Sunday, November 22, 2009

Metrigear Vector test data: modeling

Enough political commentary! Back to science and engineering.

In the MetriGear Blog they published data from a test rig in which accelerometers were rotated in a circle, and orthogonal components of acceleration were measured. Here's their plot:

Metrigear accelerometer data from their test rig

The blue curve represents data from an accelerometer measuring primarily tangential acceleration (in the direction of motion of the pedal as it rotates). The red curve represents data from an accelerometer measuring primarily radial acceleration (perpendicular to the direction of pedal motion through its circular orbit). The radial direction includes components due to gravity (the accelerometer rotates relative to the direction of gravity) and due to the centrifugal acceleration. The tangential acceleration also includes the gravity component, 90 degrees out of phase with the radial gravitational component, plus it has a component proportional to the rate of change of rotation.

This appears to be what is evident in the plot: a big centrifugal offset to the gravity signal on the radial accelerometer, the red curve, and a smaller offset of the gravity signal on the blue curve, due to the rate of change of rotation. But if you run the numbers, this offset of the blue curve is too large. Around 0.5 g, the initial value of the offset, is a much larger acceleration value than one gets rotating down to a state of rest over 17 seconds for this structure with its initial rotation rate. So another explanation is needed.

That explanation is that the accelerometers are misaligned. Around a 6 degree misalignment explains what's seen in the data. Therefore the blue curve is contaminated with radial acceleration, and the red curve is contaminated with tangential acceleration.

For the rate of change of acceleration, I assume a simple model in which frictional force is fixed, and therefore the rate of change of rotation is constant to the point the rotation ceases. This is obviously simplistic. For example, it neglects the effect of wind resistance, which presents a larger retarding force the greater the rate of rotation. But the simplifying assumption fits the data fairly well, as I'll show.

Parameters I use in my fit include:
  1. radius of rotation = 8 cm
  2. initial rate of rotation = 21.8 radians/second
  3. rate of change of rotation rate = ‒1.2 radians/second²
  4. tangent of misalignment angle: 0.1 (offset angle of 6°)

Results are plotted here, where the Metrigear data I extracted with g3data:

Simple analytic model compared with measured data

The model works quite well. The phase doesn't track perfectly, but given the simplisty of the assumed constant rate of radial deceleration, I'm pleased.

Next time, I'll look at how Metrigear could process their data to eliminate the misalignment.

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