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

Last post, I discussed how road vibration transmits a spectrum of acceleration to the frame, which excites oscillation modes in the relatively complex bike-rider system. MIT measurements showed oscillations near 30 Hz (of lesser magnitude, less in carbon fiber than in steel and Al) and near 50 Hz (of greater magnitude, more in Al than in carbon fiber or steel). I generated simulated noise spectra with, successively, a direct ground noise component at 0 Hz, then a mode at 30 Hz, then an additional mode at 50 Hz. The magnitudes of the latter two modes were based on the MIT data. Here's the curves. As you can see, the 30 Hz and 50 Hz modes have a large effect on the acceleration: simulated vibration data in frame, adding modes Now suppose I was trying to extract cadence from such accelerometer data. All of this would be considered "noise": not part of a ±1 g oscillation as the gravity vector spins in a circle relative to spindle coordinates. Essentially I want the noi...

Summary so far: a rigid body balanced on two bike tires has two primary oscillation mode: an even mode and an odd mode. The even mode is associated with the bike bouncing up and down on the two tires symmetrically. The odd mode is associated with the bike rotating on the tires: one tire compressed, the other expanded, then vice-versa. An estimation of my moment of inertia is that the root-mean-squared separation of mass from my center-of-gravity is around 39 cm. The half-wheelbase of my bike is 49.5 cm. Since the former is lower than the latter, the odd mode will have a higher oscillation frequency than the even mode, assuming the bike and body are rigid and the body is firmly attached to the bike. The issue is that the body is not rigid and it is not firmly attached to the bike. A spring-mass model for the human body has been developed which shows how vibrations are attenuated between body parts (no snickering from the crowd: that wasn't modeled). The result of this is, th...

I was curious about my moment of inertia, as this is key to comparing the even and odd modes of vibration of the bike on the tires. So I found an old photo I took of myself on the trainer, traced out my body, and created an x-bitmap image, a format which is relatively easy to manipulate in C. I used my forearm length as a crude ruler. Okay, I'm missing a bit of the foot, but don't sweat the details. I calculated the "center of mass" in the image (I put a red dot there), then the mean of the square of the distance of pixels from the center. Okay, so this isn't quite right, as the "mass per pixel" isn't the same everywhere. But good enough for blogger work. The result was the square root of the mean square distance of pixels from the center-of-mass was 39.2 cm. So there it is: 39.2 cm. Compare that with the half-wheelbase of my bike, which is 49.5 cm. This suggests that, neglecting the bicycle, the odd mode should be 60% higher in frequency than...

Last time, I analyzed a single bike tire, then assumed that tire suspended half the bicycle mass. This is equivalent to assuming the bike goes over a bump which hits each tire simultaneously, raising and lowering the bike as a unit. The result was an oscillation frequency around 10 Hz (depending on mass and tire pressure). Well, it turns out this sort of thing has been measured before. Champoux used a bike with accelerometers attached , either riding on a treadmill with a bump attached, or riding outdoors and letting the cracks on the road provide the bumps. He did further experiments with shakers attached either to the front hub, or to the hub and handlebars. Champoux on his treadmill (from his paper ) When this sort of thing is done, unless damping is extreme, you tend to see oscillations occur at near the normal oscillatory modes of the system. For example, one mode is the one I analyzed: the bike bounces up and down on its two tires together. But Champoux identified a number...

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 ...

Well, maybe not as much as one might think. Anquetil broke the hour record in 1956 with 46.159 km covered. He's arguably the best all-time in road time trials, having won Grand Prix des Nations a ridiculous 9 times over 14 years from 1953 to 1966. Taylor Phinney is reigning world pursuit champion, and more to come...