A Sequential Inspection Model based on Risk Quantitative Constraint and Component Importance

doi:10.23940/ijpe.18.12.p7.29712982

Abstract

Abstract:

Due to aerospace equipment’s need to maintain a certain degree of safety and reduce the system risk of operation, relevant maintenance and inspection strategies should be developed to meet the requirements of risk quantitative indicators. The inertial navigation system commonly used in aerospace products is taken as an example, and a sequential inspection and maintenance model based on quantitative risk constraints and component importance is proposed in this paper. Firstly, based on the quantitative constraints of the system risk and the importance of the components, the reliability constraints of the components in the inertial navigation system are determined by the fault tree analysis method. Secondly, the Wiener process is used to establish a performance degradation model for a key component of the inertial navigation system, and the expression of real-time reliability distribution is obtained with close form by use of the first-hitting time theory. The adaptive estimation method is used to estimate the unknown parameters of the model. Once the new degradation information is available, the parameters should be updated with a Bayesian equation. Thirdly, a sequential inspection model is discussed to determine the optimal intervals to satisfy the requirements for the real-time reliability at a certain time. Finally, an example of the drift data of the gyroscopes in the inertial navigation system is given to illustrate the validity of the proposed method.

Key words: quantitative constraints of risk, importance, wiener process, real-time reliability, sequential inspection

Senyang Bai, Zhijun Cheng, Qian Zhao, Xiang Jia, and Hang Yao. A Sequential Inspection Model based on Risk Quantitative Constraint and Component Importance [J]. Int J Performability Eng, 2018, 14(12): 2971-2982.

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Figures/Tables 6

Figure 1

Figure 2

Figure 3

Figure 4

Table 1

The calculation results of sequential inspection interval (${{r}^{*}}=0.903$)"

The serial number	${{t}_{ik}}/h$	${{x}_{ik}}/({}^\circ /h)$	${{a}_{ik}}$	${{D}_{ik}}$	${{Q}_{ik}}$	${{({{\sigma }^{2}})}_{ik}}$	$\Delta {{\hat{t}}_{ik}}/h$	Remarks
The initial state (${{t}_{0}}=0$)	0.00000	0.00000	0.04426	0.00017	0.00737	0.00053	4.71120	The degradation data exceeds the failure threshold 0.6 $({}^\circ /h)$ at the sixth inspection, which needs to be replaced.
${{t}_{01}}$	4.7112	0.19203	0.04419	0.00016	0.00028	0.00083	4.71120
^${{t}_{02}}$	9.4224	0.35605	0.04273	0.00012	0.00015	0.00059	3.80590
${{t}_{03}}$	13.2283	0.27352	0.04106	0.00008	0.00100	0.00434	3.38730
${{t}_{04}}$	16.6156	0.31776	0.04068	0.00007	0.00068	0.00356	3.01020
${{t}_{05}}$	19.6258	0.30006	0.04012	0.00007	0.00050	0.00285	3.58900
${{t}_{06}}$	23.2148	1.84797	——	——	——	——	——
Replace as new (${{t}_{1}}$)	23.2148	0.00000	0.04012	0.00007	0.00050	0.00285	8.75680	The degradation data exceeds the failure threshold at the fifth inspection, which needs to be replaced.
${{t}_{11}}$	31.9716	0.28173	0.03950	0.00006	0.00024	0.00104	5.26090
${{t}_{12}}$	37.2325	0.19451	0.03773	0.00005	0.00101	0.00657	4.05290
${{t}_{13}}$	41.2854	0.31754	0.03721	0.00005	0.00068	0.00546	2.60630
${{t}_{14}}$	43.8917	0.49501	0.03660	0.00005	0.00052	0.00515	0.54880
${{t}_{15}}$	44.4405	0.62656	——	——	——	——	——
Replace as new (${{t}_{2}}$)	44.4405	0.00000	0.03660	0.00005	0.00052	0.00515	7.92460	The degradation data exceeds the failure threshold at the fourth inspection, which needs to be replaced.
${{t}_{21}}$	52.3651	0.20505	0.03616	0.00005	0.00032	0.00122	6.86210
${{t}_{22}}$	59.2272	0.20363	0.03453	0.00004	0.00031	0.00273	5.85850
${{t}_{23}}$	65.0857	0.49155	0.03359	0.00004	0.00025	0.00410	0.69760
${{t}_{24}}$	65.7833	0.66387	——	——	——		——
…	…		...	…	…	…		...

Table 1

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${{R}^{*}}$	The constraint value of system reliability	${{\theta }_{k}}$	The model parameter$(a_{0k}^{(i+1)},D_{0k}^{(i+1)},Q_{k}^{(i+1)},({{\sigma }^{2}})_{k}^{(i+1)})$
${{r}^{*}}$	The constraint value of components reliability	$\Delta {{t}_{ik}}$	Inspection interval
${{I}_{i}}$	Importance of the components	${{t}_{ik}}$	Time of k^th inspection after i^th maintenance
$\Delta {{r}_{i}}$	The reliability of each component which need to be improved	${{x}_{ik}}$	Degradation data of k^th inspection after i^th maintenance
$\mu $	Drift coefficient	${{x}_{i(0:k)}}$	Set of degradation data after i^th maintenance
$\sigma $	Diffusion coefficient	${{l}_{k}}$	The remaining life of the product at the time ${{t}_{k}}$
$w$	Failure threshold	${{f}_{{{L}_{k}}\|{{X}_{0:k}}}}({{l}_{k}}\|{{X}_{0:k}})$	The PDF of the remaining life for the product
$a$	The mean of $\mu $	${{R}_{S}}(t\|{{x}_{k}})$	Real-time reliability of the product at time$t$
${{D}_{k\|k}}$	The updated variance of$\mu $	${{R}_{S}}({{l}_{ik}}\|{{X}_{0:k}})$	Real-time reliability of the remaining life for the product