Crimp - spacing Relationship
By neglecting bending resistance of yarn ,and assuming yarn cross section in fabric to be cicular Pierce(JTI 1937, T45) developed a geometrical model to determine crimp, thread spacing relationships. The lie of threads in a plain fabric under such conditions is shown in Fig 2.
Fig 2
Let d1 in mils..... denote diameter of warp
p1in mils.....denote spacing between warp threads
Θ1.....denote maximum angle of warp to plane of cloth
l1.....denote length of warp thread axis between axis of consecutive weft threads
h1.......denote maximum displacement of warp thread axis, normal to plane of cloth
c1........denote fractional crimp of warp
The above terms with subscript 2 denote the corresponding values for weft. Then
D = d1 + d2
c1 = (l1 -p2)/p2
p2 = (l1 - DΘ1)cos Θ1 +
sin Θ1
h1 = (l1 - DΘ1)sin Θ1 +
D(1 - cos Θ1)
h1 + h2 = D
Upon expanding sinΘ and cosΘ in ascending powers of Θ
c1 = (l1Θ2 - DΘ3 ...)/(l1 -l1Θ2 +DΘ3..)
This approximates to
h1 ≈ p2√2c1
In practise however, modification of the factor √2 by 4/3 gives a more accurate estimate
h1 = (4/3)p2√c1
As shown earlier, d1 = (1000/29.3)√(v/C) = 34.14√(v/C)
where d1 = diameter of yarn in mils, v = specific volume and C = Count of yarn.
Since D = h1 + h2,
D = 4/3(p2√c1 + p1√c2)
= 34.14(√(v1/C1) +√(v2/C2)
Jammed Structures
When warp is jammed, the weft starts touching the adjoining warp the moment it leaves previous warp. Length of weft is therefore made of curved wrappings made around warp
with no straight portion.In this case
l1/D = Θ1
p2/D = sin(l1/D) = (sin(1+ c1)p2/D)
h2 = 1 - cos(l1/D)
Square cloth - jammed structure
In the case of square cloth, warp and weft have the same diameter, spacing and crimp.
p1 = p2, c1 = c2,
l1 = l2 = l, h1 = h2 = 0.5
cos(l/D) = 1- (h/D) = 0.5
Θ = 600 and l/D = 1.0472 radians
p/D = sin 600 = 0.866
c = (l/p) - 1
l/p = (l/D)× (D/p) = 1.0472/.866 = 1.2092
c = 0.2092
For square cloth, warp threads leave an uncovered portion
(p - d)/p = 1 - D/2p = 0.4227.
Proportion of space not covered by warp and weft is
projection of both sets of threads and is (o.4227)2 = 0.1787
Jammed structure with race course cross sectionWhen warp or weft is jammed the threads get compressed and assume a cross section similar to that of an ellipse or race track. Race track cross section is more easily amenable to mathematical analysis.The lie of threads in jammed structure with race track cross section is shown in Fig 3.
Fig3
From Fig it is seen that E =√(F2 -h2)where F = D1 + D2, the sum of warp and weft race track radii and A is width of race track and
p = E + (A - D)
√(F2 - h2) = p - (A - D) = q
h = √(F2 - q2)
√(1 - (q1/F)2) + √(1 - (q2/F)2) = 1. From this the maximum number of picks that can be inserted into a cloth for a given ends per inch and diameter of yarn and likewise for ends per inch.
Cover Factor for close constructions
If weave is close in both directions
p2/D = sinΘ1 = sin l1/D
p1/D = sinΘ2
cosΘ1 = 1 - (h1/D)
cosΘ2 = 1 - (h2/D)
cosΘ2 = √(1 - (p1/D)2)
√(1 - (p1/D)2) + √(1 - (p2/D)2) = 1
This leads to
K1 = 28D/(p1(1 + ß))
K2 = 28D/(p2(1 + ß))
where ß = C1/C2
For a square cloth
p1 = p2 and ß = 1 and the closest construction is given by
√(1 - (p1/D)2) + √(1 - (p2/D)2) = 1
√(1 - (14/K1)2) + √(1 - (14/K2)2) = 1
Table below shows the cover factor for weft for various values of warp cover factor when both are close.| Warp Cover Factor | Weft Cover factor |
|---|
| 14.5 | 20.8 |
15 | 18.23 | 16 | 16.34 | 17 | 15.52 | 18 | 15.08 | 19 | 14.79 | 20 | 14.6 | 21 | 14.47 | 22 | 14.38 | 23 | 14.3 | 24 | 14.25 | 25 | 14.21 | 26 | 14.17 |
Normal Square Cloth
For square cloth of more open structure
h = (4/3)p√C and D = 2h
8/3(p√c) = 68.28(√(v/C)
√c = 25.59/p(√v/C) and K = 1000/p√C So
√c = 25.59K√v/1000 = .02559K√v. If v is assumed as 1.1 then
c ≈ (K(0.1)/4)2 ≈(K/4)2%
(l - x) f(l)dl)
(x-u)r(u)du 
Fig 1.
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