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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.

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 $\omega ={\omega }_0+\delta \omega$ givesThe 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.^[ref]

If $\delta \omega =\epsilon \cos \Omega t$, the mean is $-r({\omega }_0^2+{\epsilon }^2/2)$, the ripple at angular frequency $\Omega$ has amplitude $2r|{\omega }_0\epsilon |$, and the component at $2\Omega$ has amplitude $r{\epsilon }^2/2$. With $r=0.020\mathrm{m}$, ${\omega }_0=100rad/s$ and $\epsilon =1rad/s$, these are mean $-200.01m/{\mathrm{s}}^2$ 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.