= 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 $\widehat A>0$ and the relative phase is $\phi$, <phasor> subtraction gives residual amplitude
$$
E^2=(\widehat A-A)^2+4A\widehat A\sin^2(\phi/2).
$$
The 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 $\widehat A=A(1+\epsilon)$. For small $\epsilon$ and $\phi$, the leading approximation is
$$
\left(\frac EA\right)^2\approx\epsilon^2+\phi^2.
$$
This provides an approximate joint budget, rather than permission to allow the maximum phase error and maximum gain error simultaneously. For an original example, $A=4\,\mathrm{m/s^2}$ and residual limit $0.10\,\mathrm{m/s^2}$ give relative budget0.025. Assigning relative amplitude error0.015 leaves an approximate phase budget0.020rad. The exact formula, with $\widehat A=4.06\,\mathrm{m/s^2}$, 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 $g$ produces
$$
\frac{\widehat C}{C}=\left(1+\frac{\delta r}r\right)g^2.
$$
With 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 $A=4\,\mathrm{m/s^2}$ 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.<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, rotating-frame components; pp.24–25, rotation timing]{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>.
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