# DAMPING-INDUCED SELF-RECOVERY PHENOMENON IN MECHANICAL SYSTEMS WITH AN UNACTUATED CYCLIC VARIABLE

Séance du jeudi 21 février 2013, Salle L 106, 14h.

Dong Eui CHANG, Professor, Applied Mathematics, University of Waterloo, Canada

The falling cat problem has been very popular in the society of control, mechanics and mathematics since Kane published a paper on this topic in 1969. A cat, after released upside down, executes a 180-degree reorientation, all the while having a zero angular momentum. It makes use of the conservation of angular momentum that is induced by rotational symmetry in the dynamics. But if there is an external force that breaks the symmetry, then the angular momentum will not be conserved any more.

Recently, we have discovered an exciting nonlinear phenomenon for mechanical systems with one unactuated cyclic variable when the symmetry-breaking force is a viscous damping force. In this case, there arises a new conserved quantity, called damping-added momentum, in place of the original momentum map. Using this new conserved quantity, we show that the trajectory of the cyclic variable asymptotically converges back to its initial value. This phenomenon can occur even when the damping coefficient is not constant as long as the integral of the coefficient satisfies a certain condition.

The self-recovery phenomenon can be observed in a simple experiment with a rotating stool and a bicycle wheel which is a typical setup in physics classes to demonstrate the conservation of angular momentum. Sitting on the stool, one spins the wheel by his hand while holding it horizontally. The reaction torque will be created to initiate the rotational motion of the stool to the opposite direction. After some time, if the person applies a braking force halting the wheel spin, then the stool will asymptotically return to its original position, tracing back its past path regardless of the number of rotations the stool has made, provided that there is a viscous damping force on the rotation axis of the stool. This result will appear in the upcoming March issue of ASME J. Dynamic Systems, Measurement and Control.

Dong Eui CHANG, Professor, Applied Mathematics, University of Waterloo, Canada

The falling cat problem has been very popular in the society of control, mechanics and mathematics since Kane published a paper on this topic in 1969. A cat, after released upside down, executes a 180-degree reorientation, all the while having a zero angular momentum. It makes use of the conservation of angular momentum that is induced by rotational symmetry in the dynamics. But if there is an external force that breaks the symmetry, then the angular momentum will not be conserved any more.

Recently, we have discovered an exciting nonlinear phenomenon for mechanical systems with one unactuated cyclic variable when the symmetry-breaking force is a viscous damping force. In this case, there arises a new conserved quantity, called damping-added momentum, in place of the original momentum map. Using this new conserved quantity, we show that the trajectory of the cyclic variable asymptotically converges back to its initial value. This phenomenon can occur even when the damping coefficient is not constant as long as the integral of the coefficient satisfies a certain condition.

The self-recovery phenomenon can be observed in a simple experiment with a rotating stool and a bicycle wheel which is a typical setup in physics classes to demonstrate the conservation of angular momentum. Sitting on the stool, one spins the wheel by his hand while holding it horizontally. The reaction torque will be created to initiate the rotational motion of the stool to the opposite direction. After some time, if the person applies a braking force halting the wheel spin, then the stool will asymptotically return to its original position, tracing back its past path regardless of the number of rotations the stool has made, provided that there is a viscous damping force on the rotation axis of the stool. This result will appear in the upcoming March issue of ASME J. Dynamic Systems, Measurement and Control.