SIMPLIFIED ROTATING TIRE MODELS BASED ON CYLINDRICAL SHELLS WITH FREE BOUNDARY CONDITIONS

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1 F1-NVH-8 SIMPLIFIED ROTATING TIRE MODELS BASED ON CYLINDRICAL SHELLS WITH FREE BOUNDARY CONDITIONS 1 Alujevc Neven * ; Cmpllo-Dvo Nu; 3 Knd Pee; 1 Pluymes Be; 1 Ss Pul; 1 Desme Wm; 1 KU Leuven PMA Dvson Belgum; Unvesdd Mguel Henndez de Elche Spn 3 Goodye Innovon Cene Colm-Beg Luemboug KEYWORDS Te dynmcs sucul vbon ong sucues cylndcl shells smplfed models ABSTRACT A numbe of sudes hve been pevously concened wh vbons of ong es. In ode o cpue deled dynmcs of ong e comple numecl models hve been developed ccounng fo he nonhomogeneous sucul popees of he e bel nd sdewll sucul-cousc couplng nd comple geomey descpon (ncludng h of he wheel m) unde vyng oon speeds. Alhough emkble moun of dels cn be successfully modelled usng such n ppoch cn be he dffcul o pmeeze he model.e. o esblsh cle-cu elonshp beween geomecl o mel popey of e nd he coespondng chnge n he e dynmcs. Thus n lenve ppoch s employed whn he scope of hs sudy by modellng smplfed e geomees. The focus of he sudy s pu ono modellng vbons of he e bel whch could be ppomed by pessused hn cylndcl shell hvng fee boundes. Then he mode shpes nd esonnce fequences of he bel e clculed s funcons of he mel nd geomecl popees of he shell nd he speed of he shell oon. The esonnce fequences nd he mode shpes of he fee ong cylndcl shell could be used o fom foundon fo he foced vbon esponse nd hus offe possbly o couple he fee e bel o models of he sdewll nd he cvy nsde he e usng fo emple mobly-mpednce bsed mehods. INTRODUCTION The effecs of oon sgnfcnly le he dynmcl behvou of uomove es. In non-ong es vbons e chcezed by ps of velng wves h popge n oppose decons bu hve he sme speed of popgon. Ths esuls n fomon of sndng wve (vbon mode). On he cony f he e oes he wo wves vel n oppose decons wh dffeen speeds. Ths s pmly due o he Cools effecs whch le he popgon speeds of he wo wves. Alhough emkble moun of dels cn be successfully modelled usng comple numecl e models cn be he dffcul o pmeeze he model.e. o esblsh dec elonshps beween geomecl nd mel popees of e nd he coespondng chnge n s dynmcs. Thus n lenve ppoch s employed whn he scope of hs sudy by modellng smplfed e geomees. The focus of he sudy s pu ono modellng vbons of he e bel whch could be ppomed by pessused hn cylndcl shell. The vbons of shells hve been cng nees of mny eseches fo ove cenuy ncludng lso vbons of ong shell-lke sucues. The mhemcl descpon of vbons of shell-lke sucues s pehps no he smples one nd hee hve been mny shell heoes developed n he ps yng o cpue only he mos mpon feues h goven he sucul esponse of vbng shells [1]. Neveheless closed fom soluons fo fee vbon poblems of shell-lke sucues e possble only fo pcul geomees nd n combnons wh pcully convenen boundy condons. If he effecs of oon e o be suded s well hen becomes even moe dffcul o come up wh closed fom epessons fo he nul fequences. Fo emple Hung nd Soedel [] developed se of equons of moon fo ong shell nd solved he fee nd foced vbon poblem ssumng smply suppoed boundy condons. The nul fequences wee clculed s oos of chcesc polynoml whch ws shown by he uhos o be b-cubc only f he shell does no oe. On he ohe hnd ohe ypes of boundy condons hve been consdeed s well see fo emple he wok of Wbuon [3] who solved he fee vbon poblem bsed on Flügge shell equons wh ehe boh ends clmped o boh ends fee. Ths s pcully nvolved pocedue even hough he shell dd no oe. Also he oupu of he model s he lengh of shell esonng gven fequency he hn he esonnce fequency of he shell hvng cen lengh. Neveheless he cveness of he mehod s h ec mode shpes nd nul fequences cn be obned.

2 In hs ppe he mehod s developed fuhe o cove lso fo ong hn cylndcl shells hvng fee boundes. Howeve nsed of usng Flügge s equons he equons of moon developed by Hung nd Soedel [] e used nd he fee vbon poblem s solved ecly fo boh ends of he shell fee. Ths s hough o be moe useful se of boundy condons f he shell s o be used fo modellng e bel. Ths s due o n unnecessy consn of he nfne dl sffness of he sdewll f he smply suppoed boundy condons e consdeed. The esonnce fequences nd he mode shpes of he ong shell could be used o fom bss fo foced vbon esponse of he shell nd hus offe possbly o couple he fee e bel o sdewll nd he cvy models usng fo emple mobly mpednce bsed mehods. The mhemcl model s developed n he fs secon nd n emple shell s consdeed n he second secon whch s followed by conclusons secon. MATHEMATICAL MODEL The hn cylndcl shell used o ppomely model he e bel s shown schemclly n Fgue 1. h p v w u z Fgue 1: The ong cylndcl shell L The equons of moons fo used n hs sudy e hose ognlly developed by Hung nd Soedel []. In cse of fee vbon hey educe o: whee: 1 u u K v w u 1 K N K h 3 3 w w v v D 3 1 K D KN D u v w w K1 h N K h vh N w v v w w D u w w v v K N h N K h KN h w (1) () (3) = shell dus h = shell hckness L = shell lengh p = nflon pessue u = l dsplcemen v = ngenl dsplcemen w = dl dsplcemen = l coodne

3 = ngenl coodne z = dl coodne = Posson s o E = Young s modulus = mss densy N = nl enson n he ngenl decon = oon speed 3 Eh D 1 1 = bendng sffness nd Eh K = membne sffness. 1 The nl enson n he ngenl decon s gven by: N h p () whee he fs em s due o cenfugl foces nd he second em s due o he nl nflon pessue of he shell. Subsung: no he equons of moon (1)-(3) yelds: e uu cos n (5) e vv sn n (6) e ww cos n (7) n U V n W K U h n N 1 1 (8) h V W KN n K N V n KU W N K D n 1 1 V W n W n n n D n W nv h W V n N KN W V N K nku (9) (1) whee n s he numbe of ccumfeenl wves nd s fequency. The hee equons (8)-(1) cn be condensed no m fom: whee: k k k U k k k V 1 3 k k k W k n 1 n K h N 11 K 1 1 k n k K 13 k Kn1 1 k n D n h K 1 n K N N 1 (1) (11)

4 k Dn N K D n h 3 3 k K 31 k Dn 3 N K D n h 3 h n N N D 33 k K n n The deemnn of he m n Eq. (11) mus vnsh n ode fo he equons of moon o be ssfed. Assgnng vlue o he fequency nd seng he deemnn o zeo yelds bquc polynoml n : (13) whee he coeffcens e gven n he Append o hs ppe. Thus hee e egh oos of he polynoml nd so he dl componen of he dsplcemen cn be epessed s: ww ( )cos n (1) wh: W ( ) 8 B e (15) 1 whee B wh =1 8 e egh genelly comple consns. The oos of he polynoml (13) mus be clculed numeclly hs sge. As shown by Hu nd Wh [] fo he usul nge of pmees nd n 1 he oos cn be epeced o hve he fom ( p q) whee pq e el nd posve numbes. 1 1 Then he dl componen of he dsplcemen s gven by: 1 1 cosh h cos 1 3 W( ) C C sn C C sn p p q q q q e C cos C sn e C cos C sn (16) whee C e now el consns. Fom Eq. (8) - (1) nd (1): nd U k k W k k k k k k ; (17) V k k W k k k k k k (18) Subsung fo ech oo no Eqs. (17) nd (18) he epessons fo U( ) nd V( ) cn be epesened s: 1 1 C cosh C h C cos C V ( ) d d sn d d sn p q q q q d d cos C d C sn e C C cos C d C sn d d p e C C d d U( ) d d sn d d sn 1 1 C cosh C h C cos C 1 3 p q q q q d d cos d C sn d d cos d d sn p e C C d C e C C C C (19) ()

5 The consns d cn now be numeclly clculed usng Eqs. (17) nd (18) s: d V W wh 1 1 d U W wh 1 d V W wh 3 d U W wh (1) d V W wh p q 5 d V W wh p q 6 d U W wh p q 7 d U W wh p q 8 A ech end of he shell mked wh =cons. hee e fve esuln foces s shown n Fgue bu he equons of moon e of mmum fouh ode nd cn only ccommode fo fou boundy condons. Thus he Kchhoff effecve she sess esuln of he fs knd V nd he Kchhoff effecve she sess z esuln of he second knd T mus be used h ele Q o M z nd N o M especvely [1]. Fgue : The boundy foce esulns These elons e [1]: nd 1 M V Q z z () 1 T N M (3) The fou boundy condons fo shell wh boh ends fee e: N u v ;.e. w () M ;.e. w w v (5) 3 3 w v w V ;.e. z 3 (6) w v u T ;.e. h 1 h 1 (7)

6 These cn be sepely ssfed fo symmec nd n-symmec modes by ssumng he ogn n he mddle secon of he cylnde nd he boundes L. Hee he dl nd he l dsplcemen componens she he symmey o n-symmey popees nd he ccumfeenl dsplcemen componen s oppose o he dl nd l dsplcemen componens n ems of symmey o n-symmey. In cse he modes e symmec he dl componen of he dsplcemen cn be shoened by ecognsng h p e p p cosh( ) snh( ) nd ensung h he n-symmec ems n Eq. (16) vnsh. Then mus be C C C C nd C C so h Eqs. (16) (19) nd () educe o: p q p q W ( ) C cosh C cos F cosh cos F snh sn p q ( ) cosh cos cosh cos p q V d C d C d F d F d F d F snh sn p q ( ) snh sn snh cos p q U d C d C d F d F d F d F cosh sn (8) (9) (3) Subsung Eq. (8) no (1) fo he dl dsplcemen componen nd lso Eqs. (9) nd (3) no nlogue epessons fo ngenl nd l dsplcemen componens gves se of dsplcemen componens uv nd w. These cn be subsued no Eqs. ()-(7) whch fe ssumng L nd pung no m fom gves: ql pl ql pl cos cosh cos cosh 13A 1A L 1 L cosh cos 11 1 ql pl ql pl sn snh sn snh 13B 1B ql pl ql pl cos 3A cosh cos cosh A L 1 L cosh cos 1 ql pl ql pl C1 sn snh sn snh 3B B C 3 ql pl ql pl F1 sn c 33A osh sn cosh 3A L L 1 snh sn 31 3 F ql pl ql pl cos snh cos snh 33B 3B ql pl sn cosh 3A ql pl sn cosh A L L 1 snh sn 1 ql pl ql pl cos snh cos snh 3 B B (31) The coeffcens s ~ depend on he oos pq he consns d n nd nd e lsed n he Append 1 o he ppe. In ode fo he boundy condons o be ssfed he deemnn of he m n Eq. (31) mus 1 vnsh. The deemnn fe beng dvded by snh pl L pl snh cosh fo bee numecl behvou s gven by: nh b b cos sn b coh b cos sn coh b b cos cos coh nh sn coh coh nh cos sn b b b b b b b b sn cos coh b b cos b sn coh b cos sn nh bb sn 5 16 (3)

7 L L pl ql 1 whee nd 1 3 wh coeffcens b gven n he Append o he ppe. The zeoes of he lef hnd sde of Eq.(3) n fc epesen he lengh of he shell whose mn mode esones he ssumed fequency whee he lowes zeo (he smlles lengh) s fo m=1 symmec mode he ne one s mn fo m=3 symmec mode ec. In cse of he non-ong shell ( ) ssumng fequences mn yelds he sme lengh fo ehe he posve o he negve fequency howeve fo hs s no he cse snce mn he fowd nd bckwd vellng wves e now chcesed by dffeen speeds nd hus dffeen esonnce fequences. In cse of n-symmec modes mus be C C C C nd C C so h: ( ) snh sn snh p W C C F cos q F cosh p sn q 3 (33) wh sml epessons fo V( ) nd U( ). Afe eclculng he deemnn fo he n-symmec modes n epesson nlogue o Eq. (3) esuls whch cn be obned by subsung nh( ) coh( ) 1 1 nh( ) coh( ) cos( ) sn( ) nd sn( ) cos( ) no Eq. (3). Then he zeoes of he epesson 3 3 epesen he lengh of he shell whose mn mode esones he ssumed fequency whee he lowes zeo mn s fo m= n-symmec mode (Love mode) he ne one s fo m= n-symmec mode ec. The symmec Rylegh ype modes (m=-1) do eque sepe emen whch s no coveed whn hs sudy. In ode o clcule esonnce fequences fo shell wh gven lengh s necessy o ee unl he lengh esulng fom fndng zeo of he deemnn mches he el lengh of he shell o desed pecson. RESULTS Nul fequences of n emple ong shell e clculed ne nd ploed gns he ccumfeenl wvenumbe n n Fgue 3 fo non-ong shell nd fo shell ong ncesng speeds. The physcl nd 3 geomecl popees of he shell e s follows: =.1m h=.m L=.m 15kgm.5 nd E=.5 GP. The lef hnd sde bnches (negve n) show he bsolue vlues of he nul fequences fo he fowd vellng wves nd he gh hnd sde bnches (posve n) e fo he bckwd vellng wves. Ech bnch s fo m=cons. nd only he bendng wves e consdeed whch hve he lowes nul fequences nd whch e of he foemos pccl nees. As cn be seen n Fgue 3 he oon cuses symmees n he plos s he fowd vellng wves nlly decese he nul fequences fo smll oon speeds nd low ode modes (.e. m=1n=) howeve fo hghe oon speeds nd hghe mode odes he endency s h he nul fequences of boh he fowd vellng nd bckwd vellng wves ncese wh he oon speed. Ths s pedomnnly due o he enson cused by he cenfugl foces n he bel whch ncese wh squed oon speed. CONCLUSIONS Fee vbons of ong hn cylndcl shells e consdeed. A mhemcl model s developed o clcule mode shpes nd nul fequences of ong shell hvng fee ends. The boundy condons e ssfed ecly. The poposed mehod eques n ssumpon of nul fequency nd hen he clculon of he lengh of he ong shell h cn vbe h nul fequency. A numbe of nul fequences hve been clculed by eon unl he pescbed lengh s mched s funcon of he oon speed fo he emple shell. The behvou of he nul fequences s such h n pncple hey ncese wh he ncese n oon speed ecep fo some low ode modes low oon speeds. Ths s pedomnnly due o sffenng effec cused by he cenfugl foces. ACKNLOWLEDGEMENTS The esech pefomed by Neven Alujevć ws suppoed fnnclly hough n EU FP7 Me Cue Indusy- Acdem Pneshps nd Phwys (IAPP) Gn Ageemen The esech pefomed by Nu

8 Cmpllo-Dvo hs been developed whn Sho Tem Scenfc Msson funded by he COST Acon TU115 NVH nlyss echnques fo desgn nd opmzon of hybd nd elecc vehcles. Fgue 3: Nul fequences of he shell wh fee boundes fo non-ong shell nd fo ong shells wh hee dffeen oon speeds. APPENDIX The coeffcens e: 1 6 h N Kn h h N K N N K K N n h D N K n hdd N N K D N K n 3 n D h h N K hd N N K D N K n wh h 6 n 3 n n 1 1 K 8 nn 1 h N 5 n h 3 1 h 3 N 3 n 1 5 n n n D h K

9 N K h D N K N hd n 3 1N K D h K N n n h D N K 1 D hn N 1 K 1 h 1 DN K1 N hd Dn 1 K 3 N 3 KN n 1 3K 18 N 19 K1 hd1 N 1 hd3 N n K 18 N 7K 1 N 1 18 DK1 N n K hd h K N 1n 1 K D h n 8 n 1 K n D hn 1D n n K 1 N 3 n 33 n K13 n n 1 K 1 1 h D N n 8 D 1h n N 18 n D 1 5 K 6 KD D 1 h n N D K K N D 1 n The coeffcens e: s ~ nd d nnd n nd d h 1 d 1 n h d n d nd 1 3 nnd 3 n nd d 1 d nh d h n 3 3 nd 1 nd pd qd A nd pd qd 13 B nnd p q 3 A 5 3 B 6 pq

10 3 qn pd qd nq 3p q A pn pd qd n3pq p 33 B nd qd pd 16 h 1 pd n 1 d q 3 A nd pd qd 1nd pd qd n nn nd qd pd 16 n1 d p 1 qd h 1 3 B A B d pq A 6 d p q B 5 pn pd qd n3pq p 3 A n q p 3 qn pd qd 3 q 3 B nd qd pd 16 n1 d p1 qd h A nd qd pd 16 h 1 pd n1 d q B The coeffcens b e: b 1 33 B B 3 B 3 B b A B B A A B A B B A A B B A B A b b 3 A B 3 B A A 3 B A 33 B 1 3 B 1 A 13 A B B 13 A 33 B 1 A b b 6 A 13 A 3 A 1 A b 7 3 A 3 B 3 B 3 A 33 A B 33 B A B B B B B B B B B B B B B B B B A A A A A A A A A A A A A A A A B B B B B B B B B B B B B B B B b b b b 11 3 A A 3 A A A 3 A A 33 A 1 3 A 1 A 13 A A A 13 A 33 A 1 A b 1 1 B 3 A 1 A 3 B B 13 A 13 B A b 13 3 A 3 A 33 A A B A A B B A B A A B B A B A B A b b 15 3 B A 3 A B B 3 A B 33 A 1 3 A 1 B 13 B A B 3 A 1 B 33 A b B B 1 B 3 B REFERENCES [1] Soedel W.: Vbons of Shells nd Ples 3 d edon Revsed nd Epnded Mcel Dekke Inc New Yok [] Hung S.-C.; nd Soedel W.: On he foced vbon of smply suppoed ong cylndcl shells Jounl of he Acouscl Socey of Amec Vol.8 No. 1 July 1988 [3] Wbuon G. B.: Vbon of hn cylndcl shells Jounl Mechncl Engneeng Scence Vol. 7 No [] Hu W.C.L.; nd Wh T.: Vbons of Rng-Sffened Cylndcl Shells An Ec Mehod. Tech. Rep. No. 7 Conc NAS-9(6) SWRI Poj. -15 Souhwes Resech Insue Oc