Geometrical modeling of tubular braided structures using generalized rose curve

Abstract

This paper reports a geometrical modeling technique for tubular braided structures based on the generalized rose curve as the mathematical model. By analysis of the braiding process, the modeling method for the tubular braided structures is derived based on the intersection of braiding surface and helical surface. As application of this method, braided structures of diamond braid, regular braid and Hercules braid with strands and tapes are simulated using SolidWorks®, and the modeling effect is validated by two different real braided ropes with different braiding elements. This modeling method is not confined by the profiles of mandrels, and could be employed to simulate the braided structures for overbraiding bodies with varying cross-sections. As an application, braided models overbraiding a rotary hyperboloid and a bottle-like structure with strand and tape elements, respectively, are constructed.

Keywords

rose curve tubular braids mathematical models braided structure

Braided fibrous materials are finding increasing applications in aerospace, automotive, and general engineering, as their unique seamless interlaced structures prevent unraveling, as well as mitigating damage, in comparison to twisted or filament wound structures.^1,2 Braided structures are created by interlacing yarns in clockwise and anticlockwise directions in order to form a bias structure along the longitudinal axis.³ According to the interlacing patterns, braided structures can be divided into diamond, regular, and Hercules braids, and, based on the appearance and cross-section, braids can be classified as flat, tubular, or fancy braid.³ Braided structures have extensive applications, ranging from ropes, cordage, hoses, reinforcement coverings, composites tubes, and rotor hubs, to parts in automobile and aerospace industries. With the recent trend for structural light-weighting, braided structures have received extensive attention in aerospace and automotive composite structures. Compared with laminated composites, braided composites have excellent integrity due to the seamless interlaced structure.⁴

Braided structures contain two-scale geometrical features: interlacement pattern at tow scale, and macro geometry of the part (e.g. tube). Computer-aided design, together with finite element methods, creates a powerful tool for the modeling and analysis of braided structures. Accurate models are the first step and foundation to obtaining reliable simulation results. These models build the relationship between the properties and the geometry of the braids. So it is very important to provide exact descriptions and models for braided structures.

In order to create exact models, a detailed analysis of braided structures is essential. Systematic analysis of braided structures started in the 1950s. Brunnschweiler⁵ was one of the first researchers to report the mechanics of braid structures. He used idealized geometry to examine braiding angle and fiber coverage, and developed a simple equation. Zhang et al.⁶ analyzed two-dimensional-braid geometry and focused on the parameters that affect the covering factor. Both of these papers focused on the research in to the appearance of braids, and analyzed the relationship between covering factor and structural parameters. The crimp of the strand was not considered. Du and Ko⁷ developed geometric models for braids using a unit cell approach. Subsequently, Lomov et al.⁸ studied flat braids, and proposed a geometrical model by giving the paths of the yarn centerlines within a predetermined unit cell. Meanwhile, Wu et al.⁹ studied the geometry of strands in double-braided ropes, and gave the mathematical expression of strands based on detailed description of the crimp characteristics of strands. These papers paid attention to the morphology of strands in tubular braided structures and the research objects were cylindrical braids.

Guyader¹⁰ focused on the modeling of circular braiding processes, and established the relationships between the architecture of the complex-shaped braids and the process parameters by considering the transitory and steady-state process stages. Du and Popper¹¹ also developed a detailed model of the complex braiding process by over braiding a contoured mandrel. Kessels and Akkerman¹² developed a model to predict the fiber angles on complex braided preforms. Later, Akkerman et al.¹² extended the mandrels from axisymmetric structures to non-axisymmetric mandrels, and discussed the relationship between strand paths and braid angles, with the possibility of eccentric cross-sections that varied in shape and size. More recently, the braid pattern has been predicted for any arbitrary mandrel shape based on the minimum path condition, by Na.¹³ Due to the complicated morphology, this research has mainly focused on basic braiding parameters, and no mathematical models were reported.

As to the geometrical modeling technique, early work was conducted by Liao and Adanur.¹⁴ They aimed to develop yarn shapes for tubular braids by the Frenet frame approach. They created realistic three-dimensional (3D) simulations of braided fabrics with different structures around conical and other pulled preforms of diamond and regular braids, based on C++ and OpenGL, but no more information about the coordinates of the yarns was included in their paper. Then, Potluri et al.¹⁵ introduced a computer-controlled braiding machine to manufacture triaxial tubular braids, and the braided structure was simulated using virtual reality modeling language. Based on the same method, Rawal et al.^16,17 focused on producing three-dimensional shapes by braiding over contoured mandrels, which included a circular cylinder, a circular cone, an elliptical cone, a square prism, and a square pyramid. However, in order to simplify the geometrical models, these investigations did not consider the crimp of the strands. In 2012, Alpyildiz¹⁸ studied geometrical modeling of tubular braids and proposed a simple 3D model after considering the crimp of the braiding yarn. Drawings of the model for diamond, regular, and triaxial braids were given with the aid of Visual Basic and 3DSMax Studio. Based on Alpyildiz’s mathematical model, Rawal¹⁹ used VRML to visualize the models of strand trajectory on the surface of cylindrical and conical mandrels, and simulated these complicated structures. Recently, Kyosev^1,20 gave a generalized geometrical approach for modeling flat and tubular braided structures, with arbitrary floating length and filaments in the yarn based on the braiding machine. Based on this modeling method, CAD software for braiding, called Texmind Braider,²¹ was developed.

The modeling methods of both Rawal and Kyosev are realized by programming language and other rendering software. In order to make the modeling process more convenient, this paper provides a modeling method which can be easily implemented using commercial software such as SolidWorks®.

Basic assumptions and definitions of geometrical parameters

Assumptions

In order to obtain the geometry of the braided strands, it is necessary to make the following assumptions:
the projections of strands on the cross-section of braid is represented by rose curves;

the strands in the same direction are of the same structure, the difference only lies on the initial positions;

the projected curves of strands in different directions are of the opposite phases;

the structures of braids totally depend on the braiding curve—there are no relevant interactions or deformations among adjacent strands.

Definitions of geometrical parameters

The geometrical nomenclature for a traditional braiding strand used in this paper is provided, and the definitions of some key terms are specified, as follows.
Radius of braid (R). R is defined as the distance from the braiding axis to the central circle which the crimps wave around;

Amplitude of crimps (A). The amplitude of crimp refers to the maximum of amplitude of braiding strands. For braided structures, the amplitude of crimp mainly depends on the diameter of strands and the braiding patterns;

Number of crimps in a pitch (N). The number of crimps refers to the number of crimps in a pitch or braiding circle, which depend on the braiding tracks;

Braiding angle (θ). Braiding angle is defined as the angle between the braid axis and individual braid strands;

Initial position (ϕ). The initial position is the starting position of strands on the cross-section;

Pitch of braid (P). The pitch of braid refers to the distance in axial direction of a strand in a braiding circle;

Direction parameter (λ). Parameter λ is employed to characterize the direction of helix and braiding curve, for anti-clockwise, λ = 1 and for clockwise, λ = –1.

Motion analysis of braiding process

Braiding is an old technique. With the development of manufacturing technology, braiding machines have been improved greatly, while the basic braiding method has not been changed significantly. In conventional braiders, yarn carriers rotate along a circular track, with half the carriers rotating in a clockwise direction while the remaining carriers rotate in a counter-clockwise direction, similar to a maypole arrangement. As a result, the two sets of yarns interlace with each other at a bias angle to the machine axis. At the same time, the braided strands are continuously drawn out by a take-up roller to form braided structures.

Decomposition of strand motions

Motion of the carriers

On the braiding plane, carriers run following closed tracks. When carriers run along these tracks, they will run along a central circle, all the while oscillating around this circle, and interlacing with the carriers in the other direction. According to the movement of the carriers, it is reasonable to decompose their motion into two sub-motions, one a circumferential motion along the central circle, and the other a radial motion, as shown in Figure 1. These two sub-motions have different meanings for the formation of braided structures. The circumferential motion is a kind of driving motion, which ensures the strands run along the central circle and make the braiding process run smoothly, while the radial motion is the characteristic motion, which result in the interlacing of strands in the braiding process. For different braided structures with different braiding patterns, the frequency of radial motion is different.
Figure 1.
Motion decomposition of strand during braiding process.

Motion of the take-up roller

During the braiding process, the take-up roller continuously draws out the braid from the braiding zone to form the braided structures. The movement of the take-up roller provides an axial motion to the braiding strands. The speed of axial motion depends on the speed of the take-up roller, and this determines the pitch of the braided structure.

Mathematical models

Generalized rose curves

In mathematics, a rose curve is a sinusoid plotted in polar coordinates. These curves can be expressed as a pair of Cartesian parametric equations,²² as equation (1)
${x = \sin (N t) \cos (t) y = \sin (N t) \sin (t)$
(1)

For rose curves, if N is an integer, the curve will be rose-shaped. When N is even, the number of petals is 2N, while when N is odd, the number of petals is N. This is because when N is even, the concave part of the curve will form a convex part in the other side of the origin point, so the number of petals would be doubled. The generalized rose curve could be developed by adding a parameter R, as equation (2)
${x = (R + A sin (N t)) \cos (t) y = (R + A sin (N t)) \sin (t)$
(2)

There are three parameters for the generalized rose curve, so the mathematical expression of this curve could be expressed by P(R, A, N). Some generalized rose curves with different parameters are shown in Figure 2.
Figure 2.
Generalized rose curves.

From Figure 2, the following conclusions can be drawn: the number of petals of generalized rose curves when R < A is always the 2N. In this case, when N is even, as shown in Figure 2(a) and (b), the big petals surround the smaller ones, while when the N is odd, shown in Figure 2(c) and (d), the bigger petals and the small petals arrange alternately. Because there are intersections between curves and the original point, so this kind of generalized rose curve is not the desired curve in this paper. When R > A, as shown in Figure 2(e)–(h), petal number is always N, and all the petals are arranged around the circle uniformly. Compared with the figures of generalized rose curves when R < A, shown in Figure (a)–(d), the figures when R ≥ A, shown in Figure 2(e)–(h), illustrate that the curves wave around a central circle, with radius of this central circle as R, the amplitude of the wave as A, and the number of crimps as N. The properties of this kind of generalized rose curve are the same as those of the braiding track, so this kind of generalized rose curve is employed to describe the braiding curves.

Modified rose curve model

The generalized rose curve is similar to the path of braiding track, while it is obvious that the trough and crest of the generalized rose is very sharp, and may properly simulate strands with a large span, just like the strands of multi-strand braid. In order to solve this problem, a modified rose curve is introduced. Figure 3 shows the relationship between motion and the relevant curve, Figure 3(a) illustrates that in unit circle, the Y value will be changed, with the angle following a sine curve. If the unit circle is replaced by an ellipse, a modified curve will be obtained, as shown in Figure 3(b). The modified curve shows special characteristics. Its troughs and crests are almost flattened instead of sharp in the sine curve. This property is useful in meeting the requirement of braid architecture with floats.
Figure 3.
Relationship between motion and relevant curve: (a) circle and sine curve; (b) ellipse and large span curve.

The relationship between y and the angle can be obtained as shown in equation (3)
${y = \frac{ab \tan θ}{\sqrt{a 2 \tan θ 2 + b 2}} 0 << θ << π y = \frac{- ab \tan θ}{\sqrt{a 2 \tan θ 2 + b 2}} π < θ < 2 π$
(3)
where a and b are the major and minor axis of the ellipse, and $θ$ is the position angle.

The curve of this equation is along the X axis, as shown in Figure 3(b). In order to use this curve to represent the path of the braiding track, it is necessary to bend this curve to form a closed curve. This transformation is similar to the process from the sine curve to the generalized rose curve. Comparing the equations between the sine curve and generalized rose curve, it can be seen that the generalized rose curve is a kind of transformation of the sine curve; in other words, the generalized rose curve can be obtained by bending the sine curve around the origin point to form a closed curve. The parameter R is brought in as the diameter of the central circle which the curve waves around. The expression of this transformation can be shown as equation (4)
${x = (R + f (t)) \cos (t) y = (R + f (t)) \sin (t)$
(4)
where f(t) is the equation of the curve before bending. Based on this transformation and equation (3), the equation of the modified rose curve can be obtained as equation (5)
${x = (R + \frac{ab}{\sqrt{a 2 + b 2 (\cot Nt) 2}}) cos (t) y = (R + \frac{ab}{\sqrt{a 2 + b 2 (\cot Nt) 2}}) sin (t) 0 << θ << π$
(5)

${x = (R - \frac{ab}{\sqrt{a 2 + b 2 (\cot Nt) 2}}) cos (t) y = (R - \frac{ab}{\sqrt{a 2 + b 2 (\cot Nt) 2}}) sin (t) π < θ < 2 π$

In the equation of the modified rose curve, there are four parameters, R, a, b, and N. So the modified rose curve can be represented by L(R, a, b, N).

Figure 4 shows modified rose curves changed with different parameters; the following characteristics can be obtained.
R is the same as for the rose curve, which defines the radius of the curve shown as Figure 4(a) and (b).

When a > b, the trough and crest become horizontal, which is expected to represent the strand projection, shown in Figure 4(c) and (d). When a < b, the trough and crest become sharp compared to the generalized rose curve; in this case, the modified rose curve is not suitable for braiding tracks, shown as Figure 4(e) and (f).

N defines the number of crimps along the circle, shown as Figure 4(g) and (h).

Figure 4.
Modified rose curves with different parameters.
When the major axis is equal to the minor axis (a = b), the equation becomes equation (2). So the modified rose curve which is based on an ellipse is a more generalized style of the rose curve which is based on the circle. The generalized rose curve is used to represent the single strand braid and the modified rose curve can be employed to represent strands with large floats. In the following sections, the generalized rose curve is employed to describe a modeling method which is the same for the modified rose curve.

Recompose the sub-motions

The above analysis is based on the motions of carriers and take-up roller in the braiding process; the braided structures are formed by the regular motion of these two parts. In other words, the strands are the record of these motions. The projection of the braided strands on the cross-section of the braids is similar to the track. The pitch of the braids is defined by the speed of carriers and the take-up rollers. Therefore, the motions during the braiding process can be employed to describe the modeling process.

Based on the analysis of decomposition of motions, the motion of the strands is the composition of the motion of carriers and the take-up roller. So the motion of the strands can also be decomposed into circumferential motion, radial motion, and axial motion. According to the positions of these three sub-motions, they can be divided into motions in the braiding plane, including radial motion and circumferential motion, and axial motion. The motions in the braiding plane are independent of each other. Both of them are simultaneous with the axial motion, and thus they could compose with axial motion. The radial motion and the axial motion could compose a motion which always lies on a surface formed by sweeping the projected curve along the braiding axis; this surface is called the braiding surface, as shown in Figure 5(a).
Figure 5.
Relevant components: (a) braiding surface, (b) helical surface, (c) intersection of braiding curve and helical curve.

Based on the analysis of generalized rose curves, the projected curve on the cross-section of braided structures is a kind of generalized rose curve, so the equation of the projected curve can be expressed as equation (6)
${x (t) = (R + A sin (N t)) cos (t) y (t) = (R + A sin (N t)) sin (t)$
(6)

Based on the equation of the projected curve, it is possible to develop it into 3D to get the braiding surface. The parametric equation of the braiding surface is as equation (7)
${x (t) = (R + A sin (N t)) \cos (t) y (t) = (R + A sin (N t)) sin (t) z (t) = [0, h]$
(7)
where h is the height of the surface

The circumferential motion and the axial motion can compose another resultant motion that always lies on a helical surface which is formed by helically sweeping the radius of the braiding circle along the braiding axis, as shown in Figure 5(b). This surface is called the helical surface. The vector equation of the helical surface is shown in equation (8)
${x = U \cos λ (t + ϕ) y = U \sin λ (t + ϕ) z = Pt (U \geq R + A, 0 \leq t \leq 2 π)$
(8)
where, parameter λ is employed to characterize the direction of lay; for anti-clockwise, λ = 1 and for a clockwise, λ = –1. λ affects the x and y coordinates, but does not change the z coordinate. The parameter $ϕ$ is introduced as the starting point.

Because the braiding motion can be decomposed into circumferential motion, radial motion, and axial motion, these three component motions work together to form the unique braiding pattern and spatial structures. The path of the strand should satisfy these three motions at the same time, so the path of the braiding curve can be obtained by the intersection of these two different surfaces. The intersection line of these two surfaces is shown in Figure 5(c).

The equation of the braiding curve can be obtained by combining the equations of these two surfaces. Therefore, the equation of the braiding curve will be
${x (t) = (R + A sin (λ N t)) cos (t + ϕ) y (t) = (R + A sin (λ N t)) \sin (t + ϕ) z = Pt$
(9)

This equation is identical to the results of Wu,⁹ which means that it is reasonable to use the generalized rose curve to model the braiding curve.

For strands in the same direction, all strands are arranged equally around the braiding axis; the only difference lies on the initial position. For a braid with 2N strands, the number of strands in the same direction would be N. All these N strands will arrange equally around a circle, so the phase difference can be shown as equation (10)
$Δ_{ϕ} = \frac{2 π}{N}$
(10)

So, the equation of strands in the anti-clockwise direction will be expressed as
${x (t) = (R + A sin (Nt)) \cos (t + i Δ_{ϕ}) y (t) = (R + A sin (Nt)) \sin (t + i Δ_{ϕ}) z = Pt i = 1, 2 \dots N$
(11)

Then the corresponding equation of strands in the clockwise direction will be
${x (t) = (R + A \sin (- Nt)) \cos (t + i Δ_{ϕ}) y (t) = (R + A \sin (- Nt)) \sin (t + i Δ_{ϕ}) z = Pt i = 1, 2 \dots N$
(12)

Based on equations (11) and (12), the braided structure will be realized

Application

Braided structures could be obtained by taking the generalized rose curve as the mathematical model, and the intersection of the braiding surface and the helical surface as the realizing method. Based on this modeling method, not only common braided structures, but also some complicated braided structures of revolving bodies with varying cross-section, can be realized. All these geometrical models can be realized using any commercial CAD platform, such as SolidWorks®.

Geometrical modeling of strand braid

Diamond braid

Diamond braid is the simplest braid, which is formed by strands in two directions interlacing with each other one by one. For one strand, it can be obtained by the intersection of the braiding surface and the helical surface. Projected curves are the key factor in modeling braided structures, and they define the braiding patterns. Here, the generalized rose curve is employed to realize this curve, as described by equation (9).

The braiding surface can be obtained by sweeping the projected curve along the braiding axis, as shown in Figure 5(a). In SolidWorks®, the “swept surface” tool is employed to realize this step; the profile is the projected curve in the horizontal plane, and the path is the braiding axis. Sweeping the profile following the path forms the braiding surface.

The helical surface can be obtained by helically sweeping the radius of the braided structures along the braiding axis, as shown in Figure 5(b). In SolidWorks®, the “swept surface” tool is employed to realize this. The profile is the segment in the horizontal plane, and the path is the axis. Twisting the profile along the path forms the helical surface. The pitch of the helical surface defines the pitch of the strands.

The intersection of the braiding surface and the helical surface will be the curve of the braiding strand. In SolidWorks®, the tool named “intersection curve” is employed to obtain the intersection line of the two surfaces, as shown in Figure 5(c). The intersection curve has all of the characteristics of these two surfaces. Once the braiding curve is obtained, the strand can be realized by sweeping the cross-section along the braiding curve. The cross-section of the strands is not confined by the modeling method, and could be a circle, an ellipse, a polygon, or other closed curve. The braiding angle of the strands depends on the pitch of the helical surface, which can be adjusted conveniently, and if the helical surface is defined with variable pitch, then braided structures with variable pitch can be obtained.

For strands in the same direction, the spatial morphology is exactly the same, except that there is a phase difference, which is $2 π / N$ for a diamond braid with 2N strands, as shown in equation (10). So it is possible to find all the other strands by the circular pattern method. For strands in different directions, the phases of the projected curves are opposite, so the projected curves can be obtained by revolving the projected curve by $π / 2 N$ for a diamond braid with 2N strands. Based on this method, all strands can be obtained and assembled to form a diamond braid, shown in Figure 6(c).
Figure 6.
Diamond braid: (a) strand in anti-clockwise direction; (b) strand in clockwise direction; (c) diamond braid.

Regular braid

Compared with diamond braid, the regular braid and Hercules braid are of longer floating length, so the projected curve can be obtained based on the modified rose curve.

Strands of regular braid in one direction pass over or under two strands in the other direction and then pass under or over two strands. This process is repeated to form the regular braid. So the adjacent strands are a repeat unit and share the space of one strand in the diamond braid. During the modeling process, the regular braid is realized by replacing one strand in the diamond braid by two strands with a long floating length. The initial position of the two strands is on the two sides of the initial position of the diamond braid strand, and the deviation angle is $π / N$ for a regular braid with 2N strands.

In the paper published recently, Kyosev²⁰ pointed out that, in actual braiding practice, the orientation of the strands in regular braid is parallel to the product axis and not perpendicular to the braid axis, which is a common mistake in geometrical modeling of regular braid. This characteristic is reflected in this modeling method so that the adjacent strands in regular braid lie on the same braiding surface, and the difference is the initial position, shown as Figure 7(a).
Figure 7.
The repeating unit and braiding surfaces of (a) regular braid, and (b) Hercules braid.

Hercules braid

The modeling process for Hercules braid is similar with that for regular braid; the difference is that for Hercules braid, the adjacent three strands are a repeating unit. These three strands lie on the same braiding surface, shown as Figure 7(b). Compared to diamond braid, the distribution of these three strands is such that the middle strand is at the same initial position as in diamond braid, and the other two strands have a $π / N$ deviation angle in the two sides of the middle strand, for Hercules braid with 2N strands.

The simulated models of diamond braid, regular braid, and Hercules braid are shown in Figure 8.
Figure 8.
Tubular strand braids models of (a) diamond braid, (b) regular braid, and (c) Hercules braid.

Geometrical modeling for multi-strand braid

Based on single braided structures, it is simple to model multi-strand braid. The specific operation is that one strand in single braid is replaced by several strands that share the space as one strand in single braid. All the replacing strands lie on the same braiding surface as the replaced strand. Figure 9 shows the comparison between simulated rope (a) and real rope (b) with multi-strands. The simulated rope appears very similar in form to the real one in both appearance and interlacing pattern.
Figure 9.
Comparison between simulated rope (a) and real rope (b).

Geometrical modeling of braided structures with tape element

In some cases when braided structures are used as covering or decoration, the braided elements are not strands with circular or elliptical cross-section, but a very thin and wide fiber filaments or tapes. In this case, it is more reasonable to simulate the braids with tape structures.

The tape braid and strand braid are of the same interlacing patterns; the difference is just the braiding elements. In the modeling process of strand braid, the braiding curve is obtained by the intersection of the braiding surface and the helical surface. For tape braid, the braiding surface is the same, while the helical surface is replaced by two helical surfaces. These two helical surfaces pass the braiding axis and have the same path, as shown in Figure 10(a). The tape element is obtained by cutting the braiding surface with two helical surfaces, as shown in Figure 10(b).
Figure 10.
Tape braid: (a) trimming braiding surface; (b) braiding tape in anti-clockwise direction.

The middle line between these two helical surfaces is the braiding curve. The angle between these two helical surfaces determines the width of the tape, and depends on the number of crimps and the braiding pattern. The tape in the other direction can be obtained following the same method as that of strand braided structures. All strand braided structures can be realized as tape structures, following similar methods to strand braids. Common braided structures with tapes are illustrated in Figure 11.
Figure 11.
Tubular tape braids models of (a) diamond braid, (b) regular braid, and (c) Hercules braid.

Figure 12 shows the comparison between the real tape braid and simulated structures, from which it can be seen that the simulated tape braid reflects the appearance and the structure of real braid.
Figure 12.
Comparison between simulated tape rope (a) and real tape rope (b).

Braided structures of revolving body with varying cross-section

Circular braiding is a traditional technique, but with the development of composites, it has been used to produce 3D structures by braiding over complicated mandrels. Among these 3D structures, braided structures of revolving bodies with varying cross-section are the most common. For these kinds of braids, the interlacing patterns of strands in two directions are the same as in common braid, but due to the special morphology of the mandrels, the diameter of the projected curve is changed along the braiding axis, which means that the braiding surface is also changed to match the shape of the mandrels. Thus, braided structures can be obtained by replacing a cylindrical braiding surface with a braiding surface of a similar shape to the mandrel.

Taking the braid for a rotary hyperboloid as an example (shown in Figure 13), the realization of the braiding surface is the key. In SolidWorks®, the braiding surface is generated using the sweep surface tool, while in order to make the diameter of the projected curve change with the surface of mandrel, two generatrices, which are curves that can be used to generate the rotating body by a rotation method, are needed as the guide line. Once the braiding surface is obtained, a similar modeling method is employed to realize braided structures for a rotary hyperboloid, as shown in Figure 14(a) and (b). Based on the same modeling process, more complicated braided structures can be generated both with strands and tapes, as shown in Figure 14(c) and (d).
Figure 13.
Braiding surfaces of a rotary hyperboloid.

Figure 14.
Strand and tape braid models of revolving bodies: (a) strand braid of rotary hyperboloid; (b) tape braid of rotary hyperboloid; (c) strand braid of bottle structure; (d) tape braid of bottle structure.

Discussion

Based on shape similarity, this paper presented the generalized rose curve as the mathematical model of braiding strands, and discussed the influence of relevant parameters on the shape of rose curves. By comparison between braiding track and rose curve, it can be seen that only when R > A does the generalized rose curves have the same shape as the projected curve. A very similar braiding geometry can be obtained by adjusting the relevant parameters of the rose curve. For braids with longer floating length—for instance, the regular and Hercules braids—modified rose curves were developed based on the ellipse. For modified rose curves, the floating length can be adjusted to satisfy the strand shape in regular and Hercules braids.

The rose curve is the mathematical model for this modeling method, and the projected curves are obtained by the mathematical expression of the rose curve, and a modeling method based on intersection of the braiding and the helical surfaces is realized using SolidWorks®. It is necessary to mention that this modeling method is not confined to the software; other similar software, such as UG®, Pro/Engineer®, etc., can also be used to implement the simulation using the same modeling process. If secondary development is conducted based on this modeling method, a plug-in could be developed to realize the fast modeling for braids.

Conclusion

This paper presented a general geometrical modeling method for tubular braids with the generalized rose curves, based on SolidWorks®. This method, taking the generalized rose curve as the mathematical model for braiding strands, is realized by the intersection of the braiding surface and the helical surface. This modeling method can simulate strand and tape braids as diamond, regular, and Hercules braid. Different braiding pattern can be realized by the adjustment of the initial positions and phase differences. The simulated models of both strand and tape braid show good consistency in appearance and interlacing pattern. As well as cylindrical braid, braided structures of revolving bodies with varying cross-section can also be easily realized by modifying the braiding surface.

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 disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the China Scholarship Council (grant number 201200630034).

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