23.3 Reliability analysis and fault detection

Reliability analysis is crucial for understanding and improving system performance. It uses mathematical tools to calculate metrics like reliability functions, failure rates, and mean time to failure, helping engineers assess and enhance system dependability.

Probabilistic failure models, statistical fault detection, and robust design techniques form the backbone of reliability engineering. These methods allow for predicting system behavior, detecting anomalies, and creating resilient designs that can withstand failures and maintain functionality.

Reliability Analysis

Reliability metrics calculation

Top images from around the web for Reliability metrics calculation

• 

  Reliability Importance and Optimized Reliability Allocation (Analytical) - ReliaWiki View original

  Is this image relevant?

• 

  Duane Model - ReliaWiki View original

  Is this image relevant?

• 

  Duane Model - ReliaWiki View original

  Is this image relevant?

• 

  Reliability Importance and Optimized Reliability Allocation (Analytical) - ReliaWiki View original

  Is this image relevant?

• 

  Duane Model - ReliaWiki View original

  Is this image relevant?

1 of 3

Top images from around the web for Reliability metrics calculation

• 

  Reliability Importance and Optimized Reliability Allocation (Analytical) - ReliaWiki View original

  Is this image relevant?

• 

  Duane Model - ReliaWiki View original

  Is this image relevant?

• 

  Duane Model - ReliaWiki View original

  Is this image relevant?

• 

  Reliability Importance and Optimized Reliability Allocation (Analytical) - ReliaWiki View original

  Is this image relevant?

• 

  Duane Model - ReliaWiki View original

  Is this image relevant?

1 of 3

• Reliability metrics quantify system performance
  • Reliability function $R(t) = P(T > t)$ represents probability of system surviving beyond time $t$ ($T$ denotes time to failure)
  • Failure rate function $\lambda(t) = \frac{f(t)}{R(t)}$ measures instantaneous rate of failure at time $t$ ($f(t)$ denotes probability density function of $T$)
  • Mean time to failure (MTTF) $MTTF = \int_0^{\infty} R(t) dt$ represents expected time until system failure
  • Mean time between failures (MTBF) $MTBF = MTTF + MTTR$ accounts for both failure and repair times (MTTR denotes mean time to repair)
• System reliability for complex systems combines component reliabilities
  • Series systems fail when any component fails: $R_s(t) = \prod_{i=1}^n R_i(t)$ ($R_i(t)$ denotes reliability of $i$-th component)
  • Parallel systems fail only when all components fail: $R_p(t) = 1 - \prod_{i=1}^n (1 - R_i(t))$
  • k-out-of-n systems require at least $k$ functioning components out of $n$: $R_k(t) = \sum_{i=k}^n \binom{n}{i} R(t)^i (1 - R(t))^{n-i}$
• Reliability block diagrams and fault trees visually represent system reliability
  • Block diagrams show component interconnections (series, parallel, etc.)
  • Fault trees illustrate combinations of events leading to system failure (logic gates)

Probabilistic failure models

• Probability distributions model time to failure
  • Exponential distribution assumes constant failure rate: $f(t) = \lambda e^{-\lambda t}$, $R(t) = e^{-\lambda t}$
  • Weibull distribution models increasing or decreasing failure rates: $f(t) = \frac{\beta}{\eta} (\frac{t}{\eta})^{\beta-1} e^{-(t/\eta)^{\beta}}$, $R(t) = e^{-(t/\eta)^{\beta}}$
  • Lognormal distribution represents failure times with skewed distributions: $f(t) = \frac{1}{t\sigma\sqrt{2\pi}} e^{-(\ln t - \mu)^2 / (2\sigma^2)}$, $R(t) = 1 - \Phi(\frac{\ln t - \mu}{\sigma})$
• Accelerated life testing predicts reliability under normal conditions by testing at higher stress levels
  • Arrhenius model relates life to temperature: $L(T) = A e^{E_a / (kT)}$ ($L$ denotes life, $T$ denotes absolute temperature, $E_a$ denotes activation energy, $k$ denotes Boltzmann constant)
  • Inverse power law model relates life to stress: $L(S) = \frac{C}{S^n}$ ($S$ denotes stress level, $C$ and $n$ are model parameters)
• Reliability growth models track reliability improvements over time
  • Duane model assumes power-law decrease in failure rate: $\lambda(t) = \lambda_0 t^{-\alpha}$ ($\lambda_0$ and $\alpha$ are model parameters)
  • Crow-AMSAA model relates cumulative failures to time: $\Lambda(t) = (\frac{t}{\theta})^{\beta}$ ($\Lambda(t)$ denotes cumulative failure intensity, $\theta$ and $\beta$ are model parameters)

Statistical fault detection

• Hypothesis testing detects faults by comparing data to statistical models
  • Null hypothesis $H_0$ assumes no fault present
  • Alternative hypothesis $H_1$ assumes fault present
  • Type I error (false alarm) rejects $H_0$ when true
  • Type II error (missed detection) fails to reject $H_0$ when false
• Statistical process control (SPC) monitors process parameters to detect anomalies
  • Control charts compare measurements to control limits
    • Shewhart charts detect large shifts in mean or variance (3-sigma limits)
    • CUSUM charts detect small, persistent shifts in mean (cumulative sum)
    • EWMA charts detect small, gradual shifts in mean (exponentially weighted moving average)
  • Process capability indices measure process performance relative to specifications
    • $[C_p](https://www.fiveableKeyTerm:c_p) = \frac{USL - LSL}{6\sigma}$ compares specification range to process variation (USL and LSL denote upper and lower specification limits, $\sigma$ denotes process standard deviation)
    • $C_{pk} = \min(\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma})$ accounts for process centering ($\mu$ denotes process mean)
• Multivariate statistical methods diagnose faults in high-dimensional data
  • Principal component analysis (PCA) identifies main sources of variability (eigenvectors of covariance matrix)
  • Partial least squares (PLS) finds latent variables maximizing input-output covariance
  • Fisher discriminant analysis (FDA) finds linear combinations best separating fault classes

Robust system design

• Redundancy techniques improve reliability by providing backups
  • Hardware redundancy uses multiple identical components in parallel (dual processors)
  • Software redundancy uses different implementations of same functionality (N-version programming)
  • Information redundancy uses error-correcting codes or voting schemes (triple modular redundancy)
• Fault-tolerant design principles enable graceful degradation and fail-safe operation
  • Graceful degradation maintains partial functionality when faults occur (limp-home mode in vehicles)
  • Fail-safe design ensures safe state upon failure (elevator brakes)
  • Self-checking and self-repairing systems detect and recover from faults automatically (error detection and correction in memory)
• Reliability-centered maintenance (RCM) optimizes maintenance strategies
  • Identifies critical components and failure modes (FMEA)
  • Develops cost-effective maintenance plans (preventive, predictive, condition-based)
  • Balances reliability improvements with maintenance costs