How to Calculate Failure Rate from Weibull

Weibull Hazard Rate Equation:

\[ \lambda(t) = \frac{\beta}{\eta} \times \left(\frac{t}{\eta}\right)^{\beta-1} \]

Failure Rate λ(t):

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1. What is the Weibull Hazard Rate?

The Weibull hazard rate function describes the instantaneous failure rate of components or systems over time. It is widely used in reliability engineering, survival analysis, and failure prediction modeling to characterize how failure probability changes with time.

2. How Does the Calculator Work?

The calculator uses the Weibull hazard rate equation:

\[ \lambda(t) = \frac{\beta}{\eta} \times \left(\frac{t}{\eta}\right)^{\beta-1} \]

Where:

\( \lambda(t) \) — Failure rate at time t (1/time)
\( \beta \) — Shape parameter (unitless)
\( \eta \) — Scale parameter (time)
\( t \) — Time (time)

Explanation: The shape parameter β determines the failure pattern: β < 1 indicates decreasing failure rate (infant mortality), β = 1 indicates constant failure rate (random failures), and β > 1 indicates increasing failure rate (wear-out failures).

3. Importance of Failure Rate Calculation

Details: Accurate failure rate estimation is crucial for reliability engineering, maintenance planning, warranty analysis, and risk assessment. It helps predict when components are likely to fail and optimize replacement schedules.

4. Using the Calculator

Tips: Enter the shape parameter β (must be positive), scale parameter η (must be positive), and time t (must be non-negative). All parameters should use consistent time units.

5. Frequently Asked Questions (FAQ)

Q1: What does the shape parameter β represent?
A: β indicates the failure pattern: less than 1 for early failures, equal to 1 for random failures, greater than 1 for wear-out failures.

Q2: How is the scale parameter η interpreted?
A: η represents the characteristic life - the time at which approximately 63.2% of units have failed.

Q3: What are typical values for β in different applications?
A: Electronics often have β < 1, mechanical components β > 1, and random failures β ≈ 1.

Q4: Can this calculator handle different time units?
A: Yes, but all time-based parameters (η and t) must use the same units for consistent results.

Q5: What are the limitations of the Weibull model?
A: The model assumes monotonic hazard rates and may not capture complex failure mechanisms with multiple failure modes.