frekvencija segmentiranja strugotine i oblik i...

Mainstvo, 3(6), 143 – 148, (2002) S.Ekinovi,...: FREKVENCIJA SEGMENTIRANJA STRUGOTINE...

FREKVENCIJA SEGMENTIRANJA STRUGOTINE I OBLIK I DIMENZIJE SEGMENATA STRUGOTINE PRI

VISOKOBRZINSKOJ OBRADI ^ELIKA TVRDO]E 52 HRC

S.Ekinovi*, S. Dolinek**, E.Ekinovi*, * Univerzitet u Sarajevu, Mainski fakultet u Zenici, Fakultetska 1., 72000 Zenica, Bosna i Herzegovina, ** Univerzitet u Ljubljani, Mainski fakultet, Aker~eva 6, 1000 Ljubljana, Slovenija

REZIME: U radu su prikazani rezultati analize oblika strugotine pri obradi jedHRC. Obrada je vrena glodanjem u irokom dijapazonu brzina rezairok dijapazon brzina rezanja obezbe|uje obradu kako u podru~ju visokih brzina. Analizirana je frekvencija segmentiranja strugotine strugotine) oblik i dimenzije segmenta, kao i veli~ina nedeformisanih,Rezultati pokazuju da postoji ~vrsta me|uzavisnost izme|u posmatrapri obradi vosokim brzinama.

Klju~ne rije~i: Visokobrzinsko glodanje, Formiranje strugotine, Segm

FREQUENCY OF SEGMENTATION, ADIMENSIONS OF CHIP SEGMENTS A

HIGH-SPEED MACHIN

S.Ekinovi*, S. Dolinek**, E.Ekinovi*, * University of Mechanical Engineering in Zenica, Fakultetska 1., 7200** University of Ljubljana, Faculty of Mechanical Engin

Ljubljana, Slovenia

SUMMARY The results of chip shape analysis in machining of a tool steel wpaper. The machining was performed by milling in a wide range ompmin. This broad spectrum of cutting speed values ensures comachining. The analysis covered the chip segmentation frequencsegments generation), chip shape and dimensions, and also the sparts of chip segments. The results show that there is a closparameters, especially at high-speed machining.

Keywords: High-speed milling, Chip formation, Chip segmentatio

1. UVOD Visokobrzinske obrade, kao relativno nova proizvodna tehnologija je tehnologija koja omoguava veliku produktivnost, veoma dobar kvalitet obrade i dimenzionu ta~nost. Jedna od najzna~ajnijih tehnologija u okviru svih postojeih visokobrzinskih obrada, je visokobrzinsko glodanje [1],[2],[14]. Prvobitno, visokobrzinsko glodanje se uspjeno koristilo u avionskoj i automobilskoj industriji za obradu slo`enih mainskih dijelova od aluminijuma i njegovih legura.

1. INTROD High-speed mmanufacturing tgood surface qspeed milling among all highInitially, high-sin aircraft andcomplex machalloys.

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IZVORNI NAU^NI RAD

ne vrste alatnog ~elika tvrdoe 52 nja od 50 do 1500 m/min. Ovako konvencionalnih, tako i u podru~ju (frekvencija nastanka segmenata dijelova segmenata strugotine. nih parametara strugotine, naro~ito

entacija strugotine, Alatni ~elik

ND SHAPE AND T 52 HRC STEEL ING

Sarajevo, Faculty of 0 Zenica, B&H, eering, Aker~eva 6, 1000

ORIGINAL SCIENTIFIC PAPER

fith 52 HRC are presented in the cutting speeds from 50 to 1500 nventional as well as high-speed y (that is the frequency of chip ize of deformed and undeformed e relationship among these chip

n, Tool steel

UCTION

achining, being a relatively new echology, enables high productivity, very uality and dimensional accuracy. High-plays one of the most important roles -speed cutting methods [1],[2],[14]. peed milling was successfully used automotive industry for machining of ine parts made of aluminium and its


U skorije vrijeme, a zahvaljujui napretku u oblasti materijala reznih alata i tehnologija, visokobrzinsko glodanje se koristi pri obradi legiranih ~elika u otvrdnutom stanju, tj. sa tvrdoom iznad 30 HRC, pa do 60-65 HRC, [1],[2],[3],[4],[5]. Skorija istraìvanja u oblasti visokobrzinske obrade usmjerena su u nekoliko karakteristi~nih pravaca: mehanizami troenja alata [2],[6], kvalitet i integritet obra|ene povrine [7],[8], mehanizmi formiranja strugotine [1],[2],[4],[9],[10] i obrada materijala u otvrdnutom stanju [3],[7],[11],[12]. Sva ova istraìvanja u principu imaju zajedni~ki cilj, a to je da se istraè sve mogunosti primjene visokobrzinske obrade u industrijskoj praksi. U ovom radu su prikazani neki rezultati vezani za analizu segmentiranja strugotine, kao i oblik i dimenzije segmenata strugotine pri visokobrzinskom glodanju ~elika tvrdoe 52 HRC.

Recently, with the improvements in cutting tools materials and technologies, high-speed milling has also been used in machining of alloy steels in their hardened state; above 30 HRC up to 60-65 HRC, [1],[2],[3], [4],[5]. More recent high-speed machining studies direct their attention toward several characteristic areas: tool wearing mechanisnms [2],[6], surface quality and machined surface integrity [7],[8], chip formation mechanisms [1],[2],[4],[9],[10], and machining of materials in their hardened state (hard machining) [3],[7],[11],[12]. A common aim of all these studies is to explore all possibilities of high-speed machining application in industrial practice. The analysis of some results referring to segmantation frequency and shape and dimensions of the ship segments in 52 HRC steel high-speed milling is presented in this paper.

2. EKSPERIMENTALNO ISTRA@IVANJE Eksperimentalno istraìvanje je proveden u Laboratoriji za obradu rezanjem na Fakulteti za strojnitvo, Univerziteta u Ljubljani. Obrada je vrena na glodalici tip Moriseiki-Frotier. Uslovi obrade su bili: pre~nik glodala D=80 mm, dubina rezanja d=2 mm, posmak po zubu ft=0,1 mm, tip glodala SUM-UFO-4000, rezna plo~ica SFKN 12T3 A2TN - AC230, brzina rezanja v=50-1500 m/min, γ=27o, λ=-7o, κ=45o. Hemijski sastav obra|ivanog ~elika prikazan je u tabeli 1. Na slici 1 data je mikrostruktura ovog ~elika, kao i izmjerene vrijednosti mikrotvrdoe po Vickersu. Mikrostruktura je martenzitna sa eutektoidnim karbidima i zaostalim austenitom. Navedeni ~elik je obra|ivan brzina rezanja 50, 150, 300 i 1500 m/min, pri istim ostalim uslovima.

2. EXPERIMENTAL WORK The experimental work has been carried out in the Machining laboratory at the Faculty of Mechanical Engineering, University of Ljubljana. The machining was conducted on the milling machine of the Moriseiki-Frontier type. The machining conditions were: cutter diameter D=80 mm, depth of cut d=2 mm, tooth feed ft=0.1 mm, cutter type SUM-UFO-4000, cutting insert SFKN 12T3 A2TN–AC230, cutting speed v=50–1500 mpmin, γ=27°, λ=-7°, κ=45°. The chemical composition of the investigated steel is shown in the Table 1. Fig. 1 shows the microstructure and results of Vickers microhardness measurement. The microstructure is martensite with eutectoide carbides and retained austenite. The steel grade used here was machined with cutting speeds of 50, 150, 300 and 1500 mpmin, under the same above mentioned conditions.

Table 1. Chemica composition o the investigated steel l fTabela 1. Hemijski sastav ispitivanog ~elika

Chemical composition (Hemijski sastav), % Steel grade (Vrsta ~elika)/ /state (Stanje)

C Si Mn P S Cu Cr Ni Mo V Al

^.4758 (X63CrMoV51)/ /tempered (Kaljen)

0,62 1,0 0,59 0,017 0,004 0,26 5,46 0,23 1,21 0,46 0,028

100 µm 100 µm

Fig. 1. Microstructure of the steel ^.4758 (X63CrMoV51), tempered, 629 HV (52 HRC) Slika 1. Mikrostruktura ispitivanog ~elika ^.4758 (X63CrMoV51), kaljen, 629 HV (52 HRC)

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3. REZULTATI I DISKUSIJA Na slici 2 su dati mikrosnimci strugotine dobijene pri obradi. Kada je brzina rezanja v=50 m/min (slika 2.a), struktura materijala ukazuje na klasi~ni tip deformisanja sa jednoliko izduènim zrnima, ali uz pojavu bijelog sloja na unutarnjoj strani strugotine, to je posljedica termi~kog omekavanja materijala. Prosje~na izmjerena vrijednost mikrotvrdoe iznosi 660 HV, to pri uporedbi sa po~etnom vrijednou od 629 HV ukazuje na nizak stepen deformacionog o~vravanja. Pri obradi brzinom rezanja v=150 m/min, strugotina je segmentirana i ima tipi~ni testerasti (nazubljeni) oblik, slika 2.b. Jasno se uo~ava bijeli sloj kako na unu-tranjoj strani strugotine tako i izme|u segmenata. To zna~i da je prisutna pojava termi~kog omekavanja i mehanizma deformisanja. Prosje~na vrijednost izmjerene mikrotvrdoe bijelog sloja je 756 HV. Me|utim, unutranji (nedeformisani) dio segmenta strugotine ima prosje~nu mikrotvrdou samo 632 HV, to ukazuje na to da je ovdje struktura potpuno nedeformisana u odnosu na po~etno stanje materijala obratka. Pribli`no 62% povrine popre~nog presjeka segmenta strugotine je nedeformisano. Frekvencija segmenatiranja strugotine je oko 3,84 kHz. Na osnovu prethodno re~enog moè se zaklju~iti da se ve pri vrijednosti brzine rezanja od 150 m/min dolazi u podru~je u kome se moè govoriti o visokobrzinskoj obradi. Pri obradi brzinom rezanja od v=300 m/min strugotina je jo vie segmentirana, pri ~emu su segmenti manje debljine i veli~ine u odnosu na strugotinu dobijenu pri v=150 m/min, slika 2.c. Istovremeno je debljina bijelog sloja manja uz prosje~no izmjerenu mikrotvrdou u od 742 HV. Na unutranjem dijelu segmenata prosje~na mikrotvrdoa je 640 HV. U ovom slu~aju, pribli`no 40% povrine popre~nog presjeka segmenata strugotine je deformisano, a frekvencija nastanka segmenata strugotine je oko 15,6 kHz. Kona~no, pri brzini rezanja v=1500 m/min strugotina je izrazito segmentirana sa segmentima jo manje debljine i veli~ine, slika 2.d. [tavie, debljina bijelog sloja je manja nego u slu~aju obrade sa niìm brzinama rezanja, a njegova mikrotvrdoa iznosi 720 HV. Na unutranjem dijelu segmenata prosje~na mikrotvrdoa iznosi 618 HV. Slika 3 prikazuje dijagramske prikaze frekvencije segmentiranja strugotine i relativne veli~ine povrine deformisanog dijela presjeka segmenta strugotine. Sa navedenih prikaza se jasno vidi proporcionalnost izme|u navedenih parametara i brzine rezanja. U ovom slu~aju deformisano je oko 30% povrine popre~nog presjeka segmenata strugotine, a frekvencija segmenatiranja je oko 100,6 kHz.

3. RESULTS AND DISCUSION

Fig. 2 shows the microphotos of chip produced during machining. When cutting speed is v=50 mpmin (Fig. 2.a), the microstructure of material is of the classical type of deformation with uniformly elongated grains, but with appearance of the white layer on the inner side of the chip, which is the consequence of the thermal softening of the material. The average measured microhardness is 660 HV, which in relation to the initial value (629 HV) gives a low level of strain hardening. In machining at a speed of v=150 mpmin, the chip is segmented and has a typical saw-tooth shape, Fig. 2.b. Clearly visible is the white layer both on the inner side of the chip and also between the segments. It means that the appearance of the thermal softening and mechanism of deforming is present. The average measured microhardness value of the white layer is 756 HV. However, on the inner (undeformed) part of the chip segments the average microhardness is only 632 HV, which shows as a complete undeformed structure in relation to the initial state of the workpiece materials. Approximately 62% of the chip segment cross-section area is deformed. Frequency of the chip segmentation is about 3.84 kHz. On the basis of the above findings it can be concluded that the speed of v=150 mpmin is already of the range for which we can say that it is high-speed machining. In machining at a speed of v=300 mpmin, the chip is even more strongly segmented, with lower thickness and with the segments of lower dimensions in comparison with the speed of v=150 mpmin, Fig. 2.c. At the same time the thickness of the white layer is lower and its average measured microhardness is 742 HV. On the inner part of the segment the average microhardness is 640 HV. In this case, approximately 40% of the ship segment cross-section area is deformed, and the frequency of the chip segmentation is about 15.6 kHz. As a final ascertainment, with the cutting speed of v=150 mpmin the chip shape is heavily segmented with even lower thickness and magnitude of the segments, Fig. 2.d. Moreover, the thickness of the white layer is lower compared to the machining with lower speeds, its average microhardness is 720 HV. On the inner part of the segments the average microhardness is 618 HV. Finally, in this case, approximately 33% of the chip segment cross-section area is deformed, and the frequency of the chip segmentation is about 100.6 kHz. The resulting diagram on Fig. 3 shows the relationship between the frequency of chip segmentation and the relative magnitude of cross-section area of chip segment deformed part. The proportionality between these parameters and the cutting speed is evident from the particular diagrams.

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t

660 HV

White layer (Bijeli sloj)

100 µm

a) Cutting speed (Brzina rezanja), v=50 m/min

Deformed part, approx. 62% of chip segment cross-section area (Deformisani dio, cca 62% povrine presjeka segmenta strugotine), 756 HV

Undeformed part, approx. 38% of chip segment cross-section area (Nedeformisani dio, cca 38% povrine presjeka segmenta strugotine), 632 HV


100 µm

Chip segmentation frequency: 3.84 kHz Frekvencija formiranja segmenata strugotine: 3,84 kHz b) Cutting speed (Brzina rezanja), v=150 m/min


Undeformed part, approx. 60% of chip segment cross-section area (Nedeformisani dio, cca 60% povrine presjeka segmenta strugotine), 640 HV


100 µm

Chip segmentation frequency: 15.6 kHz Frekvencija formiranja segmenata strugotine: 15,6 kHz c) Cutting speed (Brzina rezanja), v=300 m/min


Undeformed part, approx. 67% of chip segment cross-section area (Nedeformisani dio, cca 67% povrine presjeka segmenta strugotine), 618 HV White layer

(Bijeli sloj) 100 µm

Chip segmentation frequency: 100.6 kHz Frekvencija formiranja segmenata strugotine: 100,6 kHz

d) Cutting speed (Brzina rezanja), v=1500 m/min

Fig.2. Cross-sections of the chip produced during machining and some results of the analysis Slika 3. Presjeci strugotine nastale pri obradi i neki rezulta i analize

4. ZAKLJUÂK U radu su prikazani rezultati eksperimentalnog ispitivanja oblika strugotine, mikrotvrdoe strugotine, frekvencije segmentiranja i veli~ine deformisanog dijela povrine popre~nog presjeka segmenta strugotine, pri obradi glodanjem ~elika X63CrMoV51 (52HRC) u dijapazonu brzina rezanja od 50 do 1500 m/min. Na osnovu dobivenih rezultata moè se konstatovati sljedee: a) Na osnovu oblika strugotine pri obradi

navedenog ~elika, moè se zaklju~iti da podru~je obrade visokim brzinama nastupa ve pri brzini rezanja od 150 m/min,

4. CONCLUSION

The results of experimental researh of the chip shape, its microhardness, chip segmentation frequency and magnitude of cross-section area of chip segments deformed parts are presented in this paper. The research refers here to the milling of steel grade X63CrMoV51 (52HRC) with cutting speed range of 50 to 1500 mpmin. The results here obtained point out to the following: a) The evaluation of the chip shape obtained during the machining of the steel grade mentioned above leads to the conclusion that high-speed machining appears when the cutting speed value is above 150 mpmin,

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High-speed machining Visokobrzinaska obrada

Conventional machiningKonvencionalna obrada

v= 150 m/min v= 300 m/min

v= 1500 m/min

Fig. 3. The frequency of chip segmenta ion and magnitude of cross-section area of chip segment deformed part in relation to the cutting speed

Slika 3. Zavisnost frekvencije segmentiranja strugotine i veli~ine deformisanog dijela popre~nog presjeka segmenta strugotine od brzine rezanja

b) Sa poveanjem brzine rezanja poveava se frekvencija segmentiranja strugotine, pri ~emu se debljina strugotine smanjuje, a tako|er i veli~ina segmenata strugotine. Istovremeno se smanjuje i ve~ina deformisanog dijela popre~nog presjeka segmenta strugotine,

b) As cutting speed increases, the chip segmentation frequency also increases, whereas the chip thickness and magnitude of the chip segments decrease. At same time, the magnitude of chip segment deformed part decreases. c) As cutting speed increases, the thermal softening of material during the process of plastic deformation becomes greater, whereas the part of chip segment cross-section area exposed to the thermal softening influence becomes lower (white layer),

c) Sa poveanjem brzine rezanja, u procesu plasti~ne deformacije sve vee u~ee zauzima termi~ko omekavanje materijala uz istovremeno smanjenje dijela popre~nog presjeka segmenta strugotine podrvrgnutom ovom omekavanju (bijeli sloj), d) Sa poveanjem frekvencije segmentiranja smanjuje se deformisani dio popre~nog presjeka segmenta strugotine i

d) As chip segmentation frequency increases, the deformed part of chip segment cross-section area decreases, and

e) Navedeni rezultati mogu biti od koristi pri visokobrzinskom glodanju otvrdnutih materijala s obzirom da umnogome rasvetljavaju mehanizam nastanka strugotine, njen oblik i frekvenciju segmetiranja.

e) These results are valid for high-speed milling of hardened materials, since they throw light on the chip-formation mechanism, its shape and the segmentation frequency.

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5. LITERATURA - REFERENCES [1] Y.Ning, M.Rahman, Y.S.Wong: Investigation of chip formation in high-speed end milling, Journal of Materials Processing Technology, Vol.113, p.p.360-367, 2001, [2] C.E.Becze, P.Clayton, L.Chen, T.I.El-Wardany, M.A.Elbestawi: High-speed five-axis milling of hardened tool steel, Internationl Journal of Machine Tools & Manufacture, Vol.40, p.p.869-885, 2000, [3] M.A.Elbestawi, L.Chen, C.E.Besze, T.I.El-Wardany: High-speed Milling of Dies and Moulds in Their Hardened State, Annals of the CIRP, Vol.46/1, p.p.57-62, 1997, [4] P.Fallbohmer, C.A.Rodriguez, T.Ozel, T.Altan: High-speed machining of cast iron and alloy steels for die and mold manufacturing, Journal of Materials Processing Technology, Vol.98, p.p.104-115, 2000, [5] H.Schuls: Hochgeschwindigkeitsbearbeitung – High-Speed Machining, Carl Hanser Verlag, 1996, [6] S.Dolinek, J.Kopa~: Mechanism and types of tool wear; some particularities in using advanced cutting materials and newest machining processes, Proceedings of the 8th International Scientific Conference on Achievements in Mechanical & Materials Engineering, p.p.185-188, 1999, [7] T.I.El-Wardany, H.A.Kishawy, M.A.Elbestawi: Surface Integrity of die Material in High Speed Hard Machining, Part 1: Microgarphical Analysis, Transactions of the ASME, Journal of

Manufacturing Science and Engineering, Vol.122, p.p.620-631, 2000, [8] T.I.El-Wardany, H.A.Kishawy, M.A.Elbestawi: Surface Integrity of die Material in High Speed Hard Machining, Part 2: Microhardness Variations and Ressidual Stresses, Transactions of the ASME, Journal of Manufacturing Science and Engineering, Vol.122, p.p.632-641, 2000, [9] S.Ekinovi, S.Dolinek, J.Kopa~, M.Godec: Transition from Conventional to High-Speed Cutting Region and Chip Formation Mechanism, accepted for publishing in Journal Strojniki Vestnik, 2002, [10] S.Dolinek, S.Ekinovi: A Contribution to the Strain-Hardening Process Analysis of Hardened Steel During High-Speed Machining, Proceedings of the AMST 2002 Conference, p.p. 137-142, Udine, 2002, [11] H.K.Tonshoff, C.Arendt, R.Ben Amor: Cutting of Hardened Steel, Annals of the CIRP, Vol.49/2, p.p.547-566, 2000, [12] G.Poulachon, A.L.Moisan: Hard Turning: Chip Formation Mechanisms and Metallurgical Aspects, Transactions of the ASME, Journal of Manufacturing Science and Engineering, Vol.122, p.p.406-412, 2000, [13] H.Schulz, T.Kneisel: Turn-Milling of Hardened Steel – an Alternative to Turning, Annals of the CIRP, Vol43/1, p.p.93-96, 1994, [14] S.Ekinovic, E.Ekinovic: High Speed Machining, Masinstvo, Vol. 4. No.1, p.p.15-50, 2000,

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Mainstvo 3(6), 149 – 156, (2002) S.Sueska: ODRE\IVANJE NAPONA U MATERIJALU…

ODRE\IVANJE NAPONA U MATERIJALU PRIMJENOM POLARIZACIJE

mr. Suad Sueska; Br~anska 7/V, 71000 Sarajevo

REZIME U radu je objanjen pojam polarizacije svjetlosti. Dat je princip i postutransparentnom materijalu primjenom polarizacije.

Klju~ne rije~i: napon, polarizacija, polariskop, izokline, izohrome.

DETERMINATION OF STRESS IN MPOLARISATION

M.Sc. Suad Sueska; Br~anska 7/V, 71000 Sarajevo

SUMMARY The study explains the phenomenon of the polarisation of light. It presfor determination of stress in transparent material by use of polarisatio

Key words: stress, polarisation, polariscope, isoclinics, isochromat

1. UVOD

Za prora~une mainskih konstrukcija i elemenata vrlo je va`no poznavati raspodjelu napona u njima. Za njihovo odre|ivanje razvijena je matemati~ka teorija elasti~nosti. Po~etkom 20. vijeka pojavljuju se i experimentalni postupci odre|ivanja napona i deformacija. Experimentalni postupci izvanredno upotpunjuju matemati~ke analize teorije elasti~nosti, koje su ipak ograni~ene ~im su konture nepravilne i uslovi na konturi slo`eniji. U ovom ~lanku je opisan experimentalni postupak odre|ivanja raspodjele napona u materijalu metodom polarizacije.

1. INTRODUC

It is very importanfor calculations of Mathematical theofor its determinatcentury, experimestress and strainextraordinarily comtheory of elasticityof irregular contconditions. The stfor determination material applying t

2. POLARIZACIJA

Polarizacija je pojava da pri prolasku kroz dvolomne kristale, ~ije opti~ke ose nisu paralelne, svjetlost gubi na intenzitetu i najslabija je ako opti~ke ose stoje okomito jedna na drugu. Nepolarizovana svjetlosti osciluje u svim ravnima pramena kroz pravac prostiranja, a vektori elektri~nog (E) i magnetnog (H) polja su me|usobno normalni i normalni na pravac prostiranja svjetlosti. Svjetlost ~iji vektori elektri~nog (E) ili magnetnog (H) polja osciluju u samo jednoj ravni je polarizovana svjetlost. Polarizovanu i nepolarizovanu svjetlost je mogue razlikovati samo posebnim filterima.

2. POLARISA

Polarisation of llight passes throptical axes are is the weakest wperpendicular. Noplane passing thrand the vectors fields are mutualon the line ofvectors electric (in just one planenonpolarised lighusing special filte

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PREGLEDNI RAD

pak za odre|ivanje napona u

ATERIAL BY
SUBJECT REVIEW
ents the principle and method n.

ics.

TION

t to know the distribution of stresses machine constructions and elements. ry of elasticity has been developed ion. At the beginning of the 20th ntal methods for determination of appeared. Experimental methods plement mathematical analyse in the , which are yet limited in the cases ours and more complex contour udy presents experimental procedure of the distribution of stresses in he method of polarisation of light.

TON OF LIGHT

ight is a phenomenon in which ough birefringent crystals, whose not parallel, loses intensity and it hen the optical axis are mutually npolarised light oscillates in every ough the line of light propagation, of electric (E) and magnetic (H) ly perpendicular and perpendicular light propagation. Light which E) or magnetic (H) fields oscillate is polarised light. Polarised and t can be diferentiated just by rs.


Svjetlost moè biti: ravanski, kru`no i elipti~no polarizovana. Ravanski polarizirana svjetlost se moè dobiti na vie na~ina: refleksijom, prelamanjem kroz prozirna tijela, prolaskom kroz prozirne kristale, pomou polarizacijskih filtera, kao i prolaskom kroz posebno konstruisane prizme. Ako dva polarizovana zraka svjetlosti (redovni i vanredni), koji izlaze iz dvolomnog materijala i osciliraju istim frekvencijama u me|usobno normalnim ravnima, relativno kasne za e=0° ili e=±π, onda se oni slaù u ravanski (linearno) polarizirani zrak. Me|utim, ako je kanjenje e=±π/2+2mπ (m=0,1,2…) i ako su redovni i vanredni zrak jednake amplitude onda se dobije kru`no polarizirani zrak. Ali ako su redovni i vanredni zrak razli~ite amplitude i relativno kasne za e=±πm/4 (m=0,1,2,…) tada se oni superponiraju u elipti~no polarizirani zrak. Dvolomni kristali (turmalin, kalcit, islandiski kalcij, liskun, herapatit) mogu polarizirati svjetlost. Ovi kristali polariziraju svjetlost u dvije me|usobno normalne ravni. Polarizirane svjetlosne zrake se nazivaju spora (redovna) i brza (vanredna) zraka jer su im brzine prostiranja u kristalu razli~ite. Zato jedna upadna zraka iz dvolomnog kristala izlazi kao dvije ortogonalno polarizovane zrake koje se prostiru paralelnim me|usobno pomaknutim pravcima. Kako su ovi kristali relativno mali za upotrebu oni su zahtjevali dodatnu opti~ku opremu i koristili su se do pojave polarizacijskih filtera. Polarizacijski filteri ili polaroidi su se pojavili 1945. Proizvode se tako da se anizotropni, sitni koloidni kristali nanesu na prozirne plasti~ne folije orijentirani svi u jednom smjeru. To omoguuje polarizaciju svjetlosti. Plo~e od polaroida mogu imati vee dimenzije (do 300mm) pa nije potrebna dodatna opti~ka oprema. Polaroidi selektiraju ravan proputanja svjetlosnog talasa apsorpcijom svjetlosnih talasa u ostalim ravnima. Ova vrsta polarizacije se naziva dihroizam. Iz polaroida izlazi polarizovana svjetlost koja osciluje samo u jednoj ravni, dok su oscilacije u svim ostalim ravnima apsorbirane u dihroi~nom materijalu. Polarizacija pomou dihroizma je 100%.

Light can be: plane, circular, and elliptically polarised. Plane polarised light can be made by number of ways: by reflection, by refraction through transparent materials, passing through transparent crystals, by polarisation filters, as well as passing through specially designed prisms. When two polarised light waves (regular and extraordinary), which come out of birefringent material and oscillate with the same frequencies in mutual perpendicular planes, are relatively displaced by e=0° or e=±π, then they are superimposed in a plane (linear) polarised wave. However, when the displacement is e=±π/2+2mπ (m=0,1,2…) and the regular and extraordinary wave have equal amplitudes, then circularly polarised wave is produced. But, when regular and extraordinary wave have different amplitudes and are relatively displaced by e=±πm/4 (m=0, 1, 2,…), then they are superimposed in an elliptically polarised wave. Birefringent crystals (tourmaline, calcite, island calcium, liskun, herapathite) can polarise light. These crystals polarise light in two mutual perpendicular planes. Polarised light waves are called slow (regular) and fast (extraordinary) wave, because their speeds of propagation in crystal are different. That is why one incident wave comes out of birefringent crystal as two orthogonal polarised waves propagating in parallel mutually displaced directions. As the crystals are relatively small for usage, they required additional optical equipment and they were used until the appearance of polarising filters. Polarising filters or polaroids appeared in 1945. They are made of anisotrop, small colloid crystals deposited on transparent plastic transparences, all with orientation in the same direction. It enables polarisation of light. Plates made of Polaroid can have bigger dimensions (up to 300mm) and additional optical equipment is not necessary. Polaroids select the plane filtering of light waves by absorption light waves in other planes. This kind of polarisation is called dichroism. Polarised light comes out of polaroid oscillating just in one plane, and the oscillations in every other plane are absorbed in dichroic material. Polarisation by dichroism is 100%.

3. FOTOELASTI^NOST

Naponi u transparentnom materijalu su uzrok promjene indexa pralamanja materijala. U optem slu~aju ovi indexi prelamanja su anizotropni pa materijal postaje dvojno prelamajui. Funkcionalna povezanost indexa prelamanja materijala i napona u materijalu je data slijedeom jedna~inom:

ni- nj=(a-b)(σi-σj)=c(σi-σj) i, j= I, II, III (1)

3. PHOTO-ELASTICITY

Stresses in transparent material are cause for change of its refractive index. In global case these refractive indices are anizotrop and material becomes birefringent. Functional interrelationship between refractive indices and stresses in material is given by the following equation:

ni- nj=(a-b)(σi-σj)=c(σi-σj) i, j= I, II, III (1)

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gdje su: c= a-b - naponsko-opti~ki koeficijent ni - index prelamanja koji ima svjetlosni

talas polariziran u pravcu glavne ose i σ I, II, III - glavni naponi

Uticaj napona na promjenu indexa prelamanja materijala se naziva elasto-opti~ki efekat. Zrak koji pada normalno na transparentnu plo~u debljine d, napregnute naprezanjima u ravni plo~e σx i σy (σI= σx, σII= σy, σIII=0) usljed djelovanje napona e se rastaviti na dvije komponente, koje e na izlasku iz plo~e relativno kasniti za:

rxy= (nx- ny)d= c(σx- σy)d (2)

gdje su: rxy – relativna retardacija c= a-b - naponsko-opti~ki koeficijent d – debljina plo~e

Ovaj zakon se naziva osnovni zakon fotoelasti~nosti. Glavna primjena stati~kog elasto-opti~kog efekta je u analizi napona. Za primjenu postupka je potrebno pripremiti transparentni model, napregnuti ga i postaviti izme|u ukrtenih polarizatora i analizatora, a zatim posmatrati izlaznu svjetlost. Specijalizirani ure|aji pomou kojih se vri posmatranje dvojnog prelamanja materijala nazivaju se polariskopi. U polariskopima se mogu promatrati prozirna tijela u ravanski ili kru`no polariziranoj svjetlosti. U slu~aju da samo posmatrano tijelo na neki na~in ne polarizira svjetlost koja kroz njega prolazi na izlazu iz analizatora se ne bi pojavila svjetlost ako su polarizator i analizator postavljeni tako da im ravni polarizacije zatvaraju ugao od 90°. Ako je prozirno tijelo, napravljeno od dvolomnog materijala, napregnuto onda e ono polarizirati svjetlost u dvije okomite ravni ~iji se smjerovi poklapaju sa smjerovima glavnih napona u materijalu, a izlazni polarizovani zraci e se fazno razlikovati za

)()(22121 σσσσ

λπ

−=−=∆ Kcd (3) λ

gdje je: ∆- fazna razlika redovnog i vanrednog zraka λ- talasna du`ina svjetlosti c- konstanta materijala modela (povrina/sila) d- debljina plo~e od dvolomnog materijala σ1; σ2- glavni naponi K=2 πcd/ λ – konstanta materijala za datu talasnu du`inu izvora. Ako je izvor svjetlosti bijeli onda se iza analizatora dobiva slika tijela sa spektralnim bojama, a ako je izvor svjetlosti monohromatski onda se dobiva slika tijela sa tamnim i svjetlim prstenovima ili prugama. Na izgled dobivenih slika uti~e vie faktora kao to su: oblik tijela, stanje naprezanja, opti~ke karakteristika materijala, temperatura…

where: c= a-b - stress-optical coefficient ni - refractive index having light wave

polarised in direction of principal axis i σI, II, III - principal stresses

Effect of stresses on change refractive index of material is called elasto-optical effect. Wave falling perpendicular on transparent plate having thickness d, stressed by stresses in plane σx i σy (σI= σx, σII= σy, σIII=0) due to stress effect will be broken into two components, which will relatively lag for: rxy= (nx- ny)d= c(σx- σy)d (2)

where: rxy - relative retardation

c= a-b - stress-optical coefficient d - plate thickness

This rule is called fundamental law of photo-elasticity. The main application of statical elasto-optical effect is in stress analysis. To use the method it is necessary to prepare transparent model, load it and put it between crossed polariser and analyser, then the observe outcoming light. Specialised apparatus, by which observations of birefringent material are made, is called polariscope. Transparent bodies in plane or circularly polarised light can be observed by the polariscopes. In case the observed body, in any way, does not polarise the light passing through it, light would not appear from analyser outlet if polariser and analyser are placed in such a way that its planes of polarisation make an angle of 90°. If transparent body, made of birefringent material, is loaded then it polarises light in two perpendicular planes whose directions coincide with directions of principal stresses in the material, and outcoming polarised beams will have the difference in phase:

)()(22121 σσσσπ

−=−=∆ Kcd

where: ∆- phase difference between regular and extraordinary wave λ- wavelength of light c- stress-optical coefficient d- thickness of plate made of birefringent material σ1; σ2- principal stresses K=2 πcd/ λ - constant of material for given wavelength of source.

If light source is white then picture of body made of spectrum colours is produced behind analyser, but if light source is monohromatic then picture of bodu with dark and shine rings or lines is produced. Manu factors affect on appearance produced pictures such as: shape of body, stress conditions, optical characteristics of material, temperature…

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Iz predhodne jednakosti se moè zaklju~iti da u nekom dijelu modela od razlike glavnih napona zavisi boja (kod bijelog izvora svjetlosti), odnosno intenzitet svjetlosti (kod monohromatskog izvora svjetlosti). Konstanta K plo~e modela se odre|uje u polariskopu na uzorku materijala plo~e napregnutom na poznata stanja naprezanja.

Ako se model ~iju raspodjelu napona treba odrediti postavi u polariskop izme|u polarizatora i analizatora ~ije su ravni polarizacije postavljene pod 90°, tada e napregnuti model dati dvije ortogonalno polarizirane zrake (redovnu i vanrednu) ~ije se oscilovanje moè izraziti slijedeim jedna~inama:

r = aK0sin(φ+∆)cosθ (4) v = aK0sin φsinθ (5)

gdje su: r- redovna zraka v- vanredna zraka a- amplituda zraka koji izlazi iz polarizatora K0- faktor apsorpcije modela (neka je K0 jednak i za redovni i za vanredni zrak) φ - ugao kojim se opisuje oscilatorno kretanje zraka (φ=2π/T) ∆- fazni pomak

Nakon prolaska kroz analizator redovna i vanredna zraka e se ravanski polarizirati i poto se fazno razlikuju za ∆ sloìt e se u zrak koji se moè opisati slijedeom jedna~inom:

w = rsinθ- vcosθ = aK0sinθcos[sin(φ+∆) - sinφ] = aK0sin2θsin(½∆)cos(φ+½∆) (6) (6) w = rsinθ- vcosθ = aK0sinθcos[sin(φ+∆)-

sinφ] = aK0sin2θsin(½∆) cos(φ+½∆) (6) gdje je w zrak nastao slaganjem polarizovanih redovnog i vanrednog zraka na izlazu iz analizatora. Svjetlost iza analizatora nestaje (w=0) u dva posebna slu~aja:

1) sin2θ=0 tj. 2θ=nπ tj. θ=½nπ (n=1, 2, …) (7) (7)

where the w is a wave produced by super-imposing of the regular and extraordinary waves coming out of the analyser. Behind analyser, light extinguishes (w=0) in two special cases:

2) sin(½∆)=0 tj. ½∆= nπ tj. ∆=2nπ (n=1, 2, …) (8)

Tamne linije (w=0) koje se dobiju prvim slu~ajem (θ=nπ/2) se nazivaju izokline, linije koje spajaju ta~ke sa jednakim smjerovima glavnih napona. Za crtanje rasporeda izoklina u modelu uzima se sredina polja izoklina kao linija izoklina. Ako se u~vrsti poloàj polarizatora i analizatora i zajedno se okrenu za izvjestan ugao slika polja izoklina e se promijeniti. Izokline se vraaju u isti poloàj nakon obrtanja za 90°.

It can be concluded from the previous equation that in any part of model the colour (white light source), or light intensity (monohromatic light source) depends on difference of principal stresses. The constant of model plate K is determined in the polariscope on sample plate of material loaded with known stress conditions. If the model, whose stress distribution is to be determined, is placed in polariscope between polariser and analyser whose polarisation planes make an angle of 90°, then the loaded model will produce two orthogonal polarised waves (regular and extraordinary) whose oscillations can be expressed by the following equations:

r = aK0sin(φ+∆)cosθ (4) v = aK0sin φsinθ

where: r- regular wave v- extraordinary wave a- amplitude of wave coming out of polariser K0- factor absorption of the model (let the K0 be equal for regular and extraordinary wave) φ- angle describing the oscillatory motion of wave (φ=2π/T) ∆- phase difference After passing through the analyser, the regular and the extraordinary wave will be plane polarised and because they have phase difference of ∆ they will superimpose in wave which can be described by the following equation:

1) sin2θ=0 => 2θ=nπ => θ=½nπ (n=1, 2, …) (7) 2) sin(½∆)=0 => ½∆= nπ => ∆=2nπ (n=1, 2, .) (8)

Dark lines (w=0) produced in the first case (θ=nπ/2) are called isoclinics, lines connecting the points with same slope of principal stresses. To draw distribution of isoclinics in the model, the middle of the isoclinic’s field is to be used as the isoclinic line. If the position of the polariser and analyser is locked and then they turned together by any angle, the picture of isoclinic’s fields will be changed. Isoclinics come back in the same position after turning for the angle of 90°.

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Linije drugog sistema (∆=2nπ) se nazivaju izohrome. To su linije koje spajaju ta~ke sa jednakom razlikom glavnih napona (σ1-σ2). Ako je izvor svjetlosti monohromatski sve ta~ke sa istom razlikom napona su na jednoj liniji, a ako je izvor bijele svjetlosti onda sve ta~ke sa istom razlikom glavnih napona imaju iste spektralne boje. U prakti~nom radu na polariskopu se i u ovom slu~aju dobiva polje izohroma, pa se za liniju izohroma uzima sredina polja izohroma. Kod polarizatora sa monohramatskim izvorom svjetlosti okretanjem spojenih polarizatora i analizatora se dobiva razlikovanje izoklina i izohroma, jer se izokline mijenjaju sa zakretanjem unakrsnih polarizatora i analizatora, dok se izohrome ne mijenjaju. Ako se u polariskopu primjeni bijeli izvor svjetlosti vrlo je lako uo~iti razliku izme|u izohroma i izoklina: izohrome su obojena polja (po ~emu su i dobile naziv), izokline ne.

Izokline su tamne linije, ili bolje rei polja, koje ~ine ta~ke modela u kojima su smjerovi glavnih napona jednaki, a koje zadovoljavaju uslov θ=nπ/2 (n=1, 2, …), tj. pravac glavnih napona modela se poklapa sa pravcem polarizacije polarizatora to dovodi do gaenja svjetlosti pri izlasku iz analizatora. Dakle, pravci glavnih napona u svim ta~kama na izoklini su jednaki (θ). Kako su komponente glavnog napona me|usobno normalne poznavanjem pravca jednog glavnog napona, poznat je i pravac drugog u nekoj ta~ci na izoklini.

Izohrome su linije jednake boje nastale gaenjem boje ~iji je cjelobrojni proizvod talasne du`ine jednak retardaciji redovnog i vanrednog zraka dobivenoj razlikom glavnih napona u modelu. Poveanjem optereenja poveava se broj izohroma. Izvorita izohroma su mjesta na kojima se generiraju izohrome (mjesta na kojima djeluju koncentrirane sile, gornja i donja vlakna tapa optereenog ~istim savijanjem). Fotografiranjem napregnutog polariziranog modela dobiju se dva podatka: raspored ta~aka sa jednakom razlikom glavnih napona u modelu i raspored ta~aka u modelu sa istim nagibom pravaca glavnih napona. Za rjeavanje ravanskih problema potrebna su tri podatka: dva glavna napona i smjer glavnih napona. Zatvaranje sistema jedna~ina za odre|ivanje pravca i smjera glavnih napona jo nije rijeeno na prikladan na~in. Slijede neki od na~ina zatvaranja sistema jedna~ina: pomou jedna~ine deformacije u smjeru debljine modela, pomou interferometra, rotiranjem modela u polariskopu, diferencni metod tangencijalnih napona koji koristi jedna~ine ravnote`e i grani~ne uslove.

The lines of the second system (∆=2nπ) are called isochromatics. These are the lines connecting the points with equal difference of the principal stresses (σ1-σ2). If light source is monochromatic, all points with equal difference of principal stresses are on one line, but if it is used source of white colour then all points with equal difference of principal stresses have the same spectrum colour. In fact, polariscope produces isochromatic field in this case too, and isochromatic line is presented by the middle of the isochromatic field. In the case of the polariser with monochromatic light source, turning the locked pola-riser and analyser produces discrimination of isoclinics and isochromatics, because the isoclinics are changed by turning the crossed polariser and analyser, but the isochromatics are not changed. If the polariscope uses white light source then it is very easy to perceive the difference between the isochromatics and isoclinics: the isochromatics are coloured fields (which are named by), the isoclinics not. The isoclinics are dark lines, or it is better to say fields, which are made of points of the model having equal slopes of principal stresses, and which fulfil the condition θ=nπ/2 (n=1, 2, …), that is the direction of principal stresses of the model coincides with the direction of polarisation of the polariser, which causes extinguishing of light coming out from analyser. So, the directions of the principal stresses are equal in all points on the isoclinic (θ). As the components of the principal stress are mutually perpendicular, by knowing direction of one principal stress, the direction of the second is also known in any point on the isoclinic. Isochromatics are lines with the same colour produced by extinction of colour having integer multiple of its wavelength equal to the retardation of the regular and extraordinary waves produced by difference of principal stresses in the model. Increase of load produces increase in the number of isochromatics. Origins of isochromatics are places where they are generated (places where concentrated forces act, upper and lower fibres of the beam loaded by pure bending). Taking the photo of a loaded polarised model gives two data: distribution of points having equal difference of principal stresses in the model, and distribution of points in the model having equal slopes of principal stresses. Three data are necessary to solve planar problem: two principal stresses and slopes of principal stresses. Closing of the equation system to determine the line and slope of the principal stresses has not been properly solved so far. Some ways of closing the equation system are the following: by the strain equation in the direction of model thickness, by interferometer, by rotation of the model in the polariscope, method differences of share stresses utilizing the equations of balance and boundary conditions.

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1) Izokline 2) Izohrome 1) Isoclinics 2) Isochromatics

Slika 1: Slike izoklina i izohroma dobivene pomou polariskopa Figure 1: Photos o isoclinics and isochroma ics made by polariscope f t

Posljednji postupak, koji se naj~ee koristi, ne odre|uje experimentalno trei podatak nego ga ra~una diferencnim jedna~inama, koje su rjeenja diferencijalne jedna~ine ravnote`e za ravansko naponsko stanje uz poznate grani~ne uslove. Aproximiranjem diferencnim jedna~inama i teoremom srednje vrijednosti uvodi se kumulativna greka koja raste sa udaljavanjem od konture. Na ovaj na~in je mogue odrediti tenzor napona u proizvoljnoj ta~ci napregnutog uzorka. Procedura ra~unanja napona u proizvoljnoj ta~ci je slijedea:

1) Odrediti polja izohroma i izoklina 2) Nacrtati osnovni pravac du` ose x i dva

pomona pravca na rastojanju ∆y i napraviti podjelu sa korakom ∆x

3) U presje~nim ta~kama dobivene mre`e odrediti parametre izohrome i izokline

4) U ta~kama ½(xi+xi+1) na pravcima a i b izra~unati tangencijalne napone po formuli: τxy=½(σ1- σ2)sin2θ’ (9)

5) Izra~unati ∆τxy/∆y po formuli: ∆τxy/∆y = (τxy(a)- τxy(b))/∆y (10)

6) Izra~unati napon σx(i) po formuli: σx(i)= σx(i-

1)- (∆τxy/∆y)∆x (11) 7) Izra~unati σy(n) po formuli: σy(n)= σx(n)-(σ1-

σ2)(n)cos2 θ’(n) (12)

The last method, the most commonly used, does not determine the third data by experiment, but it calculates the data by difference equations, being solutions of the differential equation of balance for planar stress conditions with known boundary conditions. By approximation with difference equations and the middle value theorem, cumulative error is introduced and it increases when the distance from the contour increases. The tensor of stresses in any point of loaded model can be determined by this method. The calculation procedure of the stresses in any point is the following: 1) Determine the fields of the isochromatics and

isoclinics 2) Draw main direction along x axis, and two

arbitrary directions on distance of ∆y and make division with step of ∆x

3) Determine the parameters of the isochromatics and the isoclinics in cross points of the mesh

4) Calculate share stresses in points ½(xi+xi+1) on arbitrary directions utilising equation: τxy=½(σ1- σ2)sin2θ’ (9)

5) Calculate ∆τxy/∆y by equation: ∆τxy/∆y = (τxy(a)- τxy(b))/∆y (10)

6) Calculate stress σx(i) by equation: σx(i)= σx(i-1)- (∆τxy/∆y)∆x (11)

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7) Calculate σy(n) by equation: σy(n)= σx(n)-(σ1- σ2)(n)cos2 θ’(n) (12) 8) Izra~unati glavne napone po formulama:

σ1(n)= ½[σx(n)+ σy(n)+(σ1- σ2)(n)] (13) σ2(n)= ½[σx(n)+ σy(n)-(σ1- σ2)(n)] (14)

Na ovaj na~in se moè odre|ivati napon u transparentnom opti~ki osjetljivom materijalu, ali i u modelu neke realne konstrukcije napravljenom od opti~ki osjetljivog materijala. Za ispitivani uzorak procedura odre|ivanja napona bi bila zavrena i stvarne veli~ine i pravci glavnih napona u analiziranim ta~kama elementa bi bili odre|eni. Time bi bio poznat tenzor napona u tim ta~kama. Me|utim, ako je ispitivan model napravljen od opti~ki osjetljivog materijala za odre|ivanje napona nekog elementa napravljenog od sasvim druga~ijeg materijala onda se dobivene veli~ine moraju konvertovati u one koje djeluju na stvarni element po slijedeim jedna~inama:

- napon na stvarnom elementu:

M

N

N

M

N

MMN P

Pdd

llσσ = (15) MN σσ =

- pomjeranje na stvarnom elementu:

N

M

N

M

M

NMN E

Edd

PPuu = (16) uu =

gdje su σ; l; d; P; u i E napon, duìna, debljina, sila, pomjeranje i modul elasti~nosti respektivno. Sa indexima N i M su ozna~ene veli~ine koje se odnose na stvarni element i na model respektivno. Naravno da se i pri izradi modela i konfiguriranju optereenja moralo o ovome voditi ra~una.

8) Calculate principal stresses by equations:

σ1(n)= ½[σx(n)+ σy(n)+(σ1- σ2)(n)] (13) σ2(n)= ½[σx(n)+ σy(n)-(σ1- σ2)(n)] (14)

The stress in transparent optical sensitive material can be determined in this way, but also in the model of any real construction made of optical sensitive material. For real element, the procedure of stress determination would be finished and real magnitude and directions of principal stresses in analysed points of the element would be determined. Then the tensor of the stress in the points would be known. However, if the examined model was made of optical sensitive material to determine the stresses of the element made of absolute different material, then the results have to be converted in values acting on real element utilizing equations as follows:

- The stress on real element:

M

N

N

M

N

M

PP

dd

ll

- Displacement on real element:

N

M

N

M

M

NMN E

Edd

PP

where: σ; l; d; P; u i E are: stress, length, thickness, force, displacement and modulus of elasticity respectively. The magnitudes related to real element and model are designated by the indices N i M respecfuly. Nautrally, this would have to be taken into account during the design of the model and configuration load.

4. ZAKLJUÂK

Naponi u transparentnim opti~kim elementima izazivaju polarizaciju svjetlosti. Naponi u opti~kom materijalu se mogu izraziti preko razlike indexa prelamanja nenapregnutog i napregnutog materijala, odnosno preko osnovnog zakona fotoelasti~nosti. Polariskopska metoda je pogodna za analizu napona transparentnih opti~kih elemenata. Ova metoda omoguava utvr|ivanje raspodjele razlika glavnih napona u napregnutom materijalu i pravce djelovanja glavnih napona po cijeloj povrini, a ne samo u jednoj ta~ci kao kod metode elektro-otpornih mjernih traka. Ova metoda je beskontaktna, to zna~i da samo mjerenje ne unosi nikakva naprezanja. Dodatnim prora~unom mogue je izra~unati i napon u proizvoljnoj ta~ci napregnutog elementa.

4. CONCLUSION

Stresses in transparent optical elements produce the polarisation of light. Stresses in optical material can be expressed by the difference in refractive indices of unloaded and loaded material, that is by fundamental rule of photo-elasticity. The polariscope method is suitable for stress analysis in transparent optical elements. This method enables finding the distribution differences of principal stresses in loaded material and directions of principal stresses over the whole area, and not in one point as in the method the electrical-resistance strain gauge. This mehod is contactless, it means that the measuring does not cause any stress. By additional calculation it is possible to calculate the stress in any point of the loaded element.

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5. LITERATURA - REFERENCES [1] A.W.Hendry, “Elements of Experimental Stress Analysis”, Pergamon Press, The MacMillan Company, New York, 1964 [2] Z.Kostren~i, “Teorija elasti~nosti”, [kolska Knjiga, Zageb, 1982 [3] M.Frocht, “Photoelasticity – Volume II”, John Wiley & Sons Inc., New York, 1948

[4] Б.C.Kacatkин, i drugi, “Зkcперимен-тальные Metoды Иccлeдoвaния Дeфoрмaций и Напряжений – Cправочное Пособие”, Наукова Думка, Kиев, 1981 [5] P.И.Xaиmoвa – Maльkoвa, “Metoдиka иccлeдoвaния нaпряaжeний пoляapизaциoнo-oпtичeckиm metoдom”, Издateльctвo Hayka; Mockвa, 1970

6. OZNAKE - NOMENCLATURES σi,j [Pa] - napon

- stress

c [MPa]-1 - naponsko-opti~ki koeficijent - stress-optical coefficient λ [nm] - talasna du`ina svjetlosti - wavelength of light r [nm] - relativna retardacija - relative retardation

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Mainstvo 3 (6), 157 – 168, (2002) R.Seferovi: PRILOG EKSPERIMENTALNOM ODRE\IVANJU…

PRILOG EKSPERIMENTALNOM ODRE\IVANJU OPTERE]ENJA KOD REZANJA KRU@NOM PILOM U VRU]EM STANJU

Mr. Raif Seferovi, dipl.ma.in`., KALEN Zenica

REZIME PRETHODNO SAOP[TENJE

Veliki broj mainskih sistema radi u dinami~kim uslovima. Obrada metala odsijecanjem u vruem stanju na kru`noj pili je dinami~ki proces uslovljen velikim brojem uticajnih parametara koje je teko pratiti u realnim (industrijskim) uslovima. Pri dinami~koj analizi mehani~kog sistema posebnu pa`nju treba obratiti na rad mehanizama glavnog i pomonog kretanja pile. Eksperimentalno odre|eni momenti i otpori rezanja te intenzitet i karakter njihove promjene, od presudne su va`nosti za mehani~ku stabilnost sistema.

Klju~ne rije~i: rezanje, kru`na pila, otpori rezanja

A CONTRIBUTION TO EKSPERIMENTAL DEFINITION OF HOT METAL LOAD WHEN CUTTING WITH CIRCULAR SAW

Mr. Raif Seferovi, B.Sc. Mech. Eng., KALEN Zenica

SUMMARY PRELIMINARY NOTES

Most mechanical systems work dynamically. The cutting of hot metal by circular-saw is a dynamic process and depends on many parameters, which are difficult to follow in industrial conditions. In dynamic analysis of a mechanical system a special attention should be paid to the work of mechanisms for main and auxilliary saw motion. Cutting torques and resistance as well as their magnitude and changes, experimentaly defined, are very important for stability of a mechanical system.

Key words: cutting, circular-saw, cutting resistance

1. UVOD Odsijecanje metala javlja se u tehnolokom procesu obi~no kao prva operacija i prisutna je u svim fabrikama koje se bave preradom metala [1]. Proces odsijecanja ostvaruje se u tribo-mehani~kom sistemu ~iju strukturu ~ine: rezni alat (pilni disk), predmet obrade i sredstvo za hla|enje i podmazivanje. Proces odsijecanja praen je slijedeim parametrima: pojavom otpora kretanju reznog alata kroz materijal

predmeta obrade (mehanika procesa rezanja), pojavom toplote i visokih temperatura u zoni

rezanja (termodinamika procesa rezanja), pojavom trenja u zonama dodira alata i

materijala predmeta obrade i habanja reznog alata (tribologija procesa rezanja) [2].

U istra`ivanju obrade metala odsijecanjem postoje uglavnom tri oblasti ispitivanja koje zajedno daju kvalifikaciju odre|enog procesa, kao to je i obrada u toplom stanju. Tu spadaju: mjerenje sila (otpora) pri rezanju, mjerenje temperatura, te praenje procesa habanja alata.

1. INTRODUCTION In technological process, metal cutting usually arises as the first operation and exists in all manufactures which involve metal processing [1]. The cutting process is realised in tribo-mechanical system, which consists of: cutting tool (saw disc), workpiece, coolant and lubricant. Cutting is followed by a few parameters:

• motion resistance of tool through the workpiece (mechanics of cutting),

• heat and high temperatures in cutting area (thermodynamic of cutting),

• friction of tool and workpiece in cutting area and tool wear ( tribology of cutting) [2].

There are three of testing, in research of metal treatment which together qualify a process such as hot works. These three domains are: measurinig of resisting forces, measuring of temperatures and observation of tool wearing process.

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Za mjerenje sila pri rezanju, kao i za mjerenje temperatura razvijeni su namjenski instrumenti i paralelno s njima oprema za registrovanje i obradu signala. Me|utim, za praenje habanja alata razvijene su metode, ali jo uvijek nema namjenski orjentisane opreme za kompletnu obradu podataka [3]. Za razliku od stati~ke analize pri kojoj su spoljanja dejstva veli~ine nepromjenljive u vremenu, a time i odziv strukture u vidu napona i deformacija, u dinami~koj analizi, koja odgovara realnim industrijskim uslovima, spoljanji uticaji i odgovarajui odzivi su funkcije vremena. Da bi se mogla izvriti dinami~ka analiza stanja pile za odsijecanje profila u vruem stanju u toku procesa rezanja, posebnu pa`nju treba obratiti na rad mehanizama glavnog i pomonog kretanja pile. Osnovu za to ~ine eksperimentalno odre|eni momenti i otpori rezanja. Poznavanje intenziteta i karaktera njihove promjene od presudne je va`nosti za odre|ivanje mehani~ke stabilnosti sistema [4]. Obrtni moment motora odre|uje se direktno i indirektno, odnosno kombinovanom primjenom direktne metode mjerenja, tenzometarskih mjernih traka i indirektne metode, odre|ivanjem obrtnog momenta ra~unskim putem, mjerenjem elektri~nih i neelektri~nih veli~ina kao pokazatelja optereenja elemenata mehanizma glavnog i pomonog kretanja pile u procesu rezanja obratka [5]. Obrtni moment je funkcija parametara:

M = f ( P, U, Is, n, t ) gdje je: P - aktivna snaga, [W], U - elektri~ni napon na elektromotoru, [V], Is – ja~ina struje statora motora, [A], n - brzina vrtnje (broj obrtaja) rotora elektromotora, [min-1], t - vrijeme, [s]. Mjerenjem strujnih optereenja motora u toku procesa rezanja obratka mo`e se, iz mjerenjem utvr|enih veli~ina, izra~unati moment motora i tereta sa dovoljno ta~nosti, na osnovu ~ega se odre|uju komponente otpora rezanja:

glavni otpor rezanja, T; otpor pomonog kretanja, Q; otpor prodiranja, R;

Glavni otpor rezanja T ~esto se naziva obodna ili tangencijalna komponenta rezultujueg otpora rezanja i djeluje u pravcu tangente na rezni element. Kod odsijecanja profila u vruem stanju isti iznosi [6]:

vuhspT

⋅⋅⋅⋅

=1000

[N] , (1)

gdje je: p - specifi~ni pritisak rezanja, [MPa]; s - irina rezanja, [mm]; h - visina odrezane plohe obratka, [mm]; u - brzina pomonog kretanja, [mm/s], v - brzina odsijecanja (brzina stvarnog kretanja), [m/s];

Force and temperatures are measured by fixed-instruments and equipment developed for data registration and data analysis. However, wear observation methods have been developed but there are not fixed-option instruments and equipment for total data analysis [3]. Asopposed to static analysis where both external effects and structure responses (strain and deformation) are unvariable, with dynamic analysis, in operating conditions, external effects and appropirate signal responces are variable. For an appropirate dynamic analysis of hot metal cutting with circular saw a special attention should be paid to the work of mechanisms of main and auxilliary cutting motion. Cutting torques and resistances defined by experiment make a basis for dynamic analysis. Magnitude and changes of torques and resistances are very important for defining stability of a mechanical system [4]. Motor torque could be defined directly or indirectly, i.e. by combine application of direct method (strain gauges) and indirect methods which consist of: calculaton defined torque, measuring of electric and nonelectric effects, which are indicators of main and auxilliary motion loads during cutting of the workpiece [5]. Torque is the function of the next following variables:

M=(P,U,Is, n, t ) where: P - effective power [W], U - voltage of electric motor, [V] Is - electric current of electric motor, [A], n - revolution per minute, rpm [min-1], t - time [sec.] Motor torque and load torque can be defined by measuring electric motor cutting loads. The next three resistance components are based on the following:

• main(primary) cutting resistance, T, • auxilliary motion resistance, Q,

radial feed resistance, R. Main cutting resistance force, T is often called periphery or tangential component of the resulting force and work at tangential direction of tool. By cutting hot sections this force is defined as:

vuhspT

⋅⋅⋅⋅

=1000

[N] , (1)

where : p - specific cutting pressure, [MPa]; s - cutting width, [mm]; h - height of cut sheet, [mm]; u - auxilliary motion speed, [mm/s], v - cutting speed (main motion speed), [m/s];

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a) Otpori rezanja b) Plan sila a) Cutting resistances b) Resulting forces

Slika 1. Grafi~ka prezentacija otpora pri rezanju obratka pilnim diskom sa pravim zubima Fig 1. Grafic presentation of resisting forces at streight tooth circular saw cutting

Otpori pomonog kretanja (horizontalna komponenta otpora rezanja) Q mo`e se izra~unati iz plana sila (slika 1) i iznosi:

Resisting force of auxilliary motion (horizontal component of cut resisting force) Q could be calculated from Fig.1:

−=−=−== ∑∑

=

=

=

=

zi

iiii

zi

iiH TRTRQQFQ

1112 cossinsincos ψψαα , [N] (2)

Vertikalna komponenta otpora rezanja iznosi:

+= ∑∑

=

=

=

=

zi

iiii

zi

iiV TRF

11sincos ψψ , [N] (3)

Rezultujui otpor rezanja koji optereuje vratilo pilnog diska dobija se iz izraza:

22VH FFF += , [N] . (4)

U izrazima (2) i (3) ugao α je srednji ugao kontakta pilnog diska sa obratkom, a ugao ψi ozna~ava ugao kojeg i-ti zub (u pilnom disku je ukupan broj zuba z) zaklapa sa vertikalnom osom pilnog diska. Intenzitet otpora prodiranja R, prema autoru [4] kree se u intervalu:

( ) TR ⋅= 15...8 , [N] . (5)

Vertical componet of cut resisting force is:

+= ∑∑

=

=

=

=

zi

iiii

zi

iiV TRF

11sincos ψψ , [N] (3)

Resulting cut resisting force which is loading circular saw shaft is obtained from:

22VH FFF += , [N] . (4)

In equations (2) and (3) α is the midle angle of contacts between saw and workpiece, and ψi presents the angle between "i" tooth (total number of teeth is "z") and vertical axis of circular saw. Radial motion resisting force, according to [4] is:

( ) TR ⋅= 15...8 , [N] . (5)

2. USLOVI I NA^IN IZVO\ENJA EKSPERIMENTA

2.1 Uslovi eksperimentalnih ispitivanja

Ispitivanja su izvedena prema tehnolokom procesu kontinuiranog valjanja ~eli~nih profila pri ~emu je u skladu sa teorijskim osnovama vrueg rezanja izvren izbor promjenljivih veli~ina i odre|eni nivoi njihovih variranja.

2. CONDITIONS AND MANNERS OF EXPERIMENTAL RESEARCHES

2.1 Experimental Conditions

Researches are made according to continual rolling of sections varyables and levels of their variability are selected according to hot cutting theory are selected.

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Za vrijeme eksperimentalnih ispitivanja sve promjene elektri~nih i neelektri~nih veli~ina na pogonima mehanizama glavnog i pomonog kretanja registrovane su u vidu kontinualnih vremenskih zapisa. Proces rezanja se izvodio sa konstantnom brzinom pomonog kretanja pile (u=const.) uz osiguranje ortogonalnosti rezanja. Re im obrade variran je prema sa~injenom planu eksperimenta sa brzinom glavnog kretanja v=98,1 [m/s] i brzinom pomonog kretanja u=25, 50, 75, 100 [mm/s]. Eksperimentalna ispitivanja vrena su na valja~kim proizvodima (profilima) razli~itih popre~nih presjeka i mehani~kih osobina ~iji su parametri dati u tabeli 1.

Durring the experimental researches all electric and nonelectric variables of mechanisms for main and auxilliary motion are registrated as time functions. Cutting is realised with constant auxilliary motion velocity (u=const.) and with obligatory cutting ortogonality. Cutting is realised according to a plan of experimental researches and with the following parameters: main motion speed, v=98,1 [m/s], and auxilliary motion speed, u=25,5,75,100 [mm/s]. Experimental researches are made with different rolling sections and mechanical properties.

Tabela 1. Oblici i mehani~ka svojstva materijala profila koji se odsijecaju Table 1. Forms and mechanical properties of treated sections

R.b. No. Profil obratka

Worpiece form

Dimenzije Dimension

[mm]

Materijal Material

Zatezna ~vrstoa na sobnoj temperaturi

Tensile strenght by 20oC Rm, [N/mm2]

Povrina popr. presjeka

Cross-section A, [mm2]

1 ina- rail S60x1000 ^.3108 1000 7686 2 kvadratni – square 115x115x1000 ^.0361 400 12900 3 kvadratni – square 115x115x1000 ^.3108 1000 12900 4 pljosnati - flat 60x215x1000 ^.3108 1000 12900

Izbor temperatura ispitivanja izvren je prema ritmu valjanja valjaoni~ke pruge koji ~esto nije usaglaen sa kapacitetom obradnog centra vrueg rezanja, te se deavaju slu~ajevi odrezivanja profila i na znatno ni`im temperaturama od 850 [0C]. Zbog toga su eksperimentalna ispitivanja izvedena na sljedeim temperaturama rezanja: υ=700 i 800 [oC]. Pilni disk je jedan od najva`nijih elemenata pile od kojeg u najveoj mjeri zavisi uspjean rad i stabilnost obradnog centra. U eksperimentu su koriteni novi pilni diskovi sa pravim zubima pre~nika dp=1900 [mm], debljine δ=10 [mm], proizvo|a~a Ludwig Köhler (Njema~ka), jer se pri istroenosti zubaca znatno poveavaju gubici energije, pogorava kvalitet rezanja, prouzrokuje savijanje ivica reza, pojavljuje iskrivljenje i lomljenje zubaca [7]. Materijal ovog reznog alata ima internu oznaku UH 90 U, a zateznu ~vrstou Rm=900÷1100 [N/mm2].

Research temperature choices are made according to rolling rhythm which is not often coordinated with capacity of working centre because sometimes cutting happens at temperatures lower of 850 0C. Ekspertimental researches were therefore realised at temperatures: υ=700 i 800 [oC]. Saw disc is one of the most important parts of circular saw. Successful work and stability of working centre depends on saw disc properties. In experimental researches new saw discs are used (product of Ludwig Köhler-Germany), with the following properties: strenght teeth, disc diameter dp=1900 [mm], thickness δ=10 [mm] because wearing out of saw disc teeth increases efficiency losses, decreases cutting quality, causes bending of cuting edges, and creates cause teeth curves and fractures. The internal mark of the tool material is UH 90 U and of tensile strenght Rm=900÷1100 [N/mm2].

2.2 Izvo|enje ispitivanja Pri izvo|enju eksperimentalnih ispitivanja mjerene su elektri~ne veli~ine: struja statora, aktivna snaga i napon, kao i neelektri~ne veli~ine: temperatura rezanja, brzina obrtanja, klizanje motora i pritisak hidrauli~kog medija na pristroju za pomono kretanje. Pomou ovih veli~ina indirektnom metodom odre|eni su otpori rezanja i karakter njihove promjene, na osnovu ~ega se stvaraju uslovi za prou~avanje dinami~kih oscilatornih kretanja mehanizma glavnog i pomonog kretanja pile.

2.2 The experiment Electric variables: electric current of stator, effective power and voltage, as well as nonelectric variables: cutting temperatures, revolutions, slipage of motor and pressure of hydraulic medium of mechanism for auxilliary motion are measured during the experiment. Resisting forces and their propeties are defined by indirect method and the above mentioned parameters, which create the conditions for the research of dynamic oscillations of mechanisms for main and auxilliary saw motions.

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Principijelna shema za sprovo|enje mjerenja prikazana je na slici 2 , gdje je u praznom hodu pile (slika 2a), zbog simetri~no optereenog motora mehanizma glavnog kretanja pile, snaga mjerena jednim vatmetrom koji pokazuje snagu samo jedne faze Pw, tako da je ukupna snaga praznog hoda P0=3Pw. U svim drugim slu~ajevima mjerenja su vrena indirektno (slika 2b) - preko mjernih transformatora ugra|enih u glavni razvod pogonske hale, a potom su preko QUPI pretvara~a vremenske funkcije registrovane na brzom esto-kanalnom pisa~u BRUSH-USA, maksimalne brzine trake 125 [mm/s].

The measuring principle is shown on fig 2, where by idling (Fig.2a) power is measured with one voltmeter which shows one phase power Pw due to symetricly loaded motor of the main motion mechanism. Total idling power is Po=3Pw. In all other cases the measuring is made indirectly by measuring coverters which are built in the main switch board and all time functions are registered in then fast 6-chanell writer BRUSCH (max.tape speed 125 mm/sec.) by QUIP- transducer.

a) prazan hod b) rezanje a) Idling b) Cutting

Slika 2.Principijelna ema za sprovo|enje snimanja Fig 2. Recording principle

3. REZULTATI EKSPERIMENTALNIH ISPITIVANJA

3.1. Odre|ivanje momenata i otpora rezanja U toku procesa rezanja sve mjerene veli~ine: Pm=f(t), Is=f(t), U=f(t), nm=f(t) i p=f(t) registrovane su na traci estokanalnog pisa~a. Vrijednosti ovih veli~ina o~itavane su u ta~no odre|enim vremenskim intervalima rezanja i evidentirane tabelarno. Ra~unska obrada podataka vrena je pomou software-a posebno ura|enim za ove prilike napisanog u TURBO PASCAL-u. Model prora~una je prikazan u tabeli 2 [8]. Podaci su unoeni prema ta~kama nazna~enim u tabeli 2 kao i srednji uglovi kontakta pilnog diska sa obratkom α, ~ije vrijednosti se dobiju grafi~kom metodom [8].

3. RESULTS OF EXPERIMENTAL RESEARCHES

3.1. Establishingg cutting torques and

resisting forces During the cutting all the measured variables: Pm=f(t), Is=f(t), U=f(t), nm=f(t) i p=f(t) are registered on the tape of the 6-chanell writer. Their values are read in accurate time intervals of cutting and registered in the tables. The data are processed with the aid of a softwear written in Turbo-Pascal. They are entered according to tab.2 and so are the mid-angles (α) of contact between tool and workpiece α, whose values are obtained by grafic methods [8].

Kako je kompletan prora~un baziran na metodi intervala vremena, treba naglasiti da se kontinualna kriva promjene snage, odnosno momenta, zamjenjuje stepenastom krivom (histogramom) koja ima konstantne vrijednosti ovih veli~ina u posmatranom intervalu vremena. Veli~ine Pm i nm su o~itane sa grafika na slici 3, a α u intervalima vremena od jedne sekunde. Sve ostale veli~ine dobijene su prora~unom preko formula datih u tabeli 2. Rezanje je izvedeno sa sljedeim parametrima: v=98,1 [m/s], u=25 [mm/s], ϑ=800 [0C], debljina pilnog diska δ=10 [mm].

Complete estimation is based on the time interval method. The continual curve of power changes is exchanged with the histogram that has constant values of these variables in the observed time interval. Variables Pm and nm are read from Fig.3, and angle α is read in the one second interval. All other variables are obtained by the calculation presented in Tab.2. Cutting is realised with: v=98,1 [m/s], u=25 [mm/s], ϑ=800 [0C], saw disc thickness δ=10 [mm].

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Tabela 2. Prora~un momenata i otpora rezanja Table 2. Torque and resisting forces estimation

1. UNOS PODATAKA – DATA INPUT

1.1. MEHANIZAM GLAVNOG KRETANJA PILE – MECHANISM FOR MAIN SAW MOTION

Nazivna snaga motora – Rated motor power kW Pn=132 Nazivni broj obrtaja – Rated rpm min-1 nn=986 Nazivni moment – Rated torque Nm Mm=1278 Moment inercije rotora motora – Rotors moment of inertia kgm2 Jm=15 Stepen korisnog djelovanja motora – Mechanical efficiency of motor - η=0,93 Moment inercije preostalih rotacionih masa – Rotated parts moment of inertia kgm2 Jmeh=245 Stepen korisnog djelovanja mehanizma – Mechanical eficiency of transmision - ηmeh=0,91 Pre~nik pilnog diska – Saw disc diameter m dp=1,9 Debljina pilnog diska – Saw disc thickness mm δ=10 Sinhroni broj obrtaja motora – Synhronous rpm of motor min-1 nsin=1000 Broj obrtaja rotora u praznom hodu – Rpm by idling min-1 n0=994 Otpor rotora po fazi – Rotor resistance by phasis mΩ Rr=22,5 Dodatni otpor po fazi – Additional resistance by phasis mΩ Rd=110 Snaga praznog hoda – Power by idling kW P0=18 Prenosni odnos mehanizma – Total transmision ratio - i= 1 Brzina glavnog kretanja – Main motion speed m/s v= 98,1

1.2. MEHANIZAM POMO]NOG KRETANJA PILE – MECHANISM FOR AUXILLIARY MOTION Pre~nik klipa hidrauli~kog cilindra – Piston diameter of hydraulic cylinder mm dk=80 Brzina pomonog kretanja – Auxilliary motion speed mm/s u=25;50;75;100

1.3. OBRADAK – WORKPIECE Vrsta obratka – Type - Dimenzije – Dimension mm Povrina popre~nog presjeka – Cross-section mm2 A Zatezna ~vrstoa na sobnoj temperaturi – Tensile strenght by room temperature N/mm2 Rm=400; 1000 Temperatura obratka za vrijeme rezanja – Workpiece cuttung temperatures oC υ=700;800

1.4. VELIÎNE ODRE\ENE EKSPERIMENTOM - EXPERIMENTALLY DEFINED PARAMETERS Unaprijed izabrani interval vremena rezanja obratka – Cutting time intervals defined in advance s tri=0,25; 0,5; 1 Srednji ugao kontakta pilnog diska sa obratkom u i-tom intervalu vremena – Mid-angle (α) of contacts between saw and workpiece in the time interval “i”

° αi

Pritisak u hidrauli~kom cilindru u praznom hodu pile–Hydraulic cylinder pressure by idle stroke MPa p0 Pritisak u hidrauli~kom cilindru u vremenu ti – Hydraulic cylinder pressure in time period ti MPa p Snaga koju motor uzima iz mreè u vremenu ti – Motor power in time period ti kW Pmi Broj obrtaja motora u vremenu ti – Rpm of motor in time period ti min-1 nmi Po~etni moment – Initial torque Nm Mpo~i

2. PRORAÛNI - CALCULATIONS

2.1. GUBICI SNAGE MOTORA – MOTOR POWER LOSSES

Gubici u bakru rotora i statora pri nazivnoj snazi motora – Power losses in copper of rotor and stator by rated motor power

kW 100

8,2 nCun

PP =

Gubici u èljezu statora, trenja u leìtima i ventilaciji motora – Power losses in ferrous of stator, bearing friction and motor ventilation

kW 100

2,4 nFeTV

PP =

Gubici u bakru sa dodatnim otporom za slu~aj Pm=Pn – Power losses in copper with additional resistance by Pm=Pn

kW 100

6,9 nCund

PP =

Gubici u bakru sa dodatnim otporom za slu~aj Pm≠Pn – Power losses in copper with additional resistance by Pm≠Pn

kW

2

=

n

mCundCu P

PPP

Ukupni gubici u bakru i èljezu statora, trenja u leìtima i ventilaciji – Total power losses in copper and ferrous of stator, beraing friction and ventilation

kW FeTVCuukg PPP +=

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2.2. MEHANI^KE KARAKTERISTIKE MOTORA – MECHANICAL PROPERTIES OF MOTOR

Ugaona brzina motora – Revolution speed s-1 30n

n⋅

=π

ω

Ugaona brzina motora u praznom hodu – Revolution speed by idling s-1 30

oo

n⋅=

πω

Nazivno klizanje – Rated slippage - ( )

sin

sin

nnn

S nn

−=

Stvarno klizanje zbog dodatnog otpora u kolu rotora – Real slippage caused by additional rotor resistance

- ( )

nr

drn S

RRR

S ⋅−

=`

Moment inercije svih rotacionih masa redukovan na vratilo motora – Moment of inertia of all rotated elements reduced on the motor shaft

kgm2 2iJJJ mehm ⋅+=

Elektromehani~ka vremenska konstanta bez dodatnog otpora u kolu rotora – Electro-mechanic time constant without additional rotor resistance

s n

nemeh M

SJT

⋅⋅= 0ω

Elektromehani~ka vremenska konstanta sa dodatnim otporom u kolu rotora - Electro-mechanic time constant with additional rotor resistance

s emehn

nemeh T

SS

T ⋅=`

`

Konstanta - Constant - emeh

n

Tt

e `−

Konstanta - Constant - emeh

n

T

t

e `1−

−

2.3. MOMENTI I OTPORI REZANJA – CUTTING RESISTENCE TORQUES AND FORCES

Moment motora u i-tom intervalu vremena rezanja obratka – Motor torque in “i” time period

Nm mi

ukgmimi n

PPM

−= 9550

Moment praznog hoda – Idling torque Nm O

FeTVOmi n

PPM

−= 9550

Moment tereta u i-tom periodu optereenja – Load torque in “i” loading period

za Mti>0 – for Mti>0 Nm

`

`

1 emeh

n

emeh

n

Tt

Tt

počmiti

e

eMMM

−

−

−

⋅−=

)1( −= impoč MM

za Mti<0 – for Mti<0

Nm

`

`

)1(

emeh

n

emeh

n

Tt

Tt

počmiti

e

eMMM

−

−

−⋅−=

Efektivni (stvarni) moment tereta u i-tom intervalu vremena – Effective load torque in the time period “i”

Nm otiefi MMM −=

Efektivni pritisak u hidrauli~kom cilindru u i-tom intervalu vremena – Effective pressure in hydraulic cylinder in the time period “i”

MPa oiefi ppp −=

Otpor pomonog kretanja u i-tom intervalu vremena – Auxilliary motion resistance in the time period “i”

N 4

2πkefid

pQ =

Glavni otpor rezanja u i-tom intervalu vremena – Main motion resistance in the time period “i”

N mehp

efii i

dM

T η⋅⋅=2

- 163 -


Otpor prodiranja u i-tom intervalu vremena – Radial feed resistance in “i” time period

N i

iiii

TQR

αα

cossin⋅+

=

Odnos otpora prodiranja i glavnog otpora rezanja u i-tom intervalu vremena – Ratio of radial and main resistance in “i” time period

- TiRi /

Za obradu eksperimentalno dobijenih rezultata, pored izlo`enog, potrebno je grafi~kom metodom [8] odrediti veli~ine bitne za sam proces rezanja, kao to su visina odrezane plohe "h" i put rezanja "lr". Grafi~kim integraljenjem krive "h" dobije se kriva promjene povrine popre~nog presjeka obratka. Mehanizam glavnog kretanja pile ima ugra|en zamajac. Da bi se izvrila analiza uticaja zamajnih masa na intenzitet momenta motora, a time i na sam proces rezanja, nu`no je pratiti promjenu snage i brzine vrtnje u toku i nakon zavretka rezanja tj. prelaznog procesa do trenutka kada elektromotorni pogon iz nestacionarnog stanja (n≠const.) prelazi u stacionarno stanje (n=const.). U tabeli 3 selektivno su prikazane vrijednosti momenta motora Mm i Mpo~ kod rezanja kvadratnog profila 115x115 [mm] sa: A=12900 [mm2]; Rm=1000 [N/mm2]; ϑ=800 [°C]; δ=10 [mm]; v=98,1 [m/s]; u=25 [mm/s]; i=1,0; tr=8,4[s], Σ t=15 [s], (jer je moment tereta Mt nakon rezanja jednak nuli), na osnovu kojih se crtaju grafici promjene momenata Mt i Mm kod vrueg rezanja prikazani na slici 4, a ilustruju rezanje kvadratnog profila (slika 3a). Moment praznog hoda M0 na slici 4 ozna~ava moment potreban za obrtanje svih elemenata mehanizma glavnog kretanja pile u praznom hodu, tj. za vrijeme kada je pilni disk pogonjen, a nije u kontaktu sa obratkom. Kriva promjene momenta tereta Mt ima jasno izra`en maksimum u zoni srednjeg perioda rezanja tr. U dijelu ciklusa rezanja kada je moment tereta Mt vei od momenta motora Mm, dio optereenja preuzimaju inercijalni momenti motora i zamajca. Ovo proizlazi iz smanjenja brzine motora za vrijeme vrnog optereenja. Zbog toga se dio akumulirane kineti~ke energije (povrina ome|ena dijelovima krive Mm i Mt od ta~ke C do S) predaje na vratilo motora u vidu kineti~kog momenta Mk. U periodu smanjenja optereenja kada brzina glavnog pogona raste sve do brzine praznog hoda, u zamajne mase ponovo se akumulira kineti~ka energija (povrina ome|ena dijelom krive Mm od ta~ke S do F i M0, te dijelom krive Mt do ta~ke N do S). Ovim se smanjuje neravnomjernost obrtanja za vrijeme procesa rezanja. Otpor pomonog kretanja pile odre|en je indirektnom metodom, mjerenjem pritiska u tla~nom vodu hidrocilindra u toku praznog hoda p0 = f (t) za vrijeme rezanja p = f (t), slika 4. Da bi se lake pratile promjene ovog pritiska u toku rezanja, krive 1 i 2 snimljene su sa brzinom trake pisa~a 5 [mm/s], a 3 i 4 sa 25 [mm/s]. Probni uzorci su rezani na temperaturi 700[0C]. Ispitivani uzorci bili su od ^.0361 sa Rm=400 [N/mm2] izuzev ine koja je od ^.3108 sa Rm=1000 [N/mm2].

Parameters such as height of cut section "h" and cutting displacement"lr" defined by graphic method are needed for experimental results analysis [8]. The cross section changes curve is obtained from the graphicly integrated curve "h". Flywheel is built in the mecahanism for main saw motion. For analysis of the flywheel mass effect on the motor torque intesity , as well as on the overall cutting process, the power and rotating speed during and at the end of the cutting proces needto be followed i.e. during transformation process until the moment when electromotive drive crosses from non-stationary state (n≠const.) to stationary state (n=const.) Table 3 presents the values of torques, Mm and Mpo~ of the square section cutting with the following properties: dimensions 115x115, A=12900[mm2]; Rm=1000 [N/mm2]; ϑ=800[°C]; δ=10[mm]; v=98,1[m/s]; u=25 [mm/s]; i=1,0; tr=8,4[s], Σ t=15[s] (bacause after the cutting, the load torque Mt=0). Diagrams torque (Mm and Mpo~) changes are made on the basis of the values obtained from Tab.3, and they ilustrate the square section cutting process (Fig.3). Idling motor torque Mo (Fig.4) denotes the moment necessary for rotation of all the elements of mechanisms for main saw motion, when the saw disc is driven and has no contacts with the workpiece. The curve of load torque changes Mt has a clear by stated maximum in the middle of the cutting period "tr". Due to speed reduction during the maximal loading a part of load is taken by motor and flywheel moments of inertia. At the moment of cutting period when torque Mt is bigger then motor torque Mm. A part of accumulated kinetic energy (area bounded with a portion of curve Mm and Mt from point C to point S) is transfered to the motor shaft in the form of kinetic torque (Mk). In the period of load decreasing, when motion drive speed grows up to idling speed, flywheel masses accumulate kinetic energy (area bounded with a portion of curve Mm from point S to point F and Mo, and a portion of curve Mt from point N to point S). In this way, vari-able rotation is decreased during the cutting process. Auxilliary motion resistance is defined by indirect method, by measuring of pressure on the pressure side of the hydraulic cylinder during idling p0=f(t) in the cutting process p=f(t), Fig.4. In order to following the pressure changes during the cutting more easily, curves 1 and 2 are recorded by the writer tape speed of 5[mm/s], and curves 3 and 4 with speed of 25 [mm/s]. Workpieces are cut at the temperature of 700[0C]. The treated workpieces were made of: steel ^.0361 with Rm=400[N/mm2] except for the rail which was made of: ^.3108 with Rm=1000[N/mm2].

- 164 -


p= f(t)

2bara/pod 2bara/pod

22,3min-1/pod 22,3min-1/pod

10V/pod 10V/pod

15A/pod 6A/pod

2,6kW/pod

nm= f(t)

U= f(t)

Is= f(t)

Pm=f(t)

5,2kW/pod a ) 115x115 mm b ) [ina (Rail) S 60 Rm=1000 [N/mm2] Rm=1000 [N/mm2] Slika 3. Snimci elektri~nih i neelektri~nih veli~ina u procesu rezanja

Fig.3. Electric and nonelectric parameters of the recorded cutting

Tabela 3. Rezultati prora~una Mm i MPo~ Table 3. Calculation results of Mm i MPo~

Pm [kW]

nm [min-1]

Mm [Nm]

MPo~ [Nm]

t [s]

67,6 973,0 576,5 824,6 1 47,8 977,0 396,8 576,5 1 33,8 985,0 265,9 396,8 1 27,5 989,6 206,6 265,9 1 21,8 991,7 153,2 206,6 1 20,0 992,5 136,3 153,2 1 18,3 993,5 120,3 136,3 1

0

200

400

600

800

1000

1200

1400

0 2 4 6 8 10 12 14 16

t1t2

tr(period rezanja)

M m =M k

115x115 mm Rm =1000 N/mm2

ϑ=800 °C v=98,1 m/s u=25 mm/s δ=10 mm

Mom

ent m

otor

a (M

m ) i

tere

ta (M

t ), N

m

Vrijeme t, s

M0

tS

Mm

K

N FC

Mk

0<t<t1; M t>M m; M k>0t1; M t=M tmax; M t>M m; M k>0

t2; M t=M m; M k=0tr; M t=0; M m=M k

t>tr; M t=0; M m=M k

M t

Slika 4. Konstrukcija krivih Mm=f(t) i Mt=f(t) Fig 4. Construction of curve Mm=f(t) i Mt=f(t)

- 165 -


1 2

3 4

2

4

6

0

Prazan hod pile

Pritisak p , MPa

Rezanje ( R ) R

Vrijeme t , s

R R

u=25 mm/s ; Šina S-60 u=50 mm/s ; Profil I-18 u=75 mm/s ; 115x115 mm u=100 mm/s ; 60x215 mm

2 1

3 4

Slika 5. Promjene pritiska u tla~nom vodu hidrauli~kog cilindra Fig 5. Pressure changes in the hydraulic cylinder pressure side

Poveanjem brzine pomonog kretanja pile "u" poveava pritisak p0 koji u periodu pokretanja ima izraèn oscilatorni karakter sa veim amplitudama (krive 3 i 4). Vie je uzroka ovoj pojavi. Sila potrebna za pokretanje sistema iz stanja mirovanja u stanje jednolikog kretanja vea je od sile potrebne za odràvanje ovog kretanja (koeficijent trenja mirovanja vei je od koeficijenta trenja kretanja). Sila trenja izme|u podmazanih povrina ne zavisi od materijala tijela koji se dodiruju, ve od viskoznosti maziva i brzine kretanja i proporcionalna je navedenim veli~inama.

Increase of the auxilliary motion speed "u" causes the increase of pressure po that has an expressed hunting character with bigger amplitudes (curves number 3 and 4) in the initial period. This phenomenon is occurs for many reasons. The initial force, used to move system from stationary steady to uniform motion is bigger than the main motion force (the coefficient of friction in rest is bigger than the coefficient of friction in motion). Friction force between lubricated planes is independent of material but depends on lubricant viscosity and velocity and this force is proportional with these sizes.

Sila trenja izme|u kliznih povrina, otpor vazduha i unutranje trenje usljed nepotpune elasti~nosti materijala priguuju oscilacije suporta pile u fazi pokretanja. Usljed sila priguenja period oscilovanja sistema je kratak i suport pile se sa po=const. primi~e ka rezanom metalu. Ovo je veoma vaàn podatak jer omoguava odre|ivanje efektivne vrijednosti pritiska u hidrocilindru za vrijeme rezanja obratka, ~ime se stvaraju uslovi za definisanje otpora pomonog kretanja pile. Ispitivanjem je utvr|eno da otpor pomonog kretanja pile Q zavisi od oblika obratka i njegovog poloàja u odnosu na pilni disk. Sli~an zaklju~ak vrijedi i za glavni otpor rezanja T i otpor prodiranja R ~iji dijagrami su prikazani na slikama 6 i 7. Promjene ovih otpora date su u vidu kontinualnih krivih, pri ~emu izrazi (6) i (9) predstavljaju glavne otpore rezanja T, (7) i (10) otpore pomonog kretanja Q, a (8) i (11) otpore prodiranja R. Promjene intenziteta ovih otpora date su kao funkcije polinomnog oblika treeg reda ~iji su koeficijenti korelacije "R" u svim slu~ajevima bliski jedinici. U ovim jedna~inama nezavisno promjenljiva x predstavlja visinu odrezane plohe "h" , a y odgovarajui otpor.

Friction force between sliding planes, air resistance force and inner friction caused by incomplete material elasticity damp supports vibrations during the initial movement period. Because of damping, vibration peri-od of system is short and the saw support approac-hes the workpiece with p0=const. This is very impo-rtant because it enables definition of effective pressure values in hydrocylinder during the cutting, and it crea-tes the conditions for feed resisting force definition. Experimental researches show that feed resisting force Q depends on the workpiece form and its position towards the saw disc. Similar can be concluded for the main cut resisting force T and radial resisting force whose diagrams are presented on fig. 6 and 7. Changes of these resisting forces are presented as continual curves, and equations (6) and (9) are main cut resisting force T, (7) and (10) present feed resisting force Q, equations (8) and (11) are radial resisting forces. Intensity changes of these resisting forces are presented as cub polinom functions where the correlation coefficient "R" equals 1 in almoast all samples almost 1,0. In all the equations the independent variable "x" presents the hight of cut section "h" and variable "y" presents appropirate resisting force.

- 166 -


Kod rezanja kvadratnog profila (slika 6) otpori imaju jasno izra`en maksimum u srednjem dijelu perioda rezanja. [to se ti~e komponenti otpora kod rezanja ine, njihove ekstremne vrijednosti u potpunosti se ne podudaraju sa ekstremnim vrijednostima krive promjene povrine popre~nog presjeka. Ovo se objanjava intenzivnijim hla|enjem vrata ine zbog tanjeg presjeka i relativno niskom brzinom pomonog kretanja pile "u" za vrijeme rezanja.

With the square bar steel cutting (Fig.6) the resisting force has an expressed maximum in the middle part of the cutting period. In the case of components of resisting forces by rail cutting, their extreme values do not correlate fully the extreme values of cros-section changes curve. That can be explained by more intensive cooling of rail neck due to thin section and relatively lower auxilliary motion speed "u" during the cutting.

0 200 400 600 800

1000 1200 1400 1600

0 2 4 6 8

0 50 100 150 210

T, N Q, R, Nx10

8.4 t r , s

l r , mm

y=16.407x 3 -911.57x 2 +6436.3x+659.02 R 2 =0.9453

y= - 3.7466x 3 -20.451x 2 +429.04x-27.52 R 2 =0.9801

y=23.607x 3 -773.75x 2 +4787.7x+550.51 R 2 =0.9368

115x115 mm R m =1000 N/mm 2

ϑ =800 ºC v=98.1 m/s u=25 mm/s

δ = 10 mm T Q

R

(6)

(7)

(8)

Slika 6. Grafi~ki prikaz komponenti otpora kod rezanja kvadratnog profila 115 x115 mm Fig.6. Diagram of resisting forces components in square bar (115x115) cutting

0 100 200 300 400 500 600 700 800

0 2 4 6 8 9.8 0 50 100 150 200 245

t r , s

l r , mm

T, N Q, R, Nx10 R

Q T

Šina S 60 R m =1000 N/mm 2 ϑ =800 ºC v=98.1 m/s u=25 mm/s δ= 10 mm

y=0.3127x 3 -22.905x 2 +194.98x+43.932 R 2 =0.8941

y=0.057x 3 -163.71x 2 +1579.5x+905.85 R 2 =0.8872

y=3.6354x 3 -264.93x 2 +2311.3x+1203 R 2 =0.8342

(9)

(10)

(11)

Slika 7. Grafi~ki prikaz komponenti otpora kod rezanja ine S60 Fig.7. Diagram of resisting forces components in rail S60 cutting

- 167 -


4. ZAKLJUÂK Primjena indirektne metode za odre|ivanje momenata i otpora kod odsijecanja profila u vruem stanju mogua je samo u slu~aju obrade eksperimentalno dobijenih podataka metodom vremenskih intervala. Promjene momenata i otpora rezanja definiu se kao tipi~ne karakteristike trajnih pogona sa intermitiranim optereenjem. Momenti tereta, a time i otpori rezanja, mijenjaju se i postiù maksimalne vrijednosti koje pribli`no odgovaraju trenucima stvaranja najvee strugotine. Za ravnomjeran rad glavnog pogona pile veliki zna~aj ima zamajac. Na slici 4 uo~ava se da je moment na vratilu motora za vrijeme veeg perioda rezanja manji od momenta tereta, dok je za vrijeme pauze vei, ~ime se smanjuju neravnomjernosti obrtanja primjenom asihronog motora. U mehanizmu pomonog kretanja pile povremeno se javljaju sopstvene oscilacije ne samo u reìmima uklju~enja i ko~enja, nego i djelimi~no u periodu rezanja, pogotovo tankostjenih profila; to je brzina pomonog kretanja pile vea, to su i sopstvene oscilacije mehanizma izraènije. Promjene reìma rezanja uti~u na poveanje ili smanjenje otpora rezanja. Kod rezanja obratka u trenutku maksimalnog presjeka strugotine ustanovljeno je da je R=(12...15)T za dati reìm rezanja navedenih obradaka. U daljim istraìvanjima odsijecanja ireg spektra obradaka objavljenim u radu [8] ustanovljeno je da se vrijednosti za R kreu u granicana R=(6...15)T, ~ime je potvr|en i donekle modificiran empirijski obrazac (5). Moè se zaklju~iti da je odre|ivanje momenata i otpora kod odsijecanja profila u vruem stanju dosta komplikovano i podrazumijeva veoma skupa eksperimentalna istraìvanja u realnim (industrijskim) uslovima.

4. CONCLUSION Indirect method for torque and resisting forces definition of hot section cutting is possible only by pocessing of experimental results using time interval method. Torque and resisting forces changes are defined as typical characteristics of permanent motor drive under intermittent load. Load torque and cut resisting forces change and read maximum values that approximately correspond to the momonts of biggest saw-dust. Flywheel is very important for uniform work of main saw drive. Motor shaft torque during the greater cutting period is lower than load torque (Fig.4), but during the pause, the situation is reversed, and in that way the non-uniform revolving is decreased by applying the asynhronous electric motor. In mechanism for auxilliary saw motion free vibrations sometimes appear not only by starting or breaking then but partialy by cutting, especially thin sections; bigger auxilliary motion speed causes bigger free vibrations of the mechanism. Change in the cutting regimes causes decreasing or increasing of cut resisting forces. By cutting at the moments of maximum saw-dust, cross-section is registrated R=(12...15)T for the chosen cutting regime. In the further researches of cutting with the bigger spectrum of workpieces, presented in the paper [8], it is established that values R=(6...15)T, which affirms and partially modifies the empiric equation (5). Definition of torques and resisting forces of hot section cutting is rather complicated and very expensive experimental researches in operating conditions are necessary.

5. LITERATURA - REFERENCES [1] Fedor Raji: Rezni i stezni alati, Tehni~ka knjiga, Beograd, 1984. [2] Branko Ivkovi: Teorija rezanja-Osnovi mehanike, termodinamike i ekonomije procesa rezanja, Kragujevac, 1991. [3] Malik Kulenovi: Prilog eksperimentalnoj analizi habanja reznog alata pomou mikrora~unala, II Me|unarodni nau~no-stru~ni skup "Tendencije u razvoju mainskih konstrukcija i tehnologija", Zenica, 1995. [4] Al-Shareef, K.J.H. Brandon, J.A.: On the effect of variations in the Design Parameters on the Dynamic Performance of Machine Tool Spindle-

bearing Systems, Int.Journal of Machine Tools Manufactoring, Vol.30,1990. [5] B.L.Gligorovi: Teorija i tehnika merenja I-Mehani~ko-elektri~ni mjerni sistemi, Nau~na knjiga , Beograd, 1984. [6] A.I.Celikov: Mehanizmi prokatnih stanov, Magiz, Moskva, 1946. [7] V.D.Kuznjecov: Izabranije trudi-fizika rezanja i trenija metalov i kristalov, Izdateljstvo "Nauka", Moskva, 1977. [8] R.Seferovi: Izrada modela algoritma za izbor dinami~kih kontrukcionih veli~ina pila za vrue rezanje, Magistarski rad, Mainski fakultet u Zenici, 1997.

- 168 -

Mainstvo, 3 (6), 169 – 176, (2002) S.Isi: EGZAKTNA I NUMERI^KA ANALIZA….

EGZAKTNA I NUMERI^KA ANALIZA SOPSTVENIH POPRE^NIH VIBRACIJA AKSIJALNO OPTERE]ENE

KONZOLE

Safet Isi, Univerzitet ''Dèmal Bijedi'', Mainski fakultet Mostar

REZIME

U radu je predstavljen egzaktni, analiti~ki i numeri~ki, pristup linoscilacija aksijalno optereene Ojlerove konzole. Pokazano je da sizraziti u eksplicitnom obliku, nego se moè odrediti za konkretnmetoda rjeavanja jedna~ina sa jednom nepoznatom. Ta~no antestiranje ta~nosti numeri~ke metode rjeavanja, kao i ta~nosti posNumeri~ko rjeenje odre|eno je metodom kona~nih elemenata.

Klju~ne rije~i: konzola, aksijalna sila, sopstvene popre~numeri~ko rjeenje, metod kona~nih elemen

EXACT AND NUMERICAL ANALYSIS EIGENVIBRATION OF AXIALY LO

Safet Isi, University ''Dèmal Bijedi'' Mostar, Facult

SUMMARY

In this paper exact analytical and numerical approach to linear aof axialy loaded clamped Euler column is presented. It has been not be found in explicit form. The solution could be found onlynumerical methods for solution of equation with one unknown vsolution finite elements method is described. Exact analytical sonumerical solution and the existing paproximate analytical solution.

Keywords: clamped column, axial load, transversal eigenvibrasolution, finite elements method

1. UVOD

Aksijalno optereenje mijenja sopstvene vrijednosti sopstvenih popre~nih oscilacija greda. Kad je ono jednako kriti~nom, sopstvena frekvencija postaje nula [1], [2]. Aksijalno optereenje je svakodnevno prisutno u inìnjerskim problemima (termi~ki naponi, monta`ni naponi, itd.), pa mora biti uzeto u obzir pri konstruisanju dinami~ki optereenih grednih nosa~a. Ovaj efekat, tako|e, moè biti iskoriten za upravljanje sopstvenim vrijednostima i omoguiti koritenje nosa~a u razli~itim uslovima eksploatacije bez promjene geometrijskih osobina ili materijala. U literaturi [1], [2] postoji ta~no analiti~ko rjeenje za sopstvene vrijednosti popre~nih oscilacija proste grede nepromjenljivog popre~nog presjeka optereene aksijalnom silom.

1. INTROD Axial load vibration of eigenfrequencAxial load engineering (Therefore, it dynamically loused for manuse of bechanges of g In the referesolution of uniform cross

- 169 -

IZVORNI NAU^NI RAD

earnoj analizi sopstvenih popre~nih e ta~no analiti~ko rjeenje ne moè e probleme koritenjem numeri~kih aliti~ko rjeenje iskoriteno je za tojeih pribli`nih analiti~kih rjeenja.

ne oscilacije, analiti~ko rjeenje, ata

OF TRANSVERSAL ADED COLUMN

y of Mechanical Engineering
ORIGINAL SCIENTIFIC PAPER nalyses of transversal eigenvibration shown that analytical solution could for concrete problems introducing ariable. For approximate numerical lution has been used to test the
tions, analitical solution, numerical

UCTION

changes eigenvalues of transversal beams. When this load is critical, y becomes zero [1], [2]. is an everyday phenomenon in

thermal stress, assembling stress, etc.). must be considered in design of aded beams. This effect may also be aging of eigenvalues and enable the am in different conditions without eometry or materials.

nce [1], [2] there is exact analiytical eigenvibration for simple beam with -section, loaded by axial force.


Za konzolu je u [1] dato samo aproksimativno rjeenje za najniù sopstvenu frekvenciju, bazirano na Rejlijevom koeficijentu

( )2

23.5261 5ω 1µ 14

PlBBl

= − , (1.1)

gdje su: ω - sopstvena frekvencija, l - duìna grede, P - aksijalna sila (pozitivna za pritisak), µ - masa jedini~ne duìne, B - krutost popre~nog presjeka na savijanje, pri ~emu je za oblik oscilovanja u0 = u0(x) (Sl. 1) uzeta jedna~ina elasti~ne linije konzole savijene nepromjenljivim kontinualnim popre~nim optereenjem. U ovom radu je prestavljen postupak odre|ivanja ta~nog analiti~kog rjeenje uticaja aksijalne sile na sopstvene vrijednosti popre~nih vibracija konzole konstantnog popre~nog presjeka. To rjeenje je uzeto za testiranje pribli`nih rjeenja (navedenog analiti~kog, i numeri~kog), koji se moraju primijeniti kad je ta~nu analizu teko ili nemogue izvesti, tj. kad greda nema konstantan popre~ni presjek i ima vie razli~itih oslonaca. Za numeri~ko rjeavanje navedenog problema iskoriten je jedan od naj~ee koritenih metoda - metod kona~nih elemenata (MKE).

For clamped column in [1] only aproximate solution has been presented based on Rayglah coefficient, without estimation of error

( )2

23.5261 5ω 1µ 14

PlBBl

= − , (1.1)

where: ω - eigenfrequency, l – beam length, P - axial force (positive in compression), µ - unit length of mass, B - bending stiffness of cross section, and for eigenmode of vibration u0 = u0(x) (Fig. 1) equation of elastic line of clamped column bended with uniorm distributed load has been used. The paper presents exact analytical solution of the axial force effects on egenvalues of transversal vibrations of clamped column with constant cross section. This solution has been used for test of aproximate solutions (the above presented analitycal, and numerical), which should be introduced when analytical solving is difficult (or impossible). It commonly appears when beam has non-constant cross section and has more different supports. For numerical approach, one of the most commonly used methods - finite elements method (FEM) has been used.

2. TA^NO ANALITI^KO RJE[ENJE Analiza sopstvenih popre~nih oscilacija greda podrazumijeva rjeavanje diferencijalnog problema sopstvenih vrijednosti definisanog diferencijalnom jedna~inom [1], [2]

(IV) 20 0 0 0Bu Pu uµω′′+ − = , ( 2.1 )

i grani~nim uslovima

2. EXACT ANALITICAL SOLUTION Analysis of transversal eigenvibration of axially loaded column requires solving differential eigenvalue problem defined by the forth order ordinary differential equation [1], [2]

(IV) 20 0 0 0Bu Pu uµω′′+ − = , ( 2.1 )

and boundary conditions

0 0 0 0 0(0) = (0) = ( ) = 0, ( ) = - ( )u u u l Bu l Pu′ ′′ ′′′ ′ l , ( 2.2 )

Slika 1. Mogui sopstveni oblici oscilovanja aksijalno optereene konzole. Nepoznate funkcije u0(x) moraju zadovoljiti jedna~inu (2.1) i grani~ne uslove (2.2).

Figure 1. Possible eigenmodes of vibration of axially loaded column. Unknown function u0(x) has to satisfy equation (2.1) and boundary conditions (2.2).

- 170 -


Korijeni karakteristi~ne jedna~ine diferencijalne jedna~ine ( 2.1 ) su:

2 2

1,2 142

P P kBµωλ + −

=± =± ( 2.3 )

2 2

3,4 242

P P ikBµωλ − + −

=± =±

Korijeni λ1,2 su realni, λ3,4 su imaginarni, i k2 > k1 ≥ 0. Rjeenje jedna~ine (2.1) ima oblik

The roots of characteristic equation of the differential equation ( 2.1 ) are:

2 2

1,2 142

P P kBµωλ + −

=± =± ( 2.3 )

2 2

3,4 242

P P ikBµωλ − + −

=± =±

The rotts λ1,2 are real, λ3,4 are imaginary and k2 > k1 ≥ 0. The solution of equation (2.1) has the following form

0 1 1 2 1 3 2 4 2cosh sinh cos sinu C k x C k x C k x C k= + + + x . ( 2.4 )

Uvrtavanjem rjeenja (2.4) u grani~ne uslove (2.2) dobijamo homogeni sistem linearnih algebarskih jedna~ina sa nepoznatim Ci (i = 1, ..., 4), ~ija je frekventna jedna~ina

By including the solution equation (2.4) in the boundary conditions (2.2), we obtain homogenous set of linear algebric equations with the unknown Ci (i = 1, ..., 4), whose frequent equation is

24 41 2 1 2 1 2 1 22

22 21 2 1 22

( , ) ( ) sinh sin

(2 )cosh cos 0

P PD k k k k k k k l k lBBPk k k l k lB

= + − −

+ + =

+. ( 2.5 )

Iz jedna~ina (2.3) slijedi

2 22 1

Pk k B− = . ( 2.6 )

Uz jedna~inu (2.6), jedna~ina (2.5) postaje jedna~ina sa jednom nepoznatom varijablom, ali je i dalje transcendentna, i mora biti numeri~ki rjeena za svaki novi konkretan problem.

From the equations (2.3) the following

2 22 1

Pk k B− = . ( 2.6 )

Using the (2.6) equation, (2.5) becomes an equation with one unknown varaiable, but it is still transcendent, and should be solved for every particular problem applying numerical methods.

Slika 2. Grafik funkcije f = D(k2, k1(k2)) za konkretne vrijednosti P, B, µ, i l . Korijen k2 =P/B daje trivijalno rjeenje ω = 0.

Figure 2. Graph of function f = D(k2, k1(k2)) for particular values of P, B, µ, and l . The root k2 =P/B gives trivial solution ω = 0.

- 171 -


Uvo|enjem bezdimenzinalnih parametara

i , frekventna jedna~ina

se mo`e predstaviti u obliku pogodnijem za optu analizu

2 /p Pl B= 2 4 /v lµω= B BIntroducing non-dimensional parameters

and , the frequent

equation may be expressed in the form more appropriate for general analysis

2 /p Pl B= 2 4 /v lµω=

( )

( ) 042

cos42

cosh2

42sin

42sinh2,

22222

222

=++⋅++−+

+++⋅++−−=

vppvpppv

vppvppvpvvpD (2.7)

0 2 4 6 8 100,0

0,5

1,0

1,5

2,0

2,5

12

p

v

Aproximate solution (1.1) Približno rješenje (1.1) Equation (2.7) Jednačina (2.7)

Slika 3. v-p dijagram za najni`u sopstvenu frekvenciju aksijalno pritisnute konzole konstantnog popre~nog presjeka

Figure 3. v-p diagram for axially compressed uniform column for the lowest eigenfrequency.

U ravni v-p jedna~ina (2.7) je predstavljena beskona~nim brojem krivih u prvom kvadrantu, koje monotono opadaju od vrijednosti koje odgovaraju vlastitim silama izvijanja, do minimalnih vrijednosti koje odgovaraju sopstvenim frekvencijama slobodnih oscilacija. Uvrtavanjem rjeenja jedna~ine (2.5) ili (2.7) (tj. sopstvenih frekvencija) u sistem definisan sa rjeenjem (2.4) i grani~nim uslovima (2.2), odre|ujemo nepoznate

),,( kii CxPCC = , i = 1, …, 4; i ≠ k, (2.8)

gdje je Ck proizvoljno odabran izme|u Ci i fiksiran.

In the v-p plane, the equation (2.7) is represented by infinite number of branches in first quadrant, which monotonously descend from values corresponding to buckling forces to minimum values corresponding to eigenfrequencies of free vibration. Inserting solutions of (2.5) or (2.7) (i.e. eigenfrequencies) in the system defined by (2.4) and boundary conditons (2.2), we obtain the unknown values

),,( kii CxPCC = , i = 1, …, 4; i ≠ k, (2.8)

where Ck is arbitrary choosen between Ci, and fixed.

- 172 -


To pokazuje da se sopstveni oblici oscilovanja, za razliku od proste grede, mijenjaju sa promjenom aksijalne sile (Sl. 4). Ako je aksijalna sila manja od najni`e sile izvijanja (to ima prakti~an zna~aj), samo je prvi oblik oscilovanja zna~ajnije izmijenjen.

It shows that eigenmodes change as axial load changes (Fig. 4), which is not the case with a simple beam. If axial load is less then the lowest buckling force (which has practical meaning), only first eigenmode is significantly affected.

Slika 4. Prva tri oblika oscilovanja konzole. Oblici oscilovanja se mijenjaju od a) za P = 0 do b) za kriti~nu sila izvijanja. Oblici oscilovanja su skalirani na jednaka pomjeranja slobodnog kraja konzole. Figure 4. First three eigenmodes of clamped column. Eigenmodes change from a) for P = 0 to b) when P is critical buckling load. Eigenmodes are scaled to equal displacemets of free end.

3. NUMERI^KO RJE[ENJE METODOM KONA^NIH ELEMENATA

Analizirajmo gredu proizvoljno promjenljivog popre~nog presjeka kao model podijeljen na n grednih kona~nih elemenata, kruto spojenih u ~vorovima (Sl. 5 a). Kona~ni elementi imaju nepromjenljiv popre~ni presjek. ^vorovi imaju dva stepena slobode – pomak u du` ose y i nagib ϕ (Sl. 5 b).

3. NUMERICAL SOLUTION USING FINITE ELEMENTS METHOD

Let us consider a beam of arbitrary distribution of cross section as a model divided into n finite elements, rigidly connected in nodes. Finite elements have uniform cross section (Fig. 5 a). Nodes have two d.o.f. – displacement u along axes y and rotation ϕ (Fig. 5 b).

Slika 5. a) Greda diskretizirana na kona~ne elemente, b) gredni kona~ni element. Figure 5. a) Beam divided into finite elements, b) beam finite element.

- 173 -


Potencijalna i kineti~ka energija sistema su Potencial and kinetic energy of the structure are

Tσ

1 ([ ] [ ]) 2 Pν = −d K K d , T1 [ ] 2τ = d M d& &

=

=

=

=

(3.1)

gdje su [K], [M], [Kσ] matrica krutosti, mase i geometrijska matrica i d je vektor pomaka konstrukcije. Pretpostavljamo da je d vektor pomaka koji zadovoljava grani~ne uslove. Neka on u toku vremenskog intervala [t1, t2] doìvi perturbaciju δd, koja je proizvoljna unutar intervala a i~ezava na njegovim granicama. Primjenom Hamiltonovog

principa i grani~nih uslova za

vremenski interval, dolazimo do jedna~ine ravnoteè

( )2

1

δ 0t

tdtν τ− =∫

where [K], [M], [Kσ] are structural stiffness, mass and stress stiffening matrix and d is displacement vector. We presume that d is an admissible displacement vector which satisfies boundary conditions. Let it be perturbed during time interval [t1, t2] by an amount δd, which is arbitrary inside the interval and vanishes at its ends. Applying Hamilton's principle

and boundary condition on time

interval, we obtain equilibrium equation

( )2

1

δ 0t

tdtν τ− =∫

σ([ ] [ ]) [ ] 0P− +K K d M d&& . (3.2)

Poto greda vri harmonijske oscilacije, vektor pomaka moèmo pisati kao [4]

d = d0 sin ωt (3.3) gdje je d0 oblik oscilovanja. Uvrtavanjem jedna~ine (3.3) u jedna~inu ravnoteè (3.2) dobivamo njen kona~an oblik

2σ 0([ ] [ ] ω [ ]) 0P− −K K M d . (3.4)

σ([ ] [ ]) [ ] 0P− +K K d M d&& . (3.2)

Due to the beam's harmonical oscilations, we can write displacement vector as [4]

d = d0 sin ωt , (3.3) where d0 is eigenmode of vibration. Inserting the equation (3.3) into the equilibrium equation (3.3) we obtain its final form

2σ 0([ ] [ ] ω [ ]) 0P− −K K M d . (3.4)

0 2 4 6 8 10 120,0

0,5

1,0

1,5

2,0

2,5 Analytical (Equation 2.7) Two finite elements One finite element

Analitički (Jednačina 2.7) Dva konačna elementa Jedan konačni element

p

v Slika 6. MKE aproksimacija dijagrama funkcije (2.7) za najniù soptvenu

frekvenciju konzole konstantnog popre~nog presjeka. Figure 6. FEM aproximation of function given by (2.7) for the

lowest eigenfrequency of uniform column.

- 174 -


Jedna~ina (3.4) ima oblik generaliziranog algebarskog problema soptvenih vrijednosti, i moè biti koritena za analizu stabilnosti ( za ω = 0 ) i slobodnih oscilacija grednih nosa~a ( za P = 0). Unoenjem grani~nih uslova, to je u MKE standardna operacija, moè biti koritena za analizu sopstvenih popre~nih oscilacija aksijalno optereenih greda za proizvoljne slu~ajeve oslanjanja.

The equation (3.4) has the form of generalized algebric eigenvalue problem, and may also be used both for analysis of buckling load (for ω = 0) and free vibrations (for P = 0). Introducing different boundary conditions, which is a standard FEM operation, it can be used for analysis of transversal eigenvibration of axially loaded beams with arbitrary support conditions.

2

3

4

5

6

PkritičnoAksijalna sila PPcritical

ω 1 ω 2 ω 3

Bro

j kon

ačni

h el

emen

ata

n

Num

ber

of f

inite

ele

men

ts

n

Axial Force P

Slika 7. Broj kona~nih elemenata potreban za izra~unavanje

soptvenih frekvencija sa relativnom grekom manjom od 0.1%. Figure 7. Number of finite elements neccessery to calculate

eigenfrequencies with relative error less then 0.1%.

4. ZAKLJUÂK Predstavljena je egzaktna analiti~ka i numeri~ka analiza sopstvenih popre~nih oscilacija aksijalno optereene konzole konstantnog popre~nog presjeka. Za razliku od proste grede, aksijalna sila mijenja i sopstvenu frekvenciju i oblik oscilovanja konzole. Analiti~ko odre|ivanje sopstvenih vrijednosti zahtijeva numeri~ko rjeavanje transcendentne frekventne jedna~ine za svaki konkretan problem. Pojednostavljenje odre|ivanja sopstvenih vrijednosti postiè se izraàvanjem frekventne jedna~ine u funkciji bezdimenzionalnih parametara (npr. za grafi~ko odre|ivanje soptvenih frekvencija). Pokazano je da postojea pribli`na relacija za izra~unavanje najniè sopstvene frekvencije moè biti koritena samo za aksijalne sile mnogo manje od kriti~ne sile izvijanja. Prokazani numeri~ki metod, metod kona~nih elemenata, omoguva analizu sopstvenih popre~nih vibracija greda proizvoljno promjenljivog popre~nog presjeka i uslova oslanjanja. Rezultati zadovoljavajue ta~nosti dobijaju se ve sa skromnim brojem kon~nih elemenata.

4. CONCLUSIONS An exact analytical and numerical analysis of transversal eigenvibrations of axially loaded column with constant cross section, is presented. As opposed to a simple beam, axial load changes both eigenfrequencies and eigenmodes of vibration. Analytical calculation of eigenvalues requires numerical solution of transcendent frequent equation for every particular problem. Simplification may be done expresing frequent equation in the function of nondimensional parameters (e.g. for graphic determination of eigenfrequencies). It has been shown that the existing aproximate solution for the lowest eigenfrequency may be used only for axial forcess signifficantly less then critical buckling force. The presented numerical method, the finite elements method, allows analysis of transversal aigenvibration of beam with arbitrary distribution of cross section area and with different support condition. Results with satisfying accuracy are obtained already with a small number of finite elements.

- 175 -


5. LITERATURA [1] D. Rakovi, “Teorija oscilacija”, Tree izdanje,

Gra|evinska knjiga, Beograd, 1974. [2] Ziegler Hans, ''Principles of Structural Stability'',

Blasidell Publishing Company, Watham, Massachusetts, Toronto, London, 1968.

[3] A. Pi~uga, “Uvod u metod kona~nih

elemenata”, Svjetlost Sarajevo, 1985.

[4] Cook R., Malkus D.S., Plesha M, ''Concepts and Aplication of Finite Element Analyses'', John Willey & Sons, New York, 1988.

[5] S. Isi, ''Analiza uticaja aksijalnog optereenja na

oscilovanje konstrukcija – metodom kona~nih elemenata'', magistarski rad, Mostar, 2000.

- 176 -

Mainstvo, 3 (6), 177 – 184, (2002) M.Bijedi: RAÛNANJE PRITISKA ZASI]ENJA….

RAÛNANJE PRITISKA ZASI]ENJA ÎSTIH SPOJEVA JEDNAÎNOM STANJA SOAVE-BENEDICT-WEBB-RUBINA

Dr. sci. Muhamed Bijedi, doc., Tehnoloki fakultet Tuzla, Bosna i Hercegovina

REZIME

U ovom radu dat je algoritam pro a~una pritiska zasienja ~istih spBenedict-Webb-Rubina. Kao kriterij fazne ravnoteè koriten je uslote~ne faze. Algoritam je testiran na primjeru pro a~una pritiska zasetana i propana. Dobijeni rezultati pokazuju odli~no slaganje sa pritisak zasienja u cijelom intervalu tempera ure, od trojne do kriti~ne

r

r

, t

f

l

Klju~ne rije~i: Pritisak para, jedna~ina stanja, matemati~ki model

PURE COMPOUND VAPOR PRESSURE THE SOAVE-BENEDICT-WEBB-RUBIN EQ

Muhamed Bijedi, Ph.D., Faculty of Technology, Tuzla, B

SUMMARY

In this paper an algorithm for pure compound vapor pressure calcuRubin equation of state is given. The equality o fugacities in vaporthe criteria of phase equilibrium. The algorithm is tested by calculargon, nitrogen, oxygen, ethane and propane. The results obtained ecorresponding experimenta data in the whole interval of temperature f

Key words: Vapor pressure, equation of state, mathematical mode

1. UVOD Fundamentalni kriterij fazne ravnoteè u uslovima koegzistencije pare i te~nosti svakako je jednakost temperature i pritiska pojedinih faza TV = TL

PV = PL (1) Jedan od kriterija fazne ravnoteè moè se izraziti i preko hemijskog potencijala

µV = µL (2) Mada se radi o fundamentalnoj veli~ini, hemijski potencijal nije pogodan za upotrebu u rjeavanju problema fazne ravnoteè. Zato je kao pogodna funkcija za upotrebu u prora~unu fazne ravnoteè predloèn fugacitet. Fugacitet, koji ima jedinicu pritiska, koristi se umjesto hemijskog potencijala ili slobodne energije za izraàvanje kriterija ravnoteè faza

f V = f L (3)

1. INTRODUC The fundamentaconditions of csurely the equalof each phase T

P One of criteriaexpressed by me

µAlthough fundamnot convenient equilibrium. Theras suitable funequilibrium. Fugused instead of for expressing cr

f

- 177 -

PREGLEDNI RAD

ojeva jedna~inom stanja Soavev jednakosti fugaciteta parne i ienja argona, azota, kiseonika eksperimentalnim podacima za ta~ke.

-

,

i

CALCULATION BY UATION OF STATE

osnia and Herzegovina

a

r

loi

e

e

a
SUBJECT REVIEW
f f

lation by Soave-Benedict-Webb-and liquid phases is used as tion o the vapor pressure o xhibit excellent agreement with om triple to critical point.

ls

TION

criteria of phase equilibrium in existing of vapor and liquid is ty of temperatures and pressures

V = TL V = PL (1)

of phase equilibrium can be ans of chemical potential

V = µL (2) ntal quantity, chemical potential is for solving problems of phase fore fugacity has been proposed ction for calculation of phase city, having unit of pressure, is chemical potential or free energy iteria of phase equilibrium V = f L (3)


Drugim rije~ima, pored jednakosti temperatura i pritisaka svake faze i fugaciteti obje faze moraju biti jednaki u uslovima ravnoteè. Ovaj kriterij fugaciteta ekvivalentan je jednakosti hemijskih potencijala svake faze.

By other words, besides the equality of temperatures and pressures of each phase fugacities of both phases also must be equal in phase equilibrium. This criterion of fugacity is equal to equalities of chemical potentials of each phase.

2. KOEFICIJENT FUGACITETA IZ SBWR JEDNAÎNE STANJA

Koeficijent fugaciteta Pf ~istog spoja moè se

izra~unati iz PVT podataka po slijedeoj jedna~ini

∫

−=

P

dPP

RTVRTP

f

0

1ln (4)

Jedna~ina (4) moè se preurediti u pogodniji oblik

∫∫ −=P

P

P

P

PdVdPRTP

f ln

ln 00

ln1ln (5)

Prvi ~lan sa desne strane jednakosti moè se integrirati parcijalno

( ) ∫∫ −∆= PdVPVVdP (6)

Zamjenom (6) u (5) , i uvo|enjem smjene P0V0 = RT, dobije se

∫∫ −−−=P

P

P

P

PdPdVRT

RTPVPf ln

ln 00

ln1ln (7)

Jedna~ina stanja Soave-Benedict-Webb-Rubina1 ima oblik

2. FUGACITY COEFFICIENT FROM SBWR EQUATION OF STATE

Fugacity coefficient Pf of pure compound can be

calculated from PVT data upon following equation

∫

−=

P

dPP

RTVRTP

f

0

1ln (4)

Equation (4) can be rearranged to more convenient form

∫∫ −=P

P

P

P

PdVdPRTP

f ln

ln 00

ln1ln (5)

The first article of right-hand side of equation (5) can be integrated partially

( ) ∫∫ −∆= PdVPVVdP (6)

Changing (6) into (5), and introducing P0V0= RT, gives

∫∫ −−−=P

P

P

P

PdPdVRT

RTPVPf ln

ln 00

ln1ln (7)

The Soave-Benedict-Webb-Rubin1 equation of state has the form

( ) ( )[ ]22242rc exp11 φψφψεψδψγψβψψ −+++++= TPP (8)

Parametri jedna~ine β, γ, δ, ε, i φ dati su u prilogu rada, dok je redukovana gustina

c

c

PRTρψ = (9)

Zamjenom (8) u (7), uzimajui u obzir parametre jedna~ine stanja date u prilogu rada, te integriranjem dobijenog izraza u datim granicama, kada P0 → 0, dobije se koeficijent fugaciteta ~istog spoja

Parameters of the equation β, γ, δ, ε, and φ are given in the appendix, while reduced density is

c

c

PRTρψ = (9)

Changing (8) into (7), taking into account the equation of state parameters given in the appendix, integrating of resulting expression in given limits, when P0 → 0, fugacity coefficient of pure compound is obtained

( )

−−

+−+++−−= 1exp

211

41

21ln1ln 2242 φψφψ

φεδψγψβψZZ

Pf

(10)

- 178 -


3. PRITISAK ZASI]ENJA IZ USLOVA FAZNE RAVNOTE@E

Pritisak para, P, za zadatu temperaturu, T, u uslovima ravnoteè, pri koegzistenciji pare i te~nosti, mogue je izra~unati iz kriterija ravnoteè datog jedna~inom (3). Algoritam prora~una predstavljen je dijagramom toka na slici 1. Prora~un zapo~inje zadavanjem po~etnih vrijednosti pritiska P i gustina faza ρL i ρV. Gustine te~ne i parne faze ra~unaju se iz jedna~ine (8). Poto se radi o nelinearnoj jedna~ini po nepoznatoj gustini formirana je pogodna funkcija cilja

3. VAPOR PRESSURE FROM CONDITIONS OF PHASE EQUILIBRIUM

For given temperature, T, when coexisting vapor and liquid are in equilibrium, it is possible to calculate vapor pressure, P, from criteria of equilibrium given by equation (3). An algorithm of calculation is presented by flow chart at figure 1. The calculation starts by giving initial values of pressure P and densities of phases ρL and ρV. Densities of liquid and vapor phases are calculated from equation (8). As this equation is nonlinear upon unknown density it is rearranged to give suitable objective function

( ) ( )[ ] PTP −−+++++= 22242rc exp11F φψφψεψδψγψβψψ (11)

Treba, dakle, nai takvu vrijednost gustine, ρ, pri kojoj e biti zadovoljen uslov: F < ε. Kada se izra~unaju gustine parne i te~ne faze, iz jedna~ine (10) se ra~unaju njihovi fugaciteti. Zatim se provjerava kriterij ravnoteè dat jedna~inom (3). Za tu svrhu formirana je pogodna funkcija cilja

G = f L – f V (12)

Ako je zadovoljen uslov: G<ε, prora~un je zavren, a ako nije zadovoljen, postupak se vraa na po~etak. U ovom radu je za prora~un gustine i pritiska koritena modifikovana Newtonova metoda2, pogodna za rjeavanje nelinearnih jedna~ina tipa f(x)=0. Za razliku od izvorne Newtonove metode ~ija iteraciona formula ima oblik

( )( )xfxfxx

'i1i −=+ (13)

modifikovana Newtonova metoda aproksimira derivaciju funkcije izrazom

( ) ( ) ( )i

iii'δ

δ xfxfxf −+= (14)

gdje je δi=xi·10-5 vrijednost prirasta argumenta, koja se mora uskladiti s traènom ta~nou, tako da je uvijek δi<ε. Ako je zahtijevana ta~nost prora~una ε≤10-4, onda se moè uzeti konstantna vrijednost δi=10-5. Iteraciona formula modifikovane Newtonove metode sada ima oblik

( )( ) ( )i

5ii

i5

ii1i 10

10xfxxf

xfxxx−⋅+

⋅⋅−= −

−

+ (15)

Kao to se iz algoritma sa slike 1 vidi u unutranjoj petlji se, po formuli (15), ra~unaju gustine faza dok se u vanjskoj petlji, po istoj formuli, ra~una pritisak.

So, it is necessary to find such value of density, ρ, to satisfy condition: F < ε. When densities of vapor and liquid phases are calculated, then their fugacities are calculated next from equation (10). Upon that the criteria of equilibrium given by equation (3) is checked out. For that purpose suitable objective function is prepared

G = f L – f V (12)

Having satisfied the condition: G < ε, calculation is finished, otherwise, the procedure is returned to start. In this paper modified Newton methode2, convenient for calculation of nonlinear equations f (x)=0, is used for calculation of density and pressure. In contrast to original Newton method whose iterative formula has the form

( )( )xfxfxx

'i1i −=+ (13)

modified Newton method approximates derivative of function by the expression

( ) ( ) ( )i

iii'δ

δ xfxfxf −+= (14)

where δi=xi·10-5 is value of argument increment, which has to be adjusted with required accuracy, so that is always δi<ε. If required accuracy of calculation is ε≤10-4, than constant value of δi=10-5 can be used. Iterative formula of modified Newton method now has the form

( )( ) ( )i

5ii

i5

ii1i 10

10xfxxf

xfxxx−⋅+

⋅⋅−= −

−

+ (15)

As from algorithm on figure 1 it is seen densities of phases are calculated in inner loop from formula (15) while pressure is calculated in outer loop from the same formula.

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4. REZULTATI Na slikama 2-6 grafi~ki su prikazani rezultati prora~una pritiska zasienja argona, azota, kiseonika, etana i propana SBWR jedna~inom stanja, po algoritmu sa slike 1. Eksperimentalni podaci za pritisak zasienja prikazani su kruìima, dok puna linija predstavlja izra~unate vrijednosti pritiska. Evidentno je odli~no slaganje izra~unatih vrijednosti pritiska sa eksperimentalnim podacima, u cijelom posmatranom intervalu temperature, za sve obra|ene spojeve.

4. RESULTS On figures 2-6 graphically are presented the results of vapor pressure calculation of argon, nitrogen, oxygen, ethane and propane by SBWR equation of state, upon algorithm from figure 1. Circles denote experimental data of vapor pressure, while solid line represents calculated values of pressure. Excellent agreement of calculated values of pressure with experimental data is evident in whole range of temperature for all compounds processed.

5. ZAKLJUÂK Rezultati dobijeni u ovom radu pokazuju da se jedna~ina stanja Soave-Benedict-Webb-Rubina moè uspjeno koristiti za ra~unanje pritiska zasienja ~istih spojeva. Iz datog algoritma moè se lahko napraviti kompjuterski program u nekom viem programskom jeziku, ili odgovarajui podprogram za iru upotrebu. Ta~nost dobijenih rezultata direktno ovisi od ta~nosti ra~unanja gustine parne i te~ne faze. U tom kontekstu, prednost SBWR jedna~ine stanja, u odnosu na kubne jedna~ine stanja, leì upravo u njenoj superiornosti pri ra~unanju gustine te~ne faze.

5. CONCLUSION Results obtained in this paper show that Soave-Benedict-Webb-Rubin equation of state could be successfully used for vapor pressure calculation of pure compounds. From the algorithm given it is easy to make computer program in some higher program language, or corresponding subroutine for broader use. Accuracy of the results obtained depends directly from accuracy of calculation of vapor and liquid phase densities. In that context, the advantage of SBWR equation of state, comparing to the cubic ones, lies exactly in its superiority in calculation of liquid phase density.

6. SIMBOLI - SYMBOLS P – apsolutni pritisak, bar – absolute pressure, bar R – gasna konstanta, kJ/kmolK – gas constant, kJ/kmolK T – apsolutna temperatura, K – absolute temperatue, K V – volumen, m3/kmol – volume, m3/kmol Z – faktor kompresibilnosti – compressibility factor β, γ, δ, ε, – redukovani parametri jedna~ine stanja (vidi dodatak)

– reduced parameters of the equation of state (see appendix) µ – hemijski potencijal – chemical potential ρ – gustina, kmol/m3 – density, kmol/m3 ψ – redukovana gustina – reduced density ω – Pitzerov faktor acentri~nosti – Pitzer's acentric factor

INDEKSI - SUBSCRIPTS 0 – referentno stanje – reference state c – kriti~na ta~ka – critical point r – redukovana veli~ina – reduced property

EKSPONENTI - SUPERSCRIPTS L – te~na faza – liquid phase V – parna faza

– vapor phase

7. LITERATURA - REFERENCES [1] G. S. Soave, "A noncubic equation of state for the treatment of hydrocarbon fluids at reservoir conditions", Ind. Eng. Chem. Res. 34 (1995), 3981-3994.

[2] @. Oluji, M. Gjumbir, "Program i algoritam za rjeavanje nelinearne jednad`be primjenom modificirane Newtonove metode", Kem. Ind. 30 (1981) 12, 727-729.

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[3] "Encyclopedie des Gaz", L'Air Liquide, 1978. [4] K. Ra`njevi, "Termodinami~ke tablice", [kolska knjiga, Zagreb, 1975.

[5] G. J. Van Wylen, R. E. Sonntag, "Fundamentals of classical thermodynamics", 2nd Ed., John Wiley & Sons, New York, 1978.

8. DODATAK - APPENDIX

Redukovani parametri SBWR jedna~ine stanja

ωββ

−+

−+= 3.2

r21.6

r1c

1111T

bT

b

gdje je: b1=0.2971 b2=0.422

3

r3

2

r2

r1c 111111

−+

−+

−+=

Tc

Tc

Tcγγ

gdje je: c1=-0.02663+0.06170 + 0.007792 c2=-0.00605+0.07544 - 0.061342 c3=0.00153+0.03828 + 0.011912

δ=δc/Tr

3

r3

2

r2

r1c 111111

−+

−+

−+=

Te

Te

Teεε

gdje je: e1=0.1087+0.2154ω - 0.0591ω2

e2=0.0705+0.3007ω + 0.4948ω2

e3=-0.0068+0.1858ω - 0.1157ω2

φ=0.06

βc=bZc

γc=cZc2

δc=dZc4

εc=eZc2

b=[15Zc-8+(2f2-2f3) ·exp(-f)]/3

c=[5Zc-8-3b-(1+f+f2) ·exp(-f)]/2

d=Zc-1-b-c-e(1+f) ·exp(-f)

Zc=0.2908-0.099ω+0.04ω

e=½

f=φ/Zc2

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Slika 1. Dijagram toka algoritma prora~una pritiska zasienja

Figure 1. Algorithm flow chart of saturation pressure calculation

80 90 100 110 120 130 140 150

T, K

0

5

10

15

20

25

30

35

40

45

50

P, b

ar

Slika 2. Zavisnost pritiska zasienja a gona od temperature ( eksperimentalni podacir

f t

3, — SBWR jedna~ina stanja) Figure 2. Saturation pressure of argon vs. temperature ( experimental data3, — SBWR equation o sta e)

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60 70 80 90 100 110 120 130

T, K

0

5

10

15

20

25

30

35

P, b

ar

Slika 3. Zavisnost pri iska zasienja azota od tempe ature ( eksperimentalni podacit r 4, — SBWR jedna~ina stanja)

Figure 3. Saturation pressure of nitrogen vs. temperature ( experimental data4, — SBWR equation of state)

50 60 70 80 90 100 110 120 130 140 150

T, K

0

5

10

15

20

25

30

35

40

45

P, b

ar

Slika 4. Zavisnost pritiska zasienja kiseonika od temperature ( eksperimentalni podaci4, — SBWR jedna~ina stanja) Figure 4. Saturation pressure of oxygen vs. temperature ( experimental data4, — SBWR equation o sta e) f t

- 183 -


170 190 210 230 250 270 290 310

T, K

0

5

10

15

20

25

30

35

40

45

50

P, b

ar

Slika 5. Zavisnost pritiska zasienja etana od temperature ( eksperimentalni podaci5, — SBWR jedna~ina stanja) Figure 5. Saturation pressure of ethane vs. temperature ( experimental data5, — SBWR equation o sta e) f t

190 210 230 250 270 290 310 330

T, K

0

2

4

6

8

10

12

14

16

18

P, b

ar

Slika 6. Zavisnost pritiska zasienja propana od temperature ( eksperimentalni podaci5, — SBWR jedna~ina stanja) Figure 6. Saturation pressure of propane vs. temperature ( experimental data5, — SBWR equation o sta e) f t

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Mainstvo, 3 (6), 185 – 190, (2002) H.Avduinovi: GLOBALNI TRENDOVI U LJEVARSKOJ….

GLOBALNI TRENDOVI U LJEVARSKOJ PROIZVODNJI NA KRAJU DVADESETOG VIJEKA

Hasan Avduinovi, dipl. ing., asistent, Fakultet za metalurgiju i materijale, Zenica, Travni~ka cesta 1, e-mail: [email protected]

REZIME PREGLEDNI RAD

Najzna~ajnija karakteristika ljevarske proizvodnje na kraju dvadesetog vijeka je porast proizvodnje ne-èljeznih metala i njihovih legura (posebno Al i njegovih legura). Stalna te`nja za smanjenjem teìne proizvoda auto industrije je osnovni razlog za ovu poveanu proizvodnju. Bez sumnje mi ìvimo u eri visoke tehnologije i ulaganja u modernizaciju i edukaciju zaposlenih moraju doi na prvo mjesto. Pored pomenutog, va`nu stavku u svemu ovome pretstavljaju zahtjevi na polju kvaliteta i zatite okoline (ISO 9000 i ISO 14000).

Klju~ne rije~i: proizvodnja, metal, legura, porast, ljevarstvo, tehnologija, ulaganje, modernizacija, okolina, zatita

GLOBAL TRENDS IN CASTING PRODUCTION AT THE END OF THE TWENTIETH CENTURY

Hasan Avdusinovic, B.Sc., assistant, Faculty for Metallurgy and Materials, Zenica, Travni~ka cesta 1, e-mail: [email protected]

SUMMARY SUBJECT REVIEW

The most interesting characteristic of casting production at the end of the twentieth century is increase of non-ferrous metals and their alloy production (especially Al and its alloys). Continuous attempts to reduce the weight of vehicle industry products is the main reason for such increased production. Without doubt, we live in the age of high technology and investments in modernisation and professional education of employees have to be the priority. Besides the above mentioned, other important things are demands in the field of standardisation and environment protection (ISO 9000 and ISO 14000).

Key words: production, metal, alloy, increase, casting, technology, investments, modernisation, environment, and protection

1. UVOD Ljevarstvo, kao jedna od prvih vjetina kojom je ~ovjek ovladao, igra veoma va`nu ulogu u dananjem tehnolokom razvoju. Analizirajui promjene koje se deavaju u proizvodnji raznih vrsta ljevova u svijetu zadnjih nekoliko godina a pogotovu u periodu izme|u 1995. i 2000. godine moè se vidjeti da dolazi do odre|enih zaokreta na ovom polju. Ovdje se prvenstveno misli na nagli porast interesovanja za ne-èljezne metale i njihove legure.

1. INTRODUCTION Casting, as one of the oldest human skills, has a very important role in the current technical development. If we analyse the production of different kinds of alloys, we can see that there have been some changes in this field over the last period (especially in the period from 1995 to 2000). The most important change is high increase in the interest in the non-ferrous metals and their alloys.

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2. KARAKTERISTIKE PROIZVODNJE RAZLIÎTIH VRSTA LJEVOVA

2.1. Ne-èljezni metali i njihove legure Porast proizvodnje ne-èljeznih metala i legura je jedna od osnovnih karakteristika ljevarstva danas u svijetu. U ovoj grupi metala i legura mora se posebno izdvojiti porast proizvodnje aluminijuma, magnezijuma i njihovih legura. Razlog ovome moè se nai u nagloj ekspanziji upotrebe ovih metala i njihovih legura u automobilskoj industriji. Najvei svjetski proizvo|a~i ne-èljeznih metala i legura dati su u tabeli 1. Porast proizvodnje ne-èljeznih metala i legura moè se vidjeti na primjeru Njema~ke i Francuske (dijagrami 1 i 2).

2. THE PRODUCTION CHARACTERI-STICS OF DIFFERENT KINDS OF ALLOYS

2.1. Non-ferrous metals and their alloys Increase in the production of the non-ferrous metals and their alloy is one of the basic characteristics of castings in the world today. In this group of metals and alloys it is necessary to mention especially Al, Mg and their alloys. The use of these two metals and their alloys in vehicle industry is the main reason for their increasing production. The world leader casters of non-ferrous metals and their alloys are presented in the Table 1. The diagrams 1 and 2 show the increase in the production of non-ferrous metals and their alloys in Germany and France.

Tabela 1. Najvei svjetski proizvo|a~i ne-èljeznih metala i legura (podaci iz 1999. godine) [1] Table 1. World leading producers of non-ferrous metals and their alloys in 1999, [1]

ZEMLJA/COUNTRY 1000 tona 1000 tonnes

SAD/USA 2343000 Japan/Japan 1143621 Kina/China 851399 Italija/Italy 705500

Njema~ka/Germany 692879 Francuska/France 319694 Rusija/Russia 300000

0

100000

200000

300000

400000

500000

600000

700000

800000

1996 1997 1998 1999 2000

godinayear

proi

zvod

nja,

(100

0 to

na)

prod

uctio

n, (1

000

tonn

es)

0

50000

100000

150000

200000

250000

300000

350000

400000

1996 1997 1998 1999 2000

godinayear

proi

zvod

nja,

(100

0 to

na)

prod

uctio

n, (1

000

tonn

es)

D

Dijagram 1. Proizvodnja ne-èljeznih metala i legura u Njema~koj [2]

iagram 1. The production of non-ferrous metals and their alloys in Germany, [2]

- 186 -

Dijagram 2. Proizvodnja ne-èljeznih metala i legurau Francuskoj, [3]

Diagram 2. The production of non-ferrous metals and their alloys in France, [3]


Promjene u koli~ini i vrsti proizvedenih ljevova dovode i do favorizovanja pojedinih postupaka dobivanja odljevaka iz te~nog metala. Sa porastom interesovanja za ljevanjem proizvoda od ne-èljeznih metala (a posebno Al, Mg i njihovih legura) dolaze do izraàja postupci ljevanja u matrice (die casting), ljevanje pod visokim pritiskom (high pressure casting), postupak tixo-ljevanja (thixocasting) i ostali postupci tzv. preciznog ljevanja (investment casting). Opta karakteristika svih ovih postupaka je da se dobije tankostjeni, zdrav i to je mogue dimenzionalno ta~niji odljevak sa to manje karta u proizvodnji. Da bi se ovi zahtjevi mogli ispotovati svemu ovome prethodi ra~unarska simulacija kompletnog postupka izrade odljevka.

The changes in quantity and kind of casting products leads to favouring of some casting methods. With increased interest in the non-ferrous metals and their alloys (especially in Al and Mg) die, high pressure, thixo and investment casting become the most suitable casting methods. Common characteristics of these casting methods are thin walls, soundness, near net shape and minimum of waste.

2.2 @eljezo i njegove legure Za razliku od ne-èljeznih metala i legura ljevarstvo na bazi èljeznih legura u poslijednjem periodu ima razvojni trend promjenljivog karaktera. U periodu od 1997-2000. godina primjetne su znatne oscilacije u koli~ini proizvedenih odljevaka na bazi legura èljeza, a to pokazuje i dijagram broj 3. Na ovom dijagramu su dati zbirni podaci o proizvodnji odljevaka od sivog, nodularnog i temper ljeva u pet evropskih zemalja (Njema~ka, Francuska, Engleska, Italija, Poljska i [panija). I me|u tri navedene vrste ljevova (nodularni, temper i sivi ljev) postoje razlike u kretanju godinje proizvodnje u poslijednjem periodu. Opti je stav da je koli~ina proizvedenog temper ljeva u konstantnom opadanju (glavni razlog skupi tretman termi~ke obrade), dok je proizvodnja nodularnog ljeva i dalje vrlo aktualna u svijetu zbog njegovih dobrih mehani~kih i drugih karakteristika (obradivost, sposobnost priguenja buke, i dr.) koje ga i dalje drè konkurentnim u odnosu na nove tipove legura koje se danas proizvode (razne vrste-ne èljeznih legura). U odnosu na tri navedene vrste ljevova, koli~ina proizvedenog ~eli~nog ljeva je u konstantnom opadanju u proteklih nekoliko godina. Opti stav je da se zadnjih godina veina livnica koje su proizvodile samo ~eli~ni ljev, u svijetu, gase i da se danas potrebe za ~eli~nim ljevom zadovoljavaju tako to se on proizvodi u sklopu livnica koje proizvode druge vrste ljevova na bazi èljeza ( npr. u livnicama nodularnog ljeva). Opadajui trend u proizvodnji ~eli~nog ljeva moè se vidjeti na dijagramu broj 4 na kojem su prikazani zbirni podaci proizvodnje ~eli~nog ljeva u zadnje ~etiri godine u Njema~koj, Francuskoj, Italiji, Engleskoj, Poljskoj i [paniji.

2.2 Fe and its alloys The casting of Fe and its alloys has varied in production over the last period, which is different in the case of the non-ferrous metals and their alloys. The diagram 3 shows the production in the period from 1997 to 2000 and we can see that it had oscillating character in that period. Cumulative data for the production Fe and its alloys (nodular, grey and malleable iron) in five European countries (German, France, England, Italy, Poland, and Spain) are presented on this diagram. There is a considerable difference in annual production of nodular, grey and malleable iron. The general rule is that the level of malleable iron production was continuously decreasing in the above mentioned period (the main reason for that is expensive heat treatment). The production of nodular iron is interesting because of its good mechanical and other properties (malleability, noise minimisation etc.). If we compare the production of steel casting with other kinds of alloys, we can see that it is continuously decreasing and many steel foundries have been shut down in the last few years. The current needs for making small quantities of steel in foundries of nodular or grey iron can satisfy steel casting. The decreasing trend of steel casting production is presented in diagram 4, which shows cumulative data of steel casting production in Germany, France, Italy, Poland, and Spain for last four years.

- 187 -


8500

8600

8700

8800

8900

9000

9100

9200

9300

9400

1997 1998 1999 2000

godinayear

1000

tona

1000

tonn

es

570

580

590

600

610

620

630

640

650

1997 1998 1999 2000

godinayear

1000

tona

1000

tonn

es

no(ND

(

3. M Uvo|enproizvoorganizpotrebuopstankproizvodanas livnica stru~norazvijenodlu~ujdanas organizprekvalnovih tPrimjer Slovenavidimo Analizirje da uglavnodanantehnolozatvaraTakve na lj(kalupolivnice Promatèljeznbroj livzemljamlegura livnica)

Dijagram 3. Proizvodnja odljevaka od sivog, dularnog i temper ljeva u pet evropskih zemalja jema~ka, Francuska, [panija, Italija i Poljska), [3]iagram 3. The production of grey, nodular and malleable castings in five European countries Germany, France, Spain, Italy and Poland), [3]

ODERNIZACIJA PROIZVODNJE

jem ra~unara u proizvodnju i automatizacijom dnje u livnicama dolazi do velikih promjena u acionoj struktururi livnica, a to sa sobom nosi za stalnim smanjenjem zaposlenih. Jedini vid a na svjetskom trìtu je osavremenjavanje dnje i poveanje produktivnosti. Problem koji je aktualan, a koji je vezan za prestrukturiranje i modernizaciju proizvodnje u njima, je deficit g kadra. Ovaj problem je danas izraèn u svim im zemljama jer sve manji broj mladih ljudi se e da studira na ovom polju. Pokuaji kojim se nastoji rijeiti ova problematika su ti da se uje veliki broj stru~nih seminara za obuku i ifikaciju radnika za rad u uslovima uvo|enja ehnologija i ra~unarskih sistema u livnice.

obrazovne strukture radnika zaposlenih u ~kim livnicama dat je na dijagramu 5. sa koga da je obrazovna struktura veoma nepovoljna. ajui podatke o broju i vrsti livnica primjetno dolazi da smanjenja broja livnica. To su, m, livnice koje nisu u stanju da se nose sa jim trendovima razvoja i uvo|enja savremene gije u proizvodnju, pa su prisiljene da ju svoje hale i da obustavljaju proizvodnju. livnice svoju proizvodnju su veinom bazirale udskom radu, zastarjelim tehnologijama vanja, ljevanja, i sl.). To su, naj~ee, stare ~eli~nog, sivog, temper ljeva. ra li se odnos broja livnica èljeznih i ne-ih legura, veoma je interesantan podatak da nica ne-èljeznih metala i legura u razvijenim a daleko nadmauje broj livnica èljeznih (preko pedeset posto od ukupnog broja

, to je prikazano na dijagramu 6.

3. Intauto of Thmoproof Thcodein thrseemtheAnfounuthedecatecpranThmaIf fernadefiftsitop

- 188

Dijagram 4. Zbirni podaci proizvodnje ~eli~nogljeva u zadnje ~etiri godine u Njema~koj,Francuskoj, Italiji, Engleskoj, Poljskoj i [paniji, [3] Diagram 4. Cumulative data of steel castings in

the last four years in Germany, France, Spain, Italyand Poland, [3]

MODERNISATION OF THE PRODUCTION

roduction of computers in casting production and tomatization of the processes in factories has lead great changes in organisation structure, the result which is decrease in the number of employees. e only chance to survive in world market is dern production and high level of productivity. The blem that is present today in foundries is deficit engineers of metallurgy. is problem is present in all West European untries because less and less young people cide to study metallurgy. Managers of foundries developed countries try to solve this problem ough organising a lot of meetings and scientific minars on this topic and they try to teach ploin foundries new technologies and work in conditions of automatization of processes. example of educational structure in Slovenian nders was given in diagram 5. Analysing the mber and kind of foundries, we can see that number of foundries is continuously creasing. Those are mostly old factories which nnot cope with the present trends and new hnologies and they have to be shut down. The oduction in there was based on manual work d old technologies (moulding, pouring, etc.). ose are mostly old foundries of steel, grey, lleable or nodular iron. we look at the numbers of ferrous and non-rous foundries we can see an interesting thing, mely that the number of non-ferrous foundries in veloped countries is much higher (more than y percent) than the number of ferrous ones. The uation in less developed countries is just the posite, which is presented on diagram 6.

-


Pored navedenih problema, koji su danas aktualni u ljevarstvu, bitnu stavku ~ine postavljeni zahtjevi na polju standardizacije i o~uvanja okoline. Akcenat u ovom domenu se stavlja na iplementaciju i uklapanje u okvire ISO standarda, a posebno ISO 9000 i ISO 14000. Najdalje na ovom polju se otilo na podru~ju skandinavskih zemalja (Danska, [vedska, idr) i u Italiji. Dok drugima predstoji dug put do ispunjenja postavljenih zahtjeva vezanih za kvalitet i o~uvanje ìvotne sredine.

Besides the above mentioned problems, which are present in the casting practice, another important thing is demand in the field of standardisation and environment protection. In order to satisfie these demands foundries have to implement the ISO 9000 and ISO 14000 standards. The Scandinavian countries (Denmark, Sweden, etc.), have had the most success in this field whereas other countries have to do many things to satisfy the ISO standards.

nezavrena osnovna kola/ Non-completed primary school osnovna kola i kursevi/ primary school and some courses do 2 godine usavravanja/ primary school and 2 years of vocational training do 2,5 godine usavravanja/ primary school and up to 2,5 years of vocational training srednja stru~na sprema/ secondary vocational school via stru~na sprema (inìnjeri)/ college (engineers) visoka stru~na sprema (dipl. inìnjeri)/ university degree (Bs.C.) visoka stru~na sprema (Mr., Dr.)/ university (Ms.C., Ph.D.)

Dijagram 5. Obrazovna struktura zaposlenih u Slovena~kim livnicama, [4] Diagram 5. Educational structure of employees in Slovenian foundries, [4]

0

10

20

30

40

50

60

70

80

procenat (%)percent (%)

Italy

France

USA

Germany

Croatia

Ukraine

Slovakia

China

zemljacountry

Dijagram 6. Procentualni udio livnica ne-èljeznih metala i legura u ukupnom broju livnica u pojedinim zemljama svijeta (podaci iz 1999. godine), [1]

Diagram 6. Number of the foundries of non-ferrous metals and their alloys in the overal number of foundries in some countries in 1999 expressed in percent, [1]

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4. ZAKLJUÂK Ljevarstvo, kao grana industrije, igra ulogu snabdjeva~a dijelovima irokog kruga potroa~a (auto industrija, mainogradnja, èljeznica, gra|evinarstvo idr.). Kao takva, ljevarska industrija mora da zadovolji bezbrojne zahtjeve koji se postavljaju od strane kupaca. Najbolji primjer postavljanja stalno novih zahtjeva proizvo|a~ima je automobilska industrija, koja kontinuirano uvodi promjene na svojim finalnim proizvodima. Da bi se, u to veoj mjeri, udovoljilo ovim zahtjevima potrebno je kontinuirano ulaganje u proizvodne pogone livnica kroz uvo|enje savremene opreme i ra~unarske tehnike u livnice i angaòvanje i edukacija stru~nog kadra koji moè da bude na nivou postavljenih zadataka.

4. CONCLUSION Casting, as a branch of industry, has an important role as a supplier with metal parts to other industries (vehicle industry, engineering, railway, construction industry, etc.) and in that case foundry workers have to respond to many demands to satisfy their consumers. The best example for this is vehicle industry that continuously changes the design of their products. In order to respond to these demands as much as possible continuously investment in modernisation and education of employees is necessary.

5. LITERATURA - REFERENCES

[1] G.H. Garreau, “34ème Rapport sur la

Production Mondiale des Fonderies-1999”, Hommes & Fonderie (Mars 2001),nº 312, 35-36

[2] „German report“, Foundry Trade Journal

(August 2000), 22-28

[3] L. Blanc, “Evolution de Làctivite´ dens les Industries de la Fonderia”, Homes & Fonderie (Janvier 2001), nº 310, 39-60

[4] D Vodeb, V. Pirih, M. Trebiàn, „Die

Slowenische Gießereiindustrie – Lage und Trends“, Giesserei, 88 (2001),Nr. 9, 92-98

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