On the life time of die-sets

On the life time of die-sets

Citation for published version (APA):

Buiteman, A. G. M., Doorschot, F., & Veenstra, P. C. (1975). On the life time of die-sets. (TH Eindhoven. Afd. Werktuigbouwkunde, Laboratorium voor mechanische technologie en werkplaatstechniek : WT rapporten; Vol. WT0349). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1975

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Department of mechanical engineering

ON THE LIFE TIME OF DIE-SETS

A,G,H. BUITEMAN

F. DOORSCHOT

P.C. VEENSTRA

REPORT WT 0349

EINDHOVEN UNIVERSITY PRESS (1975)

Division of production technology Eindhoven

Summary

ON THE LIFE TIME OF DIE-SETS

A.G.M. Buiteman, University of Technology EINDHOVEN

F. Doorschot, Centre for Technology, Philips Ltd EINDHOVEN P.C. Veenstra, University of Technology EINDHOVEN(l)

It is shown that the life time of die-sets used in mass-production of sheet metal components follows a Weibullian distribut-ion functdistribut-ion. Hence the behaviour of a die-set performing a given operatdistribut-ion on a given material can be characterised by the Weibull constants nand

S.

From this the number of die-sets to be ordered to manage the production volume N can be determined. Zusammenfassung

Es wird gezeigt dass die Standzahl der Werkzeuge in der Massenfertigung von Blechteilen einer Verteilung nach Weibul1 ent-spricht. Das Verhalten eines Werkzeugs kann deshalb mittels der Weibul1 konstanten n und

S

characterisiert werden.

Erweiterung dieser statistischen Methode erlaubt es die Zahl der Werkzeuge welche in die Fertigung ein zu setzen sind zu bestimmen.

Case-studies in the factory show that the life time of die-sets defined in terms of the number of products made between grindings does not follow a normal distribution.

Noting the distribution function as F(x), being the probability that a tool fails after making x products, the probabil ity of survival is

R(x) = 1 - F(x)

The probability density of F(x) being

d

f(x) "" dx F(x) the rate of failure follows to be

,(x)

=

f(x)

R{X}

(1)

(2)

Observation reveals that both f(x) and ql(x) are monotonously de-creasing functions.

This statement refers to mass production of high precision small components manufactured out of nickel-chromium steel, nickel plated steel, tungsten nickel, nickel iron and copper.

A function describing in general the behaviour as observed is the negative exponential, which in a special form is Weibull's

equation [1,2,3]

F{x}

(4)

The quantity y represents the minimum number of products made. However, as a tool can fail at the very first stroke it evidently holds y-o.

The exponent

B

determines the shape of the function whilst n is a characteristic number to be dealt with later.

With regard to the present problem it is postulated that the ljf~ time of die-sets is controlled by

and hence and F(x) _ 1 - exp { _

(~)8

}

n

(3-1 8 f(x) -..@. (~) exp { - (~) } n n n

B-1

q>(x}

=

~ (~) n n (6)

(])

Arbitrately normalizing the functions on n=l, it is clear that for B=l the failure rate is a constant.

This appears in Nature in radio-active decay.

In the ideal case from the standpoint of manufacturing 8=00 all die-sets would fail after making n products, the characteristic num-ber.

It is easy to prove that all functions F(x) intersect at the point x=n, where holds

F(x}

=

1 - - - 0.632 _e 1 (8)

This in fact is the definition of the characteristic number, being the number of products made at the point of 63.2 % tool failure. Now in order to show that the life time of die-sets obeys a Weibul1 distribution function each die-set designed to perform a particular operation is considered to be an individual member of a large hypothetic family of identical tools.

The behaviour of a limited number of members offers a sample of the overall behaviour of the family thus allowing for the calcu-lation of the cumulative percentage of failure in the whole

According to statistical routine every individual can be listed in a way corresponding to its number of products made and than be labeled by its "plotting position" [4,5,6]

i-0.3

F(xi) = . 0

4 .

100 %

J+ .

where j refers to the number of die-sets considered in the sample and i is the number of a particular tool.

An

actual example is given in table TABLE I

tool no number of plott i n9

products position % 1 131,000 9.46 2 203,000 22.97 3 284,000 36.49 4 363,000 50.00 5 454,000 68.51 6 551,000 77 .03 7 710,000 90.54

Now it follows from eqs. 1 and 5

1

In In l-F(x)

=

B

(In x - In n) (1 0)

being the base for the transformation of observations to the Weibul1 probability chart.

When using the sampling theory the argument in the left hand side corresponds to the plotting position.

Obviously when plotting the number of products x made by the die-sets against its plotting positions a straight line appears if the distribution governing the process is Weibullian.

In this way the data of table I have been plotted in fig. 1 where "age at failurel l _{equals the number of products made due to the}

constant production rate of the presses.

It is clear that the die-set involved suggests a Weibullian be-hav iour.

The Weibull chart allows for estimation of the Weibull constants. Intersection of the experimental curve with F{Xi)

=

63.2

%

gives

c C .2

3!::

_{... co}

I i

• r

_>

.-;::

lID 0 ";'Q.

E

H :::J (J

....

'" A

-5U "'

..

10 5 2

I~on

oint 1 2

- / 3

3 4 5

"

1"-l/

~

~

..,.

'""---

---

--

,""-

--

--

----

-- -

--1/ I

J

I I

~

I I

/

I I •

~V

: I I • I I

·

1

..

Life ti ... in terms of number of products .

-Fig. 1 The data of table I as plotted in the Weibul1 probability chart.

It fo 11 ows

i3

=

1 .8 ,

n

=

450 ,000.

---3

the characteristic number n

=

450.000.

A line originating from the liB-estimation point" perpendiS(ular to

the experimental curve intersects a nomographic scale at 8

=

1.8.

From the chart it is to be expected that 10

%

of the die-sets of

this class will not make more than 130.000 products and that

af-ter making 800.000 products only 6

%

of the tools survive.

In order to get the Weibul1 constants more accurately and as the behaviour of a considerable number of die-sets was to be investi-gated from data collected and recorded in the factory a computer program has been developed :

1. number the die-sets in an increasing scale according to

the number of products made between grindings.

2. calculate the plotting position F(Xi) eq.

9.

3. calculate the values

and

4.

calculate X. _I

=

In x· _I X

=

j l: X· 1 I j

5. perform the lineair regression procedure and calculate

and 6. calculate

~

y.

8

=

~

=

Y o j j - -l:(X·-X)(Y·-Y) 8 "" 1 I I J _ 2 i:(Xj-X) . 1

being a Weibull constant

8₀

n

=

exp (-

s-

+

X)

the characteristic number

j

-L(X,-X)(Y,-Y)

'I

1 I I r =[,-r-J - _ -:-:2j-_ 2J t

E(Xj-X) E(Yi-Y)

1 1

Following this procedure the life time of 18 different types of die-sets has been investigated. The number of die-sets in each category, thus being the sample size, varied between 14 and 68

as listed in table II.

The table shows the Weibull constants characterising the die-sets and the corresponding coefficient of correlation.

When the latter is compared with its critical value connected with the sample size as shown in fig. 2 it may be concluded that life time of die-sets follows very closely a Wei bull distribution.

...

c: .2

-

l1li

!

..

o Col

'0

-

c:

•

:2

-

_-

_!

(J 1.0 10 20 30 --... SaMple .Iza j

Fig. 2 The critical value of the coefficient of correlation as a function of sample size for two different values of proba-bility.

·5

c 0 (I)

.-W "C ... ro ~ c 0 10 0 N U r... III III

...

-

.-(I) ... ... 0 ... III ... 0.. _{- C} - c 10 (I) 0 - 1 0 - 10

.

-

(I) III ::l ... ::l ... ... Q)

-I Q) ..Dill ..Dill 4-r... c.. (I) 0.. .- C . - C Q) r... E

.-

>- (I) 0 Q) 0 o 0 10 0 I- ::J:u ::J:u u u V') 8822-040-2086 shearing 1.072 213.400 0.93 20 3322-060-788 blanking 2.938 77 , 1 00 0.98 35 3322-060-790 blanking 1.430 82,300 0.99 38 3322-060-942 shearing 1.063 169,700 0.99 34 blanking 1.079 206,200 0.99 33 piercing 1.499 175,900 0.96 24 deep drawing 1.264 160,500 0.99 25 notching 1.526 201,900 0.93 24 coining 0.964 250,900 0.98 25 3322-063-752 shearing 1 .256 58,500 1.00 54 pie rc i ng 1. 143 76,300 0.99 50 3322-065-324 shearing 1 .315 278,300 0.98 22 3322-066-918 shear i ng 2.280 66,800 0.98 68 3322-068-850 bending 0.848 557,300 0.98 14 3322-068-766 blanking 2.000 83,250 0.92 41 3322-080-952 blanking 1 . 182 58,000 0.99 38 3322-088-048 sheari n9 0.671 385,400 0.99 38 3322-088-694 blanking 1.274 134,000 0.98 23

6

App 1 i cat ion

Consider the I ife times Xl' X ---- Xn being mutually independant and specimen of a Weibull proBability distribution as shown. The number of products made by the die-set equals

n

Y

=

i:

X.

. • 1 1 1=

( 11)

thus requiring n grindings.

Now the total volume of production N generally outnumbers the grinding capacity of one tool and hence a number of die-sets has to be ordered. It is of major economic importance to find a means to determine the minimum number of die-sets required on the base of a given certainty not to run out of tooling during production.

Introducing a hypothetic die-set allowing for all of the n grind-ings required

n

y

=

1: X.

1=1 - I

(12 )

where ~. represents the average 1 ife time between grindings in a sample6f identical die-sets, the aim of analysis can be

formu-lated to be :

p (

Y )

N )

=

1 - a (13 )

where a stands for the uncertainty accepted to achieve the pro-duction volume

N

with the probabil ity

P

if n grindings are to be app 1 led.

When applying now the theorem of statistics stating that the sum of an adequate number of independent stochastic quantities, all of them following the same distribution law, approximately

fol-lows a normal distribution it holds with respect to

lim n+ oo X

=

~1 + ~2 + ----~n n y n (14 )

=

-~-ll 1 a p ( - - ( a )

= -

f

exp (_!x2) dx a/Vn

r:;;

-00 (15)

In the case of a Weibul1 distribution the average 1 ife time proves to be

7

r

being the Eulerian gamma function,

and

( 17)

the variance of the average life time.

From eq. 15 it is clear that the quantity a determines the sur-face area of a standard normal distribution corresponding to the

uncertainty limit a as accepted.

It can be shown numerica'lly that in the case of n ~ 6 when

writing the left-hand side of eq. 15 like

lim P(g~

a.Jn

_{+ ll)}

n+ oo

and using eqs. 14 and 13 it holds

lln + acr yn - N ~ 0

from which is easily solved

1 [2

V

2

'J

n

= -

(acr) + 2Nll - acr (acr) + 4Nll

2112 ,

( 18)

(19 )

Thus the number of grindings required to produce the quantity N is known when aplying die-sets behaving according to the Weibull constants Sand II and accepting the uncertainty a as a manufac-turing pol icy.

From the design of the die-set and the grinding routine in the factory it is known how many grindings nt can be performed before the tool is finally worn out.

The number of die-sets to be ordered to manage the production N hence equals

T=!!.

n_t (20 )

Based on the theory developed the computer program has been com-pleted. Its output is a number of curves plotting the production volume N against the number of grindings n whilst the uncertainty

a acts as a free parameter as shown in fig.

3.

In practice the method proves to be a great help to the factory management.

9

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-

I

,

-

i 1 - 1

-

1 ""'TIL "5 "i0 55 60 6$ 70 75 AO _"~ 90 ~5 100 ~

The computer output givIng the number of gri nd i ngs n requ ired for a production volume N, the uncertainty a acting as a parameter. "Prod. aantal" N in un its of 1000. "Benodigd santal s 1 i j pbeu rten" n.

1 -;. _{::x 10 % 4 2.5 %}

2 ..,. a 7.5 % 5 a 1 _%

10

References 1. Weibull W.

A statistical distribution function of wide applicability. Journ. Appl. Mech. 18, 293/297, 1951.

2. Johnson N.L., Kotz S.

Continuous univariate distributions. Houghton Miffin Cy. Boston 1970. 3. Johnson L.G.

The statistical treatment of fatigue experiments. Elsevier Publishing Cy. Amsterdam 1964.

4. Tittes E.

Ueber die Auswertung von Versuchsergebnisse mit Hilfe der Weibull Verteilung.

Bosch Tech. Ber. 4, 146/158, 1973. 5 • K i mba lIB. F .

On the choice of plotting positions on probabil ity paper. Journ. Am. Stat. Ass. 55, 546/560, 1960.

6. Mathematisch Centrum Amsterdam Report SP 30.

Different sources

7. Bain L.J., Antle E.C.

Estimation of parameters in the Weibull distribution. Technometrics 9, 4, 621/627, 1967.

8. Menon M.V.

Estimation of the shape and scale parameters of the Weibul1 distribution.