Predictive mathematical model of time saved on descents in road cycling achieved through reduction in aerodynamic drag area

Abstract

This paper presents empirically-based simulation results of descent time to completion in road cycling. A mathematical model is formulated to predict time saved on road cycling descents where a cyclist’s position is static through manipulation of aerodynamic drag area, system parameters and initial conditions. Road cyclists often adopt drastic static riding positions on long descents in order to minimize aerodynamic drag and optimize performance, measured as race time to completion. In those riding positions bicycle control is compromised and the risk of fall and injury increases. The aims of this study were to empirically determine whether there is a difference in aerodynamic drag area associated with the ‘Top-Tube’ descending riding position compared to ‘Normal’ descending riding position, and to investigate the effect of the difference on time to completion of road descents. Two elite male Australian time-trial cyclists were tested in an open jet wind tunnel. Drag force was measured using a force platform and a custom air bearing drag measurement system at 50 Hz at wind velocity of 15.6 m/s. Athletes were tested in their ‘Normal’ descending position and in the ‘Top-Tube’ position. Based on Newtonian–Lagrangian equations of motion of the cyclist–bicycle system, an analogue mathematical model as a non-linear Riccati ordinary differential equation was developed to enable prediction of velocity and time to completion of a road cycling course descent of known length and gradient as measures of performance for an athlete of known mass and empirically determined drag area in a descending position. Previously, proposed models of cycling performance have been based on physiological, anthropometric, and mechanical power output. No general closed-form time-to-completion mathematical model for cycling was found in the literature. Analytical solutions allow for a concise investigation of a dynamical system model behaviour that is not as readily available with a numerical solution. Wind tunnel testing showed up to 25% reduction in drag area for changes to cyclists riding position from their ‘Normal’ to the ‘Top-Tube’ dropped descending position. The analytical solution to the nonlinear Riccati differential equation showed that large time savings as a result of reduction of drag area can be made on road cycling race descents. In the example scenario simulated here, 29.2 s may be saved on a 5 km descent of 10% gradient with 25% reduction in aerodynamic drag area (CdA). The ‘Top-Tube’ riding position is associated with a large reduction in aerodynamic drag area in road descents compared to conventional descending riding position. Our model enables the prediction of time to completion on descents. This may assist cyclists to assess the trade-off between undertaking increased risk associated with drastic rider descending position and the potential for improved performance in the context of race tactics and strategy.