Modeling and analysis of the carrier arrangement in three-dimensional circular braiding

Abstract

In this paper, the carrier arrangement in three-dimensional (3D) circular braiding is studied. To obtain suitable arrangements, the guide grooves of carriers are defined and divided into several types. Restricted conditions of the carrier positions are derived by obeying the characteristics of the carrier motion. Based on the restricted conditions, a mathematical model is established and applied to determine the possible carrier positions. Using the model, reasonable carrier arrangements are discussed. In comparison with practical engineering, the model is proven to be a useful tool for the 3D circular braiding process and helpful for designing a 3D circular braiding machine.

Keywords

three-dimensional circular braiding carrier arrangement modeling analysis braiding process

Three-dimensional (3D) braiding is a complex formation process that interlaces strands of yarn into various shapes that usually produce hollow products. During the process of braiding the yarn, the yarns are not cut off. The interlacing motions and geometry of 3D braiding create products with a relatively complex structure that have an improved mechanical behavior over two-dimensional (2D) circular braiding in the out-of-plane direction, filament winding or laying up. With these advantages, 3D braiding is a good choice for some key structural components, such as automobile parts, aircraft parts and medical materials.

Studies on 3D circular braiding are scarce. Reviews written by Kyosev¹ and Sontag et al.² systematically introduced the braiding technique and 3D braiding, respectively. Other studies focused on some process-related parameters. van Ravenhorst and Akkerman³ investigated the 2D braiding process by proposing an inverse kinematics-based method referring to the braiding angle, which can be applied to simulate the braiding process. Guyader et al.⁴ proposed a model based on the analytical relationships between the process parameters and the geometry of the braid. In the model, they investigated several process parameters, including the kinematic parameters of the mandrel and carriers. Kyosev et al.⁵ simulated the braid structure by yarn paths, geometry of shapes, take down-speed and yarn cross-section parameters. Potluri et al.⁶ investigated a procedure for 2D braiding profiled components with complex shapes. The model focused on the geometrical parameters and the varying horn gear speed. Ravenhorst and Akkerman⁷ proposed a yarn interaction model that could simulate the yarn interaction behavior in 3D circular braiding. Ma et al.⁸ established a mathematical model of the tensioning system of the carriers, illustrating how the tensioning system works.

However, most of the studies focus on the yarn path-related parameters, and provide a limited analysis of the carrier arrangement. Some studies, for example, Ravenhorst and Akkerman,⁹ investigated the carrier occupation. Goseberg et al.¹⁰ and Lepperhoff¹¹ studied the relationship between pattern type and carrier arrangement. A software solution named TexMind Configurator explored by Kyosev¹² emulated the carrier occupation. Although the carrier occupation is influenced by a specific carrier arrangement to the fabric produced, only the two books mentioned above discussed the one-layer carrier arrangement by the observation method. Nevertheless, an increase in the number of braiding layers changes these effects.

In a 3D circular braiding machine, the carrier could travel from one layer to another by connected curves. The different connection modes of the curves lead to different braiding structures. Designers have difficulty finding a reasonable configuration that avoids carrier collision. The carrier arrangement with a single braiding layer is very simple and easy to study, whereas the carrier arrangement of a multi-layer machine is complex, and the observation method does not work effectively. Machine designers have to find a reasonable carrier arrangement by trial and error or practical experience, and they must check whether the arrangement produces the desired fabric by either producing the fabric or using the TexMind Configurator or some other software. As a result, designing a 3D circular braiding machine with more than five braiding layers is difficult. The latest 3D braiding machine has been developed by the August Herzog Company in 2012 from the 3D Braiding machine, which has five braiding layers.^13,14

To the authors’ knowledge, there is no numerical method for arranging carriers. Because carrier arrangement has a great influence on the design of 3D circular braiding, the investigation of carrier arrangement is necessary. This paper proposes a mathematical model for carrier arrangement to help engineers effectively arrange carriers in designing a 3D multi-layer braiding machine.

Mathematical modeling of carrier arrangement

To obtain a better understanding of the differences in braiding machinery, isometric views of single-layer and multi-layer 3D circular braiding machines are shown in Figures 1 and 2. Braiding layers and two to several rows of carriers are fixed on the inner ring surface. Each horn gear has four slots. Carriers on different rows move according to the guide grooves in a circular, interlacing pattern. The circular movement of carriers braids the yarns together at the ring center.

Figure 1.

A three-dimensional circular tri-axial braiding machine with a single layer: (a) global view of a single- layer machine; (b) partial view of a single-layer machine.

Figure 2.

A three-dimensional circular tri-axial braiding machine with multiple layers: (a) global view of a multi-layer machine; (b) partial view of a multi-layer machine.

Carrier arrangement problem going from a single layer to multiple layers

There are two difficulties in the design of a 3D circular tri-axial braiding machine with multiple layers: how to arrange carriers to avoid collision between carriers and how the carriers of different rows interlace.

For the first problem, Kyosev¹ concluded that for every track, the carrier arrangement could be expressed as “1 Full-1 Empty.” This method is applicable in the 3D circular tri-axial braiding machine with one layer, and the carrier arrangement could be easily deduced, as shown in Figure 3(a). However, the carrier arrangement in a multi-layer configuration is difficult to determine. In addition, the difficulty of finding a non-conflicting carrier arrangement greatly increases with an increase in the number of braiding layers.

Figure 3.

Carrier arrangement problem going from a single layer to multiple layers: (a) carrier arrangement of the single layer; (b) carrier conflict in the multi-layer arrangement.

If the single-layer carrier arrangement was employed in a multi-layer configuration directly, carrier conflict will occur. In Figure 3(b), carriers of adjacent rows will conflict with each other after the horn gears rotate a quarter round. Therefore, it is necessary to study the carrier arrangement and find an effective method for solving this problem.

For the second problem, Herzog manufactured maypole-braiding machines with switches and electromechanical drives. This type of machine can braid various types of fabric structures, but it is expensive. Another alternative is to determine the carrier arrangement first, then design the guide grooves and the other components of the machine. This type of machine has fixed guide grooves and can only weave the predefined fabric structure. The advantage is that without switches, the machine could continuously work at a high speed.

Specifically, to interlace carriers from different rows of the multi-layer machine, guide grooves consist of different segments, which differ from that of the single-layer machine. Figure 4 shows the fixed guide grooves for the corresponding carrier arrangement. According to the carrier arrangement, if a carrier needs to travel from one row to another, the corresponding guide groove is designed as a “∞“ shape. Otherwise, the guide groove is designed as an “oo” shape. In this way, the carriers of different rows interlace.

Figure 4.

Fixed guide grooves for corresponding carrier arrangements.

Modeling of paths

The path of each carrier is static and presents a periodic rule. The diagrams in Figure 5 are possible schemes of one path when the number of braiding layers $M = 3$ . In the radial direction, carriers move from one row to another forming a span of braiding layers, which is defined as the parameter s. In the peripheral direction, carriers move back and forth. If one complete back and forth motion is defined as one period z, then one path could consist of many different periods.

Figure 5.

Possible schemes of one path when $M = 3$ in one period.

Theoretically, different periods might have different values of s, which could induce various combinations. However, both the parameters s and z of one path do not vary in practical engineering. This is because if they change, an overlapping error could occur, as depicted in Figure 6. The first row and the fourth row will overlap in the fourth and sixth columns. In this situation, the actual number of paths and braiding layers will be $s < 2 M$ during the over-lapping region, which is not reasonable for path planning. Therefore, to simplify the path planning, the parameters s and z of one path do not vary.

Figure 6.

Overlapping error while s changes during different periods.

In additionally, as the 3D braiding is circular and the paths are arranged end-to–end, the total number of columns N satisfies the following equation¹

N = 2 nLCM (s i), (i = 1, 2, \dots, M)

(1)

where n is a positive integer,

LCM

is the lowest common multiple and s_i is the span of the ith row/path.

To express each path in the mathematical model, the following definitions of two basic units, which are shown in Figure 7, are presented.

Figure 7.

Definitions of basic units x and y.

Thus, every basic unit in each location can be defined as a function $h (u, v, w, ψ)$ . The value u is the unit shape (x or y), v is the order of the row, w is the order of the column and ψ is the direction of the path according to the gear rotation, which is introduced as follows where parameters α and β are opposite in direction.

We define $s i = N / λ i$ , ( $i = 1, 2, \dots, 2 M$ ), where $λ i$ is a positive number (refer to Equation (1)). As a result, the paths can be expressed in a matrix as follows

H = [η ij] 2 M \times N = [\overset{λ 1}{\overset{︷}{H 1 \dots H 1}} \dots \overset{λ i}{\overset{︷}{H i \dots H i}} \dots \overset{λ 2 M}{\overset{︷}{H 2 M \dots H 2 M}}] 2 M \times N, (i = 1, 2, \dots, 2 M, j = 1, 2, \dots, N)

(3)

where

H 1 = [h (x, 1, 1, α) \dots h (x, s 1, s 1, α) h (y, s 1, s 1 + 1, α) \dots h (y, 1, 2 s 1, α)]; H i = [h (f (i, z), m, 1, ψ (i)) \dots h (f (i, z), m + (- 1) i + 1 (s i - 1), s i, ψ (i)) \times h (f (i, z), m + (- 1) i + 1 (s i - 1), s i + 1, ψ (i)) \dots \times h (f (i, z), m, 2 s i, ψ (i))];

H 2 M = [h (y, M, 1, β) \dots h (y, M - (s 2 M - 1), s 2 M, β) h (x, M - (s 2 M - 1), s 2 M + 1, β) \dots h (x, M, 2 s 2 M, β)];

According to the geometrical relationship, the following limits can be derived

1 \leq m + (- 1) i + 1 (s i - 1) \leq 2 M

(4)

Modeling of carrier positions

The position of carriers together with the parameters α and β are defined in Table 1. In addition, the positions satisfy the equations

Table 1.

Definitions of α, β and the position of carriers

Combining Equation (5) with Table 1, a matrix C is employed to express the carrier arrangement of all the paths as

C = [D ij] 2 M \times 3 N, (i = 1, 2, \dots, 2 M, j = 1, 2, \dots, N)

(6)

where the matrix

D ij = [P_{x, y}^{1} (i, j) P_{x, y}^{2} (i, j) P_{x, y}^{3} (i, j)]

represents the configuration status of the carriers corresponding to each of the units of matrix

H

In addition, the possible conflicting relationships between units of different rows and columns are shown in Figure 8.

Figure 8.

Possible conflicting relationships between different units.

Using the above definitions and Equations (2)–(5), the possible conflicting relationships are investigated. $h (u 1, v 1, w 1, ψ 1) h (u 2, v 2, w 2, ψ 2)$ represents the logical relationships between units $h (u 1, v 1, w 1, ψ 1)$ and $h (u 2, v 2, w 2, ψ 2)$ .

For each x shaped unit, the possible conflicting relationships drawn in Figure 8 could be decomposed into the following seven types. For types i, iii, v and vii in Figure 9, the carriers in different units travel in the same direction and there is one conflicting point, which is the present intersection point. For types ii, iv and vi in Figure 9, the carriers in different units travel in the opposite direction and there are three conflicting points, the present intersection point and the future intersection points after the horn gears rotate a quarter/half round at different times.

Figure 9.

Decomposed possible conflicting relationships about unit x.

The above seven cases written in mathematical equations as $A_{ij}^{1} \sim A_{ij}^{7}$ , about unit x are as follows

h (x, i, j, α) h (x, i - 1, j - 1, α) = h (x, i, j, β) h (x, i - 1, j - 1, β) = A_{ij}^{1}

(7a)

h (x, i, j, α) h (y, i - 1, j, α) = A_{ij}^{2}

(7b)

h (x, i, j, α) h (y, i, j - 1, α) = h (x, i, j, β) h (y, i, j - 1, β) = A_{ij}^{3}

(7c)

h (x, i, j, α) h (y, i, j, β) = h (x, i, j, β) h (y, i, j, α) = A_{ij}^{4}

(7d)

h (x, i, j, α) h (y, i, j + 1, α) = h (x, i, j, β) h (y, i, j + 1, β) = A_{ij}^{5}

(7e)

h (x, i, j, β) h (y, i + 1, j, β) = A_{ij}^{6}

(7f)

h (x, i, j, α) h (x, i + 1, j + 1, α) = h (x, i, j, β) h (x, i + 1, j + 1, β) = A_{ij}^{7}

(7g)

where

A_{ij}^{1} = {(P_{x}^{1} (i, j) - 1) (P_{x}^{3} (i - 1, j - 1) - 1) \neq 0}; A_{ij}^{2} = ⋂_{k = 1, 2, 3} {(P_{x}^{k} (i, j) - 1) (P_{y}^{4 - k} (i - 1, j) - 1) \neq 0}; A_{ij}^{3} = {(P_{x}^{1} (i, j) - 1) (P_{y}^{1} (i, j - 1) - 1) \neq 0}; A_{ij}^{4} = ⋂_{k = 1, 2, 3} {(P_{x}^{k} (i, j) - 1) (P_{y}^{k} (i, j) - 1) \neq 0}; A_{ij}^{5} = {(P_{x}^{3} (i, j) - 1) (P_{y}^{3} (i, j + 1) - 1) \neq 0}; A_{ij}^{6} = ⋂_{k = 1, 2, 3} {(P_{x}^{4 - k} (i, j) - 1) (P_{y}^{k} (i + 1, j) - 1) \neq 0}; A_{ij}^{7} = {(P_{x}^{3} (i, j) - 1) (P_{x}^{1} (i + 1, j + 1) - 1) \neq 0}

Similarly, for each y-shaped unit, the decomposed possible conflicting relationships are as shown in Figure 10.

Figure 10.

Decomposed possible conflicting relationships about unit y.

The above seven cases written in mathematical equations $B_{ij}^{1} \sim B_{ij}^{7}$ , about unit y are as follows

h (y, i, j, α) h (y, i - 1, j + 1, α) = h (y, i, j, β) h (y, i - 1, j + 1, β) = B_{ij}^{1}

(8a)

h (y, i, j, β) h (y, i - 1, j, β) = B_{ij}^{2}

(8b)

h (y, i, j, α) h (x, i, j + 1, α) = h (y, i, j, β) h (x, i, j + 1, β) = B_{ij}^{3}

(8c)

h (y, i, j, α) h (x, i, j, β) = h (y, i, j, β) h (x, i, j, α) = B_{ij}^{4}

(8d)

h (y, i, j, α) h (x, i, j - 1, α) = h (y, i, j, β) h (x, i, j - 1, β) = B_{ij}^{5}

(8e)

h (y, i, j, α) h (x, i + 1, j, α) = B_{ij}^{6}

(8f)

h (y, i, j, α) h (y, i + 1, j - 1, α) = h (y, i, j, β) h (y, i + 1, j - 1, β) = B_{ij}^{7}

(8g)

where

B_{ij}^{1} = {(P_{y}^{1} (i, j) - 1) (P_{y}^{3} (i - 1, j + 1) - 1) \neq 0}; B_{ij}^{2} = ⋂_{k = 1, 2, 3} {(P_{y}^{k} (i, j) - 1) (P_{x}^{4 - k} (i - 1, j) - 1) \neq 0}; B_{ij}^{3} = {(P_{y}^{1} (i, j) - 1) (P_{x}^{1} (i, j + 1) - 1) \neq 0}; B_{ij}^{4} = ⋂_{k = 1, 2, 3} {(P_{y}^{k} (i, j) - 1) (P_{x}^{k} (i, j) - 1) \neq 0}; B_{ij}^{5} = {(P_{y}^{3} (i, j) - 1) (P_{x}^{3} (i, j - 1) - 1) \neq 0}; B_{ij}^{6} = ⋂_{k = 1, 2, 3} {(P_{y}^{4 - k} (i, j) - 1) (P_{x}^{k} (i + 1, j) - 1) \neq 0}; B_{ij}^{7} = {(P_{y}^{3} (i, j) - 1) (P_{y}^{1} (i + 1, j - 1) - 1) \neq 0}

Except for the logical relationships shown in Equations (7) and (8), other relationships are defined as the universal set, ${U}$ . From the expressions shown, the logical relationships do not conform to the multiplication exchange law, as shown in Equation (9)

h (u 1, v 1, w 1, ψ 1) h (u 2, v 2, w 2, ψ 2) \neq h (u 2, v 2, w 2, ψ 2) h (u 1, v 1, w 1, ψ 1)

(9)

Considering the paths are end-to-end, a piecewise function of columns is redefined as follows

The above replacement is equivalent to copying the Nth column to the front of the paths and the first column to the back of the paths as supplement conditions. This replacement will make a size change to Equation (3) as follows

Then, applying Equation (10) to Equations (7) and (8), the logical relationships of each unit with end-to-end paths can be derived. Finally, to avoid the interference of the carrier motion at any moment, Equations (7) and (8) are used. The different carrier positions satisfy the conditions as follows

E ij = Π_{k = 1}^{2 M} Π_{l = 0}^{N + 1} (η ij η kl), (i = 1, 2, \dots, 2 M, j = 0, 1, 2, \dots, N, N + 1)

(12)

Substituting the solutions of Equation (12) into Equation (6) and setting the initial conditions, the carrier arrangement of all paths could be expressed by the matrix C.

Numerical results and analysis of the three-dimensional circular braiding process

In the model presented above, the design parameters of the 3D braiding process are M, s and N. To obtain the best mechanical property, every carrier is expected to pass through as many braiding layers as possible. This type of process follows the principle of maximum braiding layers (PMBL). In other words, s_i is assigned as a constant M.

When $M = 1$ , the braiding process is a single-layer braiding process. With the PMBL, the design parameters are $s = [11]$ and $N = 2$ . Thus the paths of Equation (11) are

H = [h (y, 1, 0, α) h (x, 1, 1, α) h (y, 1, 2, α) h (x, 1, 3, α) h (x, 1, 0, β) h (y, 1, 1, β) h (x, 1, 2, β) h (y, 1, 3, β)]

According to Equation (10), the conditions of the carrier positions are

E ij = Π_{k = 1}^{2 M} Π_{l = 0}^{N + 1} (η ij η kl) = (η 11 η 21) (η 21 η 11) (η 12 η 22) (η 22 η 12) (η 13 η 23) (η 23 η 13) (η 14 η 24) (η 24 η 14)

(13a)

with the conditions of

D 11 = D 13; D 21 = D 23

(13b)

Substituting equations (7d and (8d) into Equation (13), it can be derived that

E = A_{11}^{4} \cap A_{12}^{4} \cap B_{11}^{4} \cap B_{12}^{4}

(14)

Providing that $D 1 = [101101]$ at the initial time, the solution to the carrier positions of Equation (6) is

C = [101101010010]

(15)

The above solution is in accordance with the carrier arrangement in a practical braiding process (Figure 1(b)), which is proof of the accuracy of the established model.

Similarly, when M increases, with the initial conditions set as provided, the corresponding carrier arrangement could be calculated by the present method.

However, different initial conditions lead to different solutions. If some carrier positions are spare, there could be various solutions. Similar presentation of single-layer braiding can be found in Kyosev.¹ For this model, when $M = 5$ , there are 700 types of carrier arrangements if the initial conditions are assigned as

D 1 = [100010000001]

(16a)

D 2 = [010000010000]

(16b)

where D₁ and D₂ represent the carrier arrangements of the first and second paths, respectively.

Conclusions

In this paper, a mathematical model of carrier arrangement was presented. By decomposing the paths into basic units, all paths with different braiding layers could be expressed in a matrix. Moreover, based on the definitions of carrier positions, along with the logical relationships of the basic units, the matrix of paths could represent the conditions of corresponding carrier positions. By gathering all the possible non-conflicting carrier positions, the carrier arrangement could be solved.

The well-known one layer braiding process was introduced to investigate the accuracy of the model. The results show that the presented method remained in accordance with practical engineering.

In conclusion, the model is helpful for engineers effectively arranging carriers in a 3D multi-layer braiding machine. This model provides researchers with a numerical method for finding all the possible non-conflicting carrier arrangements. Engineers can then study the potential fabric structures to identify structures of interest by using software or the physical production of the fabric. However, a limit to this model is that the solutions are greatly dependent on the initial conditions. If the initial conditions are not well set and some of the carrier positions are spare, the solutions can vary greatly. Therefore, the next area of research could be to improve the algorithm of the model.

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 National Natural Science Foundation of China (grant number 51475091), the Program of Shanghai Leading Talent (grant number 20141032) and the Donghua University Graduate Student Degree Thesis Innovation Fund Project (grant number CUSF-DH-D-2015099).

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