Virtual Elevation: Does Rotational Inertia Play a Role?

The method of Virtual Elevation and all its variants are derived from Newton’s 2nd Law for linear motion, neglecting rotational inertia terms.   Do those additional terms play a significant role in elevation-matching? Can we accurately measure the moment of inertia (MOI)? In this post, I’ll describe the terms and provide a method for testing MOI.

[edit: Thanks to Nicko on the SlowTwitch forum for pointing out that the added mass terms in the equations are I/R^2 instead of I/R]

Slope: The Critical Calculation

The method of Virtual Elevation is a methodology for determining the Crr and CdA of a bike-rider combination by matching a derived elevation profile against some features of a known elevation profile. Under ideal conditions involving little wind and steady rider position the method works well. But the subtlety of the calculation lies in determining the slope angle, theta, from the linear equations of motion:
F - Crr m g cos(theta) - 1/2 CdA rho v^2 - m g sin(theta) = m a

Slope angle can be computed in closed form by making a small-angle approximation:
sin(theta) = tan(theta) = theta
cos(theta) = 1

The resulting calculation for slope angle is:
theta = ( F - Crr m g - 1/2 CdA rho v^2 - m a ) / ( m g ) .

An iterative method for slope angle calculation can also be used, however it is questionable whether this is necessary for most elevation profiles.

Newton’s Second Law With Rotational Inertia

The above equation of motion neglects the rotational inertia of the front and rear wheels. Whether this is an issue is for others to decide. My aim here is to add a correction term to the existing slope calculation to incorporate rotational inertia terms. Newton’s Second Law with rotational inertia is:
F - Crr m g cos(theta) - 1/2 CdA rho v^2 - m g sin(theta) = ( m + I1/R^2 + I2/R^2 ) a

I1 and I2 are the moments of inertia (MOI) of the front and rear wheels. This modifies the small-angle slope angle formula as follows:
theta = ( F - Crr m g - 1/2 CdA rho v^2 - ( m + I1/R^2 + I2/R^2 ) a ) / ( m g ),
where I1 and I2 are the MOI of the front and rear wheels, and R is the rolling radius of the wheels.

The big question is whether I1 and I2 are large enough to significantly impact the slope calculation. On one hand, we’d like to make use of as much information as possible to stabilize the slope calculation. On the otherhand, we want to make sure not to encumber the process of applying Virtual Elevation with unnecessary steps. The test protocol is hard enough as it is, after all.

Measuring MOI: The Drop Test

A long time ago, in a galaxy far far away I had a race engineering site for competitive karting called kartuning.com.  Don’t bother looking for it — I closed the site quite some time ago due to work pressures. The only remnant of it is this article on the wayback machine that I wrote about the non-insignificant effect of rotational inertia on the dynamics of the kart (WARNING: slow to load).

I introduced the Drop Test as a way to measure the MOI of the rear axle and wheel combination of a kart. The same method could be used on a bicycle wheel. In short, the method goes like this: wrap a string with a weight at one end around the wheel axle. Let it drop and bounce back. Measure the bounce height. The Drop Test code I wrote would compute the MOI based on the original height, the bounce height, and the weight used.

Some Questions For Experienced Virtual Elevation Folk

So, can the rotational inertia terms be used to improve VE calculations? Is there any MOI data for wheelsets out there? Is it worth dusting off the original Drop Test code to compute MOI? Just let me know.