= Gear mesh frequency
{title2=$f_m=Nf_{\rm rot}$}
<Gear mesh frequency> is the rate at which successive teeth pass through a mesh. For a fixed-axis gear with $N$ teeth rotating at $f_{\rm rot}$ revolutions per second,
$$
f_m=N|f_{\rm rot}|=\frac{N|n|}{60}=\frac{N|\omega|}{2\pi},
$$
where $n$ is speed in revolutions per minute and $\omega$ is <angular velocity> in radians per second. The formula follows by counting $N$ tooth passages per complete turn. Both gears of one steady compatible mesh give the same result because their tooth-passage rates agree.<KHK Gear Nomenclature>[Basic Guide B §1.1, pp.4–5, tooth-count/speed relation of meshing gears]{ref}
For an original example, a24-tooth input rotating at1500 revolutions per minute gives $f_m=600\,\mathrm{Hz}$. Its60-tooth mate turns at600 revolutions per minute, also giving $60(600)/60=600\,\mathrm{Hz}$. The shaft rotation frequencies differ even though the mesh frequency agrees.
A <compound gear train> can have different mesh frequencies at successive stages. If that60-tooth gear is rigidly joined to a20-tooth gear driving an80-tooth output, the second mesh frequency is $20(600)/60=200\,\mathrm{Hz}$. The final output turns at150 revolutions per minute. By contrast, every mesh of an ideal fixed-axis <simple gear train> shares the same tooth-passage frequency: its intermediate gear engages both neighbors with the same tooth count and rotation rate.
A <gear vibration spectrum> may contain components at the mesh frequency and its integer multiples, together with other mechanical excitations. Measuring rotation with a <rotary encoder> lets a test compare observed frequency components with tooth-passage predictions. A spectral peak near $f_m$ does not by itself establish a structural <natural frequency> or identify a particular defect; the forcing and the system's <frequency response> both shape measured vibration.<Structural Testing Part I: Mechanical Mobility Measurements by Ole Døssing>[“Trouble-shooting”, printed pp.6–7: response peaks can arise from excitation rather than structural resonance]{ref} For trains with moving gear axes, the simple fixed-axis formula requires the appropriate relative rotational rate rather than an unqualified absolute shaft speed.
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