Computation of span length variations due to out-of-round material rolls

Abstract

It is well known that non-ideal elements such as out-of-round/eccentric material rolls affect web tension. However, the mechanism through which these non-ideal components induce tension oscillations was not clear previously. In a companion paper (Modeling and Identification of the Source of Oscillations in Web Tension) it is shown that an out-of-round/eccentric material roll produces length variations in the web span adjacent to the roll. These length variations are the main reason for oscillations in the tension signal; this was experimentally verified in the companion paper. In order to reproduce these tension oscillations in model simulations, it was necessary to include span length variations in the tension dynamics models. Given a generic profile for the out-of-round unwind roll, determination of the length of the adjacent span as a function of time as material is released from the roll is a formidable task. The focus of this paper is on finding a relationship between the shape of the out-of-round material roll and length of the span adjacent to it.

The simplest case to analyze is the length variations due to an eccentric roller. Considering the geometry of the problem, it is possible to find an expression in closed form that gives the length of the web span as function of the angular displacement of the roller. The expression for the rate of change of span length as a function of angular velocity is obtained by direct differentiation of the closed form expression. Finding closed form expressions for length of the span adjacent to an out-of-round material roll even for simple cases, such as an elliptical roll, is not trivial.

As a starting point, an elliptical material roll is taken into consideration. To find the length of the web span between the material roll and the idle roller it is necessary to find the line tangent to both of them. An analytical approach to the problem did not provide any insights into finding a closed form expression for span length as a function of angular displacement of the material roll. To overcome this problem a convex optimization problem is formulated and an efficient numerical approach is developed to obtain the common tangent to the material roll and the first idle roller. Once the common tangent is obtained, span length and rate of change of span length can be found numerically as well. The algorithm and related pertinent discussions are given. Incorporation of this algorithm into web line model simulation software will enable better correlation of model and experimental tension data.