Fault tree analysis of a hydraulic system based on the interval model using latin hypercube sampling

Abstract

There will inevitably be failures during the use of hydraulic systems in armored vehicles because of the detrimental environments in which they operate. In order to improve the reliability of such hydraulic systems, a fault tree model of top event ‘hydraulic system failure’ is established and analyzed in this study according to the system arrangement and potential fault mechanisms. To properly consider the uncertain probability of each basic failure event in the system, it is necessary to overcome the limitations of the traditional fault tree model. Accordingly, in this study, the importance of the basic event probability interval in describing the failure probability of the top event (hydraulic system failure) was calculated using Latin hypercube sampling through interval modeling. This method offers significant benefits for the reliability assessment of hydraulic systems and can be used to provide guidance for improving system reliability.

Keywords

Hydraulic system interval model fault tree analysis probability interval latin hypercube sampling

1 Introduction

The hydraulic system is among the most critical systems of armored vehicles. Due to the harsh working environment and complicated internal structure of such hydraulic systems, the use armored vehicles in training often leads to hydraulic system failure, requiring the vehicle to be parked for maintenance, consuming manpower and material resources as well as adversely affecting training [1]. With the wide application of hydraulic technology in armored vehicles, the reliability of their hydraulic systems has become an increasingly prominent concern. The breakdown of a hydraulic system can often disable an armored vehicle, directly causing it to lose its working ability and making it unable to perform its intended tasks [2]. In order to ensure that armored vehicles are able to successfully complete their objectives during combat, it is particularly important that the reliability of their hydraulic systems be studied so that appropriate preventive measures can be taken in advance to reduce the possibility of hydraulic system failure.

The fault tree analysis method is a design method that analyzes various factors that can lead to system failures by using a logic block diagram to evaluate various possible combinations of system failure causes and their occurrence probabilities. These combined occurrence probabilities are then used to calculate the overall system failure probability, which can in turn be used to inform corresponding corrective measures to improve system reliability [3 –5]. In a fault tree analysis, the most undesired fault state/event is taken as the target of the fault analysis. All possible causes and factors leading to this event are then evaluated to determine the direct causes of all possible intermediate fault events using layer-by-layer tracking until the most basic cause of the target fault is identified. Finally, the top event, intermediate events, and basic events are linked in a tree diagram using representative symbols and logic gates [6 –8]. Such fault tree analyses are widely used in the fields of aviation [9 –11], aerospace [12, 13], and transportation [14, 15].

In recent years, researchers have used fault tree analysis techniques to study the hydraulic systems of armored vehicles [16 –19]. Meng et al. [16] established a model of a tank rescue vehicle hydraulic system and studied its dynamic characteristics under typical fault conditions. Shen et al. [18] established a system reliability block diagram for the low-reliability faults of an amphibious armored vehicle hydraulic system and calculated its reliability under different working conditions.

Traditional fault tree analysis is based on the probability model, which requires sufficient data to obtain the exact probabilities of the base events [19]. Especially when evaluating an armored vehicle hydraulic system, it is difficult to accurately estimate the probability of failure as sufficient data are rarely available [20–21]. Sun et al. [22] investigated the problem of adaptive fuzzy control for a class of nontriangular structural stochastic switched nonlinear systems with full state constraints. Qiu et al. [23] studies the problem of fuzzy adaptive event-triggered control for a class of pure-feedback nonlinear systems, which contain unknown smooth functions and unmeasured states. Behrooz and Mansour [24] used two discrete optimization maps to implement fuzzy reliability analysis in the processes of reliability and fuzzy analysis. The resulting fuzzy numbers were used to reflect the fact that the determined probabilities are, in reality, an estimation of a range of probabilities, meaning that the probability of a basic event occurring is “approximately equal to some value.”

There are several well-known uncertainty quantification methods in uncertainty theory, including fuzzy sets, evidence variables, the Monte Carlo method, subset simulation, importance sampling, and convex sets. However, these methods are limited to small scopes of application. Fuzzy sets depend heavily on the choice of membership functions, requiring users to have extensive engineering experience [25]. The reliability analysis and calculation efficiency of multi-dimensional problems based on evidence variables is relatively low [26, 27]. The disadvantage of the Monte Carlo method is that it requires a large amount of calculation. The difficulty of subset simulation lies in the need to choose suitable conditional probabilities for stratification [28, 29], and the difficulty of importance sampling lies in constructing an appropriate importance sampling function [30, 31]. Convex sets, however, include interval models and ellipsoid models; an interval model provides wider applicability and is a better quantification method when faced with insufficient information describing basic events, that is, when only the upper and lower bounds of basic events can be obtained.

Indeed, an interval model only requires the boundary set of the uncertain event to describe the probability of event occurrence; the internal distribution of the event probability is not necessary [32 –35]. The interval model has been widely used as it requires significantly less data than the probability model [36 –41]. Compared with traditional fault tree analysis methods, the interval analysis method does not require a probability distribution for uncertain variables, only their ranges of operation, which are considerably easier to obtain [42]. Because this method requires relatively less data, it is ideally suited to applications with insufficient or otherwise small data sets [43]. Considering that a hydraulic system is expensive and complex while statistical data describing its operation is scarce, a fault tree analysis based on the interval analysis method is likely to produce more reasonable, and thus more beneficial results.

Due to its advantages, Latin hypercube sampling is applied to the interval simulation of the fault tree analysis in this study. Many scholars have studied Latin hypercube sampling and proposed various improved Latin hypercube sampling methods with suitable advantages for different engineering fields [44 –46]. Razi and Saman proposed progressive Latin hypercube sampling (PLHS) to sequentially generate sample points while progressively preserving the distributional properties of interest (Latin hypercube properties, space-filling, etc.) as the sample size grows [45]. Shields and Zhang proposed the Latinized partially stratified sampling (LPSS) method based on the Latinized stratified sampling (LSS) and the partially stratified sample (PSS) methods. The proposed LPSS method was observed to provide superior variance reduction for many high-dimensional applications when both low-order interactions and main effects were present [46].

This study is detailed in this paper as follows. First, according to the function, structure, and working principle of the subject armored vehicle hydraulic system, a fault tree with a top event of ‘hydraulic system failure’ is established. Next, the interval model for the fault tree analysis is applied to construct the fault tree using ‘AND gate’ and ‘OR gate’ interval operators according to the interval number algorithm. The failure probability range of the fault tree model of the subject hydraulic system is then calculated using the double Latin hypercube sampling method. This study also addresses the importance of the basic hydraulic system failure events based on the interval model.

Symbol	Definition
T	Top event
X _i	Basic event
x _and	AND gate interval operator
x _or	OR gate interval operator
φ (x_i)	State function of the i-th basic event
I_p (x_i)	Probability importance of the i-th basic event x_i
P (T)	Probability function of the fault tree _T
P (x_i)	Probability value of the i-th basic event x_i
I_Φ (x_i)	Structural importance of the basic event x_i
n	Total number of basic events
N_Φ (x_i)	Number of times that the top event changes from the failure to the normal state for all system states in which a basic event changes from the failure to the normal state.
I_K (x_i)	Critical importance degree of the i-th basic event x_i

2 Fault tree analysis based on the interval model

2.1 Fault tree structure function analysis

Assuming that there are only two potential statuses for all elements, components, and systems (normal or faulty), and that this status is independent for the failure of each component, the fault tree has n basic events with a top event T. The state of the top event T is represented by the state variable Φ, which, according to the nature of the fault tree, must be a function of the basic event state variable x_i (i = 1, 2, ⋯ , n), such that: $Φ (x) = Φ (x_{1}, x_{2}, \dots, x_{n})$ (1) $Φ (x) = {\begin{matrix} 1 & When the top event T occurs \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \\ 0 & When the top event T does not occur \end{matrix}$ (2) $x_{i} = {\begin{matrix} 1 & When the basic event i occurs \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \\ 0 & When the basic event i does not occur \end{matrix}$ (3) where Φ (x) describes the structural function of the fault tree or the fault structure function of the system. The occurrence of a fault event corresponds to the fault state, and when the fault event does not occur, the system is in its normal state.

2.2 Applying the interval model to the fault tree

The fault tree can be analyzed based on the interval model using the following definitions:

The basis in the interval model is described by letting “*”, where *∈ { + , - , · , /}, represent real binary operations on the set of real numbers. For any $[x] = [\underline{x}, \bar{x}]$ and $[y] = [\underline{y}, \bar{y}] \in I (R)$ , the binary operation on the interval set I (R) can be defined as follows:

[x] * [y] = {z | z = x * y, x \in [x], y \in [y]}

(4)

Producing the following four algorithms: ${\begin{matrix} [x] + [y] = [\underline{x} + \underline{y}, \bar{x} + \bar{y}] \\ [x] - [y] = [\underline{x} - \bar{y}, \bar{x} - \underline{y}] \\ [x] \cdot [y] = [min (\underline{x} \underline{y}, \underline{x} \bar{y}, \bar{x} \underline{y}, \bar{x} \bar{y}), max (\underline{x} \underline{y}, \underline{x} \bar{y}, \bar{x} \underline{y}, \bar{x} \bar{y})] \\ [x] / [y] = [\underline{x}, \bar{x}] \cdot [1 / \bar{y}, 1 / \underline{y}] (0 \notin [y]) \end{matrix}$ (5)

Depending on the algorithm of the interval number and the assignment of the ‘AND gates’ and ‘OR gates’ in the fault tree analysis, the ‘AND gate interval operators’ and the ‘OR gate interval operators’ can be assigned using the interval analysis of the fault tree.

The ‘AND gate interval operator’ is described by:

x_{and} \in [x_{and}^{l}, x_{and}^{u}]

(6) where:

x_{and}^{l} = min \prod_{i = 1}^{n} x_{i}

(7)

x_{and}^{u} = max \prod_{i = 1}^{n} x_{i}

(8)

and is indicated by the AND gate symbol shown in Fig. 1.

The ‘OR gate interval operator’ is described by:

x_{or} \in [x_{or}^{l}, x_{or}^{u}]

(9) where:

x_{or}^{l} = min (1 - \prod_{i = 1}^{n} (1 - x_{i}))

(10)

x_{or}^{u} = max (1 - \prod_{i = 1}^{n} (1 - x_{i}))

(11) and is indicated by the OR gate symbol shown in Fig. 2.

Fig. 1

AND gate symbol.

Fig. 2

OR gate symbol.

2.3 Fault tree model of the subject hydraulic system

In this study, an armored vehicle hydraulic system was selected as the research object in which the oil supply pressure is controlled by a relief valve. The rotation of the oil pump is driven by the engine through the transmission system and the pressurized oil is transmitted to each working mechanism through the oil filter and relief valve. The entire working cycle is powered by a clutch operation booster and planetary power steering maneuvering booster [1].

The processes of drawing the fault tree for any hydraulic system consists of three steps:

Step 1: Study the hydraulic system to determine the system function, structure, working principle, fault status, failure reasons, failure mode, impact of failure, and relevant technical information.

Step 2: Define the top event of the fault tree of the hydraulic system depending on the analysis requirements of the target system.

Step 3: Construct the fault tree of the hydraulic system, starting with its top event, then connect all possible direct causes of the top and preceding intermediate events at different levels using fault tree symbols to indicate their logical relationships until the basic events of the fault tree are determined.

In this paper, ‘hydraulic system failure’ was defined as the top event, and the subject hydraulic system was analyzed accordingly to produce the fault tree shown in Fig. 3.

Fig. 3

Fault tree of the subject hydraulic system.

In Fig. 3, X₁ indicates joint leakage, X₂ indicates cylinder leakage, X₃ indicates a pilot valve failure, X₄ indicates a large axial clearance, X₅ indicates a poor pump assembly, X₆ indicates the pump speed is too low, X₇ indicates a poor pump seal, X₈ indicates that the hydraulic fluid level in the tank is too low, X₉ indicates a clogged oil filter, X₁₀ indicates that the oil temperature is too high, X₁₁ indicates that the tubing is clogged, X₁₂ indicates a lack of hydraulic oil, X₁₃ indicates a stuck relief valve spool, and X₁₄ indicates a missing relief valve spool.

2.4 Calculation of component importance

Component importance refers to the extent to which a component contributes to the probability of top event occurrence, providing a quantitative representation of the importance of each subsystem or component in the system. Importance can be determined in terms of probability importance, structural importance, or critical importance.

2.4.1 Probability importance

The probability importance reflects the decrease in the system top event failure probability resulting from the change of a basic event from failure to normal status as follows:

$Δ P_{TOP} = P_{TOP} {φ (x_{i}) = 1} - P_{TOP} {φ (x_{i}) = 0}$ (12) where φ (x_i) =0 and φ (x_i) =1 indicate the i-th basic event as being normal and faulty, respectively.

Probability importance can be determined by calculating the partial derivatives of the fault tree structure function, written as: $I_{P} (x_{i}) = \frac{\partial P (T)}{\partial P (x_{i})}$ (13) where I_P (x_i) denotes the probability importance of the i-th basic event x_i, P (T) indicates the probability function corresponding to the structural function of the fault tree T, and P (x_i) indicates the probability of the i-th basic event x_i.

2.4.2 Structural importance

The structural importance reflects the increased proportion of top event state changes from failure to normal in all systematically possible states when a given basic event changes from the failure to the normal state, and can be described by: $I_{Φ} (x_{i}) = \frac{1}{2^{n - 1}} N_{Φ} (X_{i})$ (14) where I_Φ (x_i) is the structural importance of basic event x_i, n is the total number of basic events, and N_Φ (x_i) is the number of times that the top event changes from the failure to the normal state for all system states in which the considered basic event changes from the failure to the normal state.

The structural importance indicates the degree of importance of a basic event in the fault tree structure. The calculation of structural importance assumes that the probabilities of all basic events are the same, generally established as 0.5. From this perspective, there is no direct relationship between the physical meaning of the calculated structural importance and the actual probability of failure for basic events; structural importance simply describes the importance of the basic events within the system structure. Thus, structural importance can be represented by the following equation: $I_{Φ} (x_{i}) = [\frac{\partial P (T)}{\partial P (x_{i})}] P (x_{1}) = P (x_{2}) = \cdot \cdot \cdot P (x_{n}) = 0.5$ (15) where I_Φ (x_i) is the structural importance of the i-th basic event, P (T) indicates the failure probability of the top event T, and P (x_i) is failure probability of the i-th basic event x_i.

2.4.3 Critical importance

The critical importance is the ratio of the failure probability variation of basic event x_i to the resulting failure probability variation of the top event, reflecting the effect of the improvement of basic event state x_i on system improvement. The critical importance is defined as: $I_{K} (x_{i}) = lim_{Δ P (x_{i}) = 0} \frac{Δ P (T) / P (T)}{Δ P (x_{i}) / P (x_{i})} = \frac{\partial P (T)}{\partial P (x_{i})} . \frac{P (x_{i})}{P (T)}$ (16) where I_K (x_i) denotes the critical importance of the i-th basic event x_i, P (T) indicates the failure probability of the top event T, and P (x_i) is the failure probability of the i-th basic event x_i.

From theoretical perspective, it can be seen in Equation (16) that critical importance considers both the magnitude of the probability of a change in the basic event state and the effect of its change on the failure state of the system. Thus, the critical importance not only reflects more information about the failure state than the traditional probability importance and structural importance methods, but also identifies the weakest part of the system. Therefore, in this study, critical importance was used to evaluate the subject fault tree and its results were compared against results determined using probability importance.

3 Evaluation of hydraulic system fault tree

3.1 Solving failure probability interval based on the double latin hypercube sampling

In statistical sampling, the use of a Latin square matrix means that each row and column contains only one sample. The Latin hypercube is a generalization of the Latin square matrix in multiple dimensions, in which each hyperplane perpendicular to a dimensional axis contains at most one sample. In this way, Latin hypercube sampling represents the latest development in sampling methodology. Compared with the Monte Carlo method, the Latin hypercube method can accurately reconstruct the input distribution using fewer sampling iterations. The key to Latin hypercube sampling is to stratify the input probability distribution so that the cumulative curve is divided into equal intervals on the scale of cumulative probability [0, 1]. Then, one sample is randomly collected from each input distribution interval, or “stratified.” The sample produced when using the Latin hypercube method more accurately reflects the distribution of the input probability by forcing each sample to represent the values of each interval and then reconstructing the input probability distribution with each iteration. This technique is known as “sampling without substitution,” a process in which the number of layers of the cumulative distribution is equal to the number of iterations performed.

When theoretically deriving and calculating the probability interval based on the four interval model algorithms given in Equation (5), the solution intervals for the same expression are different depending on the order of operations. Therefore, it is necessary to simplify the expression of any complex system’s fault tree structure function during the interval operation process. However, in this application, the shortage of interval operation data makes this simplification impossible, such that an exact theoretical solution can only be obtained from strict step-by-step calculations according to the fault tree structure function, resulting in a considerable workload. However, the fault tree interval can be solved based on Latin hypercube sampling without the need for interval theory. As long as the sample size is sufficiently large, the result will be infinitesimally close to the theoretical interval value, avoiding the drawbacks of the interval operation approach and providing a new path for solving a system fault tree using the interval model.

Using the joint probability density function f_X (x) of a random variable, N samples are extracted and represented as x_j (j = 1, 2, ⋯ , N). The ratio of the number of sample points N_f falling into the failure domain F to the number of total sample points N is the estimated value of the failure probability ${\hat{P}}_{f}$ as follows [47]: ${\hat{P}}_{f} = \frac{1}{N} \sum_{j = 1}^{N} I_{F} (x_{j}) = \frac{N_{f}}{N}$ (17) where $I_{F} (x) = {\begin{matrix} 1 \begin{matrix} \end{matrix} x \in F \\ 0 \begin{matrix} \end{matrix} x \notin F \end{matrix}$ is the indicator function of the failure domain.

The determination of failure probability based on Latin hypercube sampling can be described by the following procedure:

Step 1: For the n basic event failure probability interval P_i (i ∈ [1, n]), the group M is given by the determined values of P_j1, P_j2, … P_jn (j ∈ [1, M]) in their respective probability intervals (note: j is the determined value in Steps 2 and 3), using an initial value of g₂ = 0 and g₁ = 1.

Step 2: The top event failure probability g_j is obtained by sampling each set of basic event failure probabilities P_j1, P_j2, ⋯ , P_jn. Specifically, the failure transfer function of the basic event is determined according to the fault tree structure and the ‘AND gate’ and ‘OR gate’ logic states. In the interval [0,1], the sample T_k (k ∈ [1, N]) is obtained from N random samples. Taking P_j1 as an example, where $X_{j 1}^{(k)} = 1$ if T_k ⩽ P_j1, then $X_{j 1}^{(k)} = 0$ if T_k > P_j1. Thus N sample points consisting of 0 or 1 as sample values are obtained and assumed as ( $X_{ji}^{(1)}$ , $X_{ji}^{(2)}$ , . . . , $X_{ji}^{(N)}$ ) (i ∈ [1, n]). In this study, as long as one basic event occurs, the top event also occurs because all of the fault tree connections are indicated by ‘OR gates’. Therefore, for a set of sample points ( $X_{j 1}^{(k)}$ , $X_{j 2}^{(k)}$ , . . . , $X_{jn}^{(k)}$ ) (k ∈ [1, N]), as long as one value is set to 1, then N sample points with values [0, 1] are obtained for the top event. The final event failure value is g_j = N₁/ - N where N₁ is the number of sample points that result in a value of 1 for the top event.

Step 3: If g_j > g₂, then set g₂ = g_j; if g_j < g₁, then set g₁ = g_j.

Step 4: Iterate Steps 2 and 3 M times, and the resulting g₁ value is the lower bound of the probability range of the basic events while g₂ is the upper bound for the subject fault tree stage.

The system failure probability estimation flowchart based on Latin hypercube sampling is shown in Fig. 4.

Fig. 4

Estimation of failure probability based on Latin hypercube sampling.

3.2 Basic event probability interval

The probability of a basic event occurring needs to be obtained as it is an important part of the hydraulic system fault analysis conducted in this study using the fault tree method. The basic event probability intervals of the hydraulic system fault tree, X₁–X₁₄, are shown in Table 1. Due to insufficient relevant data, it is difficult to determine the distribution characteristics of these events and thus to accurately calculate their occurrence probabilities. As a result, this study determined the basic event fault rates by adapting the interval probability method using the collected relevant information and hydraulic system-related data.

Table 1
Basic event probability intervals

Event code Probability interval Event code Probability interval

X ₁ [0.00529, 0.00539] X ₈ [0.00141, 0.00151]

X ₂ [0.00437, 0.00447] X ₉ [0.00275, 0.00285]

X ₃ [0.00364, 0.00374] X ₁₀ [0.00229, 0.00239]

X ₄ [0.00138, 0.00148] X ₁₁ [0.00259, 0.00269]

X ₅ [0.00171, 0.00181] X ₁₂ [0.00254, 0.00264]

X ₆ [0.00179, 0.00189] X ₁₃ [0.00170, 0.00180]

X ₇ [0.00110, 0.00120] X ₁₄ [0.00151, 0.00161]

Event code	Probability interval	Event code	Probability interval
X ₁	[0.00529, 0.00539]	X ₈	[0.00141, 0.00151]
X ₂	[0.00437, 0.00447]	X ₉	[0.00275, 0.00285]
X ₃	[0.00364, 0.00374]	X ₁₀	[0.00229, 0.00239]
X ₄	[0.00138, 0.00148]	X ₁₁	[0.00259, 0.00269]
X ₅	[0.00171, 0.00181]	X ₁₂	[0.00254, 0.00264]
X ₆	[0.00179, 0.00189]	X ₁₃	[0.00170, 0.00180]
X ₇	[0.00110, 0.00120]	X ₁₄	[0.00151, 0.00161]

4 Results and discussion

According to the convergence criterion of the Latin hypercube sampling method, the primary factor that affects computational accuracy is the number of inner samples (N). Moreover, the number of inner samples should be much larger than the number of outer samples (M). Considering the computing power available, the M value was set to 1000, while the N value was set to 100 000. Eventually, using Latin hypercube sampling, as presented in Section 4.1, and the event probability intervals, presented in Section 4.2, the top event failure probability interval was obtained as [0.03225, 0.03607].

4.1 Checking the rationality of the inner and outer cycle settings

The convergence of the results estimated using the proposed method should be as close as possible within the range of available computational capacity. Therefore, by using the variable-control approach to fix the value of either M or N and then gradually change the value of the non-fixed variable, a curve of the estimated results can be plotted to evaluate their convergence. The resulting curves when fixing the value of M at 1000 and when fixing the value of N at 100 000 are shown in Figs. 5 and 6, respectively, in which the open circles indicate the upper bound of the estimated range and the filled circles indicate the lower bound.

Fig. 5

Effect of the value N on the estimation results.

Fig. 6

Effect of the value M on the estimation results.

From the analysis of the influence of the values of N and M on the calculation results presented in Figs. 5 and 6, it is clear that the convergence of the estimated result is far more sensitive to the value of N.

Therefore, considering the limitations of computing power, more computing resources should be concentrated on the inner cycle to most effectively improve the accuracy of the results. Based on this finding, the values of M = 1000 and N = 100 000 were selected as being reasonable for this study.

4.2 Determining the importance degree

The probability importance of each basic event was determined using Equation (13), resulting in the importance intervals provided in Table 2.

Table 2
Probability importance intervals

Event code Probability importance interval Event code Probability importance interval

X ₁ [0.97033, 0.97158] X ₈ [0.96656, 0.96781]

X ₂ [0.96943, 0.97069] X ₉ [0.96786, 0.96911]

X ₃ [0.96872, 0.96998] X ₁₀ [0.96741, 0.96866]

X ₄ [0.96653, 0.96778] X ₁₁ [0.96770, 0.96896]

X ₅ [0.96685, 0.96800] X ₁₂ [0.96765, 0.96891]

X ₆ [0.96693, 0.96818] X ₁₃ [0.96684, 0.96809]

X ₇ [0.96626, 0.96751] X ₁₄ [0.96665, 0.96791]

Event code	Probability importance interval	Event code	Probability importance interval
X ₁	[0.97033, 0.97158]	X ₈	[0.96656, 0.96781]
X ₂	[0.96943, 0.97069]	X ₉	[0.96786, 0.96911]
X ₃	[0.96872, 0.96998]	X ₁₀	[0.96741, 0.96866]
X ₄	[0.96653, 0.96778]	X ₁₁	[0.96770, 0.96896]
X ₅	[0.96685, 0.96800]	X ₁₂	[0.96765, 0.96891]
X ₆	[0.96693, 0.96818]	X ₁₃	[0.96684, 0.96809]
X ₇	[0.96626, 0.96751]	X ₁₄	[0.96665, 0.96791]

The critical importance of each basic event was then calculated by Equation (15), with the resulting intervals as shown in Table 3.

Table 3

Critical importance intervals

Event code	Critical importance interval	Event code	Critical importance interval
X ₁	[0.14744, 0.16259]	X ₈	[0.03930, 0.04555]
X ₂	[0.12179, 0.13484]	X ₉	[0.07664, 0.08597]
X ₃	[0.10145, 0.11282]	X ₁₀	[0.06382, 0.07210]
X ₄	[0.03846, 0.04465]	X ₁₁	[0.07246, 0.08115]
X ₅	[0.04766, 0.05460]	X ₁₂	[0.07079, 0.07964]
X ₆	[0.04989, 0.05701]	X ₁₃	[0.04738, 0.05430]
X ₇	[0.03066, 0.03620]	X ₁₄	[0.04208, 0.04857]

Reviewing the results in Tables 2 and 3 with reference to Equations (13) and (15), the probability intervals of each basic event show no considerable difference, while the critical importance intervals do. The structural importance reference value is fairly small because all the logic gates in the fault tree in this case are OR gates. Therefore, there is no need to calculate any further. Furthermore, it can be seen from the critical importance intervals that X₁ (joint leakage), X₂ (cylinder internal leakage), and X₃ (pilot valve failure) have the greatest impact on the probability of system failure. As a result, the prevention of these three failure cases should be the focus of follow-up work such as maintenance, repair, and design improvement.

5 Conclusions

Considering the uncertainty of system failure probability and the failure probability statistics of the basic events in the fault tree of an armored vehicle hydraulic system, this paper proposed a method for system fault tree interval analysis. First, the fault tree was constructed by assigning AND gate and OR gate interval operators. The probability interval of each fault in the subject hydraulic system and the importance index of each basic event in the hydraulic system fault tree were then determined. The analysis results show that this method can not only help to eliminate the adverse impacts of uncertain basic event failure rates on the results of a fault tree analysis, but can also determine the failure probability interval of the top event by calculating the importance index of the basic events.

The traditional fuzzy fault tree analysis method relies heavily on the experience of engineers to correctly define the membership function. In other fault tree analysis methods, a great deal of data are required to define functions and probabilities. The proposed method was designed to improve the efficacy of fault tree analyses in which the data describing the failure probability of basic events are insufficient, and improve the efficiency of complex system evaluations in general, providing a promising tool for the evaluation of failure probabilities in many different engineering applications. The interval model adopted in this paper only requires that the upper and lower bounds of the basic event failure probabilities be defined. In this study, Latin hypercube sampling was then used to determine the failure probability of the top event, as this method converges quickly and meets the engineering requirements. In future research, we intend to integrate efficient sampling methods such as line sampling or direction sampling into the proposed fault tree analysis method to improve its analytical efficiency.

Footnotes

Acknowledgments

The authors are grateful for the support of the China Scholarship Council, Top International University Visiting Program for Outstanding Young Scholars of Northwestern Polytechnical University, and the Natural Science Foundation of Shaanxi Province (2016JQ5109, 2019JM-377).

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