Predictive Maintenance That Pays: From Vibration Data to Remaining Useful Life

Predictive maintenance pays when three conditions hold simultaneously: the failure mode develops slowly enough to be detected with useful warning, the sensing physically observes that development, and the organisation can act on the warning before the failure. Programmes fail when one of the three is missing — most often the third, which is not a technical problem at all.

The P-F interval decides whether prediction is even possible

Reliability engineering describes degradation with the P-F curve: at some point a failure becomes detectable (P), and at some later point the asset fails functionally (F). The interval between them is the entire budget available for detection, diagnosis, planning, parts and execution.

Sampling interval follows from this directly, and the constraint is stricter than intuition suggests. To be confident of catching a developing fault, the inspection interval must be at most half the P-F interval — otherwise a fault can initiate and progress to failure entirely between two inspections.

Bearing fault frequencies: the physics that makes this deterministic

Rolling-element bearing diagnosis is unusually tractable because the defect frequencies are calculable from geometry and shaft speed. A spall on the outer race is struck once per rolling element passage, producing a periodic impulse train at a frequency that can be derived, not learned.

This matters commercially. Because the target frequencies are known in advance, a diagnostic does not need a large labelled failure dataset — the hardest thing to obtain in industrial machine learning, since healthy assets by definition produce few failures. A physics-derived detector works on day one, on an asset that has never failed.

Envelope analysis, and why a raw spectrum shows nothing

An early bearing defect produces impacts with very little energy. In a raw vibration spectrum that energy is spread thinly and buried under shaft imbalance, misalignment and gear mesh, which are orders of magnitude larger. Teams look for BPFO in the raw FFT, find nothing, and conclude the sensor is inadequate.

The impacts do something more useful than appear directly: they ring the structure. Each impact excites a high-frequency structural resonance, typically several kilohertz, well above the dominant low-frequency machinery content. The resonance is therefore amplitude-modulated at the defect frequency, and demodulating it recovers a signal that is clean precisely because that band contains nothing else.

from scipy.signal import butter, sosfiltfilt, hilbert
import numpy as np

def envelope_spectrum(x, fs, band=(3000, 8000)):
    sos = butter(4, band, btype='bandpass', fs=fs, output='sos')
    resonance = sosfiltfilt(sos, x)             # isolate the ringing band
    env = np.abs(hilbert(resonance))            # demodulate the carrier
    env -= env.mean()                           # drop DC before transforming
    spec = np.abs(np.fft.rfft(env * np.hanning(len(env))))
    return np.fft.rfftfreq(len(env), 1/fs), spec

# Peak at BPFO with harmonics -> outer race defect.
# Peak at BPFI with sidebands at shaft speed -> inner race (load zone modulation).

The sideband structure is diagnostic in its own right. An inner-race defect rotates through the load zone, so its impact amplitude is modulated once per shaft revolution — producing sidebands spaced at shaft frequency around the BPFI peak. An outer-race defect is stationary in the load zone and shows harmonics without those sidebands. The spectrum distinguishes which race has failed before anything is disassembled.

Acquisition must support this. Sampling at 2.56× the highest frequency of interest is the standard practice for a reason: it places the analysis band clear of anti-alias filter roll-off. Detecting an 8 kHz resonance therefore needs at least ~20 kHz sampling, and the low-rate trending data most SCADA historians already hold will not do it. This single requirement is the most common reason an existing data estate cannot support the programme.

Remaining useful life, and what the estimate is worth

Detection is worth substantially more when paired with a defensible time-to-action. Weibull analysis of the failure population gives the baseline hazard, and the shape parameter tells you whether scheduled replacement is even a rational strategy.

That middle case is worth dwelling on. For a genuinely random failure mode, time-based replacement provides no benefit at all — it merely replaces components with an equal probability of failing. A surprising proportion of maintenance schedules address β ≈ 1 failure modes, and the analysis that reveals it usually finds savings before any sensor is installed.

The economics, stated properly

The comparison is between three strategies over the same horizon, and the honest version includes the cost of false alarms, which is the term vendors omit.

The condition C_fail ≫ C_planned is what actually determines the shortlist, and it is a property of the asset's position in the process, not of the asset. A redundant pump with a standby has a low failure cost regardless of how expensive the pump is. An unspared compressor upstream of the whole line has an enormous one. Rank candidates by consequence of failure first, and the instrumentation budget goes to the right dozen assets instead of being spread across four hundred.

Where programmes actually die

Not in the models. The two failure modes we see repeatedly are alarm fatigue — a false-positive rate that looks acceptable in percentage terms but produces enough false alarms per week that planners begin ignoring them — and the absence of a closed loop between prediction and work order. A prediction that does not become scheduled work is an observation, and observations do not reduce downtime. Integration into the CMMS is not the boring part of the project; it is the part that determines whether any of the rest was worth doing.