Nonlinear magneto-wha?

This project was for my undergraduate thesis, and it involved investigating the behavior (dynamics) of a clamped metallic beam (that displays mechanically elastic behavior) that is placed between two magnets. The beam and magnets are on a platform. By forcing the platform to vibrate, the beam vibrates and interacts with the magnets in a complex (non-linear) manner.

 

What did you do?

I improved on the existing subject by computationally verifying the theoretical equations for this system via simulations. I came up with a computational model for the system and proved that the simulations matched the theory. Furthermore, I also constructed the physical system and verified that the simulations matched the real-life system quantitatively and qualitatively.

Show me some results!

For the full details please look at my my thesis. For a more condensed summary, my poster or my presentation would be better.

The physical setup is shown below.

 

A video of the physical setup in action is given below. Notice the erratic motion of the beam.

The erratic motion of the beam is due to the nonlinear dynamics of the setup. The forced vibration of the platform causes the system to be a chaotic system . Measurements of this erratic motion are shown below. Notice the lack of noticeable periodicity in the time series, even though the vibration of the platform is periodic.

 

In order to visualize the chaotic motions, we can periodically sample the position and velocity of the beam and plot them against each other over multiple samples. In other words, we create a Poincare Map of the system. This visualization of the chaotic motions as recorded experimentally is shown below. Colder colors correspond to points sampled earlier in time.

 

For a chaotic system, the Poincare Map is actually a fractal. The Poincare Map from the simulations show this more clearly. Note the similarity between the Poincare Map from the simulations compared to that of experimental measurements.