🎓 Lesson 5
D3
Statistical Tool Life Prediction Using Weibull Distribution
The Weibull distribution is a math tool that helps predict how long a drill bit or cutter will last before it wears out, based on real-world wear data.
🎯 Learning Objectives
- ✓ Calculate Weibull parameters (k and η) from experimental tool life data using linear regression on ranked failure times
- ✓ Analyze the effect of cutting speed and feed rate on Weibull shape parameter k to assess wear mode stability
- ✓ Apply the Weibull cumulative distribution function to estimate probability of tool failure before a specified life (e.g., P(T ≤ 30 min) = ?)
- ✓ Design a statistically robust tool replacement policy using B10 life (time by which 10% of tools fail)
📖 Why This Matters
In underground hard-rock mining, unplanned cutter failures in roadheaders or TBM disc cutters cause costly downtime—averaging $12,000–$45,000/hour in lost production. Traditional 'average life' estimates ignore variability: two identical cutters may last 18 or 42 hours under the same settings. The Weibull distribution captures this scatter scientifically—enabling predictive maintenance, optimized spare inventory, and data-driven cutting parameter selection. It’s the industry standard for ISO 281 (rolling bearings) and adopted by major OEMs like Herrenknecht and Sandvik for cutter life certification.
📘 Core Principles
Tool life (T) is inherently stochastic due to microstructural heterogeneity in rock, minor variations in cutter geometry, and dynamic loading. The Weibull distribution models this via its survival function S(t) = exp[−(t/η)^k], where k (shape) indicates failure mode: k < 1 implies infant mortality (e.g., chipping on startup), k ≈ 1 suggests random failures (e.g., fatigue), and k > 1 reflects wear-dominated aging (typical for drag bits in quartzite). The scale parameter η is the characteristic life—time at which ~63.2% of tools have failed. Unlike normal distributions, Weibull handles skewed, non-negative data naturally and supports censoring (e.g., suspended tests where tools haven’t failed by test end).
📐 Weibull Parameter Estimation & Failure Probability
We use median rank regression on ranked tool life data to estimate k and η. Transforming the CDF yields a linear form: ln[−ln(1 − F_i)] = k·ln(t_i) − k·ln(η), where F_i is the cumulative probability for the i-th ranked failure. Plotting ln[−ln(1 − F_i)] vs. ln(t_i) gives slope = k and intercept = −k·ln(η). Once fitted, P(T ≤ t) = 1 − exp[−(t/η)^k] quantifies failure risk.
💡 Worked Example
Problem: Ten identical tungsten-carbide drag bits were tested in granite (UCS = 180 MPa) at 120 rpm and 0.8 mm/rev. Observed lives (min): [22, 27, 31, 35, 39, 44, 48, 53, 59, 67]. Estimate k and η, then compute P(T ≤ 40 min).
1.
Step 1: Rank failures ascending; assign median ranks F_i = (i − 0.3)/(n + 0.4) → F = [0.065, 0.165, 0.265, 0.365, 0.465, 0.565, 0.665, 0.765, 0.865, 0.965]
2.
Step 2: Compute y_i = ln[−ln(1 − F_i)] and x_i = ln(t_i); e.g., for t=22: y = ln[−ln(0.935)] ≈ −2.63, x = ln(22) ≈ 3.09
3.
Step 3: Linear regression yields slope k = 1.82, intercept = −5.94 → η = exp(−intercept/k) = exp(5.94/1.82) ≈ 26.7 min
4.
Step 4: P(T ≤ 40) = 1 − exp[−(40/26.7)^1.82] = 1 − exp[−2.04] ≈ 0.87
Answer:
The result is P(T ≤ 40 min) = 0.87 (87% failure probability), indicating high risk—suggesting parameter adjustment or proactive replacement before 40 min.
🏗️ Real-World Application
At the Antamina Mine (Peru), Sandvik engineers applied Weibull analysis to 127mm PDC bits drilling in porphyry copper ore (abrasivity index = 12.3 g/t). Using 42 field-measured bit lives (censored at 120 hrs), they found k = 2.3 and η = 89 hrs. This confirmed wear-dominated failure—validating a switch from fixed-interval (72-hr) to condition-based replacement triggered at B10 life (t_B10 = η·(−ln(0.9))^(1/k) = 89·(0.105)^(0.435) ≈ 41 hrs). Result: 31% reduction in bit-related unplanned stops and 19% lower bit cost per meter drilled (2022 Antamina Reliability Report).