Off-center braiding process for complex composite preforms based on analysis of the geometric contour model of the mandrel

Abstract

The braiding process is an important process for composite fabrication. Aiming at the problem of inner fabric folds and outer fabric sparseness in the bending of the complex geometric mandrel during braiding, a control method for off-center braiding is proposed. The mathematical description of the yarn position in the braiding process under the off-center condition is carried out, solving the differential equation for the convergence length using Euler's method to obtain the convergence length at any moment. The prediction model of the yarn on the mandrel is established to describe the spatial position of the fell point in the coordinate system of the mandrel end face and the coverage rate of the preform is solved by using the coverage rate formula under the off-center condition. The braiding experiments of the two mandrels are carried out to verify the feasibility of the prediction model and the control method. The experimental results show that the prediction model of off-center braiding can ensure that the error between the predicted value and the actual value of the braiding angle is within 5°. At the same time, off-center braiding can reduce the inner coverage rate of the bending mandrel and improve the outer coverage rate, and the error of the coverage rate is controlled within 10%. The errors are mainly caused by neglecting the slip of the yarn on the mandrel and the variation of the fell point due to the yarn interaction.

Keywords

Complex preforms off-center braiding prediction model coverage rate

Braiding as a manufacturing process for composite preforms enables the rapid formation of highly interwoven yarn structures on a complex mandrel.

The carbon fiber preform manufactured by braiding technology is easy to control, because the braiding angle and coverage can be accurately calculated by means of formulas^1,2 when the yarn is evenly distributed on the mandrel without slip and yarn damage. At the same time, it has better mechanical properties after curing with resin, and has been widely used in automobile manufacturing, construction, aerospace³ and other fields. Controlling the complex mandrel braiding process without yarn sparse, stacking and other phenomena, while ensuring the braiding angle and optimizing the coverage rate, has become one of the key problems to be solved urgently in the composite material manufacturing industry.

In the braiding process of composite preforms, yarns are formed by the cross-winding of two groups of yarns that move clockwise and counterclockwise, so the braiding angle and coverage rate of the preform have a great influence on the mechanical properties.^4,5 Whether the material has the desired tensile behavior depends on the braid angle error, and the coverage error will lead to local resin accumulation. In order to obtain the braiding angle and coverage under different braiding conditions, researchers predicted the yarn trajectory. Na et al.⁶ introduced the minimum path condition for prediction of yarn on an arbitrary mandrel. Hajrasouliha et al.⁷ proposed a theoretical model to predict the braiding angle on the constant cross-section mandrel of circular braiding of any shape. Wu et al.⁸ proposed a numerical method to predict yarn tows on an irregular mandrel, and analyzed the influence of various process parameters on yarn braiding angles. Monnot et al.⁹ proposed an improved braiding model, which was verified on a complex body framework; Wang et al.¹⁰ used the finite element method to simulate the braiding process and proposed an effective method to predict the braiding angle. After considering the force between yarns, van Ravenhorst and Akkerman¹¹ derived the convergence length of the yarn and the trajectory on the mandrel mathematically to obtain the accurate yarn drop position.

However, when braiding a complex mandrel, it is often difficult to achieve the expected value of the braiding angle or coverage rate, so researchers obtained the corresponding controlled quantity to braid a high-quality preform by solving the inverse solution of the braiding process. Du and Popper¹² analyzed the braiding process of mandrel with variable cross section, obtained the relationship between the take-up speed and the spool speed, and determined the control parameters of the machine.Van Ravenhorst and Akkerman¹³ obtained the control data of the braiding process by analyzing the yarn trajectory and using inverse kinematics. Gondran et al.¹⁴ proposed an inverse solution method to solve the take-up speed curve of the complex mandrel braiding process, and the feedback correction was carried out according to the positive kinematics solution.

At the same time, the mandrel of a special-shaped structure includes the bending mandrel, flat mandrel and so on. When braiding the mandrel of a special-shaped structure, it will happen that the mandrel does not pass the center of the guide ring, and there is an off-center value. Michaeli et al.¹⁵ described the parameters of the tube mandrel under the off-center condition, but did not take into account the variation of continuous braiding. Kessels and Akkerman¹⁶ converted the rotational speed of the spool in the braiding process with an off-center value and obtained the braiding speed of the spool relative to the center point of the mandrel after eccentricity, but this was not the real speed of yarn braiding on the mandrel. For preforms with specific process requirements, the desired fabric structure can be obtained by actively setting an off-center value for braiding. Fouladi and Jafari Nedoushan¹⁷ studied the situation in which the rectangular mandrel was not woven through the center of the guide ring, and optimized the off-center value and feed rate, but did not describe the off-center process.

Radial braiding machines are increasingly being used for the braiding of complex composite preforms with spatially curved axes, smaller components such as robot arms, prostheses and hockey sticks¹⁸ and larger components such as high-speed rail bogies and propeller blades. In the braiding process, the robot is used to clamp the mandrel and move according to the set trajectory, and it passes through the center of the braiding point plane vertically at all times.¹⁹ This braiding method can theoretically ensure the accuracy of the braiding angle and the braiding coverage rate of the preform. However, in actual production, because the axial length of the bending mandrel surface is not equal in the circumferential direction, the braiding is always perpendicular to the center of the braiding point plane, which will lead to sparse fabric and fabric accumulation of the preform. Therefore, it is difficult to design a braiding mode for a mandrel with special-shaped structure.

In this paper, an off-center braiding control method is proposed. Braiding with an off-center value can make the braided preform obtain different braiding angles and yarn axial spacing in the circumferential direction, which can adjust the overall coverage rate to a certain extent, and reduce the accumulation and sparseness of yarn. At the same time, a different braiding speed relationship from Kessel and Akkerman¹⁶ is deduced under the off-center condition. The prediction model of the off-center yarn trajectory and the calculation formula of the coverage rate are established, and then the braiding angle and coverage rate at different positions are controlled by adjusting the off-center value, which can ensure the improvement of the preform quality in actual braiding.

Mathematical model of the braiding process

Braiding process

The important components of the radial braiding machine and the braiding process of the composites are shown in Figure 1. In Figure 1(a) the main functional elements of the radial braiding machine movement process are illustrated, where the carriers that carry the spool move along the serpentine groove driven by the horn gears. As shown in Figure 1(a), the carriers of the radial braiding machine are divided into two groups, each carrying yarns at a predetermined speed $ω$ for approximately clockwise and counterclockwise circular motion. The entire braiding system is shown in Figure 1(c). The motor drives the closed-loop gear train inside the braiding machine to start working, driving the horn gears and carrier movement, and the robot clamping the mandrel continues to advance according to the designed trajectory and the calculated take-up speed. Under such cooperation, the yarn passes through the guide ring from the spool and, finally, it is braided on the surface of the mandrel to form a composite preform.

Figure 1.
(a) Main functional elements of the radial braiding machine. (b) The movement of the carrier and (c) The KUKA Robot and the YUNLU COMPOSITES 176 carrier radial braiding machine.

The braiding angle of the preform is the angle formed by the yarn and the mandrel axis, which is expressed as $α$ and shown in Figure 2. The braiding angle is related to the radius of the guide ring, the radius of the mandrel and the distance between the guide ring plane and the braiding point plane, where the braiding point plane is the plane of the yarn fell point on the mandrel and the robot trajectory is solved on the basis of the initial braiding point plane. The specific mathematical relationship is as follows¹²
$\tan α = \frac{\sqrt{R_{g}^{2} - r^{2}}}{H}$
(1)
where H is the distance between the guide ring plane and the braiding point plane, namely the convergence length; R_g is the radius of the guide ring; r is the radius of the mandrel. When controlling the braiding process, the values of braiding speed and take-up speed need to be obtained to ensure the stability of the braiding angle. The relationship between the braiding angle and speed is as follows¹⁵
$\tan α = \frac{ω \cdot r}{V}$
(2)
where $ω$ is the angular velocity of the yarn braiding on the mandrel; V is the take-up speed for the robot clamping mandrel. In the meantime, the coverage rate is described by the ratio of the area of fiber covered on the mandrel surface to the surface area of the whole mandrel. The coverage rate C of the mandrel is determined by the following equation¹
${\begin{cases} α < α_{\min} C = 1 \\ α_{\min} < α < α_{\max} C = \frac{b n_{c}}{π r \cos α} - {(\frac{b n_{c}}{2 π r \cos α})}^{2} \\ α > α_{\max} C = 1 \end{cases}$
(3)
where b is the width of a single yarn under normal conditions; n_c is the number of spools in one direction; $α_{\min} = \min (\arccos \frac{n_{c} (b + d)}{2 π r}, \frac{π}{2} - \arccos \frac{n_{c} (b + d)}{2 π r})$ ; $α_{\max} = \max (\arccos \frac{n_{c} (b + d)}{2 π r}, \frac{π}{2} - \arccos \frac{n_{c} (b + d)}{2 π r})$ ; d is the yarn thickness. The prediction of the yarn trajectory plays a very important role in the braiding work. The preform structure can be predicted according to the input process parameters, and the required control quantity can be inversed according to the expected structure. In this paper, the formation law of the fabric in the braiding process is analyzed in the case of off-center braiding, shown in Figure 2, and the yarn trajectory prediction model is established.

Figure 2.
Main parameter of off-center braiding using the YUNLU COMPOSITES 176 carrier radial braiding machine.

Model assumptions

The mathematical model proposed in this paper is based on the following assumptions for the braiding process of a mandrel with special-shaped structure under the off-center condition:
the friction and interaction force among yarns and between the yarn and the braided ring are ignored;

the cross-sectional shape of the yarn is not considered;

in the braiding process, the serpentine motion of the spool is ignored.

in the braiding process, the angular velocity of the spool moving on the chassis is equal to that of the yarn moving on the guide ring.

Off-center braiding

When the mandrel passes vertically through the braiding point plane under the off-center condition and the ratio of the braiding speed to the take-up speed remains constant, fabrics with different structures can be obtained in different directions. The braiding angle and the axial spacing of yarns in the circumferential direction of the mandrel are different, resulting in different convergence lengths and coverage rates. Therefore, it is difficult to use the classical formulas of Equations (1) and (3) to calculate the braiding angle and coverage rate.

When the off-center value e exists, the relationship between the position of the yarn on the guide ring and the position of the yarn on the drop point of the mandrel during braiding is as shown in Figure 3. The angle relationship of the braiding process position of the yarn between the guide ring and the mandrel can be obtained as follows
$θ_{g} = {\begin{cases} θ + \arccos (\frac{e \sin θ + r}{R_{g}}), clockwise yarns \\ θ - \arccos (\frac{e \sin θ + r}{R_{g}}), counter-clockwise yarns \end{cases}$
(4)
where $θ$ is the angle position of the yarn fell point in the polar coordinate system with the mandrel center as the origin; $θ_{g}$ is the angle position of the yarn on the guide ring in the polar coordinate system with the center of the guide ring as the origin.

Figure 3.
Geometrical relationship among the mandrel, yarn and guide ring under the off-center condition.

The projection length $R_{x}$ of the distance from the center of the mandrel to the guide ring on the guide ring plane in the presence of an off-center value can be deduced by the geometric relation
$R_{x} = \sqrt{{(e - R_{g} \sin θ_{g})}^{2} + {(R_{g} \cos θ_{g})}^{2}}$
(5)

At the same time, the braiding angle is related to the angular velocity of the yarn cover on the mandrel and the axial take-up speed. On the guide ring, the angular velocity of the yarn is $ω$ . However, under the off-center condition, the angular velocity $ω_{m}$ of the fabric formed by the yarn on the mandrel is $d θ / d t$ , which is calculated by Equation (6)
$ω_{m} = \frac{d θ}{d t}$

$= {\begin{cases} \frac{ω \sqrt{R_{g}^{2} - {(e \sin θ + r)}^{2}}}{\sqrt{R_{g}^{2} - {(e \sin θ + r)}^{2}} - e \cos θ}, clockwise yarns \\ \frac{ω \sqrt{R_{g}^{2} - {(e \sin θ + r)}^{2}}}{\sqrt{R_{g}^{2} - {(e \sin θ + r)}^{2}} + e \cos θ}, counter-clockwise yarns \end{cases}$
(6)

In the braiding process with an off-center value, the angular velocity $ω_{m}$ of yarn braiding on the mandrel and the distance $R_{x}$ from the center of the mandrel to the edge of the guide ring are constantly changing, so the braiding is in a continuous unstable stage.²⁰ In this stage, the take-up speed is composed of the forming speed $V_{b}$ of the fabric and the moving speed $V_{p}$ of the braiding point, then
$V = V_{b} + V_{p}$
(7)
where V is the take-up speed, and the convergence length H is constantly changing due to the movement of the braiding point. At the same time, as shown in Figure 4, the relationship between the variation length of convergence length in a very short period of time and the moving speed of the braiding point is as follows
$d H = V_{p} d t$
(8)

Figure 4.
Diagram of each parameter of an arbitrarily complex mandrel braiding process.

The following differential equations can be obtained by combining Equations (1), (2) and (7)²⁰
$\frac{d H}{d t} = V_{m} (t) - \frac{ω_{m} r H}{\sqrt{R_{x}^{2} - r^{2}}}$
(9)
where $ω_{m}$ represents the angular velocity of the yarn on the mandrel at different angle positions in the braiding process, which changes with time t at constant braiding speed $ω$ , expressed by $ω_{m} (t)$ ; $R_{x}$ is used to describe the relative position of the mandrel center in the braiding process, which changes with time t at constant braiding speed $ω$ , and is expressed by $R_{x} (t)$ ; $V_{m} (t)$ represents the speed at different positions in the circumferential direction of the mandrel when the mandrel passes through the braiding point plane. Because the axial length of the surface at the bend of the mandrel is different, as shown in Figure 4, when the robot clamps the mandrel vertically through the braiding point plane at a constant take-up speed V, because the mandrel needs different braiding lengths at the same time, the take-up speeds at different positions in the circumferential direction of the mandrel are different. The take-up speed $V_{m}$ corresponding to different positions is calculated by the following equation
$V_{m} (t) = \frac{L_{m} (t)}{L} V$
(10)
where $L_{m} (t)$ is the surface axial length corresponding to the yarn positions at different times in the braiding process using a constant braiding speed $ω$ ; L is the length of the center line of the bending mandrel. Therefore, the differential equation is converted into a first-order linear differential equation
$\frac{d H}{d t} + \frac{ω_{m} (t) r}{\sqrt{R_{x} {(t)}^{2} - r^{2}}} H = V_{m} (t)$
(11)

Equation (11) is a first-order linear non-homogeneous differential equation, the general solution of which can be obtained by the method of variation of constants, as shown in Equation (12)
$H (t) = e^{- \int \frac{ω_{m} (t) r}{\sqrt{R_{x} {(t)}^{2} - r^{2}}} d t}$

$\times (\int V_{m} (t) e^{\int \frac{ω_{m} (t) r}{\sqrt{R_{x} {(t)}^{2} - r^{2}}} d t} d t + C)$
(12)

Since Equation (12) is a general solution of Equation (11), the convergence length at any moment can be determined by setting appropriate boundary conditions, but solving Equation (12) is more complicated and the integral term in Equation (12) is difficult to solve using conventional methods; therefore, in order to determine the convergence length at any time efficiently and quickly, the differential equation (11) is solved using numerical methods. The Euler method is a typical method for solving differential equations and is the basis for many complex algorithms. Setting the time step to $δ t$ , each discrete time point can be expressed as $t_{i + 1} = t_{i} + δ t$ and $t_{0} = 0$ . Firstly, the differential equation shown in Equation (11) is transformed as follows
$H^{'} (t) = V_{m} (t) - \frac{ω_{m} (t) r}{\sqrt{R_{x} {(t)}^{2} - r^{2}}} H (t) = f (t, H (t))$
(13)

In the meantime, it can be written in the following form
$H^{'} (t_{i}) = \frac{H (t_{i} + δ t) - H (t_{i})}{δ t}$
(14)

By further solving, the following relationship can be obtained
$H (t_{i + 1}) = H (t_{i}) + δ t f (t_{i}, H (t_{i}))$
(15)

Based on the actual knitting situation, the initial condition is set to $H (t_{0}) = H_{0}$ , so that the convergence length at each discrete time point can be solved for a fixed time step. Then the coordinate value of the yarn fell point in space at each time is obtained through the yarn trajectory model.

Mathematical model of yarn trajectory

In the braiding process, because straight yarn is assumed, the interaction between yarns is ignored, so this paper analyzes the trajectory of one yarn on the mandrel.

The Cartesian coordinate system ${x, y, z}$ is established at the end face of the mandrel, and the Cartesian coordinate system ${u, v, w}$ is established at the center of the guide ring, as shown in Figure 4. The fell point of the yarn on the mandrel is expressed by p_i, and each discrete time point $t_{i}$ corresponds to a fell point p_i. Here, s_i is the unit tangent vector at the discrete point C_i on the mandrel center line at time t_i. The braiding angle $α_{i}$ is represented by the angle between $s_{i}$ and $p_{i + 1} - p_{i}$ . Since the yarn is rotated through the guide ring at an angle $ω \cdot δ t$ in time $δ t$ , and the take-up robot clamps the mandrel at a distance $V \cdot δ t$ along the axis, the mandrel is considered fixed, as in Figure 5, with the guide ring moving in the direction of the mandrel axis, which is equivalent to the mandrel being braided along the axis at any time. Under the off-center condition, in order to facilitate the calculation of the trajectory coordinates of the yarn in the mandrel coordinate system, the position q_i of the fell point in the guide ring coordinate system is obtained first. The calculation formula is as follows
$q_{i} = {[\begin{matrix} r \cos (θ_{i}^{(n)}) \\ e + r \sin (θ_{i}^{(n)}) \\ H^{(n)} (t_{i}) \end{matrix}]}^{T} [\begin{array}{l} u \\ v \\ w \end{array}]$
(16)
where $H^{(n)} (t_{i})$ represents the convergence length of the nth yarn at time $t_{i}$ , where $n = 1, 2, \dots, n_{c}$ ; $θ_{i}^{(n)}$ represents the fell point angle of the nth yarn at time $t_{i}$ in the mandrel coordinate system shown in Figure 3. Here, $θ_{i}^{(n)}$ is obtained by substituting the position $θ_{g (i)}^{(n)}$ of the nth yarn at time $t_{i}$ in the guide ring coordinate system into Equation (4) for inverse calculation, and the calculation formula of $θ_{g (i)}^{(n)}$ is as follows
$θ_{g (i)}^{(n)} = θ_{g (0)}^{(n)} + ω \cdot t_{i}$
(17)

Figure 5.
Diagram of each parameter of the yarn trajectory prediction model.

Secondly, the relationship between the guide ring coordinate system and the mandrel end coordinate system is established, and the position p_i in the mandrel end coordinate system is obtained by the coordinate system transformation of Equation (18)
${(p_{i}, 1)}^{T} = (\prod_{j = 1}^{i} T_{j}) T_{0} {(q_{i}, 1)}^{T}$
(18)
where the transformation matrix $T_{0}$ represents the transformation of the mandrel end face coordinate system relative to the guide ring coordinate system at the initial time $t_{0}$ , $T_{0} =Trans(0,0, h_{0})Rot(x_{0}, π) Rot (z_{0},- \frac{π}{2})$ . The transformation process is described as shown in Figure 6, and the composite transformation is represented by the accumulative multiplication of transformation matrix $T_{1}$ to $T_{i}$ ; the calculation formula of $T_{i}$ is as follows
$T_{i} = Trans (d \sin β_{i,} 0, d \cos β_{i}) Rot (y, - γ_{i})$
(19)
where $γ_{i}$ is the angle between the unit tangent vector s_i and s_i–1, $γ_{i} = arccos (\frac{s_{i - 1} \cdot s_{i}}{‖ s_{i - 1} \cdot s_{i} ‖})$ ; $β_{i}$ is the angle between the vector formed by the discrete point C_i_–1 pointing to C_i on the mandrel and s_i–1; d represents the linear distance between adjacent discrete points on the mandrel. Therefore, the position information of the yarn on the mandrel at any time in the braiding process can be obtained to accurately predict the yarn trajectory.

Figure 6.
Coordinate transformation diagram.

Off-center braiding control method

It is necessary to control the structure of the preform when braiding, and the coverage rate as an important structural parameter has been received attention from researchers. For the case of off-center braiding, the fabric structure of the preform is simplified and divided into independent basic units, as shown in Figure 7.

Figure 7.
Basic structural unit of the off-center braiding preform fabric.

Among them, the length $X_{n} (n = 1, 2, \dots, n_{c})$ along the circumferential direction is $(2 π r) / n_{c}$ in the case of non-eccentricity. However, under the off-center condition, this is not an average distribution, but rather is determined by the distance between the intersection points formed by the warp and weft. At the same time, the axial spacing Y_n of the yarn is determined by the braiding parameters. In the stable braiding stage of non-eccentricity, it is $(2 π V) / (ω n_{c})$ , which is difficult to obtain by formula in the off-center stage. In this paper, X_n and Y_n, the circumferential and axial spacing of different braiding units, respectively, can be obtained by the position p_i of each yarn fell point solved in the previous section, and the calculation formula of the coverage rate after eccentricity is as follows
$C = 1 - (1 - \frac{b}{Y_{n} \sin α_{min}})$

$× [(1 - \frac{b}{Y_{n} \sin α_{\min}}) + \frac{2 b (\sin α_{\max} - \sin α_{\min})}{X_{n} \sin (α_{1} + α_{2})}]$
(20)
where $a_{\min} = \min (a_{1}, a_{2})$ ; $a_{\max} = \max (a_{1}, a_{2})$ It is worth noting that this coverage rate formula is also only available at a valid braid angle range, as shown in Equation (3).

In the process of braiding the preform, fabric accumulation and sparse fabric defects are prone to occur at the outermost and innermost positions of the bending mandrel. Therefore, the coverage rate after eccentricity is calculated for these two positions. If the coverage rate at these two positions meets the requirements, the above fabric defects will not occur at other positions of the preform. Based on the braiding symmetry, the warp and weft at the top and bottom positions of the mandrel form the braided intersection $α_{1} = α_{2}$ in the braiding process. It can be seen from Equation (20) that the coverage rate at this time is mainly affected by the size of the braiding angle and the axial spacing Y_n of the yarn.

The off-center value e is substituted into Equations (4), (6) and (15) to solve the braiding angle of the yarn at the above two limit positions. At the same time, the yarn position point in the end face coordinate system of the mandrel under $y = 0, x > 0$ constraints is extracted by Equation (18). Due to off-center braiding, the braiding angle and Y_n of one side of the fabric away from the center of the guide ring can be increased. Equation (20) shows that the coverage rate will be reduced in this case, so that the off-center value e can be controlled to meet the process requirements and eliminate the internal wrinkles. The braiding angle and Y_n of the fabric near the center of the guide ring decrease, and the coverage rate can be increased.

Results and discussion

For the braiding process of the bending mandrel, the off-center braiding method is used to solve problems such as internal folds. In order to verify the correctness of the off-center braiding theory proposed in this paper, Figure 8 shows that the braiding experiments have been carried out on two mandrels with different geometric shapes. In this case, a conventional cylindrical mandrel M1 and a complex geometric curved pipe mandrel M2 with a curved centerline were used.

Figure 8.
The size of the mandrel required for the experiment.

Firstly, set initial conditions for M1 mandrel, such as the initial convergence length, the number of spools, the take-up speed and the braiding speed. The results obtained from the theoretical model are compared with the actual braiding, the magnitude of error at different positions is analyzed and compared and the causes of the error are discussed in the light of the results of the theoretical model to determine the differences between the model and the actual braiding.

Secondly, off-center braiding tests were carried out on the bending geometric mandrel M2 under different off-center values. The structural optimization under different off-center values was compared, and the feasibility of off-center braiding to solve the defects of bending mandrel fabric was analyzed. At the same time, in the case of the off-center braiding bending geometric mandrel, the numerical model proposed in this paper was compared with the actual fabric and the applicability of the analysis model was analyzed.

In this paper, MATLAB R2020b software is used for numerical analysis in the experiment. The data of off-center braiding under the braiding theory proposed in this paper are calculated and recorded. The calculation is carried out on computers equipped with AMD R7-4800H CPU, GeForce RTX™ 2060 GPU, 16GB RAM and Windows 10 operating system.

The equipment used in the experiment is a circular radial braiding machine and a KUKA six-degree-of-freedom industrial robot KR 240 R2700-prime installed on the linear guide rail. Each spool in the ring radial braiding machine moves forward according to the track of the braiding machine chassis, and carries the yarn through the yarn barrel installed on the spool. Meanwhile, the motion system of the braiding machine consists of four servo motors, two vibration motors and its controller.

The process parameters of the braided preform are measured. According to the four positions shown in Figure 3, the four fabric structures with an average distribution in the circumferential direction are analyzed, that is, the measurement of A, B, C and D, so that the theoretical and practical error values can be recorded in different directions of the mandrel. For the M2 mandrel, the fabric formed at A and D positions on the mandrel is selected for measurement because it mainly solves the problems of innermost wrinkle and outermost sparseness.

Numerical study

In this paper, the M1 mandrel is used for numerical experiments. Among them, the initial convergence length is set to 50 mm, the number of spools is 88, clockwise and counterclockwise are 44 and the radius of the guide ring is 100 mm. The braiding is carried out with the braiding angle of 60° and the downward eccentricity of 40 mm. The relative error in the experiment is determined as the difference between the braiding angle calculated by the theoretical model and the actual measured value.

The numerical analysis experiment of the M1 mandrel is carried out by using the parameters provided in Table 1, and the results shown in Figures 9 and 10 are obtained. Among them, the braiding angle of the finished yarn at different initial positions is measured, and the error between the theoretical value and the actual value is shown in Figure 9. This paper analyzes the clockwise yarn. Due to the symmetry of braiding, the conclusion is also applicable to counterclockwise yarn. In the figure, taking the clockwise yarn with A, B, C and D as the initial positions on the mandrel, the braiding angle is measured once per braiding $π /4$ on the mandrel surface. It can be seen from the figure that in the braiding process, when different yarns are near position B, the convergence length of the clockwise yarn at this position is greater than that of the counterclockwise yarn, so that the clockwise yarn is braided to the mandrel surface in advance, and the error of the braiding angle measured is large. In positions A and D, due to the symmetry of braiding, the clockwise and counterclockwise convergence lengths are equal, so the error between the results and the model calculation is small.

Table 1.
Parameters for numerical analysis of the M1 mandrel

Mandrel code Spool speed $ω$ (rad/s) Take-up speed V (mm/s) Radius of the guide ring R_g (mm) Off-center value e (mm)

M1 0.1332 4 100 –40

Figure 9.
Braiding angle errors of yarns with different initial positions on the mandrel every $π /4$ along the circumferential direction.

Figure 10.
Curves of yarn convergence length and braiding angle with time at different initial positions: (a) clockwise yarn with initial position C and counterclockwise yarn with initial position B; (b) clockwise and counterclockwise yarns with initial position A; (c) clockwise yarn with initial position B and counterclockwise yarn with initial position C and (d) clockwise and counterclockwise yarns with initial position D.

Figure 10 shows the curves of the convergence length and the braiding angle of the clockwise and counterclockwise yarns with time at different initial positions. It can be seen from the figure that due to the symmetry in the braiding process, when the initial positions of the clockwise and counterclockwise yarns are at positions A and D, the curves are the same by solving the model. Similarly, the yarns with clockwise initial position C and counterclockwise initial position B have the same results; the yarns with clockwise initial position B and counterclockwise initial position C have the same results. It is worth noting that in the results calculated by the model in this paper, after the yarn is eccentrically braided on the mandrel, the maximum angle formed by the clockwise and counterclockwise yarns of the fabric at position D of mandrel is about 144°.

Experimental study

In this paper, the M2 mandrel is used for the braiding experiment, and the robot clamping mandrel is controlled to carry out off-center braiding. Figure 11 shows the results of the M2 mandrel under three different off-center process requirements. The braiding angle is 60°, and the braiding is carried out under the off-center values of 0, –20 and –40 mm. Figure 12 shows the fabric braided by the M2 mandrel under different off-center values. It can be seen in Figure 12(a) that the internal wrinkles are obvious without eccentricity, resulting in yarn slippage and accumulation. Meanwhile, Figures 12(b) and (c) show the inner side of the mandrel under off-center braiding. Off-center braiding can improve the size of the inner braiding angle in the braiding process, reduce the coverage rate of the yarn in the inner and reduce the phenomenon of yarn pushing, so that the fabric is relatively flat. At the same time, from Figures 12(d)–(f), it can be seen that off-center braiding can reduce the braiding angle and yarn axial spacing, and improve the coverage rate. Table 2 shows the braiding angles of the inner and outer bending sides of the fabric measured by some yarns. It can be seen from the table that the braiding angles of the yarns are different under different off-center values of the mandrel. The larger the eccentric distance, the greater the braiding angle on the mandrel away from the central side of the guide ring, and the smaller the braiding angle near the central side of the guide ring. When internal wrinkles occur, the fabric is compressed, and the braiding angle at the wrinkle is forced to increase.

Figure 11.
Off-center braiding fabrics under different conditions: (a), (d) e = 0 mm inner and outer side of the bending mandrel; (b), (e) e = –20 mm inner and outer side of the bending mandrel and (c), (f) e = –40 mm inner and outer side of the bending mandrel.

Figure 12.
Errors of the off-center braiding model under different conditions: (a) e = –40 mm and (b) e = –20 mm.

Table 2.
Fabric parameters of the M2 mandrel braided under different off-center values

Parameters of inner side of mandrel
Parameters of outer side of mandrel

Yarn Braiding angle (°)
Yarn Braiding angle (°)

e = 0 mm e = –20 mm e = –40 mm e = 0 mm e = –20 mm e = –40 mm

Y_inside(1) 60.4 55.9 43.4 Y_outside(1) 59.7 61.9 63

Y_inside(4) 61.8 59.8 56.4 Y_outside (4) 58.5 59.9 60.6

Y_inside(8) 63.5 64.5 67.8 Y_outside (8) 57.2 56.7 55.9

Y_inside(12) 64.7 67.4 72.2 Y_outside (12) 58.3 54.2 52

Y_inside(16) 65.3 68.8 73.8 Y_outside (16) 56.2 54.4 49.9

Y_inside(20) 65.3 69.2 74.2 Y_outside (20) 57.7 52.9 49.8

Y_inside(24) 65.1 68.9 74.1 Y_outside (24) 56.6 54.5 51.2

Y_inside(28) 64.8 68.4 73.7 Y_outside (28) 58.1 54.7 52.3

Y_inside(32) 64.6 67.9 73.2 Y_outside (32) 57.8 52.9 52.8

Y_inside(36) 64.5 67.6 72.9 Y_outside (36) 58.9 53.5 52.9

Y_inside(40) 64.5 67.5 72.8 Y_outside (40) 56.9 54.9 53

Y_inside(44) 64.5 67.6 72.9 Y_outside (44) 58.3 54.8 52.9

When two off-center values are different, by measuring the braiding angle of the fabric formed in the direction of the inner side of the mandrel bending (position D in Figure 3) and the outer side of the mandrel bending (position A in Figure 3), the coverage rate is calculated by Equation (20). Figure 11 shows the results of the comparison between the average coverage rate of normal braiding and that of off-center braiding. It can be seen from Figure 11(a) that when the off-center value e = –40 mm, due to geometric reasons in the actual braiding, the speed of the inner mandrel passing through the braiding point plane is fast, and the yarn slips to a certain extent, which makes the axial spacing of the yarn smaller. Therefore, the actual braiding coverage rate obtained after a period of braiding can be maintained at about 82%, which is larger than the calculated value of about 74%. Similarly, the error between the theoretical coverage rate outside and the actual coverage rate outside is about 8% due to the yarn slip. Figure 11(b) is the braiding result when the off-center value e = –20 mm. Compared with Figure 11(a), the smaller the off-center value, the smaller the coverage change of the inner and outer sides of the bending mandrel.

Discussion

Usually, the ratio of angular velocity to take-up speed is constant during the braiding process, but due to the complex pattern of yarn movement in the off-center braiding case, a speed change function needs to be proposed based on the geometric relationships in the off-center braiding process, and in previous research work, Kessel and Akkerman¹⁶ suggested that the structure of the fabric changes in a complex way when braiding through a non-center point and analyzed the process. In the theory of Kessel et al., the angular velocity after off-center braiding is the speed at which the yarn rotates around the center of the mandrel at the position of the guide ring. In real braiding, the angular velocity should be the speed at which the position of the fell point rotates relative to the center of the mandrel, and the angular velocity error is negligible with larger guide rings, but it produces a large braiding angle error when the guide ring is small. The mathematical model in this paper contains the angular velocity variation function and the take-up speed variation function for the complex mandrel off-center braiding process. This is a more accurate off-center braiding model than the ‘extended two-dimensional solution.' However, as the yarn interactions are not taken into account, the theoretical model calculation process is based on braiding one yarn at a time and obtaining the final fabric structure after a cycle, while the real braiding process with multiple yarns braiding at the same time and different convergence lengths of yarns at different positions in the off-center situation will lead to an influence between adjacent yarns, resulting in a deviation in the fell point. The solution to this problem requires the prediction of the fell point, taking into account the yarn interaction during the non-stationary braiding phase, rather than using the yarn interaction model proposed by van Ravenhorst and Akkerman.¹¹

For improving the coverage of the complex mandrel, in previous work Roy et al.²¹ proposed the use of elliptical guide rings to replace the existing conventional circular guide rings for braiding. Due to the geometric factors of the elliptical shape, different braiding angles can be obtained during the braiding process, but the yarn movement during braiding with elliptical guide rings is extremely complex and expensive to produce, different sizes of guide rings are required for different mandrels and Roy et al. did not give an exact mathematical equation for the relationship. To solve this problem, setting the off-center value to change the size of the local braiding angle is low-cost and efficient. However, since in this paper it is assumed for the entire braiding process that the yarn does not slip after landing on the mandrel, in actual braiding situations the yarn tends to slip under the tension of the yarn because the off-center braided yarn does not follow the geodesic direction of the mandrel. Van Ravenhorst and Akkerman¹³ also found that the slip was more pronounced at locations with a lower coverage rate and that the slip was significantly reduced by adding axial yarns, so the harmful slip can be stopped by adding axial yarns at appropriate locations in response to the slip.

Conclusion

This paper describes the off-center braiding process based on a mathematical process. The yarn trajectory prediction model of the braiding process is established by analyzing the change of convergence length under off-center conditions. In view of the wrinkles of the bent inner fabric and the sparseness of the bent outer yarn in the process of braiding complex geometric shapes, a new braiding method is proposed. Firstly, the trajectory of the yarn is theoretically predicted for the off-center braiding process, the structural parameters of the fabric on the mandrel are solved for different off-center values and the accuracy of the model is verified to ensure that the error is kept within 5°. The braiding process can be controlled according to the model results to achieve the purpose of reducing braiding defects and improving the quality and mechanical properties of the composite material. Secondly, through the braiding experiment, the braiding angle and coverage rate of the fabric obtained by off-center braiding were analyzed. The results show that the coverage rate of the preform obtained by off-center braiding in the bending inner and outer sides is close to the expected value. Braiding under different off-center values can improve the outer coverage and reduce the inner coverage, and eliminate the defects of fabric wrinkles. Meanwhile, the larger the off-center value is, the more obvious the effect is. Therefore, the off-center braiding control method proposed in this paper uses different off-center values to adjust the take-up trajectory of the mandrel according to the actual production accuracy, and can be used as the technical basis for manufacturing high-performance composite products.

Mandrel code	Spool speed $ω$ (rad/s)	Take-up speed V (mm/s)	Radius of the guide ring R_g (mm)	Off-center value e (mm)
M1	0.1332	4	100	–40

Parameters of inner side of mandrel	Parameters of outer side of mandrel
Y_inside(1)	60.4	55.9	43.4	Y_outside(1)	59.7	61.9	63
Y_inside(4)	61.8	59.8	56.4	Y_outside (4)	58.5	59.9	60.6
Y_inside(8)	63.5	64.5	67.8	Y_outside (8)	57.2	56.7	55.9
Y_inside(12)	64.7	67.4	72.2	Y_outside (12)	58.3	54.2	52
Y_inside(16)	65.3	68.8	73.8	Y_outside (16)	56.2	54.4	49.9
Y_inside(20)	65.3	69.2	74.2	Y_outside (20)	57.7	52.9	49.8
Y_inside(24)	65.1	68.9	74.1	Y_outside (24)	56.6	54.5	51.2
Y_inside(28)	64.8	68.4	73.7	Y_outside (28)	58.1	54.7	52.3
Y_inside(32)	64.6	67.9	73.2	Y_outside (32)	57.8	52.9	52.8
Y_inside(36)	64.5	67.6	72.9	Y_outside (36)	58.9	53.5	52.9
Y_inside(40)	64.5	67.5	72.8	Y_outside (40)	56.9	54.9	53
Y_inside(44)	64.5	67.6	72.9	Y_outside (44)	58.3	54.8	52.9

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 financially supported by the National Key R&D Program of China (Grant No. 2018YFB1308800) and Fundamental Research Funds for the Central Universities (Grant No. 2232020D-30).

ORCID iDs

Qiyang Li

Xinfu Chi

Yize Sun

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Parameters of inner side of mandrel				Parameters of outer side of mandrel
Yarn	Braiding angle (°)			Yarn	Braiding angle (°)
Yarn	e = 0 mm	e = –20 mm	e = –40 mm	Yarn	e = 0 mm	e = –20 mm	e = –40 mm
Y_inside(1)	60.4	55.9	43.4	Y_outside(1)	59.7	61.9	63
Y_inside(4)	61.8	59.8	56.4	Y_outside (4)	58.5	59.9	60.6
Y_inside(8)	63.5	64.5	67.8	Y_outside (8)	57.2	56.7	55.9
Y_inside(12)	64.7	67.4	72.2	Y_outside (12)	58.3	54.2	52
Y_inside(16)	65.3	68.8	73.8	Y_outside (16)	56.2	54.4	49.9
Y_inside(20)	65.3	69.2	74.2	Y_outside (20)	57.7	52.9	49.8
Y_inside(24)	65.1	68.9	74.1	Y_outside (24)	56.6	54.5	51.2
Y_inside(28)	64.8	68.4	73.7	Y_outside (28)	58.1	54.7	52.3
Y_inside(32)	64.6	67.9	73.2	Y_outside (32)	57.8	52.9	52.8
Y_inside(36)	64.5	67.6	72.9	Y_outside (36)	58.9	53.5	52.9
Y_inside(40)	64.5	67.5	72.8	Y_outside (40)	56.9	54.9	53
Y_inside(44)	64.5	67.6	72.9	Y_outside (44)	58.3	54.8	52.9