🛠️ Predictive Maintenance — Remaining Useful Life Estimator

The underlying wear process is hidden — exactly as in a real machine, no sensor reports "wear fraction" directly. Instead, simulated vibration and temperature readings are generated from the true wear state plus independent Gaussian sensor noise, and the health index is reconstructed by normalising each sensor against its own baseline-to-failure range and combining them in a weighted sum (vibration weighted higher, since rising vibration is typically the earliest and most sensitive precursor of bearing degradation). An exponential moving average then smooths the noisy per-sample estimate into a stable health-index trend, the same way real condition-monitoring systems filter raw accelerometer and thermocouple data before trending it.

Taking the Weibull CDF F(t) = 1 – exp(–(t/η)^β) and applying the transform y = ln(–ln(1–F(t))) gives y = β·ln(t) – β·ln(η), which is exactly linear in x = ln(t). This is the classical Weibull probability-plotting technique used throughout reliability engineering: convert estimated failure fractions from condition data into (x,y) pairs, run ordinary least-squares regression to get the slope (β̂) and intercept, then recover the scale as η̂ = exp(–intercept / β̂). The simulation refits this regression continuously on a rolling window of health-index-derived wear estimates, so the fitted risk curve adapts as more operating hours of evidence accumulate.

Once β̂ and η̂ are fitted, the model failure probability at any future time is F(t) = 1 – exp(–(t/η̂)^β̂). The simulation inverts this formula to solve for the operating hour at which F(t) equals a lower "plan" threshold (half the user-set maintenance threshold) and the operating hour at which it equals the full "urgent" threshold. The interval between those two hours is the recommended maintenance window: early enough that the plan threshold gives lead time to schedule parts and downtime, late enough that the urgent threshold reflects genuinely elevated failure risk. Both bounds move live as the threshold slider, load, or sensor data change.

The regression that fits β̂ and η̂ depends on the estimated wear fraction derived from noisy sensor readings, and both the slope and intercept of a least-squares fit are sensitive to scatter in the input data, especially early in a machine's life when only a short window of hours is available. Turning up the sensor-noise slider increases the scatter in the health-index samples, which widens the range of β̂ and η̂ values the fit can produce from one run to the next, and therefore makes the projected maintenance window jump around more between refits — exactly the estimation instability that makes real condition-monitoring systems favour longer averaging windows and multiple redundant sensors over trusting any single noisy reading.

Remaining useful life depends on the trend and shape of degradation over time, not a single snapshot reading. A machine whose vibration has been flat for hundreds of hours and a machine whose vibration just started climbing sharply can show the same instantaneous value while having very different fitted β̂ (rate of acceleration) and very different projected time to cross the failure threshold. This is why the simulation fits a curve to a rolling window of history rather than thresholding the current sensor value directly — a single noisy high reading is not evidence of imminent failure, but a sustained upward trend consistent with a wear-out shape is.