BAI Yong-sheng(白永生), LI Feng(李 鋒), LI Ji(李 季), TIAN Xia(田 霞)

Department of Management Engineering, Mechanical Engineering College, Shijiazhuang 050003, China

Reliability Model for Key Devices with Compound Preventive Warranty

BAI Yong-sheng(白永生)*, LI Feng(李 鋒), LI Ji(李 季), TIAN Xia(田 霞)

DepartmentofManagementEngineering,MechanicalEngineeringCollege,Shijiazhuang050003,China

With the warranty requirements of key and important devices, the compound preventive warranty policy was introduced. For the effectiveness of compound preventive maintenance, the maintenance mode of periodic replacement with functional checks was adopted, its failure risk analysis was carried on, and the warranty reliability model was established. On the basis of above research, a numerical example was given to illustrate the engineering application and generality of the warranty reliability model.

compoundmaintenance;preventivewarranty;reliabilitymodel;potentialfailure

Introduction

Currently, high technical and new type devices are developing and producing with unprecedented speed, and playing a more and more important role in social economics. To ensure the reliability and safety of the key devices mentioned above, when the key devices are set in run, the warranty for a period is required[1].

Because of advanced performance, complex structures and expensive prices, more and more producers of key devices are carrying on preventive maintenance before failures occur. The maintenance tasks in present researches are mostly simplex[2-4], such as periodic replacement, periodic maintenance. It is fit for the devices with wear abrasion, so they can be maintained beforehand.

However, in practice, failures usually do not occur instantaneously, and they actually develop over a period of time[5]. Hence, there exist the compound maintenance tasks in practical maintenance. The compound maintenance means the maintenance mode integrating two or more kinds of maintenance types, which can make full use of the device life, reduce the failure rate effectively, and save the maintenance cost greatly[6]. For example, the maintenance policy for the transmissions of the locomotive is usually under periodic major repair with preventive minor repair some times. The researches on such type of maintenance mode are little.

Because functional check and periodic replacement are typical in practical maintenance, this paper focused on the maintenance mode of periodic replacement with functional checks, the risk was analyzed for the situation of compound maintenance, and the mathematical models were established for the warranty reliability of key devices. Furthermore, to validate the effectiveness of the models proposed, a numerical example was computed and analyzed. The researches in this paper can assist scientific decision-making to sign purchase contract or after-sale technical service contract.

1 Compound Preventive Warranty Strategy

1.1 Potential failure

Key devices are mostly high technical and new equipment, some of which may be purchased from other countries. With the requirements of safety and reliability, the warranty is being emphasized more and more. The studies on key devices warranty are very important theoretically and realistically.

In most maintenance modeling research, the items are traditionally considered to have two states: normal and failure. In 1973, Christer put forward the concept of “delay time”[7], which considered items had three states: normal, potential failure, and functional failure. If the evidence can be found that this failure process is under way, it may be possible to take relevant action to reduce or avoid the consequences caused by functional failure.

According to the items’ failure rules and characteristics, the maintenance policy of scheduled replacement singly is improved, and functional checks are carried out in the replacement period to reduce the failure hazard.

1.2 Compound preventive warranty strategy for key devices

In practical maintenance, to make the best use of device life, reduce failure rate, and save maintenance cost, the compound maintenance type that integrates major and minor repair is usually adopted. The optimization of this kind of maintenance tasks should be further studied and explored.

The maintenance process of compound maintenance is usually as follows: the major maintenance is carried out with pre-set interval, while there are several times of minor maintenance equally between the intervals. Because the major and minor maintenances are usually periodic replacement and functional check, the maintenance type of periodic replacement with functional checks is illustrated to study the compound maintenance.

The detailed process is as follows: the device is preventively replaced with the interval ofTr, and between replacements the functional checks are implemented with the interval ofTn. AndTn=Tr/k, which means there are (k-1) times of inspections before the replacement (see Fig.1). When carrying on functional checks, if a potential failure is identified, preventive maintenance should be adopted; if not, the device will continue to work until either a failure occurs or the next check. During the replacement period, if a functional failure occurs, the device should be repaired.

Fig.1 Illustration of periodic replacement with functional checks

2 Reliability Model for Compound Preventive Warranty

For key devices, the failure risk, in other words, reliability is an important index of their quality. Once a failure occurs, not only will the economic damage be caused, but the accomplishment of regular mission will be affected. Hence, the failure risk of key devices was analyzed, and the warranty reliability model was established.

2.1 Notation and assumption

(1) The run time of the device is far longer than its maintenance interval, and the average time of replacement and inspection can be ignored;

(2) The failures of the device occur independently with single failure mode;

(3) Inspection is perfect when any potential failure present would be identified at an inspection time;

(4) Either failure renewal or inspection renewal makes the device as good as new.

(5)U: the time when potential failures arise, and its probability density function (PDF) and cumulative distribution function (CDF) are denoted byg(u) andG(u), respectively.

(6)H: the delay time from potential failure to functional failure, and its PDF and CDF are denoted byf(h) andF(h), respectively.

2.2 Failure risk modeling

(1)

…

Therefore,

(2)

whereNis the number of preventive replacement beforet,N=int(t/T) andR(·) is the reliability function of the devices.

(3)

2.3 Expressions for R( Tn，Tr)

(1) WhenTr≤Tn, the event of no failure occurring requires one of the following two events.

Event 1: there is no potential failure occurring beforeTr.

Event 2: a potential failure occurs within (0,Tr) but no functional failure occurs beforeTr.

Then, it is obvious that

(4)

(2) WhenTr>Tn, the event of no failure occurring requires one of the following three events.

Event 1: there is no potential failure occurring beforeTr.

Event 2: a potential failure occurs within ((k-1)Tn,Tr) but no functional failure occurs beforeTr.

Event 3: a potential failure occurs between two successive checks, ((j-1)Tn,jTn), and this potential failure is identified at the next inspection timejTn, with no failure occurring in the replacement components over the remaining time period (jTn,Tr).

So, it can be got that

(5)

(6)

By Eqs. (1)-(6), we can get the reliability model for key devices with compound preventive warranty.

3 A Numerical Example

A simple numerical case is computed here to demonstrate and validate the reliability model for key devices with compound preventive warranty. Assuming a device has a warranty period of 5 years, and its initial time and delay time follow Weibull distributions, whose distribution parameters are listed in Table 1.

Table 1 Lifetime distribution parameters ofUandH

LifetimedistributionUHShapeparameterα1=2α2=2.5Scaleparameter/monthβ1=10β2=6

By the models established in this paper, the failure risk can be calculated with MatLab2009b, see Fig.2.

Fig.2 The three dimensional plot for failure risk

Therefore we can conclude as follows.

(1) WhenN=1, it represents that there is only periodic replacement without any functional check, and the failure risk is higher obviously. The result also shows the effectiveness of the functional check.

(2) Whenthe replacement interval is fixed, the more frequently the functional check is done, the lower of the failure risk;

(3) When the times of inspection between two successive replacements is fixed, the longer of the replacement interval, the higher of the failure risk.

4 Conclusions

Aiming to the requirements of preventive warranty for key devices, the compound preventive maintenance policy is introduced to assess the failure risk. Taking the periodic replacement with functional checks as example, the mathematical model for reliability is established, and the effectiveness is validated by a case study. Actually, the model is established only from the aspect of failure risk; if the failure consequences are evaluated by other factors, such as availability and cost, the corresponding models can also be established in the similar way. The researches on preventive warranty modeling could provide references for maintenance decision; furthermore, they are of great significance for improving decision scientificity and practical application.

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TB114 Document code: A

1672-5220(2015)01-0151-03

Received date: 2014- 08- 08

*Correspondence should be addressed to BAI Yong-sheng, E-mail: xiaobai2004@sohu.com