Detecting Mechanical Faults From Engine Sound

A rotating machine is a periodic signal generator, and a defect is a new periodicity added to it. That is what makes acoustic fault detection tractable in a way that general sound classification is not: you can compute where in the spectrum to look before you record anything.

Machinery sings at multiples of its shaft speed

Start with the shaft rate. A machine turning at 1,800 rpm has a shaft rate of 30 Hz. Nearly everything a rotating assembly emits sits at a rational multiple of that number, and the convention is to talk in orders rather than hertz: 1× is the shaft rate, 2× is twice it, and 0.5× is half.

The mapping from order to mechanism is specific. Unbalance appears at 1×. Misalignment characteristically raises 2×, often with a strong axial component. Looseness spreads energy across many integer orders and sub-harmonics. In a four-stroke engine, each cylinder fires once per two revolutions, so the firing frequency is the shaft rate times half the cylinder count: 30 Hz on a four-cylinder gives 60 Hz. A single cylinder misfiring breaks the symmetry of that pattern and deposits energy at the half-order, 15 Hz, and its odd multiples. Blade or vane passing sits at the shaft rate times the blade count. Gear mesh sits at the shaft rate times the tooth count, with sidebands spaced by the shaft rate when a tooth is damaged.

None of that requires a model. It requires knowing the machine, and that is the trade this whole field makes: a great deal of prior structure in exchange for not needing much labelled failure data, which is fortunate, because labelled failure data is precisely what nobody has.

Bearing defect frequencies, computed

Rolling-element bearings are the clearest case. A spall on one raceway is struck once per pass of a rolling element, and the pass rate follows from the geometry. With n rolling elements, ball diameter d, pitch diameter D, contact angle phi and shaft rate fr:

  BPFO = (n/2) * fr * (1 - (d/D) * cos(phi))     outer race
  BPFI = (n/2) * fr * (1 + (d/D) * cos(phi))     inner race
  BSF  = (D/(2d)) * fr * (1 - ((d/D) * cos(phi))^2)   rolling element
  FTF  = (1/2) * fr * (1 - (d/D) * cos(phi))     cage

Take the SKF 6205 deep-groove ball bearing used throughout the Case Western Reserve University Bearing Data Center’s test rig, whose published geometry is 9 balls, a 7.94 mm ball diameter and a 39.04 mm pitch diameter, with a contact angle of zero for a radially loaded deep-groove bearing. Then d/D is 0.2034, and at a 30 Hz shaft rate:

  d/D  = 7.94 / 39.04 = 0.20338

  BPFO = 4.5 * 30 * (1 - 0.20338)      = 107.5 Hz
  BPFI = 4.5 * 30 * (1 + 0.20338)      = 162.5 Hz
  BSF  = (39.04/15.88) * 30 * (1 - 0.20338^2) =  70.7 Hz
  FTF  = 0.5 * 30 * (1 - 0.20338)      =  11.9 Hz

Two details are worth carrying. First, none of these is an integer multiple of the shaft rate — the 0.2034 term guarantees that — which is why bearing faults are distinguishable from unbalance and misalignment rather than buried under them. Second, an inner-race defect moves through the load zone as the shaft turns, so its impacts are amplitude modulated at the shaft rate, and BPFI appears with sidebands at ±30 Hz. Outer-race defects are stationary in the load zone and usually show no such sidebands. The presence or absence of the sidebands is diagnostic on its own.

Why the raw spectrum hides the fault

Compute an FFT of an early-stage bearing fault and 107.5 Hz will not be there in any convincing way. The reason is that the defect does not emit a 107.5 Hz tone. It emits a very short impact 107.5 times a second, and a short impact is broadband — its energy is smeared across kilohertz, individually tiny at any one frequency, and sitting under combustion and shaft harmonics that are orders of magnitude larger.

What the impacts actually do is ring a structural resonance of the housing, typically somewhere in the low kilohertz. So the fault information is present as an amplitude modulation of a high-frequency carrier, and the classical recovery is envelope analysis, sometimes called the high-frequency resonance technique:

1. Band-pass the signal around a resonance the impacts excite — a band chosen from the machine, often several kilohertz wide. Selecting it by spectral kurtosis rather than by eye is the standard automated approach, because kurtosis is high exactly where impulsive content dominates.
2. Take the analytic signal via the Hilbert transform and keep its magnitude. That is the envelope: the slow amplitude contour of the fast carrier.
3. Remove the envelope’s DC component and take its FFT. This is the envelope spectrum, and it is where BPFO and BPFI appear as clean peaks with harmonics.

The step people skip is the demeaning in step three, which leaves a large peak at zero and can mask a low FTF component. The step people get wrong is the band in step one: choose a band with no resonance and the envelope spectrum is flat noise, which reads exactly like a healthy bearing.

The band comparison, and what it must control for

Given a healthy baseline recording and a suspect one, both at the same speed and load, the test is a ratio of envelope-spectrum energy in narrow bands centred on each computed defect frequency. Use a band of roughly ±2% of the centre frequency, wide enough to absorb slip — real bearings slip a little, so measured defect frequencies land within a percent or two of the computed ones rather than exactly on them.

  band            baseline   suspect    ratio
  ------------------------------------------------
  30.0 Hz  (1x)     0.052      0.055      1.06
  60.0 Hz  (fire)   0.048      0.049      1.02
  70.7 Hz  (BSF)    0.003      0.004      1.33
  107.5 Hz (BPFO)   0.004      0.061     15.3
  215.0 Hz (2xBPFO) 0.002      0.026     13.0
  162.5 Hz (BPFI)   0.005      0.006      1.20

  ILLUSTRATIVE. These amplitudes are not measurements;
  they show the pattern to look for, not values to expect.

The pattern, not the magnitude, is the diagnosis. BPFO and its second harmonic rise together by an order of magnitude while the shaft order, the firing frequency and the other defect frequencies are unchanged. That combination is what an outer-race defect looks like. If every band rises together, including 1× and the firing frequency, nothing is broken — the machine is running under more load, or slightly faster, and you have measured that instead.

Which is the failure mode that ruins fixed-band monitoring in the field. Defect frequencies are proportional to shaft rate, so a machine that varies its speed slides every band out from under its window. The fix is order tracking: resample the signal against a tachometer or an estimated instantaneous shaft phase so the horizontal axis is orders rather than hertz, at which point the defect frequencies are constant by construction and the bands stop moving. Without a tacho, the shaft rate can often be recovered from the signal itself by tracking the dominant low-order peak, but that estimate degrades exactly when speed varies fastest, which is when you need it.

What the microphone adds that an accelerometer does not

Everything above was developed for accelerometers bolted to a bearing housing. A microphone is a different measurement with the same mathematics and three extra problems.

• The room is in the signal. An airborne path convolves the source with a transfer function full of reflections and standing waves, so band energies depend on where the microphone was. Moving the phone half a metre can change a ratio more than a developing fault does. Baselines are only comparable if the geometry is fixed.
• Everything else in the room is also in the signal. Neighbouring machines contribute their own orders, and a nearby machine at a similar speed puts peaks in your bands that belong to it. Multiple microphones and beamforming help; a single handheld recording in a plant does not.
• Consumer capture chains destroy the ratios. Automatic gain control is a time-varying nonlinearity, which is precisely what envelope analysis is trying to measure. Lossy codecs at low bitrates discard the quiet high-frequency resonance band that carries the fault. And input clipping generates broadband harmonics that mimic impulsive content everywhere. Verify the capture chain before trusting any of this — the peak-amplitude check for clipping is the cheapest of those verifications and catches the most damage.

The compensation is that a microphone is non-contact, costs almost nothing, and can cover a whole machine rather than one bearing housing. For a screening layer that decides which machines deserve a proper vibration survey, that trade is often the right one — and it is a different job from unsupervised acoustic anomaly detection, which is what you fall back on when the machine’s geometry is unknown and no defect frequency can be computed at all.