 Centripetal amplitude-error budget

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.