STROJNIŠKI VESTNIK - University of Ljubljanalab.fs.uni-lj.si/ladisk/data/pdf/sv-02-an.pdf · Ł isothermal strain-controlled low-cycle fatigue (LCF) ... Elasto-plastic hardening

STROJNIŠKI VESTNIKJOURNAL OF MECHANICAL ENGINEERING

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Strojni�ki vestnik - Journal of Mechanical Engineering 52(2006)2, 73

73Uvodnik - Editorial

Vsebina - Contents

Strojni�ki vestnik - Journal of Mechanical Engineeringletnik - volume 52, (2006), �tevilka - number 2

Ljubljana, februar - February 2006ISSN 0039-2480

Izhaja meseèno - Published monthly

Strojni�ki vestnik - Journal of Mechanical Engineering 52(2006)2, 73Vsebina - Contents

RazpraveNagode, M., Fajdiga, M.: Modeliranje nakljuènih

termomehanskih napetostno-deformacijskihstanj

Vesenjak, M., Ren, Z., Müllerschön, H., Matthaei,S.: Raèunalni�ko modeliranje gibanja gorivain njegov vpliv na konstrukcijo rezervoarja

Borovin�ek, M., Vesenjak, M., Ulbin, M., Ren, Z.:Simulacija naleta tovornega vozila ob cestnovarnostno ograjo

Furlan, M., Rebec, R., Èernigoj, A., Èeliè, D., Èermelj,P., Bolte�ar, M.: Vibroakustièno modeliranjealternatorja

Slaviè, J., Nastran, M., Bolte�ar, M.: Modeliranje inanaliza dinamike �èetke elektromotorja

Osebne vestiDoktorat, magisterija in diplome

Navodila avtorjem

PapersNagode, M., Fajdiga, M.: Thermo-Mechanical

Modelling of Stochastic Stress-StrainStates

Vesenjak, M., Ren, Z., Müllerschön, H., Matthaei, S.:Computational Modelling of Fuel Motion andIts Interaction with the Reservoir Structure

Borovin�ek, M., Vesenjak, M., Ulbin, M., Ren, Z.:Simulating the Impact of a Truck on a Road-Safety Barrier

Furlan, M., Rebec, R., Èernigoj, A., Èeliè, D., Èermelj,P., Bolte�ar, M.: VVibro-Acoustic Modellingof an Alternator

Slaviè, J., Nastran, M., Bolte�ar, M.: Modeling andanalyzing the dynamics of an electric-motorbrush

Personal EventsDoctor�s, Master�s and Diploma Degrees

Instructions for Authors

74

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101

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126

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139

Strojni�ki vestnik - Journal of Mechanical Engineering 52(2006)2, 74-84

74 Nagode M. - Fajdiga M.

Modeliranje nakljuènih termomehanskihnapetostno-deformacijskih stanj

Thermo-Mechanical Modelling of Stochastic Stress-Strain States

Marko Nagode - Matija Fajdiga(Fakulteta za strojni�tvo, Ljubljana)

Deformacijski postopek je najbolj raz�irjena metoda napovedovanja dobe trajanja dinamiènoobremenjenih delov na podroèju malocikliène trdnosti. Uporablja se tako pri nizkih, srednjih kakor tudivisokih temperaturah, èe je temperatura med obratovanjem nespremenljiva. Deformacijski postopek jeraèunsko izjemno hiter in narekuje le uporabo analiz elastiènosti s konènimi elementi. Zaradi �tevilnihprednosti, raz�irjenosti in potencialnih mo�nosti je bil prilagojen tako, da je uporaben tudi za spremenljivetemperature. Napetostno-deformacijsko stanje je popisano s Prandtlovimi operatorji. Postopek temelji nastabilnih histereznih zankah, pri èemer lezenje ni upo�tevano. Analiziran je tudi vpliv filtriranja konicsunkov. Narejena je primerjava izsledkov raziskav z rezultati meritev in Skeltonovim modelom.© 2006 Strojni�ki vestnik. Vse pravice pridr�ane.(Kljuène besede: utrujanje termomehansko, enaèbe konstitutivne, stanja napetostno-deformacijska,modeliranje)

The isothermal strain-life approach is the most commonly used approach for determining fatiguedamage, particularly in low-cycle fatigue. It is used for low, medium and high temperatures if the temperatureremains constant during the test. Computationally, it is extremely fast and generally requires elastic finite-element analyses only. For this reason it has been adapted for variable temperatures. The local temperature-stress-strain behaviour is modelled with an operator of the Prandtl type. The hysteresis loops are supposedto be stabilized and no creep is considered. The consequences of reversal point filtering are analysed.Finally, the approach is compared to several thermo-mechanical fatigue tests and the Skelton model.© 2006 Journal of Mechanical Engineering. All rights reserved.(Keywords: thermomechanical fatigue, constitutive equations, stress strain states, modelling)

Strojni�ki vestnik - Journal of Mechanical Engineering 52(2006)2, 74-84UDK - UDC 620.179.12/.13:539.388.1Izvirni znanstveni èlanek - Original scientific paper (1.01)

0 INTRODUCTION

Many products, like internal combustion en-gines, exhaust and cooling systems, turbines, nu-clear reactors, are subjected to stochastic thermo-mechanical fatigue (TMF) loading that can causefatigue failures during their usage [1]. From the de-velopment perspective, fatigue-life prediction in thedevelopment phases before prototype or producttesting is important. Fatigue life depends primarilyon loads, applied materials, product geometry andenvironmental effects. Its approval is generally basedon the following tests ([2] and [3]):� isothermal strain-controlled low-cycle fatigue

(LCF) tests,� TMF tests on specimens and components,� thermal shock tests.

0 UVOD

�tevilni izdelki, npr. motorji z notranjimzgorevanjem, izpu�ni in hladilni sistemi, turbine,jedrski reaktorji, so v èasu uporabe obremenjeni znakljuènimi termomehanskimi obremenitvami (TMU),ki lahko privedejo do utrujenostnih razpok [1]. Zrazvojnega vidika je zato pomembna napoved dobetrajanja izdelka v razvojnih fazah pred presku�anjemprototipov oziroma izdelkov. Doba trajanja je odvisnapredvsem od obremenitev, uporabljenih materialov,oblike izdelka in okoli�èin. Za njeno dokazovanje seuporabljajo naslednji preizkusi ([2] in [3]):� deformacijsko nadzorovani malociklièni preizkus

(MCUP - LCF) pri nespremenljivi temperaturi,� termomehanski preizkus delov in izdelkov in� toplotni sunek.


75Modeliranje nakljuènih termomehanskih - Thermo-Mechanical Modelling of Stochastic

In terms of complexity, LCF tests are favouredbecause of their simplicity [4]. In addition, becausethey have been carried out for decades, the testingprocedures and data recording is standardized [2].There also exist databases with material propertiesrequired for the fatigue-life prediction of low-cycleloaded parts. The final goal is, therefore, to developa methodology of fatigue-life prediction that willenable reliable estimations by avoiding expensiveTMF and thermal shock tests. The methodology isdiscussed comprehensively in [3] to [9].

The present work is concerned with a cy-clic stress-strain response prediction suitable forvariable temperatures. Assuming that creep andtransient effects, such as cyclic hardening andcyclic softening and their effect on damage accu-mulation are negligible, the hysteresis loops aresupposed to be stabilized and the constitutiveequations for elastoplasticity are applicable forstress-strain behaviour modelling. From amongthe available models ([9] to [11]), the strain-con-trolled spring-slider model [11] with temperature-dependent Prandtl densities and temperature-in-dependent yield surfaces has been chosen. It ena-bles the extension of the isothermal strain-life ap-proach to non-isothermal problems by preservinga high computational speed.

The stress-controlled and temperature-modi-fied spring-slider model is explained in [12] and [13].However, if the strain is controlled, an equivalentbut distinct spring-slider is required.

In Section 2 the temperature-dependent cy-clic stress-strain curves and parameter assessmentis discussed. In Section 3 the temperature-modi-fied strain-controlled spring-slider model is de-veloped. In the 4th section the consequences ofreversal point filtering are analysed. In Section 5the spring-slider model is compared to severalTMF tests, and the final section lists the conclu-sions.

1 TEMPERATURE-DEPENDENT CYCLICSTRESS-STRAIN CURVES

Most metals approach a cyclically stablestate after a certain number of cycles. Cyclicallystable or half-life material properties are usuallyused in fatigue analyses. Generally speaking, thetemperature�time history is stochastic, but cy-clic stress�strain curves for only a few distincttemperatures can be found in the literature. To

MCUP se izvajajo prednostno, ker so vprimerjavi z drugimi preprostej�i [4]. Poleg tega seopravljajo �e veè desetletij, postopek testiranja inzapisovanja rezultatov testiranj pa je standardiziran[2]. Obstajajo tudi baze podatkov o materialnihlastnostih, potrebnih za napoved dobe trajanjamalociklièno obremenjenih delov. Konèni cilj je razvitimetodologijo napovedovanja dobe trajanja, ki boomogoèala zanesljive napovedi brez termomehanskihtestiranj in testiranj s toplotnim sunkom. Podroèje jepoglobljeno predstavljeno v [3] do [9].

V prispevku je obravnavan problemnapovedovanja napetostno-deformacijskih stanj obupo�tevanju spremenljive temperature obratovanja.Predpostavimo, da je vpliv lezenja in prehodnihpojavov, npr. utrjevanja ali mehèanja, neznaten.Napovedovanje napetostno-deformacijskih stanjtorej temelji na stabilnih histereznih zankah inkonstitutivnih enaèbah elastoplastiènosti. Izmedrazpolo�ljivih reolo�kih modelov ([9] do [11]) je bilizbran model vzporedno vezanih vzmeti in drsnikov[11], izra�en s Prandtlovim modelom s temperaturnoodvisnimi gostotami in temperaturno neodvisnimidrsnimi povr�inami. Model omogoèa raz�iritevdeformacijskega postopka na podroèje spremenljivihtemperatur, pri èemer se velika hitrost raèunanja nezmanj�a.

Napetostno nadzorovan in temperaturnopreoblikovan reolo�ki model je obravnavan v [12] in[13]. V primeru deformacijskega nadzora potrebujemoenakovreden, a drugaèen model.

V drugem poglavju so obravnavanetemperaturno odvisne napetostno-deformacijske krivuljein ocenjevanja parametrov. V tretjem poglavju je prikazanrazvoj temperaturno preoblikovanega in deformacijskonadzorovanega modela vzmet � drsnik. V èetrtempoglavju so analizirane posledice izloèanja konic sunkov.V petem poglavju je opisano ujemanje novega modela zrezultati preizkusov in s Skeltonovim modelom, v zadnjempoglavju pa so podani pomembnej�i sklepi.

1 TEMPERATURNO ODVISNENAPETOSTNO-DEFORMACIJSKE KRIVULJE

Ker veèina materialov preide po nekem manj�em�tevilu obremenitvenih ciklov v stabilno podroèje, seza napoved dobe trajanja navadno uporabijo materialnelastnosti, ki ustrezajo polovièni dobi trajanjapresku�anca. Temperaturni èasovni poteki obremenitevso v splo�nem nakljuèni. V literaturi je mogoèe najtinapetostno-deformacijske krivulje za le nekaj razliènih



obtain the material parameters that have not beenmeasured, linear parameter interpolation, ap-proximation or non-parametric methods can beused. The latter turned out to be the most appro-priate recently. Elasto-plastic hardening solidsare most frequently modelled by the Ramberg�Osgood relation [14]:

(1),

where E(T) is Young�s modulus and K�(T) and n�(T)are the cyclic hardening coefficient and the cyclichardening exponent respectively. Parameter interpo-lation has been dealt with in [12].

2 STRESS-STRAIN-TEMPERATUREBEHAVIOUR

It has been shown in [12] that the Masingand Memory models are not valid if the tempera-ture varies during the cycle. Masing based hisfinding on the rheological spring-slider model andassumed that the model parameters are time inde-pendent [11]. Therefore, the spring-slider modelhas been adapted for variable temperatures ([12]and [13]). The model developed so far is capableof modelling elasto-plastic hardening solids andnon-linear kinematic hardening under stress con-trol. If the strain is controlled, the rheologicalspring-slider model depicted in Fig. 1 proves tobe more convenient.

From the equilibrium in a single spring-slidersegment, the total strain e is obtained q áj je e e= + ,where the slider strain |e

qj| can never exceed the

fictive yield strain qj . The spring strain can now be

expressed as:

(2)

temperatur. Napoved napetostno-deformacijskih stanjpri temperaturah, pri katerih napetostno-deformacijskekrivulje ne obstajajo, je mogoèa z uporabo interpolacije,aproksimacije ali neparametriènih metod. Predvsemslednje so se v zadnjem èasu izkazale za najbolj�e. Zapopis elastoplastiènih lastnosti gradiv se najpogostejeuporablja Ramberg-Osgoodova enaèba [14]:

kjer je E(T) modul elastiènosti, K�(T) in n�(T) pa stautrjevalni koeficient in utrjevalni eksponent.Interpolacija je bila obravnavana v [12].

2 MODELIRANJE NAPETOSTNO-DEFORMACIJSKIH STANJ

V [12] je bilo dokazano, da Masingov model inspominski modeli niso veljavni, èe se temperatura medobremenitvenim potekom spreminja. Masing je izsledkesvojih raziskav utemeljil z reolo�kim modelom vzmet �drsnik in predpostavil, da so parametri modela neodvisniod èasa [11]. Da bi odpravili omejitve omenjenih modelov,je bil model vzmet � drsnik temperaturno spremenjen([12] in [13]). Model v sedanji obliki je primeren zamodeliranje elastoplastiènih lastnosti gradiv innelinearnega kinematiènega utrjevanja pri soèasnemnadzoru napetosti. Èe je deformacija nadzornaspremenljivka, je na sliki 1 prikazan primernej�i model zamodeliranje napetostno-deformacijskih stanj.

Iz ravnote�ja na segmentu vzmet � drsnikje mogoèe izraziti celotno deformacijo e:

q áj je e e= + . Deformacija drsnika |eqj| ne more nikoli

preseèi navidezne drsne deformacije qj.

Deformacija vzmeti

1 ( )

( , )( ) ( )

n T

g TE T K T

s s

e s

¢æ ö

= = + ç ÷¢è ø

Sl. 1. Reolo�ki model vzmet � drsnikFig. 1. Rheological spring-slider model

á á ásign( ) min{ ,| |}j j j jqe e e e e e= - - -

q0

a0(T)

q1

a1(T)

qj

aj(T)

qnq

anq(T)nq

nq

e s

qj

aj(T)

eqj(t)

eaj(t)

e(t)sj



and corresponds exactly to the play operator withthe general initial value ([15] to [17]):

(3)

for 1 20 nt t t< < < <L . Thus, the current strain state

á ( )j ite depends on the previous state á 1( )j ite - calledthe memory point. Presumably, there is no residualstrain initially, so á (0) 0je = and (0) 0js = . The stressin the spring-slider segment is then:

(4)

where ( )i iT T t= and ( )j iTa is the Prandtl density.Adding the spring-slider stresses results in the totalstress in the form known as the operator of thePrandtl type ([15] to [17]):

(5)

with temperature-dependent Prandtl densities. Theplay operator given in Eq. (3) is independent of timeand temperature. Therefore, it is modified to assureequilibrium in the spring-slider.

Let us presume that at ti-1

the spring-slidersegment j is in equilibrium and multiply Eq. (3) by

1( )j iTa - (Fig. 2). This yields:

(6),

where the memory point 2 1( ) ( )j i j it ts s- -= . If thetemperature or strain change in the interval 1[ , ]i it t- ,then the boundaries of the play operator move and:

If the equation is divided by ( )j iTa , then:

(7)

ustreza operatorju ohlapa s splo�no zaèetnovrednostjo ([15] do [17]):

za 1 20 nt t t< < < <L . Trenutna deformacija á ( )j ite

je odvisna od deformacije v predhodnem trenutku

á 1( )j ite - , imenovanem spominska toèka.Predpostavimo, da zaostalih deformacij ni. Odtodizhaja á (0) 0je = in (0) 0js = . Napetost v odsekuvzmet � drsnik je tedaj:

( )i iT T t= in ( )j iTa je Prandtlova gostota. Sse�tevanjem napetosti po posameznih odsekih jemogoèe izraziti celotno napetost s Prandtlovimioperatorji ([15] do [17]):

s temperaturno odvisnimi gostotami. Ker je operatorrege, definiran v en. (3), neodvisen od èasa intemperature, ga je treba spremeniti tako, da bozagotavljal ravnote�je v odseku vzmet � drsnik tudipri spremenljivih temperaturah.

Èe je v trenutku ti-1

j-ti odsek v ravnote�ju,pomno�imo en. (3) z 1( )j iTa - (sl. 2). Odtod izhaja:

kjer velja 2 1( ) ( )j i j it ts s- -= . Èe se v koraku 1[ , ]i it t-

spremenita temperatura ali deformacija, se mejeoperatorja ohlapa premaknejo in:

Èe nato enaèbo delimo z ( )j iTa , dobimo:

á á 1( ) max{ ( ) , min{ ( ) , ( )}}j i i j i j j it t q t q te e e e -= - +

á á( ) ( ) ( ) ( ) ( )j i j i j i j i j it E T t T ts e a e= =

q

á0

( ) ( ) ( )n

i j i j ij

t T ts a e

=

= å

1 1 1 1 1 2( ) max{( ( ) ) ( ), min{( ( ) ) ( ), ( )}}j i i j j i i j j i j it t q T t q T ts e a e a s- - - - - -= - +

Sl. 2. Ravnote�je v modelu vzmet � drsnikFig.2. Equilibrium in the spring-slider model

1( ) max{( ( ) ) ( ), min{( ( ) ) ( ), ( )}}j i i j j i i j j i j it t q T t q T ts e a e a s -= - +

1á

( )( ) max ( ) ,min ( ) ,

( )j i

j i i j i jj i

tt t q t q

T

s

e e e

a

-ì üì üï ï ïï= - +í í ýý

ï ïï ïî þî þ

(e(ti-1) - qj)aj(Ti-1) sj(ti-1) (e(ti-1) + qj)aj(Ti-1)

meje

spominska toèka

(e(ti) - qj)aj(Ti) (e(ti) + qj)aj(Ti)

sj(ti)

memory point

boundaries



q0 q1 q2 qj qnq

e

s

nq

Dq

Tk

T0

TntnT

sj(Tk)

sj(Tnt)nT

sj(T0)

and finally:

(8),

as 1 1 1( ) ( ) ( )j i j i it T ts a e- - -= . When Eq. (8) is insertedinto Eq. (5), the temperature-dependent stress-strainbehaviour can be modelled, since the temperature-modified play operator guarantees the equilibriumin the spring-slider segments at any time andtemperature.

As the temperature-dependent cyclic stress-strain curves are known, the Prandtl densities canbe precalculated. Let us disperse q 1n + fictive yieldstrains equidistantly between the zero strain and themaximum expected strain amplitude (Fig. 3). Thus,for each fictive yield strain q

j in the range

q0, ,j n= K and each temperature Tk in the range

T0, ,k n= K , the strain ( )j kTe and the stress ( )j kTs

can be determined from Eq. (1). By inserting jq and( )j kTs into Eq. (5) the Prandtl densities are obtained:

(9),

where 1 0( ) ( ) 0k kT Ts s- = = and the fictive yieldstrain class width is Dq.

The Prandtl densities can be precalculatedand stored in a table before the stress-strain-tem-perature trajectory modelling using Eqs. (8) and (5)starts. In this way a high computational speed ispreserved.

in nazadnje:

kjer je 1 1 1( ) ( ) ( )j i j i it T ts a e- - -= . Z enaèbama (8) in (5)je mogoèe modelirati napetostno-deformacijskastanja tudi v primeru nakljuèno spreminjajoèih setemperatur, saj temperaturno spremenjen operatorohlapa vedno zagotavlja ravnote�je v vseh odsekihvzmet � drsnik.

Ker so ponovitvene napetostno-deformacijskekrivulje znane, je Prandtlove gostote mogoèe izraèunativnaprej. Naj bo q 1n + navideznih drsnih deformacijenakomerno razporejenih med nièelno in najveèjoprièakovano amplitudo deformacije (sl. 3). Za vsakonavidezno drsno deformacijo q

j v obmoèju q0, ,j n= K

in vsako temperaturo Tk v obmoèju T0, ,k n= K je z en.

(1) mogoèe izraèunati deformacije ( )j kTe in napetosti( )j kTs . Prandtlove gostote izraèunamo tako, da v en.

(5) vstavimo napetosti:

kjer je 1 0( ) ( ) 0k kT Ts s- = = , �irina razreda navideznedrsne deformacije pa je Dq.

Prandtlove gostote je pametno izraèunativnaprej, jih shraniti v preglednico re�itev ter �elenato modelirati napetostno-deformacijska stanja zenaèbama (8) in (5). Tako ohranimo veliko hitrostraèunanja.

1á 1

( )( ) max ( ) ,min ( ) , ( )

( )j i

j i i j i j j ij i

Tt t q t q t

T a

a

e e e e

a

--

ì üì üï ï ïï= - +í í ýýï ïï ïî þî þ

1 1 q T

1( ) ( ( ) 2 ( ) ( )) 0, , 0, ,j k j k j k j kT T T T j n k n

qa s s s+ -= - + = =

DK K

Sl. 3. Temperaturno odvisne napetostno-deformacijske krivuljeFig. 3. Temperature dependent cyclic stress-strain curves



(a)

0,0 0,1 0,2 0,3 0,40

80

160

240

320 0,26%

1,38%

0,26%

0,19%

4,95%

s,

Nm

m-2

e, %

0

3

1

3 CONSEQUENCES OF REVERSAL POINTFILTERING

For rate-independent material behaviour andconstant temperature, only the reversal points in theload histories are needed for stress-strain trajectorymodelling and cycle counting. Load histories can thusbe compressed and the time component can be elimi-nated. The load histories are scanned online and anypoint that is not the reversal point is discharged fromthe history. Similarly, all reversal points correspond-ing to a change in amplitude smaller than a certainthreshold are filtered [17]. As history compressionresults in a considerable reduction of history lengthsthe strain-life approach is much faster than compet-ing ones. For this reason it has been used frequently.

However, if the temperature changes betweenreversal points, the position of the next point does notdepend only on the current point, but also on the tran-sition between the points. Fig. 4 shows the influenceof temperature variations on the s-e trajectory for a9Cr2Mo alloy. It starts at point 0 ( 0 270 CT = ° ), goesthrough points 1 ( 1 570 CT = ° ) and 2 ( 2 270 CT = ° )and ends at point 3 ( 3 570 CT = ° ). Let us suppose thatthe strain and temperature histories are not filtered andcalculate the stress-strain-temperature trajectory as ex-plained in the preceding section (the thick solid line).Eq. (8) ensures equilibrium in the spring-slider model atany point indicated by triangle markers. It has beenobserved that the stresses do not necessarily coincide

3 POSLEDICE IZLOÈANJA KONICSUNKOV

Èe so materialne lastnosti èasovnoneodvisne, temperature pa nespremenljive, lahkopoteke obremenitev pred zaèetkom modeliranjanapetostno-deformacijskih stanj zgostimo in takoizloèimo le zaporedje konic sunkov brez èasovnekomponente. Poteke obremenitev pregledujemosproti in izloèamo vse toèke, ki niso lokalniekstremi ter obremenitvene cikle, ki ne prispevajok zbiranju po�kodb [17]. Ker se z zgo�èevanjempoteki moèno skraj�ajo, je deformacijskipostopek numerièno bistveno hitrej�i odzahtevnej�ih postopkov in se zato tudi pogostouporablja.

Èe se temperatura med posameznimikonicami sunkov spreminja, lega naslednje koniceni veè odvisna le od lege predhodne konice, ampaktudi od prehoda med njima. Slika 4 prikazuje vplivspreminjanja temperature na krivuljo s-e za zlitino9Cr2Mo, ki se zaène v toèki 0 ( 0 270 CT = ° ), greskozi toèki 1 ( 1 570 CT = ° ) in 2 ( 2 270 CT = ° ) ter sekonèa v toèki 3 ( 3 570 CT = ° ). Predpostavimo, dasta poteka deformacij in temperatur nezgo�èeni inizraèunajmo krivuljo napetost-deformacija (glejdebelo polno èrto) po enaèbah iz prej�njegapoglavja. En. (8) zagotavlja ravnote�je v odsekihvzmet � drsnik v vseh toèkah, oznaèenih strikotniki. Izka�e se, da se izraèunane napetosti ne

(b)

-0,4 -0,3 -0,2 -0,1 0,0-380

-285

-190

-95

0

s,

Nm

m-2

e, %

0

2

Sl. 4. Vpliv temperature na spremembe s-e krivuljFig. 4. Influence of temperature variations on s-e trajectory



with those on the isothermal cyclic stress-strain curves(the dashed lines with cross markers). This way theinfluence of transitions between reversal points on thes-e curve is proved. Similar deviations are observed inthe case of the compressed load histories, where thepoint positions in the s-e plane are calculated at loadreversals only (square markers). The deviations de-pend on the memory points and the temperature varia-tions between the reversal points.

In Figs. 5 and 6 the memory-point historiesfor the q0, ,j n= K play operators (thick solid lines)bounded by their upper (thin dashed line) and lower(thin solid line) boundaries are shown. Points 0 to 3correspond to the reversal points. If the temperature

0 3 570 CT T= = = °L , the memory points coincidewith one of the boundaries or stay constant, asdepicted in Fig. 6. However, if the temperaturechanges (see Fig. 5), the memory points can jumpfrom one boundary to another or fluctuate betweenthe boundaries. This is actually why stressdeviations occur.

From Fig. 5 it can be observed that thedeviations do not depend on the length of the s-etrajectory. Whenever e changes its direction andmemory points start to approach either the lower orupper boundaries, the deviations start to decrease.As deviation are always limited, only reversal pointsshould be processed under variable temperatures, too.

ujemajo vedno z napetostmi na izotermnihnapetostno-deformacijskih krivuljah (èrtkane èrtes simboli v obliki kri�cev). S tem je dokazan vplivprehodov med konicami sunkov na krivuljo s-e.Podobna odstopanja se pojavijo v primeruzgo�èenih potekov obremenitev, kjer se lega toèkv ravnini s-e raèuna le za konice sunkov (kvadratnisimboli). Odstopanja so odvisna od spominskihtoèk in temperaturnih sprememb med konicamisunkov.

Na slikah 5 in 6 so prikazani potekispominskih toèk za q0, ,j n= K operatorjev ohlapa(debele polne èrte), omejenih z zgornjimi (tankeèrtkane èrte) in spodnjimi (tanke polne èrte) mejami.Toèke 0 do 3 ustrezajo konicam sunkov. Èe jetemperatura 0 3 570 CT T= = = °L , se spominsketoèke vedno ujemajo z eno od meja oziroma ostajajonespremenjene (sl. 6). Èe pa se temperatura spreminja(sl. 5), lahko spominske toèke preskoèijo z ene mejena drugo ali pa med njimi nihajo. To je torej razlog zaodstopanja napetosti.

S slike 5 je razvidno, da so odstopanjaneodvisna od dol�ine krivulje s-e. Ko e spremeni smerin se zaèno spominske toèke pribli�evati spodnji alizgornji meji, se zaèno odstopanja zmanj�evati. Kerostajajo odstopanja po vrednosti vedno omejena, jeizloèanje konic sunkov sprejemljivo tudi v primeruspremenljivih temperatur.

0

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Sl. 5. Spominske toèke pri spremenljivi temperaturiFig. 5. Memory point history at variable

temperature

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4 PREVERJANJE

Deformacijsko nadzorovan in temperaturnospremenjen model, sestavljen iz vzporedno vezanihodsekov vzmet � drsnik, smo primerjali z veè preizkusiTMU, ki jih je opravil Skelton [4] pri 0,6 % celotnedeformacije zlitine 9Cr2Mo. Temperaturno odvisniparametri, ki se pojavljajo v Ramberg-Osgoodovienaèbi, podatki o materialu in eksperimentalnih tehnikahso pojasnjeni v [2], [4] in [18].

Pri prikazu poteka deformacij in temperaturobièajno nana�amo deformacije na ordinatno,temperature pa na abscisno os [4]. V prispevku staobravnavani pot PKRM in zahtevna ponovitev(èrtkana èrta), prikazani na sliki 7. Medtem kopredstavlja pot PXRM 45° deltoidni cikel, izvedenproti smeri urnega kazalca, je pot PMRK enaka ciklu,izvedenem v smeri urnega kazalca. Podobno veljatudi za 135° deltoidni cikel, omejen s toèkami PKRZ.Cikla pravokotne oblike PXRZ in PKRM stapoimenovana 45° in 135° pravokotni cikel. Poti GEHFin GFHE tvorita dvojni termièni cikel ter cikel zahtevneoblike je prikazan s èrtkano èrto.

Izmerjene histerezne zanke so omejene s kri�cina slikah 8 in 9. Trikotniki oznaèujejo napetostno-deformacijske krivulje, izraèunane s Skeltonovim [4]algoritmom, debele polne èrte pa krivulje, ocenjenepo na�em algoritmu.

S slike 8 je razvidno, da je napovedobremenitvenih ciklov, izvedenih v smeri protiurnemu kazalcu, bolj�a od napovedi ciklov, izvedenih

4 VERIFICATION

The strain-controlled and temperature-modifiedspring-slider model is compared to several TMF testsconducted by Skelton [4] at a total strain range of0.6% on the 9Cr2Mo alloy. The temperature-dependent parameters associated with the Ramberg-Osgood relation as well as the material data andexperimental methods are given in [2], [4] and [18].

In deciding the temperature-strain path, theconvention is to plot strain on the vertical axis andtemperature on the horizontal axis [4]. The paperconcerns the path PKRM and the complex cycle(dashed line) given in Fig. 7. The path PXRMrepresents a 45° kite cycle performed in ananticlockwise way while the correspondingclockwise cycle is taken in the PMRX order. A similarscheme applies in the 135° kite cycle PKRZ. Theparallelogram-shaped PXRZ and PKRM are labelledas the 45° zero strain and the 135° zero strainrespectively. Finally, the anticlockwise GEHF and theclockwise GFHE bi-thermal cycles as well as thecomplex cycle given in the dashed line areconsidered.

The observed hysteresis loops from the testsare plotted as crosses in Figs. 8 and 9. The trianglemarkers and the thick solid lines denote the stress-strain trajectories modelled by the Skelton [4] andour algorithm respectively.

It is clear that in Fig. 8 the shapes of theanticlockwise loops are predicted well, whereas for

Sl. 7. Vrste TMU ciklovFig. 7. TMF cycle types



the clockwise style tests the observed loops arewider. The wider loops do not appear just becauseof non-linear cooling. Consequently, someparameters undefined so far seem to be moreinfluential. The agreement between the Skelton [4]and our algorithm is almost perfect. The method wasfinally applied to the complex cycle of Fig. 9successfully.

v smeri urnega kazalca. Pri slednjih so testnehisterezne zanke �ir�e od napovedanih. �ir�ehisterezne zanke niso le posledica nelinearnegaohlajanja, paè pa predvsem nekaterih parametrov, kidoslej �e niso bili prepoznani. Ujemanje medSkeltonovimi in na�imi rezultati je skoraj popolno.Postopek je bil nazadnje uspe�no preverjen na cikluzahtevne oblike, prikazani na sliki 9.

Sl. 9. Zahtevna TMU zankaFig. 9. Complex TMF loop

-0,35 -0,25 -0,15 -0,05 0,05 0,15 0,25 0,35

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5 SKLEP

Deformacijski postopek je alternativanapetostnemu postopku napovedovanja utrujenostnepo�kodbe, ki se uporablja predvsem, èe se pojaviteèenje. V prispevku je predstavljen in z veè TMUpreizkusi preverjen temperaturno spremenjendeformacijski postopek napovedovanja napetostno-deformacijskih stanj. Razvit in preverjen je bildeformacijsko nadzorovani model, sestavljen izvzporedno vezanih odsekov vzmet � drsnik, ki omogoèamodeliranje lokalnih temperaturno-napetostno-deformacijskih stanj. Za primer napetostnega nadzoraje bil v [12] razvit enakovreden model. Izloèanje konicsunkov je za elastoplastièna gradiva sprejemljivo. Velikahitrost raèunanja obièajnega deformacijskega postopkase ohrani. Razviti model je formalno dobro definiran zmodelom Prandtlovega tipa in pomeni posplo�enjeSkeltonovih [4] izsledkov. Èeprav predlagani model neomogoèa napovedovanja lezenja in nekaterih drugihpojavov, lahko dose�emo izbolj�ano napovedtemperaturno-napetostno-deformacijskih stanj napodlagi standardnih MCV preizkusov.

5 CONCLUSIONS

The strain-life approach is an alternative tothe stress-life approach for determining fatiguedamage, particularly when yielding occurs. Anonline temperature-modified strain-life approachhas been developed and verified through severalTMF tests. The spring-slider model has beendeveloped and verified to enable local temperature-stress-strain trajectory modelling. The model isapplicable if the strain is controlled or else the modelgiven in [12] can be used. Reversal point filteringturned out to be acceptable for elasto-plastichardening solids. The high computational speedof the classical strain-life approach can thus bepreserved. The developed model is a generalizationof the Skelton [4] findings and is formally welldefined through an operator of the Prandtl type.Although neither creep damage nor some othereffects can be assessed by the proposed approach,an improved temperature-stress-strain stateprediction based on standardized LCF tests ispossible.

6 LITERATURA6 REFERENCES

[1] Constantinescu A, Charkaluk E, Lederer G, Verger L (2004) A computational approach to thermomechanicalfatigue. Int J Fatigue, 26, 805-818.

[2] Skelton RP, Webster GA (1996) History effects on the cyclic stress-strain response of a polycrystallineand single crystal nickel-base superalloy. Materials Science and Engineering, A216, 139-154.

[3] Charkaluk E, Bignonnet A, Constantinescu A, Dang Van K (2002) Fatigue design of structures underthermomechanical loadings. Fatigue Fract Engng Mater Struct, 25, 1199-1206.

[4] Skelton RP (2004) Hysteresis, yield, and energy dissipation during thermo-mechanical fatigue of a ferriticsteel. Int J Fatigue, 26, 253-264.

[5] Ellison EG, Smith EM (1973) Predicting service life in a fatigue-creep environment. In: Carden AE, McEvilyAJ, Wells CH, editors. Fatigue at elevated temperatures. ASTM STP 520, pp 575-612.

[6] Sehitoglu H (1992) Thermo-mechanical fatigue life prediction methods. In: Mitchell MR, Landgraf RW,editors. Advances in fatigue lifetime predictive techniques. ASTM STP 1122, pp 47-76.

[7] Cai C, Liaw PK, Ye M, Yu J (1999) Recent developments in the thermomechanical fatigue life prediction ofsuperalloys. J Mater, 51(4).

[8] Chaboche JL, Gallerneau F (2001) An overview of the damage approach of durability modelling at elevatedtemperature. Fatigue Fract Engng Mater Struct, 24, 405-418.

[9] Chaboche JL (1989) Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int J Plast, 5,247-302.

[10] Krempl E, Khan F (2003) Rate (time)-dependent deformation behaviour: an overview of some propertiesof metals and solid polymers. Int J Plast, 19, 1069-1095.

[11] Conle A, Oxland TR, Topper TH (1988) Computer-based prediction of cyclic deformation and fatiguebehaviour. In: Solomon HD, Halford GR, Kaisand LR, Leis BN, editors. Low cycle fatigue. ASTM STP 942,pp 1218-1236.



[12] Nagode M, Zingsheim F (2004) An online algorithm for temperature influenced fatigue-life estimation:strain-life approach. Int J Fatigue, 26, 155-161.

[13] Nagode M, Hack M (2004) An online algorithm for temperature influenced fatigue-life estimation: stress-life approach. Int J Fatigue, 26, 163-171.

[14] Ramberg W, Osgood WR (1943) Description of stress-strain curves by three parameters. Technical noteno. 902, NACA.

Avtorjev naslov: prof.dr. Marko Nagodeprof.dr. Matija FajdigaUniverza v LjubljaniFakulteta za strojni�tvoA�kerèeva 61000 [email protected]@fs.uni-lj.si

Author�s Address: Prof.Dr. Marko NagodeProf.Dr. Matija FajdigaUniversity of LjubljanaFaculty of Mechanical Eng.A�kerèeva 6SI-1000 Ljubljana, [email protected]@fs.uni-lj.si

Prejeto: 2.11.2005

Sprejeto: 16.11.2005

Odprto za diskusijo: 1 letoReceived: Accepted: Open for discussion: 1 year


8 5Raèunalni�ko modeliranje gibanja goriva - Computational Modelling of Fuel Motion

Raèunalni�ko modeliranje gibanja goriva in njegov vpliv nakonstrukcijo rezervoarja

Computational Modelling of Fuel Motion and Its Interaction with the Reservoir

Structure

Matej Vesenjak1 - Zoran Ren1 - Heiner Müllerschön2 - Stephan Matthaei3

(1Fakulteta za strojni�tvo, Maribor; 2DYNAmore GmbH, Stuttgart-Vaihingen; 3DaimlerCrysler AG, Stuttgart)

Raèunalni�ki modeli vozil za simuliranje trkov vedno natanèneje opisujejo obna�anje resniènihvozil. Rezervoar za gorivo je eden izmed elementov vozil, katerega raèunalni�ki modeli so bili do sedaj zelopoenostavljeni. Tak�ni modeli upo�tevajo le vztrajnost mase goriva, ki je z masnimi toèkami pritrjena nasteno rezervoarja, vendar pa je vpliv gibanja goriva v rezervoarju popolnoma zanemarjen.

Prispevek opisuje nove raèunalni�ke modele, s katerimi je mogoèe simulirati deformacijo rezervoarjaza gorivo ob upo�tevanju gibanja goriva pri trku vozila. V ta namen so bile vrednotene �tiri metodesimuliranje gibanja tekoèine (Lagrange, Euler, poljubnostna Lagrange-Eulerjeva metoda - PLE inhidrodinamika zglajenih delcev - HZD) v rezervoarju preproste oblike, analizirane z eksplicitnim programomLS-DYNA. Raèunalni�ki rezultati so bili primerjani s poprej objavljenimi preizkusnimi opazovanji, prièemer je bila ugotovljena zelo dobra primerljivost med rezultati.

Najprimernej�i metodi (HZD in PLE) sta bili kasneje uporabljeni v dinamiènih simulacijah dejanskegarezervoarja za gorivo. Simulacije so pokazale, da predstavljeni modeli rezervoarja ob upo�tevanju gibanjagoriva zagotavljajo mnogo natanènej�e rezultate v primerjavi z znanimi poenostavljenimi modeli.© 2006 Strojni�ki vestnik. Vse pravice pridr�ane.(Kljuène besede: gibanje goriva, Lagrangev opis, Eulerjev opis, ALE, SPH, vplivi tekoèina - trdnina)

Computational models of vehicles for crash simulations are ever more precisely describing thebehaviour of real vehicles. A fuel-tank is a typical vehicle element that has been very simplified in thecomputational models used so far. Such models have considered only the influence of the fuel mass inertia,which was point-wise connected to the tank walls, with total neglect of the fuel motion in the tank.

This paper describes new computational models that allow for a simulation of the fuel-tank deforma-tion considering the fuel motion during a vehicle crash. For this purpose four different methods for describ-ing fluid motion (Lagrangian, Eulerian, Arbitrary Lagrange-Eulerian description - ALE, SPH) were evalu-ated on a simple reservoir problem, analysed with the explicit dynamic code LS-DYNA. The computationalresults were compared with previously published experimental observations and a good correlation of theresults was observed.

The most appropriate methods, SPH and ALE, were afterwards used in dynamic simulations of a realfuel-tank. The simulations showed that by also taking into consideration the fuel motion, the proposedcomputational models provide more accurate results in comparison with the previously used, simplifiedmodels.© 2006 Journal of Mechanical Engineering. All rights reserved.(Keywords: fuel motion, Lagrangian description, Eulerian description, ALE, SPH, fluid structure interaction)

Strojni�ki vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100UDK - UDC 621.431.37:629.063.6

004.94Izvirni znanstveni èlanek - Original scientific paper (1.01)

0 INTRODUCTION

In recent years numerical simulations in au-tomotive engineering have gained in importance.Their aim is not only the determination of the globalvehicle behaviour but also the behaviour of single

0 UVOD

V zadnjem èasu postajajo numeriènesimulacije v avtomobilizmu vse pomembnej�e. Njihovcilj ni veè zgolj doloèitev globalnega obna�anja vozil,temveè tudi natanènej�e obna�anje posameznih


8 6 Vesenjak M. - Ren Z. - Müllerschön H. - Matthaei S.

components. In the case of computational compo-nent simulation, during the designing phase of a newvehicle structure or during the design improvementsof existing vehicle designs, finite-element methodanalyses are most often used. In many cases, how-ever, simulating separate components is not sufficient.Therefore, other components or assemblies that in-teract with each other have to be included in the analy-sis. The solution process becomes more sophisticatedwhen the assembly components consist of two ormore different physical systems that interact. Whentwo or more physical systems interact with each otherand an independent solution of one system is impos-sible without a simultaneous solution of the others,such systems are known as coupled [20].

One of the vehicle components where two physi-cal systems interact is the fuel-tank. In previoussimulations the model of the tank was very simplified,since the fluid motion�s influence on the tank�s behav-iour was not considered. Only the fuel mass was takeninto account, which was distributed at discrete nodesalong the tank walls to account for the fuel inertia forces.The fuel motion during a vehicle crash, which has a largeinfluence on tank deformation and its supporting ele-ments, was thus completely neglected. Suchsimplifications were necessary due to the limitations ofthe simulation software used. However, some recent re-leases of simulation software already allow for an effec-tive solution of several physical systems simultaneously.In this paper different computational models that allowfor consideration of the fluid motion and its influence onthe structure are described and evaluated with the soft-ware LS-DYNA [10]. LS-DYNA is based on the finite-element method and it was originally designed for solv-ing structural dynamic problems. Therefore, its ability tomodel structural responses in general is well defined.However, the modelling of a coupled fluid-structure in-teraction is still quite challenging. A comparison of themethods and their applicability is illustrated on a practi-cal example, describing the fuel motion in a reservoirwith simple geometry. The computational results are fur-ther compared with the experimental measurements [11].

1 FLUID-STRUCTURE INTERACTION

One of the most popular coupled problemsin engineering is the interaction between the fluidand the structure, which consequently results in dif-ferent approaches to solving such a problem. Thedevelopment of coupled problems solution algo-rithms was also influenced by the rapid develop-

sestavnih delov. Pri razvoju konstrukcije novegavozila ali izbolj�avi konstrukcije �e znanega vozilaso za raèunalni�ko simuliranje komponentnajpogosteje uporabljene analize po metodi konènihelementov. V mnogih primerih simuliranje posameznihkomponent ni zadostno, zato morajo biti v analizovkljuèene komponente oziroma celotni sklopi, kivplivajo drug na drugega. Re�evanje problemovpostane zahtevnej�e, kadar so sestavni elementisklopa iz dveh ali veè fizikalnih sistemov, kimedsebojno vplivajo. Kadar medsebojno vplivatadva ali veè fizikalnih sistemov in je re�itevposameznega brez upo�tevanja ostalih nemogoèa,govorimo o vezanih problemih [20].

Eden izmed delov vozila, pri katerem vplivatadva fizikalna sistema, je rezervoar za gorivo. Vpreteklih simulacijah je bilo simuliranje rezervoarjevzelo poenostavljeno, saj vpliv gibanja tekoèine naobna�anje rezervoarja ni bil obravnavan. Upo�tevanaje bila le masa goriva, ki je bila porazdeljena poposameznih vozli�èih sten rezervoarja zaradiupo�tevanja vztrajnostnih sil. Gibanje goriva pri trkuvozila, ki ima velik vpliv tako na deformacijorezervoarja kakor tudi njegovih nosilnih elementov,pa je bilo popolnoma zanemarjeno. Omenjenepoenostavitve so bile neizogibne zaradi omejeneuporabe simulacijskih programskih paketov, vendarpa novej�i programski paketi omogoèajo tudiuèinkovito re�evanje veè fizikalnih sistemov hkrati.V prispevku so opisani in vrednoteni razlièninumerièni modeli, s katerimi je mogoèe opisati gibanjein vpliv tekoèine na trdnino s programskim sistemomLS-DYNA [10]. LS-DYNA temelji na metodi konènihelementov in je bila prvotno namenjena re�evanjudinamiènih problemov v mehaniki trdnin. Modeliranjeodzivov v dinamiki trdnin je zato zelo dobro razvito.Modeliranje vezanih problemov med tekoèino intrdnino pa �e vedno ostaja izziv. Primerjava metod innjihova uporabnost je ponazorjena na praktiènemprimeru gibanja tekoèine v rezervoarju s preprostogeometrijo. Z eksperimentalnimi meritvami pa soprimerjani tudi rezultati analiz [11].

1 MEDSEBOJNI VPLIV TEKOÈINE IN TRDNINE

Eden izmed najpogosteje obravnavanihvezanih problemov v in�enirski praksi jemedsebojni vpliv tekoèine in trdnine, zaradi èesarso se razvili razlièni postopki re�evanja tehproblemov. K razvoju algoritmov za raèunanjevezanih problemov je pripomogel tudi silovit razvoj



ment of computer hardware, because suchsimulations are computationally very intensive. Cou-pled problems can be solved in two ways, by usingcommercial software:· with the use of two commercial codes: one for

solving the fluid domain and one to separatelydetermine the solid-domain solution. Such codes(e.g., CFX and Nastran) are usually connected viaan interface that controls the necessary bound-ary conditions� data exchange ([4] and [19]);

· with use of a commercial software that simultane-ously solves a multi-physics problem (e.g.,ADINA, LS-DYNA, ANSYS).

The LS-DYNA code is capable of establishinga dynamic interaction between the fluid and the struc-ture with an explicit integration scheme. In LS-DYNA itis possible to describe and simulate the influence ofthe fluid on the structure, and vice versa, in two ways:(i) with contact forces (applicable for Lagrangian andSPH model) and (ii) with a leakage criteria, which ineach observed element determinates the force that isnecessary to establish equilibrium conditions betweenthe fluid and the structure (applicable for Eulerian andALE model) ([1], [6], [10] and [12]).

2 DIFFERENT DOMAIN DESCRIPTIONS

2.1 Lagrangian description

The Lagrangian formulation is usually usedfor describing solid mechanics problems. The problemis described with a large number of mass particles, wherethe motion of every single particle is observed in spaceand time (Fig. 1). The problem is exactly defined whenthe motion of all the particles is known [17].

raèunalni�ke opreme, saj so tovrstne analizeraèunsko zelo zahtevne. Vezane probleme je zuporabo razliènih programskih paketov mogoèere�evati na dva naèina:· z uporabo dveh programskih paketov: eden za

doloèitev obmoèja tekoèin in eden za loèenodoloèitev re�itve obmoèja trdnine. Tak�niprogramski paketi (npr. CFX in Nastran) so ponavadi povezani z vmesnikom, ki nadzorujeizmenjavo potrebnih podatkov o robnih pogojih([4] in [19]);

· z uporabo programskega paketa, ki omogoèahkratno re�evanje dveh ali veè vezanih fizikalnihsistemov (npr. ADINA, ANSYS, LS-DYNA).

Programski paket LS-DYNA omogoèadinamièno ugotavljanje medsebojnega vpliva medtekoèino in trdnino z eksplicitno integracijsko shemo.V LS-DYNI je vpliv tekoèine mogoèe opisati insimulirati na dva naèina: (i) s stiènimi silami (uporabnopri Lagrangevem in SPH modelu) in (ii) s kriterijemporoznosti snovi, ki v vsakem opazovanem elementudoloèi silo, ki je potrebna za vzpostavitev ravnote�jamed tekoèino in trdnino (uporabno pri Eulerjevem inmodelu PLE) ([1], [6], [10] in [12]).

2 RAZLIÈNI OPISI DOMEN

2.1 Lagrangev opis

Lagrangev opis se obièajno uporablja za opisproblemov v mehaniki trdnin. Problem se opi�e zvelikim �tevilom masnih delcev, pri èemer se opazujegibanje vsakega posameznega delca v prostoru inèasu (sl. 1). Problem je natanèno doloèen, kadarpoznamo gibanje vseh delcev [17].

Sl. 1. Lagrangev opis [17]Fig. 1. Lagrangian formulation [17]

x y

z ( )0 ,=v v v

r r r t

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The Lagrangian formulation is very simpleand easy to use for one or only a few mass particles.However, the method becomes very complicated andcomplex for a description of large number of massparticles [17].

In the Lagrangian formulation, one finite ele-ment represents the same part of the material through-out the course of the analysis. The finite-element meshis fixed to the material during the entire computationalprocess and therefore moves with the material. Figure2 illustrates the solution process of a simple fluid prob-lem using the Lagrangian formulation. It is presumedthat the loading influences only the central node. Theresult of the loading is the shift of that node in acomputational time step. If the influence of the load-ing does not stop or change, the node takes a newposition in the next time step and the mesh deformseven more, since the mesh follows the material flow.

Since the mesh follows the material, the rela-tive movement of connecting nodes can result in asignificant deformation of the finite elements. Mass,momentum and the energy are transported with themovement of the elements [2]. The mass conserva-tion equation can be, in partial differential equationform, written for the Lagrangian description as:

(1),

where r is the density and v is the velocity of thematerial (mesh). For an incompressible material it isknown that div(v) = 0. The Lagrangian descriptionfor the mass conservation can also be written in analgebraic equation as:

(2),

where J is the Jacobian between the current andreference configuration. Conservation of momentumin terms of the Lagrangian description gives:

(3),

Lagrangev naèin opisa gibanja je preprost innezahteven za uporabo, dokler gre za posameznimasni delec. V primeru prouèevanja problema zvelikim �tevilom masnih delcev pa postane izrednozahteven in zapleten [17].

Pri Lagrangevem opisu predstavlja en konènielement isti dele� materiala celoten potek analize.Mre�a konènih elementov je pritrjena na material medcelotnim raèunskim postopkom, zaradi èesar sepomika z materialom. Slika 2 prikazuje naèin re�evanjapreprostega primera tekoèine z Lagrangevim opisom.Predpostavljeno je, da obremenitev deluje le nasredinsko vozli�èe. Rezultat te obremenitve je pomikvozli�èa v raèunskem èasovnem koraku. Èe vplivobremenitve ne preneha, oz. se spremeni, vozli�èe vnaslednjem èasovnem koraku ponovno zavzamenovo lego in mre�a se vedno bolj deformira, saj sleditoku materiala.

Ker mre�a konènih elementov sledi materialu,ima relativen pomik medsebojno povezanih vozli�èizrazit vpliv na deformacijo konènih elementov. Spremikom elementov se prena�ajo tudi masa, gibalnakolièina in energija [2]. Enaèba o ohranitvi mase selahko v Lagrangevem opisu v parcialni diferencialniobliki zapi�e kot:

kjer sta r gostota in v hitrost materiala (mre�e). Zanestisljiv material je znano, da je div(v) = 0. Lagrangevopis ohranitve mase se lahko zapi�e tudi v oblikialgebrajske enaèbe kot:

kjer je J Jacobijeva matrika med trenutno inreferenèno obliko. Ohranitev gibalne kolièine je vLagrangevem opisu podana kot:

Sl. 2. Deformacija mre�e pri Lagrangevem opisuFig. 2. Mesh deformation in the Lagrangian formulation

t = 0 t = Ät t = 2 Ät

0i

i

v

t x

rr

¶¶+ × =

¶ ¶

Jr r0× =

,i jii

i

vb

t x

sr r

¶¶× = + ×

¶ ¶



x y

z ( )3 3=v

v t

( ), ,M x y z

( )1 1=v

v t

( )2 2=v

v t

where s is the Cauchy stress and b is the body force(a force per unit mass). The third equation repre-sents the energy conservation and can be, in a purelymechanical process, written as:

(4),

where u is the internal energy per unit mass.The disadvantage of the Lagrangian descrip-

tion becomes evident in cases of an extremely distortedmesh, because their formulation is always based onmesh. When the mesh is heavily distorted, the accu-racy of the formulation and hence the solution will beseverely affected. A possible option to enhance theLagrangian model is to re-mesh the problem domain.

2.2 Eulerian description

In the Eulerian formulation, which is commonlyused for solving computational fluid dynamic problems,the problem is being observed at separate points in spacethat do not follow the particles motion (Fig. 3). In onetime step, Dt, several mass particles can pass throughthe observed point. Their motion is exactly determinedat the moment of passing through that point. The fieldvariables at the observed point are time dependent.

In the Eulerian formulation the mass conser-vation is written as:

(5),

where i

i

Dv

Dt t x

¶ ¶= + ×

¶ ¶ is the total time derivativethat is physically the time rate of the change follow-ing a moving material element (sum of the local andthe convective derivative).

The conservation of momentum can be ex-pressed as:

(6).

kjer sta s Cauchy-jeva napetost in b masna sila (silana enoto mase). Tretja enaèba predstavlja ohranitevenergije in se lahko v primeru izkljuèno mehanskegapostopka zapi�e kot:

kjer je u notranja energija na enoto mase.Pomanjkljivost Lagrangevega opisa postane

razvidna v primerih s skrajno popaèeno mre�o, sajnjihov opis vedno temelji na mre�i. Popaèenost mre�evpliva na natanènost metode in zato na rezultatanalize. Sprejemljiva mo�nost za izbolj�anjeLagrangevega modela je ponovno mre�enje domeneproblema.

2.2 Eulerjev opis

Pri Eulerjevem opisu, ki se navadno uporabljaza re�evanja problemov raèunalni�ke dinamiketekoèin, se problem opazuje v doloèeni prostorskitoèki in ne sledi gibanju posameznega delca (sl. 3). Venem èasovnem koraku Dt gredo skozi toèko �tevilnimasni delci, katerih gibanje je v trenutku prehodanatanèno doloèeno. V opazovani toèki se velièinepolja spreminjajo s èasom.

Ohranitev mase je v Eulerjevem opisuzapisana kot:

kjer je i

i

Dv

Dt t x

¶ ¶= + ×

¶ ¶ snovski oziroma hitrostniodvod, ki podaja èasovno spremembo velièine inprostorsko spremembo velièine zaradinehomogenosti hitrostnega polja (vsota lokalnegain konvektivnega odvoda).

Ohranitev gibalne kolièine je izra�ena:

,i

i j

i

vu

t xr s

¶¶× = ×

¶ ¶

Sl. 3. Eulerjev opis [17]Fig. 3. Eulerian formulation [17]

0i

i

vD

Dt x

rr

¶+ × =

¶

,i jii

i

Dvb

Dt x

sr r

¶× = + ×

¶



t = 0 t = Ät

premik vozli�è /

mesh smoothing

interpolacija vmesnih

rezultatov /advection

t = Ät

And the conservation of the energy can be written as:

(7).

The basic difference between the Lagrangianand the Eulerian formulations is that in theLagrangian formulation the magnitudes x, y and zare variable coordinates of a moving particle; in theEulerian formulation those coordinates represent thesteady coordinates of the defined field point [17].

Although the Eulerian mesh in LS-DYNA ap-pears not to move or deform during the analysis, itdoes actually change its position and form, but onlywithin a single time step ([13], [14] and [16]). The rea-son for this is the use of the Lagrangian formulationin single time steps, which is much more advanced inthe LS-DYNA. The Eulerian mesh in LS-DYNA istreated in a special way (Fig. 4). To illustrate the use ofan Eulerian mesh the same example is used as at theLagrangian formulation. Because of the central nodeloading, the observed node changes its position dur-ing one computational time step (mesh deforms). Af-ter the time step the analysis stops and the followingtwo approximations are performed [5]:· mesh smoothing: all the nodes of the Eulerian

mesh that have been displaced due to loading aremoved to their original position;

· advection: the internal variables (stresses, flowfields, velocity field) for all the nodes that havebeen moved are recomputed (interpolated) so thatthey have the same spatial distribution as beforethe mesh smoothing. In this way the mesh smooth-ing does not affect the internal variable distribu-tion.

The described procedure is repeated foreach time step of the analysis and provides theanalyst with a non-movable and undeformableEulerian mesh.

In ohranitev energije se lahko zapi�e:

Temeljna razlika med Lagrangevim inEulerjevim postopkom je v tem, da so priLagrangevem opisu velièine x, y in z spremenljivekoordinate gibljivega delca. Pri Eulerjevem opisu pate koordinate pomenijo mirujoèe koordinate doloèenetoèke polja [17].

Kljub temu da se Eulerjeva mre�a med analizov LS-DYNI navidezno ne premika ali deformira, sedejansko spreminjata njena lega in oblika, vendar lev posameznem èasovnem koraku ([13], [14] in [16]).Razlog za to je uporaba Lagrangevega opisa vposameznih èasovnih korakih, ki je naprednej�a vLS-DYNI. Eulerjeva mre�a je v LS-DYNI obravnavanana poseben naèin (sl. 4). Uporaba Eulerjeve mre�e jeponazorjena na enakem primeru, ki je bil uporabljenpri Lagrangevem opisu. Zaradi obremenitev nasredinsko vozli�èe opazovano vozli�èe spremeni legov enem raèunalni�kem èasovnem koraku (mre�a sedeformira). Po èasovnem koraku se analiza ustavi inizvedeta se naslednji dva pribli�ka [5]:· premik vozli�è: vsem vozli�èem Eulerjeve mre�e, ki

so zaradi obremenitev spremenila svojo lego, seponovno doloèi izhodi�èna lega;

· interpolacija vmesnih rezultatov: vse notranjevelièine (napetost, tokovna polja, hitrostno polje),ki se nana�ajo na vozli�èa s spremenjeno lego, sointerpolirane tako, da imajo ustrezno prostorskoporazdelitev kakor pred premikom vozli�è. Takopremik vozli�è ne vpliva na porazdelitev notranjihvelièin.

Opisani postopek se ponavlja v vsehèasovnih korakih celotne analize in ponujauporabniku nepremièno in nedeformirano Eulerjevomre�o.

,i

i j

i

vDu

Dt xr s

¶× = ×

¶

Sl. 4. Deformacija mre�e pri Eulerjevem opisuFig. 4. Mesh deformation in the Eulerian formulation



PLE ALE

Lagrange Euler

pomik / displacement

material mre�a / mesh

iu

�iu

iu

� =i iu u iu

� 0=iu

hitrost / velocity


iv

�iv

iv

� =i iv v iv

� 0=iv

pospe�ek / acceleration


ia

�ia

ia

� =i ia a ia

� 0=ia

2.3 Arbitrary Lagrange-Eulerian description (ALE)

The features of the Lagrangian and Eulerian de-scriptions suggest that it would be computationally ben-eficial to combine these two descriptions so as to strengthentheir advantages and to avoid their disadvantages. Thisidea led to the development of the Arbitrary Lagrange-Eulerian formulation. In this formulation the mesh partlymoves and deforms because it follows the material(Lagrangian formulation), while at the same time the mate-rial can also flow through the mesh (Eulerian formulation).

The relationship between the convected ve-locity, c

i, the material velocity, v

i , and the mesh ve-

locity, �iv , is defined as:

(8).

Regarding the material and the mesh move-ment, the mass, the momentum and the energy con-servation equations can be written as:

(9)

(10)

(11).

From the above equations it is obvious thatif the mesh and material velocity are equal this wouldgive the Lagrangian description, and if the materialmoves and the mesh remains steady (the mesh ve-locity is equal to convected velocity) this would givethe Eulerian description (Table 1) [2].

The ALE algorithm in LS-DYNA is similar tothe described Eulerian computational procedure([13], [14] and [16]). The only difference is the meshsmoothing. In the Eulerian formulation the nodesare moved back to their original positions, while in

2.3 Poljubnostni Lagrange-Eulerjev opis (PLE)

Znaèilnosti Lagrangevega in Eulerjevegaopisa napovedujejo, da bi bilo raèunalni�ko ustreznozdru�iti omenjena opisa in poudariti njune prednostiter se izogniti njunim pomanjkljivostim. Ta zamisel jevodila do razvoja poljubnostnega Lagrange-Eulerjevega opisa (PLE). V tem opisu se mre�a lahkodeloma premika in deformira, ker sledi materialu(Lagrangev opis), hkrati pa dopu�èa, da material teèeskozi mre�o (Eulerjev opis).

Povezava med konvektivno hitrostjo ci,

hitrostjo materiala vi in hitrostjo mre�e �

iv je definiranakot:

Glede na gibanje materiala in mre�e seohranitvene enaèbe mase, gibalne kolièine in energijezapi�ejo kot:

Iz zgornjih enaèb je razvidno, da v primeru,ko sta hitrost mre�e in hitrost materiala enaka,izpeljemo Lagrangev opis. Kadar se material premikain mre�a miruje (hitrost mre�e je enaka konvektivnihitrosti), pa izpeljemo Eulerjev opis (preglednica 1)[2].

Raèunalni�ki algoritem re�evanja opisa ALEv LS-DYNI je podoben opisanemu postopkuEulerjevega opisa ([13], [14] in [16]). Razlikuje se le vpremiku vozli�è. V Eulerjevem opisu so vozli�èapremaknjena nazaj na izhodi�ène lege, pri èemer se

�i i ic v v= -

0ii

i i

vc

t x x

r rr

¶¶ ¶+ × + × =

¶ ¶ ¶

,i ji ii i

i i

v vc b

t x x

sr r r

¶¶ ¶× + × × = + ×

¶ ¶ ¶

,i

i i j

i i

vu uc

t x xr r s

¶¶ ¶× + × × = ×

¶ ¶ ¶

Preglednica 1. Primerjava kinematike PLE, Lagrangevega in Eulerjevega opisaTable 1. Comparison of the kinematics for the ALE, Lagrangian and Eulerian formulations



the ALE formulation the positions of the movednodes are calculated according to the average dis-tance to the neighbouring nodes (Fig. 5).

A similar calculation scheme is also used inother comparable codes (e.g., MSC/Dytran).

The advantage of the ALE formulation is evi-dent when a stress front needs to be followed andthe mesh is automatically refined. Another exampleis the analysis of fluid tanks, where the fluid�s move-ment inside the tank is of interest and the boundarysurface is continuously changing due to the inter-action between the fluid and tank surfaces (Fig. 6).

However, using the ALE formulation can alsoresult in a highly distorted mesh, which can intro-duce large errors in numerical simulations. In somecases the ALE formulation can encounter unex-pected terminations in the computational process,usually due to very small time steps following fromvery small, deformed Lagrangian elements or evennegative element volumes.

2.4 Smoothed Particle Hydrodynamics (SPH)

In the SPH method, the state of the system isrepresented by a set of particles (Fig. 7) that pos-

pri opisu PLE nova lega premaknjenih vozli�èizraèuna glede na povpreèno oddaljenost dososednjih vozli�è (sl. 5).

Podobna numerièna shema je uporabljena vdrugih primerljivih programskih sistemih (npr. MSC/Dytran).

Prednost Lagrange-Eulerjevega opisa sepoka�e, kadar �elimo slediti doloèeni napetosti in semora mre�a samodejno zgo�èevati. Drug primer je analizarezervoarjev s tekoèino, pri kateri je upo�tevano gibanjetekoèine v rezervoarju, mejna ploskev pa se zaradi vplivamed trdnino in tekoèino ves èas spreminja (sl. 6).

Kljub temu pa lahko uporaba PLE opisapovzroèi popaèenost elementov, kar lahko vpeljenevarne napake v numeriènih simulacijah. V doloèenihprimerih PLE opis povzroèi neprièakovano ustavitevraèunskega postopka. To se obièajno zgodi zaradizelo majhnih èasovnih korakov, kot posledica zelomajhnih deformiranih Lagrangevih elementov ali celonegativne prostornine konènih elementov.

2.4 Metoda hidrodinamike zglajenih delcev (MHZD)

Stanje sistema je pri metodi hidrodinamikezglajenih delcev (MHZD) opisano z doloèenim

Sl. 5. Deformacija mre�e pri PLE opisuFig. 5. Mesh deformation in the ALE formulation

t = 0 t = Ät t = Ät

premik vozli�è /

mesh smoothing

interpolacija vmesnih

rezultatov /advection

Sl. 6. Uporaba PLE opisaFig. 6. Applications of the ALE formulation



sess individual material properties and move accord-ing to the governing conservation equations. SPHas a meshfree, Lagrangian, particle method was de-veloped by Lucy, Gingold and Monaghan, initiallyto simulate astrophysical problems ([5], [7] to [9] and[15]). Later the SPH method was extensively studiedand extended to the dynamic response with materialstrength as well as dynamic fluid flows with largedeformations. It has some special advantages overthe traditional mesh-based numerical methods. Themost significant is the adaptive nature of the SPHmethod, which is achieved at the very early stage ofthe field variable (i.e., density, velocity, energy) ap-proximation that is performed at each time step basedon a current local set of arbitrarily distributed parti-cles. Because of the adaptive nature of the SPH ap-proximation, the formulation of the SPH is not af-fected by the arbitrariness of the particle distribu-tion. Therefore, it can handle problems with extremelylarge deformations very well. Another advantage ofthe SPH method is the combination of the Lagrangianformulation and the particle approximation.

Unlike the mesh-free nodes in other mesh-freemethods, which are only used as interpolation points,the SPH particles also carry material properties, func-tioning as both approximation points and material com-ponents. These particles are capable of moving inspace, carry all the computational information, and thusform the computational frame for solving the partialdifferential equations describing the conservation laws.The SPH formulation is divided into two key steps.The first step is the integral representation (kernel ap-proximation) of the field function, and the second is theparticle approximation (discretization). This procedureis applied to the particle differential equations to pro-duce a set of ordinary differential equations in adiscretized form with respect only to time [9].

The SPH equations for the conservation ofmass, momentum and energy can be written as:

�tevilom delcev. Le-ti posamezno vsebujejomaterialne lastnosti in se gibljejo na osnovitemeljnih ohranitvenih enaèb. MHZD je brezmre�naLagrangeva metoda delcev, ki so jo razvili Lucy,Gingold in Monaghan z zaèetnim namenom simuliratiastro-fizikalne probleme ([5], [7] do [9] in [15]).Kasneje je bila MHZD obse�no preuèena inraz�irjena za re�evanje dinamiènih odzivov trdnihmaterialov pa tudi za simuliranje toka tekoèine zvelikimi deformacijami. MHZD ima pred obièajnimibrezmre�nimi metodami doloèene prednosti.Najpomembnej�a prednost je prilagodljivost, ki jedose�ena v zaèetnem koraku pribli�ka spremenljivihvelièin (npr. gostote, hitrosti, energij). Le-ta jeizvedena v vsakem èasovnem koraku in temelji natrenutni krajevni porazdelitvi delcev s poljubno lego.Zaradi prilagodljivosti pribli�ka metoda HZD niodvisna od poljubne porazdelitve delcev. Zaraditega lahko obravnava zelo dobro probleme zizrednimi deformacijami. Naslednja prednost metodeHZD je kombinacija Lagrangevega opisa in pribli�kadelcev.

V nasprotju z drugimi brezmre�nimi metodami,pri katerih so brezmre�na vozli�èa uporabljena le kotinterpolacijske toèke, vsebujejo zglajeni delci tudimaterialne lastnosti in so tako namenjeni kotaproksimacijske toèke ter materialni elementi. Ti delciso se zmo�ni gibati v prostoru, prena�ati vseraèunalni�ke podatke in dodatno oblikovatiraèunalni�ki sistem za re�evanje parcialnihdiferencialnih enaèb, ki opisujejo ohranitvene zakone.Metoda HZD je razdeljena na dva kljuèna dela. Prvidel je predstavitev integrala funkcij polja, drugi delpa predstavlja aproksimacijo delcev. Omenjenipostopek je uporabljen za izpeljavo parcialnihdiferencialnih enaèb v navadne diferencialne enaèbev diskretizirani obliki v odvisnosti le od èasa [9].

Ohranitvene enaèbe mase, gibalne kolièinein energije MHZD lahko zapi�emo kot:

Sl. 7. Model po MHZDFig. 7. SPH model

zglajeni delec / SPH particle



(12)

(13)

(14),

where N is the number of particles in the supportdomain of the particle i; W

ij is the smoothing function of

the particle i evaluated at the particle j, and is closelyrelated to the smoothing length; and v

ij is the relative

velocity between the particles i and j [9].The main potential advantage of the SPH

technique is that it does not require an intercon-nected spatial mesh and thus avoids the problem ofmesh distortion during large deformations. Comparedwith the Eulerian description, it is more effective,since only the material�s regions of interest need tobe modelled, and not all the regions where the mate-rial might exist. With all this promise, however, theSPH technology is relatively new, compared to stand-ard mesh-based Lagrangian and Eulerian descrip-tions, with remaining known problems in the areasof the stability, consistency and conservation [15].

3 COMPARATIVE STUDY OF DIFFERENTAPPROACHES TO SOLVING A PRACTICAL

EXAMPLE

3.1 Problem description and computational model

The analysed problem consists of a closedcontainer reservoir at rest, 60% filled with waterand 40% with air (Fig. 8). The reservoir was attachedto a sled (fixed in the vertical direction) and sub-jected to a longitudinal time-dependent accelera-tion with a peak acceleration of approximately 30 gat time t = 40 ms. The time-dependent variation ofthe water surface shape and water pressure at point1 was previously measured in experimental testingof a reservoir made of PMMA plates with 30-mmthickness [11].

The reservoir was modelled with four-nodedBelytschko-Tsay shell elements with three integra-tion points through the thickness [3]. The elasticmaterial model is used for the reservoir container,with the material data corresponding to the PMMAmaterial (r = 1180 kg/m3, E = 3000 MPa and n= 0.35).Only the bottom surface of the reservoir was mod-elled as rigid. For the water and air solid elementsand SPH particles were used, depending on the ap-

kjer je N �tevilo delcev v obmoèju vpliva delca i; Wij

predstavlja gladilno funkcijo delca i, izvrednotenov delcu j, in je tesno povezana z gladilno razdaljo;v

ij pa je relativna hitrost med delcem i in delcem j

[9].Glavna mo�na prednost MHZD je v tem, da

ne potrebuje medsebojno povezane prostorskemre�e in se s tem izogne problemu popaèenostielementov pri velikih deformacijah. V primerjavi zEulerjevim opisom je uèinkovitej�a, saj je trebamodelirati le materialna obmoèja in ne vseh obmoèij,kjer bi material lahko obstajal. Kljub vsem obetom paje metoda HZD razmeroma nova, v primerjavi zobièajnim Lagrangevim in Eulerjevim opisom z mre�oelementov, z znanimi problemi na podroèjustabilnosti, zveznosti in izpolnjevanju ohranitvenihenaèb [15].

3 PRIMERJALNA �TUDIJA RAZLIÈNIHPOSTOPKOV RE�EVANJA PRAKTIÈNEGA

PRIMERA

3.1 Opis problema in raèunalni�ki model

Analizirani problem sestoji iz zaprtegarezervoarja iz poliakrilnega stekla (PMMA), vzaèetnem mirujoèem stanju, napolnjen s 60% vodein 40% zraka (sl. 8). Rezervoar je pritrjen na sani(pritrjen v navpièni smeri) in pospe�en z vzdol�nimèasovno odvisnim pospe�kom z najveèjovrednostjo pribli�no 30 g v èasu t = 40 ms. Èasovnoodvisna sprememba proste povr�ine vode in tlak vtoèki 1, v rezervoarju iz polikarilnega stekla zdebelino sten 30 mm, sta bila predhodno doloèenas preizkusom [11].

Rezervoar je modeliran z lupinskimiBelytschko-Tsayevimi lupinskimi elementi s �tirimivozli�èi in tremi integracijskimi toèkami po debelinielementa [3]. Za rezervoar je bil uporabljen elastièenmaterialni model s podatki, ki ustrezajo poliakrilnemusteklu (r = 1180 kg/m3, E = 3000 MPa in n = 0,35).Spodnja ploskev rezervoarja je bila modelirana kottoga. Za modeliranje vode in zraka so bili, odvisnood uporabljene metode, uporabljeni prostorninski

1

Niji

j ij

j i

Wm v

t x

r

=

¶¶= × ×

¶ ¶å

2 21

Nj iji i

j

j i j i

Wvm

t x

ssr r=

æ ö ¶¶= × + ×ç ÷ç ÷¶ ¶è ø

å

1

Ni j iji

j ij

j i j i

Wum v

t x

s s

r r=

× ¶¶= × × ×

¶ × ¶å



Toèka 1 (zaznavalo) Point 1 (sensor) 30

0 m

m

180

mm

1008 mm 196 mm

128 mm

x

y z

a

g

plied method. The material model Null (Type 9) [6]was used for the water (r = 1000 kg/m3 at 293 K) andthe air (r = 1 kg/m3 at 293 K) modelling. The fluiddomain was described with a material model thatneglects the deviatoric stresses. By defining a lowyield stress, a rapid transition to plasticity can beachieved (e.g., by only considering the gravitation).Under high dynamic loading, the shear forces be-come negligible in comparison with the inertial forcesof the fluid. The air was considered only in theEulerian and ALE models.

The Mie-Gruneisen (water and air) and IdealGas (only for air) equations of states have been used.The model was also loaded with the constant gravi-tational acceleration (g = 9.81 m/s2). In the Lagrangianand SPH model the automatic nodes to surface con-tact were used, and in the Eulerian and ALE modelsthe contact between fluid and structure was definedwith the keyword Constrained Lagrange in Solid ([6]and [10]).

3.2 Computational simulation data

Explicit dynamic analyses were carried outby using all four different fluid model approaches:Lagrangian, Eulerian, ALE and SPH. The models weresolved with LS-DYNA Linux Version 970. The com-putational time frame was set to 80 ms and the timestep of the simulation was defined according to thelowest resonant frequency of the structure and wasequal to 0.01 ms.

elementi in zglajeni delci. Za modeliranje vode (r =1000 kg/m3 pri 293 K) in zraka (r = 1 kg/m3 pri 293 K)je bil uporabljen konstitutivni model Null (Type 9)[6]. Domena tekoèine je opisana z materialnimmodelom, ki zanemari stri�ne napetosti. Z uporabonizke meje teèenja se dose�e hiter prehod v plastiènoobmoèje (npr. le z upo�tevanjem gravitacije). Obuporabi visokih dinamiènih obremenitev so stri�nesile tako zanemarljive v primerjavi z vztrajnostnimisilami tekoèine. Zrak je bil upo�tevan le pri Eulerjevemin modelu po PLE.

Uporabljeni sta bili Mie-Gruneisenova enaèbastanja (voda in zrak) in enaèba stanja idealnega plina(le za zrak). Celoten model je bil izpostavljennespremenljivem te�nostnem pospe�ku (g = 9,81m/s2). V Lagrangevem in MHZD modelu je biluporabljen avtomatski algoritem za doloèevanjestiènih sil, v Eulerjevem in modelu po PLE pa je bilstik med tekoèino in strukturo definiran z uporaboposebnega algoritma ([6] in [10]).

3.2 Podatki raèunalni�ke simulacije

Izvedene so bile eksplicitne dinamiène analizez uporabo �tirih razliènih modelov simuliranjatekoèine: Lagrangev, Eulerjev, PLE in MHZD. Zaizvedbo analiz je bil uporabljen program LS-DYNALinux Version 970. Opazovani èasovni interval jezna�al 80 ms, èasovni korak simulacije pa je bildoloèen glede na najni�jo resonanèno frekvencostrukture in je zna�al 0,01 ms.

Sl. 8. Izmere in zaèetne razmere analiziranega rezervoarja iz poliakrilnega steklaFig. 8. Dimensions and initial conditions of the analysed PMMA box



3.3 Computational results

The free-surface shape-prediction results ofall four dynamic simulations at the time t = 38 ms arerepresented in Figure 9. The dotted line representsthe free-surface shape observed in the experiment atthe same time instance.

From Figure 9 it is obvious that the Lagrangianand SPH models are only good for approximations of thefluid motion at the right-hand side wall, since in realitythe fluid would not retain the form of the container, whichis the case observed in simulations at the left-hand sidewall. However, this observation must be considered inview of the required computational results. In the casewhere only the impulse of the fluid towards the tank wallis needed, the deformations and deflections on the op-posite side could be neglected. The Eulerian and ALEformulations perform much better when describing theposition and the form of the water�s free surface. How-ever, this is only achieved by a dramatic increase of thecalculation times, which is not always acceptable. Fig-ure 10 represents the fluid motion in a reservoir duringthe calculation time. It is important to observe that usingthe Lagrangian formulation results in very distorted ele-ments and, consequently, large computational errors.This again confirms the fact that the Lagrangian formu-lation is unsuitable for huge deformations.

The time variation of water pressure at point1 is shown in Figure 11. The results have been deter-mined by two different approaches. In the Lagrangianand SPH models the pressure at point 1 was meas-ured with contact forces that appeared at the ob-served point. For the Eulerian and ALE model thepressure was determined by the leakage control, i.e.,by determining the force that is needed to establishequilibrium in every observed element at the bound-ary between the fluid and the reservoir wall.

3.3 Raèunalni�ki rezultati

Rezultati predvidene oblike proste povr�ineza vse �tiri simulacije so pri èasu t = 38 ms prikazanena sliki 9. Èrtkana èrta pomeni opazovano oblikoproste povr�ine v omenjenem trenutku pripreizkusu.

S slike 9 je razvidno, da sta Lagrangeva inMHZD formulacija na desni strani modela ustreznile za pribli�ek gibanja tekoèine, saj v resnici tekoèinane bi obdr�ala oblike posode, kar je razvidno izsimulacij na levi strani modela. Kljub temu pa semorajo omenjene omejitve vrednotiti glede nazahtevane raèunalni�ke rezultate. V primeru, kadarje iskan le udarec tekoèine na steno rezervoarja, sodeformacije in pomiki na nasprotni strani lahkozanemarljivi. Eulerjeva in PLE formulacija opi�etalego in obliko proste povr�ine vode mnogo bolje,kar pa je odvisno od dalj�ega raèunskega èasa(preglednica 2), kar ni vedno sprejemljivo. Potekgibanja tekoèine v rezervoarju ponazarja slika 10.Pomembno je poudariti, da se z uporaboLagrangevega modela pri zelo velikih deformacijahpojavijo zelo popaèeni elementi in posledièno velikenumeriène napake. Omenjeno ponovno potrjuje, daLagrangeva zapis ni primeren za zelo velikedeformacije.

Èasovno spreminjanje tlaka vode v toèki 1 jeprikazano na sliki 11. Rezultati so bili doloèeni nadva razlièna naèina. V Lagrangevem in MHZDmodelu je bil tlak, ki se je pojavil v toèki 1, merjen sstiènimi silami. Tlak pri Eulerjevem in PLE modelu paje bil doloèen z merili poroznosti snovi in z doloèitvijosile, ki je potrebna za vzpostavitev ravnote�ja navsakem opazovanem elementu na meji med tekoèinoin steno rezervoarja.

Sl. 9. Oblika proste povr�ine: a) Lagrangev model; b) Eulerjev model; c) PLE model; d) MHZD modelFig. 9. Shape of the free surface: a) Lagrangian model; b) Eulerian model; c) ALE model; d) SPH model

a) b)

c) d)



Very good agreement with the experimental resultswas achieved by using all four formulations. The SPH for-mulation provided very good results, especially when tak-ing into account that using this formulation results in a veryquick and uncomplicated analysis (the mesh consists onlyof SPH particles). The results of the Lagrangian formulationare very good, although the simulation terminates becauseof the high element distortion. The reason for a lower pres-sure during the ALE and Eulerian models can be attributedto the air�s influence inside the reservoir, where it acts like adamper. The drop in the pressure, observed for the Eulerianformulation, is also attributed to the need for a different airmodel (a different equation of state), which is necessary toensure a stable analysis.

Zelo dobro ujemanje z rezultati preizkusa jebilo dose�eno z uporabo vseh �tirih modelov. MHZDmodel zagotavlja zelo dobre rezultate, �e posebej, èeupo�tevamo, da uporaba te metode omogoèa zelohitro in nezahtevno analizo (mre�a sestoji le izzglajenih delcev). Rezultati Lagrangevega modela sozelo dobri, kljub temu da se simulacija predèasnoprekine zaradi velike popaèenosti elementov. Razlogza nizek izmerjen tlak pri PLE modelu in Eulerjevemmodelu se lahko pripi�e vplivu zraka v rezervoarju,ki deluje kot du�ilnik. Razlog za padec tlaka priEulerjevem zapisu pa je uporaba drugaènega modelazraka (drugaèna enaèba stanja), ki je potrebna zavzpostavitev stabilne analize.

Sl. 10. Gibanje tekoèine, modelirane s PLE opisomFig. 10. Fluid motion modelled with ALE formulation



The model size and the required CPU times for eachanalysis are listed in Table 2, to illustrate the requiredcomputational effort to solve the chosen problemwith different approaches.

The deformation of the reservoir container isnegligible, due to the thickness of the reservoir walls.To clearly illustrate the fluid-structure interaction ca-pabilities of LS-DYNA, another analysis of the sameproblem was carried with reduced container reservoirwalls to only 10% of the original thickness, i.e., 3 mm[6]. The resulting deformations of the container reser-voir due to fluid-structure interaction at different timeinstances can be observed in Figure 12.

Following from the reported simulations it isobvious that these approaches offer an alternativemeans of fluid-flow modelling and its interaction withthe structure. The advantage of using the SPH and

Velikost modelov in njihovi raèunski èasi zaposamezne analize so prikazani v preglednici 2, kiprikazuje zahtevane raèunske sposobnosti za re�itevizbranega problema z razliènimi pristopi.

Deformacije samega rezervoarja so zaradivelike debeline njegovih sten zanemarljive. Zanazoren prikaz zmo�nosti stika med tekoèino intrdnino LS-DYNE je bila izvedena ponovna analiza zistim problemom, vendar je bila debelina stenrezervoarja zmanj�ana na 10 odstotkov prvotnedebeline, tj. 3 mm [6]. Deformacije sten rezervoarja,kot rezultat stika med tekoèino in trdnino pri razliènihèasovnih korakih, so prikazane na sliki 12.

Iz opisanih simulacij je razvidno, da razliènipristopi ponujajo alternativne mo�nosti zamodeliranje toka tekoèin in njihovega vpliva nastrukturo. Prednost uporabe Lagrangevega in

Preglednica 2. Primerjava raèunskega èasaTable 2. CPU-time comparison

Skupno �tevilo / Total number

Model Vozli�èa /

Nodes

Elementi /

Elements

Raèunski èas /

CPU time

min

Lagrangev / Lagrangian 2898 2420 16

Eulerjev / Eulerian 10162 8706 225

PLE / ALE 7462 6396 260

MHZD / SPH 2898 2896 13

Tla

k p

[ba

r]

Pre

ssur

e p

[ba

r]

Èas t [ms] Time t [ms]

Eksperiment / Experiment

Lagrangev model / Lagrangian model

Eulerjev model / Eulerian model

PLE model / ALE model

MHZD model / SPH model

Sl. 11. Primerjava èasovne spremembe tlaka v toèki 1Fig. 11. Comparison of the pressure time-variation at point 1



Lagrangian models is a short pre-processing and areasonable computational time, while the ALE andEulerian models can describe the fluid motion moreaccurately. Nevertheless, the initial impact of the fluidon the reservoir wall could be simulated accuratelywith all four fluid models incorporated in LS-DYNA.

4 CONCLUSION

Four approaches to fluid-flow modelling inLS-DYNA have been presented in the paper. Differ-ent formulations (Lagrangian, Eulerian, ALE andSPH) were used to analyse the fluid motion in a de-formable reservoir, with the purpose to validate theresults in comparison with existing experimental ob-servations. Computational simulations showed thatthe fluid motion and the fluid-structure interactioncan be accurately described by applying differentalternative formulations in the LS-DYNA.

The applied models provide a basis for eco-nomical computational models that can be used foranalysing more complex problems. The results ofthe real fuel-tank with a very complex geometry,where the fluid motion�s influence was consideredwith the ALE and SPH descriptions, showed a verygood agreement with the experiments [18].

MHZD modela je kratek èas za pripravo modela insprejemljivi raèunski èas, medtem ko Eulerjev in PLEmodel opi�eta gibanje tekoèine natanèneje. Kljubtemu pa je bilo zaèetni udarec tekoèine ob stenerezervoarja mogoèe natanèno simulirati z vsemi �tirimimodeli tekoèine, ki so vkljuèeni v LS-DYNI.

4 SKLEP

V prispevku so predstavljeni �tirje postopkimodeliranja toka tekoèine v LS-DYNI. Razliènemetode (Lagrangeova, Eulerjeva, PLE in MHZD) sobile uporabljene za analizo gibanja tekoèine vdeformabilnem rezervoarju, z namenom, da bi potrdilirezultate v primerjavi z znanimi preizkusnimiopazovanji. Raèunalni�ke simulacije so pokazale, dasta gibanje tekoèine in medsebojni vpliv tekoèine intrdnine natanèno opisana z uporabo razliènih izbirnihzapisov v LS-DYNI.

Predstavljeni modeli so osnova zakomercialne raèunalni�ke modele, uporabljene zaanalizo zapletenej�ih problemov. Rezultatidejanskega rezervoarja za gorivo z zelo zahtevnogeometrijsko obliko ob upo�tevanju vpliva gibanjatekoèine s PLE in MHZD formulacijami, so pokazalizelo dobro ujemanje s preizkusi [18].

Sl. 12. Medsebojni vpliv tekoèine in trdnineFig. 12. Fluid-structure interaction

0 ms

50 ms


100 Vesenjak M. - Ren Z. - Müllerschön H. - Matthaei S.

Authors� Addresses: Matej VesenjakProf. Dr. Zoran RenUniversity of MariborFaculty of Mechanical Eng.Smetanova 172000 Maribor, [email protected]@uni-mb.si

Dr. Heiner MüllerschönDYNAmore GmbHIndustriestr. 270565 Stuttgart-Vaihingen, [email protected]

Stephan MatthaeiDaimlerChrysler AG, HPC B20970322 Stuttgart, [email protected]


[1] Anghileri, M., Castelletti, L., Tirelli, M. (2003) Fluid-structure interaction of water filled tanks during theimpact with the ground. Int. Journal of Impact Engineering.

[2] Belytschko, T., Liu, W.K., Moran B. (2000) Nonlinear finite elements for continua and structures, JohnWiley & Sons Ltd, Chichester.

[3] Bois, P.D. (2003) Crashworthiness simulation using LS-DYNA. Stuttgart.[4] Ferziger, J., Periæ, M. (1997) Computational methods for fluid dynamics. Springer Verlag, Berlin.[5] Gray, J.P., Monaghan J.J., Swift R.P. (2001) SPH elastic dynamics. Comput. Methods Appl. Mech. Engrg.

190, Elsevier Science Ltd.[6] Hallquist, J.O. (1998) LS-DYNA Theoretical manual. Livermore.[7] Idelsohn, S.R., Onate, E., Pin, F.D. (2003) A Lagrangian meshless finite element method applied to fluid-

structure interaction problems. Computers and Structures 81, Elsevier Science Ltd.[8] Lacome, L.L., Smoothed particle hydrodynamics. Part I, II.[9] Liu, G.R., Liu, M.B. (2003) Smoothed particle hydrodynamics � a meshfree particle method. World Scientific,

Singapore.[10] Livermore Software Technology Corporation. (1998) LS-DYNA Theoretical manual. Livermore.[11] Meywerk, M., Decker, F., Cordes, J. (1999) Fuel sloshing in crash simulation, EuroPAM 99.[12] Müllerschön, H. (2004) Contact modeling in LS-DYNA. DYNAmore. Stuttgart.[13] Olovsson, L. ALE and fluid-structure interaction capabilities in LS-DYNA. LSTC. Livermore.[14] Olovsson, L., Soulli, M., Do, I. (2003) LS-DYNA � ALE capabilities, fluid-structure interaction modelling.

LSTC. Livermore.[15] Schwer, L.E. (2004) Preliminary assessment of non-Lagrangian methods for penetration simulation. 8th

International LS-DYNA User Conference.[16] Souli, M. (2003) LS-DYNA advanced course in ALE and fluid/structure coupling. LSTC. Livermore.[17] �kerget, L. (1994) Mehanika tekoèin. Tehni�ka fakulteta v Mariboru in Fakulteta za strojni�tvo v Ljubljani.[18] Vesenjak, M. (2004) Methoden um Fluid-Struktur Interaktion in LS-DYNA zu simulieren. Daimler Chrysler.

Stuttgart.[19] Vesenjak, M., Ren, Z., Hriber�ek, M. (2004) Weakly coupled analysis of a blade in multiphase mixing

vessel. GAMM 2004, Dresden.[20] Zienkiewicz, O.C., Taylor, R.L. (2000) The finite element method. McGraw-Hill Ltd., London.

Naslovi avtorjev: Matej Vesenjakprof.dr. Zoran RenUniverza v MariboruFakulteta za strojni�tvoSmetanova 172000 [email protected]@uni-mb.si

dr. Heiner MüllerschönDYNAmore GmbHIndustriestr. 270565 Stuttgart-Vaihingen, Nemè[email protected]

Stephan MatthaeiDaimlerChrysler AG, HPC B20970322 Stuttgart, Nemè[email protected]

Prejeto: 6.10.2005




101Simulacija naleta tovornega vozila - Simulating the Impact of a Truck

Simulacija naleta tovornega vozila ob cestno varnostno ograjo

Simulating the Impact of a Truck on a Road-Safety Barrier

Matej Borovin�ek - Matej Vesenjak - Miran Ulbin - Zoran Ren(Fakulteta za strojni�tvo, Maribor)

Cestno varnost lahko izbolj�amo s postavitvijo cestnih varnostnih ograj, ki prepreèujejo udele�encemcestnega prometa vstop v nevarna obmoèja. Za dvig stopnje varnosti morajo biti cestne varnostne ograjeoblikovane tako, da pri naletu vozila le-to preusmerijo nazaj na cesti�èe. Med preusmerjanjem vozilamorajo s svojo deformacijo absorbirati èim veèjo kolièino kinetiène energije, da se zmanj�ajo pojemkipotnikov v vozilu. V prispevku je predstavljena raèunalni�ka simulacija naleta tovornega vozila ob cestnovarnostno ograjo. Izvedene so bile parametriène raèunalni�ke analize za oceno primernosti razliènihdodatnih elementov varnostne ograje. Za izvajanje eksplicitnih dinamiènih analiz je bil uporabljen programLS-DYNA. Raèunalni�ke simulacije dokazujejo, da je nosilnost analizirane ograje dovolj velika, da tovornjakzadr�i in preusmeri nazaj na cesti�èe. Hkrati pa se poka�e tudi ustrezno visoka stopnja akumulacije kinetièneenergije vozila v obliki deformacije cestne varnostne ograje, kar zmanj�a pojemke vozila in se ka�e vpoveèani stopnji varnosti za potnike v vozilu.© 2006 Strojni�ki vestnik. Vse pravice pridr�ane.(Kljuène besede: varnost na cestah, ograje varnostne, analize trkov, simuliranje dinamièno, LS-DYNA)

Road safety can be improved by applying appropriate road-safety barriers that prevent road-trafficparticipants from entering dangerous areas. In terms of preventing vehicles from veering off the road, aroad-safety barrier should redirect the impacting vehicle back onto the road. In the process of doing so itshould absorb as much of the vehicle�s kinetic energy as possible through its deformation, so reducing thedeceleration of the vehicle�s occupants. This paper considers a computer simulation of a truck�s impact ona steel road-safety barrier. Parametric computer simulations were used to evaluate the different additionalsafety elements of the road-safety barrier. The dynamic finite-element explicit code LS-DYNA was used forthis purpose. The computer simulations prove that the analyzed road-safety barrier design is strong enoughto retain and redirect the truck back onto the roadway. Appropriately, the high absorption of the impactingvehicle�s kinetic energy through the safety-barrier deformation is observed at the same time, which indicatesa reduction in a vehicle�s deceleration, and with that, a higher safety level for the vehicle�s occupants.© 2006 Journal of Mechanical Engineering. All rights reserved.(Keywords: road safety, road safety barrier, impact simulations, dynamic simulations, LS-DYNA)

Strojni�ki vestnik - Journal of Mechanical Engineering 52(2006)2, 101-111UDK - UDC 625.745.5:656.08

004.94:629.351:692.88Izvirni znanstveni èlanek - Original scientific paper (1.01)

0 INTRODUCTION

A high level of safety is a very importantaspect in modern road traffic. On the one hand, thecar industry is developing new, passive and activevehicle-safety systems to increase the safety of road-vehicle occupants, and on the other hand, the roadsare being made safer with the installation of moreeffective road-safety elements.

Ensuring appropriate safety levels in roadtraffic is of primary importance. One way to improve

0 UVOD

Visoka stopnja varnosti je zelo pomembna vsodobnem cestnem prometu. Na eni straniavtomobilska industrija razvija nove pasivne inaktivne varnostne sisteme za poveèanje varnostipotnikov v cestnem prometu, na drugi strani pa sevarnost cest poveèuje z uporabo uèinkovitej�ihcestnih varnostnih elementov.

Zagotovitev dovolj visoke stopnje varnostiv cestnem prometu je primarnega pomena. Eden od


102 Borovin�ek M. - Vesenjak M. - Ulbin M. - Ren Z.

road safety is to install road-safety barriers. Thepurpose of such systems is to restrain a vehicle fromentering dangerous areas, i.e., to prevent it fromveering off the road or on to the other side of theroad. This effectively reduces or completely alleviatesdamage to vehicle occupants and other road-trafficparticipants and objects.

Newly installed road-restraint systems inSlovenia have to fulfill the new Slovenian regulationrequirements for road-restraint systems and the SISTEN 1317 standard [1]. Every newly installed restraintsystem has to be certificated for the containment levelit is designed for by a full-scale crash test. Since thesetests are very expensive, it is advisable to preliminarycheck the adherence of safety-barrier design to theset regulations by means of a computer simulation.

This paper reports on a computer simulationof a truck impact into a road-safety barrier with theaim of determining the most effective safety-barrierdesign. The software system LS-DYNA ([2] and [3])was used for this purpose.

1 DESIGN REQUIREMENTS FOR ROAD-SAFETYBARRIERS

Road-safety barriers only need to be installedon road sections where the possibility of severeinjuries during vehicle impact is reduced by theinstallation of such barriers. Only certified road-safety barriers according to SIST EN 1317 may beinstalled on public roads in Slovenia.

The most frequently used type of road-safetybarriers in Slovenia are steel road-safety barriers (Fig.1). They are installed mainly on highways, main andregional roads. On roads with less traffic they areonly used in dangerous areas with a high probabilityof the vehicle veering off the road.

The main design elements of a single-sidedsteel road-safety barrier are (Fig. 1):- the guardrail � a longitudinally placed element,

the deformation of which reduces the severity ofan impact, but it should also be strong enoughnot to rupture during an impact;

- the post � carries the distance spacer and/or theguardrail and it ensures the guardrail position at acertain distance from and above the road;

- the distance spacer � ensures a certain positionof the guardrail in relation to the post,

- the end element � a part of the safety barrier placedat its beginning and end, which reduces theconsequences of a direct vehicle impact.

naèinov izbolj�anja varnosti je postavitev varnostnihograj. Namen cestnih varnostnih ograj je prepreèitizdrs vozila s ceste, torej prepreèiti izlet vozila scesti�èa ali prehod vozila na nasprotnosmernovozi�èe. S tem se prepreèijo oziroma popolnomaodpravijo mo�nosti po�kodbe potnikov v vozilu teroseb in objektov ob vozi�èu.

Novo postavljene cestne varnostne ograje vSloveniji morajo izpolnjevati zahteve novegaslovenskega pravilnika o postavitvi cestnihvarnostnih ograj in standarda SIST EN 1317 [1]. Vsakana novo postavljena ograja mora imeti uspe�noopravljen preizkus naleta za stopnjo zadr�evanja, zakatero je namenjena. Zaradi visokih cen preizku�anjvarnostnih ograj je smiselno njihovo konstrukcijopoprej preveriti z izvajanjem raèunalni�kih simulacij.

Tako so bile, z namenom, da bi izbralinajustreznej�e konstrukcijske re�itve varnostneograje, izvedene raèunalni�ke analize naletatovornjaka ob ograjo. Za izvajanje teh simulacij je biluporabljen programski paket LS-DYNA ([2] in [3]).

1 KONSTRUKCIJSKE ZAHTEVE ZA CESTNEVARNOSTNE OGRAJE

Cestne varnostne ograje morajo bilipostavljene le na odsekih, kjer je mo�nost za huj�epo�kodbe pri naletu vozila manj�a, kakor v primeru,èe ograje ne bi bilo. Na javnih cestah v Sloveniji selahko postavljajo le varnostne ograje, ki so atestiranepo standardu SIST EN 1317.

Najpogosteje uporabljena vrsta cestnihvarnostnih ograj v Sloveniji so jeklene varnostneograje (sl. 1). Postavljene so predvsem naavtocestah, glavnih in regionalnih cestah. Naredkeje prevoznih cestah se uporabljajo le nanevarnih obmoèjih, kjer je poveèana verjetnostzdrsa vozila s cesti�èa.

Glavni konstrukcijski elementi enostranskejeklene varnostne ograje so (sl. 1):- �èitnik � vzdol�ni element ograje; s svojo

deformacijo zmanj�a moè udarca, hkrati pa morabiti dovolj tog, da se med trkom ne poru�i;

- steber � nosilo distanènika in/ali �èitnika, kizagotavlja lego �èitnika na doloèeni oddaljenostiin vi�ini od vozi�èa oziroma nad njim;

- distanènik � zagotavlja doloèeno lego �èitnikaglede na steber ograje;

- zakljuèni element � del varnostne ograje, ki je nanjenem zaèetku oziroma koncu in je namenjenzmanj�anju posledic naleta vozila na ograjo.



The technical regulations for the installationof road-safety barriers on Slovenian public roadsare not yet fully operational, as they are still in theprocess of final approval by various agencies [4].

2 SIST EN 1317 REQUIREMENTS

The suitability of a safety barrier is, accordingto the SIST EN 1317 standard, determined by threemain criteria [5]:- the containment level � represents the level of

containment for different types of vehicles;- the impact severity � defined by three parameters:

the acceleration severity index (ASI), � the post-impact head deceleration (PHD) and the theoreticalhead-impact velocity (THIV),

- the deformation of the barrier.The most important ASI parameter is a

dimensionless variable, which is used as an overallmeasure of vehicle-impact consequences for thevehicle occupants. The ASI is determined as:

(1),

where xa , ya and za represent the average values ofthe acceleration of the vehicle�s center of gravity ina local coordinate system of the vehicle in a running-time interval of tD = 50 ms (Fig. 2). x�a , y�a and z�astand for the acceptable acceleration limits forindividual coordinate directions and are equal to x�a =12×g, y�a = 9×g and z�a = 10×g, where g = 9.81 m/s2.

The impact severity parameter ASI needs tobe measured and determined only for tests with low

Pravila za postavitev cestnih varnostnih ograjv okviru tehniène specifikacije za javne ceste vSloveniji �e niso popolnoma doloèena, saj jih �ezmeraj usklajujejo razliène agencije [4].

2 ZAHTEVE STANDARDA SIST EN 1317

Glede na standard SIST EN 1317 ocenjujemoustreznost varnostne ograje z ozirom na tri glavnekriterije [5]:- raven zadr�evanja vozila � pomeni raven

zadr�evanja za razliène vrste vozil;- moè udarca � se doloèa s tremi parametri: z merilom

velikosti pospe�kov (MVP - ASI), s pojemkomglave po udarcu (PGU - PHD) in s teoretiènohitrostjo glave pri udarcu (THGU - THIV);

- deformacija ograje.Najpomembnej�i parameter ASI je

brezrazse�na velièina, ki se uporablja kot splo�nomerilo za doloèanje posledic naleta vozila na potnikev vozilu. ASI je doloèen kot:

kjer xa , ya in za pomenijo povpreène vrednostipospe�kov masnega sredi�èa vozila v njegovemlokalnem koordinatnem sistemu v tekoèem èasovnemkoraku tD = 50 ms (sl. 2). x�a , y�a in z�a pomenijomejne vrednosti pospe�kov za posameznekoordinatne osi ter zna�ajo x�a = 12×g, y�a = 9×g in z�a =10×g, kjer je g = 9,81 m/s2.

Parameter moèi udarca MVP mora biti merjenin izraèunan samo za teste z vozili majhnih mas

Sl. 1. Konstrukcijski elementi in glavne mere jeklene cestne varnostne ograjeFig. 1. Main elements and regulated dimensions of a steel road-safety barrier

( ) ( )

22 2

yx z

x y z

max max 1,0 1, 4� � �

aa aASI ASI t

a a a

é ùæ öæ ö æ öê ú= = + + £é ù ç ÷ç ÷ ç ÷ë û ç ÷ê úè ø è øè øê úë û



weight vehicles (personal vehicles), which sustainsignificant deceleration during impact. However, inthis paper the ASI parameter is also used to assess atruck�s impact conditions. Since it is calculated fromfiltered coordinate acceleration data and normalizedby their limiting values, it represents a good measureof the normalized effective acceleration. Thus, it wasused here to illustrate the influence of different road-safety barrier designs on overall vehicle behavior.

3 ROAD-SAFETY BARRIERDESIGN

Simulations were used to test different designsof road-safety barrier for the containment level H1.According to the SIST EN 1317 standard such safetybarrier has to successfully sustain the impact of apersonal vehicle (TB11 test) and a truck (TB42 test).A personal vehicle has a mass of m = 900 kg, an initialvelocity of v = 100 km/h and impacts the barrier at anangle of a = 20° with regard to the guardrail. The truckhas a mass of m = 10,000 kg and impacts the barrierwith an initial velocity of v = 70 km/h at an impactangle of a = 15°. The TB11 test is used to determinethe impact-severity parameters and the TB42 test isused to test the load-carrying capability of such asystem. Both tests are equally important when makingthe final choice of road-safety barrier design.

The new H1 road-safety barrier was a designupgrade of an already-certified safety barrier for thecontainment level N2. The main components of thissafety barrier are made of construction steel S 235.The guardrail is made from 3-mm-thick metal sheetand is 4,200 mm long, where the splice length is equalto 200 mm. The distance spacer is shaped in ahexagonal form [6] to provide the highest crash-energy absorption and controllable deformation and

(osebna vozila), na katera med testom delujejo veèjipojemki. Kljub temu je bil v tem prispevku parameterMVP uporabljen tudi za analizo razmer med naletomtovornjaka. Ta parameter pomeni dobro mero zanormiran dejanski pospe�ek, saj se izraèuna iz filtriranihkoordinatnih pospe�kov in je normiran z njihovimimejnimi vrednostmi. Tako je bil tukaj uporabljen zaprikaz vpliva razliènih konstrukcijskih re�itev cestnevarnostne ograje na skupno obna�anje vozila.

3 KONSTRUKCIJSKA IZVEDBA VARNOSTNEOGRAJE

S simulacijami je bilo preizku�enih veèkonstrukcijskih re�itev cestne varnostne ograje zaraven zadr�evanja H1. Po standardu SIST EN 1317mora tak�na ograja uspe�no prestati nalet osebnegavozila (test TB11) in tovornjaka (test TB42). Osebnovozilo ima maso m = 900 kg, zaèetno hitrost v = 100km/h in trèi v cestno varnostno ograjo pod kotom a= 20° glede na postavitev �èitnika ograje. Tovornjakima maso m = 10.000 kg in trèi v varnostno ograjo zzaèetno hitrostjo v = 70 km/h pod vpadnim kotom a= 15°. Test TB11 je namenjen doloèitvi parametrovmoèi udarca, medtem ko je test TB42 namenjendoloèitvi nosilnosti cestne varnostne ograje. Obatesta sta enako pomembna za konèno odloèitev oizbiri konstrukcijske re�itve cestne varnostne ograje.

Osnova za konstrukcijo nove cestnevarnostne ograje je bila atestirana ograja za ravenzadr�evanja N2. Njeni glavni sestavni deli so v celotiizdelani iz konstrukcijskega jekla S 235. �èitnik jenarejen iz 3 mm debele ploèevine in v dol�ino meri4.200 mm, kjer je 200 mm namenjenih spoju medsosednjimi �èitniki. Distanènik ima obliko �estkotnika[6], s tem ima najveèjo zmo�nost absorpcije energijetrka in predvidljivo deformacijo, narejen pa je iz 4 mm

Sl. 2. Lokalni koordinatni sistem vozilaFig. 2. Local coordinate system of the vehicle

Sl. 3. Konstrukcija ograje s pasnico (levo) intrapeznim vodilom (desno)

Fig. 3. Design of a barrier with tension belt (left)

and wheel guidance (right)



is made from 4-mm-thick sheet metal. The posts are C-shaped with dimensions of 100 × 55 × 4 mm and 1,900mm long. The guardrails are joined together with M16screws, while all other connections are made with M10screws. All the screws are of strength class 5.8.

The basic design was additionally reinforcedin two different ways. First, with a tension belt that isconnected to the back of a distance spacer; and,second, with a wheel-guidance profile connected tothe front of the post (Fig. 3). An additional purpose ofthe wheel guidance is to hinder the direct vehicle-wheelimpact into the post, which results in smaller vehicledecelerations and lessens the severity of the impact.

The tension belt and the wheel guidance areof the same length as the guardrail and connectedwith M10 bolts. The sheet-metal thicknesses of bothreinforcements were the subject of a parametricalanalyses. Thicknesses of 3.0, 5.0, and 7.0 mm wereused for the tension belt and 2.0, 2.5, 2.8, and 3.0 mmfor the wheel-guidance profile.

4 COMPUTATIONAL SIMULATIONS

4.1 Computational model of a truck

A publicly accessible FEM model of a FordSingle-Unit Truck (Reduced Model) [7] was used as abasic truck model in the computational simulations.The truck is 8.5-m long, 3.3-m high and 2.4-m broad(Fig. 4). The FEM model is made of approximately 22,200linear elements with a reduced integration scheme [8].It consists of 1,000 solid elements, 500 beam elementsand the rest is represented by 20,700 shell elements.

The FEM model of the truck was subjectedto further changes to fulfill the SIST EN 1317standard. The vehicle mass was raised from 7,320 kgto 10,000 kg by increasing the material density of all

ploèevine. Steber je izdelan iz profila C izmer 100 × 55× 4 mm, dol�ine 1.900 mm. �èitniki so medsebojnospojeni z vijaènimi zvezami z vijaki M16, v preostalihvijaènih zvezah pa so uporabljeni vijaki M10. Vsiuporabljeni vijaki so trdnostnega razreda 5.8.

Ta osnova cestne varnostne ograje je biladodatno ojaèana na dva razlièna naèina. Enkrat zdodatno pasnico, ki je privijaèena na hrbtni deldistanènika, drugiè pa s trapeznim vodilom,privijaèenim neposredno na sprednji del stebra (sl.3). Dodaten namen trapeznega vodila je prepreèitineposredni udarec kolesa vozila ob steber ograje tertako zmanj�ati pojemke v vozilu in zmanj�ati moèudarca.

Pasnica in trapezno vodilo sta enake dol�inekakor �èitnik ograje in sta privijaèena z vijaki M10.Debeline ploèevin obeh ojaèitev so bile predmetparametriènih numeriènih analiz. Debelina ploèevineje obsegala vrednosti 3,0, 5,0 in 7,0 mm pri pasnici in2,0, 2,5, 2,8 ter 3,0 mm pri trapeznem vodilu.

4 NUMERIÈNE ANALIZE

4.1 Numerièni model tovornega vozila

Za model tovornega vozila je bil uporabljenprosto dostopen model po MKE vozila Ford SingleUnit Truck (poenostavljen) [7]. Dol�ina tovornegavozila zna�a 8,5 m, vi�ina 3,3 m in �irina 2,4 m (sl. 4.).Model po MKE sestavlja pribli�no 22.200 linearnihkonènih elementov z reducirano integracijsko shemo[8]. Od tega je okoli 1.000 prostorninskih, 500 linijskihelementov, ostalo pa predstavlja 20.700 lupinskihkonènih elementov.

Model po MKE tovornega vozila je bil nekolikospremenjen, da je ustrezal zahtevam standarda SISTEN 1317. Masa vozila je bila poveèana iz 7.320 kg na

Sl. 4. Model po MKE tovornega vozila FordFig. 4. The FEM model of a Ford single-unit truck



the vehicle parts. A rigid shell element was added atthe vehicle�s center of gravity to act as anaccelerometer and to record the displacements,velocities and accelerations in the local coordinatesystem of the truck.

4.2 Computational model of the safety barrier

The main safety barrier parts, the guardrail,the posts and the distance spacers were modeled indetail, while simplified modeling was used for the bolts.Based on previous simulations and the speed of theimpacting vehicle the length of the modeled safetybarrier was 39 m: 5 m before and 34 after the impactpoint. Along the whole length of the road-safety barrier19 posts and distant spacers were placed 2-m apart.

Shell finite elements were used to model the mainparts of the safety barrier. Posts and distance spacerswere modeled with Belytschko�Tsay shell finite elementswith three integration points through the shell thickness,while the guardrails, the tension belts and the wheel-guidance profiles were modeled using full-integrationshell elements with five integration points through thethickness to prevent �hourglassing�. The thicknessesof the shell elements were defined according to theconstruction plans of each design, as described in Section3. The bolt connections were modeled with linearHughes�Liu finite elements. The complete safety-barriermodel consisted of approximately 100,000 finite elementsand 110,000 nodes.

The material properties of the different sheet metalthicknesses were obtained experimentally. Experimentresults were used to define an isotropic bilinearelastoplastic material model with kinematic hardening(Table 1). Material failure was prescribed according to theeffective plastic deformation, which was set to 0.28.

A yield stress of 400 MPa and a limit stress of500 MPa were prescribed for the M10 bolt connectionsof strength class 5.8, while the M16 bolt connectionswere modeled with the parameters presented in Table 2.The parameters were obtained from a correlation analysis

10.000 kg, tako da je bila poveèana gostota vseh delovtovornjaka. V te�i�èe tovornega vozila je bil dodanmerilnik pospe�kov kot tog lupinski konèni element, kiomogoèa zajem pomikov, hitrosti in pospe�kov vlokalnem koordinatnem sistemu vozila.

4.2 Numerièni model ograje

Podrobneje so bili modelirani �èitnik, steberin distanènik ograje, medtem ko so bile vijaène zvezemodelirane poenostavljeno. Glede na rezultateprej�njih simulacij in hitrost vozila je bila dol�inamodelirane ograje omejena na 39 m, od tega 5 m ograjepred naletno toèko tovornega vozila in 34 m za njo.Po celotni dol�ini ograje je bilo razporejenih 19stebrov in distanènikov na medsebojni razdalji 2 m.

Glavni sestavni deli cestne varnostne ograjeso bili modelirani z lupinskimi elementi. Stebri indistanèniki so bili modelirani z lupinskimi Belytschko-Tsayevimi konènimi elementi s tremi integracijskimitoèkami po debelini, �èitniki, pasnice in trapeznavodila pa s polnimi integracijskimi lupinskimi elementis petimi integracijskimi toèkami po debelini, da je bilprepreèen pojav niène deformacijske energije.Predpisana debelina konènih elementov je ustrezalakonstrukcijski izvedbi posamezne ograje, kakor jenavedeno v 3. poglavju. Vijaène zveze so bilemodelirane z linijskimi Hughes-Liujevimi konènimielementi. Celoten model ograje je tako sestavljen izpribli�no 100.000 elementov in 110.000 vozli�è.

Materialne lastnosti ploèevin razliènihdebelin so bile doloèene s preizkusi. Glede nadobljene rezultate je bil izbran bilinearen izotropenelastoplastièni model s kinematiènim utrjevanjem(preglednica 1). Poru�itev materiala je bila predpisanaz dejansko plastièno deformacijo, ki je zna�ala 0,28.

Pri vijaènih zvezah z vijaki M10 je bilupo�tevan trdnostni razred 5.8 z mejo plastiènosti400 MPa in natezno trdnostjo 500 MPa, pri vijaènihzvezah z vijaki M16 pa so bile uporabljene vrednosti,navedene v preglednici 2. Te vrednosti so bile

Preglednica 1. Materialne lastnosti ploèevin varnostne ograjeTable 1. Material properties of sheet-metal thicknesses

d E í ys tE us

mm GPa MPa MPa MPa 2,0 2,5 2,8 3,0

190 0,29 285 696 400

4,0 5,0 7,0

200 0,29 330 969 450



of the experimental measurements and the computationalsimulation of the M16 bolt connections. This explainsthe very high ultimate effective plastic deformation, ue ,which actually accounts for the bolt and surroundingmaterial deformation.

4.3 Initial and boundary conditions

The initial and boundary conditions for thevehicle are prescribed by the SIST EN 1317 standard.The impacting vehicle for the TB42 test is positionedat an angle of 15° to the safety barrier, and the initialvelocity is equal to 70 km/h.

Continuation of the guardrail and the barrierreinforcement was modeled with spring elements (No.1 in Fig. 5) with a linear elastic characteristic and anappropriate stiffness. The soil influence was simulatedusing spring elements with an elasto-viscoplasticcharacteristic varying with depth, which was obtainedfrom previous parametric simulations [9].

Four different contact definitions were used inthe model: between the safety-barrier parts, the contactbetween the vehicle parts, the contact between theimpacting part of the vehicle and the barrier, and thecontact between the wheels of the vehicle and theground. The static and dynamic frictions in the firstthree contact definitions were set to m

s = 0.2 and m

d =

0.15, respectively. In the fourth contact definition, bothfriction coefficients were set to 0.3.

dobljene s primerjavo med preizkusi in numeriènimisimulacijami vijaènih zvez z vijaki M16. Zaradi tegaje dopustna dejanska plastièna deformacija ue takovelika, saj zajema deformacijo celotne vijaène zvezein okoli�kega materiala.

4.3 Zaèetni in robni pogoji

Robni in zaèetni pogoji za tovorno vozilo sopredpisani s standardom SIST EN 1317. Vozilo pritestu TB42 je postavljeno pod kotom 15° glede nacestno varnostno ograjo s predpisano zaèetnohitrostjo, ki zna�a 70 km/h.

Nadaljevanje �èitnika in ojaèitev ograje je bilasimulirana z vzmetnimi elementi (1 na sl. 5) ustreznetogosti in elastièno linearno znaèilnico. Vpliv zemljineje bil simuliran z vzmetnimi elementi elasto-viskoplastiènih lastnosti spremenljivih po globini,ki je bila doloèena s predhodnimi parametriènimisimulacijami [9].

V modelu so bili predpisani �tirje loèenistiki: med posameznimi deli ograje, medposameznimi deli tovornega vozila, med desnimbokom tovornega vozila in ograjo ter med kolesivozila in podlago. V prvih treh stikih je bilavrednost statiènega koeficienta trenja m

s = 0,2,

dinamiènega koeficienta trenja pa md = 0,15. V

zadnjem stiku pa je bila za oba koeficienta trenjauporabljena vrednost 0,3.

Preglednica 2. Materialni podatki za vijaène zveze z vijaki M16Table 2. Material properties for the M16 bolt connections

E í ys tE ue

GPa / MPa MPa %

190 0,29 240 0 170

Sl. 5. Robni pogoji za varnostno ograjo s stebri, zabitimi v zemljinoFig. 5. Boundary conditions for a safety barrier with posts rammed into the soil

1

2



4.4 Computational simulation properties

The computational simulation of a vehicleimpact into a road-safety barrier was performed usingthe dynamic explicit software code LS-DYNA MPPVersion 970. All the simulations were done on a PCcluster with eight Pentium-IV 3.2-GHz processorswith hyper-threading technology and a Linuxoperating system. The simulations were done forthe time interval of 1.6 s after the first contact betweenthe vehicle and the barrier. The time-step incrementwas computed with regard to the highest first-resonant frequency of the model, and was equal to1.7 ms. One simulation run on the reportedconfiguration took from 100 to 110 hours.

5 ANALYSIS OF THE COMPUTATIONAL RESULTS

Due to the similarity of the vehicle behavior fordifferent sheet-metal thicknesses of the tension beltand the wheel guidance only one vehicle-impactsequence is shown in Fig. 6. The results of all thesimulations prove that all safety-barrier designs redirectthe impacting vehicle back on to the road. The maximumsystem deformation is observed at time t = 0.8 s. Thevehicle separates from the barrier at time t = 1.5 s.

Before calculating the ASI parameter all themeasured vehicle-mass acceleration data was filteredusing a CFC 180 filter (the minimum measurementfrequency equals 1,800 Hz) according to SIST EN 1317.

4.4 Parametri raèunalni�ke simulacije

Analiza naleta vozila v cestno varnostnoograjo je bila simulirana s programskim paketom LS-Dyna Version MPP 970. Vse predstavljene analizeso bile izvajane na sistemu z osmimi procesorji IntelPentium IV 3,2GH s tehnologijo prepleta podoperacijskim sistemom Linux. Raèunalni�kesimulacije so bile izvedene v èasovnem korakut = 1,6 s, takoj po prvem stiku med vozilom in ograjo.Èasovni korak analize je bil samodejno izraèunanglede na najvi�jo prvo resonanèno frekvencocelotnega modela in je zna�al 1,7 ms. Izvajanje enesame simulacije je na opisanem sistemu trajalo med100 in 110 urami.

5 ANALIZA NUMERIÈNIH REZULTATOV

Razlike med poteki trka parametriènih analizrazliènih debelin pasnic in vodila so zanemarljive,zato je potek trka prikazan le za eno simulacijo (sl. 6).Rezultati vseh analiz ka�ejo, da vse konstrukcijskerazlièice cestne varnostne ograje vozilo uspe�nopreusmerijo nazaj na vozi�èe. Najveèja deformacijavarnostne ograje je dose�ena v èasu t = 0,8 s. Vozilose loèi od ograje v èasu t = 1,5 s.

Pred izraèunom èasovnega poteka parametraMVP so bili vsi zbrani pospe�ki filtrirani z uporabofiltra CFC 180 (najmanj�a frekvenca zbiranja podatkovzna�a 1800 Hz) kakor zahteva standard SIST EN 1317.

Sl. 6. Potek naleta tovornega vozila ob cestnovarnostno ograjo s pasnico

Fig. 6. The truck impact into a safety barrier

reinforced with a tension belt

Sl. 7. Delovna �irina (W)Fig. 7. Working width (W)



0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90

Èas [s]

AS

I

2,0 mm

2,5 mm

2,8 mm

3,0 mm

0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90

Èas [s]

AS

I

3mm

5mm

7mm

The time dependency of the ASI parameterfor the road-safety design with a tension belt is givenin Fig. 8, and for a wheel guidance in Fig. 9. Themaximum values of the ASI are presented in Table 3.In both time dependencies of the AIS parameter twodistinct peaks can be observed. One immediatelyafter the time t = 0.0 s, when the vehicle impacts thebarrier, and the second one at t = 0.2 s, when thewheel hits the barrier post. A comparison of bothfigures reveals that the average and maximum valuesof the ASI are higher for the barrier with the wheelguidance. The reason is in the greater length of theunfolded profile of the wheel guidance, which equalsl = 285.0 mm, and only l = 75.4 mm for the tensionbelt. The stiffness is additionally increased due tothe direct connection of the wheel guidance to theposts, while the tension belt is connected to thedistance spacer.

The comparison of the ASI parameter for thesame reinforcements of different thicknesses shows

Potek parametra MVP v odvisnosti od èasapri analizah cestne varnostne ograje s pasnico jeprikazan na sliki 8, pri ograji z vodilom pa na sliki 9.Najveèje dose�ene vrednosti parametra MVP sopodane v preglednici 3. Pri obeh èasovnih potekihparametra MVP sta opazna dva izrazitej�a vrha. Edentakoj v èasu t = 0,0 s, ko pride tovorno vozilo prviè vstik z ograjo in drugi v èasu t = 0,2 s, ko kolo vozilazadene steber varnostne ograje. Primerjava obeh slikpove, da so povpreène in najveèje vrednostiparametra MVP pri varnostni ograji s trapeznimvodilom veèje. Razlog za veèjo togost je veèja dol�inarazvitega profila pri trapeznem vodilu, ki zna�a l =285,0 mm in le 75,4 mm pri pasnici. Togost cestnevarnostne ograje z vodilom je dodatno poveèana �ezaradi neposredne pritrditve trapeznega vodila nastebre, medtem ko je pasnica privijaèena nadistanènik.

Primerjava poteka parametra MVP medenakimi ojaèitvami razliènih debelin poka�e manj�e

Sl. 8. Potek parametra MVP pri ograji s pasnicoFig. 8. ASI parameter for the barrier with the tension belt

Sl. 9. Potek parametra MVP pri ograji z vodilomFig. 9. ASI parameter for the barrier with the wheel guidance

MV

P /

AS

IM

VP

/ A

SI



smaller differences. The maximum values of the ASIfor the safety barrier with a tension belt are the samefor all thicknesses, while the average values are lowerfor the thinner sheet metal, as expected. The highestvalue of ASI for the barrier with the wheel guidanceis determined for the sheet-metal thickness d = 2.8mm, while the average value of ASI is the lowest atthe same time. Higher average values are recordedfor metal thicknesses d = 2.0 mm and d = 3.0 mm, butwith a lower maximum value.

The differences in working-width (Fig. 7)values for different sheet-metal thicknesses arenegligible for the barriers reinforced with the tensionbelt, while they are more pronounced for the barriersreinforced with the wheel guidance. A lower workingwidth is expected for the reinforcements with thickersheet metal, but the results show otherwise. A moreprecise analysis of the impact shows that this effectcan be attributed to the difference in the distance-spacer deformation and the barrier movement duringimpact. Comparisons of the working widths betweenboth reinforcements show smaller deformations forthe barrier reinforced with a wheel guidance. This isconsistent with previous findings.

6 CONCLUSION

Computational simulations based on aparametric, nonlinear, dynamic finite-element analysis ofa vehicle impact into a road-safety barrier were employedto evaluate different road-safety barrier designs for thecontainment level H1. The results of the parametricanalyses showed that the barrier design reinforced witha tension belt is the appropriate choice when consideringan impact severity. Because all the barrier designs with atension belt successfully redirect the vehicle back ontothe road the thinnest tension belt of thickness 3.0 mm isthe best choice in terms of cost.

razlike. Pri ograji s pasnico so najveèje dose�enevrednosti parametra MVP pri ploèevinah razliènihdebelin enake, povpreène vrednosti pa prièakovanopoka�ejo manj�e vrednosti pri tanj�i ploèevini. Priograji z vodilom je najveèja vrednost parametra MVPdose�ena pri debelini ploèevine d = 2,8 mm, kljubtemu pa je pri tej debelini povpreèna vrednostparametra MVP najmanj�a. Veèje povpreènevrednosti imata vodili debeline d = 2,0 mm in d = 3,0mm, vendar imata manj�o najveèjo vrednost.

Razlike med vrednostmi delovnih �irin (sl. 7)pri razliènih debelinah ploèevine cestne varnostneograje ojaèane s pasnico, so zanemarljive, medtemko so razlike pri cestni varnostni ograji s trapeznimvodilom nekoliko veèje. Prièakovali bi manj�odeformacijo ograje pri debelej�i ploèevini vodila, aizmerjeni rezultati ka�ejo drugaèe. Natanènej�aanaliza poteka trka poka�e, da je veèji del te razlikeposledica razliène deformacije distanènikov inpremikanja varnostne ograje med trkom. Primerjavavrednosti delovne �irine med posameznimakonstrukcijskima re�itvama pa poka�e manj�edeformacije pri ograji z vodilom. To se sklada zugotovitvijo o veèji togosti te ograje.

6 SKLEP

Za oceno razliènih konstrukcij varnostneograje za raven zadr�evanja H1 so bile izvedeneraèunalni�ke simulacije naleta tovornega vozila vcestno varnostno ograjo na podlagi parametriènihnelinearnih dinamiènih analiz po metodi konènihelementov. Rezultati parametriènih analiz ka�ejo, da jez vidika moèi udarca primernej�a cestna varnostnaograja s pasnico. Ker vse konstrukcijske re�itvevarnostne ograje s pasnico uspe�no zadr�ijo vozilona cesti�èu, je s cenovnega vidika najzanimivej�a izbiranajtanj�e pasnice debeline 3,0 mm.

Preglednica 3. Rezultati parametriènih simulacij cestnih varnostnih ograjTable 3. Results of parametric simulations of road-safety barriers

d maks MVP max ASI

Delovna �irina Working width

Razred delovne �irine Working width class

Ojaèitev Reinforcement

mm mm 3 17,5 1867 W6 5 20,0 1820 W6

Pasnica Tension belt

7 19,3 1834 W6 2,0 18,1 1519 W5 2,5 18,3 1595 W5 2,8 21,0 1682 W5

Trapezno vodilo Wheel guidance

3,0 13,9 1695 W5



Konèna ocena o primernosti cestnevarnostne ograje za raven zadr�evanja H1, pa terjatudi upo�tevanje delovne �irine in preizkus cestnevarnostne ograje z osebnim vozilom. Rezultati tehtestov ka�ejo precej manj�e vrednosti moèi udarcapri varnostni ograji s trapeznim vodilom. Pri varnostniograji s pasnico namreè pride do neposrednegaudarca kolesa osebnega vozila v steber varnostneograje, kar se ka�e na veèjih pojemkih, ki zaradi tegaprese�ejo s standardom SIST EN 1317 predpisanemejne vrednosti. Tako lahko povzamemo, da je cestnavarnostna ograja okrepljena s trapeznim vodilomnajprimernej�a konstrukcija varnostne ograje, kiizpolnjuje zahteve za raven zadr�evanja H1.

Glede na rezultate simulacij je najprimernej�acestna varnostna ograja z vodilom iz ploèevinedebeline 2,0 mm, saj pomeni najbolj�o poravnavomed povpreèno in najveèjo vrednostjo parametraMVP ter ima hkrati tudi najmanj�o delovno �irino.

For the final evaluation of the road-safetybarrier�s suitability for the containment level H1, theworking width and the impact of a personal vehiclealso had to be taken into account. The results of thesecomputations show lower values of impact severityfor the barrier with a wheel-guidance profile. In thecase of the barrier with a tension belt the wheel of thepersonal car impacts directly into the post, and thehigher accelerations that result in large impact-severityparameters exceed the limits defined in the SIST EN1317 standard. It is, therefore, concluded that thesafety-barrier design reinforced with the wheel-guidance profile is the most suitable choice to fulfillthe H1 containment-level requirements.

According to the simulation results, thesafety barrier with the 2.0-mm-thick wheel-guidanceprofile is the most appropriate design, since it offersthe best compromise between average and maximumvalues of ASI and has the smallest working width.


[1] EN 1317-1 do 1317-5 Road restraint system (1998) European Committee for Standardization, Brussels.[2] Livemore Software Technology Corporation (2003) LS-DYNA Keyword User�s Manual. Livemore.[3] Livemore Software Technology Corporation (1998) LS-DYNA Theoretical Manual. Livemore.[4] Tehnièna specifikacija za javne ceste TSC 02 (2003) Varnostne ograje, pogoji in naèini postavitve. Ministrstvo

za promet, Urad za standardizacijo in meroslovje, Direkcija republike Slovenije za ceste, oktober, 2003.[5] Vesenjak, M., Ren Z. (2003) Dinamièna analiza deformiranja cestne varnostne ograje pri naletu vozila.

IAT�03, ZSIT Slovenije � SVM in FS Ljubljana � LAVEK, Ljubljana, pp. 349-357, 2003.[6] Vesenjak, M., Ren Z. (2003) Dinamièna simulacija deformiranja cestne varnostne ograje pri naletu vozila.

Kuhljevi dnevi �03, Slovensko dru�tvo za mehaniko, Ljubljana, pp. 207-216, 2003.[7] Finite element model archive, FHWA/NHTSA National Crash Analysis Center, dostopno na WWW: http:/

/www.ncac.gwu.edu/vml/models.html [3.10.2005].[8] A finite element primer, 3rd Reprint (1992) NAFEMS, Glasgow.[9] Sennah, K., Samaan, M., Elmarakbi, A. (2003) Impact performance of flexible guardrail systems using LS-

DYNA, 4th European LS-DYNA Users Conference 2003, DYNAmore GmbH, Ulm, pp. B-III-35-43.

Naslov avtorjev: Matej Borovin�ekMatej Vesenjakdr. Miran Ulbinprof.dr. Zoran RenUniverza v MariboruFakulteta za strojni�tvoSmetanova 172000 [email protected]@[email protected]@uni-mb.si

Authors� Address: Matej Borovin�ekMatej VesenjakDr. Miran UlbinProf.Dr. Zoran RenUniversity of MariborFaculty of Mechanical Eng.Smetanova 17SI-2000 Maribor, [email protected]@[email protected]@uni-mb.si

Prejeto: 5.10.2005




112 Furlan M. - Rebec R. - Èernigoj A. - Èeliè D. - Èermelj P. - Bolte�ar M.

Vibroakustièno modeliranje alternatorja

Vibro-Acoustic Modelling of an Alternator

Martin Furlan1 - Robert Rebec1 - Andrej Èernigoj1 - Damjan Èeliè2 - Primo� Èermelj2 - Miha Bolte�ar2

(1ISKRA Avtoelektrika, �empeter pri Gorici; 2Fakulteta za strojni�tvo, Ljubljana)

V prispevku so predstavljeni osnovni prijemi pri izdelavi sklopljenega modela alternatorja zaovrednotenje vibroakustiènega odziva kot posledice razliènih vzbujanj alternatorja; vibracij motorja znotranjim zgorevanjem, neuravnote�enosti rotorja ter magnetnih sil. Prikazani sta izdelava in preverjanjestrukturnih trirazse�nih (3D) modelov metode konènih elementov (MKE) za osnovne sestavne dele; posameznepovezane sklope in za alternator kot celoto. Preverjanje strukturnih modelov je izvedeno z eksperimentalnomodalno analizo (EMA), ki je obenem namenjena za ovrednotenje du�enja ter posodobitev strukturnegamodela alternatorja. Natanèneje je opisan poseben primer izraèuna vibroakustiènega odziva kot posledicevzbujanja magnetnih sil. Pri tem je bil za ovrednotenje magnetnih sil izdelan trirazse�ni model MKE zamagnetno polje, tako da je omogoèal harmonsko analizo magnetnih sil in njihov neposreden prenos vstrukturni model MKE alternatorja. Na podlagi poznanega vzbujanja alternatorja, tj. magnetnih sil vdoloèenih obratovalnih razmerah, je bil ovrednoten in analiziran strukturni in akustièni odziv alternatorja.© 2006 Strojni�ki vestnik. Vse pravice pridr�ane.(Kljuène besede: alternatorji, vibracije, metode konènih elementov, metode robnih elementov, analize modalne)

In this paper we present the basic approaches to building a sequentially coupled model of analternator to evaluate the vibro-acoustic response resulting from different excitations, such as enginevibration, rotor unbalance and magnetic forces. Special emphasis is given to the set-up and verification ofthe three-dimensional (3D) structural finite-element (FE) model for the whole alternator and for the alternatorcomponents. The verification of the structural model was carried out using experimental modal analysis(EMA), which was also applied for the estimation of the modal damping and for the structural FE modelupdating. Special emphasis is given to the evaluation of magnetic noise, generated due to the magneticforce excitation in the alternator. To estimate the magnetic forces and their harmonic components a 3Dmagnetic FE model of the alternator was prepared. Finally, the exciting magnetic forces, calculated for thespecific operation conditions of the alternator, were transferred into a structural FE model, where thestructural and acoustic responses were calculated and analysed.© 2006 Journal of Mechanical Engineering. All rights reserved.(Keywords: alternators, vibrations, finite element methods, boundary element methods, modal analysis)

Strojni�ki vestnik - Journal of Mechanical Engineering 52(2006)2, 112-125UDK - UDC 534.83:621.43.044

519.61/.64Izvirni znanstveni èlanek - Original scientific paper (1.01)

0 INTRODUCTION

The optimisation of an alternator with aninternal ventilator was the major driving force forsetting up a coupled numerical model of the alternatorintended to evaluate its vibro-acoustic response.Additionally, this model will be used in thedevelopment process of new designs of alternators,helping to improve their technical characteristics andmake them comparable or even better than theircompetitors. In the case of the alternator,

0 UVOD

Osnovni cilj izdelave sklopljeneganumeriènega modela alternatorja za ovrednotenjevibroakustiènega odziva sta optimizacija in razvojnovih konstrukcij alternatorjev z notranjimventilatorjem, ki naj po tehniènih znaèilnostihdosegajo ali celo presegajo primerljivekonkurenène izdelke. Slednje pomeni, da prialternatorju dose�emo zadovoljiv odvod toploteoz. poveèamo izkoristek pri majhni stopnji


113Vibroakustièno modeliranje alternatorja - Vibro-Acoustic Modelling of an Alternator

improvements could be achieved with better heattransfer, which corresponds to higher efficiency,where the aerodynamic noise of the ventilators needsto be kept under control. Besides this, the alternator�sdurability should not be threatened by either anincrease in the alternator�s vibration and/or acousticresponse. Optimising the design of the alternatorfrom the thermal and structural dynamic points ofview is not enough.

When achieving the above-mentioned goal onecan meet many limitations and contradictory demandsthat make the search for the optimum design of thealternator very difficult. At this point numericalsimulations can be applied as a very effective tool tohelp the engineer. The recent development of computertechnology and numerical methods have also made itpossible to simulate very demanding multidisciplinaryproblems such as the generation of the magnetic noisein a reasonable time. The major advantage of thecomputer technology and numerical methods is thepossibility to make the optimisation and the evaluationin a virtual environment. This significantly decreasesthe number of prototypes and the quantity of theexperimental work and finally shortens the developmenttime and increases the quality of the product.

1 A COUPLED VIBRO-ACOUSTIC MODEL OFTHE ALTERNATOR

In general, the vibro-acoustic model of thealternator unites three physical problems, i.e., themagnetic, the structural dynamic and the acoustic.With regard to the physical nature of the alternator,

aerodinamiènega hrupa ventilatorja ter pri tem neogrozimo dobe trajanja alternatorja ali poveèamonjegovega vibracijskega in akustiènega odziva.Enako pomembna je optimizacija konstrukcijealternatorja z vidika magnetike, ki prav takoomogoèa izbolj�anje izkoristka ter zmanj�anjemagnetnega hrupa alternatorja, povzroèenega zmagnetnimi silami, ki vzbujajo strukturoalternatorja.

Pri doseganju omenjenega ci l ja sesreèujemo z izkljuèujoèimi se zahtevami inomejitvami, ki iskanje optimalne re�itve zeloote�ujejo. Pri iskanju takih re�itev so v velikopomoè raèunalni�ka tehnologija in numeriènemetode. Predvsem razmah raèunalni�ketehnologije v zadnjem èasu je omogoèil, da sezahtevni veèdisciplinarni problemi, npr.nastajanje hrupa, v doglednem èasu re�ijo znumeriènimi simulacijami. Bistvena prednost, kijo omogoèajo numeriène simulacije, je mo�nostuporabe optimizacije, kar zmanj�a �teviloprototipov in posledièno kolièinoeksperimentalnega dela. Skupno se to ka�e vkraj�em razvojnem èasu in veèji kakovostiizdelka.

1 SKLOPLJENI VIBROAKUSTIÈNI MODELALTERNATORJA

V splo�nem vibroakustièni model alternatorjazdru�uje tri fizikalne probleme, tj. magnetni, strukturniin akustièni problem, in njim pripadajoèe modele, kijih lahko predstavimo kot celovit � zaporedno

Sl. 1. Zaporedno povezani magnetni, strukturni in akustièni problem alternatorjaFig. 1. The sequentially coupled magnetic, structural dynamic and acoustic problem of the alternator

vibracijevibrations

Mag. polje - MKEMag. field - FEM

Meh. str. - MKEMech. str. - FEM

Zvoèno polje - MRE

Sound field - BEM

magnetne silemagnetic forces

zvoksound

obratovalni pogoji: alternator, obrem., pritrditev, vrt. hitrost, idr.operating conditions: alternator, load, support, rot. speed, etc.



each problem can be treated independently, and thenthey can all be linked into one sequentially coupledproblem, Figure 1.

To validate the presented model of thealternator it is necessary to verify it with real cases.For this reason we decided to set-up the vibro-acoustic model for the existing alternator, Figure 2.

The magnetic forces and their distribution,acting on the exterior surfaces of the rotor andstators, are calculated using the finite-elementmethod (FEM) for the magnetic field. In this case itis assumed that a consequence of the stationaryrotation of the rotor is a periodic varying of themagnetic forces caused by changes of the magneticflux densities in the air gap of the alternator.

To simplify the solution of the magnetic problemof the alternator it was necessary to make a fewassumptions. The major one relates to the statorexcitation. Here we assumed that the distribution of themagnetic forces is load independent. By assuming this,we avoid having to take into account the stator currents,so the rotor current was considered as the only magneticexcitation in the magnetic model. The distribution of themagnetic forces was calculated using a magneto-staticmodel for different rotor positions which describes thechanging of the magnetic forces during the rotation andfinally allows a calculation of their harmonic components.

The results of the changing magnetic forcesduring the rotor rotation, calculated using FEM forthe magnetic field, are used as the excitation forcesin the structural FEM model. The basic job in theprocess of the evaluation of the structural dynamicresponse is to build and to verify the structural FEMmodel. The verification of the structural FEM modelwas done using experimental modal analysis (EMA)

sklopljeni problem. Znaèilnost takega problema je,da ga lahko obravnavamo v treh loèenih zaporednihkorakih (sl. 1).

Da bi potrdili verodostojnost takopostavljenega modela, je potrebno preverjanje nadejanskih modelih. Zaradi omenjenega smo se prigradnji sklopljenega modela alternatorja odloèili, daga izvedemo za �e obstojeèi alternator (sl. 2).

Z uporabo MKE za magnetno poljeizraèunamo velikosti in porazdelitev magnetnih sil,ki delujejo na zunanje povr�ine rotorja in statorjaalternatorja. Ker vrtenje rotorja vpliva naspreminjanje gostote magnetnega pretoka v zraènire�i alternatorja, ima to za posledico periodièenznaèaj magnetnih sil.

Da bi tako zahteven model magnetnega poljalahko re�ili, smo predpostavili, da je porazdelitevmagnetnih sil neodvisna od same obremenitvealternatorja. To pomeni, da v magnetni model nismovnesli statorskih tokov, ampak le nespremenljivrotorski tok, ki pomeni magnetno vzbujanjealternatorja. Porazdelitev magnetnih sil smo izraèunaliz magnetostatiènim modelom za razliène medsebojnelege rotorja in statorja. Tako dobimo potekspreminjanja magnetnih sil pri vrtenju rotorjaalternatorja, na katerem izvedemo harmonskorazdru�itev.

Rezultate spremenljivih magnetnih sil privrtenju rotorja, izraèunanih z MKE za magnetno polje,smo uporabili kot vzbujevalne sile v strukturnemmodelu po MKE. Temeljna naloga pri doloèitvistrukturnega odziva sta bili gradnja in ovrednotenjestrukturnega modela po MKE. Strukturni model poMKE smo preverili z eksperimentalno modalnoanalizo (EMA), ki smo jo izvedli za razliène sklope in

Sl. 2. Obravnavani alternatorFig. 2. The investigated alternator



on every single part of the alternator and itssubstructures as well as on the whole structure ofthe alternator. The evaluation of the structuralresponse of the alternator FEM model, as aconsequence of the magnetic force excitation, isconsidered as a harmonic problem.

The last step in the process of the vibro-acoustic modelling of the alternator deals with thesound field as a result of the structural response,i.e., vibrations on the alternator structure. The soundfield, governing by the Helmholtz wave equation, isevaluated by the boundary-element method (BEM).Due to the numerical extensivenesses of thementioned problem [1], we limited the simulations ofthe sound field to only a few wave numbers ordiscrete frequencies. Only the frequencies with asignificant contribution to the magnetic noise or afew first harmonics of the magnetic forces are takeninto account for the sound-field calculation.

2 MAGNETIC FORCES IN THE ALTERNATOR

The evaluation of the magnetic forces wascarried out using the FEM. Because of the claw poleshape of the rotor it was necessary to build up athree-dimensional (3D) FEM model for the magneticfield. For that reason a two-dimensional FEM modelcannot be used. As a result of the cyclic symmetryof the rotor with six pole pairs, only one sixth of thecomplete alternator was modelled, Figure 3.

Taking into account the cyclic symmetry of

podsklope alternatorja ter za alternator v celoti.Ovrednotenje strukturnega odziva modela po MKEalternatorja na vzbujanje z magnetnimi silami smoobravnavali kot problem harmonskega vzbujanja, prièemer je vzbujanje podano s harmonskimikomponentami magnetnih sil.

Zadnji korak v postopku vibroakustiènegamodeliranja alternatorja obravnava zvoèno polje kotposledico strukturnega odziva. Zvoèno polje, ki jere�itev Helmholtzove valovne enaèbe, smoobravnavali z metodo robnih elementov (MRE).Glede na numerièno zahtevnost omenjenegaproblema [1] smo numeriène simulacije izvedli le zaomejeno �tevilo diskretnih vrednosti valovnih�tevil oz. frekvenc. Pri tem smo se omejili nafrekvence z najveèjim prispevkom k magnetnemuhrupu oz. na frekvence prvih nekaj harmonskihkomponent, ki nastanejo zaradi harmonskerazdru�itve magnetnih sil.

2 MAGNETNE SILE V ALTERNATORJU

Ovrednotenje magnetnih sil v alternatorjusmo izvedli z uporabo MKE. Zaradi oblike rotorja oz.rotorskih krempljastih polov smo moraliovrednotenje sil izvesti na trirazse�nem modelu, sajje omenjeno geometrijsko lastnost neposrednonemogoèe opisati z dvorazse�nim modelom. Gledena simetrièno obliko rotorja, ki ima 6 polovih parov,smo zgradili model, ki obsega le eno �estinocelotnega alternatorja (sl. 3).

Sl. 3. Pogled v trirazse�ni model po MKE alternatorja za magnetno polje (levo) in celoten alternator (desno)Fig. 3. The 3D FEM magnetic model of the alternator (left) and the complete alternator (right)



1020

3035

1

5

10

15

0

0.2

0.4

0.6

0.8

z [mm]h

FR

, h [

N]

the rotor, the magnetic forces calculation wasperformed in evenly distributed steps in a rotationover 60 degrees. The step size was chosen on thebasis of distinguishing the harmonic componentsthat arise from the discrete Fourier transform. Basedon previous practice and expertise in the modellingof electric machine noise ([1], [5] and [6]), where theinfluence of higher harmonics decrease and theirfrequencies range out of vibro-acoustic interest, forthe evaluation of magnetic forces only the first 12harmonics were chosen. From those facts it followsthat at least 24 steps must be executed to fulfil theNyquist criterion. In our case the magnetic forcescalculations during the rotation was performed insteps of 1 degree. This enables us to evaluate thehigher harmonics that contribute to a precisepresentation of the magnetic forces. Figure 4 showsthe harmonic decomposition of the magnetic forcesacting on nodes along the centre of the stator tooth.

Besides the influence of rotation on themagnetic forces, the evaluation of the operatingconditions of the alternator is also important. For thesake of simplicity we have studied only the unloadedalternator so there is no magnetic excitation on thestator side. Regarding the physical nature of thealternator, the operating conditions were defined withcurrent in the rotor winding. Finally, the magneticforces calculation was performed for different rotorcurrents. Knowing that under normal operatingconditions the rotor current is generally between 2Aand 4.2A, the appropriate rotor current wasdetermined.

Zaradi simetrije alternatorja smo vrtenjeizvedli za 60° v enakomerno razporejenih korakih. Priizbiri velikosti koraka smo se odloèili na podlagipotrebe po razloèevanju harmonskih komponent, kinastanejo pri razvoju magnetnih sil v Fourierjevovrsto. Glede na predhodne izku�nje in poznavanjekarakterja magnetnega hrupa ([1], [5] in [6]), prikaterem se vpliv vi�jih harmonskih komponentzmanj�uje in njihove frekvence segajo v frekvenènapodroèja, ki so glede vibroakustiènega odziva manjzanimiva, smo za ovrednotenje izbrali le prvih 12harmonskih komponent. Iz omenjenega izhaja, da jetreba izpeljati izraèun magnetnih sil za vsaj 24 korakov,da izpolnimo Nyquistov pogoj. V na�em primeru sobili izraèuni sil izvedeni za vrtenje v 60 korakih po 1°.To omogoèa, da ovrednotimo tudi vi�je harmonskekomponente, kar pripomore k natanènej�emuovrednotenju magnetnih sil. Slika 4 prikazujeharmonsko analizo za magnetne sile, ki delujejo vvozli�èih vzdol� sredine statorskega zoba.

Poleg vpliva vrtenja na magnetne sile je bilotreba ovrednotiti tudi vpliv razmer pri obratovanju.Zaradi zahtevnosti smo se na tem mestu omejili naobratovanje neobremenjenega alternatorja, ko nistatorskega vzbujanja. Glede na naravo delovanjaobravnavanega alternatorja smo razmere obratovanjaopredelili s tokom v rotorskem navitju. Tako smoovrednotenje magnetnih sil pri vrtenju rotorja izvedliza razliène rotorske tokove. Pri izbiri velikosti tokovsmo izhajali iz poznavanja obratovalnih razmeralternatorja in upo�tevali dejstvo, da se v obièajnihrazmerah vrednosti tokov gibljejo med 2A in 4,2A.

Sl. 4. Harmonske komponente magnetnih sil vzdol� statorskega zobaFig. 4. Harmonic components of the magnetic forces along the centre of stator tooth



The transfer of magnetic forces from themagnetic FEM model to the structural FEM modelcan be very difficult to carry out because of therebeing one way to solve the magnetic model andanother way to solve the structural model. A dissimilarrequirement for the mesh density of the magneticand structural problems is the reason why the samemesh cannot be used for the 3D FEM coupledmagneto-structural problem. The mesh densityrequired in the 3D FEM magnetic problem is toodense for solving 3D FEM structural problem. Putanother way, the mesh density of the 3D FEMstructural problem is too coarse to be used in the 3DFEM magnetic problem.

Observing the character of the magneticforces in terms of time and space distribution overthe surface of the stator and the rotor claw pole it isimpossible to entirely substitute their effect with theresulting magnetic forces and moments ([1], [5] and[6]). The latter is assumed for the affect of themagnetic forces acting on the rotor claw poles, whichare a very rigid structure, and because of theirsymmetry the magnetic forces are always incounterbalance. In contrast to the credible transferof magnetic forces, which excite the �elastic� stator,the preservation of their space distribution isrequired. Meanwhile, their time distribution ischaracterized with harmonic components.

3 STRUCTURAL RESPONSE OF THE ALTERNATOR

In the second phase of the vibro-acousticmodelling of the alternator it is necessary to determineits structural response as a consequence of theexcitation of the alternator, i.e., vibrations due to theoperation of an internal combustion engine, rotorunbalance or magnetic forces. Great stress is laid onbuilding a structural FEM model, which is introducedhere. The model checking and the final verification ofthe structural response are described with theassumption that only the response as a consequenceof the magnetic forces excitation was taken into account.With a minimum of modal updating it would also bepossible to describe the response due to vibrations ofan internal combustion engine and rotor unbalance.

The steady state operation of the alternatorwas assumed during the verification of the structuralresponse. The magnetic forces, which excite thestructure, are in this case periodic. Taking intoconsideration the mechanical structure as a low-damped linear system, our problem can be

Prenos magnetnih sil iz modela po MKE zamagnetiko v strukturni model po MKE je lahko zelozahteven predvsem zaradi razliènih postopkov re�evanjamagnetnega problema na eni in strukturnega problemana drugi strani. Razliène zahteve po diskretizaciji priobravnavi magnetike na eni strani in mehanskestrukture na drugi strani onemogoèajo uporabo enotnediskretizacije trirazse�nega modela po MKE zare�evanje sklopljenega magnetomehanskegaproblema. Gostota mre�e trirazse�nega modela poMKE, ki jo zahteva obravnava problema magnetike, jenamreè preveliko breme pri re�evanju strukturnegaproblema. Nasprotno lahko reèemo, da je diskretizacijastrukturnega modela neprimerna ali celo neuporabnaza re�evanje magnetike.

Glede na znaèaj magnetnih sil, predvsem vsmislu krajevne in èasovne porazdelitve na povr�inistatorja in krempljastih polov rotorja, je njihov uèineknemogoèe v celoti nadomestiti z rezultirajoèimi silamiin momenti ([1], [5] in [6]). Slednje smo predpostavilile za uèinek magnetnih sil, ki delujejo na krempljastepole rotorja in so znotraj zelo toge strukturekrempljastih polov zaradi simetrije vedno vravnote�ju. Nasprotno je za resnièen prenosmagnetnih sil, ki vzbujajo �elastièen� stator, potrebnoohraniti njihov krajevni znaèaj. Èasovni znaèajmagnetnih sil pa je opredeljen s harmonskimikomponentami.

3 STRUKTURNI ODZIV ALTERNATORJA

V drugi fazi vibroakustiènega modeliranjaalternatorja je treba doloèiti strukturni odziv kotposledico vzbujanja alternatorja, tj. vibracij motorjaz notranjim zgorevanjem, neuravnote�enosti rotorjaali magnetnih sil. Predstavljena je gradnjastrukturnega modela po MKE, èemur je namenjennajveèji poudarek. Opisano je preverjanje modela terkonèno ovrednotenje strukturnega odziva, pri èemersmo se omejili le na odziv kot posledico vzbujanjamagnetnih sil. Z najmanj�imi dopolnitvamistrukturnega modela je mogoèe priti tudi do odzivakot posledice vibracij motorja z notranjimzgorevanjem ali neuravnote�enosti rotorja.

Pri ovrednotenju strukturnega odziva seomejimo na ustaljeno delovanje alternatorja.Magnetne sile, ki v takem naèinu obratovanjavzbujajo strukturo alternatorja, imajo periodièenznaèaj. Ob predpostavki, da je mehanska strukturalinearen sistem z majhnim du�enjem, lahkoobravnavani problem prevedemo na problem



transformed into a forced vibration problem for thewhole spectrum of excitation, magnetic-forceharmonics. Finally, from the known response of theparticular harmonic components, the entire structuralresponse can be reconstructed.

For the verification of the structural responseof the selected alternator, a FEM was used. Thestructural FEM model had to be built and verified ona real structure first. The modelling of the wholestructure was done using a step-by-step procedure.Each part of the alternator was taken apart, modelledby means of the FEM and verified. Combining thoseFEM sub-models together was the most difficult stepin the structural FEM building up. From the necessityof the verification, various measurements of thestructural response, not only on subsystems butalso on assemblies, were carried out usingexperimental modal analysis (EMA). This wasnecessary for the verification of the structural FEMmodel, as well as for the verification of the dampingon different assemblies of the alternator.

3.1 Structural model of the alternator

With regard to the complexity of thealternator�s structure the structural FEM model wasbuilt up gradually. In this case we used alternatorsubpart definitions as they are defined in production,so the structure was divided into four basic subparts:the drive-end bracket (DEB), the rear-end bracket(REB), the stator and the rotor (Figure 5). Due to thegreat complexity of the subpart FEM models, someof them had to be built gradually, as the design soallowed. In this way we proceeded from theconstruction of very basic element FEM models, thecharacteristics of which were then experimentallyverified. The main FEM model was progressivelybuilt and experimentally verified at every couplingstep. We can say that we followed the technologicalproduction process during the building up of theFEM model of the alternator. This kind of processrequired many different prototypes and types ofexperimental work.

The final action in the structural FEM modelconstruction of the investigated alternator is thecoupling of the previously made subassemblies intoan integrity. To design the structural FEM modelconnections as realistically as possible, we shouldknow them very well. Regarding the nature of theconnection that occurs between the DEB, the REB,the stator and the rotor, we applied three different

vsiljenega nihanja za celoten spekter harmonskihkomponent magnetnih vzbujevalnih sil. Nazadnje izznanega odziva posameznih harmonskih komponentobnovimo celotni strukturni odziv.

V na�em primeru smo za ovrednotenjestrukturnega odziva na izbranem alternatorju uporabiliMKE. Prvotno je bilo treba strukturni model po MKEzgraditi ter preveriti njegovo skladnost na dejanskiizvedbi. Gradnje strukturnega modela smo se lotilikorakoma. Strukturo alternatorja smo obravnavali podelih, ki smo jih na koncu zdru�ili v celoto. Za vsakposamezni del alternatorja smo izdelali model po MKEin ga tudi preverili. Sklepno dejanje pri gradnjistrukturnega modela po MKE je bilo povezovanjeposameznih delov v celoto. Za potrebe ovrednotenjasmo izvedli razliène meritve strukturnega odziva, takoposameznih delov kakor celotnega alternatorja, prièemer smo uporabili eksperimentalno modalno analizo(EMA). To je poleg ovrednotenja strukturnega modelapo MKE omogoèilo tudi ovrednotenje du�enja zaposamezne sklope alternatorja.

3.1 Gradnja strukturnega modela alternatorja

Glede na zapletenost strukture obravnavanegaalternatorja smo strukturni model po MKE gradilipostopoma. Pri tem smo izhajali iz razdelitveobravnavanega alternatorja, ki se ponuja sama po sebi.Celotno strukturo smo razdelili v �tiri sklope: prednjile�ajni pokrov (PLP), zadnji le�ajni pokrov (ZLP),stator in rotor (sl. 5). Ker je �e vsak sklop sam po sebidovolj zahteven, je bilo treba tudi pri gradnjistrukturnega modela po MKE posamezni sklop graditiv veè korakih, kolikor je paè konstrukcija to dopu�èala.Tako smo pri gradnji strukturnega modela MKE zaposamezen sklop izhajali iz osnovnih podsestavov, kijih lahko z MKE preprosto modeliramo in v nadaljevanjus preizkusi preverimo. Postopoma smo model sklopadopolnjevali in ga sproti veèkrat preverili. Lahko reèemo,da smo pri gradnji strukturnega modela in njegovihsklopov sledili tehnolo�kemu postopku izdelavealternatorja. Tak postopek je zahteval veliko �teviloprototipov in preizkusnega dela.

Sklepno dejanje v izgradnji strukturnegamodela po MKE obravnavanega alternatorja jepovezava predhodno izdelanih sklopov v celoto.Zato da bi lahko èimbolj preprièljivo izvedli mehanskepovezave v strukturnem modelu po MKE, moramospoje dobro poznati. Glede na naravo spojev oz.mehanskih povezav, ki se pojavljajo med PLP, ZLP,statorjem in rotorjem, smo delali s tremi razliènimi



mechanical connections. First of all we defined theconnection between the rotor and both end brackets� the DEB and the REB. In our case we calculated thestiffness matrix of the bearings ([1] and [2]) and usedthem in the structural FEM model. The next type ofcoupling was used to connect the alternator as anintegrity and represents four screws between the DEBand the REB. Taking into account that the mentionedconnections couple subparts with shape and force,we modelled them by coupling all the degrees offreedom (DOFs) of the nodes situated in the proximityof the screw joint. A third different kind of connectionswas used to model the contact conditions betweenthe stator and the DEB or REB. As we have decided todescribe the mechanical structure with a linear model,we did not have the option to model the contactconditions as they should be. To ensure the structuralFEM model�s linearity, we used just connections thatcouple the nodal DOFs on the contact surfaces.

mehanskimi povezavami. Najprej smo opredelilipovezavo, ki jo predstavljata le�aja in se pojavlja vstiku med rotorjem in PLP ter ZLP. V na�em primerusmo togosti le�ajev izraèunali ([1] in [2]) ter njunelastnosti uporabili v strukturnem modelu po MKE.Drugo obliko mehanske povezave, ki povezuje PLP,ZLP in stator, ter daje alternatorju znaèaj celote,predstavljajo �tiri vijaène zveze. Glede na to, daobravnavani spoj zdru�uje okrov in prednji pokrov ssilo in obliko, smo ga v strukturnem modelu po MKEizvedli z zdru�itvijo vseh prostostnih stopenj vvozli�èih, ki se pojavljajo na mestu vijaène zveze.Tretja oblika mehanske povezave je stik, ki nastajapri naleganju med statorjem in PLP ter ZLP. Ker smose odloèili, da mehansko strukturo opi�emo zlinearnim modelom, stika nismo mogli modelirati vpravem pomenu. Zato da smo ohranili linearnoststrukturnega modela po MKE, smo ga modelirali tako,da smo spojili pomike na stiènih povr�inah.

Sl. 5. Sklopi alternatorja: PLP (zgornja vrsta), stator (srednja vrsta) in ZLP (spodnja vrsta) ter njimpripadajoèa dejanska struktura (stolpec levo), strukturni model po MKE (stolpec sredina) in model

EMA (stolpec desno)Fig. 5. Alternator subparts; drive-end bracket (upper row), stator (middle row) and rear-end bracket(lower row); and belonging real structure (left column), structural FEM model (middle column) and

EMA model (right column)



The validation of the structural FEM modelwas done by comparing the modal parameters toreal subparts (natural frequencies and mode shapes).As additional criteria, we used the modal assurancecriterion (MAC) and a visual comparison of themodal shapes, which was done by using means ofthe EMA. As a frequently used comparison criterionduring model validation, we must also mention themass of the structural FEM model. If we fulfil all thementioned criteria we can be sure that our modelproperly describes the mass and stiffness propertiesof the real structure. The issue of damping thestructural FEM model was solved by using a 1%constant-damping ratio. As we know, the FEMmodel mesh density has an influence on the accuracyof the calculation. It turns out that every doublingof the elements, irrespective of whether this isperformed on just one subpart or on the wholealternator, leads to differences between the naturalfrequencies and the measured ones. Thisdiscrepancy is less than 1% in the range 0�4 kHz.The structural FEM model of the DEB (Figure 6,Table 1) and the whole alternator (Figure 7, Table 2)are shown as examples.

Ovrednotenje strukturnega modela po MKEsmo izvajali na podlagi primerjave modalnihparametrov (lastnih frekvenc in vektorjev). Kotdodatna kriterija smo uporabili kriterij modalnegazaupanja (KMZ - MAC) ter vidno primerjavomodalnih oblik, ki nam jo omogoèa EMA.Nenazadnje tudi masa strukturnega modela po MKEgovori o resniènosti, zato smo jo vseskozi primerjaliz maso dejanske strukture. Èe so izpolnjeni vsinavedeni kriteriji, imamo zagotovilo, da model dobroopisuje masne in togostne lastnosti dejanskestrukture. Vpra�anje du�enja smo re�ili tako, da smov modelu po MKE uporabili nespremenljivirazmernik strukturnega du�enja en odstotek. Kervemo, da ima gostota diskretizacije strukturnegamodela po MKE vpliv na natanènost izraèuna, smonjen vpliv omejili takole. Vsaka podvojitev �tevilaelementov v modelu, bodisi celotnega alternatorja,sklopa ali podsestava, vodi do razlik v izraèunanihvrednostih lastnih frekvenc v obmoèju do 4 kHz, kipa so manj�e od enega odstotka. V nadaljevanju jekot primer prikazano ovrednotenje strukturnegamodela po MKE za PLP (pregl. 1 in sl. 6) ter za celotnistrukturni model alternatorja (pregl. 2 in sl. 7).

Sl. 6. Primerjava lastnih oblik PLP, strukturni model po MKE (zgoraj), model EMA (spodaj); prva (nalevi), druga (na sredini) in tretja (na desni)

Fig. 6. Comparison of the DEB mode shapes, the structural FEM model (above), the EMA model (below);first mode shape (left), second (middle), third (right)



3.2 Transfer of magnetic forces to a structuralmodel and to the response

Work proceeds with the evaluation of thestructural response of the alternator based on thepreviously built structural FEM model, while the alternatoris mounted on the internal combustion engine. The natureof the excitation forces was determined from the resultsof the magnetic forces verification using the FEM. Inorder to compare measured and calculated data, the

3.2 Prenos magnetnih sil v strukturni modelter odziv

V nadaljevanju je opisano ovrednotenjestrukturnega odziva alternatorja na podlagipredhodno zgrajenega strukturnega modela poMKE, pri èemer je alternator vpet na motor z notranjimzgorevanjem. Pri doloèitvi narave vzbujevalnih silsmo izhajali iz rezultatov, dobljenih pri ovrednotenjumagnetnih sil z MKE. Zato da bi lahko izvedli

Preglednica 1. Izraèunane in izmerjene lastne frekvence za PLPTable 1. Calculated and measured natural frequencies of DBE

MKE EMA Df

n f [Hz] n f [Hz] d [%] [%]

1 913 1 969 0,06 -5,8 2 1367 2 1457 0,04 -6,2 3 2547 3 2742 0,37 -7,1 4 2760 4 3062 0,17 -9,9 5 3107 5 3348 0,06 -7,2 6 3252 6 3693 0,07 -11,9 7 4062 7 3922 0,10 3,6

mMKE [kg] mEMA [kg] Dm [%]

0,829 0,844 -1,8

Sl. 7. Oblike strukturnega modela po MKE alternatorja; prva (levo), druga (sredina) in tretja (desno)Fig. 7. Alternator structural FEM model mode shapes; first mode shape (left), second (middle), third

(right)

Preglednica 2. Izraèunane in izmerjene lastne frekvence za alternator kot celotoTable 2. Calculated and measured natural frequencies of the alternator

MKE EMA Df

n f [Hz] n f [Hz] d [%] [%]

1 412 1 587 2,02 -29,8 2 889 2 908 0,44 -2,1 3 900 3 951 0,69 -5,4

4 1726 4 1272 1,29 35,7



primerjavo med meritvami in izraèunom, je bilo trebadoloèiti jakost magnetnih sil in njihovo osnovnofrekvenco vzbujanja. Pri tem smo se oprli na izmerjenevrednosti toka, ki so doloèale moè magnetnih sil, terna izmerjene vrednosti vrtilnih frekvenc, ki sodoloèale osnovne frekvence vzbujanja.

Ker je odziv strukture alternatorja moènoodvisen od frekvence vzbujanja in zaradi tega odvrtilne frekvence alternatorja, smo osnovnofrekvenco vzbujanja za harmonske komponentemagnetnih sil vzeli iz meritev vrtilne frekvence. Le-tase giblje med 3000 min-1 in 7500 min-1. Tako jefrekvenca osnovne harmonske komponentemagnetnih sil (�est polovih parov na rotorju) vrazliènih razmerah obratovanja nahaja v obmoèju od300 Hz do 750 Hz.

Ker med posameznimi harmonskimikomponentami enakega reda, vendar za razliènesmeri rezultirajoèih magnetnih sil, obstajajo fazniodmiki, je bilo pri prenosu magnetnih sil nastrukturni model treba upo�tevati tudi to. Faznousklajenost smo ohranili tako, da smo harmonskekomponente magnetnih sil v strukturni model poMKE vnesli v kompleksni obliki. Slika 8, na levi,prikazuje strukturni model po MKE statorjaalternatorja s prirejenimi trenutnimi vrednostmimagnetnih sil, medtem ko slika 8, na desni,prikazuje strukturni odziv kot posledicoobratovanja neobremenjenega alternatorja zaradidelovanja pete harmonske komponente magnetnihsil, katere frekvenca pri vrtilni frekvenci 3000 min-1

ustreza 1500 Hz.

intensity and the fundamental excitation frequency ofthe magnetic forces must be determined. This was doneusing the measured values of current, which determinethe intensity of the magnetic forces, and the measuredvalues of the rotational velocity, which determine thefundamental excitation frequencies.

The structural response of the alternator is stronglydependent on the excitation frequency and, consequently,on the rotational velocity of the alternator. For this reason,the fundamental excitation frequency for the harmoniccomponents of the magnetic forces was adopted from themeasurement of the rotational velocity. The latter variesfrom 3000 min-1 to 7500 min-1. So, the frequency of thefundamental harmonic component of magnetic forces (sixpole pairs on the rotor), on various working conditions,occurs in the range between 300 Hz and 750 Hz.

Between some particular harmonic componentsof the same order, but for different directions of theresulting magnetic forces, a phase shift occurs. Duringthe transfer of magnetic forces to structural model thisalso had to be considered. The harmonic componentsof the magnetic forces were entered into the structuralFEM model in a complex form, and a phase adjustmentwas preserved. Figure 8, on the left, shows the structuralFEM model of the stator (of the alternator) witharranged instantaneous values of the magnetic forces,while Figure 8, on the right, depicts the structuralresponse as a consequence of the operation of theunloaded alternator due to the effect of the 5th harmoniccomponent of the magnetic forces. The frequency ofthe 5th harmonic component at rotational velocity of3000 min-1 corresponds to 1500 Hz.

Sl. 8. Magnetne sile na statorju (na levi) ter strukturni odziv modela po MKE alternatorja kotposledice 5-te harmonske komponente magnetnih sil (na desni)

Fig. 8. Magnetic forces on the stator (left) and structural response of the FEM model of the alternatoras a consequence of the 5 th harmonic component of the magnetic forces (right)



4 AKUSTIÈNI ODZIV ALTERNATORJA

Zadnja faza vibroakustiènega modeliranjaalternatorja je bila doloèitev zvoènega polja v njegoviokolici na podlagi predhodnega poznavanjastrukturnega odziva modela MKE alternatorja. Prinumeriènem ovrednotenju, pri katerem z metodorobnih elementov (MRE) na podlagi znanegastrukturnega odziva, izraèunamo zvoèno polje.

Pri ovrednotenju zvoènega polja smo se dr�alienakih omejitev, kakor smo jih navedli priovrednotenju strukturnega odziva. Bistvena jezahteva po ustaljenem obratovanju alternatorja.Strukturni odziv oz. vibracije, ki se pri takemobratovanju pojavijo na zunanjih povr�inahalternatorja, povzroèajo zvoèno valovanje ali ti.magnetni hrup. Ob predpostavki, da mehanskostrukturo alternatorja in pojav zvoènega polja vnjegovi okolici opi�emo z linearnim modelom, lahkore�evanje obravnavanega sklopljenega problemaprevedemo na problem vsiljenega nihanja oz. nare�evanje za posamezne harmonske komponente.Tako vsaka obratovalna oblika oz. strukturni odzivposamezne harmonske komponente povzroèapripadajoèo harmonsko komponento zvoènega polja.Poznavanje zvoènega polja za vse harmonskekomponente v konèni fazi omogoèa rekonstrukcijocelotnega zvoènega polja kot posledico strukturnegaodziva alternatorja zaradi vzbujanja z magnetnimisilami.

Pri gradnji akustiènega modela MRE smoizhajali iz strukturnega modela po MKE. Celotnozunanjo povr�ino alternatorja, ki jo v strukturnemmodelu po MKE sestavljata mre�i PLP in ZLP, smoprekrili z redkej�o mre�o trikotnih robnih elementov.Pri tem je mre�a robnih elementov zgrajena tako, dasi robni elementi delijo vozli�èa strukturnega modelapo MKE, kar omogoèa preprost prenos robnihpogojev iz strukturnega modela po MKE na akustiènimodel MRE. V na�em primeru akustiènega modelaMRE je najveèja dol�ina robnega elementa manj�aod 9 mm, kar zagotavlja dovolj natanèno re�evanje vfrekvenènem obmoèju do 6,4 kHz. Za opisakustiènega medija smo vzeli lastnosti zraka vnormalnih okoli�èinah. Z namenom, da bi se seznaniliz zvoènim poljem v okolici alternatorja, smo zvoènopolje okolice ovrednotili le za nekaj izbranih primerovobratovanja. Slika 9 prikazuje zvoèno polje v okolicialternatorja pete harmonske komponentemagnetnega hrupa neobremenjenega alternatorja priobratovanju s 3000 min-1.

4 ACOUSTIC RESPONSE OF THE ALTERNATOR

In the final step of the vibro-acousticmodelling of the alternator, a sound field in itssurroundings was determined. This was obtainedfrom the known structural response of the alternatorFEM model. The numerical verification of the soundfield is basically just an upgrade in the solving ofthe structural problem, where, on the basis of knownstructural response and by means of the boundary-element method (BEM), the sound field is calculated.

During the verification of the sound field thesame restriction as at the verification of the structuralresponse was used. The demand to have a stationaryoperation of the alternator is essential. The structuralresponse during such an operation on the externalsurfaces of the alternator causes a sound radiation orthe so-called magnetic noise. Suppose that themechanical structure of the alternator and thephenomenon of the sound field in its surrounding canbe described using a linear model, the solving of thediscussed combined problem can then be transformedto a forced vibration problem or to solve a particularharmonic component. Thus, each operational shape,or the structural response at a particular harmoniccomponent, contributes a belonging component to thesound field. Knowing the sound field for each harmoniccomponent makes it possible to reconstruct the entiresound field as a consequence of the structural responseof the alternator due to the magnetic force excitations.

The basis for the acoustic BEM model was thestructural FEM model. The entire external surface of thealternator, which had been formed in the structural FEMmodel from the DEB and REB meshes, was covered witha less dense mesh consisting of triangular boundaryelements. The mesh of the boundary elements wasformed in such a way that its nodes coincided with thenodes of the structural FEM model. This enables aneasy transfer of the boundary conditions from thestructural FEM model to the acoustic BEM model. In ourcase of an acoustic BEM model, the maximum length ofthe boundary element was smaller than 9 mm, so anaccurate solving in the region of 6.4 kHz was ensured.For the description of the acoustic medium, the airproperties under regular conditions were adopted. Inorder to get to know the sound field in the surroundingsof the alternator, the sound field was evaluated for just afew selected cases of the operation. Figure 9 shows thesound field in the surroundings of the unloaded alternatorat 3000 min-1 as a consequence of the 5th harmoniccomponent of the magnetic noise.



5 SKLEP

Prispevek predstavlja postopek izdelave indeloma tudi ovrednotenja sklopljenegavibroakustiènega modela alternatorja, ki zdru�ujetri fizikalne probleme, tj. magnetni, strukturno-dinamièni ter akustièni problem. Predhodno jeovrednoten vibroakustièni odziv alternatorja kotposledica vzbujanja z magnetnimi silami. S tem jepostavljen postopek, ki omogoèa preprosto in hitroposodabljanje parametrov modela terrazpoznavanje in raziskavo vpliva kljuènihparametrov konstrukcije alternatorja na njegovetehniène znaèilnosti. Glede na dejstvo, da jeizdelava vibroakustiènega modela v zaèetni fazi,je treba izvesti dodatno preverjanje in posodobitvemodela predvsem v smislu izbolj�anja modelamehanske strukture alternatorja.

5 CONCLUSION

This paper presents the process of building, andpartly also of verifying, a coupled vibro-acoustic model ofan alternator that unites three physical problems, i.e., themagnetic, the structural dynamic and the acoustic. Thevibro-acoustic response as a result of the magneticexcitation is evaluated and it shows a preliminary study ofthe magnetic noise of the alternator. This means that wehave successfully established the process where it is easilypossible to change almost any parameters of the modeland update the response results. In addition, this makes itpossible to identify and to analyse the important designparameters of the alternator and find their influence on thetechnical characteristic of the alternator. Due to the factthat the vibro-acoustic model of the alternator is in itsearly stages, it needs additional verification and updating,especially the structural FEM model.

Sl. 9. Zvoèno polje v okolici alternatorja kot posledica 5-te harmonske komponente magnetnih sil;trenutna vrednost zvoènega tlaka (na levi) in raven zvoènega tlaka (na desni)

Fig. 9. The sound field in the surroundings of the alternator as a consequence of the 5th harmoniccomponent of the magnetic forces; the instantaneous value of sound pressure (left) and the level of sound

pressure (right)


[1] Furlan M. (2003) Karakterizacija magnetnega hrupa enosmernega elektromotorja: doktorsko delo, Univerzav Ljubljani, Fakulteta za strojni�tvo, Ljubljana.

[2] Rebec R. (2004) Karakterizacija vibracij alternatorja AAN: Diplomska naloga univerzitetnega �tudija, Univerzav Ljubljani, Fakulteta za strojni�tvo, Ljubljana.

[3] Furlan M., Èernigoj A. in Bolte�ar M. (2003) A coupled electromagnetic-mechanical-acoustic model of a DCelectric motor, Compel, 2003, letn. 22, �t. 4, str. 1155-1165.

[4] Bolte�ar M., Èeliè D., Jak�iæ N., Keber M. in Èermelj P. (2004) Analiza in modeliranje alternatorja AAN:poroèilo, Univerza v Ljubljani, Fakulteta za strojni�tvo, Ljubljana.



[5] Ramesohl I., Bauer T. in Henneberger G. (1998) Calculation procedure of the sound field caused by magneticexcitations of the claw-pole alternator, International seminar on vibations and acoustic noise of electricmachinery, Bethune.

[6] Ramesohl I., Henneberger G., Kueppers S. in Hadrys W. (1996) Three dimensional calculation of magneticforces and displacements of a claw-pole generator, IEEE Transactions on Magnetics, 32(3):1685�1688,1996.

Naslova avtorjev:dr. Martin FurlanRobert RebecAndrej ÈernigojISKRA Avtoelektrika d.d.Polje 155290 �empeter pri [email protected]@[email protected]

Damjan Èelièdr. Primo� Èermeljprof.dr. Miha Bolte�arUniverza v LjubljaniFakulteta za strojni�tvoA�kerèeva 61000 [email protected]@[email protected]

Authors� Addressses: Dr. Martin FurlanRobert RebecAndrej ÈernigojISKRA Avtoelektrika d.d.Polje 15SI-5290 �empeter pri Gorici, [email protected]@[email protected]

Damjan ÈelièDr. Primo� ÈermeljProf.Dr. Miha Bolte�arUniversity of LjubljanaFaculty of Mechanical EngineeringA�kerèeva 6SI-1000 Ljubljana, [email protected]@[email protected]

Prejeto: 9.10.2005




126 Slaviè J. - Nastran M. - Bolte�ar M.

Modeliranje in analiza dinamike �èetke elektromotorja

Modeling and analyzing the dynamics of an electric-motor brush

Janko Slaviè1 - Miha Nastran2 - Miha Bolte�ar1

(1Fakulteta za strojni�tvo, Ljubljana; 2Domel, �elezniki)

V prispevku je na kratko predstavljen Pfeiffer-Glockerjev postopek modeliranja dinamike togihteles z enostranskimi stiki. Mogoèa stièna stanja: lepenje, drsenje, trk s trenjem, sprostitev stika sepreoblikujejo na linearni komplementarni problem, ki omogoèa re�evanje veè soèasnih stiènih stanj.Predstavljeno teoretièno ozadje je prilagojeno za telesa nepravilnih oblik. Uporaba predstavljenih zamislije prikazana na dinamiènem modelu �èetke elektromotorja. Dinamièni model je definiran z veè ko 40parametri, podrobni popis geometrijske oblike pa vkljuèuje tudi povr�insko hrapavost. V numeriènempreizkusu prispevek prika�e, kako obraba in togost �èetke vplivata na njeno stabilnost.© 2006 Strojni�ki vestnik. Vse pravice pridr�ane.(Kljuène besede: dinamika teles, telesa toga, simuliranje numerièno, modeli dinamièni)

This paper briefly presents the Pfeiffer-Glocker formulation for the multibody dynamics of rigidbodies with unilateral contacts. The multiple, concurrent contact situations of stick-slip, detachment andimpact with friction are solved as a linear complementarity problem. The theory is extended toward thediscretely defined bodies of complex body shapes with nonlinearities. As shown in the numerical example ofthe electric-motor-brush dynamics the presented extensions can be used to simulate the influence of adetailed geometry, including surface roughness. The influences of brush-wear and brush-stiffness on thedynamic stability are presented from more than 40 parameters that define the brush system.© 2006 Journal of Mechanical Engineering. All rights reserved.(Keywords: multibody dynamics, rigid bodies, numerical simulations, dynamic stability)

Strojni�ki vestnik - Journal of Mechanical Engineering 52(2006)2, 126-137UDK - UDC 531.36/.38:621.313.13Izvirni znanstveni èlanek - Original scientific paper (1.01)

0 UVOD

Dinamika sistema togih teles v zadnjem èasupridobiva pozornost, saj se uporablja tako prikrmiljenju robotov, podajnih mehanizmov, kakor tudipri izdelavi navideznih prototipov in pri simuliranjunavidezne resniènosti [1]. V nasprotju s postopkisimuliranja sistemov z dvostranskimi stiki (npr.:vrtljive zveze, vodila itn.), ki so �e dobro uveljavljeni,so se naèini vkljuèevanja enostranskih stikov (npr.:trk, trk dveh teles) razvili do primerne matematièno-fizikalne doslednosti �ele v zadnjem desetletju.

V tem prispevku se bomo osredotoèili naenostranske stike, zapisane v obliki linearnegakomplementarnega problema (LKP - LCP), tak zapisje prvi uporabil Lötstedt [2]. Med pomembnej�eraziskovalce tega podroèja sodijo �e Murty[3], Baraff [4], Panagiotopoulos [5], Moreau [6],Pfeiffer in Glocker ([7] in [8]) itn.

Podrobneje si bomo ogledali enega od boljobetavnih postopkov popisa dinamike togih teles z

0 INTRODUCTION

The mathematical formulation of multibodydynamics with unilateral contacts has receivedmuch interest in recent decades. This is particularlyso for various control systems, such as the controlof robots and industrial feeding mechanisms.Furthermore, it is included into research on virtualprototyping and virtual reality [1]. Despite the factthat bilateral contacts (e.g., rotating joints, linearjoints) have been theoretically covered for severaldecades, the general formulation of unilateralcontacts (e.g., impact with the friction of twobodies) was developed to the necessarymathematical and physical consistency only in thepast decade.

This paper focuses on the research ofunilateral contacts written as a LinearComplementarity Problem (LCP), which was initiatedby Lötstedt [2]. Some of the more importantresearchers in this field are Murty [3], Baraff [4],


127Modeliranje in analiza dinamike - Modeling and analyzing the dynamics

enostranskimi stiki: to je Pfeiffer-Glockerjev postopek [7].Njuno delo pomeni matematièno pravilen in fizikalnodosleden naèin re�evanja dinamike togih teles z veèsoèasnimi stiènimi stanji. Pfeiffer in Glocker stièni problem(lepenje, drsenje, sprostitev stika in trk s trenjem)preoblikujeta na pregleden in zgo�èen zapis v oblikilinearnega komplementarnega problema. Raziskovalcasta v svojih raziskavah uvedla novo razstavitev trenja, kiv primeru odvisnih koordinat nima te�av s singularnostjoin vodi v re�itev tudi v primeru predoloèenih sistemov;kot prva sta trk s trenjem predstavila v obliki linearnegakomplementarnega problema.

Namen prispevka je prilagoditev Pfeiffer-Glockerjevega postopka za diskretno definirana telesa.V ta namen je v drugem poglavju na kratko predstavljennjun postopek simuliranja dinamike togih teles, ki temeljina komplementarnosti stiènih stanj. Uporaba v drugempoglavju predstavljenih zamisli za re�evanje diskretnodefiniranih teles je nato prikazana v tretjem poglavju nadinamiènem modelu �èetke elektromotorja z 11prostostnimi stopnjami; kot primer analize je predstavljenvpliv nekaterih parametrov modela na stabilnostdelovanja �èetke. Zadnje sledi poglavje s sklepi.

1 DINAMIKA SISTEMA TOGIH TELES KOTLINEARNI KOMPLEMENTARNI PROBLEM

Zaradi celovitosti si bomo v tem poglavju nakratko pogledali bistvene zamisli Pfeiffer-Glockerjevega postopka re�evanja sistemov togihteles z enostranskimi stiki ([7] do [9]).

Gibalne enaèbe sistema togih teles s fprostostnimi stopnjami (vkljuèujoè dvostranskestike) so:

kjer so M masna matrika, q vektor posplo�enih(generaliziranih) koordinat in h vektor posplo�enihaktivnih sil. Èe imamo v nekem trenutku mno�icostiènih toèk Ni IÎ , potem se (1) spremeni:

kjer je C

iQ posplo�ena nekonservativna sila (kotposledica stiène sile v stiku i). Naèeloma jetreba posplo�ene koordinate pri lagajat itrenutnim prostostim, ki pa so odvisne od re�itvestiènega problema. Za primer navedimo, da biza re�itev sistema z n

N stiènimi toèkami morali

Panagiotopoulos [5], Moreau [6], and Pfeiffer andGlocker ([7] and [8]).

One of the more promising and completetheories for including unilateral contacts waspresented by Pfeiffer and Glocker [7]; their theorypresents a sound and physically consistent basisfor including concurrent multiple contact situations.They were also the first to present concurrent impactswith friction as a LCP. Furthermore, theirdecomposition of stick-slip and the detachmentproblem avoids the singularity of overdefineddynamical systems.

This paper is organized as follows: thesecond section presents the Pfeiffer-Glockerformulation for simulating unilateral contactproblems in general. In addition, somemodifications for discretely defined bodies aregiven. The ideas are presented in a numericalexample of electric-motor-brush dynamics with 11degrees of freedom, given in the third section. Asan example of an analysis the stability of electric-motor-brush dynamics is studied. The last sectiongives conclusions.

1 MULTIBODY DYNAMICS AS A LINEARCOMPLEMENTARITY PROBLEM

For the sake of completeness this sectiongives a brief review of the mathematical modeling ofmultibody dynamics with unilateral contacts aspresented by Glocker and Pfeiffer ([7] to [9]).

The equations of motion for a multibodysystem with f degrees of freedom (including onlybilateral contacts) can be written as:

(1),

where M is the mass matrix, q is the vector ofgeneralized coordinates and h is the vector ofgeneralized active forces. If there is a set of Ni IÎcontact forces (as a result of unilateral contacts) thenthe equations of motion will be:

(2),

where C

iQ are the generalized, non-conservative activeforces. Note that the contact forces change the numberof degrees of freedom. In general it is not known whichdegrees of freedom disappear; this problem is usuallysolved by looking at all the possible solutions and findingthe one that is physically consistent. If there are n

N

( ) ( ) 0 ft t R, - , , = Î&& &M q q h q q

N

C f

i

i I

RÎ

- = Îå&&M q h Q



najti fizikalno komplementarno re�itev med 3 Nn

mogoèimi re�itvami [7]1. Tako iskanje ustreznere�itve postane hitro praktièno nemogoèeizvedlj ivo. Zraven tega pa je z vidikanumeriènega re�evanja zelo neprimernoneprestano prilagajanje �tevila posplo�enihkoordinat.

Kakor bomo videli pozneje, se temte�avam z uporabo LKP elegantno izognemo, sajbo �tevilo posplo�enih koordinat vedno enako�tevilu prostosti sistema brez enostranskihstikov (2).

Poglejmo si najprej povezavo relativnihstiènih sil s posplo�enimi stiènimi silami. Z uporaboJacobijeve matrike lahko normalno stièno silo F

A,N v

toèki CA na telo A (sl. 1) zapi�emo kot posplo�eno

silo:

in èe dodamo �e silo na telo B:

wN vsebuje kinematiène lastnosti stika C, l

N je stièna

amplituda sile in I oznaèuje inercialen koordinatnisistem.

Zgornja izpeljava velja za zvezno definiranatelesa v splo�nem. Kako doloèimo Jacobijevo matrikostiène toèke, èe ima telo definirano obliko glede na svojete�i�èe pa sta izpeljala Slaviè in Bolte�ar ([10] in [11]).

Analogno normalni smeri nadaljujemo stangentno stièno silo (indeks T) in izraz (2)preoblikujemo v:

Z uporabo matriènega zapisa:

gibalne enaèbe preoblikujemo v:

Stièna stanja re�imo v dveh korakih: najprejna ravni impulzov re�imo nezvezni problem trka strenjem, nato pa na ravni sil �e lepenje, drsenje alisprostitev stika. Èe ni trènih stanj, prvi korakodpade.

possible contact points with a stick-slip transition ordetachment, then there are 3 Nn possible solutions [7]1.It is clear that the search for a physically consistentcombination is time consuming. Furthermore, fornumerical simulations it is not appropriate to change theminimum number of coordinates during each time-step.

As we see later, the linear complementarityproblem (LCP) method solves this problem in an elegantway, and the number of generalized coordinates isconstant at all times. The number of generalizedcoordinates is always equal to the number of degreesof freedom of the system without unilateral contacts (2).

The real contact forces are linked with thegeneralized contact forces via the Jacobian matrix. InFigure 1 two bodies are shown, the centers of gravitybeing denoted by A and B. The normal contact force F

A,N

at point CA on the body A as a generalized contact force is:

(3)

and when including the normal force at point B:

(4)

wN includes the kinematical properties of the contact,

lN is the amplitude of the force and I denotes the

inertial frame.To adopt the formulation for bodies with

discretely defined shapes the Jacobian matrix of thecontact point can be further simplified, as shown bySlaviè and Bolte�ar ([10] and [11]).

By using a similar notation for the tangentialforce (index T), Equation (2) is rewritten as:

(5).

Or by using matrix notation:

(6)

the equations of motion are:

(7).

The contact situations are solved in two steps:in the first, the non-smooth impact with friction is solved;in the second, the stick-slip or detachment situation issolved. While the impact is solved in the impulse-domain,the stick-slip or detachment is solved in the force-domain.

A

A

T

I Cc T

A N A N C I A N

rJ l, ,

¶æ ö= = × ×ç ÷

¶è øQ F n

q

A B

c T T

N C I A C I B N N NJ J l læ öç ÷è ø

= + =Q n n w

( ) 0N

f

N N T T ii I

l lÎ

- - = Î+å&& ¡M q h w w

{ } { }i iN N T T Ni I= , = , ÎW w W w

( ) 0N f

N T

T

ll

æ öç ÷ç ÷ç ÷ç ÷è ø

- - = Î&& ¡M q h W W

1Trèna stanja so v podanem primeru izkljuèena; vsaka stiènatoèka je lahko v eni od naslednjih faz: lepenje, drsenje insprostitev stika.

1In this example the impact situations are excluded. Each ofthe contacts can be in one of the states: sticking, slipping ordetachment of the contact.



Preden v nadaljevanju podrobneje spoznamooba koraka, si poglejmo mno�ice stiènih toèk:

Mno�ica lS vkljuèuje v nekem koraku

aktivne stike, lN samo tiste z nièno normalno

relativno hitrostjo in lH mo�ne lepene stike. Stiène

mno�ice se lahko v vsakem èasovnem korakuspremenijo.

1.1 Lepenje, drsenje ali sprostitev stika

Lepenje, drsenje (in prehod med njima) insprostitev stika re�ujemo na mno�ici stiènih toèk.Gibalne enaèbe (7) in relativni stièni pospe�ki g&&

so ([7], [8] in [12]):

Indeks N oznaèuje normalno smer in Htangentno smer mo�nih lepenih stikov iz mno�ice l

H.

Indeks G oznaèuje drseèe stike (tangentna sila jeznana iz Coulombovega zakona) iz mno�ice l

N \ l

H.

Gm je diagonalna matrika koeficientov trenja.Za vsak aktiven stik Ni IÎ velja, da je relativna

normalna razdalja 0iNg = in podobno za relativno

stièno hitrost 0iNg =& . Zaradi nepredirljivosti teles

velja 0iNg ³ ; sledi, da lahko za vsak stik v normalni

smeri zapi�emo komplementarni pogoj:

All the possible contact points IG are

organized in four sets during each time-step:

(8).

The set lS contains all the closed contacts, the

set lN contains only the contacts with vanishing relative

normal velocities (stick-slip or detachment), and the set lH

contains the possibly sticking contacts. The number ofelements in the sets can change during each time-step.

1.1 Stick-slip transition or detachment

First, the stick-slip transition or detachmentproblem is solved on an impact-free set l

N. The

equations of motion (7) and the relative contactaccelerations g&& are ([7], [8] and [12]):

(9)

(10).

The index N denotes the normal direction, andthe index H denotes the tangential direction of the possiblysticking set l

H. The new index G denotes the sliding

contacts (the tangential force is known) of the set lN \ l

H

and the Gm diagonal matrix of the friction coefficients.

Each closed contact Ni IÎ is characterizedby a vanishing contact distance 0

iNg = and a normalrelative velocity 0

iNg =& . Because of the impetrabilityof the bodies 0

iNg ³ , a complementary solution foreach contact in the normal direction can be found:

Sl. 1. Stiène sileFig. 1. Contact forces

{1 2 }

{ 0} elementov/elements



G G

S SG N

N NS N

H HN T

I �n

I i I g n

I i I ng

I i I ng

= , , ,

= Î ; =

= Î ; =

= Î ; =

&

&

0N f

N G HGH

lm l

æ öç ÷æ öç ÷ç ÷ç ÷ç ÷è ø ç ÷è ø

- - + = Î&& ¡M q h W W W

N H

T

NN n nN

THH H

g w

g w

æ öç ÷ +ç ÷ç ÷ç ÷è ø

æ ö æ ö= + Îç ÷ ç ÷

è øè ø

&&&& ¡

&&

Wq

W



in tudi Ni IÎ :

Taka komplementarnost je prikazana na sliki 2.Nekaj podobnega lahko ugotovimo za vsak stik vtangentni smeri (sl. 3a), vendar moramo �e prejCoulombov zakon razdeliti na dve veji: ena zapozitivno in ena za negativno smer. Taka razdelitevin komplementarni prikaz stiènega zakona v tangentnismeri je prikazana na sliki 3b. Da bi se izognili te�avams singularnostjo, smo dejansko uporabili boljzapleteno razdelitev stika v tangentni smeri, ki je tukajzaradi pomanjkanja prostora ne bomo obravnavali [7].

Z zamudno matematièno izpeljavokomplementarni problem za vse stiène toèke vnormalni in tangentni smeri hkrati zapi�emo v obliki [7]:

kjer (14) in (15) pomenita linearni komplementarniproblem (LKP) razse�nosti n

N+4n

H in se kot neznanki

pojavljata vektorja 4{ } N Hn n+, Î¡y x . V komplementranihpogojih (15) je treba zapis yTx = 0 razumeti na ravniposameznih elementov: y

ix

i = 0 za vse i. Vektor y med

drugim vsebuje neznane stiène pospe�ke g&& in vektor x

(11)

(12)

and also Ni IÎ :

(13).

Such a complementarity is sometimes alsoreferred to as the corner law [9], and is shown inFigure 2. In the following we will also try to representthe idea of the corner law in the tangential direction.Figure 3a presents the Coulomb friction law.Figure 3b shows the friction law decomposed intotwo branches: one for positive sliding and one fornegative sliding. To avoid singularity problems, theactually used decomposition is more complicatedand is covered in [7].

By extensive mathematical manipulation [7]the normal and tangential directions are writtentogether in the form:

(14)

(15),

Equations (14) and (15) represent an LCP where thevectors 4{ } N Hn n+, Î¡y x are not known, but theycomply with the complementary conditions (15),where yTx = 0 should be understood on the elementbasis: y

ix

i = 0 for all i. Vector y includes the unknown

contact accelerations g&& , and the vector x includes

0 0 stik se ohranja/contact is maintainedii NNg l= Ù ³&&

0 0 sprostitev stika/detachment of contactii NN

g l> Ù =&&

0ii NN

g l =&&

Sl. 2. Dopolnilnost v normalni smeriFig. 2. Complementarity in the normal direction

Sl. 3. Dopolnilnost v tangentni smeri

Fig. 3. Complementarity in the tangential direction

A b= +y x

0 0 0T³ , ³ , =y x y x



neznane stiène sile l. Matriki A in b sta znani in doloèeniz masno matriko, vektorjem aktivnih sil, koeficientitrenja in kinematiko stiènih toèk/obliko teles.

1.2 Trk s trenjem

V trku sodelujejo stiki iz mno�ice lS in ga

re�ujemo na ravni impulzov, zato moramo gibalnoenaèbo (7) najprej integrirati [7], nato pa impulz vfazi kompresije s Poissonovim zakonom pove�emo zimpulzom v fazi ekspanzije. Pri tem uporabimonaslednje poenostavitve: èas trka je zanemarljivomajhen, lega teles in vse neimpulzne sile se nespremenijo, ne spremeni se tudi �tevilo stiènih toèk.

Podobno kakor smo pri lepenju, drsenju insprostitvi stika zapisali komplementarne pogoje, takolahko tudi za fazo kompresije in ekspanzije zapi�emokomplementrane pogoje. Pri èemer pa se pri trku vkomplementarnem paru namesto sil in pospe�kovpojavljajo hitrosti in impulzi. Faza se konèa, ko sorelativne stiène hitrosti v normalni smeri enake niè;takrat se zaène faza ekspanzije [7]. Podrobneje tukajne bomo �li v zapis LKP. Je pa treba izpostaviti, damoramo pri veè soèasnih trkih paziti na pogojnepredirljivosti. V fazi ekspanzije lahko namreèlokalna ekspanzija v neki toèki povzroèi prediranje vneki drugi toèki; to prepreèimo tako, da vkomplementarnem pogoju omogoèimo veèji impulzekpanzije, kakor ga dovoljuje Poissonov zakon,vendar je v tem primeru relativna stièna hitrost obkoncu ekspanzije enaka niè. Glocker [8] je pokazal,da je zakon trka v takem primeru celotno raztresen.

Omeniti velja, da je Pfeiffer-Glockerjevaformulacija ena redkih, ki omogoèa simuliranjetudi popolnega povraèljivega trka v tangentnismeri2.

2 NUMERIÈNI PREIZKUS

V nadaljevanju si bomo ogledali uporabopredstavljenih postopkov na primeru dinamike �èetkeelektromotorja (sl. 4). Dinamièni sistem sestavljajo�tiri telesa: kolektor, �èetka, vodilo in vzmet.

2.1 Definiranje dinamiènega sistema

Kolektor je definiran s polmerom,ekscentriènostjo, �tevilom lamel, povr�inskohrapavostjo Rz (sl. 4, detajl B) in �irino re�e.

the unknown contact forces l. A and b are the knownmatrix and vector defined by the mass matrix, thefriction coefficients and the contact shapes.

1.2 Impact with friction

While the stick-slip or detachment transitionis solved in the force-acceleration domain, the impactis solved in the impulse-velocity domain. Somecommon assumptions for rigid-body impacts are made:the duration of the impact is infinitely short, the waveeffects are not taken into account, during the impactall the positions and orientations, and all the non-impulsive forces and torques, remain constant.

As it is possible to write the stick-slip anddetachment problem in the form of an LCP, it is alsopossible to do it for the compression and expansionphase; the only difference is that the solution is foundin the impulse-velocity domain, and that thedecomposition of the contact law is more complicated.One of the additional complications is posed by themultiple concurrent contacts that have mutualinfluences; the physically consistent formulation needsto guarantee non-penetration at such contactsituations. As shown by Glocker [8], the Pfeiffer-Glockerformulation can deal with multiple contacts in aphysically consistent way, and it maintains the globallydissipative criteria. After the compression phase endswhen the relative contact velocities at contacts diminish,the expansion phase continues. The amount of impulsetransferred from the compression to the expansionphase is defined by the friction and Poisson laws.

More details on impacts with friction are givenin [7]; the same publication gives more details onhow to include the reversibility in the tangentialdirection2.

2 NUMERICAL EXPERIMENT

This section presents an 11-degrees-of-freedom model of electric-motor-brush dynamics,Figure 4. The dynamical system consists of fourbodies: commutator, brush, and spring holder.

2.1 Characterization of the dynamical system

The commutator is defined by the radius, theeccentricity, the number of commutator bars, the surfaceroughness, Rz, and the slot-width between the bars, Fig. 4B.

2Popolno povraèljivost v tangentni smeri ima npr. zeloelastièna �oga; medtem, ko je npr. trk pingpong �oge vtangentni smeri nepovraèljiv.

2As an example of reversibility in the tangential direction thesuper-elastic ball is usually given. On the other, hand a ping-pongball impulse in the tangential direction is completely irreversible.



�èetka elektromotorja je definirana s �irino,dol�ino, ukrivljenostjo in hrapavostjo stiènepovr�ine s kolektorjem, nagibom stiène povr�ine zvzmetjo in togostjo. Da se dose�e lokalnadeformabilnost �èetke v stiku s kolektorjem, je le-tamodelirana kot sistem �tirih togih teles. Toga telesa,ki sestavljajo �èetko, so med seboj povezana zvibroizolacijo (sl. 4, detajl A), katere parametri sopridobljeni s preizkusom. �èetka ima kot sistem togihteles 9 prostostnih stopenj.

Vodilo �èetke je definirano z dol�ino,zraènostjo glede na �èetko, zraènostjo glede nakolektor, togostjo vpetja in lego te�i�èa glede nainercialni koordinatni sistem

Ixy (sl. 4). Vodilo ima

eno prostostno stopnjo: vrtenje okoli te�i�èa.Vzmet je definirana z dol�ino, polmerom

vrtenja, togostjo, prednapetjem vzmeti in lego gledena inercialni koordinatni sistem

Ixy. Vzmet ima eno

prostostno stopnjo.Potem ko so posamezna telesa definirana, je

treba doloèiti gibalne enaèbe sistema brezenostranskih stikov in jih zapisati v matrièno

The electric-motor brush is defined by thewidth, the length, the radius of curvature of the contactsurface with commutator, the roughness of the contactsurface with the commutator, the slope of the contactsurface with the spring and with the brush stiffness.To achieve local deformability the brush is modeledas a system of four rigid bodies (9 degrees of freedom)that are connected with viscoelastic elements, Fig. 4A.The parameters of the viscoelastic elements wereobtained experimentally.

The holder of the brush is defined by thelength, the clearance between the holder and thebrush, the clearance between the holder and thecommutator, the stiffness of the attachment to thesurroundings and the position of the center of gravity,see Figure 4. The holder has 1 degree of freedom.

The spring is defined with the length, the radiusof rotation, the stiffness, the pre-stress rate and theposition. The spring has one degree of freedom.

The 11 equations of motion for the givensystem need to be written in the matrix form (1). Theequations of motion for a contact-free case can be

Sl. 4. Shematièen prikaz dinamiènega modela �èetke z 11 prostostnimi stopnjamiFig. 4. Scheme of the model with 11 degrees of freedom



obliko (1). Celoten sistem ima 11 prostostnihstopenj in je definiran z 11 gibalnimi enaèbami.Izpeljava gibalnih enaèb z uporabo Lagrangevihenaèb II. vrste je razmeroma preprosto, vendarzamudno opravilo in ga bomo tukaj zaradiobse�nosti izpustili. Popis sistema nadaljujemo sstiènimi parametri med telesi: �èetka � kolektor,�èetka � vodilo, �èetka � vzmet. Stièni parametriso: koeficient trenja m, koeficient trka v normalni e

n

in tangentni smeri et ter koeficient povraèljivosti

trka v tangentni smeri n [7].�èetka kot bistveni stièni element je narejena

iz grafit-epoksidnega materiala. Ker je kristalnare�etka grafit-epoksidnega materiala �esterokotna,so materialne lastnosti anizotropne (tako elektriènekot mehanske); poleg tega se materialne lastnostizelo spreminjajo: predvsem s temperaturo in gostototoka skozi �èetko. Pri vzpostavitvi dinamiènegamodela je tako zelo pomembno preizkusnopridobivanje ustreznih materialnih podatkov:· elastièni in du�ilni parametri v odvisnosti od

temperature in gostote toka,·· trèni parametri v odvisnosti od temperature,

gostote toka, hitrosti trka in izmere trènih teles(dol�ine �èetke),

· koeficienta trenja v odvisnosti od temperature,gostote toka in relativne hitrosti drsenja.

Vsaka od zgoraj na�tetih nalog je sama zasezahtevna naloga, vendar se je kot najbolj zahtevnaizkazala meritev koeficienta trenja v odvisnosti odtemperature in gostote toka, ki smo jo predstavili vloèeni objavi [13].

Ker je oblika togih teles definirana z robom izdiskretnih toèk, lahko kinematiène podatke trka (4)doloèimo neposredno iz oblike [11].

2.2 Vpliv spreminjanja parametrov sistema naobratovalne razmere �èetke

V prej�njem poglavju so na�teti bistveniparametri, ki definirajo obravnavani dinamièni sistem�èetke. Zaradi velikega �tevila parametrov bo tukajkot primer prikazan vpliv samo nekaterih parametrov.

2.2.1 Vpliv dol�ine �èetke na njeno stabilnost

Na sliki 5 je prikazan fazni diagram nove�èetke; opazimo, da je kotna hitrost omejena s ±30rad/s in da je �èetka praktièno vedno znotraj nagiba-7.10-3 rad do -2.10-3 rad. Kadar se �èetka obrabi,kakor je prikazano na sliki 6, pa postane �èetka

obtained relatively easily with the help of Lagrangeequations, but because this is a lengthy task it willbe omitted here. For a complete definition of thesystem the contact parameters between the brush-commutator, the brush-holder and the brush-springneed to be set. The contact parameters are as follows:the coefficient of friction m, and the coefficient ofrestitution in the normal and tangential directions, e

n

and et , respectively. If necessary, the coefficient of

reversibility in the tangential direction n needs to beset [7].

The most important body in the system isthe brush, which consists of graphite/epoxy materialwhose characteristics are highly sensitive totemperature. In addition, because the crystal latticeis hexagonal the mechanical and electrical propertiesare highly anisotropic. Because of the anisotropyand the temperature sensitivity the experimental workwas focused on determining:· the stiffness and damping properties as a function

of temperature and current density,· the impact properties as a function of temperature,

current density, impact velocity and thedimensions of the impacting bodies (length ofbrush),

· the coefficient of friction as a function oftemperature, current density and sliding velocity.

The quality of the simulation results criticallydepends on the quality of the experimental work;how we measured the coefficient of friction forvarious temperatures and current densities wepresent in a separate paper [13].The kinematical properties (4) of the contact pointsare determined automatically from the discretelydefined bodies [11].

2.2 Influence of the parameter variation on theworking conditions of the brush

The dynamical system is defined by morethan 40 parameters. As an example of influence-analysis, the influence of selected parameters on theworking conditions will be given.

2.2.1 Influence of brush-length on the dynamic stability

A phase plot of a new brush is shown inFigure 5: the angular velocity is in the range ±30 rad/s,while the angle is in the range from -7.10-3 rad to-2.10-3 rad. When the brush is at the end of its lifetimeit becomes considerably less stable, see Figure 6.



bistveno bolj nemirna: kotna hitrost sega tudi izvenpodroèja ±100 rad/s in tudi nagib �èetke je vbistveno �ir�em podroèju kakor pri novi �èetki.Obrabljena �èetka je torej bistveno bolj nemirna,kar seveda vpliva na kakovost elektriènega stika ins tem tudi na obrabo, dobo in zanesljivost delovanjaelektromotorja.

Na sliki 7 je prikazana povpreèna absolutnakotna hitrost �èetke za veè ko 1400 razliènihnumeriènih simulacij. Simulirani sistemi serazlikujejo v naboru 11 razliènih parametrov, kakorso: hrapavost kolektorja, ekscentriènost kolektorja,nagib �èetke glede na os vrtenja, togost �èetke,zraènost med �èetko in vodilom, togost napenjalnevzmeti itn.

Ker je izraba �èetke neizogibna, je dol�ina�èetke te�ko vir izbolj�av, ki pa jih lahko dose�emo sspreminjanjem vrste drugih parametrov. Izbira

The angular velocity can also be above ±100 rad/sand also a broader range of angles of the brush isobserved. The stability of the brush considerablyinfluences the quality of the electrical contact andconsequently the lifetime and the reliability of theelectric motor.

From more then 1400 simulation resultsobtained from systems with different parametervalues we can see that the brush-length has a majorinfluence on the stability, see Figure 7. Altogether,11 parameters were varied: commutator roughness,commutator eccentricity, relative position of thebrush, brush stiffness, clearance between the brushand the holder, stiffness of the spring, etc.

Because of the wear the change of the brush-length cannot be omitted, and therefore we are seekinganother property to increase the stability. One of theobvious options is the change of the brush-material, i.e.,

Sl. 5. Fazni diagram za j pri novi �èetkiFig. 5. Phase diagram for j for a new brush

Sl. 6. Fazni diagram za j pri obrabljeni �èetkiFig. 6. Phase diagram for j for a used brush at end of its lifetime



drugaènega materiala je ena od mo�nosti in na sliki 8je prikazan vpliv relativne togosti na povpreènoabsolutno kotno hitrost �èetke; prikazanih je 69rezultatov �èetke dol�ine od 10 mm do 15 mm. Oèitnoje, da bolj elastièna �èetka prispeva k stabilnosti le-te.

3 SKLEPI

V prvem delu prispevka predstavljeni postopkiso predvsem namenjeni simuliranju analitièno zapisanihproblemov in jasno izra�enih stiènih stanj; za take primerelahko kinematiène lastnosti stiènih toèk doloèimo analitiènoin vnaprej. V primeru geometrijsko zahtevnej�ih teles in vprimeru dinamiènih sistemov z geometrijskiminelinearnostmi pa se izka�e, da je doloèevanje kinematiènihlastnosti kontaktnih toèk vnaprej praktièno nemogoèe. V

the stiffness and damping parameters of the brush. Theinfluence analysis of the brush-stiffness on the brushstability at the end of its lifetime shows an increasedstability of softer materials, see Figure 8. The Figureshows the results of 69 different simulations with a brushlength ranging from 10 to 15 mm

3 CONCLUSIONS

The second section of the paper introducesthe concepts needed to simulate the dynamics ofanalytically defined systems with clear contactsituations. However, in geometrically morecomplicated and nonlinear cases the contactproperties of the contact situations cannot bedefined a priori and the extension of the conceptstoward discretely defined bodies is necessary. The

Sl. 7. Povpreèna absolutna kotna hitrost �èetke v odvisnosti od njene obrabeFig. 7. Average absolute angular velocity as a function of the length of the brush

Sl. 8. Povpreèna absolutna kotna hitrost �èetke v odvisnosti od relativne togosti �èetke(69 razliènih numeriènih simulacij)

Fig. 8. Average absolute angular velocity as a function of relative brush stiffness(69 different simulations)



tem prispevku uporabljen diskreten postopek definiranjageometrijske oblike teles pa to omejitev odpravlja, saj sekinematiène lastnosti stiènih toèk glede na geometrijskoobliko teles doloèajo sproti (v vsakem èasovnem koraku).

Predstavljeni postopki omogoèajo numeriènosimulacijo sistema togih teles z zelo zahetvnogeometrijsko obliko, katera, kakor je nakazano vnumeriènem preizkusu, vkljuèuje tudi hrapavost.

V numeriènem preizkusu predstavljen dinamiènimodel �èetke elektromotorja ima 11 prostostnih stopenj inga definira veè kot 40 razliènih parametrov. Na primeruvpliva obrabe/dol�ine �èetke in togosti �èetke je prikazano,kako lahko z numeriènim preizkusom i�èemo nove zamisliza izdelavo prototipnih izvedb in dejanskih preizkusnihpotrditev. Pri ujemanju rezultatov numeriènega indejanskega preizkusa pa je vsaj toliko kot uporaba fizikalnodosledne teorije pomembna tudi uporaba kakovostnihpreizkusnih podatkov, ki se v preizkusnih naèelih ujemajoz uporabljenimi teoretiènimi postopki. Velik pomenpredstavljenega dela je zato bil tudi na preizkusnih meritvah.

discretely defined bodies are therefore used todefine the contact parameters during each time-step.

As shown in the numerical example, theextensions for discretely defined bodies cantherefore be used to simulate the dynamics ofcomplex body shapes, including such local detailsas a geometrically exact simulation of theroughness.

The numerical experiment of the electric-motor-brush presents a rigid body model with 11degrees of freedom defined by more than 40parameters. The study of how brush-wear and brush-roughness influence the stability of the electricbrush shows how new ideas for improvements canbe found. However, the ideas have to be tested onprototypes. We believe that the quality of thesimulations critically depends on the quality of thetheoretical background and�at least as important�on the quality of the experimental work.


[1] P. Song (2002) Modeling, analysis and simulation of multibody systems with contact and friction. PhDthesis, University of Pennsylvania.

[2] P. Lötstedt (1981) Coulomb friction in two-dimensional rigid body systems. Z. Angewandte Math. Mech.,61:605�615.

[3] K.G. Murty (1988) Linear complementarity, Linear and nonlinear programming. Heldermann Verlag, Berlin.[4] D. Baraff (1992) Dynamic simulations of non-penetrating rigid bodies. PhD thesis, Cornell University, NY.[5] P.D. Panagiotopoulos (1993) Hemivariational inequalities: applications in mechanics and engineering.

Springer Verlag, Berlin, Heidelberg.[6] J.J. Moreau (1976) Sur les mesures différentialles de fonctions vectorielles et certains problemes d�évolution.

Comptes Rendus Acad. Sci. Paris, 282:837�840, Sér. A.[7] F. Pfeiffer and C. Glocker (1996) Multibody dynamics with unilateral contacts. John Wiley & Sons, Inc,

New York.[8] C. Glocker (1995) Dynamik von Starrkörpesystemen mit Reibung und Stößen. PhD Thesis, Technische

Universität München.[9] F. Pfeiffer (2003) The idea of complementarity in multibody dynamics. Archive of Applied Mechanics,

72:807�816.[10] J. Slaviè and M. Bolte�ar (2004) Dinamika diskretno definiranega sistema togih teles z enostranskimi

kontakti = Multibody dynamics of discrete bodies with unilateral contacts. In J. Korelc and D. Zupan,editors, Kuhljevi dnevi 2004, Otoèec, 30. september - 1. oktober 2004, pages 285�292. Slovensko dru�tvoza mehaniko.

[11] J. Slaviè and M. Bolte�ar (2005) Nonlinearity and non-smoothness in multi body dynamics: application towoodpecker toy. Journal of Mechanical Engineering Science, 2005. In press.

[12] R.I. Leine, C. Glocker, and D.H. Van Campen (2003) Nonlinear dynamics and modelling of some woodentoys with impact and friction. Nonlinear Dynamics, 9:25�78.

[13] J. Slaviè and M. Bolte�ar (2005) Measuring the dynamic forces to identify the friction of a graphite-coppercontact for variable temperature and current. Wear, 2005. In press.

`



Naslova avtorjev:dr. Janko Slavièprof.dr. Miha Bolte�arUniverza v LjubljaniFakulteta za strojni�tvoA�kerèeva 61000 [email protected]@fs.uni-lj.si

dr. Miha NastranDomel d.d.Otoki 214228 �[email protected]

Authors� Addressses: Dr. Janko SlavièProf.Dr. Miha Bolte�arUniversity of LjubljanaFaculty of Mechanical EngineeringA�kerèeva 6SI-1000 Ljubljana, [email protected]@fs.uni-lj.si

Dr. Miha NastranDomel d.d.Otoki 21SI-4228 �elezniki, [email protected]

Prejeto: 3.10.2005



Strojni�ki vestnik - Journal of Mechanical Engineering 52(2006)2, 138

138 Osebne vesti - Personal Events

Osebne vesti - Personal Events

Doktorat, magisterija in diplome - Doctor�s, Master�s and Diploma Degrees

Strojni�ki vestnik - Journal of Mechanical Engineering 52(2006)2, 138Osebne vesti - Personal Events

DOKTORAT

Na Fakulteti za strojni�tvo Univerze vLjubljani je z uspehom zagovarjal svojo doktorskodisertacijo:

dne 30. januarja 2006: mag. Simon Kra�na,z naslovom: �Modeliranje mehanike èlove�kegatelesa za analizo dinamike in po�kodb potnikov pritrkih vozil�.

S tem je navedeni kandidat dosegelakademsko stopnjo doktorja znanosti.

MAGISTERIJA

Na Fakulteti za strojni�tvo Univerze vMariboru so z uspehom zagovarjali svoja magistrskadela:

dne 5. januarja 2006: Janez Makov�ek, znaslovom: �Nadzor èistilnega procesa v tlaènihposodah procesne tehnike s pomoèjo analizevibracij�;

dne 6. januarja 2006: Alenka P�eniènik, znaslovom: �Sistemska ureditev pretoka materialovin informacij v proizvodnji valjev�.

S tem sta navedena kandidata doseglaakademsko stopnjo magistra znanosti.

DIPLOMIRALI SO

Na Fakulteti za strojni�tvo Univerze vLjubljani je pridobil naziv univerzitetni diplomiraniin�enir strojni�tva:

dne 25. januarja 2006: David VI�INTIN.

*

Na Fakulteti za strojni�tvo Univerze vLjubljani so pridobili naziv diplomirani in�enirstrojni�tva:

dne 13. januarja 2006: Boris GUZELJ,Primo� KOKALJ, Gorazd LAPANJA, Bo�tjan �KRLJ;

dne 27. januarja 2006: GregorCVETKOVIÈ, Janez FIREDER, Bo�tjan IR�IÈ, AndrejJESENOVEC.

Na Fakulteti za strojni�tvo Univerze vMariboru sta pridobila naziv diplomirani in�enirstrojni�tva:

dne 26. januarja 2006: Borut MEKE,Robert POLOVIÈ.


139Navodila avtorjem - Instructions for Authors

Navodila avtorjem - Instructions for Authors

Strojni�ki vestnik - Journal of Mechanical Engineering 52(2006)2, 139-140Navodila avtorjem - Instructions for Authors

Èlanki morajo vsebovati:- naslov, povzetek, besedilo èlanka in podnaslove slik v

slovenskem in angle�kem jeziku,- dvojeziène preglednice in slike (diagrami, risbe ali

fotografije),- seznam literature in- podatke o avtorjih.

Strojni�ki vestnik izhaja od leta 1992 v dveh jezikih,tj. v sloven�èini in angle�èini, zato je obvezen prevod vangle�èino. Obe besedili morata biti strokovno in jezikovnomed seboj usklajeni. Èlanki naj bodo kratki in naj obsegajopribli�no 8 strani. Izjemoma so strokovni èlanki, na �eljoavtorja, lahko tudi samo v sloven�èini, vsebovati pa morajoangle�ki povzetek.

Za èlanke iz tujine (v primeru, da so vsi avtorjitujci) morajo prevod v sloven�èino priskrbeti avtorji.Prevajanje lahko proti plaèilu organizira uredni�tvo. Èe jeèlanek ocenjen kot znanstveni, je lahko objavljen tudi samov angle�èini s slovenskim povzetkom, ki ga pripraviuredni�tvo.

VSEBINA ÈLANKA

Èlanek naj bo napisan v naslednji obliki:- Naslov, ki primerno opisuje vsebino èlanka.- Povzetek, ki naj bo skraj�ana oblika èlanka in naj ne

presega 250 besed. Povzetek mora vsebovati osnove, jedroin cilje raziskave, uporabljeno metodologijo dela,povzetekrezulatov in osnovne sklepe.

- Uvod, v katerem naj bo pregled novej�ega stanja in zadostneinformacije za razumevanje ter pregled rezultatov dela,predstavljenih v èlanku.

- Teorija.- Eksperimentalni del, ki naj vsebuje podatke o postavitvi

preskusa in metode, uporabljene pri pridobitvi rezultatov.- Rezultati, ki naj bodo jasno prikazani, po potrebi v obliki

slik in preglednic.- Razprava, v kateri naj bodo prikazane povezave in

posplo�itve, uporabljene za pridobitev rezultatov.Prikazana naj bo tudi pomembnost rezultatov inprimerjava s poprej objavljenimi deli. (Zaradi naraveposameznih raziskav so lahko rezultati in razprava, zajasnost in preprostej�e bralèevo razumevanje, zdru�eniv eno poglavje.)

- Sklepi, v katerih naj bo prikazan en ali veè sklepov, kiizhajajo iz rezultatov in razprave.

- Literatura, ki mora biti v besedilu o�tevilèenazaporedno in oznaèena z oglatimi oklepaji [1] ter nakoncu èlanka zbrana v seznamu literature. Vse opombenaj bodo oznaèene z uporabo dvignjene �tevilke1.

OBLIKA ÈLANKA

Besedilo èlanka naj bo pripravljeno v urejevalnilkuMicrosoft Word. Èlanek nam dostavite v elektronski obliki.

Ne uporabljajte urejevalnika LaTeX, saj program, skaterim pripravljamo Strojni�ki vestnik, ne uporabljanjegovega formata.

Enaèbe naj bodo v besedilu postavljene v loèenevrstice in na desnem robu oznaèene s tekoèo �tevilko vokroglih oklepajih

Papers submitted for publication should comprise:- Title, Abstract, Main Body of Text and Figure Captions

in Slovene and English,- Bilingual Tables and Figures (graphs, drawings or photo-

graphs),- List of references and- Information about the authors.

Since 1992, the Journal of Mechanical Engineeringhas been published bilingually, in Slovenian and English. Thetwo texts must be compatible both in terms of technical con-tent and language. Papers should be as short as possible andshould on average comprise 8 pages. In exceptional cases, atthe request of the authors, speciality papers may be writtenonly in Slovene, but must include an English abstract.

For papers from abroad (in case that none ofauthors is Slovene) authors should provide Slovenian trans-lation. Translation could be organised by editorial, but theauthors have to pay for it. If the paper is reviewed as scien-tific, it can be published only in English language withSlovenian abstract, that is prepared by the editorial board.

THE FORMAT OF THE PAPER

The paper should be written in the following format:- A Title, which adequately describes the content of the paper.- An Abstract, which should be viewed as a mini version of

the paper and should not exceed 250 words. The Abstractshould state the principal objectives and the scope of theinvestigation, the methodology employed, summarize theresults and state the principal conclusions.

- An Introduction, which should provide a review of recentliterature and sufficient background information to allowthe results of the paper to be understood and evaluated.

- A Theory- An Experimental section, which should provide details of

the experimental set-up and the methods used for obtain-ing the results.

- A Results section, which should clearly and concisely presentthe data using figures and tables where appropriate.

- A Discussion section, which should describe the relation-ships and generalisations shown by the results and discussthe significance of the results making comparisons withpreviously published work. (Because of the nature of somestudies it may be appropriate to combine the Results andDiscussion sections into a single section to improve theclarity and make it easier for the reader.)

- Conclusions, which should present one or more conclu-sions that have been drawn from the results and subse-quent discussion.

- References, which must be numbered consecutively in thetext using square brackets [1] and collected together in areference list at the end of the paper. Any footnotesshould be indicated by the use of a superscript1.

THE LAYOUT OF THE TEXT

Texts should be written in Microsoft Word format.Paper must be submitted in electronic version.

Do not use a LaTeX text editor, since this is notcompatible with the publishing procedure of the Journal ofMechanical Engineering.

Equations should be on a separate line in the mainbody of the text and marked on the right-hand side of thepage with numbers in round brackets.


140 Navodila avtorjem - Instructions for Authors

Enote in okraj�ave

V besedilu, preglednicah in slikah uporabljajte lestandardne oznaèbe in okraj�ave SI. Simbole fizikalnih velièinv besedilu pi�ite po�evno (kurzivno), (npr. v, T, n itn.). Simboleenot, ki sestojijo iz èrk, pa pokonèno (npr. ms-1, K, min,mm itn.).

Vse okraj�ave naj bodo, ko se prviè pojavijo,napisane v celoti v slovenskem jeziku, npr. èasovnospremenljiva geometrija (ÈSG).

Slike

Slike morajo biti zaporedno o�tevilèene in oznaèene,v besedilu in podnaslovu, kot sl. 1, sl. 2 itn. Posnete naj bodov loèljivosti, primerni za tisk, v kateremkoli od raz�irjenihformatov, npr. BMP, JPG, GIF. Diagrami in risbe morajo bitipripravljeni v vektorskem formatu.

Pri oznaèevanju osi v diagramih, kadar je le mogoèe,uporabite oznaèbe velièin (npr. t, v, m itn.), da ni potrebnodvojezièno oznaèevanje. V diagramih z veè krivuljami, morabiti vsaka krivulja oznaèena. Pomen oznake mora bitipojasnjen v podnapisu slike.

Vse oznaèbe na slikah morajo biti dvojeziène.

Preglednice

Preglednice morajo biti zaporedno o�tevilèene inoznaèene, v besedilu in podnaslovu, kot preglednica 1,preglednica 2 itn. V preglednicah ne uporabljajte izpisanihimen velièin, ampak samo ustrezne simbole, da se izognemodvojezièni podvojitvi imen. K fizikalnim velièinam, npr. t(pisano po�evno), pripi�ite enote (pisano pokonèno) v novovrsto brez oklepajev.

Vsi podnaslovi preglednic morajo biti dvojezièni.

Seznam literature

Vsa literatura mora biti navedena v seznamu nakoncu èlanka v prikazani obliki po vrsti za revije, zbornikein knjige:[1] A. Wagner, I. Bajsiæ, M. Fajdiga (2004) Measurement of

the surface-temperature field in a fog lamp usingresistance-based temperature detectors, Stroj. vestn.2(2004), pp. 72-79.

[2] Vesenjak, M., Ren Z. (2003) Dinamièna simulacijadeformiranja cestne varnostne ograje pri naletu vozila.Kuhljevi dnevi �03, Zreèe, 25.-26. september 2003.

[3] Muhs, D. et al. (2003) Roloff/Matek Maschinenelemente� Tabellen, 16. Auflage. Vieweg Verlag, Wiesbaden.

Podatki o avtorjih

Èlanku prilo�ite tudi podatke o avtorjih: imena,nazive, popolne po�tne naslove in naslove elektronske po�te.

SPREJEM ÈLANKOV IN AVTORSKE PRAVICE

Uredni�tvo Strojni�kega vestnika si pridr�ujepravico do odloèanja o sprejemu èlanka za objavo,strokovno oceno recenzentov in morebitnem predlogu zakraj�anje ali izpopolnitev ter terminolo�ke in jezikovnekorekture.

Avtor mora predlo�iti pisno izjavo, da je besedilonjegovo izvirno delo in ni bilo v dani obliki �e nikjerobjavljeno. Z objavo preidejo avtorske pravice na Strojni�kivestnik. Pri morebitnih kasnej�ih objavah mora biti SVnaveden kot vir.

Units and abbreviations

Only standard SI symbols and abbreviations shouldbe used in the text, tables and figures. Symbols for physicalquantities in the text should be written in italics (e.g. v, T, n,etc.). Symbols for units that consist of letters should be inplain text (e.g. ms-1, K, min, mm, etc.).

All abbreviations should be spelt out in full onfirst appearance, e.g., variable time geometry (VTG).

Figures

Figures must be cited in consecutive numericalorder in the text and referred to in both the text and thecaption as Fig. 1, Fig. 2, etc. Pictures may be saved inresolution good enough for printing in any common format,e.g. BMP, GIF, JPG. However, graphs and line drawings sholudbe prepared as vector images.

When labelling axes, physical quantities, e.g. t, v, m,etc. should be used whenever possible to minimise the need tolabel the axes in two languages. Multi-curve graphs should haveindividual curves marked with a symbol, the meaning of thesymbol should be explained in the figure caption.

All figure captions must be bilingual.

Tables

Tables must be cited in consecutive numerical order inthe text and referred to in both the text and the caption as Table 1,Table 2, etc. The use of names for quantities in tables should beavoided if possible: corresponding symbols are preferred to minimisethe need to use both Slovenian and English names. In addition tothe physical quantity, e.g. t (in italics), units (normal text), shouldbe added in new line without brackets.

All table captions must be bilingual.

The list of references

References should be collected at the end of thepaper in the following styles for journals, proceedings andbooks, respectively:[1] A. Wagner, I. Bajsiæ, M. Fajdiga (2004) Measurement of

the surface-temperature field in a fog lamp usingresistance-based temperature detectors, Stroj. vestn.2(2004), pp. 72-79.

[2] Vesenjak, M., Ren Z. (2003) Dinamièna simulacijadeformiranja cestne varnostne ograje pri naletu vozila.Kuhljevi dnevi �03, Zreèe, 25.-26. september 2003.

[3] Muhs, D. et al. (2003) Roloff/Matek Maschinenelemente� Tabellen, 16. Auflage. Vieweg Verlag, Wiesbaden.

Author information

The information about the authors should be enclosedwith the paper: names, complete postal and e-mail addresses.

ACCEPTANCE OF PAPERS AND COPYRIGHT

The Editorial Committee of the Journal ofMechanical Engineering reserves the right to decide whethera paper is acceptable for publication, obtain professionalreviews for submitted papers, and if necessary, require changesto the content, length or language.

Authors must also enclose a written statement thatthe paper is original unpublished work, and not under considerationfor publication elsewhere. On publication, copyright for the papershall pass to the Journal of Mechanical Engineering. The JMEmust be stated as a source in all later publications.

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