NOMENCLATURE

A

digital controller

A _rj(i)

state cumulative probability of the output signal when the output signal of jth operator is i

B

power module

B ^′

velocity feedback

C

hydraulic pump

D

hydraulic valve

E

actuator module

E ^′

position feedback

E ^″

pressure feedback

F₁

main oil circuit fault

F₂

operating system fault

F₃

return oil circuit fault

F₄

control system fault

F₅

flow fault

F₆

left flap fault

F₇

right flap fault

F₈

shell oil return fault

F₉

drive component fault

F₁₀

EDP flow fault

F₁₁

EMP flow fault

P _k

occurrence probability of the kth minimum path set

P _cj(i)

operator state probability when the state value of jth operator is i

P _rj(i)

output signal state probability when the output signal of jth operator is i

Q _k

kth minimum path set

T

the fault of the aircraft flap hydraulic system

q _i

occurrence probability of the ith bottom event

X

output signal of power module

x _i

the ith bottom event

Y

the output signal of actuator module

λ

the fault rate

2.0 PRINCIPLE OF AIRCRAFT FLAP HYDRAULIC SYSTEM

The flap hydraulic system generally consists of the following components: self-pressurized oil tank, filter, priority valve, accumulator, power drive assembly, safety valve, digital control assembly, flap screw actuator, etc. Figure 1 shows the schematic diagram of a certain type of civil aircraft flap hydraulic system. The pressure feedback of the self-pressurized oil tank through the high-pressure pipeline increases the oil suction pressure of the pump source and prevents the air suction of the pump source. The filter is mainly used to filter the solid particles and other harmful substances in the oil of the system, to ensure the pollution degree of oil within the tolerance limit of the key hydraulic components, so as to improve the reliability of the hydraulic system and extend the life of the components. When the system is under low pressure, the priority valve should ensure the oil supply and work of the key actuator to ensure flight safety. The accumulator is mainly used to reduce system pulsation, and to supply system flow in short time and large flow occasions such as interrupted takeoff, flyback and landing to ensure system work. The flap power drive assembly consists of an Engine Driven Pump (EDP) and an Electrical Motor Pump (EMP). A safety valve is installed between the EDP and the fuel tank. When the hydraulic system is extremely hot or the engine is fired, the fire shut-off valve is automatically opened; the EDP suction line is disconnected; the EDP pump is no longer working, and the possibility of engine ignition is reduced. The digital control component conducts the flap screw actuator control according to the motion command and the position feedback signal^{(Reference Wang, Mileta and Liu17)}.

Figure 1. Schematic diagram of flap hydraulic system of a certain aircraft^{(Reference OuYang, Yang, Guo and Yang18)}.

3.0 GO METHODOLOGY ANALYSIS OF AIRCRAFT FLAP HYDRAULIC SYSTEM RELIABILITY

The main analysis process of GO methodology includes establishment of GO model and completion of quantitative calculation. The corresponding modules of aircraft flap hydraulic system operate normally in accordance with the requirements of control instructions, so that the system as a whole works normally. The success probability of the system is calculated by combining the GO model diagram and the operation rules.

3.1 Solving the feedback loop of the aircraft flap hydraulic system

In system reliability analysis, the feedback loop is an important part of the aircraft flap hydraulic system, which provides real-time and accurate control and monitoring of the system. If the feedback loop cannot be effectively processed, the reliability of the system cannot be accurately evaluated, and ultimately the system performance cannot be effectively evaluated. The current practice is to split the feedback loop into several parts, and simply replace the feedback signal with a signal generator, or just ignore it, but this kind of method will inevitably bring errors and affect the calculation of reliability. In this paper, the idea of Boolean algebra is used to represent the feedback loop of the system in the form of Boolean equation, which can accurately represent the feedback signal of the aircraft hydraulic system, and improve the accuracy of reliability analysis. A schematic diagram of the feedback loop structure of the aircraft flap hydraulic system is shown in Fig. 2.

Figure 2. Structure diagram of aircraft flap hydraulic system with feedback loop^{(Reference Lan, Duan and Sang19)}.

Taking component A as an example, the set of events successfully output is represented as A _vA _w, where A _v represents the set of events successfully run by component A; A _w represents the set of all states of component A, and the representation method of other components is the same as that of component A, then the Boolean relation can be expressed as:

(1) \begin{equation}Y = {C_v}{C_w}{D_v}{D_w}{E_v}{E_w}X\end{equation}

As can be seen from Fig. 2, the input signal of B' is X; the input signal of E' and E" is Y; the input signal of or gate is A, B, E', E" so the expression of X is:

(2) \begin{align}X &= [{A_v}{A_w} + {B_v}'{B_w}'X \nonumber\\[3pt] &\quad + ({E_v}'{E_w}' + {E_v}''{E_w}'')Y\,]{B_v}{B_w}\end{align}

Simplify (1) and (2):

(3) \begin{align} X &= {A_v}{A_w}{B_v}{B_w} + {B_v}{B_w}B{'_v}B{'_w}X\nonumber\\[3pt] &\quad + {B_v}{B_w}{C_v}{C_w}{D_v}{D_w}{E_v}{E_w}(E{'_v}E{'_w} + E'{'_v}E'{'_w})X\end{align}

According to the Boolean algebra set operation A _s A _s =  A _s, 1 + A _s = 1, solve the Boolean Equation (3):

(4) \begin{align}X &= {A_v}{A_w}{B_v}{B_w} + {m_1}{B_v}{B_w}B{'_v}B{'_w}X\nonumber\\[2pt] &\quad+ {m_2}{B_v}{B_w}{C_v}{C_w}{D_v}{D_w}{E_v}{E_w}(E{'_v}E{'_w} + E'{'_v}E'{'_w})\end{align}

Where: m ₁ and m ₂ are arbitrary Boolean elements.

The GO methodology is based on system success, assuming all components of the system are successfully started, so there is:

$${A_w} = {B_w} = {C_w} = {D_w} = {E_w} = B{'_w} = E{'_w} = E'{'_w} = 1$$

And so there is

(5) \begin{align}X &= {A_v}{B_v} + {m_1}{B_v}B{'_v}\nonumber\\[2pt] &\quad+ {m_2}{B_v}{C_v}{D_v}{E_v}(E{'_v} + E'{'_v})\end{align}

Figure 2 can directly derive the expression (6) of X after E is successfully output.

(6) \begin{align}X &= {A_v}{B_v} + {A_v}{B_v}B{'_v}\nonumber\\[2pt] &\quad + {A_v}{B_v}{C_v}{D_v}{E_v}(E{'_v} + E'{'_v})\end{align}

Comparing (5) and (6), we can see that m ₁ = A _v, m ₂ = A _v, so the expressions for X and Y successfully output are respectively:

(7) \begin{align}Y &= {A_v}{B_v}{C_v}{D_v}{E_v} + {A_v}{B_v}B{'_v}{C_v}{D_v}{E_v}\nonumber\\[2pt] &\quad + {A_v}{B_v}{C_v}{D_v}{E_v}(E{'_v} + E'{'_v})\end{align}

In Equation (8): the first term represents the main path; the second term represents the ring B – B' structure; the third term represents B – C – D – E – E' and B – C – D – E – E", translated into the GO model as shown in Fig. 3.

Figure 3. Feedback loop GO model.

3.2 Constructing GO model of aircraft flap hydraulic system

The GO model is mainly composed of operators and signal streams^{(Reference Shen and Huang7)}. Combined with the schematic diagram of the aircraft flap hydraulic system and the GO signal of the feedback signal, the GO model of the aircraft flap hydraulic system is established as shown in Fig. 4.

Figure 4. GO diagram of aircraft flap hydraulic system.

Power signals, control signals, oil tanks, etc. can be represented by the type 5 signal generator operator. Controllers, sensors, hydraulic valves, hydraulic cylinders, aviation high-pressure piston pumps, etc. are considered as two-state units and can be represented by type 1 operators. The motor and the engine may have an early output signal due to an undesired external stimulus such as a short circuit of the power supply, and thus may be represented by a type 3 trigger generator. The correspondence among all components in the Fig. 4 and the GO method operator is listed in Table 1.

In Table 1, the state value 0 indicates that the unit is in an advanced state; the state value 1 indicates that the unit is in a successful state; the state value 2 indicates that the unit is in a fault state, and the probability that the unit is in a fault state is the unit failure rate λ, meaning the probability of failure of the unit in unit time; the probability that the unit is in the advanced state is P _p, that is, the probability that the unit will cause the output signal to occur in advance due to external factors.

Table 1 Aircraft flap hydraulic system operator data

3.3 GO methodology reliability operation

According to the probability analysis formula of the GO methodology, the expression of the key signal flow in the Fig. 4 is calculated as follows:

1. (1) Signal stream 8

The input signal stream of signal stream 8 is signal stream 5 and signal stream 11. Since signal stream 5 and signal stream 11 both contain signal stream 7, signal stream 7 is the common signal of signal stream 8, which needs to be corrected ^{(Reference Shen, Gao and Huang9)}.

\begin{align*}&{A_{R8}}(0) = {P_{c5}}(0) + {P_{c11}}(0)\, + \\[3pt] &{P_{c5}}(1){A_{r7}}(0) + {P_{c11}}(1){A_{r7}}(0)\\[3pt] & - {P_{c5}}(0){P_{c11}}(0)- {P_{c5}}(0){P_{c11}}(1){A_{r7}}(0)\\[3pt] &- {P_{c11}}(0){P_{c5}}(1){A_{r7}}(0) - {P_{c5}}(1){P_{c11}}(1){A_{r7}}(0)\end{align*}

\begin{align*}&{A_{R8}}(1) = {P_{c5}}(0) + {P_{c11}}(0) + \\[3pt] &{P_{c5}}(1){A_{r7}}(1) + {P_{c11}}(1){A_{r7}}(1) \\[3pt] & - {P_{c5}}(0){P_{c11}}(0) - {P_{c5}}(0){P_{c11}}(1){A_{r7}}(1)\\[4pt] &- {P_{c11}}(0){P_{c5}}(1){A_{r7}}(1) - {P_{c5}}(1){P_{c11}}(1){A_{r7}}(1)\end{align*}

\begin{align*}&{A_{R8}}(2) = 1\\[4pt] &{P_{r8}}(0) = {A_{r8}}(0)\\[4pt] &{P_{r8}}(1) = {A_{r8}}(1) - {A_{r8}}(0)\\[4pt] &{P_{r8}}(2) = {A_{r8}}(2) - {A_{r8}}(1)\end{align*}

1. (2) Signal stream 4

The signal stream 4 has a plurality of input signal streams, and each signal stream exists only on the premise of the existence of the signal stream 2 and the signal stream 8, so the signal stream 2 and the signal stream 8 are the common signals of the signal stream 4, and the common signal is corrected:

\begin{align*}a &= 1 - [1 - {P_{c6}}(0)][1 - {P_{c10}}(0)][1 - {P_{c12}}(0)]\\[3pt]b &= 1 - [1 - {P_{c6}}(1)][1 - {P_{c10}}(1)][1 - {P_{c12}}(1)]\\[3pt]{A_{r4}}(0) &= a{A_{r9}}(0) + {A_{r3}}(0) - a{A_{r3}}(0)/{A_{r2}}(0){A_{r8}}(0)\\[3pt]{A_{r4}}(1) &= b{A_{r9}}(1) + {A_{r3}}(1) - b{A_{r3}}(1)/{A_{r2}}(1){A_{r8}}(1)\\[3pt]{A_{r4}}(2) &= 1\\[3pt]{P_{r4}}(0) &= {A_{r4}}(0)\\[3pt]{P_{r4}}(1) &= {A_{r4}}(1) - {A_{r4}}(0)\\[3pt]{P_{r4}}(2) &= {A_{r4}}(2) - {A_{r4}}(1)\end{align*}

1. (3) Signal stream 21

\begin{align*}{A_{r21}}(0) &= {A_{r4}}(0){A_{r20}}(0)\\[3pt]{A_{r21}}(1) &= {A_{r4}}(1){A_{r20}}(1)\\[3pt]{A_{r4}}(2) &= 1\\[3pt]{P_{r21}}(0) &= {A_{r21}}(0)\\[3pt]{P_{r21}}(1) &= {A_{r21}}(1) - {A_{r21}}(0)\\[3pt]{P_{r21}}(2) &= 1 - {A_{r21}}(1)\end{align*}

1. (4) Signal stream 37

The signal stream 37 has a plurality of input signal streams, each of which contains a signal stream 21, so the signal stream 21 is a common signal of the signal stream 37, and the common signal is corrected:

\begin{align}{A_{r37}}(0) &= {A_{r27}}(0){A_{r30}}(0){A_{r36}}(0)/{A_{r21}}{(0)^2}\\[3pt] {A_{r37}}(1) &= {A_{r27}}(1){A_{r30}}(1){A_{r36}}(1)/{A_{r21}}{(1)^2}\\[3pt] {P_{r37}}(0) &= {A_{r37}}(0)\\[3pt] {P_{r37}}(1) &= {A_{r37}}(1) - {A_{r37}}(0)\\[3pt] {P_{r37}}(2) &= 1 - {A_{r37}}(1)\end{align}

1. (5) Signal stream 41

The signal stream 41 includes the signal stream 39 and the signal stream 40. Since both the signal stream 39 and the signal stream 40 contain the signal stream 38, the signal stream 41 needs to be corrected when calculating.

\begin{align}{A_{r41}}(0) &= {A_{r39}}(0) + {A_{r40}}(0) - {A_{r39}}(0){A_{r40}}(0)/{A_{r38}}(0)\\[3pt] {A_{r41}}(1) &= {A_{r39}}(1) + {A_{r40}}(1) - {A_{r39}}(1){A_{r40}}(1)/{A_{r38}}(1)\\[3pt] {P_{r41}}(0) &= {A_{r41}}(0)\\[3pt] {P_{r41}}(1) &= {A_{r41}}(1) - {A_{r41}}(0)\end{align}

Where, A _rj(i) is the state cumulative probability of the output signal when the output signal of jth operator is i; P _rj(i) is the operator state probability when the state value of jth operator is i; P _rj(i) is the state probability of the output signal when the output signal of jth operator is i. Combined with the data in Table 1, the calculation results are shown in Table 2.

Table 2 Feedback loop and system output probability

According to the Table 2, it is concluded that the probability of successful system output calculated by the GO model of the system with feedback loop is small, indicating that the feedback loop is more reliable to the system and considering the feedback loop of the system can accurately calculate the system reliability and effectively evaluate the system performance.

4.0 RELIABILITY ANALYSIS OF AIRCRAFT FLAP HYDRAULIC SYSTEM BASED ON FTA

The FTA method is used to analyze the aircraft flap hydraulic system mentioned above, and the fault of the aircraft flap hydraulic system is used as the top event. The fault tree is built as shown in Fig. 5.

Figure 5. Fault tree of aircraft flap hydraulic system.

In Fig. 5, F₁ is the main oil circuit fault; F₂ is the operating system fault; F₃ is the return oil circuit fault; F₄ is the control system fault; F₅ is the flow fault; F₆ is the left flap fault; F₇ is the right flap fault; F₈ is the shell oil return fault; F₉ is the drive component fault; F₁₀ is the EDP flow fault; F₁₁ is the EMP flow fault. The bottom event data is shown in Table 3.

Table 3 Data of each bottom event

Due to the large number of minimum cut sets of the fault tree shown in Fig. 5, this paper uses the minimum path set to analyze the reliability of the fault tree.

The minimum path set indicates that the top event can be prevented when all bottom events contained in a minimum path set do not occur. It can be seen that each minimal path set is a condition to ensure the failure tree top event does not occur, and a way to prevent accidents. In this sense, the minimal path set represents the security of the system. Therefore, this paper uses the probability that none of the bottom events in the minimum path set occur to calculate the reliability of the whole system in normal operation. The calculation formula is:

$${P_k} = \prod\limits_{{x_i} \in {Q_k}} {(1 - {q_i})} $$

Where P _k represents the occurrence probability of the kth minimum path set (k  = 1, 2, …, 8), namely, the system reliability; x _i ∈ Q _k represents the ith bottom event belonging to the kth minimum path set; q _i represents the occurrence probability of the ith bottom event.

According to the duality principle and the optimization idea of reliability engineering, the minimum path set with the lowest reliability is selected in all of the minimum path sets with the same number of bottom events. Also, in the minimum path sets with the closest reliability, the path set with more bottom events is selected^{(Reference Elsayed20,Reference Lu, Yuan and Xiang21)} _. Therefore, the minimum path set k of the fault tree is obtained:

$$\left\{ {{x_1}, \ldots ,{x_{14}},{x_{15}},{x_{16}},{x_{18}}, \ldots ,{x_{29}},{x_{30}},{x_{32}}} \right\}$$

After calculation, P _k = 0.999165, which can be taken as the lower limit of success probability of the system.

5.0 COMPARISON OF METHODS

In order to verify the correctness and effectiveness of the proposed method, the results of the proposed method in our study, the FTA method and the methods proposed in Refs Reference Yi, Dhillon, Dong, Shi and Jiang14–Reference Yi, Shi, Mu, Zhang, Guo and Liang16 are represented in Table 4.

Table 4 Comparison of system output probability of different GO methods and FTA

According to the above calculation method of FTA, the analysis result of FTA can be considered as the lower limit of system’s success probability. As Table 4 shows, the quantitative calculation result of the proposed method in our study is very close to the result of FTA, and a little greater than it, as well as the reliability results calculated by both methods are in the same order of magnitude, which shows that the proposed method is correct.

According to the comparison results of different GO methods dealing with feedback loop in Table 4, the result of the proposed method is same as the method presented by Yi^{(Reference Yi, Dhillon, Dong, Shi and Jiang14–Reference Yi, Shi, Mu, Zhang, Guo and Liang16)}, which shows the proposed GO methodology is feasible. But the methods presented by Yi^{(Reference Yi, Dhillon, Dong, Shi and Jiang14–Reference Yi, Shi, Mu, Zhang, Guo and Liang16)} with creating the new function operator and formulas to deal with the feedback loop in system increase the complexity of GO methodology, and may be inconvenient for engineers and researchers to use.

The comparison analysis above shows that the proposed GO method is effective and correct. However, the calculation results of the GO methods and FTA are not completely equal, because the basic concepts and algorithms of the two methods are different. Firstly, the GO methodology is a success-oriented method for reliability analysis of complex systems and includes many operators which can represent various units, while FTA is a fault-oriented method using multi-level logic chart to represent the relationship of different fault events with limited logic gate. Secondly, GO model is developed from system functional diagram and system structure diagram, reflecting the original appearance of system and the process of operation, so it can avoid the influence of engineer experience for reliability analysis while FTA is subjective and its inferences are affected by the knowledge and abilities of engineers establishing the fault tree. Moreover, the quantitative analysis of FTA is mostly based on computing minimal cut sets of the system, and computing the success(failure) probability of complex systems based on minimal cut sets is NP-hard. Therefore, the calculation results of the two methods are not equal exactly and the relative error between the proposed method and FTA is 0.0132%.