Damping-induced self-recovery phenomenon

The initial discovery of this phenomenon originates from our observation of dynamics of two-link planar manipulator when the first joint is unactuated. So, we often take this mechanical system as as example to illustrate the self-recovery phenomenon. The manipulator moves on a horizontal plane free from gravity. The second joint is actuated by a motor while the first joint is left unactuated. (In this case, the first joint is called "cyclic," meaning it does not show up in the Lagrangian). When the first joint is completely frictionless, as we know from our physics class, its angular momentum is conserved. This is what in the first video is demonstrating. It is a bit ideal to assume a mechanical joint is completely frictionless (except for when we use such a mechanism as the magnetic bearing in vacuum). It is more reasonable to assume there exists a certain amount of frictional effect, e.g. viscous damping (which is linear in velocity). If we include the viscous friction (or damping) to the first joint, the momentum will not be conserved any longer. However, there someting very interesting happens: the first joint remembers where it started, and comes right back to its initial position as the actuator in the second joint stops moving, which is demonstrated by the second video shown below.

Another interesting example is the pendulum on a cart. Unlike the classical setting where the cart is actuated and the pendulum is unactuated, we consider the opposite configuration where the pendulum is actuated with a motor while the cart is left unactuated. Now, assume that there exists a viscous damping on the wheels of the cart. As we rotate the pendulum half turn, the cart will move by the reaction force. As soon as the pendulum stops its motion, the cart converges back to its original position, thus demonstrating the self-recovery.

Below is the video of an actual experiment that demonstrates the self-recovery phenomenon of fluid combined with rigid body. The bowl moves by the reaction force caused by the rotation of the motor mounted at the center of the bowl that is controlled by RC servo. Another interesting thing about this case compared to previous ones is that the self-recovery occurs with "overshoot". Look at the markers at the bowl and the bucket, while listening to the sound of the motor. The bowl exactly comes back to its initial position after making two full turns. The whole process of self-recovery took about 2 minutes. The fluid we used is regular water mixed with some honey.