# Reliability Modelling Considering Self-Exciting Mechanisms of Shock Damage

## Abstract

**:**

_{2}among the run of ineffective shocks, then the self-exciting mechanism will be triggered and the accumulative number of effective shocks will increase by ${m}_{2}$. The system breaks down when the accumulative number of effective shocks exceeds the fixed threshold. Based on the established shock models, the reliability indices are obtained through the finite Markov chain imbedding approach. According to the operation of the system under different monitoring conditions, two types of preventive maintenance strategies are considered; then, optimization models are established, and the optimal preventive maintenance thresholds are determined. Finally, the proposed models are illustrated by numerical examples.

## 1. Introduction

## 2. Model Assumptions and Model Description

#### Problem Description

## 3. Reliability Modelling and Evaluation

#### 3.1. Reliability Modelling and Evaluation under Triggering Mechanism 1

#### 3.2. Reliability Modelling and Evaluation under Trigger Mechanism 2

## 4. Preventive Maintenance Policies

#### 4.1. Preventive Maintenance Policy 1

#### 4.2. Preventive Maintenance Policy 2

## 5. Numerical Examples

#### Reliability Analysis for the Shock Model

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notations

${X}_{i}$ | time interval between the $\left(i-1\right)$th and $i$th shocks, $i=1,2,3,\dots $ |

${N}_{1}$ | number of shocks before the system failure under self-exciting mechanism 1 |

${N}_{2}$ | number of shocks before the system failure under self-exciting mechanism 2 |

m_{1} | number of effective shocks increased under self-exciting mechanism 1 |

m_{2} | number of effective shocks increased under self-exciting mechanism 2 |

${T}_{1}$ | system lifetime under self-exciting mechanism 1 |

${T}_{2}$ | system lifetime under self-exciting mechanism 2 |

$n$ | accumulative number of effective shocks leading to the system failure |

${k}_{1}$ | number of $\delta $—ineffective shocks triggering self-exciting mechanism 1 |

${k}_{2}$ | number of $\delta $—ineffective shocks triggering self-exciting mechanism 2 |

${c}_{p}$ | preventive maintenance cost |

${c}_{f}$ | corrective maintenance cost |

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**Figure 1.**Failure process of the system under self-exciting trigger mechanism 1. (a denotes the type of shock and b denotes the shock occurrence time).

**Figure 2.**Failure process of the system under self-exciting trigger mechanism 2. (a denotes the type of shock and b denotes the shock occurrence time).

No. | Applicability | Condition | Prob. |
---|---|---|---|

1 | $0\le {u}_{1}<n-1,0\le {w}_{1}<{k}_{1}-1$ | $P\left\{{Y}_{{h}_{1}}=\left({u}_{1},{w}_{1}\right)|{Y}_{{h}_{1}-1}=\left({u}_{1},{w}_{1}\right)\right\}$ | ${q}_{2}$ |

2 | $0\le {u}_{1}<n-1,0\le {w}_{1}<{k}_{1}-1$ | $P\left\{{Y}_{{h}_{1}}=\left({u}_{1}+1,0\right)|{Y}_{{h}_{1}-1}=\left({u}_{1},{w}_{1}\right)\right\}$ | $p$ |

3 | $0\le {u}_{1}<n-1,0\le {w}_{1}<{k}_{1}-1$ | $P\left\{{Y}_{{h}_{1}}=\left({u}_{1},{w}_{1}+1\right)|{Y}_{{h}_{1}-1}=\left({u}_{1},{w}_{1}\right)\right\}$ | ${q}_{1}$ |

4 | $0\le {u}_{1}<n-1,{w}_{1}={k}_{1}-1$ | $P\left\{{Y}_{{h}_{1}}=\left({u}_{1},{w}_{1}\right)|{Y}_{{h}_{1}-1}=\left({u}_{1},{w}_{1}\right)\right\}$ | ${q}_{2}$ |

5 | $0\le {u}_{1}<n-1,{w}_{1}={k}_{1}-1$ | $P\left\{{Y}_{{h}_{1}}=\left({u}_{1}+1,0\right)|{Y}_{{h}_{1}-1}=\left({u}_{1},{w}_{1}\right)\right\}$ | $p+{q}_{1}$ |

6 | ${u}_{1}=n-1,0\le {w}_{1}<{k}_{1}-1$ | $P\left\{{Y}_{{h}_{1}}=\left({u}_{1},{w}_{1}\right)|{Y}_{{h}_{1}-1}=\left({u}_{1},{w}_{1}\right)\right\}$ | ${q}_{2}$ |

7 | ${u}_{1}=n-1,0\le {w}_{1}<{k}_{1}-1$ | $P\left\{{Y}_{{h}_{1}}={E}_{a}|{Y}_{{h}_{1}-1}=\left({u}_{1},{w}_{1}\right)\right\}$ | $p$ |

8 | ${u}_{1}=n-1,0\le {w}_{1}<{k}_{1}-1$ | $P\left\{{Y}_{{h}_{1}}=\left({u}_{1},{w}_{1}+1\right)|{Y}_{{h}_{1}-1}=\left({u}_{1},{w}_{1}\right)\right\}$ | ${q}_{1}$ |

9 | ${u}_{1}=n-1,{w}_{1}={k}_{1}-1$ | ${q}_{2}$ | |

10 | ${u}_{1}=n-1,{w}_{1}={k}_{1}-1$ | $P\left\{{Y}_{{h}_{1}}={E}_{a}|{Y}_{{h}_{1}-1}=\left({u}_{1},{w}_{1}\right)\right\}$ | $p+{q}_{1}$ |

11 | $n/a$ | $P\left\{{Y}_{{h}_{1}}={E}_{a}|{Y}_{{h}_{1}-1}={E}_{a}\right\}$ | 1 |

12 | $n/a$ | Others | 0 |

No. | Applicability | Condition | Prob. |
---|---|---|---|

1 | $0\le {u}_{2}<n-1,0\le {w}_{2}<{k}_{2}-1$ | $P\left\{{Y}_{{h}_{2}}=\left({u}_{2},0\right)|{Y}_{{h}_{2}-1}=\left({u}_{2},{w}_{2}\right)\right\}$ | ${q}_{2}$ |

2 | $0\le {u}_{2}<n-1,0\le {w}_{2}<{k}_{2}-1$ | $P\left\{{Y}_{{h}_{2}}=\left({u}_{2}+1,0\right)|{Y}_{{h}_{2}-1}=\left({u}_{2},{w}_{2}\right)\right\}$ | $p$ |

3 | $0\le {u}_{2}<n-1,0\le {w}_{2}<{k}_{2}-1$ | $P\left\{{Y}_{{h}_{2}}=\left({u}_{2},{w}_{2}+1\right)|{Y}_{{h}_{2}-1}=\left({u}_{2},{w}_{2}\right)\right\}$ | ${q}_{1}$ |

4 | $0\le {u}_{2}<n-1,{w}_{2}={k}_{2}-1$ | $P\left\{{Y}_{{h}_{2}}=\left({u}_{2},0\right)|{Y}_{{h}_{2}-1}=\left({u}_{2},{w}_{2}\right)\right\}$ | ${q}_{2}$ |

5 | $0\le {u}_{2}<n-1,{w}_{2}={k}_{2}-1$ | $P\left\{{Y}_{{h}_{2}}=\left({u}_{2}+1,0\right)|{Y}_{{h}_{2}-1}=\left({u}_{2},{w}_{2}\right)\right\}$ | $p+{q}_{1}$ |

6 | ${u}_{2}=n-1,0\le {w}_{2}<{k}_{2}-1$ | $P\left\{{Y}_{{h}_{2}}=\left({u}_{2},0\right)|{Y}_{{h}_{2}-1}=\left({u}_{2},{w}_{2}\right)\right\}$ | ${q}_{2}$ |

7 | ${u}_{2}=n-1,0\le {w}_{2}<{k}_{2}-1$ | $P\left\{{Y}_{{h}_{2}}={E}_{a}|{Y}_{{h}_{2}-1}=\left({u}_{2},{w}_{2}\right)\right\}$ | $p$ |

8 | ${u}_{2}=n-1,0\le {w}_{2}<{k}_{2}-1$ | $P\left\{{Y}_{{h}_{2}}=\left({u}_{2},{w}_{2}+1\right)|{Y}_{{h}_{2}-1}=\left({u}_{2},{w}_{2}\right)\right\}$ | ${q}_{1}$ |

9 | ${u}_{2}=n-1,{w}_{2}={k}_{2}-1$ | $P\left\{{Y}_{{h}_{2}}=\left({u}_{2},0\right)|{Y}_{{h}_{2}-1}=\left({u}_{2},{w}_{2}\right)\right\}$ | ${q}_{2}$ |

10 | ${u}_{2}=n-1,{w}_{2}={k}_{2}-1$ | $P\left\{{Y}_{{h}_{2}}={E}_{a}|{Y}_{{h}_{2}-1}=\left({u}_{2},{w}_{2}\right)\right\}$ | $p+{q}_{1}$ |

11 | $n/a$ | $P\left\{{Y}_{{h}_{2}}={E}_{a}|{Y}_{{h}_{2}-1}={E}_{a}\right\}$ | 1 |

12 | $n/a$ | Others | 0 |

**Table 3.**Optimal solution with different parameters when ${c}_{p}=2,{c}_{f}=5,\lambda =0.55,\delta =2,{m}_{1}=1$.

$\mathit{p}$ | $\mathit{n}$ | $\mathit{k}$ | $\mathit{E}\left({\mathit{N}}_{1}\right)$ | ${\mathit{N}}^{\ast}$ | $\mathit{C}\left({\mathit{N}}^{\ast}\right)$ |
---|---|---|---|---|---|

0.45 | 3 | 2 | 5.7233 | 4 | 0.5569 |

4 | 2 | 7.6310 | 6 | 0.4054 | |

4 | 3 | 8470 | 6 | 0.3841 | |

4 | 4 | 9.4388 | 7 | 0.3783 | |

5 | 2 | 9.5388 | 7 | 0.3143 | |

0.75 | 3 | 2 | 4.0910 | 3 | 0.6667 |

4 | 2 | 5.4547 | 4 | 0.5 | |

4 | 3 | 5.6590 | 4 | 0.5 | |

4 | 4 | 5.7025 | 4 | 0.5 | |

5 | 2 | 6.8184 | 5 | 0.4 |

**Table 4.**Optimal solution with different values of cost parameters when $n=5,k=2,p=0.45,\delta =2,{m}_{1}=1$.

${\mathit{c}}_{\mathit{p}}$ | ${\mathit{c}}_{\mathit{f}}$ | ${\mathit{N}}^{\ast}$ | $\mathit{C}\left({\mathit{N}}^{\ast}\right)$ |
---|---|---|---|

1.5 | 5.5 | 6 | 0.1951 |

1.5 | 6.5 | 6 | 0.1985 |

1.5 | 7.5 | 6 | 0.2073 |

2.5 | 5.5 | 7 | 0.3214 |

3.5 | 5.5 | 8 | 0.3815 |

**Table 5.**Optimal solution with different parameters when ${c}_{p}=2.5,{c}_{f}=5.5,\lambda =0.55,\delta =2,{m}_{1}=1$.

$\mathit{p}$ | $\mathit{n}$ | $\mathit{k}$ | $\mathit{E}\left(\mathit{T}\right)$ | ${\mathit{t}}^{\ast}$ | ${\mathit{C}}_{2}\left({\mathit{t}}^{\ast}\right)$ |
---|---|---|---|---|---|

0.45 | 3 | 2 | 10.9461 | 9.8307 | 0.4510 |

4 | 2 | 14.8619 | 11.2785 | 0.3327 | |

4 | 3 | 17.2837 | 13.8568 | 0.2955 | |

4 | 4 | 18.4945 | 14.7425 | 0.2836 | |

5 | 2 | 19.5388 | 13.3568 | 0.2649 | |

0.75 | 3 | 2 | 7.7819 | 7.3522 | 0.6408 |

4 | 2 | 10.5092 | 8.2760 | 0.4615 | |

4 | 3 | 10.8179 | 8.8378 | 0.4499 | |

4 | 4 | 11.1049 | 8.7688 | 0.4584 | |

5 | 2 | 13.3365 | 9.4617 | 0.3631 |

**Table 6.**Optimal solution with different values of cost parameters when $n=5,k=2,p=0.45,\delta =2,{m}_{1}=1$.

${\mathit{c}}_{\mathit{p}}$ | ${\mathit{c}}_{\mathit{f}}$ | ${\mathit{t}}^{\ast}$ | ${\mathit{C}}_{2}\left({\mathit{t}}^{\ast}\right)$ |
---|---|---|---|

1.5 | 5.5 | 9.6102 | 0.1601 |

1.5 | 6.5 | 9.3356 | 0.1703 |

1.5 | 7.5 | 8.9661 | 0.1863 |

2.5 | 5.5 | 12.8461 | 0.2345 |

3.5 | 5.5 | 19.7651 | 0.2742 |

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**MDPI and ACS Style**

Wu, Y.
Reliability Modelling Considering Self-Exciting Mechanisms of Shock Damage. *Appl. Sci.* **2022**, *12*, 10418.
https://doi.org/10.3390/app122010418

**AMA Style**

Wu Y.
Reliability Modelling Considering Self-Exciting Mechanisms of Shock Damage. *Applied Sciences*. 2022; 12(20):10418.
https://doi.org/10.3390/app122010418

**Chicago/Turabian Style**

Wu, Yaguang.
2022. "Reliability Modelling Considering Self-Exciting Mechanisms of Shock Damage" *Applied Sciences* 12, no. 20: 10418.
https://doi.org/10.3390/app122010418