Connection between longitudinal and lateral web dynamics

Abstract

Where does the entry angle come from? This paper will show that the answer to this question reveals a connection between longitudinal and lateral behavior that has gone largely unnoticed.

In beam models, entry angle refers to the angle between the tangent to the web centerline and the normal to the roller axis at the line of entry onto the roller. Whenever the entry angle becomes non-zero, a web that is moving longitudinally through a process will also move laterally on the roller in a direction that returns the entry angle to zero. If the web is modeled as a perfectly flexible string, this behavior is intuitively obvious because it bends sharply on entering a roller that is pivoted or shifted laterally. However, in the case of the most commonly used Euler-Bernoulli (E-B) beam model, the web can't make a sharp bend. If it is initially perpendicular to the roller axis, beam theory says that, provided there is no slipping, it should remain perpendicular as the roller is shifted or pivoted and thus wouldn't move. We know from experience, however, that a real moving web begins to move laterally soon after a roller pivots or shifts? So, how can this be?