Buckling instability on a rotating plane

Abstract

The initial circular shape of a rope coiling on a rotating plane loses axial symmetry and deforms continuously into two rotational order shapes, the hypotrochoid and epitrochoid. Shape transitions were experimentally induced by changing either the feeding velocity or the plane frequency. Varying the former caused the bending energy per unit length to increase significantly at the hypotrochoid-epitrochoid transition, a feature reminiscent of first-order phase transitions. Varying the latter produced comparably significant increases at both the hypotrochoid-epitrochoid transition and the epitrochoid-circle transition. In the limit of the perfectly flexible rope, the solutions to the equations of motion give either a hypotrochoid or an epitrochoid, depending on the tension relative to the inertial force. In the phase diagram, hypotrochoids emerge in the regime where the height is large, the feeding velocity is high and the plane frequency is low. Circles appear at all heights when the feeding velocity is low and the plane frequency is high. Epitrochoids emerge as an intermediate shape.