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Abstract

When the direct cabling machine produces cord, the active yarn feeder is usually used to control the balloon shape to reduce the energy consumption in the process of twisting. In order to rationally allocate the resources in the production workshop of direct cabling machine, the work proposed a prediction method of energy-consumption reduction for direct cabling machines based on balloon theory. The energy consumption of direct cabling machine with different balloon configuration parameters can be obtained. The prediction method consists of three main steps: (1) Analyze yarn force under the high velocity based on yarn balloon kinematics. (2) Take the energy consumed by the direct cabling machine corresponding to a balloon shape as the energy consumption benchmark. (3) Compare balloon shapes that need to be predicted with the referenced balloon shape to obtain the energy consumption prediction in a ratio. A mathematical balloon model was established on the MATLAB platform to test the influences of different working conditions on the prediction method. The simulation results showed that the influences of the yarn linear density, twist, and spindle speed on the method could be neglected. An experimental platform was built to test the energy consumption of the direct cabling machine under different working conditions and verify the rationality of the method. The results showed that the difference between the predicted energy consumption and the experimental results is acceptable.

Keywords

Direct cabling machine balloon theory mathematic model prediction curve energy consumption

Introduction

Energy consumption has been a major issue in the textile industry. Cords are mainly produced by the direct cabling machine, and energy consumption accounts for 4/5 of the production cost.¹ The energy consumption of the twisting process accounts for most of the energy consumption of the direct cabling machine, except for energy consumed by motor heating and wear. The yarn with pre-tension is drawn out from the transverse outlet of the yarn storage disk perpendicular to the spindle axis in the process of yarn twisting. Then, the yarn is pulled through the yarn guide fixed above the spindle. Finally, the twister located above the spindle is guided through the overflow plate.² The yarn is rotated with the spindle, which forms a balloon shape under air resistance and centrifugal force.

(1) Research on the theory of balloon

Some researchers focused their research on improving the balloon theory. Jaroslav³ applied the expanded yarn theory to calculate the reaction force, established the equation of motion of yarn under two reaction forces, and analyzed the normal reaction force composition of the free balloon system and the control balloon system. Shu⁴ used Hamiltonian principle to establish the variational equation of balloon yarn. Cave⁵ systematically investigated the effect of yarn tension on the stability of the balloon. Zhang⁶ considered the effect of Coriolis force and air resistance on the balloon and established a mathematical model of the balloon during yarn unwinding.

Another part of the research focused on applying the balloon theory to existing textile equipment Praček⁷ improved the unwinding process of cross winding bobbin and improves the unwinding stability based on the balloon theory. Chattopadhyay et al.⁸ investigated the effect of air resistance generated by vacuum pressure, Cairo roll speed and yarn linear density on the tension of the balloon during rotor spinning. Mei⁹ studied the influence of spindle angular velocity, yarn fineness and other factors on the tension of balloon of direct cabling machine, and established the fitting equation. Ni¹⁰ combined with the small-diameter balloon control ring concept and the high-speed spinning model of multi-nodule balloon control, and put forward a spinning technology for controlling open-close small diameter balloon ring. Hossain^11–13 studied yarn tension and balloon shape during ring spinning and proposed a mathematical model as a function of spindle speed. A superconducting magnetic bearing (SMB) system was also introduced to overcome the friction limitations of ring spinning frame. Ao et al.¹⁴ studied the twist and distribution of the wrapped yarn after optimizing the wrapped spinning of hollow ingot based on the balloon theory. Many scholars have improved the mathematical model of the balloon of ring spinning frame by improving the control algorithm^15,16 and considering the special factors.^17–20 It can be seen that the balloon theory is mature in the field of textile machinery, especially in the ring spinning frame, but there is a lack of balloon research on the direct cabling machine.

(2) Research on energy consumption of direct cabling machine

Researchers have studied the energy consumption of direct cabling machines. Yang et al.²¹ analyzed the structure and energy-saving technology²² of direct cabling machines and discussed factors affecting energy consumption. Cheng et al.²³ tested the tension of each process part through experiments, providing a theoretical basis for the tension control system.²⁴ The problems and advantages of existing energy-saving schemes were also discussed by analyzing the current direct cabling machines on the market. Sun et al.²⁵ took the CC3 direct cabling machine designed by Allma Company as the research object. They adopted the direct torque control strategy to design a common DC bus system based on a fully controlled rectifier. Therefore, multiple motors can utilize feedback energy for energy saving. Qiao et al.²⁶ applied a voltage regulator to an asynchronous motor. They studied the influences of electric volume changes such as input voltages, speeds, and active power consumption on the energy-saving effect. Wang et al.²⁷ studied the structures of the direct cabling machine. A method of energy saving and consumption reduction is proposed according to energy consumed by each moving part of the direct cabling machine. Some scholars study the energy consumption of straight twisting machine and the shape of gas ring according to different influencing factors.^28–30 It can be seen that the research on energy consumption of direct cabling machine mainly focuses on improving the existing structure and control scheme, but there is a lack of theoretical research on the relationship between balloon shape and energy consumption of Direct cabling machine.

In order to reasonably adjust the resource allocation of curtain production by direct cabling machine, aiming at the energy consumption changes caused by the change of balloon shape in the active yarn control device, an energy consumption prediction method with different balloon shapes was proposed based on the balloon theory. The work was organized into the following steps to predict energy consumption. Firstly, force on the yarn in the balloon shape was analyzed based on yarn balloon kinematics. Secondly, energy consumed by the direct cabling machine under a certain balloon shape was recorded as the energy consumption benchmark. Then, balloon shapes that needed to be tested were compared with the reference balloon shape. Finally, the energy consumption prediction of the direct cabling machine was obtained in a ratio. The influences of various parameters on the prediction method for direct cabling machines were obtained through numerical simulations and experimental verification. The results showed that the method has high accuracy.

Theoretical balloon model in direct cabling machines

The control system of direct cabling machine is shown in Figure 1. Two untwisted outer spindles are fixed to the upper part of the frame. Under the action of the yarn feeding motor, the outer yarn passes through the yarn tensioner and enters the storage cylinder from the bottom of the spindle through the doffing tube. After being drawn from the outlet of the storage cylinder, the outer yarn is twisted back on the storage cylinder. The outer yarn wrapped on the storage cylinder passes through the over-run plate to the twisting device's yarn guide hole. The spindle motor drives the outer yarn to rotate at high speed, forming an balloon between the twisting device and the over-run plate. At the same time, the inner yarn located in the spindle can is drawn from the tension sensor to twist with outer yarn in twisting device. The twisted yarn is fed into a tension sensor and then passed through an overfeed motor and a buffer rod to the winding area. The tension sensor will detect whether the yarn is broken in real time to ensure the quality of the yarn. The rhythmic reciprocating motion of the yarn guide makes the yarn cross wound on the reel to obtain the finished cord.

Figure 1.

Unit yarn control system of a direct cabling machine.

The outer yarn at the spindle consists of sections $A B$ and $B D$ outside the spindle can (see Figure 2). The yarn in section $A B$ rotates under spindle drive to form a balloon. The yarn in section $B D$ is wrapped on the storage cylinder and over-run plate. The yarn in section $A B$ needs to overcome friction and air resistance in the working process.

Figure 2.

Yarn path.

Figure 3 shows the motion analysis of the yarn microelement in section $A B$ . $ϕ (r, ζ, z)$ is the static coordinate system. The coordinate origin is the center-point twisting device, which is flushed with the upper part of the balloon. The positive direction of the z-axis is straight down and coincides with the center line of the spindle can. New moving coordinate system $R (r, θ, z)$ is defined at the origin of the static coordinate system. The frame rotates around the z-axis with angular velocity $ω$ . The expression of the relative position relation between two coordinate systems can be expressed as follows.

ζ = θ + ω t

(1)

where

t

is the time used;

θ

the angle between the yarn and the r-axis of the moving coordinate system.

Figure 3.

Moving coordinate system.

The yarn is in the form of a static spatial curve in a dynamic coordinate system. $e_{r}$ , $e_{θ}$ , and $e_{z}$ represent the unit radial quantity, unit angle vector, and z-axis unit vector, respectively. Point $R$ at any point in the balloon is taken, and $s$ is the yarn length from point $R$ to the origin. The expression of position vector $R$ is as follows.

R = r e_{r} + z e_{z}

(2)

where

r

is the balloon radius at point

R

The expression of unit tangent vector $τ$ at position vector $R$ is as follows.

τ = \frac{d R}{d s} = r^{'} e_{r} + r θ^{'} e_{θ} + z^{'} e_{z}

(3)

The angular speed of spindle rotations can be expressed as follows.

ω = ω \cdot e_{z}

(4)

v_{s}

is the winding speed of yarns in the coordinate system (see Equation (5)).

v_{s} = - v_{s} (r^{'} e_{r} + r θ^{'} e_{θ} + z^{'} e_{z})

(5)

Micro-segment $d s$ on the yarn air coil is taken for analysis and it receives four external force and three accelerations. The force balance equation of $d s$ is as follows.

m_{0} (a_{n} + a_{k} + a_{s}) d s = (F_{T} + F_{p u} + F_{p t} + F_{G}) d s

(6)

where

m_{0}

is the yarn linear density;

a_{n}

the centripetal acceleration of

d s

;

a_{k}

the Coriolis acceleration of

d s

;

a_{s}

the relative acceleration of

d s

;

F_{T}

the increase in yarn tension;

F_{p u}

air resistance along the

d s

normal;

F_{p t}

air resistance along the

d s

tangent;

F_{G}

gravity of

d s

The motion equations of balloon yarns can be obtained in cylindrical coordinates by combining acceleration and force equations. Equation (6) is expressed in the static coordinate system as

{\begin{cases} m_{0} [{v_{s}}^{2} r^{″} - r {(v_{s} θ^{'} + ω)}^{2}] = T (r^{″} - r {θ^{'}}^{2}) + T^{'} r^{'} + 3 m_{0} r ω^{2} + p_{n} r^{3} r^{'} θ^{'} ω^{2} \sqrt{{r^{'}}^{2} + {z^{'}}^{2}} \\ m_{0} [2 r^{'} v_{s} (v_{s} θ^{'} + ω) + r {v_{s}}^{2} θ^{″}] = T (2 r^{'} θ^{'} + r θ^{″}) + T^{'} r θ^{'} - p_{n} r^{2} ω^{2} \sqrt[3]{{r^{'}}^{2} + {z^{'}}^{2}} \\ m_{0} {v_{s}}^{2} z^{″} = T z^{″} + T^{'} z^{'} + m_{0} g + p_{n} r^{3} z^{'} θ^{'} ω^{2} \sqrt{{r^{'}}^{2} + {z^{'}}^{2}} \end{cases}

(7)

The yarn is flexible without elongation. Only the influence of the normal air resistance on the yarn is considered because normal air resistance is perpendicular to the yarn (see Eq. (8)).

\frac{d R}{d s} \cdot \frac{d R}{d s} = 1, \frac{d R}{d s} \cdot F_{p u}

(8)

The dot product between equation (7) and $d R / d s$ is used to obtain

T^{'} = - m_{0} ω^{2} r r^{'} - m_{0} g z^{'}

(9)

Equation (9) can be expressed as follows by ignoring the gravity of the element.

T^{'} = - m_{0} ω^{2} r r^{'}

(10)

Equation (10) is substituted into equation (7) to obtain the aerodynamic yarn motion equation (see Equation (11)).

{\begin{cases} r^{″} = \frac{T^{'} r^{'} + (m_{0} {v_{s}}^{2} - T) r {θ^{'}}^{2} + m_{0} r ω^{2} - 2 m_{0} ω v_{s} r θ^{'} + P_{n} r^{3} r^{'} θ^{'} ω^{2} \sqrt{{r^{'}}^{2} + {z^{'}}^{2}}}{m_{0} {v_{s}}^{2} - T} \\ θ^{″} = \frac{r θ^{'} T^{'} - 2 r^{'} θ^{'} (m_{0} {v_{s}}^{2} - T) + 2 m_{0} ω v_{s} r^{'} - P_{n} r^{3} r^{'} θ^{'} ω^{2} \sqrt{{r^{'}}^{2} + {z^{'}}^{2}}}{r (m_{0} {v_{s}}^{2} - T)} \\ z^{'} = \sqrt{1 - {r^{'}}^{2} - {(r θ^{'})}^{2}} \\ T^{'} = - m_{0} ω^{2} r r^{'} \end{cases}

(11)

The above derivation formula refers to the derivation formula of the balloon section of yarn in the literature.³¹ According to the parameters of the spindle which belongs to the direct cabling machine, the initial value of the mathematical model equation for the balloon motion is determined.

(1) The outer yarn at the top passes through the guide hole at the bottom of the twisting device, whose position determines the radius of the top balloon.

r (0) = r_{0}, θ (0) = 0, z (0) = 0

(12)

where

r_{0}

is the radius of the guide hole.

(2) The yarn comes out from the outlet of the storage cylinder and enters section $A B$ through the over-run plate. The yarn tension changes due to changes in resistance or friction, the storage cylinder will adjust the wrapping angle on it according to the principle of force balance, to ensure that the yarn from the storage cylinder into the over-run plate tension is consistent. Therefore, when working normally, the tension of the balloon is always stable, and the position of bottom of the balloon is determined by the over-run plate.

r (z_{1}) = r_{1}, z_{z} = h

(13)

where

r_{1}

is the radius of the over-run plate;

h

the balloon height.

(3) Figure 4 shows the yarn in the storage cylinder and over-run plate. There will be a wrap angle after the yarn is sent from the outlet of the storage cylinder due to the high-speed rotation of the spindle. The range of wrap angle is $0 ° \sim 360 °$ . According to the short-range line theory, the angle range of the silk thread on the storage disk can be expressed as

α_{0} = \frac{d z}{d s} = t g^{- 1} [h_{2} / (2 π r_{2} \frac{θ}{360})]

(14)

where

h_{2}

is the distance between the outlet of the storage cylinder and the bottom of the over-run plate;

r_{2}

the radius of the storage cylinder.

Figure 4.

Yarn path.

A fan-shaped surface can be obtained by spreading the surface of the over-run plate (see Figure 5). According to the short thread theory, $C B_{1}$ is the possible yarn path in a certain situation. Angle $α$ is between the line $C B_{1}$ and the line $O B_{1}$ for yarns on the over-run plate, which changes with the wrapping angles. Point $B_{1}$ is the contact point between the yarn path from point $C$ and the developed surface of the over-run plate. The cone apex angle of the over-run plate is set as $θ_{0}$ . Therefore, the apex angle of the fan-shaped surface can be obtained as

θ_{1} = π \sin θ_{0}

(15)

Figure 5.

Expansion drawing of the over-run plate.

The length of section $O B$ is $r$ , and the length of section $B C$ is $a$ . As the wrap angle increases, the path of yarn on the over-run plate is closer to $C B_{0}$ . Instead, the wrap angle increases, and the yarn path on the over-run plate is closer to $C B$ . Thus, angle range $θ (z_{1})$ during the normal operation of the direct cabling machine can be obtained (see Equation (16)).

θ (z_{1}) = α, 90 ° > α > \sin^{- 1} (\frac{r - a}{r})

(16)

(4) Yarn movement is limited by the size of the spindle can (see Figure 6). According to the above, the constraints of the balloon shape can be obtained.

r (z_{2}) > r_{3}, r (z_{3}) > r_{3}

(17)

where

r_{3}

is the radius of the spindle can;

z_{2}

the z-axis position at the top of the spindle can;

z_{3}

the z-axis position at the bottom of the spindle can.

Figure 6.

Spindle-can size.

(5) Assuming that the yarn is flexible but not extensible,

θ^{'} (0) = 0

(18a)

Energy Consumption of the Balloon

The twisting mechanism consumes the major energy of the direct cabling machine in the process of twisting the yarn and forming the package. The main energy consumption of the twisting mechanism comes from energy required for spindle rotations in addition to the energy consumption such as the heating and wear of the equipment motor. It includes energy required for spindle idling and spindle rotation, where the latter causes the yarn to form a balloon. Energy consumed by the spindle during twisting is expressed as

P_{S p i n d l e} = P_{I d l e} + P_{B a l l o o n}

(19a)

where

P_{S p i n d l e}

is energy consumed by a spindle;

P_{I d l e}

consumed by idling a spindle;

P_{B a l l o o n}

consumed by the spindle to overcome the friction and air resistance of the yarn in the balloon section when the yarn is twisted.

$P_{I d l e}$ accounts for a small proportion of $P_{S p i n d l e}$ , while $P_{B a l l o o n}$ accounts for a high proportion of $P_{S p i n d l e}$ . When the varieties of yarns and the twisting velocity are determined, the yarn balloon size is closely related to energy consumption.

If the spindle size and speed and the yarn count unchange, and the wrap angle on the storage cylinder is greater than zero, the storage cylinder will balance the yarn tension according to the changes in the wrap angle. It stabilizes the balloon shape. When the wrap angle is close to zero, there will be no yarn on the storage cylinder, which cannot control the yarn tension. Outer yarn-feeding speed is adjusted to actively control the balloon shape, which solves the failure of the control tension of the storage cylinder. Due to the small friction coefficient on the surface of the storage cylinder, the change in work caused by the reduced friction without yarn winding can be ignored compared to the situation with yarn winding on the storage cylinder.

$d s$ is used for analysis (see Figure 7). $θ$ is denoted as the angle between segment $d s$ and the horizontal line. The direction of air resistance $f$ is opposite to the direction of the circular motions of yarns. Therefore, the expression of air resistance in the normal direction can be obtained as

F_{N} = f \cdot \sin θ

(18b)

where

F_{N}

is normal air resistance.

Figure 7.

Yarn subjected to air resistance.

The expression of work done by this segment on air resistance can be obtained as

w = F_{N} \cdot d \cdot d s = f \cdot d \cdot d l

(19b)

where

d l

is the projection length of

d s

on the plane;

d

the yarn diameter.

The expression of work done by the outer yarn on air resistance is as follows.

w_{f} = \int_{0}^{l} w = \int_{0}^{l} f d \cdot d l

(20)

Work done by the segment $d s$ along path $O B$ to overcome air resistance is the same as that along path $O A$ (see Figure 7). Therefore, work done by the balloon to overcome air resistance is mainly affected by the size of the balloon projected onto the plane.

Plane cartesian coordinate system z-r is established. Figure 8 shows the projection of the yarn balloon segment on the plane. Substituted $θ = 0, θ^{'} = 0, θ^{″} = 0$ into equation (11) yields the following formula.

{\begin{cases} r^{″} = \frac{- m_{0} ω^{2} r {r^{'}}^{2} + m_{0} r ω^{2}}{m_{0} {v_{s}}^{2} - T} \\ z^{'} = \sqrt{1 - {r^{'}}^{2}} \end{cases}

(21)

Figure 8.

Plane coordinate system of the balloon.

It can be seen that the shape of yarn air coil is mainly affected by yarn density $m_{0}$ , rotary angular speed $ω$ , yarn feeding speed $v_{s}$ and yarn tension $T$ . The angular speed of rotation is determined by the speed of the spindle motor. The feed speed is affected by the winding speed and is calculated by the twist of the yarn and the speed of the spindle motor. Therefore, when the twisting mechanism of the direct cabling machine, changes the shape of the air coil by controlling the yarn tension, it only needs to consider the influence of yarn yarn density, spindle motor speed and yarn twist, since its size has been fixed. Therefore, the influence of yarn linear density, spindle motor speed and yarn twist on the known twisting mechanism of direct cabling machine should only be considered.

The air resistance of the outer yarn driven by the spindle during high-speed revolution is expressed as

f = p_{n} {v_{n}}^{2}

(22)

where

P_{n}

is the coefficient of air resistance in the normal direction;

v_{n}

the t velocity of the yarn in the normal direction, the formula is

v_{n} = ω r

Work done by the yarn in the balloon section on air resistance is calculated as follows.

W_{f} = \int_{0}^{l} f d \cdot d l = p_{n} \int_{0}^{l} ω^{2} r^{2} d \cdot d l

(23)

Energy consumption ratio

K

is used to eliminate the air-drag resistance coefficient, which cannot be obtained by ordinary methods. The energy consumption of a certain balloon shape is taken as the benchmark, which can express the energy consumption of other balloon shapes. Energy consumption ratio

K

is expressed as

K = \frac{| W_{f B} - W_{f D} |}{W_{f D}} \times 100 % = \frac{| \int_{0}^{l} {r_{B}}^{2} d l - \int_{0}^{l} {r_{D}}^{2} d l |}{\int_{0}^{l} {r_{D}}^{2} d l} \times 100 %

(24)

where

W_{f B}

is the energy consumption of the current balloon shape;

W_{f D}

the reference. Equation (24) shows that energy consumption ratio

K

is only related to the balloon shape.

Simulation

The selection of relevant parameters should meet certain practical conditions in analyzing the yarn motion model in the balloon segment, and they significantly affect the solving process of the yarn motion model (see Table 1). The yarn selected in the work is Polyester, which is a common material for cord wires.

Table 1.

Parameters used in the simulation.

Parameter	Description	Value
$ρ$	Linear density of the yarn	1500 dtex
$h$	Height of the balloon	535 mm
$r_{0}$	Radius of the guide hole	37 mm
$r_{1}$	Radius of the over-run plate	73 mm
$r_{3}$	Radius of the spindle can	210 mm
$Z_{2}$	Vertical distance from the guide hole of the twisting to the top of the spindle can	17.5 mm
$Z_{3}$	Vertical distance from the guide hole of the twisting to the bottom of the spindle can	45.5 mm

Different parameters of the balloon shape were simulated in MATLAB. The parameters of balloon in different shape are obtained by numerical simulation of equation (11). Parameters are adjusted to maximize the balloon shape. $γ_{0}$ is the angle between the top of the balloon and the central axis, and maximum balloon radius $r_{m}$ is 19.1 mm. Parameters are adjusted to minimize the balloon shape, where $γ_{1}$ is the angle between the top of the balloon and the central axis, and maximum balloon radius $r_{m}$ is 12.6 mm. In the Fig. 9, the balloon shapes are divided into 13 groups (see Table 2).

Figure 9.

Simulation of the limited balloon.

Table 2.

Grouping of the balloon shape.

Serial number	$r_{m}$ (mm)	Serial number	$r_{m}$ (mm)
1	191	8	156
2	186	9	151
3	181	10	146
4	176	11	141
5	171	12	136
6	166	13	131
7	161

The rotational speed of the direct cabling machine is generally between 7,000-9,000 r/min due to different actual processing requirements. Different yarn twists and densities and spindle speeds should be compared. Table 3 shows seven groups of simulation parameters.

Table 3.

Parameters of different groups.

Group	Spindle speed (r/min)	Yarn twist (T/m)	Yarn linear density (dtex)
1	9000	390	1500
2	8500	390	1500
3	8000	390	1500
4	7500	390	1500
5	7000	390	1500
6	9000	350	1500
7	9000	390	1400

Energy consumption was recorded as a benchmark when $r_{m}$ was 19.1mm at different rotational speeds. Different parameter groups were simulated by MATLAB (see Figure 10 for the results). Figure 10(a) shows the influence of different spindle speeds on the $K$ value under the unchanged yarn twist and yarn density. Figure 10(b) shows the influences of different yarn twists and yarn densities on the $K$ value under the constant spindle speed.

Figure 10.

Relationship between $K$ and $r_{m}$

Figure 10 shows that the relationship between $r_{m}$ and $K$ within the working range of the direct cabling machine is close to a first-order function. When the spindle, yarn twist and yarn linear density changes, the relationship curves between $r_{m}$ and $K$ changes very small, and the maximum error is close to 0.008%.Therefore, the influence of spindle speed, yarn twist and yarn linear density on the predicted curve is negligible.

The above results show that the variation curve of $r_{m}$ and $K$ calculated from a certain reference can be applied to different working conditions for the spindle assembly of a direct cabling machine with a definite structure and size.

Experiments

An experimental platform was established to study the relationship between the balloon shape and energy consumption of the direct cabling machine. TC21 direct cabling machine was the object. TC21 direct cabling machine is a product from Zhejiang Rifa Textile Machinery Co., LTD. It can be used to produce double strand symmetrical tire cord. It is a kind of industrial silk twisting equipment with many advanced technologies. It has the characteristics of large winding, short process and high quality of finished cord. Figure 11 shows the physical picture of the experimental platform, and Table 4 shows the various parameters of the direct cabling machine used in the experiment.

Figure 11.

Experimental platform.

Table 4.

Parameters used in the experiment.

Parameter	Description	Value (mm)
$h$	Balloon height	535
$r_{0}$	Guide-hole radius	37
$r_{1}$	Over-run plate radius	73
$r_{3}$	Spindle-can radius	210

One working unit of the direct cabling machine was used in the experiment to control variables. The stepping motor was used to feed the outer yarn actively, which controlled the feeding speed of the outer yarn by adjusting its speed. The infrared sensor was set at 3/5 of the overall height of the balloon. The balloon radius was the largest according to simulation results. The infrared sensor was used to measure the balloon radius based on the frequency of the yarn movement in its detection area. The speed of the yarn-feeding motor and the balloon radius were controlled near the set value after processing data collected by the control system. The power meter was connected to the working unit of the direct cabling machine to monitor the power change of the working unit used in the experiment.

Figure 12 shows the radius measurement of outer-yarn balloon in the experiment, where the spindle drives the outer yarn to form balloon. One yarn rotation will cut the infrared ray twice caused by the infrared emitter. $t_{1}$ is the time interval between two cuts. Figure 13 shows the radius calculation principle. $l_{1}$ is the distance from the spindle center to the infrared ray. Rotation angle $β$ of the yarn-cutting infrared sensor line can be expressed as follows.

β = ω \cdot t_{1}

(25)

Figure 12.

Infrared-sensor installation.

Figure 13.

Principle of measuring the balloon radius.

Therefore, the balloon radius of the outer yarn is calculated as follows.

r = \frac{l_{1}}{\cos \frac{β}{2}}

(26)

The measured radius of the balloon is compared with the set value of the control system, and the comparison result is used to adjust the speed of the yarn-feeding motor.

When the inlet velocity was not controlled in the experiment, the maximum balloon radius fluctuated between 185 and 190mm. Maximum $r_{m}$ in the experiment was set at 183 mm to keep the wrapping angle of yarn winding on the storage cylinder around 0°. The input speed was reduced until the yarn contacted the spindle can, which led to yarn breakage. Minimum $r_{m}$ for the stable operation was 133 mm.

Adjust the control parameters until the balloon shape no longer increases to get the maximum balloon shape. Adjust the parameters until the balloon shape is at the edge of the thread breakage to obtain the minimum balloon shape in normal operation. Figure 14 shows the comparison of two balloon shapes. Figure 14(a) and (b) show the maximum and minimum balloon shapes, respectively. Figure 14(c) and (d) show balloon shapes after image processing. Based on the above description, the balloon shape was divided into 11 groups (see Table 5).

Figure 14.

Balloon in the experiment.

Table 5.

Grouping of balloon shapes in the experiment.

Serial number	$r_{m}$ (mm)	Serial number	$r_{m}$ (mm)
1	183	7	153
2	178	8	148
3	173	9	143
4	168	10	138
5	163	11	133
6	158

The spindle speed range of the direct cabling machine used in the experiment can be set from 7,000 to 9,000 r/min, and subsequent experiments will be carried out within the working range. Table 6 shows the idle power of the spindle in the direct cabling machine. Different groups of parameters were set in the experiment considering the influences of the spindle speed and the yarn twist and linear density on the experiment (see Table 7).

Table 6.

Data in the experiment.

Spindle speed (r/min)	Winding speed (m/min)	Idle power (w)
9000	23	105
8500	22	99
8000	21	96
7500	19	94
7000	18	87

Table 7.

Parameters of different groups in the experiment.

Group	Spindle speed (r/min)	Yarn twist (T/m)	Yarn linear density (dtex)
1	9,000	390	1500
2	8,500	390	1500
3	8,000	390	1500
4	7,500	390	1,500
5	7,000	390	1500
6	8,000	350	1500
7	8,000	390	1260

The parameters of $r_{m}$ in the control system were used to adjust the balloon shape and a power meter was used to record the power of the direct cabling machine for 30 s after the balloon was stabilized. Average power during this period was recorded as the power of the group.

The actual power of the direct cabling machine is subtracted from idle power to obtain power consumed by balloon. The power is denoted as a benchmark to calculate $K$ at different speeds when $r_{m}$ is 183 mm (see Figure 15 for the results).

Figure 15.

K value in the experiment.

The predicted power of the direct cabling machine is obtained by adding calculated balloon power and the idling power of the direct cabling machine (see Table 8 for the results). The predicted energy consumption is compared with the experimental energy consumption in terms of errors and power fluctuations (see Figure 16 for the results). Figure 17 shows the maximum error of each group.

Table 8.

Power of the direct cabling machine obtained by the prediction.

$r_{m}$ (mm)	$P (w)$
$r_{m}$ (mm)	Group 1	Group 2	Group 3	Group 4	Group 5	Group 6	Group 7
133	365	326	274	239	203	254	256
138	383	341	286	249	211	265	267
143	402	358	300	260	220	277	279
148	422	375	314	271	229	289	291
153	443	393	328	283	238	302	304
158	466	413	343	296	248	315	318
163	488	433	359	308	259	329	332
168	512	453	375	322	269	344	346
173	536	474	392	335	280	358	361
178	562	497	409	349	292	374	377
183	587	519	427	364	303	389	393

Figure 16.

Comparison between the simulation and experiment.

Figure 17.

Maximum error of each group.

When the spindle speed and the yarn twist and linear density are changed, the energy consumption of the direct cabling machine varies significantly (see Tables 8). The energy consumption of the direct cabling machine increases with the increased spindle speed. Besides, the energy consumption of the direct cabling machine is also in direct proportion to the yarn twist and linear density.

Changing the spindle speed and the yarn twist and linear density has little effect on energy consumption ratio $K$ (see Figure 15). The trend of the $K$ value obtained by the experiments in each group is the same and close to the curve predicted by the simulation.

Errors between actual power and the predicted value in each group of experiments increase first and then decrease with decreased $r_{m}$ (see Figure 16). Compared with Figure 16(a)–(e), the error is larger in the case of high spindle speeds. Compared with Figure 16(c), (e), and (f), the changes in the yarn linear density and twist have little influence on the error. Generally, the predicted power of each experiment does not exceed the fluctuation threshold of actual power. The maximum error for each group is between 3 and 7% (see Figure 17).

Conclusions

(1) The work established the mathematical model of the yarn balloon of the direct cabling machine and boundary conditions. Through the analysis of the plane balloon, it is found that the influencing factors of the balloon shape are yarn linear density, yarn twist and spindle speed. Based on the mathematical model, the energy consumption of yarn balloon was analyzed, and a prediction method of energy consumption was proposed based on changes in balloon shapes. Finally, the accuracy of the method was verified by simulations and experiments.

(2) The Simulation and experiments showed that the spindle speed and the yarn twist and linear density had little influence on the relation curve between the maximum radius of balloon and the ratio of energy consumption obtained by the energy consumption prediction method for the spindle of the direct cabling machine with the determined structure and size. Therefore, for the same direct cabling machine structure, the energy consumption prediction curve obtained by this method has good applicability.

(3) The predicted and experimental values of energy consumption under different yarn linear density, yarn twist and spindle speed were obtained through several groups of simulations and experiments. The result showed that the curves of the $K$ value obtained by each group changing with the maximum balloon radius obtained in each group were the same and were close to the simulation curve. The maximum prediction error is mainly concentrated in the range with the maximum balloon radius of 143-163 mm, while the error is small when the maximum balloon radius is 133-138 mm and 163-183mm. The error between the predicted value and the experimental value was within the acceptable range (3%–7%). In conclusion, the method has good accuracy.

Future work

(1) ptimize the control scheme of balloon of direct cabling machine. Improve the stability of balloon control of direct cabling machine and guarantee the quality of finished product.

(2) Optimize the algorithm of control scheme and improve the accuracy of energy consumption prediction method.

(3) Applying it in the actual production process, and optimize the balloon control strategy of direct cabling machine.

(4) The amount of experimental data was increased to further verify the accuracy of the method through more experimental data.

(5) Further research is carried out to explore the resource allocation problem of the production workshop of direct cabling machine based on the research results, and optimize the resource allocation in the process of cord processing.

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: National Natural Science Foundation of China; 11872337, 91841303

ORCID iD

Hua Zhang

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The energy consumption prediction of the direct cabling machine based on balloon theory

Abstract

Keywords

Introduction

Theoretical balloon model in direct cabling machines

Energy Consumption of the Balloon

Simulation

Experiments

Conclusions

Future work

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References