ANALIZA NAPREZANJA U U^VR[^ENJU POLOVA ROTORA …mf.unze.ba/wp-content/uploads/2016/10/Masinstvo-num2-2003.pdfKlju~ne rije~i: generator s istaknutim polovima, u~vr{}enja polova, naprezanja,

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Ma{instvo 2(7), 63 – 74, (2003) N.[virig;....: ANALIZA NAPREZANJA U U^VR[ÊNJU...

ANALIZA NAPREZANJA U U^VR[ÊNJU POLOVA ROTORA GENERATORA S ISTAKNUTIM POLOVIMA

Dr.sc. Nikola [VIGIR, Fakultet elektrotehnike i ra~unarstva u Zagrebu, Sveu~ili{te u Zagrebu Doc.dr.sc. Mirko HUSNJAK, Fakultet strojarstva i brodogradnje u Zagrebu, Sveu~ili{te u Zagrebu

REZIME U radu su dani rezultati eksperimentalnih istraìvanja metodom fotoelasticimetrije rasporeda naprezanja po konturi u~vr{}enja pola rotora generatora s istaknutim polovima oblika kombinacije lastin rep-~eki} i oblika tri ~eki}a. Tako|er su odre|eni kriti~ni presjeci kao i njima faktori koncentracije naprezanja.

Klju~ne rije~i: generator s istaknutim polovima, u~vr{}enja polova, naprezanja, fotoelasticimetrijska metoda

STRESS ANALYSIS DURING ROTOR POLES FIXING OF GENERATORS WITH SALIENT POLES

Nikola [vigir, Ph.D., Faculty of Electrical Engineering and Computing, University of Zagreb Mirko Husnjak, Ph.D. Assistant Professor, Faculty of Mechanical Engineering and Shipbuilding, University of Zagreb

SUMMARY This paper gives experimental results obtained by the method of stress photoelasticity along the contours of rotor poles fixing of generators with salient poles whose shape is either swallow’s tail-hammer or three hammers. It also gives critical cross-sections and stress concentration factors.

Key words: a generator with salient poles, poles fixing, stress, photoelasticity method

1. UVOD

U fazi projektiranja i konstruiranja elektri~nih strojeva javlja se niz problema kako elektri~ke tako i mehani~ke prirode koje treba uspje{no rije{iti da bi bile zadovoljene osnovne funkcije stroja. Sinkroni stroj, koji se naj~e{}e koristi kao generator izmjeni~ne struje, predstavnik je strojeva najve}ih snaga. Izvodi se prema konstrukciji rotora ili sa cilindri~nim rotorom ili pak kao stroj s istaknutim polovima. Sam pol radi se ili u kompaktnoj izvedbi ili je napravljen od limova koji su u cjelini spojeni krajnjim polnim plo~ama. Kod velikih brzohodnih strojeva zbog velike brzine vrtnje i znatnih masa namota, javljaju se i velike centrifugalne sile koje optere}uju konstrukciju generatora, a naro~ito njegov rotor. Na rotoru s istaknutim polovima dolazi do velikih naprezanja naro~ito na mjestu u~vr{}enja samog pola na rotor. Korijen pola kojim se on u~vr{}uje na lan~ani prsten izvodi se u vi{e oblika, i to kao lastin rep, ~eki}, kombinacija lastin rep-~eki}, a kod ekstremno velikih centrifugalnih sila u~vr{}enje se izvodi oblika vi{estrukih ~eki}a.

1. INTRODUCTION

During the design stage of electrical machines a lot of problems arise. They are both electrical and mechanical and should be successfully solved in order to meet basic requirements of the machine function. A synchronous machine, which is mostly used as an a.c. generator, represents machines which are considered the most powerful. It is designed according to the rotor design or with a cylindrical rotor or as a machine with salient poles. The pole itself is produced either as a compact constru-ction or from sheets connected with pole final plates. In large high-speed machines due to rapid speeds of rotation and considerable winding masses, big centrifugal forces occur, which load the generator construction, particularly its rotor. The rotor with salient poles suffers enormous stress particularly on the spot where the pole itself is fixed to the rotor. The root of the pole by which it is fixed to the chain ring is designed in several ways, i.e. as a swallow’s tail, a hammer, a swallow’s tail-hammer combination. When extremely large centrifugal forces are present, the fixing is carried out in the shape of multiple hammers.

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Kao {to je ve} nagla{eno najsloèniji dijelovi ovih konstrukcija predstavljaju mjesta u~vr{}enja na kojima je distribucija kontaktnih naprezanja na dosjednim povr{inama nepoznata. êst je slu~aj i stati~ke neodre|enosti sklopa {to oteàva to~no definiranje optere}enja pojedinih sklopova i onemogu}ava jednostavno provo|enje izra~una metodama nauke o ~vrsto}i. Iz tog razloga provedena je u ovom radu eksperimentalna analiza naprezanja metodom fotoelasticimetrije u korijenu pola generatora s istaknutim polovima i oblika kombinacije lastin rep-~eki} s varijacijom kuta nagiba lastinog repa i oblika tri ~eki}a kako bi se {to egzaktnije odredio raspored naprezanja du` istog i definirala vr{na naprezanja po mjestu i iznosu.

As pointed out before, the most complex parts of these constructions are the spots of fixing. The contact stress distribution on contacting surfaces of these spots is unknown. Very often the assembly is not determined statically, which makes accurate definition of assembly loading rather difficult and it also prevents simple calculations by means of the science of strength. This was the reason to carry out an experimental stress analysis by means of photoelasticity method in the pole root of the generator with salient poles and in the shape of swallow’s tail-hammer with the swallow’s tail varying angle and three hammers in order to determine the stress scheme along it as accurately as possible and define peak stress according to the place and amount.

2. MATERIJAL ZA IZRADU MODELA Materijal za izradu modela mora biti opti~ki osjetljiv sa {to manjim opti~kim i mehani~kim puzanjem a uz to providan i izotropan. Mora pokazivati linearnu ovisnost deformacije u naprezanju, te proporcionalnost izme|u naprezanja i reda izokroma. Budu}i se pri optere}enju manje deformiraju povoljniji su materijali ve}e krutosti. Fotoelasti~ni materijal uz to mora biti bez unutra{njih naprezanja sa svojstvom dobre obradivosti i {to manjeg vremenskog rubnog efekta. Postavljene zahtjeve najbolje ispunjavaju materijali na bazi epoksidnih smola i poliestera. U Europi se naj~e{}e koristi araldit-B s modulom elasti~nosti E = 3,1 – 3,7 GPa, fotoelasti~nom konstantom naprezanja fσ = 103 – 120 N/cm, te dopu{tenim naprezanjem σd = 40 Mpa, kakav je kori{ten i za izradu modela u ovom radu.

2. MATERIAL FOR MODEL PRODUCTION The material for producing the model should be optically sensitive with as small optical and mechanical creeping as possible and at the same time it should be transparent and isotropic. It has to show linear dependence of deformity during the stress, as well as the proportionality between the stress and isochrome order. More rigid materials are more convenient since they are less prone to deformity during loading. Photoelastic material should be without internal stress and should have good treatment characteristics. It should also have as small time edge effect as possible. Materials based on epoxy resin and polyesters are considered the best for these purposes. The material mostly used in Europe is araldite-B with elasticity module E=3.1-3.7 Gpa, photoelastic stress constant fσσ=103–120 N/cm and the permitted stress σd = 40 MP, like the one used for producing the model described in this paper.

3. IZRADA MODELA Svi ispitivani modeli izra|eni su iz plo~e araldit-B s dopu{tenim naprezanjem σd = 40 MPa. Budu}i da to~nost rezultata u mnogome zavisi o kvaliteti izrade modela, potrebno je model izraditi vrlo precizno i to~no. U tu svrhu najprije je (za svaki model posebno) izra|ena {ablona od pleksi-stakla. Obrada modela izvr{ena je tako da se {ablona nalijepi pomo}u dvostrano ljepljive trake na plo~u araldita, te zatim tra~nom pilom izreè dio plo~e s dodatkom od nekoliko milimetara za finalnu obradu. Fina obrada izvr{ena je obodno protusmjernim glodanjem na brzohodnoj kopirnoj glodalici. Promjer prstastog glodala iznosi φ 5 mm, a broj okretaja radnog vretena n = 40000 o/min.

3. MODEL PRODUCTION All the models produced are made of an araldite-B plate with the permitted stress of σσd = 40 MP. Since the accuracy of the results depends highly on the quality of the model, the model should be produced very precisely and accurately. That is why a perspex pattern should be made for each model separately. While making the model, the pattern is first stuck by means of a two-sided sticky strip on the araldite plate and then, a part of the plate is cut by a band saw. A few millimeters should be left for further final treatment. Fine treatment is carried out by circumferential counter-directional milling on a high-speed copying milling machine. The diameter of the finger-like mill is φ5 mm, and the number of revolutions of the operating spindle n=40000 r/min.

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Za vrijeme glodanja {ablona se oslanja na centralni zatik na radnoj plo~i glodalice. Dubina se rezanja pak regulira postepenim smanjivanjem promjera zatika. Posmak se vr{i ru~no, te je nemogu}e posti}i potpunu ravnomjernost obrade. Zbog istro{enosti glodala i vibriranja vretena glodalice u modelu su se pojavila unutra{nja naprezanja te je bilo potrebno izvr{iti finu zavr{nu obradu dugotrajnim postupkom ru~nog bru{enja pomo}u finog brusnog papira, kako bi nestala zaostala naprezanja i rubni efekt. Nakon toga u modelima su simetri~no izbu{eni provrti za optere}ivanje. Provrti su izbu{eni dovoljno daleko od mjesto u~vr{}enja polova, kako ne bi do{lo do utjecaja St. Venantovog principa.

During the process of milling the pattern leans against the central pin on the working panel of the milling machine. The depth of cutting is regulated by gradual reduction of the pin’s diameter. Shift is regulated manually so it is impossible to achieve uniformity in treatment. Due to the worn milling machine and the vibration of milling machine spindle internal stresses have occurred in the model. It was, therefore necessary to apply final treatment which lasted for a long time. That is why manual polishing with fine abrasive paper was required in order to eliminate stresses and edge effect. After that symmetrical bores for loading were made on models. Bores were drilled far enough from the spot of pole fixing in order to avoid St. Venant principle.

3.1 Izrada modela pola generatora s korijenom u obliku ~eki}a, kombinacije lastin rep-~eki} i lastinog repa Modeli I - IV su izra|eni iz plo~a araldita-B debljine 8 mm, na kojima je mijenjan kut nagiba dosjednih povr{ina γ, kako je prikazano sl. 3.3. i tb.3.1. Iz iste plo~e izra|en je model V (sl. 3.2.) kod kojeg je korijen pola izveden u obliku lastinog repa.

3.1 Generator pole model with hammer like root, swallow’s tail-hammer shape and swallow’s tail Models I – IV are made from an araldite-B plate, whose thickness is 8 mm, and whose contact surface angle γ was changed (Fig. 3.3. and Table 3.1.). Model V was produced from the same plate, (Fig. 3.2.) whose pole root was made in the shape of the swallow’s tail.

3.2 Izrada modela pola generatora s korijenom u obliku tri ~eki}a Na svim modelima do sada bio je izra|en korijen pol jednostrukog oblika. Me|utim, kod ve}ih masa namota ili ve}eg broja okretaja javljaju se i ve}e centrifugalne sile koje uzrokuju naprezanja ve}a od dozvoljenih te se zato pri{lo izradi stati~ki neodre|enih nosa~a namota. U tu svrhu izra|en je model korijena pola generatora s tri ~eki}a te se ukupna centrifugalna sila raspodjeljuje na tri ~eki}a i za o~ekivati je da }e se i naprezanja po jednom ~eki}u bitno smanjiti. Dimenzije pola modela VI dane su na sl. 3.4. Model je tako|er izra|en iz plo~e araldita-B debljine 8 mm s dopu{tenim naprezanjem σdop=40 MPa.

3.2 Manufacture of the generator pole root model with three hammers All models produced so far have had a pole with one root only. However, when either winding masses or the number of revolutions increase, the centrifugal forces increase and they cause bigger stresses than those allowed. That was the reason why statically undefined winding carriers were made. Therefore, a generator pole model was made (model V) having three hammer-like roots. The centrifugal force is divided into three, which should decrease the stress in each hammer considerably. Pole model dimensions VI, are given in Fig. 3.4. The model is also made from an araldite-B panel (8mm thick) with the permitted stress, σσdop=40 MP.

Tabela 3 1. P omjenljive veli~ine za modele I – IV . r.Table 3 1. Changing values for models I – IV

Model I II III IV

γ, o 0 15 30 45

v, mm 0 2 4,5 8,25

t, mm 20 18 15,5 11,75

s1, mm 0 0,75 1,5 2,12

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Slika 3.1. Dimenzije modela pola generatora oblika

lastin rep-~eki} za modele I – IV Figure 3.1. The model size of the pole generator swallow’s tail-hammer shape for models I – IV

Slika 3.2. Dimenzije modela pola generatora oblika lastin rep, model V

Figure 3.2. The model size of the pole generator swallow’s tail, model V

Slika 3.3. Prikaz promjenjivih veli~ina za modele I – IV, kombinacija lastin rep-~eki}

Figure 3.3. The presentation of changing values for models I – IV, swallow’s tail-hammer combination

Slika 3.4. Dimenzije pola generatora s tri ~eki}a, model VI Figure 3.4. The model size of the pole generator with three hammers, model VI

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4. ISPITIVANJE MODELA

4.1 Optere}ivanje modela Kona~no obra|en model spreman je da se optereti i da se izvr{e sva potrebna mjerenja. Poloàj to~aka na modelu odre|en je pravokutnim koordinatama ~ija se jedna os poklapa sa osi simetrije modela. Urezivanje koordinatnih osi u povr{inu modela nije mogu}e zbog pojave unutra{njih naprezanja na modelu, koja bi imala negativnu posljedicu na ispitivanje. Optere}ivanje svih modela izvr{eno je u zatvorenom okvirnom nosa~u pomo}u vijka sa trapeznim navojem. Prijenos sile na modelima izveden je pomo}u posebnih ~eljusti koje se sa vijcima pri~vrste na model, a na drugoj strani ~eljusti nalazi se simetri~no izbu{ena rupa koja se spaja s prstenastim dinamometrom pomo}u zatika. Same ~eljusti su postavljene tako da budu simetri~ne sa modelom i da se sila prenosi podjednako na sve rupe modela Svi modeli optere}ivani su u dva me|usobno okomita smjera kako bi se {to vjernije simuliralo stvarno stanje naprezanja na mjestu u~vr{}enja pola. Omjer vertikalne i horizontalne sile kojom se optere}uje model odabran je tako kod svih modela, da se dobije odnos naprezanja koji odgovara stvarnom stanju naprezanja u realnim pogonskim uvjetima. Iznosi sila FV i FH kojima su optere}ivani modeli dani su u tablici 4.1. Odabran je uvijek onaj omjer sila FV / FH koji je za razmatranje pojava u modelu najpogodniji tj. za taj omjer u modelima se pojavljuje maksimalni red izokroma. Taj se red izokroma najlak{e snima u kriti~nom presjeku, budu}i bi se kod ve}ih iznosa optere}enja maksimalni red izokroma znatno pove}ao, te bi one postale toliko guste i nejasne da se ne bi mogle vi{e razlu~ivati (brojati).

4. TESTING OF THE MODEL

4.1. Loading of the model When the model is finished it is ready to be loaded and submitted to all the necessary measurements. The position of points on the model is defined by rectangular coordinates. One of their axes agrees with the model symmetry axes. Cutting in axis coordinates into the model surface is not possible because inner stresses occur in the model, which would have a negative effect on testing. The loading of all models was carried out in a closed frame carrier by means of trapeze thread screw. The transmission of force on models was carried out by means of special jaws fixed by screws to the model. On the other side of the jaw there is a symmetrically drilled hole which is connected to the ring dynamometer by means of a pin. The jaws are symmetrical with the model and the force is transmitted equally to all holes of the model. All the models are loaded in two vertical directions. In that way the simulation of the real stress on the spot where the pole is fixed is very reliable. The ratio between the vertical and horizontal force by which the model is loaded is chosen for all models, i.e. the stress ratio (σv / σh) = 1, which corresponds to the real arrangement of the generator pole fixing. Values of the forces FV and FH by which the model was loaded can be seen in Table 4.1. The force ratio, FV and FH was chosen so that it would give the most convenient way of considering the phenomena in the model, i.e. for that ratio the maximum order of isochrome appears in the models. This isochrome order is recorded easily in the critical cross-section, since at higher loading the number of maximum isochrome order is increased considerably. They become so dense and unclear that they cannot be distinguished (counted).

Tablica 4.1. Iznosi vertikalnih FV i horizontalnih FH sila optere}enja na modele Table 4.1. Amounts of vertical and horizontal (FV and FH) loading forces on models

Model I II III IV V VI

FV, N 450 450 450 450 300 1500

FH, N 460 460 460 460 310 500

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5. NAPREZANJA PO KONTURI MODELA I ODRE\IVANJE FAKTORA KONCENTRACIJE NAPREZANJA Fotoelasticimetrijska metoda je opti~ka metoda za odre|ivanje naprezanja i deformacija u tehni~kim konstrukcijama a zasniva se na piezo-opti~kom efektu tj. svojstvu materijala da u optere}enom stanju postaje opti~ki anizotropan. Fotoelasti~ni materijali su takvi materijali koji su u neoptere}enom stanju opti~ki izotropni, a kada se opterete postaju opti~ki anizotropni (dvolomni). Opti~ka anizotropnost ovisna je o veli~ini i rasporedu naprezanja, pa se mjerenjem dvoloma moè zaklju~iti o rasporedu naprezanja u modelu. Analiza anizotropnosti (dvoloma) vr{i se ure|ajima koji se nazivaju polariskopi. U njima koristimo polarizirano svjetlo tj. svjetlosni val ~ije titranje je na neki odre|eni na~in sre|eno i usmjereno. Sam eksperiment provodi se na slijede}i na~in. Iz ravne plo~e izreè se model koji je geometrijski sli~an originalnom dijelu koji se ispituje. Model se postavi u polariskop a zatim se optereti i snima, ili promatra golim okom. U vidnom polju polariskopa pojavljuju se interferencijske svijetle i tamne pruge. Analizom ovih linija izokroma zaklju~uje se o rasporedu naprezanja, odnosno deformaciji modela. Opisani eksperiment odnosi se na jednostavne probleme stati~kog optere}enja racninskih modela u elasti~nom podru~ju. Snimanje izokroma vr{eno je u kru`no polariziranom svjetlu, u svjetlom i tamnom polju, uz parametre snimanja, otvor blende 11 i ekspoziciju 8 sekundi Nakon razvijanje filma i izrada fotografija na fotografskom dokument papiru, dobiveni su snimci izokroma u tamnom i svjetlom polju, za sve modele. Na slikama 5.1 .i 5.2. prikazani su snimci izokroma u svijetlom i tamnom polju za model I. Analizom snimaka izokroma za sve ispitane modele uo~ava se da se maksimalni red izokrome javlja na rubu modela gdje se glavna naprezanja pojavljuju u radijalnom i cirkularnom smjeru: Radijalno naprezanje jednako je nuli (σ2=0) te je cirkularno naprezanje jednako:

2/, mmNhfNi σσ ⋅

= (5.1)

Kako je fσ / h = konst. slijedi da }e se za maksimalni red izokrome Ni max pojaviti maksimalno naprezanje:

2mm/N,hfN maxi

maxσ⋅

=σ (5.2)

Nominalno naprezanje dano je relacijom (5.3):

5. STRESSES ON MODEL CONTOUR AND STRESS CONCENTRATION FACTOR DEFINITION

Photoelasticity method is an optical method for determining stresses and deformities in technical structures and is based on piezo-optical effect, i.e. it is based on the property of the material which, when loaded, becomes optically anisotropic. Photoelastic materials are materials which are optically isotropic when they are not loaded, and when they are loaded they become optically anisotropic (bi-diffracted). Optical anisotropy depends on the size and arrangement of the stress. Therefore, by measuring bi-diffraction it is possible to conclude about stress arrangement in the model. The analysis of anisotropy (bi-diffraction) is carried out by means of devices called polariscopes. They use polarized light , i.e. light wave whose flickering is somehow stable and directed The experiment itself is carried out in the following way. A model is cut out of a flat plate and it is geometrically similar to the original part which is being tested. The model is put into the polariscope, it is loaded and then filmed or observed with the naked eye. Within the polariscope field of sight, interferential bright and dark lines appear. By analyzing these isochrome lines it is possible to conclude what the arrangement of the stresses is, i.e. what the deformity of the model is. The described experiment refers to simple problems of a plane stress in elastic area and for static loading. Recording of the isochromes was carried out in the circular polarized light, in the light and dark field, with the following parameters, i.e. diaphragm 11 and exposition 11s. After that the film was developed, the photographs were made following regular photograph procedure, and the photos of isochromes were obtained in both dark and light field for all the models. Fig. 5.1. and 5.2. give the isochrome photo in dark and light field for the model I. By analyzing isochrome photos for all the models examined, it can be noticed that the maximum isochrome order appears on the model edge where main stresses in radial and circular straight line are found: Radial stress equals zero (σ2 = 0) so the circular stress is equal to:

2/, mmNhfNi σσ ⋅

= (5.1)

Since fσ /h = const it is evident that for maximum isochrome order Ni max, maximum stress occurs:

2mm/N,hfN maxi

maxσ⋅

=σ (5.2)

Nominal stress is given by the relation (5.3):

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2mm/N,AFV

n =σ (5.3. )

gdje je: FV, N – vla~na sila na korijen pola A, mm2 – povr{ina presjeka vrata korijena

Poznavanjem maksimalnog i nominalnog naprezanja mogu se odrediti faktori koncentracije naprezanja αk u opasnom presjeku (I-I, II-II, sl. 5.3) pomo}u izraza (5.1.), (5.2.) i (5.3.):

ni

maxi

n

maxk N

N=

σσ

=α (5.4.)

Na dijagramima danim na slikama 5.4. do 5.9. dan je prikaz raspodjela naprezanja po konturi za sve modele.

2mm/N,AFV

n =σ (5.3. )

where: Fv, N – tensile force on the pole root A, mm2 – neck root cross section surface

When maximum and nominal stress is known, it is possible to determine the concentration stress factors αk in the dangerous cross section (I-I, II-II, Fig. 5.3) by means of the following expressions (5.1), (5.2) and (5.3):

ni

maxi

n

maxk N

N=

σσ

=α (5.4.)

In the diagrams given in figures 5.4. to 5.9., stress distribution along the contour for all the models is presented.

Slika 5.1. Snimak izokroma u tamnom polju za model I Figure 5.1. Isochrome photo in dark field for model I

Slika 5.2. Snimak izokroma u svjetlom polju za model I Figure 5.2. Isochrome photo in light field for model I

Slika 5.3. Prikaz opasnih presjeka modela V (a), i modela I-IV (b)

Figure 5.3. Presentation of dangerous cross sections of models V (a) and model (I-IV (b)

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Dijagrami su crtani po konturi modela tako da normala na konturu predstavlja ordinatu naprezanja odnosno reda izokrome. Svi dijagrami su crtani u mjerilu: - za red izokrome 10 mm = 1 red izokrome - za naprezanje 10 mm = 2 ( N/mm2) Analiza je vr{ena za sve modele po konturi na radijusima zakrivljenja u presjecima I –I i II - II (sl. 5.3). Za sve modele prema relaciji ( 5.3 ) i sl. 5.3. ( presjek I – I odnosno presjek II – II),izra~unati su iznosi nominalnih naprezanja (tb. 5.1), a iznosi maksimalnih naprezanja o~itanih iz dijagrama sa sl. 5.4. do 5.9. dani su u tablici 5.2. Faktori koncentracije naprezanja za modele I – VI izra~unati prema relaciji 5.4. dani su u tablici 5.3.

Diagrams have been drawn along the model contour so that the normal to the contour represents stress ordinate, i.e. isochrome order. All diagrams have been drawn in the following scale: - for isochrome order 10mmm = 1 isochrome order - for 10mm stress = 2 (N/mm2) The analysis was carried out for all the models along the contour on roundness radius in cross-sections I-I i II – II (Fig.5.3.). For all the models according to the relation (5.3) and Fig. 5.3. (cross section I – I and II – II), the amounts of nominal stresses have been calculated (Table 5.1.). The amounts of maximum stresses read from the diagram in Fig. 5.4. to 5.9. are given in the Table 5.2. Stress concentration factors for models I – VI have been calculated according to the relation 5.4. and are given in the Table 5.3.

Tablica 5.1. Iznosi nominalnih naprezanja σn Table 5.1. Nominal stress values, σn

Model I II III IV V VI

FV, N 450 450 450 450 300 1500

A, mm2 160 160 160 160 138 600

σn, N/mm2

2,813 2,813 2,813 2,813 2,174 2,5

Tablica 5.2. Iznosi maksimalnih naprezanja σmax Table 5.2. Maximum stress amounts, σmax

Model I II III IV V VI Napomena

7,5 6,7 6,2 5,68 10,44 1,01 Presjek I-I σmax, N/mm2

11,24 8,92 8,8 8,16 - 4,95 Presjek II-II

Slika 5.4. Raspored naprezanja i reda izokroma po konturi, za model I Figure 5.4. Stress arrangement and isochrome order along the contour for model I

Slika 5.5. Raspored naprezanja i reda izokroma po konturi, za model II Figure 5.5. Stress arrangement and isochrome order along the contour for model II

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Slika. 5.6. Raspored naprezanja i reda izokroma po konturi, za model III Figure 5.6. Stress arrangement and isochrome order along the contour for model III

Slika. 5.7. Raspored naprezanja i reda izokroma po konturi, za model IV Figure 5.7. Stress arrangement and isochrome order along the contour for model IV

Slika 5.8. Raspored naprezanja i reda izokroma po konturi, za model V

Figure 5.8. Stress arrangement and isochrome order along the contour for model V

Tablica 5.3. Faktori koncentracije naprezanja Table 5.3. Stress concentration factors, αk

Model I II III IV V VI Napomena

2,67 2,38 2,2 2,02 4,8 0,40 Presjek I-I αkI αk 4,00 3,17 3,13 2,90 - 1,98 Presjek II-II αkII

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Slika 5.9. Raspored naprezanja i reda izokroma po konturi, za model VI Figure 5.9. Stress arrangement and isochrome order along the contour for model VI

6. ZAKLJUÂK Raspodjela naprezanja na kriti~nim mjestima u~vr{}enja pola generatora, bilo u obliku lastinog repa, ~eki}a ili pak njihove kombinacije, takova je iznosa da ih treba uzeti u obzir kod mehani~kih prora~una tih elemenata. Najve}i faktor koncentracije naprezanja javlja se pri modelima kod kojih je korijen pola izra|en u obliku lastinog repa ( model V ), budu}i je kod tog oblika najizraèniji utjecaj radijusa zakrivljenosti na prijelazu tijela pola u korijen pola (sl. 5.8.). Na modelu I a posebno na modelima II – IV, faktor koncentracije naprezanja je upola manji u odnosu na model V (tablica 5.3.) {to je s jedne strane posljedica smanjenog utjecaja zareznog djelovanja (blaì je prijelaz s tijela pola na njihov korijen), a s druge strane dosjedne povr{ine izme|u korijena pola i utora u lan~anom prstenu se pove~avaju s porastom kuta γ. Kod korijena polova u obliku ~eki}a i kombinacije ~eki}-lastin rep dio naprezanja prenosio se du` vrata ~eki}a te se ukupno optere}enje ne koncentrira samo na presjek I-I (sl. 5.5.) kao pri modelu V, {to tako|er utje~e na znatno smanjenje koncentracije naprezanja na kriti~nim presjecima. Treba jo{ napomenuti da bi se dodatnim ispitivanjima trebao definirati optimalni kut nagiba γ.

6. CONCLUSION Stress arrangement in critical points of the generator pole fixing, regardless of the type of fixing, i.e. swallow’s tail, hammer or their combination, are such that their values should be taken into consideration when the strength of these elements is calculated. The biggest stress concentration factor occurs with models whose pole root has the shape of the swallow’s tail (model V), since this shape shows the influence of radius roundness on the spot where pole body becomes pole root. On model I and particularly on models II – IV, stress concentration factor decreases considerably (Table 5.3.) due to smaller influence of radius roundness (the transition from the body to the root is more temperate), whereas, on the other hand, the contact surface between the pole root and the groove in the chain ring increase as the angle γ increase. With pole roots in the shape of a hammer and the combination hammer-swallow’s tail, a portion of the stress is transmitted along the hammer neck and consequently, the total load is not concentrated on the cross section I – I only, (Fig. 5.5) as is the case with the model V. This also influences the stress concentration causing it to decrease at critical cross sections. It is also worth mentioning that additional tests should define optimal angle of the slope, γ.

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S porastom optere}enja (centrifugalne sile na originalima), treba izbjegavati izradu korijena pola u obliku lastinog repa i treba se orijentirati na korijene pola u obliku ~eki}a ili kombinacije lastin rep-~eki}, kod kojih je faktor koncentracije naprezanja manji i do 40% u odnosu na korijen oblika lastinog repa. Korijen pola s tri ~eki}a (model VI) treba se koristiti u onim slu~ajevima kada su optere}enja na korijen pola vrlo visoka tj. na korijen pola oblika jednostrukog ~eki}a izazivala bi naprezanja znatno iznad granica te~enja materijala koji se koriste za izradu polnih plo~a. Na taj na~in pove}ava se kriti~ni presjek tri puta pa se i naprezanja smanjuju u tom omjeru. Kod modela VI, faktor koncentracije naprezanja znatno se smanjuju (tablica 5.3.), u odnosu na ostale modele kod kojih je pol bio izra|en u obliku jednostrukog korijena, te se zato kod brzohodnih strojeva i velikih optere}enja uvijek mora i}i na izradu u~vr{}enja korijena pola oblika vi{estrukih ~eki}a. Rezultati ispitivanja nisu potpuno egzaktni budu}i je ispitivanje vr{eno uz aproksimaciju tj. polovi su na originalnoj konstrukciji optere}eni certifugalnim silama, dok su modeli bili optere}eni jednolikom vla~nom silom (ravninski model). Za daljnja istraìvanja predvi|a se analiza naprezanja na prostornom rotacionom modelu metodom zamrzavanja naprezanja uz simulaciju djelovanja centrifugalnih sila na u~vr{}enje pola rotora kako bi se uvjeti ispitivanju na modelu {to vi{e pribliìli stvarnim pogonskim uvjetima.

When the loading increases (centrifugal forces on the originals), the pole root in the shape of a swallow’s tail should be avoided. The design of the pole roots in the shape of a hammer or combinations of a swallow’s tail-hammer where the stress concentration factor is smaller than 40% compared to a swallow’s tail should be given priority. The pole root with three hammers (model VI) is used in the cases when the pole root loadings are very high, i.e. on a single hammer they would cause stresses far above the material limits, i.e. materials used for making pole root plates. In this way the critical cross section is increased three times. Consequently, stresses are reduced in that direction. With the model VI, stress concentration factor decreases considerably (Table 5.3.) compared to other models whose pole was made in the shape of a single root. Therefore, when fast speed machines and heavy loading are considered, it is always important to make poles with multiple hammers. Test results are not completely exact since tests were carried out with approximation, i.e. the poles on the original construction were loaded by centrifugal forces, whereas models were loaded by a uniform tensile strength (plane model). Further research should include the stress analysis on a rotary model using the freezing stress method along with centrifugal forces simulation on rotor poles fixing. In that, the way testing conditions on the model would be as close as possible to the real driving conditions.

7. LITERATURA - REFERENCES: [1] I. Alfirevi}, S. Jeci}, Fotoelasticimetrija,

Sveu~ili{na naklada Liber, Zagreb, 1983. [2] M.M. Filonenko-Borodi}, Teorija uprugosti,

Gostehizdat, Moskva-Lenjingrad, 1947. [3] M.M. Frocht, Photoelasticity, I i II, John Wiley

and Sons, New York, 1948. [4] V. Jari}, Naprezanja u elementima u~vr{~enja

generatora s istaknutim polovima (lastin rep), Elaborat TG 66.0000, Tvornica generatora – ''Rade Kon~ar'', Zagreb, 1978

[5] V. Jari}, D. Pustaji}, B. Mila{in~i}, Mehani~ki

prora~un generatora s istaknutim polovima, Tvornica generatora – ''Rade Kon~ar'', Zagreb, 1979

[6] S. Jeci}, Teorija elasti~nosti, Sveu~ili{na naknada Liber, Zagreb, 1981.

[7] S.N. Nikiforov, Teorija uprugosti i plasti~nosti,

Gosizdat, Moskva, 1955 [8] D. Pustaji}, Prora~un i dimenzioniranje

u~vr{~enja pola lastinim repom, Elaborat 60.0001, Tvornica generatora – ''Rade Kon~ar'', Zagreb, 1978

[9] D. Pustaji}, N. [vigir, Analiza naprezanja u

lastinom repu generatora s istaknutim polovima, Referat na XVII savjetovanju elektroenergeti~ara Jugoslavije, Struga, 1985, str.19-29

[10] N. [vigir, Izbor materijala polova rotora na

temelju komparativna analize naprezanja, Magistarski rad, Zagreb, 1984.

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[11] Lj. Kuterovac, Synchronous generator in unsymmetrical surge short circuits, Magistarski rad, FER, Zagreb 1995

[12] E. Wiedemann,W. Kellenberger, Konstruktion

elektrischer Maschinen, Springer-Verlag, Berlin/Heidelberg/New York, 1967

[13] N. Dizdarevi}, Lj. Kuterovac, S. Te{njak, Z.

Maljkovi}, Additional analysis of electric power system expected transient phenomena around

HEPP «Vinodol» after reconstruction, Dissertation, Faculty of Electrical Engineering and Computing, Zagreb, 1996 (Croatian)

[14] N. [vigir, Contibution to mechanical

calculation of synchronous generator with salient poles regarding transient phenomena in electric power system, doctor's thesis, Faculty of Mechanical Engineering an Naval Architecture, Zagreb, 1999 (Croatian)

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Ma{instvo 2(7), 75 – 86, (2003) M.êlebi}: EKSPERIMENTALNA ISTRA@IVANJA UTICAJA...

EKSPERIMENTALNA ISTRA@IVANJA UTICAJA HETEROGENOSTI STRUKTURE MATERIJALA NA

PONA[ANJE ZAVARENOG SPOJA SA PRSKOTINOM

Mirsad êlebi}, dipl.ing., Univerzitet “Dèmal Bijedi}” u Mostaru, Ma{inski fakultet, Mar{ala Tita bb.

REZIME U radu su prikazana i analizirana ispitivanja plo~astih epruveta sa ciljem procjene otpornosti materijala (~elik X60) prema razvoju prskotina u zavarenom spoju. Ispitivane su epruvete izva|ene iz osnovnog metala, metala zavara i zone uticaja toplote P imijenjen je koncept elastoplasti~ne mehanike loma na analizu uticaja heterogenosti strukture materijala zavarenog spoja sa prskotinom.

. r

Klju~ne rije~i: zavareni spoj, prskotina, mehanika loma

EXPERIMENTAL RESEARCH ON THE IMPACT OF MATERIAL STRUCTURE HETEROGENEITY ON THE BEHAVIOR OF A WELDED JOINT WITH CRACK

Mirsad êlebi}, B.Sc. Mech. Eng., assistant, University “Dèmal Bijedi}”, Faculty of Mechanical Engineering in Mostar, Mar{ala Tita b.b.

SUMMARY This paper describes and analyzes research on plate test tube with the aim of evaluating resistence of materials (steel X60) to appearance of cracks in welded joint. Research was conducted on the test tubes taken out from basic metal, welded part of the metal and zone influenced by heat. The concept of elastic-plastic fracture mechanic was used for analyzing impact of heterogeneity of material structure on welded joint with crack.

Key words: welded joint, crack and fracture mechanic

1. UVOD Pojam sigurnosti zavarene konstrukcije je kompleksan pojam i obuhvata ~itav niz osobina koje treba da obezbijede njenu radnu sposobnost u predvi|enim radnim uslovima i roku upotrebe. Pri procjeni sigurnosti zavarene konstrukcije treba, pored ostalog, sagledati uticaj tehnologije izrade na promjene osobina materijala i definisati poloàj i veli~inu zona sa promijenjenim osobinama. Zna~i, u svakoj zavarenoj konstrukciji se javljaju podru~ja sa izmijenjenim osobinama, tj. kriti~na podru~ja. Osnovni problem u procjeni sigurnosti zavarene konstrukcije je da se sagleda uticaj najnepovoljnijih eksploatacionih faktora na pona{anje konstrukcije na tim mjestima. To zna~i da je problem sigurnosti zavarene konstrukcije,ustvari, problem lokalne sigurnosti. Kada se defini{u poloàji, veli~ina i osobine ovih kriti~nih mjesta, onda je sljede}i korak uno{enje potrebnih veli~ina u matemati~ke izraze kojima se defini{u uslovi za po~etak loma.

1. INTRODUCTION The term safety used for a welded construction is a complex concept, which embraces a hole series of characteristics that have to provide its working capability in an expected working conditions and terms of use. During safety estimations of a welded construction in addition to other things, it is necessary to recognize the impact of technology production on its characteristics, changes of material and to define the position and the size of zones with the changed characteristics. This means that in every welded construction there are areas that are appearing with changed characteristics, i.e. critical areas. The basic problem of a safety estimation of a welded construction is to recognize the impact of the most negative exploitation factors on the behavior of construction on such places. That means that the safety problem of a welded construction is a problem of local safety. When the positions are defined, as well as dimensions and characteristics of such critical places the next step is to include the necessary dimensions into mathematical expression, where the conditions are defined as the begging of the fracture.

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Razvoj mehanike loma je omogu}io analizu prskotina u konstrukcijama. Parametri mehanike loma (faktor intenziteta napona – KI, otvaranje prskotine - COD, J integral) i standardizovane metode ispitivanja, otvorili su novu mogu}nost ocjene zna~aja prskotina u konstrukcijama. Primjena parametara mehanike loma na zavarene spojeve je prirodna, imaju}i u vidu vjerovatno}u pojave prskotina. Me|utim, pristup mehanike loma je zasnovan na homogenom kontinuumu i uniformnoj mikrostrukturi materijala, pa zbog toga nije neposredno primjenljiv na zavarene spojeve [1,2]. Nivo prihvatljivosti prskotina u zavarenim spojevima je poznat kao "podobnost za upotrebu". Dokument koji je BSI objavio pod oznakom PD 6493 "Uputstvo za neke metode za izvo|enje nivoa prihvatljivosti gre{aka u zavarenim spojevima izvedenim topljenjem" [2] razvijen je na osnovu analize mehanike loma i iskustva sa prskotinama u zavarenim konstrukcijama [3,4].

The development of the mechanical fracture has made it possible to analyze crack in constructions. The parameters of mechanical fracture (integrity factor – voltage - KI, crack opening – COD, J integral) and standardized methods of research have created a new possibility of marking the importance of crack in constructions. The application of mechanical fracture parameters on a welded joint is natural, taking into consideration the probability of the crack phenomenon. However, the approach of mechanical fracture is based on homogeny continuum and homogeny microstructure of the material, because of which it isnot directly applicable on welded joints [1,2]. The level of accessibility of the crack in welded joint is known as “fit for use”. The document that BSI has issued under the reference PD 6493 “Instruction for some methods for performing the level of mistake acceptability in welded joints made by melting” [2] developed on the basis of an analysis of fracture mechanic and experience with crack in welded joints [3,4].

2. ISPITIVANJA EPRUVETA SA PRSKOTINOM

Na bazi rezultata brojnih istraìvanja nastala su uputstva, preporuke i standardi za ocjenu veli~ine dopu{tene gre{ke u zavarenim konstrukcijama, namijenjeni inènjerskoj primjeni mehanike loma pri proizvodnji i eksploataciji zavarenih konstrukcija. Ova dokumenta izdata su u saradnji sa Me|unarodnim institutom za zavarivanje (MIZ). U dokumentima MIZ-a za primjenu mehanike loma na probleme sigurnosti zavarenih spojeva, postoje i odgovaraju}i modeli preko kojih se sklonost prskotine prema rastu moè ta~nije procijeniti i time obezbijediti siguran uvid u opasnost kojoj je konstrukcija izloèna. Za analizu zavarenih spojeva, npr. cijevi pod pritiskom, danas su najvi{e primijenjeni modeli:

- Ispitivanja plo~astih (kompleksnih) epruveta sa ciljem procjene otpornosti materijala prema razvoju prskotine.

- Ispitivanje modela cijevi pod pritiskom radi analize uticaja heterogenosti strukture materijala na pona{anje zavarenog spoja sa prskotinom.

Da bi se obezbijedila sigurnost osnovnog materijala i zavarenog spoja kao kriti~nog mjesta na cijevima pod pritiskom, potrebno je razjasniti njegovo pona{anje pri djelovanju stati~kog optere}enja (mehani~ko - tehnolo{ka ispitivanja svojstva materijala) kao i pona{anje pri djelovanju udarnog optere}enja na dijelove koji u sebi sadrè koncentraciju napona tipa zarez. Poznavanje ovih karakteristika materijala treba da, kao kona~no rje{enje, da ve}u sigurnost od pojave loma.

2. TUBE RESEARCH WITH CRACKS Based on numerous research results, instructions, recommendations and the standards have emerged for marking the allowed error created in welded constructions intended for engineering use of fracture mechanic during the production and exploitation of welded constructions. These documents have been issued in cooperation with International Institute of Welding (IIW). The IIW documents of fracture mechanic are applicable for solving the safety problems of welded joints. There are suitable models by which the growth tendency of the crack could be precisely estimated which would secure an insight into the danger the construction is exposed to. To analyze welded joints, e.g. pipes under pressure, the most applied models nowadays are:

- Research of plate (complex) tubes is done to assess material resistance according to crack development.

- Performing the research into pipe model under pressure for the purpose of analyzing the impact of material structure heterorgenity on the behavior of welded joint with crack.

To provide the safety of the basic material and welded joint as critical place on pipes under pressure, it is necessary to explain its behavior under the impact of static load (mechanical-technological research of material features), as well as behavior under the impact of load on parts containing contracted tension – type notch. The knowledge of these material characteristics would eventually lead to graeter safety against fracture appearance.

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U ovom radu su prikazani rezultati ispitivanja plo~astih epruveta sa ciljem pra}enja pona{anja prskotina u osnovnom metalu, metalu zavara i zoni uticaja toplote, odnosno provjere otpornosti materijala prema razvoju prskotine. Ispitivanja su ra|ena na niskolegiranom ~eliku povi{ene ~vrsto}e (mikrolegirani ~elik) X60 koji je namijenjen za izradu uzdu`no zavarenih cijevi izloènih visokim pritiscima i za izradu posuda pod pritiskom. Hemijski sastav i osnovne mehani~ke osobine ~elika X60 dati su u tab. 1 i 2.

In this study research results of plate tubes are shown with the aim to follow behavior of the crack in basic metal, welded metal and zone of temperature influence, i.e., to check the resistance of material towards development of crack. Research has been cenduced on low-alloy steel of increased hardness (HSLA steel) X60, its purpose is to make vertical welded pipes exposed to high pressure and dish under pressure. Chemical constitution and basic mechanical characteristics of the steel X60 are given in the Table 1 and 2.

Tabela 1. Hemijski sastav isporu~enog mikrolegiranog ~elika X60 Table 1. Chemical structure of delivered HSLA steel X60

[arà % mase - % mass

Charge C Mn P S Co V Nb

23851 0,12 0,33 0,020 0,010 0,35 0,045 0,056

Tabela 2. Mehani~ka svojstva mikrolegiranog ~elika X60 u stanju isporuke Table 2. Mechanical characteristics of HSLA steel X60 at the moment of delivery

[arà Charge

Napon te~enja Liquid tension Rp0,2 [MPa]

Zatezna ~vrsto}a Tighten hardness

Rm [MPa]

Izduènje Expense A [%]

Kontrakcija Contraction

Z [%]

Energija udara Impact energy

KV2 [J] 23851 448 596 22.7 55.6 73

Plo~aste (kompleksne) epruvete su osnovne epruvete za ispitivanje otpornosti materijala prema razvoju prskotina i definisane su dokumentima MIZ-a . Izgled i dimenzije epruvete, poloàj zavara i dimenzije prskotine pokazani su na sl. 1.. Vje{ta~ke prskotine ura|ene su elektroerozijom. Tako|e, postavljeni su lanci mjernih traka (KY11 1/120, dimenzija a = 0,6 mm, b = 1 mm, c = 7,2 mm, d = 14,5 mm firme HBM) sa unutra{nje (naspram prskotine) i sa spolja{nje strane (na strani prskotine), simetri~no u odnosu na prskotinu kao i ekstenzometar, sl. 1 . Izgled pripremljene epruvete za ispitivanje je dat na sl. 2. Kori{tene su tri epruvete sa prskotinom u osnovnom metalu (OM), metalu zavara (MZ) i zoni uticaja toplote (ZUT). Ispitivanje je sprovedeno na elektromehani~koj kidalici SCHENCK-TREBEL RM100, a za prikupljanje i obradu podataka kori{ten je vi{ekanalni ure|aj UPM 40 umreèn sa PC ra~unarom i AD konvertorom.

The plate (complex) tubes are basic tubes for research of material resistance toward the crack development, which are defined by the documents of IIW. The appearance, tube dimension, position of welded area and dimensions of the crack are presented in the Figure 1. Artificial cracks are made by electro-erosion. However, the chains of measured strips are set (KY11 1/120, dimensions a=0,6 mm, b=1 mm, c=7,2 mm, d=14,5 mm of HBM company) from the interior (opposite the crack) and exterior side (on crack side), symmetrical against the crack as well as extensometer, Figure 1. The appearance of the prepared tube for further research is given in Figure 2 Three tubes have been used with crack in a basic metal (BM), metal of welded area (MWA) and zone of temperature impact (ZTI). The research has been conducted on electro-mechanical cut machine “SCHENCK-TREBEL RM100, and for gathering and data processing multy-chanel device UPM 40 has been used connected to network with PC and AD convector.

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Exstensometer Outside strip

Inside strip

Slika 1. Plo~asta epruveta za zatezanje Figure 1. Plate test tube for straining

Tabela 3. Matrica ispitivanja plo~astih epruveta (nominalne dimenzije) Table 3. Matrix for plate test tube research (nominal dimensions)

znaka epruvete Tube mark

Duìna prskotine Crack length 2a [mm]

Debljina epruvete Tube thickness

t [mm]

Dubina prskotine Crack depth c [mm]

Odnos Ratio c/t

OM MZ ZUT

KE-1 12 5.3 3.0 0.57 + KE-2 12 5.3 3.0 0.57 + KE-3 12 5.3 3.0 0.57 +

Slika 2. Izgled pripremljene plo~aste epruvete

Figure 2. Appearance of prepared plate test tube

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Na osnovu dobijenih signala od mjernih traka, nacrtani su dijagrami zavisnosti deformacije od sile zatezanja, iskazane preko napona σ, koji su prikazani na sl. 3. za OM, sl. 4. za MZ i na sl. 5. za ZUT.

Based on the given signal from the strip, the diagrams are drawn in dependence of deformation made by straining force, shown over the tension σ, shown in the Figure 3 for (BM), Figure 4 for (MWA) and Figure 5 for (ZTI).

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a) inside strip b) spolja{nje trake b) outside strip

Slika 3. Dijagram: deformacija - napon za epruvetu KE-1 (OM) Figure 3. Diagram: deformation – tension for test tube KE-1 (BM)

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Slika 4. Dijagram: deformacija - napon za epruvetu KE-2 (MZ) Figure 4. Diagram: deformation – tension for test tube KE-2 (WM)

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a) unutra{nje trake a) inside strip

b) spolja{nje trake b) outside strip

Slika 5. Dijagram: deformacija - napon za epruvetu KE-3 (ZUT) Figure 5. Diagram: deformation – tension for test tube KE-3 (HIA)

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Dodatno je mjereno otvaranje prskotine specijalnim ekstenzometrom. Na~in postavljanja ekstenzometra na epruvetu prikazan je na sl. 6. Na osnovu pokazivanja ekstenzometra, dobijene su zavisnosti otvaranja usta prskotine (CMOD) od napona σ, i to je prikazano na sl. 7. za OM, na sl. 8. za MZ i na sl. 9. za ZUT.

It is necessary to have additional measured opening of crack, with special extensometer. The manner of setting the extensometer on a tube is shown in Figure 6. Based on extensometer results, the dependence of mouth crack opening (CMOD) is gained from tension σ�and this has been shown in Figure 7. for BM, Figure 8. for MWA and Figure 9. for ZTI.

Slika 6. Na~in postavljanja ekstenziometra na epruvetu

Figure 6. The manner of positioning extensometer on test tube

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p o

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Otvaranje prsline, µm/m

Slika 7. Dijagram: napon - CMOD za epruvetu KE-1 (OM) Figure 7. Diagram: tension – CMOD for test tube KE-1 (BM)

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Otvaranje prsline, µm

Slika 8. Dijagram: napon - CMOD za epruvetu KE-2 (MZ) Figure 8. Diagram: tension – CMOD for test tube KE-2 (WM)

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Pa

Otvaranje prsline, µm

Slika 9. Dijagram: napon - CMOD za epruvetu KE-3 (ZUT) Figure 9. Diagram: tension – CMOD for test tube KE-3 (HIA)

Pri analizi rezultata ispitivanja treba imati u vidu odstupanje stvarnih dimenzija epruveta i prskotina u odnosu na nominalne. Stvarne dimenzije epruveta, izmjerene poslije ispitivanja, date su u tab. 4.

Analyzing the research results we have to bear in mind the deviation of real tube dimensions and crack in nominal proportion. Real dimensions of tubes, measured after research are given in Table 4.

Tabela 4. Matrica ispitivanja plo~astih epruveta (stvarne dimenzije) Table 4. Matrix of plate test tubes research (real dimension)

Oznaka epruvete Tube mark

Duìna prskotine Crack lenght 2a [mm]


t [mm]


Odnos Ratio c/t

OM MZ ZUT

KE-1 12 5,2 2,75 0,52 + KE-2 12 5,1 2,85 0,56 + KE-3 12 5,2 2,90 0,56 +

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3. ANALIZA REZULTATA

Pore|enjem rezultata za maksimalni napon pri kome se kidaju plo~aste epruvete sa prskotinom u razli~itim oblastima zavarenog spoja (MZ, ZUT i OM), moè se zaklju~iti da je najve}a otpornost na rast prskotine u stati~kim uslovima u metalu zavara, a najmanja u ZUT, tab. 5. Pri tome treba svakako naglasiti da su dobijene razlike veoma male i da su sve tri epruvete pokidane mehanizmom plasti~ne deformacije, uz minimalan rast prskotine. Izgled jedne od polomljenih epruveta je dat na sl. 10.

3. ANALYZIS OF THE RESULTS

Comparing the results for maximal tension, whereby plate test tubes are cut with crack in different areas of welded joint (MWA, ZTI, BM) it can be concluded that the biggest resistance to crack growth in static conditions is in welded metal. The minimum is in ZTI – Table 5. Of course, we have to emphasize that the given differences are very small and that all three tubes are cut by mechanism of plastic deformation, with maximum growth of crack. The appearance of one of the broken tubes is shown in Figure 10.

Tabela 5. Maksimalni naponi pri lomu plo~astih epruveta Table 5. Maximal tensions during fracture of plate test tubes

Onaka epruvete Tube mark

σmax [MPa]


t [mm]


Odnso Ratio c/t

Poloàj prskotine

Crack position

KE-1 477.5 5.2 2.75 0.52 OM KE-2 465.8 5.1 2.85 0.56 ZUT KE-3 492.0 5.2 2.90 0.56 MZ

Slika 10. Izgled polomljene plo~aste epruvete

Figure 10. Appearance of broken plate test tube Osnovna karakteristika lomova za sve tri epruvete je razvoj deformacije koji je prvo i{ao u pravcu dubine epruvete (debljine epruvete), a zatim u pravcu duìne prskotine ({irine epruvete). Drugim rije~ima, prvo je dolazilo do te~enja ligamenta, a onda do te~enja neto presjeka. Pri tome je rast prskotine zanemarljiv, {to je posljedica nedovoljne o{trine njenog vrha (naro~ito u pravcu duìne) i velike otpornosti materijala na rast prskotine u stati~kim uslovima, odnosno njegove sposobnosti da se plasti~no deformi{e. Ovaj zaklju~ak vaì za sve ispitane oblasti zavarenog spoja, uklju~uju}i ZUT, koji se pona{a veoma sli~no osnovnom metalu.

The basic fracture characteristic for three tubes is development of deformation pointing first in the direction of tube depth (the thickness of tube) and then in the direction of crack length (tube breadth). In other words, it first comes to ligaments draining and then neto draining section. The crack growth is irrelevant, which is a consequence of insufficient sharpness of its peak (especially in length direction) and big material resistance to crack growth in statistic conditions, i.e. its ability to make plastic deformation. This conclusion is equal for all examined areas of welded joint, including ZTI, which has performances similar to basic metal.

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Ne{to ve}i maksimalni napon kod epruvete sa prskotinom u metalu zavara je posljedica druga~ijeg mehanizma razvoja plasti~ne deformacije u pravcu duìne prskotine. Naime, dok je kod epruveta sa prskotinom u osnovnom metalu i ZUT razvoj plasti~ne deformacije pratio preostali presjek ({to moè da se defini{e kao te~enje neto presjeka), dotle je kod epruvete sa prskotinom u metalu zavara plasti~na deformacija i{la prvo pod 45° u odnosu na pravac duìne prskotine, a zatim paralelno njemu. Ovakva deformacija je posljedica heterogenosti materijala zavarenog spoja, a posljedica ovakve deformacije je pove}an maksimalni napon kod epruvete sa prskotinom u metalu zavara. Za analizu pona{anja epruveta sa prskotinama u razli~itim oblastima zavarenog spoja, tako|e je interesantno pore|enje napona pri kome dolazi do plasti~ne deformacije i napona pri kome plasti~na deformacija prelazi 2% (opseg mjerenih traka). Navedeni naponi su odre|eni na osnovu pokazivanja mjernih traka, a rezultati ove analize su prikazani u Tabeli 6.

If there is more (maximum) tension at tube with crack in welded metal, it is a consequence of different mechanism development of plastic deformation in the direction of crack length. Namely, whereas with the tubes with crack in basic metal the ZTI plastic development deformation followed the rest of section (which it could be defined as draining of neto section), with the crack tube in welded metal the plastic deformation went first under 45o in relation to direction of crack length and then parallel. Such deformation is a consequence of heteroganic material of welded joint, a consequence of which then is increased maximum tension at tube with crack in welded metal. To analyze the behavior of tube with crack in different areas of welded joint, it is interesting to compare of tensions where plastic deformation occurs and tensions where plastic deformation crosses 2% (scope of strip). The above tensions are determined on the basis of given strips and the results of this analysis are presented in Table 6.

Tabela 6. Naponi pri ≈ 0.2% i pri 2% plasti~nih deformacija Table 6. Tensions at ≈0.2% and at 2% of plastic deformation

Oznaka epruvete Tube marks

σ0,2% [MPa]

σ2% [MPa]

Poloàj prskotine Crack position

KE-1 375 475 OM KE-2 283 428 ZUT KE-3 320 464 MZ

Na osnovu ovih rezultata moè se zaklju~iti da su oba navedena napona najve}i u osnovnom metala, a najmanji u ZUT. Drugim rije~ima, za isti nivo napona, deformacija je najve}a u ZUT, a najmanja u osnovnom metalu. Raspodjela deformacija u sve tri plo~e pokazuje da mjerne trake nisu pokrile dovoljno veliko rastojanje od prskotine, jer izmjerene deformacije u krajnjim mjernim trakama ne dostiù udaljenu deformaciju, kao {to se vidi sa sl. 11, gdje su prikazane deformacije sa unutra{nje i spolja{nje strane za neke karakteristi~ne vrijednosti udaljenog napona (deformacije). Naveden je primjer plo~aste epruvete sa zarezom u osnovnom metalu, jer je raspodjela deformacija sli~na i u metalu zavara i u ZUT. Me|utim, kako su za izra~unavanje J integrala interesantni oblast elasti~nosti i oblast plasti~nosti do 2%, a tu je pokrivenost mjernim trakama dobra, to bi i dobijeni rezultati bili dosta dobri, jer su u saglasnosti sa analiti~kim rezultatima koji se koriste zadnjih nekoliko godina.

Based on these results, we can conclude that both of the above tensions are biggest in basic metal and the smallest in ZTI. In other words, for the same level of tensions the deformation is the biggest in ZTI and smallest in basic metal. The distribution of deformations at all three plates shows that the strips did not cover enough distance from crack, because the measured deformations in final strips do not reach distant deformation, as is shown in Figure 11, where the deformations are shown from interior and exterior side, for some characteristic values of distance tensions (deformations). The example of plate test tube with notch in basic metal is set, because distribution of deformations is similar for welded metal and ZTI. However, the mathematic J integral is interesting for elasticity and plastic area up to 2 % and then there is good coverage of measured strips, the given results would then be good, because they are in accordance with analytic results which have been used for the last several years.

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0

500

1000

1500

2000

M2M1 M6M5M3 M4 M7 M9 M10M8

σ = 203,4 MPa σ = 297,9 MPa σ = 406,9 MPa σ = 470,6 MPa

Mik

rode

form

acija

, ε, µm

/m

0

5000

10000

15000

20000

25000

M2M1 M6M5M3 M4 M7 M9 M10M8

σ = 203,4 MPa σ = 297,9 MPa σ = 406,9 MPa σ = 470,6 MPa

Mik

rode

form

acija

, ε, µm

/m

Slika 11. Raspodjela deformacija - Epruveta sa prskotinom u osnovnom metalu Figure 11. Distribution of deformations – Test tube with crack in basic material

a) spolja{nje trake a) outside strip

b) unutra{nje trake b) inside strip

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4. ZAKLJUÂK Na osnovu ispitivanja plo~astih epruveta sa prskotinom moè se zaklju~iti da se sve tri razli~ite oblasti zavarenog spoja pona{aju sli~no u prisustvu prskotina. Klju~ni argument za ovaj zaklju~ak je prakti~no isti mehanizam razvoja loma, uz zanemarljiv rast prskotine. Naime, u sva tri slu~aja (tri epruvete), mehanizam loma je u osnovi bio isti: prvo razvoj plasti~ne deformacije u pravcu dubine prskotine (debljine epruvete), a zatim u pravcu duìne prskotine ({irine epruvete). U ovoj drugoj fazi razvoj plasti~ne deformacije epruveta sa prskotinom u metalu zavara je pokazao specifi~no pona{anje (linije klizanja pod 45° u odnosu na pravac duìne prskotine), {to je posljedica poloàja prskotine i heterogenosti osobina razli~itih oblasti zavarenog spoja. Na osnovu toga moè se zaklju~iti da uticaj heterogenosti strukture materijala moè da do|e do izraàja ukoliko je poloàj prskotine za to pogodan, ali da ni u tom slu~aju uticaj nije dominantan. Da uticaj mikrostrukture na pona{anje nije dominantan pokazuju i maksimalani naponi zabiljeèni pri ispitivanju epruveta, jer je njihova razlika u granicama 5%. Va`no je uo~iti razliku u maksimalnom naponu (naponu pri lomu) kod plo~astih epruveta. Naime, maksimalni napon je bio najve}i u slu~aju prskotine u metalu zavara, a najmanji u slu~aju prskotine u ZUT. Na osnovu dobijenih razlika u pona{anju plo~astih epruveta, moè se zaklju~iti da je u slu~aju “slobodne” konstrukcije (plo~e) uticaj strukture ve}i od uticaja ometene deformacije. Prema tome, smanjena otpornost na rast prskotine u ZUT ne mora uvijek da se iskaè u realnoj konstrukciji. Drugim rije~ima, postoji uticaj heterogenosti strukture materijala zavarenog spoja kako na otpornost na rast prskotine, tako i na silu rasta prskotine. Me|utim, pri tome treba imati u vidu da su dobijene razlike relativno male i da u inènjerskom smislu ne predstavljaju bitan faktor. Na osnovu svih navedenih razmatranja mogu se formulisati najva`niji, op{ti zaklju~ci ovog rada: * Pokazano je da heterogenost strukture materijala, i kao posledica toga heterogenost mehani~kih svojstava zavarenog spoja, uslovljava razli~ite sile rasta prskotine za razli~ite poloàje prskotine u zavarenom spoju, odnosno da uti~e na integritet zavarenog spoja i na taj, posredni, na~in. Me|utim, kao i u slu~aju otpornosti na rast prskotina, ove razlike nisu od primarnog zna~aja.

4. CONCLUSION Based on plate test tube with crack research we can conclude that all three different areas of welded joint are acting different in a presence of crack. The key argument for this conclusion is practically identical mechanism of crack development with negligible crack growth. However, in those three cases (three tubes) fracture mechanism is basically the same: the first development of plastic deformation in the direction of crack growth (tube thickness) and then in the direction of crack length (tube width). In the second phase, the development of tube deformation with crack in welded metal has shown specific behavior (slay ding lines under 45o in terms of the direction of crack length), which is a consequence of crack position and heteroganic virtues and areas of differences of welded joint. Based on that, we can conclude that the impact of heteroganic structure of material can come to exposure, if the crack position is fit for it. However in that case, the impact is not dominant. The maximum tension shows that the impact of microstructure on behavior is not dominant because their difference is in the limit of 5%. It is important to notice the difference in maximum tension (tension during fraction) at plate test tube. However the maximum tension was the biggest in crack case of welded metal and the smallest in crack case of ZTI. Based on the given differences in plate test tube behavior, it is possible to conclude that the case of “free” construction (plate) structure impact is bigger that impact of disrupted deformation. According to that, lower resistance on crack growth at ZTI does not have to be expressed in real construction, in other words there is an impact of heteroganic structure of welded joint material on crack growth resistance, as well as on crack growth power. However, we have to bear in mind that the given differences are relatively small and in engineering terms they are not representing a significant factor. Based on the given considerations, it is possible to formulate the most important general conclusions of this study: * It has been shown that the heterorganic material structure, as the well as consequence of the mechanical heterogeneity features of welded joint restricts different forces of crack growth for different positions in welded joint, i.e. makes them have an impact on the integrity of welded joint in such intermediary way. However, as in case of growth crack resistance, this difference does not have primary relevance.

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* Pokazano je da je mogu}e posti}i izuzetno visoku otpornost na rast prskotina u zavarenim spojevima niskolegiranih ~elika povi{ene ~vrsto}e, uporedivu sa otporno{}u na rast prskotina osnovnog metala. Pri tome je posebno va`no naglasiti da je visoka otpornost na rast prskotina postignuta kako u metalu zavara, tako i u ZUT.

* It has been shown that it is possible to achieve high resistance to crack growth in welded joints of low-alloy steel of increased hardness, comparable with resistance to crack growth in basic metal. It is necessary to emphasize that the high resistance to crack growth is achieved in welded metal as well as at ZTI.

5. LITERATURA - REFERENCES: [1] Z. BURZI], A. SEDMAK, S. SEDMAK,

“Koncept spremnosti za upotrebu posuda pod pritiskom izra|enih od niskolegiranog ~elika povi{ene ~vrsto}e”, Nau~no-tehni~ki pregled, Vol. 45, No. 4, str. 79-94, 1998.

[2] PD 6493 (1996), “Guidelines on some

methods for the derivation of acceptance levels for detects in fusion welded Joints”, London (BSI).

[3] F.M. BURDEKIN (1982), “Utvr|ivanje kriti~ne veli~ine gre{aka kod sudova pod pritiskom”, Monografija II Letnje {kole mehanike loma, Go{a-TMF Beograd, str. 151-160.

[4] S. SEDMAK, A. RADOVI], Lj. NEDELJKOVI]

(1979), “The Strength of weld in HSLA Steel after initial plastic deformation in Mechanical”, Behavior of materials (Edited by K.J. Miller and R.F. Smith), Vol. 3, Pergamon Pres, Oxford & New York p. 435-446.

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Ma{instvo 2(7), 87 – 92, (2003) T.Vasiu: METOD I ELEKTROMAGNETNI URE\AJI...

METOD I ELEKTROMAGNETNI URE\AJI ZA UTVR\IVANJE HABANJA VALJAONI^KIH VALJAKA

Dr. Sci. Teodor Vasiu, profesor, Tehni~ki fakultet - Hunedoara, 5 Revolutiei Str., 2750 Hunedoara, Rumunija

REZIME U radu su predstavljeni metoda i tri elektromagnetna ure|aja za utvr|ivanje habanja va jaoni~kih va jaka. l lKao primjena izloèni su eksperimentalni podaci i njihova obrada u vezi sa habanjem nekih valjaka za valjanje ~eli~nih traka, kvadratnih profila i okruglih profila.

Klju~ne rije~i: habanje, elektromagnetni ure|aji, valjci

METHOD AND ELECTROMAGNETIC DEVICES FOR DETERMINATION OF ROLLING CYLINDERS WEAR

Teodor Vasiu, Ph.D. lecturer, Faculty of Engineering of Hunedoara, 5 Revolutiei Str., 2750 Hunedoara, Romania

SUMMARY The paper presents a method and 3 electromagnetic devices to determine the wear of rolling cylinders. As application, we present experimental data and their processing related to wear of some rolling cylinders for steel stripes, squares and rounds.

Key words: wear, electromagnetical devices, rolls

1. UVOD U faktore koji imaju utjecaja na profil valjaka za valjanje spada tako|er i habanje. Habanje valjaka zavisi prije svega od intenziteta i trajanja njihovog kontakta sa valjanim materijalom, kao i od kvaliteta materijala od valjaka od kojih su oni napravljeni. [to je proces valjanja intenzivniji (podrazumijeva se {to su ve}i specifi~ni pritisci i {to je duè vrijeme valjanja u odnosu na prazni hod) to }e biti izraènije habanje. Kada habanje valjaka dostigne tako visoku vrijednost da se vi{e ne mogu posti}i zadovoljavaju}e tolerancije radnog dijela valjaka i valjanih proizvoda, ili izvaljana povr{ina postane nezadovoljavaju}a, radni valjci se moraju zamijeniti. Ideja o razvoju ure|aja koji omogu}ava utvr|ivanje habanja valjaoni~kih valjaka nametnula se iz potrebe mjerenja sa visokom ta~no{}u geometrijske promjene njegovog aktivnog dijela u relativno kratkom vremenskom roku, i to bez va|enja valjaka iz stalaka.

1. INTRODUCTION Among the factors that influence the rolling cylinders profile in time is also the wear. The cylinders wear depends first on the intensity and duration of contact between them and the rolling material, as well as on the quality of their fabrication material. The more intense the rolling process is (meaning that the higher the specific pressures and the longer the effective work time compared with stand-by time are), the more pronounced the wear is found to be. When the cylinders wear reaches a value so high that the tolerances of the active part and the rolled products could not be respected anymore, or the plate’s surface becomes non-satisfactory, the work cylinders have to be replaced. The idea of conceiving and achieving a device that allows to determine the rolling cylinders wear was imposed by the necessity of measuring with high precision the geometric modification of their active part, within a relatively short time, e.g. without dismantling the cylinders from the stand.

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2. RAD URE\AJA Osnovna elektri~na {ema elektromagnetnih ure|aja (slike 1, 2 i 3) sadrì magnetno jezgro M, namotaj za magnetiziranje B1 i dva mjerna namotaja B2 i B3. Impedanca za uravnoteènje Z (slike 1 i 2) i voltmetar V, ba`darenim u duìnskim mjerama, mogu se zamijeniti magnetnim tokom Sm i ampermetrom A (slika 3), tako|e ba`darenim u duìnskim mjerama.

2. DEVICES OPERATION. The principle electric diagram of electromagnetic devices (fig. 1, fig, 2, fig, 3.) includes a magnetic core M, a magnetizing coil B1 and two measuring coils B2 and B3. The balancing impedance Z (fig. 1 and 2) and voltmeter V, benchmarked in length units, could be replaced by a magnetic shunt Sm and an ampermeter A (fig. 3), also benchmarked in length units.

M

B3 B2 B1 V

Z

l 0 Rolling cylinder

Slika 1. Osnovna elektri~na {ema elektromagnetnog ure|aja koji sluì za utvr|ivanje habanja valjaka za valjanje ~eli~nih traka

Figure 1. The principle electric diagram of the electromagnetic device that serves for determining the rolling cylinders wear for steel strips

M B3 B2 B1

V

Slika 2. Osnovna elektri~na {ema elektromagnetnog ure|aja koji sluì za utvr|ivanje habanja valjaka za

valjanje kvadratnih ~eli~nih profila Figure 2. The principle electric diagram of the electromagnetic device that serves for determining the rolling

cylinders wears for square steel

Z

l 0

Rolling cylinder

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S M

A B3 B2 B1

l 0 Rolling cylinder

Slika 3. Osnovna elektri~na {ema elektromagnetnog ure|aja koji sluì za utvr|ivanj habanja valjaka za valjanje okruglih ~eli~nih profila

Figure. 3. The principle electric diagram of the electromagnetic device that serves for determining the rolling cylinders wear for round steel

Za nepohabanu povr{inu, zazor l0 je jednak nuli, a voltmetar, odnosno ampermetar pokazuje nultu vrijednost. Ako je l0>0, {to zna~i da dolazi do habanja valjka, voltmetar (ampermetar) pokazuje vrijednost koja je razli~ita od nule i koja je proporcionalna sa stepenom habanja. Geometrijske dimenzije magnetnog jezgra su odre|ene u zavisnosti od dimenzija valjka. Ure|aj, ~ija je {ema predstavljena na slici 1, koristi se za valjaonice ~eli~nih traka, a za valjke za valjanje ~eli~nih kvadratnih ili okruglih profila konstrukcijske modifikacije su prikazane na slikama 2 i 3. Detalji u vezi s na~inom ra~unanja, ba`darenja i kori{tenja gore opisanih ure|aja su obra|eni u literaturi [1].

For a non-worn surface, the entrefer l0 is null, and the voltmeter, respectively the ampermeter, indicates zero value. If l0>0, meaning that the cylinder’s wear occurs, the voltmeter (ampermeter) indication is different from zero and proportional with the wear size. The core’s geometrical dimensions are established depending on the rolling cylinder’s dimensions. The device, whose diagram is shown in fig. 1, is used for strip rolling mills and the cylinders for square steel, rounds respectively, the proper constructive modifications are shown in fig. 2, fig. 3 respectively. Details concerning the calculation, benchmarking and utilization mode of the above-described devices are shown in [1].

3. EKSPERIMENTALNI PODACI O HABANJU VALJAONI^KIH VALJAKA

U @eljezari S.C. SIDERURGICA S.A. Hunedoara mjereno je linearno habanje l0 [mm] kod valjaka za valjanje ~eli~nih traka, kvadratnih i okruglih profila, nakon razli~itih perioda rada. Prosje~ni rezultati dobiveni pri razli~itim i ponavljanim mjerenjima na temperaturi od 298 K (vodohla|eni valjci) i na oko 873 K predstavljeni su grafi~ki na slici 4.

3. EXPERIMENTAL DATA CONCERNING THE ROLLING CYLINDERS’ WEAR At S.C. SIDERURGICA S.A. Hunedoara Steel Company the linear wear l0 [mm] of steel stripes, square steel and round steel rolling cylinders, after different operating periods has been determined. The average results obtained from different and repeated measurements, at 298 K (water-cooled cylinders) and at aprox. 873 K, are shown in graphic manner in fig. 4.

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0

0,5

1

1,5

2

2,5

200 300 400 500 600 700 800 900 1000

Operating time [hours]

Lin

ear

wea

r [m

m]

Cylinder for steel stripes, t=298KCylinder for steel stripes, t=873KCylinder for square steel, t=298KCylinder for square steel, t=873KCylinder for round steel, t=298KCylinder for round steel, t=873K

Slika 4. Zavisnost habanja valjaka za valjanje od vremena rada Figure 4. Dependence of rolling cylinders wear on time

Iz slike 4 proizilazi da je variranje habanja otprilike jednako za svaki valjak u intervalu 200-600 radnih sati, bez obzira na radnu temperaturu. Moè se vidjeti i to da variranje habanja nije linearno, posebno nakon 600 sati rada. Ova ~injenica nije jo{ obja{njena s obzirom na to da mikrostruktura aktivne povr{ine valjaka za valjanje u trenutku mjerenja habanja nije bila predmet izu~avanja ovog rada.

From fig. 4 it results that the wear’s variation for each cylinder is roughly the same within the interval of 200-600 operating hours, regardless the work temperature. Also, it can be seen that the wear’s variation in time is not linear, especially after 600 hours in operation. This fact has not been explained yet, because the microstructure of the rolling cylinders’ active surface, has not been studied for the time being.

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4. OBRADA EKSPERIMENTALNIH PODATAKA

Iz literarure [3] slijedi da se linearno habanje aktivne povr{ine valjaka za valjanje moè predstaviti jedna~inom:

( ) 21

21

201

0 STS

VSFCtl c δγ

βα

τσσ

σ

= (1)

gdje je: C [-] – konstanta proporcionalnosti koja zavisi od vrste valjka i jedinice mjernog sitema koja se koristi pri ra~unanju, F [N] – normalno optere}enje na valjak; S [m2] – aktivna povr{ina valjka; t [K] – temperatura valjka u rasponu od (300-1000) K; σ [N/m2] – prekidna ~vrsto}a materijala valjka; Vc [m/s] ˘ obodna brzina valjka; T [s] – vrijeme trajanja jednog obrtaja valjka; τ [hours] – vrijeme rada; α, β, γ i δ – empirijski eksponenti. Da bi se iz jedna~ine (1) odredili nepoznati koeficijenti, kori{teni su sljede}i izrazi:

021

201

ASCt = ; 1ASF

=σ

; 221 A

S

Vc =σ

;

Tσ = A3; τδ = A4 (2) Logaritmiranjem nove jedna~ine dobiva se sljede}a jednakost: logl0=logA0+α·logA1+β·logA2+γ·logA3+δ·logA4 (3) s nepoznatim veli~inama A0 (koja uklju~uje konstantu C), α, β, γ i δ. Uz pomo} prikupljenih eksperimentalnih podataka, odre|ene su vrijednosti α = 2/3, β = 1, γ = 1 i δ = 2/3, i konstanta C=4x10-9 za ~eli~ne trake, 4,5x10-9 za kvadratne ~eli~ne profile i 5x10-9 za okrugle ~eli~ne profile.

4. PROCESSING OF EXPERIMENTAL DATA

It results from [3] that the linear wear of the rolling cylinders in the active area can be expressed by equation:

( ) 21

21

201

0 STS

VSFCtl c δγ

βα

τσσ

σ

= (1)

where: C [-] is a proportionality constant that depends on the cylinder’s type and the measuring system unit used for calculation; F [N] is the normal load on the cylinder; S [m2] is the cylinder’s active surface; t [K] cylinder’s temperature within (300-1000)0K; σ [N/m2] is the cylinder material’s shear resistance; Vc [m/s] is the cylinder’s peripheric speed; T [s] is the duration of cylinder’s single rotation; τ [hours] is the operating time; α, β, γ and δ are empiric exponents. In order to determine the coefficients from equation (1) the following notations have been used:

021

201

ASCt = ; 1ASF

=σ

; 221 A

S

Vc =σ

;

Tσ = A3; τδ = A4 (2) Making the logarithmation of the new equation, heads to: logl0=logA0+α·logA1+β·logA2+γ·logA3+δ·logA4 (3) with the unknown A0 (that includes the C constant), α, β, γ and δ. With the collected experimental data, were found the values α = 2/3, β = 1, γ = 1 and δ = 2/3, and constant C is 4×10-9 for steel stripes, 4.5×10-9 for steel squares and 5×10-9 for round steel.

- 91 -


5. ZAKLJU^CI Elektromagnetni ure|aj za ispitivanje habanja valjaka za valjanje funkcionira jednostavno, trajan je i moè ga koristiti osoblje sa prosje~nim nivom obrazovanja. Postoje tri verzije izrade, tj. za valjke koji se koriste za valjanje traka kao i za valjke za valjanje kvadratnih i okruglih profila. Osjetljivost i relativna gre{ka ure|aja su zadovoljavaju}i za potrebe proizvodnje. Upore|ivanjem vrijednosti izmjerenog habanja l0 sa izra~unatim habanjem prema jedna~ini (1) uo~ava se zadovoljavaju}e slaganje, kako pri temperaturi valjka od 298 K, tako i pri temperaturi 873 K, bez obzira na vrijednosti F, S, σ, Vc i T.

5. CONCLUSIONS The electromagnetic device for rolling cylinders wear’s investigation has a simple functioning, is solid and can be used by personnel of average education. Its construction could be in three versions, e.g.: for cylinders used for rolling of stripes, squares and rounds. The device’s sensitivity and relative error are satisfactory for the production needs. Comparing the measured wear l0 with the wear given by equation (1), we find a satisfactory accordance, both at the cylinder'’ temperature of 298 K and at 873 K, regardless the values of F, S, σ, Vc and T.

6. LITERATURA - REFERENCES [1] Gh. VASIU: “Contribution for investigation and reduction of mechanical wear at metallurgical equipment,” Doctorate thesis, Petrosani, 1975. [2] T. VASIU: “Tribology”, Copying Center of “Politehnica” University from Timisoara, 1998.

[3] * * * : "Selective documentation. Friction. Wear. Lubrication. Lubricants”. No.1, page 7-15, Bucharest, 1973.

7. POPIS OZNAKA - LIST OF SYMBOLS M – magnetno jezgro; magnetic core;

B1 – namotaj za magnetiziranje; magnetizing coil;

B2 and B3 – mjerni namotaji; measuring coils;

Z – impedenca magnetiziranja; magnetizing impedance;

V – voltmetar; voltmeter;

Sm – magnetni tok; magnetic shunt;

A – ampermetar ampermeter;

l0 – linearno habanje [mm]; linear wear [mm] ; C – konstanta proporcionalnosti [-]; proportionality constant [-] ;

F – normalna sila [N]; normal force [N];

S – aktivna povr{ina [m2]; active surface [m2];

t – temperatura [K]; temperature [K];

Vc – obodna brzina [m/s]; periphery speed [m/s];

T – vrijeme jednog obrtaja [s]; time for one rotation [s];

τ – vrijeme rada [sati]; functioning time [hours] ; α, β, γ, δ - emprijski eksponenti [-]; empirical exponents [-]; A0, A1, A2, A3 i A4 – oznake; notations

- 92 -

Ma{instvo 2(7), 93 – 104, (2003) R.Bajramovi}: ODRE\IVANJE KRITI^NE VRIJEDNOSTI...

ODRE\IVANJE KRITI^NIH VRIJEDNOSTI KOD NORMALNE I t-DISTRIBUCIJE

Doc. dr. Rasim Bajramovic, Ma{inski fakultet Sarajevo, Vilsonovo {etali{te 9, 71000 Sarajevo

REZIME Za testiranje nultih hipoteza kod statisti~ke obrade eksperimentalnih rezulta a, potrebno je porediti statisti~ke veli~ine sa kriti~nim statisti~kim v ijednostima koje se uzimaju iz odgovaraju}ih statisti~kih tabela. U ovom radu je na osnovu numeri~kih metoda dat na~in kako se odre|uju kriti~ne vrijednosti za normalnu i t-distribuciju i napravljen je odgovaraju}i softwa e u C

tr

r

t r

r t

++ programskom jeziku.

Klju~ne rije~i: nulta hipoteza, normalna distribucija, t-distribucija, kriti~ne vrijednosti, numeri~ke metode, Simpson-ovo pravilo, metoda sje~ice

DETERMINATION OF CRITICAL VALUES FOR NORMAL AND t-DISTRIBUTION

Rasim Bajramovic, Ph. D., Assistant Professor, University of Sarajevo Mechanical Engineering Faculty, 71000 Sarajevo, Bosnia&Herzegovina, Vilsonovo setaliste 9

SUMMARY

For testing null hypothesis in sta istical analysis of experimental data, it is required to compa e statistical values with critical statistical values obtained from statistical tables. Software made in C++ language and based on numerical methods is given in this paper in o der to calcula e critical values for normal and t-distribution.

Key words: null hypothesis, normal distribution, t-distribution, critical values, numerical methods, Simpson’s rule, Secant method

1. UVOD Poznavanje statistike je veoma bitno za bolje razumijevanje inìnjerskih i nau~no istraìva~kih problema te za dono{enje odgovaraju}ih zaklju~aka na osnovu eksperimentalnih podataka. Statisti~ke metode su se tradicionalno koristile u deskriptivne svrhe, za organizovanje i sumiranje numeri~kih podataka. Deskriptivna statistika se bavi tabeliranjem eksperimentalnih rezultata, njihovim predstavljanjem u grafi~kom obliku te ra~unanjem deskriptivnih pokazatelja. Savremena primjena statisti~kih metoda se u prvom redu odnosi na obavljanje statisti~ke analize i dono{enje odgovaraju}ih zaklju~aka. Sve dok se ne donose op{ti zaklju~ci na osnovu izra~unatih pokazatelja radi se samo o opisivanju onog {to je zabiljeèno u eksperimentu. U momentu kada se pored opisa do|e i do nekih op{tih zaklju~aka ulazi se u oblast statisti~ke inferencije (statisti~ka analiza i dono{enje zaklju~aka).

1. INTRODUCTION In order to understand engineering and scientific problems better and to be able to draw correct inferences from experiment, the knowledge of statistics is essential. Statistical methods are traditionally used for descriptive purposes, for organizing and summarizing numerical data. Descriptive statistics deals with tabulation of experimental data, its presentation in graphical form and calculation of descriptive measures. Modern uses of statistical methods primarily concern statistical inference. As long as people refrain from making generalization based on calculated measures, they are only describing what they have observed. But as soon as they make an inductive generalization they have passed beyond description and entered the field of inference.

- 93 -


Sva eksperimentala posmatranja su podlo`na rasipanju te se zato svi eksperimentalni rezultati moraju procjenjivati u odnosu na ovo rasipanje. Ova deklaracija je od fundamentalnog zna~aja za test zna~ajnosti, {to omogu}uje istraìva~u da procijeni da li je uo~ena razlika realna ili je samo posljedica slu~ajnosti. Mogu se posmatrati razlike izme|u dva niza posmatranja ili razlika izme|u eksperimentalne i neke hipoteti~ke distribucije. Statisti~ka hipoteza je predpostavka o distribuciji slu~ajne varijable: naprimjer da neka distribucija ima srednju vrijednost 20.3, ili da nema razlike izme|u dvije distribucije. Statisti~ko testiranje hipoteze je procedura u kojoj se koristi uzorak eksperimentalnih podataka da bi se ustanovilo da li se hipoteza moè prihvatiti ili odbiti ukoliko nije ta~na. Hipoteza koja se testira obi~no se naziva nulta hipoteza, a suprotna ovoj hipotezi se naziva alternativna hipoteza. Na osnovu eksperimentalnih rezultata odre|uje se neka od statisti~kih mjera i poredi sa kriti~nom vrijedno{}u te mjere pri nivou zna~ajnosti α (ili 100α %). Podru~je koje sadrì vrijednosti za koje se odbija hipoteza naziva se podru~je odbijanja ili kriti~no podru~je. Podru~je vrijednosti za koje se ne odbija nulta hipoteza naziva se podru~je prihvatanja. Vrlo ~esto se za α usvaja vrijednost 5%. U statisti~koj analizi postoje jednostrani i dvosstrani testovi. Kod jednostranog testa nivo zna~ajnosti α je samo na jednom repu distribucije, dok je kod dvostranog testa α /2 na jednom repu i α /2 na drugom repu distribucije. Kriti~ne vrijednosti za testiranje nulte hipoteze se uzimaju iz odgovaraju}ih statisti~kih tabela za izabrani nivo zna~ajnosti . Za mnoge statisti~ke aplikacije pravi se odgovaraju}i software, pri ~emu se kriti~ne vrijednosti daju kao ulazni podaci. Bilo bi mnogo efikasnije na}i neki na~in da se ove vrijednosti odre|uju u samom programu koriste}i odgovaraju}e numeri~ke metode te na taj na~in izbjegne upotreba tabela i mogu}ih gre{aka. U ovom ~lanku je dat numeri~ki metod i odgovaraju}i software za odre|ivanje kriti~nih vrijednosti za normalnu i t-distribuciju.

α

Experimental observations are subject to scattering, and all experimental results must be viewed in relation to the scattering. This statement is fundamental for test of significance, which enables a researcher to judge whether any observed differences are real or due to chance alone. The differences studied may be those between two comparable sets of observations or between an experimental distribution and a hypothetical distribution. A statistical hypothesis is an assumption about the distribution of a random variable; for example that a certain distribution has mean 20.3, or that there is no difference between two distributions. A statistical test of a hypothesis is a procedure in which a sample of experimental data is used to find out whether the hypothesis may be accepted or, if untrue, rejected. The hypothesis to be tested is often called the null hypothesis, and the opposite one is known as an alternative hypothesis. According to experimental data some test statistic is calculated and compared with critical value of that statistic for the significance level α (or 100α %). The region containing the values for which the hypothesis is rejected is called the rejection region or critical region. The region of values for which the hypothesis is not rejected is called the acceptance region. A frequent choice for α is 5%. There are one-sided and two-sided tests in a statistical analysis. In one-sided test, the significance level is only in one tail of the distributon, whereas in two-sided test α /2 is in one tail and

/2 in the other tail. Critical values for testing null hypothesis are taken from statistical tables for the chosen significance level α . In many statistical applications the corresponding software is made, where the critical values are a part of input. It is more convenient to find the way to calculate these values using adequate numerical methods to avoid the use of tables and possible mistakes. The numerical method and corresponding software for determination of critical values for normal and t-distribution is given in this paper.

α

α

2. NORMALNA I t-DISTRIBUCIJA Normalna ili Gaussian-ova distribucija vjerovatno}e za slu~ajnu varijablu x se daje pomo}u sljede}e funkcije gustine vjerovatno}e

−

−=2

21exp

21)(

σµ

πσxxf (1)

2. NORMAL AND t-DISTRIBUTION The normal or Gaussian probability distribution for a random variable x is given by the following probability density function

−

−=2

21exp

21)(

σµ

πσxxf (1)

- 94 -


gdje je srednja vrijednost varijable x a σ

standardna devijacija.

µ

Korisno je definisati transformisanu varijablu z=(x-)/σ , koja tako|er ima normalnu distribuciju, ali

sa srednjom vrijedno{}u nula i jedini~nom standardnom devijacijom. Ova distribucija se ~esto zove standardna normalna distribucija, a njena funkcija gustine vjerovatno}e se defini{e kao

µ

−=

2exp

21)(

2zzf Nπ

(2)

Slika 1 prikazuje grafik funkcije gustine vjerovatno}e za normalnu distribuciju u funkciji zavisnosti od obadvije varijable x i z. Kumulativna funkcija vjerovatno}e za varijablu z se daje kao

∫∞−

−=

z

N dzzzF2

exp21)(

2

π (3)

Prema konvenciji se distribucija za x i z izraàva

kao x:N( ,σ ) i z:N(0,1µ 2

2σ

2). Ovo zna~i da x ima

normalu distribuciju sa srednjom vrijedno{}u i

varijansom , a z normalnu distribuciju sa srednjom vrijedno{}u 0 i varijansom 1. Ovdje treba napomenuti da se navodi varijansa a ne standardna devijacija. Integral za odre|ivanje F

µ

N nije mogu}e izraziti preko kona~nog broja elementarnih funkcija. Zbog ovoga se kod uobi~ajene primjene ~esto koriste tabele za odre|ivanje potrebnih vrijednosti.

where is the mean of x and σ its standard

deviation.

µ

It is useful to define the transformed variable z=(x-)/σ . This is also normally distributed but has

the zero mean and unit standard deviation. This distribution is often called the standard normal distribution. Its probability density function is

µ

−=

2exp

21)(

2zzf Nπ

(2)

Figure 1 shows the graph of the normal probability density function in terms of both x and z. The cumulative probability function for z is

∫∞−

−=

z

N dzzzF2

exp21)(

2

π (3)

Conventionally, the distribution of x and z are

expressed as x:N( ,σ ) and z:N(0,1µ

2

2 2). This

means that x is normally distributed with the mean

and variance σ and z is normally distributed

with mean 0 and variance 1. It should be noted that the variance is quoted, not the standard deviation. It is impossible to express the integral for F

µ

N in terms of a finite number of elementary functions. Consequently, in ordinary use of the normal law, tables are often used to determine values of FN.

Slika 1. Povr{ine ispod krive normalne distribucije Figure 1. Areas under the normal probability curve

Ukoliko se uzme uzorak eksperimentalnih podataka veli~ine n o slu~ajnoj varijabli x koja ima normalnu

distribuciju sa nepoznatnom srednom vrijedno{}u µ

- 95 -


i poznatom varijansom σ te ako se traì da se odredi interval pouzdanosti za tada srednja

vrijednost uzorka

2

µ

2

)1,

−ix−11

n

2

21+ν

dt

∑=

=n

iix

nx

1

1 (4)

ima distribuciju N( ,σ /n) i shodno tome µ

0( : /

Nn

xzσ

µ−= (5)

U slu~aju da je varijansa σ tako|er nepoznata, tada se odre|uje varijansa uzorka s

2

2 prema izrazu

If there is a sample of experimental data of the size n on the random variable x, which is normally distributed with the unknown and known

variance , and it is required to produce a confidence interval for then the sample mean

µ2σ

µ

∑=

=n

iix

nx

1

1 (4)

has the distribution N( ,σ /n). C consequently µ 2

)1,0( : /

Nn

xzσ

µ−= (5)

In the case the variance σ is also unknown, the sample variance s

2

2 is calculated as

( )

−

−== ∑∑∑

===

2

11

2

1

22 11

1 n

ii

n

ii

n

ix

nx

nxs (6)

Varijansa uzorka s2 se koristi kao procjena

varijanse populacije σ i ako se zamijeni u jedna~ini (5) dobije se sljede}a statisti~ka mjera

/ ns

xT µ−= (7)

Ova veli~ina ne moè imati normalnu distribuciju jer ima dodatni izvor varijacije u s pored onog u varijaciji veli~ine koja ima normalnu distribiciju. Distribucija veli~ine T se zove Student-ova distribucija i ima funkciju gustine vjerovatno}e

2

1

)(

+

=

νt

Ktf (8)

Konstanta K je potrebna da bi vrijednost integrala funkcije gustine vjerovatno}e bila jednaka jedinici u granicama od − . Funkcija je uvijek u obliku zvona kao {to je i normalna distribucija, a njena {irina ovisi od parametra ν , koji se naziva broj stepeni slobode. Slika 2 prikazuje funkcije gustine vjerovatno}e Studento-ove distribucije za 1 i 5 stepeni slobode u pore|enju sa standardnom normalnom distribucijom. Vrijednost konstante K je data izrazom

∞∞ to

Γ

+

Γ=

2

21

ννπ

ν

K (9)

gdje je (x) gama funkcija bilo kog realnog broja x, definisana kao

Γ

∫∞

−−=Γ0

1)( etx tx (10)

The sample variance s2 is used as an estimate of

the population variance σ and if substituted into Equation (5), the following statistic value is obtained

2

/ ns

xT µ−= (7)

This quantity cannot have the normal distribution because it has an additional source of variation s, beyond that of the normally distributed quantity. The distribution of T is called the Student’s t distribution and has the probability density function

21

2

1

)( +

+

= ν

νt

Ktf (8)

The constant K is present to make the integral of the probability density function from come to unity. The function is always shaped as a bell, the same as the normal distribution but its width depends on the parameter ν , which is called the degree of freedom. Figure 2 shows the probability density functions of the Student’s distribution for 1 and 5 degrees of freedom compared with the standard normal distribution. The value of K is given by

∞∞− to

Γ

+

Γ=

2

21

ννπ

ν

K (9)

where (x) is the gamma function of any real number of x defined as

Γ

∫∞

−−=Γ0

1)( dtetx tx (10)

- 96 -


Postoje neke va`ne relacije gama funkcije koje su kori{tene u izradi ovog software-a

π=

ΓΓ=+Γ

21 );()1( xxx (11)

Ako je x pozitivan cijeli broj tada je Γ (x)=(x-1)! (12)

Kumulativna funkcija vjerovatno}e za t je

∫∞−

=t

dttftF )()( (13)

There are some important relations for gamma function used in designing this software

π=

ΓΓ=+Γ

21 );()1( xxx (11)

If x is a positive integer then Γ (x)=(x-1)! (12)

The cumulative probability function for t is

∫∞−

=t

dttftF )()( (13)

Slika 2. Pore|enje Student-ove disstribucije sa N(0,1) Figure 2. Compa ision of Student’s Distribution with N(0,1) r

3. MATEMATSKI MODEL Kod definisanja matematskog modela za odre|ivanje statisti~kih kriti~nih vrijednosti potrebno je prethodno objasniti neke numeri~ke metode kori{tene u ovom modelu. Problem numeri~ke integracije se sastoji u numeri~kom metodu ra~unanja sljede}eg odre|enog integrala

∫=b

a

dxxfI )( (14)

gdje su a i b poznati a f(x) funkcija gustine vjerovatno}e za razmatranu distribuciju. Geometrijski, I je povr{ina ispod krive f(x) izme|u a i b. Ovaj problem se lahko rje{ava Simpson-ovim pravilom, koje je od velike prakti~ne va`nosti jer daje dovoljno ta~ne rezultate za najve}i dio problema. Kod izvo|enja formule za Simpson-ovo pravilo koristi se kvadratna aproksimacija. Interval integracije se dijeli u paran broj jednakih podintervala (2n podintervala) du ine h=(b-a)/2n, sa grani~nim ta~kama x

bxa ≤≤

0(=a), x1, …,x2n-1,x2n(=b). Formula za Simpson-ovo pravilo se daje kao

3. MATHEMATICAL MODEL In order to define the mathematical model for calculating statistical critical values some of the numerical methods used in this model should be explained first. The problem of numerical integration is the numerical evaluation of a definite integral

∫=b

a

dxxfI )( (14)

where a and b are given and f(x) is probability density function for the given distribution. Geometrically, I is the area under the curve of f(x) between a and b. This problem can be easily solved by Simpson’s rule, which is of great practical importance because it is sufficiently accurate for most problems. The quadratic approximation is used to derive the formula for Simpson’s rule. The interval of integration is subdivided into even number of equal subintervals, say, into 2n subintervals of length h=(b-a)/2n, with end points x

bxa ≤≤

0(=a), x1,…,x2n-1,x2n(=b). The formula for Simpson’s rule is given by

( )nnn

b

a

fffffffhdxxf 212223210 42...4243

)( +++++++≈ −−∫ (15)

- 97 -


gdje je fj=f(xj). Problem nalaènja povr{ine ispod krive je prili~no jednostavan. Obrnuti problem nalaènja kriti~nih vrijednosti za datu povr{inu ispod krive je mnogo kompleksniji. Jedan na~in rje{avanja ovog problema je metodom poku{aja i gre{ke. Predpostavi se neka kriti~na vrijednost i za nju se po Simpson-ovom pravilu izra~una odgovaraju}a popvr{ina i odredi gre{ka. Poslije se predpostavi nova kriti~na vrijednost i postupak se ponavlja dok se ne dobije zadovoljavaju}e rje{enje. Ova procedura zahtijeva mnogo iteracija i uzima puno vremena. Metod dat u ovom ~lanku je vrlo brz a baziran je na problemu grani~nih vrijednosti. Da bi se mogla primijeniti ova procedura potrebno je izabrati odgovaraju}u iteracionu numeri~ku metodu za rje{enje jedna~ine f(x)=0. Za ovaj problem metoda sje~ice se pokazala kao najbolja. Metoda sje~ice se dobija iz Newton-ove metode zamjenom prvog izvoda f’(x) sa odnosom razlika

1

1 )()()('

−

−

−−

≈nn

nnn xx

xfxfxf (16)

Naredna vrijednost iteracije za x se dobije prema

)()()(

1

11

−

−+ −

−−=

nn

nnnnn xfxf

xxxfxx (17)

Geometrijski gledano, u metodi sje~ice xn+1 je presjek x-ose i sje~ice krive koja odgovora vrijednostima xn-1 i xn. Metoda sje~ice zahtijeva dvije po~etne vrijednosti, x0 i x1 ali se izbjegava odre|ivanje prvih izvoda. Funkcije gustine vjerovatno}e za standardnu normalnu i t-distribuciju su simetri~ne krive i njihove kumulativne funkcije vjerovatno}e iznose 0.5 za interval integracije − . Software dat u ovom ~lanku je baziran na ~injenici da se numeri~ka integracija vr{i od 0 do z za normalnu distribuciju odnosno 0 do t za t-distribuciju.

0 do ∞

Po~etna vrijednost u problemu grani~nih vrijednosti je poznata (iznosi 0) i treba odrediti kriti~nu vrijednost za posmatranu povr{inu. Za ovu svrhu se koristi metoda sje~ice pri ~emu ovdje treba naglasiti nekoliko ~injenica. Funkcije f(xn) and f(xn-1) u jedna~ini (17) predstavljaju odgovaraju}e povr{ine ispod krive, ne mogu se izraziti analiti~ki nego se moraju odrediti numeri~ki pomo}u Simpson-ovog pravila. Naredna vrlo bitna stvar je da se u ovam problemu ne rje{ava jedna~ina f(x)=0, nego jedna~ina f(x)=A, gdje je A zahtijevana povr{ina. Da bi se ovo uzelo u obzir treba jedna~inu (17) transformisati u jedna~inu

[ ])()(

)(1

11

−

−+ −

−−−=

nn

nnnnn xfxf

xxAxfxx (18)

Ova iterativna procedura se zavr{ava kada se postigne zadovoljavaju}a konvergencija.

where fj=f(xj). The problem of finding area under the curve is pretty simple. The opposite problem, namely that of finding critical values for the given area under the curve is a more complex one. One way to solve it is by the method of trial and error. A critical value is assumed, the corresponding area is calculated according to the Simpson’s rule, the error is determined, a new critical value is predicted and so on until the satisfactory solution is obtained. This procedure takes a lot of time. The method given in this paper is very fast and it is based on boundary value problem. To apply this procedure, adequate iteration numerical method for solving equation f(x)=0 should be chosen. In this particular problem the secant method has proven to be the best. The secant method is obtained from Newton’s method by replacing the derivative f’(x) by the difference quotient

1

1 )()()('

−

−

−−

≈nn

nnn xx

xfxfxf (16)

The next iteration value for x is determined as

)()()(

1

11

−

−+ −

−−=

nn

nnnnn xfxf

xxxfxx (17)

Geometrically, in the secant method, the xn+1 is the intersection of the x-axis and the secant of the curve corresponding to xn-1 and xn. The secant method needs two starting values, x0 and x1 but avoids the evaluation of the derivatives. The probability density functions for standard normal and t-distribution are symmetrical curves and the cumulative probability distributions for them are equal to 0.5 for the integration interval

. The software given in this paper is based on the fact that the numerical integration is performed for the interval 0 to z for the normal distribution or 0 to t for the t-distribution.

0 to∞−

The initial value in this boundary value problem is known (equal to 0) and the critical value should be determined for the required area. The secant method is used for this purpose, but a few facts should be stressed here. Functions f(xn) and f(xn-1) in the equation (17), which represent the areas under the curve, are not known analytically but must be determined numerically by the Simpson’s rule. Another very important point is that this problem does not solve the equation f(x)=0, but the equation f(x)=A, where A is the required area. In order to take this into account the equation (17) should be transformed into

[ ])()(

)(1

11

−

−+ −

−−−=

nn

nnnnn xfxf

xxAxfxx (18)

This iterative procedure is finished when satisfactory convergence is obtained.

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4. REZULTATI Na kraju ovog ~lanka kao dodatak dat je listing software u C++ jeziku baziran na datom matematskom modelu. Ovaj program uklju~uje sljede}e funkcije: funk( ) za odre|ivanje specificirane funkcije gustine vjerovatno}e; simpson( ) za numeri~ku integraciju i gamma( ) za odre|ivanje gama funkcije. Za normalnu distribuciju ulazni parametar je samo povr{ina ispod krive za interval izme|u i traène kriti~ne vrijednosti (jednostrani test), a za t-distribuciju dodatni parametar je broj stepeni slobode. Navedeni programi su potpuno testirani i rezultati pore|eni sa kriti~nim vrijednostima datim u statisti~kim tabelama. Samo dio testiranih vrijednosti je dat u sljede}im tabelama.

∞−

4. RESULTS The software in C++ language, based on the given mathematical model, is listed as the Appendix. This program includes the following functions: funk( ) for the calculation of specified probability density function; simpson( ) for the numerical integration and gamma( ) for the calculation of gamma function. For the normal distribution the input value is only the area under the curve for the interval between and the needed critical value (one-sided test) and for the t-distribution another input value is the number of degrees of freedom. The given programs have been thoroughly tested and the results compared with the critical values given in statistical tables. Only a part of tested values is given in the following tables.

∞−

NORMALNA DISTRIBUCIJA - NORMAL DISTRIBUTION

Povr{ina - Area 100(1-α )%

Statisti~ke tabele – Statistical tables Software

95 1.645 1.644854 97.5 1.960 1.959964 99 2.326 2.326348 99.5 2.576 2.575829 99.95 3.291 3.290527 99.9995 3.891 3.890592

Sljede}a tabela daje kriti~ne vrijednosti za t-distribuciju pri ~emu su vrijednosti u zagradama dobivene kori{tenjem software-a.

The next table presents the critical values for t-distribution, where the values in parentheses are obtained by the given program.

t-DISTRIBUCIJA - t-DISTRIBUTION

100(1-α )% Broj stepeni slobode

90 95 99

1 3.078 (3.077684) 6.314 (6.313752) 31.821 (31.820516) 20 1.325 (1.325341) 1.725 (1.724718) 2.528 (2.527977) 60 1.296 (1.295821) 1.671 (1.670649) 2.390 (2.390120)

Za dobivanje kriti~nih vrijednosti kori{tenjem datog software-a trebali su samo dijelovi sekunde. Slaganje rezultata navedenih u prethodne dvije tabele je izvanredno s obzirom da programski rezultati zaokruèni na tri decimale su identi~ni sa tabelarnim vrijednostima.

It took only a few parts of a second to obtain critical values using the given software. The agreement among the results given in previous two tables is perfect since the program values rounded off to three decimal places are the same as the table values.

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5. ZAKLJUÂK U oblastima inìnjerstva i eksperimentalnih nauka upotreba statistike je skoro neizbje`na u istraìva~kom radu i proizvodnji. Sa razvojem kompjuterske nauke javlja se i potreba za izradu programa koji vr{e statisti~ku analizu i dono{enje odgovaraju}ih zaklju~aka. Kriti~ne vrijednosti se uzimaju iz statisti~kih tabela i unose kao ulazni podaci. Software dat u ovom ~lanku odre|uje ove kriti~ne vrijednosti vrlo brzo i ta~no i ovaj se software moè jednostavno uklju~iti u bilo koji program statisti~ke analize. Na ovaj na~in se, pored u{tede u vremenu, smanjuju mogu}nosti gre{ke pogre{nog o~itanja ili pogre{nog uno{enja ulaznih podataka. U dodatku navedeni programi mogu se lahko transformisati u module i bilo kom drugom programu dodati. Ovaj pristup je veoma poznat kao tehnika modularnog programiranja, gdje programeri razbijaju programe u pojedine funkcije (module), pri ~emu svaka funkcija obavlja specifi~an dobro definisan zadatak. Za obavljanje ovog zadatka veoma je pogodan C++ programski jezik, koji tako|er u potpunosti podràva objektno orijentisano programiranje.

5. CONCLUSION In the fields of engineering and experimental sciences, the use of statistics is almost invariably required in research work and in production. With the development of computer science there is a need to design softwares for statistical inference. Critical values are taken from statistical tables and given as input data. The software given in this paper calculates these critical values very quickly and accurately, and it can be easily incorporated into any program for statistical inference. In this way, the possible errors due to wrong reading or wrong input are avoided. The programs listed in the appendix can simply be transformed into modules and easily added to any software. This approach is well known as modular program-design technique, where programmers break down their programs into individual functions, each of which performs a specific and well-defined task. A very convenient language for accomplishing these requirements is C++ language which also fully supports the object-oriented programming.

6. LITERATURA – REFERENCES [1] E. Kreyszig, “Advanced Engineering

Mathematics”, 6th Ed., John Wiley & Sons, New York, 1988.

[2] I. Demirdì}, “Numeri~ka Matematika”, IP

“Svjetlost” Sarajevo, 1997.

[3] R. Bajramovi}, “Savremene Statisti~ke Metode”, Predavanja na Postdiplomskom Studiju Ma{inskog Fakulteta u Sarajevu, 2001.

7. DODATAK – APPENDIX

// NDISTR.CPP Odredivanje kriticnih vrijednosti kod normalne distribucije // (problem granicnih vrijednosti) // Autor: Rasim Bajramovic Masinski fakultet Sarajevo #include<stdio.h> #include<iostream.h> #include<conio.h> #include<math.h> double funk(double x); double simpson(double a, double b); void main() { clrscr();

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double xa,xb0,xb1,pov,xb,yk0,yk1,sign; while(2) { xa=0.; sign=1.; xb0=0.2; xb1=0.3; printf("\tUnesite trazenu povrsinu: (-1 za izlaz) "); cin >> pov; if(pov==-1) break; if(pov==0.5) { printf("\n\tKriticna vrijednost je: 0.\n\n\n"); continue; } if((pov-0.5)<0) sign = -sign; pov = fabs(pov-0.5); yk0 = simpson(xa, xb0); yk1 = simpson(xa, xb1); while(1) { xb = xb1 - (yk1-pov)*(xb1-xb0)/(yk1-yk0); xb0 = xb1; xb1 = xb; yk0 = yk1; if(xb<0) { xb0 = 0.000001; xb1 = 0.00001; yk0 = simpson(xa, xb0); } yk1 = simpson(xa, xb1); if((fabs(yk1-pov)<0.00000001)) break; } // kraj while(1) printf("\n\tKriticna vrijednost je: %9.6f\n\n\n", xb*sign); } // kraj while(2) } // kraj main() double funk(double x) { return exp( -x * x/2)/sqrt(2. * 3.141592654); } double simpson(double a, double b) { int i, n = 100; double h, suma = 0, povrsina; h = (b - a) / n / 2; suma += funk(a) - funk(b); for(i = 1; i <= n; i++) suma += 4.*funk(a+(2*i-1)*h)+2.*funk(a+2*i*h);

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povrsina = fabs(h)/3.* suma; return povrsina; } // kraj simpson() // TDISTR.CPP Odredivanje kriticnih vrijednosti kod t-distribucije // (problem granicnih vrijednosti) // Autor: Rasim Bajramovic Masinski fakultet Sarajevo #include<stdio.h> #include<iostream.h> #include<conio.h> #include<math.h> double funk(double x, double K, int n); double simpson(double a, double b, double K, int ni); double gamma(int n); void main() { clrscr(); double xa,xb0,xb1,pov,xb,yk0,yk1,K,sign; int ni=1; while(2) { xa=0.; sign=1.; xb0=2.; xb1=3.; printf("\tUnesite trazenu povrsinu: (-1 za izlaz) "); cin >> pov; if(pov==-1) break; if(pov==0.5) { printf("\n\tKriticna vrijednost je: 0.\n\n\n"); continue; } if((pov-0.5)<0) sign = -sign; pov = fabs(pov-0.5); printf("\tUnesite broj stepeni slobode: "); cin >> ni; K = gamma(ni+1)/gamma(ni)/sqrt(ni*3.141592654); yk0 = simpson(xa, xb0, K, ni); yk1 = simpson(xa, xb1, K, ni); while(1) { xb = xb1 - (yk1-pov)*(xb1-xb0)/(yk1-yk0); xb0 = xb1; xb1 = xb; yk0 = yk1; if(xb<0) {

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xb0 = 0.000001; xb1 = 0.00001; yk0 = simpson(xa, xb0, K, ni); } yk1 = simpson(xa, xb1, K, ni); if((fabs(yk1-pov)<0.00000001)) break; } // kraj while(1) printf("\n\tKriticna vrijednost je: %9.6f\n\n\n", xb*sign); } // kraj while(2) } // kraj main() double funk(double x, double K, int n) { double f = K/pow(1.+x*x/n, (n+1.)/2.); return f; } // rezultat je gamma vrijednost parametra ni/2. double gamma(int ni) { int i, np; double K, dp; np = ni/2; dp = np; K = 1.; if(ni%2==1) { for(i=0; i<np; i++) K *= (dp-0.5-i); K *= sqrt(3.141592654); } else { for(i=0; i<np-1; i++) K *= (dp-1.-i); } return K; } double simpson(double a, double b, double K, int ni) { int i, n = 1000; if(ni==1) n = 10000; double h, suma = 0, povrsina; h = (b - a) / n/2; suma += (funk(a, K, ni) + funk(b, K, ni)); for(i = 0; i < n; i++) suma += 4. * funk(a+(2*i+1)*h, K, ni); for(i = 1; i < n; i++)

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suma += 2. * funk(a+2*i*h, K, ni); povrsina = fabs(h)/3.* suma; return povrsina; } // kraj simpson()

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