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Early failure modeling and analysis of CNC machine tools

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Abstract

Frequent early failures are the key factors restricting the reliability improvement of CNC (computer numerical control) machine tools. In order to eliminate the early failures of CNC machine tools as much as possible, this paper defined the early failures of CNC machine tools strictly and divided the early failures into sudden early failures and progressive early failures. According to the characteristics of sudden early failures, the corresponding reliability analysis model was established by fishbone diagram and 5M1E (man, machine, material, method, measurement, environment) method. Based on the failure time rather than failure time intervals, the reliability analysis model of progressive early failures, which can reflect the dynamic characteristics of CNC machine tool failures, was established by BBIP (bounded bathtub intensity process) method in NHPP (non-homogeneous Poisson process). The reliability evaluation indexes of progressive early failures were given, and the analysis method of the influence of the relationship between former failure and latter failure on the established reliability model was given. CNC machine tools made in China were taken as the example, and the reliability analysis models of sudden early failures and progressive early failures were established and analyzed, respectively. The conclusion that different product failures of the same model could not be analyzed by the same parameter model is obtained. The results also verify the applicability and correctness of the proposed method, which lay a foundation for the elimination of early failure and the improvement of reliability of machine tools.

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Data availability

All data generated or analyzed during this study are included in this published article.

Abbreviations

CNC:

Computer numerical control

5M1E:

Man, machine, material, method, measurement, environment

BBIP:

Bounded bathtub intensity process

NHPP:

Non-homogeneous Poisson process

HPP:

Homogeneous Poisson process

RP:

Renewal process

CMTEF:

CNC machine tools early failures

SEF:

Sudden early failures

PEF:

Progressive early failures

TTT:

Total test time

MTBF s :

Instantaneous mean time between failure

MTBF c :

Cumulative mean time between failure

S-PLP:

Superposed power law process

S-LLP:

Superposed log-linear process

t 0 :

The moment when machine tools are installed and debugged

t 1 :

Early failure time inflection point

t 2 :

Accidental failure time inflection point

t 3 :

The moment when the machine tools are scrapped

F(t):

Normal operation state fluctuation function of machine tools

δ(t):

Pulse function

t t0 :

Positive maximum fluctuation amplitude time

t t1 :

Negative maximum fluctuation amplitude time

A :

Maximum fluctuation amplitude that machine tools can bear

m :

Total number of experimental machine tools

T i :

Timing censoring time of the i-th product

n i :

Total number of failures collected in (0, Ti)

S K :

The moment when the K-th failure occurs in the time series

N :

Total number of m product failures in the statistical time

u :

Failure occurrence time

p(u):

Number of machine tools observed at time u

K :

Number of failures observed at u=SK

△t :

Very small time interval

λ(t):

Failure intensity function

λ c(t):

Cumulative failure intensity function

ω(t):

Cumulative failure intensity function

R(t):

Reliability function

λ 1(t):

Early failure process of products

λ 2(t):

Depletion failure process of products

λ 1 :

Model parameters of S-PLP

β 1 :

Model parameters of S-PLP

λ 2 :

Model parameters of S-PLP

β 2 :

Model parameters of S-PLP

α 1 :

Model parameters of S-LLP

α 2 :

Model parameters of S-LLP

γ 1 :

Model parameters of S-LLP

γ 2 :

model parameters of S-LLP

α :

Model parameters of four-parameter method

β :

Model parameters of four-parameter method

η :

Model parameters of four-parameter method

θ :

Model parameters of four-parameter method

a 1 :

Model parameters of five-parameter method

b 1 :

Model parameters of five-parameter method

c 1 :

Model parameters of five-parameter method

d 1 :

Model parameters of five-parameter method

k :

Model parameters of five-parameter method

a :

Model parameters of BBIP

b :

Model parameters of BBIP

c :

model parameters of BBIP

d :

Model parameters of BBIP

τ 0 :

Standing point of failure intensity function

τ 1 :

Standing point of the first derivative of failure intensity function

\( \hat{b} \) :

Estimated values of b

\( \hat{c} \) :

Estimated values of c

\( \hat{d} \) :

Estimated values of d

P :

Goodness of fit evaluation index

\( {N}_{S_{\mathrm{K}}} \) :

Actual cumulative failure number of CNC machine tools observed at time SK

\( {\overline{N}}_{S_{\mathrm{K}}} \) :

Expected failure number at time SK

ρ pxy :

Pearson correlation coefficient

ρ sxy :

Spearman’s rank correlation coefficient

ρ kxy :

Kendall tau rank correlation coefficient

x :

Failure time data

y :

Failure time data

σ x :

Standard deviation failure time data of x

σ y :

Standard deviation failure time data of y

d K :

Difference between the ranks of two sets of failure time data of x and y

C :

Number of elements with consistency in x and y failure time data

H 0 :

Zero hypothesis of BBIP model

H 1 :

Alternative hypothesis of BBIP model

Λ :

likelihood ratio statistic

ρ xy :

Nonlinear relationship between the first failure and the second failure

a 0 :

Coefficients of ρxy

a 1 :

Coefficients of ρxy

a 2 :

Coefficients of ρxy

b 1 :

Coefficients of ρxy

b 2 :

Coefficients of ρxy

l 0 :

Maximum value of log-likelihood function of BBIP model with the same parameters

l i :

Maximum value of log-likelihood function of i-th BBIP model with different parameters

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Funding

This work was supported by the National Natural Science Foundation of China (No. 51835001; 51705048), the National Major Scientific and Technological Special Project for “High-grade CNC and Basic Manufacturing Equipment” of China (No. 2018ZX04032-001).

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Authors and Affiliations

  1. School of Mechanical Engineering, Yangzhou University, Yangzhou, 225127, Jiangsu, China

    Yulong Li

  2. State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing, 400044, China

    Xiaogang Zhang, Yan Ran & Genbao Zhang

Authors
  1. Yulong Li
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  2. Xiaogang Zhang
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  3. Yan Ran
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  4. Genbao Zhang
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Contributions

Yulong Li analyzed the data and wrote the manuscript. Xiaogang Zhang and Yan Ran were the major contributors in collecting the data. Genbao Zhang polished the manuscript. All authors read and approved the final manuscript.

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Correspondence to Yulong Li.

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Li, Y., Zhang, X., Ran, Y. et al. Early failure modeling and analysis of CNC machine tools. Int J Adv Manuf Technol 112, 2731–2754 (2021). https://doi.org/10.1007/s00170-020-06495-0

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