Failure modes and degradation

"Nothing breaks without warning. It breaks with warning, in a channel nobody was reading, on a timescale nobody was watching."

A Retrospective AI Agent

Fault, failure, and the P-F interval

Three words that colloquial speech treats as synonyms must be kept distinct. A fault is an incipient defect: a hairline crack, a pit of corrosion, a film of fouling. A failure is the loss of a required function: the pump no longer delivers head, the gearbox no longer transmits torque. Degradation is the process that carries the asset from the first fault to functional failure. Prognostics lives entirely inside that process, which reliability engineering names the P-F interval: the time from the point \(P\), where a potential failure first becomes detectable, to the point \(F\), where the function is actually lost.

The P-F interval is the resource prognostics spends. A model can only warn you if it detects the defect after \(P\) but comfortably before \(F\), and the usable warning is not the whole interval but the slice left after your detection latency. This is why the choice of sensor is a prognostics decision, not an afterthought. On a rolling-element bearing the same defect announces itself first in ultrasonic acoustic emission (a P-F interval of months), later in high-frequency vibration (weeks), later still in audible noise and heat (days), and finally in the machine seizing (hours). Every modality trades interval length for signal-to-noise and cost. The art of the field is to instrument as early on that cascade as the budget allows, because you cannot extrapolate a curve you started sampling the day before it went vertical.

A taxonomy of failure modes and their hazard shapes

Machines do not fail in one way; they fail in a small, enumerable set of failure modes, each with its own physics, timescale, and signature. Practitioners catalog them with a Failure Modes and Effects Analysis (FMEA), which for each mode records the mechanism, the observable symptom, and the effect on function. The dominant mechanisms are worth naming because they behave differently under the sensor. Fatigue accumulates microscopic crack growth under cyclic load and tends to progress slowly then accelerate. Wear and abrasion remove material gradually and roughly linearly. Corrosion and erosion are chemically and environmentally driven. Fouling and scaling deposit material and choke flow or heat transfer. Thermal and electrical breakdown (insulation aging, dielectric failure) often show a long quiet phase and a sudden collapse.

These mechanisms map onto the classic bathtub curve of the hazard rate \(h(t)\), the instantaneous probability that an asset still alive at time \(t\) fails in the next instant. Early life shows a falling hazard (infant mortality from manufacturing and installation defects), a long flat middle of roughly constant, random hazard, and a rising tail as wear-out mechanisms dominate. A convenient parametric family is the Weibull hazard,

whose single shape parameter \(\beta\) selects the regime: \(\beta<1\) is the decreasing, infant-mortality branch, \(\beta=1\) is the memoryless constant-hazard middle (failures arrive as a Poisson process), and \(\beta>1\) is the rising wear-out tail. The distinction matters enormously for AI. A purely random, constant-hazard failure has, by construction, no degradation trajectory to learn; nothing in the past predicts it, and sensor-based prognostics cannot beat a fixed schedule. Only wear-out modes, the \(\beta>1\) tail, carry a growing signature that a model can ride. Knowing which of your failure modes are random and which are progressive tells you which are even candidates for prediction, a point revisited when hazard models arrive in Section 36.3.

Degradation as a latent state and its sensor signature

For the progressive modes, the useful abstraction is a hidden degradation state \(x(t)\), a scalar summarizing accumulated damage that we never observe directly. We observe only sensor readings \(y(t) = g(x(t)) + \varepsilon(t)\), a possibly nonlinear, noisy projection of that state. Two properties make \(x(t)\) tractable. It is typically monotone (damage rarely heals), and it is often well described by a small growth law. A widely used model for bearings, batteries, and turbine components is exponential degradation,

with a slow initial rise that sharpens near end of life, matching the empirically observed knee in most wear-out curves. The modeling task is then to invert the noisy observation back to \(x(t)\), which is a state-estimation problem of exactly the kind Part III treats; a Kalman or particle filter tracking the latent damage state, from Chapter 10, is a standard prognostic backbone.

The signature is where sensor physics re-enters. A localized bearing defect strikes the rolling elements at a geometry-determined defect frequency (ball-pass frequency outer race, and its relatives), so the earliest degradation shows up not as higher overall vibration energy but as growing energy at specific sidebands, which is precisely why the envelope and spectral tools of Chapter 7 matter. Statistical shape features help too: as isolated impacts appear, the kurtosis of the vibration waveform spikes long before its RMS amplitude moves, because kurtosis is sensitive to rare large excursions while RMS averages them away. Different modes light up different features, which is the whole reason feature engineering (Chapter 8) is the connective tissue of prognostics.

Health indicators: turning a signal into a degradation curve

A health indicator (HI) is a scalar, derived per time window from raw sensor data, that is meant to track the latent state \(x(t)\). A good HI has three measurable virtues: monotonicity (it moves in one direction as damage grows), trendability (it correlates with elapsed life across units), and prognosability (its end-of-life values cluster tightly across units so a threshold means the same thing everywhere). You do not have to eyeball these; monotonicity is quantifiable as the normalized net directionality of the differenced series,

which is \(1\) for a perfectly monotone indicator and near \(0\) for noise. The listing below computes two candidate indicators from a synthetic vibration run, RMS and kurtosis, and scores each for monotonicity, reproducing the wind-turbine finding that kurtosis is the earlier, cleaner degradation channel.

import numpy as np

rng = np.random.default_rng(0)
T = 120                                   # 120 windows over the machine's life
t = np.arange(T)
damage = np.exp(0.045 * t) - 1.0          # hidden exponential degradation x(t)

def health_indicators(win):
    rms = np.sqrt(np.mean(win**2))
    k = np.mean((win - win.mean())**4) / (win.var()**2 + 1e-12)  # kurtosis
    return rms, k

rms_hi, kurt_hi = [], []
for d in damage:
    base = rng.normal(0, 1.0, 4096)                       # background vibration
    impacts = rng.normal(0, 6.0 * d, 4096) * (rng.random(4096) < 0.02 * d)
    win = base + impacts                                  # sharp, rare defect impacts
    r, k = health_indicators(win)
    rms_hi.append(r); kurt_hi.append(k)

def monotonicity(y):
    dy = np.diff(y)
    return abs((np.sum(dy > 0) - np.sum(dy < 0)) / (len(y) - 1))

print(f"RMS      monotonicity = {monotonicity(rms_hi):.2f}")
print(f"Kurtosis monotonicity = {monotonicity(kurt_hi):.2f}")

As Listing 36.1 shows, the choice of indicator is empirical and rankable: two functions of the identical raw signal produce degradation curves of very different quality, and a one-line monotonicity score tells you which to trust. That screening step, computed on held-out run-to-failure units so the ranking does not leak, is the bridge from raw telemetry to the prediction machinery of the rest of this chapter.

from tsfresh import extract_features, select_features
X = extract_features(windows_long_df, column_id="unit", column_sort="t")
X_relevant = select_features(X.dropna(axis=1), life_fraction)   # keep trendable HIs