= Rotating-accelerometer speed-ripple term
The <rotating-accelerometer speed-ripple term> is the changing part of centripetal acceleration caused by variation in shaft speed. For a radial sensor at fixed radius $r$, the contribution is $a_r=-r\omega(t)^2$. Writing $\omega=\omega_0+\delta\omega$ gives
$$
a_r=-r\omega_0^2-2r\omega_0\delta\omega-r(\delta\omega)^2.
$$
The term linear in speed variation can be appreciable even when the ripple is small relative to mean speed. Interpreting a <rotating accelerometer> also requires its measurement axes and their changing projection of other acceleration components.<Evaluation of MEMS-Based Wireless Accelerometer Sensors in Detecting Gear Tooth Faults in Helicopter Transmissions by David G. Lewicki and others>[Printed pp.19–21, radial/tangential channel interpretation and rotating-frame spectral effects]{ref}
If $\delta\omega=\varepsilon\cos\Omega t$, the mean is $-r(\omega_0^2+\varepsilon^2/2)$, the ripple at <angular frequency> $\Omega$ has amplitude $2r|\omega_0\varepsilon|$, and the component at $2\Omega$ has amplitude $r\varepsilon^2/2$. With $r=0.020\,\mathrm m$, $\omega_0=100\,\mathrm{rad/s}$ and $\varepsilon=1\,\mathrm{rad/s}$, these are mean $-200.01\,\mathrm{m/s^2}$ and amplitudes4 and0.01m/s². Removing the mean retains both changing terms.
A synchronized shaft-speed estimate permits a model-based correction using the measured square of speed. Its accuracy is limited by <centripetal subtraction error>. Linearizing $-r\omega^2$ gives error magnitude approximately $|\omega_0^2\delta r+2r_0\omega_0\delta\omega|$. At the preceding operating point, a radius error10µm contributes0.10m/s² and a speed error0.1rad/s contributes0.40m/s² in magnitude. The signs and timing of those errors matter when subtracting signals.
This correction removes the modeled centripetal contribution, while retaining the need to account for center motion, <angular acceleration>, <specific force> and sensor orientation. In particular, speed ripple is a physical kinematic input to the channel; it need not indicate a separate structural vibration mode.
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