On Bicycles, and.... what else is there?

Consider half the rider + bike to be a point mass, suspended from the road by a spring, the bike tire. This is a classic spring-mass system . Spring-mass systems naturally resonate at an angular velocity ω₀ = sqrt[κ/M], where κ is the elastic constant of the spring (the ratio of force to displacement), and M is the total mass of the load (the bike + rider in this case). The frequency response z as a function of angular velocity ω is: z(ω) = 1 / [1 - (ω / ω₀)²] To go from angular velocity (radians per second) to frequency (oscillations per second, or Hz) divide by 2π. So well below the resonance, the frequency response is one: when riding over gradual rollers, the tire deflection barely changes. On the other hand, well above resonance, the transmission decreases proportional to the square of the frequency. In actuality, no spring is perfect: there is some energy loss with each oscillation. When this effect is included, the system becomes a "damped" spring-mass system. Whe...

I was curious how the Metrigear Vector accelerometer-based cadence extraction. might be affected by noise. Noise is basically any component of a data which is unpredictable and uninteresting. But to consider this, I'd need to consider the characteristics of the noise as sensed by the accelerometers. This would be simple to determine: go out on a ride, unclip, and coast, then download the data. Simple, sure, except I don't have a Metrigear Vector, or for that matter any other accelerometers. It's more fun to estimate stuff theoretically than measure it, anyway. Reality tends to get ugly. To model noise, I break noise down into sinusoidal components. For every frequency, the noise can then be described by two coefficients: an amplitude and a phase. The phase is easy: it is randomly chosen from the set of angles from 0 to 2π (or 0 to 360° if you're afraid of irrational numbers). That leaves amplitude. Everyone's favorite probability distribution is the normal ...