基于Weibull分布的两阶段加速寿命试验模型

doi:10.12305/j.issn.1001-506X.2025.06.22

Abstract

Abstract:

Aerospace products will go through multiple task stages in their life cycle. The existing accelerated life test (ALT) models are mainly aimed at a single task stage, so it is difficult to establish a relationship between the failure distribution of products under different stages. In this paper, two-stage ALT model with Weibull distribution is proposed. Assuming that the product life follows the Weibull distribution with different shape parameters and scale parameters in the two stages of ALT. On the basis of the tamper failure rate model, a flexible failure rate idea is introduced and two-stage ALT model is constructed to connect the failure distribution of products under different stresses. The model parameter estimation analysis is carried out based on the maximum likelihood estimation (MLE) theory under time censoring. The MLE, asymptotic confidence interval and Bootstrap confidence interval of the model parameters are obtained. The Monte Carlo simulation and the two-stage ALT data of a certain type of electrical connector are used to verify the model. The results show that the proposed model can effectively describe the life distribution of products in the two-stage ALT, reflecting the influence of the first stage test on the second stage test, whose life prediction accuracy is higher than that of the single-stage ALT model. The proposed model provides a new idea for the ALT research of products containing multiple task stages.

Key words: accelerated life test (ALT), two-stage model, Weibull distribution, failure rate, maximum likelihood estimation (MLE)

CLC Number:

Xiaodi LIU, Jianli HAN, Jianjun ZHANG, Yiqiao WU, Tianzhao MA. Two-stage accelerated life test model based on Weibull distribution[J]. Systems Engineering and Electronics, 2025, 47(6): 1941-1949.

Figures/Tables 9

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References 30

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应力组合	第一阶段振动/(g²/Hz)	第二阶段温度/℃	第二阶段拔插次数
SC1	0.2	130	13
SC2	0.4	158	11
SC3	0.6	120	9
SC4	0.8	140	7

参数	真值
r₁₀	-2
r₁₁	-1.5
σ₁	0.5
r₂₀	-3
r₂₁	2
r₂₂	-0.2
σ₂	0.4

τ	参数	MLE		Asym-CIs		Boot-CIs
τ	参数	Bias	MSE	CP	AL	CP	AL
10	r₁₀	0.005 9	0.059 5	0.960 8	1.019 2	0.951 9	0.928 6
	r₁₁	0.012 7	0.051 3	0.984 0	0.926 0	0.987 6	0.900 0
	σ₁	0.004 4	0.012 1	0.954 8	0.443 5	0.976 5	0.426 0
	r₂₀	0.043 2	0.214 4	0.998 7	8.643 9	0.985 5	0.906 7
	r₂₁	0.050 6	0.049 4	0.999 2	3.369 4	0.987 2	0.720 1
	r₂₂	0.018 0	0.000 8	0.969 9	0.178 2	0.973 2	0.072 4
	σ₂	0.028 8	0.003 4	0.996 0	0.302 0	0.991 6	0.182 1
15	r₁₀	0.005 1	0.036 6	0.981 2	0.760 9	0.973 6	0.813 7
	r₁₁	0.001 5	0.026 8	0.981 2	0.598 0	0.984 2	0.631 2
	σ₁	0.003 7	0.006 9	0.969 0	0.280 9	0.972 1	0.330 8
	r₂₀	0.071 2	0.226 9	0.998 1	9.998 8	0.984 6	0.961 6
	r₂₁	0.088 3	0.063 0	0.993 2	5.845 0	0.986 5	0.755 1
	r₂₂	0.028 4	0.001 9	0.998 9	0.553 4	0.973 2	0.100 0
	σ₂	0.033 9	0.004 1	0.997 8	0.612 2	0.974 1	0.188 8

τ	参数	MLE		Asym-CIs		Boot-CIs
τ	参数	Bias	MSE	CP	AL	CP	AL
10	r₁₀	0.005 2	0.037 5	0.971 0	0.791 6	0.962 6	0.823 9
	r₁₁	0.001 5	0.032 5	0.986 0	0.710 2	0.984 2	0.734 5
	σ₁	0.004 2	0.007 5	0.965 0	0.345 7	0.973 6	0.334 1
	r₂₀	0.041 8	0.206 9	0.999 2	6.645 4	0.986 1	0.850 0
	r₂₁	0.032 4	0.039 7	0.999 8	2.591 1	0.991 5	0.610 4
	r₂₂	0.013 0	0.000 5	0.986 0	0.140 6	0.981 2	0.058 2
	σ₂	0.021 6	0.002 3	0.980 0	0.234 4	0.973 2	0.171 3
15	r₁₀	0.004 9	0.016 2	0.976 6	0.492 6	0.964 7	0.488 5
	r₁₁	0.001 4	0.0163	0.9838	0.4975	0.9832	0.480 2
	σ₁	0.002 1	0.003 9	0.964 9	0.235 8	0.975 4	0.243 5
	r₂₀	0.054 2	0.217 7	0.998 9	8.086 9	0.987 2	0.915 0
	r₂₁	0.075 0	0.056 1	0.995 4	4.238 3	0.981 6	0.746 4
	r₂₂	0.026 8	0.001 7	0.999 0	0.410 4	0.971 9	0.096 4
	σ₂	0.031 2	0.003 2	0.996 9	0.449 2	0.975 4	0.178 9

应力组合	第一阶段振动/(g²/Hz)	第一阶段截尾时间/h	第二阶段温度/℃	第二阶段拔插次数	样本数
SC1	0.2	5.41	130	7	5
SC2	0.4	2.74	158	4	5
SC3	0.6	1.84	120	6	5
SC4	0.8	1.39	140	2	5

Two-stage accelerated life test model based on Weibull distribution

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模型	参数	MLE	95%置信区间
模型	参数	MLE	下限	上限
两阶段ALT模型	r₁₀	0.645 8	0.576 2	0.715 4
	r₁₁	-2.254 5	-2.323 7	-2.185 3
	σ₁	0.255 8	0.2269	0.284 7
	r₂₀	-24.919 6	-44.850 0	-4.983 8
	r₂₁	11.869 9	2.266 6	21.471 1
	r₂₂	-0.021 0	-0.697 3	0.655 3
	σ₂	0.453 9	0.030 6	0.869 4
单阶段ALT模型	r₁₀	0.645 8	0.576 2	0.715 4
	r₁₁	-2.254 5	-2.323 7	-2.185 3
	σ₁	0.255 8	0.226 9	0.284 7
	r₂₀	-24.742 2	-44.327 9	-5.156 6
	r₂₁	11.796 7	2.340 2	21.253 1
	r₂₂	-0.020 0	-0.694 7	0.654 7
	σ₂	0.450 0	0.031 1	0.868 9

寿命/h	本文模型	单阶段ALT	文献[30]
$\hat{\eta}_{20}$	2.642 5×10⁶	2.480 9×10⁶	2.697 9×10⁶
$\hat{t}_{0.50}$	2.237 5×10⁶	2.103 7×10⁶	—
$\hat{t}_{0.90}$	9.515 0×10⁵	9.012 0×10⁵	—
$\hat{t}_{0.999}$	1.149 0×10⁵	1.108 0×10⁵	—

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