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Univerzitet u Kragujevcu MASINSKI FAKULTET U KRAGUJEVCU KATEDRA ZA PROIZVODNO MASINSTVO

Kragujevac Srbija

31 SA VETOVANJE PROIZVODNOG v

MASINSTVA SRBIJE I eRNE GORE v v

SA MEDUNARODNIM UCESCEM

3lh CONFERENCE ONPRODUCTION ENGINEERING OFSERBIA AND MONTENEGRO

WITH FOREIGN PARTICIPANTS

~6 ~ YiCrllJ~

ZBORNIK RADOVA

PROCEEDINGS

Kragujevac 19 - 21 septembar 2006

31 SAVETOVANJE PROIZVODNOG MASINSTVA

SRBIJE I eRNE GORE v v

SA lAEDUNARODNIl1 UCESCElt Kragujevac 2006

ISBN 86-80581-92-5

ZBORNIK RADOVA

Urednici Editors

Prof dr Bogdan Nedic dipl ing Prof dr Milentije Stefanovic dipl ing Prof dr Miodrag Lazie dipl ing

Izdavac Publisher

MaSinski fakultet 34000 Kragujevac Sestre Janjic 6

Za izdavaca For publisher

Prof dr Radovan Slavkovic dekan Fakulteta

TiraZ Circulation

200 primeraka

Stampa Printed by

Graficki atelje SKVER Kragujevac

Copyright MaSinskifakultet Kragujevac 2006

Izdavanje Zbornika radova organizovanje i odr~vanje 31 Savetovanja proizvodnog m~instva Srbije i Crne Gore podr~lo je

Ministarstvo nanke i zanite zivotne sredine Republike Srbije

IT


SRBlJE I eRNE GORE SA MEBUNARODNM UCESCEM Kragujevac 2006

NOSIOCI ORGANIZAClJE ORGANIZING INSTITUTIONS

Zajednica naucno istraZivackih institucija proizvodnog mMinstva Srbije i CmeGore

MaSinski fakultet Beograd MMinski fakultet Ni~ Masectinski fakultet Kragujevac MMinski fakultet Podgorica Institut za proizvodno mMinstvo FTN Novi Sad Institut za industrijske sisteme FIN Novi Sad Tehnicki fakultet Cacak MMinski fakultet Kraljevo LOLA Institut Beograd MMinski fakultet Pristina

ORGANIZATOR ORGANIZERS

UNlVERZITET U KRAGUJEVCU MASINSKI FAKULTET KATEDRA ZA PROIZVODNO MASINSTVO Sestre Janjic 6 34000 Kragujevac Tel +381 (34) 335-990 Fax +381 (34) 333-192 web httpwwwmfkgkgacyu email mfkgmtkgkgacyu

MESTO ODRZAVANJA SYMPOSIUM VENUE

Kragujevac Hotel Sumarice 19 - 21 septembar 2006

POKROVITELJI SAVETOVANJA

Ministarsvo nauke i zasecttite ~ivotne sredine Repuhlike Srbije Grad Kragujevac Zastava automobili Kragujevac

III

1~ Prof dr r1ilisav Kalajdzi6 tv1aSinski fakultet Beograd 2 Prof dr Ljubodrag Tanovic M~inski fakultet Beograd 3 Prof dr Pavao Bojani6 Ma~inski fakultet Beograd 4 Prof dr Dragan Milutinovie Masinski fakultet Beograd 5 Prof dr Ilija Cosie Fakultet Tehni~kih nauka Novi Sad 6 Prof dr Dragoje Milikic Fakultet Tehni~kih nauka Novi Sad 7 Prof drVeIimir Todic Fakultet Tehnickih nauka Novi Sad 8 Prof dr Velibor Marinkovic M~inski fakultet NiS 9 Prof dr Dragan Domazet M~inski fakultet Nisect 10 Prof dr Milentije Stefanovic Masectinski fakultet Kragujevac 11 Prof dr Miodrag Lazi6 Masectinski fakultet Kragujevac 12 Prof dr Nedic Bogdan M~inski fakultet Kragujevac 13 Prof dr Slavko Arsovski M~inski fakultet Kragujevac 14 Prof dr Ratomir Jecmenica Tehni~ki fakultet Cacak 15 Prof dr Snefana Radonjic Tehni~ki fakultet Ca~ak 16 Prof dr Ljubodrag Dordevic M~inski fakultet Kraljevo 17 Prof dr Ljubomir Lukic Ma~inski fakultet Kraljevo 18 Prof dr Miomir Vukasojevi6 M~inski fakultet Podgorica 19 Prof dr Milan Vukcevic M~inski fakultet Podgorica 20 Dr Mirko Dapic LOLA Institut Beograd 21 Dr Vladimir Zeljkovic LOLA Institut Beograd

1 Prof dr Vladimir Mila~ic Ma~inski fakultet Beograd 2 Prof dr Joko Stanic ~inski fakultet Beograd 3 Prof dr Mileuko Jovicic M~inski fakultet Beograd 4 Prof dr Svetislav Zarie M~inski fakultet Beograd 5 Prof dr Dragutin Zelenovic Fakultet tehn nauka Novi Sad 6 Prof dr Sava Sekuli6 Fakultet Tehnickih nauka Novi Sad 7 Prof dr Ratko Gatalo Fakultet Tehnickih nauka Novi Sad 8 Prof dr Predrag Popovic M~inski fakultet Nis 9 Prof dr Brauko Ivkovic M~inski fakultet Kragujevac 10 Prof dr Branislav DevecUic M~inski fakultet Kragujevac 11 Prof dr Dusan Vukelja M~inski fakultet Kragujevac 12 Prof dr DuSan Vukelja M~inski fakultet Kragujevac 13 Prof dr Sreten Urosevic Tehnicki fakultet Cacak 14 Prof dr Vucko Mecanin M~inski fakultet Kraljevo 15 Prof dr Vuko Domazetovic M~inski fakultet Podgorica

IV

1 Prof dr Milentije Stefanovie M~inski fakultet Kragujevac 2 Prof dr Bogdan Nedie Masectinski fakultet Kragujevac 3 Prof dr LazieS Miodrag M~inski fakultet Kragujevac 4 Prof dr Slavko Arsovski M~inski fakultet Kragujevac 5 Prof dr Ratko Mitrovic M~inski fakultet Kragujevac 6 Doc dr Lazi6 Yuki6 MMinski fakultet Kragujevac 7 Doc dr Yesna Mandie M~inski fakultet Kragujevac 8 Prof dr Mo Spasic Masectinski fakultet Beograd 9 Prof dr Milosect Glavonjie M~inski fakultet Beograd 10 Prof dr Miroslav Pilipovie M~inski fakultet Beograd 11 Prof dr Ljubodrag Tanovic Masectinski fakultet Beograd 12 Prof dr Miroslav Plancak Fakultet tebnickih nauka Novi Sad 13 Prof dr Dragoje Milikic Fakultet tehnickih nauka Novi Sad 14 Prof dr Yelimir Todic Fakultet tehnickih nauka Novi Sad 15 Prof dr Miroslav Radovanovie Masectinski fakultet Nisect 16 Prof dr Dragan Domazet Masectinski fakultet Nisect 17 Prof dr Yelibor Marinkovic M~inski fakultet Nis 18 Prof dr Ljubodrag fgtordevic Masectinski fakultet Kraljevo 19 Prof dr Ljubomir Lukic Masectinski fakultet Kraievo 20 Prof dr Ratomir Jecmenica TehniCki fakultet Cacak 21 Prof dr Snefana Radonjic Tebnicki fakultet Cacak 22 Prof dr Milan Yukcevic Masectinski fakultet Podgorica 23 Prof dr Miodrag Bulatovic Masinski fakultet Podgorica 24 Dr Mirko fgtapic LOLA Institut Beograd 25 Dr Vladimir Zeljkovic LOLA Institut Beograd 26 Dr Radovan Kovacevic Herman Brown Chair Professor Southern Methodist

University Dallas Texas USA 27 Prof dr Mirko Sokovic Faculty ofMechanical Engineering Ljubljana Slovenia 28 Prof dr Ostoja Miletic Masectinski fakultet Banja Luka RS Bosna i Hercegovina 29 Prof dr Yiktor Taranenko ITSI Politechnika Lubelska Lublin Poljska 30 Prof dr Himzo fgtukic M~inski fakultet Mostar Bosna i Hercegovina

1 Prof dr Milentij e Stefanovic MFK predsednik 2 Prof dr Bogdan Nedic MFK podpredsednik 3 Mr Slobodan Mitrovic MFK tebnicki sekretar 4 Prof dr Slavko Arsovski MFK 5 Prof dr Miodrag Lazic MFK 6 Prof dr Branko Tadic MFK 7 Prof dr Goran Deve(ffic MFK 8 Doc drYesnaMandic MFK 9 Doc dr Srbislav Aleksandrovic MFK 10 Doc dr Yukic Lazic MFK 11 Doc dr Dragan Adamovic MFK 12 Doc dr Miladin Stefanovic MFK 13 Mr Milan Eric MFK 14 Mr Nada Ratkovic MFK 15 Dr Miljko Kokic Grupa Zastava vozila

v

PREDGOVOR

Prvo Savetovanje proizvodnog masinstva Jugoslavije odrZano je u Beogradu 1965 na inidjathru prof de Viadimira ~olaje kadaje i formirana Zajednica naucno-istrazivackih institucija proizvodnog mMinstva koju su sacinjavali masinski fakulteti i istraZivacki instituti iz skoro svih republiCkih centara tadMnje drZave Zajednicu proizvodnog maSinstva sca u vreme pripreme Savetovanja saciIiavaju MaSinski fakultet u Beogradu MaSinski fakultet u NiSu MaSinski fakultet u Kragujevcu MaSinski fkaultet u Podgorici Institut za proizvodno maSinstvo FTN iz Novog Sada Institut za industrijske sisteme FIN iz Novog Sada Tehnicki fakultet u Cacku MMinski fakulet u Kraljevu LOLA Institut u Beogradu i MaSinski fakultet u Pristini

31 Savetovanje proizvodnog mMinstva SCG odrZava se u Kragujevcu u organizaciji Katedre za proizvodno maSinstvo MaSinskog fakultet u Kragujevcu Prethodna Savetovanja u Kragujevcu su odrZana 1969 (5 Savetovanje) i 1985 godine (19 po redu)

I ovo Savetovanje kao i nekoliko prethodnih odrZava se u vreme intenzivnih druStvenih promena macajnih za sire aspekte proizvodnog maSinstva Vlasnicka transformacija i oZivljavanje privrede u proizvodnim oblastima posebno u metalopreradivackoj industriji na samom je pocetku Privatizacija i pokretanje proizvodnje u velikim industrijskim sistemima sprovodi se sporo i necelovito Prema drustvenim planovima zavrsetak transformacije u ovoj oblasti se najavljuje za kraj 2007 kaela bi trebalo ocekivati i macajnije pokretanje prizvodnih delatnosti

Na ovom Savetovanje organizovanom za samo godinu dana bice izlozeno 120 radova autora iz Srbije i Crne Gore i inostranstva (Ukrajina Slovacka Poljska SAD Slovenija Bosna i Hercegovina Hrvatska Makedonija) Aktivnosti na Savetovanju ce se obavljati u viSe sekcija koje obuhvataju sledece tematske oblasti Proizvodne tehnologije obradne sistemi i materijale Upravljanje proizvodnim sistemima razvoj proizvoda i CAx tehnologije Tribologiju revitalizaciju reinZinjering i odrZavanje MenadZment kvalitetom i ekoloske tehnologije

Pored osnovnog zadatka Savetovanja - upomavanje se trenutnim stanjem istraZivanja u oblasti proizvodnog maSinstva nadamo se da Ce saopsteni rezultati i diskusija na okruglom stoIu doprineti u defmisanju strategije razvoja ove izuzetno macajne oblasti za dalji privredni razvoj naSe drZave

Zahvaljujemo se svim domacim i stranim autorima clanovima recenzetskog tima kao i institucijama i pojedincima koji su doprineli kvalitetnoj relizaciji programa Savetovanja

Kragujevac 19092006 Predsednik organizacionog odbora 31 SPMSCG

Prof dr Milentije Stefanovic

Predsednik Zajednice PMSCG Prof dr Bogdan Nedic

XII

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xxv XXVI

PRODUCTION ENGINEERING

with foreign participants ~gtjevac~21092ltlQsectm

THE CONTACT PROBLEMS BASED ON THE PENALTY METHOD

S Vulovic 1) Me Zivkovic 2) N Grujovic 3) V Mandie 4)

Abstract In this paper the contact problems including frictional effects are presented The friction forces are assumed tofollow the Coulomb law with a slip criterion treated in the context ofa standard return mapping algorithm The algorithm is amenable to exact linearization and asymptotic quadratic rate of convergence can be achieved within a Newton-Raphson iterative solution scheme

Solution results for verification example are presented at the end ofthis paper Key words contact problem penalty method friction

1 INTRODUCTION

Numerical analysis of frictional contact problems has been one of the research topics of main interest in recent years Frictional contact problems arise in many application fields such as metal forming processes the impact of cars etc

The effective application of finite element contact solvers need a high degree of experience since the general robustness and stability cannot be guaranteed For this reason the development of more efficient fast and stable finite element contact discretizations is still a hot topic especially due to the fact that engineering applications become more and more complex

In this paper framework for contact problems with friction is developed based on the penalty method The penalty formulation has the advantage that it is purely geometrically based and therefore no additional degrees of freedom have to be activated or inactivated Numerical example is shown to demonstrate that the presented algorithm can be successfully applied to contact problems

2 CONTACT KINEMATICS

As the configurations of two bodies coming into the contact are not a priori known contact represents a nonlinear problem even when the continuum behaves as a linear elastic material

I) SneZana D Vulovie Faculty of Mechanical engineering Kragujevac Serbia 2) Miroslav M Zivkovic Faculty of Mechanical engineering Kragujevac Serbia

zilekgacyu 3) Nenad A Grujovic Faculty ofMechanical engineering Kragujevac Serbia 4) Vesna Mandie Faculty of Mechanical engineering Kragujevac Serbia

31 SAVETOVANJE PROIZVODNOG MASINSTVA seG

Using a standard notation in contact mechanics for each pair of contact surfaces involved in the problem we will defme slave (rgraquo and master surfaces (r~) ) Fig

The condition which must be satisfied is that any slave particle cannot penetrate the master surface

Let x be the projection point of the current position of the slave node Xk onto current position ofthe master surface r~) defined as

II =~~~~r aa(~~2) = 0 (1)

where a = 12 and aa(~l ~2) are the tangent covariant base vectors at the point x The normal gap or the penetration gN for slave node k is defmed as the distance

between current positions of this node to the master surface r~)

gN =(xk -i)middot1i (2)

where Ii refers to the normal to the master face r~) at point i (Fig 1) Normal will be

defmed using tangent vectors at the point i

(3)

Figure 1 Geometry ofthe 3D node-to-segment contact element

This gap (2) gives the non-penetration conditions as follows gN = 0 perfect contact gN gt 0 no contact gN lt 0 penetration (4)

If the analyzed problem is frictionless function (4) completely defines the contact kinematics However if friction is modeled tangential relative displacement must be introduced In that case the sliding path of the node Xk over the contact surface r~) is

described by total tangential relative displacement as

gT = ]lgTlldt = ~Itaiialldt = f~r-ta~---middot (5)pa-updt ~ ~ ~

in time interval from to to t The time derivatives ofparameter ~a in equation (5) can be computed from (1)

[7) We obtain the following result

475


(6)

where Qap = aa ap is the metric tensor in point x of the master surface rgl From the

equations (5) and (6) we can express the relative tangential velocity at the contact point

(7)

3 CONSTITUTIVE EQUATIONS FOR CONTACT INTERFACE

A contact stress vector t with respect to the current contact interface rg) can

be split into a normal and tangential part

i = tN +~ = t Ii+taia (8)

where i a is contravariant base vector The stress acts on both surfaces according to the

action-reaction principle t (fl f2) = -t in the contact point x The tangential stress

ta is zero in the case of frictionless contact In the case of contact there is condition

t lt O If there is not penetration between the bodies then relations gN gt 0 and

t = 0 hold This leads to the statements

gN 0 t 0 tgN =0 (9)

which are known as Kuhn-Tucker conditions Using the penalty method for normal stress constitutive equation can be formulated as

tN = ENgN (10)

where EN is the normal penalty parameter

In tangential direction there is difference between stick and slip As long as no sliding between two bodies occurs the tangential relative velocity is zero If the velocity is zero also the tangential relative displacement (5) is zero This state is called stick case with the following restriction

gT =0 lt=gt gT =0 (11) For stick a simple linear constitutive model can be used to describe the tangential stress

tcIt = ETgra (12)

where Er is the tangential penalty parameter

A relative movement between two bodies occurs if the static friction resistance is overcome and the loading is large enough such that the sliding process can be kept The tangential stress vector is restricted as follows

middotsl

ta =-ultNII~ill (13)

where u is friction coefficient In the simplest fonn of Coulombs law (13) u is

constant so there is no difference between static and sliding friction After the introduction of the stick and slip constraints we need to introduce

indicator to derme whether stick or slip actually take place Therefore an indicator function

(14)

476

31 SAVETOVANJE PROlZVODNOG MASINSTVA SCG

is evaluated which respect the Coulombs model for frictional interface law In the

equation (14) the frrstterm is IltTII=~tTaaaPtTP bull A backward Euler integration scheme and return mapping strategy are employed

to integrate the friction equations (14) If a state of stick is assumed the trial values of the tangential contact pressure vector tTa and the indicator function I at load step n+ 1

can be expressed in terms oftheir values at load step n as follows

t~ =tTan +ET6gTan+l =tTao +ETaap6~ (15)

trial IIt TnMal+ II I I (16)I Tn+ = - Ii tNn+l

The return mapping is completed by

taln+1 if IsO (17)ITa n+l = trial

if 1gt0

with Ii ItNo+llIia n+

tral t~+l Ilia n+1 =lit II (18)

For the both cases the penalty method can be illustrated as a group of linear elastic springs that force the body back to the contact surface when overlapping or sliding occurs

4 ALGORITHM FOR FRICTIONAL CONTACT

For solution a nonlinear equilibrium equation with inequality constraints (4) as a result ofcontact we use a standard implicit method In order to apply Newtons method for the solution system of equilibrium equation a linearization of the contact contributions is necessary Inlhis paper we do not state the linearization procedure for standard finite element formulation as well as the contact interface law for the normal and tangential part It could be found in [7

The tangent stiffness matrix for the normal contact is

KN=ENNNT (19)

The symmetric tangent stiffness matrix for stick condition is K = EraapnanPT (20)

where

n ap

-HD -H3p

N = -H2Dmiddot Tp = -H2ip = aafJTp (21)na

-H3D -H33p

-H4D -Hip

The linearization of n~alh+ gives (for details see [1])

ral ) _ [taln+1 )_ 1 [fJ Mal al f3 ] a( (22)6 Ilia n+1 - 6 Ilt~~11 -llt~~11 Sa - nra n+1Ii n+1 6fTfJ n+l

The tangent stiffness matrix for slip condition is

477

31 SAVETOVANJE PROIZVODNOG MA8INSTVA SCG

K SIIP=J1amp Ial DaNT+JlcNgNn+lamp- [SP- Ial lrlaI P]DDrT (23) T N1irn+l Ilt~1I TaPr a 1iro+lnTo+l

The second tenn the tangent matrix is non-symmetric due to the Coulombs friction can be viewed as a non-associative constitutive equation

Frictional contact algorithm using penalty method is shown in Table I

Table 1 Frictional contact algorithm using the penalty method

LOOP over all contact segment k (checkforcontact(6raquo IF gNO THEN

(the ftrst iteration) IF i=1 THEN set all active nodes to state stick t Tn + (18) compute matrix K~ck

ELSE Compute trial state t~an+1 (19) and f (20)

IF ft 0 THEN

tra 0+1 = t~a~+1 compute matrix K~ck (40) GO TO (a)

ELSE Ira n+l =J1ltNO+lln~+ compute matrix K~P (43)

ENDIF ENDIF

ENDIF (a) END LOOP

4 EXAMPLE

For purposes of comparison numerical example is taken from [6] Hence this example can be used to verifY whether the developed algorithm is able to represent stickslip behavior correctly_ An elastic block is pressed against a rough rigid foundation Simultaneously to the vertical loading the block is pulled at right side by an unifonnly nonnal stress (see Fig 2) Material constants are E = 1000 per lenght square v = 03 The properties of the contact surface have been chosen as follows

ampN =108 ampr =10 friction coefficient J1 =05 The block is discretized using 200

four-node isoparametric elements It should be noted that using developed algorithm the total load can be applied in only one step

~-a~t bull bull bull bull f bullbull bull t r 36 4

Fig 2 Initial and deformed configuration

478

31 SAVETOVANJE PROIZVODNOG MASINSTVA SCG

The computed ncnnal contact pressure lliid tangential contact stress are shown in Fig 3 and indicate good agreement between this solutions and the solutions shown in [6]

I-+-normal contact pressure --- tangential contact pressure I300

250

200

150

100

50

O~~--~~~~middot~middot~__ __~~--~~--~-r-- so 04 08 12 18 2 -r4middotaa_l_3amp 4

~100

middot150

Fig 3 Force - displacement relationship

3 CONCLUSIONS

A model for contact problem with friction based on the penalty method was presented Due to the intrinsic similarity between friction and the classical elasoshyplasticity [4] the constitutive model for friction can be constructed following the same formalism as in classical elaso-plasticity The numerical example indicates a possibility of applying the developed method in the analysis of finite deformation problems

REFERENCES

[I] Fisher KA Mortal type methods applied to nonlinear contact mechanics PhD Thesis Institut fUr Bumechanick und Numerische Mechanik Univ of Hannover Hannover 2005

[2] Grujovic N Contact problem solution by finite element methods PhD Thesis Faculty of Mech Eng Univ ofKragujevac Kragujevac 1996

[3] Kojic M R Slavkovic M Zivkovic N Grujovic The software packages PAK Faculty ofMechanical Engineering ofKragujevac Serbia and Montenegro

[4] Kojic M K J Bathe Inelastic Analysis of Solids and Structures Springer BerlinshyHeidelberg 2005

[5] Laursen T A J C Simo A continuum-based finite element formulation for the implicit solution of multibody large deformation frictional contact problems Inter J Num Meth Eng 36 3451-3485 1993

[6] Wriggers P rv Van E Stein Finite element formulation of large deformation impact-contact problems with friction Computers and Structures 37 319-333 1990

[7] Wriggers P Computational Contact Mechanics J Wiley amp Sons Ltd West Sussex England 2002

479


SRBIJE I eRNE GORE v v

SA lAEDUNARODNIl1 UCESCElt Kragujevac 2006

ISBN 86-80581-92-5

ZBORNIK RADOVA

Urednici Editors

Prof dr Bogdan Nedic dipl ing Prof dr Milentije Stefanovic dipl ing Prof dr Miodrag Lazie dipl ing

Izdavac Publisher

MaSinski fakultet 34000 Kragujevac Sestre Janjic 6

Za izdavaca For publisher

Prof dr Radovan Slavkovic dekan Fakulteta

TiraZ Circulation

200 primeraka

Stampa Printed by

Graficki atelje SKVER Kragujevac

Copyright MaSinskifakultet Kragujevac 2006

Izdavanje Zbornika radova organizovanje i odr~vanje 31 Savetovanja proizvodnog m~instva Srbije i Crne Gore podr~lo je

Ministarstvo nanke i zanite zivotne sredine Republike Srbije

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4 EXAMPLE






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200

150

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1 INTRODUCTION












II =~~~~r aa(~~2) = 0 (1)






(3)








475


(6)



(7)










gN 0 t 0 tgN =0 (9)


tN = ENgN (10)




tcIt = ETgra (12)



middotsl





(14)

476










if 1gt0







KN=ENNNT (19)


where

n ap

-HD -H3p


-H3D -H33p

-H4D -Hip




477









IF ft 0 THEN



ENDIF ENDIF

ENDIF (a) END LOOP

4 EXAMPLE






478




250

200

150

100

50


~100

middot150


3 CONCLUSIONS


REFERENCES








479



IV




v

PREDGOVOR










XII



























XV XVI












xvn












XVllI













XIX









_~~t~ bull ~M~





XX























XXII XXI


















D2




014 Mm_lilJaym 694





xxm XXIV












xxv XXVI







1 INTRODUCTION












II =~~~~r aa(~~2) = 0 (1)






(3)








475


(6)



(7)










gN 0 t 0 tgN =0 (9)


tN = ENgN (10)




tcIt = ETgra (12)



middotsl





(14)

476










if 1gt0







KN=ENNNT (19)


where

n ap

-HD -H3p


-H3D -H33p

-H4D -Hip




477









IF ft 0 THEN



ENDIF ENDIF

ENDIF (a) END LOOP

4 EXAMPLE






478




250

200

150

100

50


~100

middot150


3 CONCLUSIONS


REFERENCES








479




v

PREDGOVOR










XII



























XV XVI












xvn












XVllI













XIX









_~~t~ bull ~M~





XX























XXII XXI


















D2




014 Mm_lilJaym 694





xxm XXIV












xxv XXVI







1 INTRODUCTION












II =~~~~r aa(~~2) = 0 (1)






(3)








475


(6)



(7)










gN 0 t 0 tgN =0 (9)


tN = ENgN (10)




tcIt = ETgra (12)



middotsl





(14)

476










if 1gt0







KN=ENNNT (19)


where

n ap

-HD -H3p


-H3D -H33p

-H4D -Hip




477









IF ft 0 THEN



ENDIF ENDIF

ENDIF (a) END LOOP

4 EXAMPLE






478




250

200

150

100

50


~100

middot150


3 CONCLUSIONS


REFERENCES








479

PREDGOVOR










XII



























XV XVI












xvn












XVllI













XIX









_~~t~ bull ~M~





XX























XXII XXI


















D2




014 Mm_lilJaym 694





xxm XXIV












xxv XXVI







1 INTRODUCTION












II =~~~~r aa(~~2) = 0 (1)






(3)








475


(6)



(7)










gN 0 t 0 tgN =0 (9)


tN = ENgN (10)




tcIt = ETgra (12)



middotsl





(14)

476










if 1gt0







KN=ENNNT (19)


where

n ap

-HD -H3p


-H3D -H33p

-H4D -Hip




477









IF ft 0 THEN



ENDIF ENDIF

ENDIF (a) END LOOP

4 EXAMPLE






478




250

200

150

100

50


~100

middot150


3 CONCLUSIONS


REFERENCES








479



























XV XVI












xvn












XVllI













XIX









_~~t~ bull ~M~





XX























XXII XXI


















D2




014 Mm_lilJaym 694





xxm XXIV












xxv XXVI







1 INTRODUCTION












II =~~~~r aa(~~2) = 0 (1)






(3)








475


(6)



(7)










gN 0 t 0 tgN =0 (9)


tN = ENgN (10)




tcIt = ETgra (12)



middotsl





(14)

476










if 1gt0







KN=ENNNT (19)


where

n ap

-HD -H3p


-H3D -H33p

-H4D -Hip




477









IF ft 0 THEN



ENDIF ENDIF

ENDIF (a) END LOOP

4 EXAMPLE






478




250

200

150

100

50


~100

middot150


3 CONCLUSIONS


REFERENCES








479












xvn












XVllI













XIX









_~~t~ bull ~M~





XX























XXII XXI


















D2




014 Mm_lilJaym 694





xxm XXIV












xxv XXVI







1 INTRODUCTION












II =~~~~r aa(~~2) = 0 (1)






(3)








475


(6)



(7)










gN 0 t 0 tgN =0 (9)


tN = ENgN (10)




tcIt = ETgra (12)



middotsl





(14)

476










if 1gt0







KN=ENNNT (19)


where

n ap

-HD -H3p


-H3D -H33p

-H4D -Hip




477









IF ft 0 THEN



ENDIF ENDIF

ENDIF (a) END LOOP

4 EXAMPLE






478




250

200

150

100

50


~100

middot150


3 CONCLUSIONS


REFERENCES








479













XIX









_~~t~ bull ~M~





XX























XXII XXI


















D2




014 Mm_lilJaym 694





xxm XXIV












xxv XXVI







1 INTRODUCTION












II =~~~~r aa(~~2) = 0 (1)






(3)








475


(6)



(7)










gN 0 t 0 tgN =0 (9)


tN = ENgN (10)




tcIt = ETgra (12)



middotsl





(14)

476










if 1gt0







KN=ENNNT (19)


where

n ap

-HD -H3p


-H3D -H33p

-H4D -Hip




477









IF ft 0 THEN



ENDIF ENDIF

ENDIF (a) END LOOP

4 EXAMPLE






478




250

200

150

100

50


~100

middot150


3 CONCLUSIONS


REFERENCES








479























XXII XXI


















D2




014 Mm_lilJaym 694





xxm XXIV












xxv XXVI







1 INTRODUCTION












II =~~~~r aa(~~2) = 0 (1)






(3)








475


(6)



(7)










gN 0 t 0 tgN =0 (9)


tN = ENgN (10)




tcIt = ETgra (12)



middotsl





(14)

476










if 1gt0







KN=ENNNT (19)


where

n ap

-HD -H3p


-H3D -H33p

-H4D -Hip




477









IF ft 0 THEN



ENDIF ENDIF

ENDIF (a) END LOOP

4 EXAMPLE






478




250

200

150

100

50


~100

middot150


3 CONCLUSIONS


REFERENCES








479


















D2




014 Mm_lilJaym 694





xxm XXIV












xxv XXVI







1 INTRODUCTION












II =~~~~r aa(~~2) = 0 (1)






(3)








475


(6)



(7)










gN 0 t 0 tgN =0 (9)


tN = ENgN (10)




tcIt = ETgra (12)



middotsl





(14)

476










if 1gt0







KN=ENNNT (19)


where

n ap

-HD -H3p


-H3D -H33p

-H4D -Hip




477









IF ft 0 THEN



ENDIF ENDIF

ENDIF (a) END LOOP

4 EXAMPLE






478




250

200

150

100

50


~100

middot150


3 CONCLUSIONS


REFERENCES








479












xxv XXVI







1 INTRODUCTION












II =~~~~r aa(~~2) = 0 (1)






(3)








475


(6)



(7)










gN 0 t 0 tgN =0 (9)


tN = ENgN (10)




tcIt = ETgra (12)



middotsl





(14)

476










if 1gt0







KN=ENNNT (19)


where

n ap

-HD -H3p


-H3D -H33p

-H4D -Hip




477









IF ft 0 THEN



ENDIF ENDIF

ENDIF (a) END LOOP

4 EXAMPLE






478




250

200

150

100

50


~100

middot150


3 CONCLUSIONS


REFERENCES








479







1 INTRODUCTION












II =~~~~r aa(~~2) = 0 (1)






(3)








475


(6)



(7)










gN 0 t 0 tgN =0 (9)


tN = ENgN (10)




tcIt = ETgra (12)



middotsl





(14)

476










if 1gt0







KN=ENNNT (19)


where

n ap

-HD -H3p


-H3D -H33p

-H4D -Hip




477









IF ft 0 THEN



ENDIF ENDIF

ENDIF (a) END LOOP

4 EXAMPLE






478




250

200

150

100

50


~100

middot150


3 CONCLUSIONS


REFERENCES








479





II =~~~~r aa(~~2) = 0 (1)






(3)








475


(6)



(7)










gN 0 t 0 tgN =0 (9)


tN = ENgN (10)




tcIt = ETgra (12)



middotsl





(14)

476










if 1gt0







KN=ENNNT (19)


where

n ap

-HD -H3p


-H3D -H33p

-H4D -Hip




477









IF ft 0 THEN



ENDIF ENDIF

ENDIF (a) END LOOP

4 EXAMPLE






478




250

200

150

100

50


~100

middot150


3 CONCLUSIONS


REFERENCES








479


(6)



(7)










gN 0 t 0 tgN =0 (9)


tN = ENgN (10)




tcIt = ETgra (12)



middotsl





(14)

476










if 1gt0







KN=ENNNT (19)


where

n ap

-HD -H3p


-H3D -H33p

-H4D -Hip




477









IF ft 0 THEN



ENDIF ENDIF

ENDIF (a) END LOOP

4 EXAMPLE






478




250

200

150

100

50


~100

middot150


3 CONCLUSIONS


REFERENCES








479










if 1gt0







KN=ENNNT (19)


where

n ap

-HD -H3p


-H3D -H33p

-H4D -Hip




477









IF ft 0 THEN



ENDIF ENDIF

ENDIF (a) END LOOP

4 EXAMPLE






478




250

200

150

100

50


~100

middot150


3 CONCLUSIONS


REFERENCES








479









IF ft 0 THEN



ENDIF ENDIF

ENDIF (a) END LOOP

4 EXAMPLE






478




250

200

150

100

50


~100

middot150


3 CONCLUSIONS


REFERENCES








479




250

200

150

100

50


~100

middot150


3 CONCLUSIONS


REFERENCES








479

y'icrll'j.~ - is.fink.rsis.fink.rs/podaci/nenad_grujovic/52/31 savetovanje - the...

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