An Oldie, but a Goodie...Field Testing Frame Differences with a Power Meter

I originally posted this back in 2008 on the Slowtwitch.com forum and is an example of the types of equipment differences that can be determined with careful field testing.  Here's the link (click on the title below) to the full thread on Slowtwitch...it's long but informative, and there's some pretty good "back and forth" with a few skeptics:

Something Borrowed...Something Fast!

 
So take this guy and his position:

...and put him on this guy's bike and adjust it so that they fit identical.

Then, let the first guy test both of these "back to back" using the same 404 wheelset with PT SL hub and cover....Any guesses on what the aerodynamic drag differences (if any) one would see?

Using RChung's most excellent methodology described here: Method to the madness

Here's the results for the P2K with the 404 wheels (the wheels on the P3C above):

And here's the results for the P3C:

So...what's the bottom line?

Well...taking the same rider, the same wheels, the same basebars and brake levers, and with the seat and extensions adjusted to deliver the same positions...at basically zero yaw conditions I apparently measured a drag difference of ~.023 m^2 of CxA (or Cda, whichever you prefer - .228 m^2 for the P2K and .205 for the P3C). Using Doc C's "rule of thumb", that basically equates to ~2.5s per km of time savings.

With only an extremely small amount of crosswind however (I could just barely detect it on my skin, not enough to even move leaves on trees), the drag on both setups drops significantly, with the P3C setup dropping more at .190 m^2 vs. the P2K's .220 m^2 for a growing difference of ~.03 m^2. That translates to a difference of ~3 seconds per km...or a full 2 minutes over 40K.

Can you imagine comparing the difference between a P3C (or even the P2K) against a frame that actually increases in CxA with increasing yaw, which is actually fairly common?

Aero Field Testing using the "Chung Method" - How sensitive can it be?

As some of you may know, I've been field testing bike stuff and positioning with a power meter for 4 or 5 years now.  My method of choice is Robert Chung's "Virtual Elevation", or VE protocol, sometimes known as the "Chung Method".  He has a great presentation on it here: http://anonymous.coward.free.fr/wattage/cda/indirect-cda.pdf . When I first read Robert's info, I wrote up a spreadsheet that I've used since then to analyze everything from position changes to tire air pressure effects. Of course, since I wrote that spreadsheet for my own use, it's not exactly the most "user friendly" (Hey, I know what I'm supposed to do, I wrote it! ;-)...but, don't worry, everyone else is in luck since Andy Froncioni (the main tech guy behind Alphamantis and the ERO facility that recently opened at the indoor track in Carson) added a version of the same calculations (called "Aerolab") to the freeware power meter analysis software, Golden Cheetah.

So, the question with this type of testing usually comes down to just how sensitive can it really be...especially as compared to something like a wind tunnel?  Admittedly, there are some limitations to this type of testing, the main one being (at present time) that the results are mostly limited to zero yaw conditions, but as we saw in one of my previous blog posts, the most common yaw angles a TT'er or triathlete encounters are usually centered around zero yaw.  Using the tool to make evaluations at zero yaw still can hold a significant benefit for someone interested in improving/testing bicycle aerodynamics.

A couple years ago, Dr. Andrew Coggan published a blog post titled "A Challenge to Cycling Aerodynamicists" in which he described a field test he undertook to take up something he coined the "Tom Compton Challenge".  In short, it's an effort using known geometric shapes to try to determine the "sensitivity" of the aerodynamic field testing method.

Well, last year I discovered that my preferred field testing venue for VE runs had suffered some "traffic rerouting" that had made it much less appealing for the purpose (part of that "discovery" occurred when Andy sent his test setup to me to try and the results from my first course were very mixed due to excess vehicle interference after the nearby roads had been modified).  So, I started scouting around for an alternative course and luckily found one that is much closer to my home (I can ride there in just a few minutes) and that has laps that are significantly shorter than the old course (shorter laps = shorter test run time).  Both of these courses are best described as a sort of "extended halfpipe", an "out and back" course having a U-shaped elevation profile that allows for turnarounds to be taken at low speeds and thus avoid braking. Since identifying the new course, and having done just a few tests on it, one thing I wanted to do was to repeat the type of testing that Andy did and attempt to characterize the potential "sensitivity" of the course using the VE method.

Using Andy's setup as a guide, I set about figuring out what sorts of objects I could use for the test.  I took a quick trip to the nearby Michael's craft store and acquired some styrofoam spheres, 2", 3", and 4" in diameter.

In my garage I had an appropriate length of 1/2" diameter wooden dowel, and short work with a hand drill on the spheres and I had a setup that placed the spheres well out to the side where they should be in clear air while riding. Also shown in the pic above on the left is a small washer which I placed on the end of the dowel during the first run instead of a sphere.  I did that to act as a "cap" and make it more likely that the flow over the end of the cylinder stayed perpendicular.  Here's how the dowel and sphere setup looked after being zip-tied to the basebar of my TT bike.

The hole in each sphere ended up being a nice friction fit, so swapping between spheres was a very simple process. 

All runs were recorded with my trusty old yellow-cap PT Pro wheel mounted on the rear, with a cover in place to turn it into a de facto disc.  I prefer to use a PT for my aero field testing since it eliminates the uncertainty of variations in drivetrain resistance across the gearing, plus the PT's "coasting zero" feature allows me to have the power meter zero while soft-pedaling (i.e. turning the pedals slowly while freewheeling) down the descents of the course at least once per lap.  That helps to minimize any power meter drift during the runs.

So, with the test rig sorted out, it was time to head out to the test course and do some runs! To minimize wind and traffic effects, I prefer to head out to the course early on a Saturday or Sunday morning...before the small neighborhood that the course road services begins to wake up and starts moving around. A couple weekends ago, I headed out on a Sunday morning and rode over to my test course in 5 minutes, taking a small musette bag with the spheres, a notebook, and a couple of small tools I might need.  Starting at 6 am, I did the runs in the following order:

1. Rod only (w/washer "endplate")
2. 3" sphere
3. 2" sphere
4. 4" sphere

I mixed the runs up like that since I suspected that the "rod only" and 2" spheres may be close to the same measurement (part of the rod is covered up by the sphere) and I wanted to make sure there was a good separation between the cases.  I actually didn't sit down and calculate out the expected differences in CdA beforehand.  I wanted to first determine what the VE analysis showed as the differences from the baseline (run #1) and then see how close to the calculated values the VE runs were.  I did this because the method I use for determining the CdA using VE is a visual "leveling" procedure, and so there's a bit of "judgement" involved in determining what value best "fits" the overall plot to being level, and I didn't want that judgement being affected by any predetermined knowledge of what the expected differences should be.

Once I returned home, it was time to download the PT files into the computer and do the VE analysis.  As I described above, although it can be done in GC's Aerolab feature, I prefer to use my own home-brewed spreadsheet, mostly because I find it easier to expand the vertical scale to get a better handle on the leveling procedure, but also because I've recently added a feature that varies the on-road Crr as a function of the ambient temperature.  In order to use the spreadsheet, the following inputs are required:

1.  Total Mass - Easy to get just by stepping on a scale with bike in hand
2. Weather Conditions - this means air temp, dew point temp, and barometric pressure (to determine air density).  Luckily, there's a personal weather station listed on Weather Underground literally less than a block from my test course that has updates loaded every 5 minutes.  Using that station also allows for a cross-check on ambient wind conditions to make sure it stayed calm during the test runs.
3. Assumed Crr - For this, I use a weighted average of the front and rear tire Crr that I've determined from roller testing.  The spreadsheet then compensates for the expected Crr due to the difference in the test ambient temperature and the 20C temperature to which my Crr results are normalized.

As an example of the spreadsheet and what the VE profile plot ends up looking like, shown below is a snapshot of the first run from the testing:

OK then, let's get to the results.  Using the procedure outlined above, my best determination for the measured CdA from the runs was as follows (in order that runs were performed):

1. Rod only  = .2484 m^2
2. 3" sphere = .2498 m^2
3. 2" sphere = .2486 m^2
4. 4" sphere = .2510 m^2

Now, how does that compare to what should be expected for those shapes?  To determine that, I made a spreadsheet that calculated the expected CdA changes based on the typical values of Cd (in the Re number range of interest) for a sphere and a cylinder (sphere = 0.47, cylinder = 1.17) and their respective cross-sectional areas based on my actual measurements. As mentioned above, when I compared the "rod only" run to the sphere runs, I had to subtract the portion of the rod that was covered by the cylinder from the CdA calculation.  I then took those expected changes in CdA and added them to the measured CdA from the "rod only", or baseline run to determine the expected CdAs for the runs with the spheres.  Here's how they compared:

Run # - Sphere      Measured CdA (m^2)     Calculated CdA (m^2)   Difference (m^2)

2. - 3" sphere                  .2498                              .2501                          .0003
3. - 2" sphere                  .2486                              .2488                          .0002
4. - 4" sphere                  .2510                              .2518                          .0008


Another way of looking at it is the expected and measured differences from the baseline:

Run # - Sphere      Meas. Diff. from Baseline(m^2)     Calc. Diff. from Baseline (m^2)

2. - 3" sphere                       .0014                                            .0017
3. - 2" sphere                       .0002                                            .0004
4. - 4" sphere                       .0026                                            .0033


One last way of looking at this is from the perspective of expected change from from the previous run.  Here's how that worked out:

Run # - Sphere      Meas. Diff. from Previous (m^2)     Calc. Diff. from Previous (m^2)

2. - 3" sphere                       .0014                                            .0017
3. - 2" sphere                     -.0012                                           -.0013
4. - 4" sphere                       .0024                                            .0029

Not bad, huh?  I've always said that when using this technique I consider measurements that are within +/-.001 m^2 to be basically "tied", and the above appears to bear that assumption out as being fairly conservative.  It also gives me confidence that with careful technique I should be able to easily detect CdA differences on the order of .003-.005 m^2 and greater.

What's the Virtual CdA and Crr of the LeMond Revolution Trainer?

...and, more importantly, what's the "equivalent mass"?

Back in 2011, I was given access to a LeMond Revolution trainer.  One of the selling points of this trainer is it's "road feel", and knowing that the load was produced by air drag, I decided to see if I could figure out what the "virtual" CdA (drag coefficient) and Crr (rolling resistance coefficient) was for the trainer, just to see if it reasonably represented a typical rider and equipment.  Another part of the "road feel" is the inertial mass of the trainer's flywheel, and I was hoping I could figure out the "equivalent mass" that the flywheel represented.  In other words, what mass of a rider (plus bike) traveling down the road does the spinning flywheel mass represent?  Some of you reading this may recognize the figures from a thread I started back then on the Slowtwitch.com forum, and also one on the wattagetraining.com forum.

The majority of the load experienced by a cyclist when riding outside on level ground is produced by aerodynamic drag and the rolling resistance of the bicycle equipment (tires mostly, but bearings as well...) and is very simply represented by the following equation (assuming still air as well).

Power = Aero drag power + rolling resistance power, or

Power = (1/2 x air density x CdA x V^3) + (Mass * g * Crr * V),
 where V is the velocity of the bike and g is the gravitational constant.

This equation is handy, because if you divide both sides by V, you end up with a linear equation with respects to the variable V^2.  In other words, the equation of a line is:

Y = mx + b, or dependent variable = (slope * independent variable) + (y-intercept)

What that means is: if you are able to measure the power it takes to travel at various bike velocities, then if you plot the Power/Velocity on the y-axis and the Velocity^2 on the x-axis of a chart, you should see a line where the slope is equal to the CdA and the y-intercept is the Crr.  In fact, this technique is a common one used in field testing and is sometimes known as the "regression method", since it allows one to easily do a linear regression fit to the data.  I just thought I'd apply it to the wind trainer as well...

Here's the result of a constant cadence cassette sweep (only 9 of the 10 cogs of a 12-25 cassette, 60 rpms, with a repeat of the 53-14 at 75 rpm for some slightly higher power.) After the sweep, I did some accelerations/decelerations for the purposes of doing some inertial mass estimate.

Taking the average of the power and "virtual speed" (i.e. the result of the gearing, cadence, and assumed wheel rollout) over the last 2 minutes of each step (and the last 1 minute of the 75 rpm step), I then plotted P/V vs. V^2 for the following:

That looks pretty good!  Nice and linear, which means it basically "behaves" like a rider on the road.  Assuming an "all-up" mass of 85kg, and a rho (air density) of 1.2 kg/m^3, that y-intercept works out to represent a Crr = .0051 and the slope of the line works out to represent a CdA = .350 m^2. Sounds like a fairly "normal" road bike position (on the hoods) and Crr.  Nice.

Now...about that estimate of inertial mass. My intent was to plug the file and the calculated Crr and CdA into my VE ("virtual elevation") spreadsheet and then modify the mass entry until the small "hills" formed by the accelerations/decelerations "flatten out"...I tried that, but I realized that I needed to reconsider how varying the mass entry affects the calculated rolling resistance force.  See below.  In a perfect representation, that trace shown below would be completely level.  The fact that it's not means that something is amiss with the modeling, probably the acceleration terms.

BTW, "virtual elevation" is a technique of analyzing power files created by Robert Chung which allows one to estimate drag coefficients given a certain Crr, or vice-versa. See a presentation on the subject by Robert Chung here.

At about this time in the Slowtwitch thread, Robert asked to take a closer look at the acceleration sections near the end.  Here's what that looked like:

  

He then asked to see the VE trace in an overlay on the speed trace and so this is what I showed him:

At this point he said: "I presume you were using an all-up mass of around 80kg? So cut that in half to 40kg and double the Crr to .01. That'll keep the steady state power the same but should improve the modeling for the KE component."  

Here's what it looked like with the mass decreased by half to 42.5 kg and the Crr increased to .0105 (I played with the Crr to get the best "fit"):   

The VE changes between the gear changes at least seemed to have been "smoothed" over...but the accels/decels at the end were still a bit "spikey".
 

To that, his response was: "Cool. In the original, the VE was "countercyclical" to the speed. Now it's cyclical. I think that means halving the mass was an overcorrection. So the inertial mass is somewhere between 42.5 and 85."

I then played around with the mass and Crr a bit and here's the "best fit" I could find...and I mostly judged it off of the final "tail" which was a coast down to zero rpm. I adjusted the mass (and Crr accordingly) until the final steady-state leg and the coast down to zero had a minimum of "disjoint". This case was actually 100 lbs (45.5 kg) and a Crr of .0094 

Now...we'll see how close that is to a calculated mass based on the flywheel geometry! 

I then pulled the flywheel cover off and measured up the flywheel on the trainer. To calculate the mass moment of inertia, I decided to just throw the dimensions into Pro/E and let it do the calculations:  

Here's the mass properties output:

VOLUME = 8.0428750e-04 M^3
SURFACE AREA = 2.0893183e-01 M^2
DENSITY = 7.8887728e+03 KILOGRAM / M^3
MASS = 6.3448414e+00 KILOGRAM

CENTER OF GRAVITY with respect to _GL_FLYWHEEL coordinate frame:
X Y Z 0.0000000e+00 0.0000000e+00 2.4937030e-02 M

INERTIA with respect to _GL_FLYWHEEL coordinate frame: (KILOGRAM * M^2)

INERTIA TENSOR:
Ixx Ixy Ixz 4.2069766e-02 0.0000000e+00 0.0000000e+00
Iyx Iyy Iyz 0.0000000e+00 4.2069765e-02 0.0000000e+00
Izx Izy Izz 0.0000000e+00 0.0000000e+00 7.3130437e-02

INERTIA at CENTER OF GRAVITY with respect to _GL_FLYWHEEL coordinate frame: (KILOGRAM * M^2)

INERTIA TENSOR:
Ixx Ixy Ixz 3.8124192e-02 0.0000000e+00 0.0000000e+00
Iyx Iyy Iyz 0.0000000e+00 3.8124191e-02 0.0000000e+00
Izx Izy Izz 0.0000000e+00 0.0000000e+00 7.3130437e-02

PRINCIPAL MOMENTS OF INERTIA: (KILOGRAM * M^2)
I1 I2 I3 3.8124191e-02 3.8124192e-02 7.3130437e-02

ROTATION MATRIX from _GL_FLYWHEEL orientation to PRINCIPAL AXES:
1.00000 0.00000 0.00000
0.00000 1.00000 0.00000
0.00000 0.00000 1.00000

ROTATION ANGLES from _GL_FLYWHEEL orientation to PRINCIPAL AXES (degrees):
angles about x y z 0.000 0.000 0.000

RADII OF GYRATION with respect to PRINCIPAL AXES:
R1 R2 R3 7.7515746e-02 7.7515748e-02 1.0735906e-01 M

The value we're interested in is the the Izz value of .073 kg*m^2.

So...how do we equate this to an "equivalent mass" translating down the road? I like to do it by equating the kinetic energy of the flywheel to the kinetic energy of a bike+rider moving down the road.

Kinetic Energy of rider = Kinetic Energy of the Flywheel
KErider = KEf
1/2 x mass of rider x (velocity of bike)^2 = 1/2 x Izz x (flywheel rotational speed)^2
Mr x Vb^2 = Izz x (Wf)^2

OK...to solve this, I need to equate the flywheel rotational speed (in radians per second) to the equivalent bike velocity (in meters/second). Well, the assumption above was that the wheel rollout was 2080mm, or 1 revolution = 2*Pi radians is equivalent to 2080mm of rollout.

Wheel rotation rate = Ww = Vb x (2*Pi radians/2.080m), where Vb is in m/s, so Ww = 3.0208 x Vb.

Now, to figure the flywheel rotational rate, we need to know the pulley ratio of the drive pulley and the flywheel pulley. By my measuring, this ratio is 8:1. So, the flywheel rotational rate, Wf = 8 x Ww = 8 x 3.0208 x Vb = 24.166 x Vb.

Lastly, I'll plug this relation into the simplified KE equation above along with the calculated Izz from the solid model.

Mr x Vb^2 = Izz x (24.166 x Vb)^2
Mr = Izz x 584 = .073 x 584 = 42.6 kg

42.6 kg is equivalent to a rider weight of ~94 lbs...now, that's also not including the other rotating bits (like the pulleys, cassette, etc.)...but that's pretty darned close to the "equivalent mass" determined using the coastdown and VE above :-D  

So...in the end, what does that all mean?  Well, I'll end with one more quote by Mr. Chung: 

"Aha. Excellent. It may "coast" better than regular trainers -- but not as well as a bike on a flat road. Likewise, it appears to accelerate faster than a bike on a real road. Steady state load is reasonable, though."

Sounds good to me.