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Optimising distribution of power during a cycling time trial

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Abstract

A simple mathematical model is used to find the optimal distribution of a cyclist’s effort during a time trial. It is shown that maintaining a constant velocity is optimal if the goal is to minimise the time taken to complete the course while fixing amount of work done. However, this is usually impractical on a non-flat course because the cyclist would be unable to maintain the power output required on the climbs. A model for exertion is introduced and used to identify the distribution of power that minimises time while restricting the cyclist’s exertion. It is shown that, for a course with a climb followed by a descent, limits on exertion prevent the cyclist from improving performance by shifting effort towards the climb and away from the descent. It is also shown, however, that significant improvement is possible on a course with several climbs and descents. An analogous problem with climbs and descents replaced by headwinds and tailwinds is considered and it is shown that there is no significant advantage to be gained by varying power output. Lagrange multipliers are used solve the minimisation problems.

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Abbreviations

P:

power output (W)

Pc :

climbing power (W)

Pd :

descending power (W)

Pm :

maximum power (W)

P0 :

critical power (W)

W:

total work (J)

v:

velocity (m s−1)

vc :

climbing velocity (m s−1)

vd :

descending velocity (m s−1)

w:

headwind velocity (m s−1)

t:

time (s)

T:

time to complete course (s)

D:

duration of effort at a fixed power output (s)

x:

distance travelled (m)

xc :

climb length (m)

xd :

descent length (m)

L:

course length (m)

R:

rate of exertion

E:

exertion

m:

mass of cyclist with bicycle (kg)

g:

9.8 m s−2

c:

drag coefficient (kg m−1)

μ:

friction coefficient

s:

road grade

sc :

climbing grade

sd :

descending grade

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Authors and Affiliations

  1. Department of Mathematics, University of West Georgia, 30118, Carrollton, Georgia, USA

    Scott Gordon

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  1. Scott Gordon
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Correspondence to Scott Gordon.

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Gordon, S. Optimising distribution of power during a cycling time trial. Sports Eng 8, 81–90 (2005). https://doi.org/10.1007/BF02844006

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  • DOI: https://doi.org/10.1007/BF02844006

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