A study of an embeddable and locatable spinning system via quasi-static mechanical analysis

Abstract

In this study, a quasi-static model is built to theoretically analyze the distribution of twists and spinning tension in embeddable and locatable spun (ELS) yarn formation zone. Important equations are also derived to determine inner mechanics and external configurations of the ELS yarn formation zones 1, 2 and 3. Analysis results demonstrate that in zones 1 and 2 the tension distribution on the filament and staple strand is directly proportional to their linear mass and square of delivery speed; the larger weight causes a smaller angle between the responding component and the composite strand axis line. The angle between the composite strands 1 and 2 can be simply calculated by dividing the composite yarn velocity by composite strand velocity. Online photographs are provided to validate theoretical analysis of the ELS yarn formation zone configuration and twist distribution in zones 1 and 2.

Keywords

embeddable and locatable spinning quasi-static static model force balance yarn formation zone

An embeddable and locatable spinning system can well embed staple fibers into the yarn structure to produce smooth and strong yarns that are suitable for weaving.¹ The key factor contributing to high capacities of fiber trapping and automatic staple strand splicing lies in proper arrangements of two filaments and two staple strands in the embeddable and locatable spun (ELS) system.² Specifically, two staple strands are symmetrically fed between two filaments to form three spinning triangles (△N₁CN₂, $Δ N_{1} C_{1} {N^{'}}_{1}$ and $Δ N_{2} C_{2} {N^{'}}_{2}$ ) in the ELS yarn formation zone (Figure 1). Although embeddable and locatable spinning has been used in the Chinese worsted industry, mystery still exists for the twist and spinning tension distribution in the ELS yarn formation zone. Therefore, related research work should be conducted to clarify the law of twist and spinning tension distribution in the ELS triangles.

Figure 1.

Schematic diagram of embeddable and locatable spun yarn formation zone.¹ N₁’: nip point for strand 1; N₂’: nip point for strand 2; C: convergence point of composite spinning strands; C₁: convergence point of staple strand 1 and filament 1; C₂: convergence point of staple strand 2 and filament 2.

Many concepts of applied mathematics have been successfully introduced to analyze different spinning systems: the force and torque balance have been applied to establish a quasi-static model for two-strand spinning;³^–⁷ neural networks technology to describe yarn spinning;⁸^–¹⁰ intelligent systems to probe the position improvement and competitiveness of the textile industry;¹¹ the kinematic approach to analyze the sewing mechanisms of an overage machine;¹² the laws of the dynamics system to establish a quasi-static static model;¹³^–¹⁶ the homotopy perturbation method^17,18 to solve non-linear problems arising in yarn spinning.¹⁹ However, there have been no reports about an adequate mathematic method that is utilized to analyze the ELS system.

In this study, a quasi-static model is built to theoretically analyze the distribution of twists and spinning tension in the ELS yarn formation zone. Important equations are also derived to determine the inner mechanics and external configurations of the ELS yarn formation zone. Finally, online photos of the ELS yarn formation zone are provided to validate the corresponding theoretical analysis.

Theoretical analysis of the embeddable and locatable spun system

The dynamical character of an ELS system has a strong influence on the convergent angles; the convergent angle is a key factor influencing ELS yarn quality.¹ For an easy study, the ELS system is assumed to be stable and symmetric. Under this situation, the ELS system can be divided into three zones (i.e., zones 1, 2 and 3) shown in Figure 2. In zones 1 and 2, twist is inserted onto a filament; then the staple strand is contacted with filament F1 and they will be twisted with each other like a siro-fil spinning.

Figure 2.

Illustration of three sub-zones in embeddable and locatable spun yarn formation zone.

For ELS zones 1, 2 and 3, basic laws must be obeyed in mechanics, including force balance, mass conservation and momentum conservation. As zones 1 and 2 are symmetric, we only consider the mathematical analysis in zones 1 and 3, shown in Figures 3(a) and (b), respectively.

Figure 3.

Illustration of embeddable and locatable spun zones 1 and 3: (a) zone 1; (b) zone 3.

Analysis of zone 1

Force balance

f_{1} \cos α + f_{2} \cos β = f_{3}

(1)

f_{1} \sin α = f_{2} \sin β_{3}

(2)

where f₁ is the tension of filament 1, f₂ is the tension of staple strand 1, f₃ is the tension of composite spinning strand 1, α is the angle of filament 1 and composite spinning strand 1 axis line and β is the angle of staple strand 1 and composite spinning strand 1 axis line.

Torque balance

τ_{1} \cos α + τ_{2} \cos β + R_{1} f_{1} \sin α + R_{2} f_{2} \sin β = τ_{3}

(3)

τ_{1} \sin α + R_{1} f_{1} \cos α = τ_{2} \sin β + R_{2} f_{2} \cos β

(4)

where τ₁ is the torque of filament 1, τ₂ is the torque of staple strand 1, τ₃ is the torque of composite spinning strand 1, R₁ represents the radius of filament 1 and R₂ represents the radius of staple strand 1.

Momentum equation

R_{1}^{2} ρ_{1} v_{1}^{2} \cos α + R_{2}^{2} ρ_{2} v_{2}^{2} \cos β = R_{3}^{2} ρ_{3} v_{3}^{2}

(5)

R_{1}^{2} ρ_{1} v_{1}^{2} \sin α = R_{2}^{2} ρ_{2} v_{2}^{2} \sin β

(6)

where R₃ is the radius of composite spinning strand 1,

v_{1}

and

v_{2}

are the velocities of filament 1 and staple strand 1, respectively,

v_{3}

is the velocity of composite spinning strand 1, ρ₁ is the density of filament 1 and ρ₂ is the density of staple strand 1.

Mass conservation

R_{1}^{2} ρ_{1} v_{1} + R_{2}^{2} ρ_{2} v_{2} = R_{3}^{2} ρ_{3} v_{3}

(7)

where ρ₃ is the density of composite spinning strand 1.

From the seven equations (1)–(7), we can get the following equations:

v_{3} = \frac{R_{1}^{2} ρ_{1} v_{1} + R_{2}^{2} ρ_{2} v_{2}}{R_{3}^{2} ρ_{3}} = \frac{{av}_{1} + {bv}_{2}}{c}

(8)

\cos α = \frac{c^{2} v_{3}^{4} + a^{2} v_{1}^{4} - b^{2} v_{2}^{4}}{2 {acv}_{1}^{2} v_{3}^{2}}

(9)

\cos β = \frac{c^{2} v_{3}^{4} - a^{2} v_{1}^{4} + b^{2} v_{2}^{4}}{2 {bcv}_{2}^{2} v_{3}^{2}}

(10)

τ_{1} = \frac{\sin β}{\sin (α + β)} τ_{3} + \frac{R_{2} f_{2} \cos 2 β + R_{1} f_{1} \cos (α - β)}{\sin (α + β)}

(11)

τ_{2} = \frac{\sin α}{\sin (α + β)} τ_{3} + \frac{R_{1} f_{1} \cos 2 α - R_{2} f_{2} \cos (α - β)}{\sin (α + β)}

(12)

f_{1} = \frac{{av}_{1}^{2}}{{cv}_{3}^{2}} f_{3}

(13)

f_{2} = \frac{{bv}_{2}^{2}}{{cv}_{3}^{2}} f_{3}

(14)

where a is equal to

π R_{1}^{2} ρ_{1}

, b is equal to

π R_{2}^{2} ρ_{2}

and c is equal to

π R_{3}^{2} ρ_{3}

Distribution of spinning tension in zone 1

From Equations (13) and (14), we can get $\frac{f_{1}}{f_{2}} = \frac{{av}_{1}^{2}}{{bv}_{2}^{2}} = \frac{R_{_{1}}^{2} ρ_{1} v_{_{1}}^{2}}{R_{2}^{2} ρ_{2} v_{2}^{2}}$ . This indicates that the tension distribution on filaments and staple strands is directly proportional to the linear mass and speed square of the output component from front roller nip. In specialty, the velocities of filament 1 and staple strand 1 are equal during the conventional spinning (i. e., ν₁=ν₂) then, we obtain the following equation:

\frac{f_{1}}{f_{2}} = \frac{{av}_{1}^{2}}{{bv}_{2}^{2}} = \frac{a}{b}

(15)

Equation (15) further reveals that spinning tension distribution on the filament and staple strand is directly proportional to their dynamic linear mass.

Distribution of twists in zone 1

Previous evidence has validated that the angle of twist is directly proportional to the torque applied to the single spinning strand shaft.²⁰ From Equations (11) and (12), we can get

\frac{τ_{1}}{τ_{2}} = \frac{\sin β τ_{3} + R_{2} f_{2} \cos 2 β - R_{1} f_{1} \cos (α - β)}{\sin α τ_{3} + R_{1} f_{1} \cos 2 α - R_{2} f_{2} \cos (α - β)}

(16)

Equation (16) provides the calculation of the twist distribution ratio on spinning strand 1 and filament 1. It also indicates that more twists of composite strand C1C will cause greater twist insertion into filament 1 and staple spinning strand 1.

Configurations of spinning zone 1

Similarly, the linear speeds of filament 1 and staple strand 1 are equal during the conventional spinning (i. e., ν₁=ν₂). Then, Equations (9) and (10) can be converted as

\frac{\cos α}{\cos β} = \frac{c^{2} v_{3}^{4} + (a^{2} - b^{2}) v_{1}^{4}}{c^{2} v_{3}^{4} - (a^{2} - b^{2}) v_{1}^{4}}

(17)

If the linear mass of filament 1 is larger than that of staple spinning strand 1 (i.e., $a^{2} > b^{2}$ ), angle α will be smaller than angle β. This indicates that filament 1 is heavier than staple spinning strand 1, which will result in a smaller angle between filament 1 and the axis line of composite spinning strand 1. In the contrary, if staple spinning strand 1 is heavier than filament 1, it will cause a smaller angle between staple spinning strand 1 and the axis line of composite spinning strand 1.

Analysis of zone 2

Zones 1 and 2 are symmetrical; therefore, the distribution of spinning tensions, torques and angles on filaments and staple strands in zone 2 was similar to that in zone 1. Assuming f₆ is the tension of composite spinning strand 2, τ₆ is the torque of composite spinning strand 2, $v_{6}$ is the velocity of composite spinning strand 2, ρ₆ is the density of composite spinning strand 2, and then we can get

f_{3} = f_{6}

(18)

τ_{3} = τ_{6}

(19)

v_{3} = v_{6}

(20)

ρ_{3} = ρ_{6}

(21)

Analysis of zone 3

According to the above quasi-static analysis methods, equations can be derived that are listed below:

2 f_{3} \cos θ = f_{c}

(22)

2 τ_{3} \cos θ + 2 R_{3} f_{3} \sin θ = τ_{c}

(23)

2 R_{3}^{2} ρ_{3} v_{3}^{2} \cos θ = R_{c}^{2} ρ_{c} v_{c}^{2}

(24)

2 R_{3}^{2} ρ_{3} v_{3} = R_{c}^{2} ρ_{c} v_{c}

(25)

where f_c is the spinning tension of composite yarn, τ _c is the composite yarn torque,

v_{c}

is the composite yarn velocity, ρ _c is the composite yarn density, R_c is the composite yarn radius and θ is the angle of staple strand 1 and the composite yarn axis line.

Distribution of spinning tension in zone 3

Composite spinning strand tension can be calculated by Equation (22). Considering $0 < θ < 90 \deg$ , Equation (22) is transformed as $f_{3} > \frac{1}{2} \cdot f_{c}$ . This indicates that the spinning tension of each composite strand is larger than that of the resultant composite yarn. The combination of Equations (22) and (24) also derives the following equation:

f_{3} = \frac{R_{3}^{2} ρ_{3} v_{3}^{2}}{R_{c}^{2} ρ_{c} v_{c}^{2}} f_{c}

(26)

Equation (26) indicates that spinning tension distribution on the composite yarn and composite strand is closely related with its dynamic mass and moving speed.

Distribution of twists in zone 3

The following equation can be achieved by solving Equations (22), (23) and (24):

τ_{3} = \frac{R_{c}^{4} ρ_{c}^{2}}{4 R_{3}^{2} ρ_{3} R_{c}^{4} ρ_{c}^{2}} τ_{c} - \frac{2 R_{3}^{3} ρ_{3} \sqrt{R_{c}^{4} ρ_{c}^{2} - 4 R_{3}^{4} ρ_{3}^{2}}}{4 R_{3}^{2} ρ_{3} R_{c}^{4} ρ_{c}^{2}} f_{c}

(27)

Thus, twist distribution on the composite yarn and composite strand can be calculated by Equation (27). In particular, the greater the twist density on the composite yarn, the more severe the upstream twist passing onto the composite strands.

Configurations of spinning zone 3

The aforementioned analysis illustrates that composite strands 1 and 2 are symmetrical in theory. However, the triangle (2θ) between composite strands 1 and 2 is determined by some spinning parameters. Firstly, the triangle in spinning zone 3 is associated with the spinning tension distribution on the composite yarn and strands according to Equation (22). In addition, Equation (28) can be derived from a combination of Equations (24) and (25). Equation (28) indicates that the triangle in spinning zone 3 can be simply calculated by dividing composite yarn velocity by composite strand velocity:

\cos θ = \frac{v_{c}}{v_{3}}

(28)

Online pictorial proof

For validation of the above theoretical analysis, Ne 32^S carded (denoted as 32CZ) and Ne 140^S combed (denoted as 140CmZ) cotton yarns with Z twists are used to conduct the S-twist plying on an embeddable and locatable ring spinning frame, which is illustrated in Figure 4. In detail, two Ne32^S CZ yarns are dyed black; two Ne140^S CmZ yarns are dyed red. Then each of them is combined with its corresponding original color yarn as one of the ELS components. This is done to observe the dynamic twist transmit and distribution distinctly in the ELS triangle.

Figure 4.

Illustration of experimental embeddable and locatable spun components’ arrangement design for validation.

Online ELS triangle configurations and dynamic twist distributions are shown in Figure 5. In the Figure 5(a) are three online ELS configuration photographs captured randomly by an IXUS 75-type Canon digital camera during a spinning process with 6.75 m/min front roller speed and 1180 revolutions/min (denoted as rpm) spindle speed. The other three ELS configuration photographs in Figure 5(b) were taken by the same camera during another spinning process with 6.75 m/min front roller speed and 1530 rpm spindle speed. All online photos illustrate the thicker combined Ne32CZ component is almost in the same straight line with the pre-twisted composite plied yarn, while the finer Ne140CmZ component is bending at the convergence with the thicker combined Ne32CZ component. This phenomenon is in agreement with the theoretical analysis of ELS yarn formation zones 1 and 2 configurations. In particular, the twist density on the finer Ne140CmZ component is much lower than that on the thicker Ne32CZ component, which can be seen in Figures 5(a) and 5(b). This result validates the above theoretical analysis of the twist distribution in the ELS formation zone.

Figure 5.

Online photos of embeddable and locatable spun yarn formation zone configuration: (a) under a lower twisting speed; (b) under a higher twisting speed.

Conclusion

A self-contained quasi-static model is built to theoretically analyze the distribution of twists and spinning tension in the ELS yarn formation zone. For an easy analysis, the ELS yarn formation zone is divided as three zones (i.e., zones 1, 2 and 3). Important equations are derived to determine the distribution of twists and spinning tension and external configurations of the ELS yarn formation zones 1, 2 and 3. Analysis results indicate that the tension distribution on filaments and staple strands is directly proportional to the linear mass and speed square of the output component from the front roller nip; more twist on the composite strand will cause a larger twist insertion ratio into the filament and staple strand in zone 1. In particular, filament 1 is heavier than staple spinning strand 1, which will result in a smaller angle between filament 1 and the axis line of composite spinning strand 1; contrarily, if the staple spinning strand is heavier than filament 1, it will cause a smaller angle between staple spinning strand 1 and the axis line of composite spinning strand 1. The distribution of spinning tensions, torques and angles on filaments and staple strands in zone 2 is similar to that in zone 1. The triangle between composite strands 1 and 2 in spinning zone 3 is not only associated with the spinning tension distribution on the composite yarn and strands, but it can also be simply calculated by dividing the composite yarn velocity by the composite strand velocity.

Online photos of the ELS yarn formation zone validate the theoretical analysis of ELS yarn formation zones 1 and 2 configurations. The twist distribution in the ELS formation zone is also verified based on the online inspection of ELS formation zones.

Footnotes

Acknowledgement

The authors acknowledge the support of Hui Liu and Shengli Feng in the Jihua 3542 Textile Mill (Xiangyang, China) for their assistance with the original color yarn manufacture. We also thank Sibin Hu (the vice professor in the Art Dyeing Laboratory of Wuhan Textile University) for his assistance in dyeing of the original color yarns.

Funding

This research was supported by the innovation funding (NO. 101-06-0019040) from Donghua University, Songjiang District, Shanghai, China.

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