letnik/volume 52 - {t./no. 2/06 - str./pp. 73-140 Ljubljana, feb./Feb. 2006, zvezek/issue 490 STROJNIŠKI VESTNIK JOURNAL OF MECHANICAL ENGINEERING cena 800 SIT 770039 248001 ISSN 0039-2480 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 73 Vsebina - Contents Vsebina - Contents Strojniški vestnik - Journal of Mechanical Engineering letnik - volume 52, (2006), številka - number 2 Ljubljana, februar - February 2006 ISSN 0039-2480 Izhaja mesečno - Published monthly Razprave Papers Nagode, M., Fajdiga, M.: Modeliranje naključnih Nagode, M., Fajdiga, M.: Thermo-Mechanical termomehanskih napetostno-deformacijskih Modelling of Stochastic Stress-Strain stanj 74 States Vesenjak, M., Ren, Z., Miillerschon, H., Matthaei, Vesenjak, M., Ren, Z., Miillerschon, H., Matthaei, S.: S.: Računalniško modeliranje gibanja goriva Computational Modelling of Fuel Motion and in njegov vpliv na konstrukcijo rezervoarja 85 Its Interaction with the Reservoir Structure Borovinšek, M., Vesenjak, M., Ulbin, M., Ren, Z.: Borovinšek, M., Vesenjak, M., Ulbin, M., Ren, Z.: Simulacija naleta tovornega vozila ob cestno Simulating the Impact of a Truck on a Road- varnostno ograjo 101 Safety Barrier Furlan, M., Rebec, R, Černigoj, A., Čelič, D., Čermelj, Furlan, M., Rebec, R, Černigoj, A., Čelič, D., Čermelj, P., Boltežar, M.: Vibroakustično modeliranje P, Boltežar, M.: VVibro-Acoustic Modelling alternatorja 112 of an Alternator Slavic, J., Nastran, M., Boltežar, M.: Modeliranje in Slavic, J., Nastran, M., Boltežar, M.: Modeling and analiza dinamike ščetke elektromotorja analyzing the dynamics of an electric-motor 126 brush Osebne vesti Personal Events Doktorat, magisterija in diplome 138 Doctor’s, Master’s and Diploma Degrees Navodila avtorjem 139 Instructions for Authors Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 74-84 UDK - UDC 620.179.12/.13:539.388.1 Izvirni znanstveni članek - Original scientific paper (1.01) Modeliranje naključnih termomehanskih napetostno-deformacijskih stanj Thermo-Mechanical Modelling of Stochastic Stress-Strain States Marko Nagode - Matija Fajdiga (Fakulteta za strojništvo, Ljubljana) Deformacijski postopek je najbolj razširjena metoda napovedovanja dobe trajanja dinamično obremenjenih delov na področju malociklične trdnosti. Uporablja se tako pri nizkih, srednjih kakor tudi visokih temperaturah, če je temperatura med obratovanjem nespremenljiva. Deformacijski postopek je računsko izjemno hiter in narekuje le uporabo analiz elastičnosti s končnimi elementi. Zaradi številnih prednosti, razširjenosti in potencialnih možnosti je bil prilagojen tako, da je uporaben tudi za spremenljive temperature. Napetostno-deformacijsko stanje je popisano s Prandtlovimi operatorji. Postopek temelji na stabilnih histereznih zankah, pri čemer lezenje ni upoštevano. Analiziran je tudi vpliv filtriranja konic sunkov. Narejena je primerjava izsledkov raziskav z rezultati meritev in Skeltonovim modelom. © 2006 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: utrujanje termomehansko, enačbe konstitutivne, stanja napetostno-deformacijska, modeliranje) The isothermal strain-life approach is the most commonly used approach for determining fatigue damage, particularly in low-cycle fatigue. It is used for low, medium and high temperatures if the temperature remains 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 supposed to 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) 0 UVOD 0 INTRODUCTION Številni izdelki, npr. motorji z notranjim zgorevanjem, izpušni in hladilni sistemi, turbine, jedrski reaktorji, so v času uporabe obremenjeni z naključnimi termomehanskimi obremenitvami (TMU), ki lahko privedejo do utrujenostnih razpok [1]. Z razvojnega vidika je zato pomembna napoved dobe trajanja izdelka v razvojnih fazah pred preskušanjem prototipov oziroma izdelkov. Doba trajanja je odvisna predvsem od obremenitev, uporabljenih materialov, oblike izdelka in okoliščin. Za njeno dokazovanje se uporabljajo naslednji preizkusi ([2] in [3]): • deformacijsko nadzorovani malociklični preizkus (MCUP - LCF) pri nespremenljivi temperaturi, • termomehanski preizkus delov in izdelkov in • toplotni sunek. Many products, like internal combustion engines, exhaust and cooling systems, turbines, nuclear reactors, are subjected to stochastic thermo-mechanical fatigue (TMF) loading that can cause fatigue failures during their usage [1]. From the development perspective, fatigue-life prediction in the development phases before prototype or product testing is important. Fatigue life depends primarily on loads, applied materials, product geometry and environmental effects. Its approval is generally based on the following tests ([2] and [3]): • isothermal strain-controlled low-cycle fatigue (LCF) tests, • TMF tests on specimens and components, • thermal shock tests. 74 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 74-84 MCUP se izvajajo prednostno, ker so v primerjavi z drugimi preprostejši [4]. Poleg tega se opravljajo že več desetletij, postopek testiranja in zapisovanja rezultatov testiranj pa je standardiziran [2]. Obstajajo tudi baze podatkov o materialnih lastnostih, potrebnih za napoved dobe trajanja malociklično obremenjenih delov Končni cilj je razviti metodologijo napovedovanja dobe trajanja, ki bo omogočala zanesljive napovedi brez termomehanskih testiranj in testiranj s toplotnim sunkom. Področje je poglobljeno predstavljeno v [3] do [9]. V prispevku je obravnavan problem napovedovanja napetostno-deformacijskih stanj ob upoštevanju spremenljive temperature obratovanja. Predpostavimo, da je vpliv lezenja in prehodnih pojavov, npr. utrjevanja ali mehčanja, neznaten. Napovedovanje napetostno-deformacijskih stanj torej temelji na stabilnih histereznih zankah in konstitutivnih enačbah elastoplastičnosti. Izmed razpoložljivih reoloških modelov ([9] do [11]) je bil izbran model vzporedno vezanih vzmeti in drsnikov [11], izražen s Prandtlovim modelom s temperaturno odvisnimi gostotami in temperaturno neodvisnimi drsnimi površinami. Model omogoča razširitev deformacijskega postopka na področje spremenljivih temperatur, pri čemer se velika hitrost računanja ne zmanjša. Napetostno nadzorovan in temperaturno preoblikovan reološki model je obravnavan v [12] in [13]. V primeru deformacijskega nadzora potrebujemo enakovreden, a drugačen model. V drugem poglavju so obravnavane temperaturno odvisne napetostno-deformacijske krivulje in ocenjevanja parametrov. V tretjem poglavju je prikazan razvoj temperaturno preoblikovanega in deformacijsko nadzorovanega modela vzmet - drsnik. V četrtem poglavju so analizirane posledice izločanja konic sunkov. V petem poglavju je opisano ujemanje novega modela z rezultati preizkusov in s Skeltonovim modelom, v zadnjem poglavju pa so podani pomembnejši sklepi. 1 TEMPERATURNO ODVISNE NAPETOSTNO-DEFORMACIJSKE KRIVULJE Ker večina materialov preide po nekem manjšem številu obremenitvenih ciklov v stabilno področje, se za napoved dobe trajanja navadno uporabijo materialne lastnosti, ki ustrezajo polovični dobi trajanja preskušanca. Temperaturni časovni poteki obremenitev so v splošnem naključni. V literaturi je mogoče najti napetostno-deformacijske krivulje za le nekaj različnih In terms of complexity, LCF tests are favoured because of their simplicity [4]. In addition, because they have been carried out for decades, the testing procedures and data recording is standardized [2]. There also exist databases with material properties required for the fatigue-life prediction of low-cycle loaded parts. The final goal is, therefore, to develop a methodology of fatigue-life prediction that will enable reliable estimations by avoiding expensive TMF and thermal shock tests. The methodology is discussed comprehensively in [3] to [9]. The present work is concerned with a cyclic stress-strain response prediction suitable for variable temperatures. Assuming that creep and transient effects, such as cyclic hardening and cyclic softening and their effect on damage accumulation are negligible, the hysteresis loops are supposed to be stabilized and the constitutive equations for elastoplasticity are applicable for stress-strain behaviour modelling. From among the available models ([9] to [11]), the strain-controlled spring-slider model [11] with temperature-dependent Prandtl densities and temperature-independent yield surfaces has been chosen. It enables the extension of the isothermal strain-life approach to non-isothermal problems by preserving a high computational speed. The stress-controlled and temperature-modified spring-slider model is explained in [12] and [13]. However, if the strain is controlled, an equivalent but distinct spring-slider is required. In Section 2 the temperature-dependent cyclic stress-strain curves and parameter assessment is discussed. In Section 3 the temperature-modified strain-controlled spring-slider model is developed. In the 4th section the consequences of reversal point filtering are analysed. In Section 5 the spring-slider model is compared to several TMF tests, and the final section lists the conclusions. 1 TEMPERATURE-DEPENDENT CYCLIC STRESS-STRAIN CURVES Most metals approach a cyclically stable state after a certain number of cycles. Cyclically stable or half-life material properties are usually used in fatigue analyses. Generally speaking, the temperature–time history is stochastic, but cyclic stress–strain curves for only a few distinct temperatures can be found in the literature. To Modeliranje naključnih termomehanskih - Thermo-Mechanical Modelling of Stochastic 75 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 74-84 temperatur. Napoved napetostno-deformacijskih stanj pri temperaturah, pri katerih napetostno-deformacijske krivulje ne obstajajo je mogoča z uporabo interpolacije, aproksimacije ali neparametričnih metod. Predvsem slednje so se v zadnjem času izkazale za najboljše. Za popis elastoplastičnih lastnosti gradiv se najpogosteje uporablja Ramberg-Osgoodova enačba [14]: e = g(s,T) obtain the material parameters that have not been measured, linear parameter interpolation, approximation or non-parametric methods can be used. The latter turned out to be the most appropriate recently. Elasto-plastic hardening solids are most frequently modelled by the Ramberg– Osgood relation [14]: E(T) yK'(T) 1n'(T) (1), kjer je E(T) modul elastičnosti, K’(T) in n’(T) pa sta utrjevalni koeficient in utrjevalni eksponent. Interpolacija je bila obravnavana v [12]. 2 MODELIRANJE NAPETOSTNO-DEFORMACIJSKIH STANJ V [12] je bilo dokazano, da Masingov model in spominski modeli niso veljavni, če se temperatura med obremenitvenim potekom spreminja Masing je izsledke svojih raziskav utemeljil z reološkim modelom vzmet -drsnik in predpostavil, da so parametri modela neodvisni odčasa [11]. Da bi odpravili omejitve omenjenih modelov, je bil model vzmet - drsnik temperaturno spremenjen ([12] in [13]). Model v sedanji obliki je primeren za modeliranje elastoplastičnih lastnosti gradiv in nelinearnega kinematičnega utrjevanja pri sočasnem nadzoru napetosti. Če je deformacija nadzorna spremenljivka je na sliki 1 prikazan primernejši model za modeliranje napetostno-deformacijskih stanj. Iz ravnotežja na segmentu vzmet - drsnik je mogoče izraziti celotno deformacijo e: e = eqj+ eaj. Deformacija drsnika |e | ne more nikoli preseči navidezne drsne deformacije qj. Deformacija vzmeti where E(T) is Young’s modulus and K’(T) and n’(T) are the cyclic hardening coefficient and the cyclic hardening exponent respectively. Parameter interpolation has been dealt with in [12]. 2 STRESS-STRAIN-TEMPERATURE BEHAVIOUR It has been shown in [12] that the Masing and Memory models are not valid if the temperature varies during the cycle. Masing based his finding on the rheological spring-slider model and assumed that the model parameters are time independent [11]. Therefore, the spring-slider model has been adapted for variable temperatures ([12] and [13]). The model developed so far is capable of modelling elasto-plastic hardening solids and non-linear kinematic hardening under stress control. If the strain is controlled, the rheological spring-slider model depicted in Fig. 1 proves to be more convenient. From the equilibrium in a single spring-slider segment, the total strain e is obtained e = eqj + eaj, where the slider strain |eqj can never exceed the fictive yield strain qj . The spring strain can now be expressed as: e$= e-sign(e-e^min{qj,|e-e^ |} (2) sj f _t_e(t) eaj(t)f-paj(T) eqj(t)f qj te ia0(T)- Ea1(T) r D q0 D nuni nntaniiiitaš q1 aj(T) fs an(T) qnqq Sl. 1. Reoloski model vzmet - drsnik Fig. 1. Rheological spring-slider model 76 Nagode M. - Fajdiga M. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 74-84 ustreza operatorju ohlapa s splošno začetno and corresponds exactly to the play operator with vrednostjo ([15] do [17]): the general initial value ([15] to [17]): eaj(ti) = max{e(ti)-qj, min{e(ti) + qj,eqj(ti-1)}} (3) za 0 < t1 < t2 < ¦¦ ¦ < t . Trenutna deformacija eaj(ti) for 0 < t1 < t2 < ¦¦ ¦ < t . Thus, the current strain state je odvisna od deformacije v predhodnem trenutku eaj (ti) depends on the previous state eaj (ti 1) called ej (ti1), imenovanem spominska točka. the memory point. Presumably there is no residual Predpostavimo, da zaostalih deformacij ni. Odtod strain initially, so eaj (0) = 0 and sj (0) = 0. The stress izhaja eaj(0) = 0 in sj(0) = 0. Napetost v odseku in the spring-slider segment is then: vzmet - drsnik je tedaj: Ti=T(ti) in a j (Ti) je Prandtlova gostota. S where Ti=T(ti) and a (Ti) is the Prandtl density. seštevanjem napetosti po posameznih odsekih je Adding the spring-slider stresses results in the total mogoče izraziti celotno napetost s Prandtlovimi stress in the form known as the operator of the operatorji ([15] do [17]): Prandtl type ([15] to [17]): s(ti) = nq aj(T)eaj(ti) (5) s temperaturno odvisnimi gostotami. Ker je operator with temperature-dependent Prandtl densities. The rege, definiran v en. (3), neodvisen od časa in play operator given in Eq. (3) is independent of time temperature, ga je treba spremeniti tako, da bo and temperature. Therefore, it is modified to assure zagotavljal ravnotežje v odseku vzmet - drsnik tudi equilibrium in the spring-slider. pri spremenljivih temperaturah. Let us presume that at ti-1 the spring-slider Če je v trenutku ti-1 j-ti odsek v ravnotežju, segment j is in equilibrium and multiply Eq. (3) by pomnožimo en. (3) z aj (Ti -1) (sl. 2). Odtod izhaja: aj (Ti-1) (Fig. 2). This yields: s j (ti-1) = max{(e (ti-1) - qj)aj (Ti -1),min{(e (ti -1) + qj)aj (Ti -1), sj (ti-2)}} (6), kjer velja s (ti-2) = s (ti-1). Če se v koraku [ti -1, ti] where the memory point s (ti-2) = s (ti-1). If the spremenita temperatura ali deformacija, se meje temperature or strain change in the interval [ti 1,ti], operatorja ohlapa premaknejo in: then the boundaries of the play operator move and: sj (ti) = max{(e(ti ) - qj)aj (Ti),min{(e(ti) + qj)aj (Ti), sj (ti -1)}} Če nato enačbo delimo z aj (Ti), dobimo: If the equation is divided by aj (Ti), then: ej(ti ) = maxLti)-q,min|e(ti) + qj , aj(Ti)}J (7) spominska točka i memory point (e(ti-1) - qj)aj(Ti-1) sj(ti-1) (e(ti-1) + qj)aj(Ti-1) ' meje \ l ~\ * boundaries sj(ti) Sl. 2. Ravnotežje v modelu vzmet - drsnik Fig.2. Equilibrium in the spring-slider model Modeliranje naključnih termomehanskih - Thermo-Mechanical Modelling of Stochastic 77 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 74-84 in nazadnje: and finally: eaj (ti ) = max e (ti )-qj, min e (ti ) + qj , aj(Ti) e«j (t) (8), kjer je sj (ti -1) = aj (Ti-1)e (ti-1). Z enačbama (8) in (5) je mogoče modelirati napetostno-deformacijska stanja tudi v primeru naključno spreminjajočih se temperatur, saj temperaturno spremenjen operator ohlapa vedno zagotavlja ravnotežje v vseh odsekih vzmet - drsnik. Ker so ponovitvene napetostno-deformacijske krivulje znane, je Prandtlove gostote mogoče izračunati vnaprej. Naj bo n +1 navideznih drsnih deformacij enakomerno razporejenih med ničelno in največjo pričakovano amplitudo deformacije (sl. 3). Za vsako navidezno drsno deformacijo qj v območju j = 0,K,n in vsako temperaturo Tk v območju k = 0,K, nT je z en. (1) mogoče izračunati deformacije ej(Tk) in napetosti sj (Tk). Prandtlove gostote izračunamo tako, da v en. (5) vstavimo napetosti: as sj(ti-1) = aj(Ti-1)e (ti-1). When Eq. (8) is inserted into Eq. (5), the temperature-dependent stress-strain behaviour can be modelled, since the temperature-modified play operator guarantees the equilibrium in the spring-slider segments at any time and temperature. As the temperature-dependent cyclic stress-strain curves are known, the Prandtl densities can be precalculated. Let us disperse nq +1 fictive yield strains equidistantly between the zero strain and the maximum expected strain amplitude (Fig. 3). Thus, for each fictive yield strain qj in the range j = 0,K,nq and each temperature Tk in the range k = 0,K,nT, the strain ej(Tk) and the stress sj (Tk) can be determined from Eq. (1). By inserting qj and sj (Tk) into Eq. (5) the Prandtl densities are obtained: aj(Tk) = —(sj+1(Tk)-2sj(Tk) + sj kjer je s-1 (Tk) = s0 (Tk) = 0, širina razreda navidezne drsne deformacije pa je Dq. Prandtlove gostote je pametno izračunati vnaprej, jih shraniti v preglednico rešitev ter šele nato modelirati napetostno-deformacijska stanja z enačbama (8) in (5). Tako ohranimo veliko hitrost računanja. 1(Tk)) j=0,K,n k = 0,K,nT (9), where s- 1(Tk) = s0(Tk) = 0 and the fictive yield strain class width is Dq. The Prandtl densities can be precalculated and stored in a table before the stress-strain-temperature trajectory modelling using Eqs. (8) and (5) starts. In this way a high computational speed is preserved. sj(T0) j(TntT) - YJ'-V Tk] / t? /) T ntT ' Dq <--------—»• q0 q1 q2 qj Sl. 3. Temperaturno odvisne napetostno-deformacijske krivulje Fig. 3. Temperature dependent cyclic stress-strain curves 78 Nagode M. - Fajdiga M. e Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 74-84 3 POSLEDICE IZLOČANJA KONIC SUNKOV Če so materialne lastnosti časovno neodvisne, temperature pa nespremenljive, lahko poteke obremenitev pred začetkom modeliranja napetostno-deformacijskih stanj zgostimo in tako izločimo le zaporedje konic sunkov brez časovne komponente. Poteke obremenitev pregledujemo sproti in izločamo vse točke, ki niso lokalni ekstremi ter obremenitvene cikle, ki ne prispevajo k zbiranju poškodb [17]. Ker se z zgoščevanjem poteki močno skrajšajo, je deformacijski postopek numerično bistveno hitrejši od zahtevnejših postopkov in se zato tudi pogosto uporablja. Če se temperatura med posameznimi konicami sunkov spreminja, lega naslednje konice ni več odvisna le od lege predhodne konice, ampak tudi od prehoda med njima. Slika 4 prikazuje vpliv spreminjanja temperature na krivuljo a-sza zlitino 9Cr2Mo, ki se začne v točki 0 (T0 = 270 °C), gre skozi točki 1 (T1 = 570 °C) in 2 (T2 = 270 °C) ter se konča v točki 3 (T3 = 570 °C). Predpostavimo, da sta poteka deformacij in temperatur nezgoščeni in izračunajmo krivuljo napetost-deformacija (glej debelo polno črto) po enačbah iz prejšnjega poglavja. En. (8) zagotavlja ravnotežje v odsekih vzmet - drsnik v vseh točkah, označenih s trikotniki. Izkaže se, da se izračunane napetosti ne 3 CONSEQUENCES OF REVERSAL POINT FILTERING For rate-independent material behaviour and constant temperature, only the reversal points in the load histories are needed for stress-strain trajectory modelling and cycle counting. Load histories can thus be compressed and the time component can be eliminated. The load histories are scanned online and any point that is not the reversal point is discharged from the history. Similarly, all reversal points corresponding to a change in amplitude smaller than a certain threshold are filtered [17]. As history compression results in a considerable reduction of history lengths the strain-life approach is much faster than competing ones. For this reason it has been used frequently. However, if the temperature changes between reversal points, the position of the next point does not depend only on the current point, but also on the transition between the points. Fig. 4 shows the influence of temperature variations on the as trajectory for a 9Cr2Mo alloy. It starts at point 0 (T0 = 270 °C), goes through points 1 (T1 = 570 °C) and 2 (T2 = 270 °C) and ends at point 3 (T3 = 570 °C). Let us suppose that the strain and temperature histories are not filtered and calculate the stress-strain-temperature trajectory as explained in the preceding section (the thick solid line). Eq. (8) ensures equilibrium in the spring-slider model at any point indicated by triangle markers. It has been observed that the stresses do not necessarily coincide (a) 320 0,26%—0,26% (b) 240 _ 160 _ 80 0,0 ,1,38% 0 -95 -190 _ -285 _ 0,1 0,2 0,3 0,4 -380 0 -0,4 -0,3 -0,2 -0,1 0,0 e, % e, % Sl. 4. Vpliv temperature na spremembe s-e krivulj Fig. 4. Influence of temperature variations on s-e trajectory 0 Modeliranje naključnih termomehanskih - Thermo-Mechanical Modelling of Stochastic 79 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 74-84 0,45 0,30 0,15 _ 0,00 0 skok jump -0,15 -0,30 -0,45 0 5 10 15 20 25 30 35 40 45 50 55 ti-1 Sl. 5. Spominske točke pri spremenljivi temperaturi Fig. 5. Memory point history at variable temperature ujemajo vedno z napetostmi na izotermnih napetostno-deformacijskih krivuljah (črtkane črte s simboli v obliki križcev). S tem je dokazan vpliv prehodov med konicami sunkov na krivuljo as. Podobna odstopanja se pojavijo v primeru zgoščenih potekov obremenitev, kjer se lega točk v ravnini vračuna le za konice sunkov (kvadratni simboli). Odstopanja so odvisna od spominskih točk in temperaturnih sprememb med konicami sunkov. Na slikah 5 in 6 so prikazani poteki spominskih točk za j = 0,K, nq operatorjev ohlapa (debele polne črte), omejenih z zgornjimi (tanke črtkane črte) in spodnjimi (tanke polne črte) mejami. Točke 0 do 3 ustrezajo konicam sunkov. Če je temperatura T0 = L = T3 = 570 °C , se spominske točke vedno ujemajo z eno od meja oziroma ostajajo nespremenjene (sl. 6). Če pa se temperatura spreminja (sl. 5), lahko spominske točke preskočijo z ene meje na drugo ali pa med njimi nihajo. To je torej razlog za odstopanja napetosti. S slike 5 je razvidno, da so odstopanja neodvisna od dolžine krivulje as. Ko s spremeni smer in se začno spominske točke približevati spodnji ali zgornji meji, se začno odstopanja zmanjševati. Ker ostajajo odstopanja po vrednosti vedno omejena, je izločanje konic sunkov sprejemljivo tudi v primeru spremenljivih temperatur. 0,45 0,30 0,15 _ 0,00 < -0,15 -0,30 -0,451 — ¦ — ¦ — ¦ — ¦ — ¦ — ¦ - -V- ¦ — ¦ — ¦ — ¦ — I 0 5 10 15 20 25 30 35 40 45 50 55 ti-1 Sl. 6. Spominske točke pri nespremenljivi temperaturi Fig. 6. Memory point history at constant temperature with those on the isothermal cyclic stress-strain curves (the dashed lines with cross markers). This way the influence of transitions between reversal points on the o-^curve is proved. Similar deviations are observed in the case of the compressed load histories, where the point positions in the emplane are calculated at load reversals only (square markers). The deviations depend on the memory points and the temperature variations between the reversal points. In Figs. 5 and 6 the memory-point histories for the j = 0,K, nq play operators (thick solid lines) bounded by their upper (thin dashed line) and lower (thin solid line) boundaries are shown. Points 0 to 3 correspond to the reversal points. If the temperature T0=L = T3= 570 °C, the memory points coincide with one of the boundaries or stay constant, as depicted in Fig. 6. However, if the temperature changes (see Fig. 5), the memory points can jump from one boundary to another or fluctuate between the boundaries. This is actually why stress deviations occur. From Fig. 5 it can be observed that the deviations do not depend on the length of the as trajectory. Whenever s changes its direction and memory points start to approach either the lower or upper boundaries, the deviations start to decrease. As deviation are always limited, only reversal points should be processed under variable temperatures, too. 80 Nagode M. - Fajdiga M. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 74-84 4 PREVERJANJE 4 VERIFICATION Deformacijsko nadzorovan in temperaturno spremenjen model, sestavljen iz vzporedno vezanih odsekov vzmet - drsnik, smo primerjali z večpreizkusi TMU, ki jih je opravil Skelton [4] pri 0,6 % celotne deformacije zlitine 9Cr2Mo. Temperaturno odvisni parametri, ki se pojavljajo v Ramberg-Osgoodovi enačbi, podatki o materialu in eksperimentalnih tehnikah so pojasnjeni v [2], [4] in [18]. Pri prikazu poteka deformacij in temperatur običajno nanašamo deformacije na ordinatno, temperature pa na abscisno os [4]. V prispevku sta obravnavani pot PKRM in zahtevna ponovitev (črtkana črta), prikazani na sliki 7. Medtem ko predstavlja pot PXRM 45° deltoidni cikel, izveden proti smeri urnega kazalca, je pot PMRK enaka ciklu, izvedenem v smeri urnega kazalca. Podobno velja tudi za 135° deltoidni cikel, omejen s točkami PKRZ. Cikla pravokotne oblike PXRZ in PKRM sta poimenovana 45° in 135° pravokotni cikel. Poti GEHF in GFHE tvorita dvojni termični cikel ter cikel zahtevne oblike je prikazan s črtkano črto. Izmerjene histerezne zanke so omejene s križci na slikah 8 in 9. Trikotniki označujejo napetostno-deformacijske krivulje, izračunane s Skeltonovim [4] algoritmom, debele polne črte pa krivulje, ocenjene po našem algoritmu. S slike 8 je razvidno, da je napoved obremenitvenih ciklov, izvedenih v smeri proti urnemu kazalcu, boljša od napovedi ciklov, izvedenih 0,4 0,3 0,2 0,1 0,0 -0,1 -0,2 -0,3 The strain-controlled and temperature-modified spring-slider model is compared to several TMF tests conducted by Skelton [4] at a total strain range of 0.6% on the 9Cr2Mo alloy. The temperature-dependent parameters associated with the Ramberg-Osgood relation as well as the material data and experimental methods are given in [2], [4] and [18]. In deciding the temperature-strain path, the convention is to plot strain on the vertical axis and temperature on the horizontal axis [4]. The paper concerns the path PKRM and the complex cycle (dashed line) given in Fig. 7. The path PXRM represents a 45° kite cycle performed in an anticlockwise way while the corresponding clockwise cycle is taken in the PMRX order. A similar scheme applies in the 135° kite cycle PKRZ. The parallelogram-shaped PXRZ and PKRM are labelled as the 45° zero strain and the 135° zero strain respectively. Finally, the anticlockwise GEHF and the clockwise GFHE bi-thermal cycles as well as the complex cycle given in the dashed line are considered. The observed hysteresis loops from the tests are plotted as crosses in Figs. 8 and 9. The triangle markers and the thick solid lines denote the stress-strain trajectories modelled by the Skelton [4] and our algorithm respectively. It is clear that in Fig. 8 the shapes of the anticlockwise loops are predicted well, whereas for -0,4 250 300 350 400 450 500 550 600 T, °C Sl. 7. Vrste TMU ciklov Fig. 7. TMF cycle types Modeliranje naključnih termomehanskih - Thermo-Mechanical Modelling of Stochastic 81 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 74-84 (a) 400 300 200 100 0 -100 -200 -300 -400 (b) 400 300 200 100 0 -100 -200 -300 -400 -0,35 -0,25 -0,15 -0,05 0,05 0,15 0,25 0,35 e, % -0,35 -0,25 -0,15 -0,05 0,05 0,15 0,25 0,35 e, % Sl. 8. TMU 135° zanka (a) proti smeri urnega kazalca (b) v smeri urnega kazalca Fig. 8. TMF loop 135° kite (a) anticlockwise, (b) clockwise v smeri urnega kazalca. Pri slednjih so testne histerezne zanke širše od napovedanih. Širše histerezne zanke niso le posledica nelinearnega ohlajanja, pač pa predvsem nekaterih parametrov, ki doslej še niso bili prepoznani. Ujemanje med Skeltonovimi in našimi rezultati je skoraj popolno. Postopek je bil nazadnje uspešno preverjen na ciklu zahtevne oblike, prikazani na sliki 9. 350 300 250 200 150 100 50 0 -50 -100 -150 -200 -250 -300 -350 -400 the clockwise style tests the observed loops are wider. The wider loops do not appear just because of non-linear cooling. Consequently, some parameters undefined so far seem to be more influential. The agreement between the Skelton [4] and our algorithm is almost perfect. The method was finally applied to the complex cycle of Fig. 9 successfully. -0,40 -0,30 -0,20 -0,10 0,00 0,10 0,20 e, % Sl. 9. Zahtevna TMU zanka Fig. 9. Complex TMF loop 82 Nagode M. - Fajdiga M. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 74-84 5 SKLEP 5 CONCLUSIONS Deformacijski postopek je alternativa The strain-life approach is an alternative to napetostnemu postopku napovedovanja utrujenostne the stress-life approach for determining fatigue poškodbe, ki se uporablja predvsem, če se pojavi damage, particularly when yielding occurs. An tečenje. V prispevku je predstavljen in z več TMU online temperature-modified strain-life approach preizkusi preverjen temperaturno spremenjen has been developed and verified through several deformacijski postopek napovedovanja napetostno- TMF tests. The spring-slider model has been deformacijskih stanj. Razvit in preverjen je bil developed and verified to enable local temperature- deformacijsko nadzorovani model, sestavljen iz stress-strain trajectory modelling. The model is vzporedno vezanih odsekov vzmet - drsnik, ki omogoča applicable if the strain is controlled or else the model modeliranje lokalnih temperaturno-napetostno- given in [12] can be used. Reversal point filtering deformacijskih stanj. Za primer napetostnega nadzora turned out to be acceptable for elasto-plastic je bil v [12] razvit enakovreden model. Izločanje konic hardening solids. The high computational speed sunkov je za elastoplastična gradiva sprejemljivo. Velika of the classical strain-life approach can thus be hitrost računanja običajnega deformacijskega postopka preserved. The developed model is a generalization se ohrani. Razviti model je formalno dobro definiran z of the Skelton [4] findings and is formally well modelom Prandtlovega tipa in pomeni posplošenje defined through an operator of the Prandtl type. Skeltonovih [4] izsledkov. Čeprav predlagani model ne Although neither creep damage nor some other omogoča napovedovanja lezenja in nekaterih drugih effects can be assessed by the proposed approach, pojavov, lahko dosežemo izboljšano napoved an improved temperature-stress-strain state temperaturno-napetostno-deformacijskih stanj na prediction based on standardized LCF tests is podlagi standardnih MCV preizkusov. possible. 6 LITERATURA 6 REFERENCES [I] Constantinescu A, Charkaluk E, Lederer G, Verger L (2004) A computational approach to thermomechanical fatigue. Int J Fatigue, 26, 805-818. [2] Skelton RP, Webster GA (1996) History effects on the cyclic stress-strain response of a polycrystalline and 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 under thermomechanical loadings. Fatigue Fract Engng Mater Struct, 25, 1199-1206. [4] Skelton RP (2004) Hysteresis, yield, and energy dissipation during thermo-mechanical fatigue of a ferritic steel. Int J Fatigue, 26, 253-264. [5] Ellison EG, Smith EM (1973) Predicting service life in a fatigue-creep environment. In: Carden AE, McEvily AJ, 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 of superalloys. J Mater, 51(4). [8] Chaboche JL, Gallerneau F (2001) An overview of the damage approach of durability modelling at elevated temperature. 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 properties of metals and solid polymers. Int J Plast, 19, 1069-1095. [II] Conle A, Oxland TR, Topper TH (1988) Computer-based prediction of cyclic deformation and fatigue behaviour. In: Solomon HD, Halford GR, Kaisand LR, Leis BN, editors. Low cycle fatigue. ASTM STP 942, pp 1218-1236. Modeliranje naključnih termomehanskih - Thermo-Mechanical Modelling of Stochastic 83 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 74-84 [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 note no. 902, NACA. Avtorjev naslov: prof.dr. Marko Nagode prof.dr. Matija Fajdiga Univerza v Ljubljani Fakulteta za strojništvo Aškerčeva 6 1000 Ljubljana marko.nagode@fs.uni-lj.si matija.fajdiga@fs.uni-lj.si Author’s Address: Prof.Dr. Marko Nagode Prof.Dr. Matija Fajdiga University of Ljubljana Faculty of Mechanical Eng. Aškerčeva 6 SI-1000 Ljubljana, Slovenia marko.nagode@fs.uni-lj.si matija.fajdiga@fs.uni-lj.si Prejeto: Received: 2.11.2005 Sprejeto: Accepted: 16.11.2005 Odprto za diskusijo: 1 leto Open for discussion: 1 year 84 Nagode M. - Fajdiga M. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100 UDK - UDC 621.431.37:629.063.6 004.94 Izvirni znanstveni članek - Original scientific paper (1.01) Računalniško modeliranje gibanja goriva in njegov vpliv na konstrukcijo rezervoarja Computational Modelling of Fuel Motion and Its Interaction with the Reservoir Structure Matej Vesenjak1 - Zoran Ren1 - Heiner Miillerschon2 - Stephan Matthaei3 (1Fakulteta za strojništvo, Maribor; 2DYNAmore GmbH, Stuttgart-Vaihingen; 3DaimlerCrysler AG, Stuttgart) Računalniški modeli vozil za simuliranje trkov vedno natančneje opisujejo obnašanje resničnih vozil. Rezervoar za gorivo je eden izmed elementov vozil, katerega računalniški modeli so bili do sedaj zelo poenostavljeni. Taksni modeli upoštevajo le vztrajnost mase goriva, ki je z masnimi točkami pritrjena na steno rezervoarja, vendar pa je vpliv gibanja goriva v rezervoarju popolnoma zanemarjen. Prispevek opisuje nove računalniške modele, s katerimi je mogoče simulirati deformacijo rezervoarja za gorivo ob upoštevanju gibanja goriva pri trku vozila. V ta namen so bile vrednotene štiri metode simuliranje gibanja tekočine (Lagrange, Euler, poljubnostna Lagrange-Eulerjeva metoda - PLE in hidrodinamika zglajenih delcev - HZD) v rezervoarju preproste oblike, analizirane z eksplicitnim programom LS-DYNA. Računalniški rezultati so bili primerjani s poprej objavljenimi preizkusnimi opazovanji, pri čemer je bila ugotovljena zelo dobra primerljivost med rezultati. Najprimernejši metodi (HZD in PLE) sta bili kasneje uporabljeni v dinamičnih simulacijah dejanskega rezervoarja za gorivo. Simulacije so pokazale, da predstavljeni modeli rezervoarja ob upoštevanju gibanja goriva zagotavljajo mnogo natančnejše rezultate v primerjavi z znanimi poenostavljenimi modeli. © 2006 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: gibanje goriva, Lagrangev opis, Eulerjev opis, ALE, SPH, vplivi tekočina - trdnina) Computational models of vehicles for crash simulations are ever more precisely describing the behaviour of real vehicles. A fuel-tank is a typical vehicle element that has been very simplified in the computational 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 deformation considering the fuel motion during a vehicle crash. For this purpose four different methods for describing fluid motion (Lagrangian, Eulerian, Arbitrary Lagrange-Eulerian description - ALE, SPH) were evaluated on a simple reservoir problem, analysed with the explicit dynamic code LS-DYNA. The computational results were compared with previously published experimental observations and a good correlation of the results was observed. The most appropriate methods, SPH and ALE, were afterwards used in dynamic simulations of a real fuel-tank The simulations showed that by also taking into consideration the fuel motion, the proposed computational models provide more accurate results in comparison with the previously used, simplified models. © 2006 Journal of Mechanical Engineering. All rights reserved. (Keywords: fuel motion, Lagrangian description, Eulerian description, ALE, SPH, fluid structure interaction) 0 UVOD 0 INTRODUCTION V zadnjem času postajajo numerične simulacije v avtomobilizmu vse pomembnejše. Njihov cilj ni več zgolj določitev globalnega obnašanja vozil, temveč tudi natančnejše obnašanje posameznih In recent years numerical simulations in automotive engineering have gained in importance. Their aim is not only the determination of the global vehicle behaviour but also the behaviour of single 85 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100 sestavnih delov. Pri razvoju konstrukcije novega vozila ali izboljšavi konstrukcije že znanega vozila so za računalniško simuliranje komponent najpogosteje uporabljene analize po metodi končnih elementov. V mnogih primerih simuliranje posameznih komponent ni zadostno, zato morajo biti v analizo vključene komponente oziroma celotni sklopi, ki vplivajo drug na drugega. Reševanje problemov postane zahtevnejše, kadar so sestavni elementi sklopa iz dveh ali več fizikalnih sistemov, ki medsebojno vplivajo. Kadar medsebojno vplivata dva ali več fizikalnih sistemov in je rešitev posameznega brez upoštevanja ostalih nemogoča, govorimo o vezanih problemih [20]. Eden izmed delov vozila, pri katerem vplivata dva fizikalna sistema, je rezervoar za gorivo. V preteklih simulacijah je bilo simuliranje rezervoarjev zelo poenostavljeno, saj vpliv gibanja tekočine na obnašanje rezervoarja ni bil obravnavan. Upoštevana je bila le masa goriva, ki je bila porazdeljena po posameznih vozliščih sten rezervoarja zaradi upoštevanja vztrajnostnih sil. Gibanje goriva pri trku vozila, ki ima velik vpliv tako na deformacijo rezervoarja kakor tudi njegovih nosilnih elementov, pa je bilo popolnoma zanemarjeno. Omenjene poenostavitve so bile neizogibne zaradi omejene uporabe simulacijskih programskih paketov, vendar pa novejši programski paketi omogočajo tudi učinkovito reševanje več fizikalnih sistemov hkrati. V prispevku so opisani in vrednoteni različni numerični modeli, s katerimi je mogoče opisati gibanje in vpliv tekočine na trdnino s programskim sistemom LS-DYNA [10]. LS-DYNA temelji na metodi končnih elementov in je bila prvotno namenjena reševanju dinamičnih problemov v mehaniki trdnin. Modeliranje odzivov v dinamiki trdnin je zato zelo dobro razvito. Modeliranje vezanih problemov med tekočino in trdnino pa še vedno ostaja izziv. Primerjava metod in njihova uporabnost je ponazorjena na praktičnem primeru gibanja tekočine v rezervoarju s preprosto geometrijo. Z eksperimentalnimi meritvami pa so primerjani tudi rezultati analiz [11]. 1 MEDSEBOJNI VPLIV TEKOČINE IN TRDNINE Eden izmed najpogosteje obravnavanih vezanih problemov v inženirski praksi je medsebojni vpliv tekočine in trdnine, zaradi česar so se razvili različni postopki reševanja teh problemov. K razvoju algoritmov za računanje vezanih problemov je pripomogel tudi silovit razvoj components. In the case of computational component simulation, during the designing phase of a new vehicle structure or during the design improvements of existing vehicle designs, finite-element method analyses are most often used. In many cases, however, simulating separate components is not sufficient. Therefore, other components or assemblies that interact with each other have to be included in the analysis. The solution process becomes more sophisticated when the assembly components consist of two or more different physical systems that interact. When two or more physical systems interact with each other and an independent solution of one system is impossible without a simultaneous solution of the others, such systems are known as coupled [20]. One of the vehicle components where two physical systems interact is the fuel-tank. In previous simulations the model of the tank was very simplified, since the fluid motion’s influence on the tank’s behaviour was not considered. Only the fuel mass was taken into account, which was distributed at discrete nodes along the tank walls to account for the fuel inertia forces. The fuel motion during a vehicle crash, which has a large influence on tank deformation and its supporting elements, was thus completely neglected. Such simplifications were necessary due to the limitations of the simulation software used. However, some recent releases of simulation software already allow for an effective solution of several physical systems simultaneously. In this paper different computational models that allow for consideration of the fluid motion and its influence on the structure are described and evaluated with the software LS-DYNA [10]. LS-DYNA is based on the finite-element method and it was originally designed for solving structural dynamic problems. Therefore, its ability to model structural responses in general is well defined. However, the modelling of a coupled fluid-structure interaction is still quite challenging. A comparison of the methods and their applicability is illustrated on a practical example, describing the fuel motion in a reservoir with simple geometry. The computational results are further compared with the experimental measurements [11]. 1 FLUID-STRUCTURE INTERACTION One of the most popular coupled problems in engineering is the interaction between the fluid and the structure, which consequently results in different approaches to solving such a problem. The development of coupled problems solution algorithms was also influenced by the rapid develop- 86 Vesenjak M. - Ren Z. - Mullerschon H. - Matthaei S. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100 računalniške opreme, saj so tovrstne analize računsko zelo zahtevne. Vezane probleme je z uporabo različnih programskih paketov mogoče reševati na dva načina: . z uporabo dveh programskih paketov: eden za določitev območja tekočin in eden za ločeno določitev rešitve območja trdnine. Takšni programski paketi (npr. CFX in Nastran) so po navadi povezani z vmesnikom, ki nadzoruje izmenjavo potrebnih podatkov o robnih pogojih ([4] in [19]); . z uporabo programskega paketa, ki omogoča hkratno reševanje dveh ali več vezanih fizikalnih sistemov (npr. ADINA, ANSYS, LS-DYNA). Programski paket LS-DYNA omogoča dinamično ugotavljanje medsebojnega vpliva med tekočino in trdnino z eksplicitno integracijsko shemo. V LS-DYNI je vpliv tekočine mogoče opisati in simulirati na dva načina: (i) s stičnimi silami (uporabno pri Lagrangevem in SPH modelu) in (ii) s kriterijem poroznosti snovi, ki v vsakem opazovanem elementu določi silo, ki je potrebna za vzpostavitev ravnotežja med tekočino in trdnino (uporabno pri Eulerjevem in modelu PLE) ([1], [6], [10] in [12]). ment of computer hardware, because such simulations are computationally very intensive. Coupled problems can be solved in two ways, by using commercial software: . with the use of two commercial codes: one for solving the fluid domain and one to separately determine the solid-domain solution. Such codes (e.g., CFX and Nastran) are usually connected via an interface that controls the necessary boundary conditions’ data exchange ([4] and [19]); . with use of a commercial software that simultaneously solves a multi-physics problem (e.g., ADINA, LS-DYNA, ANSYS). The LS-DYNA code is capable of establishing a dynamic interaction between the fluid and the structure with an explicit integration scheme. In LS-DYNA it is possible to describe and simulate the influence of the fluid on the structure, and vice versa, in two ways: (i) with contact forces (applicable for Lagrangian and SPH model) and (ii) with a leakage criteria, which in each observed element determinates the force that is necessary to establish equilibrium conditions between the fluid and the structure (applicable for Eulerian and ALE model) ([1], [6], [10] and [12]). 2 RAZLIČNI OPISI DOMEN 2.1 Lagrangev opis Lagrangev opis se običajno uporablja za opis problemov v mehaniki trdnin. Problem se opiše z velikim številom masnih delcev, pri čemer se opazuje gibanje vsakega posameznega delca v prostoru in času (sl. 1). Problem je natančno določen, kadar poznamo gibanje vseh delcev [17]. 2 DIFFERENT DOMAIN DESCRIPTIONS 2.1 Lagrangian description The Lagrangian formulation is usually used for describing solid mechanics problems. The problem is described with a large number of mass particles, where the motion of every single particle is observed in space and time (Fig. 1). The problem is exactly defined when the motion of all the particles is known [17]. M0 ( x0,y0,z0 ) t = t, M (x, y, z) Sl. 1. Lagrangev opis [17] Fig. 1. Lagrangian formulation [17] Računalniško modeliranje gibanja goriva - Computational Modelling of Fuel Motion 87 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100 J__i__L 4 t = 0 t =At t = 2At Sl. 2. Deformacija mreže pri Lagrangevem opisu Fig. 2. Mesh deformation in the Lagrangian formulation Lagrangev način opisa gibanja je preprost in nezahteven za uporabo, dokler gre za posamezni masni delec. V primeru proučevanja problema z velikim številom masnih delcev pa postane izredno zahteven in zapleten [17]. Pri Lagrangevem opisu predstavlja en končni element isti delež materiala celoten potek analize. Mreža končnih elementov je pritrjena na material med celotnim računskim postopkom, zaradi česar se pomika z materialom. Slika 2 prikazuje način reševanja preprostega primera tekočine z Lagrangevim opisom. Predpostavljeno je, da obremenitev deluje le na sredinsko vozlišče. Rezultat te obremenitve je pomik vozlišča v računskem časovnem koraku. Če vpliv obremenitve ne preneha, oz. se spremeni, vozlišče v naslednjem časovnem koraku ponovno zavzame novo lego in mreža se vedno bolj deformira, saj sledi toku materiala. Ker mreža končnih elementov sledi materialu, ima relativen pomik medsebojno povezanih vozlišč izrazit vpliv na deformacijo končnih elementov. S premikom elementov se prenašajo tudi masa, gibalna količina in energija [2]. Enačba o ohranitvi mase se lahko v Lagrangevem opisu v parcialni diferencialni obliki zapiše kot: The Lagrangian formulation is very simple and easy to use for one or only a few mass particles. However, the method becomes very complicated and complex for a description of large number of mass particles [17]. In the Lagrangian formulation, one finite element represents the same part of the material throughout the course of the analysis. The finite-element mesh is fixed to the material during the entire computational process and therefore moves with the material. Figure 2 illustrates the solution process of a simple fluid problem using the Lagrangian formulation. It is presumed that the loading influences only the central node. The result of the loading is the shift of that node in a computational time step. If the influence of the loading does not stop or change, the node takes a new position in the next time step and the mesh deforms even more, since the mesh follows the material flow. Since the mesh follows the material, the relative movement of connecting nodes can result in a significant deformation of the finite elements. Mass, momentum and the energy are transported with the movement of the elements [2]. The mass conservation equation can be, in partial differential equation form, written for the Lagrangian description as: dt dx (1), kjer sta p gostota in v hitrost materiala (mreže). Za nestisljiv material je znano, da je div(v) = 0. Lagrangev opis ohranitve mase se lahko zapiše tudi v obliki algebrajske enačbe kot: where p is the density and v is the velocity of the material (mesh). For an incompressible material it is known that div(v) = 0. The Lagrangian description for the mass conservation can also be written in an algebraic equation as: P-J = P0 (2), kjer je J Jacobijeva matrika med trenutno in where J is the Jacobian between the current and referenčno obliko. Ohranitev gibalne količine je v reference configuration. Conservation of momentum Lagrangevem opisu podana kot: in terms of the Lagrangian description gives: p i = — + P'bi dt dx (3), 88 Vesenjak M. - Ren Z. - Mullerschon H. - Matthaei S. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100 kjer sta s Cauchy-jeva napetost in b masna sila (sila na enoto mase). Tretja enačba predstavlja ohranitev energije in se lahko v primeru izključno mehanskega postopka zapiše kot: where s is the Cauchy stress and b is the body force (a force per unit mass). The third equation represents the energy conservation and can be, in a purely mechanical process, written as: r- 8u dt s 8vi dx (4), kjer je u notranja energija na enoto mase. Pomanjkljivost Lagrangevega opisa postane razvidna v primerih s skrajno popačeno mrežo, saj njihov opis vedno temelji na mreži. Popačenost mreže vpliva na natančnost metode in zato na rezultat analize. Sprejemljiva možnost za izboljšanje Lagrangevega modela je ponovno mreženje domene problema. 2.2 Eulerjev opis Pri Eulerjevem opisu, ki se navadno uporablja za reševanja problemov računalniške dinamike tekočin, se problem opazuje v določeni prostorski točki in ne sledi gibanju posameznega delca (sl. 3). V enem časovnem koraku Dt gredo skozi točko številni masni delci, katerih gibanje je v trenutku prehoda natančno določeno. V opazovani točki se veličine polja spreminjajo s časom. Ohranitev mase je v Eulerjevem opisu zapisana kot: Dr D 8 8 kjer je Dt = Jt + vi-^ snovski oziroma hitrostni odvod, ki podaja časovno spremembo veličine in prostorsko spremembo veličine zaradi nehomogenosti hitrostnega polja (vsota lokalnega in konvektivnega odvoda). Ohranitev gibalne količine je izražena: where u is the internal energy per unit mass. The disadvantage of the Lagrangian description becomes evident in cases of an extremely distorted mesh, because their formulation is always based on mesh. When the mesh is heavily distorted, the accuracy of the formulation and hence the solution will be severely affected. A possible option to enhance the Lagrangian model is to re-mesh the problem domain. 2.2 Eulerian description In the Eulerian formulation, which is commonly used for solving computational fluid dynamic problems, the problem is being observed at separate points in space that do not follow the particles motion (Fig. 3). In one time step, Dt, several mass particles can pass through the observed point. Their motion is exactly determined at the moment of passing through that point. The field variables at the observed point are time dependent. In the Eulerian formulation the mass conservation is written as: 8vi 8x 0 (5), D 8 8 where Dt = Jt + vi ^ is the total time derivative that is physically the time rate of the change follow-ing a moving material element (sum of the local and the convective derivative). The conservation of momentum can be expressed as: Dvi = si, j + b ' Dt 8xi r' i M(x,y,z) (6). x Sl. 3. Eulerjev opis [17] Fig. 3. Eulerian formulation [17] Računalniško modeliranje gibanja goriva - Computational Modelling of Fuel Motion 89 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100 P- In ohranitev energije se lahko zapiše: Du Dt Temeljna razlika med Lagrangevim in Eulerjevim postopkom je v tem, da so pri Lagrangevem opisu veličine x, y in z spremenljive koordinate gibljivega delca. Pri Eulerjevem opisu pa te koordinate pomenijo mirujoče koordinate določene točke polja [17]. Kljub temu da se Eulerjeva mreža med analizo v LS-DYNI navidezno ne premika ali deformira, se dejansko spreminjata njena lega in oblika, vendar le v posameznemčasovnem koraku ([13], [14] in [16]). Razlog za to je uporaba Lagrangevega opisa v posameznih časovnih korakih, ki je naprednejša v LS-DYNI. Eulerjeva mreža je v LS-DYNI obravnavana na poseben način (sl. 4). Uporaba Eulerjeve mreže je ponazorjena na enakem primeru, ki je bil uporabljen pri Lagrangevem opisu. Zaradi obremenitev na sredinsko vozlišče opazovano vozlišče spremeni lego v enem računalniškem časovnem koraku (mreža se deformira). Po časovnem koraku se analiza ustavi in izvedeta se naslednji dva približka [5]: . premik vozlišč: vsem vozliščem Eulerjeve mreže, ki so zaradi obremenitev spremenila svojo lego, se ponovno določi izhodiščna lega; . interpolacija vmesnih rezultatov: vse notranje veličine (napetost, tokovna polja, hitrostno polje), ki se nanašajo na vozlišča s spremenjeno lego, so interpolirane tako, da imajo ustrezno prostorsko porazdelitev kakor pred premikom vozlišč. Tako premik vozlišč ne vpliva na porazdelitev notranjih veličin. Opisani postopek se ponavlja v vseh časovnih korakih celotne analize in ponuja uporabniku nepremično in nedeformirano Eulerjevo mrežo. And the conservation of the energy can be written as: (7). The basic difference between the Lagrangian and the Eulerian formulations is that in the Lagrangian formulation the magnitudes x, y and z are variable coordinates of a moving particle; in the Eulerian formulation those coordinates represent the steady coordinates of the defined field point [17]. Although the Eulerian mesh in LS-DYNA appears not to move or deform during the analysis, it does actually change its position and form, but only within a single time step ([13], [14] and [16]). The rea-son for this is the use of the Lagrangian formulation in single time steps, which is much more advanced in the LS-DYNA. The Eulerian mesh in LS-DYNA is treated in a special way (Fig. 4). To illustrate the use of an Eulerian mesh the same example is used as at the Lagrangian formulation. Because of the central node loading, the observed node changes its position during one computational time step (mesh deforms). After the time step the analysis stops and the following two approximations are performed [5]: . mesh smoothing: all the nodes of the Eulerian mesh that have been displaced due to loading are moved to their original position; . advection: the internal variables (stresses, flow fields, velocity field) for all the nodes that have been moved are recomputed (interpolated) so that they have the same spatial distribution as before the mesh smoothing. In this way the mesh smoothing does not affect the internal variable distribution. The described procedure is repeated for each time step of the analysis and provides the analyst with a non-movable and undeformable Eulerian mesh. premik vozlišč/ mesh smoothing interpolacija vmesnih rezultatov /advection t = 0 t =At t =At Sl. 4. Deformacija mreže pri Eulerjevem opisu Fig. 4. Mesh deformation in the Eulerian formulation 90 Vesenjak M. - Ren Z. - Mullerschon H. - Matthaei S. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100 2.3 Poljubnostni Lagrange-Eulerjev opis (PLE) 2.3 Arbitrary Lagrange-Eulerian description (ALE) Značilnosti Lagrangevega in Eulerjevega opisa napovedujejo, da bi bilo računalniško ustrezno združiti omenjena opisa in poudariti njune prednosti ter se izogniti njunim pomanjkljivostim. Ta zamisel je vodila do razvoja poljubnostnega Lagrange-Eulerjevega opisa (PLE). V tem opisu se mreža lahko deloma premika in deformira, ker sledi materialu (Lagrangev opis), hkrati pa dopušča, da material teče skozi mrežo (Eulerjev opis). Povezava med konvektivno hitrostjo c, hitrostjo materiala v in hitrostjo mreže vi je definirana kot: Glede na gibanje materiala in mreže se ohranitvene enačbe mase, gibalne količine in energije zapišejo kot: The features of the Lagrangian and Eulerian descriptions suggest that it would be computationally beneficial to combine these two descriptions so as to strengthen their advantages and to avoid their disadvantages. This idea led to the development of the Arbitrary Lagrange-Eulerian formulation. In this formulation the mesh partly moves 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, the material velocity, v , and the mesh ve-locity, vi, is defined as: vi (8). Regarding the material and the mesh movement, the mass, the momentum and the energy conservation equations can be written as: dp dp dvi — + ci-!— + p- — 8t dx dx dv dv P i + P'ci i dt dx da du p-Tt +p-ci Iz zgornjih enačb je razvidno, da v primeru, ko sta hitrost mreže in hitrost materiala enaka, izpeljemo Lagrangev opis. Kadar se material premika in mreža miruje (hitrost mreže je enaka konvektivni hitrosti), pa izpeljemo Eulerjev opis (preglednica 1) [2]. Računalniški algoritem reševanja opisa ALE v LS-DYNI je podoben opisanemu postopku Eulerjevega opisa ([13], [14] in [16]). Razlikuje se le v premiku vozlišč. V Eulerjevem opisu so vozlišča premaknjena nazaj na izhodiščne lege, pri čemer se dxi du dv +p-bi dx dx (9) (10) (11). From the above equations it is obvious that if the mesh and material velocity are equal this would give the Lagrangian description, and if the material moves and the mesh remains steady (the mesh velocity is equal to convected velocity) this would give the Eulerian description (Table 1) [2]. The ALE algorithm in LS-DYNA is similar to the described Eulerian computational procedure ([13], [14] and [16]). The only difference is the mesh smoothing. In the Eulerian formulation the nodes are moved back to their original positions, while in Preglednica 1. Primerjava kinematike PLE, Lagrangevega in Eulerjevega opisa Table 1. Comparison of the kinematics for the ALE, Lagrangian and Eulerian formulations PLE ALE Lagrange Euler pomik / displacement material mreža / mesh material mreža / mesh material mreža / mesh ui vi vi ai ai ui = ui vi vi = vi ai ai = ai u i hitrost / velocity vi=0 pospešek / acceleration ai ai =0 Računalniško modeliranje gibanja goriva - Computational Modelling of Fuel Motion 91 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100 premik vozlišč/ mesh smoothing interpolacija vmesnih rezultatov /advection t = 0 t =At t =At Sl. 5. Deformacija mreže pri PLE opisu Fig. 5. Mesh deformation in the ALE formulation pri opisu PLE nova lega premaknjenih vozlišč izračuna glede na povprečno oddaljenost do sosednjih vozlišč (sl. 5). Podobna numerična shema je uporabljena v drugih primerljivih programskih sistemih (npr. MSC/ Dytran). Prednost Lagrange-Eulerjevega opisa se pokaže, kadar želimo slediti določeni napetosti in se mora mreža samodejno zgoščevati. Drug primer je analiza rezervoarjev s tekočino, pri kateri je upoštevano gibanje tekočine v rezervoarju, mejna ploskev pa se zaradi vpliva med trdnino in tekočino ves čas spreminja (sl. 6). Kljub temu pa lahko uporaba PLE opisa povzroči popačenost elementov, kar lahko vpelje nevarne napake v numeričnih simulacijah. V določenih primerih PLE opis povzroči nepričakovano ustavitev računskega postopka. To se običajno zgodi zaradi zelo majhnih časovnih korakov, kot posledica zelo majhnih deformiranih Lagrangevih elementov ali celo negativne prostornine končnih elementov. 2.4 Metoda hidrodinamike zglajenih delcev (MHZD) the ALE formulation the positions of the moved nodes are calculated according to the average distance to the neighbouring nodes (Fig. 5). A similar calculation scheme is also used in other comparable codes (e.g., MSC/Dytran). The advantage of the ALE formulation is evident when a stress front needs to be followed and the mesh is automatically refined. Another example is the analysis of fluid tanks, where the fluid’s movement inside the tank is of interest and the boundary surface is continuously changing due to the interaction between the fluid and tank surfaces (Fig. 6). However, using the ALE formulation can also result in a highly distorted mesh, which can introduce large errors in numerical simulations. In some cases the ALE formulation can encounter unexpected terminations in the computational process, usually due to very small time steps following from very small, deformed Lagrangian elements or even negative element volumes. 2.4 Smoothed Particle Hydrodynamics (SPH) Stanje sistema je pri metodi hidrodinamike In the SPH method, the state of the system is zglajenih delcev (MHZD) opisano z določenim represented by a set of particles (Fig. 7) that pos- :::::;:..;::::: •••i*:::*-! •'............:: iSi «^iiiil!. Sl. 6. Uporaba PLE opisa Fig. 6. Applications of the ALE formulation 92 Vesenjak M. - Ren Z. - Mullerschon H. - Matthaei S. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100 številom delcev. Le-ti posamezno vsebujejo materialne lastnosti in se gibljejo na osnovi temeljnih ohranitvenih enačb. MHZD je brezmrežna Lagrangeva metoda delcev, ki so jo razvili Lucy, Gingold in Monaghan z začetnim namenom simulirati astro-fizikalne probleme ([5], [7] do [9] in [15]). Kasneje je bila MHZD obsežno preučena in razširjena za reševanje dinamičnih odzivov trdnih materialov pa tudi za simuliranje toka tekočine z velikimi deformacijami. MHZD ima pred običajnimi brezmrežnimi metodami določene prednosti. Najpomembnejša prednost je prilagodljivost, ki je dosežena v začetnem koraku približka spremenljivih veličin (npr. gostote, hitrosti, energij). Le-ta je izvedena v vsakem časovnem koraku in temelji na trenutni krajevni porazdelitvi delcev s poljubno lego. Zaradi prilagodljivosti približka metoda HZD ni odvisna od poljubne porazdelitve delcev. Zaradi tega lahko obravnava zelo dobro probleme z izrednimi deformacijami. Naslednja prednost metode HZD je kombinacija Lagrangevega opisa in približka delcev. V nasprotju z drugimi brezmrežnimi metodami, pri katerih so brezmrežna vozlišča uporabljena le kot interpolacijske točke, vsebujejo zglajeni delci tudi materialne lastnosti in so tako namenjeni kot aproksimacijske točke ter materialni elementi. Ti delci so se zmožni gibati v prostoru, prenašati vse računalniške podatke in dodatno oblikovati računalniški sistem za reševanje parcialnih diferencialnih enačb, ki opisujejo ohranitvene zakone. Metoda HZD je razdeljena na dva ključna dela. Prvi del je predstavitev integrala funkcij polja, drugi del pa predstavlja aproksimacijo delcev. Omenjeni postopek je uporabljen za izpeljavo parcialnih diferencialnih enačb v navadne diferencialne enačbe v diskretizirani obliki v odvisnosti le od časa [9]. Ohranitvene enačbe mase, gibalne količine in energije MHZD lahko zapišemo kot: sess individual material properties and move according to the governing conservation equations. SPH as a meshfree, Lagrangian, particle method was developed by Lucy, Gingold and Monaghan, initially to simulate astrophysical problems ([5], [7] to [9] and [15]). Later the SPH method was extensively studied and extended to the dynamic response with material strength as well as dynamic fluid flows with large deformations. It has some special advantages over the traditional mesh-based numerical methods. The most significant is the adaptive nature of the SPH method, which is achieved at the very early stage of the field variable (i.e., density, velocity, energy) approximation that is performed at each time step based on a current local set of arbitrarily distributed particles. Because of the adaptive nature of the SPH approximation, the formulation of the SPH is not affected by the arbitrariness of the particle distribution. Therefore, it can handle problems with extremely large deformations very well. Another advantage of the SPH method is the combination of the Lagrangian formulation and the particle approximation. Unlike the mesh-free nodes in other mesh-free methods, which are only used as interpolation points, the SPH particles also carry material properties, functioning as both approximation points and material components. These particles are capable of moving in space, carry all the computational information, and thus form the computational frame for solving the partial differential equations describing the conservation laws. The SPH formulation is divided into two key steps. The first step is the integral representation (kernel approximation) of the field function, and the second is the particle approximation (discretization). This procedure is applied to the particle differential equations to produce a set of ordinary differential equations in a discretized form with respect only to time [9]. The SPH equations for the conservation of mass, momentum and energy can be written as: zglajeni delec / SPH particle Sl. 7. Model po MHZD Fig. 7. SPH model Računalniško modeliranje gibanja goriva - Computational Modelling of Fuel Motion 93 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100 dt j =1 SviN dt j =1 mj dui N dt dWij mj-vij------ rirj m---------- • v =1 Pi-Pj dxi (12) (13) (14), kjer je Nštevilo delcev v območju vpliva delca i; Wij predstavlja gladilno funkcijo delca i, izvrednoteno v delcu j, in je tesno povezana z gladilno razdaljo; v pa je relativna hitrost med delcem i in delcem j [9]. Glavna možna prednost MHZD je v tem, da ne potrebuje medsebojno povezane prostorske mreže in se s tem izogne problemu popačenosti elementov pri velikih deformacijah. V primerjavi z Eulerjevim opisom je učinkovitejša, saj je treba modelirati le materialna območja in ne vseh območij, kjer bi material lahko obstajal. Kljub vsem obetom pa je metoda HZD razmeroma nova, v primerjavi z običajnim Lagrangevim in Eulerjevim opisom z mrežo elementov, z znanimi problemi na področju stabilnosti, zveznosti in izpolnjevanju ohranitvenih enačb [15]. 3 PRIMERJALNA ŠTUDIJA RAZLIČNIH POSTOPKOV REŠEVANJA PRAKTIČNEGA PRIMERA 3.1 Opis problema in računalniški model Analizirani problem sestoji iz zaprtega rezervoarja iz poliakrilnega stekla (PMMA), v začetnem mirujočem stanju, napolnjen s 60% vode in 40% zraka (sl. 8). Rezervoar je pritrjen na sani (pritrjen v navpični smeri) in pospešen z vzdolžnim časovno odvisnim pospeškom z največjo vrednostjo približno 30 g v času t = 40 ms. Časovno odvisna sprememba proste površine vode in tlak v točki 1, v rezervoarju iz polikarilnega stekla z debelino sten 30 mm, sta bila predhodno določena s preizkusom [11]. Rezervoar je modeliran z lupinskimi Belytschko-Tsayevimi lupinskimi elementi s štirimi vozlišči in tremi integracijskimi točkami po debelini elementa [3]. Za rezervoar je bil uporabljen elastičen materialni model s podatki, ki ustrezajo poliakrilnemu steklu (p= 1180 kg/m3, E = 3000 MPa in v= 0,35). Spodnja ploskev rezervoarja je bila modelirana kot toga. Za modeliranje vode in zraka so bili, odvisno od uporabljene metode, uporabljeni prostorninski where N is the number of particles in the support domain of the particle i; W ij is the smoothing function of the particle i evaluated at the particle j, and is closely related to the smoothing length; and vij 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 interconnected spatial mesh and thus avoids the problem of mesh distortion during large deformations. Compared with the Eulerian description, it is more effective, since only the material’s regions of interest need to be modelled, and not all the regions where the material might exist. With all this promise, however, the SPH technology is relatively new, compared to standard mesh-based Lagrangian and Eulerian descriptions, with remaining known problems in the areas of the stability, consistency and conservation [15]. 3 COMPARATIVE STUDY OF DIFFERENT APPROACHES TO SOLVING A PRACTICAL EXAMPLE 3.1 Problem description and computational model The analysed problem consists of a closed container reservoir at rest, 60% filled with water and 40% with air (Fig. 8). The reservoir was attached to a sled (fixed in the vertical direction) and subjected to a longitudinal time-dependent acceleration with a peak acceleration of approximately 30 g at time t = 40 ms. The time-dependent variation of the water surface shape and water pressure at point 1 was previously measured in experimental testing of a reservoir made of PMMA plates with 30-mm thickness [11]. The reservoir was modelled with four-noded Belytschko-Tsay shell elements with three integration points through the thickness [3]. The elastic material model is used for the reservoir container, with the material data corresponding to the PMMA material (r = 1180 kg/m3, E = 3000 MPa and n= 0.35). Only the bottom surface of the reservoir was modelled as rigid. For the water and air solid elements and SPH particles were used, depending on the ap- 94 Vesenjak M. - Ren Z. - Mullerschon H. - Matthaei S. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100 elementi in zglajeni delci. Za modeliranje vode (p = 1000 kg/m3 pri 293 K) in zraka (p= 1 kg/m3 pri 293 K) je bil uporabljen konstitutivni model Null (Type 9) [6]. Domena tekočine je opisana z materialnim modelom, ki zanemari strižne napetosti. Z uporabo nizke meje tečenja se doseže hiter prehod v plastično območje (npr. le z upoštevanjem gravitacije). Ob uporabi visokih dinamičnih obremenitev so strižne sile tako zanemarljive v primerjavi z vztrajnostnimi silami tekočine. Zrak je bil upoštevan le pri Eulerjevem in modelu po PLE. Uporabljeni sta bili Mie-Gruneisenova enačba stanja (voda in zrak) in enačba stanja idealnega plina (le za zrak). Celoten model je bil izpostavljen nespremenljivem težnostnem pospešku (g = 9,81 m/s2). V Lagrangevem in MHZD modelu je bil uporabljen avtomatski algoritem za določevanje stičnih sil, v Eulerjevem in modelu po PLE pa je bil stik med tekočino in strukturo definiran z uporabo posebnega algoritma ([6] in [10]). 3.2 Podatki računalniške simulacije Izvedene so bile eksplicitne dinamične analize z uporabo štirih različnih modelov simuliranja tekočine: Lagrangev, Eulerjev, PLE in MHZD. Za izvedbo analiz je bil uporabljen program LS-DYNA Linux Version 970. Opazovani časovni interval je znašal 80 ms, časovni korak simulacije pa je bil določen glede na najnižjo resonančno frekvenco strukture in je znašal 0,01 ms. plied method. The material model Null (Type 9) [6] was used for the water (r = 1000 kg/m3 at 293 K) and the air (r = 1 kg/m3 at 293 K) modelling. The fluid domain was described with a material model that neglects the deviatoric stresses. By defining a low yield stress, a rapid transition to plasticity can be achieved (e.g., by only considering the gravitation). Under high dynamic loading, the shear forces become negligible in comparison with the inertial forces of the fluid. The air was considered only in the Eulerian and ALE models. The Mie-Gruneisen (water and air) and Ideal Gas (only for air) equations of states have been used. The model was also loaded with the constant gravitational acceleration (g = 9.81 m/s2). In the Lagrangian and SPH model the automatic nodes to surface contact were used, and in the Eulerian and ALE models the contact between fluid and structure was defined with the keyword Constrained Lagrange in Solid ([6] and [10]). 3.2 Computational simulation data Explicit dynamic analyses were carried out by using all four different fluid model approaches: Lagrangian, Eulerian, ALE and SPH. The models were solved with LS-DYNA Linux Version 970. The computational time frame was set to 80 ms and the time step of the simulation was defined according to the lowest resonant frequency of the structure and was equal to 0.01 ms. <^ a g V & "" Točka 1 (zaznavalo)/ " Point 1 (sensor) [ 128 mm zk y 1008 mm 196 mm i y x Sl. 8. Izmere in začetne razmere analiziranega rezervoarja iz poliakrilnega stekla Fig. 8. Dimensions and initial conditions of the analysed PMMA box Računalniško modeliranje gibanja goriva - Computational Modelling of Fuel Motion 95 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100 3.3 Računalniški rezultati 3.3 Computational results Rezultati predvidene oblike proste površine za vse štiri simulacije so pri času t = 38 ms prikazane na sliki 9. Črtkana črta pomeni opazovano obliko proste površine v omenjenem trenutku pri preizkusu. S slike 9 je razvidno, da sta Lagrangeva in MHZD formulacija na desni strani modela ustrezni le za približek gibanja tekočine, saj v resnici tekočina ne bi obdržala oblike posode, kar je razvidno iz simulacij na levi strani modela. Kljub temu pa se morajo omenjene omejitve vrednotiti glede na zahtevane računalniške rezultate. V primeru, kadar je iskan le udarec tekočine na steno rezervoarja, so deformacije in pomiki na nasprotni strani lahko zanemarljivi. Eulerjeva in PLE formulacija opišeta lego in obliko proste površine vode mnogo bolje, kar pa je odvisno od daljšega računskega časa (preglednica 2), kar ni vedno sprejemljivo. Potek gibanja tekočine v rezervoarju ponazarja slika 10. Pomembno je poudariti, da se z uporabo Lagrangevega modela pri zelo velikih deformacijah pojavijo zelo popačeni elementi in posledično velike numerične napake. Omenjeno ponovno potrjuje, da Lagrangeva zapis ni primeren za zelo velike deformacije. Časovno spreminjanje tlaka vode v točki 1 je prikazano na sliki 11. Rezultati so bili določeni na dva različna načina. V Lagrangevem in MHZD modelu je bil tlak, ki se je pojavil v točki 1, merjen s stičnimi silami. Tlak pri Eulerjevem in PLE modelu pa je bil določen z merili poroznosti snovi in z določitvijo sile, ki je potrebna za vzpostavitev ravnotežja na vsakem opazovanem elementu na meji med tekočino in steno rezervoarja. The free-surface shape-prediction results of all four dynamic simulations at the time t = 38 ms are represented in Figure 9. The dotted line represents the free-surface shape observed in the experiment at the same time instance. From Figure 9 it is obvious that the Lagrangian and SPH models are only good for approximations of the fluid motion at the right-hand side wall, since in reality the fluid would not retain the form of the container, which is the case observed in simulations at the left-hand side wall. However, this observation must be considered in view of the required computational results. In the case where only the impulse of the fluid towards the tank wall is needed, the deformations and deflections on the opposite side could be neglected. The Eulerian and ALE formulations perform much better when describing the position and the form of the water’s free surface. However, this is only achieved by a dramatic increase of the calculation times, which is not always acceptable. Figure 10 represents the fluid motion in a reservoir during the calculation time. It is important to observe that using the Lagrangian formulation results in very distorted elements and, consequently, large computational errors. This again confirms the fact that the Lagrangian formulation is unsuitable for huge deformations. The time variation of water pressure at point 1 is shown in Figure 11. The results have been determined by two different approaches. In the Lagrangian and SPH models the pressure at point 1 was measured with contact forces that appeared at the observed point. For the Eulerian and ALE model the pressure was determined by the leakage control, i.e., by determining the force that is needed to establish equilibrium in every observed element at the boundary between the fluid and the reservoir wall. Sl. 9. Oblika proste površine: a) Lagrangev model; b) Eulerjev model; c) PLE model; d) MHZD model Fig. 9. Shape of the free surface: a) Lagrangian model; b) Eulerian model; c) ALE model; d) SPH model 96 Vesenjak M. - Ren Z - Miillerschon H. - Matthaei S. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100 Sl. 10. Gibanje tekočine, modelirane s PLE opisom Fig. 10. Fluid motion modelled with ALE formulation Zelo dobro ujemanje z rezultati preizkusa je bilo doseženo z uporabo vseh štirih modelov. MHZD model zagotavlja zelo dobre rezultate, še posebej, če upoštevamo, da uporaba te metode omogoča zelo hitro in nezahtevno analizo (mreža sestoji le iz zglajenih delcev). Rezultati Lagrangevega modela so zelo dobri, kljub temu da se simulacija predčasno prekine zaradi velike popačenosti elementov. Razlog za nizek izmerjen tlak pri PLE modelu in Eulerjevem modelu se lahko pripiše vplivu zraka v rezervoarju, ki deluje kot dušilnik. Razlog za padec tlaka pri Eulerjevem zapisu pa je uporaba drugačnega modela zraka (drugačna enačba stanja), ki je potrebna za vzpostavitev stabilne analize. Very good agreement with the experimental results was achieved by using all four formulations. The SPH formulation provided very good results, especially when taking into account that using this formulation results in a very quick and uncomplicated analysis (the mesh consists only of SPH particles). The results of the Lagrangian formulation are very good, although the simulation terminates because of the high element distortion. The reason for a lower pressure during the ALE and Eulerian models can be attributed to the air ’s influence inside the reservoir, where it acts like a damper. The drop in the pressure, observed for the Eulerian formulation, is also attributed to the need for a different air model (a different equation of state), which is necessary to ensure a stable analysis. Računalniško modeliranje gibanja goriva - Computational Modelling of Fuel Motion 97 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100 -Eksperiment / Experiment - Lagrangev model / Lagrangian model ¦ Eulerjev model / Eulerian model PLE model / ALE model -MHZD model / SPH model 10 :o 30 40 60 SO Čas t [ms] Time t [ms] Sl. 11. Primerjava časovne spremembe tlaka v točki 1 Fig. 11. Comparison of the pressure time-variation at point 1 Velikost modelov in njihovi računski časi za posamezne analize so prikazani v preglednici 2, ki prikazuje zahtevane računske sposobnosti za rešitev izbranega problema z različnimi pristopi. Deformacije samega rezervoarja so zaradi velike debeline njegovih sten zanemarljive. Za nazoren prikaz zmožnosti stika med tekočino in trdnino LS-DYNE je bila izvedena ponovna analiza z istim problemom, vendar je bila debelina sten rezervoarja zmanjšana na 10 odstotkov prvotne debeline, tj. 3 mm [6]. Deformacije sten rezervoarja, kot rezultat stika med tekočino in trdnino pri različnih časovnih korakih, so prikazane na sliki 12. Iz opisanih simulacij je razvidno, da različni pristopi ponujajo alternativne možnosti za modeliranje toka tekočin in njihovega vpliva na strukturo. Prednost uporabe Lagrangevega in The model size and the required CPU times for each analysis are listed in Table 2, to illustrate the required computational effort to solve the chosen problem with different approaches. The deformation of the reservoir container is negligible, due to the thickness of the reservoir walls. To clearly illustrate the fluid-structure interaction capabilities of LS-DYNA, another analysis of the same problem was carried with reduced container reservoir walls to only 10% of the original thickness, i.e., 3 mm [6]. The resulting deformations of the container reservoir due to fluid-structure interaction at different time instances can be observed in Figure 12. Following from the reported simulations it is obvious that these approaches offer an alternative means of fluid-flow modelling and its interaction with the structure. The advantage of using the SPH and Preglednica 2. Primerjava računskega časa Table 2. CPU-time comparison Model Skupno število / Total number Računski čas / Vozlišča / Elementi / CPU time Nodes Elements min Lagrangev / Lagrangian 2898 2420 16 Eulerjev / Eulerian 10162 8706 225 PLE / ALE 7462 6396 260 MHZD / SPH 2898 2896 13 98 Vesenjak M. - Ren Z. - Mullerschon H. - Matthaei S. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100 ,___--__-I__-_-_- liliiPlP 0 ms ._._,., . i i l Hit :;iiL: :iiCEnnc_ --t.: ill M M Ml ¦ -H . . . '"qrr|i 1* * I1 n ffcri /Ff? ^R^rtT ^ !', ::: ,-.,„.,, :Ti 1 1 " ^ ;;;:;! i; &± j : :¦! :. ': ._ t 50 ms Sl. 12. Medsebojni vpliv tekočine in trdnine Fig. 12. Fluid-structure interaction MHZD modela je kratek čas za pripravo modela in sprejemljivi računski čas, medtem ko Eulerjev in PLE model opišeta gibanje tekočine natančneje. Kljub temu pa je bilo začetni udarec tekočine ob stene rezervoarja mogoče natančno simulirati z vsemi štirimi modeli tekočine, ki so vključeni v LS-DYNI. 4 SKLEP V prispevku so predstavljeni štirje postopki modeliranja toka tekočine v LS-DYNI. Različne metode (Lagrangeova, Eulerjeva, PLE in MHZD) so bile uporabljene za analizo gibanja tekočine v deformabilnem rezervoarju, z namenom, da bi potrdili rezultate v primerjavi z znanimi preizkusnimi opazovanji. Računalniške simulacije so pokazale, da sta gibanje tekočine in medsebojni vpliv tekočine in trdnine natančno opisana z uporabo različnih izbirnih zapisov v LS-DYNI. Predstavljeni modeli so osnova za komercialne računalniške modele, uporabljene za analizo zapletenejših problemov. Rezultati dejanskega rezervoarja za gorivo z zelo zahtevno geometrijsko obliko ob upoštevanju vpliva gibanja tekočine s PLE in MHZD formulacijami, so pokazali zelo dobro ujemanje s preizkusi [18]. Lagrangian models is a short pre-processing and a reasonable computational time, while the ALE and Eulerian models can describe the fluid motion more accurately. Nevertheless, the initial impact of the fluid on the reservoir wall could be simulated accurately with all four fluid models incorporated in LS-DYNA. 4 CONCLUSION Four approaches to fluid-flow modelling in LS-DYNA have been presented in the paper. Different formulations (Lagrangian, Eulerian, ALE and SPH) were used to analyse the fluid motion in a de-formable reservoir, with the purpose to validate the results in comparison with existing experimental observations. Computational simulations showed that the fluid motion and the fluid-structure interaction can be accurately described by applying different alternative formulations in the LS-DYNA. The applied models provide a basis for economical computational models that can be used for analysing more complex problems. The results of the real fuel-tank with a very complex geometry, where the fluid motion’s influence was considered with the ALE and SPH descriptions, showed a very good agreement with the experiments [18]. Računalniško modeliranje gibanja goriva - Computational Modelling of Fuel Motion 99 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 85-100 5 LITERATURA 5 REFERENCES [I] Anghileri, M., Castelletti, L., Tirelli, M. (2003) Fluid-structure interaction of water filled tanks during the impact with the ground. Int. Journal of Impact Engineering. [2] Belytschko, T, Liu, W.K., Moran B. (2000) Nonlinear finite elements for continua and structures, John Wiley & Sons Ltd, Chichester. [3] Bois, P.D. (2003) Crashworthiness simulation using LS-DYNA. Stuttgart. [4] Ferziger, J., Peric, 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, FD. (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. [II] Meywerk, M., Decker, F, Cordes, J. (1999) Fuel sloshing in crash simulation, EuroPAM 99. [12] Miillerschon, 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] Škerget, L. (1994) Mehanika tekočin. Tehniška fakulteta v Mariboru in Fakulteta za strojništvo v Ljubljani. [18] Vesenjak, M. (2004) Methoden um Fluid-Struktur Interaktion in LS-DYNA zu simulieren. Daimler Chrysler. Stuttgart. [19] Vesenjak, M., Ren, Z., Hriberšek, 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 Vesenjak Authors’ Addresses: Matej Vesenjak prof.dr. Zoran Ren Prof. Dr. Zoran Ren Univerza v Mariboru University of Maribor Fakulteta za strojništvo Faculty of Mechanical Eng. Smetanova 17 Smetanova 17 2000 Maribor 2000 Maribor, Slovenia m.vesenjak@uni-mb.si m.vesenjak@uni-mb.si ren@uni-mb.si ren@uni-mb.si dr. Heiner Miillerschon Dr. Heiner Miillerschon DYNAmore GmbH DYNAmore GmbH Industriestr. 2 Industriestr. 2 70565 Stuttgart-Vaihingen, Nemčija 70565 Stuttgart-Vaihingen, Germany hm@dynamore.de hm@dynamore.de Stephan Matthaei Stephan Matthaei DaimlerChrysler AG, HPC B209 DaimlerChrysler AG, HPC B209 70322 Stuttgart, Nemčija 70322 Stuttgart, Germany stephan.matthaei@daimlerchrysler.com stephan.matthaei@daimlerchrysler.com Prejeto: 6. .2005 Sprejeto: 16. Odprto za diskusijo: 1 leto Received: 10 Accepted: 11.2005 Open for discussion: 1 year 100 Vesenjak M. - Ren Z. - Miillerschon H. - Matthaei S. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 101-111 UDK - UDC 625.745.5:656.08 004.94:629.351:692.88 Izvirni znanstveni članek - Original scientific paper (1.01) Simulacija naleta tovornega vozila ob cestno varnostno ograjo Simulating the Impact of a Truck on a Road-Safety Barrier Matej Borovinšek - Matej Vesenjak - Miran Ulbin - Zoran Ren (Fakulteta za strojništvo, Maribor) Cestno varnost lahko izboljšamo s postavitvijo cestnih varnostnih ograj, ki preprečujejo udeležencem cestnega prometa vstop v nevarna območja. Za dvig stopnje varnosti morajo biti cestne varnostne ograje oblikovane tako, da pri naletu vozila le-to preusmerijo nazaj na cestišče. Med preusmerjanjem vozila morajo s svojo deformacijo absorbirati čim večjo količino kinetične energije, da se zmanjšajo pojemki potnikov v vozilu. V prispevku je predstavljena računalniška simulacija naleta tovornega vozila ob cestno varnostno ograjo. Izvedene so bile parametrične računalniške analize za oceno primernosti različnih dodatnih elementov varnostne ograje. Za izvajanje eksplicitnih dinamičnih analiz je bil uporabljen program LS-DYNA. Računalniške simulacije dokazujejo, da je nosilnost analizirane ograje dovolj velika, da tovornjak zadrži in preusmeri nazaj na cestišče. Hkrati pa se pokaže tudi ustrezno visoka stopnja akumulacije kinetične energije vozila v obliki deformacije cestne varnostne ograje, kar zmanjša pojemke vozila in se kaže v povečani stopnji varnosti za potnike v vozilu. © 2006 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: varnost na cestah, ograje varnostne, analize trkov, simuliranje dinamično, LS-DYNA) Road safety can be improved by applying appropriate road-safety barriers that prevent road-traffic participants from entering dangerous areas. In terms of preventing vehicles from veering off the road, a road-safety barrier should redirect the impacting vehicle back onto the road. In the process of doing so it should absorb as much of the vehicle’s kinetic energy as possible through its deformation, so reducing the deceleration of the vehicle’s occupants. This paper considers a computer simulation of a truck’s impact on a steel road-safety barrier. Parametric computer simulations were used to evaluate the different additional safety elements of the road-safety barrier. The dynamic finite-element explicit code LS-DYNA was used for this purpose. The computer simulations prove that the analyzed road-safety barrier design is strong enough to retain and redirect the truck back onto the roadway. Appropriately, the high absorption of the impacting vehicle’s kinetic energy through the safety-barrier deformation is observed at the same time, which indicates a 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) 0 UVOD 0 INTRODUCTION Visoka stopnja varnosti je zelo pomembna v sodobnem cestnem prometu. Na eni strani avtomobilska industrija razvija nove pasivne in aktivne varnostne sisteme za povečanje varnosti potnikov v cestnem prometu, na drugi strani pa se varnost cest povečuje z uporabo učinkovitejših cestnih varnostnih elementov. Zagotovitev dovolj visoke stopnje varnosti v cestnem prometu je primarnega pomena. Eden od A high level of safety is a very important aspect in modern road traffic. On the one hand, the car industry is developing new, passive and active vehicle-safety systems to increase the safety of road-vehicle occupants, and on the other hand, the roads are being made safer with the installation of more effective road-safety elements. Ensuring appropriate safety levels in road traffic is of primary importance. One way to improve 101 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 101-111 načinov izboljšanja varnosti je postavitev varnostnih ograj. Namen cestnih varnostnih ograj je preprečiti zdrs vozila s ceste, torej preprečiti izlet vozila s cestišča ali prehod vozila na nasprotnosmerno vozišče. S tem se preprečijo oziroma popolnoma odpravijo možnosti poškodbe potnikov v vozilu ter oseb in objektov ob vozišču. Novo postavljene cestne varnostne ograje v Sloveniji morajo izpolnjevati zahteve novega slovenskega pravilnika o postavitvi cestnih varnostnih ograj in standarda SIST EN 1317 [1]. Vsaka na novo postavljena ograja mora imeti uspešno opravljen preizkus naleta za stopnjo zadrževanja, za katero je namenjena. Zaradi visokih cen preizkušanj varnostnih ograj je smiselno njihovo konstrukcijo poprej preveriti z izvajanjem računalniških simulacij. Tako so bile, z namenom, da bi izbrali najustreznejše konstrukcijske rešitve varnostne ograje, izvedene računalniške analize naleta tovornjaka ob ograjo. Za izvajanje teh simulacij je bil uporabljen programski paket LS-DYNA ([2] in [3]). 1 KONSTRUKCIJSKE ZAHTEVE ZA CESTNE VARNOSTNE OGRAJE Cestne varnostne ograje morajo bili postavljene le na odsekih, kjer je možnost za hujše poškodbe pri naletu vozila manjša, kakor v primeru, če ograje ne bi bilo. Na javnih cestah v Sloveniji se lahko postavljajo le varnostne ograje, ki so atestirane po standardu SIST EN 1317. Najpogosteje uporabljena vrsta cestnih varnostnih ograj v Sloveniji so jeklene varnostne ograje (sl. 1). Postavljene so predvsem na avtocestah, glavnih in regionalnih cestah. Na redkeje prevoznih cestah se uporabljajo le na nevarnih območjih, kjer je povečana verjetnost zdrsa vozila s cestišča. Glavni konstrukcijski elementi enostranske jeklene varnostne ograje so (sl. 1): -ščitnik - vzdolžni element ograje; s svojo deformacijo zmanjša moč udarca, hkrati pa mora biti dovolj tog, da se med trkom ne poruši; - steber - nosilo distančnika in/ali ščitnika, ki zagotavlja lego ščitnika na določeni oddaljenosti in višini od vozišča oziroma nad njim; - distančnik - zagotavlja določeno lego ščitnika glede na steber ograje; - zaključni element - del varnostne ograje, ki je na njenem začetku oziroma koncu in je namenjen zmanjšanju posledic naleta vozila na ograjo. road safety is to install road-safety barriers. The purpose of such systems is to restrain a vehicle from entering dangerous areas, i.e., to prevent it from veering off the road or on to the other side of the road. This effectively reduces or completely alleviates damage to vehicle occupants and other road-traffic participants and objects. Newly installed road-restraint systems in Slovenia have to fulfill the new Slovenian regulation requirements for road-restraint systems and the SIST EN 1317 standard [1]. Every newly installed restraint system has to be certificated for the containment level it is designed for by a full-scale crash test. Since these tests are very expensive, it is advisable to preliminary check the adherence of safety-barrier design to the set regulations by means of a computer simulation. This paper reports on a computer simulation of a truck impact into a road-safety barrier with the aim of determining the most effective safety-barrier design. The software system LS-DYNA ([2] and [3]) was used for this purpose. 1 DESIGN REQUIREMENTS FOR ROAD-SAFETY BARRIERS Road-safety barriers only need to be installed on road sections where the possibility of severe injuries during vehicle impact is reduced by the installation of such barriers. Only certified road-safety barriers according to SIST EN 1317 may be installed on public roads in Slovenia. The most frequently used type of road-safety barriers in Slovenia are steel road-safety barriers (Fig. 1). They are installed mainly on highways, main and regional roads. On roads with less traffic they are only used in dangerous areas with a high probability of the vehicle veering off the road. The main design elements of a single-sided steel road-safety barrier are (Fig. 1): - the guardrail – a longitudinally placed element, the deformation of which reduces the severity of an impact, but it should also be strong enough not to rupture during an impact; - the post – carries the distance spacer and/or the guardrail and it ensures the guardrail position at a certain distance from and above the road; - the distance spacer – ensures a certain position of the guardrail in relation to the post, - the end element – a part of the safety barrier placed at its beginning and end, which reduces the consequences of a direct vehicle impact. 102 Borovinšek M. - Vesenjak M. - Ulbin M. - Ren Z. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 101-111 Sl. 1. Konstrukcijski elementi in glavne mere jeklene cestne varnostne ograje Fig. 1. Main elements and regulated dimensions of a steel road-safety barrier Pravila za postavitev cestnih varnostnih ograj v okviru tehnične specifikacije za javne ceste v Sloveniji še niso popolnoma določena, saj jih še zmeraj usklajujejo različne agencije [4]. 2 ZAHTEVE STANDARDA SIST EN 1317 Glede na standard SIST EN 1317 ocenjujemo ustreznost varnostne ograje z ozirom na tri glavne kriterije [5]: - raven zadrževanja vozila - pomeni raven zadrževanja za različne vrste vozil; - moč udarca - se določa s tremi parametri: z merilom velikosti pospeškov (MVP - ASI), s pojemkom glave po udarcu (PGU - PHD) in s teoretično hitrostjo glave pri udarcu (THGU - THIV); - deformacija ograje. Najpomembnejši parameter ASI je brezrazsežna veličina, ki se uporablja kot splošno merilo za določanje posledic naleta vozila na potnike v vozilu. ASI je določen kot: ASI [ASI(t)] kjer ax,ay in a pomenijo povprečne vrednosti pospeškov masnega središča vozila v njegovem lokalnem koordinatnem sistemu v tekočem časovnem koraku At = 50 ms (sl. 2). ax ,ay in a pomenijo mejne vrednosti pospeškov za posamezne koordinatne osi ter znašajo a= 12-g, a = 9-g in a = 10.g, kjer je g = 9,81 m/s2. Parameter moči udarca MVP mora biti merjen in izračunan samo za teste z vozili majhnih mas The technical regulations for the installation of road-safety barriers on Slovenian public roads are not yet fully operational, as they are still in the process of final approval by various agencies [4]. 2 SIST EN 1317 REQUIREMENTS The suitability of a safety barrier is, according to the SIST EN 1317 standard, determined by three main 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 theoretical head-impact velocity (THIV), - the deformation of the barrier. The most important ASI parameter is a dimensionless variable, which is used as an overall measure of vehicle-impact consequences for the vehicle occupants. The ASI is determined as: 1, 0 (1, 4) (1), where ax, ay and a represent the average values of the acceleration of the vehicle’s center of gravity in a local coordinate system of the vehicle in a running-time interval of At = 50 ms (Fig. 2). ax ,ay and a stand for the acceptable acceleration limits for individual coordinate directions and are equal to ax = 12-g, a = 9-g and a = 10-g, where g = 9.81 m/s2. The impact severity parameter ASI needs to be measured and determined only for tests with low Simulacija naleta tovornega vozila - Simulating the Impact of a Truck 103 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 101-111 Sl. 2. Lokalni koordinatni sistem vozila Sl. 3. Konstrukcija ograje s pasnico (levo) in Fig. 2. Local coordinate system of the vehicle trapeznim vodilom (desno) Fig. 3. Design of a barrier with tension belt (left) and wheel guidance (right) (osebna vozila), na katera med testom delujejo večji weight vehicles (personal vehicles), which sustain pojemki. Kljub temu je bil v tem prispevku parameter significant deceleration during impact. However, in MVP uporabljen tudi za analizo razmer med naletom this paper the ASI parameter is also used to assess a tovornjaka. Ta parameter pomeni dobro mero za truck’s impact conditions. Since it is calculated from normiran dejanski pospešek, saj se izračuna iz filtriranih filtered coordinate acceleration data and normalized koordinatnih pospeškov in je normiran z njihovimi by their limiting values, it represents a good measure mejnimi vrednostmi. Tako je bil tukaj uporabljen za of the normalized effective acceleration. Thus, it was prikaz vpliva različnih konstrukcijskih rešitev cestne used here to illustrate the influence of different road- varnostne ograje na skupno obnašanje vozila. safety barrier designs on overall vehicle behavior. 3 KONSTRUKCIJSKA IZVEDBA VARNOSTNE 3 ROAD-SAFETY BARRIER OGRAJE DESIGN S simulacijami je bilo preizkušenih več Simulations were used to test different designs konstrukcijskih rešitev cestne varnostne ograje za of road-safety barrier for the containment level H1. raven zadrževanja H1. Po standardu SIST EN 1317 According to the SIST EN 1317 standard such safety mora takšna ograja uspešno prestati nalet osebnega barrier has to successfully sustain the impact of a vozila (test TB11) in tovornjaka (test TB42). Osebno personal vehicle (TB11 test) and a truck (TB42 test). vozilo ima maso m = 900 kg, začetno hitrost v = 100 A personal vehicle has a mass of m = 900 kg, an initial km/h in trči v cestno varnostno ograjo pod kotom a velocity of v = 100 km/h and impacts the barrier at an = 20° glede na postavitevščitnika ograje. Tovornjak angle of a = 20° with regard to the guardrail. The truck ima maso m = 10.000 kg in trči v varnostno ograjo z has a mass of m = 10,000 kg and impacts the barrier začetno hitrostjo v = 70 km/h pod vpadnim kotom a with an initial velocity of v = 70 km/h at an impact = 15°. Test TB11 je namenjen določitvi parametrov angle of a = 15°. The TB11 test is used to determine moči udarca, medtem ko je test TB42 namenjen the impact-severity parameters and the TB42 test is določitvi nosilnosti cestne varnostne ograje. Oba used to test the load-carrying capability of such a testa sta enako pomembna za končno odločitev o system. Both tests are equally important when making izbiri konstrukcijske rešitve cestne varnostne ograje. the final choice of road-safety barrier design. Osnova za konstrukcijo nove cestne The new H1 road-safety barrier was a design varnostne ograje je bila atestirana ograja za raven upgrade of an already-certified safety barrier for the zadrževanja N2. Njeni glavni sestavni deli so v celoti containment level N2. The main components of this izdelani iz konstrukcijskega jekla S 235. Ščitnik je safety barrier are made of construction steel S 235. narejen iz 3 mm debele pločevine in v dolžino meri The guardrail is made from 3-mm-thick metal sheet 4.200 mm, kjer je 200 mm namenjenih spoju med and is 4,200 mm long, where the splice length is equal sosednjimi ščitniki. Distančnik ima obliko šestkotnika to 200 mm. The distance spacer is shaped in a [6], s tem ima največjo zmožnost absorpcije energije hexagonal form [6] to provide the highest crash- trka in predvidljivo deformacijo, narejen pa je iz 4 mm energy absorption and controllable deformation and 104 Borovinšek M. - Vesenjak M. - Ulbin M. - Ren Z. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 101-111 pločevine. Steber je izdelan iz profila C izmer 100 x 55 x 4 mm, dolžine 1.900 mm. Ščitniki so medsebojno spojeni z vijačnimi zvezami z vijaki M16, v preostalih vijačnih zvezah pa so uporabljeni vijaki M10. Vsi uporabljeni vijaki so trdnostnega razreda 5.8. Ta osnova cestne varnostne ograje je bila dodatno ojačana na dva različna načina. Enkrat z dodatno pasnico, ki je privijačena na hrbtni del distančnika, drugič pa s trapeznim vodilom, privijačenim neposredno na sprednji del stebra (sl. 3). Dodaten namen trapeznega vodila je preprečiti neposredni udarec kolesa vozila ob steber ograje ter tako zmanjšati pojemke v vozilu in zmanjšati moč udarca. Pasnica in trapezno vodilo sta enake dolžine kakor ščitnik ograje in sta privijačena z vijaki M10. Debeline pločevin obeh ojačitev so bile predmet parametričnih numeričnih analiz. Debelina pločevine je obsegala vrednosti 3,0, 5,0 in 7,0 mm pri pasnici in 2,0, 2,5, 2,8 ter 3,0 mm pri trapeznem vodilu. 4 NUMERIČNE ANALIZE 4.1 Numerični model tovornega vozila Za model tovornega vozila je bil uporabljen prosto dostopen model po MKE vozila Ford Single Unit Truck (poenostavljen) [7]. Dolžina tovornega vozila znaša 8,5 m, višina 3,3 m in širina 2,4 m (sl. 4.). Model po MKE sestavlja približno 22.200 linearnih končnih elementov z reducirano integracijsko shemo [8]. Od tega je okoli 1.000 prostorninskih, 500 linijskih elementov, ostalo pa predstavlja 20.700 lupinskih končnih elementov. Model po MKE tovornega vozila je bil nekoliko spremenjen, da je ustrezal zahtevam standarda SIST EN 1317. Masa vozila je bila povečana iz 7.320 kg na is made from 4-mm-thick sheet metal. The posts are C-shaped with dimensions of 100 x 55 x 4 mm and 1,900 mm long. The guardrails are joined together with M16 screws, while all other connections are made with M10 screws. All the screws are of strength class 5.8. The basic design was additionally reinforced in two different ways. First, with a tension belt that is connected to the back of a distance spacer; and, second, with a wheel-guidance profile connected to the front of the post (Fig. 3). An additional purpose of the wheel guidance is to hinder the direct vehicle-wheel impact into the post, which results in smaller vehicle decelerations and lessens the severity of the impact. The tension belt and the wheel guidance are of the same length as the guardrail and connected with M10 bolts. The sheet-metal thicknesses of both reinforcements were the subject of a parametral analyses. Thicknesses of 3.0, 5.0, and 7.0 mm were used for the tension belt and 2.0, 2.5, 2.8, and 3.0 mm for the wheel-guidance profile. 4 COMPUTATIONAL SIMULATIONS 4.1 Computational model of a truck A publicly accessible FEM model of a Ford Single-Unit Truck (Reduced Model) [7] was used as a basic 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 22,200 linear elements with a reduced integration scheme [8]. It consists of 1,000 solid elements, 500 beam elements and the rest is represented by 20,700 shell elements. The FEM model of the truck was subjected to further changes to fulfill the SIST EN 1317 standard. The vehicle mass was raised from 7,320 kg to 10,000 kg by increasing the material density of all Sl. 4. Model po MKE tovornega vozila Ford Fig. 4. The FEM model of a Ford single-unit truck Simulacija naleta tovornega vozila - Simulating the Impact of a Truck 105 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 101-111 10.000 kg, tako da je bila povečana gostota vseh delov tovornjaka. V težišče tovornega vozila je bil dodan merilnik pospeškov kot tog lupinski končni element, ki omogoča zajem pomikov, hitrosti in pospeškov v lokalnem koordinatnem sistemu vozila. 4.2 Numerični model ograje Podrobneje so bili modelirani ščitnik, steber in distančnik ograje, medtem ko so bile vijačne zveze modelirane poenostavljeno. Glede na rezultate prejšnjih simulacij in hitrost vozila je bila dolžina modelirane ograje omejena na 39 m, od tega 5 m ograje pred naletno točko tovornega vozila in 34 m za njo. Po celotni dolžini ograje je bilo razporejenih 19 stebrov in distančnikov na medsebojni razdalji 2 m. Glavni sestavni deli cestne varnostne ograje so bili modelirani z lupinskimi elementi. Stebri in distancniki so bili modelirani z lupinskimi Belytschko-Tsayevimi končnimi elementi s tremi integracijskimi točkami po debelini, ščitniki, pasnice in trapezna vodila pa s polnimi integracijskimi lupinskimi elementi s petimi integracijskimi točkami po debelini, da je bil preprečen pojav nične deformacijske energije. Predpisana debelina končnih elementov je ustrezala konstrukcijski izvedbi posamezne ograje, kakor je navedeno v 3. poglavju. Vijačne zveze so bile modelirane z linijskimi Hughes-Liujevimi končnimi elementi. Celoten model ograje je tako sestavljen iz približno 100.000 elementov in 110.000 vozlišč. Materialne lastnosti pločevin različnih debelin so bile določene s preizkusi. Glede na dobljene rezultate je bil izbran bilinearen izotropen elastoplastični model s kinematičnim utrjevanjem (preglednica 1). Porušitev materiala je bila predpisana z dejansko plastično deformacijo, ki je znašala 0,28. Pri vijačnih zvezah z vijaki M10 je bil upoštevan trdnostni razred 5.8 z mejo plastičnosti 400 MPa in natezno trdnostjo 500 MPa, pri vijačnih zvezah z vijaki M16 pa so bile uporabljene vrednosti, navedene v preglednici 2. Te vrednosti so bile the vehicle parts. A rigid shell element was added at the vehicle’s center of gravity to act as an accelerometer and to record the displacements, velocities and accelerations in the local coordinate system 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 in detail, while simplified modeling was used for the bolts. Based on previous simulations and the speed of the impacting vehicle the length of the modeled safety barrier was 39 m: 5 m before and 34 after the impact point. Along the whole length of the road-safety barrier 19 posts and distant spacers were placed 2-m apart. Shell finite elements were used to model the main parts of the safety barrier. Posts and distance spacers were modeled with Belytschko–Tsay shell finite elements with three integration points through the shell thickness, while the guardrails, the tension belts and the wheel-guidance profiles were modeled using full-integration shell elements with five integration points through the thickness to prevent “hourglassing”. The thicknesses of the shell elements were defined according to the construction plans of each design, as described in Section 3. The bolt connections were modeled with linear Hughes–Liu finite elements. The complete safety-barrier model consisted of approximately 100,000 finite elements and 110,000 nodes. The material properties of the different sheet metal thicknesses were obtained experimentally. Experiment results were used to define an isotropic bilinear elastoplastic material model with kinematic hardening (Table 1). Material failure was prescribed according to the effective plastic deformation, which was set to 0.28. A yield stress of 400 MPa and a limit stress of 500 MPa were prescribed for the M10 bolt connections of strength class 5.8, while the M16 bolt connections were modeled with the parameters presented in Table 2. The parameters were obtained from a correlation analysis Preglednica 1. Materialne lastnosti pločevin varnostne ograje Table 1. Material properties of sheet-metal thicknesses d E v cry Et mm GPa 2,0 2,5 2,8 3,0 4,0 5,0 7,0 190 200 MPa 0,29 0,29 285 330 MPa 696 969 su u MPa 400 450 106 Borovinšek M. - Vesenjak M. - Ulbin M. - Ren Z. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 101-111 Preglednica 2. Materialni podatki za vijačne zveze z vijaki M16 Table 2. Material properties for the M16 bolt connections E I v I sy GPa / MPa 190 0,29 240 0 170 dobljene s primerjavo med preizkusi in numeričnimi simulacijami vijačnih zvez z vijaki M16. Zaradi tega je dopustna dejanska plastična deformacija e tako velika, saj zajema deformacijo celotne vijačne zveze in okoliškega materiala. 4.3 Začetni in robni pogoji Robni in začetni pogoji za tovorno vozilo so predpisani s standardom SIST EN 1317. Vozilo pri testu TB42 je postavljeno pod kotom 15° glede na cestno varnostno ograjo s predpisano začetno hitrostjo, ki znaša 70 km/h. Nadaljevanje ščitnika in ojačitev ograje je bila simulirana z vzmetnimi elementi (1 na sl. 5) ustrezne togosti in elastično linearno značilnico. Vpliv zemljine je bil simuliran z vzmetnimi elementi elasto-viskoplastičnih lastnosti spremenljivih po globini, ki je bila določena s predhodnimi parametričnimi simulacijami [9]. V modelu so bili predpisani štirje ločeni stiki: med posameznimi deli ograje, med posameznimi deli tovornega vozila, med desnim bokom tovornega vozila in ograjo ter med kolesi vozila in podlago. V prvih treh stikih je bila vrednost statičnega koeficienta trenja ms = 0,2, dinamičnega koeficienta trenja pa md = 0,15. V zadnjem stiku pa je bila za oba koeficienta trenja uporabljena vrednost 0,3. MPa eu % of the experimental measurements and the computational simulation of the M16 bolt connections. This explains the very high ultimate effective plastic deformation, e , which actually accounts for the bolt and surrounding material deformation. 4.3 Initial and boundary conditions The initial and boundary conditions for the vehicle are prescribed by the SIST EN 1317 standard. The impacting vehicle for the TB42 test is positioned at an angle of 15° to the safety barrier, and the initial velocity is equal to 70 km/h. Continuation of the guardrail and the barrier reinforcement was modeled with spring elements (No. 1 in Fig. 5) with a linear elastic characteristic and an appropriate stiffness. The soil influence was simulated using spring elements with an elasto-viscoplastic characteristic varying with depth, which was obtained from previous parametric simulations [9]. Four different contact definitions were used in the model: between the safety-barrier parts, the contact between the vehicle parts, the contact between the impacting part of the vehicle and the barrier, and the contact between the wheels of the vehicle and the ground. The static and dynamic frictions in the first three contact definitions were set to ms = 0.2 and md = 0.15, respectively. In the fourth contact definition, both friction coefficients were set to 0.3. 2 Sl. 5. Robni pogoji za varnostno ograjo s stebri, zabitimi v zemljino Fig. 5. Boundary conditions for a safety barrier with posts rammed into the soil Simulacija naleta tovornega vozila - Simulating the Impact of a Truck 107 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 101-111 4.4 Parametri računalniške simulacije Analiza naleta vozila v cestno varnostno ograjo je bila simulirana s programskim paketom LS-Dyna Version MPP 970. Vse predstavljene analize so bile izvajane na sistemu z osmimi procesorji Intel Pentium IV 3,2GH s tehnologijo prepleta pod operacijskim sistemom Linux. Računalniške simulacije so bile izvedene v časovnem koraku t = 1,6 s, takoj po prvem stiku med vozilom in ograjo. Časovni korak analize je bil samodejno izračunan glede na najvišjo prvo resonančno frekvenco celotnega modela in je znašal 1,7 ms. Izvajanje ene same simulacije je na opisanem sistemu trajalo med 100 in 110 urami. 5 ANALIZA NUMERIČNIH REZULTATOV Razlike med poteki trka parametričnih analiz različnih debelin pasnic in vodila so zanemarljive, zato je potek trka prikazan le za eno simulacijo (sl. 6). Rezultati vseh analiz kažejo, da vse konstrukcijske različice cestne varnostne ograje vozilo uspešno preusmerijo nazaj na vozišče. Največja deformacija varnostne ograje je dosežena v času t = 0,8 s. Vozilo se loči od ograje v času t = 1,5 s. Pred izračunom časovnega poteka parametra MVP so bili vsi zbrani pospeški filtrirani z uporabo filtra CFC 180 (najmanjša frekvenca zbiranja podatkov znaša 1800 Hz) kakor zahteva standard SIST EN 1317. Sl. 6. Potek naleta tovornega vozila ob cestno varnostno ograjo s pasnico Fig. 6. The truck impact into a safety barrier reinforced with a tension belt 108 4.4 Computational simulation properties The computational simulation of a vehicle impact into a road-safety barrier was performed using the dynamic explicit software code LS-DYNA MPP Version 970. All the simulations were done on a PC cluster with eight Pentium-IV 3.2-GHz processors with hyper-threading technology and a Linux operating system. The simulations were done for the time interval of 1.6 s after the first contact between the vehicle and the barrier. The time-step increment was computed with regard to the highest first-resonant frequency of the model, and was equal to 1.7 ms. One simulation run on the reported configuration took from 100 to 110 hours. 5 ANALYSIS OF THE COMPUTATIONAL RESULTS Due to the similarity of the vehicle behavior for different sheet-metal thicknesses of the tension belt and the wheel guidance only one vehicle-impact sequence is shown in Fig. 6. The results of all the simulations prove that all safety-barrier designs redirect the impacting vehicle back on to the road. The maximum system deformation is observed at time t = 0.8 s. The vehicle separates from the barrier at time t = 1.5 s. Before calculating the ASI parameter all the measured vehicle-mass acceleration data was filtered using a CFC 180 filter (the minimum measurement frequency equals 1,800 Hz) according to SIST EN 1317. Sl. 7. Delovna širina (W) Fig. 7. Working width (W) Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 101-111 0,70 0,60 0,50 0,40 0,30 0,20 0,10 0,00 A 5 y/w kv^fW« t !t JV pw: VV^W W5 3mm 5mm 7mm ^%^CyM^ 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 Čas [s] Sl. 8. Potek parametra MVP pri ograji s pasnico Fig. 8. ASI parameter for the barrier with the tension belt 0,70 0,60 0,50 0,40 0,30 0,20 0,10 0,00 -» %e ^,r m 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 0,80 0,90 Čas [s] Sl. 9. Potek parametra MVP pri ograji z vodilom Fig. 9. ASI parameter for the barrier with the wheel guidance Potek parametra MVP v odvisnosti od časa pri analizah cestne varnostne ograje s pasnico je prikazan na sliki 8, pri ograji z vodilom pa na sliki 9. Največje dosežene vrednosti parametra MVP so podane v preglednici 3. Pri obeh časovnih potekih parametra MVP sta opazna dva izrazitejša vrha. Eden takoj v času t = 0,0 s, ko pride tovorno vozilo prvič v stik z ograjo in drugi v času t = 0,2 s, ko kolo vozila zadene steber varnostne ograje. Primerjava obeh slik pove, da so povprečne in največje vrednosti parametra MVP pri varnostni ograji s trapeznim vodilom večje. Razlog za večjo togost je večja dolžina razvitega profila pri trapeznem vodilu, ki znaša l = 285,0 mm in le 75,4 mm pri pasnici. Togost cestne varnostne ograje z vodilom je dodatno povečana še zaradi neposredne pritrditve trapeznega vodila na stebre, medtem ko je pasnica privijačena na distančnik. Primerjava poteka parametra MVP med enakimi ojačitvami različnih debelin pokaže manjše The time dependency of the ASI parameter for the road-safety design with a tension belt is given in Fig. 8, and for a wheel guidance in Fig. 9. The maximum values of the ASI are presented in Table 3. In both time dependencies of the AIS parameter two distinct peaks can be observed. One immediately after the time t = 0.0 s, when the vehicle impacts the barrier, and the second one at t = 0.2 s, when the wheel hits the barrier post. A comparison of both figures reveals that the average and maximum values of the ASI are higher for the barrier with the wheel guidance. The reason is in the greater length of the unfolded profile of the wheel guidance, which equals l = 285.0 mm, and only l = 75.4 mm for the tension belt. The stiffness is additionally increased due to the direct connection of the wheel guidance to the posts, while the tension belt is connected to the distance spacer. The comparison of the ASI parameter for the same reinforcements of different thicknesses shows Simulacija naleta tovornega vozila - Simulating the Impact of a Truck 109 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 101-111 razlike. Pri ograji s pasnico so največje dosežene vrednosti parametra MVP pri pločevinah različnih debelin enake, povprečne vrednosti pa pričakovano pokažejo manjše vrednosti pri tanjši pločevini. Pri ograji z vodilom je največja vrednost parametra MVP dosežena pri debelini pločevine d = 2,8 mm, kljub temu pa je pri tej debelini povprečna vrednost parametra MVP najmanjša. Večje povprečne vrednosti imata vodili debeline d = 2,0 mm in d = 3,0 mm, vendar imata manjšo največjo vrednost. Razlike med vrednostmi delovnih širin (sl. 7) pri različnih debelinah pločevine cestne varnostne ograje ojačane s pasnico, so zanemarljive, medtem ko so razlike pri cestni varnostni ograji s trapeznim vodilom nekoliko večje. Pričakovali bi manjšo deformacijo ograje pri debelejši pločevini vodila, a izmerjeni rezultati kažejo drugače. Natančnejša analiza poteka trka pokaže, da je večji del te razlike posledica različne deformacije distančnikov in premikanja varnostne ograje med trkom. Primerjava vrednosti delovne širine med posameznima konstrukcijskima rešitvama pa pokaže manjše deformacije pri ograji z vodilom. To se sklada z ugotovitvijo o večji togosti te ograje. 6 SKLEP Za oceno različnih konstrukcij varnostne ograje za raven zadrževanja H1 so bile izvedene računalniške simulacije naleta tovornega vozila v cestno varnostno ograjo na podlagi parametričnih nelinearnih dinamičnih analiz po metodi končnih elementov. Rezultati parametričnih analiz kažejo, da je z vidika moči udarca primernejša cestna varnostna ograja s pasnico. Ker vse konstrukcijske rešitve varnostne ograje s pasnico uspešno zadržijo vozilo na cestišču, je s cenovnega vidika najzanimivejša izbira najtanjše pasnice debeline 3,0 mm. smaller differences. The maximum values of the ASI for the safety barrier with a tension belt are the same for all thicknesses, while the average values are lower for the thinner sheet metal, as expected. The highest value of ASI for the barrier with the wheel guidance is determined for the sheet-metal thickness d = 2.8 mm, while the average value of ASI is the lowest at the same time. Higher average values are recorded for metal thicknesses d = 2.0 mm and d = 3.0 mm, but with a lower maximum value. The differences in working-width (Fig. 7) values for different sheet-metal thicknesses are negligible for the barriers reinforced with the tension belt, while they are more pronounced for the barriers reinforced with the wheel guidance. A lower working width is expected for the reinforcements with thicker sheet metal, but the results show otherwise. A more precise analysis of the impact shows that this effect can be attributed to the difference in the distance-spacer deformation and the barrier movement during impact. Comparisons of the working widths between both reinforcements show smaller deformations for the barrier reinforced with a wheel guidance. This is consistent with previous findings. 6 CONCLUSION Computational simulations based on a parametric, nonlinear, dynamic finite-element analysis of a vehicle impact into a road-safety barrier were employed to evaluate different road-safety barrier designs for the containment level H1. The results of the parametric analyses showed that the barrier design reinforced with a tension belt is the appropriate choice when considering an impact severity. Because all the barrier designs with a tension belt successfully redirect the vehicle back onto the road the thinnest tension belt of thickness 3.0 mm is the best choice in terms of cost. Preglednica 3. Rezultati parametričnih simulacij cestnih varnostnih ograj Table 3. Results of parametric simulations of road-safety barriers Ojačitev Reinforcement Pasnica Tension belt Trapezno vodilo Wheel guidance d 3 17,5 1867 maks MVP max ASI 2,0 18,1 1519 2,5 18,3 1595 2,8 21,0 1682 3,0 13,9 1695 Delovna širina Working width 5 20,0 1820 W6 7 19,3 1834 W6 Razred delovne širine Working width class ___________ W6 W5 W5 W5 W5 110 Borovinšek M. - Vesenjak M. - Ulbin M. - Ren Z. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 101-111 Končna ocena o primernosti cestne varnostne ograje za raven zadrževanja H1, pa terja tudi upoštevanje delovne širine in preizkus cestne varnostne ograje z osebnim vozilom. Rezultati teh testov kažejo precej manjše vrednosti moči udarca pri varnostni ograji s trapeznim vodilom. Pri varnostni ograji s pasnico namreč pride do neposrednega udarca kolesa osebnega vozila v steber varnostne ograje, kar se kaže na večjih pojemkih, ki zaradi tega presežejo s standardom SIST EN 1317 predpisane mejne vrednosti. Tako lahko povzamemo, da je cestna varnostna ograja okrepljena s trapeznim vodilom najprimernejša konstrukcija varnostne ograje, ki izpolnjuje zahteve za raven zadrževanja H1. Glede na rezultate simulacij je najprimernejša cestna varnostna ograja z vodilom iz pločevine debeline 2,0 mm, saj pomeni najboljšo poravnavo med povprečno in največjo vrednostjo parametra MVP ter ima hkrati tudi najmanjšo delovno širino. For the final evaluation of the road-safety barrier’s suitability for the containment level H1, the working width and the impact of a personal vehicle also had to be taken into account. The results of these computations show lower values of impact severity for the barrier with a wheel-guidance profile. In the case of the barrier with a tension belt the wheel of the personal car impacts directly into the post, and the higher accelerations that result in large impact-severity parameters exceed the limits defined in the SIST EN 1317 standard. It is, therefore, concluded that the safety-barrier design reinforced with the wheel-guidance profile is the most suitable choice to fulfill the H1 containment-level requirements. According to the simulation results, the safety barrier with the 2.0-mm-thick wheel-guidance profile is the most appropriate design, since it offers the best compromise between average and maximum values of ASI and has the smallest working width. 7 LITERATURA 7 REFERENCES [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] Tehnična specifikacija za javne ceste TSC 02 (2003) Varnostne ograje, pogoji in načini postavitve. Ministrstvo za promet, Urad za standardizacijo in meroslovje, Direkcija republike Slovenije za ceste, oktober, 2003. [5] Vesenjak, M., Ren Z. (2003) Dinamična 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) Dinamična simulacija deformiranja cestne varnostne ograje pri naletu vozila. Kuhljevi dnevi ’03, Slovensko društvo 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 Borovinšek Authors’ Address: Matej Borovinšek Matej Vesenjak Matej Vesenjak dr. Miran Ulbin Dr. Miran Ulbin prof.dr. Zoran Ren Prof.Dr. Zoran Ren Univerza v Mariboru University of Maribor Fakulteta za strojništvo Faculty of Mechanical Eng. Smetanova 17 Smetanova 17 2000 Maribor SI-2000 Maribor, Slovenia ren@uni-mb.si ren@uni-mb.si m.vesenjak@uni-mb.si m.vesenjak@uni-mb.si ulbin@uni-mb.si ulbin@uni-mb.si matej.borovinsek@uni-mb.si matej.borovinsek@uni-mb.si Prejeto: Sprejeto: Odprto za diskusijo: 1 leto 5.10.2005 Received: Accepted: 16.11.2005 Open for discussion: 1 year Simulacija naleta tovornega vozila - Simulating the Impact of a Truck 111 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 112-125 UDK - UDC 534.83:621.43.044 519.61/.64 Izvirni znanstveni članek - Original scientific paper (1.01) Vibroakustično modeliranje alternatorja Vibro-Acoustic Modelling of an Alternator Martin Furlan1 - Robert Rebec1 - Andrej Černigoj1 - Damjan Čelič2 - Primož Čermelj2 - Miha Boltežar2 (1ISKRA Avtoelektrika, Šempeter pri Gorici; 2Fakulteta za strojništvo, Ljubljana) V prispevku so predstavljeni osnovni prijemi pri izdelavi sklopljenega modela alternatorja za ovrednotenje vibroakustičnega odziva kot posledice različnih vzbujanj alternatorja; vibracij motorja z notranjim zgorevanjem, neuravnoteženosti rotorja ter magnetnih sil. Prikazani sta izdelava in preverjanje strukturnih trirazseznih (3D) modelov metode končnih elementov (MKE) za osnovne sestavne dele; posamezne povezane sklope in za alternator kot celoto. Preverjanje strukturnih modelov je izvedeno z eksperimentalno modalno analizo (EMA), ki je obenem namenjena za ovrednotenje dušenja ter posodobitev strukturnega modela alternatorja. Natančneje je opisan poseben primer izračuna vibroakustičnega odziva kot posledice vzbujanja magnetnih sil. Pri tem je bil za ovrednotenje magnetnih sil izdelan trirazsežni model MKE za magnetno polje, tako da je omogočal harmonsko analizo magnetnih sil in njihov neposreden prenos v strukturni model MKE alternatorja. Na podlagi poznanega vzbujanja alternatorja, tj. magnetnih sil v določenih obratovalnih razmerah, je bil ovrednoten in analiziran strukturni in akustični odziv alternatorja. © 2006 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: alternatorji, vibracije, metode končnih elementov, metode robnih elementov, analize modalne) In this paper we present the basic approaches to building a sequentially coupled model of an alternator to evaluate the vibro-acoustic response resulting from different excitations, such as engine vibration, rotor unbalance and magnetic forces. Special emphasis is given to the set-up and verification of the three-dimensional (3D) structural finite-element (FE) model for the whole alternator and for the alternator components. 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 model updating. Special emphasis is given to the evaluation of magnetic noise, generated due to the magnetic force excitation in the alternator To estimate the magnetic forces and their harmonic components a 3D magnetic FE model of the alternator was prepared. Finally, the exciting magnetic forces, calculated for the specific operation conditions of the alternator, were transferred into a structural FE model, where the structural 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) 0 UVOD 0 INTRODUCTION Osnovni cilj izdelave sklopljenega numeričnega modela alternatorja za ovrednotenje vibroakustičnega odziva sta optimizacija in razvoj novih konstrukcij alternatorjev z notranjim ventilatorjem, ki naj po tehničnih značilnostih dosegajo ali celo presegajo primerljive konkurenčne izdelke. Slednje pomeni, da pri alternatorju dosežemo zadovoljiv odvod toplote oz. povečamo izkoristek pri majhni stopnji The optimisation of an alternator with an internal ventilator was the major driving force for setting up a coupled numerical model of the alternator intended to evaluate its vibro-acoustic response. Additionally, this model will be used in the development process of new designs of alternators, helping to improve their technical characteristics and make them comparable or even better than their competitors. In the case of the alternator, 112 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 112-125 aerodinamičnega hrupa ventilatorja ter pri tem ne ogrozimo dobe trajanja alternatorja ali povečamo njegovega vibracijskega in akustičnega odziva. Enako pomembna je optimizacija konstrukcije alternatorja z vidika magnetike, ki prav tako omogoča izboljšanje izkoristka ter zmanjšanje magnetnega hrupa alternatorja, povzročenega z magnetnimi silami, ki vzbujajo strukturo alternatorja. Pri doseganju omenjenega cilja se srečujemo z izključujočimi se zahtevami in omejitvami, ki iskanje optimalne rešitve zelo otežujejo. Pri iskanju takih rešitev so v veliko pomoč računalniška tehnologija in numerične metode. Predvsem razmah računalniške tehnologije v zadnjem času je omogočil, da se zahtevni večdisciplinarni problemi, npr. nastajanje hrupa, v doglednem času rešijo z numeričnimi simulacijami. Bistvena prednost, ki jo omogočajo numerične simulacije, je možnost uporabe optimizacije, kar zmanjša število prototipov in posledično količino eksperimentalnega dela. Skupno se to kaže v krajšem razvojnem času in večji kakovosti izdelka. 1 SKLOPLJENI VIBROAKUSTIČNI MODEL ALTERNATORJA V splošnem vibroakustični model alternatorja združuje tri fizikalne probleme, tj. magnetni, strukturni in akustični problem, in njim pripadajoče modele, ki jih lahko predstavimo kot celovit - zaporedno improvements could be achieved with better heat transfer, which corresponds to higher efficiency, where the aerodynamic noise of the ventilators needs to be kept under control. Besides this, the alternator’s durability should not be threatened by either an increase in the alternator’s vibration and/or acoustic response. Optimising the design of the alternator from the thermal and structural dynamic points of view is not enough. When achieving the above-mentioned goal one can meet many limitations and contradictory demands that make the search for the optimum design of the alternator very difficult. At this point numerical simulations can be applied as a very effective tool to help the engineer. The recent development of computer technology and numerical methods have also made it possible to simulate very demanding multidisciplinary problems such as the generation of the magnetic noise in a reasonable time. The major advantage of the computer technology and numerical methods is the possibility to make the optimisation and the evaluation in a virtual environment. This significantly decreases the number of prototypes and the quantity of the experimental work and finally shortens the development time and increases the quality of the product. 1 A COUPLED VIBRO-ACOUSTIC MODEL OF THE ALTERNATOR In general, the vibro-acoustic model of the alternator unites three physical problems, i.e., the magnetic, the structural dynamic and the acoustic. With regard to the physical nature of the alternator, magnetne sile magnetic forces Mag. polje - MKE Mag. field - FEM vibracije vibrations Meh. str. - MKE Mech. str. - FEM • Zvočno polje - MRE Sound field - BEM. zvok sound K /K /K /|\ /|\ /|\ /|\ /|\ /|\ /|\ /|\ /K /|\ obratovalni pogoji: alternator, obrem., pritrditev, vrt. hitrost, idr. operating conditions: alternator, load, support, rot. speed, etc. Sl. 1. Zaporedno povezani magnetni, strukturni in akustični problem alternatorja Fig. 1. The sequentially coupled magnetic, structural dynamic and acoustic problem of the alternator Vibroakustično modeliranje alternatorja - Vibro-Acoustic Modelling of an Alternator 113 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 112-125 Sl. 2. Obravnavani alternator Fig. 2. The investigated alternator sklopljeni problem. Značilnost takega problema je, da ga lahko obravnavamo v treh ločenih zaporednih korakih (sl. 1). Da bi potrdili verodostojnost tako postavljenega modela, je potrebno preverjanje na dejanskih modelih. Zaradi omenjenega smo se pri gradnji sklopljenega modela alternatorja odločili, da ga izvedemo za že obstoječi alternator (sl. 2). Z uporabo MKE za magnetno polje izračunamo velikosti in porazdelitev magnetnih sil, ki delujejo na zunanje površine rotorja in statorja alternatorja. Ker vrtenje rotorja vpliva na spreminjanje gostote magnetnega pretoka v zračni reži alternatorja, ima to za posledico periodičen značaj magnetnih sil. Da bi tako zahteven model magnetnega polja lahko rešili, smo predpostavili, da je porazdelitev magnetnih sil neodvisna od same obremenitve alternatorja. To pomeni, da v magnetni model nismo vnesli statorskih tokov, ampak le nespremenljiv rotorski tok, ki pomeni magnetno vzbujanje alternatorja. Porazdelitev magnetnih sil smo izračunali z magnetostatičnim modelom za različne medsebojne lege rotorja in statorja. Tako dobimo potek spreminjanja magnetnih sil pri vrtenju rotorja alternatorja, na katerem izvedemo harmonsko razdružitev. Rezultate spremenljivih magnetnih sil pri vrtenju rotorja, izračunanih z MKE za magnetno polje, smo uporabili kot vzbujevalne sile v strukturnem modelu po MKE. Temeljna naloga pri določitvi strukturnega odziva sta bili gradnja in ovrednotenje strukturnega modela po MKE. Strukturni model po MKE smo preverili z eksperimentalno modalno analizo (EMA), ki smo jo izvedli za različne sklope in each problem can be treated independently, and then they can all be linked into one sequentially coupled problem, Figure 1. To validate the presented model of the alternator 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 and stators, are calculated using the finite-element method (FEM) for the magnetic field. In this case it is assumed that a consequence of the stationary rotation of the rotor is a periodic varying of the magnetic forces caused by changes of the magnetic flux densities in the air gap of the alternator. To simplify the solution of the magnetic problem of the alternator it was necessary to make a few assumptions. The major one relates to the stator excitation. Here we assumed that the distribution of the magnetic 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 magnetic excitation in the magnetic model. The distribution of the magnetic forces was calculated using a magneto-static model for different rotor positions which describes the changing of the magnetic forces during the rotation and finally allows a calculation of their harmonic components. The results of the changing magnetic forces during the rotor rotation, calculated using FEM for the magnetic field, are used as the excitation forces in the structural FEM model. The basic job in the process of the evaluation of the structural dynamic response is to build and to verify the structural FEM model. The verification of the structural FEM model was done using experimental modal analysis (EMA) 114 Furlan M. - Rebec R. - Černigoj A. - Čelic D. - Čermelj P. - Boltežar M. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 112-125 podsklope alternatorja ter za alternator v celoti. Ovrednotenje strukturnega odziva modela po MKE alternatorja na vzbujanje z magnetnimi silami smo obravnavali kot problem harmonskega vzbujanja, pri čemer je vzbujanje podano s harmonskimi komponentami magnetnih sil. Zadnji korak v postopku vibroakustičnega modeliranja alternatorja obravnava zvočno polje kot posledico strukturnega odziva. Zvočno polje, ki je rešitev Helmholtzove valovne enačbe, smo obravnavali z metodo robnih elementov (MRE). Glede na numerično zahtevnost omenjenega problema [1] smo numerične simulacije izvedli le za omejeno število diskretnih vrednosti valovnih števil oz. frekvenc. Pri tem smo se omejili na frekvence z največjim prispevkom k magnetnemu hrupu oz. na frekvence prvih nekaj harmonskih komponent, ki nastanejo zaradi harmonske razdružitve magnetnih sil. 2 MAGNETNE SILE V ALTERNATORJU Ovrednotenje magnetnih sil v alternatorju smo izvedli z uporabo MKE. Zaradi oblike rotorja oz. rotorskih krempljastih polov smo morali ovrednotenje sil izvesti na trirazsežnem modelu, saj je omenjeno geometrijsko lastnost neposredno nemogoče opisati z dvorazsežnim modelom. Glede na simetrično obliko rotorja, ki ima 6 polovih parov, smo zgradili model, ki obsega le eno šestino celotnega alternatorja (sl. 3). on every single part of the alternator and its substructures as well as on the whole structure of the alternator. The evaluation of the structural response of the alternator FEM model, as a consequence of the magnetic force excitation, is considered as a harmonic problem. The last step in the process of the vibro-acoustic modelling of the alternator deals with the sound field as a result of the structural response, i.e., vibrations on the alternator structure. The sound field, governing by the Helmholtz wave equation, is evaluated by the boundary-element method (BEM). Due to the numerical extensivenesses of the mentioned problem [1], we limited the simulations of the sound field to only a few wave numbers or discrete frequencies. Only the frequencies with a significant contribution to the magnetic noise or a few first harmonics of the magnetic forces are taken into account for the sound-field calculation. 2 MAGNETIC FORCES IN THE ALTERNATOR The evaluation of the magnetic forces was carried out using the FEM. Because of the claw pole shape of the rotor it was necessary to build up a three-dimensional (3D) FEM model for the magnetic field. For that reason a two-dimensional FEM model cannot be used. As a result of the cyclic symmetry of the rotor with six pole pairs, only one sixth of the complete alternator was modelled, Figure 3. Taking into account the cyclic symmetry of Vibroakustično modeliranje alternatorja - Vibro-Acoustic Modelling of an Alternator 115 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 112-125 Zaradi simetrije alternatorja smo vrtenje the rotor, the magnetic forces calculation was izvedli za 60° v enakomerno razporejenih korakih. Pri performed in evenly distributed steps in a rotation izbiri velikosti koraka smo se odločili na podlagi over 60 degrees. The step size was chosen on the potrebe po razločevanju harmonskih komponent, ki basis of distinguishing the harmonic components nastanejo pri razvoju magnetnih sil v Fourierjevo that arise from the discrete Fourier transform. Based vrsto. Glede na predhodne izkušnje in poznavanje on previous practice and expertise in the modelling karakterja magnetnega hrupa ([1], [5] in [6]), pri of electric machine noise ([1], [5] and [6]), where the katerem se vpliv višjih harmonskih komponent influence of higher harmonics decrease and their zmanjšuje in njihove frekvence segajo v frekvenčna frequencies range out of vibro-acoustic interest, for področja, ki so glede vibroakustičnega odziva manj the evaluation of magnetic forces only the first 12 zanimiva, smo za ovrednotenje izbrali le prvih 12 harmonics were chosen. From those facts it follows harmonskih komponent. Iz omenjenega izhaja, da je that at least 24 steps must be executed to fulfil the treba izpeljati izračun magnetnih sil za vsaj 24 korakov, Nyquist criterion. In our case the magnetic forces da izpolnimo Nyquistov pogoj. V našem primeru so calculations during the rotation was performed in bili izračuni sil izvedeni za vrtenje v 60 korakih po 1°. steps of 1 degree. This enables us to evaluate the To omogoča, da ovrednotimo tudi višje harmonske higher harmonics that contribute to a precise komponente, kar pripomore k natančnejšemu presentation of the magnetic forces. Figure 4 shows ovrednotenju magnetnih sil. Slika 4 prikazuje the harmonic decomposition of the magnetic forces harmonsko analizo za magnetne sile, ki delujejo v acting on nodes along the centre of the stator tooth. vozliščih vzdolž sredine statorskega zoba. Besides the influence of rotation on the Poleg vpliva vrtenja na magnetne sile je bilo magnetic forces, the evaluation of the operating treba ovrednotiti tudi vpliv razmer pri obratovanju. conditions of the alternator is also important. For the Zaradi zahtevnosti smo se na tem mestu omejili na sake of simplicity we have studied only the unloaded obratovanje neobremenjenega alternatorja, ko ni alternator so there is no magnetic excitation on the statorskega vzbujanja. Glede na naravo delovanja stator side. Regarding the physical nature of the obravnavanega alternatorja smo razmere obratovanja alternator, the operating conditions were defined with opredelili s tokom v rotorskem navitju. Tako smo current in the rotor winding. Finally, the magnetic ovrednotenje magnetnih sil pri vrtenju rotorja izvedli forces calculation was performed for different rotor za različne rotorske tokove. Pri izbiri velikosti tokov currents. Knowing that under normal operating smo izhajali iz poznavanja obratovalnih razmer conditions the rotor current is generally between 2A alternatorja in upoštevali dejstvo, da se v običajnih and 4.2A, the appropriate rotor current was razmerah vrednosti tokov gibljejo med 2A in 4,2A. determined. z [mm] Sl. 4. Harmonske komponente magnetnih sil vzdolž statorskega zoba Fig. 4. Harmonic components of the magnetic forces along the centre of stator tooth 116 Furlan M. - Rebec R. - Černigoj A. - Čelic D. - Čermelj P. - Boltežar M. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 112-125 Prenos magnetnih sil iz modela po MKE za magnetiko v strukturni model po MKE je lahko zelo zahteven predvsem zaradi različnih postopkov reševanja magnetnega problema na eni in strukturnega problema na drugi strani. Različne zahteve po diskretizaciji pri obravnavi magnetike na eni strani in mehanske strukture na drugi strani onemogočajo uporabo enotne diskretizacije trirazsežnega modela po MKE za reševanje sklopljenega magnetomehanskega problema. Gostota mreže trirazsežnega modela po MKE, ki jo zahteva obravnava problema magnetike, je namreč preveliko breme pri reševanju strukturnega problema. Nasprotno lahko rečemo, da je diskretizacija strukturnega modela neprimerna ali celo neuporabna za reševanje magnetike. Glede na značaj magnetnih sil, predvsem v smislu krajevne in časovne porazdelitve na površini statorja in krempljastih polov rotorja je njihov učinek nemogoče v celoti nadomestiti z rezultirajočimi silami in momenti ([1], [5] in [6]). Slednje smo predpostavili le za učinek magnetnih sil, ki delujejo na krempljaste pole rotorja in so znotraj zelo toge strukture krempljastih polov zaradi simetrije vedno v ravnotežju. Nasprotno je za resničen prenos magnetnih sil, ki vzbujajo “elastičen” stator, potrebno ohraniti njihov krajevni značaj. Časovni značaj magnetnih sil pa je opredeljen s harmonskimi komponentami. 3 STRUKTURNI ODZIV ALTERNATORJA V drugi fazi vibroakustičnega modeliranja alternatorja je treba določiti strukturni odziv kot posledico vzbujanja alternatorja, tj. vibracij motorja z notranjim zgorevanjem, neuravnoteženosti rotorja ali magnetnih sil. Predstavljena je gradnja strukturnega modela po MKE, čemur je namenjen največji poudarek. Opisano je preverjanje modela ter končno ovrednotenje strukturnega odziva, pri čemer smo se omejili le na odziv kot posledico vzbujanja magnetnih sil. Z najmanjšimi dopolnitvami strukturnega modela je mogoče priti tudi do odziva kot posledice vibracij motorja z notranjim zgorevanjem ali neuravnoteženosti rotorja. Pri ovrednotenju strukturnega odziva se omejimo na ustaljeno delovanje alternatorja. Magnetne sile, ki v takem načinu obratovanja vzbujajo strukturo alternatorja, imajo periodičen značaj. Ob predpostavki, da je mehanska struktura linearen sistem z majhnim dušenjem, lahko obravnavani problem prevedemo na problem The transfer of magnetic forces from the magnetic FEM model to the structural FEM model can be very difficult to carry out because of there being one way to solve the magnetic model and another way to solve the structural model. A dissimilar requirement for the mesh density of the magnetic and structural problems is the reason why the same mesh cannot be used for the 3D FEM coupled magneto-structural problem. The mesh density required in the 3D FEM magnetic problem is too dense for solving 3D FEM structural problem. Put another way, the mesh density of the 3D FEM structural problem is too coarse to be used in the 3D FEM magnetic problem. Observing the character of the magnetic forces in terms of time and space distribution over the surface of the stator and the rotor claw pole it is impossible to entirely substitute their effect with the resulting magnetic forces and moments ([1], [5] and [6]). The latter is assumed for the affect of the magnetic forces acting on the rotor claw poles, which are a very rigid structure, and because of their symmetry the magnetic forces are always in counterbalance. In contrast to the credible transfer of magnetic forces, which excite the “elastic” stator, the preservation of their space distribution is required. Meanwhile, their time distribution is characterized with harmonic components. 3 STRUCTURAL RESPONSE OF THE ALTERNATOR In the second phase of the vibro-acoustic modelling of the alternator it is necessary to determine its structural response as a consequence of the excitation of the alternator, i.e., vibrations due to the operation of an internal combustion engine, rotor unbalance or magnetic forces. Great stress is laid on building a structural FEM model, which is introduced here. The model checking and the final verification of the structural response are described with the assumption that only the response as a consequence of the magnetic forces excitation was taken into account. With a minimum of modal updating it would also be possible to describe the response due to vibrations of an internal combustion engine and rotor unbalance. The steady state operation of the alternator was assumed during the verification of the structural response. The magnetic forces, which excite the structure, are in this case periodic. Taking into consideration the mechanical structure as a low-damped linear system, our problem can be Vibroakustično modeliranje alternatorja - Vibro-Acoustic Modelling of an Alternator 117 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 112-125 vsiljenega nihanja za celoten spekter harmonskih komponent magnetnih vzbujevalnih sil. Nazadnje iz znanega odziva posameznih harmonskih komponent obnovimo celotni strukturni odziv. V našem primeru smo za ovrednotenje strukturnega odziva na izbranem alternatorju uporabili MKE. Prvotno je bilo treba strukturni model po MKE zgraditi ter preveriti njegovo skladnost na dejanski izvedbi. Gradnje strukturnega modela smo se lotili korakoma. Strukturo alternatorja smo obravnavali po delih, ki smo jih na koncu združili v celoto. Za vsak posamezni del alternatorja smo izdelali model po MKE in ga tudi preverili. Sklepno dejanje pri gradnji strukturnega modela po MKE je bilo povezovanje posameznih delov v celoto. Za potrebe ovrednotenja smo izvedli različne meritve strukturnega odziva, tako posameznih delov kakor celotnega alternatorja, pri čemer smo uporabili eksperimentalno modalno analizo (EMA). To je poleg ovrednotenja strukturnega modela po MKE omogočilo tudi ovrednotenje dušenja za posamezne sklope alternatorja. 3.1 Gradnja strukturnega modela alternatorja Glede na zapletenost strukture obravnavanega alternatorja smo strukturni model po MKE gradili postopoma. Pri tem smo izhajali iz razdelitve obravnavanega alternatorja, ki se ponuja sama po sebi. Celotno strukturo smo razdelili v štiri sklope: prednji ležajni pokrov (PLP), zadnji ležajni pokrov (ZLP), stator in rotor (sl. 5). Ker je že vsak sklop sam po sebi dovolj zahteven, je bilo treba tudi pri gradnji strukturnega modela po MKE posamezni sklop graditi v več korakih, kolikor je pač konstrukcija to dopuščala. Tako smo pri gradnji strukturnega modela MKE za posamezen sklop izhajali iz osnovnih podsestavov, ki jih lahko z MKE preprosto modeliramo in v nadaljevanju s preizkusi preverimo. Postopoma smo model sklopa dopolnjevali in ga sproti večkrat preverili. Lahko rečemo, da smo pri gradnji strukturnega modela in njegovih sklopov sledili tehnološkemu postopku izdelave alternatorja. Tak postopek je zahteval veliko število prototipov in preizkusnega dela. Sklepno dejanje v izgradnji strukturnega modela po MKE obravnavanega alternatorja je povezava predhodno izdelanih sklopov v celoto. Zato da bi lahko čimbolj prepričljivo izvedli mehanske povezave v strukturnem modelu po MKE, moramo spoje dobro poznati. Glede na naravo spojev oz. mehanskih povezav, ki se pojavljajo med PLP, ZLP, statorjem in rotorjem, smo delali s tremi različnimi transformed into a forced vibration problem for the whole spectrum of excitation, magnetic-force harmonics. Finally, from the known response of the particular harmonic components, the entire structural response can be reconstructed. For the verification of the structural response of the selected alternator, a FEM was used. The structural FEM model had to be built and verified on a real structure first. The modelling of the whole structure was done using a step-by-step procedure. Each part of the alternator was taken apart, modelled by means of the FEM and verified. Combining those FEM sub-models together was the most difficult step in the structural FEM building up. From the necessity of the verification, various measurements of the structural response, not only on subsystems but also on assemblies, were carried out using experimental modal analysis (EMA). This was necessary for the verification of the structural FEM model, as well as for the verification of the damping on different assemblies of the alternator. 3.1 Structural model of the alternator With regard to the complexity of the alternator’s structure the structural FEM model was built up gradually. In this case we used alternator subpart 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 the great complexity of the subpart FEM models, some of them had to be built gradually, as the design so allowed. In this way we proceeded from the construction of very basic element FEM models, the characteristics of which were then experimentally verified. The main FEM model was progressively built and experimentally verified at every coupling step. We can say that we followed the technological production process during the building up of the FEM model of the alternator. This kind of process required many different prototypes and types of experimental work. The final action in the structural FEM model construction of the investigated alternator is the coupling of the previously made subassemblies into an integrity. To design the structural FEM model connections as realistically as possible, we should know them very well. Regarding the nature of the connection that occurs between the DEB, the REB, the stator and the rotor, we applied three different 118 Furlan M. - Rebec R. - Černigoj A. - Čelic D. - Čermelj P. - Boltežar M. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 112-125 Sl. 5. Sklopi alternatorja: PLP (zgornja vrsta), stator (srednja vrsta) in ZLP (spodnja vrsta) ter njim pripadajoča 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) mehanskimi povezavami. Najprej smo opredelili mechanical connections. First of all we defined the povezavo, ki jo predstavljata ležaja in se pojavlja v connection between the rotor and both end brackets stiku med rotorjem in PLP ter ZLP. V našem primeru - the DEB and the REB. In our case we calculated the smo togosti ležajev izračunali ([1] in [2]) ter njune stiffness matrix of the bearings ([1] and [2]) and used lastnosti uporabili v strukturnem modelu po MKE. them in the structural FEM model. The next type of Drugo obliko mehanske povezave, ki povezuje PLP, coupling was used to connect the alternator as an ZLP in stator, ter daje alternatorju značaj celote, integrity and represents four screws between the DEB predstavljajo štiri vijačne zveze. Glede na to, da and the REB. Taking into account that the mentioned obravnavani spoj združuje okrov in prednji pokrov s connections couple subparts with shape and force, silo in obliko, smo ga v strukturnem modelu po MKE we modelled them by coupling all the degrees of izvedli z združitvijo vseh prostostnih stopenj v freedom (DOFs) of the nodes situated in the proximity vozliščih, ki se pojavljajo na mestu vijačne zveze. of the screw joint. A third different kind of connections Tretja oblika mehanske povezave je stik, ki nastaja was used to model the contact conditions between pri naleganju med statorjem in PLP ter ZLP. Ker smo the stator and the DEB or REB. As we have decided to se odločili, da mehansko strukturo opišemo z describe the mechanical structure with a linear model, linearnim modelom, stika nismo mogli modelirati v we did not have the option to model the contact pravem pomenu. Zato da smo ohranili linearnost conditions as they should be. To ensure the structural strukturnega modela po MKE, smo ga modelirali tako, FEM model’s linearity, we used just connections that da smo spojili pomike na stičnih površinah. couple the nodal DOFs on the contact surfaces. Vibroakustično modeliranje alternatorja - Vibro-Acoustic Modelling of an Alternator 119 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 112-125 Ovrednotenje strukturnega modela po MKE The validation of the structural FEM model smo izvajali na podlagi primerjave modalnih was done by comparing the modal parameters to parametrov (lastnih frekvenc in vektorjev). Kot real subparts (natural frequencies and mode shapes). dodatna kriterija smo uporabili kriterij modalnega As additional criteria, we used the modal assurance zaupanja (KMZ - MAC) ter vidno primerjavo criterion (MAC) and a visual comparison of the modalnih oblik, ki nam jo omogoča EMA. modal shapes, which was done by using means of Nenazadnje tudi masa strukturnega modela po MKE the EMA. As a frequently used comparison criterion govori o resničnosti, zato smo jo vseskozi primerjali during model validation, we must also mention the z maso dejanske strukture. Če so izpolnjeni vsi mass of the structural FEM model. If we fulfil all the navedeni kriteriji, imamo zagotovilo, da model dobro mentioned criteria we can be sure that our model opisuje masne in togostne lastnosti dejanske properly describes the mass and stiffness properties strukture. Vprašanje dušenja smo rešili tako, da smo of the real structure. The issue of damping the v modelu po MKE uporabili nespremenljivi structural FEM model was solved by using a 1% razmernik strukturnega dušenja en odstotek. Ker constant-damping ratio. As we know, the FEM vemo, da ima gostota diskretizacije strukturnega model mesh density has an influence on the accuracy modela po MKE vpliv na natančnost izračuna, smo of the calculation. It turns out that every doubling njen vpliv omejili takole. Vsaka podvojitev števila of the elements, irrespective of whether this is elementov v modelu, bodisi celotnega alternatorja, performed on just one subpart or on the whole sklopa ali podsestava, vodi do razlik v izračunanih alternator, leads to differences between the natural vrednostih lastnih frekvenc v območju do 4 kHz, ki frequencies and the measured ones. This pa so manjše od enega odstotka. V nadaljevanju je discrepancy is less than 1% in the range 0-4 kHz. kot primer prikazano ovrednotenje strukturnega The structural FEM model of the DEB (Figure 6, modela po MKE za PLP (pregl. 1 in sl. 6) ter za celotni Table 1) and the whole alternator (Figure 7, Table 2) strukturni model alternatorja (pregl. 2 in sl. 7). are shown as examples. Sl. 6. Primerjava lastnih oblik PLP, strukturni model po MKE (zgoraj), model EMA (spodaj); prva (na levi), 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) 120 Furlan M. - Rebec R. - Černigoj A. - Čelic D. - Čermelj P. - Boltežar M. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 112-125 Preglednica 1. Izračunane in izmerjene lastne frekvence za PLP Table 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. Izračunane in izmerjene lastne frekvence za alternator kot celoto Table 2. Calculated and measured natural frequencies of the alternator MKE f [Hz] 1 2 3 4 412 889 900 1726 1 2 3 4 EMA f [Hz] 3.2 Prenos magnetnih sil v strukturni model ter odziv V nadaljevanju je opisano ovrednotenje strukturnega odziva alternatorja na podlagi predhodno zgrajenega strukturnega modela po MKE, pričemer je alternator vpet na motor z notranjim zgorevanjem. Pri določitvi narave vzbujevalnih sil smo izhajali iz rezultatov, dobljenih pri ovrednotenju magnetnih sil z MKE. Zato da bi lahko izvedli 587 908 951 1272 d [%] 2,02 0,44 0,69 1,29 Df [%] -29,8 -2,1 -5,4 35,7 3.2 Transfer of magnetic forces to a structural model and to the response Work proceeds with the evaluation of the structural response of the alternator based on the previously built structural FEM model, while the alternator is mounted on the internal combustion engine. The nature of the excitation forces was determined from the results of the magnetic forces verification using the FEM. In order to compare measured and calculated data, the Vibroakustično modeliranje alternatorja - Vibro-Acoustic Modelling of an Alternator 121 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 112-125 primerjavo med meritvami in izračunom, je bilo treba določiti jakost magnetnih sil in njihovo osnovno frekvenco vzbujanja. Pri tem smo se oprli na izmerjene vrednosti toka, ki so določale moč magnetnih sil, ter na izmerjene vrednosti vrtilnih frekvenc, ki so določale osnovne frekvence vzbujanja. Ker je odziv strukture alternatorja močno odvisen od frekvence vzbujanja in zaradi tega od vrtilne frekvence alternatorja, smo osnovno frekvenco vzbujanja za harmonske komponente magnetnih sil vzeli iz meritev vrtilne frekvence. Le-ta se giblje med 3000 min1 in 7500 min 1. Tako je frekvenca osnovne harmonske komponente magnetnih sil (šest polovih parov na rotorju) v različnih razmerah obratovanja nahaja v območju od 300 Hz do 750 Hz. Ker med posameznimi harmonskimi komponentami enakega reda, vendar za različne smeri rezultirajočih magnetnih sil, obstajajo fazni odmiki, je bilo pri prenosu magnetnih sil na strukturni model treba upoštevati tudi to. Fazno usklajenost smo ohranili tako, da smo harmonske komponente magnetnih sil v strukturni model po MKE vnesli v kompleksni obliki. Slika 8, na levi, prikazuje strukturni model po MKE statorja alternatorja s prirejenimi trenutnimi vrednostmi magnetnih sil, medtem ko slika 8, na desni, prikazuje strukturni odziv kot posledico obratovanja neobremenjenega alternatorja zaradi delovanja pete harmonske komponente magnetnih sil, katere frekvenca pri vrtilni frekvenci 3000 min1 ustreza 1500 Hz. intensity and the fundamental excitation frequency of the magnetic forces must be determined. This was done using the measured values of current, which determine the intensity of the magnetic forces, and the measured values of the rotational velocity, which determine the fundamental excitation frequencies. The structural response of the alternator is strongly dependent on the excitation frequency and consequently on the rotational velocity of the alternator. For this reason, the fundamental excitation frequency for the harmonic components of the magnetic forces was adopted from the measurement of the rotational velocity. The latter varies from 3000 min1 to 7500 min-1. So, the frequency of the fundamental harmonic component of magnetic forces (six pole pairs on the rotor), on various working conditions, occurs in the range between 300 Hz and 750 Hz. Between some particular harmonic components of the same order, but for different directions of the resulting magnetic forces, a phase shift occurs. During the transfer of magnetic forces to structural model this also had to be considered. The harmonic components of the magnetic forces were entered into the structural FEM model in a complex form, and a phase adjustment was preserved. Figure 8, on the left, shows the structural FEM model of the stator (of the alternator) with arranged instantaneous values of the magnetic forces, while Figure 8, on the right, depicts the structural response as a consequence of the operation of the unloaded alternator due to the effect of the 5th harmonic component of the magnetic forces. The frequency of the 5th harmonic component at rotational velocity of 3000 min1 corresponds to 1500 Hz. Sl. 8. Magnetne sile na statorju (na levi) ter strukturni odziv modela po MKE alternatorja kot posledice 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 alternator as a consequence of the 5 th harmonic component of the magnetic forces (right) 122 Furlan M. - Rebec R. - Černigoj A. - Čelic D. - Čermelj P. - Boltežar M. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 112-125 4 AKUSTIČNI ODZIV ALTERNATORJA Zadnja faza vibroakustičnega modeliranja alternatorja je bila določitev zvočnega polja v njegovi okolici na podlagi predhodnega poznavanja strukturnega odziva modela MKE alternatorja. Pri numeričnem ovrednotenju, pri katerem z metodo robnih elementov (MRE) na podlagi znanega strukturnega odziva, izračunamo zvočno polje. Pri ovrednotenju zvočnega polja smo se držali enakih omejitev, kakor smo jih navedli pri ovrednotenju strukturnega odziva. Bistvena je zahteva po ustaljenem obratovanju alternatorja. Strukturni odziv oz. vibracije, ki se pri takem obratovanju pojavijo na zunanjih površinah alternatorja, povzročajo zvočno valovanje ali ti. magnetni hrup. Ob predpostavki, da mehansko strukturo alternatorja in pojav zvočnega polja v njegovi okolici opišemo z linearnim modelom, lahko reševanje obravnavanega sklopljenega problema prevedemo na problem vsiljenega nihanja oz. na reševanje za posamezne harmonske komponente. Tako vsaka obratovalna oblika oz. strukturni odziv posamezne harmonske komponente povzroča pripadajočo harmonsko komponento zvočnega polja. Poznavanje zvočnega polja za vse harmonske komponente v končni fazi omogoča rekonstrukcijo celotnega zvočnega polja kot posledico strukturnega odziva alternatorja zaradi vzbujanja z magnetnimi silami. Pri gradnji akustičnega modela MRE smo izhajali iz strukturnega modela po MKE. Celotno zunanjo površino alternatorja, ki jo v strukturnem modelu po MKE sestavljata mreži PLP in ZLP, smo prekrili z redkejšo mrežo trikotnih robnih elementov. Pri tem je mreža robnih elementov zgrajena tako, da si robni elementi delijo vozlišča strukturnega modela po MKE, kar omogoča preprost prenos robnih pogojev iz strukturnega modela po MKE na akustični model MRE. V našem primeru akustičnega modela MRE je največja dolžina robnega elementa manjša od 9 mm, kar zagotavlja dovolj natančno reševanje v frekvenčnem območju do 6,4 kHz. Za opis akustičnega medija smo vzeli lastnosti zraka v normalnih okoliščinah. Z namenom, da bi se seznanili z zvočnim poljem v okolici alternatorja, smo zvočno polje okolice ovrednotili le za nekaj izbranih primerov obratovanja. Slika 9 prikazuje zvočno polje v okolici alternatorja pete harmonske komponente magnetnega hrupa neobremenjenega alternatorja pri obratovanju s 3000 min1. 4 ACOUSTIC RESPONSE OF THE ALTERNATOR In the final step of the vibro-acoustic modelling of the alternator, a sound field in its surroundings was determined. This was obtained from the known structural response of the alternator FEM model. The numerical verification of the sound field is basically just an upgrade in the solving of the structural problem, where, on the basis of known structural response and by means of the boundary-element method (BEM), the sound field is calculated. During the verification of the sound field the same restriction as at the verification of the structural response was used. The demand to have a stationary operation of the alternator is essential. The structural response during such an operation on the external surfaces of the alternator causes a sound radiation or the so-called magnetic noise. Suppose that the mechanical structure of the alternator and the phenomenon of the sound field in its surrounding can be described using a linear model, the solving of the discussed combined problem can then be transformed to a forced vibration problem or to solve a particular harmonic component. Thus, each operational shape, or the structural response at a particular harmonic component, contributes a belonging component to the sound field. Knowing the sound field for each harmonic component makes it possible to reconstruct the entire sound field as a consequence of the structural response of the alternator due to the magnetic force excitations. The basis for the acoustic BEM model was the structural FEM model. The entire external surface of the alternator, which had been formed in the structural FEM model from the DEB and REB meshes, was covered with a less dense mesh consisting of triangular boundary elements. The mesh of the boundary elements was formed in such a way that its nodes coincided with the nodes of the structural FEM model. This enables an easy transfer of the boundary conditions from the structural FEM model to the acoustic BEM model. In our case of an acoustic BEM model, the maximum length of the boundary element was smaller than 9 mm, so an accurate solving in the region of 6.4 kHz was ensured. For the description of the acoustic medium, the air properties under regular conditions were adopted. In order to get to know the sound field in the surroundings of the alternator, the sound field was evaluated for just a few selected cases of the operation. Figure 9 shows the sound field in the surroundings of the unloaded alternator at 3000 min1 as a consequence of the 5th harmonic component of the magnetic noise. Vibroakustično modeliranje alternatorja - Vibro-Acoustic Modelling of an Alternator 123 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 112-125 Sl. 9. Zvočno polje v okolici alternatorja kot posledica 5-te harmonske komponente magnetnih sil; trenutna vrednost zvočnega tlaka (na levi) in raven zvočnega tlaka (na desni) Fig. 9. The sound field in the surroundings of the alternator as a consequence of the 5th harmonic component of the magnetic forces; the instantaneous value of sound pressure (left) and the level of sound pressure (right) 5 SKLEP 5 CONCLUSION Prispevek predstavlja postopek izdelave in This paper presents the process of building, and deloma tudi ovrednotenja sklopljenega partly also of verifying, a coupled vibro-acoustic model of vibroakustičnega modela alternatorja, ki združuje an alternator that unites three physical problems, i.e., the tri fizikalne probleme, tj. magnetni, strukturno- magnetic, the structural dynamic and the acoustic. The dinamični ter akustični problem. Predhodno je vibro-acoustic response as a result of the magnetic ovrednoten vibroakustični odziv alternatorja kot excitation is evaluated and it shows a preliminary study of posledica vzbujanja z magnetnimi silami. S tem je the magnetic noise of the alternator. This means that we postavljen postopek, ki omogoča preprosto in hitro have successfully established the process where it is easily posodabljanje parametrov modela ter possible to change almost any parameters of the model razpoznavanje in raziskavo vpliva ključnih and update the response results. In addition, this makes it parametrov konstrukcije alternatorja na njegove possible to identify and to analyse the important design tehnične značilnosti. Glede na dejstvo, da je parameters of the alternator and find their influence on the izdelava vibroakustičnega modela v začetni fazi, technical characteristic of the alternator. Due to the fact je treba izvesti dodatno preverjanje in posodobitve that the vibro-acoustic model of the alternator is in its modela predvsem v smislu izboljšanja modela early stages, it needs additional verification and updating, mehanske strukture alternatorja. especially the structural FEM model. 6 LITERATURA 6 REFERENCES [1] Furlan M. (2003) Karakterizacija magnetnega hrupa enosmernega elektromotorja: doktorsko delo, Univerza v Ljubljani, Fakulteta za strojništvo, Ljubljana. [2] Rebec R. (2004) Karakterizacija vibracij alternatorja AAN: Diplomska naloga univerzitetnega študija, Univerza v Ljubljani, Fakulteta za strojništvo, Ljubljana. [3] Furlan M., Černigoj A. in Boltežar M. (2003) A coupled electromagnetic-mechanical-acoustic model of a DC electric motor, Compel, 2003, letn. 22, št. 4, str. 1155-1165. [4] Boltežar M., Čelič D., Jakšič N, Keber M. in Čermelj P. (2004) Analiza in modeliranje alternatorja AAN: poročilo, Univerza v Ljubljani, Fakulteta za strojništvo, Ljubljana. 124 Furlan M. - Rebec R. - Černigoj A. - Čelič D. - Čermelj P. - Boltežar M. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 112-125 [5] Ramesohl I., Bauer T. in Henneberger G. (1998) Calculation procedure of the sound field caused by magnetic excitations of the claw-pole alternator, International seminar on vibations and acoustic noise of electric machinery, Bethune. [6] Ramesohl I., Henneberger G., Kueppers S. in Hadrys W. (1996) Three dimensional calculation of magnetic forces and displacements of a claw-pole generator, IEEE Transactions on Magnetics, 32(3):1685–1688, 1996. Naslova avtorjev:dr. Martin Furlan Authors’ Addressses: Dr. Martin Furlan Robert Rebec Robert Rebec Andrej Černigoj Andrej Černigoj ISKRA Avtoelektrika d.d. ISKRA Avtoelektrika d.d. Polje 15 Polje 15 martin.furlan@iskra-ae.com martin.furlan@iskra-ae.com robert.rebec@iskra-ae.com robert.rebec@iskra-ae.com andrej.cernigoj@iskra-ae.com andrej.cernigoj@iskra-ae.com Damjan Čelič Damjan Čelič dr. Primož Čermelj Dr. Primož Čermelj prof.dr. Miha Boltežar Prof.Dr. Miha Boltežar Univerza v Ljubljani University of Ljubljana Fakulteta za strojništvo Faculty of Mechanical Engineering Aškerčeva 6 Aškerčeva 6 1000 Ljubljana SI-1000 Ljubljana, Slovenia damjan.celic@fs.uni-lj.si damjan.celic@fs.uni-lj.si primoz.cermelj@fs.uni-lj.si primoz.cermelj@fs.uni-lj.si miha.boltezar@fs.uni-lj.si miha.boltezar@fs.uni-lj.si Prejeto: Sprejeto: Odprto za diskusijo: 1 leto 9.10.2005 Received: Accepted: 16.11.2005 Open for discussion: 1 year Vibroakustično modeliranje alternatorja - Vibro-Acoustic Modelling of an Alternator 125 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 126-137 UDK - UDC 531.36/38:621.313.13 Izvirni znanstveni članek - Original scientific paper (1.01) Modeliranje in analiza dinamike ščetke elektromotorja Modeling and analyzing the dynamics of an electric-motor brush Janko Slavic1 - Miha Nastran2 - Miha Boltežar1 (1Fakulteta za strojništvo, Ljubljana; 2Domel, Železniki) V prispevku je na kratko predstavljen Pfeiffer-Glockerjev postopek modeliranja dinamike togih teles z enostranskimi stiki. Mogoča stična stanja: lepenje, drsenje, trk s trenjem, sprostitev stika se preoblikujejo na linearni komplementarni problem, ki omogoča reševanje več sočasnih stičnih stanj. Predstavljeno teoretično ozadje je prilagojeno za telesa nepravilnih oblik. Uporaba predstavljenih zamisli je prikazana na dinamičnem modelu ščetke elektromotorja. Dinamični model je definiran z več ko 40 parametri, podrobni popis geometrijske oblike pa vključuje tudi površinsko hrapavost. V numeričnem preizkusu prispevek prikaže, kako obraba in togost ščetke vplivata na njeno stabilnost. © 2006 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: dinamika teles, telesa toga, simuliranje numerično, modeli dinamični) This paper briefly presents the Pfeiffer-Glocker formulation for the multibody dynamics of rigid bodies with unilateral contacts. The multiple, concurrent contact situations of stick-slip, detachment and impact with friction are solved as a linear complementarity problem. The theory is extended toward the discretely defined bodies of complex body shapes with nonlinearities. As shown in the numerical example of the electric-motor-brush dynamics the presented extensions can be used to simulate the influence of a detailed geometry, including surface roughness. The influences of brush-wear and brush-stiffness on the dynamic 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) 0 UVOD Dinamika sistema togih teles v zadnjem času pridobiva pozornost, saj se uporablja tako pri krmiljenju robotov, podajnih mehanizmov, kakor tudi pri izdelavi navideznih prototipov in pri simuliranju navidezne resničnosti [1]. V nasprotju s postopki simuliranja sistemov z dvostranskimi stiki (npr.: vrtljive zveze, vodila itn.), ki so že dobro uveljavljeni, so se načini vključevanja enostranskih stikov (npr.: trk, trk dveh teles) razvili do primerne matematično-fizikalne doslednosti šele v zadnjem desetletju. V tem prispevku se bomo osredotočili na enostranske stike, zapisane v obliki linearnega komplementarnega problema (LKP - LCP), tak zapis je prvi uporabil Lotstedt [2]. Med pomembnejše raziskovalce tega področja sodijo še Murty [3], Baraff [4], Panagiotopoulos [5], Moreau [6], Pfeiffer in Glocker([7] in [8]) itn. Podrobneje si bomo ogledali enega od bolj obetavnih postopkov popisa dinamike togih teles z 0 INTRODUCTION The mathematical formulation of multibody dynamics with unilateral contacts has received much interest in recent decades. This is particularly so for various control systems, such as the control of robots and industrial feeding mechanisms. Furthermore, it is included into research on virtual prototyping and virtual reality [1]. Despite the fact that bilateral contacts (e.g., rotating joints, linear joints) have been theoretically covered for several decades, the general formulation of unilateral contacts (e.g., impact with the friction of two bodies) was developed to the necessary mathematical and physical consistency only in the past decade. This paper focuses on the research of unilateral contacts written as a Linear Complementarity Problem (LCP), which was initiated by Lotstedt [2]. Some of the more important researchers in this field are Murty [3], Baraff [4], 126 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 126-137 enostranskimi stiki: to je Pfeiffer-Glockerjev postopek [7]. Njuno delo pomeni matematično pravilen in fizikalno dosleden način reševanja dinamike togih teles z več sočasnimi stičnimi stanji. Pfeiffer in Glocker stični problem (lepenje, drsenje, sprostitev stika in trk s trenjem) preoblikujeta na pregleden in zgoščen zapis v obliki linearnega komplementarnega problema. Raziskovalca sta v svojih raziskavah uvedla novo razstavitev trenja, ki v primeru odvisnih koordinat nima težav s singularnostjo in vodi v rešitev tudi v primeru predoločenih sistemov; kot prva sta trk s trenjem predstavila v obliki linearnega komplementarnega problema. Namen prispevka je prilagoditev Pfeiffer-Glockerjevega postopka za diskretno definirana telesa. V ta namen je v drugem poglavju na kratko predstavljen njun postopek simuliranja dinamike togih teles, ki temelji na komplementarnosti stičnih stanj. Uporaba v drugem poglavju predstavljenih zamisli za reševanje diskretno definiranih teles je nato prikazana v tretjem poglavju na dinamičnem modelu ščetke elektromotorja z 11 prostostnimi stopnjami; kot primer analize je predstavljen vpliv nekaterih parametrov modela na stabilnost delovanja ščetke. Zadnje sledi poglavje s sklepi. 1 DINAMIKA SISTEMA TOGIH TELES KOT LINEARNI KOMPLEMENTARNI PROBLEM Zaradi celovitosti si bomo v tem poglavju na kratko pogledali bistvene zamisli Pfeiffer-Glockerjevega postopka reševanja sistemov togih teles z enostranskimi stiki ([7] do [9]). Gibalne enačbe sistema togih teles s f prostostnimi stopnjami (vključujoč dvostranske stike) so: Panagiotopoulos [5], Moreau [6], and Pfeiffer and Glocker ([7] and [8]). One of the more promising and complete theories for including unilateral contacts was presented by Pfeiffer and Glocker [7]; their theory presents a sound and physically consistent basis for including concurrent multiple contact situations. They were also the first to present concurrent impacts with friction as a LCP Furthermore, their decomposition of stick-slip and the detachment problem avoids the singularity of overdefined dynamical systems. This paper is organized as follows: the second section presents the Pfeiffer-Glocker formulation for simulating unilateral contact problems in general. In addition, some modifications for discretely defined bodies are given. The ideas are presented in a numerical example of electric-motor-brush dynamics with 11 degrees of freedom, given in the third section. As an example of an analysis the stability of electric-motor-brush dynamics is studied. The last section gives conclusions. 1 MULTIBODY DYNAMICS AS A LINEAR COMPLEMENTARITY PROBLEM For the sake of completeness this section gives a brief review of the mathematical modeling of multibody dynamics with unilateral contacts as presented by Glocker and Pfeiffer ([7] to [9]). The equations of motion for a multibody system with f degrees of freedom (including only bilateral contacts) can be written as: M(q,t)q-h(q,q,t) = 0 e Rf (1), kjer so M masna matrika, q vektor posplošenih (generaliziranih) koordinat in h vektor posplošenih aktivnih sil. Če imamo v nekem trenutku množico stičnih točk ieIN, potem se (1) spremeni: where M is the mass matrix, q is the vector of generalized coordinates and h is the vector of generalized active forces. If there is a set of i e IN contact forces (as a result of unilateral contacts) then the equations of motion will be: Mq-h = YQiC e Rf (2), kjer je QC posplošena nekonservativna sila (kot posledica stične sile v stiku i). Načeloma je treba posplošene koordinate prilagajati trenutnim prostostim, ki pa so odvisne od rešitve stičnega problema. Za primer navedimo, da bi za rešitev sistema z nN stičnimi točkami morali where QC are the generalized, non-conservative active forces. Note that the contact forces change the number of degrees of freedom. In general it is not known which degrees of freedom disappear; this problem is usually solved by looking at all the possible solutions and finding the one that is physically consistent. If there are nN Modeliranje in analiza dinamike - Modeling and analyzing the dynamics 127 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 126-137 najti fizikalno komplementarno rešitev med 3nN mogočimi rešitvami [7]1. Tako iskanje ustrezne rešitve postane hitro praktično nemogoče izvedljivo. Zraven tega pa je z vidika numeričnega reševanja zelo neprimerno neprestano prilagajanje števila posplošenih koordinat. Kakor bomo videli pozneje, se tem težavam z uporabo LKP elegantno izognemo, saj bo število posplošenih koordinat vedno enako številu prostosti sistema brez enostranskih stikov (2). Poglejmo si najprej povezavo relativnih stičnih sil s posplošenimi stičnimi silami. Z uporabo Jacobijeve matrike lahko normalno stično silo F N v točki C na telo A (sl. 1) zapišemo kot posplošeno silo: dIrCA dq possible contact points with a stick-slip transition or detachment, then there are 3nN possible solutions [7]1. It is clear that the search for a physically consistent combination is time consuming. Furthermore, for numerical simulations it is not appropriate to change the minimum number of coordinates during each time-step. As we see later, the linear complementarity problem (LCP) method solves this problem in an elegant way, and the number of generalized coordinates is constant at all times. The number of generalized coordinates is always equal to the number of degrees of freedom of the system without unilateral contacts (2). The real contact forces are linked with the generalized contact forces via the Jacobian matrix. In Figure 1 two bodies are shown, the centers of gravity being denoted by A and B. The normal contact force FA N at point CA on the body A as a generalized contact force is: Qc A,N in če dodamo še silo na telo B: A,N=JTCA/InA-XN (3) and when including the normal force at point B: QcN={JCTAInA+JCTBInB\XN=wNXN (4) wN vsebuje kinematične lastnosti stika C, 1 je stična amplituda sile in I označuje inercialen koordinatni sistem. Zgornja izpeljava velja za zvezno definirana telesa v splošnem. Kako določimo Jacobijevo matriko stične točke, če ima telo definirano obliko glede na svoje težišče pa sta izpeljala Slavičin Boltežar([10] in [11]). Analogno normalni smeri nadaljujemo s tangentno stično silo (indeks T) in izraz (2) preoblikujemo v: Mq-h-^{wNXN + w wN includes the kinematical properties of the contact, XN 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 the contact point can be further simplified, as shown by Slavicand Boltežar ([10] and [11]). By using a similar notation for the tangential force (index T), Equation (2) is rewritten as: Z uporabo matričnega zapisa: gibalne enačbe preoblikujemo v: WN = {wN}, Or by using matrix notation: WT={wT}, ieIN the equations of motion are: M q&& -h-(WN WT f (5). (6) (7). Stična stanja rešimo v dveh korakih: najprej na ravni impulzov rešimo nezvezni problem trka s trenjem, nato pa na ravni sil še lepenje, drsenje ali sprostitev stika. Če ni trčnih stanj, prvi korak odpade. 1Trčna stanja so v podanem primeru izključena; vsaka stična točka je lahko v eni od naslednjih faz: lepenje, drsenje in sprostitev stika. 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 is solved. While the impact is solved in the impulse-domain, the stick-slip or detachment is solved in the force-domain. 1In this example the impact situations are excluded. Each of the contacts can be in one of the states: sticking, slipping or detachment of the contact. 128 Slavic J. - Nastran M. - Boltežar M. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 126-137 F =-F Sl. 1. Stične sile Fig. 1. Contact forces Preden v nadaljevanju podrobneje spoznamo oba koraka, si poglejmo množice stičnih točk: IG ={1,2,.., nG} In ={izIS;gN=0} All the possible contact points I are IH ={ieIN;gT=0} Množica lS vključuje v nekem koraku aktivne stike, lN samo tiste z nično normalno relativno hitrostjo in lH možne lepene stike. Stične množice se lahko v vsakem časovnem koraku spremenijo. 1.1 Lepenje, drsenje ali sprostitev stika Lepenje, drsenje (in prehod med njima) in sprostitev stika rešujemo na množici stičnih točk. Gibalne enačbe (7) in relativni stični pospeški g so ([7], [8] in [12]): organized in four sets during each time-step: nS elementov/elements n elementov/elements nH elementov/elements (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 of elements in the sets can change during each time-step. 1.1 Stick-slip transition or detachment First, the stick-slip transition or detachment problem is solved on an impact-free set lN. The equations of motion (7) and the relative contact accelerations g are ([7], [8] and [12]): Mq-h wn+wgmg W )(l n g& &N g& &H WN WT q + wN wH (9) (10). Indeks N označuje normalno smer in H tangentno smer možnih lepenih stikov iz množice lH. Indeks G označuje drseče stike (tangentna sila je znana iz Coulombovega zakona) iz množice lN \ lH. m je diagonalna matrika koeficientov trenja. G Za vsak aktiven stik i e IN velja, da je relativna normalna razdalja gN =0 in podobno za relativno stično hitrost = 0. Zaradi nepredirljivosti teles velja gN > 0; sledi, da lahko za vsak stik v normalni smeri zapišemo komplementarni pogoj: The index N denotes the normal direction, and the indexH denotes the tangential direction of the possibly sticking set lH. The new index G denotes the sliding contacts (the tangential force is known) of the set lN \ lH diagonal matrix of the friction coefficients. Each closed contact ieIN is characterized by a vanishing contact distance gN =0 and a normal relative velocity gN i = 0. Because of the impetrability of the bodies gN > 0, a complementary solution for each contact in the normal direction can be found: and the m G Modeliranje in analiza dinamike - Modeling and analyzing the dynamics 129 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 126-137 0a/LN i >0 gni 0A N stik se ohranja/contact is maintained sprostitev stika/detachment of contact (11) (12) in tudi i e IN : and also i e IN : N i N i Taka komplementarnost je prikazana na sliki2. Nekaj podobnega lahko ugotovimo za vsak stik v tangentni smeri (sl. 3a), vendar moramo še prej Coulombov zakon razdeliti na dve veji: ena za pozitivno in ena za negativno smer. Taka razdelitev in komplementarni prikaz stičnega zakona v tangentni smeri je prikazana na sliki 3b. Da bi se izognili težavam s singularnostjo, smo dejansko uporabili bolj zapleteno razdelitev stika v tangentni smeri, ki je tukaj zaradi pomanjkanja prostora ne bomo obravnavali [7]. Z zamudno matematično izpeljavo komplementarni problem za vse stične točke v normalni in tangentni smeri hkrati zapišemo v obliki [7]: K =0 (13). Such a complementarity is sometimes also referred to as the corner law [9], and is shown in Figure 2. In the following we will also try to represent the idea of the corner law in the tangential direction. Figure 3a presents the Coulomb friction law. Figure 3b shows the friction law decomposed into two branches: one for positive sliding and one for negative sliding. To avoid singularity problems, the actually used decomposition is more complicated and is covered in [7]. By extensive mathematical manipulation [7] the normal and tangential directions are written together in the form: (14) (15), y = Ax+b y > 0, x > 0, yT x = 0 kjer (14) in (15) pomenita linearni komplementarni problem (LKP) razsežnosti n+4nH in se kot neznanki pojavljata vektorja {y,x} e RnN+4nH. V komplementranih pogojih (15) je treba zapis yTx = 0 razumeti na ravni posameznih elementov: yxi = 0 za vse i. Vektor y med drugim vsebuje neznane stične pospeške g in vektorx Equations (14) and (15) represent an LCP where the vectors {y, x}eC+4nH are not known, but they comply with the complementary conditions (15), where yTx = 0 should be understood on the element basis: yxi = 0 for all i. Vector y includes the unknown contact accelerations g, and the vector x includes 9Ti -poi Ani ° ;" +^0i Ani Ati Sl. 2. Dopolnilnost v normalni smeri Fig. 2. Complementarity in the normal direction 9Ti = ^T-j - 3t 9Ti -/iOi ANi ° 0 At +/i0i Ani Q^ At i,:. Sl. 3. Dopolnilnost v tangentni smeri Fig. 3. Complementarity in the tangential direction Ni 130 Slavic J. - Nastran M. - Boltežar M. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 126-137 neznane stične sileAMatrikiA in b sta znani in določeni z masno matriko, vektorjem aktivnih sil, koeficienti trenja in kinematiko stičnih točk/obliko teles. 1.2 Trk s trenjem V trku sodelujejo stiki iz množice lS in ga rešujemo na ravni impulzov, zato moramo gibalno enačbo (7) najprej integrirati [7], nato pa impulz v fazi kompresije s Poissonovim zakonom povežemo z impulzom v fazi ekspanzije. Pri tem uporabimo naslednje poenostavitve: čas trka je zanemarljivo majhen, lega teles in vse neimpulzne sile se ne spremenijo, ne spremeni se tudi število stičnih točk. Podobno kakor smo pri lepenju, drsenju in sprostitvi stika zapisali komplementarne pogoje, tako lahko tudi za fazo kompresije in ekspanzije zapišemo komplementane pogoje. Pri čemer pa se pri trku v komplementarnem paru namesto sil in pospeškov pojavljajo hitrosti in impulzi. Faza se konča, ko so relativne stične hitrosti v normalni smeri enake nič; takrat se začne faza ekspanzije [7]. Podrobneje tukaj ne bomo šli v zapis LKP. Je pa treba izpostaviti, da moramo pri več sočasnih trkih paziti na pogoj nepredirljivosti. V fazi ekspanzije lahko namreč lokalna ekspanzija v neki točki povzroči prediranje v neki drugi točki; to preprečimo tako, da v komplementarnem pogoju omogočimo večji impulz ekpanzije, kakor ga dovoljuje Poissonov zakon, vendar je v tem primeru relativna stična hitrost ob koncu ekspanzije enaka nič. Glocker [8] je pokazal, da je zakon trka v takem primeru celotno raztresen. Omeniti velja, da je Pfeiffer-Glockerjeva formulacija ena redkih, ki omogoča simuliranje tudi popolnega povračljivega trka v tangentni smeri2. 2 NUMERIČNI PREIZKUS V nadaljevanju si bomo ogledali uporabo predstavljenih postopkov na primeru dinamike ščetke elektromotorja (sl. 4). Dinamični sistem sestavljajo štiri telesa: kolektor, ščetka, vodilo in vzmet. 2.1 Definiranje dinamičnega sistema Kolektor je definiran s polmerom, ekscentričnostjo, številom lamel, površinsko hrapavostjo Rz (sl. 4, detajl B) in širino reže. 2Popolno povračljivost v tangentni smeri ima npr. zelo elastična žoga; medtem, ko je npr. trk pingpong žoge v tangentni smeri nepovračljiv. the unknown contact forcesAA and b are the known matrix and vector defined by the mass matrix, the friction coefficients and the contact shapes. 1.2 Impact with friction While the stick-slip or detachment transition is solved in the force-acceleration domain, the impact is solved in the impulse-velocity domain. Some common assumptions for rigid-body impacts are made: the duration of the impact is infinitely short, the wave effects are not taken into account, during the impact all the positions and orientations, and all the non-impulsive forces and torques, remain constant. As it is possible to write the stick-slip and detachment problem in the form of an LCP, it is also possible to do it for the compression and expansion phase; the only difference is that the solution is found in the impulse-velocity domain, and that the decomposition of the contact law is more complicated. One of the additional complications is posed by the multiple concurrent contacts that have mutual influences; the physically consistent formulation needs to guarantee non-penetration at such contact situations. As shown by Glocker [8], the Pfeiffer-Glocker formulation can deal with multiple contacts in a physically consistent way, and it maintains the globally dissipative criteria. After the compression phase ends when the relative contact velocities at contacts diminish the expansion phase continues. The amount of impulse transferred from the compression to the expansion phase is defined by the friction and Poisson laws. More details on impacts with friction are given in [7]; the same publication gives more details on how to include the reversibility in the tangential direction2. 2 NUMERICAL EXPERIMENT This section presents an 11-degrees-of-freedom model of electric-motor-brush dynamics, Figure 4. The dynamical system consists of four bodies: commutator, brush, and spring holder. 2.1 Characterization of the dynamical system The commutator is defined by the radius, the eccentricity, the number of commutator bars, the surface roughness, Rz, and the slot-width between the bars, Fig.4B. 2As an example of reversibility in the tangential direction the super-elastic ball is usually given. On the other, hand a ping-pong ball impulse in the tangential direction is completely irreversible. eling and analyzing the dynamics 131 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 126-137 Sl. 4. Shematičen prikaz dinamičnega modela ščetke z 11 prostostnimi stopnjami Fig. 4. Scheme of the model with 11 degrees of freedom Ščetka elektromotorja je definirana s širino, dolžino, ukrivljenostjo in hrapavostjo stične površine s kolektorjem, nagibom stične površine z vzmetjo in togostjo. Da se doseže lokalna deformabilnost ščetke v stiku s kolektorjem, je le-ta modelirana kot sistem štirih togih teles. Toga telesa, ki sestavljajo ščetko, so med seboj povezana z vibroizolacijo (sl. 4, detajl A), katere parametri so pridobljeni s preizkusom. Ščetka ima kot sistem togih teles 9 prostostnih stopenj. Vodilo ščetke je definirano z dolžino, zračnostjo glede na ščetko, zračnostjo glede na kolektor, togostjo vpetja in lego težišča glede na inercialni koordinatni sistem xy (sl. 4). Vodilo ima eno prostostno stopnjo: vrtenje okoli težišča. Vzmet je definirana z dolžino, polmerom vrtenja, togostjo, prednapetjem vzmeti in lego glede na inercialni koordinatni sistem xy. Vzmet ima eno prostostno stopnjo. Potem ko so posamezna telesa definirana, je treba določiti gibalne enačbe sistema brez enostranskih stikov in jih zapisati v matrično The electric-motor brush is defined by the width, the length, the radius of curvature of the contact surface with commutator, the roughness of the contact surface with the commutator, the slope of the contact surface with the spring and with the brush stiffness. To achieve local deformability the brush is modeled as a system of four rigid bodies (9 degrees of freedom) that are connected with viscoelastic elements, Fig. 4A. The parameters of the viscoelastic elements were obtained experimentally. The holder of the brush is defined by the length, the clearance between the holder and the brush, the clearance between the holder and the commutator, the stiffness of the attachment to the surroundings 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 radius of rotation, the stiffness, the pre-stress rate and the position. The spring has one degree of freedom. The 11 equations of motion for the given system need to be written in the matrix form (1). The equations of motion for a contact-free case can be 132 Slavic J. - Nastran M. - Boltežar M. Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 126-137 obliko (1). Celoten sistem ima 11 prostostnih stopenj in je definiran z 11 gibalnimi enačbami. Izpeljava gibalnih enačb z uporabo Lagrangevih enačb II. vrste je razmeroma preprosto, vendar zamudno opravilo in ga bomo tukaj zaradi obsežnosti izpustili. Popis sistema nadaljujemo s stičnimi parametri med telesi: ščetka - kolektor, ščetka - vodilo, ščetka - vzmet. Stični parametri so: koeficient trenja e koeficient trka v normalnien in tangentni smeri s ter koeficient povračljivosti trka v tangentni smeriv[7]. Ščetka kot bistveni stični element je narejena iz grafit-epoksidnega materiala. Ker je kristalna rešetka grafit-epoksidnega materiala šesterokotna, so materialne lastnosti anizotropne (tako električne kot mehanske); poleg tega se materialne lastnosti zelo spreminjajo: predvsem s temperaturo in gostoto toka skozi ščetko. Pri vzpostavitvi dinamičnega modela je tako zelo pomembno preizkusno pridobivanje ustreznih materialnih podatkov: . elastični in dušilni parametri v odvisnosti od temperature in gostote toka, .trčni parametri v odvisnosti od temperature, gostote toka, hitrosti trka in izmere trčnih teles (dolžine ščetke), . koeficienta trenja v odvisnosti od temperature, gostote toka in relativne hitrosti drsenja. Vsaka od zgoraj naštetih nalog je sama zase zahtevna naloga, vendar se je kot najbolj zahtevna izkazala meritev koeficienta trenja v odvisnosti od temperature in gostote toka, ki smo jo predstavili v ločeni objavi [13]. Ker je oblika togih teles definirana z robom iz diskretnih točk, lahko kinematične podatke trka (4) določimo neposredno iz oblike [11]. 2.2 Vpliv spreminjanja parametrov sistema na obratovalne razmere ščetke V prejšnjem poglavju so našteti bistveni parametri, ki definirajo obravnavani dinamični sistem ščetke. Zaradi velikega števila parametrov bo tukaj kot primer prikazan vpliv samo nekaterih parametrov. 2.2.1 Vpliv dolžine ščetke na njeno stabilnost Na sliki 5 je prikazan fazni diagram nove ščetke; opazimo, da je kotna hitrost omejena s ±30 rad/s in da je ščetka praktično vedno znotraj nagiba -7 . 103 rad do -2 . 10-3 rad Kadar se ščetka obrabi, kakor je prikazano na sliki 6, pa postane ščetka obtained relatively easily with the help of Lagrange equations, but because this is a lengthy task it will be omitted here. For a complete definition of the system the contact parameters between the brush-commutator, the brush-holder and the brush-spring need to be set. The contact parameters are as follows: the coefficient of friction M, and the coefficient of restitution in the normal and tangential directions, en and et , respectively. If necessary, the coefficient of reversibility in the tangential direction vneeds to be set [7]. The most important body in the system is the brush, which consists of graphite/epoxy material whose characteristics are highly sensitive to temperature. In addition, because the crystal lattice is hexagonal the mechanical and electrical properties are highly anisotropic. Because of the anisotropy and the temperature sensitivity the experimental work was 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 the dimensions of the impacting bodies (length of brush), . the coefficient of friction as a function of temperature, current density and sliding velocity. The quality of the simulation results critically depends on the quality of the experimental work; how we measured the coefficient of friction for various temperatures and current densities we present in a separate paper [13]. The kinematical properties (4) of the contact points are determined automatically from the discretely defined bodies [11]. 2.2 Influence of the parameter variation on the working conditions of the brush The dynamical system is defined by more than 40 parameters. As an example of influence-analysis, the influence of selected parameters on the working conditions will be given. 2.2.1 Influence of brush-length on the dynamic stability A phase plot of a new brush is shown in Figure 5: the angular velocity is in the range ±30 rad/s, while the angle is in the range from -710-3 rad to -2 . 103 rad. When the brush is at the end of its lifetime it becomes considerably less stable, see Figure 6. Modeliranje in analiza dinamike - Modeling and analyzing the dynamics 133 Strojniški vestnik - Journal of Mechanical Engineering 52(2006)2, 126-137 bistveno bolj nemirna: kotna hitrost sega tudi izven The angular velocity can also be above ±100 rad/s področja ±100 rad/s in tudi nagib ščetke je v and also a broader range of angles of the brush is bistveno širšem področju kakor pri novi ščetki. observed. The stability of the brush considerably Obrabljena ščetka je torej bistveno bolj nemirna, influences the quality of the electrical contact and kar seveda vpliva na kakovost električnega stika in consequently the lifetime and the reliability of the s tem tudi na obrabo, dobo in zanesljivost delovanja electric motor. elektromotorja. From more then 1400 simulation results Na sliki 7 je prikazana povprečna absolutna obtained from systems with different parameter kotna hitrost ščetke za več ko 1400 različnih values we can see that the brush-length has a major numeričnih simulacij. Simulirani sistemi se influence on the stability, see Figure 7. Altogether, razlikujejo v naboru 11 različnih parametrov, kakor 11 parameters were varied: commutator roughness, so: hrapavost kolektorja, ekscentričnost kolektorja, commutator eccentricity, relative position of the nagib ščetke glede na os vrtenja, togost ščetke, brush, brush stiffness, clearance between the brush zračnost med ščetko in vodilom, togost napenjalne and the holder, stiffness of the spring, etc. vzmeti itn. Because of the wear the change of the brush- Ker je izraba ščetke neizogibna, je dolžina length cannot be omitted, and therefore we are seeking ščetke težko vir izboljšav, ki pa jih lahko dosežemo s another property to increase the stability. One of the spreminjanjem vrste drugih parametrov. Izbira obvious options is the change of the brush-material, i.e., Sl. 5. Fazni diagram za