Variational approach for analysis of harmonic vibration and stabiligy of moving panels

In this paper, the stability of a simply supported axially moving elastic panel (plate undergoing cylindrical deformation) is considered. A complex variable technique and bifurcation theory are applied. As a result, variational equations and a variational principle are derived. Analysis of the variational principle allows the study of qualitative properties of the bifurcation points. Asymptotic behaviour in a small neighbourhood around an arbitrary bifurcation point is analyzed and presented. It is shown analytically that the eigenvalue curves in the (ω, V0) plane cross both the ω and V0 axes perpendicularly. It is also shown that near each bifurcation point, the dependence ω(V0) for each mode approximately follows the shape of a square root near the origin. The obtained results complement existing numerical studies on the stability of axially moving materials, especially those with finite bending rigidity. From a rigorous mathematical viewpoint, the presence of bending rigidity is essential, because the presence of the fourth-order term in the model changes the qualitative behaviour of the bifurcation points. The results are applicable to both axially moving panels and axially moving beams.