The complex case of aerodynamics and weight in racing

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Nereth
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Joined: Wed Jan 18, 2023 10:18 am

by Nereth

... ok, well first of all let's think through it intuitively so we know what answer we expect. If you are pedalling on the flat, do you even notice a speed oscillation? I don't. I would guess it's less than 0.1 kph each way. Is there a big power difference in aero drag between say, 35.1 and 35? Not huge. And certainly not a lot bigger than the drag benefit between 34.9 and 35, which is the other side of that coin. So no, we don't expect much.

What about while climbing? Maybe if you're climbing at 60 rpm, you notice it, usually you're doing that because you're out of gears, and if you're in your easiest gear at 60 rpm you're probably doing on the order of 10kph. Now is there a power difference? Well... The relative change of speed will be much higher. Perhaps 5 percent? I don't know - but at 10 kph climbing at reasonable w/kg, almost none of your power goes into aero, and the speed variation doesn't really effect the efficiency of the other power sinks. So, minor again.

That said if you still want to do the maths, there are two options. Either we approximate to a pulse energy input or square wave at cadence x2 legs frequency, and I can show most of the maths here, or we go one level more realistic, to a sine wave power input, and I'll be doing it in excel and mostly showing results. The former will show a larger wobble and thus represent the worst case, which I expect to still be negligible, but it's easier and would thus demonstrate the point more completely, so it's the method I would choose if i was doing it for work. The latter generates some pretty graphs but ultimately will be harder for anyone to follow along with.

Which would you prefer?

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RimClencher
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by RimClencher

A worst case with the easiest maths seems best for demonstrating if it is a silly point. I don't really know what exactly which parameters need to be assumed, but I'll try a few. So, at 10 kph, 5% speed variation is 0.5 kph. That seems very conservative to me for climbing out the saddle on a steep grade at low cadence. I think 2 kph either side is more realistic. So speed varies between 8 kph and 12 kph. Given a 40T/30T setup, 10 kph gives a cadence of around 60. So speed varies between 8 and 12 kph 60 times a second. As it's the worst and simplest case, let's assume the rider and bike is a single system varying between 8 and 12 kph, ignoring rotational weight for simplicity, and assuming a heavier climber at 69 kg. As far as the pedal stroke, keeping it simple let's use a square wave with a 100%/0% power input/dead zone. Let's say the dead zone is 40% of the pedal stroke, so the square wave is 0.6 sec at 100%/0.4 sec at 0%.

So we are comparing how much extra power this guy needs to keep up with the same guy climbing with no speed variation, which I guess is what all the online calculators currently assume.

Edit: Whoops, at a cadence of 60 speed would vary between 8 and 12 kph 120 times a second, and the square wave is 0.3 sec at 100%/0.2 sec at 0%.

BdaGhisallo
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by BdaGhisallo

RimClencher wrote:
Sun Jan 22, 2023 7:27 am
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.

It can be quite significant.

I don't know if you recall the Australian rider, Brad McGee. He was a very good time trialist and his coaches wanted to transform him into a GT contender. They tried over a few seasons and, despite his prodigious power output, could never translate his power into GT podium contending form. The closest he got was 8th in the 2004 Giro.

After much investigation, his coaches came to the conclusion that the pedaling dynamics required to climb well were different to those required to time trial well. While McGee could absolutely blitz a time trial, he just couldn't adapt his pedaling style to make full use of that power when it came to climbing that required a fuller pedal stroke. So some riders can deal with that different forces acting on them when climbing, compared to riding on the flat, a lot more effectively than others.
Last edited by BdaGhisallo on Sun Jan 22, 2023 12:39 pm, edited 1 time in total.

RimClencher
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by RimClencher

BdaGhisallo wrote:
Sun Jan 22, 2023 11:18 am
After much investigation, his coaches came to the conclusion that the pedaling dynamics required to climb well were different to those required to time trial well. While McGee could absolutely blitz a time trial, he just could adapt his pedaling style to make full use of that power when it came to climbing that required a fuller pedal stroke. So some riders can deal with that different forces acting on them when climbing, compared to riding on the flat, a lot more effectively than others.
Thinking about how riders obsess over even mm changes in saddle position, and how saddle setback also varies with gradient, I can see how someone who has trained a lifetime for time trial would have trouble adjusting. Makes the GC riders who can manage both much more impressive.

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fa63
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by fa63

RimClencher wrote: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.
viewtopic.php?p=60824#p60824

TL;DR version - it doesn't really matter :-)

Nereth
Posts: 250
Joined: Wed Jan 18, 2023 10:18 am

by Nereth

RimClencher wrote:
Sun Jan 22, 2023 10:46 am
A worst case with the easiest maths seems best for demonstrating if it is a silly point.
Ok, so if that's the preference, what I'll do is a very first order approximation here, then maybe as I'm waiting for/recovering from surgery to this bone tomorrow, I have a fun idea for a spreadsheet to calculate speed vs time given varying inputs e.g. sinusoidal power or a sudden sprint. We can see how close the first order approximation is.

For this, the assumptions we'll use are as follows;

4 losses:
- Gravity given by vertical speed*m*g,
- drag given by 0.5*CDa*Rho*v^3,
- drivetrain losses of a constant percent of power output,
- rolling resistance given by m*g*Crr
Our example situation can be:
- 72kg cyclist+ 8kg bike system, i.e. 80kg
-3.5 w/kg i.e. 252W
- Crr 0.006
- CDa 0.34
- Rho(air)=1.25kg/m^3
- Drivetrain efficiency 2%
- Gradient 8%
- 60 Cadence
Key simplifications:
- Pulse power (i.e. impulse) input at 2xcadence=2Hz. This will increase variation and cause increased losses over say, a sinusoid.
- I'm not doing any calculus at all, that's a problem for the excel sheet. I'll just use an energy method to get an approximate max and min speed around those pulses.
- I'm not integrating drag over time. In this model we would get roughly a sawtooth wave, but we can just assume 50%time at max speed, 50% at min (again serves to drive power up) and calculate if the average power requirement of those two is higher than that of a smooth run at the same speed.

Right, so, average speed for a smooth run:

First lets make gradient more convenient. Grade is expressed as rise over horizontal run, not rise over road length. So lets change that.

Hypotenuse of an 8% grade triangle is sqrt*(.08^2+1^2)=1.003195.
With that, our vertical rise for a given ROAD speed, becomes 8/1.003195=7.975%.

Seems like a small nitpick, but I suspect we'll be splitting hairs real hard later with our conclusion, so it's important.

Average speed Vave is now:

252=[(80*g*0.07975*Vave)+(80*g*0.006*Vave)+(0.5*Vave^3*0.34*1.25)]*0.98
Solvimg for Vave (https://www.wolframalpha.com/input?i=so ... 29%5D*0.98)= 3.6655m/s, which is 13.12 kph.

For context, airodynamic drag represents 10.47W of our 252W at this speed. I think you should already see where this is going.

Right, now we want to figure out how hard we oscillate about this for our pulsing case.

Let's use a rough energy method. When we have power off between pulses, it's going to be 252w*0.98 efficiency=247.0W, not getting to the ground for 0.5 seconds. That's 123.48J. If we neglect for a second the fact that aerodynamic drag is changing during that period (drag represents a whopping ~5 of these joules so yeah, who cares), at the end of that 0.5s period, we'll be going at a speed that represents 123.48J less kinetic energy. That's equation 1:

KEmax-KEmin=123.48J

The second equation is, we said we would assume a sawtooth wave (will be very close to that since only air drag deviates from linear decel, and it's only a few % of our total drag), so for our average speed to be the same as our smooth case, we have to oscillate evenly around our smooth case speed, i.e.:

(Vmax+Vmin)/2=3.6655

First equation, when we sub in the kinetic energy formula, becomes;

0.5*80*Vmax^2-0.5*80*Vmin^2=123.48, so rearranged, Vmax=sqrt((123.48+0.5*80*Vmin^2)/(0.5*80))

Sub into eqn 2;

(Vmin+sqrt((123.48+0.5*80*Vmin^2)/(0.5*80)))/2=3.6655
Solve on wolfram alpha to save time, Vmin= 3.455m/s, or 12.438kph
Vmax is the other side of that coin where our average smooth speed is still the average - 3.876m/s or 13.95 kph.

So the first finding is, even if we do perfect pulse input, speed variation is at MOST ~1.5kph range here, or, +/-0.75kph around a centrepoint.

What's the impact on total power to maintain that pulsing, if it were a square wave? (Again it's triangular so this is highly boosting the power requirement)

Solving the drag part of that power equation above two more times, for these two speeds (only the drag part is nonlinear with speed, so you can solve the others, but they won't show a difference between pulsed and smooth,for our asssumed sources of drag):

Drag at Vmax:
(0.5*3.876^3*0.34*1.25)=12.37
Drag at Vmin:
(0.5*3.455^3*0.34*1.25)=8.764
Average drag over the square wave = (12.37+8.764)/2=10.565W

Our drag for our smooth case, you may recall, was (0.5*3.6655^3*0.34*1.25)=10.47W

Net difference, 10.565-10.47=0.0995W. And I think that will half or less when solved continuously (numerically or with calculus) and a sine wave input.

Thus, for our 3.5w/kg rider on an 8% slope, the effect of non smooth pedalling is less than 0.1W. The speed variation was much higher than I thought (albeit for impossibly pulsed input), but the power difference was still, and I quote (somewhat cheekily);
Nereth wrote:
Sun Jan 22, 2023 7:50 am
on the order of less than a tenth of a watt.
Q.E.D.

justkeepedaling
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by justkeepedaling

A lot of our discussion here also neglects wind on a climb

RimClencher
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by RimClencher

Hmm, 8% seems far from worst case to me. It's just past the point where conventional wisdom already says a climbing bike is probably quicker than an aero bike. Something over a 12% gradient would be better. Watts may need to increase, but 3.5 w/kg is beginner level anyway. A VAM of 1600 seems reasonable for an elite climber.

RimClencher
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Joined: Tue Jan 14, 2014 1:00 am

by RimClencher

fa63 wrote:
Sun Jan 22, 2023 1:46 pm
RimClencher wrote: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.
viewtopic.php?p=60824#p60824

TL;DR version - it doesn't really matter :-)
Hmm, okay, I'll have to read into it.

Edit: Looks like they are modeling a 3% slope, which is far from the kind of worst case scenario we are thinking of?

Nereth
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by Nereth

RimClencher wrote:
Sun Jan 22, 2023 4:30 pm
Hmm, 8% seems far from worst case to me. It's just past the point where conventional wisdom already says a climbing bike is probably quicker than an aero bike. Something over a 12% gradient would be better. Watts may need to increase, but 3.5 w/kg is beginner level anyway. A VAM of 1600 seems reasonable for an elite climber.
You have enough info in my post to re-solve at any grade or speed or other set of parameters you like, at your convenience. I leave that to you. But it should be clear that 0.1w is not going to grow into anything significant for any reasonable set of conditions.

My next contribution, if I'm sufficiently bored, shall be a speed vs time calculator.

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fa63
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Location: Atlanta, GA, US

by fa63

RimClencher wrote:
fa63 wrote:
Sun Jan 22, 2023 1:46 pm
RimClencher wrote: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.
viewtopic.php?p=60824#p60824

TL;DR version - it doesn't really matter :-)
Hmm, okay, I'll have to read into it.

Edit: Looks like they are modeling a 3% slope, which is far from the kind of worst case scenario we are thinking of?
You must not have made it to the end; the poster also ran the numbers for a 10% slope.

RimClencher
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by RimClencher

You're right.
But doesn't the poster also run all the simulations at a cadence of 90? At a higher cadence, the dead zone during the pedal stroke would last for a shorter duration of time, so the system would have less time to decelerate. We are thinking of a worst case, so a cadence of about 60 is probably what we need.

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fa63
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by fa63

Tadej Pogacar goes up 10% climbs at 90 rpm:

https://www.velonews.com/events/vuelta- ... na-part-1/

For normal folks like us, none of this really matters anyways and it is all mental gymnastics, isn't it? :-)
RimClencher wrote:
Sun Jan 22, 2023 11:23 pm
You're right.
But doesn't the poster also run all the simulations at a cadence of 90? At a higher cadence, the dead zone during the pedal stroke would last for a shorter duration of time, so the system would have less time to decelerate. We are thinking of a worst case, so a cadence of about 60 is probably what we need.


RimClencher
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by RimClencher

fa63 wrote:
Sun Jan 22, 2023 11:31 pm
Tadej Pogacar goes up 10% climbs at 90 rpm:

https://www.velonews.com/events/vuelta- ... na-part-1/

For normal folks like us, none of this really matters anyways and it is all mental gymnastics, isn't it? :-)
Mental gymnastics until someone tests it :lol: As normal folks are more likely to climb steep slopes at 60 rpm, they would be interested in seeing the power loss due to this effect compared to a smoother power input/higher cadence pedal stroke. Since weight is a big factor in this phenomenon too, if there is a significant effect, normal weight weenies should be interested in seeing what a 1 kg difference does to it. So lots of potential interest for normal folks. Just need to see the worst case tested, then we will know.

Nereth
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by Nereth

Well, however silly this is (and it is silly), I did build a RK4 cycling speed solver that can plot speed versus time for time/position varying inputs (power/grade/etc) yesterday, so when I get home from surgery, I can plot you some pretty graphs.

What would you like, 60rpm, 350w, 12 percent, 80kg system? Or more?

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