= Gear vibration order
{title2=$q=f/f_{\rm rot}$}
<Gear vibration order> expresses the frequency of a measured vibration component relative to a specified shaft's rotation rate:
$$
q=\frac f{f_r},\qquad f_r=\frac{|n|}{60}.
$$
Here $f$ is frequency in hertz and $n$ is shaft speed in revolutions per minute. The dimensionless order is cycles per shaft revolution. It must name the reference shaft: the same <gear mesh frequency> has different orders relative to the two gears in one mesh. <Order tracking> follows such rotation-related components as speed changes.<ordertrack: Track and extract order magnitudes from vibration signal by MathWorks>[Description and orderlist input: extraction of requested orders using a supplied shaft RPM history]{ref}
For a24-tooth input at1500 revolutions per minute, $f_r=25\,\mathrm{Hz}$ and mesh frequency is600Hz, so mesh order is24. Its60-tooth mate turns at10Hz, giving mesh order60 relative to that shaft. If the input rises to1800 revolutions per minute, mesh frequency becomes720Hz but input-relative mesh order remains24. An unrelated structural <natural frequency> at600Hz would instead move from order24 to order20 as reference speed increases.
<Angular resampling> makes the distinction explicit. Let $\theta(t)=2\pi\int_0^t f_r(u)\,du$ be shaft angle. A tooth-periodic illustrative signal $a(t)=A\cos[N\theta(t)]$ can spread across many fixed-frequency bins during acceleration, while as a function of angle it remains $a(\theta)=A\cos(N\theta)$, with order $N$. Enough angle samples are needed to resolve the desired tooth-related oscillations. This derivation describes the resampling approach; not every <order tracking> algorithm uses that implementation.
Rotation-synchronous <amplitude modulation> can produce neighboring <modulation sidebands>: a24th-order component modulated once per input revolution has orders23 and25. Modulation associated with another shaft need not give integer spacing in the chosen reference. Orders identify kinematic relationships, while amplitudes still depend on loads, mounting and <frequency response>. A high mesh order or a sideband is therefore not itself a diagnosis of damaged teeth.<Evaluation of MEMS-Based Wireless Accelerometer Sensors in Detecting Gear Tooth Faults in Helicopter Transmissions by David G. Lewicki and others>[Printed p.21, rotating measurement direction produces shaft-spaced sidebands; pp.24–25, rotation-synchronous processing from a reconstructed timing signal]{ref}
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