Bifurcation and Stability of a Whirling Transported Thread

The problem of a rotating transported thread or yarn occurs in textile applications such as high-speed spinning and unwinding from a cylindrical package. In case of low tension the yarn is known to buckle and fly out under the action of centrifugal forces. The motion initially appears fixed when viewed from a rotating coordinate frame and is known as ballooning (see [1], which also contains references to the early literature). We study the problem by modelling the yarn as a flexible string drawn at constant speed through two axially aligned guide eyes a constant distance apart. Large deformations give rise to geometrical nonlinearities in the equations. If airdrag is neglected then exact solutions for the shape of the whirling thread are obtained (in terms of elliptic integrals and functions) and an infinite family of ballooning modes, with increasing number of nodes, are found to bifurcate from the straight thread. In the limit of zero angular velocity the results reproduce the classical critical speeds in the literature on buckling of axially moving materials such as belts, paper sheets, magnetic tapes and power transmission chains (see, for instance, [2]). As an application of our results we further consider the unwinding of yarn from a package. For this problem we merely have to change the boundary conditions, but we find, rather surprisingly, that in the absence of airdrag no lift-off solutions exist. In the case with airdrag we numerically compute some solutions and find critical curves in parameter space for their existence.