Flexural–torsional buckling of misaligned axially moving beams. I. Three-dimensional modeling, equilibria, and bifurcations

The equations of motion for the flexural–flexural–torsional–extensional dynamics of a beam are generalized to the field of axially moving continua by including the effects of translation speed and initial tension. The governing equations are simplified on the basis of physically justifiable assumptions and are shown to reduce to simpler models published in the literature. The resulting nonlinear equations of motion are used to investigate the flexural–torsional buckling of translating continua such as belts and tapes caused by parallel pulley misalignment. The effect of pulley misalignment on the steady motion (equilibrium) solutions and the bifurcation characteristics of the system are investigated numerically. The system undergoes multiple pitchfork bifurcations as misalignment is increased, with out-of-plane equilibria born at each bifurcation. The amount of misalignment to cause buckling and the post-buckled shapes are determined for various translation speeds and ratios of the flexural stiffnesses in the two bending planes. Increasing translation speed decreases the misalignment necessary to cause flexural–torsional buckling. In Part II of the present work, the stability and vibration characteristics of the planar and non-planar equilibria are analyzed.