A model-based optimal reliability evaluation method for numerical control machine tools

Abstract

The application discloses a kind of based on model optimization's numerical control machine tool reliability evaluation method, belong to numerical control machine tool technical field;Including the following steps: numerical control machine tool fault data division;Grouping processing of fault data;Model parameter estimation based on least square method and maximum likelihood method;Model hypothesis test based on K-S method;Model optimization based on correlation coefficient method;Reliability evaluation.The application first divides according to subsystem to numerical control machine tool original fault data, on this basis, carry out subsystem fault data grouping processing;Subsequently based on least square method and maximum likelihood method respectively, and utilize K-S test method to complete model hypothesis test;Further, carry out model optimization by correlation coefficient method, finally realize the reliability evaluation of system.

Description

Technical Field

[0001] This invention belongs to the field of CNC machine tool reliability assessment technology, and relates to a reliability assessment method based on model optimization. Specifically, it involves the classification of original fault data of CNC machine tools, grouping and processing of fault data, model parameter estimation based on least squares method and maximum likelihood method, model hypothesis testing based on KS method, model optimization based on correlation coefficient method, and reliability assessment. Background Technology

[0002] A reliability model is a prerequisite for reliability assessment. It is based on machine tool failure data and uses mathematical statistics to calculate relevant reliability indicators. Currently, a common approach is to pre-define a uniform distribution model and then estimate and verify the model parameters. However, in reality, a single distribution model is difficult to accurately reflect the different distribution characteristics of machine tool subsystem failure data, leading to significant differences in subsequent reliability assessment results. Therefore, further research is needed on the applicability of distribution models. Summary of the Invention

[0003] The present invention is achieved using the following technical solution, which is described below in conjunction with the accompanying drawings:

[0004] A model-based reliability assessment method for CNC machine tools includes the following steps:

[0005] Step 1: Divide the fault data based on the subsystem partitioning of the CNC machine tool;

[0006] Step 2: Group the fault data using the Stochastic empirical formula;

[0007] Step 3: Model parameter estimation;

[0008] Step 4: Perform model hypothesis testing using the KS method;

[0009] Step 5: Optimize the model using the correlation coefficient method;

[0010] Step 6: Conduct reliability assessment using the mean time between failures (MTBF) with different distribution functions.

[0011] The specific method for step one is as follows:

[0012] Taking into account industry consensus, machine tool structural characteristics, and the functions of each part, CNC machine tools are divided into 11 subsystems, and fault data are divided according to the system to which they belong.

[0013] The specific method for grouping and processing fault data based on the Stochastics empirical formula in step two is as follows:

[0014] (1) Determining the number of groups k

[0015] After obtaining the fault data, to facilitate discussion of the distribution type of fault interval times, it is necessary to first calculate the time interval between every two faults. Secondly, these interval times need to be divided into several groups to better observe their distribution patterns. The group size should be chosen appropriately; it should not be too large or too small, otherwise it will lead to imbalance in the fault data within each interval, thus increasing the fitting error. Based on the Sturgess empirical formula, the number of groups k is determined as follows:

[0016] (1)

[0017] Where, n f This represents the total number of subsystem failures.

[0018] Calculate Then, the number of groups k is rounded up according to the actual situation. Then, the fault interval time is divided into k groups on average, and the number of fault interval points falling in each group is counted. A group statistics table is drawn, as shown in Table 1.

[0019] In Table 1, t max Δt is the maximum fault interval time. i– Let Δt be the left endpoint of the grouping interval. i+ This is the right endpoint of the grouping interval. t i The midpoint of the grouped intervals, n i f is the number of fault data in the i-th group. i Let F be the probability density value of the i-th group. i f is the cumulative distribution function value for the i-th group. i and F i The calculation formula is as follows:

[0020] (2)

[0021] (3)

[0022] Table 1 Fault Data Grouping

[0023]

[0024] The model parameter estimation method in step three is as follows:

[0025] The least squares method is used to estimate the parameters of the exponential, Weibull, normal, and log-normal distributions. For the gamma distribution, the maximum likelihood method is used to estimate the gamma distribution parameters.

[0026] (1) Exponential distribution

[0027] Assume that the fault data of the CNC machine tool subsystem follows an exponential distribution, and the probability density function is as follows: The cumulative distribution function is , where λ is the exponential distribution parameter and t is the failure interval. Taking the logarithm of both sides of the cumulative distribution function, we have .make , Then you can get .

[0028] Suppose there are n data pairs (x i y i (satisfies linear relationship) Based on the least squares method, the coefficients a0 and a1 of the linear relationship can be obtained as follows: (4)

[0029] The relationship between the unknown parameter λ of the exponential distribution and the regression coefficients a0 and a1 obtained by the least squares method is as follows: .

[0030] (2) Weibull distribution

[0031] Assume that the fault data of the CNC machine tool subsystem follows a Weibull distribution, with the probability density function being... The cumulative distribution function is In the formula, β is the shape parameter (β>0); η is the scale parameter (η>0), and γ is the position parameter (γ≥0). Taking the logarithm of the probability density function twice, we get... ,make , , can be obtained .

[0032] The coefficients β and η can be obtained by least squares linear regression:

[0033] (5)

[0034] (3) Normal distribution

[0035] Assume that the fault data of the CNC machine tool subsystem follows a normal distribution, with the probability density function being... The cumulative distribution function is In the formula, σ is the scale parameter and μ is the location parameter. This can be converted into a standard normal distribution. The left quantile of the standard normal distribution is obtained. ,make , Then a linear relationship can be constructed. .

[0036] The coefficients μ and σ can be obtained by least squares linear regression:

[0037] (6)

[0038] (4) Log-normal distribution

[0039] Assume that the fault data of the CNC machine tool subsystem follows a log-normal distribution, and the probability density function is: The cumulative distribution function is In the formula σ l μ is the scale parameter. l These are location parameters. They can be converted to a standard normal distribution. The left quantile of the standard normal distribution is obtained. ,make , Then a linear relationship can be constructed. .

[0040] The coefficients μ and σ can be obtained by least squares linear regression:

[0041] (7)

[0042] (5) Gamma distribution

[0043] Assume that the fault data of the CNC machine tool subsystem follows a gamma distribution, with the probability density function being... The cumulative distribution function is , for Let a be the shape parameter and b be the scale parameter. Construct the likelihood function. For easier calculation, it can be converted to .

[0044] The coefficients a and b can be obtained by solving the following formula:

[0045] (8)

[0046] The model hypothesis testing method in step four is as follows:

[0047] Given n test data points in ascending order, and based on the assumed distribution function, calculate the cumulative distribution function value F(t) for each fault data point. i ), and F(t) i ) and empirical distribution function F n (t i In a comparative study of the values ​​of ), the maximum absolute value of the difference between the two is the test statistic D. n The observed values, the observed values ​​D n Critical value D of the KS test n,α Comparative studies show that if the following conditions are met, the null hypothesis is true; otherwise, the null hypothesis is false.

[0048] (9)

[0049] in F(x) is the assumed cumulative distribution function; F n (x) is the assumed cumulative distribution function. ;D n,α The critical value for the KS test can be obtained from a table.

[0050] The model optimization method in step five is as follows:

[0051] The correlation coefficient method uses the linear correlation coefficient R to represent the degree of linear correlation between measurement data. A larger R value indicates a higher linearity between the measured data and a better linear fit, thus determining the final reliability model. The correlation coefficient R is calculated as follows:

[0052] (10)

[0053] Where x i Let i be the value of the i-th data sample; y is the sample mean; i This represents the i-th value of the fitted model; This represents the mean of the fitted model values.

[0054] For nonlinear relationships, the goodness-of-fit index R is used. NL The calculation formula is as follows:

[0055] (11)

[0056] Among them, y i Let i be the value of the i-th sample data; This represents the mean of the fitted model values.

[0057] The reliability calculation method in step six is ​​as follows:

[0058] Based on the probability density function of the determined distribution model, the point estimate of the MTBF for each subsystem can be calculated. Let the mean time between failures (MTBF) for the five distribution functions be MTBF, ... e MTBF w MTBF n MTBF l and MTBF g The calculation formula is as follows.

[0059] (12)

[0060] (13)

[0061] (14)

[0062] (15)

[0063] (16). Attached Figure Description

[0064] Figure 1 Flowchart for the implementation of this method Detailed Implementation

[0065] The invention will now be described in detail with examples.

[0066] Step 1: Divide the CNC machine tool into subsystems. The division results are shown in Table 2. Then, classify the fault data to obtain the fault information table of the measuring instrument system, as shown in Table 3.

[0067] Table 2: Subsystem and fault data classification for CNC machine tools.

[0068]

[0069] Table 3: Measuring instrument system fault data.

[0070]

[0071] Step 2: The fault data is grouped and processed using the Stochastic empirical formula. Formulas (1), (2), and (3) are used for calculation to obtain the fault data grouping table 4.

[0072] Table 4: Fault Data Grouping Table

[0073]

[0074] Step 3: Based on the fault data of the measuring instrument system, perform parameter estimation using formulas (4), (5), (6), (7), and (8). The obtained parameter estimates are shown in Table 5.

[0075] Table 5: Estimated Parameters of the Measuring Instrument System Failure Interval Time Model

[0076]

[0077] Step 4: Use the KS method to perform hypothesis testing on the model, and use formula (9) to test the d values ​​of different distribution models of the measuring instrument system. i The calculation process is shown in Table 6.

[0078] Table 6: KS Test for Different Distribution Models of Measuring Instrument Systems i Calculation process

[0079]

[0080] Referring to the critical value table for the KS test, when n>35, and the significance levels are 0.01, 0.05, and 0.10 respectively, D... n,α The calculation formula is shown below. The significance level is usually set at 0.05, and the KS test is performed under a 95% confidence level.

[0081]

[0082] The observed values ​​D of the KS test under different models of the measuring instrument system can be calculated using the above formula. n As shown in Table 7.

[0083] Table 7: KS Test for Different Distribution Models of Measuring Instrument Systems n value

[0084]

[0085] Step 5: Calculate the goodness-of-fit index according to Equation (10) to optimize the model. At the same time, use Equation (11) to perform nonlinear goodness-of-fit tests on the gamma distribution and the distribution with good linear goodness-of-fit. The results are shown in Tables 8 and 9.

[0086] Table 8: Goodness-of-Mouth Indicators of Five Distribution Models for Measuring Instrument Systems

[0087]

[0088] Table 9: Goodness-of-fit test of nonlinear fitting of measuring instrument system

[0089]

[0090] By comparing the goodness indices of various distribution models, the goodness index R of the gamma distribution is... NL The value is larger, so the gamma distribution fits better, and the gamma distribution is chosen as the reliability model.

[0091] Step 6: Select formula (16) based on the reliability model of gamma distribution to calculate the MTBF of the measuring instrument system. The calculated value is 662h.

Claims

1. A reliability assessment method for CNC machine tools based on model optimization, characterized in that: Includes the following steps: Step 1: Divide the fault data based on the subsystem partitioning of the CNC machine tool; Step 2: Group the fault data using the Stochastic empirical formula; Step 3: Model parameter estimation; Step 4: Perform model hypothesis testing using the KS method; Step 5: Optimize the model using the correlation coefficient method; Step 6: Conduct reliability assessment using the mean time between failures (MTBF) with different distribution functions.

2. The reliability assessment method for CNC machine tools based on model optimization according to claim 1, characterized in that: The specific method for step one is as follows: Taking into account industry consensus, machine tool structural characteristics, and the functions of each part, CNC machine tools are divided into 11 subsystems, and fault data are divided according to the system to which they belong.

3. The reliability assessment method for CNC machine tools based on model optimization according to claim 1, characterized in that: The specific method for grouping and processing fault data based on the Stochastics empirical formula in step two is as follows: (1) Determining the number of groups k; After obtaining the fault data, to facilitate discussion of the distribution type of fault interval times, the time interval between every two faults is calculated; these interval times are divided into several groups, and the number of groups k is determined according to the Sturgess empirical formula: （1）； Where, n f The total number of subsystem failures; Calculate Then, the number of groups k is rounded up according to the actual situation. Then, the fault interval time is divided into k groups on average, the number of fault interval points falling in each group is counted, and a group statistics table is drawn, as shown in Table 1. In Table 1, t max Δt is the maximum fault interval time. i– Let Δt be the left endpoint of the grouping interval. i+ The right endpoint of the grouping interval; t i The midpoint of the grouped intervals, n i f is the number of fault data in the i-th group. i Let F be the probability density value of the i-th group. i f is the cumulative distribution function value of the i-th group; i and F i The calculation formula is as follows: （2）； （3）； Table 1 Fault Data Grouping 。 4. The CNC machine tool reliability assessment method based on model optimization according to claim 1, characterized in that: The model parameter estimation method in step three is as follows: The least squares method is used to estimate the parameters of the exponential, Weibull, normal, and log-normal distributions; for the case of the gamma distribution, the maximum likelihood method is used to estimate the gamma distribution parameters. (1) Exponential distribution; Assume that the fault data of the CNC machine tool subsystem follows an exponential distribution, and the probability density function is as follows: The cumulative distribution function is Where λ is the exponential distribution parameter and t is the failure interval; taking the logarithm of both sides of the cumulative distribution function, we have ; make , Then you can get ; Suppose there are n data pairs (x i y i (satisfies linear relationship) Based on the least squares method, the coefficients a0 and a1 of the linear relationship can be obtained as follows: (4) The relationship between the unknown parameter λ of the exponential distribution and the regression coefficients a0 and a1 obtained by the least squares method is as follows: ; (2) Weibull distribution; Assume that the fault data of the CNC machine tool subsystem follows a Weibull distribution, with the probability density function being... The cumulative distribution function is In the formula, β is the shape parameter (β>0); η is the scale parameter; and γ is the position parameter. Taking the logarithm of the probability density function twice, we get... ,make , ,have to ; The coefficients β and η are obtained by least squares linear regression: （5）； (3) Normal distribution; Assume that the fault data of the CNC machine tool subsystem follows a normal distribution, and the probability density function is: ， The cumulative distribution function is , In the formula, σ is the scale parameter and μ is the location parameter; this is converted to a standard normal distribution. The left quantile of the standard normal distribution is obtained. ,make , Then a linear relation can be constructed. ; The coefficients μ and σ are obtained by least squares linear regression: （6）； (4) Log-normal distribution; Assume that the fault data of the CNC machine tool subsystem follows a log-normal distribution, and the probability density function is: The cumulative distribution function is In the formula σ l μ is the scale parameter. l These are location parameters; they can be converted to a standard normal distribution. The left quantile of the standard normal distribution is obtained. ,make , Then a linear relationship can be constructed. ; The coefficients μ and σ are obtained by least squares linear regression: （7）； (5) Gamma distribution; Assume that the fault data of the CNC machine tool subsystem follows a gamma distribution, with the probability density function being... The cumulative distribution function is , for Let a be the shape parameter and b be the scale parameter; construct the likelihood function. For easier calculation, it can be converted to ; The coefficients a and b are obtained by solving the following formula: （8）。 5. The reliability assessment method for CNC machine tools based on model optimization according to claim 1, characterized in that: The model hypothesis testing method in step four is as follows: Given n test data points in ascending order, and based on the assumed distribution function, calculate the cumulative distribution function value F(t) for each fault data point. i ), and F(t) i ) and empirical distribution function F n (t i In a comparative study of the values ​​of ), the maximum absolute value of the difference between the two is the test statistic D. n The observed values, the observed values ​​D n Critical value D of the KS test n,α Comparative studies show that if the following conditions are met, the null hypothesis is true; otherwise, the null hypothesis is false. （9）； in F(x) is the assumed cumulative distribution function; F n (x) is the assumed cumulative distribution function. ; D n,α This is the critical value for the KS test.

6. The reliability assessment method for CNC machine tools based on model optimization according to claim 1, characterized in that: The model optimization method in step five is as follows: The correlation coefficient method uses the linear correlation coefficient R to represent the degree of linear correlation between measurement data. The larger the R value, the higher the linearity between the detection data and the better the linear fit, thus determining the final reliability model. The correlation coefficient R is calculated as follows: （10） Where x i Let i be the value of the i-th data sample; y is the sample mean; i This represents the i-th value of the fitted model; This represents the mean of the fitted model values; For nonlinear relationships, the goodness-of-fit index R is used. NL The calculation formula is as follows: （11）。 7. The reliability assessment method for CNC machine tools based on model optimization according to claim 1, characterized in that: The reliability calculation method in step six is ​​as follows: Based on the probability density distribution function of the determined distribution model, the point estimate of the MTBF for each subsystem is calculated. Let the mean time between failures (MTBF) for the five distribution functions be MTBF, ... e MTBF w MTBF n MTBF l and MTBF g The calculation formula is as follows; （12）； （13）； （14）； （15）； （16）。

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