Maintainability assurance of plastic deformation tools.
Key words: reliability modeling, simulation, maintainability assurance.
The intense global competition and the rapid progress in technology enlarged the scope of the notion of quality, to include aspects concerning the time varying performance of the systems. Systems must meet customer's expectations not just when they are placed in service, but for the expected life of the systems.
The general property of a system to conserve its performance in time became the specific notion that is reliability. The concept of reliability, developed based on electronic technology, was readily generalized to describe any technical system and became one of the features most desired by customers at the beginning of twenty-first century. The data concerning the reliability measures of plastic deformation tools are reduced. This situation it is due to the following causes:
a) the high degree of individualization for each tool;
b) the causes which can lead to the presence of the failure are numerous, so that the establishment of the distribution function it is difficult;
c) the great number of data which must be processed impose the existence of a data registration system, used by a personnel informed about the investigation methods;
d) the difficulty in transferring the experience existing in other field, due to the different phenomena, which influence their reliability.
In order to take into account the maintenance actions, the concept of reliability must be understood as the overall capacity of a system to accomplish a specified task. For plastic deformation tools, their effectiveness depend not only on their reliability, but also on the characteristics of maintenance actions.
Within this framework, the aim of this paper it is to perform the reliability modeling of the plastic deformation tools and the maintainability assurance by inspection planning and renewal policies.
2. RELIABILITY MODELING OF THE PLASTIC DEFORMATION TOOLS
The mathematical reliability modeling of plastic deformation tools it is based on the concept of failure, when at least one of the tool performances exceeds its tolerance limits.
In order to detect the failure of the tool, a data acquisition system was developed. The system it is based on the deformation during the plastic process. Comparing the deformation value with their reference value, the failure can be detected. Two resistive strain gages, an analog strain indicator, the PCI-1200 board acquisition, a connector block that used the first channel of the board acquisition and a personal computer, compose the data acquisition system. The software to read the data was developed in C++ language and performed the board acquisition settings and the registration of the experimental data on the PC hard-disk. The signal analysis and processing were performed using Matlab software.
The essential step in mathematical reliability modeling of plastic deformation tools it is the adoption of the distribution law that will model the reliability, taking into account the physical interpretation and experimental data and it is based on the theory of hypothesis testing (Meeker & Hamada, 1995). According to this theory, the null hypothesis H0 regarding the distribution law and the alternate hypothesis H1 which excluded the distribution law selected by the H0 hypothesis are formulated. A decision between the two hypotheses is taking according to a goodness-of-fit test and is affected by the a and [beta] risks. One of the most used a goodness-of-fit test is the Kolmogorov-Smirnov test, which use the time to failure of the tool under the observation. For this test, the distribution law is accepted if and only if (Catuneanu& Mihalache, 1989):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where F(t) and [??](t) are the true and estimated cdf and e1-a (n) is the 1-[alpha] percentile of the Kolmogorov-Smirnov distribution. The parameters of the proposed distribution are unknown and must be estimated from the experimental data. The most common method is the least squares estimation, which will be used in this paper.
A numerical evaluation of the of plastic deformation tools reliability may be performed by simulation. The method consists of generating possible states of components according to their reliability functions and evaluating the reliability of each combination of individual states. The ratio between the resulting number of plastic deformation tools good state and the total number of simulations is an estimate of the reliability function (Catuneanu& Mihalache, 1989):
[[??].sub.S] = ([N.summation over (i=1)][S.sub.i])/N (2)
where [S.sub.i] is the value of the structural function for simulation i and N is the number of simulations. The accuracy of this an estimate increases with the number of simulations.
3. MAINTAINABILITY ASSURANCE OF PLASTIC DEFORMATION TOOLS
For the reliability assurance of the tools of plastic deformation tools the inspection planning and renewal policy were proposed. In order to planning the inspection, the percentile of the time to failure [X.sub.[gamma];t] may be used (Radu & Baban, 1999). This is the solution of equation:
[R(t + x)/R(t)] = [gamma] (3)
where t is the number of pieces realized previously by the tool, x is the supplementary pieces and [gamma] is the probability that the supplementary pieces will be realized in good conditions. If the number of pieces realized previously is known and taking a minimal value of reliability function, from relation (3) results the x number of pieces after the inspection of degradations of the tools must be done.
The best reliability assurance methods are those that prevent failures. One of these methods is the renewal policies, which consists in preventive renewals of the system prior to failure. The preventive renewal eliminates the cumulative wear (the system became "good as new") and thus prolongs the time to failure of the system. A renewal policy is specified by the scheduled times of preventive renewals. In the event of systems failure, renewals are carried out at random times, whereas preventive renewals may be may be periodic or non-periodic events, according to the type of renewal policy (Catuneanu& Mihalache, 1989; Elsayed, 1996).
Several criteria may be used in formulating renewal policies. The first criterion used in this paper is that the operational reliability should exceed a specific lower value:
R(T)[greater than or equal to][R.sub.0] (4)
The second criterion is the minimization of the average maintenance cost rate. The maintenance cost of the system is considered equal to unity if the system fails before it reaches age x and equal to a in the opposite case. Dividing the average maintenance cost by the mean lifetime, we get:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
Minimizing expression (5), the equation of the age x* when preventive renewal should be performed is obtained:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
The average maintenance cost of renewal policy must be compared to the average maintenance cost when no preventive renewals are carried out. In this case, the average maintenance cost is:
[c.sub.FRP] = 1/m (7)
4. CASE STUDY
The theoretical considerations were applied for a cold plastic die. The time-to-failure was achieved using the data acquisition system. During the cold plastic deformation process, the deformation of tools can be measured. Comparing the deformation value with the deformation value when the tool is good and when it fails, the time-to-failure can be registered.
Using the data acquisition system, the following times to failure of the tool were obtained (in cycles): 19356, 17845, 18954, 4125, 16589, 13698, 15428, 14876, 23649, 20369, 22198, and 21587.
In order to perform the reliability assurance, the distribution law associated with the failure mechanisms of tools must be established. We have assumed that after each failure, the tool it is repaired and became "good as new". The association of a distribution law with the specific failure mechanism of this active element must be sustained by physical interpretation and the experimental data (which have the last word). Two laws were proposed to describe the time to failure of the tools: power and alpha (Dorin et al., 1994; Park&Kim, 1992).
A Kolmogorov-Smirnov goodness-of-fit test was performed to adopt the distribution law. The risk of the first order was adopted at [alpha]=0.20. applying the Kolmogorov-Smirnov goodness-of-fit test, the power law was adopted. The parameters of the power law were also computed: [[delta].sub.est]=1.478, [b.sub.est]=28187.506.
The possible values of the state vector of cold plastic deformation tools were obtained by simulation, using the Monte Carlo method. The simulation of state vector and the estimation of the reliability function was computed by the integrated system, too. The minimal reliability measure resulted as [R.sub.min]=0.80.
Taking into account a minimum level of reliability [R.sub.0]=0.80, the number of supplementary cycles after which the inspection planning must be achieved, was computed: X0,8;0 = 9496 cycles. The time for next verification can be established: [X.sub.0,8; 14697] = 5548 cycles and so on.
The time of preventive renewals taking into account the minimum reliability level criterion was computed at: T=x=9496 cycles. Suppose that the cost of a preventive renewal it is 40% of the cost of a renewal upon failure (a=0.40) and R0=0.80, the age x* after which the preventive renewal should be performed was established: x*=17573 cycle. The average maintenance cost rate for ARP and FRP was also computed: [c.sub.ARP]=3.48589.10-5 [cycles.sup.-1], [c.sub.FRP]=5.9378.10-5 cycles-1, so that [c.sub.ARP] <[c.sub.FRP]. Thus, the average maintenance cost of the ARP policy it is less than in the case of FRP policy, but in this case the reliability function becomes R(15408)=0,51.
The present development of the plastic deformation tools it is characterized by a continuous growth of their complexity, conducting to a more complex process of reliability analysis. In this paper, the reliability modeling was performed using the Kolmogorov-Smirnov goodness-of-fit test. A numerical evolution of the reliability was performed by simulation, using Monte Carlo method. For the inspection planning, the percentile of the time to failure was proposed. The renewal policies were designed using the minimum reliability level and the average maintenance cost rate minimization criteria. The computation of the average maintenance cost was also performed.
In conclusion, the adoption of the reliability model of the time-to-failure tools of plastic deformation tools allows the estimation of the reliability measures of these tools. Using these measures, the maintainability assurance can be achieved by inspection planning and renewal policies.
Catuneanu,V.M. & Mihalache,A. (1989). Reliability Fundamentals, Elsevier Press, ISBN 0-444-98879-3, Amsterdam.
Dasgupta,A. & Pecht, M. (1991). Material failure mechanism and damage models, IEEE Transaction on Reliability, vol.40, no.5, pp. 531-536, ISSN 0018-9529.
Dorin, A.C., Isaic-Maniu, A., Voda, V.Gh. (1994). Statistical problems in reliability theory (in Romanian), Editura Economica, ISBN 973-96487, Bucuresti.
Elsayed, A.E. Reliability Engineering(1996). Addison Wesley Longman, ISBN: 0-201-63481-3, New York.
Meeker,W.Q. & Hamada,M. (1995): Statistical tools for the rapid development and evaluation of high-reliability products, IEEE Transaction on Reliability, vol. 44, no.2, pp.187-198, ISSN 0018-9529.
Park,W.J., Kim,Y.G. (1992). Goodness-of-fit Tests for the Power-law process, IEEE Transaction on Reliability, vol. 41, no.1, pp.107-111, ISSN 0018-9529.
Radu, I.E. & Baban, C.F., (1999). Reliability Engineering (in Romanian), Editura Universitatii din Oradea, ISBN 973-9416-58-9, Oradea.
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|Author:||Baban, Calin Florin; Baban, Marius; Radu, Ioan Eugen|
|Publication:||Annals of DAAAM & Proceedings|
|Date:||Jan 1, 2007|
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