Modelling of a cyclist's power for time trials on a velodrome

Abstract

We formulate a phenomenological model to study the power applied by a cyclist on a velodrome\, -- \,for individual timetrials\, -- \,taking into account the straights, circular arcs, connecting transition curves and banking. The dissipative forces we consider are air resistance, rolling resistance, lateral friction and drivetrain resistance. Also, power can be used to increase the kinetic and gravitational potential energy. Herein, to model a steady ride\, -- \,as expected for individual timetrials\, -- \,we assume a constant centre-of-mass speed, while allowing the cadence and power to vary during a lap. Hence, the kinetic energy is constant and the only mechanical energy whose change we need to consider is the increase of gravitational potential energy due to raising the centre of mass upon exiting each curve. The effect of dissipative forces is examined at each point of the lap; the effect of conservative forces is examined as an average. The latter is a small\, -- \,albeit not negligible\, -- \,part of the total power, and its inclusion within a model is a novelty presented herein. It increases the model's empirical adequacy. Following derivations and justifications of expressions that constitute this mathematical model, we present a numerical example. We show that the cadence and power vary slightly during a steady ride. In other words, a constant centre-of-mass speed entails nearly constant cadence and power, as expected for a steady ride and as supported by measurements. Also, we examine changes in the required power due to changes of various quantities, such as air density at a velodrome, laptime and several others, as well as the model sensitivity to input errors. Furthermore, we examine the effects on the required power of slight and gradual changes in speed, which are pertinent to individual time trials.