The complex case of aerodynamics and weight in racing

[Hexsense]

My guess approximation:
Total resistance = Vertical component + Horizontal component.
airF = air resistance which scale by cube of the speed
rolF = rolling resistance which scale linear by speed


Case 0 : To maintain speed riding on flat:
\ Vertical component ~= 0 + some loss from poor quality road causing vertical vibration.
\ Horizontal component ~= airF + rolF pretty much dominate all the force.

Case 1: To accelerate (regardless if it's up hill or flat)
\ Vertical component ~= sin(grade) * (acceleration rate + 9.8 m/s^2) * weight
\ Horizontal component ~= cos(grade) * acceleration rate * weight + airF + rolF
,where airF and rolF is pretty much 0 if we start accelerating from a stop.

Case 2: To maintain speed and keep going up steep hill:
\ Vertical component ~= sin(grade) * ( 9.8 m/s^2) * weight contribute to ~90% of total force up steep hill
\ Horizontal component ~= airF + rolF is approximately 10% of total force if the hill is steep.

When I think of it this way, I see that airF and rolF is in all the equations. However, while weight is negligible in case 0, it matter a lot for case 1 and 2.

[Nereth]

Correct, and that's how the vector maths does work.

On your climbing case, you can use the equation from your acceleration case, set acceleration to zero (leaving G), and for any reasonable grade (e.g. 6%+) at moderate speeds (e.g. <20kph), you will find air resistance is very low, thus leaving 80-90% of the power going into climbing and rolling resistance (both of which scale linearly with weight), hence my initial estimate.

You can solve for air resistance as well for more accuracy if desired. I do this periodically on 5minute+, 8%+ grades on windless days here to check power meter slope calibration, taking into account change in speed and rolling resistance also and you usually get within 1-1.5% of power meter, more error than that and its time to hang a calibration weight off the power meter to check for error.

Edit: I see you editted your post before I sent this and basically developed your formulae further

[Singular]

I should acknowledge that other posters beat me to the punch with a couple of these points. Well played, other posters!

...but good things take time, and that was mot likely both the longest and most well-written post to appear on WW for a long, long time. Chapeau!

[pmprego]

A lot of math competition and then, when any type of mountainous/hilly route comes, the Pros use shallow rims. And I'm talking the Pro who ride up those hilly at 25kmh.

And don't come saying the Pro don't care about science and stuff. Just look at how they did the tour down under tt under regular road bikes.

[maxim809]

how important is aerodynamics for the cyclist in the middle of the pack?

In general, the biggest and taller you are the more it matters.

[RustyCage]

The aero vs weight discussion has been covered a few times on the inter webs. But I just wanted to reflect a few bits of wisdom...

1. Fit and comfort for aero positioning can dominate anything done to the bike. Practice can improve time spent in more aero positions.
2. Helmets are big in aero, but heat management maybe a larger driver depending on the temperature/speed.
3. Outside of pure hill climbs, it doesn't make sense not to have aero considerations, particularly the front wheel. "Aero" setups can be had at sub-7kg.
4. Rolling resistance should be gauged against flatting.

[Steve Curtis]

GCN did a vid today on this very topic. Yes I know it's GCN, and everyone on the forum is far more intelligent than them, but it has been covered.

[youngs_modulus]

I promised to come back in a few hours after I'd checked my own math, and it's been longer than that. While I may have some insight into the disconnect between myself and Nereth on that math, I'll save that for another post.

Frankly, I'm horrified by the tone of my response to Nereth. There's simply no excuse for that kind of condescension, and I'm mortified that I responded that way.

Nereth, I apologize for addressing you in such an ugly, condescending way. Whatever our differences, the tenor of my reply crossed a line. I'm sorry.

I also apologize to the people in this thread and the community at large. I've been here since 2006, which easily long enough to know that my tone was simply counterproductive. I regret it. I really enjoy this community, and I want to help make it better, not worse.

This is not an excuse, but rather an attempt to learn from my mistake: when I wrote that, I had just pulled an all-nighter for work. I'm also dealing with something extraordinarily stressful (but non-tragic!) in my personal life. I wasn't thinking clearly and obviously my personal filter was broken.

But so what? I would never speak to anyone that way in real life, so I absolutely should have known better than to do so here, regardless of my own circumstances.

I'm sorry, everyone. I can do better, and I will.


Jason

[warthog101]

^^It takes person of good character to respond with an apology like that and I think you are being a bit hard on yourself.

[Nereth]

All good, my own initial post wasn't intended to be inflammatory, but it nevertheless was extremely blunt (I reallydo have a broken collar bone right now so I am not too motivated to post longer-than-strictly-required messages with my one good hand). I can understand how that message could have been triggering. My bad.

My second message was more sassy but to be fair, you sort of triggered me by that point too . I would also point out my second argument in my second message is flawed and misleading, though it (my second point) is based on a nugget of value regarding proportions, it is objectively wrong as written and the value is too hard to extract.

To put some time into the initial issue now, since I think we're on a better foot, my first post should have contained this:

1) The dimensionsional analysis of power goes like this; the units are energy/second, energy can break into force times distance, so in iso units it's N*m/s, ie force times speed.

2) the force of gravity doesn't directly oppose the direction of motion though, but runs at an angle to it dictated by the grade. Your sin of angle the becomes useful in finding EITHER the proportion of speed acting vertically, OR the proportion of gravity acting along the direction of ground speed. So we align our force and speed vectors.

3) the force of gravity is, more specifically, M*g. Mass of rider and bike times 9.81.

4) the speed is your ground speed, in Meters per second for convenience.

5) so combining the above, we get, sin(angle)*(M*g)*(m/s).

Solve that and you get energy into climbing as you intended.

But, you can't get the proportion of overall power lost for a change in these terms unless you solve the other components, or just assume they are zero - which they tend towards, for steep slopes. Which is why I originally thought you had picked a 10 percent slope - to save some effort that doesn't make much difference to the final result.

[Nereth]

Sorry, to expand further on your original calculation, once you've assumed most of the energy is going into climbing you don't need to multiply by sin of angle a second time, since that's already been done in the implied maths that let you say you're climbing up a hill at 500w and all that is going into climbing and the other losses are negligible. At that point the only term changing is the mass term (if we're holding speed constant), so 2.6 percent loss in weight is a direct 2.6 percent loss in required power.

[fa63]

I teach undergraduate and graduate level engineering courses, and I find that it is not uncommon for engineers get tangled up in the numbers and forget common sense. It happens to me as well; it is what it is

As a rule of thumb, once a climb gets above 7-8%, gravity dominates and it is pretty much all about W/kg. So when kg goes down by some amount, Watts also go down by the same amount to maintain a given speed.

[RimClencher]

I wonder if anyone has looked into the fact that riders do not apply constant power throughout the pedalstroke. A variable power input will create accellerations/decelerations in a system, so the heavier a given system the greater the momentum and the greater the effect of variable power input on overall speed. As an example, think about climbing out the saddle on a steep slope in a relatively high gear. You accelerate through the downstroke, then deccelerate slightly until you accelerate again through the downstroke, and so on. Obviously, we can't assume the whole rider + bike system is accelerating/deccelerating at the same rate either, because the rider and bike can move independently of eachother. A rider may climb out the saddle in a way that isolates their own movement from that of the bicycle, so their center of gravity barely accelerates/decelerates through the pedal stroke relative to the acceleration/decceleration of the bike. Even when seated and particularly when grinding, riders try to move their body to carry momentum through the dead spot. Seems like something that could be significant, or at least warrants investigation.

[Nereth]

All the losses are linear with speed, except Aero drag, aero drag is roughly cubic so that will go up from speed oscillations, but the speed delta over pedal stroke is so small I imagine the effect would be on the order of less than a tenth of a watt. Math is doable but largely I don't think anyone needs me to do it to convince them of that?

A peripherally related, but much larger effect is that slower cadence, e.g. 80 instead of 90, churns up less air, with less intensity, and can cause actual watts worth of savings. As does holding a more stationary upper body, from what I've heard.

[RimClencher]

but the speed delta over pedal stroke is so small I imagine the effect would be on the order of less than a tenth of a watt. Math is doable but largely I don't think anyone needs me to do it to convince them of that?

Well, I need you to.