Probability distribution of machining center failures

Reliability Engineering and System SaJety 50 ( 1995 ) 12 l - 125 ELSEVIER

0951-8320(95)00070-4

© 1995 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0951-8320/95/$9.50

Probability distribution of machining center failures Jia Yazhou, Wang Molin Mechanical Engineering Department, Jilin University of Technology, Changchun 130025, P.R. China

&

Jia Zhixin Mechanical Engineering Department, Harbin Institute of Technology, Harbin 150001, P.R. China (Received l0 April 1993; accepted 23 June 1995)

Through field tracing research for 24 Chinese cutter-changeable CNC machine tools (machining centers) over a period of one year, a database of operation and maintenance for machining centers was built, the failure data was fitted to the Weibull distribution and the exponential distribution, the effectiveness was tested, and the failure distribution pattern of machining centers was found. Finally, the reliability characterizations for machining centers are proposed.

tracing record, we searched for the rules and distribution of failure of the machining centers, analysed the characteristics in estimating the reliability of machine tools, provided the basis for more research on the reliability of CNC machine tools.

1 INTRODUCTION The CNC machine tool is the main piece of equipment for m o d e r n metal machining. It is the basis of the flexibile manufacturing system (FMS) and the computer integrated manufacturing system (CIMS), and plays an increasingly important role in manufacturing automation. A CNC machine tool which has cutter-base and manipulator, and can change cutter automatically, is called a machining center. It has a higher automatic degree and is the main type of CNC machine tool. With the completeness of the machining center's function, its constitution is getting more and more complex, and the failure probability in running also increases. So the problem of reliability becomes more and more important. The high automation and complexity of machining centers has strengthened the necessity and urgency of improving their reliability. However, the research for the reliability of machining centers has been very rare until now. Although most machining centers fail frequently whilst running, little is known about the reasons and rules of failures, and thus the efficient automatic equipment cannot work well. Under the hot competition in the machine tool market, it appears more urgent to study the reliability of m o d e r n machine tools, t'2 In this paper, through the field

2 DATA COLLECTION Data collection is the basis of failure analysis. The more detailed and truly the data are collected, the more accurately the rules and reasons of failure are analysed. Reliability data come mainly from field experiments and experiments in the laboratory. As for CNC machine tools, which are very large and expensive, the data should come mainly from workshops. 3 Through field tracing records for 24 machining centers over a period of one year, we primarily built a database of operation and maintenance for machining centers. The database includes the following information: (1) numbering of machine tools (3) type of machine tools (5) the customer (7) date of putting into operation 121

(2) name of machine tools (4) manufactory (6) report number (8) hour-meter reading

122

Jia Yazhou, Wang Molin, Jia Zhixin

(9) down time (11) failure part (13) failure effect.

(10) failure codes (12) repair time

given by F(t)=l-exp[-(O)

The above information is only part of the needed message. The database is primary. It can be expanded, with more information added, and it provides basic data for studying the reliability of machining centers. 3 FAILURE

DISTRIBUTION

FITTING

Based on the database of operation and maintenance, one can analyse the rules of failure distribution. As known, the Weibull distribution is most useful for the lifetime test, 4 its cumulative distribution function

No. 1 26"3 1240.9 2905-8

153"2 1396"0 3047'2

No. 2 76"3 136-2 628"2 744-2 1886.0 1929"9 2829"2 2877"0

= 3478.5(h) 283.4 355-5 898-0 943-1 2464.3 2513.8 3354.7

Here no. is the numbering of machining centers, r is the failure number. To represents the end time of equipment test, the data in the table is the time of failure occurred (i.e. h o u r - - m e t e r reading). Rewrite eqn (1) and take the logarithm, then 1

In I n - 1 -

- b l n t - b In 0

504.9 1901.8 3664.0

569.9 2160.6 3697.0

763-9 2832.1 3744.4

429.0 1377.9 2576.0

492.2 1573.9 2649.2

566.2 1792.1 2759.0

The linear regressive equation is given by f = A + Bx

(2)

F(t)

i -0.3 r+0.4 .

(xi

for .

i=1,2, . .

(4)

where A and B are estimators of the linear regressive parameters. Parameter B is given by

B =

Usually one can use median rank to evaluate F(t) approximately, as given by Ref. 4: F(t~)-

(1)

where b is the shape parameter (Weibull gradient) and 0 is the characteristic life parameter. Since the Weibull distribution is a series of functions, if b -- 1, it's the exponential distribution; b = 2 , it's the Rayleigh distribution; b = 3.57, it's very similar to the Gauss distribution. At first, we filled the field failure data with the Weibull distribution. In this paper, we have fitted the data of 24 machining centers with the Weibull distribution respectively. We give the failure data of 2 machining centers to illustrate the fitting procedure.

r = 27 To = 3894(h) 201"8 247.0 340.9 407.3 1446"2 1494.4 1534.4 1867.8 3126"7 3207.0 3300.2 3418.4 r = 32 To 174-5 221.2 775-6 842.8 1983-6 2403.6 3187"7 3308.7

h]

. r.

-

-

y)

i=l

, (x,

(5)

-

i=1

(3) In (t)

Then make the Weibull probability paper according to eqn (2) and plot the couples

-2 I

99.9

-1 I

0 I

1 I

2

3 I

2

63.2 --

1

0

(ln ti, In In 1 _~:(ti) ) •

-1~' k~

10.0 Figures 1 and 2 show the results of plotting the failure data of the above two machining centers on Weibull probability paper; the lines are fitted using the linear regressive method. 5 This part of the work was done by a computer. Using a special software package, the distribution coefficient, the linear regressive parameters and the correlation parameters for statistical testing are worked out.

-z _'

f.h

-4 1,0-

-5 -6

0.1

O. 1

I

* r I l lltl

I

I

I I

1.0

Ill

10.0

I

I

I

I

-7

100.0

t (x 1 0 0 )

Fig. 1. Weibull distribution of the failure data of the no. 1 machine center.

123

M a c h i n i n g center f a i l u r e s

Table 1. Linear regressive parameters

In (t)

99.9

-2 I

-i I

63.2 -

10.0 - -

y 0

1.

2

3

4

1.0--

I

0.1

0.1

I

I I I IIII

i

I

2

No.

A

B

0

S

S,,

0

1 2

-7-165 -8.241

0-939 1"126

2059-434 1690.071

0.976 0"985

0-487 0-448

-1

I I I Jill

1.0

~

I

10.0

I I IIII

-

-1~.

-

R, -2 - -

-

-3

-

-4

-

-5

-

-6 -7

100. 0

t (x 1 0 0 )

Fig. 2. Weibull distribution of the failure data of the no. 2 machine center. here, £ and )7 are the means of variables x~ and y~, that is: .~ = -

Xi

ri_l

Table 1 shows the results of applying linear regressive processing to the failure data of the previously mentioned two machining centers. From the data in the table we can see that ISI > S., so the effectiveness of linear correlation is very significant. We fitted the 24 machining centers in the Weibull probability paper, and each time we got ISl > S., that is to say, the failure distribution of machining centers fits the Weibull distribution. Meanwhile, the values of shape parameter b are all very near to 1, this means that this type of distribution perhaps fits the special case of Weibull distribution--the exponential distribution.

1

y= -~,

ri-1

Yi

4 THE EXPONENTIAL

DISTRIBUTION

TEST

and parameter A is given by (6)

A : f - B£.

The goodness coefficient: 5

of fit is evaluated

by correlation

From eqn (1), we can see that if the shape parameter b = 1, the biparameter Weibull distribution is simplified to the single-parameter exponential distribution. The distribution function is: F ( t ) = 1 - exp -

(Xi -- £ ) ( y i - f )

(11)

i=1

s -

(7)

[ ~ (xi- £)2 i=, ~ (Yi- y)2] Note that the absolute value of correlation coefficient ISI < 1. The larger the value of ISI, the more significant the linear relation is. T h e r e is a critical value of correlation coefficient S,~ according to each significance level a respectively, which depends on the value of freedom ( r - 2 ) only. For a =0.01, the critical value of the correlation coefficient can be obtained by 2.576 & - Vv + 3

(8)

where y = r - 2. In general, if ISI > S~, one can draw the conclusion that the linear relation is significant. Rewriting eqn (2) gives y = b x - b In 0.

(9)

Comparing eqn (9) with (4), we obtain b = B,

0 = exp(-A/B),

(10)

where the Weibull distribution parameters b and 0 can be found through linear regressive parameters A and B.

where 0 is the distribution average, i.e. the mean time between failures (MTBF), its reciprocal A is called the failure rate, that is, A = 1/0. Rewriting eqn (11) and taking the logarithm gives In - -

1

1 - F(t)

At.

(12)

F(ti) is evaluated by eqn (3), and each couple

/,, ln / is plotted in the rectangular coordinates, the lines fitted out by the linear regressive method. Figures 3 and 4 show the results of fitting the data of the previously mentioned two machining centers with the exponential distribution. The linear regressive parameters are listed in Table 2. A and B in the table are parameters of linear regressive equation. B is equal to A of the exponential distribution; parameter 0 = 1/A = 1 / B , the meaning of S and S~ are similar to Table 1. From Table 2, we can also get ISl>S~, so the effectiveness of linear correlation is very significant. That means the failure distribution of machining centers fits the exponential

Jia Yazhou, Wang Molin, Jia Zhixin

124

Table 3. Goodness of fittest

3.0

No.

a

.v~

Ze

x,,

1 2

0"1 0.1

37"8805 46-3629

66-0885 77-6998

71.8091 83-3267

2.5-

2.0

;-

1.5-

Z 2 >Z~ ,/2(2r), reject the hypothesis, as the failure = 1.0 -

rate is most p r o b a b l y decreasing. F o r c~ = 0.1, a test for t h e g o o d n e s s of fit for the failure data of the machining centers gives the results shown in Table 3, where x t = Z2,/2(2r), x,, = Z~ ,/2(2r). F r o m Table 3 we can see that test statistics fall b e t w e e n two fractiles of the chi-square distribution:

~-

0.5 0 - "d ~ •" / f I • 0 1000

I 2000

I 3000

I 4000

t (h)

Fig. 3. Exponential distribution of the failure data of the no. 1 machine center. 3.0

2, .g

.5/

Z~/2(2r) < Z 2 < Z~ ,/2(2r). T h e test for the o t h e r machining centers all fit the a b o v e inequality. So we can say, with confidence, that the failure distribution of machining centers fits the exponential distribution.

2.0

5 CONCLUSIONS

1.5

W e have shown that the failure pattern for machining centers fits the exponential distribution. Based on it, one can estimate the reliability. T h e selection of reliability characterizations of e q u i p m e n t d e p e n d s on the rules of failure distribution. T h e m e a n time b e t w e e n failures ( M T B F ) is one of the most used characterizations, its m a t h e m a t i c a l definition being the expectation of the running time T b e t w e e n failures. Its probability density function due to the exponential distribution is given by

1.0 0.5 0 0

1000

2000

3000

I 4000

t (h) Fig. 4. Exponential distribution of the failure data of the no. 2 machine center.

f ( t ) = Ae a,: distribution with constant failure rate. T a k i n g a further step, we test the g o o d n e s s of fit for the a b o v e conclusion. A c c o r d i n g to the e q u i p m e n t reliability test I E C 605-6, 6 the test statistic is

1

its m a t h e m a t i c a l expection is defined by: v

E(T) =

tAe A' dt = 1/A.

(15)

E q u a t i o n (15) is used to estimate the M T B F , i.e. M T B F = 1/A.

.£, T* Z 2 = 2 ~.~ In - i

(14)

(13)

~t"

w h e r e r is the total failure n u m b e r , T* is the total r u n n i n g time of the e q u i p m e n t , 77, is the cumulative r u n n i n g time at the ith failure. If ZZ,/2(2r) and Z~ ,/2(2r) are set to be the fractiles of the Z 2 distribution whose f r e e d o m is 2r and significance level is a, then if Z 2 < Z2/2(2r), reject the hypothesis, as the failure rate is m o s t p r o b a b l y increasing; and if

T h e failure rate A can be estimated by r

Ai

(16)

1

where r is the sum total of failure of all machining centers; ti is the actual r u n n i n g time of the ith machining center during inspection: n is the n u m b e r of tested machining centers. So, the m e a n time b e t w e e n

Table 2. Linear regressive parameters of the exponential distribution fitting

No.

A

B

0

S

S.

1 2

-0.177 -0.202

0-0000627 0-000765

1594-969 1306.982

0.897 0-913

0.487 0-448

Machining center failures failures can be estimated by 1 M T B F --- - ~ ti

(17)

ri=l

Equation (17) can be developed only if the failure rate is constant, i.e. in the exponential distribution case. So the study on the rules of failure distribution of machining centers provides the basis of reliability estimation.

REFERENCES 1. Yazhou, J. & Zhixin, J., Fatigue load and reliability design of machine-tool components. Int. J. Fatigue, 15

125

(1993) 47-52. 2. Yazhou, J., Sense of urgency for enhancing the reliability of machine tools. Machine Tools, 4 (1992) 27-30 (in Chinese). 3. Keller, A. Z. & Kamath, A. R. R., Reliability analysis of CNC machine tools, Reliab. Engng 3 (1982) 449-473. 4. Kapur, K. C. & Lamberson, L. R., Reliability in Engineering Design, John Wiley & Sons, NY, 1977, pp. 291-295. 5. Walpole, R. E. & Myers, R. H., Probability and statistics for Engineers and scientists, Macmillan Publishing Co. Inc., NY, 1978, pp. 280-308. 6. Test for the validity of a constant failure rate assumption. Part 6, Amendment 1, IEC 605-6 (1986), 1989. 7. Billinton, R. & Allan, R. N., Reliability Evaluation of Engineering Systems, Plenum Press, London, 1983, pp. 149-156.