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Abstract

Braiding carriers, which are the important parts of a braiding machine, have the functions such as carrying braiding materials, controlling tension of carbon fiber, and driving carbon fiber movement. During the braiding process, two groups of carbon fibers braided in clockwise and counter clockwise direction contact each other and form relative motion, which causes friction and fuzzing. In order to improve this situation, the structural parameters of the carriers need to be optimized. In this paper, the kinematics and dynamics models were established based on the structure of braiding carriers. The micro-element method was used to analyze the relationship between the fiber length released from the yarn barrel, the rotation angle of the lever, and the tension of the carbon fiber. To limit the fluctuant range of carbon fiber tension, and to alleviate the fluffing phenomenon caused by the two groups of carbon fiber in contact with each other, antlion algorithm was used to optimize the structural parameters of braiding carriers. The simulation results showed that the tension of the carbon fiber can meet the processing requirements by adjusting the starting angle of each stage of carrier, the length of lever, the elastic coefficient of springs, and pre-compression of springs. It can be known that the structural parameters of braiding carriers optimized by antlion algorithm could meet the requirement of carbon fiber tension.

Keywords

Parameter optimization antlion algorithm braiding carrier dynamic model

Introduction

Braiding is a traditional processing method. To obtain a thicker, wider, and stronger product or to cover the outline (over-braiding), three or more diagonal strings are interlaced to the product axis (parallel to the longest dimension of the resulting product) [1]. The products obtained by braiding are more resistant to impact, delamination, and creep than products obtained by other processing methods. Moreover, braiding has good copying effect, which means that the necessary preform can be obtained by adjusting the shape of the mandrel.

With the development of composite materials, the combination of braiding process and carbon fiber can obtain carbon fiber preforms with corrosion resistance, high temperature resistance, high strength, high modulus, and light weight. Then the preforms can be enhanced by resin transfer molding (RTM) [1,2], vacuum-assisted resin transfer molding (VARTM) [3,4], or resin film infusion (RFI) [5] techniques to obtain the final machined parts. Through the prior art, carbon fiber can be used for braiding to obtain T-beams, I-beams, box beams, cross beams, F-beams, cylinders, conical shells, circular/rectangular/variable cross-section, and other shapes of prefabricated parts. The reinforced braiding preforms can be applied in various industrial fields such as plates, beams, shafts, aircraft propeller blades [6], automobile bumpers [7,8], artificial bones [9,10], baseball bats [11,12], bicycles [13], and other structural parts. Military aircraft manufactured by carbon fiber braiding preforms can meet the requirements of appearance, light weight, and preventing radar detection [14]; and it is seen that missile warheads manufactured by carbon fiber braiding preforms have better external shape and chemical stability. Due to the broad application prospects of carbon fiber preforms, many scholars have conducted related research on the braiding process.

Braiding carriers have many functions such as reserving carbon fiber, releasing carbon fiber, adjusting the length, and controlling tension. During the process of braiding, braiding carriers compensate for the length of carbon fiber to ensure the stability of the carbon fiber tension and meet the processing requirements. Carbon fiber tension plays an important role in the quality of the braiding products. If the carbon fiber tension is too large, it will cause carbon fiber damage or even fracture. If the carbon fiber fluctuates too much, the structure of the braiding products will be uneven and the mechanical properties will become poor. Based on the importance of carriers, Ma et al. [15] and Kyosev [16,17] have conducted studies on certain types of carriers.

As the carriers move along the chassis track during the braiding process, the length of the catenary (from the exit point of carriers to the braiding point) continuously changes, which makes the tension of carbon fiber fluctuate. In order to obtain a braiding product, which has more uniform structure and better mechanical properties, the structural parameters of carriers with double springs-lever are optimized. The optimized carriers can control the tension of carbon fiber within a specified range, and the tension can be adjusted according to different braiding materials. The results can provide a reasonable reference for the optimal design of carriers in the future.

Carrier modeling

Braiding process and carrier structure

Braiding process

The chassis of the horizontal inner braiding machine has 72 horn gears. A horn gear has four slots. The carriers are installed as 1F1E method (a gap is set between two adjacent carriers, which move in the same direction). About 144 carriers are driven during braiding. According to the movement direction of carriers, the carriers can be divided into group $CA$ (counterclockwise direction) and group $CB$ (clockwise direction). There are 72 carriers in group $CA$ and group $CB$ , respectively. As shown in Figure 1, the carriers in $CA$ group move counterclockwise, and the carriers in $CB$ group move clockwise. Thereby, the carbon fiber released by two groups of carriers are mutually overlapped and wound on the surface of the mandrel.

Figure 1.

Braiding process.

Figure 2.

Release speed of carriers.

Assuming that the braiding task is a cylindrical braided tube, if the radius of the braiding mandrel and the width of the carbon fiber can be known, then equation (1) can be used to calculate the braiding angle at 100% coverage. For the ease of reading and understanding, the notation is attached at the end of the article.

2 π r_{0} = 72 \frac{d_{c}}{cos α}

(1)

The radius of the braiding chassis can be got when the radius of the horn gears is known

R = \frac{r}{tan (\frac{360^{\circ}}{72 \times 2})}

(2)

The distance (H) between the braiding point and the center of the chassis can be obtained, based on the radius of braiding chassis and the braiding angle.

H = \frac{R}{tan α}

(3)

In order to ensure that the carbon fiber is evenly wound along the braiding angle, there is a ratio between the moving speed of the mandrel and the angular velocity of the horn gears. The speed of the braiding mandrel is obtained by the following equation

tan α = \frac{2 π r_{0} w_{c}}{72 π v_{0}}

(4)

Using equations (1) to (4), the length of the catenary between the braiding point and exit point of the carrier can be evaluated as

\begin{matrix} {\begin{matrix} l_{d A 1} = \sqrt{R^{2} + {[r cos (w_{c} t)]}^{2} - r_{0}^{2} + {[H - r sin (w_{c} t)]}^{2}} \\ l_{d A 2} = \sqrt{R^{2} + {[r cos (- w_{c} t)]}^{2} - r_{0}^{2} + {[H - r sin (- w_{c} t)]}^{2}} \\ l_{dB 1} = \sqrt{R^{2} + {[r cos (\frac{π}{2} - w_{c} t)]}^{2} - r_{0}^{2} + {[H - r sin (\frac{π}{2} - w_{c} t)]}^{2}} \\ l_{dB 2} = \sqrt{R^{2} + {[r cos (w_{c} t - \frac{π}{2})]}^{2} - r_{0}^{2} + {[H - r sin (w_{c} t - \frac{π}{2})]}^{2}} \end{matrix} \end{matrix}

(5)

The derivative of the catenary length of $C A_{1}$ , $C A_{2}$ , $C B_{1}$ , and $C B_{2}$ versus time can be obtained by the following equation

\begin{matrix} {\begin{matrix} v_{d A 1} = \frac{d l_{d A 1}}{dt} = \frac{- w_{c} r^{2} sin (w_{c} t) cos (w_{c} t) - w_{c} r cos (w_{c} t) [H - r sin (w_{c} t)]}{\sqrt{R^{2} + [r cos (w_{c} t)] 2 - r_{0}^{2} + [H - r sin (w_{c} t)] 2}} \\ v_{d A 2} = \frac{d l_{d A 2}}{dt} = \frac{w_{c} r^{2} sin (- w_{c} t) cos (- w_{c} t) + w_{c} r cos (- w_{c} t) [H - r sin (- w_{c} t)]}{\sqrt{R^{2} + [r cos (- w_{c} t)] 2 - r_{0}^{2} + [H - r sin (- w_{c} t)] 2}} \\ v_{dB 1} = \frac{d l_{dB 1}}{dt} = \frac{w_{c} r^{2} sin (\frac{π}{2} - w_{c} t) cos (\frac{π}{2} - w_{c} t) + w_{c} r cos (\frac{π}{2} - w_{c} t) [H - r sin (\frac{π}{2} - w_{c} t)]}{\sqrt{R^{2} + [r cos (\frac{π}{2} - w_{c} t)] 2 - r_{0}^{2} + [H - r sin (\frac{π}{2} - w_{c} t)] 2}} \\ v_{dB 2} = \frac{d l_{dB 2}}{dt} = \frac{- w_{c} r^{2} sin (w_{c} t - \frac{π}{2}) cos (w_{c} t - \frac{π}{2}) - w_{c} r cos (w_{c} t - \frac{π}{2}) [H - r sin (w_{c} t - \frac{π}{2})]}{\sqrt{R^{2} + [r cos (w_{c} t - \frac{π}{2})] 2 - r_{0}^{2} + [H - r sin (w_{c} t - \frac{π}{2})] 2}} \end{matrix} \end{matrix}

(6)

The speed of carbon fiber wound on the surface of mandrel can be calculated by the movement speed of the braiding mandrel based on equation (4)

v_{f} = \frac{dlf}{dt} = \frac{v_{0}}{cos α}

(7)

Combining equations (6) and (7), the carbon fiber release speed of carriers is known during the braiding process

v_{l} = v_{f} + v_{d}

(8)

Carrier structure

As shown in Figure 3, the carbon fiber is unwound from the spool, then through the tube, pulley 1, pulley 2, and the exit point successively. When the speed at the exit point is positive (carrier needs release carbon fiber), the carbon fiber pulls up the right end of the lever and the springs are compressed. If the rotation angle of the lever is between −30 ° and $0^{\circ}$ , the lever will only compress spring 1. If the rotation angle of the lever is more than $0^{\circ}$ , the lever will compress spring 1 and spring 2 at the same time. When the rotation angle of the lever exceeds $20^{\circ}$ , the left end of the lever depresses rod 2 so that the stop pin is pulled into the carrier body. At this moment, the spool loses the constraint of the stop pin. The spool spins to release the carbon fiber due to the tension of the carbon fiber. When the speed at the exit point is negative (carrier needs retract carbon fiber), the length of the carbon fiber increases between pulley 1 and the exit point. The clockwise rotation of the lever causes the compression length of the springs to decrease, and then the carbon fiber tension decreases.

Figure 3.

Carrier structure.

In order to analyze the tension change of the carbon fiber, according to the carrier structure, the path diagram of carbon fiber is shown in Figure 4. Point A is the unwinding point of the carbon fiber; $B_{1} B_{2}$ is the arc of carbon fiber wound on the tube; $C_{1} C_{2}$ is the arc of carbon fiber wound on pulley 1; $D_{1} D_{2}$ is the arc of carbon fiber wound on pulley 2; point E is the exit point of the carrier; point $O_{1}$ is the center of the bottom of the spool; point $O_{2}$ is the center of the bottom of the tube; point $O_{3}$ is the center of pulley 1; point $O_{4}$ is the center of pulley 2.

Figure 4.

Path diagram of carbon fiber: (a) diagram of carrier; (b) plane projection of the carbon fiber path; (c) top view of the carbon fiber path.

Kinematic model

A kinematic model of carrier is established based on Figure 4. In order to facilitate the subsequent calculation, the carbon fiber is divided into $l_{1}$ and $l_{2}$ between the unwinding point A and the exit point E, and the length of carbon fiber released from the spool is set as $l_{s}$ . The rotation angle of the lever is $θ_{1}$ in the initial state. When the rotation angle of the lever is $θ_{2}$ , the lever starts to be subjected to spring 1 and spring 2. When the rotation angle of the lever is $θ_{3}$ , the stop pin is retracted into the carrier body. When the rotation angle of the lever is between $θ_{1}$ and $θ_{3}$ , the length of $l_{1}$ does not change and the length of $l_{2}$ changes. When the rotation angle of the lever is greater than $θ_{3}$ , the spool releases the carbon fiber, and the length of $l_{1}$ , $l_{2}$ , and $l_{s}$ changes. Therefore, the working process of the carrier is divided into two stages, namely the regulating stage and the releasing stage. The spool is stationary at the regulating stage. The spool releases the carbon fiber at the releasing stage.

l_{1} = l_{AB 1} + l_{B 1 B 2} + l_{B 2 C 1} + l_{C 1 C 2}

(9)

l_{2} = l_{C 2 D 1} + l_{D 1 D 2} + l_{D 2 E}

(10)

l_{s} = \frac{r_{f} \cdot β}{sin γ}

(11)

Modeling at regulating stage

The length of $l_{2}$ can be calculated based on the carrier geometry

l_{2} = \sqrt{K_{1} + K_{2} sin (θ + K_{3})} + π r_{l} + \sqrt{K_{4} + K_{5} sin (θ + K_{6})}

(12)

where

\begin{matrix} {\begin{matrix} K_{1} = {(x_{O 3} - x_{O 4})}^{2} + l_{r}^{2} + y_{O 3}^{2} + z_{O 3}^{2} - 2 r_{l}^{2} \\ K_{2} = - 2 l_{r} \sqrt{(y_{O 3}^{2} + z_{O 3}^{2})} \\ K_{3} = \arcsin \frac{y_{O 3}^{2}}{\sqrt{(y_{O 3}^{2} + z_{O 3}^{2})}} \\ K_{4} = {(x_{E} - x_{O 4})}^{2} + l_{r}^{2} + y_{E}^{2} + z_{E}^{2} - r_{l}^{2} \\ K_{5} = - 2 l_{r} \sqrt{(y_{E}^{2} + z_{E}^{2})} \\ K_{6} = \arcsin \frac{y_{E}^{2}}{\sqrt{(y_{E}^{2} + z_{E}^{2})}} \end{matrix} \end{matrix}

(13)

The reduction of $l_{2}$ is equivalent to the length of carbon fiber released from the exit point E at the regulating stage

l_{2} (t) = l_{2} (t - dt) - v_{l} \cdot dt

(14)

Modeling at releasing stage

The horizontal distance of $O_{1} O_{2}$ and $O_{2} O_{3}$ can be calculated based on the coordinates of $O_{1}$ , $O_{2}$ , and $O_{3}$

p_{O 1 O 2} = \sqrt{{(x_{O 1} - x_{O 2})}^{2} + {(y_{O 1} - y_{O 2})}^{2}}

(15)

p_{O 2 O 3} = \sqrt{{(x_{O 2} - x_{O 3})}^{2} + {(y_{O 2} - y_{O 3})}^{2}}

(16)

Carbon fiber is wrapped around the tube to form an angle. $∠ B_{1} O_{2} B_{2}$ is the projection angle of this angle on the horizontal plane. $∠ B_{1} O_{2} B_{2}$ can be calculated based on $p_{O 1 O 2}$ and $p_{O 2 O 3}$ .

\begin{matrix} ∠ B_{1} O_{2} B_{2} = 2 π - \arctan \frac{y_{O 2} - y_{O 1}}{x_{O 2} - x_{O 1}} \\ - \arccos \frac{r_{al}}{p_{O 2 O 3}} - \arctan \frac{y_{O 3} - y_{O 2}}{x_{O 3} - x_{O 2}} - \arccos \frac{r_{al} + r_{f}}{p_{O 1 O 2}} \end{matrix}

(17)

Point N is the foot of point A in the horizontal plane of $O_{3}$ . The length of $l_{AN}$ can be obtained. $l_{AN}$ is the altitude difference from point A to point $O_{3}$ .

l_{AN} = | z_{A} - z_{O 3} |

(18)

Equations (15) to (17) are substituted into equation (19) to obtain the length of $l_{NO 3}$ .

\begin{matrix} l_{NO 3} = \sqrt{(\frac{p_{O 1 O 2} \cdot r_{f}}{r_{f} + r_{al}}) 2 - r_{f}^{2}} + \sqrt{(\frac{p_{O 1 O 2} \cdot r_{al}}{r_{f} + r_{al}}) 2 - r_{al}^{2}} \\ + r_{al} ∠ B_{1} O_{2} B_{2} + \sqrt{p_{O 2 O 3}^{2} - r_{al}^{2}} \end{matrix}

(19)

According to calculations, the height of point A has an effect on the calculation result of $∠ C_{1} O_{3} C_{2}$ , as shown in the following equation

\begin{matrix} {\begin{matrix} ∠ C_{1} O_{3} C_{2} = \frac{π}{2} - \arcsin \frac{l_{AN}}{\sqrt{l_{AN}^{2} + l_{NO 3}^{2} - r_{l}^{2}}} & z_{A} \geq z_{O 3} + r_{l} \\ ∠ C_{1} O_{3} C_{2} = \frac{3 π}{2} - \arctan \frac{l_{NO 3}}{l_{AN}} - \arccos \frac{r_{l}}{\sqrt{l_{AN}^{2} + l_{NO 3}^{2} - r_{l}^{2}}} & z_{A} < z_{O 3} + r_{l} \end{matrix} \end{matrix}

(20)

To sum up, the length of $l_{1}$ is as follows

l_{1} = \sqrt{l_{AN}^{2} + l_{NO 3}^{2} - r_{l}^{2}} + r_{l} ∠ C_{1} O_{3} C_{2}

(21)

At this stage, the spool starts to release the carbon fiber due to the tension of the carbon fiber. The unwinding length of carbon fiber can be calculated according to equation (11). The analysis shows that the length changes of $l_{1}$ , $l_{2}$ , and $l_{s}$ should be equal to the release length of the carbon fiber at the exit point at the releasing stage.

l_{s} (t) - l_{1} (t) - l_{2} (t) = v_{l} \cdot dt + l_{s} (t - dt) - l_{1} (t - dt) - l_{2} (t - dt)

(22)

Dynamic model

The tension of carbon fiber is analyzed according to the working principle of the carrier. For ease of calculation, the dynamic model is built according to the working stage of the carrier.

Tension at the regulating stage

The spool is restrained by the stop pin and remains stationary at the regulating stage. The demand is met by adjusting the length of $l_{2}$ at the exit point E. When the release speed of the carbon fiber at the exit point is known, the rotation angle of the lever can be calculated by using equations (12) to (14) at any time of the regulating stage, and the values of spring force 1 and spring force 2 can be further obtained. Finally, the tension of the carbon fiber is determined at the regulating stage.

According to the spatial positions of spring 1 and spring 2, the value of spring force 1 and spring force 2 can be calculated

\begin{matrix} {\begin{matrix} F_{s 1} = k_{s 1} [d_{p 1} + s_{1} (tan θ - tan θ_{1})] if θ > θ_{1} \\ F_{s 1} = 0 if θ \leq θ_{1} \end{matrix} \end{matrix}

(23)

\begin{matrix} {\begin{matrix} F_{s 2} = k_{s 2} [d_{p 2} + s_{2} (tan θ - tan θ_{2})] if θ > θ_{2} \\ F_{s 2} = 0 if θ \leq θ_{2} \end{matrix} \end{matrix}

(24)

According to equations (23) and (24), when θ is greater than $θ_{1}$ and less than $θ_{2}$ , the lever is only subjected to spring force 1, and this state is called the small tension state. When θ is greater than $θ_{2}$ , the lever is subjected to spring force1 and spring force 2, and this stage is called the large tension stage.

According to the force analysis of the lever, the torque of the lever is 0 at equilibrium stage. Since pulley 2 is a movable pulley, the tension of the carbon fiber on the left of pulley 2 should be equal to the tension on the right side of pulley 2. The torque equation of the lever is established to obtain the tension of carbon fiber.

F_{T} = \frac{F_{s 1} \cdot s_{1} + F_{s 2} \cdot s_{2}}{2 l_{r} cos θ}

(25)

Tension at the releasing stage

When the rotation angle of the lever is greater than $θ_{3}$ , the stop pin is retracted into the carrier body, and the spool starts to release the carbon fiber. Since the tube and pulley 1 are fixed pulleys, the tension of the carbon fiber acting on the spool is equal to the tension of the carbon fiber in the $C_{2} D_{1}$ and $D_{2} E$ sections. According to the inertia theorem, the angular acceleration of the spool can be calculated, thereby obtaining the rotation angle of the spool

F_{T} (t) \cdot r_{f} = J \overset{··}{β} (t)

(26)

where J is the rotational inertia of the spool.

J = \frac{m_{f} r_{f}^{2}}{2}

(27)

The acceleration obtained from equation (26) can be used to determine the rotational speed and angle of the spool

\overset{\cdot}{β} (t) = \overset{\cdot}{β} (t - 1) + \overset{··}{β} Δ t

(28)

β (t) = \overset{\cdot}{β} (t) Δ t + \frac{\overset{··}{β}}{2} Δ t^{2}

(29)

According to equation (29), the rotation angle of the spool is known at any time of the releasing stage, so the length of carbon fiber released from the spool in unit time can be obtained

d_{ls} = \frac{r_{f} \cdot [β (t) - β (t - 1)]}{sin γ}

(30)

Antlion optimization algorithm

The antlion algorithm [18,19] was a bionic algorithm proposed by Australian scholar Seyedali Mirjalili in 2015. The antlion algorithm is mainly to simulate the foraging behavior of antlions. The antlion algorithm requires initialization prior to the calculation so that each ant and each antlion get a position within a limited range. The initial positions of antlions and ants are substituted into the fitness function to obtain the fitness value, and the fittest antlion is set as the elite. In order to obtain a global optimal solution, each ant walks randomly to form a new generation of ants within a limited range. Then the positions of the new generation of ants are substituted into the fitness function to obtain the fitness value of the new generation of ants. If the fitness value of the new generation of ants is better than that of the corresponding antlions, the position of the antlion is replaced by the position of the ant. If the fitness value of the new generation antlion is better than that of the elite, the position of the elite is replaced by the position of the antlion. Ants and antlions will continue to try new positions until they meet the optimization requirement. The position of ultimate elite represents the optimization solution. Because the antlion algorithm has better searching efficiency and convergence accuracy, Hadidian-Moghaddam et al. [20], Ali et al. [21], and Mirjalili et al. [19] used the antlion algorithm for optimization.

To get the best solution, the objective function need be set first. Assume that substituting the best solution into the objective function can get the global minimum. For satisfying this condition, there are i ants and j antlions within the specified range. The positions of the ants and antlions correspond to a solution. Substituting the ants' and ant–lions' position vector into the objective function, we can obtain the corresponding fitness value. The steps involved in ALO algorithm are described as follows:

Step 1: Set the initial position of the ants and antlions and select the initial elite.

The initial positions of i ants and j antlions are randomly selected within the specified range. The fitness values of ants and antlions are calculated respectively. And the optimal antlion is set as elite.

Step 2: Limit the movement range of ants.

Each antlion builds a trap based on its own fitness. When an ant falls into a trap, the antlion kicks away the sand in the center of the trap to avoid the ant running away. In order to imitate this process, equations (31) to (33) are used to simulate the ever-reducing radius of the trap, and then the optimization range of ants is obtained in g-generation optimization

c^{g} = \frac{c^{g}}{I}

(31)

d^{g} = \frac{d^{g}}{I}

(32)

I = 1 + 10^{\frac{wg}{G}}

(33)

c_{i}^{g} = {Antlion}_{j}^{g} + c^{g}

(34)

d_{i}^{g} = {Antlion}_{j}^{g} + d^{g}

(35)

where w = 2, when g > 0.1G; w = 3, when g > 0.5G; w = 4, when g > 0.75G; w = 5, when g > 0.9G; and w = 6, when g > 0.95G.

Step 3: Update positions of ants.

Assuming that the ants walk randomly within a limited range, the walk of the ants is set using a random function (36) to obtain equation (37). To avoid ants from exceeding the scope of optimization, the equation (38) is used to standardize the random walk of ants

\begin{matrix} r (g) = {\begin{matrix} 1 & if rand > 0.5 \\ 0 & if rand \leq 0.5 \end{matrix} \end{matrix}

(36)

X (g) = [0, cumsum (2 r (g_{1}) - 1), cumsum (2 r (g_{2}) - 1), \dots, cumsum (2 r (g_{n}) - 1)]

(37)

X_{i}^{g} = \frac{(X_{i}^{g} - a_{i}) \times (d_{i}^{g} - c_{i}^{g})}{(b_{i} - a_{i})} + c_{i}^{g}

(38)

Step 4: Get ants' fitness.

The positions of the new generation of ants are substituted into the fitness function to obtain the fitness value of each new ant.

Step 5: Update antlion.

If the fitness value of ant is higher than that of antlion, the position of ant is assigned to antlion

{Antlion}_{j}^{g} = {Ant}_{i}^{g} if f ({Ant}_{i}^{g}) < f ({Antlion}_{j}^{g})

(39)

Step 6: Update the elite.

In the optimization process, the elite is set as the optimal solution. When an antlion's fitness value is higher than the elite's fitness value, the antlion is set as new elite and its position will be preserved. The fittest antlion can affect the movements of all ants. The following equation can be obtained based on an assumption that each ant randomly walks around an antlion selected by roulette and elite

{Ant}_{i}^{g} = \frac{R_{A}^{g} + R_{E}^{g}}{2}

(40)

Step 7: Repeat step 2 until the termination condition is satisfied.

Figure 5 is drawn for easy understanding, which is the flowchart of proposed ALO algorithm.

Figure 5.

Flowchart of the proposed ALO algorithm.

Optimizing structural parameters of carriers using antlion optimization algorithm

Optimization objectives

Braiding carriers can adjust the tension of the carbon fiber by the rotation angle of the lever and the length of the carbon fiber unwinding from the spool. However, the fluctuation of the carbon fiber tension causes the defects in the shape and texture of braiding products. The carbon fiber braided clockwise and the carbon fiber braided counterclockwise touch and move relatively to each other, which causes friction and fluff. The degree of friction is affected by the tension of the carbon fiber. The greater the tension is, the more obvious the fluffing is. In order to obtain better quality and more compact braiding products, the antlion algorithm can be used to optimize the structural parameters of the braiding carriers.

The optimization objectives are as follows:

Reduce the phenomenon where the two groups of carbon fiber braided in different directions are at the large tension state;

Make the tension of the carbon fiber fluctuate around the required values;

Avoid altofrequent and sharp fluctuations of the carbon fiber tension.

Select optimization parameters

Based on the kinematics model and dynamics model, the tension of the carbon fiber is mainly affected by spring 1 and spring 2. The operating time of spring 1 and spring 2 is related to the working stage of the carriers, which is determined by $θ_{1}$ , $θ_{2}$ , $θ_{3}$ , and $l_{r}$ . Therefore, some structural parameters of the braiding carriers are selected to optimize, which are $d_{p 1}$ (pre-compression of spring 1), $d_{p 2}$ (pre-compression of spring 2), $k_{s 1}$ (elastic coefficient of spring 1), $k_{s 2}$ (elastic coefficient of spring 2), $θ_{1}$ (the starting angle of the lever into the regulating stage), $θ_{2}$ (the starting angle at which the lever is affected by spring 1 and spring 2), $θ_{3}$ (the starting angle of the lever into the releasing stage), and $l_{r}$ (the length of the lever).

Optimization ranges of the parameters

In order to get the optimal parameters, it was necessary to expand the range of the lever rotation angle. As shown in Figure 3,

θ_{1}

θ_{2}

, and

θ_{3}

were related to the structure of the carrier body. The minimum value of

θ_{1}

and the maximum value of

θ_{3}

were selected according to the size of the carrier body.

l_{r}

was positively correlated with carbon fiber length adjustment ability. If the value of

l_{r}

was excessive, the carrier would collide with the carrier that moves in the opposite direction in the working process.

k_{s 1}

and

k_{s 2}

were set to an appropriate range, respectively.

d_{p 1}

and

d_{p 2}

appropriately increased the selection range based on the original values. In summary, the optimization ranges of the parameters are shown in Table 1.

Table 1.

Optimization range of the parameters.

	$d_{p 1}$ (mm)	$d_{p 2}$ (mm)	$k_{s 1}$ (N/m)	$k_{s 2}$ (N/m)	$θ_{1}$	$θ_{2}$	$θ_{3}$	$l_{r}$ (mm)
Upper	50	50	200	250	−30 °	$20^{\circ}$	$30^{\circ}$	60
Lower	0	0	50	50	−50 °	$0^{\circ}$	$20^{\circ}$	35

Results and discussions

Pre-optimization simulation

As shown in Table 2, the initial structural parameters of the braiding carrier were substituted into the kinematics model and dynamic model. The release speed of the carrier is shown in Figure 2. Figure 6 is drawn according to the simulation results.

Figure 6.

Angle of the lever and carbon fiber tension before optimization: (a) angle of the lever before optimization; (b) carbon fiber tension before optimization.

Table 2.

Structural parameters of braiding carrier before optimization.

Parameter	$d_{p 1}$ (mm)	$d_{p 2}$ (mm)	$k_{s 1}$ (N/m)	$k_{s 2}$ (N/m)	$θ_{1}$	$θ_{2}$	$θ_{3}$	$l_{r}$ (mm)
Value	40	15	140	180	−30 °	$0^{\circ}$	$20^{\circ}$	58

As shown in Figure 6, the carriers entered a stable working stage after 4 s in the simulation, so the simulation results of the carbon fiber tension were analyzed in detail from 4 s to 5 s. As shown in Figure 6(a), θ was greater than 0 from 4 s to 5 s. The carriers used two springs to regulate the tension of the carbon fiber. As shown in Figure 6(b), under the simultaneous action of two springs, the tension of the carbon fiber carried by $C A_{1}$ , $C A_{2}$ , $C B_{1}$ , and $C B_{2}$ oscillated between 0.6 and 0.75 N from 4 s to 5 s. The four carbon fibers were at the large tension stage. At this moment, there were several contact points in the catenary segment between the carbon fiber braided in the clockwise and counterclockwise directions. And the relative motion at the contact point caused fluff on the surface of the carbon fiber, which seriously affected the structure, quality, and mechanical properties of the braiding products. Therefore, it was necessary to improve the structure of the braiding carriers, and the fluffing phenomenon of the carbon fiber was mitigated by changing the tension fluctuations of the carbon fibers.

In addition to reducing carbon fiber fluff by optimizing structural parameters of carriers, it was also possible to adjust the fluctuation range of the carbon fiber tension for different braiding tasks and materials. Before the braiding work, the calculation was conducted to obtain the carbon fiber fluctuant interval required for processing. And the structural parameters of braiding carriers were calculated to satisfy the requirement, thereby improving the qualification ratio of braiding products.

Optimization results

The simulation by MATLAB was carried out for the previously proposed optimization objectives. According to different optimization requirements, the structural parameters of the braiding carriers were optimized by the antlion algorithm. The time from 4 s to 5 s is divided into 100 unit times, so $t = 4 + n \times 0.01 (n = 1, \dots, 100)$ .

Optimization objective (a): Reducing the overlap time

The fluffing phenomenon was exacerbated by the fact that the two carbon fibers with large tension produced the friction at the same time. Therefore, the objective function was the overlap time in which the carbon fibers braided clockwise and counterclockwise were all at the large tension stage. If the length of the overlap time was 0, then it could be shown that at least one of the two carbon fibers that generated friction was at the small tension state from 4 s to 5 s. Fluffing could be alleviated. The objective function is shown as in the following equation. The optimized structural parameters are shown in Table 3.

FITNES S_{1} = \sum_{n = 1}^{100} Time (n)

(41)

Table 3.

Optimized structural parameters to reduce the overlap time.

Parameter	$d_{p 1}$ (mm)	$d_{p 2}$ (mm)	$k_{s 1}$ (N/m)	$k_{s 2}$ (N/m)	$θ_{1}$	$θ_{2}$	$θ_{3}$	$l_{r}$ (mm)	$FITNES S_{1}$
Before optimization	40	15	140	180	−30 °	$0^{\circ}$	$20^{\circ}$	58	100
After optimization	28.51	23.66	122.97	118.67	−50 °	$20^{\circ}$	$20^{\circ}$	54.47	0

If carbon fiber braided in clockwise and carbon fiber braided in counterclockwise were at the large tension stage, then $Time (n) = 1$ ; else, $Time (n) = 0$ .

Optimization objective (b): Regulating the tension of the carbon fiber

Different braiding tasks and materials might have different requirements for carbon fiber tension values. Therefore, the antlion algorithm was used to optimize the structural parameters to regulate the fluctuant range of the carbon fiber tension in the braiding process. According to Figure 6, the carbon fiber would be at the large tension state after the carriers entered a stable working state. The carbon fiber was set to fluctuate around 0.8 N at the large tension state, and around 0.5 N at the small tension state. The objective function was shown as equation (42), and the optimized structural parameters are as shown in Table 4.

FITNES S_{2} = \sum_{n = 1}^{100} | F_{T} (n) - {\bar{F}}_{T} |

(42)

Table 4.

Optimized structural parameters to regulate the carbon fiber tension.

Parameter	$d_{p 1}$ (mm)	$d_{p 2}$ (mm)	$k_{s 1}$ (N/m)	$k_{s 2}$ (N/m)	$θ_{1}$	$θ_{2}$	$θ_{3}$	$l_{r}$ (mm)	$FITNES S_{2}$
Before optimization	40	15	140	180	−30 °	$0^{\circ}$	$20^{\circ}$	58	13.04
After optimization	47.94	35.85	119.36	83.32	−30.06 °	$15 . 59^{\circ}$	$20^{\circ}$	51.02	0.85

Optimization objective (c): Comprehensive optimization

Based on the optimization results presented above, a new optimization objective (reduce the number of fluctuations) was added. By optimizing the parameters to reduce the overlap time, regulate tension of carbon fiber, and reduce the number of fluctuations at the same time, the objective function was shown as in equation (43). The optimized structural parameters are shown in Table 5.

\begin{matrix} FITNES S_{3} = w_{1} \sum_{n = 1}^{100} Time (n) + w_{2} \sum_{n = 1}^{100} | F_{T} (n) - {\bar{F}}_{T} | \\ + w_{3} \sum_{n = 1}^{100} (co n_{A 1} (n) + co n_{A 2} (n) + co n_{B 1} (n) + co n_{B 2} (n)) \end{matrix}

(43)

Table 5.

Optimized structural parameters to meet optimizations.

Parameter	$d_{p 1}$ (mm)	$d_{p 2}$ (mm)	$k_{s 1}$ (N/m)	$k_{s 2}$ (N/m)	$θ_{1}$	$θ_{2}$	$θ_{3}$	$l_{r}$ (mm)	$FITNES S_{3}$
Before optimization	40	15	140	180	−30 °	$0^{\circ}$	$20^{\circ}$	58	21.30
After optimization	48.50	34.98	83.48	56.90	−30.51 °	$17 . 31^{\circ}$	$20^{\circ}$	35.44	7.20

If the stage of carbon fiber tension changed, then $con (n) = 1$ ; else $con (n) = 0$ .

Result analysis

To calculate the carbon fiber tension values, the optimized structural parameters obtained from the optimization objectives (a), (b), and (c) were substituted into the model of carrier. The simulation results are shown in Figure 7.

Figure 7.

Comparison of simulation results: (a) pre-optimization simulation; (b) reduce the overlap time; (c) regulating the tension of the carbon fiber; (d) comprehensive optimization.

Analysis of optimization results based on objective (a)

The optimization results based on the optimization objective (a) are shown in Figure 7(b). Figure 7(b) and (a) are compared and analyzed. Before optimization, $θ_{2} = 0^{\circ}$ , $θ_{3} = 20^{\circ}$ . The rotation angle of the lever was greater than $θ_{2}$ from 4 s to 5 s, so the spool continued to be subjected to the two springs. When the rotation angle of the lever reached $θ_{3}$ , the spool lost the limitation of the stop pin and the spool was subjected to the carbon fiber tension. The spool started to rotate and the carbon fiber was released, then the carbon fiber tension was lowered. If the carbon fiber released at the exit point of the carrier did not satisfy the braiding requirement, the lever was pulled up again and then the spool released carbon fiber. Therefore, a waveform diagram as shown in Figure 7(a) can be obtained. At the peak of curve, the spool was subjected to carbon fiber tension many times, so the zigzag pattern was formed.

After optimization, $θ_{2} = θ_{3} = 20^{\circ}$ . When the angle of the lever was greater than $20^{\circ}$ , the lever was subjected to two spring forces and the spool lost the limitation of the stop pin to release carbon fiber. Therefore, the carbon fiber tension rose rapidly when it reached $20^{\circ}$ . Because the carbon fiber was released, the rotation angle of the lever reduced to make the rotation angle less than $20^{\circ}$ , and the carbon fiber tension was immediately lowered. By optimization, the two groups of carbon fibers that were in contact with each other could be completely avoided obvious fluffing which was caused by two groups of carbon fiber with the large tension. However, there were many fluctuations in the tension of carbon fiber, which also affected the quality of the braiding products.

Analysis optimization results based on objective (b)

The optimization results based on the optimization objective (b) are shown in Figure 7(c). Figure 7(c) and (a) are compared and analyzed. After optimization, the pre-compression and elastic coefficient of the two springs were adjusted. The spring force was related to the compression of the spring. If the carbon fiber was only at the large tension stage, the fluctuation of the carbon fiber tension was large as shown in Figure 7(a). Therefore, by changing the value of $θ_{2}$ to distribute the carbon fiber tension at two stages, it was possible to achieve a small fluctuant range of the carbon fiber tension around 0.5 N at the small tension state and around 0.8 N at the large tension state.

Analysis optimization results based on objective (c)

The carbon fiber tension generated multiple large fluctuations based on the optimization objective (a). A new optimization objective (reducing the number of large fluctuations of the carbon fiber tension) was introduced. $w_{1}$ , $w_{2}$ , and $w_{3}$ were introduced as the weights of the optimization objectives. The optimized carbon fiber tension was shown in Figure 7(d). Through the antlion algorithm optimization, the objective was achieved: the overlap time was greatly reduced, which represented the two groups of the carbon fiber in contact with each other were at the large tension stage; the carbon fiber tension fluctuated around 0.8 N at the large tension state, and around 0.5 N at the small tension state; the number of carbon fiber tension fluctuations was reduced relative to objective (a).

According to different braiding tasks and materials, $w_{1}$ , $w_{2}$ , and $w_{3}$ can be adjusted to obtain the results that met the processing requirements.

Conclusions

The antlion algorithm was used to optimize the structural parameters of the braiding carrier. The kinematics model and dynamics model based on the structure of the double springs-lever type carrier obtained the relationship between the relevant structural parameters and the carbon fiber tension. Then the optimization results were compared and analyzed through MATLAB software. In conclusion, it could be known that:

By changing $θ_{1}$ , $θ_{2}$ , and $θ_{3}$ , the time during which the carbon fiber tension was at the large tension state or the small tension state can be adjusted. Thereby, the fluffing phenomenon of the carbon fiber surface was reduced, which was caused by the two groups of the carbon fibers in contact with each other at the large tension stage.

The pre-compression and the elastic coefficient of springs combined with the starting angle of the carrier state could maintain the carbon fiber tension within a small range around the required value during the stable working.

By introducing the objective weights, the degree of the different optimization objectives could be adjusted. The optimized structural parameters could meet the requirement of the braiding task and material.

In short, the structural parameters of the braiding carriers optimized by antlion algorithm could improve the quality and mechanical properties of the braiding products.

Declaration of conflicting interests

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

Footnotes

Funding

The author(s) 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 no. 51475091], the rolling support plan for the excellent innovation team by Ministry of Education of the People's Republic of China [IRT_16R12], and the Donghua University Graduate Student Degree Thesis Innovation Fund Project [Grant no. CUSF-DH-D-2017064].

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Optimizing structural parameters of carbon fiber braiding carriers based on antlion optimization algorithm

Abstract

Keywords

Introduction

Carrier modeling

Braiding process and carrier structure

Braiding process

Carrier structure

Kinematic model

Modeling at regulating stage

Modeling at releasing stage

Dynamic model

Tension at the regulating stage

Tension at the releasing stage

Antlion optimization algorithm

Optimizing structural parameters of carriers using antlion optimization algorithm

Optimization objectives

Select optimization parameters

Optimization ranges of the parameters

Results and discussions

Pre-optimization simulation

Optimization results

Optimization objective (a): Reducing the overlap time

Optimization objective (b): Regulating the tension of the carbon fiber

Optimization objective (c): Comprehensive optimization

Result analysis

Analysis of optimization results based on objective (a)

Analysis optimization results based on objective (b)

Analysis optimization results based on objective (c)

Conclusions

Declaration of conflicting interests

Footnotes

Funding

References