## Tuesday, October 6, 2009

### Metrigear Vector Pt 5: Interpretation of Torques

Now for torques....

There's three torque modes for each pedal, or six for the pair, which I've described by using sums and differences of the corresponding torques for each pedal. I'll focus here on the torques for each pedal.

First, there's Tx. This is the torque driving the pedal spindle to screw into the crank arm. I say screw into because the handedness of my coordinate system is the opposite of the handedness of the threads: positive torque is counterclock for the right pedal and counter-clockwise for the left pedal, looking from the outward side of the pedal. I hope I got that right ... Since the pedal is spinning relative to the spindle, and this induces a force, this torque should generally be somewhat positive. How positive depends on how effective the pedal bearings are.

Work can be calculated as the integral of torque with respect to angle. So <Txl + Txr> × cadence, where <> denotes an average with respect to rotation angle (not time), is proportional to the power being lost to bearing friction, at least assuming pedal deflection is small (if the pedal deflection were large, it becomes too complicated for my simple level of analysis). In other words, it assume the pedal and crank are mostly rigid.

A further complication is that the average of the torque with respect to the angle can't be determined. The relevant angle is the angle of the pedal body with respect to the spindle (and therefore with respect to the crank arm). We can determine the angle of the spindle (and crankarm) from the accelerometers, but we then need to assume an orientation for the pedal body relative to space. For example, assume the pedal body is always at a fixed angle, for example always horizontal. If the pedal body is at an angle which changes throughout the pedal motion, as it always will be, then its bearings will sometimes be spinning faster, sometimes slower, than the crank arm. This will introduce an error.

Moving onward: there's two other torque components for each pedal: Ty and Tz. With the right pedal at 3 o'clock and with the pedal body horizontal, Tz is similar to Tx in that it describes rotation about an axis which is nominally free. As the foot rotates about the pedal body, on an X-series Speedplay there should be only a small restoring force (the "pedaling on ice" feeling people love or hate about these pedals). On a Zero, there should be a restoring force dependent on the tension setting.

However, it gets more complicated. One is the Vector measures a torque, but not the angular motion, and you need both to calculate a power. It has no idea what the shoe is doing relative to the pedal body.

But another issue is the assumption that the pedal body is horizonal. Of course, it could be at any angle. Again, the vector has no idea. So I'm not sure how useful Tz turns out to be.

Again with the horizontal pedal body at 3 o'clock, Ty would be a measure of how uniform the propulsive force is between the inside and the outside of the pedal body, or at least how uniformly this applied force is transmitted to the inside and outside bearings at the spindle. Interesting? I don't know. But things become more complicated as the shoe rotates and as the pedal body rotates. Whoa -- all too much for my small brain!

So that's the torques. About the only interesting torque mode to me is Txl + Txr, which may be able to estimate power loss in the pedal bearings. That would be really cool.