Study on the coiler drive mechanism of the capacity increasing function

Abstract

Keywords

eccentric belt drive mechanism kinematics analysis coiler density of per sliver lap increasing the capacity

In textile factories, the sliver can is used to deposit and transport sliver in the process of carding, combing and drawing. Increasing the capacity of the sliver can will reduce the frequency of changing the sliver can; hence, the working efficiency can be improved. It is therefore necessary to study how to increase the sliver can’s capacity.^1–8 Many methods of increasing the capacity of the sliver can have been investigated. Chen et al.^9–11 established the relationship between the center hole diameter of the sliver with the diameter of the sliver can and the width of sliver, and then derived the value of the center hole diameter when the volume of the sliver is greatest. Zhou¹² discussed the sliver overlapping in the densest zone around the hole of the can coil package, and derived the equations of the maximum coil package to determine the optimal eccentricity between the coiler and sliver can center. However, the above two methods cannot increase the capacity of the sliver can. Huang¹³ proposed a hold-down mechanism that can press the sliver tight and increase the number of sliver layers in the same can height, leading to the increase in the capacity of the sliver can. However, this kind of hold-down mechanism is complex and not easy to maintain. When the pressure head compresses the sliver column, the sliver will be damaged by the pressure head and thus decrease the quality of sliver. Gao et al.^14–16 proposed an reciprocating turntable coiler (RTC) coiling system, a device for increasing the capacity of the sliver can. The principle of the RTC coiling system is that the sliver can rotates around its axis and at the same time reciprocates slightly. This improves the sliver storage density along the radial direction of the sliver can, which consequently increases the capacity of the sliver can. However, in order to meet the sliver can’s motion, a driving mechanism that makes the sliver can reciprocate must be added. This not only increases the production cost, but also reduces the reliability of the machine. The Rieter Company^17,18 designed a new coiling system and compared it with the common coiling system; the features of this new device are that the sliver can is rectangular and the motion of the sliver can is the combination of reciprocating linear and clearance swing. This device can also increase the capacity of the sliver can. The new coiling system is simple and compact, although it does not have the capability to prevent impurity accumulation in the inclined tube.

Different from above studies, a new kind of coiler driving method is proposed in this study. It is based on an eccentric belt drive mechanism rather than any complicated mechanism. This new mechanism has the advantages of low cost, simple structure and convenient maintenance. This application not only can increase the capacity of the sliver can, but also has the capability of preventing impurity accumulation of the inclined tube.

The kinematic analysis of the eccentric belt drive mechanism

Firstly, we denote that point O is the rotational center of the driven pulley, and point O₂ is the geometric center of the driven pulley. The distance between point O and point O₂ is e (i.e. OO₂ = e). Point O₁ is defined as the rotational and geometric center of the driving pulley. At the initial position of this mechanism, points O, O₁ and O₂ are on the same line, and point O₂ is on the left-hand side of point O, as shown in Figure 1.

Figure 1.

Geometric center of the driven pulley O₂ locates above the x-axis.

The geometric center O₂ of the driven pulley is above the x-axis

Suppose φ is the angle between line OO₂ and the negative direction of the x-axis. Clockwise is defined as the positive direction. The tangent point between the belt and the driving pulley is denoted as point A. The tangent point between the belt and the driven pulley is point B. Line AC is parallel to line O₁O₂ and intersects with line O₂B at point C, as shown in Figure 1.

In this study, the elastic sliding of the belt drive mechanism was ignored. The radius of the driving pulley and the driven pulley were denoted as r₁ and r₂, respectively. We also denoted the distance of line OO₁ as a, the distance of line O₁O₂ as b, the distance of line OB as r, the linear velocity of the driving pulley and the driven pulley as V and V₂, respectively, and the angular velocity of the driven pulley as $ω_{2}$ .

From ΔOO₁O₂ in Figure 1, the cosine law gives

a 2 + e 2 - 2 ae \cos (π - φ) = b 2 b = \sqrt{a 2 + e 2 + 2 ae \cos φ}

(1)

b 2 + e 2 - 2 be \cos β = a 2

(2)

Combining equations (1) and (2), we get

\cos β = \frac{e + a \cos φ}{\sqrt{a 2 + e 2 + 2 ae \cos φ}}

(3)

and

\sin β = \frac{a \sin φ}{\sqrt{a 2 + e 2 + 2 ae \cos φ}}

(4)

In ΔABC

\sin γ = \frac{BC}{b} = \frac{r_{2} - r_{1}}{\sqrt{a 2 + e 2 + 2 ae \cos φ}}

(5)

\cos γ = \frac{\sqrt{a 2 + e 2 + 2 ae \cos φ - (r_{2} - r_{1}) 2}}{\sqrt{a 2 + e 2 + 2 ae \cos φ}}

(6)

In ΔOO₂B

e 2 + r_{2}^{2} - 2 {er}_{2} \cos (α + β) = r 2 r = \sqrt{e 2 + r_{2}^{2} - 2 {er}_{2} \cos (α + β)}

(7)

r_{2}^{2} + r 2 - 2 {rr}_{2} \cos ϕ = e 2 \cos ϕ = \frac{r_{2} - e cos (α + β)}{\sqrt{e 2 + r_{2}^{2} - 2 {er}_{2} cos (β - α)}}

(8)

because

α = \frac{π}{2} - γ

, then

cos (α + β) = \cos (∠ β + \frac{π}{2} - γ) = \sin (γ - β) = \sin γ \cos β - \cos γ \sin β

(9)

Substituting equations (3)–(6) into equation (9), we can achieve

cos (α + β) = \frac{{(r_{2} - r_{1}) (e + a \cos φ) - a \sin φ \sqrt{a 2 + e 2 + 2 ae \cos φ - (r_{2} - r_{1}) 2}}}{a 2 + e 2 + 2 ae \cos φ}

(10)

Also we have

V_{2} = V \cos ϕ

(11)

ω_{2} = \frac{V_{2}}{r}

(12)

Substituting equations (7), (8), (10) and (11) into equation (12), the relationship between

ω_{2}

and φ on the geometric center of the driven pulley O₂ above the x-axis (i.e.

0 \leq φ \leq π

) can be described as

ω_{2} = \frac{V \cos ϕ}{\sqrt{e 2 + r_{2}^{2} - 2 {er}_{2} cos (α + β)}} = \frac{V (r_{2} - e k_{1})}{e 2 + r_{2}^{2} - 2 {er}_{2} k_{1}}

(13)

where

The geometric center O₂ of the driven pulley is below the x-axis

From ΔOO₁O₂ in Figure 2, the cosine law gives

a 2 + e 2 - 2 ae \cos (φ - π) = b 2 b = \sqrt{a 2 + e 2 + 2 ae \cos φ}

(14)

b 2 + e 2 - 2 be \cos β = a 2

(15)

Combining equations (14) and (15), we get

\cos β = \frac{e + a \cos φ}{\sqrt{a 2 + e 2 + 2 ae \cos φ}}

(16)

and

\sin β = \frac{- a \sin φ}{\sqrt{a 2 + e 2 + 2 ae \cos φ}}

(17)

Figure 2.

Geometric center of the driven pulley O₂ locates below the x-axis.

In ΔABC

\sin γ = \frac{BC}{b} = \frac{r_{2} - r_{1}}{\sqrt{a 2 + e 2 + 2 ae \cos φ}}

(18)

\cos γ = \frac{\sqrt{a 2 + e 2 + 2 ae \cos φ - (r_{2} - r_{1}) 2}}{\sqrt{a 2 + e 2 + 2 ae \cos φ}}

(19)

In ΔOO₂B

e 2 + r_{2}^{2} - 2 {er}_{2} \cos (β - α) = r 2 r = \sqrt{e 2 + r_{2}^{2} - 2 {er}_{2} \cos (β - α)} r_{2}^{2} + r 2 - 2 {rr}_{2} \cos ϕ = e 2

(20)

\cos ϕ = \frac{r_{2}^{2} + r 2 - e 2}{2 {rr}_{2}} = \frac{r_{2} - e \cos (β - α)}{\sqrt{e 2 + r_{2}^{2} - 2 {er}_{2} \cos (β - α)}}

(21)

Because

α = \frac{π}{2} - γ

, then

cos (β - α) = \cos (β - \frac{π}{2} + γ) = \sin (β + γ) = \sin γ \cos β + \cos γ \sin β

(22)

Substituting equations (16)–(19) into equation (22), we can achieve

cos (β - α) = \frac{{(r_{2} - r_{1}) (e + a \cos φ) - a \sin φ \sqrt{a 2 + e 2 + 2 ae \cos φ - (r_{2} - r_{1}) 2}}}{a 2 + e 2 + 2 ae \cos φ}

(23)

Also, we have

V_{2} = V \cos ϕ

(24)

ω_{2} = \frac{V_{2}}{r}

(25)

Substituting equations (20), (21), (23) and (24) into equation (25), the relationship of geometric center of the driven pulley O₂ below the x-axis (i.e.

π < φ \leq 2 π

) between

ω_{2}

and φ can be described by equation (26)

ω_{2} = \frac{V_{2}}{r} = \frac{V \cos ϕ}{\sqrt{e 2 + r_{2}^{2} - 2 {er}_{2} \cos (β - α)}} = \frac{V (r_{2} - e k_{2})}{e 2 + r_{2}^{2} - 2 {er}_{2} k_{2}}

(26)

where

k_{2} = \frac{{(r_{2} - r_{1}) (e + a \cos φ) - a \sin φ \sqrt{a 2 + e 2 + 2 ae \cos φ - (r_{2} - r_{1}) 2}}}{a 2 + e 2 + 2 ae \cos φ}

In summary, because

k_{1} = \frac{{(r_{2} - r_{1}) (e + a \cos φ) - a \sin φ \sqrt{a 2 + e 2 + 2 ae \cos φ - (r_{2} - r_{1}) 2}}}{a 2 + e 2 + 2 ae \cos φ}

k_{2} = \frac{{(r_{2} - r_{1}) (e + a \cos φ) - a \sin φ \sqrt{a 2 + e 2 + 2 ae \cos φ - (r_{2} - r_{1}) 2}}}{a 2 + e 2 + 2 ae \cos φ}

k_{1} = k_{2}

k = k_{1} = k_{2}

and therefore the relationship between

ω_{2}

and φ can be expressed by equation (27) uniformly

ω_{2} = \frac{V_{2}}{r} = \frac{V (r_{2} - ek)}{e 2 + r_{2}^{2} - 2 {er}_{2} k}

(27)

According to equation (27), given the value of the relevant parameters, the curve describing the relationship between $ω_{2}$ and φ was plotted with MATLAB® (Math Works Inc., USA), as shown in Figure 3(a). Points M and N are denoted as the lowest and highest points of the curve, respectively. $φ_{1}$ and $φ_{2}$ are their abscissa values correspondingly. With the change of angle φ, the tangent point between the belt and the driving pulley changes, as does the tangent point between the belt and the driven pulley; hence, the difference between $φ_{2}$ and $φ_{1}$ does not equal $π$ (i.e. $Δ = φ_{2} - φ_{1} \neq π$ ). The initial installation location of the eccentric belt drive mechanism can be adjusted so that the maximum angular velocity $ω_{2 max}$ of the driven pulley could be achieved under the initial movement of the machine, as shown in Figure 3(b). From Figure 3, we see that when the driving pulley rotates at a constant speed, the driven pulley rotates at a periodically variable speed. With the change of rotation angle (φ), the angular velocity ( $ω_{2}$ ) also changes, and the periodic is $2 π$ .

Figure 3.

Relationship between $ω_{2}$ and φ: (a) initial condition; (b) after adjustment.

Application of the eccentric belt drive mechanism on the coiler

Chen and Liu¹⁹ proved that the density of sliver along the diameter direction of the sliver can is not uniform, and the highest winding density is located in the circular ring near the center of the sliver can, which is called the region of the hard core. The utilization ratio of the sliver can mainly depends on the number of sliver layers near the center of the sliver can.

A belt drive mechanism with the driven pulley eccentrically installed was used as the drive for the coiler in the present study. It ensures that the coiler rotates quickly when the coiler outlet approaches the center of the sliver can, and rotates slowly as the coiler outlet approaches the sliver can wall. The following analysis proves that this kind of drive mechanism can increase the capacity of the sliver can, while the sliver density of the hard core zone decreases.

The density of per sliver lap at the diameter direction of the sliver can

The slivers form a cycloid shape in the sliver can because of the coordination motion of the coiler and the sliver can.^20–24 However, the rotational speed of the sliver can of the coiler is very low compared with the rotational speed of the coiler. To simplify the calculation, we ignored the rotational speed of the sliver can and the cycloid shape was approximately taken as a circle.

As shown in Figure 4(a), we denoted the center of the sliver can as the center of a circle, the radius of the outlet of the inclined tube as r, the radius of a series of concentric circles as R, the inner and outside radius of the ring as R₁ and R₂, respectively, the width of the sliver as d and the eccentricity between the coiler and the can as e₁.

Figure 4.

Intersection of the circle of radius R and the sliver ring: (a) $(r - e_{1} + d / 2) < R < (r + e_{1} - d / 2)$ ; (b) $(r - e_{1} - d / 2) \leq R \leq (r - e_{1} + d / 2)$ ; (c) $(r + e_{1} - d / 2) \leq R \leq (r + e_{1} + d / 2)$ .

The two points at which a circle with radius of R intersects with the approximate circle are P₁ (x₁, y₁) and P₂ (x₂, y₂), respectively, and $Δ θ$ ( $Δ θ = θ_{2} - θ_{1}$ ) is the central angle corresponding to the arc P₁P₂.Then, the required time of depositing the sliver corresponding to the angle of $Δ θ$ is described by

Δ t = Δ θ / ω_{2}

(28)

where

ω_{2}

is the angular velocity of the driven pulley. The density of the sliver at radius R in the sliver can is proportional to

2 Δ t / 2 π R T

; therefore, the density of sliver ρ is

ρ = N 2 Δ t / 2 Δ t T T / 2 Δ t / 2 Δ t T T 2 π R 2 π R = N Δ t / π RT

(29)

where N is total number of cycles of the sliver can and T is the period of the coiler with a cycle. So the sliver density at the diameter direction of the sliver can depends on the density per sliver lap (DPSL)

ρ_{o}

ρ_{0} = Δ t / π RT

(30)

A rectangular coordinate system was established, as shown in Figure 4(a). Regarding the geometric center Q of the sliver can as the origin of coordinates, the sliver’s inner side and outside trajectory equations are

{(x_{1} - e_{1}) 2 + y_{1}^{2} = R_{1}^{2} (x_{2} - e_{1}) 2 + y_{2}^{2} = R_{2}^{2}

(31)

where

R_{1} = r - d / 2

and

R_{2} = r + d / 2

. In addition, we can also achieve the following equations

{x_{1}^{2} + y_{1}^{2} = R 2 x_{2}^{2} + y_{2}^{2} = R 2

(32)

From equations (31) and (32), the coordinate values of points P₁ and P₂ can be achieved as follows

{x_{1} = (e_{1}^{2} + R 2 - (r - d / 2) 2) / 2 e_{1} y_{1} = \sqrt{R 2 - x_{1}^{2}}

and

{x_{2} = (e_{1}^{2} + R 2 - (r - d / 2) 2) / 2 e_{1} y_{2} = \sqrt{R 2 - x_{2}^{2}}

The intersection position between the circle with radius R and the sliver ring changes along with the change of R; therefore, three cases are listed bellow to calculate the DPSL $ρ_{o}$ .

Case 1

When $(r - e_{1} + d / 2) < R < (r + e_{1} - d / 2)$ , as shown in Figure 4(a), the circle of radius R intersects with the sliver circle at points P₁ and P₂ respectively

Δ θ = θ_{2} - θ_{1} ρ_{0} = Δ θ / π R ω_{2} T

(33)

Case 2

When $(r - e_{1} - d / 2) \leq R \leq (r - e_{1} + d / 2)$ , as shown in Figure 4(b), the circle of radius R intersects with the sliver circle at point P₃

Δ θ = θ_{3} ρ_{0} = θ_{3} / π R ω_{2} T

(34)

Case 3

When $(r + e_{1} - d / 2) \leq R \leq (r + e_{1} + d / 2)$ , as shown in Figure 4(c), the circle of radius R intersects with the sliver circle at point P₄

Δ θ = θ_{4} ρ_{0} = θ_{4} / π R ω_{2} T

(35)

We can achieve the arbitrary values of $ρ_{0}$ with equations (33)–(35).

DPSL with the two coiler drive mechanism

The coiler is driven by a concentric driven pulley

Take the RBS-D30 drawing frame produced by the Rieter Company as an example, where r = 110 mm, e = 60 mm, D = 350 mm, d = 10 mm, T = 2/9 s and $ω_{2} = 9 π rad / s$ .²⁵ The DPSL at the sliver’s diameter direction plotted with MATLAB® (Math Works Inc., USA) is shown in Figure 5.

Figure 5.

Density per sliver lap (DPSL) with concentric installation of the driven pulley.

In Figure 5, point H represents the maximum point of the curve, which indicates the highest winding density located at this point. The radius of the corresponding point H is equal to the sum of the radius of the blowhole and the width of the sliver. However, after point H, with the increase of the sliver can’s radius, the value of DPSL drops sharply and reaches the minimum point L; the lowest winding density locates near point L. When R is approaching the wall of the sliver can, the value of DPSL gradually increases. We can establish, therefore, that the density of the sliver along the diameter direction of the sliver can is not uniform, which causes the problem of low utilization ratio of the sliver can. The result we achieved is consistent with Chen and Liu’s¹⁹ conclusion.

The coiler is driven by an eccentric driven pulley

According to Figure 3(b), set the initial position of the coiler reasonably to ensure that when the sliver outlet of the coiler approaches the center of the sliver can, the rotational speed of the coiler is high, and when the sliver outlet of the coiler approaches the sliver can wall, the rotational speed of the coiler is slow.²⁶ In this way, the utilization ratio of the sliver can is improved. Similarly, the DPSL at the sliver’s diameter direction plotted with MATLAB® (Math Works Inc., USA) is as shown in Figure 6.

Figure 6.

Density per sliver lap (DPSL) with eccentric installation of the driven pulley.

In Figure 6, because the coiler rotates with the fastest speed when the coiler outlet approaches the center of the sliver can, the sliver deposited at the center of the sliver can decreases, as does the value of DPSL of the hard core zone. Hence, the capacity of the sliver can will increase.

Discussion

We combine the above two curves into one coordinate system, as shown in Figure 7. From Figure 7(b), we achieved the maximum DPSL decrease from 0.3795 (constant speed) to 0.3537 (variable speed). Because the DPSL of the hard core zone decreases, the sliver can could hold more layers of sliver. We established, from Figure 7(c), that the maximum DPSLs of the two cases are 0.2283 (variable speed) and 0.2202 (constant speed). The maximum DPSL with concentric installation of the driven pulley is larger than that with eccentric installation of the driven pulley; however, the utilization ratio of the sliver can mainly depends on the DPSL near the center of the sliver can. This means the capacity of the sliver can will increase. The capacity increment of the sliver can is

δ = [(ρ_{o max} \times r) / (ρ_{o max}^{*} \times r) - 1] \times 100 % = [(0.3795 \times 120) / (0.3537 \times 120) - 1] \times 100 % \approx 7.29 %

where r is the radius of the outlet of the inclined tube,

ρ_{o max}

is the maximum DPSL with concentric installation of the driven pulley and

ρ_{o max}^{*}

is the maximum DPSL with eccentric installation of the driven pulley.

Figure 7.

Comparing the DPSL of two cases: (a) overall comparison drawing; (b) local enlarged drawing of area B1; (c) local enlarged drawing of area B2.

The above results were calculated by taking the eccentricity (e) of the driven pulley as 6 mm (i.e. e = 6 mm). With the different values of eccentricity (e), the values of capacity increment (δ) are also different. Taking eccentricity (e) as 4, 5, 6, 7, 8, 9 and 10 mm, respectively, the capacity increments of the sliver can’s (δ) corresponding eccentricity (e) are 5.41%, 6.36%, 7.29%, 8.27%, 9.27%, 9.81% and 10.51%, respectively. According to the above calculated values of these points, the relationship between δ and e was plotted with MATLAB® (Math Works Inc., USA), as shown in Figure 8. Figure 8 shows that, with the increase of eccentricity (e), the capacity increment (δ) of the sliver can also increases. From Figure 8, we also found that the relationship between the eccentricity and the capacity increment is not linear.

Figure 8.

The capacity increment of different eccentricities.

However, the value of eccentricity (e) should not be too large as it could aggravate the vibration of the coiler and worsen the machine’s work condition. In order to minimize the vibration of this mechanism at the working condition with a high speed, the optimal eccentricity (e) must be confirmed. To come up with the optimal eccentricity (e), some factors, such as accidental drafting of sliver, the capacity of the sliver can, the vibration of this mechanism and so on, need to be considered comprehensively. Therefore, much work needs to be done. In the next stage of our research, we will set up an experimental platform to carry out relevant experiments to confirm the optimal condition with mechanism running and optimal eccentricity (e), and to verify the results achieved in this study.

Conclusions

In this study, a new belt drive mechanism with the driven pulley eccentrically installed was proposed. It has the advantages of low cost, simple structure and convenient maintenance. The relationship between the angular velocity ( $ω_{2}$ ) and the rotation angle (φ) of the driven pulley was derived, and the feature of the mechanism is that, when the driving pulley rotates at a constant speed, the driven pulley rotates at a periodically varying speed with a period of $2 π$ .

Driven by this mechanism, the coiler can also achieve a motion with periodically varying speed. Following the motion rule of the driven eccentric pulley, the initial position of the driven pulley should be appropriately adjusted to ensure that the coiler rotates quickly when the coiler outlet approaches the center of the sliver can, and rotates slowly as the coiler outlet approaches the sliver can wall. This mechanism can not only increase the capacity of the sliver can, but also has the capability of preventing impurity accumulation of the inclined tube.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Science and Technology Support Program of China (grant number 2011BAF08B03).

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Study on the coiler drive mechanism of the capacity increasing function

Abstract

Keywords

The kinematic analysis of the eccentric belt drive mechanism

The geometric center O2 of the driven pulley is above the x-axis

The geometric center O2 of the driven pulley is below the x-axis

Application of the eccentric belt drive mechanism on the coiler

The density of per sliver lap at the diameter direction of the sliver can

Case 1

Case 2

Case 3

DPSL with the two coiler drive mechanism

The coiler is driven by a concentric driven pulley

The coiler is driven by an eccentric driven pulley

Discussion

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References

The geometric center O₂ of the driven pulley is above the x-axis

The geometric center O₂ of the driven pulley is below the x-axis