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Mathematical modeling and numerical simulation of yarn behavior in a modified ring spinning system
Applied Mathematical Modelling 35 (2011) 139–151
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Mathematical modeling and numerical simulation of yarn behavior in a modified ring spinning system H.B. Tang a,b, B.G. Xu b,*, X.M. Tao b, J. Feng b a b
School of Automobile, Chang’an University, Xi’an 710064, China Institute of Textiles and Clothing, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
a r t i c l e
i n f o
Article history: Received 1 December 2009 Received in revised form 6 May 2010 Accepted 24 May 2010 Available online 1 June 2010 Keywords: Dynamic model Ring spinning Yarn path Twist of yarn
a b s t r a c t In the paper, yarn dynamical behavior and twist distribution in a modified ring spinning system are investigated. Equations of motion and twist wave propagation are used to obtain the numerical solutions of yarn path, yarn tension and twist distribution in steady state. It is observed that yarn path in the twisting zone has several classic modes corresponding to the yarn tension, and all of the yarn paths are approximately planar curves rather than spatial curves. The angular velocities of yarn at the twisting device are given as well as the twist of yarn in the modified ring spinning system. Experiments are conducted to evaluate the yarn paths and twist distributions under consideration. The theoretical and experimental results have a good agreement. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction Ring spinning has been the major manufacturing system for spun yarns. Yarn balloon generated in the spinning machine affects the manufacture process and quality of yarn significantly. Previous studies [1–6] investigated the balloon formation and dynamic analysis. In a conventional ring spinning frame, there are two yarn balloons, i.e. that between the yarn guide and traveler as well as that between the traveler and winding point on the bobbin. The latter can be assumed as a straight line since its distance is rather small. On the basis of previous research, Fraser [6] has established the detailed mathematical model and boundary conditions of yarn balloon. In his work, many numerical examples of a balloon between the yarn guide and traveler are given. Recently, the stability of ring spinning has been an attractive subject [7–9]. On the other hand, twist propagation in a yarn is a fundamental topic [10–12], since it impacts on the quality of yarn seriously. Miao and Chen [13] proposed a method of which the governing equation of twist distribution of a straight yarn is a wave equation. Fraser and Stump [14] further carried out the research on the twist of yarn in a ring spinning balloon. They presented a new derivation of the wave equation and theoretically proved that the twist distribution is independent of the yarn path in a balloon. That means the twist propagation of yarn could be calculated regardless of the yarn path in a spinning system. Considering twist blockage by friction, Guo et al. [15] and Xu and Tao [16] have proposed mechanical models to evaluate the twist distribution in rotor spinning numerically. Detailed expressions of twist waves in a ring spinning system were presented by Tang et al. [17] according to the general solutions of the twist wave equation and boundary conditions. Continuous developments have taken place in the ring spinning sector. Recently a modified ring spinning system has been invented and investigated [18–22] to produce a low torque and soft handle singles yarn. Compared with the conventional ring spinning frames, this new system is furnished with an extra false twisting device, thus an extra balloon resulted from the false twisting device is present. However, the dynamic behavior and twist distribution in this system have not been * Corresponding author. Tel.: +852 27664544; fax: +852 27731432. E-mail address: [email protected] (B.G. Xu). 0307-904X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2010.05.013
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studied up to now. Therefore, the investigations associated with the new system will be conducted in the work. The equation of motion of yarn and boundary conditions are given. Then the twist wave equation and general solutions are presented, and the twist distribution of the entire system is formulated theoretically. Afterwards, the numerical examples of the balloon are demonstrated. Next, the physical experiments of the modified ring spinning system are described and the experimental results are illustrated and discussed.
2. Theoretical In this section, dynamic characteristics and twist distribution of the modified ring spinning frame will be investigated theoretically. Four assumptions are made. First, the yarn is an inextensible and flexible cylinder. Secondly, the yarn has a circular cross section and uniform linear density. Thirdly, the shape of the cross section of yarn is unchanged under deformation. Finally, the relationship between the torque and twist of yarn is of linearity. A modified ring spinning system is shown in Fig. 1. A yarn is delivered from the front rollers A. Then the yarn will pass through a pin, a false twisting device. The pin has a hollow cylinder which is rotating around its symmetry axis. A small hook B is mounted at the bottom of the pin. The yarn contacts the pin at the internal upper edge, and then it moves forward, being wound on the hook one turn. When the pin rotates, the yarn wound on the hook will be rotating with the identical angular velocity. The first section of the whole system is from the front rollers to the hook. Note that the yarn path in this section could be treated as a straight line owing to the small distance. Next, the yarn passes to the yarn guide C from the hook. The balloon is present in this section, which is resulted from the extra twisting device. The next section contains the yarn path from the yarn guide C to the traveler D. The balloon in this section has been comprehensively analyzed in the published literature [6]. The final section is from the traveler D to the winding point E on the bobbin, which could be viewed as a straight line as well. Accordingly, the authors will pay more attention to the balloon from the hook to the yarn guide in this study.
Fig. 1. Schematic diagram of a modified ring spinning system.
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2.1. Dynamic model of the balloon In what follows, the dynamic model of the balloon will be established theoretically. The balloon from the hook to the yarn guide can be depicted in Fig. 2. A cylindrical coordinate system (r, h, z) is employed to specify the position of a point. The origin o of the coordinates is set to the location of the hook. It is noticeable that the point o and the yarn guide C lie on the same central line of the system. The coordinate axis ez is coincident with the axis z shown in Fig. 2 and the cylindrical coordinates is rotating around axis ez with an angular velocity identical to that of the pin. The yarn path is expressed as R(s, t), where the variables s and t stand for the arc length and time respectively. And h is the height of the balloon. Here d means the small deviation between the coordinate origin o and the point from which the yarn leaves the hook. Consider an infinitesimal section of yarn. The equation of motion of the balloon can be expressed as the following [6]
D2 R DR m þ 2xez þ x2 ez ðez RÞ Dt Dt 2
! ¼
@ @R T þ F; @s @s
ð1Þ
where m is the linear density of the yarn, x denotes the angular velocity of the pin. Here T is the tension of the yarn, and F means the air drag force applied on the yarn. The expression F is formulated as
F ¼ Dn jm n jm n ;
ð2Þ
where mn denotes the velocity component of yarn of which the direction corresponds to the binormal of the yarn path. And Dn is the air drag coefficient. The operator D/Dt represents the material derivative of a function with respect to time t which is given by
D @ @ ¼ þV ; Dt @t @s
ð3Þ
where V is the delivery velocity of the yarn. In the left hand side of Eq. (1), the first term means the acceleration of the yarn in the rotating coordinates. And the second and third terms stand for the Coriolis and centripetal accelerations of the yarn, respectively. For the purpose of simplifying the mathematical model, only the steady state case will be considered in the following study. To interpret the results clearly, the dimensionless parameters are employed, which are given by
R¼
R ¼ r er þ zez ; a
s s ¼ ; a
mn ¼
m T F ; T¼ ; F¼ ; xa0 mx2 a2 mx2 a
ð4Þ
where a is the radius of the traveler ring. In terms of the dimensionless scheme, the governing equation of the balloon under the steady state condition is derived as 2
X
! 2 d R dR d dR 1 þ F; þ ez ðez RÞ ¼ þ 2X e z T ds2 ds ds ds
where X(X = xa/V) means the angular velocity parameter. The dimensionless air drag F is given by
Fig. 2. The balloon from the hook to the yarn guide.
ð5Þ
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F¼
1 p jmn jmn ; 16 0
ð6Þ
where coefficient p0 = 16Dna/m. Forming the dot product of Eq. (5) with dR=ds, one can obtain the differential equation as
dT dr ¼ r : ds ds
ð7Þ
Solving Eq. (7), one can gain
1 T ¼ T 0 r2 ; 2
ð8Þ
where T 0 could be specified as the tension of yarn at the yarn guide. Generally, the first two terms of Eq. (5) are small, because X1 and X2 are of the order less than 3% and 1%. They will be omitted in the following study. The components of the governing equation are formulated as [6]
1 n ; T 0 12 r 2 ðr 00 r h02 Þ ¼ r þ rr02 16 p0 ðr 2r 0 h0 Þm 3 p0 0 00 0 1 2 0 2 0 1 n ; T 0 2 r ð2r h þ r h Þ ¼ r r h þ 16 r m 1 n ; T 0 12 r 2 z00 ¼ rr 0 z0 16 p0 ðr 2 h0 z0 Þm
ð9Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ r r 02 þ z02 . The inextensibility where the prime means the derivative with respect to the dimensionless arc length s, and m condition of yarn is given by
ðr 0 Þ2 þ ðr h0 Þ2 þ ðz0 Þ2 ¼ 1: The coordinate values of the balloon at the hook and yarn guide are determined as follows.
d rð0Þ ¼ d ¼ ; a
zð0Þ ¼ 0;
hð0Þ ¼ 0;
h0 ð0Þ ¼ 0;
r ðs1 Þ ¼ 0;
¼ h: zðs1 Þ ¼ h a
ð10Þ
Note that s1 means the total arc length of the balloon. The parameters s1 and hðs1 Þ will be obtained as a part of the solutions of the governing equations. The dynamic model of the balloon is experienced in Eq. (9) with the boundary condition Eq. (10). Furthermore, numerical methods will be applied for solving the model, which will be discussed in Section 3. 2.2. Twist distribution of the system Twist is the key factor influencing the quality of yarn. The general twist propagation equation was formulated by [17]
@ 2 Nðs; tÞ @ 2 Nðs; tÞ @ 2 Nðs; tÞ 2 þ V V 20 þ 2V ¼ 0; 2 @s@t @s2 @t
ð11Þ
where N(s, t) means the rotating angle of the cross section of yarn at position s and time t, and V0 is the propagation velocity of twist of yarn, which is stated as
V0 ¼
sffiffiffiffiffiffiffi GI : J
Here G and I are the shear modulus and polar moment of inertia respectively, and J means the unit moment of inertia. By separation of variable, the general solutions of Eq. (11) have the forms
N1 ðs; tÞ ¼ eiðxtþk1 sþc1 Þ ;
N2 ðs; tÞ ¼ eiðxtþk2 sþc2 Þ ;
ð12Þ
where i is the imagery unit of a complex number. In addition, the constants c1 and c2 stand for the initial phases of the rotating yarn physically. Here k1 and k2 denote the wavenumbers which are given by,
k1 ¼
x V0 V
;
k2 ¼
x V0 þ V
:
Given that V0 is greater than V, N1 means the twist wave propagating in the negative s direction, and N2 is along the positive s direction. Generally, the twist propagation velocity is much greater than the delivery velocity of yarn in ring spinning systems. In a certain twisting zone, the final twist wave will consist of three components, the original twist wave and two waves arisen from the two boundaries of the zone. The angular velocity of yarn at points A, B and D (see Fig. 1) are given by
xA ¼ 0; xB ¼ x; xD ¼ x1 : At the winding point E, the angular velocity xE of yarn is equivalent to that xD at the traveler to calculate the twist of yarn, i.e. xE = x1. According to the boundary conditions, the twist waves in the zones could be formulated.
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H.B. Tang et al. / Applied Mathematical Modelling 35 (2011) 139–151 Table 1 The twist of yarn of the modified ring spinning system. Zone
AB
BD
DE
Twist
x 2pV
x1 2pV
2pV
x1
Table 2 The torques applied on the yarn by twisting devices. Device
A
B
D
E
Torque
GI x V
GI ðx1VxÞ
0
GI xV1
The twist of yarn of the modified ring spinning system can be achieved as shown in Table 1. Furthermore, the torques applied on the yarn by twisting devices are also reached as listed in Table 2. Generally, the angular velocity of the pin is larger than that of the traveler. Accordingly, it can be found from Tables 1 and 2 that the twist of yarn in zone AB is much greater than that in the other zones, and the torque acted on the yarn by the pin is the largest one among the twisting devices. It should be emphasized that the friction force between the yarn and mechanical parts will produce a little influence on the twist of yarn in practice.
3. Numerical computation In this section, the dynamic model of the balloon will be computed numerically according to governing equations (Eq. (9)) and boundary conditions (Eq. (10)) of the balloon. It is manifest that the dynamic model is a two-point boundary value problem of ordinary differential equations (ODEs) mathematically. To solve the ODEs, the shooting method will be applied in the work. First, a Runge–Kutta method with variable steps is employed to solve the initial value problem. Then the Secant scheme is used to satisfy the boundary conditions by adjusting the trial initial values. In this study, variables s1 and r0 ð0Þ will be adjusted until the boundary conditions are met for a given tension T 0 . The parameter values used in numerical computation is shown as follows. The air drag coefficient p0 is equal to 4.0 [6]. At ¼ 3:0 and the beginning of, a special case of the balloon is illustrated under the conditions T 0 ¼ 0:3125; h d ¼ 0 as shown in Fig. 3. It is noticeable that the yarn path of the balloon is just a straight line in the case. The phenomenon is no doubt caused by the boundary condition d ¼ 0. Note that the trail value r0 ð0Þ ¼ 0:01 is always used as the initial value of numerical computation. In the following numerical computation, the small deviation d is specified as 0.01 and the height of the balloon is set to 3.0. Fig. 4(a) to (f) show the yarn paths of the balloon in terms of different tensions. When the tension T 0 is increasing, the number of curve peaks is decreasing. These cases represent several classic modes of the yarn path of the balloon. Each curve of the yarn paths is approximately located in one plane (x–z plane). That is to say, the yarn paths of the balloon are almost
¼ 3:0, and Fig. 3. The yarn path of the balloon with T 0 ¼ 0:3125; h d ¼ 0.
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¼ 3:0 and (a) T 0 ¼ 0:03125, (b) T 0 ¼ 0:0625, (c) T 0 ¼ 0:09375, (d) T 0 ¼ 0:3125, (e) T 0 ¼ 0:625, (f) Fig. 4. The yarn paths of the balloon with d ¼ 0:01; h T 0 ¼ 1:25.
planar curves rather than spatial curves. To evaluate this attribute of the balloon, the maximum distance Dmax between the yarn and x–z plane is calculated. That means the yarn path will be entirely situated between the planes y = Dmax and y = Dmax. The distances Dmax of these cases are given as the following, (a) 1.8873e 4, (b) 4.7682e 4, (c) 6.5707e 4,
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145
¼ 3:0 and (I) T 0 ¼ 0:03125, (II) T 0 ¼ 0:0625, (III) T 0 ¼ 0:09375, (IV) T 0 ¼ 0:3125, (V) Fig. 5. Planar curves of yarn paths of the balloon with d ¼ 0:01; h T 0 ¼ 0:6250, (VI) T 0 ¼ 1:2500.
(d) 1.1899e 4, (e) 9.1148e 5 and (f) 1.6853e 4. Consequently, it is reasonable that these yarn paths of the balloon could be viewed as the planar curves approximately. To illustrate these numerical examples clearly, the planar curves of yarn paths of the balloon are re-plotted shown in Fig. 5. According to the yarn paths of the balloon, the tension distributions of yarn are further acquired from Eq. (8), as shown in Fig. 6, where the horizontal and perpendicular axes stand for the coordinate z and the ratio T=T 0 respectively. The number of curve peaks is equal to the number of points where r ¼ 0 obviously. Next, the yarn paths of the balloon are figured out in terms of different tensions T 0 under the conditions ¼ 5:0, which are shown in Fig. 7. Moreover, the corresponding tension distributions of yarn in the balloon d ¼ 0:04 and h shown in Fig. 8 are also reckoned. In Figs. 7 and 8, it is demonstrated that the conclusions acquired above are also applicable. Further, the experiments of the modified ring spinning system will be carried out to assess the method in the next section. 4. Experimental Experiments were conducted to measure yarn tensions, yarn paths and twist distributions in the modified ring spinning system. Comparison with numerical calculation, based on the dynamic model of balloon and kinetic model of twist distribution, will be repeated.
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¼ 3:0 and (I) T 0 ¼ 0:03125, (II) T 0 ¼ 0:0625, (III) T 0 ¼ 0:09375, (IV) T 0 ¼ 0:3125, (V) Fig. 6. Spatial tension distributions of the balloon with d ¼ 0:01; h T 0 ¼ 0:6250, (VI) T 0 ¼ 1:2500.
4.1. Yarn tension and path The ring spinning is a complicated process, and its dynamic models are always simplified to some extent. Therefore, the experimental exploration on the yarn paths of the balloon will be performed in a qualitative approach. Digital Tension Meter was used to measure the yarn tension of the balloon at region BC in Fig. 1, and then the signals of tensions were sent to a linked computer for recording. Images of dynamic yarn paths were captured by a high speed camera as shown in Fig. 9. The experimental set-up consists of a high speed camera, lighting and a tripod, and the camera and lighting are mounted on the tripod. Further, a sampling frequency of 8113 frames per second is adopted to capture the images of yarn in the ring spinning process. The experimental parameters are as the following. Yarn linear density: 29.5 gm/km Twist of yarn: 440 turns/m Rotational speed of the Spindle: 10,000 rpm Rotational speed of the Pin: 25,000 rpm Total time of measurement: 160 s In terms of different masses of the traveler, three sets of the experiment are conducted. The yarn tensions of those cases are given in Table 3, and the corresponding photos of yarn paths of the balloon are displayed in Fig. 10. Note that the axis ez of yarn paths in numerical computation points to the vertical down direction in the experimental images. Table 3 shows that the measured yarn tension of the balloon at region BC decreases if the traveler mass decreases but not following a linear relationship. Further, Fig. 10 (a), (b) and (c) is similar to Fig. 5(VI), (V) and (IV) as well as Fig. 7 (VI), (V) and
H.B. Tang et al. / Applied Mathematical Modelling 35 (2011) 139–151
147
¼ 5:0 and (I) T ¼ 0:090, (II) T ¼ 0:186, (III) T ¼ 0:330, (IV) T ¼ 0:906, (V) T ¼ 1:500, Fig. 7. Planar curves of yarn paths of the balloon with d ¼ 0:04; h 0 0 0 0 0 (VI) T 0 ¼ 5:000.
(IV) qualitatively, respectively. Meanwhile, the peak number of the yarn paths increases when the yarn tension of the balloon decreases, which agrees with the numerical results. From the physical experiments, it is demonstrated that the yarn paths of the balloon from the hook to the yarn guide possess different modes indeed in terms of the corresponding yarn tensions. 4.2. Spatial twist distribution of yarn The high speed camera system which was used to capture the above balloon images was also employed to measure the twist distribution of yarn. Additionally, a piece of scale paper of 1 mm 1 mm was put behind the yarn as a reference frame for the calculation of actual yarn twist values. The twist Tw of yarn can be obtained from the images by
TW ¼
1 ; hp
where hp means the pitch of the thread. The pitches of threads were gauged directly from the images of yarn. For each spinning zone, ten images of yarn were used to calculate the average of yarn twist. Combed roving with a linear density of 534 gm/km was used as the raw material. And the mass variation of the roving was 4.26%. In the following experiments, the angular velocity of the spindle was set to 10,000 rpm and the delivery velocity V of yarn was equal to 22.72 m/s.
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¼ 5:0 and (I) T 0 ¼ 0:090, (II) T 0 ¼ 0:186, (III) T 0 ¼ 0:330, (IV) T 0 ¼ 0:906, (V) T 0 ¼ 1:500, Fig. 8. Spatial tension distributions of the balloon with d ¼ 0:04; h (VI) T 0 ¼ 5:000.
Fig. 9. The experimental set-up to capture images of yarn during spinning.
Two cases k = 2.5 and k = 2.9 were investigated where k is the angular velocity ratio of the pin to the spindle. The images of yarn in twisting zones AB, BC and CD (see Fig. 1) of the two cases are illustrated in Figs. 11 and 12, respectively. And the
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H.B. Tang et al. / Applied Mathematical Modelling 35 (2011) 139–151 Table 3 Measured parameters of yarn balloon with different traveler masses. Balloon shape
Balloon height (mm)
Traveler mass (g)
Mean measured yarn tension (cN)
Case 1 Case 2 Case 3
30 30 30
0.11 0.04 0.02
13.328 6.468 4.214
Fig. 10. Yarn balloon with different traveler masses: (a) case 1, (b) case 2, (c) case 3.
measured twist distributions of yarn are listed in Table 4. Note that the yarn twist of zone DE is the yarn twist of final product measured by a regular de-twisting device. The theoretical yarn twist of zone BC coincides with that of zone CD, since there is no additional twisting considered between the two zones. Nevertheless, the measured yarn twists of the two zones are not identical to each other because of the twist blockage effect of yarn guide [23]. Yarn twist blockage emerges when the yarn slides over the yarn guide, which will influence the twist distribution of yarn.
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Fig. 11. Images of yarn twist of the balloon in zones AB, BC and CD, k = 2.5.
Fig. 12. Images of yarn twist of the balloon in zones AB, BC and CD, k = 2.9.
Table 4 Comparisons of twist distributions of yarn between the experimental and theoretical results (see Table 1). k
Twisting zone
Measured twist of yarn (turns/m)
Theoretical value (turns/m)
Error (%)
2.5
AB BC CD DE
1019.4 486.8 584.8 411.9
1100.4 440.1 440.1 440.1
7.95 9.59 24.74 6.85
2.9
AB BC CD DE
1297.4 500.0 667.2 424.2
1276.4 440.1 440.1 440.1
1.62 11.98 34.04 3.75
Table 4 demonstrates a good agreement between the experimental and theoretical values in zones AB and DE. But the errors of comparisons of yarn twists in zones BC and CD are relatively large, especially in zone CD. This discrepancy is mainly due to the ignorance of yarn twist blockage of yarn guide in the kinetic model. 5. Concluding remarks In this study, the dynamic properties of a spun yarn produced in a modified ring spinning system, including the balloon shapes, yarn tension and twist distributions, have been theoretically investigated. To begin with, the equation of motion of yarn balloon between the false twister and yarn guide is formulated in theory. Then the boundary conditions of the balloon are assigned. Furthermore, the twist of yarn in the modified ring spinning system is studied. Afterward, the dynamic model of the balloon is solved numerically. The shooting method is employed to gain the numerical solutions of the governing equations. The numerical results suggest that the yarn paths of the balloon possess several classic modes on the curve shape. In addition, all of the yarn paths of the balloon are planar curves instead of spatial curves approximately. The experiments of the modified ring spinning system are conducted to verify the theoretical work and numerical result. First, three cases of yarn paths of the balloon are examined. The comparisons of yarn paths show that the experimental
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curves agree with the theoretical ones qualitatively. Then the experiments on the twist distributions of yarn of two cases show that the measured twists of yarn in the first and final zones agree well with the theoretical results. However, the errors of comparisons of yarn twists in the middle zones are relatively large because of the ignorance of yarn twist blockage by yarn guide in the present theoretical treatment. Acknowledgments The authors wish to acknowledge the funding support from The Hong Kong Polytechnic University and the Hong Kong Research Grants Council (Project No: PolyU 5325/08E) for the work reported here. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Mathematical modeling and numerical simulation of yarn behavior in a modified ring spinning system H.B. Tang a,b, B.G. Xu b,*, X.M. Tao b, J. Feng b a b
School of Automobile, Chang’an University, Xi’an 710064, China Institute of Textiles and Clothing, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
a r t i c l e
i n f o
Article history: Received 1 December 2009 Received in revised form 6 May 2010 Accepted 24 May 2010 Available online 1 June 2010 Keywords: Dynamic model Ring spinning Yarn path Twist of yarn
a b s t r a c t In the paper, yarn dynamical behavior and twist distribution in a modified ring spinning system are investigated. Equations of motion and twist wave propagation are used to obtain the numerical solutions of yarn path, yarn tension and twist distribution in steady state. It is observed that yarn path in the twisting zone has several classic modes corresponding to the yarn tension, and all of the yarn paths are approximately planar curves rather than spatial curves. The angular velocities of yarn at the twisting device are given as well as the twist of yarn in the modified ring spinning system. Experiments are conducted to evaluate the yarn paths and twist distributions under consideration. The theoretical and experimental results have a good agreement. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction Ring spinning has been the major manufacturing system for spun yarns. Yarn balloon generated in the spinning machine affects the manufacture process and quality of yarn significantly. Previous studies [1–6] investigated the balloon formation and dynamic analysis. In a conventional ring spinning frame, there are two yarn balloons, i.e. that between the yarn guide and traveler as well as that between the traveler and winding point on the bobbin. The latter can be assumed as a straight line since its distance is rather small. On the basis of previous research, Fraser [6] has established the detailed mathematical model and boundary conditions of yarn balloon. In his work, many numerical examples of a balloon between the yarn guide and traveler are given. Recently, the stability of ring spinning has been an attractive subject [7–9]. On the other hand, twist propagation in a yarn is a fundamental topic [10–12], since it impacts on the quality of yarn seriously. Miao and Chen [13] proposed a method of which the governing equation of twist distribution of a straight yarn is a wave equation. Fraser and Stump [14] further carried out the research on the twist of yarn in a ring spinning balloon. They presented a new derivation of the wave equation and theoretically proved that the twist distribution is independent of the yarn path in a balloon. That means the twist propagation of yarn could be calculated regardless of the yarn path in a spinning system. Considering twist blockage by friction, Guo et al. [15] and Xu and Tao [16] have proposed mechanical models to evaluate the twist distribution in rotor spinning numerically. Detailed expressions of twist waves in a ring spinning system were presented by Tang et al. [17] according to the general solutions of the twist wave equation and boundary conditions. Continuous developments have taken place in the ring spinning sector. Recently a modified ring spinning system has been invented and investigated [18–22] to produce a low torque and soft handle singles yarn. Compared with the conventional ring spinning frames, this new system is furnished with an extra false twisting device, thus an extra balloon resulted from the false twisting device is present. However, the dynamic behavior and twist distribution in this system have not been * Corresponding author. Tel.: +852 27664544; fax: +852 27731432. E-mail address: [email protected] (B.G. Xu). 0307-904X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2010.05.013
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studied up to now. Therefore, the investigations associated with the new system will be conducted in the work. The equation of motion of yarn and boundary conditions are given. Then the twist wave equation and general solutions are presented, and the twist distribution of the entire system is formulated theoretically. Afterwards, the numerical examples of the balloon are demonstrated. Next, the physical experiments of the modified ring spinning system are described and the experimental results are illustrated and discussed.
2. Theoretical In this section, dynamic characteristics and twist distribution of the modified ring spinning frame will be investigated theoretically. Four assumptions are made. First, the yarn is an inextensible and flexible cylinder. Secondly, the yarn has a circular cross section and uniform linear density. Thirdly, the shape of the cross section of yarn is unchanged under deformation. Finally, the relationship between the torque and twist of yarn is of linearity. A modified ring spinning system is shown in Fig. 1. A yarn is delivered from the front rollers A. Then the yarn will pass through a pin, a false twisting device. The pin has a hollow cylinder which is rotating around its symmetry axis. A small hook B is mounted at the bottom of the pin. The yarn contacts the pin at the internal upper edge, and then it moves forward, being wound on the hook one turn. When the pin rotates, the yarn wound on the hook will be rotating with the identical angular velocity. The first section of the whole system is from the front rollers to the hook. Note that the yarn path in this section could be treated as a straight line owing to the small distance. Next, the yarn passes to the yarn guide C from the hook. The balloon is present in this section, which is resulted from the extra twisting device. The next section contains the yarn path from the yarn guide C to the traveler D. The balloon in this section has been comprehensively analyzed in the published literature [6]. The final section is from the traveler D to the winding point E on the bobbin, which could be viewed as a straight line as well. Accordingly, the authors will pay more attention to the balloon from the hook to the yarn guide in this study.
Fig. 1. Schematic diagram of a modified ring spinning system.
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2.1. Dynamic model of the balloon In what follows, the dynamic model of the balloon will be established theoretically. The balloon from the hook to the yarn guide can be depicted in Fig. 2. A cylindrical coordinate system (r, h, z) is employed to specify the position of a point. The origin o of the coordinates is set to the location of the hook. It is noticeable that the point o and the yarn guide C lie on the same central line of the system. The coordinate axis ez is coincident with the axis z shown in Fig. 2 and the cylindrical coordinates is rotating around axis ez with an angular velocity identical to that of the pin. The yarn path is expressed as R(s, t), where the variables s and t stand for the arc length and time respectively. And h is the height of the balloon. Here d means the small deviation between the coordinate origin o and the point from which the yarn leaves the hook. Consider an infinitesimal section of yarn. The equation of motion of the balloon can be expressed as the following [6]
D2 R DR m þ 2xez þ x2 ez ðez RÞ Dt Dt 2
! ¼
@ @R T þ F; @s @s
ð1Þ
where m is the linear density of the yarn, x denotes the angular velocity of the pin. Here T is the tension of the yarn, and F means the air drag force applied on the yarn. The expression F is formulated as
F ¼ Dn jm n jm n ;
ð2Þ
where mn denotes the velocity component of yarn of which the direction corresponds to the binormal of the yarn path. And Dn is the air drag coefficient. The operator D/Dt represents the material derivative of a function with respect to time t which is given by
D @ @ ¼ þV ; Dt @t @s
ð3Þ
where V is the delivery velocity of the yarn. In the left hand side of Eq. (1), the first term means the acceleration of the yarn in the rotating coordinates. And the second and third terms stand for the Coriolis and centripetal accelerations of the yarn, respectively. For the purpose of simplifying the mathematical model, only the steady state case will be considered in the following study. To interpret the results clearly, the dimensionless parameters are employed, which are given by
R¼
R ¼ r er þ zez ; a
s s ¼ ; a
mn ¼
m T F ; T¼ ; F¼ ; xa0 mx2 a2 mx2 a
ð4Þ
where a is the radius of the traveler ring. In terms of the dimensionless scheme, the governing equation of the balloon under the steady state condition is derived as 2
X
! 2 d R dR d dR 1 þ F; þ ez ðez RÞ ¼ þ 2X e z T ds2 ds ds ds
where X(X = xa/V) means the angular velocity parameter. The dimensionless air drag F is given by
Fig. 2. The balloon from the hook to the yarn guide.
ð5Þ
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F¼
1 p jmn jmn ; 16 0
ð6Þ
where coefficient p0 = 16Dna/m. Forming the dot product of Eq. (5) with dR=ds, one can obtain the differential equation as
dT dr ¼ r : ds ds
ð7Þ
Solving Eq. (7), one can gain
1 T ¼ T 0 r2 ; 2
ð8Þ
where T 0 could be specified as the tension of yarn at the yarn guide. Generally, the first two terms of Eq. (5) are small, because X1 and X2 are of the order less than 3% and 1%. They will be omitted in the following study. The components of the governing equation are formulated as [6]
1 n ; T 0 12 r 2 ðr 00 r h02 Þ ¼ r þ rr02 16 p0 ðr 2r 0 h0 Þm 3 p0 0 00 0 1 2 0 2 0 1 n ; T 0 2 r ð2r h þ r h Þ ¼ r r h þ 16 r m 1 n ; T 0 12 r 2 z00 ¼ rr 0 z0 16 p0 ðr 2 h0 z0 Þm
ð9Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ r r 02 þ z02 . The inextensibility where the prime means the derivative with respect to the dimensionless arc length s, and m condition of yarn is given by
ðr 0 Þ2 þ ðr h0 Þ2 þ ðz0 Þ2 ¼ 1: The coordinate values of the balloon at the hook and yarn guide are determined as follows.
d rð0Þ ¼ d ¼ ; a
zð0Þ ¼ 0;
hð0Þ ¼ 0;
h0 ð0Þ ¼ 0;
r ðs1 Þ ¼ 0;
¼ h: zðs1 Þ ¼ h a
ð10Þ
Note that s1 means the total arc length of the balloon. The parameters s1 and hðs1 Þ will be obtained as a part of the solutions of the governing equations. The dynamic model of the balloon is experienced in Eq. (9) with the boundary condition Eq. (10). Furthermore, numerical methods will be applied for solving the model, which will be discussed in Section 3. 2.2. Twist distribution of the system Twist is the key factor influencing the quality of yarn. The general twist propagation equation was formulated by [17]
@ 2 Nðs; tÞ @ 2 Nðs; tÞ @ 2 Nðs; tÞ 2 þ V V 20 þ 2V ¼ 0; 2 @s@t @s2 @t
ð11Þ
where N(s, t) means the rotating angle of the cross section of yarn at position s and time t, and V0 is the propagation velocity of twist of yarn, which is stated as
V0 ¼
sffiffiffiffiffiffiffi GI : J
Here G and I are the shear modulus and polar moment of inertia respectively, and J means the unit moment of inertia. By separation of variable, the general solutions of Eq. (11) have the forms
N1 ðs; tÞ ¼ eiðxtþk1 sþc1 Þ ;
N2 ðs; tÞ ¼ eiðxtþk2 sþc2 Þ ;
ð12Þ
where i is the imagery unit of a complex number. In addition, the constants c1 and c2 stand for the initial phases of the rotating yarn physically. Here k1 and k2 denote the wavenumbers which are given by,
k1 ¼
x V0 V
;
k2 ¼
x V0 þ V
:
Given that V0 is greater than V, N1 means the twist wave propagating in the negative s direction, and N2 is along the positive s direction. Generally, the twist propagation velocity is much greater than the delivery velocity of yarn in ring spinning systems. In a certain twisting zone, the final twist wave will consist of three components, the original twist wave and two waves arisen from the two boundaries of the zone. The angular velocity of yarn at points A, B and D (see Fig. 1) are given by
xA ¼ 0; xB ¼ x; xD ¼ x1 : At the winding point E, the angular velocity xE of yarn is equivalent to that xD at the traveler to calculate the twist of yarn, i.e. xE = x1. According to the boundary conditions, the twist waves in the zones could be formulated.
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H.B. Tang et al. / Applied Mathematical Modelling 35 (2011) 139–151 Table 1 The twist of yarn of the modified ring spinning system. Zone
AB
BD
DE
Twist
x 2pV
x1 2pV
2pV
x1
Table 2 The torques applied on the yarn by twisting devices. Device
A
B
D
E
Torque
GI x V
GI ðx1VxÞ
0
GI xV1
The twist of yarn of the modified ring spinning system can be achieved as shown in Table 1. Furthermore, the torques applied on the yarn by twisting devices are also reached as listed in Table 2. Generally, the angular velocity of the pin is larger than that of the traveler. Accordingly, it can be found from Tables 1 and 2 that the twist of yarn in zone AB is much greater than that in the other zones, and the torque acted on the yarn by the pin is the largest one among the twisting devices. It should be emphasized that the friction force between the yarn and mechanical parts will produce a little influence on the twist of yarn in practice.
3. Numerical computation In this section, the dynamic model of the balloon will be computed numerically according to governing equations (Eq. (9)) and boundary conditions (Eq. (10)) of the balloon. It is manifest that the dynamic model is a two-point boundary value problem of ordinary differential equations (ODEs) mathematically. To solve the ODEs, the shooting method will be applied in the work. First, a Runge–Kutta method with variable steps is employed to solve the initial value problem. Then the Secant scheme is used to satisfy the boundary conditions by adjusting the trial initial values. In this study, variables s1 and r0 ð0Þ will be adjusted until the boundary conditions are met for a given tension T 0 . The parameter values used in numerical computation is shown as follows. The air drag coefficient p0 is equal to 4.0 [6]. At ¼ 3:0 and the beginning of, a special case of the balloon is illustrated under the conditions T 0 ¼ 0:3125; h d ¼ 0 as shown in Fig. 3. It is noticeable that the yarn path of the balloon is just a straight line in the case. The phenomenon is no doubt caused by the boundary condition d ¼ 0. Note that the trail value r0 ð0Þ ¼ 0:01 is always used as the initial value of numerical computation. In the following numerical computation, the small deviation d is specified as 0.01 and the height of the balloon is set to 3.0. Fig. 4(a) to (f) show the yarn paths of the balloon in terms of different tensions. When the tension T 0 is increasing, the number of curve peaks is decreasing. These cases represent several classic modes of the yarn path of the balloon. Each curve of the yarn paths is approximately located in one plane (x–z plane). That is to say, the yarn paths of the balloon are almost
¼ 3:0, and Fig. 3. The yarn path of the balloon with T 0 ¼ 0:3125; h d ¼ 0.
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¼ 3:0 and (a) T 0 ¼ 0:03125, (b) T 0 ¼ 0:0625, (c) T 0 ¼ 0:09375, (d) T 0 ¼ 0:3125, (e) T 0 ¼ 0:625, (f) Fig. 4. The yarn paths of the balloon with d ¼ 0:01; h T 0 ¼ 1:25.
planar curves rather than spatial curves. To evaluate this attribute of the balloon, the maximum distance Dmax between the yarn and x–z plane is calculated. That means the yarn path will be entirely situated between the planes y = Dmax and y = Dmax. The distances Dmax of these cases are given as the following, (a) 1.8873e 4, (b) 4.7682e 4, (c) 6.5707e 4,
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¼ 3:0 and (I) T 0 ¼ 0:03125, (II) T 0 ¼ 0:0625, (III) T 0 ¼ 0:09375, (IV) T 0 ¼ 0:3125, (V) Fig. 5. Planar curves of yarn paths of the balloon with d ¼ 0:01; h T 0 ¼ 0:6250, (VI) T 0 ¼ 1:2500.
(d) 1.1899e 4, (e) 9.1148e 5 and (f) 1.6853e 4. Consequently, it is reasonable that these yarn paths of the balloon could be viewed as the planar curves approximately. To illustrate these numerical examples clearly, the planar curves of yarn paths of the balloon are re-plotted shown in Fig. 5. According to the yarn paths of the balloon, the tension distributions of yarn are further acquired from Eq. (8), as shown in Fig. 6, where the horizontal and perpendicular axes stand for the coordinate z and the ratio T=T 0 respectively. The number of curve peaks is equal to the number of points where r ¼ 0 obviously. Next, the yarn paths of the balloon are figured out in terms of different tensions T 0 under the conditions ¼ 5:0, which are shown in Fig. 7. Moreover, the corresponding tension distributions of yarn in the balloon d ¼ 0:04 and h shown in Fig. 8 are also reckoned. In Figs. 7 and 8, it is demonstrated that the conclusions acquired above are also applicable. Further, the experiments of the modified ring spinning system will be carried out to assess the method in the next section. 4. Experimental Experiments were conducted to measure yarn tensions, yarn paths and twist distributions in the modified ring spinning system. Comparison with numerical calculation, based on the dynamic model of balloon and kinetic model of twist distribution, will be repeated.
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¼ 3:0 and (I) T 0 ¼ 0:03125, (II) T 0 ¼ 0:0625, (III) T 0 ¼ 0:09375, (IV) T 0 ¼ 0:3125, (V) Fig. 6. Spatial tension distributions of the balloon with d ¼ 0:01; h T 0 ¼ 0:6250, (VI) T 0 ¼ 1:2500.
4.1. Yarn tension and path The ring spinning is a complicated process, and its dynamic models are always simplified to some extent. Therefore, the experimental exploration on the yarn paths of the balloon will be performed in a qualitative approach. Digital Tension Meter was used to measure the yarn tension of the balloon at region BC in Fig. 1, and then the signals of tensions were sent to a linked computer for recording. Images of dynamic yarn paths were captured by a high speed camera as shown in Fig. 9. The experimental set-up consists of a high speed camera, lighting and a tripod, and the camera and lighting are mounted on the tripod. Further, a sampling frequency of 8113 frames per second is adopted to capture the images of yarn in the ring spinning process. The experimental parameters are as the following. Yarn linear density: 29.5 gm/km Twist of yarn: 440 turns/m Rotational speed of the Spindle: 10,000 rpm Rotational speed of the Pin: 25,000 rpm Total time of measurement: 160 s In terms of different masses of the traveler, three sets of the experiment are conducted. The yarn tensions of those cases are given in Table 3, and the corresponding photos of yarn paths of the balloon are displayed in Fig. 10. Note that the axis ez of yarn paths in numerical computation points to the vertical down direction in the experimental images. Table 3 shows that the measured yarn tension of the balloon at region BC decreases if the traveler mass decreases but not following a linear relationship. Further, Fig. 10 (a), (b) and (c) is similar to Fig. 5(VI), (V) and (IV) as well as Fig. 7 (VI), (V) and
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¼ 5:0 and (I) T ¼ 0:090, (II) T ¼ 0:186, (III) T ¼ 0:330, (IV) T ¼ 0:906, (V) T ¼ 1:500, Fig. 7. Planar curves of yarn paths of the balloon with d ¼ 0:04; h 0 0 0 0 0 (VI) T 0 ¼ 5:000.
(IV) qualitatively, respectively. Meanwhile, the peak number of the yarn paths increases when the yarn tension of the balloon decreases, which agrees with the numerical results. From the physical experiments, it is demonstrated that the yarn paths of the balloon from the hook to the yarn guide possess different modes indeed in terms of the corresponding yarn tensions. 4.2. Spatial twist distribution of yarn The high speed camera system which was used to capture the above balloon images was also employed to measure the twist distribution of yarn. Additionally, a piece of scale paper of 1 mm 1 mm was put behind the yarn as a reference frame for the calculation of actual yarn twist values. The twist Tw of yarn can be obtained from the images by
TW ¼
1 ; hp
where hp means the pitch of the thread. The pitches of threads were gauged directly from the images of yarn. For each spinning zone, ten images of yarn were used to calculate the average of yarn twist. Combed roving with a linear density of 534 gm/km was used as the raw material. And the mass variation of the roving was 4.26%. In the following experiments, the angular velocity of the spindle was set to 10,000 rpm and the delivery velocity V of yarn was equal to 22.72 m/s.
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¼ 5:0 and (I) T 0 ¼ 0:090, (II) T 0 ¼ 0:186, (III) T 0 ¼ 0:330, (IV) T 0 ¼ 0:906, (V) T 0 ¼ 1:500, Fig. 8. Spatial tension distributions of the balloon with d ¼ 0:04; h (VI) T 0 ¼ 5:000.
Fig. 9. The experimental set-up to capture images of yarn during spinning.
Two cases k = 2.5 and k = 2.9 were investigated where k is the angular velocity ratio of the pin to the spindle. The images of yarn in twisting zones AB, BC and CD (see Fig. 1) of the two cases are illustrated in Figs. 11 and 12, respectively. And the
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H.B. Tang et al. / Applied Mathematical Modelling 35 (2011) 139–151 Table 3 Measured parameters of yarn balloon with different traveler masses. Balloon shape
Balloon height (mm)
Traveler mass (g)
Mean measured yarn tension (cN)
Case 1 Case 2 Case 3
30 30 30
0.11 0.04 0.02
13.328 6.468 4.214
Fig. 10. Yarn balloon with different traveler masses: (a) case 1, (b) case 2, (c) case 3.
measured twist distributions of yarn are listed in Table 4. Note that the yarn twist of zone DE is the yarn twist of final product measured by a regular de-twisting device. The theoretical yarn twist of zone BC coincides with that of zone CD, since there is no additional twisting considered between the two zones. Nevertheless, the measured yarn twists of the two zones are not identical to each other because of the twist blockage effect of yarn guide [23]. Yarn twist blockage emerges when the yarn slides over the yarn guide, which will influence the twist distribution of yarn.
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Fig. 11. Images of yarn twist of the balloon in zones AB, BC and CD, k = 2.5.
Fig. 12. Images of yarn twist of the balloon in zones AB, BC and CD, k = 2.9.
Table 4 Comparisons of twist distributions of yarn between the experimental and theoretical results (see Table 1). k
Twisting zone
Measured twist of yarn (turns/m)
Theoretical value (turns/m)
Error (%)
2.5
AB BC CD DE
1019.4 486.8 584.8 411.9
1100.4 440.1 440.1 440.1
7.95 9.59 24.74 6.85
2.9
AB BC CD DE
1297.4 500.0 667.2 424.2
1276.4 440.1 440.1 440.1
1.62 11.98 34.04 3.75
Table 4 demonstrates a good agreement between the experimental and theoretical values in zones AB and DE. But the errors of comparisons of yarn twists in zones BC and CD are relatively large, especially in zone CD. This discrepancy is mainly due to the ignorance of yarn twist blockage of yarn guide in the kinetic model. 5. Concluding remarks In this study, the dynamic properties of a spun yarn produced in a modified ring spinning system, including the balloon shapes, yarn tension and twist distributions, have been theoretically investigated. To begin with, the equation of motion of yarn balloon between the false twister and yarn guide is formulated in theory. Then the boundary conditions of the balloon are assigned. Furthermore, the twist of yarn in the modified ring spinning system is studied. Afterward, the dynamic model of the balloon is solved numerically. The shooting method is employed to gain the numerical solutions of the governing equations. The numerical results suggest that the yarn paths of the balloon possess several classic modes on the curve shape. In addition, all of the yarn paths of the balloon are planar curves instead of spatial curves approximately. The experiments of the modified ring spinning system are conducted to verify the theoretical work and numerical result. First, three cases of yarn paths of the balloon are examined. The comparisons of yarn paths show that the experimental
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curves agree with the theoretical ones qualitatively. Then the experiments on the twist distributions of yarn of two cases show that the measured twists of yarn in the first and final zones agree well with the theoretical results. However, the errors of comparisons of yarn twists in the middle zones are relatively large because of the ignorance of yarn twist blockage by yarn guide in the present theoretical treatment. Acknowledgments The authors wish to acknowledge the funding support from The Hong Kong Polytechnic University and the Hong Kong Research Grants Council (Project No: PolyU 5325/08E) for the work reported here. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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