approach to analyse of reaction forces in ring–traveller … · approach to analyse of reaction...

APPROACH TO ANALYSE OF REACTION FORCES IN RING–TRAVELLER SYSTEM

Jaroslav Beran, Miroslava Konečna

Technical University of Liberec, Department of Textile Machine Design, Studentska 2, 461 17 Liberec, Czech Republic, tel. +420485353171Corresponding author, e-mail: [email protected]

1. Introduction

The ring–traveller system belongs to the basic parts of ring twisting or spinning frames. The traveller circulates on the ring, and its motion provides both for the formation of the twist in the yarn and for the winding of yarn upon the bobbin. Depending on the character of the seating, determined by the shapes of the ring and the traveller, as a rule, it is a case of seating with one or two reaction forces. The driving force of the traveller is the yarn tension, generated as a result of the balance of forces acting on the traveller during the winding process of the yarn [1], [2]. The ballooning of yarn, which is an accompanying phenomenon of the winding process, exerts a considerable influence on the reaction forces in the ring–traveller system. As a rule, the yarn creates two rotating curves, I and II (Fig. 1), separated mutually by the traveller.

Speed of the traveller is one of the basic limiting factors in the increase of the output of ring frames. The traveller is pressed to the ring flange by a force that is the resultant of the forces acting on the traveller. During the winding process, the normal reaction force between the ring and the traveller produces a friction force and a contact pressure, which exert a considerable influence on the wear of the traveller, and consequently, on its service life. High traveller speed leads to higher contact pressure (up to 35 N/mm2) and generation of heat. There are some works in which friction forces and heat generated were reduced using a rotating ring with magnetic or air bearing [3], [4], [5]. However, these new concepts of ring spinning have not been implemented in new textile machinery. The objective of this study is to present the approach to analyse of reaction forces and contact pressure for ring–traveller system and to put forward recommendations for optimization of the shape of the traveller to increase its service life.

Abstract:

The paper presents the approach to analyse of reaction forces and contact pressure for ring–traveller system in twisting machine. The theory of the ballooning yarn has been adapted for the calculation of reactions, and the motion equation is developed for traveller with two reaction forces. The components of normal reaction forces are analyzed for system with free balloon and control one. The effect of the balloon control ring is shown to be significant, reducing the reaction forces of traveller or increasing the speed of spindle. Contact pressure generated between the ring and the traveller is analyzed using Hertzian contact theory. The article also presents the recommendations for optimization of the traveller shape to increase its service life.

Keywords:

ring-traveller system, reaction forces, contact pressure.

Figure 1. Schema of ring twisting

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AUTEX Research Journal, Vol. 17, No 3, September 2017, DOI: 10.1515/aut-2016-0028 © AUTEX

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the z-axis is identical with the axis of the bobbin (see Fig. 5) and with the axis ζ of the coordinates system Oραζ too. With respect to the above-defined system of coordinates Oραζ, the system Orϕz rotates with the angular velocity identical with the traveller speed. We shall assume that every point of the curve of ballooning yarn is described by means of cylindrical coordinates r(s, t), ϕ(s, t), and z(s, t), which are the function of the time t and the yarn length s measured along the curve of the yarn from the guide eye. Vector of yarn tension T(s, t) in the ballooning yarn is given by the unit vector

of the tangent to the curve of the ballooning yarn. The yarn tension Tb is acting against the vector τ(sA−, t), and the yarn tensions acting on the traveller can be expressed as follows:

Tb = −T(sA−, t), Tn = T(sA+, t), (2)

2. Equation of the traveller motion

In the literature, a relatively small attention is given to the motion of the traveller on the ring, except the studies concerned with yarn ballooning, in which the equation of the traveller motion constitutes one of the boundary conditions of the analyzed process, for example, Refs [6], [7], and others. During its circulation on the ring, the traveller performs a complicated motion during the winding process, consisting of a number of partial motions, namely, the longitudinal motion given by the motion of the ring rail, the rotational motion on the ring, and the turning with respect to the ring, depending on the character of the traveller seating on the ring. Generally, it is a complicated problem; therefore, this paper focuses on the rotational motion of the traveller on the ring, which is decisive for the reaction forces and contact pressure in the ring–traveller system. The equation of the traveller motion has been established under the following assumptions. The traveller is considered to be a mass point. The mass of the traveller is concentrated in its center of gravity; because of practical reasons, the path of the center of gravity has been substituted with a circle of the radius equal to the inner radius of the ring. Due to the concentration of the mass into the center of gravity of the traveller, the moments of the forces acting on the traveller are not taken into account for the solution. The solution does not include high-frequency effects caused by the oscillation of the spindle, mounting, and manufacturing inaccuracies. In ring twisting and spinning, the spindle is rotated at a constant rate and the traveller is dragged around the ring by the yarn. Due to the frictional force between the ring and traveller, the required difference in speed between the spindle and traveller is automatically adjusted. Mechanism of twist insertion to the yarn, when the spindle is stationary, can be found in Ref [8].

There are two types of traveller seating, one is with one reaction force and the other is with two reaction forces, examples of these travellers are shown in Figs. 2 and 3, respectively. The equation of the traveller motion for the seating with one reaction force has been published, for example, in the works [1], [2], and [6]. This paper focuses on vertical ring with the seating with two reaction forces. In view of the above-mentioned assumptions, it is a solution to the forced movement of the mass point related to a circle, under the effect of the three-dimensional system of forces passing through one point (Fig. 4). We shall introduce a system of cylindrical coordinates Oραζ. The beginning O of the coordinate system is in the center of the guide eye, and the axis ζ is identical with the axis of the ring. The equation of the traveller can be written as

bm 021nb FGRRTTa +++++= , (1)

where m is the mass of the traveller, a is the acceleration of the traveller, Tb is the yarn tension at the bottom of the balloon, Tn is the yarn tension from the yarn balloon between wound bobbin and traveller (winding tension), R1 and R2 are the reaction forces of the ring, G is the weight of the traveller, and F0b is the air drag.

We shall introduce a system of cylindrical coordinates Orϕz, in which the beginning O is in the center of the guide eye and

Figure 2. Traveller with two reaction forces

Figure 3. Traveller with one reaction force


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where s = sA(t) is the length of the yarn between guide eye and traveller. Then the components of the yarn tensions are:

(3)

The winding tension can be estimated by using the Euler equation as follows:

(4)

where f1 is the friction coefficient between yarn and traveller and γ is the wrap angle, and it is defined by the vectors Tn and Tb. The equation of motion (1) might be broken down into the following component equations, using equations (3) and (4):

(5)

where α is the angular velocity of the traveller, α is the angular acceleration of the traveller, ζ is the acceleration of the ring rail, rp is the ring radius, N1 and N2 are the normal components of reaction forces of the ring, Fs1 and Fs2 are friction forces, and g is the gravity acceleration. The traveller is in contact with two surfaces of the ring. We shall assume that the coefficient of friction is the same for both the pairs of contact surfaces. Consequently, we can write

(6)

where f is the coefficient of friction between the traveller and the ring. The air drag is expressed by the following relation:

(7)

where c is the dimensionless co-efficient of air drag, S is the surface of the largest cross-section of the traveller related to the aerodynamic drag, and ρ is the air density.

The ring–rail position is considered as constant in the presented model. Due to other constraint r=rp, the traveller is left with one degree of freedom, representing the movement of traveller along a circle. The proper equation of motion is then obtained after excluding the reaction forces N1 and N2 from the system of equations (5):

(8)

The derived equation requires the solution of both the sections I and II of the ballooning yarn. Under certain assumptions, the calculation can be simplified in the following manner. With ring twisting or spinning, the distribution motion of the yarn along the axis of the package is often a slow process. It means that the curve II of the ballooning yarn between the traveller and the package comes close to a planar curve and 0

sz

Ass

=

∂∂

+=.

Furthermore, we shall substitute this curve with a straight line, which is tangent to the rotary surface of the bobbin and passes through the center of gravity of the traveller. The components of the yarn tension Tn can then be expressed as follows:

(9)

where the angle β can be determined from Fig. 4. The equation of the traveller motion (8) is then converted into the following simplified form:

(10)

Figure 4. Forces acting in the ring–traveller system



where µo is the linear yarn density, l is unloading length of the yarn, and For, Foϕ, and Foz are the components of the air drag force related to the unit length of the yarn. The air drag force vector is given by the relation

(12)

where ρ is the air density, D is the effective yarn diameter, and vn is the normal component of yarn velocity.

The calculation of the curve of the ballooning yarn with the ring frames leads to a boundary problem for the system of ordinary nonlinear differential equations. For free balloon in the point A1 (guide eye), the following boundary conditions can be written for s = 0:

r(0) = ro, ϕ(0) = 0, z(0) = 0, ϕ´(0) = 0, (13)

where ro is the radius of the guide eye. In the point A2 (traveller), the following boundary conditions apply for s = sA:

r(sA) = rp, z(sA) = h, (14)

where h is the balloon height (Fig. 5). Another boundary condition on the traveller is given by the equation of the traveller motion by excluding time from equation (10):

.

(15)

3. Model of the yarn balloon

An exact solution of the curve of the ballooning yarn in the general three-dimensional case is difficult. Practically, it is not feasible to achieve a simple analytical expression for this curve. Under certain simplifying assumptions, the curve of the ballooning yarn can be established by numerical solution, for example, Refs [9–11]. During the twisting or spinning process, the boundary conditions given by the package formation are changing. Changes in yarn tension and balloon shape further dependent on the yarn kind were analyzed in Ref [12]. Even if a number of yarn processes are generally declared to be quasi-stationary, wholly stationary conditions do not exist really. The choice of a suitable approximation of the yarn ballooning depends on the problem under consideration. The approximation by a quasi-stationary process is satisfactory for some cases of ring spinning or twisting and much less acceptable for the majority of yarn unwinding processes [13]. Very close to the stationary boundary conditions are the conditions in the process of two-for-one twisting [10]. Moreover, close to them is the process of winding upon the cylindrical surface of the package in ring twisting, when the radius of the bobbin changes in time only after the whole layer has been wound, and by a very small value only, and at the same time, the velocity of the ring rail is considerably lower in comparison with the speed of yarn through the balloon. This approximation has also been chosen for the following analysis of reaction forces in the ring–traveller system.

Following are the assumptions defining the stationary mathematical model of the ballooning yarn:- the yarn is perfectly flexible and linear elastic,- the yarn has uniform mass linear density,- the deformation of the yarn is not influenced neither by

relaxation nor by creeping phenomena, and it does not depend on the temperature,

- the component of the air drag force acting normal to the thread line is only considered, and the coefficient of the air drag is constant along the whole curve of the ballooning yarn,

- the angular velocity of the yarn ω is constant,- the velocity of yarn through the balloon (sliding velocity) w is

constant,- the boundary conditions are independent on the time.

The effect of the tangential component of the air drag upon the yarn is neglected. This assumption has been studied, for example, in Ref. [14].

The component equations of the balance of the yarn element ds expressed in the cylindrical coordinate system Orϕz (Fig. 5), which rotates around the z-axis at the angular velocity ω, developed in Ref. [14], are of the following forms:

Figure 5. Ballooning yarn

(11)



Newton–Raphson scheme is used, and the iteration is finished if the boundary conditions (14) and (15) on the traveller are fulfilled. The proposed algorithm of the solution can also be applied to the controlled balloon with ring limiter.

5. Results of the analysis

The derived equation of the traveller motion and the mathematical model of the ballooning yarn are used for determining the reaction forces and contact pressure in the ring–traveller system. After excluding the time, from the simplified component equations of motion (5), the normal reaction forces N1 and N2 can be expressed as follows:

(17)

.

For the parameters of a heavy twisting ring frame, the dependencies of both normal reaction forces on the traveller speed under the following assumptions have been analyzed. Both the friction coefficients are constant for the chosen interval of traveller speed, which is to be analyzed, and the number of yarn twists per unit length is constant. Normal forces for the free balloon and the balloon with ring limiter are first analyzed by using the same traveller mass for the maximum winding radius (Fig. 7). This mass causes the required maximum diameter of the yarn balloon. By using a balloon ring limiter, the normal forces are negligibly reduced in this case; however, the maximum radius of the balloon is reduced considerably. If reducing the traveller mass to such a value when the maximum radius of the balloon reaches its original value again, the

The controlled balloon with one ring limiter can by divided into three regions. Two regions of ballooning curve develop in the air and are subject to the same acting forces as free balloon. The third region of yarn is moving on the surface of the ring limiter. The length of yarn in this region is very much smaller than the total length of the ballooning yarn. Then the length of this region can be neglected. In this case, the ring limiter is modeled as a frictionless geometrical circle constraint (Fig. 6) on the yarn path in the balloon. On this circle constraint, the following boundary conditions can be written for s = s1:

r(s1) = rom, z(s1) = hom. (16)

where hom is the distance between balloon limiter and guide eye, rom is radius of circle constraint (Fig. 6), and s1 is the length of the yarn between guide eye and circle constraint. Another boundary condition is given by the equilibrium equation of the yarn tension on the upper side of limiter, the yarn tension on the lower side of limiter, and the reaction force with normal and friction components [16].

Figure 6. Model of the ring limiter

4. Numerical solution

The system of nonlinear differential equations (11) is rewritten to a system of the first-order differential equations. The boundary problem is solved by the multiple shooting method in which the character of the boundary conditions allows converting the solution into a problem with initial conditions. In our case, the sequence of initial problems with different values r´(0) and T(0) is solved. The Runge–Kutta method has been used to integrate system of differential equations, and because the value of sA is not known in advance, the boundary condition z(sA) = h is used for stopping the process of integration. In this way, the value of sA is determined. For adjusting the approximations of the initial values T(0) and r´(0), the iteration method based on the

Figure 7. Reaction forces (free and controlled balloon, max. winding radius)



Figure 8. Reaction forces (controlled balloon, max. winding radius) Figure 9. Reaction forces (free balloon, min. winding radius)

Figure 10. Reaction forces (controlled balloon, min. winding radius)

normal reaction force N1 is reduced considerably, more than triply, compare the results in Figs. 7 and 8. Concurrently, the yarn tension drops down favorably. The reduction of the normal force N2 constitutes a 2,5-multiple approximately. An analogous drop of the normal forces is also reached on the minimum winding radius (Figs. 9 and 10). From the standpoint of the loading of the traveller, using a balloon limiter into the twisting line allows to increase the spindle speed. For example, the value of the normal force N1, corresponding to 3,800 rpm. with free balloon, is achieved for a controlled balloon at a value as high as 7,000 rpm with reduced traveller mass. Furthermore, results of analysis show that the normal reaction force N1 in the radial direction is significantly higher than the reaction force N2 in the direction of the ring axis except in the case of controlled balloon and minimum winding radius (Fig. 10).

Well-known results from the ring spinning/twisting show that the yarn tension in the balloon and the reaction forces are proportional to the square of the traveller speed [2], [9], [10]. The values calculated using equation (17) are corresponding to this thesis.

Contact pressure, which is generated between the ring and the traveller during winding, leads to the wear of the traveller and the generation of heat. When the spindle speed is increased, the friction work between the ring and the traveller causes thermal damage and fails. Assume the shape of a traveller with two cylindrical contact configurations: (1) at point (a) (Fig. 11), there is cylinder on cylinder contact configuration; and (2) at the point (b), there is cylinder on flat plane contact configuration. When two cylindrical parts with parallel axis are in contact under a force, the contact line turns to area which is a narrow rectangle. The Hertz

theory of elastic contact [17] is used for the analysis of contact pressure with assumptions made in this theory. For example, each body is considered as an elastic solid, the significant dimensions of the contact are small compared with the dimension of each body, and the contact surfaces are assumed to be frictionless so that only a normal pressure is transmitted between them.



The results presented in Figs. 12 and 13 show that the contact pressure between the traveller and the ring is linear with increasing traveller speed, which follows from the Hertz theory. Contact pressure in place (b) is higher than that in place (a) for a given cylinder length in places (a) and (b), although the ratio of normal forces is opposite. This is caused by the different size of the contact surfaces. Contact pressure in the ring–traveller system for the free balloon and for the control one are compared in Figs. 12 and 13. By using a control balloon, the contact pressure in point (a) and point (b) is reduced approximately 1.9 times and 1.6 times, respectively. In this case, the speed of spindles can be increased by up to 60%, whereby the contact pressure remains the same.

Figure 11. Contact configuration of point (a) andpoint (b)

Figure 12. Contact pressure for maximum winding radius

There is some possibility for an optimization of the contact pressures in the ring–traveller system, because the pressure at contact points (a) and (b) is generally different, namely, by the effect of different loads and different size of the contact surfaces on the traveller. Using this presented approach to analyse of reaction forces would be achieved a state at which both the contact surfaces might be loaded with same contact pressure. For the optimization of traveller shape, in this paper, geometric parameter is used – cylinder length at contact point (b). Result of optimization is presented in Fig. 14. The same pressure is achieved using a cylinder with a length of 3.4 mm, which is corresponding to the traveller width. The contact pressure in point (b) is reduced to a value at the point (a). Optimal value is independent on the traveller speed.

6. Conclusions

This paper describes the development of the motion equation for traveller with two reaction forces. Using the mathematical model of the yarn balloon, the reaction forces in the ring-traveller system has been analyzed. By introducing a balloon ring limiter into the twisting line, the reaction forces can be reduced to 40% or even to 30% with respect to the magnitude of reaction forces with a free balloon, bringing a substantial reduction of contact pressure between the ring and the traveller as a consequence. On the other hand, the introduction of a balloon limiter into the twisting line allows increasing the speed of spindles by up to 60% without increasing the contact pressure. Presented results show that contact pressure between traveller and ring depending on traveller speed is a linear function unlike reaction forces that are proportional to the square of traveller speed. The approach presented in this paper can also be used for the optimization of the contact pressures in the ring–traveller system with the purpose of achieving the same loading of both



the contact areas of the traveller, thereby increasing its service life. Commonly used travellers with two reaction forces have different size of the contact areas with the ring and different magnitude of both reaction forces. The optimal traveller shape

Figure 13. Contact pressure for minimum winding radius

Figure 14. Optimization of contact pressure

with the same contact pressure in contact areas can be found by numerical solution in order to increase its service life. In this paper, the optimization of traveller shape with two cylindrical contact configuration is presented.



Acknowledgment

The paper was supported by the subvention of Specific University Research by Ministry of Education CZ and by the subvention of the Project OP VaVpI Centre for Nanomaterials, Advanced Technologies and Innovation CZ.1.05/2.1.00/01.0005.

References

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