Reliability of simple pneumatic systems.

Nanasi, Tibor

1. INTRODUCTION

Pneumatic systems have reputation of extremely high reliability when compared to hydraulic or mechanical equipment. Producers estimate the life cycle of various types of valves, which are key elements of pneumatic systems, up to as high numbers as fifty million of cycles (Smith, 2001). Production lines are usually designed without any redundancy except for situations, where extremely high requirements are laid on the safety of the system (Birolini, 1999). Basic reliability model of such systems has usually the form of series system when expressed by conventional reliability block diagram. High number of elements in serial layout has destroying effect on the overall reliability even in case of excellent reliability of individual element. Conventional reliability model gives conservative estimate of the global reliability of the complete system. Moreover, individual components are in most cases modelled as two state elements which are either fully functional or in failure. However, real subsystems of technical systems may admit many intermediate partially functional states between the two extremes (Papazoglou, 1998). This is manifested by the existence of several different failure modes and failure mechanisms that contribute to the loss of functionality. Another obvious fact is, that complex systems are sufficiently robust in withstanding the failure of those subsystems, that are not absolutely vital for their functionality (Kececioglu, 2002a; Dekys et al., 2004).

[FIGURE 1 OMITTED]

More realistic alternative to conventional analysis are methods, which allow for multi-state nature of elements and subsystems, resulting in less conservative and more realistic reliability estimates. In this paper elements of simple pneumatic productive system are modelled via Markov process to allow for multi-state nature of the possible failure mechanism. Numerical results demonstrate slight improvement over the conventional analysis.

2. MULTISTATE FAILURE MODEL

Let us consider the textbook example of simple pneumatic production system, see Fig. 1. To simplify the analysis, the pneumatic press is decomposed to two subsystems. Connecting links and the air supply are neglected in our simplified analysis. First subsystem is the pair of identical 3/2 valves with pushbuttons (marked as 3 in Fig. 1), the second one is the linear actuator together with the controlling 5/2 valve buttons (marked as 1 and 2, respectively, in Fig. 1). The valve of the first subsystem is required to be open while the push-button is pressed by the operator and to be closed when push-button released. Internal state of the push-button should be changed due to the action of the operator and the valve should react appropriately. Possible states of the first subsystem under above simplifications are summarised in Table 1 together with the evaluation of the functionality of the system.

Bracketed values at the second column and rows 6 and 7 are acceptable, when spurious opening or spurious closure of the valve is acceptable. Typical case would be a single mission system. If many cycles of steady faultless operation are required, then the values in bracket should be refused. States of the first subsystem can be defined as follows:

STATE 1: Nr.1 &2 (fully functional system)

STATE 2: Nr. 6 & 7 (failed valve and push-button)

STATE 3: Nr. 5 & 8 (failed push-button)

STATE 4: Nr. 3 & 4 (failed valve).

To set up the corresponding Markov process, the transition probabilities between states are to be introduced. For system without repairs the appropriate failure rates are for assumed failure modes as follows:

[[lambda].sub.13]--failure rate of the push-button

[[lambda].sub.14]--failure rate of the valve

[[lambda].sub.42]--failure rate of the push-button after the valve failed

[[lambda].sub.32]--failure rate of the valve after push-button failed

[[lambda].sub.12]--failure rate because of unspecified cause.

[FIGURE 2 OMITTED]

We remind, that failure rates [[lambda].sub.ij] express the conditional probabilities ([[lambda].sub.ij] [DELTA]t) of the occurence of a failure between time t and t + [DELTA]t provided that failure did not happen until time t.

3. NUMERICAL RESULTS

Markovian model corresponding to state diagram in Fig. 2 results in Chapman-Kolmogorov differential equations (Birolini, 1999; Kececioglu, 2002b), which are a set of four coupled first order linear differential equations for probabilities [P.sub.i](t) of states i=1,2,3,4.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The Chapman-Kolmogorov system was solved by direct numerical integration for initial conditions

[P.sub.1](0) = 1, [P.sub.2](0) = 0, [P.sub.3] (0) = 0, [P.sub.4] (0) = 0,

and for transient probabilities [[lambda].sub.13] = [[lambda].sub.42] = 0.1 [10.sup.-6], [[lambda].sub.14] = [[lambda].sub.32] = 0.9 [10.sup.-6] and [[lambda].sub.12] = 0.09 [10.sup.-6] estimated ad hoc according to data given in (Smith, 2001). Fig. 3 shows probabilities [P.sub.1](t), [P.sub.1](t) + [P.sub.2](t), [P.sub.1](t) + [P.sub.2](t) + [P.sub.3](t) and [P.sub.1](t) + [P.sub.2](t) + [P.sub.3](t) ) + [P.sub.4](t)=1 as they are changing with the time. For convenience the time scale is running in thousands of hours and the transient probabilities [[lambda].sub.ij] are given in failures per million hours.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

The thick curve in Fig. 4 corresponds to the traditional reliability function [P.sub.1](t), thin line gives [P.sub.2](t), lower dashed curve is the [P.sub.3](t) and the upper long dashed curve corresponds to the time dependence of the probability [P.sub.4](t) of state 4. These illustrative results demonstrate that allowing for multi-states has the potential of increasing the conservative estimate of reliability of systems. The formulation the Markov process for the second subsystem is not treated here, as it is a complete analogy of the first subsystem except for different numerical values of transitional probabilities.

4. CONCLUSION

The present paper demonstrates the usefullness of treating the components of pneumatic systems as multi-state elements. For properly defined states and corresponding transition probabilities it is possible to set up Markov process leading to Chapman-Kolmogorov equations. Key element of such analysis is the realistic definition of individual states, which is always highly dependent on detailed knowledge of actual failure modes of the system under consideration. Due to lack of detailed experimental data future research might be focused on development of consistent methods to estimate the transitional probabilities.

The author greatly acknowledges the financial support for this work by the research projects KEGA-2/4154/06 and AV-4/0102/06

5. REFERENCES

Birolini, A. (1999). Reliability Engineering. Theory and Practice. 3rd Edition, Springer-Verlag, ISBN 3-540-663851, Berlin

Dekys, V.; Saga, M. & Zmindak, M. (2004). Dynamika a spol'ahlivost' mechanickych sustav, VTS ZU, ISBN 80969165-2-1, Zilina

Kececioglu, D. (2002a). Reliability Engineering Handbook Vol. 1, DEStech Publications, ISBN 1-932078-00-2, Lancaster, USA

Kececioglu, D. (2002b). Reliability Engineering Handbook Vol. 2, DEStech Publications, ISBN 1-932078-01-2, Lancaster, USA

Papazoglou, I. A. (1998). Functional Block Diagrams and Automated Construction of Event Trees. Reliability Engineering and System Safety, Vol. 61, pp. 185-214

Smith, D. J. (2001). Reliability, Maintainability and Risk. Practical Methods for Engineers, 6th Edition, Butterworth Heinemann, ISBN 0-7506-5168-7, Oxford

Tab 1. Possible states of 3/2 valve.

Nr. Functionality Operator's Button's Valve
 of the system action state opened

1 1 1 1 1
2 1 0 0 0
3 0 0 0 1
4 0 1 1 0
5 0 1 0 0
6 0 (1) 1 0 1
7 0 (1) 0 1 0
8 0 0 1 1