Centripetal speed-gain error

A centripetal speed-gain error is an error in a calculated centripetal correction caused by a multiplicative scale error in the measured angular velocity. For true centripetal magnitude $C=r{\omega }^2$ and indicated speed $\hat{\omega }=g\omega$, the computed magnitude with correct radius isWriting $g=1+ϵ$ gives exact relative error $2ϵ+{ϵ}^2$. The leading relative error is therefore twice the speed-scale error, rather than equal to it.

For an original example, let $r=0.020\mathrm{m}$ and $\omega =100rad/s$, giving $C=200m/{\mathrm{s}}^2$. A1% high speed indication gives $\hat{C}=204.02m/{\mathrm{s}}^2$, an overestimate4.02m/s². Subtracting that value from a radial reading would leave an error of the opposite sign if the true motion were otherwise modeled correctly. The example illustrates sensitivity; it is not a measured calibration error of a particular sensor.

Radius error combines with speed gain. If the indicated radius is $r+\delta r$, thenTo first order, relative error is $\delta r/r+2ϵ$. Apparent cancellation between these terms at one condition does not separately identify either calibration parameter. An independent radius measurement or speed reference helps distinguish their contributions.

An additive speed offset $b$ behaves differently:Measurements at several speeds can distinguish the quadratic scale contribution from the linear and constant offset contributions, subject to the remaining model and measurement errors. Near zero true speed, a relative-error description becomes unsuitable even though the absolute expression remains meaningful.

The same gain factor applies to the first-order ripple correction $2r{\omega }_0\epsilon$ when both mean speed and ripple amplitude share that speed scale. The resulting mismatch enters the centripetal amplitude-error budget, while relative channel timing enters the centripetal phase-error tolerance. A primary rotating MEMS-sensor experiment identifies rotating-frame components and reconstructs a rotation reference, making the physical need for consistent speed and timing information concrete.^[ref]

At a specified operating point, the sensitivity coefficient to speed is $\partial C/\partial \omega =2r\omega$. A speed standard uncertainty therefore contributes approximately $2r|\omega |u(\omega )$ to the correction's standard uncertainty, with radius uncertainty and any covariance included separately.^[ref]