As I've emphasized in previous blog posts, Vector's motto of "Direct your Forces" makes it clear the ephasis is on force, not just power. Power is a single scalar quantity for each pedal, but force contains many more components. Indeed, since the pedal rotation varies around the pedal stroke, and since the orientation of the pedal inspindle into which the Vector is installed cannot predicted, the Vector must measure multiple force components and extract the one which contributes to propulsion. Providing users with not just this propulsive force, but non-propulsive forces as well, may be beneficial to pedal stroke optimization or bike fit, as I've discussed.
Key to all of this is being able to measure these forces and torques. Recall each pedal has three axes, and force components can be applied to the pedal along each of these axes, and additionall torque components can be applied which attempt to rotate the pedal about each of these three axes. So is measuring each of these components possible?
The Metrgear is a module inserted into the spindle of the pedal, with piezoelectric sensors which measure the bending of the spindle. The bending moment on the spindle, relative to a given point on the spindle, is an applied torque: a force times a distance. Before continuing, consider the following schematic of a pedal:
Force diagram of pedal
Starting from the left, the threads of the spindle are considered to be screwed into a crank arm, which is considered rigid. Real cranks are not rigid, nor are the bicycles to which they attached. However, for the purpose of measuring forces on the pedal, that should not be important: as long as the pedal is prevented from accelerating, for a given set of applied forces, the bending of the spindle will be the same whether the crank or rest of the bicycle is rigid or compliant.
Then I assume two idealized torque sensors. These measure the bending moment on the pedal from their respective positions, which are within the spindle distances x1 and x2 from the crank arm. The force moments they measure are labeled τ1 and τ2.
Next are the applied forces, which are assumed on the pedal body. Of course forces may be applied all along the pedal body, since the cleat may be in contact with the entire pedal body surface. However, it simplifies things if it is assumed force is applied at only two discrete points. These points are assumed to be at distances x3 and x4 from the crank arm. The magnitude of these forces are labeled F1 and F2.
There are also forces applied to the threads. Consider the case where the spindle is not accelerating. Then if there were no force on the threads, by Newton's second law, the spindle would accelerate or undergo a state change like cracking or heating. So the crank must induce forces opposite to the forces applied to the pedal body, with a torque which cancels that applied to the pedal body. If the pedal were not attached to the bike, for example, but only to the cleat, pushing on the pedal will just accelerate the pedal.
Of course, the spindle is accelerating: it is rotating, which implies continuous acceleration, and the bike may be changing speed or orientation or direction, all form of acceleration which apply as well to the spindle. So these forces do not really balance. Since the spindle has non-zero mass, it takes net force to accelerate the spindle. Were the pedal and spindle zero mass, things would be simpler. Then the forces and torques would need to balance, even with acceleration.
Further with the static loading assumption is that the bending of the spindle is changing only slowly. For example, whack the pedal without a rider touching it and the pedal will oscillate back and forth for awhile, like a pendulum attached to springs. Obviously in this case there is no rider power -- the rider isn't even touching the pedals. Yet the spindle is bending. Typically this sort of thing can be neglected if the analysis is averaged over times sufficient that the pedal would oscillate frequently over the averaging time. That assumption seems valid in this case.
So many assumptions. Typically in the analysis of bending cantilevers (see this example, after Bernouilli) the forces at the constrained end (in this case, the threads) are not considered in detail. Rather they are treated as an idealized constraint. In his case, that the position and orieniation of the spindle are fixed. For example, in the diagram here, that the coordinates and slope of the spindle are both held at zero at the threads.
Well, that's already too much for one blog post. To be continued.