (15)
L c ,o; -0^ ,2 UTC = GI [J {-^ i + cj dx] (30) P -0 ' "c
GI 2 2 or UTC = —r^ L(a + a + a oi ) 3 c s z s z
Therefore, total energy of the loop was given by
UTOT = UBS + UBZ + UBC + UTS + UTZ + UTC (31)
On substituting the values of individual terms in equation (31), the
total energy UTOT could be expressed as
UTOT = f(Q Q 3, p, T, E, G, I, I L) 27
where T, E, G, I, and I were yarn parameters (fixed), and L was the knitted fabric parameter (fixed). Q , Q , P and p were variable loop parameters. Product EI was bending rigidity of the yarn and product GI was torsional rigidity of the yarn. For testing purposes, EI was taken to be equal to 1.0 and GI was taken to be equal to 0.5. P 28
CHAPTER V
RESULTS MD DISCUSSION
5.1 Experimental Results
Polyester yarn was twisted to six levels of twist. Three levels of twist were obtained on a Saco-Lowell spinomatic spinning frame, namely
7.7, 9.8 and 14.7 turns per inch (nominal). Twist levels 12.0 and 16
(nominal) were introduced with a S.K.F. spinster 82 spinning frame. The
S.K.F. machine could not be run at less than 16,000 rpm and during twist ing the traveller became too hot. Many times the yarn was burnt. A physical change in the properties of the yarn was expected at these twist levels. The nominal and actual twist levels are given in Table 2.
Table 2. Actual and Nominal Twists per Inch
Actual Nominal Machine Used
9.1 7.7 Saco-Lowell 11.8 9.8 Saco-Lowell 14.0 12.0 S.K.F. 82 16.2 14.7 Saco-Lowell 16.8 16.0 S.K.F. 82
The yarn was conditioned for 72 hours in standard atmosphere at
70''F and 65% relative humidity. Samples were knitted on the F. A.K. machine,
Fabrics could not be made with 24 inches per course except with yarn with
-^^j€- 29
9.1 turns per inch. The yarns with higher twists broke at the needles.
Spirality of the fabrics was measured with a large protractor using a
magnifying glass with a lamp. The results are shown in Table 3. As sus
pected, yarns twisted on the S.K.F. spinster 82 suffered change in the
I physical properties and samples made from those yarns showed less spiral
ity as compared to the samples made from the yarns twisted on the Saco-
Lowell spinomatic.
As can be seen from Table 3, spirality increased with increasing
loop length at each twist level to a certain point after which the un
balanced torque in the fabric distorted the samples into rope like struc
tures and spirality measurements were not possible and valid. With yarns
at higher twist levels (above 16 turns per inch) stable fabrics could be
made with only one loop length, namely 0.14 inch (corresponding to 28
inches per course). With higher lengths, the fabrics distorted into
ropes and with lower loop lengths, fabric production was not possible.
Lower twist levels gave a wider range of loop lengths for stable distor-
1 tion free fabrics.
I 5.2 Theoretical Results
The total energy expression for a knitted loop in a minimum energy
configuration, developed in Chapter IV, was minimized by using Q.A.S.
loaded on a PDP-10 computer at Emory University. The following initial
I values were used.
Free parameters:
P = 10° Q = 10"
Q = 15" p = 0.0625 inch 30
Table 3. Spirality in Degrees
Inches Twists per Inch per ' Course 9.1 ' 11.8 14.0 16.2 16.8
24 9.7 0 0 0 0
28 13.6 17.7 12.7^ . 29.7 21.9*'^
32 15.2 22.5 16.7^ * *
36
40
44 * * * * *
0 Samples could not be made due to high yarn twist.
t Change in the physical properties of the yarn was suspected due to high rpm on the S .K.F, machine which occasionally caused melting of yarn during twisting.
''^ Samples were so much distorted that measurement of spirality was not possible. • 31
Assigned parameters:
L = 0.28 inch T = 8 turns per inch
EI = 1.0 GI = 0.5 P
The results are summarized in Table 4, It was observed that the total energy of the loop came out to be negative. On examination it was found that this was due to the negative value of Q . The value of course spac ing was 212 inches, which was not reasonable.
Table 4. Minimum Loop Energy
UBS 2.6337 E - 02 UBZ 6.3899 E + 03 UBC -1.8686 E + 03 UTS 1.3019 E + 06 UTZ 2.3163 E + 11 UTC -2.3163 E + 11 i TOT -1.0771 E + 06 ' ' \ P 7.8491 E + 01 "^ 78° j Q -5.8909 E + 01 ^ 59° s Q 9.000 E + 01 =^ 90° z CSPG 2.1236 E + 02 ^ 212 inches
In the total energy expressions, the course spacing term appeared in the denominator and no restriction was applied to its value. In the process of minimization, it was natural for the Q.A.S. to increase the value of course spacing in an effort to minimize total energy. To over come the above difficulties, the following modifications were made. All 32
angles, that is Q , Q and P, were forced to be positive and varying be
tween 0 and TT/2. Course spacing was restricted by
CSPG = 5.3 X cosP where 5.3 was the value of K for relaxed fabrics. c
The total loop energy was minimized for different loop lengths and
initial yarn twists. The values of helix angles, spiralities and course
spacings with various levels of twists and loop lengths are shown in Fig
ures 10, 11, 12, and 13. Initial values in each case were Q - 10° Q = 10° s z P - 10° CSPG = 0.0625 inch
As expected from the theoretical model, Q was always greater than
Q indicating that the right leg of the loop buckled to dissipate the
strain energy due to twist. At lower loop lengths, both Q and Q in
creased with increase of twist. With higher loop lengths, the angles in
creased with twist to a certain point and then decreased with the increase
of twist, though Q was still higher than Q . The decrease in helix
angles with increasing twist might have a bearing on the experimental
observation of spirality in polyester fabrics (Table 3), which shows that
spirality increased with twist to a certain point beyond which there was
so much of energy in the fabric that the whole fabric was distorted into
a rope like structure and spirality and helix angles had no meaning.
It was interesting to note that P, the angle of inclination was
constant for a particular loop length and was independent of turns per
inch. Further, the helix angles were so small that practically both legs 33
0.0700
0.0600
0.0500
03 cu u 0.0400 tuO 0) Q d •H S 0.0300 i—i bO
0.0200
0.0100 LP =.J.,5^24°- t — - """
P = 0.055 0.055 0.055
0.0000 12 16
Turns per Inch
Figure 10. Angles Q , Q , P and Course Spacing p for Loop Length L = 0.28 Inch 34
0.0700
\ ' 0.0600
0.0500 -
CO 0)
P = 15.23° P = 15.23°
"'^--.^^ P = 15.25° 0.0200; I '''" p = 0.068 p = 0.068 ^^^ Q s p = 0.069 0.0100
1 1 8 12 16
Turns per Inch
Figure 11. Angles Q , Q , P and Course Spacing p for Loop Length L = 0.35 Inch 35
0.0700
0.0600
0.0500 z
0) O) U 0.0400 bO Q) P d •H 0.0300 Ml C <1 P = 15.25° P = 15.25°
4 0.0200 P = 15.25' p = 0.082 p = 0.082
p = 0.082 0.0100
0.0000 > 1 8 12 16
. Turns per Inch
Figure 12. Angles Q , Q , P and Course Spacing p for Loop Length = 0.42 Inch 36
0.0700
0.0600
0.0500
w 0.0400
c •H 0.0300 w (U 1—1 bO C
0.0200 •>« Q L = 0.35 s • Q L = 0.28 s 0.0100 • ^
0.0000 12 16 Turns per Inch
Figure 13. Comparison of Angles Q , Q , P for Loop Lengths 0.23, 0.35 and 0.42 Inch 37
of the loop would be straight without any significant helix formation, which suggested that a square loop would be formed for minimum energy configuration. In an actual fabric, however, the loops are not square.
Formation of square loops (theoretically) can be avoided by taking into consideration the friction at the interlocking points, which was neglec ted in the development of the energy equation. Also in the development of the energy expression, only torque and curvature compatability at the interlocking points was taken into consideration. In addition to torque and curvature, slope compatability should also be taken into account. 38
CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS
Conclusions
1. Spirality increased with twist per inch in the yarn.
2. Spirality increased with loop length in the knitted struc tures. '
3. At low twist levels, stable fabrics were obtained over a wide range of loop lengths.
4. At high twist levels, not many loop lengths yielded a stable fabric. Fabrics would distort into rope like structures with large loop lengths.
5. There was a definite range of combination of loop lengths and turns per inch, above which stable fabrics could not be made. Below that range fabric production was not possible on the F.A.K.
6. In the development of theoretical expressions for predicting spirality, friction at interlocking points should not be neglected.
7. A more elaborate treatment is desired to develop mathematical relations concerning spirality.
Recommendations
It is suggested that a further study of the problem be conducted by taking into consideration the friction at the interlocking points.
Compatability of the slopes at the interlocking points should also be 39
considered along with the curvature and torsion compatability.
It could be interesting to develop a method for following the loop
shape in space between the interlocking points and see if they followed
the assumed, one complete, circular helix path. 40
APPENDICES 41
APPENDIX A
GEOMETRY OF THE HELIX
A circular helix is a curve generated by a point moving in a circle of radius R, such that when it has rotated through an angle 2rr, it rises to a height H, called the pitch of the helix as shown in Figure 14. This curve lies on the surface of a circular cylinder of radius R. The angle of helix Q is the angle which the helix makes with any generator of the cylinder.
If the surface of the cylinder is slit along one complete helix, a right angle triangle will be obtained whose base will be 2TTR and height
H and the angle Q of the helix as shown. The parameters of the helix are
H = pitch of the helix
R = radius of the helix
Q = angle of the helix
L = length of the curve along one complete helix
Radius R and length L are given by
L = H sec Q
L sin Q R = 2TT .
Coordinates of any point on the helix are given by
X = R cosG
y = R sinG
z = R 0 cot Q 42
2TTR
Figure 14. Geometry o£ Helix 43
where 9 is the angle of rotation of the point generating the helix, from the X-axis. The curvature of the helix is given by
2 2 2 2 . . ,2 , , .1 . ,2 d s K = -^:^ '"^^ 3 '^^^' ^ (32) (i) where s is the length along the helix.
2 ,2 , ,1 = -R sine ; ^ := -R cosG ; (^) = R^co! de ^dG
2.2 ^ - R cose ; ^ ^= -R sine ; (^) ^ R^si2.2n , ^^ de^ ^de^'^
2 2 , ,2 dz „ . _ d z ^Q ~ R cotQ ; —2 " 0 dG^ eMe ) =°
Also
dz\2 i = y(i) ^(le) ^dG( /
R cos e + R sin G + R cos Q / = R cosec Q
dG^
Substituting in (32) 44
s/X 2. ^2 . 2 K = V R COS + R'sin'e + 0-0 2 2 R cosec Q or 45
APPENDIX B
CHANGE OF TWIST DUE TO BUCKLING (15)
Consider a loop AB, which can be a circle, as seen in Figure 15(a), of a twistless rubber rod or yarn. In this position there is no twist in
the rod. If we keep the end B fixed and lift the end A, without rotating the ends with respect to each other, the rod will, first take the shape of an S helix. Figure 15(b), before it is straightened out, Figure 15(c).
During this process of straightening, the rod will have acquired one full turn of S twist over the length AB. Now, if the straightened length AB be buckled into an S helix, it will lose some of the acquired twist. If we keep on buckling, it will keep on losing the S twist by gaining Z twist until it loses the one turn of S twist, i.e., gains one turn of Z twist, when end A reaches B, as they were originally.
Similarly, if A is kept stationary, while B is raised without ro
tation relative to A, it will first form a Z helix. Figure 15(d), and then straighten out. Figure 15(e), acquiring one Z twist over the whole length
AB. If this length AB be buckled into a Z helix, it will lose the Z
twist by gaining an S twist in the process of buckling. When the length
AB is buckled to the extent that it forms a circle, it will have lost one complete turn of Z twist or gained one turn of S twist. In other words, if a rod or yarn is buckled into an S helix, it gains Z twist; and if it
is buckled into a Z helix it gains an S twist. 46
X t
B B IA A (c) Cb) (d) (e)
Figure 15. Change in Twist Due to Buckling
2TTR
* <•*
Figure 16. Helix 47
Now consider a helix of pitch H, radius R and angle Q, as shown in Fig- ure 16.
For this helix
tanQ - -jj2TT-R
Now 1/H = turns/unit length = T
= turns/inch if H is in inches
tanQ = 2TTRT
From the geometry of a helix (Appendix A)
L = arc length/pitch = 7^^
1 sinQ L 2TTR
and cosQ = H/L
If a straight yarn of length L is buckled into a helix of pitch H and radius R, then the change in twist over the total length L is
^ . . L-H change m twist = —jr— so that if H = L, there is no change in twist (no buckling)
H = 0, change in twist is one twist
change in twist/unit length = —r- L 4(-!) or / sinQ change in twist/unit length = „ (1- cosQ) ZTTK 48
APPENDIX C .
LAWSON-HEMPHILL F. A. K.
Lawson-Hemphill Fabric Analysis Knitters are designed to control the knitting process variables precisely. The positive feed device is calibrated to feed exactly the same amount of yarn on each revolution of the cylinder. A servo mechanism controls the yarn tension automatically.
There are three gear ratios on this machine to accommodate a wide range of length of yarn per course as follows:
Gear Ratio Inches per Course
2:1 6 to 12
4:1 12 to 24
8:1 24 to 48
Selection of Meterhead Circumference
Select the gear ratio appropriate to the desired inches per course.
Divide the required inches per course by the selected gear ratio. The quotient rounded off to the nearest two decimal places gives the required meterhead circumference in inches. When the meterhead is fully closed, the circumference is three inches and when fully open, the circumference is six inches. Adjust the meterhead to the required circumference.
Example
Required inches per course =40 inches
Gear ratio selected =8:1 49
Dividing 40 by 8 gives -^ = 5.00
Therefore the setting of the meterhead is five inches
•r -^
••'«•* 50
APPENDIX D
QUESTION ANSWERING SYSTEM
The "Question Answering System" developed by Dr. M. V. Konopasek of the School of Textile Engineering is an attempt to make the computer behave more intelligently and take over a share of the information pro cessing chores. With this method, an ordinary computer user does not have to become an expert in bridging or by-passing computer deficiencies.
This system has been applied in developing and using a non-procedural conversational knowledge oriented language (Q.A.S.) on sets of algebraic equations
The basic features of the system are as follows:
1. The basic properties of the mathematical model are defined by assigning names.
2. The equations concerning the mathematical model are expressed as a sequence of Fortran statements using the above variable names.
3. Questions are formulated by setting up input subsets on the set of properties and inserting input values and specifying output sub sets. The computer accepts simple messages headed by an operation code; for instance, AI, RI, AO, RO, for Assigning or Removing the properties to or from the Input or Output subsets; EX for proceeding with the solu tion of the model and printing the results, etc.
The operation codes are followed by an argument consisting of the property code, and value when applicable. The computer responds by 51
printing out the answers (results or messages indicating erroneous user
entries or inconsistencies in the problem solving specifications includ
ing, for instance, a prompt stating which property should be added to or
removed from the input subset if the problem is under or overdetermined).
{ ,..-..
%^ 52
BIBLIOGRAPHY
Literature Cited
1. Davis, W., Edwards, C. H., and Stanbury, G. R. , "Spirality in Knitted Fabrics," Journal of Textile Institute, 2^, T122 (1934),
2. Davis, W., Edwards, C. H., and Stanbury, G. R. , "Spirality in Knitted Fabrics: II. Cotton," Journal of Textile Institute, 26, T103 (1935).
3. Nutting, T. S., "Spirality in Weft Knitted Fabrics," Hosiery and Allied Trade Research Association, Nottingham, England, Harta Note #13 (December 1971).
4. Lord, P. R., Mohamed, M. H., and Ajgaon kar, D. B., The Per rorm- ance of Open End, Twist less, and Ring Yarns in Weft Knitted Fabrics," Textile Research Journal, 44(6), 405 (1974).
5. Doyle, P. J,, "Fundamental Aspects of the Design of Knitted Fab rics," Journal of the Textile Institute, 4^, P561 (1953).
6. Munden, D. L., "The Geometry and Dimensional Properties of Plain Knit Fabrics," Journal of the Textile Institute, 5£, T448 (1959).
7. Chamberlain, W., Hosiery Yarns and Fabrics, ^, 107 (1949). r^ '->, 8. Shinn, W. E., "An Engineering Approach to Jersey Fabric Construc tion," Textile Research Journal, 25^, 270 (1955). 9. Pierce, F. T., "Geometrical Principles Applicable to the Design of Functional Fabrics," Textile Research Journal, 1_7^(3), 123 ^^ (1947).
%0. Leaf, G. A. V. and Glaskin, A., "The Geometry of a Plain Knitted :i* Loop," Journal of Textile Institute, 46^, T587 (1955).
11. Leaf, G. A. V., "Models of Plain Knitted Loop," Journal of the Textile Institute, T49 (1960).
12. Postel, R. and Munden, D. L., "Analysis of Dry Relaxed Knitted Loop Configuration," Journal of the Textile Institute, 5^, 329 (1967).
13. Shanahan, W. J. and Postel, R. , "A Theoretical Study of the Plain- Knitted Structure," Textile Research Journal, 40^, 656 (1970). 53
BIBLIOGRAPHY (Concluded)
14. Axelrad, D. R., Strength of Materials for Engineers; Sir Issac Pitman and Sons Ltd., Melbourne/Great Britain, 1959.
15. Tayebi, A., Unpublished lecture notes, School of Textile Engineer ing, Georgia Institute of Technology (1975).
Other References
Chi, M., "Analysis of Multi-wire Strands in Tension and Combined Tension and Torsion," Developments in Theoretical and Applied Mechanics, 1_, 529 (1974).
Marin, J., Strength of Materials; Macmillan Co., New York, 1948.
Moullin, E. B., Pye, D. R. and Southwell, R. V., Editors, An Introduction to the Theory of Elasticity for Engineers and Physicists; Oxford University Press, London, 1941.
Prescot, J., Applied Elasticity; Longmans, Green & Co., London, 1924.