Optimization analysis of carrier-track collision in braiding process

Yao Lingling; Song Wei; Pan Wei; Lu Yapeng

doi:10.1515/aut-2024-0006

Article Open Access

Optimization analysis of carrier-track collision in braiding process

Yao Lingling , Song Wei , Pan Wei and Lu Yapeng

Published/Copyright: December 2, 2024

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From the journal AUTEX Research Journal Volume 24 Issue 1

Abstract

The relationship between the speed of the driver plate and the collision of the spindle and the track is studied in this work. It is theoretically analyzed that the spindle will collide with the track in different degrees in the process of motion, and the impact is more intense due to the vibration of the mechanism. First, the dynamic equation of the spindle-track model is established. Then, fireworks algorithm is used to optimize the speed of the driver plate, to reduce the collision and impact times of the spindle on the track, so as to reduce the severe collision times between the spindle and the track, resulting in the wear of mechanical parts and affecting the braiding effect. Finally, it is concluded that the number of collisions increases sharply with the increase in recovery coefficient when the random disturbance is not considered; second, the number of collisions decreases with the appropriate increase in friction coefficient when the random disturbance is not considered; and third, the number of collisions increases sharply with the increase in random disturbance.

Keywords: 3D braiding machine; collision dynamics; optimization

1 Introduction

With the rapid development of braiding technology and composite material technology, researchers have combined braiding technology with composite material processing technology to develop braided structural composite materials with excellent performance. Its superiority lies in its non-layered overall structure, which is highly valued by researchers.

In the braiding process, a certain degree of gap is designed in order to prevent the slider from getting stuck inside the track. Scholars have conducted relevant research on the braiding track due to the influence of the track on the movement of the spindle. Kyosev et al. [1,2] studied the fit between spindle and track. Liu et al. [3] optimized the braiding track and reduced the motion impact of the spindle during the transition stage. Li and Mao [4] designed a three-dimensional guide rail to replace the groove track, reducing the collision impact of the spindle during movement. In the actual process of movement, the spindle will collide with the track to a certain extent. For example, Hu [5] and Zhang Yujing studied the collision between the spindle and the track. Meng et al. [6] analyzed the related issues of spindle dynamics based on contact collision theory.

Although there are few research achievements on the collision between the spindle and the track, there is certain reference value for collision problems in other fields. Wang [7] analyzed the collision of mechanical systems with gaps. Jiang et al. [8] discovered relevant collision issues during the impact process. Brach [9] used algebraic methods to solve related collision problems. James et al. [10] predicted the movement of the ball after hitting the board. Al-Thairy and Wang [11] studied the variation in rigid body velocity during the collision processes. Liang [12] studied the collision situation considering friction. Chai [13] analyzed the collision problem of engineering machinery systems with gaps. Wang [14] analyzed the collision problem between beams, flexible supports, rods, etc. Guoguang et al. [15,16,17,18] analyzed the collision issues related to mechanical systems. Meng et al. [19] proposed the concept of contact collision for moving spindles. Fu et al. [20] used the Newton Euler equation to analyze the collision problems of multi-body systems. Zhenjie et al. [21,22,23] used the momentum impulse method to analyze the collisions of multi-body systems. Jianyao et al. [24] conducted an analysis of flexible collision problems.

In the analysis process of contact collision, research in the field of contact force models is very important. Therefore, a large number of scholars have conducted research on the establishment of contact force models in contact collision. In 1882, Hertz’s classic research “On the Contact of Elastic Solids” has always been the fundamental theory in the field of contact mechanics and has been highly favored by scholars. The contact force model proposed by Johnson [25] is derived based on pure elastic theory and is suitable for calculating contact forces in small loads and large clearances. However, in actual collision processes, energy loss is inevitable. And there is a certain deviation between the calculated values of the Hertz contact force model and the actual values. In order to address the issue of not considering energy loss during collision in the Hertz contact force model, Zukas et al. [26] proposed the Kelvin–Voigt model to address the aforementioned issues in the Hertz contact force model. However, in the calculation of the Kelvin–Voigt contact force model, negative values may occur due to discontinuous calculations and inaccurate damping calculations. The Hunt–Crossley contact force model proposed by Hunt and Crossley [27] addresses the shortcomings of the Kelvin–Voigt contact force model in calculating contact forces. The linear damping part of the Kelvin–Voigt contact force model has been improved, and the improved Hunt–Crossley contact force model is more consistent with the actual situation in calculating contact forces. Lankarani and Nikravesh [28] proposed the Lankarani–Nikravesh (L-N) model, which is used for collision situations with high recovery coefficients. Zhiying and Qishao [29] established the Zhiying–Qishao contact force model, which is suitable for situations without recovery coefficient limitations. Flores et al. [30] proposed a Flores contact force model based on the Hunt–Crossley model, which is suitable for situations without recovery coefficient constraints. Gonthier et al. [31] proposed a Gonthier contact force model based on the L-N model, which is suitable for situations without recovery coefficient constraints and small gaps.

Consider that there is a certain gap between the slider and the track during the braiding process, which leads to collision. First, the dynamic equation of the spindle-track model was established based on the Lagrange equation. Then, the fireworks algorithm was used to optimize the speed of the driver plate to reduce the number of collisions and impacts between the spindle and the track, thereby reducing the wear of mechanical components caused by severe collisions between the spindle and the track and affecting the braiding effect. Finally, MATLAB simulation was used to analyze the impact of the dial speed on the spindle-track collision under specific parameter conditions.

2 Theoretical analysis of circular braiding machine

Figure 1 shows the circular 3D braiding machine, which consists of robot, track plate, motor, yarn carrier, gear, driving plate, rack, braided ring, and so on. The number of driving plates is 352, 88 for axial yarns, 704 for braided yarns, and 11 for motors in the circular 3D braiding machine. The traction distance of the core is controlled by a traction motor and the six axes of the robot are controlled by six motors. In the braiding process, gears are driven by four main motors on the braiding rack, which leads to the rotation of the driving plate on track plate. In the braiding process, the core is clamped on the robot and the axis of the mandrel must be perpendicular to the braided surface.

Figure 1

The diagram of braiding process.

Multi-ring braiding equipment mainly consists of braiding chassis, braiding ring, spindle, etc. The axis coordinate system is established on the plane of the braided disc as shown in Figure 2. h _gr,in represents the distance between the center of the braiding ring and the center of the bobbin surface. h _gr,out represents the distance between the center of the other braiding ring and the center of the bobbin surface. r _sp represents the radius of braiding chassis plane. Assuming that the spindle starts from BB₁ in the track and moves counterclockwise through EE₁ and FF₁ toward DD₁, it is mainly divided into two parts of motion, namely, a smooth circular motion part and an impact transition part.

Figure 2

The coordinate system diagram of braiding chassis.

When the spindle of the braiding equipment moves in the circular phase, i.e., the BE and FD segments, the theoretical speed and acceleration of its movement are

(1) V = r 1 ω ,

(2) a = a n = r 1 ω 2 .

When the spindle of the braiding equipment moves in the linear transition phase, i.e., the EF segments, the theoretical speed and acceleration of its movement are

(3) V = r 1 ω cos 2 ( φ 2 − φ 1 ) ,

(4) a = a n i + a τ j = a n 2 + a τ 2 + 2 a n a τ cos ( φ 2 − φ 1 ) a n = r 1 ω 2 cos ( φ 2 − φ 1 ) a τ = d V d t = 2 r 1 ω 2 sin ( φ 2 − φ 1 ) cos 3 ( φ 2 − φ 1 ) .

According to equation (4), the theoretical acceleration of the spindle is

(5) a = r 1 ω 2 cos 3 ( φ 2 − φ 1 ) cos 4 ( φ 2 − φ 1 ) + 4 sin 4 ( φ 2 − φ 1 ) ,

where r ₁ represents the radius of the track, ω represents the rotation angular velocity of the spindle, φ ₂ − φ ₁ = ωt, a _n represents the centripetal acceleration, and a _τ represents the tangential acceleration.

The theoretical speed and acceleration curve of spindle movement by using MATLAB is shown in Figure 3.

Figure 3

Theoretical velocity and acceleration curve of spindle movement. (a) Theoretical velocity curve of spindle movement and (b) theoretical acceleration curve of spindle movement.

During the transition stage, the collision impact of the spindle and track is relatively large, as shown in Figure 3.

According to the above analysis, the Hunt–Crossley model and fireworks algorithm were used to optimize parameters of the collision model when establishing the dynamic model of the spindle motion process, reducing the number of collisions between the spindle and track. Taking the spindle, driver plate, two sliders as a multi-body system, the structure diagram of the spindle and track system is established as shown in Figure 4.

Figure 4

The structure diagram of the spindle and track system.

The dynamic model of the spindle movement is established according to Figure 4. Figures 1, 2 and 4 are sourced from the multi-ring braiding equipment jointly developed by Donghua University and Xuzhou Henghui braiding Machinery Co., Ltd. During the braiding process, the center point of the braiding chassis is taken as the origin of the coordinate system O _i. The direction of x _i and y _i is shown in Figure 4. O _c represents the center point of the spindle base. The coordinate system direction of spindle base is shown as x _c, y _c. O _s1 and O_s2 are, respectively, the center points of the slider. The coordinate system direction of the slider is shown as x _s1, y _s1, x _s2, and y _s2. The coordinates of the spindle base are (x _c, y _c). The coordinates of the slider are (x _s1, y _s1), (x _s2, y _s2). The distance between O _c and O _s1 is C _s1. The distance between O _c and O _s2 is O _s2, in addition, C _s1 = C _s2 = L. The angle between the line connecting the origin of the coordinate system O and the center point of the spindle base O_c and the x _i-axis direction of the coordinate system is α. The angle between the line connecting O _i O _c and O _i O _s2 is θ, as shown in Figure 4.

(6) x s 1 = x c − L · cos 90 ° − α − 1 2 · θ ,

(7) y s 1 = y c − L · sin 90 ° − α − 1 2 · θ ,

(8) x s 2 = x c + L · cos 90 ° − α + 1 2 · θ ,

(9) y s 2 = y c + L · sin 90 ° − α + 1 2 · θ ,

where α represents the angle that varies with time, and θ represents a constant value.

Take the first-order differential of the equation of x _c, y _c, x _s1, y _s1, x _s2, and y _s2 to obtain the following equations of V _xc, V _yc, V _xs1, V _ys1, V _xs2, and V _ys2:

(10) V xc = x c ,

(11) V y c = y c ,

(12) V xs 1 = x c − L · α · sin 90 ° − α − 1 2 · θ ,

(13) V y s 1 = y c − L · α · cos 90 ° − α − 1 2 · θ ,

(14) V xs 2 = x c + L · α · sin 90 ° − α + 1 2 · θ ,

(15) V ys 2 = y c − L · α · cos 90 ° − α + 1 2 · θ .

According to the above description, the kinetic energy of the system is further written as

(16) T = 1 2 · m c · [ ( x c ) 2 + ( y c ) 2 ] + 1 2 · J c · α 2 + 1 2 · m s 1 · x c − L · α · sin 90 ° − α − 1 2 · θ 2 + y c + L · α · cos 90 ° − α − 1 2 · θ 2 + 1 2 · J s 1 · α 2 + 1 2 · m s 2 · x c + L · α · sin 90 ° − α + 1 2 · θ 2 + y c − L · α · cos 90 ° − α + 1 2 · θ 2 + 1 2 + J s 2 · α 2 .

According to equation (16),

(17) ∂ T ∂ ( x c ) = m c · x c + m s 1 · x c − L · α · sin 90 ° − α − 1 2 · θ + m s 2 ∙ x c + L · α · sin 90 ° − α + 1 2 · θ ,

(18) d d t ∂ T ∂ x c = ( m c + m s 1 + m s 2 ) · x c + ( m s 1 + m s 2 ) · L · α · cos ( 90 ° − α ) · sin 1 2 · θ + ( m s 1 + m s 2 ) · L · α 2 · sin ( 90 ° − α ) · sin 1 2 · θ ,

(19) ∂ T ∂ ( y c ) = m c · y c + m s 1 · y c + L · α · cos 90 ° − α − 1 2 · θ + m s 2 ∙ y c · L · α · cos 90 ° − α + 1 2 · θ ,

(20) d d t ∂ T ∂ ( y c ) = ( m c + m s 1 + m s 2 ) · y c + ( m s 1 + m s 2 ) · L · α · sin ( 90 ° − α ) · sin 1 2 · θ − ( m s 1 + m s 2 ) ∙ L ∙ α 2 ∙ cos ( 90 ° − α ) ∙ sin ∙ 1 2 ∙ θ .

During the movement of the spindle, there are two pairs of contact pairs, namely, one contact pair between the guide block and the track, and the other contact pair between the spindle and the track. In order to maintain relative sliding, there will be gaps between the two contact pairs, which can lead to collision, vibration, and friction, affecting the dynamic performance of the spindle.

Substituting equations (17)–(20) in the Lagrange equation, the final dynamic equation for the collision between the spindle and the track is obtained as follows:

(21) m c · x c + x c · ( m s 1 + m s 2 ) + ( m s 1 + m s 2 ) · L · α ∙ cos ( 90 ° − α ) · sin ( 1 2 ∙ θ ) + ( m s 1 + m s 2 ) ∙ L ∙ α 2 ∙ sin ( 90 ° − α ) ∙ sin 1 2 ∙ θ = F g ( x , x ) − f x ( x ) − T 1 ∙ cos φ − F a + δ F * ,

(22) m c ∙ y c + y c ∙ ( m s 1 + m s 2 ) + ( m s 1 + m s 2 ) ∙ L ∙ α ∙ sin ( 90 ° − α ) ∙ sin 1 2 ∙ θ − ( m s 1 + m s 2 ) ∙ L ∙ α 2 ∙ cos ( 90 ° − α ) ∙ sin 1 2 ∙ θ = F h ( y , y ) − f y ( x , y ) + T 1 ∙ sin φ + δ F * ,

where F _g(x,x) and F _h(y,y) are contact forces in both directions, and f _x(x) and f _y(x,y) are friction forces in both directions. The centrifugal force is F _a = m·r _c·ω². The total mass of the spindle is m = 1.2 kg. α represents the angle between OO _c and the x-axis direction of the coordinate system. The coordinates of the spindle base are x _c = r _c·cosα and y _c = r _c·sinα. T ₁ represents the yarn tension. The mass of slider 1 is m _s1 = 0.3 kg. The mass of slider 2 is m _s2 = 0.3 kg. The distance between O_c and O _s1 is C _s1. The distance between O _c and O _s2 is C _s2. L = C _s1 = C _s2 = 8.4012 mm, and r _c = 65mm.

3 Optimization of the collision between the spindle and track and analysis of influencing factors

3.1 Fireworks algorithm principle

3.1.1 Explosive operation

Assume that the fireworks population is {x _1, x ₂,…,x _N} during initialization. The calculation of explosion intensity and amplitude is the most important operation when fireworks explode. In the algorithm, the explosion intensity of fireworks is defined as the number of sparks that appear around each fireworks explosion. The explosion intensity is

(23) S i = m · Y max − f ( x i ) + ε ∑ i = 1 N ( Y max − f ( x i ) ) + ε ,

where S _i is the number of sparks that appear around the fireworks when they explode, m is the total number of sparks, Y _max is the worst fitness value, ε is a very small number, and f(x _i) is the fitness value of the fireworks x _i.

The number of sparks needs to be limited by a function to prevent the number from being too large or too small. The function is

(24) Ŝ i = round ( a · m ) , if S i < a · m round ( b · m ) , if S i > b · m round ( a · m ) , otherwise ,

where Ŝ _i represents the number of sparks after restriction, 0 < a < b < 1 round represents a rounding function.

The explosion amplitude is

(25) A i = Â f ( x i ) − Y min + ε ∑ i = 1 N ( f ( x i ) − Y min ) + ε ,

where A _i represents the explosion amplitude of the fireworks. Â represents the total explosion amplitude. Y _min represents the optimal fitness value. The definitions of f(x _i) and ε are the same as above.

The purpose of the displacement operation is to cause the individual fireworks to move a certain distance to generate a variation spark, whose function is

(26) x i k = x i k + U ( − A i , A i ) ,

where U(−A _i, A _i) represents a group of random numbers within a certain explosion amplitude range. x i k is the kth dimension value of x _i.

3.1.2 Mutation operation

The design of mutation operation is based on Gaussian distribution, and the definition of mutation is

(27) x i k = x i k · g ,

where the definition of x i k is the same as above. g represents a random value that follows a Gaussian distribution with a mean and variance of 1.

3.1.3 Mapping operations

The purpose of the mapping operation is to ensure that each spark appears within the feasible region. The definition of a mapping operation is

(28) x i k = X LB , k + x i k % ( X UB , k − X LB , k ) ,

where x i k represents the position of spark beyond the feasible region. X _UB,k and X _LB,k are, respectively, the upper and lower bounds of the feasible region. % represents a surplus operation.

3.1.4 Selection strategy

Euclidean distance is

(29) R ( x i ) = ∑ j = 1 k d ( x i , x j ) = ∑ j = 1 k ‖ x i − x j ‖ ,

where d(x _i, x _j) represents the Euclidean distance between x _i and x _j. R(x _i) represents the sum of the distance between x _i and other individuals. k represents the total number of fireworks and sparks except for x _i.

Selection strategy of the fireworks algorithm is designed based on roulette method, and the selected probability of x _i is p(x _i)

(30) p ( x i ) = R ( x i ) ∑ j = 1 k R ( x j ) ,

where R(x _i) represents the sum of distance between x _i and other individuals. R(x _j) represents the sum of distance between x _j and other individuals.

3.2 Optimization objective and optimization parameter setting

3.2.1 Optimization goals setting

During the braiding process, the number of collisions between the spindle and track can be reduced by adjusting the speed of driver plate. Severe collision between the spindle and track can lead to the wear of mechanical components and affect the braiding quality. To improve the braiding quality and reduce wear of mechanical components, a fireworks algorithm is used to optimize the speed of driver plate to reduce the number of collisions between the spindle and track.

The optimization objective is to reduce the number of collisions between the spindle and the track. The optimization objective function is as shown in equation (33).

(31) X = F g ( x , x ̇ ) − f x ( x ̇ ) − T 1 ∙ cos φ − F a + δ F * − ( m s 1 + m s 2 ) ∙ L ∙ α ̈ ∙ cos ( 90 ° − α ) ∙ sin 1 2 ∙ θ − ( m s 1 + m s 2 ) ∙ L ∙ α ̇ 2 ∙ sin ( 90 ° − α ) ∙ sin 1 2 ∙ θ / ( m c + m s 1 + m s 2 ) ,

(32) A = findpeaks ∫ 0 t ∫ 0 t X d t d t ,

(33) fitness = num 1 + num 2 ,

where num1 = numel (find(A ≥ 0.003 m)), num2 = numel (find(A ≤ 0.001 m)),

num1 represents the number of peaks with collision displacement in the direction of x greater than or equal to 0.003 m, num2 represents the number of peaks with collision displacement in the direction of x less than or equal to 0.0001 m.

3.2.2 Optimization parameters selecting

According to the safe speed range of main motor for braiding equipment without random disturbance, the speed range is 0–1.875 × 10³ rpm. In actual production, the rated speed of the main motor is selected as 2,000 rpm, and the optimal range of the driver plate is 0–18 rad/s.

3.3 Analysis of the collision between the spindle and track

3.3.1 Effect of the speed of the driver plate on the collision between the spindle and track under different recovery coefficient conditions

The recovery coefficient refers to the ratio of the separation velocity of two objects along the normal direction of the contact before and after the collision to the approaching velocity without considering random disturbances. When the recovery coefficient is 0, it is a completely inelastic collision; when the recovery coefficient is 0–1, it is an inelastic collision; and when the recovery coefficient is taken as 1, it is a fully elastic collision, so the value of the recovery coefficient is between 0 and 1. Assuming that there is a small recovery coefficient of C _r1 = 0.1, C _r2 = 0.1, friction coefficient of µ ₁ = 0.1, µ ₂ = 0.1, and speed of the driver plate of 0 rad/s < w < 9 rad/s, the number of collisions is 5. As the speed of the driver plate increases to 9 rad/s < w < 18 rad/s, the number of collisions between the spindle and track increases significantly. For example, when taking C _r1 = 0.1, C _r2 = 0.1, µ ₁ = 0.1, µ ₂ = 0.1, w = 3 rad/s, and C _r1 = 0.1, C _r2 = 0.1, µ ₁ = 0.1, µ ₂ = 0.1, w = 10 rad/s, the displacement curve of the spindle in the x-direction when there is collision between the spindle and track is shown in Figure 5.

Figure 5

The displacement curve of spindle in the x-direction when there is collision between the spindle and track at C _r1 = 0.1, C _r2 = 0.1, w = 3 rad/s, and w = 10 rad/s.

The number of collisions increases as the recovery coefficient increases without considering random disturbances. Assuming that there is a large recovery coefficient of C _r1 = 0.9, C _r2 = 0.9, friction coefficient of µ ₁ = 0.1, µ ₂ = 0.1, and speed of the driver plate of 0 rad/s < w < 2 rad/s, and the number of collisions is 12. Assuming that there is a large recovery coefficient of C _r1 = 0.9, C _r2 = 0.9, and speed of the driver plate of w > 2 rad/s, the number of collisions between the spindle and track increases significantly. For example, when taking C _r1 = 0.9, C _r2 = 0.9, w = 1.9 rad/s and C _r1 = 0.9, C _r2 = 0.9, w = 5.8 rad/s, the displacement curve of the spindle in the x-direction when there is collision between the spindle and track is shown in Figure 6.

Figure 6

The displacement curve of spindle in x-direction when there is collision between the spindle and track at C _r1 = 0.9, C _r2 = 0.9, w = 1.9 rad/s, and w = 5.8 rad/s.

As can be seen from Figures 5 and 6, the number of collisions increases as the recovery coefficient C _r1 and C _r2 increases without considering random disturbances. Due to C _r1 = 0–1, C _r2 = 0–1, it is assumed that C _r1 = 0.9, C _r2 = 0.9, when the speed of the spindle in braiding equipment is 0 rad/s < w < 2 rad/s, the number of collisions between the spindle and track is relatively small, i.e., 12. At the same time, it is assumed that C _r1 = 0.9, C _r2 = 0.9, when the speed of the driver plate is w > 2 rad/s, the number of collisions between the spindle and track increases significantly.

3.3.2 Effect of the speed of the driver plate on the collision between the spindle and track under different dynamic friction coefficient conditions

The displacement curve of the spindle in the x-direction when the collision between the spindle and track at the friction coefficient of µ ₁ = 0.1, µ ₂ = 0.1, and the recovery coefficient of C _r1 = 0.1, C _r1 = 0.1 is shown in Figure 5. Meanwhile, as the friction coefficient is µ ₁ = 0.4, µ ₂ = 0.4, the recovery coefficient is C _r1 = 0.1, C _r2 = 0.1, the speed of the driver plate in braiding machine is 0 rad/s < w < 13 rad/s, and the number of collisions is 5. And with the increase in the speed of the driver plate in braiding machine w > 13 rad/s, the number of collisions between the spindle and track increases significantly. For example, when taking µ ₁ = 0.4, µ ₂ = 0.4, C _r1 = 0.1, C _r2 = 0.1, w = 9 rad/s and µ ₁ = 0.4, µ ₂ = 0.4, C _r1 = 0.1, C _r2 = 0.1, w = 17 rad/s, the displacement curve of the spindle in the x-direction when there is collision between the spindle and track is shown in Figure 7.

Figure 7

The displacement curve of spindle in the x-direction when there is collision between the spindle and track at µ ₁ = 0.4, µ ₂ = 0.4, C _r1 = 0.1, C _r2 = 0.1, w = 9 rad/s, and w = 17 rad/s.

As the friction coefficient increases to µ ₁ = 0.4, µ ₂ = 0.4, the recovery coefficient to C _r1 = 0.9, C _r2 = 0.9, the speed of the driver plate in braiding machine to 0 rad/s < w < 3.1 rad/s, the number of collisions is 5. And with the increase in the speed of the driver plate in braiding machine w > 3.1 rad/s, the number of collisions between the spindle and track increases significantly. For example, when taking µ ₁ = 0.4, µ ₂ = 0.4, C _r1 = 0.9, C _r2 = 0.9, w = 2.7 rad/s and µ ₁ = 0.4, µ ₂ = 0.4, C _r1 = 0.9, C _r2 = 0.9, w = 11.6 rad/s, the displacement curve of the spindle in the x-direction when there is collision between the spindle and track is shown in Figure 8.

Figure 8

The displacement curve of spindle in the x-direction when there is collision between the spindle and track at µ ₁ = 0.4, µ ₂ = 0.4, C _r1 = 0.9, C _r2 = 0.9, w = 2.7 rad/s, and w = 11.6 rad/s.

As can be seen from Figures 5 and 7, the degree of the collision improves with an appropriate increase in the dynamic friction coefficient µ ₁ = 0.4, µ ₂ = 0.4, and the recovery coefficient C _r1 = 0.1, C _r2 = 0.1 without considering random disturbances. Due to C _r1 = 0–1, C _r2 = 0–1, it is assumed that C _r1 = 0.9, C _r2 = 0.9, when the speed of the spindle in braiding equipment is 0 rad/s < w < 2 rad/s, the number of the collisions between the spindle and track is relatively small, i.e., 12. At the same time, it is assumed that C _r1 = 0.9, C _r2 = 0.9, when the speed of the driver plate is w > 2 rad/s, the number of collisions between the spindle and track increases significantly. As can be seen from Figures 6 and 8, assuming that random disturbances are not considered and the recovery coefficient takes a larger value, i.e., C _r1 = 0.9, C _r2 = 0.9, the degree of collision between the spindle and the rail will also improve with an appropriate increase in the dynamic friction coefficient of µ ₁ = 0.4, µ ₂ = 0.4.

3.3.3 Effect of the speed of the driver plate on the collision between the spindle and track under different random disturbances

Assuming that there is a small random perturbation of δ F ⁎ = 0.1 ⋅ cos ( t ) , a friction coefficient of µ ₁ = 0.1, µ ₂ = 0.1, the recovery coefficient of C _r1 = 0.1, C _r2 = 0.1, and speed of the driver plate of 0 rad/s < w < 8 rad/s, the number of collisions is 5. As the speed of the driver plate increases to w > 8 rad/s, the number of collisions between the spindle and track increases significantly. For example, when taking δ F ⁎ = 0.1 ⋅ cos ( t ) , µ ₁ = 0.1, µ ₂ = 0.4, C _r1 = 0.1, C _r2 = 0.1, w = 3.1 rad/s and C _r1 = 0.1, C _r2 = 0.1, w = 12 rad/s, the displacement curve of the spindle in the x-direction when there is collision between the spindle and track is shown in Figure 9.

$Figure 9 The displacement curve of spindle in the x-direction when there is collision between the spindle and track at δ F ⁎ = 0.1 ⋅ cos ( t ) {\delta }_{F}^{\ast }=0.1\cdot \hspace{.25em}\cos (t) , C r1 = 0.1, C r2 = 0.1, w = 3.1 rad/s, and w = 12 rad/s.$

Figure 9

The displacement curve of spindle in the x-direction when there is collision between the spindle and track at δ F ⁎ = 0.1 ⋅ cos ( t ) , C _r1 = 0.1, C _r2 = 0.1, w = 3.1 rad/s, and w = 12 rad/s.

Assuming that there is a small random perturbation of δ F ⁎ = 0.1 ⋅ cos ( t ) , a friction coefficient of µ ₁ = 0.1, µ ₂ = 0.1, the recovery coefficient of C _r1 = 0.9, C _r2 = 0.9, and speed of the driver plate of 0 rad/s < w < 1.9 rad/s, the number of collisions is 12. And as the recovery coefficient is C _r1 = 0.9, C _r2 = 0.9, speed of the driver plate is w > 1.9 rad/s, the number of collisions between the spindle and track increases significantly. For example, when taking δ F ⁎ = 0.1 ⋅ cos ( t ) , µ ₁ = 0.1, µ ₂ = 0.1, C _r1 = 0.9, C _r2 = 0.9, w = 1.8 rad/s and δ F ⁎ = 0.1 ⋅ cos ( t ) , µ ₁ = 0.1, µ ₂ = 0.1, C _r1 = 0.9, C _r2 = 0.9, w = 4.4 rad/s, the displacement curve of the spindle in the x-direction when there is collision between the spindle and track is shown in Figure 10.

$Figure 10 The displacement curve of spindle in the x-direction when there is collision between the spindle and track at δ F ⁎ = 0.1 ⋅ cos ( t ) {\delta }_{F}^{\ast }=0.1\cdot \hspace{.25em}\cos (t) , C r1 = 0.9, C r2 = 0.9, w = 1.8 rad/s, and w = 4.4 rad/s.$

Figure 10

The displacement curve of spindle in the x-direction when there is collision between the spindle and track at δ F ⁎ = 0.1 ⋅ cos ( t ) , C _r1 = 0.9, C _r2 = 0.9, w = 1.8 rad/s, and w = 4.4 rad/s.

Assuming that there is a small random perturbation of δ F ⁎ = 0.1 ⋅ cos ( t ) , a friction coefficient of µ ₁ = 0.4, µ ₂ = 0.4, the recovery coefficient of C _r1 = 0.1, C _r2 = 0.1, and speed of the driver plate of 0 rad/s < w < 11.9 rad/s, the number of collisions is 5. And as the speed of the driver plate increases to w > 11.9 rad/s, the number of collisions between the spindle and track increases significantly. For example, when taking δ F ⁎ = 0.1 ⋅ cos ( t ) , µ ₁ = 0.4, µ ₂ = 0.4, C _r1 = 0.1, C _r2 = 0.1, w = 10.8 rad/s and δ F ⁎ = 0.1 ⋅ cos ( t ) , µ ₁ = 0.4, µ ₂ = 0.4, C _r1 = 0.1, C _r2 = 0.1, w = 16.9 rad/s, the displacement curve of the spindle in the x-direction when there is collision between the spindle and track is shown in Figure 11.

$Figure 11 The displacement curve of spindle in the x-direction when there is collision between the spindle and track at δ F ⁎ = 0.1 ⋅ cos ( t ) {\delta }_{F}^{\ast }=0.1\cdot \hspace{.25em}\cos (t) , µ 1 = 0.4, µ 2 = 0.4, C r1 = 0.1, C r2 = 0.1, w = 10.8 rad/s, and w = 16.9 rad/s.$

Figure 11

The displacement curve of spindle in the x-direction when there is collision between the spindle and track at δ F ⁎ = 0.1 ⋅ cos ( t ) , µ ₁ = 0.4, µ ₂ = 0.4, C _r1 = 0.1, C _r2 = 0.1, w = 10.8 rad/s, and w = 16.9 rad/s.

Assuming that there is a small random perturbation of δ F ⁎ = 0.1 ⋅ cos ( t ) , a friction coefficient of µ ₁ = 0.4, µ ₂ = 0.4, the recovery coefficient of C _r1 = 0.9, C _r2 = 0.9, and speed of the driver plate of 0 rad/s < w < 3 rad/s, the number of collisions is 5. And as the speed of the driver plate increases to w > 3 rad/s, the number of collisions between the spindle and track increases significantly. For example, when taking δ F ⁎ = 0.1 ⋅ cos ( t ) , µ ₁ = 0.4, µ ₂ = 0.4, C _r1 = 0.9, C _r2 = 0.9, w = 2.6 rad/s and δ F ⁎ = 0.1 ⋅ cos ( t ) , µ ₁ = 0.4, µ ₂ = 0.4, C _r1 = 0.9, C _r2 = 0.9, w = 16 rad/s, the displacement curve of the spindle in the x-direction when there is collision between the spindle and track is shown in Figure 12.

$Figure 12 The displacement curve of spindle in the x-direction when there is collision between the spindle and track at δ F ⁎ = 0.1 ⋅ cos ( t ) {\delta }_{F}^{\ast }=0.1\cdot \hspace{.25em}\cos (t) , µ 1 = 0.4, µ 2 = 0.4, C r1 = 0.9, C r2 = 0.9, w = 2.6 rad/s, and w = 16 rad/s.$

Figure 12

The displacement curve of spindle in the x-direction when there is collision between the spindle and track at δ F ⁎ = 0.1 ⋅ cos ( t ) , µ ₁ = 0.4, µ ₂ = 0.4, C _r1 = 0.9, C _r2 = 0.9, w = 2.6 rad/s, and w = 16 rad/s.

Assuming that there is a large random perturbation of δ F ⁎ = cos ( t ) , a friction coefficient of µ ₁ = 0.1, µ ₂ = 0.1, recovery coefficient of C _r1 = 0.1, C _r2 = 0.1, and speed of the driver plate of 0 rad/s < w < 7.3 rad/s, the number of collisions is 5. And as the speed of the driver plate increases to w > 7.3 rad/s, the number of collisions between the spindle and track increases significantly. For example, when taking δ F ⁎ = cos ( t ) , µ ₁ = 0.1, µ ₂ = 0.1, C _r1 = 0.1, C _r2 = 0.1, w = 3.6 rad/s and δ F ⁎ = cos ( t ) , µ ₁ = 0.1, µ ₂ = 0.1, C _r1 = 0.1, C _r2 = 0.1, w = 13.3 rad/s, the displacement curve of the spindle in the x-direction when there is collision between the spindle and track is shown in Figure 13.

$Figure 13 The displacement curve of spindle in the x-direction when there is collision between the spindle and track at δ F ⁎ = cos ( t ) {\delta }_{F}^{\ast }=\hspace{.25em}\cos (t) , C r1 = 0.1, C r2 = 0.9, w = 3.6 rad/s, and w = 13.3 rad/s.$

Figure 13

The displacement curve of spindle in the x-direction when there is collision between the spindle and track at δ F ⁎ = cos ( t ) , C _r1 = 0.1, C _r2 = 0.9, w = 3.6 rad/s, and w = 13.3 rad/s.

Assuming that there is a large random perturbation of δ F ⁎ = cos ( t ) , a friction coefficient of µ ₁ = 0.1, µ ₂ = 0.1, recovery coefficient of C _r1 = 0.9, C _r2 = 0.9, and speed of the driver plate of 0 rad/s < w < 2 rad/s, the number of collisions is 12. And as the speed of the driver plate increases to w > 2 rad/s, the number of collisions between the spindle and track increases significantly. For example, when taking δ F ⁎ = cos ( t ) , µ ₁ = 0.1, µ ₂ = 0.1, C _r1 = 0.9, C _r2 = 0.9, w = 1.8 rad/s and δ F ⁎ = cos ( t ) , µ ₁ = 0.1, µ ₂ = 0.1, C _r1 = 0.9, C _r2 = 0.9, w = 4.4 rad/s, the displacement curve of the spindle in the x-direction when there is collision between the spindle and track is shown in Figure 14.

$Figure 14 The displacement curve of spindle in the x-direction when the collision between the spindle and track at δ F ⁎ = cos ( t ) {\delta }_{F}^{\ast }=\hspace{.25em}\cos (t) , C r1 = 0.9, C r2 = 0.9, w = 1.8 rad/s, and w = 4.4 rad/s.$

Figure 14

The displacement curve of spindle in the x-direction when the collision between the spindle and track at δ F ⁎ = cos ( t ) , C _r1 = 0.9, C _r2 = 0.9, w = 1.8 rad/s, and w = 4.4 rad/s.

Assuming that there is a large random perturbation of δ F ⁎ = cos ( t ) , a friction coefficient of µ ₁ = 0.4, µ ₂ = 0.4, recovery coefficient of C _r1 = 0.9, C _r2 = 0.9, the number of collisions between the spindle and track increases significantly.

As can be seen from Figures 9 and 13, when there is a small recovery coefficient of C _r1 = 0.1, C _r2 = 0.1 and a dynamic friction coefficient of µ ₁ = 0.1, µ ₂ = 0.1, the number of collisions between the spindle and track increases significantly. Meanwhile, as can be seen from Figures 10 and 14, when there is a large recovery coefficient of C _r1 = 0.9, C _r2 = 0.9 and a dynamic friction coefficient of µ ₁ = 0.1, µ ₂ = 0.1, the number of collisions between the spindle and track increases significantly. And as can be seen from Figures 11 and 12, when there is a small recovery coefficient of C _r1 = 0.1, C _r2 = 0.1, a dynamic friction coefficient of µ ₁ = 0.4, µ ₂ = 0.4, and when there is a large recovery coefficient of C _r1 = 0.9, C _r2 = 0.9, a dynamic friction coefficient of µ ₁ = 0.4, µ ₂ = 0.4, the number of collisions between the spindle and track increases significantly with the increase in random perturbation.

To sum up, no matter what the parameter conditions are, with the increase in random disturbance, the collision between the spindle and track intensifies.

4 Conclusion

In this work, the fireworks algorithm to optimize the speed of the driver plate under specific parameter conditions is studied in order to reduce the number of collisions and impacts between the spindle and the track. This further reduces the severe collision between the spindle and the track, which leads to the wear of mechanical components and affects the braiding effect. The main contents are summarized as follows:

The number of collisions increases sharply with the increase in recovery coefficient when the random disturbance is not considered; simultaneously, the spindle material is selected comprehensively based on the optimal recovery coefficient under different working conditions.
The number of collisions decreases with the appropriate increase in friction coefficient when the random disturbance is not considered.

Funding information: The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Applied Basic Research Programs of Changzhou (Grant number CJ20235047), Major Program of Natural Science Research in Higher Education Institutions in Jiangsu Province (Grant no. 21KJA460004), and National Natural Science Foundation of China (Grant no. 52375101).
Author contributions: The overall research and writing were completed by Yao Lingling. The overall correction was completed by Song Wei. The overall correction and and modification was completed by Pan Wei. The overall correction was completed by Lu Yapeng.
Conflict of interest: The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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Received: 2024-06-19

Revised: 2024-09-22

Accepted: 2024-09-25

Published Online: 2024-12-02

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/aut-2024-0006

Keywords for this article

3D braiding machine; collision dynamics; optimization

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