3.2.1. Parameters of failure rate model

For torpedoes, technical parameter data is an important indicator. This study uses the service life and design information to estimate the parameters of the inherent failure rate model, in which the degradation function is ${\lambda _a}(t)$.

Assume that the initial reliability is ${R_0}$, the failure threshold is ${R_{\text{out}}}$ and the life in the standard storage environment is ${T_{\text{use}}}$ years. Without considering the impact of detection and repair on the inherent failure rate (${R_b}(t) = 1$), the parameters should make the model meet the following conditions:

1. 1) The failure rate shows an increasing trend during the service life,

2. 2) In the early and middle periods of life, the failure rate increases slowly. In the later period, it increases faster. The failure rate function approaches or reaches the inflection point when service ends,

3. 3) Within the time range $[0,\text{ }{T_{\text{use}}}]$, Equation (2.1) should satisfy $R({T_{\text{use}}}) \approx {R_{\text{out}}}$ based on the Equations (3.4)– (3.6).

Parameter adjustment is similar to PID adjustment. According to Equation (3.1), the average failure rate inherently is ${\bar{\lambda }_{gu}} ={-} \ln ({R_{\text{out}}}/{R_0})/{T_{\text{use}}}$.

Firstly, use the value of ${\bar{\lambda }_{gu}}$ to estimate $a,\text{ }b$ roughly. The progress is:

• • Let $a = 0.01$ and take the b form 1. With the increase of b value, the curve moves downwards until the average failure rate inherently ${\bar{\lambda }_{gu}}$ is between the first inflection point and the flat area.

• • Then adjust the a value to control the shape until the length of the gentle area in the middle is close to life ${T_{\text{use}}}$.

• • Take the point near the minimum value of the bathtub curve as the storage start time, and reversely take ${T_a} - {T_{\text{use}}}$ as the end time ${T_b}$.

Secondly, adjust the values of a and b exactly. The progress is:

• • Making $\lambda (t),\text{ }{T_b} < t \le {T_a}$ to satisfy 3), as shown in Figure 6(a).

• • Let $T = {T_a}$, using the parameter T to adjust the time in order to obtain the inherent failure rate function model shown in Equation (3.4), as shown in Figure 6(b) that the horizontal axis is the service time and the vertical axis is failure rate.

Figure 6. Failure rate curve

Assuming that the initial reliability is $R({t_i}^ + )$ and the environmental coefficient is ${\pi _E}$ in a stage, the reliability caused by performance degradation is:

(3.7)\begin{equation}{R_a}(t) = R({t_i}^ + ){e^{ - \int_{{t_i}}^t {{\pi _E}{\lambda _a}(t)dt} }}\end{equation}

When the torpedo in the first stage of service life, $R({t_i}^ + ) = {R_0}$.

3.2.2. Detection parameters of failure rate

The torpedo belongs to repairable components. Regular detection is necessary for long-term storage. The reliability ${R_b}(t)$ determined by the increase in failure rate due to detection is

(3.8)\begin{equation}\begin{aligned} {R_b}(t) &= {e^{[ - {\Delta _\lambda }_{bk}(t - {t_k})]}}\\ {\Delta _\lambda }_{bk} &= {\lambda _b}(k) - {\lambda _b}(k - 1)\\ {\lambda _b}(k) &= {\lambda _0}{(k + 1)^\beta } \end{aligned}\end{equation}

where ${\lambda _0}$ is inherent failure rate at the initial moment, $k$is detection times, ${\lambda _b}(k)$ is failure rate after the $k\text{th}$ detection (Li et al., Reference Li, Wang and Zhang2006), ${t_k}$is the time interval from the initial moment to the last detection and $\beta$ reflects the increasing trend of the failure rate with the detection times. For the convenience of the following description, this study defines it as the detection parameters of failure rate.

According to the technical manual of torpedoes, regular detection is required for long-term storage. Assuming that the detection cycle is $\tau$ years, the reliability declines from ${R_0}$ to ${R_{\text{out}}}$ during the service life without considering the impact of performance degradation. In this study, the $\lambda (t = 0)$ is taken as the initial failure rate ${\lambda _0}$, and the detection parameter $\beta$ is estimated.

When the life of components is ${T_{\text{use}}}$ in the standard storage environment, detection is carried out every $\tau$ year, the initial reliability is ${R_0}$, and the failure threshold is ${R_{\text{out}}}$, then the max number of detection in the service life is

(3.9)\begin{equation}{k_{\max }} = [{T_{\text{use}}}/\tau ]\end{equation}

where [] indicates rounding, and the corresponding reliability is:

(3.10)\begin{equation}{R_{\text{out}}} = {R_0}{\text{e}^{ - [{\lambda _0} + \sum\nolimits_{i = 1}^{{k_{\max }} - 1} {{\lambda _b}(i)} ]\tau }}\end{equation}

We can get the parameter value $\hat{\beta }$ by solving Equation (3.10). If we consider the impact of performance degradation, the actual value $\beta$ is less than $\hat{\beta }$. Then the range is: $0 < \beta < \hat{\beta }$.

3.3.1. Relationship of the adjacent stage

At the time ${t_i}$ of the $i\text{th}$ detection, the personnel tests the components and the detection information is obtained. If a failure is found, the staff will repair it. If there is no failure, the component will continue to be stored. At the time ${t_i}$ of the $i\text{th}$ detection, the personnel tests the components and the detection information is obtained. If a failure is found, the staff will repair it. If there is no failure, the component will continue to be stored. When the end of the previous stage is $t_{i}^{-}$, the start of the next stage is $t_{i}^{+}$, the reliability of the detection moment is ${R_{ti}}$, the environmental coefficient is 1, the reliability of products that only under the condition of performance degradation is ${R_{guti}}$, then the relationship of reliability before and after detection is

(3.11)\begin{equation}R(t_{i}^+) = \begin{cases} R_{ti} & {\text{no failure}}\\ {{R_{\text{guti}}} = {R_0}{\text{e}^{ - \int_0^{{t_i}} {{\lambda_a}(t)dt} }}} & {\text{repairable failure}} \end{cases} \end{equation}

3.3.2. Reliability estimation at detection moment

In this study, failure rate function, detection information and Bayes (Han, Reference Han2016; Wan et al., Reference Wan, Chen, Ouyang and Zhang2018; Li et al., Reference Li, Zhu X and Li2019a) are applied to estimate the reliability at the detection moment. And these pieces of information are defined that the stage before the ith detection is i and the initial reliability is $R({t_{i - 1}}^ + )$. Then in stage i, the value range of failure rate is

(3.12)\begin{align}&\left( {{\lambda_{\min }}(t) = \lambda ({t_{i - 1}}) + \int_{{t_{i - 1}}}^t {{\pi_E}} \frac{{{\lambda_a}(t)}}{{dt}}} \right) < \lambda (t)\notag\\ &\quad < \left( {\lambda ({t_{i - 1}}) + \int_{{t_{i - 1}}}^t {{\pi_E}} \frac{{{\lambda_a}(t)}}{{dt}} + {\Delta_\lambda }_{bk} = {\lambda_{\max }}(t)} \right),{t_{i - 1}} \le t < {t_i}\end{align}

The reliability in stage i is $R(t) = R({t_{i - 1}}^ + ){e^{ - \int_{{t_{i - 1}}}^t {{\lambda _{\max }}(t)dt} }}$.

In the $i\text{th}$ detection, the number of samples is ${n_i}$, and the number of failures is ${f_i}$. Under the binomial distribution and square loss, the Bayes assessment of reliability at the $i\text{th}$ detection moment is (Han, Reference Han2005)

(3.13)\begin{equation}{\hat{R}_i} = \int_{{R_{i1}}}^{{R_{i2}}} {\left( {\frac{{{R^{{n_i} + q - {r_i}}}{{(1 - R)}^{{r_i}}}}}{{\int_{{R_{i1}}}^{{R_{i2}}} {{R^{{n_i} + q - {r_i} - 1}}{{(1 - R)}^{{r_i}}}dR} }}} \right)} dR\end{equation}

where ${R_{i1}} = R({t_{i - 1}}^ + ){e^{ - \int_{{t_{i - 1}}}^t {{\lambda _{\min }}(t)dt} }}$, ${R_{i2}} = R({t_{i - 1}}^ + ){e^{ - \int_{{t_{i - 1}}}^t {{\lambda _{\max }}(t)dt} }}$, $q = {R_{i2}}/\text{1} - {R_{i2}}$.

The failure rate and reliability in stage i are corrected by the evaluation result ${\hat{R}_i}$ at the $i\text{th}$ detection moment and the increment of failure rate at the $(i - 1)\text{th}$ detection moment is used to update the detection parameter $\hat{\beta }$. The process is as follows:

1. 1) Calculate the average failure rate ${\bar{\lambda }_i} ={-} \ln ({\hat{R}_i}/R({t_{i - 1}}^ + ))/({t_i} - {t_{i - 1}})$ in the stage i.

   1. Calculate the average failure rate ${\bar{\lambda }_{\text{Mini}}} = (\int_{{t_{i - 1}}}^{{t_i}} {{\lambda _{\min }}(t)dt} )/({t_i} - {t_{i - 1}})$ in the stage i without considering the impact of the $(i - 1)\text{th}$ detection.

   2. Calculate the actual increment ${\Delta _{{\lambda _{i - 1}}}} = {\bar{\lambda }_i} - {\bar{\lambda }_{Mini}}$ of failure rate caused by the $(i - 1)\text{th}$ detection.

2. 2) In the stage i, the failure rate is ${\lambda _i}(t) = {\lambda _{\min }}(t) + {\Delta _{{\lambda _{i - 1}}}}$, the reliability is ${R_{i2}} = R({t_{i - 1}}^ + ){e^{ - \int_{{t_{i - 1}}}^t {{\lambda _i}(t)dt} }}$.

3. 3) Calculate the increment ${\lambda _b}(i - 1) - {\lambda _0} = {\lambda _0}{(i)^\beta } - {\lambda _0} = \sum\nolimits_{k = 1}^{i - 1} {{\varDelta _{{\lambda _{k - 1}}}}}$ of failure rate caused by the $(i - 1)\text{th}$ detection relative to the initial failure rate, the new parameter values $\hat{\beta }$ is obtained by solving this equation.