= Centripetal phase error
A <centripetal phase error> is a mismatch in the phase of a time-varying centripetal correction relative to the signal being corrected. It is a frequency-specific form of <centripetal timing error>. A constant centripetal magnitude has no changing phase to misalign, but speed ripple gives an oscillating contribution. For <angular velocity> $\omega(t)=\omega_0+\varepsilon\cos\Omega t$, expansion of $r\omega(t)^2$ to first order in $\varepsilon$ gives an oscillation of amplitude $A=2r\omega_0\varepsilon$ when $r,\omega_0,\varepsilon$ are positive.
Let the true oscillating correction be $A\cos\theta$ and the applied correction $A\cos(\theta-\phi)$. Their difference is
$$
E=A[\cos(\theta-\phi)-\cos\theta]
=2A\sin(\phi/2)\sin(\theta-\phi/2),
$$
so its amplitude is $2A|\sin(\phi/2)|$. Equivalently, <phasor> subtraction gives magnitude $|A(e^{-i\phi}-1)|$. Even perfectly matched amplitudes do not guarantee cancellation when their phases differ.
For an original example, $r=0.020\,\mathrm m$, $\omega_0=100\,\mathrm{rad/s}$ and $\varepsilon=1\,\mathrm{rad/s}$ give $A=4\,\mathrm{m/s^2}$. A phase lag0.10rad leaves amplitude $8\sin(0.05)\approx0.3998\,\mathrm{m/s^2}$. The <centripetal phase-error tolerance> for residual amplitude at most $0.10\,\mathrm{m/s^2}$, choosing the small phase-error range around zero, is
$$
|\phi|\le2\arcsin\!\left(\frac{0.10}{2A}\right)\approx0.02500\ \mathrm{rad}.
$$
For a pure delay $\phi=\Omega\tau$ at $\Omega=20\,\mathrm{rad/s}$, this corresponds to about1.25ms. The same delay gives a larger phase error at a higher frequency.
A primary rotating-sensor experiment compared internal rotating MEMS sensors with stationary transmission-housing accelerometers; its analysis distinguishes rotating-frame spectral components and reconstruction of a rotation reference.<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, rotating-frame spectra; pp.24–25, timing reconstruction]{ref} The calculation above is an illustrative correction model, not a measured error attributed to that apparatus. A whole-period delay cancels one ideal periodic component, but does not generally align transients or other frequencies.
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