# Reliability Modeling and Evaluation Method of CNC Grinding Machine Tool

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## Abstract

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## Featured Application

**The proposed research aims to provide one accurate and reliable evaluation method for the subsystem and system of electromechanical equipment.**

## Abstract

## 1. Introduction

## 2. Modeling of Subsystem

#### 2.1. Reliability Models

#### 2.2. Reliability Data Grouping

#### 2.3. Probability Density Function Fitting

_{i}, f

_{i}). The probability density function can be fitted according to the data pair using the least squares method or maximum likelihood method for each subsystem by different distribution type, and the point estimation and interval estimation of the parameters of the probability density function can be executed [14].

#### 2.3.1. Exponential Distribution

_{i}and $\mathrm{ln}[1-F({t}_{i})]$ are linear and the observed value of parameters λ can be calculated by Equation (8).

#### 2.3.2. Weibull Distribution

_{i}are linear and the observed value of parameters m and η can be calculated by Equation (10).

#### 2.3.3. Normal Distribution

_{i}is the lower fractile of standard normal distribution and can be calculated by the inverse function of the standard normal distribution. If the sample data meet normal distribution, z

_{i}and t

_{i}are linear and the observed value of parameters μ and σ can be calculated by Equation (13).

#### 2.3.4. Logarithmic Normal Distribution

_{i}and lnt

_{i}are linear and the observed value of parameters μ and σ can be calculated by Equation (16).

#### 2.3.5. Gamma Distribution

#### 2.4. Model Optimization

- Calculate the adjacent points’ slope of sample data and fitted curve, as shown in Equation (19).$$\{\begin{array}{l}{K}_{i}=\frac{\widehat{F}({t}_{i+1})-\widehat{F}({t}_{i})}{{t}_{i+1}-{t}_{i}}\hfill \\ {K}_{ji}=\frac{{F}_{j}({t}_{i+1})-{F}_{j}({t}_{i})}{{t}_{i+1}-{t}_{i}}\hfill \end{array}$$
- $\widehat{F}({t}_{i})$ is the function value of the ith sample data,
- ${K}_{ji}$ is the slope of the ith and (i+1)th sample data of the jth fitting function, and
- ${F}_{j}({t}_{i})$ is the function value of the jth fitting function when the independent variable equals to the value of the ith sample data.

- Calculate the adjacent points’ median of sample data and fitted curve, as shown in Equation (20).$$\{\begin{array}{l}{V}_{i}=\frac{\widehat{F}({t}_{i+1})+\widehat{F}({t}_{i})}{2}\hfill \\ {V}_{ji}=\frac{{F}_{j}({t}_{i+1})+{F}_{j}({t}_{i})}{2}\hfill \end{array}$$
- Calculate the absolute value of difference of slope and median, as shown in Equation (21).$$\{\begin{array}{l}D{K}_{ji}=\left|{K}_{i}-{K}_{ji}\right|\hfill \\ D{V}_{ji}=\left|{V}_{i}-{V}_{ji}\right|\hfill \end{array}$$
- Maximum image processing of slope and median, as shown in Equation (22).$$\{\begin{array}{c}R{K}_{ji}=D{K}_{ji}/\mathrm{max}(D{K}_{ji})\\ R{V}_{ji}=D{V}_{ji}/\mathrm{max}(D{V}_{ji})\end{array}$$
- Calculate the gray correlation between sample data and fitted function, as shown in Equation (23).$${R}_{kvj}=\frac{1}{n-1}{\displaystyle \sum _{i=1}^{n-1}R{K}_{ji}}\frac{1}{n-1}{\displaystyle \sum _{i=1}^{n-1}R{V}_{ji}}$$
- Normalization. In Equation (23), R
_{kvj}is the similarity and closeness between the jth fitted curve and the sample data. The smaller the value of R_{kvj}, the better the fitting. Usually, the big gray correlation degree indicates the good fitting, and its value is between 0 and 1, which needs to be normalized, as shown in Equation (24).$${R}_{gj}=\frac{\mathrm{min}({R}_{kvj})}{{R}_{kvj}+\rho \mathrm{min}({R}_{kvj})}$$

## 3. Reliability Evaluation Method

#### 3.1. Monte Carlo Simulation

_{1}, X

_{2}, …, X

_{N}, the arithmetic mean of the random variable X is shown in Equation (25).

_{α}is any value.

- α is the significant level,
- σ is the standard deviation of the random variable
**X**, and - μ
_{α}is the quantile of the standard normal distribution.

#### 3.2. Random Data Generation

#### 3.3. Reliability Evaluation Algorithm

_{L}years, the average annual maintenance time of the life cycle is T

_{M}, the annual average normal downtime is T

_{S}, and then the normal working time in the life cycle of the CNC grinding machine T

_{W}is as shown in Equation (34).

_{i}is the MTBF of the ith subsystem.

_{i}generated by the ith subsystem can be calculated by Equation (36).

**T**

_{i}of the ith subsystem is shown in Equation (37).

_{i}is the amount of random data of the ith subsystem.

## 4. Case Study

#### 4.1. Modeling of Subsystem

#### 4.2. System Reliability Evaluation

#### 4.2.1. Observed Value

_{s}

^{o}, the amount of CNC grinding machine is n

_{m}, the year of failure data collecting is y

_{r}, the day of failure data collecting each year is d

_{r}, the work time each day is h

_{r}, the amount of failures is n

_{f}, and the mean maintenance time is t

_{ma}, the observed value of system reliability can be calculated by Equation (41).

- MTBF
_{s}^{o}is the observed value of system reliability of the CNC grinding machine, - C
_{L}is the lower confidence coefficient, and - C
_{U}is the upper confidence coefficient.

#### 4.2.2. Whole Fitting Method

#### 4.2.3. Monte Carlo Simulation

#### 4.3. Comparison and Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Cumulative distribution curve of each subsystem. CNC—computer numerical control; MTBF—mean time between failures. (

**a**) base; (

**b**) CNC; (

**c**) spindle; (

**d**) feeder; (

**e**) electrical; (

**f**) hydraulic; (

**g**) cooler; (

**h**) lubrication; (

**i**) header; (

**j**) gauge.

**Figure 4.**Probability density curve of time between failures of total machine. MTBF—mean time between failures.

**Figure 6.**Point estimation comparison of different time and quantity of the computer numerical control (CNC) grinder.

Group Number | t_{i}_{–} | t_{i} | t_{i}_{+} | n_{i} | f_{i} |
---|---|---|---|---|---|

1 | 0 | t_{m}/2k | t_{m}/k | n_{1} | f_{1} |

2 | t_{m}/k | 3t_{m}/2k | 2t_{m}/k | n_{2} | f_{2} |

… | … | … | … | … | … |

k | (k-1)∙t_{m}/k | (2k-1)∙t_{m}/2k | t_{m} | n_{k} | f_{k} |

_{m}: the maximal time between failures; t

_{i}

_{–}: the left end of the ith time interval; t

_{i}

_{+}: the right end of the ith time interval; t

_{i}: the median of the ith time interval; n

_{i}: the fault number of the ith time interval; f

_{i}: the probability density of the ith time interval, which can be calculated by Equation (6). In Equation (6), Δt is the time interval length.

**Table 2.**Reliability models and parameters of subsystems. CNC—computer numerical control; MTBF—mean time between failures.

Subsystem | Distribution Type | Parameters |
---|---|---|

base | exponential | λ = 6.61 × 10^{–5} |

CNC | gamma | a = 6.81 × 10^{–1}, b = 9.41 × 10^{3} |

spindle | gamma | a = 7.81 × 10^{–1}, b = 9.68 × 10^{3} |

feeder | gamma | a = 1.539, b = 1.31 × 10^{3} |

electrical | Weibull | β = 3.84 × 10^{–1}, η = 6.58 × 10^{2} |

hydraulic | gamma | a = 7.57 × 10^{–1}, b = 1.36 × 10^{4} |

cooler | lognormal | μ_{l} = 7.011, σ_{l} = 1.644 |

lubrication | gamma | a = 6.05 × 10^{–1}, b = 1.26 × 10^{4} |

header | Weibull | β = 5.62 × 10^{–1}, η = 1.65 × 10^{3} |

gauge | gamma | a = 8.44 × 10^{–1}, b = 9.25 × 10^{3} |

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**MDPI and ACS Style**

Liu, Y.; Peng, H.; Yang, Y.
Reliability Modeling and Evaluation Method of CNC Grinding Machine Tool. *Appl. Sci.* **2019**, *9*, 14.
https://doi.org/10.3390/app9010014

**AMA Style**

Liu Y, Peng H, Yang Y.
Reliability Modeling and Evaluation Method of CNC Grinding Machine Tool. *Applied Sciences*. 2019; 9(1):14.
https://doi.org/10.3390/app9010014

**Chicago/Turabian Style**

Liu, Yongjun, Hua Peng, and Yong Yang.
2019. "Reliability Modeling and Evaluation Method of CNC Grinding Machine Tool" *Applied Sciences* 9, no. 1: 14.
https://doi.org/10.3390/app9010014