The T-S FTA is an analysis method that considers the impact of multiple fault levels on the system. It has IF-THEN rules and describes inter-event connections with T-S gates. When the input event fault state of the IF-THEN rule is two-state and the output event satisfies the traditional fault tree logic gates, the T-S Fault tree degenerates to the traditional fault tree. The traditional fault tree is a special case of the T-S Fault tree. The T-S Fault tree model is shown in Fig. 1a, where X1, X2, and X3 are bottom events, M1 is the middle event, M2 is the top event, and T-S gate 1 and T-S gate 2 are T-S fuzzy gates.

Considering the fuzziness of the event fault state, the T-S FTA uses the fuzzy number to describe the event fault state. For example, the fault state of the T-S gate input event xi(i=1,2,…,n) is a fuzzy number xi[ai](ai=1,2,…,ki). The fault state of the output event y is a fuzzy number y[bj](bj=1,2,…,nj). Then the fault logic relationship between the events can be described by the T-S gate rules. Rule l(l=1,2,…,r): If the input event x1 fault state is x1[a1], input event x2 fault state is x2[a2],… , and input event xn fault state is xn[an], the probability of output event fault state y[bj] is Pl(y[bj]). Among them, r is the total number of T-S rules that satisfy r=k1k2…kn=∏i=1n⁡ki.

The Bayesian network is a directed acyclic network composed of a directed acyclic graph and a conditional probability table, as shown in Fig. 1b. Where X1, X2, and X3 are the root nodes, Y1 is the intermediate node, and Y2 is the leaf node. X1, X2, and X3 are the parent nodes of Y1, and Y2 is the child node of Y1. The nodes are connected by directed edges and the strength of the connection is determined by the conditional probability P. P consists of conditional probability parameters between variables. The probability distribution P(X) of the BN in Fig. 1b is: (1)P(X)=P(X1,X2,X3)=∏i=13⁡P(Xi|Pa(Xi))where Pa(Xi) is the parent node set of nodes Xi. When Pa(Xi) is an empty set, P(Xi|Pa(Xi)) is the prior probability P(Xi) of node Xi.

The T-S FTA solves the problem of uncertainty about the failure mechanism and the degree of failure in complex systems. However, it is computationally complex and cannot be reasoned backward. And BN is superior to T-S FTA in computational analysis [17]. However, it is more difficult to construct BN directly.

The fault tree analysis method based on the fuzzy reliability model is relatively mature. However, in engineering practice, it is often difficult to obtain sufficient data to determine the probability density function or fuzzy affiliation function of parameters [30]. As a mathematical method to deal with uncertainty inference problems, D-S evidence theory quantifies the degree of confidence and likelihood of propositions. It captures the unknown and uncertainty of the problem better than the traditional probability theory. It provides a way to obtain input data in fault tree analysis.

D-S evidence theory is a theory of information fusion [31]. It expresses the uncertainty problem through the belief function and the plausibility function [32]. Define the discernment framework Θ, which consists of several completely mutually incompatible elements. Denote A as any subset of the discernment framework Θ. Define the basic creditability allocation function for the evidence i to be mi:2Θ→[0,1] (2Θ is a powerful set of Θ) satisfying: (2){m(∅)=0∑A⊆Θ⁡m(A)=1

Define the belief degree Bel: 2Θ→[0,1] and plausibility degree Pl: 2Θ→[0,1]. When ∀A⊆Θ, and A≠∅, there are Bel(A)=∑B⊆A⁡m(B), Pl(A)=∑B∩A≠∅⁡m(B), Bel(A)=1−Pl(A).

According to the D-S synthesis rule, the synthesis rule for n pieces of independent evidence on the identification frame Θ as shown in Eq. (3): (3)m(A)=(m1⊕m2⊕⋯⊕mn)(Ai)=1K∑A1∩A1∩⋯∩An=A⁡∏j=1n⁡mj(Ai)

Among them, K is the normalization factor, which is used to measure the degree of conflict between evidence, and can be obtained by Eq. (4): (4)K=∑A1∩A1∩⋯∩An≠∅∏j=1n⁡mj(Ai)=1−∑A1∩A1∩⋯∩An=∅⁡∏j=1n⁡mj(Ai)

The D-S theory of evidence cannot resolve serious conflicts and complete conflicts in the evidence. And the more the number of elements in a subset, the greater the ambiguity of the subset. HM is a convex set model. It has the advantages of continuous parameter variation, simple model structure, and easy correlation analysis. It can better compensate for the lack of too-conservative analysis results of the Interval model [22], as shown in Fig. 2.

The HM is a method of reflecting the deviation of a random variable based on the distance between its equivalent unit hypersphere coordinate origin and the failure surface of a structure in normalized vector space. If the random variable Xi∈[xiL,xiU](i=1,2,⋯,n) in the set of random variables X, where xiL and xiU are the lower bound and upper bound of the value of Xi, respectively. Then the HM of the set of random variables X is described as: (5)[X1−x10x1,⋯,Xn−xn0xnr]T⋅[X1−x10x1,⋯,Xn−xn0xnr]≤1where, xi0=xiL+xiU2, xir=−xiL+xiU2 are the nominal value and the deviation of xi, respectively.

The multistate series-parallel system is shown in Fig. 4. Construct its T-S fault tree as shown in Fig. 5a, where G1 denotes “AND” gates. G2 and G3 denote “OR” gates. The Fault occurrence state, Fault in fuzzy uncertainty state, and Fault non-occurrence state of the system and components are denoted by A1,(A1,A2), and A2. The failure rate of the component Xi(i=1,2,3,4) in the Fault occurrence state is 5.0×10−5/h, 1.0×10−5/h, 1.5×10−5/h, and 1.2×10−5/h, respectively. The failure rate of components in the Fault in fuzzy uncertainty state is 9.0×10−5/h, 1.0×10−4/h, 2.8×10−5/h, and 3.6×10−5/h, respectively.

The T-S FTA of the multistate series-parallel system was converted into a BN as shown in Fig. 5b. The T-S gate rules are converted into a conditional probability table. For the G1 gate, see Table 4. Conditional probability tables are the same for G2 and G3, see Table 5. Table 4 rule 1 indicates that under the condition that the S1 state is A1, S2 state is A1, the probability of S0 state A1 is 1 and the probability of (A1,A2) and A2 is 0. Other rules can be followed in this way.

First, the multistate series-parallel system is analyzed by the traditional Bayesian network method. Based on the conditional probability table and the exact failure rate of each root node, the probability of the Fault occurrence state of the leaf node is calculated to be P(S0=A1)=298299.7969×10−14, the probability of the Fault in fuzzy uncertainty state is P(S0=(A1,A2))=738060.0823×10−14, and the probability of the Fault non-occurrence state is P(S0=A2)=999999.9963×10−6. The multistate series-parallel system was analyzed using the interval Bayesian network and the method proposed in this paper. Table 6 shows the failure rates at each root node.

Probability at each state of leaf nodes was obtained by the Bayesian inference algorithm. The comparative analysis results of the traditional Bayesian network, the interval Bayesian network, and the method proposed in this paper are shown in Table 7.

The system failure rate is assumed exponentially distributed. The reliability curves of the multistate series-parallel system under the three methods are obtained as shown in Fig. 6.

As can be seen from Table 7 and Fig. 6, the traditional Bayesian analysis results using exact failure rates are within the HE-BN method interval results. Also, the length of the analysis result interval for HE-BN is smaller and closer to the exact value than the interval Bayesian results. It proves that the present method can solve the problem of insufficient failure data. It can also make up for the relatively conservative calculation results of the Interval model, which is more in line with engineering practice.

Using the HE-BN method, the posterior probability of each root node is calculated by Eq. (16) when the multistate series-parallel system is in the fault occurrence state. The posterior probability results are sorted by the interval number sorting method based on the order relationship. For the interval number A[a−,a+], define the measurement mθ(a)=(1−θ)a−+θa+ of the interval numbers, θ∈[0,1]. The larger the value of mθ(a), the larger the corresponding interval number. After sorting them by the interval number sorting method [33], a comparison with traditional Bayesian results is shown in Fig. 7.

As can be seen from Fig. 7, the method proposed in this paper is consistent with the results of the posterior probability ranking of the root nodes obtained from the traditional Bayesian method. The results are X1>X3>X4>X2. X1 is the most important and X2 is the least important, which corresponds to the actual situation. The failure rate of component X1 is the highest and the failure rate of X2 is the lowest, which proves the feasibility of this method.

The Shibata-type coupler consists of a coupler head, coupler knuckle, uncoupling air cylinder, coupler body, and coupler yoke [34]. The specific structure of the Shibata-type coupler is shown in Fig. 8.

As can be seen from Fig. 8, the structure of the Shibata-type coupler can be divided into two parts: the coupler and the buffer. The coupler body includes a coupler head, coupler body, coupler yoke, and other structures. The buffer is composed of the horizontal pin, vertical pin, frame joint, rubber pile, baffle, buffer frame, and bracket.

When two couplers are attached, the convex cone of one of the couplers is inserted into the concave cone of the body of the other coupler. The side of the convex cone presses against the tongue of the coupler in the concave cone and turns 40° counterclockwise. When the two couplers are fully connected, the convex cone is no longer pressing on the latch and the latch returns to the closed position. The automatic coupling is completed. To de-couple, the driver operates the uncoupling valve or manually pushes the uncoupling lever to turn the coupler tongue counterclockwise to the unlocked position. The two couplers can then be released.

The T-S FTA model is constructed based on the failure of the Shibata-type coupler system as the top event, as shown in Fig. 9. And transformed into a BN as shown in Fig. 10 according to the method in Fig. 3. The event names represented by each node are shown in Table 8.

The coupler system discernment framework is Θ = {Fault occurrence, Fault non-occurrence, Fault in fuzzy uncertainty state}, which is recorded as Θ = {A1, A2, (A1,A2)}. Using the root node X1 as an example, calculate the initial failure probability interval. The expert evaluation data for root node X1 of the coupler system is shown in Table 9. Substitute it into Eqs. (6) and (7) to obtain the evidence conflict value k1 and the initial failure probability interval P(X1) of root node X1. k1=0.00000929×0.99997089+0.00001345×0.99997822=41.5998×10−7 P(X1)=[Bel,Pl]=[0.00001345×0.00000929+0.00001345×0.00001249+ 0.00000929×0.000015661−4.15998×10−6,(0.00001345+0.00001566)×(0.00000929+0.00001249)1−4.15998×10−6]=[4.3843×10−10,6.3403×10−10]

The initial failure probability interval for each root node is obtained by the same calculation as above. The root node failure probability interval after the HM constraint is obtained from Eq. (14). See Table 10 for part of this.

The T-S gate rule of the coupler system is derived from historical data and expert experience. The conditional probability table of the BN is obtained according to the method in Fig. 3. The conditional probability table for the intermediate node Y2 is shown in Table 11 as an example. Where rule 1 indicates that under the condition that X28 has a fault state of A2 and X29 has a fault state of A2, the possibility that Y2 fault state being A2 is 1, the possibility that Y2 is (A1,A2) and A1 is 0. Other rules can be used in this way.

According to the failure probability interval of X28 and X29 and Table 11, the failure probability interval of Y2 in various states can be obtained by using Eq. (15) as follows: P(y2−A1)=∑x28,x29⁡P(x28,x29;y2=A1)=∑x28,x29⁡P(y2=A1|x28,x29)×P(x28)×P(x29)=[1.3611×10−12,1.4528×10−12] P(y2=(A1,A2))=∑x28,x29⁡P(x28,x29;y2=(A1,A2))=∑x28,x29⁡P(y2=(A1,A2)|x28,x29)×P(x28)×P(x29)=[9.1862×10−13,9.6967×10−13] P(y2−A2)=∑x28,x29⁡P(x28,x29;y2=A2)=∑x28,x29⁡P(y2=A2|x28,x29)×P(x28)×P(x29)=[99999999.9998×10−8,99999999.9998×10−8]

The above results show that Y2 has a small probability of a fault and a fault-indeterminate state and a high probability of no fault. This is consistent with the actual situation. Based on the constructed BN, the probability intervals of failure for leaf node T with fault states A1, (A1,A2), and A2 were found using the method and combined with the conditional probability tables of the nodes. The reliability curve of the hook system can be obtained as shown in Fig. 11. As can be seen in Fig. 11, the reliability of the coupler system is in the range [0.7873, 0.8505] after 1×108 h of operation. Based on the results of the reliability assessment of the coupler system, a basis can be provided for subsequent design and maintenance work. P(T=A1)=[1.6663×10−10,1.4425×10−9] P(T=(A1,A2))=[1.0585×10−9,1.4525×10−9] P(T=A2)=[99999.9997×10−5,99999.9998×10−5]

The posterior probability of the root node xi for leaf node T with fault states A1 and (A1,A2) can be calculated according to Eq. (16). After sorting them by the interval number sorting method [33], the posterior probability is shown in Fig. 12.

As can be seen from Fig. 12, when the Shibata-type coupler system fails, the root node posterior probabilities are ordered as follows:

X2>X16>X9>X10>X1>X3>X17>X4>X15>X14>X20>X21>X24>X13>X11>X12>X18>X22>X7>X27>X25>X29>X23>X19>X5>X6>X8>X28>X26.

When the fault of the Shibata-type coupler system is in a fuzzy uncertainty state, the posterior probability of the root node is sorted as follows:

X16>X2>X1>X9>X10>X17>X3>X4>X15>X14>X20>X24>X21>X13>X11>X12>X18>X22>X7>X27>X25>X29>X23>X19>X5>X6>X8>X26>X28.

A comparison of the failure rates from CRH2 and CRH30A (L) rolling stock operational data [35] with the results obtained from a reliability analysis method based on T-S FTA and HE-BN is shown in Fig. 13.

It can be seen from Fig. 13 that in the operation data of CRH2 and CRH380A (L) multiple units, the failure rate of key components of the Shibata-type coupler system is ranked as follows: air duct connector, coupler body, uncoupling air cylinder, electrical connector, horizontal and vertical pin, buffer frame, coupler lock, coupler tongue, uncoupling lever. The results are consistent with the sequence when the coupler system fault state is A1 in the reliability analysis method of the T-S FTA and HE-BN, which verifies the correctness and feasibility of the method.

The greater the posterior probability, the greater the impact of the component on the system failure, which is the weak link that the system should focus on. From the analysis results in Figs. 12 and 13, it can be seen that when the coupler system is in a (A1,A2) state, coupler yoke pin fracture and coupler yoke pin bolt fracture have the greatest impact on the system. This is followed by the Air pipeline leak and uncoupling air cylinder failure. Therefore, these components should be inspected and repaired first when the system is in a half-fault state or has reached the preventive maintenance time point to ensure longer system life. When the system is in the A1 fault state, the Air pipeline leak has the greatest impact on the system, followed by the Coupler yoke pin fracture, coupler yoke pin bolt fracture, and the Coupler body crack, and the impact of the Non-standard operation of the maintenance personnel is the smallest.

The structure of a bogie bearing for a moving train is shown in Fig. 14 and includes the inner ring, the outer ring, the rolling element, and the cage. The inner ring fits with the shaft and rotates with it. The outer ring fits into the bearing housing and plays a supporting role. The rolling element is evenly distributed between the inner ring and outer ring with the help of the cage. The bogie axle box bearing supports the static load of the vehicle and the longitudinal and transverse impact of the vehicle in operation and other dynamic loads, and its failure mode can be classified as rust, discoloration, surface plastic deformation, fatigue spalling and pitting failure, cracks and defects in five categories.

A T-S fault tree with bearing failure as the top event was constructed for a type of double-row tapered roller bearing used in the bogie of a moving train, as shown in Fig. 15. It is transformed into a BN as shown in Fig. 16. The meanings represented by the symbols in Fig. 16 are shown in Table 12.

Define the identification framework of the bearing system as Θ = {Fault occurrence, Fault in fuzzy uncertainty state, Fault non-occurrence}, denoted as Θ = {A1, (A1,A2), A2}. Substitute the expert evaluation data of the root nodes of the bearing Bayesian network into Eqs. (6) and (7) to obtain the initial failure probability interval of each root node. The failure probability interval of the root node after the HM constraint is obtained from Eq. (14) and is shown in Table 13.

The T-S gate rule for the bearing system is derived from historical data and expert experience, and the conditional probability table for the BN is obtained according to the method in Fig. 3. Take the conditional probability table of the intermediate node M1 as an example, see Table 14.

Based on the constructed BN and the conditional probability tables of the nodes, the probability intervals of failure for leaf node T with fault states A1, (A1,A2), and A2 were found. A reliability curve for the bearing system can be obtained, as shown in Fig. 17. As can be seen in Fig. 17, the bearing system reliability interval is [0.8793, 0.8975] after 1954 h of system operation, and the reliability decreases sharply within 40,000 h of operation. Based on the results of the reliability assessment of the bearing system, a basis for subsequent design and maintenance work can be provided. P(T=0)=[9.9987×10−1,9.9989×10−1] P(T=0.5)=[5.4848×10−5,6.5295×10−5] P(T=1)=[5.5327×10−5,6.5835×10−5]

The posterior probability of the root node xi for leaf node T with fault states A1 and (A1,A2) can be calculated according to Eq. (16). After sorting them by the interval number sorting method [33], they are shown in Fig. 18.

As can be seen from Fig. 18, when the bearing system failure occurs, the root node posterior probability ranking is as follows: X1>X2>X11>X10>X9>X4>X7>X6>X8>X5>X3.

When the failure of the bearing system is in a state of fuzzy uncertainty, the root node posterior probabilities are ordered as follows: X2>X1>X11>X10>X9>X4>X7>X6>X8>X5>X3.

As can be seen in Fig. 18, X1,X2, and X8 are of greater importance. The main faults of these three basic events are poor lubrication, seal failure, and excessive impact loads. Therefore, the following improvement measures can be prioritized to improve the reliability of this bearing system.

Strengthen and improve the bearing lubrication technology, and bearing seal device maintenance, to ensure that the bearings are subject to good lubrication. In bearing maintenance, develop on-site applicable maintenance and cleaning cycles, improve the operational level of maintenance of bearings, improve maintenance conditions, and ensure the quality of bearing maintenance. In terms of bearing design, improve the structure of the bearing to increase the bearing’s impact resistance and improve the internal force condition of the bearing.