3.1. Reliability prediction by FRPN

#### 3.1.1. Method

On the basis of algorithm provided by Gao [19], the simulation can be operated automatically.

2. If there are many places to one transition like AND gate in FTA model, the upper truth value will be the minimum; if there are many places to many transitions like OR gate in

4. The truth degree vector <sup>θ</sup> <sup>¼</sup> ð Þ <sup>θ</sup>1; <sup>θ</sup>2; <sup>⋯</sup>θ<sup>n</sup> <sup>T</sup> shows the fuzzy possibility of the faults.

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

dij ¼ max 1 ≤ k ≤ p

<sup>T</sup> ⊗ γ<sup>k</sup> ⊕ θ <sup>k</sup> � � � �

<sup>γ</sup><sup>k</sup>þ<sup>1</sup> <sup>¼</sup> <sup>γ</sup><sup>k</sup> <sup>⊕</sup> <sup>O</sup> <sup>⊗</sup> <sup>μ</sup> � � <sup>θ</sup><sup>k</sup>þ<sup>1</sup> <sup>¼</sup> <sup>θ</sup><sup>k</sup> <sup>⊕</sup> ½ � ð Þ <sup>O</sup> � <sup>C</sup> <sup>⊗</sup> <sup>r</sup>

which reflects the status of the components in the mechanical system. The FRPN model is

1. The FRPN model is constructed by the places and logical connections which match the

<sup>¼</sup> 1m�<sup>1</sup> � <sup>θ</sup><sup>k</sup> <sup>γ</sup><sup>k</sup> <sup>¼</sup> 1m�<sup>1</sup> � <sup>γ</sup><sup>k</sup>

θ k

(

<sup>m</sup>�<sup>1</sup> <sup>¼</sup> <sup>1</sup><sup>m</sup>�<sup>1</sup> � ð Þ <sup>I</sup> <sup>þ</sup> <sup>H</sup> <sup>T</sup> <sup>⊗</sup> <sup>γ</sup><sup>k</sup>

dij ¼ max aij; bij

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

aik � bkj

, n � 1 shows the propagation of the faults in model. If the

� � (9)

� � (10)

(11)

(12)

(13)

⊕ H<sup>T</sup> ⊗ γ<sup>k</sup> ⊕ θ<sup>k</sup> � � � �

1. If one transition is fired, the token will be sent to the upper place.

FTA model, the upper truth value will be the maximum.

⋯γ<sup>n</sup> � �<sup>T</sup>

The PRPN model takes advantage of the following maximum algebra

element γ<sup>i</sup> ¼ 1, the place p<sup>i</sup> will get the token.

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

<sup>m</sup>�<sup>1</sup> <sup>¼</sup> <sup>1</sup><sup>m</sup>�<sup>1</sup> � <sup>I</sup>

The marking and truth degree vectors can be obtained by

(

suitable to describe the status transition in a mechanical system because

properties of mechanical systems with multiple components.

μk

8 < :

rk

The following are the main rules:

60 Petri Nets in Science and Engineering

3. The vector γ ¼ γ1; γ2…γ<sup>i</sup>

respectively, such that

in which

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 denotes to

Figure 1. Main process of reliability prediction.

$$
\lambda\_p = \sum\_{i=1}^{\text{T}} N\_i \cdot \lambda\_{\text{Ci}} \pi\_{\text{Qi}} \tag{14}
$$

where λ<sup>p</sup> is the final predicted failure rate, λGi and πQi are the indexes which indicate importance and quality of the components [28].

#### 3.1.2. Case study

We take the deployable solar array used in spacecraft as an example. The running process of a typical deployable solar array is shown in Figure 2, which is widely used for power supply in the spacecraft nowadays. In general, the entire running process includes three stages, i.e. the deployable solar array is first folded, then deployed in the orbit and finally oriented to the sun to generate power for satellite.

In general, the mechanical system of the solar array consists of seven kinds of mechanisms [29–31], i.e. the hold-down and release mechanism, the solar panel, the driving mechanism, the deployable mechanism, the locking mechanism, the synchronization mechanism, and the orientation mechanism, as shown in Figure 3. Torsion spring is often chosen to drive the solar array, the closed cable loop (CCL) is used as the synchronization mechanism, and the stepping motor or servo motor is carried to orient to the sun. The driving mechanism, the deployable mechanism and the locking mechanism are always integrated into the hinge. Therefore the five main mechanisms of the solar array include hold-down and release mechanism, the solar panel, the hinge, the synchronization mechanism and the orientation mechanism.

We use R<sup>1</sup> to R<sup>5</sup> to represent the reliability of the five mechanisms, respectively. Then the reliability of the mechanical system can be calculated as follows:

$$R = R\_1 R\_2 R\_3 R\_4 R\_5 \tag{15}$$

In the phase of conceptual design, designers should divide the reliability of the system into the five main parts. The following section introduces a new method of reliability apportionment which focuses on how to get the predicted values of Rið Þ i ¼ 1; 2; 3; 4; 5 to meet the requirement of the design standard. We build an FRPN model for the mechanical system of the solar array

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By the method shown in Figure 1, we can measure the complexity of the ith place (CP) as a number of Ni, the final truth degree of the ith place (FTD) as λGi, and the environmental factor

(Figure 4). Table 1 shows the markers and events of FRPN model [32, 33].

Figure 3. Mechanisms in a spacecraft solar array.

Figure 4. The FRPN model of the solar array for reliability prediction.

Figure 2. Operating principle of a deployable solar array.

Figure 3. Mechanisms in a spacecraft solar array.

<sup>λ</sup><sup>p</sup> <sup>¼</sup> <sup>X</sup> T

tance and quality of the components [28].

to generate power for satellite.

3.1.2. Case study

62 Petri Nets in Science and Engineering

i¼1

where λ<sup>p</sup> is the final predicted failure rate, λGi and πQi are the indexes which indicate impor-

We take the deployable solar array used in spacecraft as an example. The running process of a typical deployable solar array is shown in Figure 2, which is widely used for power supply in the spacecraft nowadays. In general, the entire running process includes three stages, i.e. the deployable solar array is first folded, then deployed in the orbit and finally oriented to the sun

In general, the mechanical system of the solar array consists of seven kinds of mechanisms [29–31], i.e. the hold-down and release mechanism, the solar panel, the driving mechanism, the deployable mechanism, the locking mechanism, the synchronization mechanism, and the orientation mechanism, as shown in Figure 3. Torsion spring is often chosen to drive the solar array, the closed cable loop (CCL) is used as the synchronization mechanism, and the stepping motor or servo motor is carried to orient to the sun. The driving mechanism, the deployable mechanism and the locking mechanism are always integrated into the hinge. Therefore the five main mechanisms of the solar array include hold-down and release mechanism, the solar

We use R<sup>1</sup> to R<sup>5</sup> to represent the reliability of the five mechanisms, respectively. Then the

panel, the hinge, the synchronization mechanism and the orientation mechanism.

reliability of the mechanical system can be calculated as follows:

Figure 2. Operating principle of a deployable solar array.

Ni � λGiπQi (14)

R ¼ R1R2R3R4R<sup>5</sup> (15)

In the phase of conceptual design, designers should divide the reliability of the system into the five main parts. The following section introduces a new method of reliability apportionment which focuses on how to get the predicted values of Rið Þ i ¼ 1; 2; 3; 4; 5 to meet the requirement of the design standard. We build an FRPN model for the mechanical system of the solar array (Figure 4). Table 1 shows the markers and events of FRPN model [32, 33].

By the method shown in Figure 1, we can measure the complexity of the ith place (CP) as a number of Ni, the final truth degree of the ith place (FTD) as λGi, and the environmental factor

Figure 4. The FRPN model of the solar array for reliability prediction.


Table 1. Markers and events of FRPN model for reliability prediction.

(EF) of the ith place as πQi. Some details can be checked in [32]. We collected the actual reliability data (lifetime of mechanical systems) of the solar arrays in a group of satellites from 1950s to 2000s provided by [34]. The results show that all of the predicted reliability lies in the interval of the operation data, which demonstrates the correctness of FRPN-based model for reliability prediction. Figure 5 validates the predicted reliability by using the four selected time: 0.025 106 h, 0.05 106 h, 0.075 106 h and 0.1 106 h. Some more details can be checked in [34].

reliability apportionment approaches including equal distribution method, Alins distribution method and algebra distribution method are widely used in the early stage of the reliability design [35, 36]. However, these methods have some limitations. It is obvious that dividing the system reliability into those of the subsystems equally may ignore the diversity of the components. Although the Alins distribution method and the algebra distribution method involve the importance or complexity of the different units, they are heavily dependent on the existing data and engineering experience which are scare in the early stage of the reliability design. Here we propose an FRPN-based method for reliability apportionment to solve the problems

Figure 6. Procedures for reliability apportionment by FRPN. The FRPN model is used in the first and second steps and

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discussed above. This method includes the following steps (Figure 6):

Figure 5. Comparison between the predicted reliability and real reliability at selected phases.

the following two steps use the fuzzy comprehensive evaluation.

#### 3.2. Reliability apportionment by FRPN

#### 3.2.1. Method

After reliability prediction in the conceptual design phase, the engineer should start reliability apportionment that acts when a product is in the stage of initial design. The conventional

Figure 5. Comparison between the predicted reliability and real reliability at selected phases.

(EF) of the ith place as πQi. Some details can be checked in [32]. We collected the actual reliability data (lifetime of mechanical systems) of the solar arrays in a group of satellites from 1950s to 2000s provided by [34]. The results show that all of the predicted reliability lies in the interval of the operation data, which demonstrates the correctness of FRPN-based model for reliability prediction. Figure 5 validates the predicted reliability by using the four selected time: 0.025 106 h,

array

After reliability prediction in the conceptual design phase, the engineer should start reliability apportionment that acts when a product is in the stage of initial design. The conventional

0.05 106 h, 0.075 106 h and 0.1 106 h. Some more details can be checked in [34].

P<sup>15</sup> Fault of the gear in the reducer \_ P<sup>30</sup> Fault of the mechanical system of the solar

3.2. Reliability apportionment by FRPN

Table 1. Markers and events of FRPN model for reliability prediction.

Marker Event Truth

P<sup>1</sup> Harsh thermal environment in

P<sup>2</sup> Vacuum and micro-gravity environment in space

64 Petri Nets in Science and Engineering

P<sup>3</sup> Fault of the grease used in hinges between panels

P<sup>4</sup> Impact caused by particles in

P<sup>6</sup> Fault of the main driving torsion

P<sup>7</sup> Fault of the reserved driving torsion spring

P<sup>8</sup> Fault of the driving pin in the

P<sup>9</sup> Fault of the side wall of the

P<sup>11</sup> Fault of the reserved locking

P<sup>12</sup> Fault of the locking pin of the

P<sup>13</sup> Fault in the mechanical part of the stepping motor

P<sup>14</sup> Fault in the electronic part of the stepping motor

space

space

spring

hinge

hinge

spring

hinge

degree

P<sup>5</sup> Fault of the brass gasket 0.5 P<sup>20</sup> Fault of the deployable mechanism \_

P<sup>10</sup> Fault of the main locking spring 0.8 P<sup>25</sup> Fault of the hold-down and release

Marker Event Truth

0.9 P<sup>16</sup> Fault of the bearing in the reducer 0.4

0.7 P<sup>19</sup> Fault of the driving mechanism \_

0.6 P<sup>21</sup> Fault of the locking mechanism \_

0.6 P<sup>22</sup> Fault of the steel wire 0.7

\_ P<sup>23</sup> Fault of the stepping motor \_

\_ P<sup>24</sup> Fault of the transmission system \_

0.5 P<sup>26</sup> Fault of the solar panels \_

0.5 P<sup>27</sup> Fault of the hinges \_

0.3 P<sup>28</sup> Fault of the synchronization mechanism \_

0.2 P<sup>29</sup> Fault of the orientation mechanism \_

mechanism

0.6 P<sup>17</sup> Fault of the electronic arcing of the holddown and release mechanism

0.4 P<sup>18</sup> Fault of the cutter of the hold-down and release mechanism

degree

0.7

0.7

\_

\_

3.2.1. Method

Figure 6. Procedures for reliability apportionment by FRPN. The FRPN model is used in the first and second steps and the following two steps use the fuzzy comprehensive evaluation.

reliability apportionment approaches including equal distribution method, Alins distribution method and algebra distribution method are widely used in the early stage of the reliability design [35, 36]. However, these methods have some limitations. It is obvious that dividing the system reliability into those of the subsystems equally may ignore the diversity of the components. Although the Alins distribution method and the algebra distribution method involve the importance or complexity of the different units, they are heavily dependent on the existing data and engineering experience which are scare in the early stage of the reliability design. Here we propose an FRPN-based method for reliability apportionment to solve the problems discussed above. This method includes the following steps (Figure 6):


3.2.2. Case study

mechanism

space

markers and events of the FRPN model [32].

P<sup>10</sup> Electronic arcing of the hold-down and release mechanism

P<sup>11</sup> Cutter of the of the hold-down and release

P<sup>13</sup> Vacuum and micro-gravity environment in

Table 2. Markers and events of FRPN model for reliability apportionment.

Marker Event Truth

mechanical system itself [33].

3.3.1. Method

3.3. Reliability analysis by FRPN

We take the spacecraft solar array as an example to conduct the reliability apportionment by using the FRPN model (Figure 3). According to the operational principle of array mechanical systems of a solar array, we build an FRPN model for reliability apportionment of spacecraft solar array. The graphical representation of this model is shown in Figure 7. Table 2 shows the

degree

P<sup>1</sup> Grease used in hinges between panels 0.4 P<sup>14</sup> Particles in space — P<sup>2</sup> Brass gasket 0.5 P<sup>15</sup> Driving mechanism — P<sup>3</sup> Main deriving torsion spring 0.6 P<sup>16</sup> Deployable mechanism — P<sup>4</sup> Reserved driving torsion spring 0.6 P<sup>17</sup> Locking mechanism — P<sup>5</sup> Driving pin in the hinge — P<sup>18</sup> Steel wire 0.7 P<sup>6</sup> Side wall of the hinge — P<sup>19</sup> Stepping motor 0.2 P<sup>7</sup> Main locking spring 0.8 P<sup>20</sup> Transmission system 0.6

P<sup>8</sup> Reserved locking spring 0.5 P<sup>21</sup> Hold-down and release

P<sup>9</sup> Locking pin of the hinge 0.5 P<sup>22</sup> Solar panels —

P<sup>12</sup> Harsh thermal environment in space 0.9 P<sup>25</sup> Orientation mechanism —

Marker Event Truth

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Reliability Evaluation for Mechanical Systems by Petri Nets

mechanism

0.7 P<sup>23</sup> Hinges —

0.7 P<sup>24</sup> Synchronization mechanism —

— P<sup>26</sup> Mechanical system of the solar array

degree

67

—

—

From Figure 7, the FRPN model of solar array includes 13 bottom places- P1, P2, P3, P4, P7, P8, P9, P10, P11, P12, P18, P<sup>19</sup> and P20. And P21, P22, P23, P24, and P<sup>25</sup> represent the subsystems (Table 2). The final reliability apportionment results are illustrated in Figure 8 under the system reliability of 0.9, 0.99 and 0.999. In this figure, RS represents the reliability of the system and Rið Þ i ¼ 21; 22; 23; 24; 25 expresses the reliability of the five key subsystems. The reliability apportionments are shown in Figure 8. By using the FRPN based model, the system reliability can be allocated considering the environmental factors and the intrinsic connection in the

Reliability analysis happens in the stage that the mechanical system has been built physically. By using the FRPN model, we can analyze the reliability of the system with the following steps:

Figure 7. The FRPN model of the solar array for reliability apportionment.

Figure 8. The reliability apportionment of the five key components of solar array. The reliability system is equal to 0.9, 0.99 and 0.999 respectively.


Table 2. Markers and events of FRPN model for reliability apportionment.

#### 3.2.2. Case study

1. Decompose the mechanical system;

4. Fuzzy comprehensive evaluation;

5. Reliability apportionment.

66 Petri Nets in Science and Engineering

0.99 and 0.999 respectively.

2. Build the FRPN model of the mechanical system;

Figure 7. The FRPN model of the solar array for reliability apportionment.

3. Analyze the three aspects including the complexity of one component during propagation

Figure 8. The reliability apportionment of the five key components of solar array. The reliability system is equal to 0.9,

of the faults, the importance of one component and the working environment;

We take the spacecraft solar array as an example to conduct the reliability apportionment by using the FRPN model (Figure 3). According to the operational principle of array mechanical systems of a solar array, we build an FRPN model for reliability apportionment of spacecraft solar array. The graphical representation of this model is shown in Figure 7. Table 2 shows the markers and events of the FRPN model [32].

From Figure 7, the FRPN model of solar array includes 13 bottom places- P1, P2, P3, P4, P7, P8, P9, P10, P11, P12, P18, P<sup>19</sup> and P20. And P21, P22, P23, P24, and P<sup>25</sup> represent the subsystems (Table 2). The final reliability apportionment results are illustrated in Figure 8 under the system reliability of 0.9, 0.99 and 0.999. In this figure, RS represents the reliability of the system and Rið Þ i ¼ 21; 22; 23; 24; 25 expresses the reliability of the five key subsystems. The reliability apportionments are shown in Figure 8. By using the FRPN based model, the system reliability can be allocated considering the environmental factors and the intrinsic connection in the mechanical system itself [33].

#### 3.3. Reliability analysis by FRPN

#### 3.3.1. Method

Reliability analysis happens in the stage that the mechanical system has been built physically. By using the FRPN model, we can analyze the reliability of the system with the following steps:


#### 3.3.2. Case study

We also take the spacecraft solar array as a case for reliability analysis. Figure 9 shows the FRPN model of the spacecraft solar array for reliability analysis and Table 3 represents markers and events [37].

Define θ<sup>i</sup> as the truth degree of the bottom place pi

Table 3. Markers and events of FRPN for reliability analysis.

P<sup>18</sup> Vibration of panels induced by thermal

deformation

Marker Event Marker Event

P<sup>12</sup> Fault of CCL P<sup>15</sup> Fault of the transmission unit

P<sup>8</sup> Electronic arcing is out of service P<sup>17</sup> Vibration caused by clearances of hinges

Table 4. Solar array classification ranks of the fault model.

Table 5. Fault rank of the bottom places and their truth degree.

Improvement measures

different characters, like stiffness

Table 6. Improvement measures.

of particles, and make adjustment with expedition

Bottom place

possibility of the event is higher, which means the fault occurs much easier. Table 4 demonstrates the ranks, occurrence, and truth degrees of the bottom places. According to the characteristics of

P<sup>9</sup> Fault of the cutters P<sup>7</sup> Bad thermal characteristic of honeycomb materials

Rank I II III IV V VI VII Occurrence Very low Low Fairly low Moderate Fairly high High Very high Truth degree 0.1 0.3 0.4 0.5 0.6 0.8 1.0

Marker of bottom places P<sup>1</sup> P<sup>2</sup> P<sup>3</sup> P<sup>4</sup> P<sup>5</sup> P<sup>6</sup> P<sup>7</sup> Rank VII III V V VI V VI Truth degree 1.0 0.4 0.6 0.4 0.8 0.6 0.8 Marker of bottom places P<sup>8</sup> P<sup>9</sup> P<sup>13</sup> P<sup>14</sup> P<sup>15</sup> P<sup>16</sup> P<sup>17</sup> Rank V V V II IV VI VI Truth degree 0.4 0.6 0.8 0.3 0.5 0.8 0.8

P<sup>1</sup> The thermal environment in space is the crucial factor of the failure. Some approaches to improve the

P<sup>16</sup> That happens occasionally. There is no effective measure to avoid particles in space, maybe only two ways:

reliability of the system. (1) Investigate the temperature in space precisely where the solar array works and sum the rules; (2) use new material that is fit for the change of the temperature in space; (3) research the temperature impact on the structure, and optimize the structure of the crucial part of the system P<sup>13</sup> (1) Test the torsion spring on the ground, then find the torque-angle curve to know the characteristics of the torsion spring more deeply; (2) test the performance of the whole system, using torsion springs with

(1) make the structure stronger; (2) make the system more agile to detect the vibration caused by the impact

, θ<sup>i</sup> ∈½ � 0; 1 . A higher value indicates that the

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P<sup>16</sup> Impact caused by particles in space

Figure 9. The FRPN model of solar array for reliability analysis.



Table 3. Markers and events of FRPN for reliability analysis.

1. Decompose the mechanical system.

68 Petri Nets in Science and Engineering

4. Calculate the truth degree of top place.

Figure 9. The FRPN model of solar array for reliability analysis.

Marker Event Marker Event

P<sup>24</sup> Failure of the solar array system P<sup>1</sup> Harsh thermal environment in space

P<sup>22</sup> Fault of orientation to the sun P<sup>5</sup> Insufficient preload of the cable P<sup>23</sup> Other faults of mechanical system P<sup>6</sup> Poor thermal characteristic of the cable

P<sup>11</sup> Insufficient preload of the torsion spring P<sup>14</sup> Fault of the motor

P<sup>19</sup> Fault of the unlock-mechanism P<sup>2</sup> Fault of the grease used in hinges between panels P<sup>20</sup> Faults during deployment process P<sup>3</sup> Insufficient torque of the main torsion spring P<sup>21</sup> Faults during locking process P<sup>4</sup> Insufficient torque of the reserved torsion spring

P<sup>10</sup> Deadlocking in hinges P<sup>13</sup> Inappropriate driving torque of the locking torsion

spring

3.3.2. Case study

events [37].

2. Build the FRPN model of the mechanical system.

the system, operation data and engineering experience

3. Get the truth degrees of the bottom places according to the characteristics of the faults in

We also take the spacecraft solar array as a case for reliability analysis. Figure 9 shows the FRPN model of the spacecraft solar array for reliability analysis and Table 3 represents markers and

5. Use the cosine matching function (CMF) to analyze reliability of the system.

Define θ<sup>i</sup> as the truth degree of the bottom place pi , θ<sup>i</sup> ∈½ � 0; 1 . A higher value indicates that the possibility of the event is higher, which means the fault occurs much easier. Table 4 demonstrates the ranks, occurrence, and truth degrees of the bottom places. According to the characteristics of


Table 4. Solar array classification ranks of the fault model.


Table 5. Fault rank of the bottom places and their truth degree.


Table 6. Improvement measures.

the faults in the system, operation data and engineering experience [9]. Table 5 represents the fault rank of the bottom places and their truth degrees.

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We can get the results of reliability analysis by using the method in Section 3.3.1. According to the results, we can evaluate the importance of bottom places in the FRPN model. Some details can be checked in [37]. To improve the system reliability, we should propose some approaches to enhance the weak links. Table 6 shows some improvement measures for the mechanical system of a spacecraft solar array.
