Effect of mass transfer on multi-span lateral dynamics of a uniform web

Abstract

This paper will show that the acceleration equation used in multi-span lateral dynamic models is a consequence of mass transfer between spans1. Mass transfer effects fully account for the equation currently used in Euler-Bernoulli models and provides an analytical pathway to an acceleration equation that incorporates shear deformation. It also ties together contributions from three other researchers - John Shelton, who pioneered the use of beam theory in models of lateral web dynamics, Lisa Sievers, who proposed the principle of continuity of bending angle and Richard Benson, who was the first to publish an acceleration equation that correctly incorporates shear deformation.

The acceleration equation is applied in conjunction with the normal entry rule to convert information about web shape to a time-based differential equation. Several versions of the acceleration equation have been proposed that include one or more terms to account for shear. All but Benson's lead to results that either contradict observed web behavior or else fail to provide meaningful solutions.

Consideration of mass transfer arises from the moment of force that develops in a web when it is displaced by an upstream disturbance or by movement of a roller that is transporting it. Any moment at the entry to a roller causes the longitudinal tension to vary in a linear fashion across the width of the web. Analysis of the effect of this tension profile on mass flow leads to: 1) The acceleration equation that is currently used for models without shear deformation. 2) A new understanding of why this equation works and improved insight into how multi-span systems behave 3) An acceleration equation for models that include shear deformation 4) Identification of a new mechanism that can cause micro-slip at the point of entry onto a roller. 5) Justification for the zero-moment steady state boundary condition at the downstream roller.