1. Introduction

Some mechanical systems experience complicated environment which may continuously influence the reliability and availability. For instance, the spacecraft solar arrays are one of the most vital links to satellite mission success because providing reliable power over the anticipated mission life is critical to all satellites [1–3]. Although the faults have been reduced in the last few years by some measures, it still affects the longevity of the satellite severely, and faults of mechanical system occupy a large proportion of all the anomalies [3]. As a result, it is necessary for mechanical systems to evaluate reliability in different stages, including conceptual design of mechanical system, initial design and system improvement. The tasks of reliability

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

evaluation in these stages are defined as reliability prediction, reliability apportionment and reliability analysis, respectively. Many methodologies such as reliability block diagram (RBD), failure mode effect analysis (FMEA) and fault tree analysis (FTA) are widely used in reliability evaluation for electronic systems [4–6]. Recently, a number of papers reported the methodologies that use these models to evaluate reliability of the mechanical systems [7–9]. However, there still has some obstacles needed to be overcome for reliability evaluation of mechanical systems. Generally, three tasks should be accomplished, including reliability prediction, reliability apportionment and reliability analysis. We summarize the defects of previous research from the three aspects mentioned above.

reliability apportionment and reliability analysis. Some cases are included to illustrate the

A great volume of literature combines fuzzy reasoning and Petri net to accomplish the fault diagnosis and reliability analysis [25–27]. Gao presented an FRPN model and proposed a fuzzy reasoning algorithm based on matrix equation expression [19]. An FRPN model can be

, <sup>1</sup> � n; (1)

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R ¼ f g r1;r2…r<sup>m</sup> , 1 � m; (2)

I : P � R ! f g 0; 1 , n � m (3)

O : P � R ! f g 0; 1 , n � m (4)

H : P � R ! f g 0; 1 , n � m (5)

<sup>θ</sup> <sup>¼</sup> ð Þ <sup>θ</sup>1; <sup>θ</sup>2; <sup>⋯</sup>θ<sup>n</sup> T, <sup>θ</sup><sup>i</sup> <sup>∈</sup> ½ � <sup>0</sup>; <sup>1</sup> , n � <sup>1</sup> (6)

rj : C ¼ diag cf g <sup>1</sup>; c2…c<sup>25</sup> , 1 � m: (8)

, n � 1 (7)

defined as an 8-tuple model instead of the basic 5-tuple Petri net model [19].

P ¼ p1; p2…pn

γ : P ! f g 0; 1 , γ ¼ γ1; γ2…γ<sup>n</sup>

<sup>T</sup>

effectiveness of the methods.

2. Fuzzy reasoning Petri net

1. Places, namely, a set of propositions,

3. Directed arcs propositions to rules,

4. Directed arcs from rules to propositions,

5. Complementary arcs from positions to rules,

2. Transitions,

6. Truth degree vector:

7. Marking vector:

8. Confidence of

For reliability prediction, there are currently four main ways of reliability prediction for mechanical systems [10–12], including the similar product method (SPM), correction coefficient method (CCM), analysis of physics reliability method (APR) and parts count reliability prediction (PCRP). However, in the phase of conceptual design stage for one complex mechanical system, there has no enough experimental data or field record because the machine is not physically built. Moreover, APR is based on the physical failure mechanism which cannot be clearly identified in the conceptual design stage.

For reliability apportionment, there are two important issues needed to be addressed, i.e. how to describe the relationship among the different components and how to overcome data deficiency problem in the early stage of design [13–16]. It is usually hard to describe the factors of one mechanical system by the binary logic because the state cannot be simply classified into function or failure. Further, since the lack of system reliability data is a commonly encountered case in the initial stage of design, the reliability apportionment merely based on mathematics may not be feasible.

For reliability analysis, the FTA model has been widely employed as a powerful technique to evaluate the safety and reliability of complex systems by many scholars [17–19]. However, FTA has some limitations in reliability analysis. Firstly, in FTA, the probabilities of basic events must be known before analysis, but the designers can hardly obtain the probability of each fault because the conventional reliability test of the solar array mechanical system is difficult to carry out [19]. Secondly, it is not easy for FTA to conduct further quantitative analysis automatically due to the lack of effective means of mathematical expression. Thirdly, FTA cannot find the weak links of the system precisely, describe the propagation of fault and represent the characteristics of the system before and after improvement. In the literature, fuzzy reasoning is an effective method to solve the above problems [20].

The Petri net is one of the mathematical modeling approaches for the description of distributed systems, which consists of places, transitions, and directed arcs [21–23]. Many extensions to the Petri nets have been successfully applied in analyzing reliability of mechanical systems [24]. The fuzzy reasoning Petri net is a mathematical and graphical combined tool that can build a complex system with a variety of logical connections by using fuzzy reasoning, which may fit for building the reliability model for mechanical systems and evaluating reliability of them [20]. As a result, the primary objective of this chapter is to introduce the FRPN based models to evaluate the reliability of mechanical systems, including reliability prediction, reliability apportionment and reliability analysis. Some cases are included to illustrate the effectiveness of the methods.

#### 2. Fuzzy reasoning Petri net

A great volume of literature combines fuzzy reasoning and Petri net to accomplish the fault diagnosis and reliability analysis [25–27]. Gao presented an FRPN model and proposed a fuzzy reasoning algorithm based on matrix equation expression [19]. An FRPN model can be defined as an 8-tuple model instead of the basic 5-tuple Petri net model [19].

1. Places, namely, a set of propositions,

$$P = \{p\_1, p\_2...p\_n\}, 1 \times n;\tag{1}$$

2. Transitions,

evaluation in these stages are defined as reliability prediction, reliability apportionment and reliability analysis, respectively. Many methodologies such as reliability block diagram (RBD), failure mode effect analysis (FMEA) and fault tree analysis (FTA) are widely used in reliability evaluation for electronic systems [4–6]. Recently, a number of papers reported the methodologies that use these models to evaluate reliability of the mechanical systems [7–9]. However, there still has some obstacles needed to be overcome for reliability evaluation of mechanical systems. Generally, three tasks should be accomplished, including reliability prediction, reliability apportionment and reliability analysis. We summarize the defects of previous research

For reliability prediction, there are currently four main ways of reliability prediction for mechanical systems [10–12], including the similar product method (SPM), correction coefficient method (CCM), analysis of physics reliability method (APR) and parts count reliability prediction (PCRP). However, in the phase of conceptual design stage for one complex mechanical system, there has no enough experimental data or field record because the machine is not physically built. Moreover, APR is based on the physical failure mechanism which cannot be

For reliability apportionment, there are two important issues needed to be addressed, i.e. how to describe the relationship among the different components and how to overcome data deficiency problem in the early stage of design [13–16]. It is usually hard to describe the factors of one mechanical system by the binary logic because the state cannot be simply classified into function or failure. Further, since the lack of system reliability data is a commonly encountered case in the initial stage of design, the reliability apportionment merely based on mathematics

For reliability analysis, the FTA model has been widely employed as a powerful technique to evaluate the safety and reliability of complex systems by many scholars [17–19]. However, FTA has some limitations in reliability analysis. Firstly, in FTA, the probabilities of basic events must be known before analysis, but the designers can hardly obtain the probability of each fault because the conventional reliability test of the solar array mechanical system is difficult to carry out [19]. Secondly, it is not easy for FTA to conduct further quantitative analysis automatically due to the lack of effective means of mathematical expression. Thirdly, FTA cannot find the weak links of the system precisely, describe the propagation of fault and represent the characteristics of the system before and after improvement. In the literature, fuzzy reasoning is

The Petri net is one of the mathematical modeling approaches for the description of distributed systems, which consists of places, transitions, and directed arcs [21–23]. Many extensions to the Petri nets have been successfully applied in analyzing reliability of mechanical systems [24]. The fuzzy reasoning Petri net is a mathematical and graphical combined tool that can build a complex system with a variety of logical connections by using fuzzy reasoning, which may fit for building the reliability model for mechanical systems and evaluating reliability of them [20]. As a result, the primary objective of this chapter is to introduce the FRPN based models to evaluate the reliability of mechanical systems, including reliability prediction,

from the three aspects mentioned above.

58 Petri Nets in Science and Engineering

clearly identified in the conceptual design stage.

an effective method to solve the above problems [20].

may not be feasible.

$$R = \{r\_1, r\_2 \dots r\_m\}, 1 \times m;\tag{2}$$

3. Directed arcs propositions to rules,

$$I: P \times R \to \{0, 1\}, n \times m \tag{3}$$

4. Directed arcs from rules to propositions,

$$O: P \times R \to \{0, 1\}, n \times m \tag{4}$$

5. Complementary arcs from positions to rules,

$$H: P \times R \to \{0, 1\}, n \times m \tag{5}$$

6. Truth degree vector:

$$\boldsymbol{\theta} = (\theta\_1, \theta\_2, \dots, \theta\_n)^T, \theta\_i \in [0, 1], n \times 1 \tag{6}$$

7. Marking vector:

$$\gamma: P \to \{0, 1\}, \gamma = \left(\gamma\_1, \gamma\_2 \dots \gamma\_n\right)^T, n \times 1 \tag{7}$$

8. Confidence of

$$r\_{\rangle}: \mathbb{C} = \text{diag}\{c\_1, c\_2...c\_{25}\}, 1 \times m. \tag{8}$$

On the basis of algorithm provided by Gao [19], the simulation can be operated automatically. The following are the main rules:


The PRPN model takes advantage of the following maximum algebra

1. ⊕ : A ⊕ B ¼ D, where A, B and D are all m � n dimensional matrices, such that

$$d\_{\vec{\eta}} = \max\{a\_{\vec{\eta}}, b\_{\vec{\eta}}\} \tag{9}$$

2. The FRPN model can describe the fault propagation in mechanical system by fuzzy reasoning, which can describe the properties of mechanical systems accurately.

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3. The FRPN model is based on an iteration algorithm, so the status transition can be easily tracked, which may be useful for examining the fault propagation and fault severity in the

For evaluating the reliability of a mechanical system, one should complete a series of work including reliability prediction in the stage of conceptual design, reliability apportionment in the stage of initial design, and reliability analysis in the stage of system improvement. The following subsections will illustrate the method of how to evaluate reliability by FRPN models.

Reliability prediction acts when a product is in the stage of conceptual design. Here we introduce a method of reliability prediction of mechanical systems. This method includes the following steps (Figure 1). First, we will build an FRPN model of the mechanical system by its working principle and the logical connections among the components. Second, we get three key values which characterize quantity, importance and quality of the components in the mechanical system. Third, we will arrive at the reliability prediction result by parts count reliability prediction (PCRP). Finally, the reliability prediction formula of mechanical system

system.

3.1.1. Method

denotes to

3. Reliability evaluation by FRPN

3.1. Reliability prediction by FRPN

Figure 1. Main process of reliability prediction.

2. ⊗ : A ⊗ B ¼ D, where A, B and D are m � p, p � n and m � n-dimensional matrices respectively, such that

$$d\_{\vec{\eta}} = \max\_{1 \le k \le p} \left\{ a\_{\vec{a}k} \cdot b\_{\vec{\eta}} \right\} \tag{10}$$

The firing and control vectors are stated as follows [19]:

$$\begin{cases} \mu\_{m \times 1}^{k} = \mathbf{1}\_{m \times 1} - (I + H)^{T} \otimes \overline{\boldsymbol{\gamma}}^{k} \\ \rho\_{m \times 1}^{k} = \mathbf{1}\_{m \times 1} - \left( I^{T} \otimes \left( \overline{\boldsymbol{\gamma}}^{k} \oplus \overline{\boldsymbol{\mathcal{O}}}^{k} \right) \right) \oplus \left( H^{T} \otimes \left( \overline{\boldsymbol{\gamma}}^{k} \oplus \boldsymbol{\mathcal{O}}^{k} \right) \right) \end{cases} \tag{11}$$

in which

$$\begin{cases} \overline{\theta}^k = \mathbf{1}\_{\mathbf{m} \times 1} - \theta^k \\ \overline{\gamma}^k = \mathbf{1}\_{\mathbf{m} \times 1} - \gamma^k \end{cases} \tag{12}$$

The marking and truth degree vectors can be obtained by

$$\begin{cases} \gamma^{k+1} = \gamma^k \oplus [O \otimes \mu] \\ \theta^{k+1} = \theta^k \oplus [(O \cdot \mathbb{C}) \otimes \rho] \end{cases} \tag{13}$$

which reflects the status of the components in the mechanical system. The FRPN model is suitable to describe the status transition in a mechanical system because

1. The FRPN model is constructed by the places and logical connections which match the properties of mechanical systems with multiple components.

