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5TECHNICAL SCIENCES
3D MODELLING OF UNTWISTED MULTIFILAMENT THREADS, CURVED IN A KNITTED LOOP
Ielina T.,
Candidate of technical sciences Kyiv National University of Technologies and Design
Galavska L., Doctor sciense
Kyiv National University of Technologies and Design Nemirovich-Danchenko Str., 2, Kyiv, Ukraine Ausheva N., Doctor sciense National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute
Dzikovich T. Candidate of technical sciences Kyiv National University of Technologies and Design
Abstract
To improve the mathematical software for three-dimensional knitwear modelling systems at the micro-level, an algorithm for shifting filaments within reserved volumes has been developed. The algorithm increases the accuracy of the model of knitwear structures, made of untwisted multifilament threads, considering arbitrary lateral shifts of the filaments within their reserved boundaries.
Keywords: Knit, 3D modelling, untwisted multifilament thread, algorithm
Introduction. The development of three-dimensional modelling of textile materials is based on algorithmic and mathematical means of describing their form at one of the next levels: fibre - elementary thread - complex thread - fabric - textile shell. The choice of an optimal level of three-dimensional models of textile materials is being discussed both in papers devoted to modelling of textile reinforcement of polymer composites [1-2], and in ones that address the issue of modelling of weaved and knitted fabric structures [35]. Improving the accuracy of 3D modelling of textile structures can be reached due to a detailed mathematical description of the yarn structure. Description of the geometry of twisted complex threads in the knitted structure is considered in the works [6-7]. An approach to the three-dimensional modeling of woven structures, made of multifilament threads, at the level of separate filaments is examined in [8]. A mathematical software for the description of the geometry of a plain knit stitch, made of twisted ply yarn and wrap spun yarn is suggested in [9].
Results and discussions.
Considering separate filaments in the three-dimensional model of a knitted loop becomes possible, provided a mathematical apparatus of description of surfaces, which limits the part of the space, occupied by every individual filament, is elaborated. It is known that the size and shape of the yarn cross-section changes along its axial line. We assume that along the axial line of an untwisted multifilament thread, several segments of decreasing and increasing diameter are present (fig. 1). Let's a part of the thread l, is divided into h segments. Points to, ti, ... tj-i, tj, ... 4 - are characteristic points of the axial line of the thread, in which local maximum or minimum values of the thread width are detected. Each segment g, is limited by cross-sections ej-i and ej, taken at characteristic points tJ-i and ti respectively. The cross-section in characteristic points can be denoted as characteristic cross-sections. The length of the segment gjis assigned as dlj. It's assumed as well, that the thread is composed of n filaments, that have an unchangeable cross-section with diameter df. Let's assign as Oj. the central point of the i-s filament in the co-ordinate system of the j-s cross-section of the thread.
Figure 1. Projection of a part of yarn with a varied cross-section on a plane
Let's suppose that in cross-sections, where the thread has a minimum diameter Dmm - filaments are located as tightly as possible (Fig. 2a). The increase in the diameter may occur unevenly. In general, a cross-section shape of an unfolded multifilament thread can
be approximated by an ellipse with a major Pmaj and minor Pmin axis (Fig. 2b), which can take a form of a circle in particular cases, e.g., in characteristic points of the minimal thread width.
a B C
Figure 2. Cross-section of a thread, consisting of 32 filaments, with compact (a) and scattered (b) placement of the filaments; an arbitrary shift of the filament in the limits of its bounding rectangle (c)
The transition from a round to an elliptical form can be realized by increasing the distance between the individual filaments, keeping their mutual position unchanged. If all filaments' axis line traces remain in the central points of their boundaries, optical porosity will be significantly higher than it could be in a real thread. This may lead to discrepancies of the results of permeability simulation and visualization quality. At the points with a loose placement, the filaments' cross section can shift without crossing the boundary as shown in Fig. 2b. Let's determine a reserved rectangle as a rectangular area, belonging to the cross section ej, which limits a possible displacement of a separate filament i's cross-section. It's dimensions ax x ay (Fig 2c) can be assessed as (1) and (2):
ax. = a + dXj = df + dxj
(1)
ay. = a + djj = df + djj (2)
Then an arbitrary shift of the filament cross-section along the axis of the abscise and the ordinate in the
co-ordinate system of the j-s cross-section of the thread remains within:
0 < tx < (axj - df) 0 < ty < (ayj - df).
The key assumptions used in this algorithm are:
• all filaments have a round shape of cross-section with a constant diameter;
• tangents to the axis lines of separate filaments, built at the point of their intersection with the characteristic cross-sections, are parallel to the tangent line, built to the axial line of the thread at the appropriate characteristic points.
In Fig. 3 a block diagram of the algorithm is shown. The algorithm collects a set of data, containing coordinates of the centres of the filaments' cross-sections, distributed in the area of the appropriate thread cross-section ej, considering an arbitrary lateral shift of the filaments within their reserved boundaries.
Calculate ar ; a.
i=1
Calculate coordinates of the centers of the filament's cross sections o;. * oi;1 * within the thread cross sections ey* and ey-t*
no m <l) n
Array of start and end points of the filaments' axis lines in the section g¡*
( End )
Figure 3. The algorithm offorming a data set, that contains coordinates of the start and end points of the filaments ' axis lines, in the j-s segment of the thread * variables, that contain mathematical descriptions of graphical objects
Let's assign the sets of the reserved rectangles in the key cross sections ej and ej.i as:
RJ = ft!'
rh
-..'in }'
(3)
and
Rj-i = (rj-i
....Tj-ln }. (4)
The elements of these sets are graphical objects, containing their mathematical descriptions. The use of the algorithm increases the realism of the 3D model. This can be seen in Figure. 4.
Figure 4. 3D model of a segment of a multifilament untwisted thread, generated with the use of the algorithm
In the structure of the knit, the axial line of thread has a more complicated configuration.
It is possible to assume that there are some characteristic points of a knitted loop, where the thread remains in the most compressed state (points Ci, C2, C3 and C4 in Fig. 5). The cross sections in these points are marked as ci, C2, C3, C4 respectively. It is assumed as well, that at these points the filaments are inserted into a circular shape contour. A set of circular shaped cross sections can be described as:
C = {Cl, C2..
.Ck},
(5)
projection of the thread on the fabric plane has the greatest width. In Fig. 5 these are points Ei, E2, E3, E4, E5. We assume that at these points the filaments are located within elliptical boundary, ei, e2, e3, e4 and e5. Accordingly, we can say, that within a repeated unit, a set of elliptical cross-sections of type e corresponds to the zones of the sparsest distribution of the filaments.
E = {ei, e2........eq}, (6)
where q is the number of the cross-sections of type e in the repeated unit of the thread (for a plain knit loop
q=5).
where k is the number of cross-sections of type c in a repeated unit of the thread (in the simplest case -in a plain knit loop k=4).
There are also characteristic points, in which the
The indicated points divide a section of the thread l, knitted in a plain knit stitch, in h segments, marked here as gj.
i<j < h, h=k+q-i.
a 6
Figure 5. Placement of characteristic points on the axial line of a plain knit stitch on the projection of its technical face (a) and technical back (b) to the fabric plane.
In Fig. 6 a 3D model of a thread segment is shown. It consists of 32 filaments, limited by cross sections, built at points ('3 and E4.
a b
Figure 6. 3D model of a multifilament thread section, limited by cross sections, built in adjacent characteristic points (a); a segment of one filament inside its reserved volume (b)
The algorithm of building a three-dimensional model of a plain knit stitch, made of a multifilament thread, is shown in Fig. 7. As the input data for construction, the algorithm uses:
• the equation of the thread axial line;
• coordinates of the characteristic points Ei (Xe ,, Ye,, Zen), C,(Xc,, Yc,, Zc,), E2(Xe2, Ye2, Ze2), C2(Xc2, Yc2, Zc2), E3(Xe3, Ye3, Ze3), C3(Xc3, Yc3, Zc3), E4(Xe4,
Ye4, Ze4), C4(Xc4, Yc4, Zc4), E5(Xe5, Ye5, Ze5);
• number of filaments n;
• and diameter of each individual filament df. From the input data, the geometric description of
the characteristic cross-section of the thread is processed, using standard mathematical formulae and the method, explained in [14].
Figure 7.
Algorithm of construction of 3D model of a knitted loop made of a multifilament thread, at the micro-level
For each segment of the thread, the algorithm is executed, and a new cycle begins, where the number of repetitions equals the number of thread filaments. After the completion of the first filament, the transition to the next one is performed. When all the filaments are
checked, the transition to the next segment is performed, and after processing all segments, the model generates.
Fig. 8 represents a 3D model of a plain knit loop, made of multifilament thread. The number of filaments is 420. Diameter of one filament is 0,025 mm.
Figure 8. Visualization of a knitted structure at the filament level (micro-level)
Application of the algorithm of 3D modelling of knitted structures, made of untwisted multifilament threads, considering lateral displacement of filaments within the reserved volumes, increases the accuracy and realism of the model and simplifies its processing by the means of CFD analysis.
Conclusions
During the study, an algorithm of the lateral displacement of filaments within the reserved volumes was developed, intended to increase the accuracy of three-dimensional geometric models of knitted structure, made of untwisted multifilament threads.
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