analiza naprezanja u ucvršcenju polova rotora generatora s ...

Ma{instvo 2(7), 63 – 74, (2003)
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ANALIZA NAPREZANJA U U^VR[^ENJU 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 istra`ivanja 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 tailhammer
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 construction
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 najslo`eniji dijelovi ovih
konstrukcija predstavljaju mjesta u~vr{}enja na
kojima je distribucija kontaktnih naprezanja na
dosjednim povr{inama nepoznata. ^est je slu~aj i
stati~ke neodre|enosti sklopa {to ote`ava 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 f σ = 103 – 120 N/cm, te
dopu{tenim naprezanjem σ d = 40 Mpa, kakav je
kori{ten i za izradu modela u ovom radu.
3. IZRADA MODELA
Svi ispitivani modeli izra|eni su iz plo~e araldit-B s
dopu{tenim naprezanjem σ d = 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 pleksistakla.
Obrada modela izvr{ena je tako da se {ablona
nalijepi pomo}u dvostrano ljepljive trake na plo~u
araldita, te zatim tra~nom pilom izre`e 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.
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 f σ σ=103–120 N/cm and the permitted stress
σ d = 40 MP, like the one used for producing the
model described in this paper.
3. MODEL PRODUCTION
All the models produced are made of an araldite-B
plate with the permitted stress of σσ d = 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 highspeed
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.
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.
3.2 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 σ dop =40 MPa.
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 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 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, σσ dop =40 MP.
Tabela 3. 1. Promjenljive veli~ine za modele I – IV
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
s 1 , 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. Polo`aj
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 F V i F H kojima su optere}ivani modeli
dani su u tablici 4.1. Odabran je uvijek onaj omjer
sila F V / F H 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 (σ v / σ h ) = 1, which
corresponds to the real arrangement of the
generator pole fixing.
Values of the forces F V and F H by which the model
was loaded can be seen in Table 4.1. The force
ratio, F V and F H 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 F V i horizontalnih F H sila optere}enja na modele
Table 4.1. Amounts of vertical and horizontal (F V and F H ) loading forces on models
Model I II III IV V VI
F V , N 450 450 450 450 300 1500
F H , 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
mo`e 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 izre`e 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 (σ 2 =0) te je
cirkularno naprezanje jednako:
σ
N i
⋅ f
2
= σ
, N / mm (5.1)
h
Kako je f σ / h = konst. slijedi da }e se za
maksimalni red izokrome N i max pojaviti maksimalno
naprezanje:
N i max
⋅ fσ
2
σ
max
= , N / mm (5.2)
h
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 (bidiffracted).
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 (σ 2 = 0) so the circular
stress is equal to:
σ
N i
⋅ f σ
2
= N
(5.1)
h
, / mm
Since f σ /h = const it is evident that for maximum
isochrome order N i max , maximum stress occurs:
N i max
fσ
2
σ
max
= , N / mm (5.2)
h
⋅
Nominal stress is given by the relation (5.3):
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σ = F V
2
n
, N / mm
(5.3. )
A
gdje je:
F V , N – vla~na sila na korijen pola
A, mm 2 – povr{ina presjeka vrata korijena
Poznavanjem maksimalnog i nominalnog naprezanja
mogu se odrediti faktori koncentracije naprezanja
α k u opasnom presjeku (I-I, II-II, sl. 5.3) pomo}u
izraza (5.1.), (5.2.) i (5.3.):
σ
N
=
max i max
α
k
=
(5.4.)
σn
Ni
n
Na dijagramima danim na slikama 5.4. do 5.9. dan
je prikaz raspodjela naprezanja po konturi za sve
modele.
F 2
σ V
n
= , N / mm
(5.3. )
A
where:
F v , N – tensile force on the pole root
A, mm 2 – neck root cross section surface
When maximum and nominal stress is known, it is
possible to determine the concentration stress factors α k
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):
σ
N
=
max i max
α
k
=
(5.4.)
σn
Ni
n
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/mm 2 )
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.
Tablica 5.1. Iznosi nominalnih naprezanja σ n
Table 5.1. Nominal stress values, σ n
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/mm 2 )
The analysis was carried out for all the models
along the contour on roundness radius in crosssections
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.
Model I II III IV V VI
F V , N 450 450 450 450 300 1500
A, mm 2 160 160 160 160 138 600
σ n ,
N/mm 2
2,813 2,813 2,813 2,813 2,174 2,5
Tablica 5.2. Iznosi maksimalnih naprezanja σ max
Table 5.2. Maximum stress amounts, σ max
Model I II III IV V VI Napomena
σ max , N/mm 2 7,5 6,7 6,2 5,68 10,44 1,01 Presjek I-I
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, α k
Model I II III IV V VI Napomena
α k
2,67 2,38 2,2 2,02 4,8 0,40 Presjek I-I α kI
4,00 3,17 3,13 2,90 - 1,98 Presjek II-II α kII
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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. ZAKLJU^AK
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 najizra`eniji 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
(bla`i 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 istra`ivanja 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 pribli`ili 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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