Road noise sensitivity analysis with respect to suspension geometry

Transcription

1 Rod noise sensiiviy nlysis wih respec o suspension geomery Fumihiko KOSAKA 1 ; Hiroki MIZUNO ; Tsuyoshi INOUE 3 ; Kenro TAKAGI 4 1, MITSUBISHI MOTORS ORPORATION, Jpn 3, 4 Ngoy Universiy, Jpn ABSTRAT In he vehicle developmen, he impornce of reducing rod noise is incresing ecuse of he incresing populriy of uie elecric vehicle. The suspension plys n imporn role on he mechnism of he srucure orn rod noise. As he counermesure for reducing rod noise, low siffness suspension ush is ofen used from he poin of view of virion insulion. However, in generl, o use he low siffness suspension ush mkes he driving siliy worse. From such reson, i hs een difficul o chieve he compiiliy eween NVH nd driving siliy high uliy. Therefore, he oher wys re reuired for reducing he rod noise. As one of he wys, he rod noise reducion y he suspension geomery is presened in his pper. In order o find he suspension geomery which reduces he rod noise, he sensiiviy nlysis is used. For he sensiiviy nlysis wih respec o he posiion of he suspension rm connecing poins, he suspension model is simplified. The rms re modeled s rigid links wheres he vehicle ody, ire nd shock sorer re modeled s modl model. The suspension geomery which reduces he rod noise is oined wih he sensiiviy nlysis. Keywords: Suspension, Sensiiviy, Rod noise I-INE lssificion of Sujecs Numer(s): 46, INTROUTION Recenly, he impornce of reducing rod noise of he vehicle is incresing ecuse of he increse populriy of uie elecric vehicle. On he mechnism of srucure orn rod noise, he suspension plys n imporn role. As ypicl counermesure for reducing rod noise in he vehicle developmen, low siffness ush re ofen used from he poin of view of virion insulion. However, in generl, his mkes he hndling siliy worse. The suspension design which chieves he compiiliy eween NVH nd hndling siliy high uliy is reuired. In spie of such circumsnce, here re no so mny lierures wrien ou he rod noise reducion y he suspension. This would e ecuse of he difficulies in conrolling he virion ehvior of he suspension. The virion energy rnsmis hrough he muliple connecion poins which he suspension is conneced o he vehicle ody. When one of he connecion poins highly conriues o he energy rnsmission, i my e hough h reducing he virion of his poin will reduce he rod nose. However, some counermesures for his poin migh increse he virions he oher connecion poins ecuse he locl counermesure my ffec he glol suspension virion ehvior. Thus, he counermesures h consider he rde-off mong he virions of he ech connecion poins re reuired. The counermesures for reducing he rod noise would e ou he suspension ushes or he suspension geomery. In generl, he suspension ushes could e modified he ler sge of he vehicle developmen, however, he suspension geomery is fixed he erly sge nd could no e le o e modified he ler sge. Therefore, in his pper, for incresing he possiiliy of he compiiliy high uliy, he focus is given o he rod noise reducion y he suspension geomery. 1 fumihiko.kosk@misuishi-moors.com hiroki1.mizuno@misuishi-moors.com 3 inoue@nuem.ngoy-u.c.jp 4 kgi@nuem.ngoy-u.c.jp Iner-noise 14 Pge 1 of 1

2 Pge of 1 Iner-noise 14 For finding he suspension geomery h reduces he rod noise, he sensiiviy nlysis (1) wih respec o he suspension connecion poins is presened in his pper. As n ojecive funcion, he inpu power o he vehicle is used from he fc h he sisicl spce verged comprmen cviy SPL is proporionl o he inpu power from he poin of view of power lnce (). In order o mke he sensiiviy nlysis esy, he suspension rms re modeled s rigid link wheres he ody, he ire nd he shock sorer re modeled s modl model. The suspension geomery h reduces he inpu power is oined y he shpe opimizion sed on he sensiiviies. Then, i is presened h he rigid links modeling is suile for he design of he suspension geomery he erly sge of he vehicle developmen y invesiging he chrcerisic of he suspension virion ehvior.. MOELING The mechnism of srucure orn rod noise cn e hough s followings. Firs, he ires re excied y he forced displcemen due o he roughness of he rod surfce. Then, he virions rnsmi o he vehicle ody hrough he suspension susysems. The ech inpu power from he rod surfce o ech ires is sisiclly uncorreled. Therefore, he inpu power from he ech ire cn e considered seprely. In his pper, he rod noise from rer lef side muli-link suspension is considered. On he ypicl ending rigidiy of he suspension rms nd siffness of he ushes, he rms ehve like rigid ody in mny of he muli-link suspension susysem virion mode shpes of which he nurl freuencies re round freuencies of ineres (e.g. round 15 Hz). Therefore, in order o mke he invesigion ou he reducion of he rod noise due o he configurion of suspension esy, he suspension model is simplified s rigid links conneced wih ushes. On he oher hnd, he vehicle ody, he sorer nd he ire susysems hve deformed virion mode shpes he freuencies of ineres so h hese re modeled s modl model which re idenified eiher numericlly or experimenlly. The schemic of he modeling is shown in Fig. 1. Figure 1 schemic of proposed modeling The suspension rm consiss of he four rigid componens which re knuckle, lower rmpper rm nd oe conrol rm s shown in Fig. 1. The rigid componens re conneced wih ushes poins nmed KL, KU nd KT. Then, he suspension rms re conneced o he vehicle ody wih ushes four poins which re nmed LB, UB, TB nd KB. The sorer connecs he lower rm nd he vehicle ody poin LA nd AB, respecively. A he poin AB, he sorer is rigidly conneced o he vehicle ody wheres i is conneced o he lower rm wih ush he poin LA. The posiion of he poin LA is deermined y inernlly dividing secion from he poin LB o KL y 3:1. The ire is rigidly conneced o he knuckle poin nmed W nd excied y he forced displcemen poin nmed G. Pge of 1 Iner-noise 14

3 Iner-noise 14 Pge 3 of 1 3. GOVERING EQUATIONS 3.1 egree Of Freedom Suspension In one rigid componen, coordines re ken ino ccoun he cener of grviy nd ech connecion poins. For exmple, he knuckle rigid componen hs following displcemen vecor u, kcl kcl T T T T T T { u u u u u u } T u =, (1) kclw kclkl kclku where u kclw kclkl kclku kclkt kclkb nd ukclo re he displcemen vecors he poin W, KL, KU, KT, KB nd he cener of grviy of he knuckle componen, respecively. Ech displcemen vecors connecion poins nd cener of grviy hs rnslion nd roion OFs in resin coordine (i.e. { u v w θ θ θ } T for ech poin displcemen vecors), so h x y he ol numer of OF of knuckle componen is 36. The displcemen vecor of he suspension susysem is expressed y he rigid componens displcemen vecors s z kclkt T T T T { u u u u } T kcl lwr upr oe kclkb =, () where u lwr upr nd u oe re he displcemen vecor of he lower rm componen, he upper rm componen nd he oe conrol rm componen, respecively, like h of he knuckle componen expressed in En. (1). For displcemen vecors u lwr upr nd u oe, he coordines re ken cener of grviies nd oh ends of ech rms. In ddiion, he coordine he poin LA is ken for he lower rm componen. So, he ol numer of OF of he suspension susysem displcemen vecor is 96. Euion () cn e reduced o he displcemen vecor represened only independen displcemen vecors of he ech rigid componens due o he geomericl consrin of he rigid links s kclo = β o, (3) o o T T T T { u u u u } T =, (3) kclw lwro where β is rigid elemen mrix h expresses he linerized geomericl consrin of he rigid links. Therefore, he rigid elemen mrix is he funcion of he geomery of he suspension susysem. In his pper, he displcemen vecors he poin W nd cener of grviies of he lower rmpper rm nd oe conrol rm re ken s independen displcemen vecors. The numer of OF of independen displcemen vecor is 4. o 3.1. Vehicle ody The displcemen vecor of he vehicle ody u is expressed y he modl coordine in he glol coordine under he condiion of eing uncoupled wih he suspension susysem s u = φ ξ, (4) where vehicle ody. upro φ is he mode shpe mrix of he vehicle ody nd Tire As well s he cse of he vehicle ody, he displcemen vecor of he ire oeo ξ is he modl mpliude vecor of he u is lso expressed y he modl coordine in he glol coordine under uncoupled condiion s u = φξ, (5) where φ is he mode shpe mrix of he ire nd ξ is he modl mpliude vecor of he ire. Iner-noise 14 Pge 3 of 1

4 Pge 4 of 1 Iner-noise Asorer The configurion of he sorer depends on h of he lower rm componen so h he modl coordine of he sorer hs o e expressed in he locl coordine nd is rnsformed o he glol coordine y he rnsformion mrix s ( l ) u = Tφ ξ, (6) where u is he displcemen vecor of he sorer in he glol coordine nd T is he ( l ) rnsformion mrix from he locl coordine o he glol coordine. φ is he mode shpe mrix of he sorer in he locl coordine, where he superscrip ( ) mrix is descried in he locl coordine. l denoes h he mode shpe ξ is he modl mpliude vecor of he sorer. 3. ouplings The coupling wih he ush is descried s he force o he coupled susysem. For exmple, he coupling force eween he vehicle ody nd he lower rm poin LB is descried s f LB lwrlb ( g ) ( u u ) = f = K, (7) LB where f LB nd f lwrlb re he force o he vehicle ody nd o he lower rm he poin LB, ( ) respecively. K g LB is he siffness mrix of he ush he poin LB. The superscrip ( g ) denoes h he siffness mrix is expressed in he glol coordine. u LB nd u lwrlb re he displcemen vecor of he vehicle ody nd he lower rm he poin LB, respecively. The rigid conneced displcemen couplings re considered y consrin euions. The consrin euions of he sysem re followings. u AB u AB =. (8) u W u kclw =. (8) v G = 1. (8c) Euions (8) nd (8) re he rigid connecion coupling he poin AB nd W respecively. Euion (8c) is he consrin euion he poin G, which descries he forced displcemen inpu. In his pper, he focus is on he inpu power round 15 Hz. In such freuencies, i is known h he conriuion from he y direcionl forced displcemen v hs high conriuion o he rod noise so h uni y direcionl displcemen he poin G is considered s inpu. 3.3 Euion of Moion The euion of moion of he sysem cn e wrien considering he couplings ll he connecion poins s LB lwrlb G c c c sym. c ξ ξ ξ =, (9) λ u where,, nd re he dynmic siffness mrixes of he ech susysems which ke nd re he coupling ino ccoun he siffness of he ushes he connecion poins. mrixes due o he ushes eween he suspension nd he vehicle ody, he suspension nd he sorer, respecively. c, c, c nd c re he Jcoin mrixes of he consrin euions nd λ is he Lgrnge muliplier. u is he inpu vecor due o he forced displcemen of Pge 4 of 1 Iner-noise 14

5 Iner-noise 14 Pge 5 of 1 En. (8c). By considering he rigid link relion of En. (3), he euion of moion of he sysem cn e finlly oined s x = F. (1) = T β c β T c c β sym. T c β β, ξ ξ x = ξ, o λ F = (1) u 3.4 Inpu Power o he Body The Inpu power o he vehicle ody Π is expressed y he summion of he inpu powers he ech connecion poins s LB UB TB KB AB Π = Π + Π + Π + Π + Π, (11) where he superscrip denoes he connecion poin from which he inpu power is supplied. As n exmple, he inpu power from he connecion poin LB is wrien s LB jω * Π = flb ulb. (1) The superscrip * denoes complex conjuge nd ω is ngulr freuency. Susiuing Ens. (), (4) LB nd (7) ino En. (1), Π cn e expressed in funcion of ξ nd. For he connecion poin UB, TB nd KB, he inpu power he connecion poin cn e oined in he sme mnner s he poin LB. The inpu power from he poin AB cn e expressed in he modl coordine of he vehicle ody s T * ( λ) ξ AB jω * jω Π = fab uab = c. (13) In he expnsion of En. (13), i is used h he consrin force he connecion poin AB cn e expressed using he Lgrnge muliplier s λ. Susiuing Ens. (1) nd (13) ino En. (11), T c nd king ino ccoun he firs euion of En. (9), he ol cive inpu power o he vehicle ody is expressed s follows. 1 H Re { Π } = ξ Rξ, (14) R ij ω ωiη i, =, ( i = j) ( i j) where he superscrip H denoes conjuge rnspose., (14) ω i nd i η re he nurl ngulr freuency nd he modl dmping loss fcor of he i h vehicle ody mode, respecively. The form of En. (14) is eul o he dissipion power wihin he vehicle ody susysem. This indices he euliy of he inpu power nd he dissipion power, so clled power lnce. 3.5 Sensiiviy Anlysis For finding he suspension geomery which reduces he inpu power o he vehicle ody, he sensiiviy nlysis wih respec o he posiion of he connecion poin is used. The posiion of he poin KL, KU nd KT re ken s design vrile γ in his pper. From he fc h he mrix R in En. (14) is no funcion of design vrile γ, he sensiiviy cn e wrien s Iner-noise 14 Pge 5 of 1

6 Pge 6 of 1 Iner-noise 14 From En. (1), Re γ { Π} ξ is pr of x so h H ξ = Re Rξ γ / γ. (15) ξ cn e oined from x / γ. Firs order sensiiviy of x cn e led from En. (1) y considering h F is no funcion of design vrile, x 1 = x. (16) γ γ The derivive 4. RESULTS / γ cn e clculed lgericlly or numericlly. 4.1 Vlidiy of Modeling For he proposed rigid rm modeling, he concenred msses of he ech rigid componens re esimed from he deil 3 geomery. For he knuckle rigid componen, he msses of he rke roor nd he cliper re considered in ddiion o h of he knuckle rms. In Fig., he comprison of he inpu power eween he deil FE model nd he proposed rigid rm modeling is shown. Less difference cn e seen eween he wo modeling especilly round freuencies of lrge inpu power (e.g. from 1 Hz o Hz). This indices h he influence of he ending rigidiies of he rms cn e negleced o descrie he virion ehvior of he sysem he freuencies of ineres. The shpes of he suspension rms re neiher srigh nor uniform cross secion. However, for he simplified cse, i is considered h he rms re ssumed o hve he cener of grviy he geomericl cener (i.e. he rm shpes re ssumed o e srigh em wih uniform cross secion). There is lso less difference eween he simplified nd he exc cse. This indices h he deil mss disriuion could no e imporn unless he drsic mss disriuion is considered. In oher word, he rod noise could e prediced wihou he deil informion of he rm shpes. This chrcerisic cn mke pplicion of he modeling possile he erly sge of he vehicle developmen in which he deil rm shpes re no decided. Figure Vlidiy of he proposed rigid rm suspension modeling. Inpu power compued from deil FE model (lue dshed), from he proposed simplified rigid rm modeling (red solid), from he proposed rigid rm modeling wih exc mss disriuion propery (red dshed). 4. Sensiiviy Anlysis By he rigid rm suspension modeling, he sensiiviy nlysis cn e conduced using Ens. (15) nd (16). The x, y, z posiions of he poin KL, KU nd KT re ken s design vriles (i.e. olly nine design vriles), nd he overll of inpu power from 1 Hz o Hz is ken s n ojecive Pge 6 of 1 Iner-noise 14

7 Iner-noise 14 Pge 7 of 1 funcion. Wih he design vriles ken in his pper, he derivive / γ of En. (16) is le o e evlued wihou unceriny, excep he pr of he derivive of he mss mrix of he suspension susysem. The derivives of he mss mrix of he suspension susysem is expressed s ( g ) ( l ) T M T ( ) M l T T ( l ) T = M T + T T + TM. (17) γ γ γ γ ( g ) ( l ) where M, M re mss mrixes of suspension susysem. The superscrip denoes he coordine in which he mss mrix is expressed. T is rnsformion mrix from locl coordine o glol. On he righ hnd side of En. (17), he firs nd he hird erms descrie he conriuion from geomericl chnge wih consn mss properies, while he second erm descries he conriuion from he mss chnge of he rigid rms hemselves due o heir lengh chnges. In generl, he shpes of he suspension rm componens cn e rirry (e.g. n rm wih pered cross secion, curved rm nd so on). Therefore, he derivives of he mss mrixes of he second erm of En. (17) cnno e deermined uniuely. In order o clcule he derivives, one nominl rm shpe is considered h he rm hs srigh wih uniform cross secion ssumed s simplified cse in he rigid rm modeling of Sec For clculing he derivives of he knuckle componen mss mrix, i is ssumed h he mss increses proporionl o knuckle rm lengh exension he connecion poin. The increse re is esimed s 4 kg per uni lengh from he cul mss disriuion of he knuckle. The sensiiviy resuls re shown in Fig. 3 wih he oh cses in which he derivives of he mss mrixes re ken ino ccoun or no. The negive sensiiviy shown in Fig. 3 mens h he inpu power is incresed wih he negive direcion posiion chnge (e.g. replcing z coordine of he poin KL long -z xis increses inpu power). In Fig. 3, less difference cn e seen eween he oh cses eiher wih or wihou he considerion of he mss mrixes derivives. This indices h he mss propery chnges could no e imporn. This chrcerisic is lso convenien for he pplicion he erly sge of vehicle developmen. Figure 3 Inpu power sensiiviy wih respec o he posiions of he connecion poins KL, KU nd KT. The derivives of he rms msses re no ken ino ccoun (lue), re ken ino ccoun (red). 4.3 Shpe Opimizion Opimized resuls Using he sensiiviy nlysis, shpe opimizion cn e conduced. From he resul of Sec. 4.1 nd 4., he rms re simplified s srigh wih uniform cross secion nd he mss mrixes derivives re negleced in he sensiiviy nlysis. Besides, he mss properies of he rms on heir principl xes re kep consn during he opimizion process (i.e. ech rigid componens respecively hs he sme mss propery s he iniil shpes on heir locl principl xes he influences due o he geomericl rnsformions re ken ino ccoun). The flowchr of he Iner-noise 14 Pge 7 of 1

8 Pge 8 of 1 Iner-noise 14 opimizion is shown in Fig. 4. MMA (3) ws used s n opimizion mehod. Figure 4 Flowchr of he opimizion () () (c) (d) (e) Figure 5 The opimizion resuls: () () (c) omprison of he geomery in ech plne views eween he iniil (lue dshed) nd he opimized (red solid). The design spces re shown in cyn pch. (d) The opimizion hisory. (e) omprison of he inpu power eween he iniil (lue dshed), he opimized wihou mss chnge (red solid) nd he opimized wih mss chnge (red dshed). Pge 8 of 1 Iner-noise 14

9 Iner-noise 14 Pge 9 of 1 The opimizion resuls re shown in Fig. 5. The design spces were ken -5 mm o 5mm from he poin KL, KU nd KT iniil posiions for ll direcions, which re shown in Figs. 5()-(c) s cyn sure pch. I is oserved h he opimizion is well converged from he opimizion hisory of Fig. 5(d). In Fig. 5(e), i cn e found h he inpu power is reduced in he freuencies of opimizion rge. The influence of he mss chnges due o he rm lengh chnges is lso invesiged. The inpu power h considers he mss chnges he opimizion configurion is lso ploed in Fig. 5(e). The mss chnge of he rigid rms is clculed from he ssumpion used in Sec. 4.. The wo specr re lmos he sme so h i is confirmed h he ssumpion of keeping he mss properies of he rigid componens consn is vlid Inpu power conriuion The opimizion resuls cn e dignosed y he inpu power conriuion. The inpu power conriuions from he ech connecion poins re ploed in Fig. 6 for he iniil configurion nd he opimized. Figures 6() nd 6() show he mgniude of he inpu power wheres Figs. 6(c) nd 6(d) show he posiive/negive conriuion y he normlized inpu power expressed in he following euion. As n exmple, he normlized inpu power from he poin LB s LB UB TB KB AB ( Π + Π + Π + Π + Π ) Π is wrien LB LB Π nor = Π /. (18) LB nor () () (c) (d) Figure 6 Inpu power from ech connecion poins: () Inpu power in he cse of he iniil in db scle. () Inpu power in he cse of he opimized in db scle. (c) Normlized inpu power in he cse of he iniil. (d) Normlized inpu power in he cse of he opimized. Iner-noise 14 Pge 9 of 1

10 Pge 1 of 1 Iner-noise 14 The negive normlized inpu power mens h he energy of he vehicle ody is dissiped y he exciion he connecion poin due o he opposie direcion of he force nd he virion. The normlized inpu power cn e used s he indicor for undersnding he mechnism of he inpu power reducion. The decrese of posiive normlized inpu power mens h he inpu power reducion could e chieved y he reducion of virion velociy of he rigid rm iself wheres he increse of he negive inpu power conriuion mens h he inpu power reducion could e chieved y cncelion mong he inpu power from ech connecion poins. From Figs. 6() nd 6(), he reducion of he inpu power over enire freuencies is mosly chieved y reducing he conriuion from he poin TB. Around 15 Hz, he inpu power reducion from he poin UB lso conriues o he inpu power reducion. In Figs. 6(c) nd 6(d), he signs nd he negive conriuion of he normlized inpu power hve sme endency eween he iniil nd he opimized over enire freuencies (excep round 15 Hz). This indices h he inpu power reducion could e chieved y he reducing he virion of he highly conriued rigid suspension rm wihou incresing he virion of he oher rms. 5. ISUSSION The opimized suspension geomery shown in Figs. 5()-(c) my no chieve he compiiliy eween NVH nd hndling siliy high uliy (e.g. cmer complince migh e incresed due o he smll z direcionl widh). In order o chieve he purpose, limiions ou hndling siliy should e considered in he opimizion process. If n opiml resul doesn reduce he inpu power enough y king ino ccoun he limiions, dding he siffness of he ushes o he design vriles migh improve he resuls (in such cse, considering weigh funcion for he sensiiviies would e necessry ecuse of he difference in sensiiviy uni eween he rm connecion posiion nd siffness of ush). 6. ONLUSIONS The sensiiviy nlysis wih respeced o he connecion posiions ws conduced y modeling he suspension s rigid rms. The modeling ws vlided y compring he inpu power wih h clculed y he deil FE model. The opimized suspension geomery h minimizes he inpu power ws clculed using he shpe opimizion sed on he sensiiviy nlysis. In he sensiiviy nlysis nd he shpe opimizion, i ws found h he influence of he mss chnges ws no imporn for he inpu power compred o h of geomericl chnge. Therefore he suspension rms could e modeled s srigh wih uniform cross secion rigid em. This chrcerisic is suile for he pplicion he erly sge of he vehicle developmen in which he suspension geomery cn e only modified. REFERENES 1. Hug EJ, hoi KK, Komkov V. esign sensiiviy nlysis of srucurl sysems. Acdemic Press, Inc.; Kdomsu K, Iwng Y. Viro-cousic rnsfer funcion nd driving-poin conducnce for vehicle ody. Trnscions of he Jpn Sociey of Mechnicl Engineering. ; 68(67): Kriser S. The mehod of moving sympoes - new mehod for srucurl opimizion. Inernionl Journl for Numericl Mehods in Engineering. 1987; 4: Pge 1 of 1 Iner-noise 14