= Centripetal timing error
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 $\widehat C(t)=r\omega(t-\tau)^2$, with positive lag $\tau$. The correction error is
$$
E(t)=\widehat C(t)-C(t)\approx-2r\omega(t)\dot\omega(t)\tau
$$
for 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.<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 signals and rotating-frame frequency components; pp.24–25, rotation-timing reconstruction]{ref}
With $r=0.020\,\mathrm m$, $\omega=100\,\mathrm{rad/s}$, $\dot\omega=10\,\mathrm{rad/s^2}$ and lag5ms, the first-order error is $-0.20\,\mathrm{m/s^2}$. Both channels can be individually accurate while their combination is mistimed.
For periodic speed ripple $\omega=\omega_0+\varepsilon\cos\Omega t$, keep terms first order in $\varepsilon$. The correction-error amplitude is
$$
E_{\rm amp}=4r|\omega_0\varepsilon\sin(\Omega\tau/2)|.
$$
This is a <centripetal phase error> with phase offset $\Omega\tau$. For $r=0.020\,\mathrm m$, $\omega_0=100\,\mathrm{rad/s}$, $\varepsilon=1\,\mathrm{rad/s}$, $\Omega=20\,\mathrm{rad/s}$ 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.
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