Unified model for longitudinal and lateral web dynamics
Brown, Jerald L.
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In "The Effect of Mass Transfer on Multi-Span Lateral Dynamics of Uniform Webs" , it is shown that behavior of Shelton's lateral dynamic beam model  can be explained as the interaction of the normal entry equation and mass transfer between spans. The implications of mass transfer are discussed further in "The Connection between Longitudinal and Lateral Web Dynamics" .In this paper, the mass transfer idea is generalized and used to develop a dynamic model that combines lateral and longitudinal (tension) behavior. Nonlinear elasticity theory is used to model the web as a two-dimensional membrane in a state of plane stress. Boundary conditions at the downstream roller are: 1) the normal entry equation, used in lateral models, and 2) the continuity equation, used in tension models.Results from the new model, such as lateral force, lateral position, face angle and slope are shown to agree closely with the static and dynamic beam models developed and tested by Shelton . Comparisons are also made with the dynamic beam model from reference  that includes the effect of shear.Information that is unique to the two-dimensional model is presented and discussed, such as: 1. Transient disturbances in tension at the downstream roller caused by pivoting and shifting of rollers; 2. Transient gradients in lateral velocity that can cause wrinkling; 3. The effect of tension change on lateral position.