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The existing standard reliability models for computerized numerical control (CNC) machine tools are not satisfactory and they fall short of predicting failure rates or lifetime of key functional parts of CNC machine tools. This is attributed to two reasons: the small sample size of failure data and a large truncated ratio of the censored failure data. Improved correction method (ICM), maximum likelihood estimation (MLE), and empirical maximum likelihood estimation (EMLE) are presented and compared with each other in this study. In order to improve the shortage of reliability models developed by the traditional methods, an improved maximum likelihood estimation method (IMLE) is proposed through enlarging censored failure data. Moreover, the correction factors of mean ratio to extend censored time are designed, by which the censored failure data can be close to the true time between failures (TBF). Furthermore, a solution method of correction factors considering amount of calculation is proposed to meet the requirements of calculation precision. Finally, verification by the orthogonal experiment is simulated to verify the proposed model. The verifying test results show that the proposed method can be applied in reliability modelling for not only CNC machine tools but also the key functional parts of CNC machine tools.

With the rapid development of automatic control and information technologies, computerized numerical control (CNC) machine tools, such as lathe (turning), milling, and boring machine tools, have become important manufacturing equipment in the manufacturing industry.

Key performance indicators on reliability and life-cycle modelling approaches are various for different purposes. Many scholars predicted remaining useful life (RUL) for cutting tools according to its wear-processing during the real cutting process [

According to the characteristics of the failure data of CNC machine tools, the reliability model with the variable of time between failures (TBF) can be divided into two categories. One is the complete failure data model and the other is the incomplete failure data model. The reliability modelling method of CNC machine tools with the incomplete failure data can make full use of the test information and improve the accuracy of the reliability model, so it has been paid more and more attention [

However, traditional methods cannot be well suitable to reliability modelling for CNC machine tools due to the small size sample failure data with lots of censored data. For this problem, researchers have proposed many reliability modelling methods based on Bayesian theory, such as Bayesian reliability inference method [

This paper is devoted to establish an improved maximum likelihood estimation (IMLE) method for reliability modelling of CNC machine tools with lots of censored failure data. The remainder of this paper is organized as follows: in Section

Failure of one system is the state of the system that does not perform the specified function. Failure of CNC machine tools mainly includes two meanings. One is that the machining task cannot be completed normally, and the other is that the processing precision cannot reach the predetermined requirements.

Due to the limitation of the test sample size and test time, there are three kinds of failure data. They are precensored failure data, complete failure data, and postcensored failure data. The precensored failure data is a set of time intervals from the beginning of the test to the first failure and the complete failure data refers to a set of time intervals between two continuous failures. The postcensored failure data is a set of time intervals from the last failure to the end of the test. If the tested products do not fail during the trial, the postcensored failure data is the whole test time. The precensored failure data and postcensored failure data are both right censored data. The censored failure data mentioned in this paper belongs to the right censored data, which is also called type I censored data.

In order to simplify the reliability modelling process, it is assumed that the CNC machine tools tracked at the same time are from the same type and same batch of machine tools. Moreover, the CNC machine tools tracked in the field test are regarded as working under the same external conditions including temperature and humidity. Internal conditions, mainly working conditions, have a great impact on the reliability of CNC machine tools. As usual, the main working conditions are divided into heavy load, medium load, and light load groups. If the difference of working conditions of the tracked machine tools is large, we should divide them into two or more groups for making one group work in the similar working conditions. So the machine tools from the same group are regarded as working in the same working conditions. As CNC machine tools are typical mechatronic systems, it is always assumed that they can be repaired as new as the ones before they failed. So we mainly focus on the statistical reliability modelling method for CNC machine tools.

The failure data of CNC machine tools collected from the field tracking test is the basis of reliability modelling. After preliminary processing, a set of incomplete failure data is obtained. Then, according to the relative frequency histogram of the failure data, Weibull distribution is selected as the distribution of TBF of CNC machine tools [

Reliability modelling procedure of CNC machine tools.

Because of the short test time and small batches of CNC machine tools, the failure data obtained in the field tracking test is usually small size sample. Therefore, two reliability models developed by the ICM and MLE methods have the disadvantage of instability. In order to improve the accuracy of the model, the precensored failure data is always treated as the complete failure data when applying the MLE method. This method is renamed as the empirical maximum likelihood estimation (EMLE) method. The three methods are introduced as follows.

It is assumed that the field tracking test has been carried out under the same conditions for

The failure data collected in the field tracking test can be used to calculate the reliability of discrete

In order to improve the accuracy of the model, the precensored failure data is always treated as the complete failure data when there is a large truncated ratio. Hence, the modelling method, namely, empirical maximum likelihood estimation method, can be described as

The solution of MTBF in the EMLE method is the same as that of the MLE method.

Verification by the orthogonal experiment is simulated to verify the proposed model.

In order to judge the accuracy of the modelling curves obtained by different modelling methods, goodness of the modelling curve is proposed and it can be computed by

In order to compare the accuracy of modelling curves with different methods vividly,

Explanatory diagram for three levels of

In Figure

In order to compare the modelling methods mentioned above, simulation experiments are carried out with MATLAB under the conditions that 10 CNC machine tools are traced. The field test usually lasts two to three months, which is about 1800 hours, so the simulation test time is selected 1500 and 2000 h, respectively. The machine tools tracked in the field test are generally in the random failure period, which means

Test parameters for 8 simulation conditions.

Number | Test time | Ideal parameters | MRCT | |
---|---|---|---|---|

| | |||

1 | 1500 | 0.8 | 1200 | 0.78 |

2 | 0.8 | 1600 | 0.83 | |

3 | 1.2 | 1200 | 0.78 | |

4 | 1.2 | 1600 | 0.86 | |

| ||||

5 | 2000 | 0.8 | 1200 | 0.70 |

6 | 0.8 | 1600 | 0.79 | |

7 | 1.2 | 1200 | 0.69 | |

8 | 1.2 | 1600 | 0.78 |

RCT calculated by (

Considering the calculation speed and simulation accuracy, reliability modelling process is simulated 100 times by ICM, MLE, and EMLE methods, respectively. Then

The procedure for modelling simulation in MATLAB.

Comparison of the ICM, MLE, and EMLE methods.

In the bottom of Figure

So, from Table

The EMLE method is suitable for a large truncated ratio in a certain extent. But when the test time is short and there is a large truncated ratio

It is difficult to gain the best values of

Flow chart of solutions of

The incomplete failure data from 10 CNC machine tools simulated for 1500 hours is used to establish a reliability model, where

Test simulation conditions.

Test time | Ideal parameters | Number of CNC machine tools | |
---|---|---|---|

| | ||

1500 | 1.2 | 1600 | 10 |

Values of

Number | Parameters | |
---|---|---|

| | |

1 | 1.5 | 1.5 |

2 | 2 | |

3 | 2.5 | |

| ||

4 | 2 | 1.5 |

5 | 2 | |

6 | 2.5 | |

| ||

7 | 2.5 | 1.5 |

8 | 2 | |

9 | 2.5 |

It can be found from Figure

Comparison of reliability models established by the IMLE method in different

Comparison of

In the bottom of Figure

Figure

Comparison of modelling between EMLE and IMLE method.

From Figure

The number of CNC machine tools from different field tracking tests is changeable. And the range of parameters of Weibull distribution is always really large with

Test parameters for four-factor and four-level orthogonal simulation conditions.

Number | Number of machine tools | Test time | Ideal parameters | MRCT | |
---|---|---|---|---|---|

| | ||||

1 | 5 | 1000 | 800 | 0.8 | 0.81 |

2 | 5 | 1400 | 1200 | 1.2 | 0.80 |

3 | 5 | 1800 | 1600 | 1.6 | 0.82 |

4 | 5 | 2200 | 2000 | 2.0 | 0.84 |

5 | 10 | 1000 | 1200 | 1.6 | 0.90 |

6 | 10 | 1400 | 800 | 2.0 | 0.67 |

7 | 10 | 1800 | 2000 | 0.8 | 0.84 |

8 | 10 | 2200 | 1600 | 1.2 | 0.74 |

9 | 20 | 1000 | 1600 | 2.0 | 0.95 |

10 | 20 | 1400 | 2000 | 1.6 | 0.91 |

11 | 20 | 1800 | 800 | 1.2 | 0.57 |

12 | 20 | 2200 | 1200 | 0.8 | 0.67 |

13 | 50 | 1000 | 2000 | 1.2 | 0.94 |

14 | 50 | 1400 | 1600 | 0.8 | 0.84 |

15 | 50 | 1800 | 1200 | 2.0 | 0.73 |

16 | 50 | 2200 | 800 | 1.6 | 0.49 |

Verification results of the IMLE method.

As shown in Table

Seen from conditions of 6, 8, 11, 12, 15, and 16, when

Therefore, the IMLE method can compensate the instability of the curves of the reliability model established by the EMLE method when

The IMLE method is used to evaluate the reliability of the key functional parts of CNC machine tools in order to verify it. The key functional parts are components that have high failure rate or that have poor maintainability. For example, tool rest has a higher failure rate compared to other functional parts in CNC machine tools, so it is regarded as a key functional part. Another example is motorized spindle because of its poor maintainability [

In order to validate whether the IMLE method can be applied to the key functional parts of CNC machine tools, tool rest, whose MTBF is about 5000 h, is selected as one example. So the simulation parameters are set as

Comparison of simulation with large scale parameter of Weibull distribution.

It can be seen from Figure

Due to great changes of

Test parameters for three-factor and three-level orthogonal simulation conditions.

Number | Ideal parameters | Test time | MRCT | |
---|---|---|---|---|

1 | 0.8 | 3000 | 6000 | 0.68 |

2 | 0.8 | 4500 | 5000 | 0.82 |

3 | 0.8 | 6000 | 4000 | 0.90 |

4 | 1.2 | 3000 | 5000 | 0.71 |

5 | 1.2 | 4500 | 4000 | 0.87 |

6 | 1.2 | 6000 | 6000 | 0.86 |

7 | 1.6 | 3000 | 4000 | 0.77 |

8 | 1.6 | 4500 | 6000 | 0.77 |

9 | 1.6 | 6000 | 5000 | 0.91 |

Application of IMLE method in 5 key functional parts.

As shown in Figure

Application of IMLE method in 10 key functional parts.

As shown in Figure

Combining Figure

There is a large limitation when the IMLE method is used to estimate MTBF of CNC machine tools, but it truly improves modelling accuracy in the large MRCT condition. When it is applied to the key functional parts of CNC machine tools, the improvement of modelling accuracy is not really large due to the fact that values of

In this work, we have presented a novel improved maximum likelihood estimation method (IMLE) through enlarging censored failure data. Moreover, the correction factor of mean ratio to extend censored time has be designed. Verification by the orthogonal experiment has been simulated to verify the proposed model. It is concluded the following.

There are no conflicts of interest related to this paper.

This research is financially supported by National Natural Science Foundation of China (Grant nos. 51505186 and 51675227), Jilin Province Excellent Researcher Foundation (20170520103JH), Key Research and Development Plan of Jilin Province (20180201007GX), and CNC First Generation of Guangdong Province (2013B0110304006). The authors thank the authors of this paper’s references whose works have contributed greatly to the completion of this thesis.