Modeling and analysis of carbon fiber carrier tension systems used in 3D braiding machine

Abstract

Carbon fiber braids are widely applied in various industrial fields as structural components. However, using braiding machines to braid carbon fibers is an emerging technology, but some technical problems must be solved. For instance, carbon fibers are easily fluffed when using traditional carriers; with increasing braiding speed, the machine fiercely vibrates. To investigate these problems, an improved mechanical structure of the carrier is designed, which reduces the friction between the carbon fiber and guide hole and changes the carbon fiber storage quantity. Based on the carrier, the corresponding mathematical model of the tensioning system is established by the dynamical equations, and numerical analyses and simulations are carried out. Moreover, the validation of the established mathematical model is verified. According to the results of the simulation, the fluctuation in the tensioning force is smaller than that in previous studies, and the trend is in accordance with previous research results. As a result, the present model provides a better description of the dynamic process of carbon fiber braiding.

Keywords

Carbon fiber carrier tensioning system braiding machine

Introduction

Braiding machines are widely used to form textile products, ropes/tubes, and other structural or composite components. The traditional braided fabric consists of yarns. However, along with the greater demands in aerospace, vehicle and other industrial engineering applications, new types of braided fabric are needed. Braiding machines are urgently needed to braid carbon fibers, glass fibers, or ceramic fibers for use as braiding materials to form products. However, these new types of fibers are not easily braided because their mechanical performance is quite different from that of yarns. For instance, carbon fibers are easy to fluff and fracture, if a traditional braiding machine is used. Glass and ceramic fibers are fragile to braid. Braiding machines must be investigated and changed to use these fibers.

Some studies have been performed on the braiding machines or braiding process.^1,2 Guyader et al.³ studied the adapted relationships between the process parameters and the braid shape by considering the transitory and steady-state process stages with differential geometrical methods. Potluri et al.⁴ investigated the rapid development of braided preforms by using a geometrical simulation, which consists of the number of yarn carriers, the rotational speed of carriers, the take-up speed and the effective perimeter of the cross-section of a mandrel. Bigaud et al.⁵ studied the main parameters of the braiding process and the mechanical properties of the 3D braided composites. Zhou et al.⁶ analyzed the transverse impact responses of a 3D circular braided composite from the aspect of different braiding angles. Heieck et al.⁷ calculated the cover factor of braided carbon fiber preforms and proposed a method for calculating the cover factors. Van Ravenhorst and Akkerman⁸ proposed an inverse kinematics-based procedure to simulate the braid angle and later established a model that includes the yarn interaction, assuming an axisymmetric biaxial process with a cylindrical mandrel and Coulomb friction, and obtained the automatic generation of machine control data. Recently, these authors have published a procedure for relating the braid patterns to spool patterns, which could simulate and optimize some features of the braiding processes.⁹ Goseberg et al.¹⁰ and Lepperhoff¹¹ also analyzed the relationship between the carrier arrangement and the different pattern types. Kyosev and colleagues^12,13 presented modeling algorithms and process simulation results for 3D braiding machines.

Most of the above studies intended to establish the relationship between the general processing characteristics and the parameters of braids. In addition, some important factors affect the properties of braids, such as the yarns’ processing behavior and the tensioning system. Ebel et al.¹⁴ carried out a bench test to investigate the yarn damaging mechanisms of the processing behavior of different yarns. Mulvihill et al.¹⁵ and Cornelissen et al.¹⁶ studied the friction between carbon fibers and guide parts. Branscomb studied the yarns’ processing behavior of the net shape preforms of different yarns.

In fact, yarn damage is caused by rewinding, the spool, the carrier, yarn nets, and so on. The inner structure of the carrier itself also influences the carbon braided fabric and cannot be ignored. In particular, while braiding carbon fibers, the yarn tension, the amount of releasing/receiving yarn in each loop, and their fluctuations could result in severe yarn damage. However, few studies have focused on these aspects. Ma et al.¹⁷ investigated the traditional carrier by separating the braiding process into several kinematic regions. However, no systematic modeling has been established. In the article, the braking system of the carrier is not clearly revealed, which could control the carbon fiber to not fluff or fracture. In particular, when analyzing the releasing region, the carrier system is simplified to a model, which does not reflect how the inner structure influences the tensioning system. In fact, the releasing region is very complex, but important to designers. However, the model is not suitable for studying braiding carbon fibers because the end point of the carrier is fixed with a yarn hole, which could damage the carbon fibers.

Improving the traditional carrier and establishing a carbon fiber–suited model is worthy and has practical value. This article aims to establish systematic kinetic equations of an improved carrier to illustrate the tensioning system.

Mathematical modeling

Mechanism of carrier

Commonly, a yarn carrier has a guide hole at the exit side, leading the yarn out from the carrier. As discussed above, this setting is unreasonable for carbon fibers. To braid carbon fibers, an improved mechanical structure of the carrier is designed, as shown in Figure 1. In this structure, the exit side is replaced by two rollers and some other components. One of the rollers is fixed on the rocker, and the other is pulled by two extension springs. The rocker could rotate around a pin fixed on the carrier with a bidirectional torsion spring. All springs have different pretension forces according to the parameters of the tensioning system.

Figure 1.

Improved mechanical structure of the carbon fiber–suited carrier.

To investigate the mechanical principle of the tensioning system, the corresponding mechanical schematic diagram is depicted in Figure 2. As shown, a carbon fiber is released by the spool, which can rotate around the rotation shaft fixed on the carrier base. Then, the carbon fiber orderly passes over roller1, pulley1, pulley2, and roller2. The fiber will be removed from pulley3 and will ultimately arrive at the braiding point. Roller1, pulley1, and pulley3 are fixed on the carrier base. Pulley2 is a movable pulley fixed on a rotatable linkage. Roller2 can move along the groove on the rocker restrained by spring3. Figure 2 describes the initial status of the tensioning system. At this moment, there is no tensioning force in the carbon fiber. The spool is ratchet-locked. A clearance between the linkage and slider1 is set.

Figure 2.

Schematic diagram of the tensioning system with a zero tensioning force.

Commonly, the carbon fiber is constrained by a robotic or mechanical arm at the end point. Accompanying the braiding, pulley2 and roller2 are lifted, and spring2 and spring3 are stretched. While the linkage turns at an angle $φ_{1}$ , touching slider1, spring1 is compressed and is gradually removed from the pawl of the spool. Once slider1 is completely removed, the spool will rotate and release the carbon fiber. Furthermore, as the spool rotates, the tensioning force in the carbon fiber will decrease. All springs will return to their initial status. Thus, slider1 will go upward, locking the spool again. When the spool is locked, the release of the carbon fiber is stopped. Consequently, the process is a periodic cycle in this way.

Dynamical equations of the tensioning system

To analyze the tensioning system, a schematic diagram of the tensioning system with maximal tensioning force is drawn in Figure 3. Because the dynamic characteristics of this tensioning system are periodic, only one period will be investigated. Here, the rotational point of linkage is defined as the original point. The other parameters are assigned as follows: $T (t)$ is the tensioning force; $t$ is the time; $m_{1}$ is the mass of slider1; $m_{2}$ is the mass of slider2; $m_{3}$ is the mass of roller2; $J_{0}$ is the rotational inertia of the spool; $J_{1}$ is the rotational inertia of the linkage and pulley2; $J_{2}$ is the rotational inertia of the rocker, roller1 and roller2; $r_{1}$ is the radius of the pulleys; $r_{2}$ is the radius of roller2; $R_{0}$ is the radius of the spool; $h$ is the tooth height of the pawl; $h_{0}$ is the clearance between the linkage and slider1; $R_{1}$ is the length of the center distance between pulley2 and the original point; $R_{2}$ is the length of the center distance between roller2 and the rotational point of the rocker; $e_{1}$ is the horizontal distance from spring1 to the original point; $e_{2}$ is the horizontal distance from spring2 to the original point; $λ_{i}$ ( $i = 1, 2, 3, 4$ ) are the preload deflections of the springs; $l_{1}$ is the length of the center distance between pulley1 and the original point; $l_{2}$ is the length of the center distance between pulley1 and the rotational point of the rocker; $l_{3}$ is the length of the center distance between the rotational point of the rocker and the original point; $l_{4}$ is the length of the center distance between pulley3 and the original point; $x_{1} (t)$ is the length of the center distance between pulley1 and pulley2; $x_{2} (t)$ is the length of the center distance between pulley2 and roller2; $x_{3} (t)$ is the length of the center distance between roller2 and pulley3; $θ_{i}$ stands for the angle between $l_{i}$ and the horizontal axis $(i = 1, 2, \dots, 4)$ ; $φ_{0} (t)$ is the rotational angle of the spool; $φ_{1} (t)$ is the rotational angle of the linkage; $φ_{2} (t)$ is the rotational angle of the rocker; and $k_{i}$ stands for the stiffness of spring1, spring2, spring3 and spring4, $(i = 1, 2, \dots, 4)$ , respectively.

Figure 3.

Schematic diagram of the tensioning system with the maximal tensioning force.

Generally speaking, the dynamic motion of the carrier system could be divided into three periods: the preparation period, the releasing period, and the tightening period. During the preparation period, the tensioning force $T$ of the carbon fiber starts from zero and ends at a certain value $C_{1}$ . At the end of the experiment, the spool is released. Then, $T$ fluctuates quickly until the spool is locked. the next period is the tightening period. During this period, $T$ starts at a certain value $C_{2}$ and ends at $C_{1}$ . $C_{i}$ ( $i = 1, 2$ ) are constant values depending on the system parameters. In this way, the cycle repeats. It can be easily induced that while braiding, the amount of carbon fiber released from spool is either stored among pulley1 to pulley3 or carried away to the braiding point. According to this principle, the following equation can be obtained

\begin{array}{l} R_{0} φ_{0} (t) - q (t) = \sqrt{x_{1} {(t)}^{2} - 4 r_{1}^{2}} + \sqrt{\frac{x_{2} {(t)}^{2}}{4 - r_{1}^{2}}} \\ + \sqrt{\frac{x_{2} {(t)}^{2}}{4 - r_{2}^{2}}} + \sqrt{\frac{x_{3} {(t)}^{2}}{4 - r_{1}^{2}}} \\ + \sqrt{\frac{x_{3} {(t)}^{2}}{4 - r_{2}^{2}}} - X_{0} \end{array}

(1)

where $q (t)$ is the time-varying braiding length and $X_{0}$ is the initial amount of carbon fiber stored among pulley1 to pulley3.

Slider1, slider2, linkage, roller2, and the rocker are separated from the system (Figure 4), and according to Newton’s law of motion, the following dynamical equations can be obtained

2 T \sin (α_{1}) R_{1} - (F_{1} e_{1} + F_{2} e_{2}) = J_{1} {\overset{\cdot\cdot}{φ}}_{1}

(2a)

k_{1} δ_{1} - F_{1} = m_{1} {\overset{\cdot\cdot}{δ}}_{1}

(2b)

k_{2} δ_{2} - F_{2} = m_{2} {\overset{\cdot\cdot}{δ}}_{2}

(2c)

2 T \sin (α_{2}) (R_{2} + δ_{3}) - (M (t_{k}) + k_{4} φ_{2} (t)) = J_{2} {\overset{\cdot\cdot}{φ}}_{2}

(2d)

J_{0} {\overset{\cdot\cdot}{φ}}_{0} (t) = T R_{0}

(2e)

2 T \cos (α_{2}) - k_{3} (δ_{3} + λ_{3}) = m_{3} {\overset{\cdot\cdot}{δ}}_{3}

(2f)

where $α_{1}$ is the angle between $R_{1}$ and the resultant force of pulley2 stressed by the carbon fiber; $α_{2}$ is the angle between $R_{2}$ and the resultant force of roller2 stressed by the carbon fiber; $F_{1}$ and $F_{2}$ are the forces of slider1 and slider2 to linkage, respectively; $M (t_{k})$ is the initial force in spring4; and $δ_{i}$ is the stretched/compressed length of springs ( $i = 1, 2, 3$ ).

Figure 4.

Force analysis diagram of the key parts.

By analyzing the geometrical relationships of the components (Figure 5), $α_{1}$ and $α_{2}$ can be derived in the following equations

\begin{array}{l} α_{1} = π - \frac{1}{2} \arccos \geq ({\overset{⇀}{s}}_{1} - {\overset{⇀}{s}}_{5}, {\overset{⇀}{s}}_{3} + {\overset{⇀}{s}}_{6} - {\overset{⇀}{s}}_{5}) \\ - \arcsin (\frac{l_{1} \sin (θ_{1} - ϕ_{1} - φ_{1})}{| {\overset{⇀}{s}}_{1} - {\overset{⇀}{s}}_{5} |}) \end{array}

(3a)

\begin{array}{l} α_{2} = π - \arccos ({\overset{⇀}{s}}_{6}, {\overset{⇀}{s}}_{3} + {\overset{⇀}{s}}_{6} - {\overset{⇀}{s}}_{5}) \\ - \frac{1}{2} \arccos ({\overset{⇀}{s}}_{3} + {\overset{⇀}{s}}_{6} - {\overset{⇀}{s}}_{4}, {\overset{⇀}{s}}_{3} + {\overset{⇀}{s}}_{6} - {\overset{⇀}{s}}_{5}) \end{array}

(3b)

δ_{1} = e_{1} (\tan (ψ_{1} (t_{k})) - \tan (ψ_{1} (t_{k}) - φ_{1})) δ (t - t_{02})

(3c)

δ_{2} = e_{2} (\tan (ψ_{2} (t_{k})) - \tan (ψ_{2} (t_{k}) - φ_{1})) δ (T - k_{2} λ_{2})

(3d)

M (t_{k}) = k_{4} (λ_{4} + φ_{2} (t_{k}))

(3e)

where $δ$ is the Dirac operator and $t_{02}$ is the moment when linkage and slider1 contact with each other

\begin{array}{l} {\overset{⇀}{s}}_{1} = (l_{1} \cos θ_{1}, l_{1} \sin θ_{1}); \\ {\overset{⇀}{s}}_{2} = (l_{3} \cos θ_{3} - l_{1} \cos θ_{1}, l_{3} \sin θ_{3} - l_{1} \sin θ_{1}); \\ {\overset{⇀}{s}}_{3} = (l_{3} \cos θ_{3}, l_{3} \sin θ_{3}); \\ {\overset{⇀}{s}}_{4} = (l_{4} \cos θ_{4}, l_{4} \sin θ_{4}); \\ {\overset{⇀}{s}}_{5} = (R_{1} \cos (φ_{1} + ϕ_{1}), R_{1} \sin (φ_{1} + ϕ_{1})); \\ {\overset{⇀}{s}}_{6} = ((R_{2} + δ_{3}) \cos (φ_{2} + ϕ_{2}), - (R_{2} + δ_{3}) \sin (φ_{2} + ϕ_{2})) \end{array}

$ϕ_{1}$ and $ϕ_{2}$ denote the initial angle of the linkage and the rocker referring to the horizontal axis and $ψ_{1} (t_{k})$ and $ψ_{2} (t_{k})$ represent the initial angle between slider1 and slider2 and original point in each period, respectively.

Figure 5.

Geometrical relationships of the components.

Substituting equations 3(a) to (e) into equations 2(a) and (b), the following relationship can be obtained

\begin{array}{l} 2 T R_{1} f (φ_{1}, φ_{2}) - k_{1} e_{1} (e_{1} h_{1} (φ_{1}) + λ_{1}) - k_{2} e_{2} \\ (e_{2} h_{2} (φ_{1}) + λ_{2}) = J_{1} {\overset{\cdot\cdot}{φ}}_{1} + m_{1} {\overset{\cdot\cdot}{δ}}_{1} e_{1} + m_{2} {\overset{\cdot\cdot}{δ}}_{2} e_{2} \end{array}

(4a)

\begin{array}{l} 2 T (R_{2} + δ_{3}) g (φ_{1}, φ_{2}) - k_{4} \\ (λ_{4} + φ_{2} (t_{k}) + φ_{2} (t)) = J_{2} {\overset{\cdot\cdot}{φ}}_{2} \end{array}

(4b)

where

\begin{array}{l} f (φ_{1}, φ_{2}) = \sin α_{1}; \\ g (φ_{1}, φ_{2}) = \sin α_{2}; \\ h_{1} (φ_{1}) = \tan (ψ_{1} (t_{k})) - \tan (ψ_{1} (t_{k}) - φ_{1}); \\ h_{2} (φ_{1}) = \tan (ψ_{2} (t_{k})) - \tan (ψ_{2} (t_{k}) - φ_{1}) \end{array}

On the contrary, the storage length of the carbon fiber is divided into three parts, which can be expressed as

x_{1} (t) = | s_{5} - s_{1} |

x_{2} (t) = | s_{6} + s_{3} - s_{5} |

x_{3} (t) = | s_{4} - s_{3} - s_{6} |

(5)

Therefore, the right side of equation (1) could be expressed as a function of $φ_{1}$ and $φ_{2}$ based on equation (5). Here, the function is defined as $p (φ_{1}, φ_{2})$ . Combining all the equations above, the dynamical equations of the carrier system could be derived.

It can be easily observed that the process consists of several piecewise functions. To clearly illustrate the process, a schematic graph of the tensioning force is depicted in Figure 6. Dynamical equations of each period will be discussed.

Figure 6.

Schematic graph of the tensioning force.

To solve the dynamical equations, the boundary conditions of each period are discussed as follows. Because the traction manipulator keeps moving at a constant speed $v$ , the function $q (t)$ could be rewritten as $v t$ .

In the preparation period, once the tensioning force overcomes the preload force of either spring1 or spring4, that is, $T = \min (k_{2} λ_{2}, k_{4} λ_{4}) / (2 R_{2} g (0, φ_{2} (0)))$ , the carrier starts to work. Commonly, $k_{2} λ_{2} > k_{4} λ_{4}$ , which means that rocker moves first. Meanwhile, as the spool does not rotate, the amount of carbon fiber for braiding all comes from storage. Thus

- v t = p (0, φ_{2})

(6a)

2 T R_{2} g (0, φ_{2}) - k_{4} (λ_{4} + φ_{2}) = J_{2} {\overset{\cdot\cdot}{φ}}_{2}

(6b)

where ${\overset{⇀}{s}}_{6} = (- R_{2} \sin (φ_{2} + ϕ_{2}), R_{2} \cos (φ_{2} + ϕ_{2}))$ .

With the initial conditions, $φ_{2} (0) = 0$ ; ${\dot{φ}}_{2} (0) = 0$ ; $T (0) = k_{4} λ_{4} / (2 R_{2} g (0, φ_{2} (0)))$ .

When $T (t_{01}) = k_{2} λ_{2} / 2 R_{2} g (0, φ_{2} (0))$ , the linkage begins to rotate. The dynamical equations are changed into the following forms

- v t = p (φ_{1}, φ_{2})

(7a)

2 T R_{1} f (φ_{1}, φ_{2}) - k_{2} e_{2} (e_{2} h_{2} (φ_{1}) + λ_{2}) = J_{1} {\overset{\cdot\cdot}{φ}}_{1} + m_{2} {\overset{\cdot\cdot}{δ}}_{2} e_{2}

(7b)

2 T R_{2} g (φ_{1}, φ_{2}) - k_{4} (λ_{4} + φ_{2}) = J_{2} {\overset{\cdot\cdot}{φ}}_{2}

(7c)

where $h_{2} (φ_{1}) = \tan ψ_{2} - \tan (ψ_{2} - φ_{1})$ ; $(φ_{2} + ϕ_{2}), R_{2} \cos (φ_{2} + ϕ_{2}))$ .

With the initial conditions, $φ_{1} (t_{01}) = 0$ ; ${\dot{φ}}_{1} (t_{01}) = 0$ ; $φ_{2} (t_{01})$ , ${\dot{φ}}_{2} (t_{01})$ are determined by the solutions of equation (7).

When $e_{1} (\tan ψ_{1} - \tan (ψ_{1} - φ_{1} (t_{02}))) = h_{0}$ , the linkage and slider1 contact each other. The dynamical equations are changed into

- v t = p (φ_{1}, φ_{2})

(8a)

\begin{array}{l} 2 T R_{1} f (φ_{1}, φ_{2}) - k_{1} e_{1} (e_{1} h_{1} (φ_{1}) + λ_{1}) \\ - k_{2} e_{2} (e_{2} h_{2} (φ_{1}) + λ_{2}) = J_{1} {\overset{\cdot\cdot}{φ}}_{1} + m_{1} {\overset{\cdot\cdot}{δ}}_{1} e_{1} + m_{2} {\overset{\cdot\cdot}{δ}}_{2} e_{2} \end{array}

(8b)

2 T R_{2} g (φ_{1}, φ_{2}) - k_{4} (λ_{4} + φ_{2}) = J_{2} {\overset{\cdot\cdot}{φ}}_{2}

(8c)

where ${\overset{⇀}{s}}_{6} = (- R_{2} \sin (φ_{2} + ϕ_{2}), R_{2} \cos (φ_{2} + ϕ_{2}))$ ; $\tan (ψ_{1} - φ_{1} (t_{02})) - \tan (ψ_{1} - φ_{1}); h_{2} (φ_{1}) = \tan ψ_{2} - \tan (ψ_{2} - φ_{1}) .$ $\tan (ψ_{2} - φ_{1})$

With the initial condition, $φ_{1} (t_{02})$ could be solved by $h_{0};$ ${\dot{φ}}_{1} (t_{02})$ , $φ_{2} (t_{02})$ , ${\dot{φ}}_{2} (t_{02})$ are determined by the solutions of equations 8(a) to (c).

When $T = k_{3} λ_{3} / (2 \cos α_{2})$ , spring3 initiates a positive displacement and should be taken into account. As a result, the dynamical equations are changed into

- v t = p (φ_{1}, φ_{2})

(9a)

\begin{array}{l} 2 T R_{1} f (φ_{1}, φ_{2}) - k_{1} e_{1} (e_{1} h_{1} (φ_{1}) + λ_{1}) \\ - k_{2} e_{2} (e_{2} h_{2} (φ_{1}) + λ_{2}) = J_{1} {\overset{\cdot\cdot}{φ}}_{1} + m_{1} {\overset{\cdot\cdot}{δ}}_{1} e_{1} + m_{2} {\overset{\cdot\cdot}{δ}}_{2} e_{2} \end{array}

(9b)

2 T (R_{2} + δ_{3}) g (φ_{1}, φ_{2}) - k_{4} (λ_{4} + φ_{2}) = J_{2} {\overset{\cdot\cdot}{φ}}_{2}

(9c)

2 T \cos (α_{2}) - k_{3} (δ_{3} + λ_{3}) = m_{3} {\overset{\cdot\cdot}{δ}}_{3}

(9d)

where ${\overset{⇀}{s}}_{6} = (- R_{2} \sin (φ_{2} + ϕ_{2}), R_{2} \cos (φ_{2} + ϕ_{2}))$ ; $= \tan (ψ_{1} - φ_{1} (t_{02})) - \tan (ψ_{1} - φ_{1}); h_{2} (φ_{1}) = \tan (ψ_{2}) -$ tan(ψ₂ - ϕ₁).

With the initial conditions, $φ_{1} (t_{03})$ , ${\dot{φ}}_{1} (t_{03})$ , $φ_{2} (t_{03})$ , and ${\dot{φ}}_{2} (t_{03})$ are determined by the solutions of equations 9(a) to (d).

When $e_{1} (\tan (ψ_{1} - φ_{1} (t_{02})) - \tan (ψ_{1} - φ_{1} (t_{1}))) = h$ , the spool is unlocked, and the system turns into the releasing period. The dynamical equations can be derived as

\begin{array}{l} R_{0} φ_{0} (t) - v t = \sqrt{x_{1} {(t)}^{2} - 4 r_{1}^{2}} \\ + \sqrt{\frac{x_{2} {(t)}^{2}}{4 - r_{1}^{2}}} + \sqrt{\frac{x_{2} {(t)}^{2}}{4 - r_{2}^{2}}} \\ + \sqrt{\frac{x_{3} {(t)}^{2}}{4 - r_{1}^{2}}} + \sqrt{\frac{x_{3} {(t)}^{2}}{4 - r_{2}^{2}}} - X_{0} \end{array}

(10a)

\begin{array}{l} 2 T R_{1} f (φ_{1}, φ_{2}) - k_{1} e_{1} (e_{1} h_{1} (φ_{1}) + λ_{1}) \\ - k_{2} e_{2} (e_{2} h_{2} (φ_{1}) + λ_{2}) = J_{1} {\overset{\cdot\cdot}{φ}}_{1} + m_{1} {\overset{\cdot\cdot}{δ}}_{1} e_{1} + m_{2} {\overset{\cdot\cdot}{δ}}_{2} e_{2} \end{array}

(10b)

2 T (R_{2} + δ_{3}) g (φ_{1}, φ_{2}) - k_{4} (λ_{4} + φ_{2}) = J_{2} {\overset{\cdot\cdot}{φ}}_{2}

(10c)

J_{0} {\overset{\cdot\cdot}{φ}}_{0} (t) = T R_{0}

(10d)

2 T \cos (α_{2}) - k_{3} (δ_{3} + λ_{3}) = m_{3} {\overset{\cdot\cdot}{δ}}_{3}

(10e)

where $h_{1} (φ_{1}) = \tan (ψ_{1} - φ_{1} (t_{02})) - \tan (ψ_{1} - φ_{1})$ and $h_{2} (φ_{1}) = \tan (ψ_{2}) - \tan (ψ_{2} - φ_{1})$ .

With the initial conditions, $φ_{0} (t_{1}) = 0$ , ${\dot{φ}}_{0} (t_{1}) = 0$ , and $φ_{1} (t_{1})$ could be solved by $h$ ; and ${\dot{φ}}_{1} (t_{1})$ , $φ_{2} (t_{1})$ , ${\dot{φ}}_{2} (t_{1})$ , and $T (t_{1})$ are determined by the solutions of equations 9(a) to (d).

When $e_{1} (\tan (ψ_{1} - φ_{1} (t_{02})) - \tan (ψ_{1} - φ_{1} (t_{1}))) = h - ε$ , the spool is locked again. Here, $ε$ is a parameter that is dependent on the backlash of the pawl and slider1. The tensioning system turns into the tightening period. The dynamical equations are the same as in equations 9(a) to (d). However, the initial conditions are $φ_{1} (t_{2})$ and can be solved by $h - ε;$ ${\dot{φ}}_{1} (t_{2})$ , $φ_{2} (t_{2})$ , ${\dot{φ}}_{2} (t_{2})$ , and $T (t_{2})$ are determined by the solutions of equations 10(a) to (e).

The tightening period passes, and the system meets the initial condition of the releasing period, that is, $e_{1} (\tan (ψ_{1} - φ_{1} (t_{02})) - \tan (ψ_{1} - φ_{1} (t_{1}))) = h$ , and the spool is unlocked again. The tensioning system cycle repeats.

Numerical results and discussion

According to practical engineering, the parameters are assigned as follows: $k_{1} = 2.315 N / m m;$ $k_{2} = 2.315 N / m m;$ $k_{3} = 6.75 N / m m;$ $k_{4} = 125.62 N \cdot m m / r a d;$ $λ_{1} = 3 m m,$ $λ_{2} = 6 m m$ ; $λ_{3} = 1 m m$ ; $λ_{4} = 0^{\circ}$ ; $r_{1} = r_{2} = 12 m m;$ $m_{1} = m_{2} = 0.003 k g$ ; $m_{3} = 0.008 k g$ ; $R_{0} = 25 ~ 50 m m$ ; $v = 2 ~ 5 m m / s$ ; $h = 1.9 m m$ ; $h_{0} = 1.4 m m$ ; $R_{1} = 58 m m$ ; $l_{1} = 105 m m$ ; $l_{2} = 60 m m$ ; $l_{3} = 169 m m$ ; $l_{4} = 1384 m m$ ; $e_{1} = 7.5 m m$ ; $e_{2} = 6.5 m m$ ; $J_{0} = 48.5 k g . m m^{2}; J_{1} = 10.5 k g . m m^{2};$ $J_{2} = 20.5 k g . m m^{2}$ $θ_{1} = {65.3}^{\circ}$ ; $θ_{2} = {71.8}^{\circ}$ ; $θ_{3} = {70.5}^{\circ}$ ; $θ_{4} = {39.3}^{\circ}$ ; $ψ_{1} = {58.5}^{\circ}$ ; $ψ_{2} = {40.5}^{\circ}$ ; and $ϕ_{1} = 30^{\circ}$ . To investigate the mathematical model established above, a computer program for the nonlinear dynamic equations is written. Here, $R_{0} = 50 m m$ and $v = 4 m m / s$ are set. With these conditions, the equations could be solved by the ode45 method in MATLAB. The results are shown in the following diagrams in Figures 7 to 10.

Figure 7.

The value of the tensioning force, $T (t) .$

Figure 8.

The rotational angle of the linkage, $φ_{1} (t) .$

Figure 9.

The rotational angle of the rocker, $φ_{2} (t) .$

Figure 10.

The defection of spring, $δ_{3} (t) .$

From Figures 7 to 10, initially, the yarn tension has to overcome the prestress of Spring4, which is approximately 0.63 N. From this time, the rocker starts to move, and the yarn begins to be dragged forward.

During the period from 0 to t01, the tensioning force climbs until it overcomes the prestress of Spring2, which is approximately 0.89 N. From t01, the linkage starts to move, pushing Slider2 to move upward.

During the period of t01 to t02, the tensioning force gently changes between [0.87 N, 0.96 N] until the rocker reaches its extreme position (–0.07 rad). This singular point leads to a quick drag of the yarn, causing a sudden tension drop from 0.96 N to 0.93 N. In addition, the rotational angle of the linkage changes from 0.055 rad to 0.027 rad.

During the period from t02 to t03, the tensioning force changes between [0.94 N, 0.95 N] until it overcomes the prestress of Spring3, which is approximately 0.94 N, causing a sudden tension climb from 0.94 N to 1.6 N. From t03, roller2 and roller3 start to move.

During the period from t03 to t1, the tensioning force fluctuates and gradually rises from 1.63 N to 2.14 N until it overcomes the prestress of spring1. The linkage pushes slider1 to move downward. At time t1, slider1, and the pawl are separated, and the spool starts to release the yarn.

During the period of t1 to t2, the tensioning force drops to 1.64 N, then rises to 2.04 N, and drops to 1.63 N. After the fluctuation, the releasing is completed.

During the period from t2 to t3, the change in the tensioning force is the same as that from t03 to t1, and then it loops.

The tensioning force exhibits sudden changes while the external conditions are altered. These changes bring vibration to the traction manipulator. Every sudden change in the tensioning force will lead to a shock to the traction manipulator. Thus, with an increasing braiding speed, the frequency of the shocks increases. Because there are hundreds of carriers in a large braiding machine, the shock cannot be ignored. For example, in a large braiding machine with a 440-carrier, the shock is approximately 400 N.

To verify the validation of the present model, dashed lines connecting each initial and end dot of each period were drawn in Figure 7. The trend of the tensioning force is in accordance with the experimental results in the literature.¹ Compared to the traditional carrier, the fluctuating value of the tensioning force in the present carrier falls to a ratio of 10.9%, which is obviously smaller than that of the abovementioned article. However, the fluctuating frequency climbed. Furthermore, sudden changes can be found in Figures 8 and 9. However, the rotational angle of the linkage and rocker do not indicate sudden changes because the braiding speed is assumed to be constant in the dynamical equations, which is a fluctuating value in practical engineering.

Conclusion

In this article, an improved carrier was proposed to adapt the braiding for carbon fibers. The mathematical model was established and verified by numerical analysis and previous studies. The results show that the present model reduces the fluctuating value of the tensioning force. Moreover, the tensioning force exhibits sudden changes if the traction manipulator maintains a constant braiding speed. This conclusion provides a good explanation for the vibration of traction manipulators in practical engineering. If the vibration is reduced, then the braiding speed could be increased, leading to improved productivity.

Therefore, the next study could focus on optimizing the parameters to reduce the sudden changing value of the tensioning force. Alternatively, a tension controller could be added at the end of every carrier. On this occasion, the braiding speed will change with time. This change means that the braided pitch length changes, which will influence the mechanical properties of the carbon fiber preform. Designers should check whether this change is permitted.

Footnotes

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.

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 Program of Jiangsu Innovative Talent under grant [20171122] and the Fundamental Research Funds for the Central Universities under grant [2232019D3-31].

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