Generalized geometric modeling of tubular and flat braided structures with arbitrary floating length and multiple filaments

Abstract

This paper provides a critical overview about the state of the art in the area of three-dimensional modeling of braided structures. It gives a generalized geometrical approach for modeling braided structures with arbitrary floating length and filaments in the yarn. The approach is tested with large set of structures of different types. Subsequently, one of the simulated geometries is compared with the real geometry of braided tube.

Keywords

braided structures tubular flat three dimensions geometric modeling multifilament model

Braided structures are widely used as ropes, tubes, composites, medical and other applications. The prediction of their properties is the subject of several investigations, which gives partial solution for some specific cases, based on some assumption about the geometry of the yarns in the braid. There are several works that try to model the yarn orientation in the braid in the general case in order to provide good initial data for more application-specific mechanical calculations. Unfortunately, these proceedings do not yet cover all possible braided structures. This paper gives a critical overview of the recent works and methods used for modeling of braided structures and adjacently presents a unified approach for geometric modeling of braided structures with arbitrary floating length and yarns in a group.

State of the art

In this research only the related to the macroscale three-dimensional (3D) geometry works will be discussed. Because of the limited space, models that solely deal with unit cells of braided (and woven) fabrics are not considered. Moreover, one of the problems of some models, which is addressed in the third section in this paper, appears at the macro level even if the unit cell is correctly modeled, but wrongly oriented in the macro space.

In one of the most cited papers about 3D modeling, Liao and Adanur¹ sweep a two-dimensional (2D) simple closed contour $c (u), u \in [0, M]$ along a regular 3D curve in the 3D space $γ (s), s \in [0, L]$ with non-vanishing curvature in order to receive tubular surface as the sweep object:

G (s, u) = γ (s) + c_{1} (u) n (s) + c_{2} (u) b (s)

(1)

where

n (s)

and

b (s)

denote the normal and binormal vectors of

γ (s),

respectively. They suggest modifying the Frenet frame so that the sweep object is valid in a general case, being able to handle trajectories γ that consist of planar and nonplanar segments as long as the curvature of each nonplanar segment is nonzero. This method in its digitalized form (where the normals and binormals are computed based on a discrete data set) presents the basic principle of the modeling of yarn surfaces of the textile structures. By implementing this method in C++ and using OpenGL, the authors have created realistic 3D simulations of braided fabrics with different structures around conical and other pulled preforms of diamond and regular braids. How they exactly computed the coordinates of the yarns is not explained in the paper. The derivation of the equations for the yarn paths of open braids is given by Rawal et al.,^2,3 where the yarn undulations are not considered. For instance, for overbraiding of a cylinder they are to be found as follows:

x = r \cdot cos (θ) y = \pm r \cdot sin (θ) z = r \cdot θ \cdot cot (α)

(2)

where r is the nominal radius of the braid and

θ = ω \cdot t

is the angle of the investigated point, depending on the angular velocity of the carrier ω around the product axis.

Kyosev⁴ and Kyosev et al.⁵ used an extended version of these equations, considering the undulations of the radius r on the places of contact points:

r = r_{mandrel} (z) + d_{yarn} \pm 0.5 \cdot d_{yarn}

(3)

The profile of the mandrel is defined as a function in a discretized form $r_{mandrel} (z)$ . Before the calculation of each crossing point of the yarns is done, the stability conditions of the yarns over the mandrel are checked. This conducted check is based on the equation for winding bodies from Proshkov,⁶ relating to the geodesic angle of the curve $τ_{G}$ and the limit friction angle $ɛ_{\max}$ , which is determined from the static friction coefficient between the yarn and mandrel μ and the braiding angle α:

\tan α \leq \tan τ_{G} \leq \tan ɛ_{max} = μ

(4)

In the case that the yarns are stable positioned and do not slip around the crossing points, the complete set of yarn paths is generated and connected using splines, in a similar way to that reported by Pastore et al.,⁷ who used Bezier curves, or Bogdanovich,⁸ Bogdanovich et al.⁹ and Lomov.¹⁰ This algorithm is tested for parts that have rotational symmetry. In this case, the stability condition has to be tested once per set of crossing points with the same Z-coordinate. The geometry of braids using the elliptical cross-section of the yarns is created as presented in Figure 1.

Alpyildiz¹¹ uses pure analytical description of the curves of the yarn path to describe the equations for the yarn undulation in detail. For a regular braid the regions of the floating are fixed to the amount of $\pm a / 2 .$ The region between them is presented with a sinus-function, so that the equation for the radius r for the yarns, in which carriers move in a counter-clockwise direction, looks like

r (θ) = {\frac{a}{2}; 0 < θ < k_{1} β / 2 \frac{a}{2} \cdot \sin (\frac{2 π}{β} θ + \frac{π}{2}); \frac{k_{1} β}{2} < θ < (k_{1} + 1) β / 2 - \frac{a}{2}; (k_{1} + 1) β / 2 < θ < (2 k_{1} + 1) β / 2 \frac{a}{2} \cdot \sin (\frac{2 π}{β} θ + \frac{π}{2}); (2 k_{1} + 1) β / 2 < θ < (2 k_{1} + 1) β

(5)

For the ones that drive in a clockwise direction, the sign of a/2 and the phase in the sinus function regions is changed.¹¹

Rawal et al.¹² extend the equations for the case of a mandrel of conical cross-section and parts that consist of a combination of cylindrical and conical regions.

The above cited papers concentrate on the modeling of three main braiding structures – diamond, regular and Hercules-braided structures – covering biaxial and triaxial braids up to a floating length of three. Unit cells with arbitrary floating length can be created with Wisetex.¹³ Nevertheless, it is the responsibility of the user of Wisetex to obtain the correct topology (from the braiding point of view) and the correct orientation of the unit cell into the macro geometry of the braided product. The models in Kyosev¹⁴ represent the structures with arbitrary floating length correctly, but the explanation of how they were created is not to be found. Furthermore, the structures in Kyosev¹⁴ do not represent braids with several yarns in a group realistically. As explained there, these have more space between the yarns and thus can be used for understanding the braid, but not to execute extended calculations.

There is as well a commercial computer-aided design (CAD) software for braiding named TexMind Braider,^15,16 which seems to cover all kind of tubular and flat braided structures, including the ones with multiple yarns in a group. Public information about the models used has not been available until now.

Three-dimensional models of braids with floating length of two (regular braids)

In the modeling works of several authors^1,5,11,12 appears a small mistake in the models of the regular braids only, recognizable by experienced braiders only. These cannot be produced on any “classical normal” maypole braiding machine with horn gears. The arrow on Figure 1 points to very well visible ridges that are perpendicular and not parallel to the product axis. Figure 2(a) visualizes once more schematically the orientation of the ridges in the above-mentioned papers. The visible yarn pieces, which build the visible ridges, are placed in square mesh as this is done usually during the drawing of the draft of the structure.^14,17–19 In a standard machine each horn gear rotates in only one direction. All the carriers moving outside of one horn gear are building a vertical ridge, parallel to the take-off speed, which is parallel to the product axis as well (Figure 2(b)). Thus, during the analysis of the braids, the ridges are counted and their number gives the number of the horn gears. In order to produce horizontal ridges, the horn gears have to move the carriers alternating: one carrier in one direction, the other in the opposite direction, etc. (Figure 2(a)). This configuration is only possible on 3D braiding machines with individual drives of the horn gears, which are of very limited amount. It is not possible for the maypole braiding machines, due to the fact that all horn gears are connected with gears and solely move without changing the rotation direction during the production.

Topologically, the unit cells of both structures are equivalent, simply one structure is rotated at 90°. Technologically, a braid with unit cell A (Figure 2) is not identical to a braid with unit cell B at the macro level. The models of the structures with horizontal ridges cannot be used for color patterning of tubular braids. Furthermore, their use for mechanical calculations can lead to some differences in the mechanical behavior.

After analyzing the modeled pictures and comparing the proper yarn positions with the modeled ones, it was possible to detect the positions where the yarns were placed on the wrong side. These positions are marked and illustrated in Figure 3(a) with a thick dotted line.

It can be seen that on track 1 (Figure 3(b)), every second yarn is placed with a phase shifting, which corresponds to the moving of the carrier on two slots around the horn gear. Counting the yarns in this track not consequently, but instead into two groups, where the second has a phase change of angle, corresponding to angle π of the horn gears, might be a possible solution for the correction of the mathematical models for the regular braids.

The yarns of track 2 (Figure 3) are once in on the correct side and once on the wrong side in their cells. Alternating as well is the starting position – one yarn is starting with proper position and one is starting with a wrong position.

Writing the correct phases and numerations of the mentioned models is not the goal of this paper, because the approach presented in the next sections is more intuitive and generalized. The aim is to present a common method for the calculations of braids with any floating length per ridge and any number of yarns in a group (= filaments in the yarn). Being independent of the braiding architecture, this method allows implementation in industrial software as it is not limited to one or other type of braid.

Generalized model for yarn path of a braid with arbitrary floating length

For this model is considered a part of a generalized braid (tubular or flat) with ridges with different floating length (Figure 4). This part consists of five visible ridges, where the first, second and the fifth have a floating length of two, while the third ridge has a floating length of five and the forth has a floating length of one. The sequence of the floating lengths per ridge can be written as 2:2:5:1:2. The part of the machine that is used for the production of such a braid is presented in the bottom part of Figure 4, following the main braiding equation as a rule for the floating length FL:¹⁴

FL = [\frac{Number_slots_per_horn_gear}{Repeat_of_the_carrier_arrangement}]

(6)

Considering the most common arrangement of 1 full 1 empty, the repeat of the arrangement is 2, so the horn gears should have 4,4,10, 2 and 4 slots for this braiding part.

Considering yarn 1 in Figure 4, it floats first (starting from z = 0) over two yarns, then under five, over one and under two. This floating corresponds to the carrier motion – the floating over each yarn corresponds to the carrier motion on the angle, equal to one segment, in the case of full occupation of the machine (full occupation means that all possible carrier positions are filled with carriers and correspond to a 1 full 1 empty arrangement along the track¹⁴).

In this way for yarn 1 has to be created the same number of key points as the number of yarns in the opposite track, as all these yarns will be crossed. If the current ridge is visible, the key point has positive Y coordinate; if the ridge is hidden (the yarn is under another) then the key point has a negative Y coordinate. Each ridge with floating length of two has two key points on the same side, a ridge with a floating length of five has five points and so on. For yarn 1 the first ridge has a floating length of two, so the first two points have a positive Y coordinate; the following ridge is hidden and has a floating length of five – there are five points with a negative Y coordinate; the next ridge is visible and has a floating length of one and the last is hidden and has a floating length of two (Figure 5(b)).

In Figure 5 it can be seen that the next yarn starts its motion with an opposite phase. Such specific rules can be derived only for specific floating lengths and not for the general case.

An algorithm for the calculation of the coordinates of the key points of a tubular or flat braid will be very similar to the algorithms for a graphical analysis of the braided structures, presented by Kyosev¹⁴and Engels.¹⁹ This algorithm uses inverse braiding, which means that the horn gears are staying and the carriers are jumping from slot to slot, summarized as follows.

Initialize the braiding machine, considering the (minimal) size of the horn gears for the wished floating length of their ridges. The number of slots is stored in a vector ${BM}_{i}, i \in [1, N_{HG}]$ .

Initialize the rotation direction of the horn gears ${HGR}_{i}, i \in [1, N_{HG}]$ .

Initialize the track(s) of this machine as a list of slots as elements T(i_hg, i_slot), where the carrier slots in one track, denoted with their horn gear number i_hg and the slot number i_slot, are stored.

Fill the tracks with carriers according to the given carrier arrangement, for instance A = [1 0], one full, one empty. The carrier arrangement can be derived from the braiding equation.

Determine which slots are currently moving the carriers, which are building visible yarn parts and which are building hidden ones. The connection line between the horn gear centers can be used as the visibility limit.

Create the mesh of cells in a manner that provides every two slots of one horn gear corresponding to one column. The mesh size – defined by cell height and cell length – is easily determinable by using the number of yarns in the braid, the width or the diameter of the braid and the braiding angle.¹⁴

For each carrier:

go through its track and get the X and Z coordinates of the corresponding cells. If the current slot is visible, set the Y coordinate to +a, if it is not visible to –a, and in case of a turning cell (for flat braids) to zero. The variable a can be adjusted depending on the used model for the yarn cross-section and depending on the type of the braid – bi- or triaxial. For triaxial braids the yarn configuration of the inlay yarn has to be considered in the variable a.

For flat braids the x coordinate remains on the same plane and for tubular braids it has to be rolled over a cylinder surface, using it as a counter variable for the division of the central angle.

Any refinements about the form of the curve inside the unit cell can be done by using well-known published methods referring to geometric relations or minimization of the energy of the yarn.

The described algorithm for two yarns is visualized on Figure 5. The carrier with yarn 1 moves at the beginning along the visible slots, then along five invisible slots, one visible and two invisible. The corresponding Y coordinate is presented in Figure 5(b).

Implementation

The complete algorithm is implemented in C++ programming language. A Graphical User Interface for editing the properties of the braids is created with the wxWidgets library²⁰ and the main sample data is stored as Extensible Markup Language (XML) files. The images are rendered using Visualization Toolkit VTK.

Model results

Figure 6 presents modeled flat braided fabrics with 3, 5 and 13 yarns and a floating length of one; Figures 7(a) and (b) illustrate flat braids with floating lengths of two and tree. The fancy braid, modeled in Figure 7(c) is used often as an accessory made from leather and consists of ridges with changing floating length 2:3:2:6:6:2:3:2. Tubular braids with floating lengths of 2, 3 and 4 and well-visible vertical ridges are presented in Figure 8. The braids with floating lengths of 3 and 4 in this figure are not common for rope production, but are becoming more so because they provide less fiber crimp, which is important for carbon and glass fibers, for instance.

Extension to multifilament models

Once the yarn paths of the braid are calculated, multiple filaments in each yarn can be added, as all these work as a group. For each point of the yarn paths a modified¹ Frenet frame t,p₀,q₀ is generated and saved, which defines the orientation of the yarn in the untwisted state. Rotating the modified normal and binormal (p₀,q₀) around the tangential vector at each point to an angle γ, a twist to this cross-section can be applied. For each single filament of the cross-section a set of coordinates of their centers c_i is defined and oriented according to the orientation of the p-q vectors of this cross-section (Figure 9). In this way different configurations of multifilament yarns can be created – with flat, circular, elliptical and any other cross-section. Some aspects about the orientation of the yarn cross-section under consideration of its geometrical moment of inertia can be found in any book about strength of materials or with some yarn considerations, for instance in Kyosev, Angelova and Kovar.²¹

Simulation results of the multifilament models

Figure 10 presents a modeled tubular braid with floating length of two and six wires in each group and the photo of such a braid from the water hose, from which it is possible to see that the simulated geometry is very close to the real one. Figure 11 presents another tubular braid with floating length of three and for wires in a group, and three regular braids (floating length of two) with six filaments distributed on the circular yarn cross-section with no twist (Figure 11(b)), twist 200 m^–1 in the S and Z directions for each track (Figure 11(c)) and for 400 m^–1 (Figure 11(d)). The geometry of the single filaments can be a good starting point for extended mechanical calculations.

Discussion

The presented method allows quick and stable generation of the geometry of the tubular and flat braided structures with intuitive connection of the input data and algorithm to the real braiding machine and process. This reduces the risk of errors of the model, but requires extended understanding of the braiding process, which is not always the case for persons who are able to develop mathematical models and vice versa.

The described method concentrates only on the definition of several important key points of the yarn path. If only these points are used, the models are of small size and computational shorter time. The accuracy of the path between these points is reduced as well, because these paths are approximated with polynomials and no geometric or mechanic rules for contact are applied there. Pure geometrical contact detection and removal algorithms can be applied in these places, but in such a case the curves can lose their smoothness and in some cases are even less accurate than the not corrected curves. Really meaningful improvement of the accuracy of the models can be achieved only if the complete mechanical data of the yarns is used and some kind of mechanical equilibrium is reached, as for instance the implemented algorithms in the software Wisetex.¹³ Such a development is actually no longer in the scope of geometrical modeling and is beyond the limits of this paper.

Conclusions

The critical overview of the works about the geometrical modeling of braided structures pointed out a small inaccuracy of some existing models for the special case of regular braids. This inaccuracy is explained and, furthermore, a way for its correction is presented. A generalized model for the calculation of the yarn paths of tubular and flat braids, using the knowledge of the braiding process, is presented and tested on several different braided structures. One of them is optically compared with a real structure. The concept of the extension of the algorithm for a multifilament model is given and tested on braids with flat and circular distribution of filaments with and without twist.

Footnotes

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

References

Liao

Adanur

. A novel approach to three-dimensional modeling of interlaced fabric structures. Text Res J 1998; 68: 841–847.

Rawal

Potluri

Steele

. Geometrical modeling of the yarn paths in three-dimensional braided structures. J Ind Text 2005; 35: 115–135.

Rawal

Potluri

Steele

. Prediction of yarn paths in braided structures formed on a square pyramid. J Ind Text 2007; 36: 221–226.

Kyosev Y. Model generator for tubular braided fabrics. In: Lomov SV (ed.) proceedings of finite element modelling of textiles and textile composites, St.-Petersburg, Russia, 26–28 September 2007.

Kyosev Y, Brücken A, Tillmanns A, et al. Modeling of the geometry and numerical calculation of the elasticity parameters of 3d braided shapes. In: Advani SG and Gillespie JW (eds) Recent advances in textile composites: proceedings of the 9th international conference on textile composites (TEXCOMP9), Lancaster, PA: DEStech Publications, 2008, pp.255–263.

Proshkov A. Mechanisms for yarn winding (topics of design): in Russian, original title: Прошков, А.Ф. Механизмы раскладки нити (вопроcы проектирования), Москва, Легпромбытиздат. Moscow: Legpromizdad, 1986.

Pastore CM, Birger A and Clyburn E. Geometrical modelling of textile reinforcements. In: Poe CC and Harris CE (eds) Mechanics of textile composites conference. NASA, Hampton, Virginia Hampton, 1995, pp.597–623.

Bogdanovich

. Three-dimensional analysis of anisotropic spatially reinforced structures. Compos Manuf 1993; 4: 173–186.

Bogdanovich AE, Pastore CM and Birger A. Analysis of composite shallow shell structure reinforced with textiles. In: Hamelin P and Verchery G (eds) Textile composites in Building construction, Part 2. Paris, Pluralis, 1992, pp.35–44.

10.

Lomov

Parnas

Bandyopadhyay Ghosh

. Experimental and theoretical characterization of the geometry of two-dimensional braided fabrics. Text Res J 2002; 72: 706–712.

11.

Alpyildiz

. 3D geometrical modelling of tubular braids. Text Res J 2012; 82: 443–453.

12.

Rawal

Gupta

Saraswat

. Geometrical modeling of near-net shape braided preforms. Text Res J 2015; 85: 1055–1064.

13.

Lomov SV, et.al. Wisetex. Department MTM, KU Leuven, 2012.

14.

Kyosev Y. Braiding technology for textiles: principles, design and processes: Woodhead Publishing Series in Textiles No. 158. 1st ed. Cambridge: Woodhead Publishing Limited, 2014.

15.

Kyosev Y. TexMind Braider, Mönchengladbach, www.texmind.com (2012).

16.

Kyosev Y. Computer aided colour and structural design of braided structures. In: Chen X and Hearle J (eds) CD-ROM proceedings of the first international conference on digital technologies for the textile industries, Manchester, 5–6 September. Manchester: TexEng Software Ltd, 2013.

17.

Lepperhoff B. Die flechterei. 3rd ed. Eugen G. Leuze Verlag, Saulgau/Württenberg, 1953.

18.

Goseberg F, Brabender K and Moeller P. Textiltechnik maschinengeflechte: ausbildungsmittel unterrichtshilfen. Frankfurt am Mai: Arbeitgeberkreis Gesamttextil, 1981.

19.

Engels H. Handbuch der schmaltextilien. Die Flechttechnologie. Teil I: maschinen und verfahren zur erzeugung konventioneller geflechte. Mönchengladbach, Fachbereich Textil- und Bekleidungstechnik, Hochschule Niederrhein,1994.

20.

Smart J and Roebling R, Zeitlin V, et al. wxWidgets cross platform GUI Library: wxWidgets, 2015.

21.

Kyosev

Angelova

Kovar

. 3D modelling of plain weft knitted structures from compressible yarn. Res J Text Apparel 2005; 9: 88–97.