Theoretical study on travelling web dynamics and instability under non-homogeneous tension

Problems of dynamics and stability of a moving web, travelling between two rollers at a constant velocity, are studied using analytical approaches. Transverse vibrations of the web are described by a partial differential equation that includes the centrifugal force, in-plane tension, elastic reaction and nonstationary inertial terms. The model of a thin elastic plate subjected to bending and non-homogeneous tension is used to describe the bending moment and the distribution of membrane forces. The stability of the plate is investigated with the help of studies of small out-of-plane vibrations. The influence of linearly distributed in-plane tension on the characteristics of the web vibrations is studied. The static forms of instability are investigated using numerical methods. It can be concluded that inhomogeneities in the applied tension may significantly decrease the critical web velocities and even small inhomogeneities in the tension may have a large effect on the divergence forms.