extra~e values. Asimptote distribution of minimQ~ values type

III is called t'Teibull distribution.

803

In order to solve a problem of extreme values a general appli­

cation methodology of G~~el prognosis model is discussed, 14,5,6,71.

Algori~~ of forming GQ~el prognosis model of extreme values

Algorit~~ of Gumbell prognosis model of extra~e values, shown

on Fig.2 consists of following algorithm steps (procedure),la I: .1. Choice of prognosis model, on the basis of interactive dia­

logue user-computer: m =1 prognosis model of HINIMUIvI, m =2 pro-p p gnosis model of MAXIl"IUH;

.2. Reading sample dimensions n~25 and values of discrete acci­

dental variable xk (k=1,2, ••• ,n) of a sample according to acci­

dental distribution;

.3. Arrangig values of accidental variable xk (k=l,2, ••• ,n) of a

sa~ple according to decreasing set (distribution) if m =1 (at p

MINI~ml prognosis) or according to increasing set (distribution)

if m =2 (at l-lAXIHUl'1 prognosis); p

.4. Forming tabular concept of determining ~~pirical points in

extreme and probability paper on the basis of accidental variab­

le value xk (k=1,2, ••• ,n) and probability ¢k(y), that is standa­

rdized variable Yk(k=1,2, ••• ,n);

.5. Esti~ating the center of scattering x and standard deviation

ax of a sample;

-.6. Adopting mean value Yn and standard deviation an in functi-on of sample size n, according to data placed in data set;

.7. Evaluation of value parameters ~ and q of theoretical stra­ight line in extra~e and probability paper;

a 'h i 1 t i ht Ii f th h 1 (1.'f • • Form1.ng t eoret ca s ra g ne 0 e s ape: x=q-a- Y m =1) or x=q+!.y (if m =2); pap

.9. Forming a field of confidence of theoretical model for si­

gnificance-level 5%, on the basis of mean square fault q(yp)

p-th values of standard variable, in function; y, ¢(y) and n,

according to data placed in data set;

804

.10. Determining a degree of agreement e~pirica1 and theoreti­cal model. If empirical model is not sufficiently in agreement

with correspodent theoretical one samples are raade homogenous

by the method of abandoning definite boundary a~pirica1 results

(data) from a sa..-:tp1e and bv foming a sample xk (k==1,2, ••• ,n), StQ.rt

z.

Yes

Fig.2. A1gorith of

<= Mean fa!ua,.e fQu.lt G(Y ... J in funcllon .f p,..ba6;lii,/ and sa.mple ""Ium~ n

of extreme values

so that m<n, and repeating a complete procedure. If after that

empirical model is not sufficiently in agrea~ent with theoreti­

cal one, the dimensions of the sample n are increasing. On the contrary, the fol1m'ling algorithm step is taken •

• 11. Printing output processing results in the fom of a concept

805

of determining a~pirical pOints, theoretical straight line and

the field of confidence of theoretical straight line in extra~e probability paper.

Progr~~ realization of G~~bel prognosis model of extreme values

According to algorithm shmm on Fig. 2 a progr~~ packet is formed

for Gumbel prognosis model of extreme values coded in FORTRAN

77 for compute system EI-Hone~lell 6/95, designed on modulated

principle. Program packet consists of main progrfu~, six general subprogram and t1rlO independent programs for. forming data sets

(one for auxiliary values Yn and an and another one for mean

square fault a(y ),191. The main program controls six general p

subprogra~s, and contains just a choice of Gumbel prognosis mo-

del of extreme values on the basis of interactive dialogue user­

computer, reading sample n dimensions and values of descrete ac­

cidental s~~plevariable xk (k=I,2, ••• ,n) according to accidental

schedu~e and calls for subprogr~s. Each general subprogra~ ma­

kes one independent unit, that is one mode, which is not bigger

than one page of the list.

Estimate of up tIDe minm.u.~ and dot,m time maxLrnu.T!t of building machines

Bulldozer TG-50 is a typical machine systa~ represented by com­

ponents and relation fu~ong them. Structural and functional link

a~ong components provides criterion function in correspondent

exploiting conditions and in a planned time. The ability to pe­rforn criterion functions in that exploitation period is called bulldozer OPERATING ABILITY. Only until bulldozer is in its ba­sic state UP TIr,lE criterion function is provided. However, du­

ring the phase of real exploitation bulldozer is exposed to di­fferent influances according to the intensity, direction and character, what leads to deviation from the level of criterion

function, that is, to reduction of bulldozer operating ability.

Bulldozer is in the second basic state-OOlm TIME if the necess­

ary interdependence of components in the system is disbalanced,

as well as their characteristics and significance. In that case

it is necessary to return the system in the state UP TIME by

corrective maintenance or repair. Basic states, as a quantita­

tive indicator of bulldozer TG-50 functional quality in the sh­

ape of t~e picture of states are given on Fig.3 131.

806

During real exploitation of bulldozer TG-50 E'.a.nufactured in "14

October" Krusevac/Yugoslavia industry, nachine stroke is identi­

fied as a critical unit. Time picture of the state in the shape

~uPtime:::::::-----.-__

1TTT1TIIIIllImmlfftl1rTfTnl~11111111 ~ t [time U'litj

cown time

Fig.3. General time picture of systen state

of alternating intervals UP TIME II03h I; DO~'J1'1 TDlE I h I of machine

stroke (Table 1).

Table 1. Time picture of the bulldozer TG-50 machine stroke state -1

Up time [10 3hJ ,,~14 2,001 1,8o~ 0,.81 f,~58 2,115

Down time [h J 2,30 ~,oo '!,4s Z,~o 3,05 -f,~5

2 Up time [103h J 2,2" 1158'f fz,G3(o 3,'1'':15 f,'f?li 11°12-

Down time [h 1 2180 2,50 2,00 0,80 2,30 2,10

3 Up time lfOlh 1 2,H3 2,~03 3,'f01 2,880 2,SH ','!l1 Down time [h] 2,40 4,90 -1,00 ?I,I5 2,25 3,10

if Up -time [1O Sh J 2,S'f4 3,45~ 2,422 3,153 ~,'Ja 1,421

Down time [h] 2,50 -1,50 '?I,la 2,305 1,1S '!,25

5 Up time [10"h] 2,!J'1 2,''''2 4,250 1,2" ',34'f 2)022-

Down Time [h J 2,'0 2,05 ?115o 2,35 -1,20 2)45

6 Up time f101h J 1,552 1,84'" 2,'05 2,521 /,-153 Down time f h ] 2,lo 2,15 3,25 1,"fo 2,40

Processing empirical data on the cO:!lputer EI-EoneY'.;ell 6/95,

according to developed prograr:l packet on FORT?~lI.2\T 77, prognosis

model of ~iINIHU"'l UP TLiE of bulldozer r:lachine stroke is forr:led

(Fig.4). Erepirical distribution with probability 95% correspo­

nds to theoretical straight line in extrene and probability

paper: X=2,6469-0,7449·Y a.ccording to this model it is possib­

le to deternine reliability of lU~TL'-lU:'l UP TElE realization to

breakc10vln machine stroke and to define quantitatively a part of

the sru~ple where those conditions are not applied. Bulldozer

TG-50 vTill be UP TP,m Nith 90% of probability: 0,9706 moto h

970,6 h, \'Thich does not apply to every tenth datum in the sa­

mple size N=35. The prognosis does not apply to any of 31 data

in the exa.mined sample.

prognosis model of :'L~.xr-m;'l DONN THm of bulldozer TG-50 (Fig. 5),

with 95% interval of confidence has the shape X=1,9735+0~Ol6·Y.

According to this model, with probability of 90%, maxinwu ti3e

of corrective maintenance Viill be 3,327 h, it does not refer to

the sa:-ne quantitative sai"lple part as \.,rell as vlith the model of

minimwn prognosis •.

,!. U1

.!. o

I

P U1

o

o U1

,... C>

!" U1

0..02 - -o.o.~

0.08 0.10 0..15

0..20. --D.25 --0..30 0..35 O~o. -.-O.~~

o..51 0.5. O.b(

0.6

0.70

0..75

0.00

0.85

068 -/ 0..90 .

0..92

o.9~

0.9 V

0.96

0.97

0..98

'& ~

'< '< ~

.- - -f -i -. i -- =1

-I --- - -I -- -I---

-i- 1--

~ - - -I

~I- -.--I-I---j

1-- -i-I-

-1- -7 -i-V-I -i,L I-'T, A I

-~ -I-T-/ i

-V 0

7 . I-- .

II " "-x

" N

0-~

'" .a I

C>

:.... 0 ~ ....

~ -< --

... o

.;-.

J

!IO II 0

V II. I II

~: II I

.:.v. :"/ /

II --

-./ ..- . +.

/ I 0

0

II 0 I' ,oJ I

V 1-

. -- . -- .

807

.... .... 0. i.n

1.0.2 0

lD4

1.10.

- 1.25 - -

-'-- -1- 1.50

2.00.

250.

lOo.

4.0.0.

5.0.0.

6.0.0. 7.00

--' 8.00. 9.0.0.

.l- moo. --- I- 12.0.0.

15.00 -I-

18.0.0 -- I- 2000

--I- 25.00

30.00

40.00

- 50.00

t-3 ~

x ~

Fig.4. Variability dependence and HINIl1UH UP TIME of bulldozer TG-50 machine stroke

808

I P U1

<:>

0.02 -.--0 0..0.4 -- -I- -~ O.nB - - --1-0..10. -.- -1- 0 0..15 -- - - --\ r\. 0..20 .- - - - 1- -,-0..25 -i-0.3 1\ 0.35 - --0.40 - -- - -0..4 -- ---n~ - - ---0..5 0..6C -- --OJ)

0..71 -- --I-

0.7! - - - - ---I-

o.eo-0.85 -- --

0.66 --- --0.90 . -- .

0..92 -- -.-- -0..94 -- -- -- --I-

0..9~ -.- . - - -- -i-

0.96 -- - .- - --

0.97 -- ---

098 - -- -- - -1-1-

"&

'< q ~

DOWN TIME. (hI NNNUI!-"LLlUI

~~~gt;;~~ It

I !

0 f\ 0

1\ 1\ 1\ ~\

1\ \ 0 o '\

\ 1\ '\ ~ 1\

i\ f\ " \ 0r\ \ i', '\

~ \ \0 I~

1\ '< ! i\ \

. ,.... ~- . ~-~ . r- .

II 1\ x -" -\

i-- ;(;

\ .... -w U1 . <:> g c;: -< 0

I

><

1.02

1.0.4

1.10

1.25

1.50

2.00

2.50

3.0.0.

4.0.0

5.00.

6.00. 7.00 B.o.O 9.0.0. 10.0. o

0.

00

o o

12.0

15.

18.0. 200.

'\ 25 00

00 30..

40.0

SO.

o 00

>-:3 ~

>: ~

Fig. 5. Variability dependence and MAXI:'lUM DOIVN TIME of bulldozer TG-50 machine stroke

809

Conclusion

Esti::,ate of reliability c;laracteristics and r:1aintenance of mac­

hine stroke as a critical subsyste@ of bulldozer TG-50 is done

on the basis of ti~<1e picture of state from exploitation.For the

r~odel of asinptote distribution of extrep..e values type I (Gurr.bel

distribution) in progracQ language FoaTAAN 77 a progran packet is

formed 'dhich is used to process e:-apirical data by computer. Ob­

tained results nake the prognosis of ~UNElU>i UP TDIE values and

HA..'GllU:i Dmm TIm: values possible, "lith corresponding probabili­

ty, as ':,ell as a quantitative part of a grou? of data the prog­

nosis stands for. The created prognosis Dodel makes objective

limiting of continuous Hork and ninimizing the t:L-ne of correc­

tive 'c'.aintenance TG-50 bulldozer possible.

References

1. Gu .. -nbel,E.J.: Statistics of Extre!'les, ColtLnbia University press, New York, 1962.

2. GalaC'bos,J.: The Asy::'.ptotic Theory of Extrel1\e Order Statisti­cs; John 11iley and Sons, Ne"" York, 1978.

3. Todorovic J., Zelenovic D.: Syst~~ efectivity in mechanical engineering (in Serbo-croat), Naucna knjiga, Belgrade, 1981.

4. LisoHski, Z.: J..pplication of Gumbel theory \·,hile estiillating rail vehicles reliability, Zagadnienia tarcia zazicia i srna­rO':Taniil No.10, )I/arsha':" 1972.

5. LisO\·,ski,Z.: Freitting, pitting and spalling as wear forms of rail vehicles elenents, Zagadnienia eksploataciji maszyn No.4, i'Jarsha'il, 1972.

6. Papic,L.: Possibilities of applying Gumbel prognosis model of extreme values (in Serbo-croat), SY:'l-OP-IS '87, Herceg Novi, 1987.

7. Papic,L.: Reliability analysis of complex systems by apply­ing extreme values statistics (in Serbo-croat), SIPT '88, Cavtat, 1988.

8. Dasic P., Papic L.: Computers application of G~~bel progno­sis model of extreille values (in Serbo-corat), Racunarstvo u nauci i obrazovanju No.3., Belgrade, 1988.

9. Dasic,P.: FORTRAN 77 in the field of production engineering, part II (in Serbo-croat), Krusevac, 1988.