arXiv subjects

Raphaël A. Slawinski

Publications and source records attributed to Raphaël A. Slawinski.

6 recordsLinked to original sources

arXiv2026

On the brachistochrone problem for cycling ascents

VAM ({\it velocità ascensionale media}) is a measurement that quantifies a cyclist's climbing ability. We show that to minimize the time to attain a given height gain\, -- \,which is tantamount to maximizing VAM\, -- \,a cyclist should climb as steep a constant-grade hill as possible. Apart from the power-to-weight ratio, the limit of steepness is imposed by such factors as the efficiency of pedalling, which is related to feasible cadence, maintaining balance, preventing lifting of the front, and skidding of the rear, wheel. In an appendix, we discuss steepness constraints due to pedalling efficiency. The article itself is focused on consequences of the power available to the cyclist, which can be viewed as a necessary condition to examine other aspects of climbing strategy. We show that\, -- \,for given start and end points, and for any fixed average-power constraint\, -- \,the brachistochrone, which is the trajectory of minimum ascent time, is the straight line connecting these points, covered with a constant speed, which along such a line is equivalent to a constant power. This is in contrast to the classical solution of a descent brachistochrone under gravity, which is a cycloid along which the speed is not constant.

physics.pop-ph↗

arXiv2025

On minimizing cyclists' ascent times: Part II

We formulate an optimization of a bicycle ascent time under the constraints of the average, maximum, and minimum powers. In contrast to the first part of this study, we do not restrict the departure to flying starts with an initial speed determined by the model and its optimization. We allow for various initial speeds, from a standstill to a launched start. We accomplish this by generalizing the discontinuous piecewise constant speed model to a continuous piecewise linear speed model. Regardless of the initial speed, steepness or profile of the ascent the optimal strategy tends to a constant ground speed, in agreement with the conclusion of the previous, more restricted, formulation. This new formulation allows us to compare various initial-speed strategies and, hence, has a direct application to competitive cycling. Notably, in timetrials composed of flat and steep sections, it helps one decide whether or not to change bicycle, which requires stopping and restarting, from one that is more appropriate for flats to one that is more appropriate for uphills.

physics.class-ph↗

arXiv2025

On minimizing cyclists' ascent times

We prove that, given an average power, the ascent time is minimized if a cyclist maintains a constant ground speed regardless of the slope. Herein, minimizing the time is equivalent to maximizing -- for a given uphill -- the corresponding mean ascent velocity (VAM: velocità ascensionale media), which is a common training metric. We illustrate the proof with numerical examples, and show that, in general, maintaining a constant instantaneous power results in longer ascent times; both strategies result in the same time if the slope is constant. To remain within the athlete's capacity, we examine the effect of complementing the average-power constraint with a maximum-power constraint. Even with this additional constraint, the ascent time is the shortest with a modified constant-speed -- not constant-power -- strategy; as expected, both strategies result in the same time if the maximum and average powers are equal to one another. Given standard available information -- including level of fitness, quantified by the power output, and ascent profile -- our results allow to formulate reliable and convenient strategies of uphill timetrials.

physics.class-ph↗

arXiv2023

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

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.

physics.class-ph↗

arXiv2021

On modelling bicycle power for velodromes: Part II Formulation for individual pursuits

We model the instantaneous power on a velodrome--as applied to individual pursuits and other individual time trials--taking into account its straights, circular arcs, and connecting transition curves. The forces opposing the motion are air resistance, rolling resistance, lateral friction and drivetrain resistance. We examine the constant-cadence and constant-power cases, and discuss their results, including an examination of an empirical adequacy of the model. We also examine changes in the kinetic and potential energy.

physics.pop-ph↗

arXiv2020

On modelling bicycle power for velodromes: Part I: Formulation for individual pursuits

For a moving bicycle, the power can be modelled as a response to the propulsion of the centre of mass of the bicycle-cyclist system. On a velodrome, an accurate modelling of power requires a distinction between the trajectory of the wheels and the trajectory of the centre of mass. We formulate and examine an individual-pursuit model that takes into account the aforementioned distinction. In doing so, we provide details of the invoked physical principles and mathematical derivations, with an emphasis on their limitations. We assume that a velodrome consists of two parallel straights and two semicircular arcs. We neglect the effects of the track inclination along the straights and assume the track inclination along the curves to be constant. For either segment, we consider two distinct black-line speeds. For the latter, the lean-angle expression is derived based on a noninertial frame of the cyclist. Among conclusions quantified by this model is the fact that a constant-cadence approach to an individual pursuit does not minimize the required power.

physics.pop-ph↗