Reliability modeling of the bivariate deteriorating product with both monotonic and non-monotonic degradation paths

doi:10.23919/JSEE.2021.000083

Abstract

Abstract:

Fiber optical gyroscope (FOG) is a highly reliable navigation element, and the degradation trajectories of its two accuracy indexes are monotonic and non-monotonic respectively. In this paper, a flexible accelerated degradation testing (ADT) model is used for analyzing the bivariate dependent degradation process of FOG. The time-varying copulas are employed to consider the dynamic dependency structure between two marginal degradation processes as the Wiener process and the inverse Gaussian process. The statistical inference is implemented by utilizing an inference function for the margins (IFM) approach. It is demonstrated that the proposed method is powerful in modeling the joint distribution with various margins.

Key words: fiber optical gyroscope, bivariate accelerated degradation testing, Wiener process, inverse Gaussian process, time-varying copulas, dependence

Fuqiang SUN, Hongxuan GUO, Jingcheng LIU. Reliability modeling of the bivariate deteriorating product with both monotonic and non-monotonic degradation paths[J]. Journal of Systems Engineering and Electronics, 2021, 32(4): 971-983.

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Stress level	Temperature/°C	Sample size	Number of specimens
S₁	55	3	FG-01, FG-02, FG-03
S₂	70	3	FG-04, FG-05, FG-06
S₃	80	3	FG-07, FG-08, FG-09

Family	C(x, y)	Θ(θ)
Normal copula	${\varPhi _{{\theta _t}}}\left[ {{\varPhi ^{ - 1}}(x),{\varPhi ^{ - 1}}(y)} \right]$	$\dfrac{ {1 - {{\rm{e}}^{ - \theta } } } }{ {1 + {{\rm{e}}^{ - \theta } } } }$
Gumbel copula	$\exp \left\{ { - {{\left( {{{\left( { - \ln (1 - x)} \right)}^{{\theta _t}}} + {{\left( { - \ln \left( {1{\rm{ - }}y} \right)} \right)}^{{\theta _t}}}} \right)}^\frac{1}{{{\theta _t}}}}} \right\}$	1+θ²
Clanton copula	${\left( {{x^{ - {\theta _t}}} + {y^{ - {\theta _t}}} - 1} \right)^{ - \frac{1}{{{\theta _t}}}}}$	θ²?0.999
Frank copula	$- \dfrac{1}{ { {\theta _t} } }\ln \left( {1 + \dfrac{ {\left( { {{\rm{e}}^{ - {\theta _t}x} } - 1} \right)\left( { {{\rm{e}}^{ - {\theta _t}y} } - 1} \right)} }{ { {{\rm{e}}^{ - {\theta _t} } } - 1} } } \right)$	θ

Parameter	Estimator	Parameter	Estimator
$\hat \sigma $	3.315 2e?06	$\hat \lambda $	0.009 0
$\hat r$	1.504 6	$\hat \gamma $	0.998 5
${\hat a_0}$	?22.938 8	${\hat b_0}$	2.467 8
${\hat a_1}$	5.003 3	${\hat b_0}$	8.665 1

Time-varying copula	Parameter’s estimator	AIC	Rank
Normal	[1.452 9, 0.056 9, ?2.498 9]	?908.82	2
Gumbel	[1.236 4, 0.0170, ?3.013 5]	?1 002.12	1
Clayton	[?1.381 0, 0.345 3, ?1.880 3]	?888.27	3
Frank	[1.467 3 0.929 4, ?4.995 3]	?480.04	4

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