Coupled Rigid Body Dynamics with Application to Diving

Abstract

Platform and springboard diving is a sport involving athletes falling or jumping into a pool of water, usually while performing acrobatic manoeuvres. At the highest level it challenges the physical laws of gravity as athletes try to outperform each other by executing more sophisticated dives. With a mathematical model we are able to assist the athletes and coaches by providing some insight into the mechanics of diving, which hopefully gives them an edge during competition. In this thesis we begin with an introduction to rigid body dynamics and then extend the results to coupled rigid bodies. We generalise Euler's equations of motion and equations of orientation for rigid bodies to be applicable for coupled rigid bodies. The athlete is represented as a mathematical model consisting of ten simple geometric solids, which is used to conduct three projects within this thesis. In the first project we look at somersaults without twists, which provides a significant reduction as the model becomes planar. The equations of motion and equations of orientation reduce from vector form to a single scalar differential equation for orientation, since angular momentum is conserved. We digitise footage of an elite diver performing 107B (forward 3.5 somersault in pike) from the 3m springboard, and feed that data into our model for comparison between the theoretically predicted and observed result. We show that the overall rotation obtained by the athlete through somersault is composed of two parts, the major contribution coming from the dynamic phase and a small portion from the geometric phase. We note that by modifying the digitised dive slightly we can leave the dynamic phase intact, but change the geometric phase to provide a small boost in overall rotation. The technique involved in doing so is not practical for actual diving though, so we move away from this idea and devise another way of optimising for the overall rotation. We find that by shape changing in a particular way that takes slightly longer than the fastest way of moving into and out of pike, the overall rotation achieved can be improved by utilising the geometric phase. In the second project we use the model to simulate divers performing forward m somersaults with n twists. The formulas derived are general, but we will specifically look at 5132D, 5134D, 5136D, and 5138D (forward 1.5 somersaults with 1, 2, 3, and 4 twists) dives. To keep the simulation as simple as possible we reduce the segment count to two by restricting the athlete to only using their left arm about the abduction-adduction plane of motion. We show how twisting somersaults can be achieved in this manner using this simple model with predetermined set of motor actions. The dive mechanics consist of the athlete taking off in pure somersaulting motion, executing a shape change mid-flight to get into twist position, perform twisting somersaults in rigid body motion, and then executing another shape change to revert the motion back into pure somersaulting motion to complete the dive. In the third and final project we use our model to show how a 513XD dive (forward 1.5 somersaults with 5 twists) is performed. This complicated dive differs from all currently performed dives in that once the diver initiates twist in the somersaulting motion via shape change, they need to perform another appropriately timed shape change to speed up the twist rather than stopping the twist, and only then is five twists obtainable with practical parameters. Such techniques can be found in aerial skiing where the airborne time is longer, but our theory shows that it may also be applicable to platform and springboard diving too. To date, no athlete has ever attempted a 513XD in competition, nor does the International Swimming Federation (FINA) cover dives with five twists in their degree-of-difficulty formula. Our theory shows that 513XD dive is theoretically possible, and with extrapolation we estimate it would have a degree-of-difficulty of 3.9.