= Gear vibration spectrum
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>.<Structural Testing Part I: Mechanical Mobility Measurements by Ole Døssing>[Printed pp.5–7, distinction between signal and system analysis; p.14, response spectrum as force spectrum times frequency response]{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>:
$$
a(t)=A[1+\mu\cos(2\pi f_rt)]\cos(2\pi f_mt).
$$
The product-to-sum identity gives a600Hz component of amplitude $A$ and <modulation sidebands> at575 and625Hz, each of amplitude $\mu A/2$. For $A=1\,\mathrm{m/s^2}$ and $\mu=0.20$, each sideband has peak amplitude $0.10\,\mathrm{m/s^2}$. <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.<Evaluation of MEMS-Based Wireless Accelerometer Sensors in Detecting Gear Tooth Faults in Helicopter Transmissions by David G. Lewicki and others>[Printed p.21, equations(1)–(2), rotating-frame sidebands; Figure20, printed p.23]{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.
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