COR Estimation
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Assume the shaker table is a rigid square with center O. Assume the instantaneous velocities v1 and v2 at points r1 and r2 are known, as shown below:

The goal is to recover C, the instantaneous center of rotation (COR) of the table, and w, the angular velocity of the table about C. Without loss of generality, assume r1+r2 = O. It can be shown that

 Recall that the waveform sent to each motor is, up to a scaling factor, of type:

where w0 is the angular frequency (not to be confused with w, above, which denotes the table's instantaneous angular velocity). In practice, the two 2-axis sensors installed at r1 and r2 will be recovering a total of four accelerations.

Because the table is a rigid body, all accelerations will be a scaled version of the above waveform.

Consider a generic representation of the above waveform:

To recover the four coefficients for each accelerometer (a total of 16 coefficients needs to be recovered), we (1) sample each of the four analog streams, and (2) dot each sequence of samples with sin(wt), cos(wt), sin(wt), and cos(2wt). The velocity waveforms are the integral of the recovered acceleration waveforms, so they will be identical in form to the above. In fact, simple acceleration yields:

The quantities v1 and v2 needed to recover the COR will have to be split into their 1st- and 2nd-harmonic components v1st and v2nd; up to a common scaling factor (1/w0), the corresponding magnitudes are given by:

So by taking the four estimates for v1st (along each measured acceleration axis), the COR can be estimated. This should coincide with the same calculation if v2nd is used.

The sign of v1st and v2nd -- in the 1d friction-induced force sense -- is determined by the phase of the 2nd harmonic relative to the first one, namely, that phase must lie in the [0,2*pi] interval. This condition can be expressed algebraically by:

 

© 2000 Dan S. Reznik, <dreznik@cs.berkeley.edu>