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Inspired by earlier discussions here about air resistance and rolling resistance, I decided to do some tests and calculations. If you don't have access to a wind tunnel or a wheel-test laboratory machine, then you could instead do the following, as far as I have concluded:

1. Find two long slopes with the very same type of surface (same road perhaps). The angles need to be different - bigger difference is better. The angles must also be close to constant for such a long distance that you can find the equilibrium speed.

2. A day with no wind, find the equilibrium speed for rolling down each of the slopes. It will take some experimenting to find it, because you will need to enter the slope with a speed close to the equilibrium speed to get necessary time to make sure you are not accelerating or retarding any more. Take several readings and calculate the average value.

3. Measure the slope angles. I put one end of a 1 m steel rod to the road, held it horizontal with a water-level, and measured the distance from the other rod end to the road with a vernier caliper. Then I repeated at several places on the slope, and calculated the average. There may be easier ways for those with more appropriate tools.

4. Find the total mass for you, your gear, and your bike.

5. Put in all values in the formulas given in the Excel pic below.

Once you have calculated your air resistance coefficient CwA, then you "only" need to repeat the above procedure for one slope to find that terrain's rolling resistance coefficient. When you have both the air and rolling coefficients, you can calculate your power and where it goes at any given speed and slope for that terrain.

My tests were made with a hardtail with 29x2.1" IRC Mythos XC II - FK front and RK rear, a bit worn but not very much. Tire pressure was my normal - 26 psi front and rear. I am 1.85 m (6'1) and have my handlebar 12 cm (4.7") under the saddle horizontal. My torso was at about 30 degrees angle from the horizontal.

The formulas were derived from the constant speed relationship Drive Power = Climb Power + Rolling power + Air Power: P = m*g*sin(x)*v + m*g*Cr*v + p*v^3*CwA/2.

Did I miss anything?
 

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air resistance has nothing to do with mass, im not sure how it works in this example, or why it doesn't just drop out...
 

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anden said:
Inspired If you don't have access to a wind tunnel or a wheel-test laboratory machine, then you could instead do the following, as far as I have concluded:

My tests were made with a hardtail with 29x2.1" IRC Mythos XC II - FK front and RK rear, a bit worn but not very much. Tire pressure was my normal - 26 psi front and rear. I am 1.85 m (6'1) and have my handlebar 12 cm (4.7") under the saddle horizontal. My torso was at about 30 degrees angle from the horizontal.

The formulas were derived from the constant speed relationship Drive Power = Climb Power + Rolling power + Air Power: P = m*g*sin(x)*v + m*g*Cr*v + p*v^3*CwA/2.

Did I miss anything?
"If you don't have access to a wind tunnel "

Lucky for me, my local park has a coin operated "wind tunnel" right at the main trail head.
Me and all my friends always put in a couple quarters prior to our rides.....test at various winds and then tweak our helmet angles and sunglasses, to make sure we've optimized our gear for lowest wind resistance. I now tape my ears prior to every ride. :)

I like the numbers, but then again I'm weird about numbers even if I don't completely understand the formula.....
I do have one problem with just how accurate you can be with your torso at a 30 degree angle. Of all the factors, this one would seem to outweigh all the others and yet it is the most difficult to be certain about unless you had some type of device limiting your forward lean and holding it in postion. Of course with dozens of multiple runs perhaps you could come to some average.

Also, with you running your tires only at 26 psi this adds to the portion of the rolling resistance. I am usually closer to 40 psi.
Still, at only 19 mph we can see that 71% of the resistance is from air.....yet not that many riders seem to pay so much attention to conciously becoming more aerodynamically efficient. Even at 12 mph its about 50/50 there are lots of mild climbs where you are doing 12 mph and yet folks aren't thinking too much about being aero.

I tried to find something I just read a couple days ago about the TDF where they said even at (I think) 12 mph going up a climb that 15% of the total power was going to overcome wind resistance......(but they might have said 18 mph.....I'll try to find it)
 

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Anden you old dog well done. The idea to eliminate first rolling resistance then air resistance coefficients from the equations using 2 separate runs on the same surface but different slope is ingenious. You didn't miss anything but here's a less crude version of that Cr equation in your spreadsheet:

Cr = [v1^2*sin(x2)-v2^2*sin(x1)]/(v1^2-v2^2)

Although I'm sure you didn't miss this either but did you calibrate your speedo to the air pressure you are running WITH you sitting on the bike (Its a 2 person job)? Although I know it seems like a PITA I have measured it once for some tests of mine and tire deflection (especially at a low 26 PSI in fact in pic below you can see my back tire deflects 9 mm when I sit on it) is significant enough to give you notable difference in speed readings. Here's the example:


You'll notice that, at first glance, results seem to indicate that as the air pressure in the tires is dropped speed down the slope increases. However once this is adjusted for the correct wheel diameter at a correct air pressure (taken with me sitting on the bike) you see that there is practically no difference in speed between different air pressures.

My speedo is on the back wheel so this is more significant for my tests (as back wheel carries more weight and compresses tire more) but granted you too should still account for this even if yours is on the front especially since your speedo measures speed down to 0.1 kph.

Good news is that you don't have to redo the experiment. Take a measurement of your speedo wheel diameter at 26 PSI WITH you sitting on the bike. See what wheel diameter was input in the speedo when you took the results and work out 'adjustment factor' like above which just multiplies the measured speed and gives you adjusted correct speed. Please do this if you haven't already!!!!!

In any case send me the spreadsheet over to s2007208[AT]student.rmit.edu.au

I'll try to replicate the experiment but with 2 fireroad slopes and 1 asphalt. Although I have speedo only accurate to 1 kph that shouldn't make a difference averaged over many runs like you did it.
 

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you guys are sick;)

still interesting to see the amount of rolling resistance and windresistance changing with higher speeds though.

i'll do my next rides with my skinsuit then;) on my roadbike that's worth one full km/h in top speed. but that's at speeds around 40-45 km/h.

then again i have some testing results of german roadie magazine TOUR who tested 12 different tubular tires. the difference from fastest to slowest would also mean about 1.5 km/h difference at the same power output. so while the fastest tubular at 300 wats would result in 42.5 km/h the slowest tubular would make it to 41.0 only. we are talking about race tubulars here which all are suggested for high-end machinery!



 

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Discussion Starter · #6 · (Edited)
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Boj,

I have the speed sensor at the front wheel and measured the circumference by laying a 3 m steel measuring tape on the floor, sitting on the bike with the front air valve at 0 mm, and rolling the bike until the air valve is again at the lowest position. The valve was then at 2232 mm. I have the feeling that this would be at least as accurate as starting out from the wheel's radius. My speedometer's precision is 1 km/h at the main display but 0.1 km/h at the 'max' recording display. I had both up simultaneously, and tried to always approach the equilibrium from below, to read at 0.1 km/h precision.

Good to hear that you will also try this, and I am looking forward to see your posted results. My number one recommendation is to really take the time when you are looking for the steeper of the two slopes, to find one that has a constant angle for a long distance. My slacker road section had constant angle for a couple of hundred meters, which in combination with low speed made it easy to accurately find the equilibrium. The steeper slope was shorter, and with three times the speed it was much more challenging to get reliable readings. Recommendation number two is patience...

Chester,

Helmet, sunglasses and ears: Thanks, I just had the feeling that something was missing in my aero tuning...

About the torso angle: Maybe it was 40 degrees, or 20. I haven't tried to measure it accurately, since it is not needed in the calculations. The only important thing about the body is to be in the same position at every reading, and that I was. I mentioned my length and riding position just to give you an idea wether your CwA would be bigger or smaller than mine. Same with tire pressure - although it affects the results, it's not needed in the calculations. The calculations show you the CwA and Cr at the tire pressure and body position that you were using at the experiment. One place where the governing formula is explained is http://www.analyticcycling.com/Glossary_Disc.html. My formulas are the result of combining the two equations you get when putting in constants for the two different runs in the governing formula.

I have also been thinking about the possible "unawareness" of air resistance. Compare that, and our also little understanding of rolling resistance, with our bike weight spreadsheets with 0.1% margin of error...

Those TdF riders probably have much lower rolling resistance as well as lower air resistance. 0.25 seems to be a common estimate for CwA for road riders in drop bars position, and under 0.008 for Cr. Put in those numbers together with the climb angle and speed in the constant speed power relationship, and you'll find where the TdF riders' power go.

Jm,

The air resistance coefficient does not depend on mass, that's true. But determining the air resistance coefficient with this method, involves the mass. The mass unit drops out since you have a density in the denominator.

All,

I have been thinking after the tests that I perhaps didn't reach the equilibrium at the steeper road, which was probably too short to give accurate readings. I recall that I often got the max reading at the end of the section. If, just for example, equilibrium would be 45 km/h rather than 41.8, then the coefficients would fall out as follows (which is also a hint about the effects of non-accurate readings):

CwA : 0.42 (versus 0.49)
Cr(road) : 0.011 (versus 0.010)
Cr(gravel) : 0.022 (versus 0.020)

Below is another illustration of the result (by "% climb" I mean vertical travel divided by horizontal travel).

Edit: I made a typo mistake when putting in the numbers in the gravel&grass track Cr calculation: The Cr for the gravel track should be 0.020 (not 0.029). I have corrected accordingly the numbers in the 45 km/h values above, and the graph below.
 

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No hi-jack intended, if you fele it is I'll delete myself or close as you wish Anden :)

I missed this thread, but now Anden linked to it from the 29" forum. I don't quite understand it yet, but am impressed.
I'll soon have access (my own) to a 29" Powertap rear wheel. I'd like to use it to test myself as well as bikes, drafting efficiency, etc.
Any way I could put it to use to come up with useful and reliable data?
I wonder if reading will be precise enough to power require to roll at a given speed.
Also I'd like to se how significant tire width is at higher speeds. My gut feeling says Big Apple 28x2.35" tires are way fast up to pretty serious speed, roadies keep insisting on using <23mm, even on the Champs Elisées.
Also interested in stuff like ideal tire pressures to roll fast on surfaces like beach, muddy cross course, and soft fire roads. Plus the effect of tire width.
 

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2 unknowns, 1 equation

It seems for each test you have 2 unknowns (Cr, CwA) and 1 equation. To determine air resistance, you assume the rolling resistance is the same at a higher speed on the steeper slope because the surface is the same. If that is what you've done I think that intruduces error. Rolling resistance goes up with speed. Again if that is what you've done, then image setting air resistance the same for 2 different surfaces at the same speed.

I do really appreciate your test. Air resistance is overlooked by mountain bikers in favor of low mass, and low rolling resistance, but it seems we need to pay more attention to it.

anden said:
Inspired by earlier discussions here about air resistance and rolling resistance, I decided to do some tests and calculations. If you don't have access to a wind tunnel or a wheel-test laboratory machine, then you could instead do the following, as far as I have concluded:

1. Find two long slopes with the very same type of surface (same road perhaps). The angles need to be different - bigger difference is better. The angles must also be close to constant for such a long distance that you can find the equilibrium speed.

2. A day with no wind, find the equilibrium speed for rolling down each of the slopes. It will take some experimenting to find it, because you will need to enter the slope with a speed close to the equilibrium speed to get necessary time to make sure you are not accelerating or retarding any more. Take several readings and calculate the average value.

3. Measure the slope angles. I put one end of a 1 m steel rod to the road, held it horizontal with a water-level, and measured the distance from the other rod end to the road with a vernier caliper. Then I repeated at several places on the slope, and calculated the average. There may be easier ways for those with more appropriate tools.

4. Find the total mass for you, your gear, and your bike.

5. Put in all values in the formulas given in the Excel pic below.

Once you have calculated your air resistance coefficient CwA, then you "only" need to repeat the above procedure for one slope to find that terrain's rolling resistance coefficient. When you have both the air and rolling coefficients, you can calculate your power and where it goes at any given speed and slope for that terrain.

My tests were made with a hardtail with 29x2.1" IRC Mythos XC II - FK front and RK rear, a bit worn but not very much. Tire pressure was my normal - 26 psi front and rear. I am 1.85 m (6'1) and have my handlebar 12 cm (4.7") under the saddle horizontal. My torso was at about 30 degrees angle from the horizontal.

The formulas were derived from the constant speed relationship Drive Power = Climb Power + Rolling power + Air Power: P = m*g*sin(x)*v + m*g*Cr*v + p*v^3*CwA/2.

Did I miss anything?
 

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if you reach terminal velocity, you would have a second equation, Fg = Fa + Fr, right?

i read somewhere that air resistance had more effect with the width of the shape rather than height; narrower bars were better than wider bars despite rider height...

for rolling resistance, wouldn't you use the normal force? mgcos(x)Cr
 

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very cool, thanks

anden:
excellent experiment, your graphs greatly resemble those i have seen in full-blown scientific experiments. nice job! drives the point home about how technique (ie. lance armstrong) is really the best way to go faster, not equipment!

drunkle:
technicall only the overall area affects air resistance, although in practice turbulent effects (which cannot be predicted) come into play. i would doubt at these low speeds that the shape of the rider (ie narrow vs wide bars) actually makes a difference provided the area is the same.
also, the normal force will have zero affect on rolling resistance, as that resistance is parallel to the road surface (and perpendicular to the normal force).
 

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ferday said:
anden:
excellent experiment, your graphs greatly resemble those i have seen in full-blown scientific experiments. nice job! drives the point home about how technique (ie. lance armstrong) is really the best way to go faster, not equipment!

drunkle:
technicall only the overall area affects air resistance, although in practice turbulent effects (which cannot be predicted) come into play. i would doubt at these low speeds that the shape of the rider (ie narrow vs wide bars) actually makes a difference provided the area is the same.
also, the normal force will have zero affect on rolling resistance, as that resistance is parallel to the road surface (and perpendicular to the normal force).
what i'm saying is that some magazine or other had made the claim that overall area was not an acceptable generalization, that specific parameters (width vs height) mattered. with arms out wider (and a shorter torso), maybe a rider suffers more from the "parachute" effect of wider spaced arms than he benefits from being short... anyway, i cant recall the actual article so oh well.

as far as rolling resistance and normal force... if you increase the weight on the tires, rolling resistance increases. the force of drag is parallel to the road, but the coefficient of drag should be a function of force normal to the surface...
 

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Discussion Starter · #12 ·
Cloxxki,

If you can find an entirely flat surface and hold constant your power input and speed with at least two or three numbers precision, then it becomes much easier to find the coefficients, with one reservation: you would need to know the drivetrain efficiency. But you may set 0.95, according to damonrinard.com/aero/formulas.htm

You need to make two runs, at two quite different speeds. For example, make one as fast you can hold steady and one at half that speed. You may want to make several runs at each speed and use averages.

Then just put the two speeds (m/s), the two powers (W), and the drivetrain efficiency in these formulas (reservation: the derivation was a bit quick):

Cr = eta*(P2-P1*v2^3/v1^3)/(m*g*v2*(1-v2^2/v1^2))

CwA = 2*eta*((P2/v2)-(P1/v1))/(v2^2-v1^2)/p

If you can have enough control on the speeds and powers, you can put on any tires you like and see how Cr and CwA changes.



Motivated,

It's one original equation, yes, but it has four variables. By putting in two different sets of speed and slope angle, you get two different equations with the same two variables, CwA and Cr. Solving those two equations gives the formulas for CwA and Cr. Rolling resistance is not assumed to be the same at different speeds.



drunkle,

I guess that what you call terminal velocity is what I called "equilibrium speed" - the speed at which you don't accelerate any more. If not all the derivations where based on the assumption of equilibrium speed, then they would also need to assume how much power goes to change of speed, which would be difficult.

Cw and A is hard to separate for a bike rider. Also, there is little reason to do it for the purpose of this experiment. The CwA (=Cw*A) in this experiment is only valid for the exact case that is measured, and can hardly be used to assume any other CwA's.

You are right about the missing cos(x) for the rolling resistance. Correct formulas are:
CwA = 2*m*g/p*(sin(x2)-sin(x1)*cos(x2)/cos(x1))/(v2^2-v1^2*cos(x2)/cos(x1))
Cr = (sin(x2)-v2^2/v1^2*sin(x1))/(cos(x2)-cos(x1)*v2^2/v1^2)

Luckily for my first, incorrect formulas and their results, the error starts at the fourth and fifth significant digit of the CwA and Cr :) For this level of accuracy, it seems that the cos(x)'s can be omitted.



ferday,

Glad you liked it. The results make me too think about the importance of the "engine" versus the different losses.
 

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anden said:
You are right about the missing cos(x) for the rolling resistance. Correct formulas are:
CwA = 2*m*g/p*(sin(x2)-sin(x1)*cos(x2)/cos(x1))/(v2^2-v1^2*cos(x2)/cos(x1))
Cr = (sin(x2)-v2^2/v1^2*sin(x1))/(cos(x2)-cos(x1)*v2^2/v1^2)

Luckily for my first, incorrect formulas and their results, the error starts at the fourth and fifth significant digit of the CwA and Cr :) For this level of accuracy, it seems that the cos(x)'s can be omitted.
heh, good enough for government work!
 
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