Propagation of longitudinal tension in a slender moving web

Abstract

To date, most of the theoretical work on longitudinal web behavior has been directed at the problem of controlling average tension. Very little attention has been given to the subject of this paper - propagation of tension within a span.

The model presented here is based on the one-dimensional wave equation, modified for a moving medium. Boundary conditions are developed that, for the first time, incorporate tension and mass transfer on rolling supports. The P.D.E. is solved analytically using Laplace transforms.

A number of phenomena are described that will be of interest to process designers and troubleshooters. These can be used to explain existing tension problems, whose causes may have been unrecognized in the past, and to anticipate problems that will appear as line speeds are increased. Among these are:

1. Propagation of strain discontinuities when draw is increased suddenly.

2. Amplification of repetitive strain disturbances due to strain reflection and reinforcement.

3. Damping of solitary strain disturbances.

4. Alteration of longitudinal resonant frequencies by transport motion.

Another important use of the model is to serve as a necessary step toward more advanced models that include out-of-plane motion, viscoelasticity and aerodynamics.

The model is tested by comparing it to the currently accepted O.D.E. model. At large time scales, where propagation phenomena are imperceptible, the two models are in good agreement.