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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.

David G. Lewicki is a researcher whose documented work addresses gear dynamics, durability and vibration-based detection of tooth damage. His1986 NASA technical paper, Predicted Effect of Dynamic Load on Pitting Fatigue Life for Low-Contact-Ratio Spur Gears by David G. Lewicki, studied how dynamically varying tooth loads change predicted gear surface pitting fatigue life. Its parametric calculations showed why a static transmitted-load estimate alone can miss important speed-dependent effects.^[ref]

The scientific distinction is between the mean transmitted torque and the changing tooth-contact force. Even at the same mean torque, a mesh can have different peak loads as operating speed approaches a structural natural frequency. The1986 study linked these changing dynamic loads to life predictions for low-gear contact ratio spur gears; it was a model-based investigation, rather than a universal measured life law.

Lewicki later coauthored a2015 study with Nicholas A. Lambert and Robert S. Wagoner comparing rotating accelerometers on a helicopter-transmission pinion with stationary housing sensors. That experimental work connects the dynamic-load problem to the practical question of which gear vibration spectrum features reveal damage. The1986 title page gives his affiliation as the Propulsion Directorate at Lewis Research Center; the2015 report lists Glenn Research Center.^[ref]

Nicholas A. Lambert coauthored research on rotating accelerometers for gear damage detection with David G. Lewicki and Robert S. Wagoner. Their2015 NASA report identifies Lambert and Wagoner with Ridgetop Group and Lewicki with Glenn Research Center. The contribution documented there is a joint experimental evaluation of embedded rotating sensing against conventional stationary housing measurements.^[ref]

The apparatus used a microelectromechanical system acceleration-sensing package on the rotating pinion. Its radial channel and two tangential channels did not measure identical combinations of motion. With the reported sensor orientation, adding the tangential channels isolated rotational acceleration, while subtracting them isolated the shared lateral contribution. This makes the experiment an example of choosing measurement geometry to separate motions, rather than interpreting every channel as the same gear vibration spectrum.^[ref]

An idealized algebraic version writes the channels as $T_1=u+r\alpha$ and $T_2=-u+r\alpha$, where $u$ is one projected translational acceleration, $r$ is sensor radius and $\alpha$ is angular acceleration. Then $(T_1+T_2)/(2r)=\alpha$ while $(T_1-T_2)/2=u$. Sensor placement, polarity and matching measurement calibration are what permit those cancellations. This illustrative decomposition explains the measurement principle; it does not assign sole authorship of the apparatus to one coauthor.

Robert S. Wagoner is a coauthor of experimental research using rotating accelerometers to detect damage in transmission gears. In the2015 report with David G. Lewicki and Nicholas A. Lambert, his affiliation is Ridgetop Group. The experiment compared embedded microelectromechanical system sensors with stationary housing sensors while an intentionally notched pinion was run toward tooth failure.^[ref]

A central signal-processing problem was separating repeatable shaft-related motion from unrelated vibration. The study reconstructed a rotation timing signal from a strong periodic component of the measured data and used it for time-synchronous averaging. Its comparison with a measured timing reference tested whether that reconstruction preserved the rotation-related waveform.^[ref]

The underlying averaging idea can be illustrated by records $a_j(\theta )=g(\theta )+{\eta }_j(\theta )$ aligned at the same shaft angle. Averaging $M$ revolutions preserves the repeatable waveform $g$; independent zero-mean noise with common variance ${\sigma }^2$ has mean-record variance ${\sigma }^2/M$. Timing error or changing operating conditions breaks those assumptions. This distinguishes detecting a repeatable vibration feature from proving that it uniquely identifies a tooth defect. The report found that fault indicators generally responded near actual tooth fracture, limiting the demonstrated early-warning interval.^[ref]

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.