Reliability model of series systems with multiple shock sources subject to dependent competing failure processes using phase-type distribution

ABSTRACT

In this paper, we introduce the concept of a series system with two components and three shock sources considering degradation to build a reliability model. Sources 1 and 2 affect components 1 and 2, respectively. Source 3 covers both components. Both components are subject to dependent competing failure processes (DCFPs). A general reliability model of the n-component series system with m-shock sources subject to DCFPs is derived. The phase-type distribution method is applied to calculate the reliability of the hard failure process. The time lag among shocks follows the continuous phase-type distribution (PH_c). The lifetime and system reliability properties are discussed based on the phase-type distribution. The dependence of shock sources is also considered according to the proposition of phase-type distribution (PH). Finally, an application example and sensitivity analysis of micro-electro-mechanical systems (MEMS) oscillators subject to various shock models are presented to illustrate the developed reliability models.

KEYWORDS:

• Reliability
• (DCFPs) dependent competing failure processes
• phase-type distribution
• series system

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notation

Hk	=	Soft failure threshold of the kth component
Yj	=	Catastrophic failure threshold subject to the jth shock source
R(t)	=	Reliability function of the system by time t
Rk(t)	=	Reliability function of the kth component by time t
Cj(t)	=	Number of shocks produced by the jth shock source by time t
ij	=	Number of shocks of the jth shock source
$T_k^{\left(i_j\right)}$	=	Lifetime of the kth component accumulated to the ij shock
Xk(t)	=	Continuous degradation of the kth component at time t
Xs(t)	=	Total degradation level
βk	=	Degradation rate of the kth component
λj	=	Intensity of random shocks from the jth source
$S_{j{i}_j}$	=	Shock damage magnitudes of the ij shock load caused by the jth source
$X_{j{i}_j}$	=	Interval time between the ij and (ij +1) shocks from the jth source
Sjsum(t)	=	Degradation damage increment caused by $i_{th}^j$ shocks under jth shock source
$P_{\mathrm{e}}\left\{min\left(T_1,T_3\right)>t\right\};P_{\mathrm{c}}\left\{min\left(T_1,T_3\right)>t\right\};P_{\mathrm{r}}\left\{min\left(T_1,T_3\right)>t\right\}$	=	For the extreme shock model, the cumulative shock model, or run shock model, the survival function of the hard failure process
Pj	=	For the extreme shock model, the probability that shock magnitudes produced by the jthsource are beyond Dj
$P_{j{i}_j}$	=	For the cumulative shock model, the probability that the sum of the ij shock magnitudes exceeds Dj
Pjr	=	For the run shock model, the probability that the shock magnitudes produced by the jth source are critical
$W_{j{i}_j}$	=	Magnitude of the ij shock load caused by the jth source
N	=	Total number of transfers until the Markov chain enters the absorption state
Q	=	Transition matrix
NHFk(t)	=	No hard failure occurs for the kth component at time t
NSFk(t)	=	No soft loss happens for the kth component at time t

Acronyms

MEMS	=	Micro-Electro-Mechanical Systems
DCFPs	=	Dependent competing failure processes
PHd	=	Discrete phase-type distribution
PHc	=	Continuous phase-type distribution
PH	=	Phase-type distribution
CTCM	=	Continuous time Markov chain

Additional information

Funding

This research was funded by the National Natural Science Foundation of China Project (51605083, 12072069); supported by China Scholarship Council Visiting Scholars Project (201906085037) and the Fundamental Research Funds for the Central Universities N2203007.

Notes on contributors

Hao Lyu

Hao Lyu was born in China in 1982. He received the BS and MS degrees in School of Automobile from the Chang’an University, China, in 2010 and the Ph.D. degree in mechanical engineering from Northeastern University, China, in 2014. From 2014 to now, he was a lecturer at Northeastern University. In 2020, he is a visiting scholar at CALCE, UMD. He is the author of one book, more than 10 articles, and more than 10 inventions. His research interests include calculation reliability methods, mechanical reliability, dynamic reliability, mechanical vibration, and vehicle reliability.

Hongchen Qu

Hongchen Qu received the BS degree in mechanical engineering from the University of Science and Technology Liaoning, Anshan, China, in 2019. He is currently pursuing an MS degree in mechanical engineering at Northeastern University, Shenyang, China. His main research interests include theory and method of mechanical and dynamic reliability.

Shuai Wang

Shuai Wang received a BS degree in mechanical engineering from the Taiyuan University of Science and Technology, Taiyuan, China, in 2019. He is currently pursuing an MS degree in mechanical engineering at Northeastern University, Shenyang, China. His main research interests include theory and method of mechanical and dynamic reliability.

Li Ma

Li Ma received a B.Eng. degree in vehicle engineering from the Liuzhou Institute of Technology, Liuzhou, China, in 2019. She is currently pursuing an M.Eng. degree in mechanical engineering at Northeastern University, Shenyang, China. Her main research interests include theory and method of mechanical reliability and mechanical performance degradation modeling.

Zaiyou Yang

Zaiyou Yang received an MS degree in mechanical engineering from Guangxi University of Science and Technology, Guangxi, China, in 2017. He is currently pursuing a Ph.D. in mechanical engineering at Northeastern University, Shenyang, Liaoning, China. His research interests include reliability model and simulation, reliability analysis, and optimization.