Research on two dimensional Wiener stochastic degradation model based on the wear model

One dimensional Wiener degradation process is often used to describe the degradation of product performance. However one dimensional Wiener degradation process doesn’t sufficiently consider the relevance of multiple degradation factors and wear process, which lead to inaccurate results. To overcome these problems, a new two dimensional Wiener stochastic degradation model is proposed, which applys to the products on wear process and stochastic degradation process. Combining wear model and two dimensional Wiener stochastic degradation model, a new reliability analytical form is obtained by constructing the FokkerPlanck equation. Then using the relation among wear volume, degradation characteristic lifetime and drift parameter, parameters of two dimensional Wiener degradation model on the basic of wear model can be estimated. Compared with the existing approaches, the proposed method can effectively improve accuracy. Finally, a case study is illustrated the application and advantages of the proposed method.


Introduction
Products subjected to the action of the external environment, the material properties of the product will gradually change. And because of the randomness of the external environment and the material properties, the degradation of the product is also random. So stochastic process [1] is usually used to describe the degradation process of product performance. Wiener process [2][3][4][5] is the most often used method to describe the degradation of product performance. Tang [6] et al estimated failure time distribution by a Wiener degradation model based on intermediate data. Wang [7] used degradation data of Wiener processes with random effects to study reliability level of the bridge. Peng [8] et al proposed Bayesian method to assess the reliability of the products with Wiener process degradation. Nicolai [9] et al proposed an application to coatings on steel structures by comparing models for measurable deterioration with Wiener processes. Balka [10] et al reviewed and implemented the cure models based on first hitting times of Wiener processes. Si [11] et al estimated remaining useful life based on a nonlinear diffusion degradation process with monitoring degraded signal. Most of the studies on degradation of product performance are based on one dimensional Wiener process degradation, without taking into account the multiple degradation and the physical mechanism of degradation.
Wear is one of the common failure types of components. When the wear occurs, the features of the product can be reduced or failed, reliability and safety will be lost if products are continued to put into a Corresponding author : zhanghui1133018@163.com 1 2 { ( ) ( ( ), ( )) , 0} T X t X t X t t   [18], namely, the degradations of products are binary. At the time t, it follows:  is the correlation coefficient. Assume 0   , which means the two marginal degradation process

Reliability Assessment
X t t  exceeds its failure threshold, product fails. The lifetime of the product can be represented as: Then the reliability function ( ) R t is derived:  are the solution of the equation below: When 0,1 , and density   will be obtained. If 0,1 x and 0,2 x in formula (5) are regarded as the performance degradation when product run to a certain time, then formula (5) is actually the reliability of the continue running time of product, and ( ) 1 ( ) is the distribution of the remaining lifetime.

The performance degradation data model
Assuming a total of N products of a degradation test, at the time of 1 2 , , 3 Method for estimating parameter  and 2 

Maximum likelihood estimation
Using the maximum likelihood estimation(MLE) method to get the estimated values of Estimation of the correlation coefficient: In this paper, in order to measure the two degradations at the same time, the equal interval measurement is adopted, which is so called stable measurement, i.e. ij t t independent and identically distributed bivariate normal samples, the estimations of Estimation of the correlation coefficient:

Archard Wear Model
The Archard wear model is a classic wear calculation model by Professor J.F.Archard [20]. The model is verified by experiment and it has a good practicability. The model is as follows: Where V and S are the wear volume and relative slip distance, respectively, n F is the normal force, K is the wear coefficient, H is the Brinell hardness. Whereas a and b are unknown parameters,  is a variable.
Archard [21]wear model is a linear model, it can be seen that wear volume is a linear function of time t, considering the randomness in the process of wear, the Archard wear model is combine with the Markov model to form a new wear model: Where  is the wear volume, 0  and 1  are determined coefficient,  is a random diffusion, and   2 0, N   , t is the characteristic lifetime.
We use T to represent the lifetime of the product, so its expectation is Bring them into the formula (21), the reliability function   R t and distribution function   F t of the model proposed in this paper are obtained:

The example of reliability assessment of steam turbine pump
Steam turbine is the power device of ship, and steam turbine pump is the key part of the steam turbine as the output device of lubricating oil. Once a fault occurs, it will affect the whole ship power system. Now a 1000h reliability test data of the steam turbine pump is used as an example to verify the rationality and accuracy of the method proposed in this paper. concepts of life evaluation of key parts. In order to master the wear of the key parts of the test process completely and accurately, reliability test program requires precise measurements of the thickness and size of the key parts before and after test. In this paper the retarder rotor support bearing clearance and the support bearing temperature are selected as key parameters to assess the steam turbine pump lifetime, because these data can reflect the working state of the steam turbine pump. Reliability and lifetime assessment of key parts is an important part of describing the reliability level and service life of the whole machine. At the same time, it also provides reference for the determination of maintenance cycle and maintenance level.
In accordance with the requirements of the test program, the steam turbine pump must be dismantled for inspection when total running time reaches 100, 200, 300, 400, 500, 600, 700, 800, 900 and 1000 hours. Table1 is the data of retarder rotor support bearing clearance and the support bearing temperature. According to the design requirements of the bearing, when the support bearing temperature is higher than 80℃, it is considered that the bearing fails, namely, the failure threshold of the support bearing temperature is 80℃. When the bearing clearance is greater than 0.3mm, it is considered failure, that is to say, the failure threshold of the bearing clearance is 0.3mm. And the reliability index of the steam turbine pump is that the reliability is 0.7 when the working time reach 10000h.

Reliability assessment of steam turbine pump
We use three methods to assess the reliability of steam turbine pump, respectively, one dimensional Wiener degradation model based on the maximum likelihood method, two dimensional Wiener degradation model based on the maximum likelihood method and the method we proposed where 1  and 2 1  are parameters to support bearing temperature, 2  and 2 2  are parameters to reducer rotor support bearing clearance and  is the correlation coefficient between the two.

One dimensional Wiener degradation model based on MLE
The parameters of the support bearing temperature and the reducer rotor support bearing clearance are estimated by MLE, results are shown in Table 2  Bringing estimated results of parameter into one dimensional Wiener degradation model, the reliability of support bearing temperature is 0.8370 in 10000h, and the reliability of reducer rotor support bearing clearance is 0.8032 in 10000h. Because when one of the two parameters reaches its failure threshold, the steam turbine pump fails. So choose the result which reach its failure threshold first as reliability assessment result of steam turbine pump. In this example, the assessment of reducer rotor support bearing clearance is taken as result of steam turbine pump, namely, the reliability assessment of steam turbine pump is 0.8032 when the working time reach 10000h. The reliability curve is shown in Figure 1.   Stable measurement is used to measure two parameters. Considering the correlation between these two parameters, the results of 1

Two dimensional Wiener degradation model based on MLE
 and  are derived by bringing the data of two parameters into formula (10)- (12). Results are shown in Table 3 Bringing the parameter estimator into formula (5), the reliability of steam turbine pump is obtained, it is 0.6376 at 10000h. The reliability curve is shown in Figure 2.

Two dimensional Wiener degradation model based on wear model
Bringing the data of two parameters into formula (12)  Bringing the parameter estimator into formula (22), the reliability of steam turbine pump at 10000h is obtained, which is 0.6920. The reliability curve is shown in Figure 3.

Comparison and analysis
Taking parameters estimator of three methods into their reliability functions respectively, the reliabilities at 10000h, are obtained, then calculate the relative error based on index. As shown in the Table 5, the reliability curves of the three methods are shown in the Figure 4.  Comparing and analyzing the reliability calculation results and reliability curves of the three methods, it shows: the relative error of one dimensional Wiener degradation model based on MLE is the maximum which is 14.74%, followed by two dimensional Wiener degradation model based on MLE which is 8.91%. The minimum is the method proposed in this paper, which is 1.14%. It can be seen that the method proposed in this paper is the closest to the real index. From the Fig.4, the reliability falling speed becomes faster over time. As time goes by, support bearing temperature showed a fluctuating upward trend, which will increase the wear of metal components. Then reliability will decrease. Therefore, the method proposed in this paper is more in line with the actual situation.

Conclusion
(1) In practical engineering, more than one parameter affects the life of the product. When assess the reliability of the product, only considering one parameter to estimate the whole product is not comprehensive. The evaluation results of the multivariate degradation process are more in line with the actual situation than one dimensional degradation process. (2) In this paper, from the point of view of two performance parameters, combined with the theory of random degradation, a model of two dimensional Wiener stochastic degradation model is proposed based on the wear model for the products which have a wear process and uncertain stochastic degradation processes. The two dimensional Wiener degradation model is optimized by the method proposed in this paper in a certain extent. (3) From the result of steam turbine pump reliability assessment, compared to the results of one dimensional Wiener degradation model based on MLE and two dimensional Wiener degradation model based on MLE, the method proposed in this paper has the minimum relative error which is 1.14% on the reliability 0.7 at 1000h. Thus the method proposed in this paper is more in line with the actual situation and more reliable.