LI Xinrong()， CHANG Weiwei()， WANG Shengze()， LAN Tianbao()

1 School of Mechanical Engineering， Tianjin Polytechnic University， Tianjin 300387， China2 College of Mechanical Engineering， Donghua University， Shanghai 201620， China

Abstract： The problems of the bobbin capacity and the high speed unwinding yarn taking off the ring or not during the winding of the ring spinning machine were dealt with. The mathematical model of the ring winding machine motion law was analyzed and optimized. Through the analysis and study of the movement rule of bottom bobbins and the ring spinning frame spool forming， the mathematical model of active bottom bobbins forming movement was established. Also， the mathematical model of bobbins formation was simulated and optimized. The concept of “controlled nail” was put forward. On this basis， the mathematical model of the current tube forming motion was optimized. The experimental results demonstrated that the theory of “controlled nail” met the high-speed spinning winding theory. The theory of “controlled nail” not only did the capacity of the bobbin increase， but also the spinning speed and efficiency are both improved， reducing the high-speed winder winding off. This research could provide a theoretical basis for the development of a new ring spinning frame.

Key words： ring spinning frame; active ring rail lifting; bottom of bobbin forming; cam; pin

Introduction

The ring spinning machine winding forming system includes the ring rail， the yarn balloon， the elevating mechanism of the yarn guide and the high speed driving mechanism of the spindle. The corresponding functions are the twisting， winding and forming of the spool finishing[1-2]. Therefore， the movement law of the winding forming mechanism directly affects the forming and quality of the spool. At present， the ring spinning frame forming form is divided into three types. One type is the mechanical forming cam and mechanical forming pin mechanism， which are utilized in the old machine of domestic spinning machine[1]. The second type is an electronic cam plus mechanical forming convex nail forming mechanism[3-4]. The third type is the positive winding forming mechanism[5]. The mechanical forming cam mechanism includes the concussion and pause[6]， which are liable to cause yarn breakage[7-8]. In addition the shape of the pipe bottom depends on the position of the forming pin, which is heavily dependent on experience without a specific criterion[9]. The electronic cam forming， is to utilize the servo motor， including the function electronic cam， instead of the original mechanical forming cam[10]. In contrast， the original pull-type elevating mechanism and mechanical forming pin mechanism have not changed[10]. To a certain extent， the concussion and pause of the forming cam mechanism is solved， whereas the problem of instability and easy chatter of the pull-type elevating mechanism has not been solved[11]. Also， the tube bottom shape not only depends on the experience， but also it depends on a non-uniform standard. The active winding forming mechanism is a servo motor instead of a mechanical forming cam and a mechanical forming pin， whereas the turbine worm mechanical transmission is utilized， instead of the pull-type elevating mechanism. This is to solve the original mechanism existence problem， as well as the wide range of fluctuations of the yarn tension. Moreover， it reduces the rate of the yarn breakage and leads the bottom of the tube into a positive control[12-13].

In this paper，the mathematical model of the bottom forming movement was established. The mathematical model analysis of the pipe bottom forming was simulated and optimized. A new concept- control nail was proposed， which satisfied the high-speed spinning winding theory and provided the theoretical basis for the development of the new ring spinning machine.

1 Analysis of Winding Forming of Ring Spinning Machine

The winding and forming process of the spool is divided into the tube bottom forming and the tube winding. The structure is presented in Fig. 1， which consisted of two parts: the bottom of the tube and the tube body. Firstly， the bottom of the tube was finished and consequently the bottom of the pipe was finished.

Fig.1 Tube structure diagram

In Fig.1，mrepresents the level of each layer of yarn，lis the winding height，r0describes the radius of the empty tube， andRdenotes the maximum winding radius.

1.1 Analysis of winding process

In the bottom of the pipe forming stage， the first layer of the yarn was winded on the empty tube， whereas consequently each layer of the yarn was substantially covered with the yarn layer on the front layer， finally ascending layer by layer. With the winding of the layers， the yarn diameterdxand the forming angleγx/2 increased. When the tube diameterdxand the forming angleγx/2 reached the maximum value， the pipe bottom was finished.

According to the geometric knowledge， the analysis of Fig. 1 demonstrated that as the winding started， the level increase of themand the winding heightlincreased layer by layer. Following the end of tube forming，bothmandlwere converted to constants. Consequently， the normal level of appreciation and the winding height completed the winding of the tube and the tube top[14].

1.2 Main winding parameters

In order to constitute the winding both hierarchical and methodical， the spool should be winded in the form of a spiral with equal normal pitch[15-16].

1.2.1Normalpitchhn

The normal pitchhnis the shortest distance between adjacent yarn loops.

1.2.2MaximumwindingradiusR

The maximum winding radius is also called the package radius. Although the low package radius increased the spindle speed， the yarn falls moved up in frequency. That will reduce the labor productivity. In addition， the package radius is too high， and it is not easy to sustain high-speed winding. At the same time, the power consumption is high. In contrast， the number of falling yarns is reduced， along with the number of times of the yarn. Finally， the production efficiency is improved. Generally， theΦ42×180 orΦ35×180 package is utilized[8].

Fig.2 Equal normal thread pitch helix

1.2.3Emptytuberadiusr0

The empty tube radius is the minimum winding radius. The general empirical data 2r0are equal to the 0.4 to 0.5 times ring diameter[11].

1.2.4Formingangleγ

The appropriate forming angle is selected. The significantly low package capacity will be reduced. If the yarn is high-sized， it is quite easy to unwind. Generally，γis in the range of 10°-15°[12].

1.2.5Yarnlengthofspiralwithequalnormalpitch

By the principles of textile science，it could be observed as follows[17].

The length of the winding layer can be defined as

(1)

The length of the binding layer can be defined as

(2)

where，Ris the radius of the tube，r0is the bobbin radius，h1is the pitch of the winding layer，h2is the pitch of the bond layer pitch andγis the forming angle.

1.3 Winding motion analysis

Following winding， the yarn was the spiral with equal normal pitch in the conical surface， as presented in Fig. 3. The parametric equation can be written as[11]

x=(R-Z×tanγ)cosθ,

y=(R-Z×tanγ)sinθ,

hz=hρcosγ,

(3)

where，Ris the radius of the tube，γis the forming angle of the spool， generallyγis in the range of 10°-15°， andhzis the projection of the normal pitchhρon theZaxis.

Fig. 3 Schematic diagram of yarn spiral

2 Mathematical Model of Ring Winding Movement in The Spinning Machine

Through the test analysis，the winding process must utilize the full range cross-winding method with the equal normal pitch， to adapt to the high-speed unwinding and high-speed spinning. The equal pitch winding mathematical model will be divided into a multi-segment yarn， as well as into sub-sections for precise control[14]. With the two sections consideration as an example， the model is divided into the bottom forming section and the pipe forming section.

2.1 Mathematical model of normal forming section of ring spinning machine

2.1.1Optimizationofwindingpitch

Subsequently to several tests and comparisons， the pitch of the winding layer should be adjusted to 5 times the diameter of the yarn， along with thek1between 0.9-1.3. On the basis of this， the equations for the adjustment optimization of the winding pitch was obtained[4].

(4)

(5)

where，Nmis the number of cotton yarns，h1for the winding layer pitch，h2for the bond layer pitch，inis the cam ratio (inis in the rage of 2-3) andk1is the winding pitch factor (k1is in the rage of 0.9-1.3).

2.1.2Shortliftparameters

According to the forming requirements and the geometrical rules of the spiral of the yarn,

(6)

where，vhis the ring lift speed，vis the yarn winding speed， which can be calculated from the measured value.hnis normal pitch which can be calculated by the yarn count.rkis the current winding radius. The following calculation is

(7)

where，lis the current yarn length of the winding，Lis the total length of the lap，Ris the tube radius andr0is the bobbin radius.

According to the cotton brochureit could be observed as

(8)

According to Eq. (8)， the ring speed can be calculated as

(9)

In the low-sized yarn stage， theγincreased from 0°to the maximum， consequently retaining the maximum value. In addition， the maximum value range was 10°-15°.

2.1.3Analysisoflevelincrease

From the theory of spinning and geometric knowledge， for each level of height[4],

(10)

where，mis the distance of level increase，k2is the winding density (g/cm3) andk2is in the rage of 0.56-0.62，k3is the level increase parameters， which is in the rage of 0.9-1.1.γis the forming angle， whereas the range of the forming angle is 10°-15°.

At the end of the pipe forming stage， the height of the level increase should be gradually increased， subsequently converted to a constant.

2.2 Mathematical model of bottom bobbins forming section of ring spinning machine

2.2.1Mathematicalmodelofactivebottombobbinsformingsectionofringspinningmachine

The spinning machine bottom bobbins forming not only increases the tube yarn capacity， but also improves the production efficiency. Through the ring short drive and distance changes of the level increase on each floor to complete the bottom bobbins forming， the tube with a higher capacity and a good shape could be obtained.

The normal forming section of the short drive distance isH，mis the height of the level increase of the normal stage.Hiis the winding height at theilayer，miis the height of the level increase at theilayer. Following analysis， the variableHiandmiwere handled and stored in a table， as presented in Table 1.

Table 1 Bottom forming height and the level of appreciation

The initial short drive value was set toH1， the starting value of the level increasem1.nis the number of ring movements required for the bottom forming of the pipe. Consequently， the mathematical model of the bottom bobbins forming can be established.

(11)

Hi=H1+(i-1)ΔH，

(12)

where ΔHis the increment of a short drive， and Δmis the increment of a level increase，i=1， 2， …，n.

Through the consideration of Eqs.(4) and (5)， Eq.(10) can be rewritten as

(13)

In addition，

(14)

mi=m1+(i-1)Δm.

(15)

In summary， the mathematical model of bottom bobbins forming can be expressed as

(16)

where，His short drive distance at the normal forming stage， which is generally set up at 46 mm;mis height of the level increase at the normal forming stage;i=1, 2, …，n.

The parameter valuesHiandmiwere adjusted for each grade of the ring. In this manner， theHiandmiincreased to the fixed value in the bottom bobbins forming stage， which was in the line with the requirements of the bottom bobbins forming.

2.2.2nvaluecalculation

Fig.4 Bottom bobbins shape schematic

In Fig.4，ais the height of the pipe bottom，his the height of the truncated cone，y=f(x) is the shape curve of the bottom of the pipe，Ris the radius of the full pipe，r0is the average radius of the bobbin，Gis the weight of the yarn at the end of the bottom bobbins forming，Vis the winding volume at the end of the bottom bobbins forming，k2is the winding density andk2=0.56 g/cm3.

(17)

πr2(a+h).

(18)

Through the consideration of Eqs.(17) and (18)，ncan be rewritten as

(19)

The yarn was set to 18.2 tex， the diameter of the yarn was set to 0.17 mm， the height of the level increase of the ring was set to 0.32 mm， the diameter of the spool was set to 39 mm， whereas the outer diameter of the bobbin was set to 23 mm.

From Fig.4，a=25 mm，h=46 mm， the coordinate ofAwas (11.5， 0)， the coordinate ofBwas (19.5，25). The curve equation was set as

y=bx2+cx+d,

(20)

when theAandBwere satisfied， the curve equation could be rewritten as

y1=0.1x2-13.225.

(21)

Through Eq. (18)，V1， the winding volume at the end of the bottom bobbins forming， can be rewritten as

26 182 mm3.

When the cam lift ratio is 1∶3， the transmission ratio forming cam to the front roller isI=92.54. Following the table checked，d=27 mm.

According to Eq. (19)，n1is written as

2.2.3Establishmentofmathematicalmodelforbobbinsformingcurveofringspinningmachine

The mathematical model can be written as

(22)

In summary， the bobbins forming became active and controllable through the mathematical model. The concept of the “controlled nail” was put forward. Through the process parameters input to adjust the bobbins forming of the ring spinning machine， the “controlled nail” met the high-speed spinning winding theory and the capacity of the bobbin increased. As a result， the bottom forming was well-stacked[18].

3 Simulation Analysis and Optimization of Bobbins Forming Curve

3.1 Simulation analysis

Similarly to the yarn during modeling，m1=0.12 mm andn=100.

Δm=(0.32-0.12)/100=0.002 mm

mi=0.12+0.002(i-1)，i=1， 2， …， 100.

According to formula (22)， formula (23) is

(23)

Through the curve fitting of the data point with Matlab， the shape of the bobbins forming curve could be obtained， as they2presented in Fig. 5. The curve equation following fitting is

y2=0.156 2x2-2.106x+3.558.

3.2 Optimization and verification

Δm=0.002 mm， which was quite low. Consequently， relatively higher agency requirements were required. It did not have the universal applicability. Therefore，m1and Δmwere adjusted through repeated testing. With the assumption the curve of the entire pipe was divided intotsegments， the starting and increment values of the level increase were fixed， whereas the bottom curve was drawn with thetlow-sized line.m1was set to 0.12 mm， so

Δm=(0.32-0.12)/t,

t=20.

Table 2 was consequently obtained.

Table 2 Comparison for the appreciation of the bottom of the pipe(t=20)

According to formula (23)， through the curve fitting of the data point with Matlab， the shape of the bobbins forming curve was obtained， as they3presented in Fig. 5. The curve equation following fitting is

y3=0.156 2x2-2.156x+4.137.

According to the data in Table 3 and formula (23)， through the curve fitting of the data point with Matlab， the shape of the bobbins forming curve was obtained， as they4presented in Fig.5. The curve equation following fitting is

y4=0.156 2x2-2.218x+4.866.

Table 3 Comparison for the appreciation of the bottom of the pipe(t=10)

According to the data in Table 4 and formula (23)， through the curve fitting of the data point with Matlab， the shape of the bobbins forming curve was obtained， as they5presented in Fig. 5. The curve equation following fitting is

y5=0.156x2-2.337x+6.305.

Table 4 Comparison for the appreciation of the bottom of the pipe(t=5)

Fig.5 Comparison of tube bottom curves following level appreciation optimization

The deviation foriginated from the original curve and the curve following segmentation. The questions for the appropriate number of sections as well as the change in the volume of the winding higher impact on the number of windings remained. These required further analysis and verification.

Based on the mathematical model， the shape curvey2of the bottom of the pipe was obtained by the second order polynomial fitting.

y2=0.1562x2-2.106x+3.558.

The value ofawas calculated as

a=21.9 mm.

According to formula (22)， the winding volume at the end of the bottom bobbins formingV2can be calculated.

Also， according to Eq. (19)，n2was calculated as

The analysis and optimization of the bobbins forming curve shape are given as follows.

(1)t=20，y3=0.1562x2- 2.156x+4.137. The values ofa，V3， andn3were calculated as

a=21.5 mm，

(2)t=10，y4=0.1562x2- 2.218x+4.866. The values ofa，V4， andn4were calculated as

a=21 mm，

(3)t=5，y5=0.156x2- 2.337x+6.305. The values of a，V5， andn5were calculated as

a=20 mm,

Since the aforementioned analysis demonstrated that the deviation was not high， the number of winding formingn2was equal ton1，n3andn4， being only two times different fromn1. Therefore， the errors were almost negligible. In contrast， the curvey3corresponded to 20 sub-processing， whereas the increment of a level increase was 0.01. The accuracy requirements for the elevating mechanism were still higher; consequently， the curvey3was discarded. Curvesy4andy5were corresponded to the same number of bobbins forming. In contrast， the curvey4corresponded to 10 sub-processing， whereas the increment of a level increase was 0.02. The curvey5corresponded to 5 sub-processing， whereas the increment of a level increase was 0.04. Apparently， if the curvey5could be considered from the institutional point of view， the agency accuracy requirements were slightly lower， whereas compared to the experimental yarn tube curve shape， the curvey5obtaining the bottom curve shape was relatively full to meet the requirements of spinning. In addition， according to the different varieties of spinning yarn， different values of a level increase could be selected. If the mechanism accuracy is sufficient， they4could be considered， in order for a higher capacity of the bobbin to be achieved in the case of relatively few segments.

4 Conclusions

(1) The mathematical model of the bobbins forming was established， which led to the forming of the bobbins to being independent on the mechanical forming pin. The concept of “controlled nail” was demonstrated. It satisfied the winding theory of high speed spinning.

(2) The mathematical model of the bobbins forming was optimized through simulation analysis and the ideal forming effect could be obtained.

(3) The active winding forming elevating mechanism made the “controlled nail” application accurate， for the original spinning frame mechanism to be highly simplified. It provided the the oretical support for the new generation of intelligent spinning frame design.

(4) Subjected to the experimental conditions， the aforementioned data should be appropriate to be selected and dealt with for different varieties of yarn.