In my work, I am using a vibration table to test the resonant properties and survivability of a structure under different shaking regimes. The table is driven by an eccentrically weighted rotary motor which excites vibrations in the Y and Z axes (green and blue respectively in the image below).
Using 3-axis accelerometers on the table and at various points on the structure, I can record the excitation and response vibrations in each axis (example plot at 20Hz below).
However, two sinusoids of differing amplitudes and phases define an elliptical polarization (the same data as above is plotted parametrically below).
Based on the answers to other Stack Exchange questions (here and here), I know that the parametric form of an ellipse, centered at the origin and rotated by an angle $\theta$ is $$ a_y(t) = A\cos\big(2\pi ft\big)\cos\big(\theta\big) - B\sin\big(2\pi ft\big)\sin\big(\theta\big) $$ $$ a_z(t) = A\cos\big(2\pi ft\big)\sin\big(\theta\big) + B\sin\big(2\pi ft\big)\cos\big(\theta\big) $$ where $A$ and $B$ are the major and minor radii respectively.
My question is, how can I convert my axes-aligned sinusoids $$ a_y(t) = Y\sin\big(2\pi ft\big) $$ $$ a_z(t) = Z\sin\big(2\pi ft + \phi\big) $$ into the above form, such that I can immediately read off the rotation angle and radii?
My goal is to know the direction and magnitude of the major radius of the acceleration ellipse, so that I can better align my structure in the direction of strongest vibration.
What I've Tried So Far
Based on the process discussed in this article, I’ve attempted to simply set the two forms equal to one another and solve for $A$, $B$, and $\theta$ as functions of $Y$, $Z$, and $\phi$. However, that process has yielded inconsistent results, and none of them seem to agree with the ellipse as I’ve plotted it.
I also have a gut feeling that the major and minor radii correspond to the eigenvalues/eigenvectors of some transformation matrix, but I have no idea how to go about constructing said matrix.
Numerically, the sinusoids I've plotted above have parameters $ Y = 0.7847g $, $ Z = 1.0498g $, and $ \phi = 0.9537 $. The major and minor radii are approximately $R = 1.1805g$ and $r = 0.5696g$ respectively, and the rotation angle is approximately $\theta = 1.0330$.