= Rotating-accelerometer centripetal term
{title2=$-r\omega^2$}
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 gives
$$
\mathbf v=r\omega\mathbf e_t,\qquad
\mathbf a=r\dot\omega\mathbf e_t-r\omega^2\mathbf e_r.
$$
It 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.<Evaluation of MEMS-Based Wireless Accelerometer Sensors in Detecting Gear Tooth Faults in Helicopter Transmissions by David G. Lewicki and others>[Figure7, printed p.7; “MEMS Results”, printed p.19, sensor directions and channel combinations]{ref}
At $r=0.020\,\mathrm m$ and $\omega=100\,\mathrm{rad/s}$, the radial contribution is $-200\,\mathrm{m/s^2}$. 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 $\omega(t)=\omega_0+\varepsilon\cos\Omega t$, then
$$
-r\omega(t)^2=-r\omega_0^2-2r\omega_0\varepsilon\cos\Omega t
-\frac{r\varepsilon^2}{2}[1+\cos(2\Omega t)].
$$
For the preceding radius, $\omega_0=100\,\mathrm{rad/s}$ and ripple amplitude $\varepsilon=1\,\mathrm{rad/s}$ give mean $-200.01\,\mathrm{m/s^2}$, 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.
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