Estimating drag without a power meter

[Fred on a stick]

Here are speed data from two coastdowns I did a couple of days ago. The
speeds are in km/h at one second intervals. As I said, these were coast
downs so power, of course, was zero. I forgot to weigh myself and the bike
when I got home so I'm guessing the all-everything mass to be around 86 kg.
I didn't measure air density but I'm thinking it must be around 1.17 kg/m^3.
And the total drop from entry to exit of the test section was about 5
meters. Besides not measuring my mass or the air density, there was a tiny
amount of wind in my face during the test runs -- but let's ignore all those
problems and simply assume wind was zero and everything was exactly as noted
above.

Run 1:
15.2 15.8 16.3 16.9 16.9 17.4 17.7 18.1 18.5 18.9 18.6 17.9 17.9 18.6 19.7
20.0 20.9 21.6 22.5 22.5 23.4 23.7 23.5 24.3 25.1 25.7 26.0 25.6 25.3 24.6
24.2 23.9 23.5 23.1 22.9 22.6 22.3 22.3 22.0 21.9 21.7 21.8 21.4 20.7 20.9

Run 2:
26.6 26.6 26.6 26.5 26.6 26.8 26.1 25.4 25.6 26.3 26.8 27.3 27.9 28.4 29.1
28.7 29.6 29.8 30.6 30.3 29.7 29.2 28.7 28.3 27.4 27.2 26.7 26.1 25.8 25.5
25.3 25.0 24.5 24.2 23.9

Here's the challenge: based on these data, estimate my CdA and Crr. Show
your work, and state any additional assumptions you may need.

This was on my commuter bike and I was wearing street clothes so don't give
me crap about my drag numbers.

Bonus question: it's easy to figure out the average slope over the test
section. What was the maximum slope (nearest 0.1% is okay)?

[thirty-six]

On Oct 20, 2:00 am, "Fred on a stick"

Shaved legs and face, carmine lipstick, 12" hair and 4" heels will
score you at least 4/10. DDs and a sweet voice will get you to 6.
The rest is in your movement, let's see the vid.

[Anton Berlin]

Do this experiment in a practical way.

1. Take your bike to the stop of a 10-12 story building.

2. Wrap a 1/4" steel cable loop around your ball sack snugly.

3. Attach the other end (about 70 meters) to an accurate pull scale
with memory - like this one http://www.topac.com/afg.html

4. Attach other end to fixed location (chimney. steel beam, rooftop
water tank, etc)

5. Ride off roof and let gravity take the lead (these are exciting
times)

6. Sometime before you hit the pavement you will get a reading for
your max drag (right before your balls rip off )

7. if alive proceed to the earlier post and began your drag queen
career.

[Andre Jute]

On Oct 20, 2:00 am, "Fred on a stick"
<anonymous.cow...@address.invalid> wrote:

The required formulae are in my article
"How a cyclist can discover his horsepower and Cd with no tools except
his bike and a road, and use the information to improve his
performance"
at
http://www.audio-talk.co.uk/fiultra/BICYCLE%20TECH%20--%20basic%20cycling%20parameters.html

Andre Jute
Helping little old ladies across the road whether they want to go or
not

[RicodJour]

> at http://www.audio-talk.co.uk/fiultra/BICYCLE%20TECH%20--%20basic%20cyc...

I'm not good with formul-a, or even formul-b. Be a sport and just use
Fred's data to show everyone how it's done.

I asked politely. Now you have to. It's the law.

R
Chief Wheedler

[Dan O]

> > athttp://www.audio-talk.co.uk/fiultra/BICYCLE%20TECH%20--%20basic%20cyc...

>
> I'm not good with formul-a, or even formul-b. Be a sport and just use
> Fred's data to show everyone how it's done.
>

Sorry, I didn't understand the strings of numbers and what they mean
or their units, etc. - I guess it's supposed to fit some template that
serious bicycle racers all know and use when they get tired of
"training".

If I was racing, and wanted to win races, I just wouldn't coast hardly
at all - except on the steeper descents.

> I asked politely. Now you have to. It's the law.
>

I don't stop at stop signs, either.

[RicodJour]

On Oct 22, 11:41 am, Dan O <danover...@gmail.com> wrote:
> On Oct 21, 10:23 pm, RicodJour <ricodj...@aol.com> wrote:
> > On Oct 21, 11:09 pm, Andre Jute <fiult...@yahoo.com> wrote:

FOAS mentioned the units earlier in the post. I like the strings of
numbers because they're easier to ignore than a single number.

> If I was racing, and wanted to win races, I just wouldn't coast hardly
> at all - except on the steeper descents.

I was responding to he that be Jute, but I don't think FOAS is looking
to win races on his commuter. I think he's asking for assistance in
his exercise in futility. ;) As if no one else here has ever done
that.

> > I asked politely. Now you have to. It's the law.
>
> I don't stop at stop signs, either.

It's okay. I'm sure you've found an equally slippery way around that
pesky transfer of momentum thing. ;)

R

[Fred on a stick]

RicodJour wrote:

> I was responding to he that be Jute, but I don't think FOAS is looking
> to win races on his commuter. I think he's asking for assistance in
> his exercise in futility. ;) As if no one else here has ever done
> that.

Actually, no, I was looking for entertainment. Maybe assistance in my
entertainment but that sounds kind of sordid and, now that I think of it,
intriguing.

Anyway, I already calculated the CdA and Crr from these data.

[thirty-six]

On Oct 22, 6:11 pm, "Fred on a stick"

Do it again after a bottle of whisky.

[Fredmaster of Brainerd]

On Oct 22, 10:11 am, "Fred on a stick"

Your CdA is 0.36 m^2 ( +/- 0.09 or so), your Crr is 0.0065,
and the maximum slope on the course is 0.00 or 0%, although
I don't know how you expected anyone to calculate it to
0.1% with data of this quality.

Thanks,
Fredmaster Ben

[Fred on a stick]

Fredmaster of Brainerd wrote:

> Your CdA is 0.36 m^2 ( +/- 0.09 or so), your Crr is 0.0065,
> and the maximum slope on the course is 0.00 or 0%, although
> I don't know how you expected anyone to calculate it to
> 0.1% with data of this quality.

Killjoy. There are lots of problems with the data (as I noted) but if we
take them as given you get an exact solution for CdA and Crr, and for the
slopes. I just meant that I didn't need to see a number for slope
like -4.1234567%.

[Fredmaster of Brainerd]

On Oct 22, 5:05 pm, "Fred on a stick"

<anonymous.cow...@address.invalid> wrote:
> Fredmaster of Brainerd wrote:
> > Your CdA is 0.36 m^2 ( +/- 0.09 or so), your Crr is 0.0065,
> > and the maximum slope on the course is 0.00 or 0%, although
> > I don't know how you expected anyone to calculate it to
> > 0.1% with data of this quality.
>
> Killjoy. There are lots of problems with the data (as I noted)

I didn't mean that as an insult to the data. It seems about
as good as one should expect to get with bicycle-computer
type instrumentation and in the presence of real world
effects (small but variable wind, etc).

> but if we
> take them as given you get an exact solution for CdA and Crr, and for the
> slopes. I just meant that I didn't need to see a number for slope
> like -4.1234567%.

"Exact solution"

I don't know what this term means [*],
can you explain it further?

[*] in the context of fitting a model to data that include
real world effects: wind, digitization noise, etc.

The way I did it is probably not ideal (differentiating and
integrating the velocity time series); it's not explicitly
fitting a model, although there is a model implied of
course. It yields a large number of noisy estimates
of CdA which have to be averaged over to get a value
and error estimate, which IMO is more useful than an
exact solution. Presumably any method that yields a
single number should then allow you to calculate
residuals off the model, from which an error estimate
could be backed out, in some non-trivial way.

Fredmaster Ben

[Andre Jute]

You buncha spoilsports! Rico was going to show me how to do it... Now
I'll never know.

[RicodJour]

On Oct 22, 10:11 pm, Andre Jute <fiult...@yahoo.com> wrote:
> On Oct 23, 2:26 am, Fredmaster of Brainerd <bjwei...@gmail.com> wrote:

Encourage commuters?! What, are you nuts? FOAS' bike probably has
_fenders_!

Besides, you're the one with the formul-A. The Pope never misses a
chance to pontificate, I figured you'd be the same. :)~

The only times I ran were when a coach was yelling at me or I was
being chased. Numbers are like that - I only touch them under
duress. They're antiquated and unnecessary in a modern world.

If I need to go faster I buy a new bike. The carbon fiber gremlins do
the calculations for me. It's a formul-A, but it works!

R

[Fred on a stick]

Fredmaster of Brainerd wrote:
> On Oct 22, 5:05 pm, "Fred on a stick"

>> but if we
>> take them as given you get an exact solution for CdA and Crr, and
>> for the slopes. I just meant that I didn't need to see a number for
>> slope
>> like -4.1234567%.
>
> "Exact solution"
>
> I don't know what this term means [*],
> can you explain it further?
>
> [*] in the context of fitting a model to data that include
> real world effects: wind, digitization noise, etc.
>
> The way I did it is probably not ideal (differentiating and
> integrating the velocity time series); it's not explicitly
> fitting a model, although there is a model implied of
> course. It yields a large number of noisy estimates
> of CdA which have to be averaged over to get a value
> and error estimate, which IMO is more useful than an
> exact solution. Presumably any method that yields a
> single number should then allow you to calculate
> residuals off the model, from which an error estimate
> could be backed out, in some non-trivial way.

You remind me of that apocryphal story about von Neumann being posed that
puzzle about the infinite series.

There's a full information estimate: integrate the power equation up to a
work equation. Plug in the info given and you'll end up with two equations
with two unknowns so you can solve directly. Then plug the Crr and CdA back
into the power equation and solve for the slopes -- or, do as I do and
integrate the slopes up to an elevation profile. That's what I've been
calling the "virtual elevation" since it's in the presence of wind and
errors in meaurement and squirming around and transitory irregularities. If
the two virtual elevation profiles "fit" well (and there are a handful of
ways to evaluate goodness-of-fit) then you're golden.

As for directly assessing the variability of the estimates, I've been doing
something akin to what you're describing by setting up data windows of a
particular length and estimating the parameters based on that, then sliding
the data window one element forward and re-estimating. It's sort of like a
windowed jackknife. You end up with a large number of (relatively) noisy
estimates. There's high serial correlation but that's a pretty standard
problem and we know how to deal with that. Under benign test conditions
(with a power meter, not with coast downs, but the method is the same) the
precision of the CdA estimate was pretty damn good.

[Andre Jute]

I'm not a commuter, Rico, so you couldn't possibly encourage me. I'm
far, far worse: I'm a recreational cyclist who doesn't even pretend to
be training for some non-existent event. And I haven't kept an
automobile since 1992, so I'm greener too than all these wankers who
presume to lecture us on saving the environment. I actually do what
they only preach...

As for doing math, you're a funny man. It's interesting generating a
formula to measure something. (And I was very well paid, several
times, for generating that one. It's from my book for special car
builders.) But I leave repetitive scutwork like math to flunkies, aka
engineers. There's always an eager beaver trying to show how much
smarter he is than his betters, just straining at the leash to get at
the calculator. Check out DougC, absolutely quivering to prove that my
bike is no good because he and the other cheapskates never heard of
the maker -- and failing to grasp that that is an added attraction on
top of the manifold qualitative advantages of the bike.

As for the Pope, maybe the next time I need a PR man I'll consider
him. But he's getting on a bit, and the general feasting and drinking
and womanizing of the PR trade will probably wipe him for good, which
is why I don't do my own PR any more. I wouldn't want that on my
conscience.

Andre Jute
Scenes from a Bizarre Life -- tentative title of my memoirs

[critposer]

http://www.cyclebanter.com/showthread.php?t=204228

[Fredmaster of Brainerd]

On Oct 22, 7:54 pm, "Fred on a stick"

<anonymous.cow...@address.invalid> wrote:
>
> You remind me of that apocryphal story about von Neumann being posed that
> puzzle about the infinite series.

People say that to me about von Neumann all the time.
It must be because I'm gnomic, Hungarian, and not a genius.
Two out of three ain't bad.

> There's a full information estimate: integrate the power equation up to a
> work equation. Plug in the info given and you'll end up with two equations
> with two unknowns so you can solve directly. Then plug the Crr and CdA back
> into the power equation and solve for the slopes -- or, do as I do and
> integrate the slopes up to an elevation profile. That's what I've been
> calling the "virtual elevation" since it's in the presence of wind and
> errors in meaurement and squirming around and transitory irregularities. If
> the two virtual elevation profiles "fit" well (and there are a handful of
> ways to evaluate goodness-of-fit) then you're golden.

What I did, which wasn't sensible, essentially created the slope
profile,
but it involved numerically differentiating the velocity time series,
which of course amplifies noise. Actually what I did was smooth
the data slightly and compute dv/dt, and integrate to get s(t),
s=distance.
You then have an equation of motion for each run, which involves
Crr + slope, but you know slope(s) should be equal between
the runs, and plotting it showed that it was, although noisy.
So subtracting the two time series yields
dv/dt_2 - dvdt_1 = -0.5 CdA rho/m (v_2^2 - v_1^2)

v_1 is velocity in the first run, and so on. This allows a
measurement
of CdA at each timestep (not quite independent since the data were
smoothed).
It's noisy, of course, but the scatter tells you something - although
possibly
more about the noisy method than the statistical error.

Your method makes a lot more sense. I would rephrase it as
you're just using a work-energy equation: you know the start
and end kinetic and potential energy, and you want to compute
the work done by rolling resistance and drag. The work done
by rolling resistance is the same between runs, it's just
Crr * mg * total distance. You can compute the work done by
drag, up to the CdA factor, from the velocity time series.

Where I went wrong is that often when you do a drag problem,
you only know start and end quantities, and conservation of
energy isn't helpful, and you have to integrate something to
solve the motion. Here, you gave us the velocity time series,
so that part is already done.

> As for directly assessing the variability of the estimates, I've been doing
> something akin to what you're describing by setting up data windows of a
> particular length and estimating the parameters based on that, then sliding
> the data window one element forward and re-estimating. It's sort of like a
> windowed jackknife. You end up with a large number of (relatively) noisy
> estimates. There's high serial correlation but that's a pretty standard
> problem and we know how to deal with that. Under benign test conditions
> (with a power meter, not with coast downs, but the method is the same) the
> precision of the CdA estimate was pretty damn good.

Basically you're breaking each run into a set of sub-runs (ok,
overlapping
because the jackknife allows that). If you can be sure to have the
start and end points match between a subunit of run 1 and run 2,
then you do the same method. If you had confidence in the elevation
profile, than any two subsets allow a calculation, even if they aren't
matched in start and end point. I found that the start to end PE
change
was significantly more than the KE change, so the elevation is
important
if you did that.

Of course, it would be nice to have several full trials, and then you
could
estimate CdA and Crr several times from different pairs. This would
also
give a handle on how much scatter is caused by uncontrolled
differences
(small wind, clothing, phase of the moon).

Fredmaster Ben

[atriage]

On 24/10/2011 05:45, Fredmaster of Brainerd wrote:


> (small wind, clothing, phase of the moon).
>

Whether or not it's a Tuesday.

--

[Fred on a stick]

Fredmaster of Brainerd wrote:
> I would rephrase it as
> you're just using a work-energy equation

Yup.

> you know the start
> and end kinetic and potential energy, and you want to compute
> the work done by rolling resistance and drag. The work done
> by rolling resistance is the same between runs, it's just
> Crr * mg * total distance. You can compute the work done by
> drag, up to the CdA factor, from the velocity time series.
>
> Where I went wrong is that often when you do a drag problem,
> you only know start and end quantities, and conservation of
> energy isn't helpful, and you have to integrate something to
> solve the motion. Here, you gave us the velocity time series,
> so that part is already done.

Yup, again. The classic way to solve for CdA and Crr if you happen to have a
power meter is to hold constant speed and do this on a constant (usually
zero) slope so the PE and KE parts go away. However, we have the velocity
time series so we don't actually need to zero out the KE component.

Yeah, so the velocity time series gives us a way to calculate the KE
component. As you said, if you have confidence in an elevation profile you
can get the PE component. What I've been suggesting in the case where you
have a power meter is to ride laps or loops of a course. When you get back
to the start point you know the PE is zero. So I actually do a few laps or
loops, as you surmised. When I want to directly assess the variability of
the estimates, I do the work-energy equation over a one-lap window, then
slide the window over one interval (making an adjustment to get the lap
length right so the PE zeroes out), lather, rinse, repeat. As in the coast
down example, if you vary the speed/power over the laps it improves the
estimation.

Here's a little example. It was three laps (out-and-back) along a U-shaped
"half-pipe" test course. We were checking out the difference between two
front brakes. Top panel is speed in m/s, middle panel is the "virtual
elevation" based on speed and power, bottom panel is the estimated CdA over
one-lap long intervals.
http://anonymous.coward.free.fr/simkins-egg/R/mach-egg.png

If I hadn't split up the laps separately and considered the entirety of the
three laps then the PE over the entire three laps would be zero but then I
wouldn't have enough info to get separate estimates of Crr and CdA. Here's a
plot that illustrates that problem: each panel shows various combinations of
Crr and CdA that get you back to the same "elevation" at the end of the
three laps but you can see "misfit" and an asymmetry in the putative
elevation profiles for the top and bottom panel. (I didn't calculate the
error-minimizing Crr and CdA this way -- it's just to show how being off
affects the virtual elevation profiles).
http://anonymous.coward.free.fr/simkins-egg/R/halfpipe-crr-cda.png