 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 $\hat{C}(t)=r\omega (t-\tau {)}^2$, 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.