Centripetal phase-error tolerance

A centripetal phase-error tolerance bounds the relative phase mismatch allowed when subtracting an oscillating centripetal correction. It translates an acceptable residual acceleration into a phase requirement and, for a known frequency, a timing requirement. With equal true and correction amplitudes $A>0$, a centripetal phase error $\varphi$ leaves residual amplitude $2A|\sin (\varphi /2)|$. For a budget $0\le E_{\max }<2A$, the permitted principal mismatch around zero isFor a pure time lag at angular frequency $\Omega >0$, divide this bound by $\Omega$ to obtain the lag tolerance. An allowable phase at one frequency is not a universal allowable delay for every spectral component.

For an original example, $A=4m/{\mathrm{s}}^2$ and $E_{\max }=0.10m/{\mathrm{s}}^2$ give a bound of about0.02500rad. At $\Omega =20rad/s$ this is approximately1.25ms; at $200rad/s$ it is approximately0.125ms. The tenfold change follows from the frequency dependence of phase delay, without changing the acceleration budget.

A centripetal amplitude-error budget is needed if the applied correction amplitude $\hat{A}$ differs from $A$. Phasor subtraction givesEven at zero phase error, the residual is at least $|\hat{A}-A|$. If that exceeds $E_{\max }$, timing improvement alone cannot meet the requirement. Otherwise, for positive amplitudes and a budget smaller than their sum, the allowed principal phase isWith $A=4$, $\hat{A}=3.96$ and budget0.10 in consistent acceleration units, the phase bound falls to about0.02303rad, or1.15ms at20rad/s.

The tolerance is a property of the correction model and the chosen operating range. A primary rotating MEMS-sensor study analyzes rotating-frame frequency components and reconstruction of a rotation reference; these are practical reasons to retain channel timing and phase information.^[ref] The numerical tolerances here are illustrative design calculations, not measured limits asserted for that apparatus.

A centripetal amplitude-error budget allocates the permitted error in an oscillating centripetal correction between amplitude mismatch and phase mismatch. If the true correction amplitude is $A>0$, the applied amplitude is $\hat{A}>0$ and the relative phase is $\varphi$, phasor subtraction gives residual amplitudeThe first term remains even with perfect timing. The second consumes part of the same acceleration-error budget; their separate limits cannot both be set to the entire allowed residual.

Write $\hat{A}=A(1+ϵ)$. For small $ϵ$ and $\varphi$, the leading approximation isThis provides an approximate joint budget, rather than permission to allow the maximum phase error and maximum gain error simultaneously. For an original example, $A=4m/{\mathrm{s}}^2$ and residual limit $0.10m/{\mathrm{s}}^2$ give relative budget0.025. Assigning relative amplitude error0.015 leaves an approximate phase budget0.020rad. The exact formula, with $\hat{A}=4.06m/{\mathrm{s}}^2$, gives a slightly smaller bound of about0.01985rad.

A centripetal speed-gain error illustrates how the mismatch can arise. For $C=r{\omega }^2$, using radius $r+\delta r$ and a speed reading scaled by $g$ producesWith correct radius and $g=1.01$, the correction is2.01% high. In a small-ripple model whose constant speed and ripple amplitude are both scaled by the same sensor gain, the oscillating correction amplitude has this same gain factor. For $A=4m/{\mathrm{s}}^2$ that alone gives0.0804m/s² amplitude mismatch. The constant centripetal term also needs its own error check; budgeting only the ripple does not control it.

Rotating-frame components and the rotation reference are experimentally consequential: a primary MEMS-sensor study compares rotating transmission measurements with stationary housing accelerometers and reconstructs a timing reference.^[ref] The budgets above are original illustrative calculations. Actual limits require the relevant operating speeds, radii, frequency range and measurement calibration, alongside the centripetal phase-error tolerance.

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]