Incorporation of Non-Linear and Quasi-Linear Hydraulic Mount Formulations into a Vehicle Model

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1 Incopoaion of Non-Linea and Quasi-Linea Hydaulic Moun Foulaions ino a Vehicle Mol Copyigh 007 SAE Inenaional Song He and Rajenda Singh Acousics and Dynaics Laboaoy The Ohio Sae Univesiy ABSTRACT This pape copaaively evaluaes easueen-based quasi-linea and ue non-linea (echanical and fluid ype) ols of hydaulic engine ouns and exaines hei dynaic effecs wihin he conex of a siplifd half-vehicle syse. A non-linea appoxiae ol is also veloped o povi ipoved insigh ino he coupling effecs. The poposed ol is validaed by copaing pedicions wih hose fo a ue non-linea fluid ol. When ebedd ino he vehicle syse, hydaulic oun efficnly povis high apliusensiive daping and unes he engine bounce o. Poposed ol conceps could be effecively uilized o exaine linea and non-linea vehicle esponses in boh ie and fequency doains. INTRODUCTION Ove he pas wo cas, significan eseach [-] has been conduced on he dynaics of non-linea hydaulic engine ouns. Much of he lieaue [4-9] focuses on is fequency- and exciaion apliusensiive popes on a vice level, which is necessay bu no sufficn fo he syse vwpoin. This pape ais o fill in his void by incopoaing linea, quasi-linea o non-linea oun ols ino a siplifd, ye easonable, half-vehicle ol. This will pei a copaaive evaluaion of he copeing oling ehods in he conex of vehicle dynaics. I is of couse assued ha he oun popes as easued by he elasoe es achines can be dcly ipoed ino vehicle ols even hough hee ae key diffeences beween he elasoe es ehod and insiu vehicle ouning syse. In fac, any paciiones use ad hoc ehods o bing in easued daa (o paaees) in a vehicle ol. Accodingly, he following objecives ae esablished: Fis, popose a siplifd half-vehicle ol ha would incopoae copeing oun ols (based on elasoe es daa). Modal sudy of he vehicle syse, based on he quasi-linea ol, will help o answe he key quesion ha ceain vehicle os could be effecively uned due o a coupling beween hydaulic oun and ohe vehicle coponens. Second, velop a non-linea appoxiae INCORORATION OF HYDRAULIC MOUNT FORMULATION(S) INTO HALF-VEHICLE MODEL Table Copaison of oun sub-syse ols Mol Type Moun (Fig.) Analysis Doain Moling Effos Need DOF of Fig. (a) Linea Rubbe (b) Tie & Modal ol fo he fee couple ype hydaulic oun o povi ipoved insigh ino he coupling effecs. The swiching echanis will be scibed in es of a cleaance ype non-lineaiy. Thid, esiae effecive paaees via he appoxiae ehod fo educed oling effos. The poposed ol will hen be validaed by copaing pedicions wih a ue fluid ype ol (based on ie doain soluions [5]). Finally, he ue non-linea fluid ol will be uilized o exaine non-linea dynaic vehicle esponses such as he supehaonics. Quasi- Linea Hydaulic (c) Tie & Modal Tue Non- Linea (Fluid) Hydaulic (d) Tie & Feq. (FFT) Non-Linea (Appoxiae ehod) Hydaulic (e) Tie & Feq. (FFT) Minial Minial High Moae 6 DOF 6+ DOF 6+ DOF 6+ DOF Fig. (a) shows a siplifd 6 DOF half-vehicle ol wih displaceens only in he veical dcion x i (i =,,6). Luped asses o 6 coespond o he ineia eleens of he engine, fon vehicle body, cenal body, ea body, fon wheel & axle and ea wheel & axle, especively. Coodinaes x 7 and x 8 epesen he igid bases a he fon and ea s. Siffness eleens inclu body flexue, suspensions and s; hese ae foulaed using he Voigh ol (siffness and daping eleens in paallel), as shown in Fig. (a). Alenae engine oun sub-syses could hen be incopoaed ino he vehicle ol. These ae suaized in Table including: (i) ubbe oun

2 scibed by he Voigh ol (esuling in a 6 DOF linea vehicle ol), as shown in Fig. (b); (ii) quasilinea hydaulic oun foulaion [3] (leading o a 7 DOF quasi-linea vehicle ol in Fig. (c)); and (iii) non-linea fluid o appoxiae oun foulaions (ylding 8 DOF non-linea vehicle ols), as shown by Figs. (d-e). Maheaical scipions of such nonlinea ols will be given lae in his pape. (a) 6 DOF half-vehicle ol incopoaing oun sub-syse; (b) ubbe oun sub-syse; (c) quasi-linea hydaulic oun sub-syse; # F x = x + x F x # k b q d C q i #d I i R i #i b k k b b k x x + x () I d R d # C k b F ( ) T (d) fluid ol of he non-linea hydaulic oun sub-syse; F ( ) T (e) appoxiae ol of he non-linea hydaulic oun sub-syse. Fig. Vehicle syse and sub-syse exaples. Also, see Table.

3 MODAL ANALYSIS BASED ON QUASI-LINEAR MODEL Fis, analyical odal analysis is conduced fo he ubbe oun of Fig. (b) as benchak. Given noinal ass, siffness and daping values of Table, he following six vehicle os ae obained: () vehicle bounce; () engine bounce; (3) ea wheel hop; (4) fon wheel hop; and (5) and (6) ae vehicle body pich and beaing (hough he discee ol does no have enough esoluion fo hese). The naual fequencs f n and esiaed odal daping aios ζ ae lised in Table 3. Hee, we ae os concened abou he engine bounce o of Fig., which shows soe coupling wih ea body and axle. Also, his o is found o be os sensiive o he paaees of he oun sub-syse. Table Paaees of half-vehicle ol of Fig. (a) Mass (kg) (Poweain) 5 (Fon Body) 0 3 (Wheel+Axle) 45 4 (Cene Body) 70 5 (Rea Body) 40 6 (Wheel+Axle) 75 Eleen k (N/) c (N-s/) PT oun k 300* b 50* Fon suspension k 3 c 3 00 Fon k 3 00 c 3 40 Body flexue k 4,000 c 4 75 Body flexue k 45,800 c Rea suspension k 56 6 c Rea k 6 00 c 6 40 *Baseline values fo he ubbe oun Table 3 Vibaion os of 6 DOF half-vehicle ol wih ubbe oun of Fig. (b) Inx Mo Type f n (Hz) ζ (%) Vehicle bounce..6 Engine bounce Rea wheel hop Fon wheel hop Vehicle pich Body bending Fig. Engine bounce o wih ubbe oun subsyse. Un ceain opeaional condiions, he quasi-linea foulaion [3] of Fig. (c) leads o a luped paaee epesenaion of he hydaulic oun sub-syse in es of equivalen sping, ass, dape eleens. Thus, odal analysis and fequency esponse suds could be cad ou in he conex of vehicle dynaics. This should povi alenae and apid evaluaion of he uning capabiliy of hydaulic oun. An exaple case is povid hee using he quasi-linea paaees esiaed fo one hydaulic oun (signaed as D) a X = wih k = 344 N/, k = 379 N/, b = 4 N/s, b = 300 N/s and = 37.6 kg. Table 4 liss he esuling naual fequencs f n and esiaed odal daping aios ζ. The engine bounce o of Fig. wih ubbe oun now sees o be ansfoed ino wo new coupled os as shown in Fig. 3. The es of vehicle os eain alos unaffeced. Table 4 Mos of half-vehicle ol wih hydaulic oun scibed by a quasi-linea ol of Fig (c) Inx Mo Type fn (Hz) ζ (%) Vehicle bounce..6 Coupled o I (in phase) Rea wheel hop Fon wheel hop Vehicle pich Coupled o II (ou of phase ) Body bending

4 Fig. 3 Coupled engine bounce os of half-vehicle ol wih hydaulic oun sub-syse of Fig. (c) foulaed using he quasi-linea ol. (a) In-phase o; (b) Ou-of-phase o. Obseve he following: Fis, and vibae in phase fo he o piced by Fig. 3(a), while and ae ou of phase fo he ohe coupled o of Fig. 3(b). Second, he axiu aplius of new os coespond o he effecive ineia ack ass insead of he engine ineia, so ha he ineia ack esonance o is doinan ove he engine bounce o fo boh cases. Thid, since ineia ack is inenionally signed o povi high fluid daping, he ineia ack esonance o is associaed wih high viscous daping, as clealy shown by he high odal daping aios fo hese wo coupled os. Fo his specific case, ζ inceases 4 and 0 ies fo he wo new engine bounce os. This eveals he supeioiy of a hydaulic oun ove he convenional ubbe oun in conolling engine bounce esonances. By coelaing he echanical eleens o fluid paaees [3], he quasilinea ol could be efficnly used by vehicle signes o quanify he oun specificaions. Fig. 4 Acceleance speca (a) Acceleance A /F a engine given foce exciaion fo engine; (b) Acceleance A /F 3 a fon body given foce exciaion fo fon wheel. Key:, ubbe oun;, hydaulic oun using quasi-linea foulaion;, hydaulic oun using dc invesion ehod. Nex, fequency esponse funcions (esponses of displaceen X(f), velociy V(f) o acceleaion A(f) given uni foce exciaion F(f)) ae ived by using he quasilinea foulaion. These ae hen copaed o hose yld by he dc invesion ehod given easued dynaic siffness daa (as benchak) o quanify he eos induced by he quasi-linea esiaion algoih. Fig. 4(a) shows he A /F engine acceleance given uniapliu engine foce exciaion. Likewise, Fig. 4(b) shows he A /F 3 fon body acceleance given uniapliu fon wheel foce exciaion. Copaed wih he ubbe oun, he hydaulic oun sufficnly conols he engine bounce esonance aound 8 Hz in Fig. 4(a), as well as he vehicle pich esonance aound 6 Hz in Fig. 4(b).

5 Copaison esuls of Fig. 4 iply ha hydaulic oun can be uilized o povi no only high daping fo engine oion conol, bu i also povis addiional coupling beween vehicle sub-syses. Such coupling could be useful in solving soe vehicle vibaion pobles (such as he pich esonance) ha ae encouneed in sub-syses away fo he engine ouns. Mino discepancs exis beween he quasilinea ol pedicions and he dc invesion ehod due o he appoxiaion pocess [3] and he assupion a in he pocess [3,8], i.e. F T () F(). FLUID VS. APPROXIMATE NON-LINEAR MODELS OF HYDRAULIC MOUNTS In a ecen pape [3], we poposed a quasi-linea appoxiae ol ha uilizes seady sae dynaic siffness easueens o consuc fequency- and exciaion apliu-sensiive oun ols un ceain opeaing condiions. Copaed wih a ue nonlinea fluid ol, i significanly educes he oling effecs and is capable of poviding a quick assessen of he augened daping and ineia effecs [3,4]. Ye, he quasi-linea ol is no sufficnly accuae o scibe he non-linea esponses especially fo he couple echanis. Theefoe, an ipoved nonlinea appoxiae ol is veloped as follows wih an ai o capue he on-off swiching acions of he couple. We will eploy a cleaance ype eleen, while inheiing effecive paaees fo he quasilinea ol fo educed oling effos. FLUID SYSTEM FORMULATION The non-linea fluid ol of Fig. 5 is bfly inoduced hee o ive he appoxiae ol as well as fo he sake of copaison. Refe o [5,6] fo ailed scipion of he ol and is expeienal validaion. k b # # q d I d R d C F x = x + x q i #d I i R i #i b k The viual diving poin foce F() could be fined as follows whee x() is he pison displaceen;, b and k ae he ass, daping and siffness of ubbe eleen; p () is he dynaic pessue of he op chabe and A is he effecive pison aea. F = ɺɺ x + b xɺ + k x + A p () Coninuiy equaions fo he op and boo chabes yld: A xɺ q q = C ( p ) pɺ, () i d i d q + q = C ( x ) pɺ. (3) Hee, C (p ) is he uli-sad op chabe copliance; C (x ) is he boo chabe copliance sensiive o ean displaceen x ; q i () and q d () ae he volueic flow aes hough he ineia ack (i) and couple (d), especively. Moenu equaions fo he couple and ineia ack yld he following, whee I d and I i ae he ineia of fluid coluns; R i (q i ) and R d (q d ) ae he nonlinea esisances: p p = I qɺ + R ( q ) q, (4) d d d d d p p = I qɺ + R ( q ) q (5) i i i i i Noe ha Eq. (4) dicaes he coupled sae when he couple gap is open, and Eq. (5) is doinan ove he coupled sae wih he couple gap closes. The dynaic coponen of foce F T () ansied o he igid base is ived and is elaed o F() as follows: F = k x + b xɺ + A p = F ɺɺ x (6) T NON-LINEAR APPROXIMATE METHOD An appoxiae echanical ol of Fig. 6 is veloped fo ipoved unsanding of he coupling effecs of a fee couple oun. k b F k x b x () x + x # C k b F ( ) T FT Fig. 5 Non-linea fluid ol of he hydaulic oun wih ineia ack and couple. Fig. 6 Non-linea appoxiae ol of he hydaulic oun wih ineia ack and couple

6 Noe ha all non-linea paaees such as C (p ) ae oled as a funcion of he noinal exciaion apliu X. The swiching echanis is scibed in es of a cleaance ype non-lineaiy and he effecive paaees as scibed in [3] could be dcly incopoaed ino he appoxiae ol. Given equaions (-6) and paaees of Table 5, he nonlinea appoxiae ol of Fig. 6 is ived as follows, whee x () is he (absolue) displaceen of he ineia ack fluid ass, and x () is he elaive displaceen beween he effecive asses and. The absolue displaceen of is heefoe x () + x (). Table 5 Physical and effecive paaees of he appoxiae ol of Fig. 6 Paaees Physical value Effecive value Ineia ack fluid ass Decouple ass Ineia ack daping Decouple daping Ineia ack displaceen Decouple displaceen Decouple gap lengh Top chabe fluid siffness Boo chabe fluid siffness b i = Ai I i = A I i d = Ad I d = A I d = A i = Ai Ri i b d = Ad Rd b = A Rd qi d xi = A qd d xd = A l g i d b R qi d x = A x l k k qd d = A = l g Ad A = A / C = A / C The effecive (bu viual) dynaic foce a he diving poin of Fig. 6 is ived as: [ x( ) x x ( )] F = ɺ x + k x( ) + b xɺ + k (7) In he coupled sae, he couple gap is open such ha 0 < x () < l. Fluid flows ainly hough he couple gap since R i >> R d (o b >> b ). Consequenly, q i ( ) << q d (), o xɺ << xɺ and ɺɺ x << ɺɺ x. Fluid equaions (-4) ae ewien in es of he echanical syse sybols of Table 5 as follows: [ ] [ ] ɺɺ x + bxɺ + k x + x x, + k x + x = 0 coupled sae ( 0 < x < l ) (8a) Since ɺɺ x ɺɺ x + ɺɺ x, Eq. (8a) is appoxiaed by Eq. (8b), which dicaes he dynaics of duing he coupled sae, when a elaive oion xɺ exiss beween and. [ ɺɺ ɺɺ ] ɺ x + x + b x + k [ x + x], + k [ x + x x] = 0 coupled sae ( 0 < x < l ) (8b) Eqs. (4-5) also lead o he syse equaion as follows: ɺɺ x + b xɺ = ɺɺ x + b xɺ, coupled sae ( 0 < x < l ) (9a) Ove he lowe fequency an, say up o 50 Hz, assue ha he couple viscous foce is doinan so ha b xɺ >> ɺɺ x. Eq. (9a) is hen appoxiaed by Eq. (9b) coesponding o he govening equaion of duing he coupled sae. ɺɺ x + b xɺ b xɺ = 0, coupled sae ( 0 < x < l ) (9b) Duing he coupled sae, he couple gap is closed so ha x = 0 o x = l. No fluid flows hough he couple gap, i.e. q d () = 0 o x ɺ = 0 and ɺɺ x = 0. This ipls no elaive oion exiss beween and. Eqs (, 3, 5) ae conveed ino: ɺɺ x + b xɺ + k [ x x] + k x = 0, coupled sae (x = 0 o l ) (0a) Since >>, Eq. (0a) is appoxiaed by Eq. (0b), which dicaes he dynaic oion of cobined asses + while oving ohe afe eaching eihe he op o boo sop. ( ) + ɺɺ x + b xɺ + k [ x x] + k x = 0, coupled sae (x = 0 o l ) (0b) Finally, fo boh saes, F T () is conveed fo Eq. (6) o he following: F = k x + b xɺ + k x x x () T [ ]

7 The swiching echanis beween he coupled and coupled saes is conveed ino a cleaance ype nonlineaiy in he appoxiae ol of Fig. 6: When he exciaion apliu of x() is sall enough so ha avels wihou eaching he op o boo sop ( 0 < x < l ), is coupled fo he oun syse. Thus he syse is in a sof sae dicaed by Eqs. (8b) and (9b) ylding educed siffness and daping. Un highe exciaion aplius, when eaches he op o boo sop (x = 0 o l ), oves along wih and he syse is scibed by Eq. (0b). The ineia ack is hus coupled ino he syse poviding high fluid esisance b. Noe ha ou quasilinea ol [3] is essenially a liiing case of his when b (o R d ) appoaches infiniy. The copessive conac foce F c () beween and is foulaed as: ɺɺ x + F F, x = l Fc = ɺɺ x F + F, x = 0 whee F k [ x x ] (a) = and F = kx ae he effecive elasic foces analogous o he op and boo chabe pessue foces p ( ) A and p ( ) A. Since he couple is ypically signed like a flow conol valve, is ineial foce ɺɺ x is negligible copaed wih he elasic foce. Hence, he swiching condiion fo he coupled o he coupled sae is conveed fo Eq. (a) o he following: F F < 0, x = l F F > 0, x = 0 (b) COMPARISON BETWEEN TWO NONLINEAR MODELS The non-linea appoxiae ol is nex copaed wih a validaed non-linea fluid ol [5,6] in boh ie and fequency doains. Assuing a sinusoidal displaceen x() wih f = 5 Hz and X = (p-p), Fig. 7(a) pics he dynaic chabe pessues p () and p () ha ae pediced by he fluid ol. And, Fig. 7(b) shows he conac foce F () and F () wavefos as pediced by he appoxiae ol. Obseve ha pediced wavefos shae essenially he sae chaaceisics. The spikes in p () o F () coespond o he ie insans when he couple is closed in he fluid syse o when is coupled wih in he appoxiae syse; he fla egions in p () o F () ae associaed wih he opeaion condiions such ha he couple is open o when loses conac wih in he coupled sae. The p () and F () wavefos ae copaaively soohe due o he fac ha C >> C in he fluid syse, o k >> k in he appoxiae ol. Fig. 7 Dynaic esponses given sinusoidal displaceen exciaion wih X =.0 and f = 5 Hz: (a) p () and p () pediced by he fluid ol; (b) elasic conac foces F () and F () pediced by he appoxiae ol. Key:, p () o F ();, p () o F (). In Fig. 8, siulaed dynaic siffness daa ae copaed. They ae fo he fluid and appoxiae ols un haonic exciaions wih aplius anging fo 0.3 o 3 (p-p) in up o 50 Hz. Nealy inical dynaic siffness speca ae obained, which confis ha he appoxiae ol wih cleaance non-lineaiy ined capues he couple swiching echanis of he fluid ol in Fig. 5.

8 ( + ) ɺɺ x + b [ xɺ xɺ ] + k [ x x ] A p = F sin ( π f + φ ) e e e (3) The coninuiy equaion fo he boo chabe eains he sae as Eq. (3), and he coninuiy equaion fo he op chabe now chans o: [ ] A xɺ xɺ q q = C ( p ) pɺ (4) i d Govening equaions fo he couple and ineia ack dynaics eain he sae as Eqs. (4) and (5), especively, while he ansied foce is now: [ ] [ ] F = k x x + b xɺ xɺ + A p (5) T Dynaics of he fon body is dicaed by: ( ) + [ ( ) ( )] + [ ɺ ( ) ɺ ( )] ( ) 3 [ ( ) 3( )] 4 [ ( ) 4( )] [ ɺ ( ) ɺ ( )] [ ɺ ( ) ɺ ( )] 0 ɺɺ x k x x b x x + A p + k x x + k x x + c x x + c x x = (6) The es of he govening equaions no dcly involved wih he hydaulic oun sub-syse could be easily ived. Fig. 8 Dynaic siffness pediced using (a) fluid ol wih fee couple; (b) appoxiae ol wih cleaance non-lineaiy. Key:, X = 0.3 (p-p);, X = 0.5 (p-p);, X = (p-p);, X = (p-p);, X = 3 (p-p). INCOPORATION OF NON-LINEAR MODELS INTO VEHICLE SYSTEM Nex, non-linea hydaulic oun ols ae incopoaed ino he vehicle syse and copaed wih linea and quasi-linea foulaions. Key govening equaions of he non-linea vehicle syse wih fluid ype oun ae given as follows; copaable vehicle foulaions including he non-linea appoxiae oun could also be ived by cobining Eqs. (8) o () wih ohe vehicle coponens in a siila anne. Wih efeence o Figs. (a) and (d), whee he posiive dcions of x i () ae aken o be upwad, he engine dynaics is scibed as: Vehicle esponses ae calculaed in boh ie and fequency doains given engine foce exciaions. Since highe haonics ae no easily disinguished fo he ie doain esponses, only he fequency speca ae pesened hee in o o copae linea and quasilinea vehicle ols. Figs. 9-0 copae esuls of vehicle ass displaceens o acceleaions given sinusoidal engine foce wih he apliu of 000 N a 0 Hz. In Fig. 9, he engine esponse is shown o be os sensiive o he ouning syse, and he engine bounce esonance a 8.4 Hz (wih ubbe oun) is educed up o 40 db by using he hydaulic oun. The vehicle pich esonance aound 5 Hz is also affeced due o a coupling beween hydaulic oun and ohe vehicle coponens. Given he sae sinusoidal exciaion wih apliu of 000 N a 0 Hz, Fig. 0 copaes he dynaic vehicle acceleaions pediced by using he quasi-linea foulaion and by using he non-linea fluid ol (wih fixed couple), especively. Obseve ha: (i) The slopes of he acceleaion speca pediced by boh ols ach well wih each ohe. (ii) Boh ols capue he basic vehicle esonances which wee pediced by he odal analysis. Recall Table 4 fo he naual fequencs and esiaed odal daping aios coesponding o hese esonances. (iii) The wheel hop esonance aound Hz is doinaing he dynaic esponses, and is supe-haonics a he uliples of he fundaenal fequency ae pediced only by he nonlinea fluid (o appoxiae) ol. This confis he necessiy of incopoaing non-linea hydaulic oun foulaion ino he vehicle syse fo exaining nonlinea phenoena such as he supe-haonics.

9 Fig. 9 Pediced vehicle displaceen speca (db e ) given sinusoidal engine foce exciaion wih an apliu of 000 N a 0 Hz. Key:, linea ol wih ubbe oun;, quasi-linea ol wih hydaulic oun. Fig. 0 Pediced vehicle acceleaion speca (db e /s ) given sinusoidal engine foce exciaion wih an apliu of 000 N a 0 Hz. Key:, quasi-linea ol wih hydaulic oun;, non-linea fluid ol wih fixed couple.

10 CONCLUSION This pape has copaaively evaluaed linea, quasilinea and ue non-linea (appoxiae and fluid ype) ols of hydaulic engine ouns and exained hei dynaic effecs in a siplifd half-vehicle syse. A non-linea appoxiae ol is veloped o povi ipoved insigh ino he coupling effecs. Hydaulic oun is shown o be highly efficn in poviding high daping and in uning vehicle dynaics. Poposed vehicle ols could be effecively uilized o exaine vehicle esponses in boh ie and fequency doains. Ou vehicle ols (in alenae fos) could seve as he plafo fo fuhe eseach ino ipoved esiaions of daping and siffness ols, bee pedicions in ie doain as well as efineens of he vehicle sign and inegaions issues. The fundaenal quesion of ipoing easued daa (given displaceen exciaion and blocked bounday in he elasoe es ehod) and incopoaing he in a syse wih ealisic boundas and foce exciaions is ye o be esolved. We ae exaining his issue and will povi soe guidance in fuue papes. ACKNOWLEDGMENTS We acknowled he expeienal effos of M. Tiwai and J. Soenson fo 999 o 00. Those suds wee financially suppoed by he Fod Moo Copany. We hank D. J. H. Lee fo his coens and sugsions. REFERENCES. R. Singh, G. Ki and P. V. Ravinda, Linea analysis of auooive hydo-echanical oun wih ephasis on couple chaaceisics, J. Sound Vib. 58, 9-43 (99).. G. Ki and R. Singh, Sudy of Passive and Adapive Hydaulic Engine Moun Syses wih Ephasis on Nonlinea Chaaceisics, J. Sound Vib. 79, (995). 3. S. He and R. Singh, Esiaion of Apliu and Fequency Depenn Paaees of Hydaulic Engine Moun Given Liied Dynaic Siffness Measueens, Noise Conol Eng. J., 53(6), 7-85, (005). 4. J. E. Colgae, C. T. Chang, Y. C. Chiou, W. K. Liu and L. M. Kee, Moling of a Hydaulic Engine Moun Focusing on Response o Sinusoidal and Coposie Exciaions, J. Sound Vib., 84(3), (995). 5. M. Tiwai, H. Adiguna and R. Singh, Expeienal Chaaceizaion of a Nonlinea Hydaulic Engine Moun, Noise Conol Eng. J. 5(), (003). 6. H. Adiguna, M. Tiwai and R. Singh, Tansn Response of a Hydaulic Engine Moun, J. Sound Vib. 68, 7-48 (003). 7. Y. Yu, N. G. Naganahan and R. V. Dukkipai, A Lieaue Revw of Auooive Vehicle Engine Mouning Syses, J. Dynaic Sys, Measueen, and Conol, 3(), (00). 8. J. H. Lee, M. S. Bae and K. J. Ki, Liiaions of Mechanical Mol Wih Luped Mass in Repesening Dynaic Chaaceisics of Hydaulic Moun, SAE Pape (003). 9. N. Tsujiuchi, T. Koizui and K. Yaazaki. Vibaion Analysis of Engine Suppoed by Hydaulic Mouns, SAE Pape (003). 0. T. Jeong and R. Singh, Inclusion of Measued Fequency-and Apliu-Depenn Moun Popes in Vehicle o Machiney Mols, J. Sound Vib. 45, (00).. S. B. Choi and H. J. Song, Vibaion Conol of a Passen Vehicle Uilizing a Sei-Acive ER Engine Moun, J. Vehicle Sys. Dynaics, 37(3), 93-6, (00). CONTACT Pofesso Rajenda Singh Acousics and Dynaics Laboaoy Cene fo Auooive Reseach and NSF I/UCRC Sa Vehicle Conceps Cene The Ohio Sae Univesiy Eail: singh.3@osu.edu Phone: Websie:

ÖRNEK 1: THE LINEAR IMPULSE-MOMENTUM RELATION Calculate the linear momentum of a particle of mass m=10 kg which has a. kg m s

ÖRNEK 1: THE LINEAR IMPULSE-MOMENTUM RELATION Calculate the linear momentum of a particle of mass m=10 kg which has a. kg m s MÜHENDİSLİK MEKANİĞİ. HAFTA İMPULS- MMENTUM-ÇARPIŞMA Linea oenu of a paicle: The sybol L denoes he linea oenu and is defined as he ass ies he elociy of a paicle. L ÖRNEK : THE LINEAR IMPULSE-MMENTUM RELATIN