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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 $\omega (t)={\omega }_0+\epsilon \cos \Omega t$, expansion of $r\omega (t{)}^2$ to first order in $\epsilon$ gives an oscillation of amplitude $A=2r{\omega }_0\epsilon$ when $r,{\omega }_0,\epsilon$ are positive.

Let the true oscillating correction be $A\cos \theta$ and the applied correction $A\cos (\theta -\varphi )$. Their difference isso its amplitude is $2A|\sin (\varphi /2)|$. Equivalently, phasor subtraction gives magnitude $|A(e^{-i\varphi }-1)|$. Even perfectly matched amplitudes do not guarantee cancellation when their phases differ.

For an original example, $r=0.020\mathrm{m}$, ${\omega }_0=100rad/s$ and $\epsilon =1rad/s$ give $A=4m/{\mathrm{s}}^2$. A phase lag0.10rad leaves amplitude $8\sin (0.05)\approx 0.3998m/{\mathrm{s}}^2$. The centripetal phase-error tolerance for residual amplitude at most $0.10m/{\mathrm{s}}^2$, choosing the small phase-error range around zero, isFor a pure delay $\varphi =\Omega \tau$ at $\Omega =20rad/s$, 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.^[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.

Centripetal subtraction error is the residual error left when a modeled centripetal contribution is removed from a rotating accelerometer signal. For the positive magnitude $C=r{\omega }^2$, small errors in radius and angular speed giveThe residual's sign depends on whether the correction is overestimated or underestimated. A correct model also needs synchronized speed information and the appropriate sensor-axis projection; a primary rotating-sensor experiment documents the importance of radial/tangential directions and rotating-frame effects.^[ref]

Using first-order propagation of uncertainty, the standard uncertainty satisfiesThe covariance term must be included when the estimates are correlated.^[ref] For $r=0.020\mathrm{m}$, $\omega =100rad/s$, $u_r=10\mu \mathrm{m}$ and $u_{\omega }=0.1rad/s$, the two uncorrelated contributions have magnitudes0.10 and0.40m/s², giving $u_C\approx 0.4123m/{\mathrm{s}}^2$.

A centripetal timing error can remain even with accurate individual channels. If the speed used for correction lags by $\Delta t$, then $\Delta \omega \approx -\dot{\omega }\Delta t$. At the same radius and speed, acceleration rate $\dot{\omega }=10rad/{\mathrm{s}}^2$ and lag5ms produce correction error approximately $-0.20m/{\mathrm{s}}^2$. Channel timing therefore belongs in the measurement model. These linearized formulas describe small errors; they do not replace accounting for other motion or specific force contributions.

A centripetal timing error occurs when the speed used to correct a rotating accelerometer sample refers to another time. For centripetal magnitude $C(t)=r\omega (t{)}^2$, let the correction use $\hat{C}(t)=r\omega (t-\tau {)}^2$, with positive lag $\tau$. The correction error isfor small lag. It is one contribution to centripetal subtraction error. A primary rotating-sensor experiment shows why channel direction, rotation reference and phase matter when interpreting motion spectra.^[ref]

With $r=0.020\mathrm{m}$, $\omega =100rad/s$, $\dot{\omega }=10rad/{\mathrm{s}}^2$ and lag5ms, the first-order error is $-0.20m/{\mathrm{s}}^2$. Both channels can be individually accurate while their combination is mistimed.

For periodic speed ripple $\omega ={\omega }_0+\epsilon \cos \Omega t$, keep terms first order in $\epsilon$. The correction-error amplitude isThis is a centripetal phase error with phase offset $\Omega \tau$. For $r=0.020\mathrm{m}$, ${\omega }_0=100rad/s$, $\epsilon =1rad/s$, $\Omega =20rad/s$ and $\tau =0.005\mathrm{s}$, the amplitude is about0.3998m/s². Its small-lag approximation is0.40m/s².

The relevant timing is relative between the channels and any filtering stages. A common shift of both timestamps leaves that relative delay unchanged. Rate estimates formed from a time window also need their effective time stated, rather than assuming the newest recorded timestamp represents an instantaneous speed at the accelerometer sample.