A Geometrical Analysis of Moire Finising
By Takeshi Takeuchi, Member; TMSJ
Sakai Textile Industry Co., Ltd.
Abstract
A study of the unit moire has shown that the manufacturing methods of the unit moire can be classified into two classes. Various moire pat terns obtainable are: (1) Using no-wave rules mT=nS or m=0 and n=0 Horizontal straight strips S=T Vertical straight strips m=0 or n= O Slanting strips (2) Using wave rulesS =T and m=0 Symmetrical vertical straight strips S=T and m*0 Unsymmetrical vertical straight strips S*T and m=0 Symmetrical curve strips S•‚T and m•‚O Unsymmetrical curve strips where S is the inverse number of counts of weft in fabric A, and T is the inverse number of counts of weft in fabric B, in is the tangent of inclination of weft in fabric A, n is the tangent of the angle of in clination of weft in fabric B.
Introduction Case (b) The floating weft of fabric A, presses The ordinary method of moire finishing on the floating warp of fabric B, becomes used is to put together a piece each of two flattened and shines more brightly than the fabrics of proper densities of warp; and weft, other areas when unit moire of fabric B is pass one of them between two rollers under heavy tension, then put the two pieces together, formed. Unit moire of fabric A is explained one on top of the other and press them at a similarly. high temperature. The moire line thus obtain ed is a series of unit moire. In our study, we have interested ourselves not only in the con struction of unit moire of weft but also in its relation to warp.
1. Construction of Unit Moire
Consider the following two types of fabric: (a) Warp : Filament yarn with no twist Weft : Hard twisted filament yarn (b) Warp: Filament yarn with no twist Weft : Filament yarn with no twist Moire finishing is generally used for fila ment fabrics. Let us study the construction of unit moire. Case (a) This fabric is twisted hard; the weft is, therefore, bent as seen in Figure 1. When weft P changes its shape, warp length between P and R is longer than that between P and Q. The weft fabric B is pressed on the warp of fabric B. Therefore, the portion of warp between P and R gains in luster and constitutes Fig. 1 one part of unit moire. 62
Fig. 4
Equation[s] of wefts of fabric A: y = mx + bk (1) Equation of wefts of fabric B: y = nx + ck (2) where k is positive integers, and rn= tan a, n tanƒÀ. Inverse number S of weft counts of fabric Fig. 2 A, and inverse number T of weft counts of fabric B are shown by the following equations: 2. Shapes of Moire Lines S = bk bk-1, T = Ck C1-1 Equation of moire line is In the case of (a), the moire lines are very thin, since the weft is bent arch-like over two or three warp yarns only. In the case of (b), the moir lines are very wide, because the flat parts (The straight lines b0 and c0, are assumed to of warp are laid out widely. The moire lines pass through origin, and K is positive inte are made by connecting the junctions of weft gers.) lines of both fabrics and, therefore, are The inclination angle r of moire lines is analyzed geometrically by count, shape and given by slope of weft. The moire lines of both cases (a) and (b) are generally classified into the following two kinds: (1) Moire line produced without using wave rules Width B of moire line is the width of direction x: (2) Moire line produced by using wave rules [2] (it being assumed that in both fabrics A and B the warp and weft densities are regular and their tensions uniform). However, if both the warp and weft are of yarns with no twist, the width of moire line is equal to the width of an area occupied, by two or three warp yarns. Width TV of moire line
The wave rule is shown in Figure 3.
2-1. Moire Line of Kind (1) Figure 4 shows moire line of kind (1) produced without using wave rules. a is the angle between the weft of fabric A and the positive direction of axis x. /i is the angle between the weft of fabric B and the posi tive direction of axis x. Warps of both fabrics A and B are parallel with axis y. Fig. 5 63 is the width of direction y (Figure 5) : (iii) m=0 Inclination angle
In what follows several cases are discussed as special cases of the general equations derived Width B of moire line is the width of direction above. x:
(i) n.S= mT
Since tan r =0, inclination angle is zero, Width W of moire line is the width of width B is infinite, that is, full width of cloth. direction y: Width
(See Figure 8.) (See Figure 6.) (iv) n=0 (ii) S=T Inclination angle: When the counts of the weft of fabric A are equal to counts of the weft of fabric B tan r=??. Inclination angle is 90 degrees. Width B of moire line is the width of direc tion x:
Width W of moire line is the width of direc tion y: Width W of moire line is the width of direc W=?? (9) tion p: (See Figure 7.)
(See Figure 8.) (v) m=0 and n=0 It is entirely the same as (i).
Fig. 6
Fig. 8
2-2. Moire Line of Kind (2) Now as to moire line of kind (2)--pro duced by using wave rules. Since fabric B is passed over the wave rules under high ten sion, its weft yarns are deformed into wave lines, presumable sine curves. The moire pat tern given by putting fabrics A and B together, one on top of the other, is analyzed as follows: Fig. 7 64
If x1, and x2 are the roots of the pair of the simultaneous equations, the width of the moire lines is given by (x1-x2) Assuming in the above pair of equations that
Fig. 9
Equation of weft of fabric A : y=mx+bn. Equation of weft of fabric B : Width (direction y) is determined by the following two equations:
Here 7n is the slope of the weft of fabric A bn, cn, is the ordinate of the weft of fabric B, a is the maximum height of wave line. By substituting a certain value of x into the equations, the values of y2 and Y2 will be obtain ed. Thus the width is given by:
Eliminating n from the above equations, equation of moire lines is determined, i.e.: (See Figure 10.)
K is positive integers, including zero. The highest or lowest point of the moire line is determined from the following equation:
Examining d2y/dx2 in a range, 0??(x/a)??n, we have a maximum when S > T, and a minimum when S < T. In another range, ƒÎ??(x/a)??2ƒÎ, we have a minimum when S > T, and a maximum when Fig. 10 S Width B (direction x) : Maximin height of wave: Width w (direction ;y) : W=?? (29) K is positive integers, including zero. Width (direction x) : The stripes pass the point x/a =ƒÎ/2, 3ƒÎ/2, 5ƒÎ/2, etc. (See Figure 13) Fig. 11 The stripes are symmetrical in each repetition and the centers of symmetry line on lines x/a=ƒÎ/2, 3ƒÎ/2, 5ƒÎ/2, etc. (ii) S=T and m•‚0 Equation of moire line : Width (direction x) : Fig. 13 Width (direction y) : W (iv) S•‚T and m•‚0 =??(See Figure 12) This is the most common case and the equation of moire line, width, and maximum height is given by the general equation men tioned above. The moire line, when S > T, has the same shape as when S < T, but is shifted a half-wave length. The inclination of the moire line, when m > 0, has the same magni tude as when rn < 0, but the direction is in versed. Conclusion The foregoing study assumes that counts, density and orientation of warp and weft are in perfect evenness and that the yarns are under uniform tension. This is not the actual case. However, the state of individual weft yarns prior to moire-finishing can be known from Fig. 12 moire-finished state. Conversely, it is also possible to visualize the moire pattern from The stripes are unsymmetrical for the term the state of individual weft yarns of fabrics m(x/a); the patterns of stripes are repeated on to be finished. lines given by x/a=ƒÎ/2, 3ƒÎ/2, 5ƒÎ /2, etc. Literature cited (iii) S•‚T and m=O Equation of moire line: [ 1 ] T. Takeuchi; Physics of Textile, Published by S anseido, Japan, p. 49 [2] B. Mihira; P.ulletinn of the Textile Research Institute, Japan, Vol. 9, p. 110