Power Variation
Strategies in
Cycling Time
Trials
Louis Grisez
Overview
 Abstract
 Introducing
Cycling
 Creation of a Mathematical Model
 Initial Conditions
 Calibration
 Results
Abstract

The ultimate goal of cycling time trials

Need for a pacing strategy

Energy Depletion

Mean work rate and various pacing strategies.

This research is a verification of the steady state
approximation used in "Power variation strategies for
cycling time trials: A differential equation model" by author,
Graeme P.Boswell

Seeks to improve the optimal pacing strategy found previously
using a steady-state approximation.

Energy based differential equation

Model will be used in theoretical courses with varying
conditions and cyclists of different masses.
An Introduction to Cycling





Competitive event rode in the fastest way possible
National, world, and Olympic championship events
Races are a primarily measurement of athletic ability
over a period of time.
Only known tactic:
Suggested method: uniform work output rate


Refined method: Altered power output



Assumption: constant conditions of power output as well
as resistive forces
Models altering course conditions such as road gradient
This is done as an improved method compared to
the steady state approximation in reducing the time
needed to complete the course.
Previous studies (Gordon, 2005; Swain 1997) have
shown the validity of variable power output
approximations.
The Steady State

Accelerations are assumed to be instantaneous (Swain 1997).


Showed that the refined method had a time saving effect as high
as 8.3 percent compared to the constant work rate.




instantaneous acceleration: do not directly apply to road cycling
This problem was later improved by Atkinson's improved calibration
model.
Kinetic energy and speed were observed at one second intervals.
Jumps in velocity were approximated as continuous acceleration.


This actually increased with the growing incline of the track.
Problem concerning these assumptions


Considered being in steady state
changes in velocity between the transitions were ignored
The variable pacing strategy used by both Swain and Atkinson
increased the mean power beyond the constant power
approximation by as high as 10 percent due to the ascending time
being greater than the descending time. As a burden, this pacing
strategy may be overestimated.
Mathematical Model

Cyclist must apply power to the drive chain.

The combined speed of the bicycle is a
function of the difference between P and Δ



P is applied power
Δ is power of resistive forces
Three Resistive forces



Gravity
Rolling
Wind
Mathematical Model
 Rate
of change of kinetic energy
Mathematical Model
 Assuming
there are no changes in the
rider's velocity due to varying grades,
surfaces, wind strength and applied
power, the previous equation becomes
 This
is the steady state
Initial Conditions

Initial data points, given by G. Boswell to be:



Initial data points used




x(0)=0
v(0)=0
v(0)=1
Small enough to be negligible in terms of final
output.
This is also the first instance in which procedure
differed from that of previous work done by G.
Boswell.
Assuming there are no changes in the rider's
velocity due to varying grades, surfaces, wind
strength and applied power, equation $(5)$ can
be written as
Calibration
 Constants
need to be defined
 Assumption:

Rider does not refer to tuck position
 Cyclists

were pulled behind vehicles
Eliminates the variability of power output
and small changes in velocity
 Human
Athletic Performance
Human Athletic Performance

Oxygen limitation


Average oxygen intake of 3.7


Events last 2-3 hours
(Swain,1997)
𝐿𝑖𝑡𝑒𝑟𝑠
𝑀𝑖𝑛𝑢𝑡𝑒
Human biomechanical performance

25% efficient
Courses
 10
5
[km] flat
[km] uphill followed by 5 [km] downhill
 Alternating
1 [km] sections
 Alternating
0.5 [km] sections
 Variable
gradients of: 5, 10, 15%
 Variable power of:
5, 10, 15%
Pacing Strategies
 Research
separates from G. Boswell
 Initial:


Increase of power as a percent
Decrease using
 Final:


Increase of power as a percent
Decrease of power as same percent
Solution Methods
 Use

of MATLAB 2012
Numerical Solver ode45
 Set
of 4 integrated for loops
Results of Change in KE
 Compared


to data found by G. Boswell
2% error
Less time for cyclist to complete course
Results of Change in KE
Results of Change in KE
Results of Change in KE
Progress on Steady State
 Use
of ode solver to model the original
steady state approximation
 Function
is to create comparable data to
variable pacing strategy
 Results

show a very high error
MATLAB solver error
 Possible

benefits
Time saving could be potentially over 10%
Conclusion
 Cycling
as a whole
 Relevance of a pacing strategy
 Results
 Future Improvements
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Di Prampero, P.E., Cortelli, G. Mognomi, P., & Saibene, F. (1979). Equation of motion of a
cyclist. Journal of Applied Physiology, 47, 201-206.
Gordon, S (2005). Optimizing distribution of power during a cycling time trial. Sports
Engineering, 8, 81-90.
Graeme P. Boswell (2012): Power variation strategies for cycling time trials: A differential
equation model, Journal of Sports Sciences, 30:7, 651-659.
Martin, J. C., Gardner, A. S., Barras, M., & Martin, D. T. (2006) Modelling sprint cycling using
field-derived parameters and forward integration. Medicine and Science in Sports
and Exercise, 3, 592-597.