Modeling Giant Swings

This goal of this Bowdoin College project is to create a mathematical model for a common gymnastics skill.

A giant swing is one of the fundamental tricks on the women's uneven parallel bars and the men's high bar. Its progression can be see in the video on the right.

This motion can be understood through extending the well-established simple pendulum. After using Newtonian and Lagrangian dynamics to develop a few possible models, we move forward with the single forced pendulum with friction. We aim to select parameters of a periodic forcing function so that the gymnast starts in a handstand with a small amount of angular velocity and ends her swing in the same position after one period of the forcing function. Ideally, a solution would resemble the physically reasonable behavior shown in the phase plane below.

Solutions to this boundary value problem can be found using Newton's method applied to forcing function parameters and adjusted for increased dimensions and a moving endpoint of evaluation. After discussing differential equations theory, energy conservation, and chaos, solutions under a variety of periodic forcing functions are found using ODE architect, Matlab, and Mathematica. Unfortunately, even the more well-behaved solutions are not as physically realistic as we might hope, as shown in the phase plane below. Furthermore, in acquiring these solutions it becomes apparent that this method may require initial parameter estimates very close to a local solution for Newton's method to converge in a computationally reasonable manner.

Further work in this area could include:

- Comparing other candidate models and forcing functions to acquire a physically reasonable solution
- Developing a global search to find regions close to a solution from which starting point parameter values could be picked
- Quantifying the number of expected solutions

Role: Honors thesis completed under the guidance of Prof. Matthew Killough. [Full thesis text]