Numerical Estimation of Stresses in Railway Axles Using a Train-Track Interaction Model

Documen downloaded om: This pape mus be ci ed as: The inal publica ion is a ailable a Copy igh Addi ional In o ma ion h p://dx.doi.o g/10.1016/j.ij a igue.2012.07.006 h p://hdl.handle.ne /10251/78196 Else ie Ma ínez Casas, J.; Mazzola, L.; Baeza González, LM.; B uni, S. (2013). Nume ical Es ima ion o S esses in Railway Axles Using a T ain-T ack In e ac ion Model. In e na ional Jou nal o Fa igue. 47:18-30. doi:10.1016/j.ij a igue.2012.07.006. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Nume ical Es ima ion o S esses in Railway Axles Using a T ain- T ack In e ac ion Model Au ho s: José Ma ínez-Casas1, Lau a Mazzola2, Luis Baeza1, S e ano B uni2 A ilia ion: 1Cen o de In es igación en Tecnología de Vehículos, Uni e sidad Poli écnica de Valencia, Camino de Ve a s/n, 46022 Valencia - Spain 2Dipa imen o di Meccanica, Poli ecnico di Milano, Via La Masa 1, 20156 Milano, I aly E-mail co esponding au ho : s e ano.b [email protected] Abs ac The a igue design o ailway axles equi es ha he s esses a ising in he axle in eal se ice a e accu a ely quan i ied. This pape desc ibes a me hod o compu e he dynamic s esses a ising in ailway axles as he e ec o ain- ack in e ac ion, based on he nume ical simula ion o he dynamic in e ac ion be ween a lexible wheelse and a lexible ack. The wheelse is modelled as a lexible o a ing body using an Eule ian app oach, whe eas ack is ega ded as an in ini e pe iodic sys em wi h he ail modelled as a Timoshenko beam es ing on disc e e elas ic suppo s, conside ing he ine ia associa ed wi h he sleepe s. The pape p esen s an applica ion o he p oposed p ocedu e o he calcula ion o he dynamic s esses caused in he axle by di e en ypes o geome ic impe ec ion occu ing on he wheel and ail su aces, conside ing he cases o a single ha monic ail co uga ion, andom ail oughness and a wheel la . Keywo ds Railway ehicles, wheelse design, wheelse axle a igue, dynamic loads, ain- ack in e ac ion, ail oughness, wheel la *Manusc ip Click he e o iew linked Re e ences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 1. In oduc ion Railway axles du abili y is a key issue in designing and co ec ly main aining ailway ehicles, o ensu e ha he highes sa e y s anda ds a e me and, a he same ime, o op imise li e-cycle cos s om a sys em poin o iew, i.e. conside ing no only he ehicle bu also he in e ac ing in as uc u e. F om he single poin o iew o a igue esis ance, he axle design should end owa ds inc easing he size o educe s esses, bu his componen also ep esen s a signi ican con ibu ion o he wheelse un-sp ung mass, which, on he con a y, shall be minimised o educe he gene a ion o dynamic o ce a wheel- ail con ac and hence damage in he ack and in he wheels, especially o high-speed ains. A p esen , wheelse axles a e designed o in ini e a igue li e, howe e a small numbe o axle ailu es due o a igue con inues o be epo ed, some imes wi h ca as ophic consequences. One o he easons which ha e been p oposed o explain hese un o eseen ailu es is ha he loads assumed by he s anda d o he pu pose o a igue design e i ica ion do no ully e lec he ac ual se ice loads o he ehicle, which a e la gely depending on he ehicle design pa ame e s (e.g. unsp ung masses, suspension s i ness and damping, …) and on he ehicle se ice p o ile [1, 2]. The p ecise knowledge o se ice loads is also pi o al o he de ini ion o app op ia e in e als o he non- des uc i e inspec ion o ailway axles [3, 4], which a e nowadays pa o s anda d main enance p ac ice o emo e om se ice c acked axles be o e ailu e. De ailed in o ma ion on he se ice loads is o en de i ed expe imen ally, using ins umen ed wheelse s [5 - 7]. This app oach howe e p esen s some d awbacks: i s o all, he physical measu e is no applicable a he design s age o a new ehicle, se ing-up and unning he es s is expensi e and demanding due o he ha sh measu ing en i onmen and ex ensi e es campaigns a e equi ed o co e all se ice condi ions (di e en lines and speeds, ope a ion a a e / ull load, new / wo n wheel p o iles, …). Fu he mo e, he expe imen al measu e o he se ice load is no mally 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 no able o ully explain he causes o ex eme loads, which esul in s ess peaks mos ly a ec ing he a igue esis ance o he axle and c ack p opaga ion. Fo his eason, in he las yea a emp s ha e been made o de ine p ocedu es based on he use o mul i-body models o a ailway ehicle o ain se , o p o ide a nume ical es ima e o he se ice loads [6 – 8]. This app oach can be used o complemen measu emen s, e.g. add essing se ice condi ions no co e ed by he es s and o p o ide a be e physical unde s anding o he ac o s in luencing he se ice spec a, including (among o he s) ehicle design pa ame e s, he condi ion o he wheel and ail su aces and he se ice scena io. To he Au ho s’ knowledge, hese app oaches ha e howe e un il now been de eloped conside ing he wheelse as a igid body. Hence, axle s esses can be co ec ly compu ed only in he low- equency ange (app oxima ely 0- 20 Hz), whe eas hei high- equency componen s, mainly esponsible o he occu ence o s ess peaks, a e excluded om he analysis [8]. This pape aims he e o e a es ablishing a me hod o de ine nume ically he s esses in a ailway axle, including he high equency componen s, as unc ion o ehicle- ack pa ame e s and o he ehicle’s unning condi ion, he eby ex ending he scope and de ail o he exis ing nume ical app oaches o p edic he wheelse se ice loads. To his end, a model o a lexible wheelse in e ac ing wi h a lexible ack in angen ack is de i ed and used o simula e ypical se ice scena ios o a high-speed ailway ehicle. The wheelse is modelled as a lexible o a ing body, using an Eule ian app oach o ake ad an age om axial symme y and modal syn hesis o educe he size o he p oblem. The ack is modelled as an in ini e pe iodic sys em, wi h he ail modelled as a Timoshenko beam es ing on disc e e elas ic suppo s, conside ing he ine ia associa ed wi h he sleepe s. The nume ical simula ion conside s he e ec o geome ic i egula i ies appea ing on he ail and wheel su aces, allowing o conside he e ec o e.g. ail co uga ion, ail dips, wheel polygonalisa ion, wheel la s e c. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 T ain- ack in e ac ion models including a de ailed desc ip ion o he wheelse as an elas ically de o mable body ha e been p oposed by se e al au ho s. Some models, e.g. [9] a e de ined in he equency domain and de i e he dynamic o ces a wheel- ail con ac by combining he wheelse and ack equency esponse unc ions wi h he assumed ail co uga ion. This app oach is inhe en ly linea and does no allow o conside he ails as disc e ely suppo ed. Modelling app oaches which inco po a e a disc e e-suppo ack model and he e ec o nonlinea i ies in he ack and in wheel– ail con ac ha e been de eloped in Re s. [10–14]: in his case, he ain- ack model is de ined in he ime domain and he wheelse and ack models a e de ined using he ini e elemen me hod, in oducing a educ ion o he deg ees o eedom using mode supe posi ion. These app oaches ha e been mos ly applied o he s udy o damage phenomena in he ack and a wheel- ail in e ace, and he e o e ocus on he de ini ion o wheel- ail con ac o ces a he han axle s esses. The e o e, a simpli ied model o he wheelse axle (some imes based on beam elemen s) can be used. Fu he mo e, he e ec o wheelse o a ion is neglec ed o conside ed wi h some app oxima ion, whe eas his e ec may impo an ly a ec he wheelse esonance condi ions, which in u n can be a cause o ex eme s ess peaks a ising in he axle: he p esen pape aims he e o e a de eloping a modelling and simula ion app oach speci ically ailo ed on he p edic ion o axle s esses The pape desc ibes he wheelse - ack in e ac ion model de eloped and he p ocedu e o compu e he dynamic s esses in he axle. Resul s o he nume ical p ocedu e a e p esen ed o di e en exci a ion cases including ail co uga ion ha ing di e en wa eleng h and a wheel la , and conside ing he e ec o wheelse speed. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 2. THE VEHICLE-TRACK INTERACTION MODEL The ehicle – ack in e ac ion model, see Figu e 1, is de ined adop ing a sub-s uc u ing echnique [15, 16], acco ding o which he whole sys em is di ided in o subs uc u es: he ehicle, he ails and he ail suppo s. Fo each subs uc u e, he equa ions o mo ion a e w i en sepa a ely, and in e ac ion e ec s be ween ehicle and ack a e ep esen ed by he wheel- ail con ac o ces and by he o ces gene a ed a he ailpads [15]. The simula ion app oach is de eloped conside ing he mo ion o he wheelse in angen ack. Gi en he ange o equencies add essed, he ehicle model is con ined o one wheelse wi h p ima y suspension, see sec ion 2.1. The ib a ion o he lexible wheelse is exp essed using Eule ian modal coo dina es, see Sec ion 3, aking ad an age o m he axial symme y o he body. Fu he mo e he modal app oach is chosen o desc ibe he wheelse mo ion in o de o educe he compu a ional e o equi ed by he simula ion. The ack is modelled by means o a cyclic app oach, which p o ides some bene i wi h espec o classical ack modelling o he ini e one. The sub-s uc u ing echnique is s ill adop ed he e o simpli y he sys em modelling; a de ailed desc ip ion o he ack is p o ided in sec ion 2.2. 2.1 The Vehicle Model The dynamic s esses a ising in he wheelse axle a e mos ly ela ed wi h ain- ack in e ac ion e ec s in he equency ange abo e 20 Hz, which a e exci ed by sho wa eleng h geome ic impe ec ions in he wheel and ail p o iles and by singula i ies such as ail dips and wheel la s. In his equency ange he dynamics o he sp ung masses (bogie ame and ca body) a e e ec i ely isola ed om he mo ion o he un-sp ung masses (wheelse s and axle boxes) on accoun o he mechanical il e in oduced by he suspensions. The e o e, he ehicle model used in his pape conside s one single wheelse , modelled as an elas ically lexible body, and he p ima y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 suspension, ep esen ed using isco-elas ic lumped pa ame e elemen s. The s a ic load associa ed wi h he g a i a ional o ces ac ing on he bogie and on he ca body masses is ep esen ed by wo s a ic o ces applied h ough he p ima y suspension on he wo sides o he wheelse . 2.2 The T ack Model The ack is modelled by means o a cyclic app oach, whe e a ini e sec ion o he ack is de ined, in oducing cyclic bounda y condi ions a he ends o he model, hence he ack, see Figu e 2, can be in e p e ed as an in ini e ack, nego ia ed by an in ini e se o iden ical ehicles, uni o mly dis ibu ed in such a way ha each ehicle is se a a cons an dis ance L apa om he adjacen ones. Due o he pe iodici y o he s uc u e and o he loading condi ions, he s udy is educed o a single sec ion ha ing ini e leng h L, whose alue is se la ge enough o a oid in e ac ion be ween he ehicles. The model adop ed o he di e en ack componen s is illus a ed in Figu e 3: he ails a e modelled as Timoshenko beams, including bending de o ma ions in e ical/la e al di ec ions, as well as o sional de o ma ions. Rail ib a ion is in oduced in e ms o modal supe posi ion o he uncons ained ail wi h cyclic bounda y condi ions, hence esul ing in o a se o de-coupled 1-d.o. . equa ions. Mo eo e due o he symme y o he sub-sys em wi h espec o he ack cen eline, only one ail is modelled in he p esen s udy. The disc e e ail suppo s a e in oduced in he o m o lumped pa ame e sys ems, see Figu e 3 igh . The ail pads a e modelled as lumped isco-elas ic elemen s gene a ing he in e ac ion o ces be ween he ails and he sleepe s, ep esen ed as lumped masses. Ballas dynamics is neglec ed he e, being no ele an o he s ess analysis on he wheelse , bu he equi alen ballas s i ness and damping a e accoun ed o by means o lumped sp ing and dashpo elemen s. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 2.3 The Model o Wheel-Rail Con ac Fo ces The ack model is coupled wi h he lexible wheelse ia he wheel- ail con ac o ces, which a e exp essed as unc ion o he ela i e wheel- ail displacemen and eloci y in he con ac pa ch. In he nume ical model p esen ed he e, he heo y o He z is adop ed o de ine he no mal con ac o ce componen and FASTSIM [17] is used o de ine he angen ial con ac o ces as unc ion o he no mal con ac o ce and o he c eepage componen s. A each ime s ep, he displacemen and eloci y o he wheel in he con ac poin a e ob ained using Eq. (1) e alua ed a he con ac poin posi ion. 3. THE FLEXIBLE WHEELSET MODEL In o de o model he kinema ics o he lexible wheelse , wo con igu a ions (unde o med and de o med) a e de ined. The unde o med con igu a ion is associa ed wi h he spinning eloci y o he wheelse (a cons an angula eloci y  ). The de o med con igu a ion conside s he lexibili y and small igid solid displacemen s. The displacemen ield ela es he de o med con igu a ion wi h he unde o med con igu a ion as i will be shown in Eq. (1). The coo dina es ha a e implemen ed in he wheelse model do no ollow he ma e ial poin s o he solid which is he commones p ocedu e in Mechanics, ne e heless hey a e associa ed wi h spa ial poin s (Eule ian app oach). Le u an Eule ian ec o coo dina e in a ixed coo dina e ame. Any p ope y o he solid ),( u  co esponds o he ma e ial poin o he solid whose unde o med con igu a ion is in he spa ial poin u a ins an . Following his c i e ion, he displacemen ield is de ined by means o he ollowing o mula: ),( uwu  , (1) whe e is he inal posi ion o he pa icle, and w is he displacemen s associa ed wi h lexibili y and small igid body displacemen s. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 The coo dina e ame is chosen so ha he spin o a ion is in he second axle. The ollowing ma ices a e de ined as ollows:             001 000 100 J ;            100 000 001 E . (2) The angula eloci y enso e i ies: JΩ                 00 000 00 ~ ; and EΩΩ 2 ~~   . (3) The eloci y due o he igid body spinning is:   uuJuΩ ~ ~ T 321 ΩΩ  , (4) whe e   T 13 0 ~uu u . The eloci y o he pa icle is compu ed h ough he ma e ial de i a i e o , and ha is           ii i ii i ii iu uΩΩ u u w wuJ w w uwu ~ D D D D D D . (5) In o de o calcula e he kine ic ene gy o he solid, he squa e o he pa icle eloci y Eq. (5) has o be ob ained. I has he ollowing o mula: . ~ 2 ~ 22 ~~ D D D D T T 2T T 2TT2 T                                              ii i ii i ii i ii i u uΩ u uΩΩ u u u uΩΩ w wuJ w uJw ww wwuEu   (6) The nex modal app oach is adop ed: )()(),( quΦuw  , (7) being )(uΦ he mode shape unc ions ma ix o he ee-bounda y wheelse . The small igid displacemen s o he solid a e conside ed in his app oach h ough he igid modes o he wheelse . I mus be poin ed ou ha he mode shape unc ions do no depend on ime since he o a ion o he solid does no change he mode shapes unc ions in ixed coo dina es, because he axial symme y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 ampli ude o he dynamic o ce luc ua ion is app oxima ely 81 kN, and is he e o e conside ably lowe han o he 60 mm wa eleng h (c . Fig. 6), on accoun o he ail showing a g ea e mobili y in his esonance condi ion. Also in his case, he peak-peak o ce ampli ude is nea ly p opo ional o he co uga ion ampli ude. The lowe sub igu e shows he e ical con ac o ce plo ed s. he a elled dis ance o a co uga ion wa eleng h exci ing he i s o wa d bending mode o he wheelse . In his case, he peak-peak ampli ude o he con ac o ce o he la ge co uga ion ampli ude is 82.2 kN, again signi ican ly lowe han o he 60 mm wa eleng h. In his second case howe e he inc ease o he peak-peak o ce ampli ude wi h he co uga ion ampli ude is signi ican ly less han p opo ional, and also he wa e o m o he con ac o ce is di e en o he wo co uga ion ampli udes: his shall be asc ibed o he di e en impo ance o he sleepe -passing e ec in he wo cases, a he han o he e ec o non-linea i ies. In Figu e 9, he y-axis no mal s ess in he axle is shown o he same cases as in Fig. 8. Fo he co uga ion wa eleng h exci ing he pinned-pinned esonance o he ack, no signi ican di e ence is obse ed wi h espec o he cases epo ed in Figu e 7 o he same speed (300 km/h), and again an inc ease o he co uga ion ampli ude lea es una ec ed he ampli ude o he axial s ess. Howe e , when he co uga ion wa eleng h exci ing he wheelse esonance and he la ge co uga ion ampli ude is conside ed, he s ess ampli ude becomes 33% highe han in Figu e 7 (despi e he con ac o ce being lowe in his case han o he 60 mm wa eleng h) and de ia es signi ican ly o m he sinusoidal wa e o m. This esul is jus i ied by he e ec o wheelse lexibili y playing a pa icula ly impo an ole in he case conside ed, on accoun o one mode o ib a ion being exci ed in esonance. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 A use ul pa ame e o quan i y he ele ance o dynamic e ec s a ec ing he wheel- ail con ac o ces and he s esses in he axle is he dynamic ac o , de ined as he a io be ween he maximum dynamic alue o he quan i y conside ed and he co esponding s a ic alue. Fo he con ac o ce, he dynamic ac o Q k is de ined acco ding o he ollowing equa ion: s max,d QQ Q k (20) wi h max,d Q he maximum alue o he con ac o ce and s Q he s a ic wheel load. Fo he s ess in he axle he dynamic ac o  k is de ined as: max,s max,d k     (21) wi h max,d  maximum dynamic s ess in he ma e ial poin conside ed and max,s  he co esponding maximum s ess unde he ac ion o o a ing bending p oduced by he s a ic loads ac ing on he wheelse . In Figu e 10 he dynamic ac o s Q k and  k a e epo ed as unc ion o he ehicle speed o he di e en ypes o ail co uga ion p esen ed abo e, conside ing a co uga ion ampli ude en imes g ea e han he ISO 3095 limi ; in his case, also he esul s o a co uga ion wa eleng h exci ing he second o wa d bending mode o he wheelse a e p esen ed. Fo he con ac o ce dynamic ac o Q k , in he en i e speed ange conside ed, he la ges alues a e ob ained o he co uga ion wa eleng h exci ing he i s o wa d bending mode o he wheelse , wi h a maximum alue close o 1.8. A local maximum appea s o mos o he co uga ion cases conside ed o speeds in he 125- 135 km/h ange, his is due o a esonance e ec associa ed wi h he sleepe passing equency ha exci es P2 equency (see P1 and P2 equencies in Re . [15]).. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 As a as he s ess dynamic ac o  k is conce ned, e y simila ends and alues a e ob ained o he case wi h no oughness, o oughness wa eleng h exci ing he pinned-pinned equency, leading o low alues o s ess ampli ica ion (max. alue below 1.2). Only O he wise, in he case o he o co uga ion wa eleng h exci ing one o he wheelse he bending esonances o he wheelse he s ess dynamic ac o is signi ican ly highe and can be eaches a maximum alue close up o abou 1.8 o he case o co uga ion wa eleng h exci ing , wi h he esonance o he i s o wa d and up o 1.4 app oxima ely o co uga ion wa eleng h exci ing bending mode leading o la ge dynamic s esses han o he second o wa d bending mode. The ends wi h speed a e close o mono onically inc easing, and he e ec o he esonance associa ed wi h he sleepe passing is less e iden han in he end o he Q k dynamic ac o . These esul s sugges ha he bending s esses in he axle a e highly a ec ed by he wa eleng h o ail co uga ion, and ha combina ions o ain speed and co uga ion wa eleng h leading o a esonance o a bending mode may be especially ele an in iew o axle esis ance o a igue. Pa icula ly signi ican in iew o eal applica ions is he case o ail co uga ion, a o m o i egula wea o he ail o en appea ing in ailway sys ems and cha ac e ised quasi-ha monic wea pa e ns de eloping on he ail head in longi udinal di ec ion [18]. Co uga ion wa eleng h can ange om 50 mm o less up o mo e han 1m in he case o hea y haul co uga ion, and ypical wea dep h alues a e in he ange o some en hs o millime e. To quan i y he ele ance o a igue o ail co uga ion, we no e ha he design me hods p esc ibed o ailway axles by he Eu opean S anda ds EN13103 and EN13104 [19, 20] assume a 1.25 dynamic ampli ica ion ac o on he loads gene a ed a he p ima y suspension. Since he calcula ion o he bending s esses is hen based on s a ic equilib ium, also he bending s esses a e magni ied by 25% wi h espec o he s a ic case, whe eas he esul s in Figu e 10 sugges ha la ge dynamic ac o s migh apply when a pa icula combina ion o co uga ion wa eleng h and ain speed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 exci es one bending mode. Fo ins ance, a sinusoidal ail co uga ion ha ing app oxima ely 350 mm wa eleng h and 0.15 mm dep h (i.e. en imes he ISO3095 limi o he conside ed wa eleng h) would p oduce a dynamic s ess ampli ica ion close o 40%. I shall be poin ed ou howe e ha he equi alen s a ic loads p esc ibed by EN13103/104 also include la e al wheel- ail con ac o ces due o cu ing, which a e no included in he analysis p esen ed he e. 4.2 Resul s o andom ail co uga ion and o a wheel la The simple exci a ion cases conside ed in Sub-sec ion 4.1 allow o poin ou he in luence o di e en dynamic e ec s on ain- ack in e ac ion and on he s esses in he wheelse , bu do no ep esen ealis ic exci a ion cases. In his Sub-sec ion, wo sou ces o exci a ion ypically occu ing in se iced ains a e conside ed: andom ail oughness and a wheel la . Random oughness akes place on he ail head on accoun o di e en causes: geome ic impe ec ions associa ed wi h he manu ac u ing and ins alla ion o he ail, i egula wea caused by ain passage and, o la ge wa eleng hs, non-uni o m ack se lemen due o pe manen de o ma ions in he soil and ballas (in his case, he e m ― ack i egula i ies‖ is used ins ead o co uga ion). S udies ha e shown ha ail oughness and ack i egula i y ake he o m o s a iona y andom p ocesses cha ac e ised by hei powe spec al densi y, which can be de ined based on he quali y o ack main enance [2181]. On he o he hand, wheel la s a e localised de ec s occu ing on he wheel su ace as he esul o ull slip o he wheel, ypically caused by poo ly adjus ed o aul y b akes [1922]. Figu e 11 shows he e ical con ac o ce plo ed s. he a elled dis ance o he ain speeds o 100 and 300 km/h, conside ing he exci a ion p oduced by andom ail co uga ion (ampli ude co esponding o he ISO 3095 limi ). The esul s ob ained conside ing a igid and lexible wheelse model a e compa ed. Impo an dynamic luc ua ions o he con ac o ce a e e idenced, wi h he maximum dynamic alue co esponding o app oxima ely 1.6 imes he s a ic load. The di e ences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 be ween he esul s ob ained o he igid and lexible wheelse a e ela i ely small in his case, and do no a ec signi ican ly he maximum con ac o ce alue. Figu e 12 p esen s he e ical con ac o ce s. a elled dis ance o a ehicle a ec ed by a wheel la . The calcula ion has been ca ied ou o 50 and 300 km/h by means o he igid and enhanced wheelse models. A se e e dynamic e ec is obse ed, consis ing o he comple e loss o con ac be ween he wheel and he ail, ollowed by an impac leading o a maximum alue o he con ac o ce which is be ween 3 and 4 imes he s a ic load and inally by a ansien ib a ion. The di e ences be ween he esul s o he igid and lexible wheelse model a e small in e ms o du a ion o he con ac loss and o maximum o e loading, bu he ansien ollowing he impac is a ec ed qui e ema kably by wheelse lexibili y, as demons a ed by he di e en equency con en s o he con ac o ce signal, see pa icula ly he zoomed iew on he igh side o he igu e. The y-axis s esses o he andom ail co uga ion and wheel la exci a ion cases a e shown in Figu e 13. In he andom co uga ion case, he dynamic luc ua ions o he s ess componen a e ela i ely low and lead o a maximum peak-peak s ess ampli ude a ound 91.2 MPa, co esponding o a dynamic ampli ica ion wi h espec o he maximum s ess p oduced by he s a ic load  k o 1.18 app oxima ely. When he wheel la de ec is conside ed, he esul s a e highly a ec ed by he ehicle speed, wi h la ge dynamic e ec s aking place a low speed (wi h a dynamic ac o  k o 1.34 app oxima ely) on accoun o he epea ed loss o con ac o he wheel o he ail, and wi h a educ ion o he dynamic s esses a highe speed. Finally, in Figu e 14 he dynamic ac o s Q k and  k a e epo ed o he andom co uga ion and wheel la as unc ion o he ehicle speed. The con ac o ce dynamic ac o Q k is nea ly mono onically inc easing up o a maximum alue close o 1.8 o andom co uga ion exci a ion, whe eas in case o wheel la exci a ion he dynamic ac o is much highe , in he ange o 2.8-4.2, 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 wi h la ge alues occu ing a lowe speeds: his is because a low speed he du a ion o he con ac loss caused by he wheel la is longe and hence he ollowing impac is la ge . The s ess dynamic ac o  k o he andom co uga ion exci a ion case is almos mono onically inc easing wi h speed, wi h a maximum alue close o 1.3. Fo he wheel la exci a ion case, a comple ely di e en end is obse ed, he s ess dynamic ac o  k being ini ially dec easing wi h he speed and hen inc easing abo e 200 km/h. The maximum alue o he s ess dynamic ac o o his exci a ion case is ob ained a he lowes speed conside ed in he analysis and is sligh ly below 1.5. As in he case o exci a ion gene a ed by a single-ha monic ail co uga ion, i is in e es ing o obse e ha in some cases he dynamic ac o  k exceeds he 1.25 alue assumed in EN13103/104. Fo wheel la exci a ion, his happens a low speeds (below 75 km/h) which a e ypical e.g. o eigh applica ion whe eas o andom ail oughness he 1.25 alue is exceeded only a e y high speeds abo e 300 km/h, which a e only ele an o e y high speed ains. I shall be s essed howe e ha he esul s shown in Figu e 14 depend on he ampli ude o he de ec s being conside ed and, in case o mo e se e e i egula i ies, la ge dynamic s esses shall be expec ed. I is also in e es ing o poin ou ha by using a s a ic calcula ion o de i e he bending s esses, S anda ds EN13103/104 inhe en ly imply a p opo ionali y be ween he con ac o ces and he s esses, whe eas he esul s in igu es 10 and 14 show ha he s ess dynamic ac o  k is gene ally lowe (some imes much lowe ) han he dynamic ac o o he e ical con ac o ce Q k . This ci cums ance is due o he ine ia o ces gene a ed in he wheels and o he di e en magni ica ion o wheelse lexible modes p oduced when esonance condi ions occu and sugges s ha a me hod based on a s a ic calcula ion could be no ully sui ed o es ima e se ice s esses in he axle. Dynamic ain- ack in e ac ion models such as he one p oposed in his pape could be en isaged as a means o de i e a mo e ealis ic es ima e o se ice s esses in he axle, bu his 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 equi es he s udy o be ex ended o conside he e ec o cu ing, which is en isaged as a nex de elopmen o he wo k desc ibed he e. . 5. CONCLUSIONS Dynamic e ec s may be ex emely impo an in de e mining he a igue esis ance o ailway axles, and need o be p ope ly conside ed in he axle design p ocess. This pape has p esen ed a me hod o he nume ical es ima ion o he dynamic s esses in ailway axles, based on he simula ion o he in e ac ion be ween a lexible wheelse and a lexible ack. The me hod is p esen ly limi ed o he case o angen ack unning o he wheelse , bu he ex ension o he cu ing condi ion is en isaged as a nex s ep o he esea ch, o conside he addi ional axle loading due o wheel- ail con ac o ces in a cu e. Resul s o he nume ical p ocedu es we e p esen ed in he pape , conside ing he dynamic exci a ion caused by ail co uga ion and by a wheel la o di e en wheelse speed alues. The esul s clea ly show ha dynamic e ec s may lead o a signi ican dynamic ampli ica ion o he s esses in he axle, which is in some cases close o 70%. Howe e , he ac ual ele ance o dynamic e ec s a ec ing axle s esses is s ongly depending upon he ype o exci a ion and he ehicle speed. When single ha monic co uga ion is conside ed, he dynamic ampli ica ion o axle s esses is ela i ely low excep in he case when he co uga ion wa eleng h exci es he bending modes o he wheelse . In he case o andom mul i-ha monic ail co uga ion, he dynamic ampli ica ion ac o s a e ela i ely low because he exci a ion is sp ead o e a wide ange o equencies, hus educing he impo ance o esonance e ec s. In all ail co uga ion cases conside ed, ega dless he wa e o m o he exci a ion, he dynamic s ess ampli ica ion ends o inc ease wi h he ehicle speed, so ha he mos c i ical case is ob ained o high-speed applica ions. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Wheel la exci a ion also appea s o be c i ical in iew o he axle a igue li e, because high dynamic ampli ica ion ac o s up o 45-50% a e ob ained, bu on accoun o he localized na u e and small size o his de ec in his case he mos se e e dynamic exci a ion occu s a low speeds, a which he equency o he exci a ion gene a ed by he la exci es he bending esonances o he axle. Fo he wheel la case, he e o e, he mos c i ical case appea s o be ha o eigh ca s, which may be o en a elling a low speed. O e all, he analyses p esen ed in his pape show ha dynamic e ec s a e essen ial o co ec ly es ima e he wheelse ’s a igue li e. Mo e esea ch and alida ion is needed o ans e he indings p esen ed he e in o axle design p ac ices. Hope ully howe e he pape has highligh ed an impo an a ea o u u e esea ch, in iew o u he imp o ing he eliabili y o ailway anspo . Acknowledgemen s The au ho s g a e ully acknowledge he suppo o his wo k p o ided by he P ojec TRA2010- 15669 (Minis e io de Ciencia e Inno ación) and TRA2007-67167 (Minis e io de Educación y Ciencia-FEDER). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Re e ences [1] R. A. Smi h, Railway a igue ailu es: an o e iew o a long s anding p oblem, Ma e ialwissenscha und We ks o echnik 36:11 (2005) 697-705 [2] V. G ubisic, G. Fische , P ocedu e o eliable du abili y alida ion o ain axles, Ma e ialwissenscha und We ks o echnik 37:12 (2006) 973-982 [3] S. Be e a, M. Ca boni Va iable ampli ude a igue c ack g ow h in a mild s eel o ailway axles: Expe imen s and p edic i e models, Enginee ing F ac u e Mechanics 78 (2011) 848- 862. [4] M. Luke, I. Va olomee , K. Lü hepol, A. Esde s, Fa igue c ack g ow h in ailway axles: Assessmen concep and alida ion es s, Enginee ing F ac u e Mechanics 78 (2011) 714– 730. [5] V. G ubisic, G. Fische , Bemessung on Radsa zwellen – Ein lussg ssen und Vo gehen bei de Auslegung. Be ich FB-226, LBF, Da ms ad ; 2005. [6] S. Al i, F. B aghin, S. B uni, Nume ical and expe imen al e alua ion o ex eme loads o imp o ed wheelse design, Vehicle Sys em Dynamics, 46 S (2008) 431-444. [7] A. S. Wa son, K. Timmis, A me hod o es ima ing ailway axle s ess spec a, Enginee ing F ac u e Mechanics 78 (2011) 836–847. [8] S. B uni, R. Co adi, L. Mazzola, Wheel ail con ac o ces as inpu o op imal and obus axle design, 8 h In e na ional Con e ence on Con ac Mechanics and Wea o Rail/Wheel Sys ems (CM2009), Fi enze, I aly, Sep embe 15-18, 2009. [9] D.J. Thompson, C.J.C. Jones Thompson A e iew o he modelling o wheel/ ail noise gene a ion, Jou nal o Sound and Vib a ion, 231 (2000), pp. 519–536. [10] J.C.O. Nielsen, A. Igeland, Ve ical dynamic in e ac ion be ween ain and ack—in luence o wheel and ack impe ec ions, Jou nal o Sound and Vib a ion 187 (5) (1995) 825–839. [11] G. Diana, F. Cheli, S. B uni, A. Collina, Expe imen al and nume ical in es iga ion on subway sho pi ch co uga ion, Vehicle Sys em Dynamics, 29 S, (1998) 234-245. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 [12] J.C.O. Nielsen, J. Osca sson, Simula ion o dynamic ain_ ack in e ac ion wi h s a e- dependen ack p ope ies, Jou nal o Sound and Vib a ion 275 (2004) 515–532. [13] J. Fayos, L. Baeza, J. E. Ta ancón and F. D. Denia, An Eule ian coo dina e-based me hod o analysing he s uc u al ib a ions o a solid o e olu ion o a ing abou i s main axis, Jou nal o Sound and Vib a ion 306 (2007) 618–635 [14] L. Baeza, J. Fayos, A. Roda and R. Insa, High equency ailway ehicle- ack dynamics h ough lexible o a ing wheelse s, Vehicle Sys em Dynamics 46 (2008) 647 – 659 [15] Baeza, L., Huajiang, O., A ailway ack dynamics model based on modal subs uc u ing and cyclic bouda y condi ion, Jou nl o Sound and V ib a ion 330 (2011) 75-86. [16] L. Baeza, A. Roda, J.C.O. Nielsen, Railway ehicle/ ack in e ac ion analysis using a modal subs uc u ing app oach, Jou nal o Sound and Vib a ion 293 (2006) 112–124 [17] J.J. Kalke , A as algo i hm o he simpli ied heo y o olling con ac , Vehicle Sys em Dynamics 11 (1982) 1–13. [18] S. L. G assie, J. Kalousek, Rail co uga ion: cha ac e is ics, causes and ea men s, P oc. Ins n. Mech. Eng s. Pa F – Jou nal o Rail and Rapid T ansi , 207 (1993), 57-68. [19] EN 13103 Railway applica ions - Wheelse s and bogies – Non-powe ed axles - Design me hod, CEN, B ussels, Ap il 2001. [20] EN 13104 Railway applica ions - Wheelse s and bogies - Powe ed axles - Design me hod, CEN, B ussels, Ap il 2001. [1821] ORE B 176: Bogies wi h s ee ed o s ee ing wheelse s. Repo No. 1: Speci ica ions and p elimina y s udies, Vol. 2, Speci ica ion o a bogie wi h imp o ed cu ing cha ac e is ics. ORE, U ech 1989. [2219] A. Johansson, J.C.O. Nielsen, Ou -o - ound ailway wheels — wheel- ail con ac o ces and ack esponse de i ed om ield es s and nume ical simula ions, P oc. Ins n. Mech. Eng s. Figu e 3: De ail o he ack model. Le : model o some sleepe bays. Righ :sleepe and ailpad model. Figu e 3 Figu e 4: Fini e elemen model o he lexible wheelse . Figu e 4 Figu e 5: S udied sec ion o wheelse and spa ial poin whe e he s esses ha e been calcula ed. y z z x Figu e 5 Figu e 6: Wheel- ail con ac o ce when he ehicle ci cula es on a co uga ed ack wi h co uga ion wa eleng h 60 mm. Abo e: e ec o speed and o wheelse lexibili y o co uga ion ampli ude co esponding o he ISO 3095 limi . Below: e ec o co uga ion ampli ude o speed 300 km/h and lexible wheelse . 0 0.2 0.4 0.6 0.8 1 1.2 52 54 56 58 60 62 64 66 68 70 Rail oughness Dis ance x [m] Ve ical con ac o ce [kN] Flexible wheelse V = 100 km/h Flexible wheelse V = 300 km/h Rigid wheelse V = 300 km/h 0 0.2 0.4 0.6 0.8 1 1.2 10 20 30 40 50 60 70 80 90 100 110 Rail oughness Dis ance x [m] Ve ical con ac o ce [kN] ISO 3095 10 x ISO 3095 Figu e 6 Figu e 7: y-axis no mal s ess in he s udied sec ion o he axle (c . Fig. 5) when he ehicle ci cula es on a co uga ed ack wi h co uga ion wa eleng h 60 mm. Le : wo ehicle speeds, co uga ion ampli ude co esponding o he ISO 3095 limi . Righ : wo co uga ion ampli udes ( he esul s a e undis inguishable).               Ro a ed angle [ ad]  y [MPa] V = 100 km/h V = 300 km/h               Ro a ed angle [ ad]  y [MPa] ISO 3095 10 x ISO 3095 Figu e 7 Figu e 8: Wheel- ail con ac o ce when he ehicle ci cula es on a co uga ed ack wi h di e en co uga ion wa eleng hs. Two co uga ion ampli udes a e conside ed, co esponding o he ISO 3095 limi and 10 imes he ISO 3095 limi . Abo e: he co uga ion wa eleng h exci es he pinned-pinned mode o he ail. Below: he co uga ion wa eleng h exci es he i s o wa d bending mode o he wheelse . 0 0.2 0.4 0.6 0.8 1 1.2 10 20 30 40 50 60 70 80 90 100 110 Rail oughness Dis ance x [m] Ve ical con ac o ce [kN] ISO 3095 10 x ISO 3095 0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6 10 20 30 40 50 60 70 80 90 100 110 Rail oughness Dis ance x [m] Ve ical con ac o ce [kN] ISO 3095 10 x ISO 3095 Figu e 8 Figu e 9: y-axis no mal s ess in he s udied sec ion o he axle (c . Fig. 5) when he ehicle ci cula es on a co uga ed ack a 300 km/h speed. Le : he co uga ion wa eleng h exci es he pinned-pinned mode o he ail. Righ : he co uga ion wa eleng h exci es he i s o wa d bending mode o he wheelse .               Ro a ed angle [ ad]  y [MPa] ISO 3095 10 x ISO 3095           Ro a ed angle [ ad]  y [MPa] ISO 3095 10 x ISO 3095 Figu e 9 Figu e 10: Dynamic ac o s Q k ( e ical con ac o ce, uppe sub igu e) and  k (y-axis s ess, lowe sub igu e) o di e en exci a ion cases and ehicle eloci ies. The oughness ampli ude is en imes highe he ISO 3095 limi . 050 100 150 200 250 300 350 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Vehicle eloci y [km/h] Dynamic/s a ic con ac o ce a io No oughness Co uga ion exci es pinned-pinned equency Co uga ion exci es 1s o wa d mode Co uga ion exci es 2nd o wa d mode 050 100 150 200 250 300 350 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Vehicle eloci y [km/h] Dynamic/s a ic s ess a io No oughness Co uga ion exci es pinned-pinned equency Co uga ion exci es 1s o wa d mode Co uga ion exci es 2nd o wa d mode Figu e 10 Figu e 11: Wheel- ail con ac o ce when he ehicle ci cula es a 100 and 300 km/h speeds on a andomly co uga ed ack. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 10 20 30 40 50 60 70 80 90 100 Rail oughness Dis ance x [m] Ve ical con ac o ce [kN] Flexible wheelse V = 100 km/h Flexible wheelse V = 300 km/h Rigid wheelse V = 300 km/h Figu e 11 Figu e 12: Wheel- ail con ac o ce when he ehicle ci cula es a 50 (abo e) and 300 km/h (below) speeds on a pe ec ly e en ail in p esence o a 50 mm wheel la . 0 0.5 1 1.5 0 50 100 150 200 250 Dis ance x [m] Ve ical con ac o ce [kN] View A Flexible wheelse V = 50 km/h Rigid wheelse V = 50 km/h 0.1 0.2 0.3 0.4 0.5 0 50 100 150 200 250 Dis ance x [m] Ve ical con ac o ce [kN] Zoomed iew A 0 0.5 1 1.5 0 20 40 60 80 100 120 140 160 180 Dis ance x [m] Ve ical con ac o ce [kN] View A Flexible wheelse V = 300 km/h Rigid wheelse V = 300 km/h 0.1 0.2 0.3 0.4 0.5 0.6 0 20 40 60 80 100 120 140 160 180 Dis ance x [m] Ve ical con ac o ce [kN] Zoomed iew A Figu e 12