Ghosh, Tushar K
ABSTRACT
In recent years, the interest in studying balloon formation during unwinding has been rekindled by recent publication of several significant papers on the dynamic analysis of moving yams in rotational modes. The renewed interest can be partially attributed to the availability of powerful computing tools needed to solve problems of this kind. This paper is the first of a series of papers reporting results of ongoing research at NCSU. In this paper, apart from a critical evaluation of some recently published work, the physics of unwinding as proposed by earlier publications is examined through extensive parametric studies. That is, two parameters of high practical importance, balloon shape and unwinding tension, are calculated as functions of the direction of unwinding (from front-to-back and back-to-front), wind angle, residual tension in the yam on the package, and the coefficient of yam-package drag. In addition, limitations of a two-region analysis are addressed.
Over-end unwinding of yam is required in many textile processes (e.g., winding, doubling, twisting, warping, etc.) for it permits high speed at low energy input. Possible speeds, however, are limited by yam breakage. Understanding the dynamics of unwinding is therefore critically important to process set-up and control. In 1992, we (Fraser, Ghosh, and Batra (FGB) [7]) modeled over-end unwinding of yarn from the central region of the package (excluding the two ends) going from frontto-back of the package as a two-region problem. Since then, we (Ghosh et al. [8-10]) have discussed the role of various unwinding parameters on balloon shape and tension distribution in a number of presentations. More recently, Kong et al. [14] studied the variation in yam tension during unwinding using a modified FGB two-- region model in conjunction with their experimental observations of cyclic fluctuation. Clark et al. [5] followed this with analytical modeling of some of the phenomena observed by Kong et al. More on this later.
In our current series of papers, we explore further the role of several critical parameters and some unresolved issues in detail. In Part I, we carry out a parametric study of the FGB [7] model to reveal the influence of package parameters on unwinding tension and balloon shapes.
Background
In over-end unwinding, the yam is withdrawn through a guide eye 0 located along the axis of a stationary package (see Figure 1). Thus far in the literature, the path of the moving yam has been assumed to consist of two regions characterized by different features. In region 1, from the guide eye 0 to the lift-off point L, the yam moves in air. It rotates about the package axis, forming a "balloon." In region 2, from the lift-off point L to the unwind point U, the yam slides on the surface of the package away from its original stationary position.
A review of the previous analyses of the problem is given in reference 7. For present purposes, we deem an abbreviated account of the evolution of the analysis as appropriate. Padfield [22-24] developed equations of motion (EoM) of yarn in the balloon region (region 1). For slow speed unwinding, Padfield obtained solutions (balloon shapes and the corresponding yam tension distributions) for a certain assumed set of boundary conditions at the lift-off point, including yarn wind angle sigma = 0. Nonsteady-state unwinding from packages with nearly zero wind angles, she proposed, could be approximated by a series of steady-state boundary value problems with modified boundary conditions. Both Booth [4] and Padfield [23] recognized that the yam slides on the package before entering the balloon region, and they derived the EoM in the sliding region (region 2). Booth, in fact, obtained closed form solutions of the yarn tension distribution in region 2, provided the boundary conditions at the lift-off point were known; in practice, they are not. Kothari and Leaf [15-19] extended Padfield's work by obtaining a large number of numerical solutions, and using these results, they obtained much simpler correlational equations relating the process and package parameters. The idea was to make Padfield's work more accessible to practitioners in the industry, who had only slide rules or at best mechanical calculators on which to rely.
Nearly forty years later, we (FGB [7]) developed an integrated approach to couple the analysis of the two regions, from guide eye to unwind point. In doing so, we recognized that the problem involves two time scales. One relates to the periodic nature of unwinding: the duration of unwinding yarn from front-to-back and back-- to-front of the package constituted a period tau for a layer of radius c. The second relates to transient effects within this period. We used a perturbation expansion of the equations of motion to remove the time dependence from the Oth order equations of motion, but we accounted for the time dependence by specifying a boundary condition at the unwind point. Using this model, we [7] examined the shape and tension distribution along the path through a whole unwinding period for small wind angles.
In 1999, Kong et al. [14] presented a two-region model as well as experimental data on unwinding of cylindrical packages. Their model, a modified version of the FGB two-region model, calculates tension variation during unwinding of a cylindrical package. However, the calculation requires specification of the lift-off angle sigma^sub l^ as a boundary condition. In practice, this cannot be known a priori, which limits the practical value of their analysis. Nevertheless, their experimental observations are in general agreement with their theoretical results as well as those we presented previously (Ghosh et al. [8-10]). Their conclusion that "the unwinding direction on the package greatly influences balloon tension" in large measure is questionable. It is based on experimental tension measurements illustrated in their Figure 8 (not reproduced here). The illustration shows that unwinding tension achieves the expected minima-maxima as it unwinds from front-to-back and back-to-front. Further, the time span from a minima to maxima, roughly corresponding to unwinding from front to back, is fairly consistently smaller than the time span from a maxima to minima, roughly corresponding to unwinding from back-- to-front. Seen thus, their Figure 8 suggests a longer duration of unwinding (implying slower balloon rotation) from back-to-front relative to the duration of unwinding from front-to-back.
In Figure 8 of their publication, Kong et al. showed three consecutive synchronization pulses (dashed lined), or two consecutive time intervals, as defining the time taken by one full rotation of the balloon, which implies unwinding of one coil in either direction. Following the tension trace in their Figure 8, it is easy to count 3 or 3.5 coils in the front-to-back direction in contrast to 5-5.5 coils in the back-to-front direction. This suggests that during unwinding, the wind angle on the package from front-to-back differs from that in the back-to-front direction. This phenomenon is plausible with suitably designed "precision winders." Thus the difference in time duration of tension minima to maxima and then maxima to minima observed by Kong et al. can be largely accounted for by the different winding angles (or the number of coils) in the two directions. There is one other contribution to the different durations of unwinding from front-to-back relative to back-to-front, which we discuss next.
Visualize unwinding from front-to-back: In the body of the package, the unwind point proceeds along coils with a constant helix angle until it gets close to the end zone at the back. In the end zone, the wind angle starts decreasing to zero and builds up to a final negative value, indicating reversal in the direction of the helix. Simultaneously, the residual tension in the yam decreases to zero and builds back up to a steady value, even if the helix direction is reversed. The balloon height during the same process continues to increase, but not necessarily all the way to the back of the package; the decrease in residual tension would, at some point, permit the unwind point to jump from the positive helix side to the negative helix side. As such, the lift-off point may never reach the bottom of the package. The peak value of the unwind tension need not correspond to the unwind point or lift-off point, being located at the bottom of the package.
By the same token, visualize the unwinding from back-to-front: First note that the wind angle at lift-off is positive, while that at the unwind point is negative (see Figure 2). As a result (and as we shall show), the length between unwind and lift-off points dragging over the package is larger than in the front-to-back case. The situation persists until we reach the front end zone. Here, it is possible to have the lift-off point reach the front end of the package while the unwind point is still further away from the front end. The residual tension at the unwind point may be what it was in the body of the package or less. The boundary values at this stage change character. The lift-off point distance from the guide eye remains constant for some time. During this time, the distance of the unwind point from the guide eye continues to decrease, equals that of the lift-off point, and then increases again. During this same period, the residual tension goes through a minimum (zero) and increases again. The time interval ends when yam dynamic equilibrium approaches the conditions where the lift-off point starts moving away from the front end as unwinding from front-to-back proceeds. Previously, this phenomenon was referred to as the "lift-off point falls off the package" or "the lift-off point is on the virtual package"; these descriptions create an erroneous view of the physics of what goes on.
Next and most recently, Clark, Fraser, and Stump [5] modeled over-end unwinding of cylindrical packages, assuming the thread possessed bending and torsional rigidity, e.g., a monofilament. They showed that the effects of bending and torsional rigidity are highly localized near the unwind point. In fact, for all practical purposes the flexible thread model of the FGB analysis holds. Given that they reached this conclusion assuming the monofilament to be an isotropic material, it would be even more true of mutifilament yams of highly anisotropic fibers.
Clark et al. [5] also attempted to explain the time difference between unwinding from front-to-back versus back-to-front reported by Kong et al. and discussed earlier in this section. Their analysis is mathematically elegant, but it suffers from two weaknesses: First, it assumes a constant wind angle all through the package, including the ends. As such, it ignores changes in the kinematics of unwinding due to varying wind angles at the front and back ends of the package. Second, it also ignores the difference in wind angles in the body of the package in the two directions, as mentioned earlier.
Neither Kong et al. [14] nor Clark et al. [5] have addressed two fundamental limitations of the FGB model: First, the FGB model assumes that the yam tension at the unwind point is completely determined by the dynamics of unwinding exclusive of the characteristics of the package. As a result, the tension at the unwind point becomes a function of the balloon height and the radius of the package and has no apparent relation to the residual tension in the yarn in the package. Second, the model does not account for differences in "back-to-front" and "front-to-back" unwinding of the package; more on this later.
Note that the notion of residual tension or residual stress in wound yam and fabric packages is not new. It has been investigated and discussed in various contexts in the past by Nakashima [20], Beddoe [3], Neal [21], Vicentini [27], Batra et al. [2], Shintaku et al. [25-26], and Ghosh et al. [11-13]. Suffice it to note here that while the concept is real and physically meaningful, there are no reliable (convenient or practical) methods to measure, or estimate, the residual tension (or stress) due to the discrete-continuum nature of the package and the nonlinear, viscoelastic, and anisotropic nature of the materials involved.
In the context of unwinding, practitioners of the art report that considerable differences in unwinding characteristics of "hard" and "soft" packages are distinguished by the difficulty or ease of squeezing the package. We translate that to mean that hard packages imply considerable residual tension in the yam well into the depth (radial) of the package, or positive (extensional) hoop stress well into the depth of the package. If so, the squeezing action induces additional tension in the yarn or extensional hoop stress in the package, which is resisted by the yarn. Such packages may also be characterized by the "piddling" phenomenon. That is, if the package is left standing on its end on a flat surface, the free end of the yarn will start unwinding by itself and drop in the form of loose coils on the flat surface until cohesive forces due to surface finish and changed residual tension in the subsequent layers of the yam overcome this behavior. The reverse is assumed to be true of soft packages.
If this argument holds, the residual tension at the unwind point, at the time it enters the unwinding zone (region 2), would then become the driving parameter that determines the unwinding tension.
Boundary Value Problem Formulation
We have fully explained the derivation of the equations of motion, the perturbation scheme, the nondimensionalization, as well as the methods of solving the resulting boundary value problem, in reference 7, so we will not repeat them here. To facilitate interpretation of the results, definitions of nondimensional entities of the problem are given in the Appendix. All parameters used in this paper are nondimensional. In this study, however, in contrast to the FGB methodology, we obtain the solutions by iterating on the field equations of regions 1 and 2 simultaneously, satisfying the boundary conditions at the guide eye as well as at the unwind point and the continuity conditions at the lift-off point connecting regions 1 and 2. In Fps [7], we used a closed form solution (Equation 64, FGB [7]) of the equations of yam motion on the package to calculate the position of the unwind point on the package.
ACKNOWLEDGEMENT
The research reported in this paper was supported by grants from the National Textile Center funded by the U.S. Department of Commerce. We also acknowledge the intellectual and moral support of the Textured Yam Association of America and Mr. Joe Plasky.
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Manuscript received May 10, 2000; accepted December 11, 2000.
TUSHAR K. GHOSH, SUBHASH K. BATRA, AND A. S. MURTHY
College of Textiles, North Carolina State University, Raleigh, North Carolina 27695, U.S.A.
