Gear mesh frequency ($f_m=N{f}_{\mathrm{rot}}$)

Gear mesh frequency is the rate at which successive teeth pass through a mesh. For a fixed-axis gear with $N$ teeth rotating at $f_{\mathrm{rot}}$ revolutions per second,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.^[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.^[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.

A gear vibration spectrum describes how a measured gearbox motion signal is distributed over frequency. An accelerometer on a housing supplies a response spectrum; a rotary encoder supplies a rotation reference. Gear mesh frequency predicts tooth-passage components, but their observed amplitudes also depend on applied loads, sensor direction and the mechanical frequency response. A response peak alone therefore does not identify a structural natural frequency.^[ref]

For an original example, a24-tooth gear at1500 revolutions per minute rotates at $f_r=25\mathrm{Hz}$ and has mesh frequency $f_m=600\mathrm{Hz}$. Consider an illustrative acceleration signal with amplitude modulation:The product-to-sum identity gives a600Hz component of amplitude $A$ and modulation sidebands at575 and625Hz, each of amplitude $\mu A/2$. For $A=1m/{\mathrm{s}}^2$ and $\mu =0.20$, each sideband has peak amplitude $0.10m/{\mathrm{s}}^2$. Gear vibration order expresses these frequencies relative to the chosen shaft: the central order is24 and the neighboring orders23 and25. These positions follow from the constructed signal, without requiring a damaged tooth.

A primary apparatus example makes that ambiguity concrete. NASA's2015 transmission experiment compared rotating accelerometers on a pinion shaft with stationary housing sensors. The rotating radial channel showed sidebands at1854 and2060Hz around a1957Hz mesh frequency, spaced by the103Hz shaft rate. The report derives this shift from the rotating measurement direction and discusses sums and differences of tangential channels. A measurement-frame effect can therefore create sidebands as well as a mechanical defect.^[ref]

A useful test records speed, load, mounting and sensor orientation with each spectrum. If speed changes appreciably within one record, fixed-frequency lines can spread; comparisons should then account for the changing rotation rate. Record duration also controls Fourier frequency resolution. Sideband changes are evidence to interpret alongside a baseline and the test conditions, rather than a unique diagnosis from one spectral peak.

Gear vibration order expresses the frequency of a measured vibration component relative to a specified shaft's rotation rate: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.^[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.^[ref]