Rotating accelerometer

A rotating accelerometer is an accelerometer whose sensing axes rotate with a moving component, such as a gear shaft. It can measure motion close to its source, but its output combines the point's actual acceleration with a changing projection direction. Treating its channels as fixed laboratory axes can therefore change the interpretation of a gear vibration spectrum. A NASA transmission experiment mounted a microelectromechanical system sensor package on a pinion and compared it with stationary housing measurements.^[ref]

For a rigid shaft rotating in a plane, let its center have inertial acceleration ${\mathbf{a}}_O$. An attached point at radius $r$ haswhere ${\mathbf{e}}_r,{\mathbf{e}}_t$ rotate with the shaft. The radial contribution is the rotating-accelerometer centripetal term. At constant $\omega =100rad/s$ and $r=0.020\mathrm{m}$, its magnitude is $200m/{\mathrm{s}}^2$ even without vibration of the center. For the ideal specific force convention, the measured vector is ${\mathbf{a}}_P-\mathbf{g}$ projected onto each sensor axis, so gravitational acceleration can also produce a changing projection.

Suppose a laboratory acceleration $A\cos (2\pi f_{m}t)$ projects onto a sensor direction rotating at $f_r$. The product contains components at $f_m-f_r$ and $f_m+f_r$:This measurement geometry can produce modulation sidebands without a damaged tooth. The NASA radial channel showed1854 and2060Hz around a1957Hz mesh frequency with103Hz rotation. Paired tangential channels also allowed different motion contributions to be separated by sums and differences.^[ref] Correct interpretation therefore specifies mounting radius, orientation, measurement calibration and rotation timing, as well as sensor bandwidth.

The rotating-accelerometer centripetal term is the inward acceleration contribution $-r{\omega }^2$ at a sensor rigidly attached a distance $r$ from a rotation axis. With outward radial unit vector ${\mathbf{e}}_r$ and tangential vector ${\mathbf{e}}_t$, differentiating the rotating position givesIt is a real kinematic acceleration, rather than an electronic zero error. A rotating accelerometer measures the relevant projection of specific force, so center motion and gravitational acceleration must also be included when interpreting an actual channel. A primary rotating-sensor experiment explicitly separates radial and tangential measurement contributions.^[ref]

At $r=0.020\mathrm{m}$ and $\omega =100rad/s$, the radial contribution is $-200m/{\mathrm{s}}^2$. It is steady in the rotating radial coordinate even though its direction continuously turns in a laboratory frame. Reversing rotation preserves this term because $\omega$ is squared; the tangential term instead follows angular acceleration.

A rotating-accelerometer speed-ripple term remains when speed is not constant. If $\omega (t)={\omega }_0+\epsilon \cos \Omega t$, thenFor the preceding radius, ${\omega }_0=100rad/s$ and ripple amplitude $\epsilon =1rad/s$ give mean $-200.01m/{\mathrm{s}}^2$, fundamental ripple amplitude4m/s² and second-ripple amplitude0.01m/s². Subtracting a record's mean removes only the steady part. A spectral component can therefore arise from speed variation in this kinematic term, which should be checked against the measured rotation history before attributing it to another vibration source.

The rotating-accelerometer speed-ripple term is the changing part of centripetal acceleration caused by variation in shaft speed. For a radial sensor at fixed radius $r$, the contribution is $a_r=-r\omega (t{)}^2$. Writing $\omega ={\omega }_0+\delta \omega$ givesThe term linear in speed variation can be appreciable even when the ripple is small relative to mean speed. Interpreting a rotating accelerometer also requires its measurement axes and their changing projection of other acceleration components.^[ref]

If $\delta \omega =\epsilon \cos \Omega t$, the mean is $-r({\omega }_0^2+{\epsilon }^2/2)$, the ripple at angular frequency $\Omega$ has amplitude $2r|{\omega }_0\epsilon |$, and the component at $2\Omega$ has amplitude $r{\epsilon }^2/2$. With $r=0.020\mathrm{m}$, ${\omega }_0=100rad/s$ and $\epsilon =1rad/s$, these are mean $-200.01m/{\mathrm{s}}^2$ and amplitudes4 and0.01m/s². Removing the mean retains both changing terms.

A synchronized shaft-speed estimate permits a model-based correction using the measured square of speed. Its accuracy is limited by centripetal subtraction error. Linearizing $-r{\omega }^2$ gives error magnitude approximately $|{\omega }_0^2\delta r+2r_0{\omega }_0\delta \omega |$. At the preceding operating point, a radius error10µm contributes0.10m/s² and a speed error0.1rad/s contributes0.40m/s² in magnitude. The signs and timing of those errors matter when subtracting signals.

This correction removes the modeled centripetal contribution, while retaining the need to account for center motion, angular acceleration, specific force and sensor orientation. In particular, speed ripple is a physical kinematic input to the channel; it need not indicate a separate structural vibration mode.

Centripetal subtraction error is the residual error left when a modeled centripetal contribution is removed from a rotating accelerometer signal. For the positive magnitude $C=r{\omega }^2$, small errors in radius and angular speed giveThe residual's sign depends on whether the correction is overestimated or underestimated. A correct model also needs synchronized speed information and the appropriate sensor-axis projection; a primary rotating-sensor experiment documents the importance of radial/tangential directions and rotating-frame effects.^[ref]

Using first-order propagation of uncertainty, the standard uncertainty satisfiesThe covariance term must be included when the estimates are correlated.^[ref] For $r=0.020\mathrm{m}$, $\omega =100rad/s$, $u_r=10\mu \mathrm{m}$ and $u_{\omega }=0.1rad/s$, the two uncorrelated contributions have magnitudes0.10 and0.40m/s², giving $u_C\approx 0.4123m/{\mathrm{s}}^2$.

A centripetal timing error can remain even with accurate individual channels. If the speed used for correction lags by $\Delta t$, then $\Delta \omega \approx -\dot{\omega }\Delta t$. At the same radius and speed, acceleration rate $\dot{\omega }=10rad/{\mathrm{s}}^2$ and lag5ms produce correction error approximately $-0.20m/{\mathrm{s}}^2$. Channel timing therefore belongs in the measurement model. These linearized formulas describe small errors; they do not replace accounting for other motion or specific force contributions.

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.

A centripetal phase error is a mismatch in the phase of a time-varying centripetal correction relative to the signal being corrected. It is a frequency-specific form of centripetal timing error. A constant centripetal magnitude has no changing phase to misalign, but speed ripple gives an oscillating contribution. For angular velocity $\omega (t)={\omega }_0+\epsilon \cos \Omega t$, expansion of $r\omega (t{)}^2$ to first order in $\epsilon$ gives an oscillation of amplitude $A=2r{\omega }_0\epsilon$ when $r,{\omega }_0,\epsilon$ are positive.

Let the true oscillating correction be $A\cos \theta$ and the applied correction $A\cos (\theta -\varphi )$. Their difference isso its amplitude is $2A|\sin (\varphi /2)|$. Equivalently, phasor subtraction gives magnitude $|A(e^{-i\varphi }-1)|$. Even perfectly matched amplitudes do not guarantee cancellation when their phases differ.

For an original example, $r=0.020\mathrm{m}$, ${\omega }_0=100rad/s$ and $\epsilon =1rad/s$ give $A=4m/{\mathrm{s}}^2$. A phase lag0.10rad leaves amplitude $8\sin (0.05)\approx 0.3998m/{\mathrm{s}}^2$. The centripetal phase-error tolerance for residual amplitude at most $0.10m/{\mathrm{s}}^2$, choosing the small phase-error range around zero, isFor a pure delay $\varphi =\Omega \tau$ at $\Omega =20rad/s$, this corresponds to about1.25ms. The same delay gives a larger phase error at a higher frequency.

A primary rotating-sensor experiment compared internal rotating MEMS sensors with stationary transmission-housing accelerometers; its analysis distinguishes rotating-frame spectral components and reconstruction of a rotation reference.^[ref] The calculation above is an illustrative correction model, not a measured error attributed to that apparatus. A whole-period delay cancels one ideal periodic component, but does not generally align transients or other frequencies.

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]