What is the optimal power variation for a cyclist on a varied race course?

[etotheipi]

I just thought up a little problem and wondered whether anyone could advise as to how to go about it!

On a flat course, suppose a cyclist might be able to maintain 300W for around an hour. This gives a total allowed energy expenditure of $1080t \text{ kJ}$, where $t$ is measured in hours. However, the height of a particular time trial course varies along its length, and let's suppose that the surface can be modeled by a function $h(x)$, which we can choose later on. I want to try and figure out how the cyclist's power output should vary with $x$ according to the gradient of the slope, in order to minimise the time taken to complete the course (i.e. $x=0$ to $x=d$). I will, initially, only assume a resistive term of $kv^{2}$, for pushing the air out of the way.

I don't know what approach would be most appropriate for solving such an optimisation problem. The trajectory of the cyclist is parameterised by

$\vec{r} = \begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}x \\ h(x)\end{bmatrix}$

such that
$\frac{d\vec{r}}{dt} = \begin{bmatrix}\frac{dx}{dt} \\ h'(x)\frac{dx}{dt}\end{bmatrix}$
We also know that $P(x) = \vec{F} \cdot \vec{v} = \begin{bmatrix}F_x \\ F_y\end{bmatrix} \cdot \begin{bmatrix}\frac{dx}{dt} \\ h'(x)\frac{dx}{dt}\end{bmatrix} = F_x(x) \frac{dx}{dt} + F_y(x) h'(x)\frac{dx}{dt}$

and we could perhaps also put $F_x = m\ddot{x}$ and $F_y = m\ddot{(h(x))}$. However, this all looks like it's going to get pretty messy quite quickly and I don't even know if it is possible to get anything out of this working.

If it turns out that it is a bit fiddly to do analytically, I wonder whether this problem would lend itself more to a computational solution? Thanks for your advice!

 

[sophiecentaur]

On a flat course, suppose a cyclist might be able to maintain 300W for around an hour. This gives a total allowed energy expenditure of 1080t kJ1080t kJ1080t \text{ kJ}, where ttt is measured in hours.

It strikes me that the cyclist has to be modeled very well if you want a meaningful answer to this. The 300W constant output may not even be realistic because a real cyclist may find it easier to vary the output and achieve a higher mean. But that would involve psychology and physiology too and could be for a future analysis.
I suggest that an Energy based calculation could be easier (either approach should be valid.) If you work assuming a 300W output, fixed for the whole journey then you can (you have to) assume perfect optimal gearing and the 300J in any time interval (second) will consist of the gravitational potential plus the kv² term plus any change in Kinetic Energy (the remains of the 300J. The correct gearing would let him ascend vertically, even, and reach terminal velocity on any particular gradient.
Starting from cold, I'd be inclined to use a piecewise linear numerical approach, involving a sample terrain (gradients and heights to mimic a known course) and calculate the resulting changes in height and speed, second by second. I've done that sort of thing to calculate satellite orbits and it gives a convincing looking graphical result.

 

[etotheipi]

It strikes me that the cyclist has to be modeled very well if you want a meaningful answer to this.

Yeah, when I think about it there is going to need to be a lot of abstraction! Wind strengths and directions, rolling resistances, temperatures, psychology even...

Starting from cold, I'd be inclined to use a piecewise linear numerical approach

This seems like a good starting point! I suppose I could start setting up a numerical approach on Excel with a time-step of something like 1 second, and see how far I can get. An energy approach also seems like a nice way of getting around some of the more tedious force analysis.

As for the terrain, maybe it will be easiest to consider an "easy" shape like a parabola first.

Thanks for your help!

 

[hutchphd]

Are you limiting
(1) the maximum instantaneous power to be 300W or
(2) the average power to be 300W for the duration but allowing variation?
The answer for (1) is I think obvious (pedal at 300W) so I don't understand the question you wish to resolve...
My initial instinct for (2) is that maintaining a constant speed by varying power within overall total energy constraints is the optimal least time answer...I have not worked it out and leave it to you (for now) to consider.

 

[etotheipi]

Are you limiting
(1) the maximum instantaneous power to be 300W or
(2) the average power to be 300W for the duration but allowing variation?
The answer for (1) is I think obvious (pedal at 300W) so I don't understand the question you wish to resolve...
My initial instinct for (2) is that maintaining a constant speed by varying power within overall total energy constraints is the optimal least time answer...I have not worked it out and leave it to you (for now) to consider.

Yes I am considering option 2, constraining the total energy expenditure to $300t\text{ W}$. The constant speed idea is interesting and I hadn't considered that as a possibility!

 

[sophiecentaur]

Constant speed would imply a longer overall time. Would that be better in a race?
You couldn’t have a shorter time than at your max power all the time. Assuming your body can deliver, that would be best. If vsquared losses are high then pushing hard downhill could perhaps wear you out but that’s getting much more complicated and would need the rider’s abilities to be well characterized.
The simplest case is the one to deal with first.

 

[hutchphd]

But in case (2) the proposed constraint is not max power but limited total energy expenditure (corresponding to 300W average power). Then the cyclist in me knows to use more power going uphill (>300W) and less downhill (<300W)...I think the optimal is in fact a constant speed throughout but that is a guess. I suggest that @etotheipi try to show this by analysis if possible

 

[etotheipi]

But in case (2) the proposed constraint is not max power but limited total energy expenditure (corresponding to 300W average power). Then the cyclist in me knows to use more power going uphill (>300W) and less downhill (<300W)...I think the optimal is in fact a constant speed throughout but that is a guess. I suggest that @etotheipi try to show this by analysis if possible

Sure, I started the spreadsheet on Saturday but have become a little preoccupied with schoolwork so will hopefully get back to it when things quieten down a bit. I have no idea myself what the optimal strategy is, however I'll try out all of these suggestions once I get the model working and will report back!

 

[sophiecentaur]

Vsquared losses will always be there and there will always be a terminal velocity even when power is ‘left over’ from climbing. If the stipulation is 300W max continuous then you have to do max all the time for max average speed. But cyclists are not just machines so that stipulation may just not be at all realistic.