Spatially Dependent Transfer Functions for Web Lateral Dynamics in Roll-to-Roll Manufacturing

In this paper we derive spatially dependent transfer functions for web span lateral dynamics which provide web lateral position and slope as outputs at any location in the span; the inputs are guide roller displacement, web lateral position disturbances from upstream spans, and disturbances due to misaligned rollers. This is in sharp contrast to the existing approach where only web lateral position response is available on the rollers. We describe the inherent drawbacks of the existing approach and how the new approach overcomes them. The new approach relies on taking the 1D Laplace transform with respect to the temporal variable of both the web governing equation and the boundary conditions. One can also obtain the web slope at any location within the web span with the proposed approach. A general span lateral transfer function, which is an explicit function of the spatial position along the span, is obtained first followed by its application to different intermediate guide configurations.