**4. Case Study**

The sub-tree of friction torque fault was the research object selected in this paper. The drop of voltage and current would slow down the speed of the flywheel, which would lead to a friction torque fault. The friction torque fault is also directly related to the shaft temperature (source used in this article from NASA). There were voltage, current, speed, shaft and friction moment data in this. One group of them could be chosen for the experiment. After selecting the data, they needed to be preprocessed. After the normalization of the data, fuzzy operator formula and ER fusion were used to obtain the data as the real value.

The fault diagnosis principle of the FFBRB flywheel system proposed in this paper included four parts: First, this paper normalized the collected data to make the data more accurate in practical application. Second, the normalized data were input into the fuzzy fault tree of the flywheel system, and the fuzzy probability of the intermediate event and the top event is calculated according to the corresponding formula. Third, this paper mapped the fuzzy fault tree to the BRB through the transformation space of the Bayesian network, so that the analysis process of the fuzzy fault tree corresponded to the inference process of BRB, and the input and output of the fuzzy fault tree correspond to the input and output of BRB, respectively. Finally, the data were handed over to the BRB for processing to realize the one-to-one correspondence between the BRB optimized value and the real value.

### *4.1. Construction of the FFBRB Fault Diagnosis Model*

### 4.1.1. The Fault Tree of the Friction Torque Fault of the Flywheel System Is Constructed

In the following description, the fault tree of the flywheel friction torque fault is preliminarily constructed to sort out the logical relationship between each fault event and determine the cause of the fault. The friction torque fault tree is shown in Figure 9:

**Figure 9.** Friction torque fault tree.

In the fuzzy fault tree graph of the case, the triangle fuzzy number is marked to limit the probability of each event within a range. This paper marked the meanings of each symbol in the fault tree below in advance to better describe the problem. The meanings of specific symbols are shown in Table 1.



4.1.2. FFTA Is Mapped to the BRB Using the Bayesian Network as a Bridge

After the establishment of the fault tree, this paper used the bridge of the Bayesian network to map the fault tree of FFTA to several different BRBS, so that the transformation from FFTA to BRB is perfectly realized, and the FFBRB model can be initially established. The relationship between the transformed Bayesian network graph and BRB is shown in Figure 10.

**Figure 10.** FFTA to BRB Bayesian network transformation diagram.

4.1.3. Determining the Fuzzy Number of Occurrence Probability of Bottom Event and Top Event

This step first needed to determine the trigonometric fuzzy number of the occurrence probability of the bottom event, and then calculate the trigonometric fuzzy number of the occurrence probability of the top event by using the formulas of fuzzy operators under different logic gates. The failure probability of the bottom event corresponds to the input of the BRB, and the occurrence probability of the top event corresponds to the output of the BRB, which is ready for the subsequent processing of the BRB program.

According to the previous introduction, corresponding data are divided into three groups (*a*1, *a*2, *a*), (*m*1, *m*2, *m*) and (*b*1, *b*2, *b*) according to the rules before. The data of the three groups are carried into the subsequent BRB, respectively, for fault diagnosis.

Triangulation fuzzy numbers of event probability in the BRB2 experiment are listed in Table 2 for reference.

**Remark 3.** *Each event in the above table only captures the data listed in article 10, from the data in the floating range there is a probability value of 10% of the incident left and if the interval data value is less than zero, the table is down to zero, if the data interval right value is greater than 1, the table down to 1, so the data that are limited to 0 to 1 can better describe probability*.

4.1.4. Built Initial Belief Rules

$$\begin{aligned} \text{If } & \mathbf{x}\_1 \text{ is } A\_1 \land \mathbf{x}\_2 \text{ is } A\_2\\ \text{Then result is } & \{ (Top\_1, \beta\_1), (Top\_2, \beta\_2), (Top\_3, \beta\_3), (Top\_4, \beta\_4) \} \\ \text{with rule weight } \theta\_1, \theta\_2, \dots, \theta\_K \\ \text{and attribute weight } \delta\_1, \delta\_2 \end{aligned} \tag{23}$$

The initialization of BRB requires belief rule construction. In this case, the belief rule construction of BRB is as above.


**Table 2.** Trigonometric fuzzy number of event probability in FFTA.

4.1.5. Set Reference Points and Values

In the BRB, it needed to set the reasonable reference values for the program to work properly. In this case, this paper set four reference points and reference values for each attribute, noting that the first reference value is an upper bound and the last reference value is a lower bound. The setting of reference values in BRB is shown in Table 3 above. The four numbers from left to right indicate the Very High(G), High(H), Middle(M), and Low(L) possibility of an event. The reference setting of BRB is shown in Table 3.

**Table 3.** Reference value of data in BRB.


**Remark 4.** *When the median value of triangle fuzzy number interval of event occurrence probability is 0, the reference value of the lower bound of the interval is set as a number approaching 0, because the probability of an event cannot be negative*.

### *4.2. Training and Optimization of the FFBRB Model*

4.2.1. Optimized Parameters and Results

Data show the optimized data of BRB2 (*b*1, *b*2, *b*), and the optimized parameters in BRB are shown in Table 4.

In Tables 4–6, the optimized rule weights are expressed as RuleWF and the optimized output belief degree is expressed as BeliefF. The results of the optimization of the upper and lower bounds of the interval and the median of the interval are listed.


**Table 4.** Optimized parameters table in BRB2r.

Table 4 is the optimal value of the upper bound of the interval, Table 5 is the optimal value of the ideal value of the interval, and Table 6 is the ideal value of the lower bound of the interval.


**Table 5.** Optimized parameters table in BRB2m.

To avoid data redundancy, only four bits of data are reserved in Tables 4–6. As the same, the optimized rule weights are expressed as RuleWF and the optimized output belief degree is expressed as BeliefF.


**Table 6.** Optimized parameters table in BRB2l.
