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.

What's up with those funky rings...?

With the successes of Bradley Wiggins in 2012, there has been a lot of interest shown lately on the non-round rings he's been riding.  These rings are nothing new, and non-round rings have been dabbled with for a very long time, but the objective data (physical tests, as opposed to mathematical modeling) on whether or not these rings allow for a greater power output than round rings has been mostly in the negative.  Yet, there are typical reports of riders seeing 15-20W (or more) increases in their power output after mounting them on their bikes.  For a while, it's been speculated that some of these reports may be the result of an artificial inflation due to the way that some power meters calculate their power readings, and mathematical analyses along with user reports have suggested that the "real world" matches the artificial inflation theory.

What is this "artificial inflation" theory, you might ask?  Well, to understand the theory, first one has to understand how bicycle power meters measure and calculate power.  Simply put, for the most common power meters on the market (I'm talking SRM, Quarq, and PowerTap) there is a torque measurement that is averaged over a certain "window" that is then multiplied by the angular velocity to get power, i.e.

Power = Torque x Angular Velocity

However, the power output of a cyclist isn't constant torque, like with an electric motor, for example.  Instead, torque input to the crankarms of a bike is pulsed, with the majority of the torque being applied in a fairly narrow range during the pedal downstroke.  As such, if one plots torque vs. pedal position over a single pedal revolution, you'll typically see a sinusoidal-type plot with 2 "peaks" per pedal revolution.  Because of this, in order to better represent the true power output of a cyclist, the SRM and Quarq power meters have been designed to use what's known as an "event-based" power calculation algorithm.  What this means is that the torque is averaged over a complete pedal revolution (no matter how long it takes) and then the average torque is multiplied by the average angular velocity over that single pedal revolution.  This way, the only measure of angular velocity is a simple magnet/reed switch construction whereby each crank trigger signifies that 360 degrees of pedal rotation have been accomplished in the time between magnet pulses on the reed switch.  The over-riding assumption of this architecture is that the crank velocity does not vary significantly from the average during the pedal revolution, and with the high inertia of a cyclist traveling on the road at any reasonable speed, this is a fairly accurate assumption...with round rings that is.  If tension is maintained in the chain of the bicycle during the pedal stroke, then by definition the crank rotational velocity will be fairly steady.  If one where to be able to measure the instantaneous rotational velocity and the instantaneous torque and calculated the instantaneous power, then if you averaged all of those instantaneous power calculations over the complete pedal revolution, the result would be very close to averaging the torque over the complete pedal cycle and multiplying by the average rotational velocity.  This is why PMs have been designed this way.

Now then, how does this factor into non-round rings possibly causing artificially inflated power values?  Well, the way that most non-round rings are designed to "work" (one exception being the old Shimano Biopace rings) is that the effective chainring size is increased during the times that the legs are applying the highest amount of torque (and, since the system needs to be symmetric, also during the time when the leg is rising back into position in the "recovery" portion.  But, I digress..) and the effective chainring size is decreased during the portions of the pedal stroke across the top and bottom of the cycle. This causes the instantaneous rotational speed of the crank to vary in a roughly sinusoidal manner (dependent on the shape of the ring) above and below the average crank rotational velocity.

For a given wheel speed and for equivalent number of teeth on the chainring, the average rotational velocity of a non-round ring will be the same as for a round ring.  This is true because in order for the teeth to be at a spacing of 1/2", that means that the circumferences of a non-round and round ring of the same tooth count are equivalent.  However, with a non-round ring that means that, by design, the rotational velocity of the crank is being manipulated such that the rotational speed of the crank is slowed during the downstroke (and upstroke portions as well due to symmetry) to a value below the average

So, what does that mean?  In short, it means that during those times that the largest torques are being applied, the rotational velocity is lower than the average over the complete pedal revolution.  Since the majority of the power is produced in the downstroke portion of the pedal cycle, this means that power reported will tend to be over-reported, hence the term "artificial power inflation".  Get it?

So, that's the theory...how does it end up working in reality?  Well, in short, reality matches the theory fairly well and by my measurements, using Osymetric rings on a Quarq power meter will cause the Quarq to report a power value ~2.7-3.5% higher than reality.

Here's some testing I did near the end of 2012:

The setup: 

Test performed on a LeMond Revolution trainer to maintain a rear "wheel speed" similar to riding on the road (i.e. high flywheel inertia)
 
54T Round and 54T Osymetric both mounted on the same Quarq with the Osymetric in place of the inner ring. This was done to eliminate changes in drivetrain losses due to using different cogs. Calibration shows the Round ring (mounted on the outer position) reads 1% high and the Osymetric (mounted on the inner position) reads 1% low and the torque slope was set to the average value between the 2 rings. Power values from the Quarq reported below were corrected based on the calibration. Zero offset was checked before the first pair of runs, before the 2nd set of runs, and then at the finish. Offset did not move by more than 5 counts (512, 507, 511) which is equivalent to 1.3W @ 80 rpm. Runs were ~4 minutes in length.

Runs in order of performance:
- 54x15 gear, 80 rpm (36.2 km/hr indicated)

• Round Ring (Quarq - corrected) = 262.0W, LeMond Power Pilot = 249.5W, HR=160 bpm
• Osymetric    (Quarq - corrected) =271.3W, LeMond Power Pilot = 250.3W, HR=166 bpm

- 54x16 gear, 80 rpm (34.1 km/hr indicated)

• Osymetric  (Quarq - corrected) = 227.5W, LeMond Power Pilot = 212.3W, HR=158 bpm
• Round Ring (Quarq - corrected) =221.6W, LeMond Power Pilot = 212.0W, HR=161 bpm

So, in brief, based on just this limited test, there actually DOES appear to be a "non-round ring inflation factor" of 2.7%-3.5%. (BTW, Dan Connelly estimated a 4% "inflation" based on a 20% within pedal stroke speed variation here (http://djconnel.blogspot.com/...uniform-cadence.html) and by my measurements the Osymetric ring would cause a 20% crank speed variation assuming a constant rear wheel speed.

I followed up the trainer testing rides outside where I ran a PT wheel on the same bike as the Quarq/Osymetric setup with the large ring being the non-round ring and the small ring being round.  The artificial inflation with the non-round ring was observed there as well with the difference between the power as measured by the Quarq as compared to the PT being ~4% higher when in the non-round ring under a continous effort.
 
So, there you have it...my suspicion is that a good majority of the power "improvements" claimed with non-round rings are merely mis-measurements of the actual power.  It's important to note that this sort of power inflation is not anything that can be "calibrated out".  It will be present even if the PM torque calibration is perfect since it's a result of the power calculation algorithm itself and ANY power meter that employs an "event-based" calculation as described above will suffer from it.

Now then...on the question of whether or not non-round rings have the potential to actually increase pedaling power output...well, I have my opinions on that, but I'll leave that for another blog post....