Mechanical system reliability analysis using a combination of graph theory and Boolean function

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Abstract

A new method based on graph theory and Boolean function for assessing reliability of mechanical systems is proposed. The procedure for this approach consists of two parts. By using the graph theory, the formula for the reliability of a mechanical system that considers the interrelations of subsystems or components is generated. Use of the Boolean function to examine the failure interactions of two particular elements of the system, followed with demonstrations of how to incorporate such failure dependencies into the analysis of larger systems, a constructive algorithm for quantifying the genuine interconnections between the subsystems or components is provided. The combination of graph theory and Boolean function provides an effective way to evaluate the reliability of a large, complex mechanical system. A numerical example demonstrates that this method an effective approaches in system reliability analysis.

Introduction

Reliability has become a key factor in the design and operation of today's large, complex, and expensive mechanical systems. The integrity of modern mechanical systems is strongly dependent upon the durability and reliability of the components. However, reliability theory depends heavily on an understanding of failure physics modeling and on the techniques of probability and statistics. Thus, mathematical reliability models play a very important role in reliability analysis. In today's reliability analysis perhaps the most pervasive technique is that of estimating the reliability of a system in terms of the reliability of its components. In fact, reliability predictions for complex systems typically begin with predictions of the probabilities of mission success for the components in a system, and the component predictions are combined in accordance with a logic model that describes how the components interact in a system [11]. Then, the result is a predicted mission success probability for the system. Therefore, it is of great importance to have practical algorithms which efficiently predict the reliability of complex systems, and which also give useful design information with respect to individual units. For this reason a substantial number of formal approaches, such as Fault Tree Analysis (FTA) and Failure Mode Effects and Criticality Analysis (FMECA), in the area of reliability, have been carried out for mechanical systems.

The FTA incorporates the desired consideration for mechanical systems in terms of the topology of a system and interactions; therefore, it usually is used as a system reliability model in finding the important modes of failure in a system, and in the assessment of first occurrence probabilities of the top event of a system. The method is mathematically correct; however, it requires extensive calculations for a complex fault tree. General speaking, for a sequence of N events there will be 2N branches of the tree. Although the number may be reduced by eliminating impossible branches, this computational processing requirement may still be beyond the capability of available machines [9], [12]. Also, discrepancies still exist between theoretical reliability estimation and actual failure observed in practice. Intuitively, it might appear that this poor correlation is because the model is not a good functional representation of the real system [3]. In such analysis it is frequently assumed that the component failure is mutually independent, whereas in reality, this is often not the case. Therefore, it is necessary to replace the simple reliability models with more sophisticated models that take into account the interactions of component failures. In another words, FTA identifies the possible causes of a particular failure and is useful for troubleshooting at any level from component to system, while the reliability assessment needs a high degree of effort [7], [9], [12]. Since it differs itself from the approach to the problem and the scope of the analysis, FTA may be looked upon as an alternative to the use of reliability block diagrams in determining system reliability in terms of the corresponding components.

On the other hand, FMECA is one of the most widely employed techniques for enumerating the possible modes by which components may fail, and for tracing through the characteristics and consequences of each mode of failure on the system as a whole. It allows the assessment of the probability of a failure occurrence as well as the effect of a failure. The quantitative assessment permits relative ranking of failure risks and provides input to other analyses. The method is an analytical technique that ensures all possible failure modes of a system have been addressed [7], [9], [11], [12], [13]. It is primarily qualitative in nature, although some estimations of failure probabilities are often included. The emphasis in FMECA is usually on the basic physical phenomena that can cause a device to fail. Therefore, it often serves as a suitable starting point for enumerating and understanding the failure mechanisms before the progression of accidents when they pass through several stages and analyze the effects of component redundancies on system safety. Hence, for quantifying system behavior, other approaches, such as event-tree or FTA, are often combined as a supplement to FMECA methods.

In this paper, a system-reliability model based on graph theory and Boolean function is proposed to formulate a system equation, characteristic of the reliability of the system. The methodology proposed incorporates the graph theory for system level reliability and Boolean analysis for interactions. Therefore, this method considers not only the topology (structure) of the system, but also effectively incorporates interactions between components. Through the graph theory, a binary argument can be used for considering the connection between the components. Hence, an algorithm, which uses graph theory for the cause and effect relationship between components and assessing reliability of a complex system, is proposed. The graph theory is used to reflect the logical relationship among various fault events. Nevertheless, the structure of the system is not explicitly evident as the only logical relationship among various fault events in a mechanical system; the physical interconnection is also important [3]. Therefore, the problem of how to evaluate the physical interconnection is induced. In order to quantify the physical interconnections, a new methodology based on a Boolean Equation, Shannon's formula, has also been proposed in this paper. Consequently, the physical interconnection is calculated in a straightforward fashion.

Obviously, even though the variables have only two possible values and functions with these same two values, there exists the almost unlimited possibility of combining many functions of many variables through many stages of modern engineering systems that lends Boolean analysis its own typical complexity in theory and practice [1]. For instance, for one variable X only, there are four Boolean functions, namely two constants, φ1(X)=0, φ2(X)=1, and two logics, φ3(X)=X, and φ4(X)=X̄. For an n-component system, it still is a huge computational problem. However, in the proposed method because the use of Boolean analysis is limited to examine failure interactions between the components, followed with estimations of how to incorporate such failure dependencies into the analysis of a system, only the partial components will be involved in each physical interconnection computation. Instead of the system graph, a sub-graph that contains related components only will be used. Therefore, the algorithm will deduce the combinatorially appropriate tree for the system in a rational level; A decomposition of a system graph will be performed so that the number of events in a sub-tree will be extremely reduced. It would benefit from efficient algorithms.

In applying the proposed methodology, the quantification of mechanical system reliability has been obtained effectively. The proposed method provides not only a new algorithm, but also a strategic point that allows decomposing the original system as several small subsystems for reducing the complexity of its analysis. Moreover, various system parameters can also be naturally incorporated into graph models, and existing mathematical results and algorithms in graph theory and Boolean function can be effectively utilized to an advantage for failure consideration.

Section snippets

Mechanical system graph for reliability assessment

As the terminology suggests, see Appendix A [2], a graph is not usually thought of as an ordered pair, but as a collection of vertices, some of which are joined by edges. It is then a natural step to simulate a mechanical system by drawing a picture of a graph. According to the definition, a mechanical system is one of many real-world objects that can conveniently be described by means of a diagram consisting of a set of points together with lines joining certain pairs of these points. In fact,

Basic concept of assessing system reliability using the graph theory and Bryant tree

For system success, all n-components must operate successfully. The reliability of the system is then the intersection of each component success [10].PS=P(X1∩X2∩X3∩⋯∩Xn)=P(X1)P(X2|X1)P(X3|X1X2)⋯P(Xn|X1X2X3…Xn−1)where Xi is signified as the successful event of component i, and P(Xi|X1X2Xi−1) is the conditional probability, which is the reliability of component i evaluated under that components 1,2,…,i−1 are operating. For instance, a simple n-component system that is considered as a series

An application of the methodology

Consider a mechanical system S, which consists of six components, denoted by Ci. Each component is assumed as a basic event Xi. The structure of this system expressed by using a reliability block diagram is shown in Fig. 7. The corresponding system graph is depicted in Fig. 8.

Assume the reliability of each component in the system S is Ri, and let ωij be the physical interconnection between component Ci and Cj. The component reliability matrix RC, and the adjacency matrix A of this graph and its

Conclusions

Traditional FTA and FMECA techniques for mechanical system reliability analysis are tedious for the complex systems. Moreover, they can only provide an approximate result in some cases. To improve the efficiency of the analysis even further, a new analysis method of mechanical system reliability based on graph theory and Boolean function has been studied in this paper. The use of graph theoretic modeling to perform the mechanical system reliability is a reliable and efficient approach. The

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