The rolling suitcase instability: a coupling between translation and rotation

A two-wheel suitcase or trolley can exhibit undamped rocking oscillations from one wheel to the other when pulled fast enough. We study this instability both experimentally—with a toy model of a suitcase rolling on a treadmill—and theoretically. The suitcase oscillates only if a finite perturbation is applied. This is because intrinsic dissipation occurs when the supporting wheel switches. When unstable, the suitcase either increasingly rocks until overturning or reaches a stable limit cycle. The friction force at the rolling wheels constrains wheels to roll without slipping. This constraint imposes a coupling between the translational motion and the three-dimensional rotational motion of the suitcase that drives the rocking instability. The same behaviours are observed in the experiments and in the simulations. The asymptotic scaling laws we observe in the simulations are explained by means of a simplified model where the coupling force is explicit.