The major goal of this research was to develop the theoretical framework necessary to address the question of optimal chainring shape for endurance cycling. Three separate theoretical developments were performed during this research. First, an energy analysis technique was developed and used to evaluate the muscular mechanical energy expenditure (MMEE) of cycling with a non-circular chainring that was hypothesized to reduce the work done during endurance cycling (thereby increasing efficiency). It was determined that the proposed chainring did not reduce MMEE, and in fact resulted in an increase in MMEE relative to cycling with a conventional circular chainring. Thus, a new optimization framework was developed. As the first step in the optimization development, a theoretical investigation evaluated the contribution of non muscular components to the force applied to the pedal because of the intrinsic relationship between chainring shape and the dynamics of the pedalling motion. The pedal force analysis showed that non-muscular contributions to the pedal force were extensive, and that changes in the total pedal force with cadence were substantial and could not be predicted with a simple linear scaling law. As a result, dynamic optimization techniques were determined necessary to adequately assess the question of optimal chainring shape and the final theoretical development consisted of establishing the dynamic optimization framework necessary to optimize chainring shape.

Two example applications of the dynamic optimization revealed that the technique produced non-circular chainring shapes that improved performance (defined as the value of a simple cost function) relative to the conventional circular chainring during endurance cycling. The dynamic optimization technique has enabled this study to be the first to show that the theoretical cost of endurance cycling could be reduced at a given workload and cadence by using a non-circular chainring. The result of the dynamic optimization is a simulation of pedalling with a non-circular chainring that uses inputs which cause a lower value of the cost function than the inputs for optimally pedalling a circular chainring as determined using conventional optimal control techniques. While the results of the example applications provide interesting insight into the dynamics of the pedalling process, they are only the first steps taken towards the eventual goal of determining a chainring shape that could improve performance for endurance cycling and do not produce a chainring design for implementation. The use of the dynamic optimization approach is encouraged as part of an iterative design process which should advance fundamental understanding of pedalling with non-circular chainrings, while advancing understanding of conventional endurance cycling in the process.