Abstract
For the system degradation process undergoing a sudden change, optimal maintenance policies were developed using the cumulative damage model and twostage degradation modeling. Single shock damage value and the number of shock times are assumed to be normal distribution and homogeneous Poisson process, respectively. On this basis, average longrun cost rate of a renewal cycle was modeled with considering the probabilities of corrective, preventive and continuous monitoring, respectively. In order to develop an optimal policy, four types of maintenance policies (i.e., global, timedepended, adaptive and simplified adaptive policies) were analyzed with different alarm thresholds and interinspection time. Influence analysis of different parameters for maintenance policy was given, where different maintenance policies were compared in terms of average longrun cost rate. In addition, the impacts of degradation model parameters (i.e., changepoint distribution, shock strength, shock frequency) on the average longrun cost rate were analyzed. Finally, maintenance policy for gearbox degradation experiment was analyzed in case study.
1. Introduction
Performance degradation is a common phenomenon in many systems, especially in mechanical and structural systems. Deterioration modeling plays a more and more important role in maintenance decisionmaking. Many researchers have mainly studied intensively systems degradation with stationary processes to optimize maintenance problems [14]. However, the degradation processes for many systems are nonstationary due to internal mechanism or external environment influences etc. [5]. For example, some systems are deteriorating in a process of two stages [511], where the degradation rate is usually small in the first stage and large in the second stage.
In order to study system degradation, it is necessary to establish corresponding degradation model. The deterioration process model with independent random increment is divided into two types, continuous time model and cumulative damage model. The continuous time model [12, 13, 14] presents system degradation in terms of continuous time stochastic process. The most representative models for continuous time model are Brownian motion model and Gamma process model. Whereas, in the cumulative damage model [15, 16, 17], it is assumed that system degradation process is discrete. The model describes the degradation process by cumulating a number of random increments caused by damage in the system operation. Existing studies for twostage degraded system mainly focused on continuous time model. Application of cumulative damage model to twostage degradation process was seldom investigated.
Due to the discrepancy of different deterioration modes for a twostage degraded system, maintenance decisionmaking methods for singlestage degradation process system cannot be applied to twostage deteriorating mode system. In order to solve the problem, In order to solve the problem, studies have been done and a number of maintenance strategies were developed. An activation zone method for maintenance decisionmaking is presented by Saassouh [6] with considering the random change of degradation rate, but the degradation rate change time is assumed to be continuous and perfectly monitored. Ponchet [12] developed two maintenance decisionmaking methods with and without considering the deterioration mode change in system degradation processes, respectively. The results of numerical example show that it can bring considerable benefits if a policy with changeable thresholds was used. A predictive maintenance policy for a system with two deterioration mode based on process data was proposed by Zhao [18], and the maintenance actions were implemented based on different reliability thresholds. Some maintenance policies for twostage deteriorating mode systems have been presented, but no much study has been done to investigate the performance of different maintenance polices with different thresholds and inspection intervals.
In this paper, degradation modeling and maintenance decisionmaking methods for twostage deteriorating mode system based on cumulative damage model will be investigated. The main contributions of this study are: (a) Cumulative damage model is used for twostage degradation modeling, and it shows system degradation rate change through different shock strengths and shock frequencies. (b) An optimal policy of a twostage degraded system is developed by analyzing and comparing four types of maintenance policies (global, timedepended, adaptive, simplified adaptive).
The remainder of this paper is organized as follows. Section 2 is devoted to twostage deteriorating modeling based on cumulative damage model. Section 3 studies four kinds of maintenance policies and analyzes maintenance policy evaluation method. Numerical examples are used to analyze the influence of different factors on maintenance policies in Section 4. Conclusions are made in Section 5.
2. Twostage deteriorating modeling
2.1. System description
The considered system is assumed to be an observable system which degrading stochastically. The degradation level at time $t$ is supposed to be presented by a random variable $Y\left(t\right)$ [19, 20]. The system degradation process is an increasing stochastic process with initial state $Y\left(t\right)=0$. System will be declared as failed when deterioration level $Y\left(t\right)$ exceeds a failure threshold ${Y}_{f}$ (namely $Y\left(t\right)\ge {Y}_{f}$). ${T}_{f}$ is defined as system failure time. System failed does not mean that the system cannot work, but implies that it’s economic and safety impacts will be unacceptable if it still in operation.
Fig. 1Twostage degradation process
System degradation rate changes at time ${t}_{c}$ during a life cycle (as shown in Fig. 1). It means that the parameters of system degradation process undergo transitions at changepoint ${t}_{c}$. The system is supposed to be in a nominal degradation mode and denoted by ${M}_{1}$ before ${t}_{c}$. After ${t}_{c}$, the system degradation mode evolves to an accelerated mode ${M}_{2}$. Degradation rate is usually small in the first stage ${M}_{1}$ and large in the second stage ${M}_{2}$. Therefore, system degradation process can be modeled by two stochastic processes under the same law but with different parameters [6].
Most physical degradation observation and the property of the Levy processes have shown that system deterioration can be thought as the accumulation of large numbers of small shocks [21], and deterioration level can be defined as the sum of damage values due to shock. When the system is in mode ${M}_{k}$, damage values ${x}_{ki}$ ($k=$ 1, 2; $i=$ 0, 1, 2, …) are assumed to be normally distribution, ${x}_{ki}~N\left({u}_{k},{\sigma}_{k}^{2}\right)$. ${x}_{ki}$ is a constant in mode ${M}_{k}$ (see Fig. 1). Random variable ${N}_{S}$ represents the number of shock times during time [0, ${t}_{s}$] and it obeys to homogeneous Poisson distribution with strength ${\lambda}_{k}$ in mode ${M}_{k}$. Therefore, the probability of shock times ${N}_{S}$ just as $n$ in mode ${M}_{k}$ is:
For the convenience of expression, in this paper ${x}_{ki}$ is denoted as shock strength and $1/{\lambda}_{k}$ is denoted as shock frequency. It is not difficult to find that shock strength and shock frequency decide the size of degradation rate.
2.2. Degradation level modeling
In the twostage degradation process, the value of damage time ${t}_{s}$ may be in the first stage ($0\le {t}_{s}\le {t}_{c}$) or in the second stage (${t}_{s}>{t}_{c}$). The calculation methods of degradation level are not alike in different degradation stage. According to the above notation, degradation level at time ${t}_{s}$ can be written as:
where ${Y}_{{t}_{s}}^{k}$stands for degradation level in mode ${M}_{k}$, ${I}_{\left\{E\right\}}=1$if $E$ is true and ${I}_{\left\{E\right\}}=0$ otherwise. When $0\le {t}_{s}\le {t}_{c}$, the degradation level is:
In this case, the probability of shock times equal to $n$ within time [0, ${t}_{s}$] is the same to Eq. (1). Due to every shock damage is independently and unrelated, it can be known from the characters of normal distribution (the sum of normal distribution parameters still in line with normal distribution) that $Y$ is obey to normal distribution, namely:
When ${t}_{s}>{t}_{c}$, degradation level consists of the damage in first stage ${M}_{1}$ and the damage in second stage ${M}_{2}$. In this case, degradation time length in the first stage is ${t}_{c}$, ${t}_{s}\u2013{t}_{c}$ to the second stage. Hence, the system degradation level is:
Similar to the Eq. (4), system degradation level is in line with normal distribution, there is:
Because shocks between the two stages are independently, the probability of the number of shock times just as $m$ in mode ${M}_{1}$ and equal to $n$ in mode ${M}_{2}$ is:
2.3. Reliability modeling
In engineering practice, the changepoint ${t}_{c}$ for degradation rate is not fixed, but is distributed in a certain time interval. Shown as Fig. 1, the time distribution interval of changepoint ${t}_{c}$ for degradation rate is [${t}_{A}$, ${t}_{B}$], in other words, the deteriorating mode may change at any time from ${M}_{1}$ to ${M}_{2}$ when system works during time [${t}_{A}$, ${t}_{B}$].
System reliability is the probability for degradation level $Y$ less than failure threshold ${Y}_{f}$ when damage time is ${t}_{s}$. Shock strength, shock frequency and changepoint should be considered in reliability modeling, which are main factors in cumulative damage model. Reliability modeling for twostage degraded system is specific expressed as following.
When $0\le {t}_{s}\le {t}_{c}$, system reliability is:
$=\sum _{m=0}^{\infty}\mathrm{\Phi}\left(\frac{{Y}_{f}m{\mu}_{1}}{\sqrt{m}{\sigma}_{1}}\right)\cdot \frac{{\left({\lambda}_{1}{t}_{s}\right)}^{m}}{m!}\cdot {e}^{{\lambda}_{1}{t}_{s}}.$
When ${t}_{s}>{t}_{c}$, system reliability is:
$=\sum _{m=0}^{\infty}\sum _{n=0}^{\infty}\mathrm{\Phi}\left(\frac{{Y}_{f}\left(m{\mu}_{1}+n{\mu}_{2}\right)}{\sqrt{m{\sigma}_{1}^{2}+n{\sigma}_{2}^{2}}}\right)\cdot P\left({N}_{1}=m,{N}_{2}=m\right)$
$=\sum _{m=0}^{\infty}\sum _{n=0}^{\infty}\mathrm{\Phi}\left(\frac{{Y}_{f}\left(m{\mu}_{1}+n{\mu}_{2}\right)}{\sqrt{m{\sigma}_{1}^{2}+n{\sigma}_{2}^{2}}}\right)\cdot \frac{{\lambda}_{1}^{m}\cdot {\lambda}_{2}^{n}}{m!\cdot n!}\cdot {\int}_{{t}_{A}}^{{t}_{B}}\left({e}^{{\lambda}_{1}\tau {\lambda}_{2}\left({t}_{s}\tau \right)}{\tau}^{m}{\left(t\tau \right)}^{n}g\left(\tau \right)\right)d\tau ,$
where $g\left(t\right)$ is the probability density distribution function for changepoint during time [${t}_{A}$, ${t}_{B}$].
3. Maintenance policies
Research of maintenance decisionmaking is one of the focuses for system degradation modeling. Conditionbased maintenance policy widely uses in various systems, which is structured according to the information available through online monitoring [12]. In order to reduce maintenance costs, preventive maintenance actions take place before system failure by monitoring. In other words, suitable monitoring method and maintenance policy can help to improve the efficiency and profitability of a system.
The selections of alarm threshold and interinspection time are the keys to maintenance policy. According to different alarm thresholds and interinspection times, this paper considers four kinds of maintenance decisionmaking methods. The first kind of method is global maintenance policy. There are just one alarm threshold and one interinspection time in this method, which are constants and never change. The second kind of method is timedepended maintenance policy. There are also one alarm threshold and one interinspection time in this method, but the interinspection time is change with system working time. The next kind of method is adaptive maintenance policy. Different alarm thresholds and interinspection times corresponding to different degradation rates, that is say, there are two alarm thresholds and two interinspection times in this method. The finally kind of method is simplified adaptive maintenance policy. This method is similar to adaptive maintenance policy, but it just has one interinspection time.
In the framework of this study, there are three possible maintenance actions, inspection, preventive maintenance and corrective maintenance, respectively. System is perfectly monitored through periodic monitor, and system state restores to be as good as new after preventive maintenance or corrective maintenance with negligible time.
3.1. Global maintenance policy
In order to show the importance of considering the changes of system degradation rate, traditional maintenance decisionmaking method is presented in the first place, which called global maintenance policy. The method just defines a single alarm threshold ${Y}_{A}$ and a single interinspection time $\u2206T$, as done in Dieulle et al. [22]. It is not difficult to find that the method only pay attention to system degradation level and ignore the degradation rate.
The possible maintenance actions which can put into practice after inspection time ${T}_{i}$ are defined as follows:
• If $Y\left({T}_{i}\right)<{Y}_{A}$, do nothing and the system is left as it is until next inspection time ${T}_{i+1}={T}_{i}+\u2206T$.
• If ${Y}_{A}\le Y\left({T}_{i}\right)<{Y}_{f}$, the system is too badly deteriorated so it is necessary to perform preventive maintenance.
• If $Y\left({T}_{i}\right)\ge {Y}_{f}$, the system is considered as failed and it has to be performed corrective maintenance.
The rule of global maintenance policy is illustrated in Fig. 2.
Fig. 2Global maintenance policy
3.2. Timedepended maintenance policy
As the degradation rate in mode ${M}_{2}$ larger than in mode ${M}_{1}$, the interinspection time should be shorter and shorter in term of work time. This kind of maintenance decisionmaking method called timedepended maintenance policy. For example, the interinspection time of $i$th monitor is $\u2206{T}_{i}$, the next interinspection time is $\u2206{T}_{i+1}$, $\u2206{T}_{i+1}=q\xb7\u2206{T}_{i}$ and $q<1$.
The possible maintenance actions which can put into practice after inspection time ${T}_{i}$ are defined as follows:
• If $Y\left({T}_{i}\right)<{Y}_{A}$, do nothing and the system is left as it is until next inspection time ${T}_{i+1}={T}_{i}+\u2206{T}_{i+1}$.
• If ${Y}_{A}\le Y\left({T}_{i}\right)<{Y}_{f}$, the system is too badly deteriorated so it is necessary to perform preventive maintenance.
• If $Y\left({T}_{i}\right)\ge {Y}_{f}$, the system is considered as failed and it has to be performed corrective maintenance.
The rule of timedepended maintenance policy is illustrated in Fig. 3.
Fig. 3Timedepended maintenance policy
3.3. Adaptive maintenance policy
According to the characteristics of system degradation rate suddenly changes from nominal mode ${M}_{1}$ to accelerated mode ${M}_{2}$, Saassouh et al. [6, 12] put forward adaptive maintenance policy. The method is different with global maintenance policy, it considers system degradation level and degradation rate. As a result, this maintenance decisionmaking method is more responsive to systems with twostage deteriorating mode.
The alarm threshold and interinspection time for adaptive maintenance policy are defined as follows:
Set ${Y}_{nom}$ as the alarm threshold and $\u2206{T}_{nom}$ is the interinspection time for nominal degradation mode ${M}_{1}$. When the inspection time ${T}_{i}$ is less than changepoint ${t}_{c}$ (${T}_{i}\le {t}_{c}$), the possible maintenance actions which can put into practice are defined as follows:
• If $Y\left({T}_{i}\right)<{Y}_{nom}$, do nothing and the system is left as it is until next inspection time ${T}_{i+1}={T}_{i}+\u2206{T}_{nom}$.
• If ${Y}_{nom}\le Y\left({T}_{i}\right)<{Y}_{f}$, the system is too badly deteriorated so it is necessary to perform preventive maintenance.
• If $Y\left({T}_{i}\right)\ge {Y}_{f}$, the system is considered as failed and it has to be performed corrective maintenance.
Set ${Y}_{acc}$ as the alarm threshold and $\u2206{T}_{acc}$ is the interinspection time for accelerated degradation mode ${M}_{2}$. When the inspection time ${T}_{j}$ is greater than changepoint ${t}_{c}$ (${T}_{j}>{t}_{c}$), the possible maintenance actions which can put into practice are defined as follows:
• If $Y\left({T}_{j}\right)<{Y}_{acc}$, do nothing and the system is left as it is until next inspection time ${T}_{j+1}={T}_{j}+\u2206{T}_{acc}$.
• If ${Y}_{acc}\le Y\left({T}_{j}\right)<{Y}_{f}$, the system is too badly deteriorated so it is necessary to perform preventive maintenance.
• If $Y\left({T}_{j}\right)\ge {Y}_{f}$, the system is considered as failed and it has to be performed corrective maintenance.
As the degradation rate for mode ${M}_{2}$ is greater than mode ${M}_{1}$, so the maintenance policy parameters ${Y}_{acc}<{Y}_{nom}$ and $\u2206{T}_{acc}<\u2206{T}_{nom}$. The rule of adaptive maintenance policy is illustrated in Fig. 4.
Fig. 4Adaptive maintenance policy
3.4. Simplified adaptive maintenance policy
Adaptive maintenance policy so complex that not suitable for engineering application. Therefore, adaptive maintenance policy is simplified in application by some researchers [12]. Interinspection time $\u2206T$ is a constant value and never changes in simplified adaptive maintenance policy.
The maintenance policy rule (alarm threshold, possible maintenance action) is similar to adaptive maintenance policy, the only difference is that: no matter $Y\left({T}_{i}\right)<{Y}_{nom}$ or $Y\left({T}_{i}\right)<{Y}_{acc}$, the next inspection time always ${T}_{i+1}={T}_{i}+\u2206T$ (namely $\u2206{T}_{nom}=\u2206{T}_{acc}=\u2206T$).
3.5. Maintenance policy evaluation
3.5.1. Evaluation method
Maintenance cost occurs when a maintenance action is performed. In this study the average longrun cost rate over an infinite time span is used to evaluate maintenance policy. As it has assumed that system state restores to as good as new if a preventive/corrective maintenance action performed, renewal reward theory [23] can be used to calculate the average longrun cost rate as follows:
where $C\left(t\right)$ is the total maintenance cost at time $t$, $T$ is the average time length of a renewal cycle.
The total maintenance cost in a renewal cycle $T$ can be expressed as follows:
The expected time length of a renewal cycle $T$ is written as:
Adaptive maintenance policy is the most complex method relative to other three maintenance policies, which parameters obtained more difficult. In this paper, parameters obtained method of adaptive maintenance policy are mainly analyzed, parameters for other three maintenance policy can also be obtained as this method.
3.5.2. Probability of corrective maintenance
If any one event of the following events (${A}_{C1}$, ${A}_{C2}$, ${A}_{C3}$) occurs, system is considered as failure. That is to say, system needs corrective replacement and it will cause corrective maintenance cost ${C}_{C}$. Take the event ${A}_{C1}$ as a example, system degradation process is in stage ${M}_{1}$ (${T}_{z1}<{T}_{z}\le {t}_{c}$), if the degradation level $Y\left({T}_{z1}\right)<{Y}_{nom}$ for $(z1)$th inspection and $Y\left({T}_{z}\right)>{Y}_{f}$ for $z$th inspection, corrective maintenance action will be performed.
${A}_{C2}=\left\{Y\left({T}_{z1}\right)<{Y}_{acc}\bigcap Y\left({T}_{z}\right)\ge {Y}_{f}\bigcap {t}_{c}<{T}_{z1}\le {T}_{z}\right\},$
${A}_{C3}=\left\{Y\left({T}_{z1}\right)<{Y}_{nom}\bigcap Y\left({T}_{z}\right)\ge {Y}_{f}\bigcap {T}_{z1}<{t}_{c}\le {T}_{z}\right\}.$
The probability for a corrective maintenance in a renewal cycle is expressed as:
System cumulative damage distribution is the probability for system degradation level $Y$ less than a certain value $y$ when shock time is ${t}_{s}$. The formula of cumulative damage distribution in stage ${M}_{1}$ can be denoted as:
Therefore, the density function of cumulative damage distribution in stage ${M}_{1}$ is:
The specific analytic formula of the probability for corrective maintenance event ${A}_{C1}$ is expressed as follows:
$=\sum _{z=1}^{\infty}P\left(Y\left({T}_{z1}\right)<{Y}_{nom},Y\left({T}_{z}{T}_{z1}\right)\ge {Y}_{f}Y\left({T}_{z1}\right)\right)\cdot P\left({T}_{z}\le {t}_{c}\right)$
$=\sum _{z=1}^{\infty}P\left(\sum _{i=0}^{N\left({T}_{z1}\right)}{x}_{1i}<{Y}_{nom},\sum _{j=0}^{N\left({T}_{z}{T}_{z1}\right)}{x}_{1j}\ge {Y}_{f}\sum _{i=0}^{N\left({T}_{z1}\right)}{x}_{1i}\right)\cdot P\left({T}_{z}\le {t}_{c}\right)$
$=\sum _{z=1}^{\infty}\sum _{m=0}^{\infty}\sum _{n=0}^{\infty}P\left(\sum _{i=0}^{m}{x}_{1i}<{Y}_{nom},\sum _{j=0}^{n}{x}_{1j}\ge {Y}_{f}\sum _{i=0}^{m}{x}_{1i}\right)\cdot P\left(N\left({T}_{z1}\right)=m\right)\cdot P\left(N\left({T}_{z}{T}_{z1}\right)=n\right)\cdot P\left({T}_{z}\le {t}_{c}\right)$
$=\frac{1}{2\pi}\sum _{z=1}^{\infty}\sum _{m=0}^{\infty}\sum _{n=0}^{\infty}{\int}_{0}^{{Y}_{nom}}{\int}_{{Y}_{f}u}^{{Y}_{f}}{{e}^{\frac{{\left(um{\mu}_{1}\right)}^{2}}{2m{\sigma}_{{}_{1}}^{2}}}}^{\frac{{\left(\tau n{\mu}_{1}\right)}^{2}}{2n{\sigma}_{{}_{1}}^{2}}}d\tau du\cdot P\left(N\left({T}_{z1}\right)=m\right)\cdot P\left(N\left({T}_{z}{T}_{z1}\right)=n\right)\cdot P\left({T}_{z}\le {t}_{c}\right),$
where, $P\left(N\left({T}_{z1}\right)=m\right)$ is the probability of the number of shock times just as $m$ during time $\left[0,{T}_{z1}\right]$, $P\left(N\left({T}_{z}{T}_{z1}\right)=n\right)$ is the probability of the number of shock times equal to $n$ within time $\left[{T}_{z1},{T}_{z}\right]$, $P\left({T}_{z}\le {t}_{c}\right)$ is the probability for system degradation in the first stage ${M}_{1}$.
The probability for corrective maintenance events ${A}_{C2}$, ${A}_{C3}$ can be expresses as follows:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}=\frac{1}{\sqrt{8{\mathrm{\pi}}^{3}}}\sum _{\mathrm{z}=1}^{\mathrm{\infty}}\sum _{\mathrm{m}=0}^{\mathrm{\infty}}\sum _{\mathrm{n}=0}^{\mathrm{\infty}}\sum _{\mathrm{l}=0}^{\mathrm{\infty}}\left(\begin{array}{c}{\int}_{0}^{{\mathrm{Y}}_{\mathrm{a}\mathrm{c}\mathrm{c}}}{\int}_{0}^{{\mathrm{Y}}_{\mathrm{a}\mathrm{c}\mathrm{c}}\mathrm{u}}{\int}_{{\mathrm{Y}}_{\mathrm{f}}\mathrm{u}\mathrm{\omega}}^{\mathrm{\infty}}{{\mathrm{e}}^{\frac{{\left(\mathrm{u}\mathrm{m}{\mathrm{\mu}}_{1}\right)}^{2}}{2\mathrm{m}{\mathrm{\sigma}}_{{\mathrm{}}_{1}}^{2}}}}^{\frac{{\left(\mathrm{\omega}\mathrm{n}{\mathrm{\mu}}_{2}\right)}^{2}}{2\mathrm{n}{\mathrm{\sigma}}_{{\mathrm{}}_{2}}^{2}}\frac{{\left(\mathrm{\tau}\mathrm{n}{\mathrm{\mu}}_{2}\right)}^{2}}{2\mathrm{n}{\mathrm{\sigma}}_{{\mathrm{}}_{2}}^{2}}}\mathrm{d}\mathrm{\tau}\mathrm{d}\mathrm{\omega}\mathrm{d}\mathrm{u}\cdot \\ \mathrm{P}\left(\mathrm{N}\left({\mathrm{t}}_{\mathrm{c}}\right)=\mathrm{m}\right)\cdot \mathrm{P}\left(\mathrm{N}\left({\mathrm{T}}_{\mathrm{z}1}{\mathrm{t}}_{\mathrm{c}}\right)=\mathrm{n}\right)\cdot \mathrm{P}\left(\mathrm{N}\left({\mathrm{T}}_{\mathrm{z}}{\mathrm{T}}_{\mathrm{z}1}\right)=\mathrm{l}\right)\cdot \mathrm{P}\left({\mathrm{T}}_{\mathrm{z}1}>{\mathrm{t}}_{\mathrm{c}}\right)\end{array}\right),$
$=\frac{1}{\sqrt{8{\pi}^{3}}}\sum _{z=1}^{\infty}\sum _{m=0}^{\infty}\sum _{n=0}^{\infty}\sum _{l=0}^{\infty}\left(\begin{array}{c}{\int}_{0}^{{Y}_{nom}}{\int}_{{Y}_{f}u}^{\infty}{\int}_{{Y}_{f}u\omega}^{\infty}{{{e}^{\frac{{\left(um{\mu}_{1}\right)}^{2}}{2m{\sigma}_{{}_{1}}^{2}}}}^{\frac{{\left(\omega m{\mu}_{1}\right)}^{2}}{2m{\sigma}_{{}_{1}}^{2}}}}^{\frac{{\left(\tau n{\mu}_{2}\right)}^{2}}{2n{\sigma}_{{}_{2}}^{2}}}d\tau d\omega du\cdot \\ P\left(N\left({T}_{z1}\right)=m\right)\cdot P\left(N\left({t}_{c}{T}_{z1}\right)=n\right)\cdot P\left(N\left({T}_{z}{t}_{c}\right)=l\right)\cdot P\left({T}_{z1}<{t}_{c}\le {T}_{z}\right)\end{array}\right).$
3.5.3. Probability of preventive maintenance
If any one event of the following events (${A}_{P1}$, ${A}_{P2}$, ${A}_{P3}$) occurs, it is considered that preventive replacement needs to be performed and it will cause preventive maintenance cost ${C}_{P}$. When a twice continuous monitoring just happened before and after the changepoint for degradation rate (${T}_{z1}<{t}_{c}\le {T}_{z}$), if the degradation level $Y\left({T}_{z1}\right)<{Y}_{nom}$ for ($z1$)th inspection and $Y\left({T}_{z}\right)\ge {Y}_{f}$ for $z$th inspection, preventive maintenance action will be performed.
${A}_{P2}=\left\{Y\left({T}_{z1}\right)<{Y}_{acc}\bigcap {Y}_{acc}\le Y\left({T}_{z}\right)<{Y}_{f}\bigcap {t}_{c}<{T}_{z1}\le {T}_{z}\right\},$
${A}_{P3}=\left\{Y\left({T}_{z1}\right)<{Y}_{nom}\bigcap {Y}_{acc}\le Y\left({T}_{z}\right)<{Y}_{f}\bigcap {T}_{z1}<{t}_{c}\le {T}_{z}\right\}.$
The probability for a preventive maintenance in a renewal cycle is expressed as:
The probability for preventive maintenance events ${A}_{P1}$, ${A}_{P2}$, ${A}_{P3}$ can be written as follows:
$=\frac{1}{2\pi}\sum _{z=1}^{\infty}\sum _{m=0}^{\infty}\sum _{n=0}^{\infty}{\int}_{0}^{{Y}_{nom}}{\int}_{{Y}_{nom}u}^{{Y}_{f}u}{{e}^{\frac{{\left(um{\mu}_{1}\right)}^{2}}{2m{\sigma}_{{}_{1}}^{2}}}}^{\frac{{\left(\tau n{\mu}_{1}\right)}^{2}}{2n{\sigma}_{{}_{1}}^{2}}}d\tau du\cdot P\left(N\left({T}_{z1}\right)=m\right)P\left(N\left({T}_{z}{T}_{z1}\right)=n\right)P\left({T}_{z}\le {t}_{c}\right),$
$=\frac{1}{\sqrt{8{\pi}^{3}}}\sum _{z=1}^{\infty}\sum _{m=0}^{\infty}\sum _{n=0}^{\infty}\sum _{l=0}^{\infty}\left(\begin{array}{c}{\int}_{0}^{{Y}_{acc}}{\int}_{0}^{{Y}_{acc}u}{\int}_{{Y}_{acc}u\omega}^{{Y}_{f}u\omega}{{e}^{\frac{{\left(um{\mu}_{1}\right)}^{2}}{2m{\sigma}_{{}_{1}}^{2}}}}^{\frac{{\left(\omega n{\mu}_{2}\right)}^{2}}{2n{\sigma}_{{}_{2}}^{2}}\frac{{\left(\tau n{\mu}_{2}\right)}^{2}}{2n{\sigma}_{{}_{2}}^{2}}}d\tau d\omega du\cdot \\ P\left(N\left({t}_{c}\right)=m\right)\cdot P\left(N\left({T}_{z1}{t}_{c}\right)=n\right)\cdot P\left(N\left({T}_{z}{T}_{z1}\right)=l\right)\cdot P\left({T}_{z1}>{t}_{c}\right)\end{array}\right),$
$=\frac{1}{\sqrt{8{\pi}^{3}}}\sum _{z=1}^{\infty}\sum _{m=0}^{\infty}\sum _{n=0}^{\infty}\sum _{l=0}^{\infty}\left(\begin{array}{c}{\int}_{0}^{{Y}_{nom}}{\int}_{{Y}_{acc}u}^{{Y}_{f}u}{\int}_{{Y}_{acc}u\omega}^{{Y}_{f}u\omega}{{{e}^{\frac{{\left(um{\mu}_{1}\right)}^{2}}{2m{\sigma}_{{}_{1}}^{2}}}}^{\frac{{\left(\omega m{\mu}_{1}\right)}^{2}}{2m{\sigma}_{{}_{1}}^{2}}}}^{\frac{{\left(\tau n{\mu}_{2}\right)}^{2}}{2n{\sigma}_{{}_{2}}^{2}}}d\tau d\omega du\cdot \\ P\left(N\left({T}_{z1}\right)=m\right)\cdot P\left(N\left({t}_{c}{T}_{z1}\right)=n\right)\cdot P\left(N\left({T}_{z}{t}_{c}\right)=l\right)\cdot P\left({T}_{z1}<{t}_{c}\le {T}_{z}\right)\end{array}\right).$
3.5.4. Probability of continuous monitoring
If any one event of the following events (${A}_{I1}$, ${A}_{I2}$) occurs, the system is left as it is until next inspection time and it will cause monitoring cost ${C}_{I}$.
${A}_{I2}=\left\{Y\left({T}_{z}\right)<{Y}_{acc}\bigcap {T}_{z}>{t}_{c}\right\}.$
The probability for system left until next inspection in a renewal cycle can be expressed as:
The probability for continuous monitoring events ${A}_{I1}$, ${A}_{I2}$ can be written as follows:
$=\frac{1}{\sqrt{2\pi}}\sum _{z=0}^{\infty}\sum _{m=0}^{\infty}{\int}_{0}^{{Y}_{nom}}{e}^{\frac{{\left(um{\mu}_{1}\right)}^{2}}{2m{\sigma}_{{}_{1}}^{2}}}du\cdot P\left(N\left({T}_{z}\right)=m\right)\cdot P\left({T}_{z}\le {t}_{c}\right),$
$=\frac{1}{2\pi}\sum _{z=0}^{\infty}\sum _{m=0}^{\infty}{\int}_{0}^{{Y}_{acc}}{\int}_{0}^{{Y}_{acc}u}{e}^{\frac{{\left(um{\mu}_{1}\right)}^{2}}{2m{\sigma}_{{}_{1}}^{2}}\frac{{\left(\tau m{\mu}_{2}\right)}^{2}}{2m{\sigma}_{{}_{2}}^{2}}}d\tau du\cdot P\left(N\left({t}_{c}\right)=m\right)\cdot P\left(N\left({T}_{z}{t}_{c}\right)=n\right)\cdot P\left({T}_{z}>{t}_{c}\right).$
Average number of monitoring actions in a renewal cycle $T$ is:
3.5.5. Expected time length of renewal cycle
As Eq. (14) shown, expected time length of renewal cycle $E\left[T\right]$ is affected by system life ${T}_{f}$ and average work time length ${T}_{P}$ when system ends with preventive replacement. If system faults, it is considered that system will not work any time. Therefore, the expected time length ${T}_{f}$ when system ends with corrective maintenance is the time interval for degradation level from initial value 0 to failure threshold ${Y}_{f}$. However, the expected time length ${T}_{P}$ when system ends with preventive maintenance is different. System will no longer work if monitoring shows preventive replacement should be performed, so system life when system ends with preventive maintenance is times of interinspection time $\u2206T$.
System fault occurs in the second stage when system degradation with twostage mode. System life ${T}_{f}$ is affected by shock strength ${x}_{1i}$, ${x}_{2j}$ and changepoint ${t}_{c}$. The system mean time to failure is:
$=\sum _{m=0}^{\infty}\sum _{n=0}^{\infty}\mathrm{\Phi}\left(\frac{{Y}_{f}\left(m{\mu}_{1}+n{\mu}_{2}\right)}{\sqrt{m{\sigma}_{1}^{2}+n{\sigma}_{2}^{2}}}\right)\cdot \frac{{\lambda}_{1}^{m}\cdot {\lambda}_{2}^{n}}{m!\cdot n!}\cdot {\int}_{{t}_{A}}^{{t}_{B}}{\int}_{\tau}^{\infty}\left({e}^{{\lambda}_{1}\tau {\lambda}_{2}\left(t\tau \right)}\cdot {\tau}^{m}\cdot {\left(t\tau \right)}^{n}\cdot g\left(\tau \right)\right)dtd\tau .$
The average system life when system ends with preventive maintenance is:
$+P\left({A}_{P3}\right)\left({t}_{c}+{\int}_{0}^{\Delta {T}_{nom}}\omega {f}_{\omega}d\omega \right).$
Fig. 5The definition of ω
Where ${z}_{p1}$ is the number of monitoring in mode ${M}_{1}$ for event ${A}_{p1}$, ${z}_{p2}$ is the number of monitoring in mode ${M}_{2}$ for event ${A}_{p2}$, the time length from ${t}_{c}$ to next inspection time is $w$ (as shown in Fig. 5), ${f}_{w}$ is the distribution density function of w during [${T}_{z1}$, ${T}_{z}$].
4. Influence analysis of different parameters for maintenance policy
This section aims to find some characteristics for twostage deteriorating mode system: (a) Making a comparison of the average longrun cost rate for different maintenance policies in the same situation in order to raise the awareness of monitoring method. (b) Studying the influence of parameters in the degradation model and average longrun cost model, for the purpose of improving the understanding in twostage deteriorating modeling and developing an optimal maintenance policy.
4.1. Choice of parameters values
In this study, the time distribution of changepoint ${t}_{c}$ is assumed to follow uniform distribution. In order to emphasize the influence of distribution of changepoint ${t}_{c}$, different uniform distribution parameters are considered as follows:
• Changepoint of twostage degradation mode: ${t}_{c}~U\left(\text{1,}\text{}\text{200}\right)$.
• Early changepoint of twostage degradation mode: ${t}_{c}~U\left(\text{1,}\text{}\text{100}\right)$.
• Middle changepoint of twostage degradation mode: ${t}_{c}~U\left(\text{50,}\text{}\text{150}\right)$.
• Late changepoint of twostage degradation mode: ${t}_{c}~U\left(\text{100,}\text{}\text{200}\right)$.
The upper limit of changepoint distribution 200 is considered that a majority of system failures occur in degradation mode ${M}_{2}$ and seldom in degradation mode ${M}_{1}$. Early and late time distributions present the first and second half of full changepoint distribution, respectively.
In order to make the influence analysis of different parameters more close to the actual situation of gearbox degradation process, the selection of failure threshold ${Y}_{f}$ is based on the actual value of gearbox lifecycle experiment in Section 5. Therefore, the failure threshold is evaluated as ${Y}_{f}=$ 10000 g^{2} in this study. Meanwhile, for the purpose of ensuring the credibility of optimization results for global maintenance policy (the optimization results will be regarded as a basis of comparison), the unit maintenance costs are evaluated as other literatures [1, 12, 24], so ${C}_{I}$ = 5, ${C}_{P}$ = 50, ${C}_{C}$ = 100.
4.2. Influence of maintenance policy
Fig. 6EC∞ is affected by adaptive maintenance policy parameters
a)$E\left({C}_{\infty}\right)$ is affected by ${Y}_{nom}$ and ${T}_{acc}$
b)$E\left({C}_{\infty}\right)$ is affected by ${Y}_{acc}$ and ${T}_{acc}$
c)$E\left({C}_{\infty}\right)$ is affected by ${T}_{nom}$ and ${T}_{acc}$
For the purpose of studying on the influence of different monitoring methods, the four kinds of conditionbased maintenance policies (global, timedepended, simplified adaptive and adaptive) presented in Section 3 are assessed. Because adaptive maintenance policy is the most complex in the four methods, an example focusing on adaptive policy analyzing is presented to show the approach of obtaining minimal average longrun maintenance cost rate $E\left({C}_{\infty}\right)$. When ${x}_{1i}~N$(10, 20^{2}), ${x}_{2j}~N$(40, 80^{2}) and ${\lambda}_{1}={\lambda}_{2}=1$, $E\left({C}_{\infty}\right)$ is affected by adaptive maintenance policy parameters ${Y}_{nom}$, ${Y}_{acc}$, ${T}_{nom}$, ${T}_{acc}$, as shown in Fig. 6. It illustrates that ${Y}_{nom}$, ${Y}_{acc}$, ${T}_{nom}$, ${T}_{acc}$ should be considered at the same time when optimizing $E\left({C}_{\infty}\right)$. Contour map (Fig. 7) shows $E\left({C}_{\infty}\right)$ for different values of ${Y}_{acc}$ and ${T}_{acc}$ under adaptive maintenance decision when ${Y}_{nom}$ = 8000, $\u2206{T}_{nom}$ = 70. $E\left({C}_{\infty}\right)$ in the same contour are equal. It can be seen that optimal parameter values which minimize the cost rate (${E}_{4}\left({C}_{\infty}\right)$ = 0.3547) are ${Y}_{acc}$ = 7000 and $\u2206{T}_{acc}$ = 37 when ${Y}_{nom}$ = 8000, $\u2206{T}_{nom}$ = 70.
Fig. 7EC∞ under adaptive maintenance policy parameters (Ynom = 8000, ∆Tnom = 70)
As shown in Table 1, for the global maintenance policy, the optimal interinspection time $\Delta {T}^{1}$ falls in between $\Delta {T}_{nom}^{4}$ and $\Delta {T}_{acc}^{4}$ for adaptive maintenance policy. The situation of $\Delta {T}^{3}$ for simplified adaptive maintenance policy is in between $\Delta {T}_{nom}^{4}$ and $\Delta {T}_{acc}^{4}$, too. $\Delta {T}^{2}$, the first interinspection time for timedepended maintenance policy, is the largest optimal interinspection time for the four kinds of conditionbased maintenance policies.
The expected costs for the four kinds of maintenance policies are different because they are impacted by alarm thresholds and interinspection times. As the monitoring method for global maintenance policy does not consider the influence of degradation rate changes, taking the expected cost ${E}_{1\left({C}_{\infty}\right)}$ of global maintenance policy as a basis of comparison, the expected cost ${E}_{2\left({C}_{\infty}\right)}$ for timedepended maintenance policy have a decrease of 0.0286, the optimal value is 7.69 % of ${E}_{1\left({C}_{\infty}\right)}$. As shown in Table 1, it is obviously that the expected costs for other three maintenance policies are optimized compare with the average cost for global maintenance policy. Adaptive maintenance policy is better than simplified adaptive maintenance policy. Timedepended maintenance policy is the best replacement strategy for the system with twostage degradation process.
4.3. Influence of changepoint distribution
$E\left({C}_{\infty}\right)$ for different changepoint distributions are computed, see results in Table 1. It can be noticed that the decreases of expected costs for different maintenance policies (timedepended, simplified adaptive, adaptive) are 3.94 %, 1.59 %, 3.14 %, respectively, when the changepoint is in early distribution $U$(1, 100). The impact of average longrun maintenance costs rate when the changepoint is in middle distribution $U$(50, 150) and late distribution $U$(100, 200) are also given in Table 1. These results show that more the time of changepoint occurs late, more the maintenance policies have a decrease on $E\left({C}_{\infty}\right)$.
As seen previously, decreases of 7.69 %, 2.71 %, 4.68 % can be obtained respectively for different maintenance policies when the changepoint distribution is $U$(1, 200), and they are 6.53 %, 2.61 %, 4.34 %, respectively for the case that the changepoint distribution is $U$(50, 150). It is not difficult to find that the decreases for former are larger than latter. Meanwhile, in the two situations, the mean value of changepoint distribution is the same, both equal to 100. Therefore, it is shown that more profits can be obtained in using different maintenance policies for a larger interval of time distribution.
The results show that the distribution of changepoint impact on expected costs for different maintenance policies. More late for changepoint occurs and more a large time interval of changepoint distribution, more benefits can be obtained.
4.4. Influence of shock strength
In degradation modeling based on cumulative damage model, degradation rate is decided by shock strength and shock frequency. The degradation rate is in direct proportion to shock strength and shock frequency. Hence, the degradation rate can be expressed by shock strengths if the shock frequencies are the same.
The influence of different maintenance policies and changepoint distribution have been analyzed for a twostage degraded system in Table 1 when ${x}_{1i}~N$(10, 20^{2}), ${x}_{2j}~N$(40, 80^{2}). The same computed results with ${x}_{1i}~N$(10, 20^{2}) and ${x}_{2j}~N$(20, 40^{2}) are shown in Table 2. The degradation rate size of mode ${M}_{2}$ is four times superior than the mode ${M}_{1}$ in Table 1, while it is twice in Table 2. From Table 1 and Table 2, it can be known that the decreases are more considerable in Table 1. That is say, the profits is more considerable when degradation rate changes more significantly between mode ${M}_{2}$ and mode ${M}_{1}$.
Table 1Influence of different maintenance policies and changepoint time distribution (x1i~N(10, 202), x2j~N(40, 802), λ1=λ2= 1)
Time distribution  Policy structure  Optimal parameters  Expected cost  Impact 
${t}_{c}\text{~U}\left(\text{1, 200}\right)$  Global  ${Y}_{A}^{1}=\text{4700}$, $\mathrm{\Delta}{T}^{1}=\text{66}$  ${E}_{1}\left({C}_{\mathrm{\infty}}\right)=\text{0.3721}$  
Timedepended  ${Y}_{A}^{2}=\text{5300}$, $\mathrm{\Delta}{T}^{2}=\text{111}$, $$$q=\text{0.66}$  ${E}_{2}\left({C}_{\mathrm{\infty}}\right)=\text{0.3435}$  0.0286 (7.69 %)  
Simplified Adaptive  ${Y}_{nom}^{3}=\text{7750}$, ${Y}_{acc}^{3}=\text{5250}$, $\mathrm{\Delta}{T}^{3}=\text{60}$  ${E}_{3}\left({C}_{\mathrm{\infty}}\right)=\text{0.3620}$  0.0101 (2.71 %)  
Adaptive  ${Y}_{nom}^{4}=\text{8000}$, $\mathrm{\Delta}{T}_{nom}^{4}=\text{70}$, ${Y}_{acc}^{4}=\text{7000}$, $\mathrm{\Delta}{T}_{acc}^{4}=\text{37}$  ${E}_{4}\left({C}_{\mathrm{\infty}}\right)=\text{0.3547}$  0.0174 (4.68 %)  
${t}_{c}~U\left(\text{1, 100}\right)$  Global  ${Y}_{A}^{1}=\text{5300}$, $\mathrm{\Delta}{T}^{1}=\text{62}$  ${E}_{1}\left({C}_{\mathrm{\infty}}\right)=\text{0.4268}$  
Timedepended  ${Y}_{A}^{2}=\text{7100}$, $\mathrm{\Delta}{T}^{2}=\text{84}$, $q=\text{0.66}$  ${E}_{2}\left({C}_{\mathrm{\infty}}\right)=\text{0.4100}$  0.0168 (3.94 %)  
Simplified Adaptive  ${Y}_{nom}^{3}=\text{7750}$, ${Y}_{acc}^{3}=\text{5000}$, $\mathrm{\Delta}T=\text{66}$  ${E}_{3}\left({C}_{\mathrm{\infty}}\right)=\text{0.4200}$  0.0068 (1.59 %)  
Adaptive  ${Y}_{nom}^{4}=\text{8400}$, $\mathrm{\Delta}{T}_{nom}^{4}=\text{68}$, ${Y}_{acc}^{4}=\text{6800}$, $\mathrm{\Delta}{T}_{acc}^{4}=\text{38}$  ${E}_{4}\left({C}_{\mathrm{\infty}}\right)=\text{0.4134}$  0.0134 (3.14 %)  
${t}_{c}~U\left(\text{50, 150}\right)$  Global  ${Y}_{A}^{1}=\text{4700}$, $\mathrm{\Delta}{T}^{1}=\text{74}$  ${E}_{1}\left({C}_{\mathrm{\infty}}\right)=\text{0.3598}$  
Timedepended  ${Y}_{A}^{2}=\text{6100}$, $\mathrm{\Delta}{T}^{2}=\text{95}$, $q=\text{0.72}$  ${E}_{2}\left({C}_{\mathrm{\infty}}\right)=\text{0.3363}$  0.0235 (6.53 %)  
Simplified Adaptive  ${Y}_{nom}^{3}=\text{7750}$, ${Y}_{acc}^{3}=\text{5000}$, $\mathrm{\Delta}{T}^{3}=\text{69}$  ${E}_{3}\left({C}_{\mathrm{\infty}}\right)=\text{0.3504}$  0.0094 (2.61 %)  
Adaptive  ${Y}_{nom}^{4}=\text{7800}$, $\mathrm{\Delta}{T}_{nom}^{4}=\text{76}$, ${Y}_{acc}^{4}=\text{7000}$, $\mathrm{\Delta}{T}_{acc}^{4}=\text{37}$  ${E}_{4}\left({C}_{\mathrm{\infty}}\right)=\text{0.3442}$  0.0156 (4.34 %)  
${t}_{c}~U\text{(}\text{100, 200}\text{)}$  Global  ${Y}_{A}^{1}=\text{4300}$, $\mathrm{\Delta}{T}^{1}=\text{76}$  ${E}_{1}\left({C}_{\mathrm{\infty}}\right)=\text{0.3104}$  
Timedepended  ${Y}_{A}^{2}=\text{5000}$, $\mathrm{\Delta}{T}^{2}=\text{125}$, $q=\text{0.69}$  ${E}_{2}\left({C}_{\mathrm{\infty}}\right)=\text{0.2810}$  0.0294 (9.47 %)  
Simplified Adaptive  ${Y}_{nom}^{3}=\text{7500}$, ${Y}_{acc}^{3}=\text{5000}$, $\mathrm{\Delta}{T}^{3}=\text{69}$  ${E}_{3}\left({C}_{\mathrm{\infty}}\right)=\text{0.3017}$  0.0087 (2.80 %)  
Adaptive  ${Y}_{nom}^{4}=\text{8400}$, $\mathrm{\Delta}{T}_{nom}^{4}=\text{84}$, ${Y}_{acc}^{4}=\text{6600}$, $\mathrm{\Delta}{T}_{acc}^{4}=\text{43}$  ${E}_{4}\left({C}_{\mathrm{\infty}}\right)=\text{0.2942}$  0.0162 (5.22 %) 
Table 2Influence of different maintenance policies and changepoint time distribution (x1i~N(10, 202), x2j~N(20, 402), λ1=λ2= 1)
Time distribution  Policy structure  Optimal parameters  Expected cost  Impact 
${t}_{c}~U\left(\text{1, 200}\right)$  Global  ${Y}_{A}^{1}=\text{6600}$, $\mathrm{\Delta}{T}^{1}=\text{89}$  ${E}_{1}\left({C}_{\mathrm{\infty}}\right)=\text{0.2395}$  
Timedepended  ${\text{Y}}_{\text{A}}^{\text{2}}\text{=}\text{6500}$, $\mathrm{\Delta}{T}^{2}=\text{105}$, $q=\text{0.96}$  ${E}_{2}\left({C}_{\mathrm{\infty}}\right)=\text{0.2297}$  0.0098 (4.09 %)  
Simplified Adaptive  ${Y}_{nom}^{3}=\text{8000}$, ${Y}_{acc}^{3}=\text{7000}$, $\mathrm{\Delta}{T}^{3}=\text{84}$  ${E}_{3}\left({C}_{\mathrm{\infty}}\right)=\text{0.2348}$  0.0047 (1.96 %)  
Adaptive  ${Y}_{nom}^{4}=\text{8000}$, $\mathrm{\Delta}{T}_{nom}^{4}=\text{95}$, ${Y}_{acc}^{4}=\text{7000}$, $\mathrm{\Delta}{T}_{acc}^{4}=\text{75}$  ${E}_{4}\left({C}_{\mathrm{\infty}}\right)=\text{0.2304}$  0.0091 (3.80 %)  
${t}_{c}~U\left(\text{1,}\text{}\text{100}\right)$  Global  ${Y}_{A}^{1}=\text{6600}$, $\Delta {T}^{1}=\text{92}$  ${E}_{1}\left({C}_{\mathrm{\infty}}\right)=\text{0.2513}$  
Timedepended  ${Y}_{A}^{2}=\text{6100}$, $\mathrm{\Delta}{T}^{2}=\text{126}$, $q=\text{0.96}$  ${E}_{2}\left({C}_{\mathrm{\infty}}\right)=\text{0.2427}$  0.0086 (3.42 %)  
Simplified Adaptive  ${Y}_{nom}^{3}=\text{8000}$, ${Y}_{acc}^{3}=\text{6750}$, $\mathrm{\Delta}{T}^{3}=\text{91}$  ${E}_{3}\left({C}_{\mathrm{\infty}}\right)=\text{0.2484}$  0.0029 (1.15 %)  
Adaptive  ${Y}_{nom}^{4}=\text{8000}$, $\mathrm{\Delta}{T}_{nom}^{4}=\text{85}$, ${Y}_{acc}^{4}=\text{7500}$, $\mathrm{\Delta}{T}_{acc}^{4}=\text{65}$  ${E}_{4}\left({C}_{\mathrm{\infty}}\right)=\text{0.2440}$  0.0073 (2.99 %)  
${t}_{c}~U\left(\text{50,}\text{}\text{150}\right)$  Global  ${Y}_{A}^{1}=\text{6900}$, $\mathrm{\Delta}{T}^{1}=\text{95}$  ${E}_{1}\left({C}_{\mathrm{\infty}}\right)=\text{0.2314}$  
Timedepended  ${Y}_{A}^{2}=\text{6700}$, $\mathrm{\Delta}{T}^{2}=\text{114}$, $q=\text{0.90}$  ${E}_{2}\left({C}_{\mathrm{\infty}}\right)=\text{0.2217}$  0.0097 (4.19 %)  
Simplified Adaptive  ${Y}_{nom}^{3}=\text{7750}$, ${Y}_{acc}^{3}=\text{6500}$, $\mathrm{\Delta}{T}^{3}=\text{99}$  ${E}_{3}\left({C}_{\mathrm{\infty}}\right)=\text{0.2272}$  0.0042 (1.82 %)  
Adaptive  ${Y}_{nom}^{4}=\text{8000}$, $\mathrm{\Delta}{T}_{nom}^{4}=\text{90}$, ${Y}_{acc}^{4}=\text{7500}$, $\mathrm{\Delta}{T}_{acc}^{nom}=\text{65}$  ${E}_{4}\left({C}_{\mathrm{\infty}}\right)=\text{0.2228}$  0.0086 (3.72 %)  
${t}_{c}~U\left(\text{100, 200}\right)$  Global  ${Y}_{A}^{1}=\text{6000}$, $\mathrm{\Delta}{T}^{1}=\text{110}$  ${E}_{1}\left({C}_{\mathrm{\infty}}\right)=\text{0.2125}$  
Timedepended  ${Y}_{A}^{2}=\text{7300}$, $\mathrm{\Delta}{T}^{2}=\text{128}$, $q=\text{0.81}$  ${E}_{2}\left({C}_{\mathrm{\infty}}\right)=\text{0.2012}$  0.0113 (5.32 %)  
Simplified Adaptive  ${Y}_{nom}^{3}=\text{8500}$, ${Y}_{acc}^{3}=\text{6250}$, $\mathrm{\Delta}{T}^{3}=\text{106}$  ${E}_{3}\left({C}_{\mathrm{\infty}}\right)=\text{0.2080}$  0.0045 (2.12 %)  
Adaptive  ${Y}_{nom}^{4}=\text{8000}$, $\mathrm{\Delta}{T}_{nom}^{4}=\text{95}$, ${Y}_{acc}^{4}=\text{7000}$, $\mathrm{\Delta}{T}_{acc}^{4}=\text{70}$  ${E}_{4}\left({C}_{\mathrm{\infty}}\right)=\text{0.2042}$  0.0083 (3.91 %) 
4.5. Influence of shock frequency
As the previously analysis, $E\left({C}_{\infty}\right)$ for different shock frequencies can be obtained when shock strengths are the same. When ${x}_{1i}~N$(10, 20^{2}), ${x}_{2j}~N$(40, 80^{2}), ${t}_{c}\in $[50, 150], shock frequency parameters are ${\lambda}_{1}={\lambda}_{2}=$ 0.5, ${\lambda}_{1}={\lambda}_{2}=$ 1, ${\lambda}_{1}={\lambda}_{2}=$ 2, respectively, $E\left({C}_{\infty}\right)$ obtained in Table 3. As results obtained in Section 4.2, it can be known that adaptive maintenance policy is better than simplified adaptive maintenance policy, timedepended maintenance policy is the best replacement strategy for the system with twostage degradation process.
Further analysis, $E\left({C}_{\infty}\right)$ for different maintenance policies are respectively 2.64 %, 4.78 %, 8.28 % when degradation model parameters ${\lambda}_{1}=$1, ${\lambda}_{2}=$0.5, ${x}_{1i}~N$(10, 20^{2}), ${x}_{2j}~N$(40, 80^{2}) and respectively 5.38 %, 1.38 %, 2.54 % when degradation model parameters ${\lambda}_{1}=$1, ${\lambda}_{2}=$2, ${x}_{1i}~N$(10, 20^{2}), ${x}_{2j}~N$(40, 80^{2}) as shown in Table 3. The degradation rate size of mode ${M}_{2}$ is eight times superior than the mode ${M}_{1}$ in the former, while it is twice in the latter. It can be known from the results that adaptive replacement policy is always better than simplified adaptive maintenance policy, especially under the situation that degradation rate undergoes change hugely. But the timedepended monitor method is no suitable for a system which the degradation rate in mode ${M}_{2}$ is significantly larger than the degradation rate in mode ${M}_{1}$.
Table 3Influence of different maintenance policies and shock frequencies (x1i~N(10, 202), x2j~N(40, 802), tc∈ [50, 150])
Time distribution  Policy structure  Optimal parameters  Expected cost  Impact 
${\lambda}_{1}={\lambda}_{2}=\text{0.5}$  Global  ${Y}_{A}^{1}=\text{4200}$, $\mathrm{\Delta}{T}^{1}=\text{54}$  ${E}_{1}\left({C}_{\mathrm{\infty}}\right)=\text{0.5305}$  
Timedepended  ${Y}_{A}^{2}=\text{6300}$, $\mathrm{\Delta}{T}^{2}=\text{67}$, $$$q=\text{0.65}$  ${E}_{2}\left({C}_{\mathrm{\infty}}\right)=\text{0.5019}$  0.0286 (5.39 %)  
Simplified Adaptive  ${Y}_{nom}^{3}=\text{7000}$, ${Y}_{acc}^{3}=\text{4500}$, $\mathrm{\Delta}{T}^{3}=\text{50}$  ${E}_{3}\left({C}_{\mathrm{\infty}}\right)=\text{0.5152}$  0.0153 (2.88 %)  
Adaptive  ${Y}_{nom}^{4}=\text{7400}$, $\mathrm{\Delta}{T}_{nom}^{4}=\text{52}$, ${Y}_{acc}^{4}=\text{6400}$, $\mathrm{\Delta}{T}_{acc}^{4}=\text{26}$  ${E}_{4}\left({C}_{\mathrm{\infty}}\right)=\text{0.5046}$  0.0259 (4.88 %)  
${\lambda}_{1}={\lambda}_{2}=\text{1}$  Global  ${Y}_{A}^{1}=\text{4700}$, $\mathrm{\Delta}{T}^{1}=\text{74}$  ${E}_{1}\left({C}_{\mathrm{\infty}}\right)=\text{0.3598}$  
Timedepended  ${Y}_{A}^{2}=\text{6100}$, $\mathrm{\Delta}{T}^{2}=\text{95}$, $q=\text{0.72}$  ${E}_{2}\left({C}_{\mathrm{\infty}}\right)=\text{0.3363}$  0.0235 (6.53 %)  
Simplified Adaptive  ${Y}_{nom}^{3}=\text{7750}$, ${Y}_{acc}^{3}=\text{5000}$, $\mathrm{\Delta}{T}^{3}=\text{69}$  ${E}_{3}\left({C}_{\mathrm{\infty}}\right)=\text{0.3504}$  0.0094 (2.61 %)  
Adaptive  ${Y}_{nom}^{4}=\text{7800}$, $\mathrm{\Delta}{T}_{nom}^{4}=\text{76}$, ${Y}_{acc}^{4}=\text{7000}$, $\mathrm{\Delta}{T}_{acc}^{4}=\text{37}$  ${E}_{4}\left({C}_{\mathrm{\infty}}\right)=\text{0.3442}$  0.0156 (4.34 %)  
${\lambda}_{1}={\lambda}_{2}=\text{2}$  Global  ${Y}_{A}^{1}=\text{6000}$, $\mathrm{\Delta}{T}^{1}=\text{107}$  ${E}_{1}\left({C}_{\mathrm{\infty}}\right)=\text{0.2168}$  
Timedepended  ${Y}_{A}^{2}=\text{6900}$, $\mathrm{\Delta}{T}^{2}=\text{160}$, $$$q=\text{0.69}$  ${E}_{2}\left({C}_{\mathrm{\infty}}\right)=\text{0.2051}$  0.0117 (5.40 %)  
Simplified Adaptive  ${Y}_{nom}^{3}=\text{7500}$, ${Y}_{acc}^{3}=\text{6500}$, $\mathrm{\Delta}{T}^{3}=\text{103}$  ${E}_{3}\left({C}_{\mathrm{\infty}}\right)=\text{0.2135}$  0.0033 (1.52 %)  
Adaptive  ${Y}_{nom}^{4}=\text{8000}$, $\mathrm{\Delta}{T}_{nom}^{4}=\text{110}$, ${Y}_{acc}^{4}=\text{7000}$, $\mathrm{\Delta}{T}_{acc}^{4}=\text{80}$  ${E}_{4}\left({C}_{\mathrm{\infty}}\right)=\text{0.2097}$  0.0071 (3.27 %)  
$\begin{array}{l}{\lambda}_{1}=\text{1}\\ {\lambda}_{2}=\text{0.5}\end{array}$  Global  ${Y}_{A}^{1}=\text{4500}$, $\mathrm{\Delta}{T}^{1}=\text{53}$  ${E}_{1}\left({C}_{\mathrm{\infty}}\right)=\text{0.4841}$  
Timedepended  ${Y}_{A}^{2}=\text{5800}$, $\mathrm{\Delta}{T}^{2}=\text{71}$, $q=\text{0.69}$  ${E}_{2}\left({C}_{\mathrm{\infty}}\right)=\text{0.4713}$  0.0128 (2.64 %)  
Simplified Adaptive  ${Y}_{nom}^{3}=\text{7000}$, ${Y}_{acc}^{3}=\text{4500}$, $\mathrm{\Delta}{T}^{3}=\text{45}$  ${E}_{3}\left({C}_{\mathrm{\infty}}\right)=\text{0.4611}$  0.0230 (4.78 %)  
Adaptive  ${Y}_{nom}^{4}=\text{8200}$, $\mathrm{\Delta}{T}_{nom}^{4}=\text{59}$, ${Y}_{acc}^{4}=\text{5800}$, $\mathrm{\Delta}{T}_{acc}^{4}=\text{26}$  ${E}_{4}\left({C}_{\mathrm{\infty}}\right)=\text{0.4440}$  0.0401 (8.28 %)  
$\begin{array}{l}{\lambda}_{1}=\text{1}\\ {\lambda}_{2}=\text{2}\end{array}$  Global  ${Y}_{A}^{1}=\text{6800}$, $\mathrm{\Delta}{T}^{1}=\text{100}$  ${E}_{1}\left({C}_{\mathrm{\infty}}\right)=\text{0.2323}$  
Timedepended  ${Y}_{A}^{2}=\text{6400}$, $\mathrm{\Delta}{T}^{2}=\text{155}$, $$$q=\text{0.75}$  ${E}_{2}\left({C}_{\mathrm{\infty}}\right)=\text{0.2198}$  0.0125 (5.38 %)  
Simplified Adaptive  ${Y}_{nom}^{3}=\text{8250}$, ${Y}_{acc}^{3}=\text{6750}$, $\mathrm{\Delta}{T}^{3}=\text{98}$  ${E}_{3}\left({C}_{\mathrm{\infty}}\right)=\text{0.2291}$  0.0032 (1.38 %)  
Adaptive  ${Y}_{nom}^{4}=\text{8300}$, $\mathrm{\Delta}{T}_{nom}^{4}=\text{104}$, ${Y}_{acc}^{4}=\text{6500}$, $\mathrm{\Delta}{T}_{acc}^{4}=\text{94}$  ${E}_{4}\left({C}_{\mathrm{\infty}}\right)=\text{0.2264}$  0.0059 (2.54 %) 
5. Case study
A case study is carried out for a gearbox deterioration modeling and decisionmaking on maintenance using experiment data. In the case study, a gearbox lifecycle experiment has done to obtain the degradation data that a gearbox ran from new to failure. The experiment rig is shown in Fig. 8, where four accelerometers are fitted onto the casing of gearbox to record vibration data. In the experiment, the sampling frequency is 20 kHz. Lots of equalspaced vibration monitoring performed in the test process. Each vibration monitoring provides a date file collected in 2 seconds at every 5 minutes, twelve groups of date files are collected in every hour. The magnetic brake provide about 22.5 times of the rated torque of gearbox in order to accelerate the test and reduce the lifetime of gearbox.
Fig. 8Experiment rig (1 – load, 2 – accelerometers, 3 – sensor of speed and torque, 4 – electromotor, 5 – test bed, 6 – gearbox system)
Fig. 9Gear after experiment
Fig. 10Special frequency band energy of vibration signal
The total experimental time is 450 hours, the gear after experiment is shown in Fig. 9. As shown in Fig. 10, the special frequency band energy of vibration signal presents that degradation process of gearbox is obviously twostage process. Using linear fitting analysis, degradation parameters of gearbox are obtained as follows: ${x}_{1i}~N$(12, 25^{2}), ${x}_{2i}~N$ (76, 130^{2}), ${t}_{c}\in $[0, 450], ${\lambda}_{1}={\lambda}_{2}=$ 1. Meanwhile, the failure threshold is evaluated as ${Y}_{f}$ = 10000g^{2}.
Based on the proposed model and maintenance policies, optimal results for different maintenance policies of gearbox are shown in Table 4. Because the degradation rate of second stage for twostage deteriorating mode system is faster than the first stage, the alarm threshold and interinspection time for the second stage should be smaller than the first stage. The initial interinspection time of timedepended maintenance policy $\u2206{T}^{2}$ is larger than the interinspection time of global maintenance policy $\u2206{T}^{1}$, but interinspection time of timedepended maintenance policy is smaller and smaller with working time, as a result the expected cost ${E}_{2}\left({C}_{\infty}\right)$ make a decrease of 0.0108 from ${E}_{1}\left({C}_{\infty}\right)$. The alarm thresholds of the first stage for adaptive and simplified adaptive maintenance policy ${Y}_{nom}^{4}$, ${Y}_{nom}^{3}$ are both larger than alarm threshold of global maintenance policy ${Y}_{A}^{1}$, but the alarm thresholds of the second stage ${Y}_{acc}^{4}$, ${Y}_{acc}^{3}$ are both smaller than ${Y}_{A}^{1}$. Meanwhile, the interinspection times $\u2206{T}^{1}$, $\u2206{T}^{2}$ both between $\mathrm{\Delta}{T}_{nom}^{4}$ and $\mathrm{\Delta}{T}_{acc}^{4}$. These phenomena conform to the conjecture in modeling. If use adaptive or simplified adaptive maintenance policy, the average longrun cost can reduce 9.29 %, 6.43 %, respectively. It can be seen that adaptive maintenance policy is the best method for gearbox.
Table 4Optimal results for different maintenance policies
Policy structure  Optimal parameters  Expected cost  Impact 
Global  ${Y}_{A}^{1}=\text{6400}$, $\Delta {T}^{1}=\text{130}$  ${E}_{1}\left({C}_{\infty}\right)=\text{0.2971}$  
Timedepended  ${Y}_{A}^{2}=7500$, $\mathrm{\Delta}{T}^{2}=142$, $q=0.72$  ${E}_{2}\left({C}_{\infty}\right)=\text{0.2863}$  0.0108 (3.64 %) 
Simplified Adaptive  ${Y}_{nom}^{3}=\text{7200}$, ${Y}_{acc}^{3}=\text{6100}$, $\mathrm{\Delta}{T}^{3}=\text{115}$  ${E}_{3}\left({C}_{\infty}\right)=\text{0.2780}$  0.0191 (6.43 %) 
Adaptive  ${Y}_{nom}^{4}=\text{8000}$, $\Delta {T}_{nom}^{4}=\text{138}$, ${Y}_{acc}^{4}=\text{6000}$, $\Delta {T}_{acc}^{4}=\text{103}$  ${E}_{4}\left({C}_{\infty}\right)=\text{0.2695}$  0.0276 (9.29 %) 
6. Conclusions
This paper is meant to investigate degradation modeling and maintenance decisionmaking methods for twostage deteriorating mode system, where the degradation rate is usually small in the first stage and large in the second stage. To this purpose, degradation level modeling and reliability modeling based on cumulative damage model are studied at first place, then four kinds of maintenance policies (global, timedepended, adaptive, simplified adaptive) are studied and evaluated through their average longrun cost rate. The four kinds of maintenance policies are differentiated from alarm threshold and interinspection time.
Moreover, influence analysis of different parameters for maintenance policy is studied and proves that: (a) It is necessary to consider degradation process undergoing a sudden change in maintenance policy, suitable maintenance policy can help to improve system efficiency. (b) It is obvious that the average longrun cost rate is impacted by changepoint distribution, shock strength and shock frequency.
The case study of degradation data analysis for gearbox lifecycle experiment shows that degradation process of gearbox presents obviously twostage feature. In addition, it is helpful to reduce the average maintenance cost by choosing appropriate maintenance policy.
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