Mathematical Problems in Engineering

Volume 2016, Article ID 9842607, 11 pages

http://dx.doi.org/10.1155/2016/9842607

## Reliability Evaluation of NC Machine Tools considering Working Conditions

^{1}College of Mechanical Science and Engineering, Jilin University, Changchun 130025, China^{2}College of Mechanical Engineering, Beihua University, Jilin 132013, China

Received 16 November 2015; Accepted 17 February 2016

Academic Editor: Yuming Qin

Copyright © 2016 Hongzhou Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Reliability evaluation is the basis for reliability design of NC machine tools. Since traditional reliability evaluation methods do not consider the working conditions’ effects on reliability, there is a great error of a result of a traditional method compared with an actual value. A new reliability evaluation model of NC machine tools is proposed based on the Cox proportional hazards model, which describes the mathematical relation between the working condition covariates and the reliability level of NC machine tools. Firstly, the coefficients of working condition covariates in the new reliability evaluation model are estimated by the partial likelihood estimation method; secondly, the working condition covariates which have no effects on the reliability of NC machine tools are eliminated by the likelihood ratio test; then parameters of the baseline failure rate function are estimated by the maximum likelihood estimation method. Thus, the reliability evaluation model of NC machine tool is obtained under different working conditions and the reliability level of NC machine tools is obtained. Case study shows that the proposed method could establish the relation between the working condition covariates and the reliability level of NC machine tools, and it would provide a new way for the reliability evaluation of NC machine tools.

#### 1. Introduction

With the rapid development of high-speed and high-precision technologies, NC machine tools are becoming the main equipment for advanced manufacturing technology [1], and so they have been applied widely in many industries and regions. Kinds of machining workpieces are also very different from user to user. Meanwhile, the atmosphere pressure and temperature vary greatly from one region to another. Therefore, NC machine tools are usually in different working conditions [2]. Generally, different working conditions have different impacts on reliability of products [3], which have been confirmed by many researches [4–10]. Hu studied the influence of various road conditions on reliability of cars, and the strengthening coefficients based on the strengthening road conditions were obtained [4]. The reliability model of power system considering weather conditions was established and the weather conditions were divided into 3-state or 2-state weather model [5, 6]. In order to evaluate the reliability of electronic system, hybrid stochastic Petri net was used to establish the reliability model. And the reliability levels of the system under different temperature and voltage were obtained [7]. Chen et al. studied the failure physics equation, which is generalized Eyring model, of aerospace electrical connectors; the multiple-stress accelerated test scheme is adopted; then the reliability levels of aerospace electrical connectors under different temperature and vibration stresses were obtained [8]. Besides, some scholars established relations between the environmental stresses and reliability levels of some products by carrying out accelerated life tests [11, 12]. Li et al. established the relationship between working conditions (the speed and load) and reliability of harmonic driver by accelerated life test [11]. Nogueira et al. studied temperature, humidity, and current impacts on reliability of high luminosity AlGaInP LEDs by accelerated life test [12]. Thus, it can be deduced from the above studies that working conditions would also affect the reliability of NC machine tools. Different working conditions cause different reliability levels, and the more different the working conditions are, the more obvious the disparity of the reliability levels is.

Reliability is one of the most important indicators of measuring the performance of NC machine tools, which has always been the researching focus by scholars [13–17]. Given that the result of reliability evaluation is influenced by many factors, methods including the Markov model [18], Petri net [19], Monto Carlo method [20], and Bayesian method [21] are applied to reliability evaluation of NC machine tools. However, for the present, the relation between reliability of the NC machine tools and the working conditions has not been established, which causes error in the result of reliability evaluation of NC machine tools compared with the actual value. Therefore, to decrease the evaluation error, it is of great engineering significance to establish the relation between the reliability of NC machine tools and the working conditions, which is also a hard problem in reliability research on NC machine tools.

In different research areas, there are many models to describe the relations between the reliability and the working conditions. Commonly seen models, besides [4–12], include Arrhenius Model [22], Inverse Power Law Model [23], and Cumulative Exposure Model [24], as well as other models [25]. However, each of them is usually aimed at a specific type of product. For the NC machine tool is a typical mechanic-electric-hydraulic system, the above methods are difficultly used to establish the relationship between working conditions and reliability.

The proportional hazards model is a life statistical model, which is usually used for survival analysis in medicine field [26, 27], and can describe relation between patients and influencing factor. Thus, a new reliability evaluation method for NC machine tools based on the Cox proportional hazards model is proposed in this paper. Working conditions of NC machine tools are taken as covariates, and then the relation between the reliability level of NC machine tools and the covariates is established based on the proposed model. The coefficients of working condition covariates of the proposed model are estimated by the partial likelihood method; the covariates which have no effects on the reliability of NC machine tools are eliminated by the likelihood ratio test; the parameters of baseline failure rate function are estimated by the maximum likelihood estimation. Then, the reliability model of NC machine tools is developed under different working conditions. A batch of NC machine tools is taken as the research object for case study, where the impact laws of environment temperature, cutting fluid, number of tool changes, and cutting force on the reliability of NC machine tools are researched, respectively. Finally, the feasibility of the proposed method is validated in the case study.

#### 2. Reliability Model of NC Machine Tools considering Working Conditions

##### 2.1. Proportional Hazards Model

The proportional hazards model was developed in 1972 by Cox, a British statistician. This model considers the relation between failure rate and covariates, which is defined as [28]where is the time between failures (TBF, a random variable) of NC machine tools, is the vector of working condition covariates, which affects the failure rate of NC machine tools, is the th covariate, such as cutting force, environment temperature, number of tool changes, or vibration, and is function of working condition covariates.

In general, can be expressed aswhere is the vector of ’s coefficients, which reflect the covariates’ influences on the failure rate function, and is the coefficient of . When , it indicates that catalyzes the machine tools to fail; when , has no effects on the failure rate of machine tools; when , depresses the machine tools to fail. is the baseline failure rate of NC machine tools, that is, the failure rate function when ; represents the failure rate function of NC machine tools under covariate .

Equation (3) is equivalent to the following equation:

The failure rate function of NC machine tools under the covariate is ; thus,

According to (4) and (5), then

Therefore,

Assume that the probability density function (PDF) of NC machine tools’ TBF under covariate vector is , and corresponding reliability function is . According to (7), of NC machine tools iswhere is the reliability function of NC machine tools under covariate vector .

can be expressed by

TBF of NC machine tools is generally considered to follow two-parameter Weibull distribution [2, 13, 21, 29, 30]. Suppose that, under working condition covariate , the failure rate function can be expressed by

Therefore, (7) is equivalent towhere is the shape parameter, ; is the scale parameter, .

So

Therefore, the PDF of NC machine tools’ TBF considering the working conditions can be expressed as

Then, MTBF of NC machine tools under covariate vector can be obtained bywhere is Gamma function.

##### 2.2. Parameter Estimation

There are several parameters and coefficients in (11), which are and and ). Thus, a two-step estimation method to estimate these parameters and coefficients is employed. Firstly, is estimated by the partial likelihood estimation method [31]; then, and are estimated by the maximum likelihood estimation method.

###### 2.2.1. Covariate Coefficients Estimation

From (7), we can get

Assuming that there are failures, the th failure of NC machine tools can be expressed as (, , ), where is time between the th failure and the th failure; is an indicator variable of datum ; when , is noncensoring time and when , is censoring time.

Equation (16) is dimensional column vector, which indicates that there are covariates in the th failure of NC machine tools:

Therefore, the partial likelihood function is given bywhere is the number of failures whose TBF is equal to . If is the set of failures of NC machine tools at time , then . is the sum of working condition covariates of failures, and so when ; is the set of data that NC machine tools do not fail and there is no censoring at .

Now take the logarithm of both sides in (17); then

Take the first derivative with respect to and let it be equal to zero, sowhere is the th element in .

There is no analytical solution to (19). So Newton-Raphson iterative method is applied to estimating the parameters [32].

The second partial derivatives of consist of the order matrix , of which the elements arewhere .

So can be estimated by Newton-Raphson numerical algorithm.

###### 2.2.2. Elimination of No Impacting Covariate

In order to eliminate the covariates which have little or no impact on the reliability of NC machine tools, the likelihood ratio test is used and the procedure is shown in Figure 1 [33].