A yarn hanging model for estimating the applied initial warp tension in the multi-layer weaving process

Abstract

Carbon fiber warp yarns tend to hang due to the gravity in the multi-layer weaving process, which leads to chaotic shedding and impairs fabric quality. The hanging shape of carbon fiber warp yarns is mainly determined by the applied initial warp tension in the weaving process, and excessive warp tension increases the friction between the yarn and the heald frame, resulting in yarn wear. A yarn hanging model based on catenary theory was established to estimate the applied minimum initial warp tension that could ensure clear shedding in the multi-layer weaving process. The relationship between the warp hanging shape and various weaving process parameters (warp tension, yarn specifications and the size of shedding) was obtained. According to the weaving conditions, the applied initial yarn tension could be estimated using the model before manufacturing. Multi-layer yarn hanging experiments were conducted using different specification carbon fibers and yarn tension, and the theoretical predictions and experimental results were compared. The results showed that the yarn hanging model could well simulate the actual hanging characteristics of carbon fiber warp yarn under different tension. The research results provide a tool for estimating the applied initial warp tension in the multi-layer weaving process.

Keywords

Warp tension yarn hanging shedding shape multi-layer weaving process carbon fiber

Textile composites with three-dimensional (3D) reinforced woven fabric have the advantages of high specific strength, high specific modulus and good toughness, which are widely used in aerospace, construction and automotive fields.^1,2 The multi-layer weaving process is often used to prepare various 3D woven performs for carbon fiber-reinforced composites due to the simple, productive efficiency, low cost and design ability.³ The initial warp tension is the tension applied to the warp yarn before the shedding motion in the multi-layer weaving process. The applied initial warp tension is the most important technological parameter, because the initial warp tension is the key to controlling the yarn hanging and yarn wear in the multi-layer weaving process.^4,5

The warp tension determines the shedding shape and the breakage rate of warp yarns. Warp yarns with low tension or high tension lead to defects during the weaving process, especially in multi-layer weaving processes with a large number of warp yarns.⁶ The applied initial warp tension is the basis for the various warp tension changes during weaving. Due to repeated friction of the warp yarns with the heald frame in the 3D woven fabric weaving process, the excessive initial warp tension applied will increase the friction between the warp yarns and the heald frame, which further increases yarn wear.^7,8 In addition, the warp yarn tends to hang in the weaving process when the applied initial warp tension is low, which will cause chaotic shedding and yarn contact friction.⁹ The initial warp tension applied to the warp yarn in the multi-layer weaving process should ensure clear multi-layer shedding while reducing warp yarn wear. Some researchers have studied the relationship between warp tension and fabric structure and mechanical properties.^10–12 However, the initial warp tension applied in the weaving process of 3D woven fabrics is mostly based on practical experience and lacks a relevant calculation process, which makes the test period long and costly.^13,14 Different structures of 3D woven fabrics require different specifications of carbon fibers and weaving parameters. Setting the initial warp tension only by practical experience is poorly flexible.

The hanging shape of carbon fiber warp yarn under yarn tension in the weaving process could be studied based on catenary theory. Carbon fibers are slender and flexible materials, and their hanging shape in the weaving process is mainly determined by yarn tension and gravity.¹⁵ Catenary refers to a kind of curve, which is the curve shape of a flexible and inextensible chain with uniform thickness and mass fixed at both ends under gravity.¹⁶ Wang¹⁷ used catenary theory to calculate the drape of cable with a certain spanning distance and rigidity at minimal maximum tension. Liu and Liu¹⁸ and Zhou and Chen¹⁹ used catenary theory to calculate the maximum hanging for suspension bridges under gravity. Zhang and Yang²⁰ established a model for the variation of conveyor belt tension based on the catenary theory to study the law of conveyor belt tension variation, and proved the validity of the model by examples. Although the above work has studied the drape shape of slender and flexible materials under gravity and tension, it is not completely suitable for the warp yarns in the multi-layer weaving process due to the complex weaving conditions.

The study endeavors to estimate the applied initial warp tension in the multi-layer weaving process. The hanging trajectory of warp yarns in the multi-layer weaving process is catenary, and a yarn hanging model based on catenary theory is established to analyze the relationship between the carbon fiber warp hanging and weaving parameters. The hanging characteristics and shedding shape of carbon fibers under different tensions are simulated using the model. The reasonability of the yarn hanging model will be verified by carbon fiber warp yarn hanging experiments.

Analysis of warp hanging in the multi-layer weaving process

The multi-layer weaving process is obtained by improving the two-dimensional (2D) weaving process. As shown in Figure 1, warp yarns are evenly arranged in the multi-layer weaving process. In the first step, the two groups of warp yarns are interlaced to form multi-layer sheds under the lifting of the heald frame, introducing the weft yarn at the corresponding shedding after the shed is formed. In the second step, the operation of the first step is repeated, but the two groups of warp yarns move in an opposite direction from the first. After repeating the above operations and coordinated deformation, the warp and weft yarns are interwoven to form a 3D woven fabric.

Figure 1.

The multi-layer weaving process.

Carbon fiber warp yarn is a slender and flexible material, if the applied initial warp tension in the multi-layer weaving process is not appropriate, the carbon fiber warp yarn naturally hangs under gravity. As shown in Figure 2, the slack carbon fiber warp yarns form two catenaries between the cloth-fell, the heald frame and the yarn exit, which seriously deviates from the ideal warp yarn path in the multi-layer weaving process, resulting in chaotic shedding and difficulty in weft insertion. The hanging shape of warp yarn is determined by warp tension, and the initial warp tension is the basis for the various warp tension changes in the weaving process. Large hanging carbon fiber warp yarns tend to cause friction with each other in the weaving process, which may cause damage to the carbon fiber surface and should be avoided. It is undesirable to increase blindly the applied initial warp tension to reduce the warp hanging, which increases the friction between the yarn and the heald frame and leads to yarn wear. Therefore, carbon fiber warp yarns should be applied with appropriate initial warp tension to create a clear shedding while reducing yarn wear.

Figure 2.

The yarn trajectory in the multi-layer weaving process.

Mathematical model

Basic assumptions

Yarn stiffness is neglected. The yarn produces no bending moments and only bears the axial tensile load when the yarn hanging is due to gravity.

Yarn density is uniform, twist is ignored.

Yarn hanging model

The trajectory of warp yarns is a space curve in the multi-layer weaving process. In order to facilitate the analysis, the yarn trajectory is projected on the $yoz$ plane. The simplified diagram of the trajectory of warp yarn when the shedding reached maximum is shown in Figure 3(a). The warp yarn ABD is straddled between the cloth-fell point A, the heddle-carrying yarn point B and the yarn exit point D. The point C is the lowest point between the yarn AB segments. Establish the plane right angle coordinate system $yoz$ with A as the coordinate origin, and the coordinates of each point are given as $A (0, 0)$ , $B (y_{B}, z_{B})$ , $C (y_{C}, z_{C})$ and $D (y_{D}, 0)$ .

Figure 3.

(a) Schematic diagram of carbon fiber warp yarn trajectory and (b) The force analysis of the yarn micro-element segment.

The yarn trajectory ABD can be divided into two catenaries, the catenary AB and the catenary BD. $δ$ is the vertical distance from the actual trajectory of the yarn to the ideal yarn trajectory. $α$ and $β$ , respectively, are the angles of inclination of the tangents of yarn segments AB and BD at point B. $f$ is the friction between the yarn and the heddle eye. The force analysis of the yarn micro-element segment $d l$ is shown in Figure 3(b). $T$ and $T + d T$ , respectively, are the tensile forces at the ends of the yarn micro-element $d l$ , $q$ is the mass per unit length of the yarn, and the gravity of the yarn micro-element segment is $qgdl$ in the vertical downward direction. Decompose $T$ into T_y and $T_{z}$ along the $y$ and $z$ axes, and $θ$ as the angle between $T$ and T_y.

The equilibrium relationship between the tensions is as follows:

{\begin{cases} T_{z} + d T_{z} - qgdl - T_{z} = 0 \\ \frac{T_{z}}{T_{y}} = \tan θ \end{cases}

(1)

Simplify the equation (1) to obtain:

{\begin{cases} \frac{d T_{z}}{d l} = q g \\ T_{z} = T_{y} \tan θ = T_{y} \frac{d z}{d y} \end{cases}

(2)

The mass per unit length of carbon fibers is:

q = \frac{N_{tex}}{10^{6}}

(3)

where the unit is kg/m, and

N_{tex}

is the yarn linear density.

Taking the derivative of the system of equations (2) results in:

{\begin{cases} \frac{d T_{z}}{d y} = T_{y} \frac{d^{2} z}{d y^{2}} \\ \frac{d^{2} z}{d y^{2}} = \frac{qgdl}{T_{y} d y} \end{cases}

(4)

Take the yarn micro-element $d l$ , and its corresponding coordinate micro-element is $d y$ and $d z$ , their relationship can be expressed by the Pythagorean theorem as:

d l = \sqrt{d y^{2} + d z^{2}} = \sqrt{1 + {(\frac{d z}{d y})}^{2}} d y

(5)

Combining the system of equations (4) and (5), we can get:

\frac{d^{2} z}{d y^{2}} = z ″ = \frac{q g}{T_{y}} \sqrt{1 + z^{' 2}}

(6)

and integrating equation (6) twice, we can obtain:

z = \frac{T_{y}}{q g} \frac{e^{\frac{q g}{T_{y}} y + C_{1}} - e^{- (\frac{q g}{T_{y}} y + C_{1})}}{2} + C_{2} = \frac{T_{y}}{q g} [\cosh \frac{q g}{T_{y}} (y + C_{1})] + C_{2}

(7)

Equation (7) is the general form of the yarn hanging equation, where $C_{1}$ and $C_{2}$ are constants. For the yarn catenary $A B$ , using the initial conditions $A (0, 0)$ and $B (y_{B}, z_{B})$ can be deduced the hanging equation $z_{1}$ as follows:

z_{1} = \frac{T_{y}}{q g} [\cosh \frac{q g}{T_{y}} (y + y_{C}) - \cosh \frac{q g y_{C}}{T_{y}}]

(8)

where

y_{c}

is the horizontal coordinate of the lowest point C, and it can be expressed as:

y_{c} = \frac{y_{B}}{2} - \frac{T_{y}}{q g} \sinh^{- 1} (\frac{q g z_{B}}{2 T_{y} \sinh \frac{q g y_{B}}{2 T_{y}}})

(9)

Similarly, the hanging equation $z_{2}$ of the $B D$ section of the yarn can be obtained as follows:

z_{2} = \frac{T_{0}}{q g} [\begin{array}{l} \cosh \frac{q g}{T_{0}} (y + \frac{y_{B} + y_{D}}{2} - \frac{T_{0}}{q g} \sinh^{- 1} \\ \frac{z_{B} q g}{2 T_{0} \sinh \frac{q g (y_{B} - y_{D})}{2 T_{0}}}) \\ - \cosh \frac{q g}{T_{0}} (\frac{y_{D} - y_{B}}{2} + \frac{T_{0}}{q g} \sinh^{- 1} \\ \frac{z_{B} q g}{2 T_{0} \sinh \frac{q g (y_{B} - y_{D})}{2 T_{0}}}) \end{array}]

(10)

where

T_{0}

is the horizontal tension on the

B D

section of the warp yarn. And the horizontal tension on the

A B

and

B D

sections of the warp yarn have the following relationship:

T_{0} = T_{y} + f

(11)

\begin{array}{l} {\begin{cases} f = μ (T_{0} | \tan β | + T_{y} | \tan α |) \\ \tan α = z_{1}_{(y_{B})}' = \sinh \frac{q g}{T_{y}} (y_{B} - y_{C}) \\ \tan β = z_{2}_{(y_{B})}' = \sinh \frac{q g}{T_{0}} (y_{B} - \frac{y_{B} + y_{D}}{2} + \frac{T_{0}}{q g} \sinh^{- 1} \\ \frac{z_{B} q g}{2 T_{0} \sinh \frac{q g (y_{B} - y_{D})}{2 T_{0}}}) \end{cases} \end{array}

(12)

when the applied warp tension

T_{0}

and the friction coefficient

μ

between the yarn and the heald frame are small,

f

can be ignored to simplify the calculation, and the horizontal tensions of the

A B

and

B D

sections of the carbon fiber warp is as follows:

T_{0} = T_{y}

(13)

The yarn trajectory is the spatial catenary in the weaving process, and the calculated data points projected on the $yoz$ plane need to be converted into 3D spatial coordinates. Determine the starting point of a section of carbon fiber warp as $A' (x_{1}, y_{1}, z_{1})$ and the ending point as $B' (x_{2}, y_{2}, z_{2})$ . The calculated 2D data points for each yarn in the $yoz$ plane are transformed into 3D spatial homogeneous coordinates of the form $[\begin{matrix} x & y & z & 1 \end{matrix}]$ , where $x = 0$ . Then each 2D data point can be transformed into 3D spatial coordinate points $(x', y', z')$ by using the rotation translation matrix M, and the rotation translation matrix M is as follows:

[\begin{matrix} x' & y' & z' & 1 \end{matrix}] = [\begin{matrix} x & y & z & 1 \end{matrix}] M

(14)

M = [\begin{matrix} 0 & 0 & 0 & 0 \\ \sin θ & \cos θ & 0 & 0 \\ 0 & 0 & 1 & 0 \\ Δ x & Δ y & Δ z & 1 \end{matrix}]

(15)

where

Δ x = x_{1}

Δ y = y_{1}

and

Δ z = z_{1}

, and

θ

can be obtained as:

θ = {\begin{cases} \arctan | \frac{x_{2} - x_{1}}{y_{2} - y_{1}} | x_{2} > x_{1} \\ - \arctan | \frac{x_{2} - x_{1}}{y_{2} - y_{1}} | x_{2} < x_{1} \end{cases}

(16)

Assuming the 450 tex twist-free carbon fiber warp applied 0.01 N and 0.05 N tension in the multi-layer weaving process, and the distance between the eyes is 0.1 m, the distance $L_{1}$ from the cloth-fell to the heald frame is 1 m, and the distance $L_{2}$ from the heald frame to the yarn exit is 1.5 m. The spatial trajectory of the carbon fiber warp yarn applied with tension of 0.05 N when shedding reaches maximum is shown in Figure 4.

Figure 4.

Sketch of the carbon fiber warp yarn hanging path.

The red and blue lines in Figure 4, respectively, represent the two groups of warp yarns in the multi-layer weaving process in Figure 1, and there are five columns of warp yarns in the figure. As shown in Figure 4, each shed is formed by five columns of warp yarns at the same level in the multi-layer weaving process, and the trajectory of each column of warp yarns at the same level are not parallel. When the number of warp yarns (number of columns) in the weaving process is large, the larger the warp hanging, the smaller the shed formed by the final projection, and the more difficult the weft insertion.

The warp yarn trajectory was projected to the $yoz$ plane in order to get a clearer view of the effect of the applied warp tension on the carbon fiber warp hanging and the shedding shape. The trend of the carbon fiber warp yarn hanging height $H$ with the distance $L$ from the cloth-fell to the measurement point when the shedding reaches maximum is shown in Figure 5. Figure 5(a) and (b) shows the hanging path of a single group of multi-layer carbon fiber warp yarns applied a yarn tension of 0.01 N and 0.05 N. Figure 5(c) and (d) shows the shed shape formed by two groups of multi-layered warp yarns with a distance of 0.5 m when applying 0.01 N and 0.05 N yarn tension, respectively.

Figure 5.

The yarn hanging path and shedding shape at different applied tensions: (a) T = 0.01N; (b) T = 0.05N; (c) T = 0.01N; (d) T = 0.05N.

As shown in Figure 5(a) and (b), when the distance $L$ from the cloth-fell to the measurement point is constant, the carbon fiber warp yarns with different applied tension will produce different hanging, and the yarn hanging decreases with the increasing applied yarn tension. The shedding shape is shown in Figure 5(c) and (d), when the carbon fiber warp yarns have a larger hanging, the carbon fiber warp yarns on two groups of multi-eye heddles with a certain distance form a smaller shedding after staggered up and down. The above results are consistent with the hanging characteristic of carbon fiber warp yarns on a multilayer loom described in the literature.²¹

The greater the applied yarn tension is, the closer the carbon fiber trajectory is to the ideal trajectory in the shedding motion, and the clearer the shed is, as shown in Figure 5. However, the partial fiber will break due to the large shedding displacement and repeated yarn friction in the fabricating process when the applied yarn is too large.⁸ The yarn hanging model could estimate the applied yarn tension based on the weaving conditions and the path of yarn hanging required for clear shedding. The yarn tension estimated by the model can ensure that the shedding is clear and reduce yarn wear as much as possible, as the tension is the minimum necessary for a clear shedding. The yarn hanging model provides an effective tool for selecting loom tension.

Experiment

Experimental materials

To verify the validity of the yarn hanging model, carbon fiber warp yarn hanging experiments were conducted using HTA-6K and HTA-12K carbon fibers produced by Toho Tenax. The carbon fiber parameters are shown in Table 1.

Table 1.

The carbon fiber parameters

Specification	Twist	No. of monofilaments	Monofilament diameter (μm)	Linear density (tex)	Tensile modulus (GPa)	Tensile strength (MPa)
HTA-6K	Untwist	6000	7.0	400	235	3920
HTA-12K	Untwist	12,000	7.0	800	235	3920

Experimental equipment and methods

The schematic diagram of the multi-layer weaving process is shown in Figure 6. In order to simulate the hanging path of carbon fiber warp yarns in the multi-layer weaving process, experimental equipment was built as shown in Figure 7. In the equipment, carbon fiber warp yarns are pressed on a liftable test stand by a compression device to simulate cloth-fell, and the other end is held in place by a layering frame, and the yarn ends were hung with weights to simulate the yarn tension control device. The height of the cloth-fell, the distance $L$ between the cloth-fell and the layering frame and the yarn layer spacing can be changed, and the weight of the yarn end can also be changed to simulate yarn hanging at different applied warp tensions.

Figure 6.

Sketch of the multi-layer weaving process.

Figure 7.

Experimental equipment.

In the carbon fiber warp hanging experiments, carbon fiber warp yarns of different layers were randomly selected to measure the hanging height in each hanging experiment. After setting the distance L from the layering frame to the cloth-fell, the yarn layer spacing and the yarn tension (provided by the weights), the hanging height of the warp yarn was measured and recorded every 0.1 m along the cloth-fell to the layering frame with a ruler, and each measurement point was measured five times and averaged.

Results and discussion

The carbon fiber warp yarns of HTA-6K and HTA-12K were applied 0.02 N and 0.05 N tensions in the experiments. Tables 2 –5 show a part of the hanging height of carbon fiber warp yarns obtained through theoretical calculation and experiment.

Table 2.

Experimental and theoretical hanging height of HTA-6K with yarn tension T 0.02 N applied when L is 1.0 m

Measuring position (L/m)	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
Experiment result (H/m)	0.020	0.042	0.065	0.092	0.120	0.151	0.181	0.215	0.251	0.3
Theoretical calculation result (H/m)	0.021	0.043	0.067	0.094	0.122	0.153	0.185	0.220	0.256	0.3
Error	4.5%	2.3%	3%	1.6%	1.6%	1.3%	2.2%	2.3%	1.5%	0

Table 3.

Experimental and theoretical hanging height of HTA-6K with yarn tension T 0.05 N applied when L is 1.0 m

Measuring position (L/m)	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
Experiment result (H/m)	0.025	0.054	0.077	0.106	0.132	0.161	0.190	0.225	0.255	0.3
Theoretical calculation result (H/m)	0.026	0.052	0.080	0.108	0.137	0.167	0.197	0.229	0.261	0.3
Error	4%	3.7%	3.8%	1.8%	3.7%	3.7%	3.6%	1.7%	2.3%	0

Table 4.

Experimental and theoretical hanging height of HTA-12K with yarn tension T 0.02 N applied when L is 1.0 m

Measuring position (L/m)	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
Experiment result (H/m)	0.011	0.025	0.045	0.071	0.092	0.125	0.160	0.195	0.242	0.3
Theoretical calculation result (H/m)	0.012	0.027	0.046	0.069	0.096	0.128	0.163	0.203	0.247	0.3
Error	9.1%	8%	2.2%	2.8%	4.3%	2.4%	1.8%	4.1%	2.1%	0

Table 5.

Experimental and theoretical hanging height of HTA-12K with yarn tension T 0.05 N applied when L is 1.0 m

Measuring position (L/m)	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
Experiment result (H/m)	0.023	0.045	0.070	0.097	0.124	0.155	0.182	0.219	0.252	0.3
Theoretical calculation result (H/m)	0.022	0.046	0.072	0.099	0.127	0.157	0.189	0.223	0.258	0.3
Error	4.3%	2.2%	2.8%	2%	2.4%	1.3%	3.8%	2.7%	2.4%	0

As can be seen from Tables 2 –5, the error of the hanging height of carbon fiber warp yarns obtained through theoretical calculation and experiment is basically less than 5%. In order to increase the reliability of the comparison results, eight carbon fiber warp yarn hanging experiments were conducted using different experimental conditions, and two carbon fiber warp yarns were randomly selected as the measurement objects in each experiment. Figure 8 shows the hanging shape of all the carbon fiber warp yarns simulated by the theoretical model and the curve fitted to the experimental data measured by the experiment projected on the $yoz$ plane. In Figure 8(a) and (b), respectively, is shown the theoretical and actual hanging shape of HTA-6K carbon fiber warp yarns with applied tension of 0.02 N and 0.05 N at the distance of the cloth-fell from the heald frame $L = 0.8 m$ and $L = 1.0 m$ . Figure 8(c) and (d), respectively, show the theoretical and actual hanging shape of HTA-12K carbon fiber warp yarns with applied tension of 0.02 N and 0.05 N at the distance of the cloth-fell from the heald frame $L = 0.8 m$ and $L = 1.0 m$ .

Figure 8.

Theoretical and actual hanging shape of carbon fiber warp yarns: (a) HTA-6K carbon fibers applied 0.02 N tension; (b) HTA-6K carbon fibers applied 0.05 N tension; (c) HTA-12K carbon fibers applied 0.02 N tension; (d) HTA-12K carbon fibers applied 0.05 N tension.

It can be concluded from the comparison of the theoretical hanging curve and the actual hanging curve of carbon fiber yarn under each weaving condition in Figure 8 that the yarn hanging model could simulate the hanging characteristics of carbon fiber warp yarns relatively well. However, the warp yarn is not hanging statically in the weaving process, rather it moves up and down together with the heald frame. The warp tension also changes dynamically in the weaving process and increases significantly during the shedding movement and the beating-up.¹⁴ It can be seen from the yarn hanging model that the higher the warp tension, the smaller the hanging of the warp yarn. The initial warp tension is the basis for the various warp tension changes in the weaving process, which is smaller than the warp tension in the shedding motion. Therefore, the carbon fiber warp hanging during the shedding motion is smaller than the static hanging with the initial tension applied. If yarn wear is within the acceptable range, it is feasible to use the yarn tension calculated from the yarn hanging model as the initial warp tension according to the carbon fiber warp hanging required for clear shedding. In future work, we will try to establish a yarn dynamic hanging model to analyze the influence of heald frame motion on the yarn hanging and the applied initial warp tension.

Conclusions

The initial warp tension is the key to controlling warp hanging and warp wear in the multi-layer weaving process. In this research, a yarn hanging model was developed based on catenary theory that could estimate the applied minimum initial warp tension according to weaving conditions, which could greatly reduce the test cycle and cost of setting the initial warp tension in the weaving process. The warp hanging and shedding shapes under different yarn tension were simulated using the yarn hanging model, and the result showed that the warp hanging increases with the increase of yarn density and decreases with the increase of the initial warp tension. The yarn hanging experiment was carried out with different specification carbon fibers. The experimental results showed that the error between the predicted warp hanging height and the experimental results under different tension is basically less than 5%. The reasonability of the yarn hanging model was verified by experiment. The research results provide a tool for selecting the applied initial warp tension in the multi-layer weaving process.

Footnotes

Declaration of conflicting Interests

The author(s) declared that they do not have any commercial or associative interests that represent a conflict of interest in connection with the work submitted.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Wensuo Ma

References

Mathes

The composites industry: plenty of opportunities in heterogeneous market. Reinforced Plastics 2018; 62: 44–51.

Bilisik

Multiaxis three-dimensional weaving for composites: a review. Text Res J 2012; 82: 725–743.

Gao

Chen

A review of multi-scale numerical modeling of three-dimensional woven fabric. Composit Struct 2021; 263: 113685.1–113685.10.

Schindler

Bauder

Wolfrum

, et al. Engineering of three-dimensional near-net-shape weave structures for high technical performance in carbon fibre-reinforced plastics. J Eng Fibers Fabrics 2019; 14: 1–16.

Tourlonias

Bueno

M-A.

Experimental simulation of friction and wear of carbon yarns during the weaving process. Composit Part A Appl Sci Manufact 2016; 80A: 228–236.

Kim

HA.

Effect of fabric structural parameters and weaving conditions to warp tension of aramid fabrics for protective garments. Text Res J 2018; 88: 987–1001.

Wang

Boussu

, et al. Investigation of the strength loss of HMWPE yarns during manufacturing process of 3D warp interlock fabrics. Appl Composit Mater 2021; 29: 357–371.

Leng

Huang

Heald frame motion trajectory based on minimum warp friction in the shedding process of three-dimensional woven fabrics. Text Res J 2022; 92: 2424–2432.

Gong

Shilin

, et al. Characteristics of warp yarn hanging path under weaving tension. J Text Res 2016; 37: 32–36 [in Chinese].

10.

de Vasconcelos

Marcicano

JPP

Sanches

. Influence of yarn tension variations before the positive feed on the characteristics of knitted fabrics. Text Res J 2015; 85: 1864–1871.

11.

Ying

, et al. Effect of the frictional coefficient and pre-tension on stress distribution of fabric during the weaving process. Text Res J 2018; 88: 904–912.

12.

Süle

Investigation of bending and drape properties of woven fabrics and the effects of fabric constructional parameters and warp tension on these properties. Text Res J 2012; 82: 810–819.

13.

Mebrate

Gessesse

Zinabu

Effect of loom tension on mechanical properties of plain woven cotton fabric. J Nat Fibers 2020; 2: 1–6.

14.

Yao

Adjustment method of loom tension for mini-jacquard fabric using double beams. Cotton Text Technol 2016; 44: 78–80 [in Chinese].

15.

Lee

Effect of weaving damage on the tensile properties of three-dimensional woven composites. Composit Struct 2002; 57: 405–413.

16.

Behroozi

A fresh look at the catenary. Eur J Phys: A journal of the European Physical Society 2014; 35: 55007-1–55007-11.

17.

Wang

CY.

The optimum elastica catenary cable. Structures 2020; 28: 878–880.

18.

Liu

Design of hanger tension and cable configuration for self-anchored suspension bridges. China Civil Engng J 2008; 41: 79–83 [in Chinese].

19.

Zhou

Chen

Iterative nonlinear cable shape and force finding technique of suspension bridges using elastic catenary configuration. J Engng Mechan 2019; 145: 5.

20.

Zhang

Yang

Research of variation characteristic of conveyor belt of vehicle group chain conveyor based on elastic catenary theory. J Mech Transmis 2020; 44: 34–38, 45 [in Chinese].

21.

Yang

Detection method of warp tension in multilayer loom for carbon fiber fabric. J Text Res 2016; 37: 137–141 [in Chinese].