Geometrical model for a newly designed three-dimensional functional rib-knitted cord

Abstract

A functional rib-knitted cord and the knitting machine for its production were designed having Turkish Patent Institute Applications No: 2011/00438 and No: 2011/00483 respectively. Functions such as mechanical, thermal etc. can be cultivated using the designed cord. Our interest here is to introduce this newly designed cord and to give a geometrical model for this cord. The model is created and drawn to scale by using the 3DS-Max computer graphical program. The model is the general one and can be applied to various applications of the structure of various tightness values; therefore, the direct comparison of the model with any set of experimental findings, depending on the application area of the cord, is left for the future. However, the assumptions about obtaining spirally connected loops of the model are observed to be correct, as the loop shapes are similar when we compare them with the real cord knitted as samples.

Keywords

knitted cord tubular rib fabric knitted loop geometrical model functional cord

Traditional fancy yarn technology uses various techniques, including knitted cords. Some forms of ply yarns and cords, are emerging as preferred yarns in the technical textile market to cultivate some technical functions (i.e. conductivity, thermal comfort, mechanical performance, etc.). Some machine companies have designed machines to produce these types of ply and cord yarns. An alternative new knitted cord was aimed to be designed by the authors to be used for cultivating various technical functions in the technical textile industry and a patent application for this functional rib-knitted cord was made with the TPE (Turkish Patent Institute; Application No: 2011/00438).¹

Knitting of the designed cord

Cord will be knitted as on a very small diameter circular rib knitting machine. The machine will have eight cylinder needles and four dial needles; the needle arrangement is given in Figure 1.

Figure 1.

Knitting needle arrangement of the designed cord and its knit notations: (a) needle arrangement; (b) knit notation of the rib course; (c) knit notation of the rib course and plain course.

Needle selection can be done on the cylinder by, for example, using long and short butt needles, thus odd numbered or even numbered needles on the cylinder can be selected to knit one row (the odd and even numbered needles are seen in Figure 1(a)). Dial needles will be all long butt needles, as shown in Figure 1(a). The basic design of the cord will be composed of knitting one rib course using the long butt needles on the cylinder and on the dial by having a knitting system and then knitting a plain course by using the short butt needles on the cylinder by having a second knitting system.

Knit notations of these courses are given in Figures 1(b) and (c).

For cultivating more functions, some additional inlay yarns can be added to the basic design given in Figure 1. These inlay yarns are as follows (examples of inlay placements can be seen in Figures 2–5).

The helical weft inlay yarn, which is placed between the cylinder and the dial needles and lies in the rib course as seen in Figures 2(a) and (b). This helical weft inlay yarn may be placed without including plain course (Figure 2(a)) or with including plain course (Figure 2(b)).

The central core yarn (thread, wire, tube, mandrel, etc.) can be placed at the center of the structure to be laid along the cord, as shown in Figure 3.

Warp inlay yarns can be placed i) between the dial loops of the rib course and the cylinder plain row loops (as in Figure 4(a)) or ii) with including helical weft inlay yarn and without plain row (as in Figure 4(b)).

Figure 2.

Knit notation of cord structure and helical weft inlay yarn: (a) without plain course; (b) with plain course.

Figure 3.

The basic structure of cord and central core yarn.

Figure 4.

Designed cord with warp inlay yarns: (a) with plain course; (b) without plain course instead with helical weft inlay yarns.

Figure 5.

Designed cord structure with all the inlay yarns.

The cord can also be constructed including all the inlay yarns (warp, central core and helical weft yarns), as shown in Figure 5.

In order to knit such cords and also small diameter tubular knitted fabrics, a very small diameter circular rib knitting machine was also designed with TPE Application No: 2011/00483.²

Some samples of the present cord were knitted on this patented four-gauge machine; photographs of the samples are given in Figure 6. The samples were knitted by using cotton yarns of 4 × 60 tex for the rib course. For the plain course samples two different types of yarns were used: cotton yarns of 3 × 75 tex (Figure 6(a) – green-colored yarns) and acrylic yarns of 80 tex (Figure 6(b) – gray-colored yarns). The loop lengths for the rib course and the plain course of the two samples are 15.4 and 19.5 mm, respectively. The course-spacing of both of the samples is 11 courses/cm, while the wales per cm of the samples are 2.8 and 3.9 for the cord (Ø 8.5 mm) with green-colored cotton yarns (Figure 6(a)) and for the cord (Ø 10 mm) with gray-colored acrylic yarns (Figure 6(b)) respectively. These samples were manufactured only for demonstration purposes.

Figure 6.

Photographs of the designed cord from the samples knitted on the designed machine.³ (It is to be noted that during knitting of these samples, only two basic rows, namely the rib and plain knit rows as in Figure 1(c), are used.).

The presented cord can be designed so that it can be a basic structural element of the technical textiles for obtaining various functions. Some of the possible technical functions of the designed cord are summarized and given in Appendix A.

Since the present cord structure is a new one, the physical or geometrical properties of it have not been fully explored yet. To start with, a geometrical model of the structure is aimed to be given here for the following reasons.

To introduce such a new structure.

In accordance with increasing demands in parallel with increasing technical textile products, some engineering software programs, such as WISETEX,⁴ are also developed to predict the designed technical function properties (i.e. thermal, mechanical, etc.) of the technical textile products. These software programs, in turn, use the geometrical model of the structure as input data for building the assumptions for predicting the required technical function of the product. This was the second reason to create a geometrical model of the structure in the present work. As the presented cord is proposed to be used as a technical textile itself or as a yarn used in the manufacture of the technical textile, there will be the need to predict its properties using such software programs before the actual manufacturing. Thus, there is the need for the geometrical model of the presented cord.

As far as we know, there is no geometrical model created in the literature for spirally connected plain and/or rib loops situated in tubular weft-knitted structures. Therefore, the third aim is to give models for spirally connected plain and rib weft-knitted loops in tubular fabrics by modeling this newly designed cord structure.

There are geometrical models created for knitted fabrics by previous researchers. In this context, Chamberlain,⁵ Pierce,⁶ Leaf and Glaskin,⁷ Leaf,⁸ Munden,⁹ Postle,¹⁰ Smirfitt,¹¹ Natkanski,¹² Kurbak and Ekmen,¹³ Kurbak and Soydan,¹⁴ Kurbak,¹⁵ Kurbak and Alpyıldız,¹⁶ Kurbak and Soydan,¹⁷ Kurbak and Amreeva,¹⁸ Alpyıldız and Kurbak¹⁹ and Kurbak and Alpyıldız^20–22 were given geometrical models for knitted fabrics.

Among them, the models of Kurbak and his colleagues were seen to be most suitable the basis of the present modeling of the designed cord. therefore, the present model will be created based on these.^13–22

As was mentioned above (aim (c)), Kurbak and Ekmen¹³ gave a model for widthwise curlings of a plain knit structure to be used for obtaining models for tubular fabrics and also for obtaining models of rib fabrics. Kurbak and Soydan¹⁷ obtained the models of 2 × 2, 3 × 3, 4 × 4 and 5 × 5 balanced rib fabrics using Kurbak and Ekmen’s work, for the widthwise curling parts of the structures. In those studies, modeling of spirally connected loops in a tubular fabric was left for future works. So, in addition to the above-mentioned aims, the present work is conducted to compensate for this in Kurbak and colleagues’ works, in addition to the other two main aims mentioned above (aims (a) and (b))

Creation of a geometrical model for the cord

A general model of the cord is created here including all the inlay yarns (weft or warp); therefore, the geometry of the basic structure is adjusted so that it can accommodate the whole of any inlay yarns. All of the yarns are assumed to have a circular cross-section and no yarn swellings occurred.

Geometrical model for the rib course

Assumptions

The cord can be manufactured from only rib courses because each course is spirally connected, as can be seen in Figure 7(a), where ‘c’ is the course-spacing.

It is assumed that the loop arms have equal length with each other and are at the same vertical level, thus spirality can be obtained by only the inclining of the connecting parts of the rib structure for simplicity reasons, as can be seen in Figure 7(b).

According to this assumption, one turn will be as the first loop is at the zero level, the second loop is at the c/8 level and so on; these levels are given in Figure 7(a).

It is seen that the model of a quarter part of a rib row is enough to draw the whole row. A quarter part of the rib row is given in Figure 8. In order to geometrically model the rib row, only a quarter of the rib row will be needed to start the modeling.

From Figure 8, the below equation is obtained by using this assumption:

\tan ξ = \frac{c / 8}{h_{xy}}

(1)

where

h_{xy}

is the component of the length of the rib connection part in the xy plane. This will be explained further in the next section.

The loop heads are assumed to be circular in shape. It is reasonable to have this assumption because the rib row moves around a center, thus making a slacker structure. However, the model is planned to be versatile enough to consider elliptical loop heads as well, as in Kurbak’s models.^13–22

The rest of the structure in Figure 8 is modeled by assuming the models of Kurbak and his colleagues^13–22 as the basic works. In particular, loop arms are modeled that are similarly to those in the 1 × 1 rib model given by Kurbak.¹⁵

It is assumed that the widths and the heights of all loops (cylinder, dial, plain row, rib row loops) are equal to each other.

Wale jammings are assumed to occur for the dial loops. Since the loops are not touching each other at their maximum width points, calculations would be lengthy; therefore, an empirical method is considered here to satisfy this assumption, as given in the following.

Figure 7.

Schematic drawings of the rib course with vertical loop levels: (a) upper view; (b) an inner rib loop with rib connection parts ( zero level, second level (after spirality)).

Figure 8.

Schematic drawings of the unit rib row (a quarter part of the rib course).

The upper view of the 3DS-Max drawing of the 1 × 1 rib loop model with minimum wale-spacing (i.e. if we take w = 4d, a model is obtained as the two arms of a loop that is also touching) is obtained using Kurbak.¹⁵ Four upper loop shapes are obtained by extracting them from the 1 × 1 rib model. Then they are placed on an arranged drawing, given in Figure 7(a).

During uniting the model shape geometries, they were so adjusted that the dial loops were touching each other. With this placement, for a cord with normal tightness, the q value is obtained in terms of yarn diameter as q = 2.5d.

Some abbreviations used in the equations are as follows: W = wale-spacing; d = yarn diameter; q = radius of dial; p = radius of cylinder.

The parameter p in Figure 9 is taken as p = 5.75d.

Figure 9.

Schematic drawings of the upper views of the unit rib and plain courses.

A distance p – q = 5.75d − 2.5d = 3.25d is obtained between the large and the small circles. The openings between the cylinders and dial loops are adjusted so that we can place helical weft inlay yarn and warp inlay yarns there. Normally without inlay yarns, eight cylinder loops (rib and plain loops) can be assumed to be touching to each other (as seen in Figure 6).

Creation of the model

The model is created by having circular loop heads according to assumption (ii) and by wrapping the yarn arms and rib connecting parts on circular cylinders as in Kurbak’s 1 × 1 rib model.¹⁵ The model of a quarter part of the 1 × 1 rib row is given in Figure 10.

Figure 10.

The present model application of the unit rib course.

From the geometry in Figure 10(b), the following equations are obtained:

h_{xy} = \sqrt{[(p - w / 4) \frac{\sqrt{2}}{2} - \frac{w}{4}]^{2} + [(p + w / 4) \frac{\sqrt{2}}{2} - q]^{2}}

(2)

χ = \tan^{- 1} [\frac{(p - \frac{w}{4}) \frac{\sqrt{2}}{2} - \frac{w}{4}}{(p + \frac{w}{4}) \frac{\sqrt{2}}{2} - q}]

(3)

β = χ - \frac{π}{4}

(4)

The length of cylinder 2 in figure 10(a) is obtained as

h_{2} = \sqrt{h_{xy}^{2} + (\frac{c}{8})^{2}}

(5)

where h_xy is the projection of the imaginary cylinder length.

In Kurbak’s models,^13–22 yarn arms and also rib connecting parts are modeled by having imaginary elliptical cylinders (see Figure 10(a)) and wrapping the yarn axis around these cylinders using the equation

z = a' S_{1}^{2} + b' S_{1} + c'

(6)

where z is the wrapping length of the cylinder, S₁ is the elliptical arc length on the cross-section of the imaginary elliptical cylinders and

a'

b'

and

c'

are constants.

For Kurbak’s 1 × 1 rib model¹⁵ and also for the present model, the imaginary cylinders were taken with circular cross-section, therefore Equation (6) becomes

z = a' (r θ)^{2} + b' (r θ) + c'

(7)

where

S_{1} = r θ

(θ = wrapping angle, r = radius of the imaginary cylinder).

The application of Equation (7) on an imaginary circular cylinder is given elsewhere;^15,20 therefore, they are not given again. The only results of them are given here as follows:

A_{1 i} = \frac{4 h_{i}}{δ_{max i} (u_{1 i} - u'_{1 i})} - \frac{(u_{1 i} + u'_{1 i})}{(u_{1 i} - u'_{1 i})}

(8)

ϕ_{1 i} = δ_{max i} \frac{- A_{1 i} \mp \sqrt{A_{1 i}^{2} + 1}}{2}

(9)

u_{2 i} = \frac{(u_{1 i} - u'_{1 i}) δ_{max i} / 2 + (u_{1 i} + u'_{1 i}) ϕ_{1 i}}{2 ϕ_{1 i}}

(10)

a'_{i} = \frac{u_{2 i} - u_{1 i}}{2 r (δ_{max i} / 2 - ϕ_{1 i})}

(11)

b'_{i} = u_{1 i}

(12)

b ″_{i} = u'_{1 i}

(13)

c'_{i} = 0

(14)

where h_i is the length of the wrapping cylinder (see Figure 11),

u_{1 i} = \tan α_{1 i}

, where

α_{1 i}

is the helix angle at point A in Figure 11. This angle is defined between the yarn axis and the circumference of the cross-section of the cylinder.

u'_{1 i} = \tan α'_{1 i}

, where

α'_{1 i}

is the helix angle at point B in Figure 11. This angle is also defined similarly to angle

α_{1 i}

δ_{max i}

is the maximum wrapping angle (see Figure 11).

u_{2 i} = \tan α_{2 i}

, where

α_{2 i}

is the maximum helix angle at point C in Figure 11.

ϕ_{1 i}

is the distance between

δ_{max i} / 2

and the place of the maximum helix angle

α_{2 i}

(point C in Figure 11).

Figure 11.

Application of the parabolic curve (Equation (7)) on a circular cylinder.

We have seen that if the parameters h_i, $u_{1 i}$ , $u'_{1 i}$ and $δ_{max i}$ are known, the other parameters $u_{2 i}$ , $ϕ_{1 i}$ and $a'_{i}$ can be found from Equations (8)–(14).

Parameters h_i and $δ_{max i}$ can be found from the geometry in Figure 10. Parameters $u_{1 i}$ and $u'_{1 i}$ can be calculated from the curvature equalities as given in the following.

In order to calculate $u_{1 i}$ and $u'_{1 i}$ , $α_{1 i}$ at A and $α'_{1 i}$ at B (Figure 12) shall be found. For a circular loop head, the radius, a, is given by

a = \frac{w}{4} + \frac{d}{2}

(15)

Figure 12.

Continuously connecting the rib loop curve parts.

The radius of curvature of a circle is its radius, therefore

ρ_{A} = a

(16)

The radius of curvature $ρ'_{A}$ for the helix at the same point A can be written as

ρ'_{A} = \frac{r}{\cos^{2} α_{1 i}}

(17)

where r is the radius of the cylinder. When we equalize these two radii of curvatures, the equation below is obtained:

a = \frac{r}{\cos^{2} α_{1 i}}

(18)

From Equation (18), $α_{1 i}$ is written as

α_{1 i} = \tan^{- 1} \sqrt{\frac{a}{r} - 1}

(19)

It can be assumed that the radius of the cylinder is equal to half of the yarn diameter:

r = d / 2

(20)

When we put Equation (15) and Equation (20) into Equation (19), we obtain $α_{1 i}$ as

α_{1 i} = \tan^{- 1} \sqrt{\frac{w}{2 d}}

(21)

On the other hand, at point B in Figure 12, the axes of two cylinders with the same radius r cross each other. When we equalize the radius of curvatures at B we obtain an equation as

\frac{r}{\cos^{2} α'_{1 i}} = \frac{r}{\cos^{2} α_{1, i + 1}}

(22)

From Equation (22), $α_{1, i}$ must be equal to $α_{1, i + 1}$ :

α'_{1, i} = α_{1, i + 1}

(23)

From the geometry given in Figure 12(b)

α'_{1, i} = α_{1, i + 1} = φ / 2

(24)

where φ is the angle between the cylinder axes. If the inclination angle of the axis of cylinder with the subscript ‘i + 1’ is ξ, then

α'_{1 i} = α_{1, i + 1} = (π / 2 - ξ) / 2

(25)

is obtained.

The parameters

h_{i}, δ_{max i}, α_{1 i}

and

α'_{1 i}

for the quarter part of rib course, which can be seen in Figure 9, are obtained in a similar way as above and are given in Table 1.

Table 1.

Calculation formulae of the imaginary cylinders used in the quarter part of the rib course

Cylinder No. (i)	h_i	$δ_{max i}$	$α_{1 i}$	$α'_{1 i}$
Cylinder 1	c	$π + β$	$\tan^{- 1} (\sqrt{w / (2 d)})$	$(\frac{π}{2} - ξ) / 2$
Cylinder 2	$h_{2} = \sqrt{h_{xy}^{2} + (\frac{c}{8})^{2}}$	π	$(\frac{π}{2} - ξ) / 2$	$(\frac{π}{2} + ξ) / 2$
Cylinder 3	c	$\frac{5 π}{4} + β$	$(\frac{π}{2} + ξ) / 2$	$\tan^{- 1} (\sqrt{w / (2 d)})$
Cylinder 4	c	$\frac{5 π}{4} + β$	$\tan^{- 1} (\sqrt{w / (2 d)})$	$(\frac{π}{2} - ξ) / 2$
Cylinder 5	$h_{5} = \sqrt{h_{xy}^{2} + (\frac{c}{8})^{2}}$	π	$(\frac{π}{2} - ξ) / 2$	$(\frac{π}{2} + ξ) / 2$
Cylinder 6	c	$π + β$	$(\frac{π}{2} + ξ) / 2$	$\tan^{- 1} (\sqrt{w / (2 d)})$

By taking

w = 4 d

and

c = 4 d

, all the parameters are calculated and are given in Table 2.

Table 2.

Evaluation of the parameters of the imaginary cylinders given in Table 1

Cylinder No.	h_i	$δ_{_{max i}}^{(\circ)}$	$α_{1 i}^{(0)}$	$α_{2 i}^{(0)}$	$α {' (0)}_{1 i}$	$ϕ_{1 i}^{(0)}$
Cylinder 1	4d^a	181.061	54.735	76.063	37.982	9.721
Cylinder 2	3.314 d	180.000	37.982	67.307	52.018	17.558
Cylinder 3	4 d	226.061	52.018	74.241	54.735	−28.045
Cylinder 4	4 d	226.061	54.735	71.728	37.982	18.449
Cylinder 5	3.314 d	180.000	37.982	60.403	52.018	−30.833
Cylinder 6	4 d	181.061	52.018	75.058	54.735	−2.502

d is the effective yarn diameter.

Using the evaluated parameters in Table 2 and transforming all local coordinate systems to the XYZ main coordinate system, the model of cord composed of only the rib course is drawn by using the 3DS-Max computer graphical program; this drawing is given in Figure 13.

Figure 13.

The present modeling of the rib course of the designed cord drawn to scale by using the 3DS-Max computer graphical program.

Geometrical model for the plain knit course

It is seen that a quarter part of the plain knit row is also enough to model the whole plain knit course.

The head and the arms of the plain loop are modeled as in the rib course. A circular loop head is taken with the radius, a, and loop arms are wrapped around imaginary circular cylinders with radii r and have lengths equal to course-spacing (c). The maximum wrapping angle ( $δ_{max}$ ) is taken that is equal to $π - β_{1}$ (Figure 14).

Figure 14.

Schematic drawing of the plain course (upper view).

The inclination of the plain knit course within the structure is considered only during modeling of the loop feet for simplicity reasons. The two arms of the plain loop are assumed to be same as each other and they are in the same vertical level, which was the case for the rib course. Since the plain row will be placed between the outside loops of the rib row given in Figure 7(a), the vertical level of the plain loop in Figure 14 will be at the same vertical level with the inner loops of the rib row as in the $c / 8$ level in Figure 7(a). The next plain loop in Figure 14 will be in the $\frac{c}{8} + \frac{2 c}{8} = \frac{3 c}{8}$ level. Therefore, there will be a distance of $\frac{2 c}{8}$ between two plain loops throughout the structure length.

Modeling of plain loop feet

Plain loop foot together with adjacent loop arms are given in Figure 15.

Figure 15.

Crossing of a three-dimensional object and a plane surface to obtain the loop leg curve of the plain course: (a) definition of the uv plain surface; b) the cross-section of the defined three-dimensional object, which is laying in z direction.

For modeling of the loop foot the following steps are taken.

A surface plane is defined as uv in Figure 15 that makes an angle $(\frac{π}{2} - α'_{1})$ with the $(- z)$ direction and also makes an angle γ with the x direction. The straight line KL lies in the defined uv surface.

A cylindrical object with a cross-section like a race track is defined that is laying in the z direction, as seen in Figure 15(b). The cross-section of this object is composed of two halves of the ellipse at two ends, which are just touching the imaginary cylinders from the outside, and there is a rectangular part in the middle of the two halves of the ellipses.

Curvatures of the curves wrapped on the cylindrical object and on the imaginary cylinders are equalized at their touching points (at K and at L₁ in Figures 15(a) and (b))

The curve representing the loop foot is obtained by intersecting the defined uv plane and the defined KMLN cross-sectioned cylindrical object given in Figure 15(b).

In general, the touching points of the defined cylindrical object and the imaginary cylinders change, as seen in Figures 16 and 17, according to the tightness of the structure, the radius of the tubular structure itself and if the plain course is knitted directly on top of a cylindrical mandrel or not.

Figure 16.

Cross-section of the defined three-dimensional object, which is tangent to the imaginary yarn arm cylinders at $K'$ and $L'$ .

Figure 17.

Loop foot curve of the plain row when it is knitted on a mandrel with radius R.

In Figure 16, the touching points of the imaginary cylinders and the defined cylindrical object are $K'$ and $L'$ instead of K and L.

The cross-section of the defined cylindrical object can also be changed to obtain the required curve for the plain loop feet. For example, when we knit such a plain knit on a cylindrical mandrel, the cross-section of the cylindrical object to obtain the loop foot can be defined as given in Figure 17.

However, these works should be searched further during the real applications. We assumed here that the touching points of the imaginary cylinders and the defined cylindrical object are the points K and L in Figure 15(b). The shape of the object is also taken as in Figure 15(b), for simplicity reasons.

Calculations of the above arrangements are too lengthy to give here, therefore they are given in Appendix B and only the results are given in the following.

The loop foot curve for the left-hand side of the plain loop in Figure 15 is given by the following equations.

For the straight line part:

\begin{matrix} x = x \\ y = - b_{1} \\ z = - b_{1} \tan α'_{1} + x \tan γ \end{matrix}

(26)

where x changes between zero and

(A - a_{1})

For the elliptical part:

\begin{matrix} x = a_{1} \sin ϕ + (A - a_{1}) \\ y = b_{1} \cos ϕ \\ z = (A - a_{1}) \tan γ + a_{1} \tan γ \sin ϕ + b_{1} \tan α'_{1} \cos ϕ \end{matrix}

(27)

where ϕ is the variable to draw the curve, which changes from

π / 2

to π.

The loop foot curve for the right-hand side of the plain loop in Figure 15 is given by the following equations.

For the elliptical part:

\begin{matrix} x' = - (A - a_{1}) + a_{1} \sin ϕ \\ y' = b_{1} \cos ϕ \\ z' = - (A - a_{1}) \tan γ + a_{1} \tan γ \sin ϕ + b_{1} \tan α'_{1} \cos ϕ + 2 \frac{c}{8} \end{matrix}

(28)

where the variable ϕ changes from π to

3 π / 2

For the straight line part:

\begin{matrix} x' = - x' \\ y' = - b_{1} \\ z' = x' \tan γ - b_{1} \tan α'_{1} + 2 \frac{c}{8} \end{matrix}

(29)

where

x'

changes between

(A - a_{1})

and zero.

The parameters a₁ and b₁ are the major and minor radii of the elliptical cross-section of the defined cylindrical object (Figure 15(b)).

The parameter a₁ is taken here as

a_{1} = 1.5 d

(30)

The parameter b₁ is obtained from the curvature equalities of the curves at points K and L in Figures 15(a) and (b) as

b_{1} = \sqrt{a_{1} r .} \cdot \sqrt[4]{1 + \cos^{2} α'_{1} \tan^{2} γ}

(31)

Calculations to obtain Equation (31) are also given in Appendix B.

The parameter r is given by

r = d / 2

(32)

where d is the yarn diameter.

The parameter γ is the inclination angle of the line KL in Figure 15(a), which is measured from the x direction. This parameter can be given by

\tan γ = \frac{2 c / 8}{2 A}

(33)

The angle $α'_{1}$ is the helix angle at point K in Figure 15. This angle is also used to define the uv plane. The uv plane, in turn, is used to obtain the loop foot equations. The angle $α'_{1}$ is assumed to be equal to 30° in this work for the reason that the plain row stays within the rib course and it should be above the rib connection parts. The helix angle $α'_{1}$ , which is more than 30°, would make a loop foot curve that could not be accommodated in the rib loops and the crossing of it with the rib connection parts would be inevitable.

The following equations are also written from the given geometry:

β_{1} = 45^{\circ}

(34)

S = p = 5.75 d

(35)

A = \frac{\sqrt{2}}{2} (S - w / 4) + \frac{d}{2}

(36)

T = \frac{\sqrt{2}}{2} (S + w / 4)

(37)

The parameters to draw the plain knit course are given in Table 3.

Table 3.

Evaluation of the parameters to draw the unit plain course of the designed cord

Parameter	A	a ₁	b ₁	$β_{1}^{\circ}$	$γ^{\circ}$	$α'_{1}$
3.859 d	1.5 d	0.869 d	45	7.383	30
Parameter	h_i	$δ_{max i}^{(0)}$	$α_{1 i}^{(0)}$	$α_{2 i}^{(0)}$	$α'_{1 i}^{(0)}$	$ϕ_{1 i}^{(0)}$
Cylinder 1	4 d	135	54.735	80.269	30	5.841
Cylinder 2	4 d	135	54.375	80.269	30	5.841

Since all the parameters of the plain row are known, it can be drawn by transforming the local coordinate systems to the main XYZ coordinate system. For the drawing, c = 4d and w = 4d are taken as these values that were also used for the rib course.

The drawing of the plain row of the structure using the 3DS-Max computer graphical program is given in Figure 18(a). A plain knitted cord with vertical stripes can also be obtained by using the same model, as can be seen in Figure 18(b). It is to be noted here that the designed knitting machine can knit the structure given in Figure 18(b) as well.

Figure 18.

The present modeling of the plain row of the designed cord, drawn to scale by using the 3DS-Max computer graphical program: (a) plain course of the designed cord; (b) a color-striped tubular structure only using two plain courses, one of which is modeled as in Figure 18(a).

Construction of the present cord structure

The model of the present patented cord is given by the model of the rib row (in Figure 13), together with the model of the plain knit row (in Figure 18(a)). The 3DS-Max drawing of the present cord model is given in Figure 19(a). Figure 19(b) shows the basic structure, central core yarn and helical weft inlay yarn. Figures 20(a) and (b) show the present cord with warp inlay yarns:

including the plain row; and

not including plain row, and instead including the helical weft inlay yarn.

Figure 19.

The present modeling of the designed cord using the 3DS-Max program: (a) the basic structure, which is composed of the rib and the plain courses; (b) the basic structure plus helical weft inlay yarn and central core yarn.

Figure 20.

The present modeling of the designed cord with warp inlay yarns, drawn to scale by using 3DS-Max: (a) including the plain course; (b) not including the plain course and instead including helical weft inlay yarn.

The present cord, including all the inlaid yarns, is given in Figure 21.

Figure 21.

The present modeling of the designed cord including all the inlay yarns, drawn to scale by using the 3DS-Max computer graphical program.

Conclusion

A functional three-dimensional (3D) rib-knitted cord was designed with TPE Application No: 2011/00438¹ and the knitting machine for its production was also designed with TPE Application No: 2011/00483.²

In this paper, firstly the knitting instructions of the designed cord and its properties are explained and secondly a general geometrical model including all the inlay yarns for this cord is given. The model is drawn to scale by using the 3DS-Max computer graphical program. There were three reasons for the generation of the present geometrical models:

to introduce such a new structure;

if one wants to use the present structure for cultivating a technical function and one wants to predict the cultivated technical function beforehand in an engineering software, a geometrical model of the structure is needed as input;

to realize the modeling of spirally connected loops in weft-knitted plain and rib tubular fabrics, which is lacking in the previous studies.

Since the created model is a general one, the direct comparison of it with a set of experimental results is left for the future works on any special application of the structure. The structure changes for cultivating different functions, as explained in Appendix A. Tightnesses, course-spacings and wale-spacings can be adjusted directly (as they are the input data) by the present model, but some modifications should be applied if some inlay yarns are missing and also if any swellings occur on the yarn used.

At first glance of the model and the real samples in Figure 6, it is seen that the assumptions of the model on the spirally connected loops seem to be correct. Because the loop shapes are similar in both the real fabrics and the model, and because no asymmetries or tilts on the loops are observed on the real samples in Figure 6, this was also assumed on the model created.

Footnotes

Acknowledgment

The authors would like to thank to the BSc students Hasan Durna and Hakan Arı for their contributions in the initial drawings.

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

References

Kurbak A and Soydan AS. Üç Boyutlu Fonksiyonel Bir Örme Kordon. Patent application 2011/00438, Turkey, 2011.

Soydan AS and Kurbak A. Çok Küçük Çaplı Yuvarlak Rib Örme Makinesi. Patent application 2011/00483, Turkey 2011.

Soydan AS. Küçük Çaplı Yuvarlak Örme Kumaşlar ve Üretim Makinaları Hakkında Bazı Çalışmalar. PhD Thesis, Dokuz Eylül University, İzmir, Turkey, 2011.

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