# Failure Rates for Aging Aircraft

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Conceptual Framework for Aging and Systems Failure

_{0}, and subsequent cycles beginning at status defined by the repair line. A representation of the cycles of degradation and repair for an aircraft is shown in Figure 1.

## 3. Models for Patterns in the Number of Failures

#### 3.1. Linear Approximation to Failure Rate between Repairs

#### 3.2. Power Law for Failure Rate between Repairs

_{j}= time between failures (j − 1) and j time between failures (j − 1) and j, the cumulative distribution for t

_{j}is given by:

## 4. Model Estimation

#### 4.1. Linear Failure Rate Model

- ${X}_{1}=$ indicator for the first $\tau :$ repair in interval,
- ${X}_{2}=$ indicator for the ${k}^{th}$ $\delta :$ repair in interval, $k\ge 1,$
- ${X}_{3}=$ difference between the squared number of repairs, ${k}_{2}^{2}-{k}_{1}^{2},$
- ${X}_{4}=$ the difference in residual times ${r}_{2}-{r}_{1},$
- ${X}_{5}=$ the difference in squared residual times ${r}_{2}^{2}-{r}_{1}^{2},$
- $\epsilon =$ random error.

- (1)
- $H:{\beta}_{2}-{\beta}_{1}>0\text{}\Rightarrow $ block repair is incomplete
- (2)
- $H:{\beta}_{3}>0\text{}\Rightarrow $ repair fraction is decreasing
- (3)
- $H:{\beta}_{4}>0\text{}\Rightarrow $ failure rate is increasing
- (4)
- $H:{\beta}_{5}>0\text{}\Rightarrow $ an accelerated failure rate.

#### 4.2. Power Law Analysis

## 5. Conclusions

- (1)
- The failure rate is increasing with use within a phase between scheduled maintenance D checks.
- (2)
- The failure rate grows with the number of scheduled maintenance phases.
- (3)
- The repair at scheduled maintenance is not to “good as new”.
- (4)
- The failure rate within a repair phase depends on the number of prior failures.

## Author Contributions

## Conflicts of Interest

## References

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Event Name | Frequency |
---|---|

Navigation/communication equipment | 30 |

Declared emergency/priority | 243 |

Decompression/pressurization | 55 |

Door/canopy openings indications | 59 |

Electrical problem | 95 |

Engine—malfunction | 211 |

Engine failure | 58 |

Engine oil problem | 60 |

Engine shut down | 86 |

Flight control systems | 44 |

Flight instrument failure | 22 |

Fuel | 70 |

Hydraulic problem | 42 |

Landing gear | 172 |

Other operational incident | 117 |

Precautionary landing | 89 |

Smoke/fumes—aircraft | 53 |

Windshield/window (aircraft) | 25 |

Source | Sum of Squares | DF | ${\mathit{F}}_{\mathit{o}}$ | ${\mathit{P}}_{\mathit{o}}$ |
---|---|---|---|---|

Regression | 615.79 | 4 | 96.22 | 0.000 |

Error | 963.00 | 217 | - | - |

Variable | Parameter | Estimate | ${\mathit{T}}_{\mathit{j}}$ | ${\mathit{P}}_{\mathit{j}}$ |
---|---|---|---|---|

X_{1} | b_{1} | −1.54 | −6.52 | 0.000 |

X_{2} | b_{2} | 6.39 | 5.03 | 0.000 |

X_{4} | b_{4} | 0.12 | 6.02 | 0.000 |

X_{5} | b_{5} | 0.02 | 10.86 | 0.000 |

Order of UL | Mean Time between | Standard Deviation of Time | Number of UL’s |
---|---|---|---|

1 | 2186.3 | 653.2 | 19 |

2 | 1135.1 | 760.7 | 15 |

3 | 582.7 | 485.7 | 12 |

4 | 101.0 | 136.2 | 5 |

5 | 96.0 | 128.7 | 2 |

Failure | β—Shape | ϕ—Scale | λ—Rate |
---|---|---|---|

1 | 3.65 | 2419.67 | 0.00018 |

2 | 1.39 | 1273.36 | 0.00079 |

3 | 0.57 | 603.21 | 0.00129 |

4 | 0.41 | 40.99 | 0.00884 |

5 | 0.35 | 103.28 | 0.00557 |

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**MDPI and ACS Style**

MacLean, L.; Richman, A.; Hudak, M.
Failure Rates for Aging Aircraft. *Safety* **2018**, *4*, 7.
https://doi.org/10.3390/safety4010007

**AMA Style**

MacLean L, Richman A, Hudak M.
Failure Rates for Aging Aircraft. *Safety*. 2018; 4(1):7.
https://doi.org/10.3390/safety4010007

**Chicago/Turabian Style**

MacLean, Leonard, Alex Richman, and Mark Hudak.
2018. "Failure Rates for Aging Aircraft" *Safety* 4, no. 1: 7.
https://doi.org/10.3390/safety4010007