# Failure time prediction for mechanical device based on the degradation sequence

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## Abstract

Mechanical devices are playing a crucial role in modern industry. With the ever-growing demands of multiple function and high performance, the unpredicted failures of mechanical device might greatly increase maintenance cost during its lifetime. As a key state indicator of mechanical device, the degradation of some important performance provides substantial information for failure prognosis. More and more attention has been paid to the degradation-based failure time prediction. However, even mechanical devices of the same type might show greatly diverse degradation processes under different working environments. It is still a challenge to identify global degradation pattern and then predict the failure time of a specific mechanical device based on its degradation sequence. This paper proposes a novel approach for failure time prediction with the degradation sequence of mechanical device. The proposed approach combines the exponential regression and parametric empirical Bayesian (PEB) technology. Firstly, exponential regression is adopted to represent the local degradation pattern and then local failure time observations can be computed. Secondly, according to the rule that local failure time observations manifest, appropriate prior assumption is made and the posterior distribution is estimated by PEB technology. Herein, two prior assumptions are considered, including the exchangeable PEB and linear PEB case. The global failure time distribution can be predicted with the estimated prior and posterior distribution. Finally, three case studies are implemented to validate the proposed approach, including the simulation case, crack case and precision case of machine tool.

## Keywords

Failure time prediction Degradation sequence Mechanical device Exponential regression Parametric empirical Bayes## List of symbols

- \(t_i\)
Time of the \(i\)th degradation measurement

- \(t_{i:j}\)
Measurement time from \(t_i \) to \(t_j\)

- \(y_i \)
Degradation measurement at \(t_i \)

- \(y_{i:j} \)
Degradation measurements corresponding to \(t_{i:j} \)

- \(n\)
Total number of degradation measurements

- \(y_D, T_D \)
Failure threshold and actual failure time respectively

- \(q_0 \)
Minimum number of measurements for exponential regression and \(q_0 \) is set to 3

- \(k\)
Measurement index ranging from \(q_0 +1\) to \(n\)

- \(q\)
Measurement index ranging from 1 to \(k-q_0 +1\)

- \(x_k^q \)
Predicted failure time by \(\hbox {et}_k^q \)

- \(\uptheta _k \)
Random variable of the local failure time at \(t_k \)

- \({\uptheta }_0\)
Random variable of the global failure time

- \(x_k \)
Local failure time observation at \(t_k \) also the observation of \({\uptheta }_k \)

- \(\sigma _k ^{2}\)
Observation variance of \(x_k \) for \({\uptheta }_k \)

- \(x_{i:j}\)
Local failure time observations at \(t_{i:j} \)

- \(N({\uptheta },\sigma ^{2})\)
Normal distribution with mean \({\uptheta }\) and variance \(\sigma ^{2}\)

- \(X\sim N({\uptheta },\sigma ^{2})\)
X is normally distributed with mean \({\uptheta }\) and variance \(\sigma ^{2}\)

- \(\overline{x} \)
Mean of all \(x_k \)

- \(f(x_k |\theta _k )\)
Conditional probability of \(x_k \) given \(\theta _k \)

- \(p_0 \)
Minimum number of local failure time observations required by linear PEB and \(p_0 \) is set to 4

- \(p\)
Index ranging from \(q_0 +1\) to \(n-p_0 +1\)

Probability density function

- \(\hbox {npdf}(x,\mu ,V)\)
Probability density of \(x\) on \(N\left( {\mu ,V} \right) \)

## Notes

### Acknowledgments

The authors are grateful to the Technical Editor and all Reviewers for their valuable and constructive comments. The research is supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 51375181, 51105156, and the National 973 Basic Research Program of China under Grant No. 2011CB706803.

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