"Damping-Induced Self-Recovery Phenomenon in Mechanical Systems"

Dong Eui Chang, Department of Applied Mathematics, University of Waterloo

The falling cat problem has been very popular in control, mechanics and mathematics since Kane and Sher 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. However 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 a unactuated cyclic variable in the case where 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 condition. 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 hand while holding it horizontally. A reaction torque will be created to initiate a rotational motion of the stool in 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, as if it has a memory, tracing back its past path regardless of the number of turns the stool has made, provided that there is a viscous damping force on the rotation axis of the stool.

We have also discovered the self-recovery phenomenon in 2D incompressible viscous fluid flows. As a corollary, we will give a "dynamic" explanation of the famous experiment by G.I. Taylor on the "kinematic" reversibility of low-Reynolds-number flows.

In this talk, several videos will be played to show this fascinating phenomenon of self-recovery in mechanical systems.

Dong Eui Chang is associate professor in the department of applied mathematics at the university of Waterloo in Canada. He obtained a PhD in control & dynamical systems at Caltech. His research interests are in control and mechanics.

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