YAN Jing(晏 晶)，YIN Heng-su(尹珩蘇)，ZHOU Jie(周 杰)，LI Yan-feng(李彥鋒) ，HUANG Hongzhong(黃洪鐘)

School of Mechatronics Engineering，University of Electronic Science and Technology of China，Chengdu 611731，China

Introduction

The electrical system is one of the most important subsystems in CNC machine tool，so it is important to analyze the reliability of the electrical system.The traditional system reliability analysis method，fault tree analysis (FTA)method，based on static logic and static failure mechanism is not applicable because the electrical system is very complex.Moret et al.used the FT and Markov model to deal with periodic task with dynamic behavior［1-3］.However，the scale of the state space shows exponential growth as the system scale increases.Patterson-Hine and Dugan gave three modularization approaches to deal with DFT in order to improve analysis efficiency about dynamic fault tree (DFT)［4］.

DFT analysis method described in this paper can solve the problem above.In this method，DFT first should be pretreated to get a simplified FT;then we modularize FT to get the independent static subtrees and dynamic subtrees.We use binary decision diagram to analyze static subtrees， while an approximation algorithm will be used to deal with dynamic subtrees.Therefore，the problem of state explosion caused by Markov Model can be solved.When the scale of each subtree is smaller than the system scale，the analysis efficiency can be improved greatly.

In this paper，the application of DFT analysis method was researched in system reliability modeling and analysis.Static subtrees analysis method based on binary decision diagram(BDD)［5］and dynamic subtrees analysis method based on an approximation algorithm were also introduced.Finally，we established the DFT model about the electrical system of a certain type of CNC machine tool，after which the reliability analysis based on DFT analysis method was carried out.And we proved the usefulness of DFT analysis method when dealing with complex FT.

1 FT Analysis Method

FT is a graphical design method like the event tree，which is presented in inverted shape［6］.FTA is developed from 1960s，and the main point is not only to find all the combinations of the fault causes that make system failure occur，but also to find the occurrence probability of system failure by analyzing the factors that can cause product failure［7］.Generally，F(xiàn)TA is an effective evaluation method of complex systems’reliability and safety［8-10］.However，in the system with dynamic characteristics， FTA neither can provide appropriate qualitative description， nor can obtain the quantitative impact of a part accurately.

2 DFT Analysis Method

DFT analysis method introduces new logic gates to characterize the dynamic characteristics.DFT analysis method is an effective way to solve the reliability analysis of DFT.

2.1 Pretreatment

Generally，before modularization for a large-scale DFT，some pretreatments should be done first to downsize the scale of FT.There are two commonly used pretreatment technologies:merger of similar gates and optimization of repeated events［11］.

2.2 Modularization

We will apply modularization thinking in the DFT analysis，and the independent static and dynamic subtrees can be found.Finally，we integrate the results of each module to obtain the systematic analysis results.We carry on Deep-First-Left-Most(DFLM)search twice to find independent subtrees［11］.

3 Analysis Method for Static and Dynamic Subtrees

3.1 Qualitative analysis for static subtrees

Through conversion from FT into BDD，all the failure modes of the system have been obtained.In a BDD diagram，cut sets can be gotten from the paths starting from the root node to the leaf node“1”.

3.2 Quantitative analysis for static subtrees

Suppose the BDD diagram contains bottom events x1，x2，…，xn，and the corresponding failure probabilities are q1，q2，…，qn.We assume that the backtracking set of the leaf node 1 is:

Then the occurrence probability of the top event is［13］:

3.3 Qualitative analysis for dynamic sub-tree

DFT is usually analyzed by Markov model.However，the establishment of Markov model and finding the minimal cutsequence is quite time-consuming.We introduce a method better than the Markov model［11］.

The dynamic constraint of the dynamic gate consists of the logical constraint and the time constraint.For example，for a cold spare gate (CSP)，whose bottom events include X1and X2，the logical constraint relation of CSP gate is“and”，so we convert the CSP gate into AND gate.For static FT，the minimal cut set can be obtained:(x1，x2).The minimal cut sequence of the DFT can be obtained by adding the time sequence relationship between X1and X2:(x1→x2).

3.4 Quantitative analysis for dynamic subtree

In 2003，Amari et al.proposed a new method for computing the occurrence probability of top event of DFT［12］.In most cases，this method does not need to turn FT into a Markov chain.Amari et al.gave a detailed calculation method about Priority-AND gate.However，other gates aren't given any specific explanation.Therefore， we will give a detailed calculation description about Priority-AND gate［13］.

The top event occurs only when the two bottom events occur in the order from left to right.So the Priority-AND gate is a special case of sequence enforcing gate.Assuming that the distribution of the occurrence time of the bottom event A and B are G1(t)and G2(t)respectively，the distribution of the top event occurrence time G (t)is:

4 Synthesis of DFT Modules

Supposing that the bottom events of a dynamic FT are x1，x2，…，xn，the structure function is f = f1(x1，x2，…，xn).We can get the dynamic subtrees D1，D2，…，Diand the static subtrees M1，M2，…，Mjby modularizing DFT.So DFT can be expressed as［11］:

5 Reliability Analysis for Integrated Closed Center Frame

5.1 DFT of ICCF system

The electrical system is the core subsystem of CNC machine tool，which controls the running state of machine tool.The electrical system coordinates all parts to complete the machining task，which is a complex logic control circuit and consists of lots of relays and other electronic components.The electrical system refers to all electrical components and some sensors connecting lines，inverters，solenoid valves，switches，etc.So it is crucial to strengthen the reliability of electrical system.

When it is necessary to keep some relative movement between the lathe tools，center frame plays an auxiliary supporting role to make the process more quickly and accurately.The reliability of center frame system affects the accuracy of machining parts significantly.Therefore，center frame system is one of the key subjects in reliability analysis of electrical system in CNC machine tool.Because of the limited space，we only list the analysis of the integrated closed center frame (ICCF)system in this paper，the schematic diagram of the ICCF system is shown in Fig.1.Among them，“switches”includes four limit switches，four self-reset buttons with a light，three two position option switches with a light，and a push-pull type tight stop buttons with a lamp “relays”includes five relays;and“contactors”includes two contactors.

Fig.1 Schematic diagram of ICCF system

In order to establish FT easily，we coded all events.First，we assume that the components and the system only have two states:working and failure，and the failure distribution of parts are exponential.The events codes and failure rates of bottom events are shown in Tables 1 and 2，respectively.Then，“the failure of integrated closed center frame”is set as top event F.We can build the FT of ICCF，as shown in Fig.2.Assuming the failures between the various components are mutually independent，the ICCF system can be analyzed.

Table 1 Codes of the ICCF system

Table 2 Failure rate of bottom event (λ)

Fig.2 FT of ICCF System

5.2 Reliability analysis of ICCF system

(1)Pretreatment

First we need to do some pretreatments to get a simplified FT，which is shown in Fig.3.

(2)Modularization

Search the FT usingDFLM strategy twice，and the first and second search results are shown in Tables 3 and 4 respectively.

Fig.3 Simplified FT of ICCF system

Table 3 Traversal result for the first time

Table 4 Traversal result for the second time

From the search results，we can get the static sub-modules:Gl，G2，G3，G4，G5，G6and the dynamic sub-module:G7.Therefore，analyze G1，G2，G3，G4，G5and G6by BDD method，while analyze G7by the approximate algorithm.

1)Sub-module G1and its corresponding BDD graph(shown in Fig.4)

Backtrack the BDD diagram，we can get five paths with end node 1，which are shown as follows:

Then，the disjoint expression of the top event G1is:

We can obtain the occurrence probability of top event G1easily by substituting into the failure rates of the bottom events from Table 2 (assuming that the task time t=600 h):P1=0.014 9.

Fig.4 Sub-module G1 and the corresponding BDD

Similar to the analysis process of G1，we can get the results about sub-module G2，G3，G4，G5，G6.

2)The MCS of sub-module G2:T2= {{X6}，{X7}}The disjoint expression of the top event G2is:

We can obtain the occurrence probability of top event G2:P2=0.002 8.

3)The MCS of sub-module G3:T3= {{X8}，{X9}}

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The disjoint expression of the top event G3is:

The occurrence probability of top event G3:P3=0.000 06.

4)The MCS of sub-module G4:T4= {{X10}，{X11}，{X12}，{X13}}

The disjoint expression of the top event G4is:

The occurrence probability of top event G4is P4=0.003 6.

5)The MCS of sub-module G5:T5= {{X14}，{X15}，{X16}，{X17}}

The disjoint expression of the top event G5is:

The occurrence probability of top event G5is P5=0.002 4.

6)The MCS of sub-module G6:T6= {{X18}，{X19}，{X20}}

The disjoint expression of the top event G6is:

The occurrence probability of top event G6is P6=0.001 8.

7)Sub-module G7(shown in Fig.5)

We can get the minimum cut sequence of G7by the method above，which is T7= {X21→X22}

The life of thetwo parts obeys exponential distribution，so the failure rate are

Then we can get the life distribution function of the two parts:

From Eq.(3)，we can get:

Fig.5 Sub-module G7

Table 5 lists the calculation results of occurrence probability of top event G7(P7)with three different step lengths(assuming that task time t= 600 h).

Table 5 Occurrence probability of top event G7

This calculation results approximate analytical solution of Markov model with reducing step gradually.Taking into account the existence of the other uncertainties in the system，h can be taken as 1 h，so P7= 5.385 6 ×10-7.

8)Synthesis of DFT module

Set G1，G2，G3，G4，G5，G6，G7as the new bottom events，so the original FT can be shown in Fig.6.

Fig.6 FT of F

The corresponding BDD is shown in Fig.7.

Fig.7 BDD of F

So the minimal cut sequence of the original FT can be expressed by:T = {{G1}，{G2}，{G3}，{G4}，{G5}，{G6}，{x20}，{G7}}.

Substitute the MCS of each module，we can get Top event

Then backtrack BDD，we can get the disjoint expression of the top event G:

By substituting into the failure rates of each module，the occurrence probability P of top event F can be obtained as P =0.025 9.

6 Conclusions

This paper studied the system reliability modeling and evaluation based on DFT analysis method.The complicated FT is modularized to get the independent static subtrees and dynamic subtrees.BDD and an approximation algorithm analysis method are used to analyze static subtrees and dynamic subtrees respectively.When the scale of each subtree is smaller than the system scale，the analysis efficiency can be improved greatly.This paper established the reliability model of the integrated closed center frame of CNC machine tool electrical system，and then DFT analysis method was applied to dealing with this model.The analysis process of the ICCF has shown that DFT analysis method can effectively solve the problem of reliability analysis and evaluate the reliability of complicated engineering systems with dynamic failure characteristics.

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