Vibration Analysis of Powertrain Mounting System with a Combination of Active and Passive Isolators with Spectrally-varying Properties

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1 Copigh 9 SE Innaional Vibaion nali of Powain Mouning Sm wih a Combinaion of civ and Paiv Iolao wih Spcall-vaing Popi Ja-Yol Pa and Rajnda Singh h Ohio Sa Univi BSRC Mo of h pio wo on aciv mouning m ha bn conducd in h con of a ingl dg-offdom vn hough h vhicl powain i a i dg-of-fdom iolaion m. W o ovcom hi dficinc b popoing a nw i dgof-fdom analical modl of h powain m wih a combinaion of aciv and paiv moun. ll iffn and damping lmn conain pcallvaing popi and w amin powain moion whn cid b an ocillaing oqu. wo mhod a dvlopd ha dcib h moun lmn via a anf funcion (in Laplac domain. Nw analical fomulaion a vifid b compaing h fqunc pon wih numical ul obaind b h dic invion mhod (bad on Voig p moun modl. Eignoluion of a pcall vaing mouning m a alo pdicd b nw modl. Compl ignvalu poblm fomulaion wih pcall-vaing popi povid a clo mach wih pimnal ul han h al ignvalu fomulaion wih fquncindpndn moun. Givn h pcal vaianc in h moun popi a impl oll mod dcoupling chm i uggd fo h powain iolaion m. hn h ol of aciv pah i claifid b compaion wih no acuao opaion. Muli-dimnional moion (pciall coupling a pdicd and in paicula h ffc of aciv moun paam uch a h oinaion angl locaion and acuao inpu a invigad fom h moion coupling ppciv. INRODUCION Pio anal of igid bod iolaion m including vibaion anmiibili naual fqunc placmn and moion dcoupling udi [1 7] aum ha h iffn ( and vicou damping (c popi a conan. Howv al-lif moun lmn inhnl (and om vn b dign hibi conidabl fqunc- and ampliud-dpndnc p = ( ω and c= c( ω wh ω i h angula fqunc (ad/ and i h ampliud of ciaion [8 11]. h igid bod mouning m a picall modld in m of a i dg of fdom (6-DOF inial bod ha i uppod b i-aial iolaion lmn a 3 o 4 locaion along wih a igid foundaion [1 7]. Yu Naganahan and Duipai [1] H and Singh [13] and Jong and Singh [14] hav uggd ha i i nca o incopoa h pcall-vaing popi in h ngin iolaion m modl. Sval aciv o ma ngin mouning dvic [15 19] hav bn dignd o duc noi and vibaion pciall und high dnamic oqu condiion. pical dign ciia [1 7] includ dcoupling of powain moion and moion conol. Moion conol i achivd hough ducion in onan pa naual fqunc placmn ducd vibaion anmiibili and incad acouic comfo. Evn hough h powain (igid bod i a 6 dg-of-fdom (DOF iolaion m mo of analical wo ha bn conducd in h con of ingl-dg-of-fdom m [15 19]. In val ca high damping oluion hav bn implmnd o conol onanc( a low fqunci and hn mo complian moun hav bn ough a high fqunci o minimi h foc anmid in h h Engining Ming Boad ha appovd hi pap fo publicaion. I ha uccfull compld SE p viw poc und h upviion of h ion ogani. hi poc qui a minimum of h (3 viw b indu p. ll igh vd. No pa of hi publicaion ma b poducd od in a ival m o anmid in an fom o b an man lconic mchanical phoocoping coding o ohwi wihou h pio win pmiion of SE. ISSN Poiion and opinion advancd in hi pap a ho of h auho( and no ncail ho of SE. h auho i oll ponibl fo h conn of h pap. SE Cuom Svic: l: (inid US and Canada l: (ouid US Fa: CuomSvic@a.og SE Wb dd: hp:// Pind in US

2 ba; hi impli an inoducion of fquncdpndn iolao hough paiv adapiv o aciv man. h abov-mniond dign ma no aif h powain moion dcoupling conidaion and hu h lcion of an aciv moun in h con of a 6- DOF m main an mpiical cinc. W o ovcom hi dficinc b popoing a nw analical modl ha will amin a combinaion of aciv and paiv moun. h liau on muli-dg-of-fdom aciv iolaion m i pa. Fo ampl Gadonio Ellio and Pinningon [] Kim and L [1] and Roon and Singh [] hav limid hi anal o 3-DOF m (.g. anv-aial-pich moion o pich-oll-bounc moion. lo pio ach hav ampd o minimi h foc anmid ino h igid o complian foundaion wihou coniding h mulidimnional moion conol and coupling iu. In paicula Gadonio Ellio and Pinningon [] and Kim and L [1] hav uggd ha paiv and aciv moun paam hould b popl lcd pio o h iolaion conol poblm pciall a h low fqunci (a up o 5 H. hi aicl nd h pio muli-dg-of-fdom aciv iolao wo [ ] b focuing on h ignoluion coupling dnamic and moion conol iu. PROBLEM FORMULION Figu 1 illua a pical 6-DOF igid bod iolaion m compod of an inial bod (ngin and anmiion a igid ba (chai and h o fou lina im invaian aciv and paiv moun uch ha ach iolaion lmn could b abiail placd a an io poin and oind in an dicion. h dnamic chaaciic of iolao a gnall pnd in m of h compl-valud co poin dnamic iffn K(j ω fom non-onan dnamic [8] wh j i h imagina uni. Figu how ampl maumn of ( ω and c( ω fo wo ampl ca (ubb and hdaulic ngin moun. W could mbd h o imila i ( ω and ci ( ω popi in on o mo moun lmn of Fig. 1. h govning quaion of h mouning m in fqunc domain (ω a a follow wh q ( ω i h dnamic diplacmn vco and f ( ω i h nal ciaion (foc/oqu vco [ ω M+ j ωc( ω + K( ω] q( ω = f ( ω (1 H M i h inial (ma mai and K ( ω and C ( ω a h iffn and vicou damping maic. Fo an innal combuion powain m h main ciaion f ( com fom h pulaing oqu ha i gnad b muli-clind ngin and i could b ih piodic (und ad a o anin (und a-up o wiching condiion. Fqunc pon could b numicall calculad b h dic invion mhod (dignad in hi aicl a Mhod I wh w could impl u diffn and c valu a ach fqunc (niall a loo-up abl chm. Howv Mhod I in Eq. (1 canno dicl lad o h analical modal panion fo pon pdicion and moion coupling anal. In od o ovcom hi limiaion Jong and Singh [14] had fomulad val pcal ignvalu poblm and hn uggd modal uppoiion pocdu. Howv hi mhod i valid fo a pcial cla of poblm (wh h dnamic iffn i givn in a pcific fom and hi appoimaion canno b ndd o a mo gnal pcall-vaing vibaing m. f ( KP( K( ( = K ( K ( 4 Powain Ocillaing oqu ( CG q( K ( diplacd b q( Fig. 1 Muli-dg-of-fdom powain iolaion m wih on aciv moun and h paiv moun. Each moun i dcibd b i-aial lmn wih fqunc dpndn iffn ( ω and damping c( ω popi. H Ki ( i = 3 and 4 ( = n: numb of moun and K( a h dnamic iffn m of paiv and aciv lmn (in a pcific dicion pcivl. (a (b Fig. Moun ampl wih pcall-vaing iffn and vicou damping popi. Maud daa fo ubb and hdaulic moun a compad pcivl wih anf funcion (F modl givn b Eq. (. (a Siffn pca ( ω = R[ K(j ω] ; (b Vicou damping pca c( ω = Im[ K(j ω]/ ω. K: K ( 3

3 maud daa fo a hdaulic moun wih an ciaion ampliud of = 1.5 mm ; maud daa fo a ubb moun; cond od F fo h hdaulic moun; oh od F fo h ubb moun ( = 8 N mm c = 3 N m. h co-poin dnamic iffn (in Laplac domain of a pical ngin moun a a cain ciaion ampliud und a pcific man load could b givn b h following pion [8]: b 3 αb + L + α3 + α + α1+ α Ki ( =. ( a βa + L + β + β1+ β H a and b a h od of dnominao and numao pcivl and h cofficin α and β a dmind b pimnal o numical (analical mhod flcing h innal fluid ucu and ubb maial popi. Obv ha Eq. ( mach wll wih h pimnal ul of Fig. fo hdaulic (wih a = b = 3 and ubb (wih a = b = 1 moun. Emplomn of Eq. ( o h powain moun in Fig. 1 i dignad h a Mhod II. f vificaion b compaion wih pimnal and numical ul hi fomulaion will b ud h fo vibaion and dcoupling anali of powain mouning m wih combinaion of aciv and paiv moun wih pcall-vaing popi. civ and paiv moun modl ha a dpicd in Fig. 3. Effciv conol of powain moion i nial ov h low fqunc ang up o 5 H inc h igid bod mod ignificanl domina and could coupl wih oh vhicl m mod [ 1]. cuao diplacmn in aciv moun i adjud inuoidall [3 4] b a diplacmn acuao j( ω + φ ( = wh i aciv diplacmn ampliud ω angula aciv diplacmn inpu fqunc and φ pha angl wih pc o h nal oqu ciaion. Conidaion of conan and φ (a a im would allow u o ma h aciv moun a analicall acabl in h con of a mulidg-of-fdom iolaion m. Ou modl will analicall amin h paam of aciv and paiv moun (uch a iffn damping and aciv inpu hi locaion and oinaion angl. nali i limid o onl h ngin oqu ciaion. Conid a 6-DOF iolaion m coniing of a igid bod (wih powain ma m and inia I i and j = und an ocillaing oqu ( ( and 4 i-aial moun which a aumd o b aachd o a igid ba. Ou of h on i an aciv moun (givn b ubcip wih dnamic iffn K( = f [ KP ( K( ( ( ] in a pcific dicion wh i h Laplac vaiabl ( i h powain diplacmn and ( i h acuao diplacmn. h oh h a paiv dvic wih dnamic ij iffn Ki ( i = 3 4 in a pcific dicion. Each moun lmn (in an dicion i aumd o hav fqunc-dpndn iffn ( ω and vicou damping c( ω popi. Govning quaion in mai fom a a follow wh ( impli h Laplac domain q ( = [ θ ]( θ θ i h diplacmn vco and f ( i h nal foc vco (pimail h oqu ( ciaion [ M+ K(]( q = f ( (3 wh Μ i h ma mai (powain ma and inia and K ( i h iffn mai ha includ h anf funcion (dnamic iffn modl of aciv and paiv moun. Paiv moun a modld a hown in Fig. 3(a in m of h co poin iffn [8 13] K( = F(/ (. H K ( could b ih analicall availabl fom a fluid moun modl [8] o pimnall maud b a non-onanc moun [13 5]. hi p of anf funcion modl i valid in h low fqunc ang (a up o 5 H. F ( (a K ( ( K ( P F ( (b ( K ( Fig. 3 Paiv and aciv ngin moun givn diplacmn inpu (. (a Paiv moun; (b civ moun wih pion diplacmn p inpu ( (. In boh ca foc anmid o h igid ba i F (. Siffn m a givn in h Laplac domain. h chif objciv of hi aicl i o inviga h compl ignoluion and fqunc pon of h powain m of Fig. 1 whn cid b an ocillaing oqu wih on aciv moun and 3 paiv moun. h aciv moun modl of Fig. 3(b i fi fomulad fo fluid pion p aciv dvic uch a a hdaulic ngin moun [1 3-5]. Ou mhod will b validad b compaing analical pdicion in fqunc domain wih h dic invion (numical mhod wh w could impl u diffn and c valu a ach fqunc. h cond objciv i o amin muli-dicional moion conol and vibaion iolaion iu in h con of a muli-dg-offdom mouning m. Gnnau [6] amind fou aciv iolaion chm and concludd ha an acuao ucu in (

4 paalll wih a ubb lmn i h mo pfd dign a h aciv acuao hould b dignd o gna onl h dnamic foc and h aic foc hould b povidd b h ubb lmn. Bad on hi concp a nw analical modl fo aciv moun wih acuao diplacmn inpu i popod in Fig. 3(b. Sinc hi wo i limid o h low fqunc gim h co poin anf funcion ( K ( concp i alo applid o pn h dnamic pop of an aciv moun. In hi modl h foc anmid ino h igid ba ( F ( coni of a paiv foc ( FP ( and an aciv foc ( F( : F( = FP( + F(. (4 h individual anf funcion KP ( and K( of h paiv (pima and aciv (conda pah a dfind b h following: FP ( F( KP ( = and K( ( = (. (5a b H ( i h powain diplacmn pincipal dicion componn of h aciv moun and ( i h acuao diplacmn in Laplac domain. Fo a ingl-dg-of-fdom iolaion m of Fig. 4 h govning quaion i: [ m + KP (] ( = F( K( (. (6 H h aciv foc F( = K( ( can b viwd a an addiional nal ciaion inc i ac indpndn of (. hfo h ignoluion and paiv dnamic a govnd b h paiv pah ( KP (. Whil an adapiv moun i uuall dignd o hav a la wo paiv anf funcion dpnding on dignad opaing condiion [5] onl on paiv anf funcion i aignd o an aciv moun. hfo daild dnamic anali of h paiv lmn in an aciv mouning m i cucial. K ( P F ( m ( K ( ( Fig. 4 Singl-dg-of-fdom m wih an aciv moun. wo aciv moun concp wh h acuao lmn i in paalll wih h paiv ubb lmn will b amind. Fi conid an aciv hdaulic dvic a hown in Fig. 5. h aciv moun coni of h conol volum (upp and low chamb a dignad b #1 and # pcivl and h inia ac i pnd b #i. h momnum quaion fo ubb ma m i: m && ( = F ( ( c & ( p 1(. (7 h coninui quaion fo h low and upp chamb a: & ( qi( = Cp& 1 1( + & ( (8 qi ( = Cp& (. (9 h momnum quaion fo h inia ac i: p1( p( = Iiq& i( + Rq i i(. (1 Sinc F( = F( ( = FP( + F( in h low fqunc ang [8] h anf funcion fo hi aciv moun a divd fom Eq. (7 - (1 a follow: FP ( α + α1+ α KP ( = = m + c+ + ( β + β1r i + β F( α + α1+ α K( = = ( β + β1r i + β (11a b α = CI i α 1 = CR i α = (11c- β = ICC i 1 β 1 = CC 1 Ri β = C1 + C. (11f-h pion dcoupl ( acuao m C 1 p 1 ( ba ubb F( #1:upp chamb C p ( R i I i q i ( inia ac: #i #: low chamb c / F ( / ( Fig. 5 Schmaic of an aciv ngin moun bad on hdaulic moun. H C 1 and C a h complianc of upp and low chamb p ( and 1 p ( a h chamb pu Ri I i and qi ( a h fluid ianc inanc and flow a in h inia ac i h quivaln pion aa of ubb i h acuao pion aa and and c a h ubb iffn and vicou damping m (Voig modl aumd.

5 MULI-DEGREE-OF-FREEDOM ISOLION SYSEM WIH CIVE ND PSSIVE MOUNS W conid h powain iolaion m (Fig. 1 wih on o wo aciv moun and wo o h ubb moun. h following h coodina m a ud: inial fnc coodina (YZ g fid a h gound wih i oigin a h aic quilibium (a h cn of gavi CG along wih local moun coodina (YZ m i which a paalll o (YZ g and pincipal moun coodina (YZ mp i who pincipal a a no paalll o (YZ g wh ubcip i ( = 1 L n i h moun ind and n i h numb of moun. Paiv ubb moun fomulad b Ki( = i + c i a dcibd b h i-aial ping and vicou (o ucual damping lmn; h iffn valu a aumd o b conan and inniiv o h ciaion ampliud. Convl aciv moun a dcibd b KP ( and K( a dvlopd in h pviou cion. Onl h oqu ciaion i conidd in hi aicl vn hough an ciaion foc can b applid o h igid powain. h diplacmn of h im-invaian inial bod (of dimnion i a aumd o b mall and h diplacmn vco q ( = [ θ θ θ] ( i pd b h anlaional and angula diplacmn of h cn of gavi (CG. h govning quaion a fomulad in mai fom a hown blow wh q& ( and q&& ( a h vloci and acclaion vco pcivl: Mq&& ( + Cq& ( + Kq( = f( + f (. (1 H M i inial (ma mai K i h iffn mai C i h vicou damping mai and f ( i h nal ciaion (foc/oqu vco. H f ( i h acion foc gnad b h aciv moun; wi i a f ( = f P ( + f ( wh f P ( and f ( a h foc fom h paiv and aciv pah pcivl. Equaion (1 bcom: Mq&& ( + Cq& ( + Kq( = f( + fp ( + f (. (13 h acion foc f P ( and f ( a h um of foc gnad b ach aciv moun lmn and h a pd a follow wh N i h numb of aciv moun: N N fp ( = f P ( and f ( = f (. (14 = 1 = 1 Uili h anf funcion modl of Eq. (5 o pn h paiv and aciv pah of N aciv moun a follow: FP ( L fp ( KP ( = = = 1 L N (15a P ( L P ( F ( L f ( K ( = = = 1 L N. (15b ( L ( H L i h Laplac anfom. No ha fp ( and f ( a paiv and aciv pah foc pcivl in a pcific dicion of an aciv moun componn whil P ( and ( a inial bod diplacmn in h aciv moun oinaion dicion and aciv inpu diplacmn in im domain. h local moun acion foc fp ( and f ( a pnd in h global (YZ g coodina in m of h moun paam and hi oinaion angl and locaion; h inial bod diplacmn P ( i found bad on h inmaic of iolaion m. h uling dflcion q mi ( a ach moun i a follow bad on h igid foundaion aumpion: qmi ( = [ I Lmi ] q ( (16 i i L mi = i (17 w m. Uing h Eul angl a givn b ( θi ϕi φ i fo i-h moun h oaional mai Θ g mi i found b oaing abou (YZ g a in h qunc of Y and Z [3]. Racion foc in h i-h moun in h global coodina m i obaind b a anfomaion fom h local moun coodina and h uling acion foc i: fgm i ( fm i ( I fgm i( = = = mi ( gm iθ ( mi m i ( f. (18 f f Lmi f ( = Θ f ( Eq. (18 bcom: Sinc mi gm i mpi I fgmi( = Θgmifmp i (. (19 Lmi Bad on h fac ha h anmid (oupu acion foc fp ( and f ( hough aciv moun a dcibd in h (YZ mpi coodina a f P mp i ( = fp ( and f ( mp i = f ( P ( f ( o h global coodina m a pd uing Eq. (19 a follow: fp ( I fp ( = fp gmi ( = gmi Θ Lmi (a f ( I f ( = f gmi ( = gmi Θ Lmi (b wh f ( = L K ( ( = 1 L N. long wih q ( = Θ q ( h diplacmn of h mpi θ gmi θ

6 i-h moun wih pc o h pincipal moun coodina (YZ mpi i pd a follow: qmp i ( Ι Lmi qmpi( = = gmi ( mp iθ ( Θ q. (1 q I Sinc h inial diplacmn P ( o an aciv moun fom igid bod moion ( q ( i o b on of h pincipal dicion of i-h moun componn P ( i obaind b finding a coponding vco lmn in q mp i ( of Eq. (1 a follow: P ( = q mp i ν ( ν = o o. ( h uling inial diplacmn in h dicion of h aciv moun componn i now compll dcibd in m of h oinaion angl i locaion and igid bod moion wihou inoducing an addiional vaiabl fo ilf. W appl h inv Laplac anfomaion o conv Eq. (15 o im domain fomulaion and obain h quaion a: a( = b P ( (3 wh b auming ha a pical anf funcion ( KP ( i aumd o b pnd a KP ( = ( α + α1 + α /( β + β1 + β bad on h fac ha an aciv hdaulic moun i modld in Eq. (11 whn m and c a ngligibl in low fqunc ang [8] a( = a1 L an ( a( = L ( α + α1 + α P ( = 1 L N (4a bp ( = bp 1 L bp N ( bp ( = L ( β + β1 + β FP ( = 1 L N. (4b No ha ( in Eq. (4 i: P Ι L = Θ Q mi P ( gmi ( I. (5 Fo an ammic mouning m Eq. (13 and (3 a pandd uing h powain m inmaic dvlopd abov and h govning quaion wih aciv and paiv moun a pnd in an ndd fom a follow: m &&( + cmq& ( + mq( = ( f + ( f + f ( fp1 L f P N (6a m &&( + cmq& ( + mq( = ( f + ( f + f ( fp1 L f P N (6b m &&( + cmq& ( + mq( = ( f + ( f + f ( fp1 L f P N (6c I && θ ( + I && θ ( + I && θ ( + cm θ q& ( + ( mθ q (6d = ( f + ( f + M ( f L f θ θ θ P1 P N I && θ ( + I && θ ( + I && θ ( + cm θ q& ( + ( mθ q = ( f θ + ( f ( 1 f θ + M θ f P L P N (6 I && θ ( + I && θ ( + I && θ ( + cm θ q& ( + ( mθ q = ( f + ( f + M ( f L f (6f θ θ θ P1 P N ( b fd ( L fd ( ( ( = b ( fp1 L fp1 L fp N L f P N h( = 1 L N. (6g Combining Eq. (6a g and auming o iniial condiion in h inv Laplac anfom in Eq. (6g w ambl h following ndd govning quaion (in mai fom fo h mouning m wih aciv moun: Mq&& ( + Cq& ( + Kq ( = f(. (7 H M C and K a ndd m maic a dfind blow: M M = M C C1 P C = and P C P C3 P K K1 P K =. K P K3 P Endd ciaion and diplacmn vco a f( = f ( + f( σ P( = f ( + f( σp1 ( L σp N ( q( = q ( f P( wh fp ( = fp1 ( L fp N (. Sinc h acion foc fp ( in aciv moun dpnd on h inial diplacmn P ( h a mbddd a addiional lmn in h ndd diplacmn vco q (. h mainl ac a addiional foc lmn in boh pandd diplacmn and nal foc vco. Obv ha maic M C and K a pcall-invaian vn hough h a no mmic du o an amm in h pandd fomulaion. W mplo hi aciv iolaion m modl fo fuh anal. o appl h compl modal mhod o a nonconvaiv dic m Eq. (7 i ca in h a-pac fi od m fom [3] a: p& ( + Bp( = g( (8 wh h a vco p ( and ciaion vco g ( a dfind a: q& ( f p( = ( ( g( = q (9a b and m maic and B a dfind a: M = K C K B = K. (3a b

7 h compl ignvalu poblm aociad wih Eq. (8 i λ U + BU = (31 wh λ C ( = 1 3 L ( N + N i h -h compl-valud ignvalu (includ boh al and imagina pa du o vicou damping and U i h -h a-pac compl-valud ignvco. h compl ignvco U a U [ ] = λu u wh u i h configuaion (phical pac ignvco ha aifi h following ignvalu poblm: [ λ M + λ C + K] u =. o dvlop an panion hom fo ammic ignm (non-lf-adjoin dic m an addiional ignvalu poblm fo h adjoin ignm mu b dfind a: λ V + BV = (3 in which V i h -h ignvco of h adjoin m (in a pac ha i in h fom of [ ] V = λv v. Bad on h bi-ohogonal pop: VU = δ VBU = λδ = 1 3 L ( N+ N wh δ i h Kon dla funcion h modal panion hom i now applicabl o ou aciv powain mouning m. uming follow: jω g( = G h hamonic pon i a jω p( = U (j ωi Λ VG (33 wh U= u1 u L u ( N+ N 1 u( N+ N V= v1 v L v ( N+ N 1 v( N+ N ( λ1 λ λ( 1 λ + ( + Λ = diag L N N N N. h fqunc pon funcion givn hamonic oqu ciaion (wih uni ampliud a calculad uing h modal panion hom and compad wih ho compud uing mhod I. RESULS ND DISSCUSSION In h focalid mouning m a hown in Fig. 6 an inial coodina m i chon o b h am a h pincipal coodina m and laic cn li on on of h pincipal a a h ai. Ocillaing oqu i aumd o b in h dicion. I i h mo did ca fo h mouning m in m of laic ai focaliaion o oqu oll ai dcoupling dign inc i would ild a compl dcoupling givn h oqu ciaion. Bad on h popod m modl and compl ignvalu fomulaion ignoluion fo a focalid aciv mouning m of Fig. 1 and 6 a fi analicall amind givn h following powain paam: Ma m = 1.5 g ; momn of inia (g m 1.65 I = I =.43 I =.54 ; inia YY ZZ poduc (g m IY = IZ = IYZ =. Popi and locaion of h ubb moun a: iffn a = 8 N mm ; iffn a aio L( = a/ b =.5; damping c a = 3 N m ; damping a aio Lc( = ca/ cb =.5; moun oinaion φ = ; moun locaion in h -dicion 1 = = 318 mm 3 = 4 = 318 mm ; moun locaion in h -dicion 1 = 3 = 198 mm = 4 = 198 mm ; and moun locaion in h -dicion 1 = = 3 = 4 = 94 mm. h aciv moun of Fig. 5 i now placd a locaion #1 and paam of Eq. (14 a givn a follow [1] whn m and c a ngligibl [8] ov low fqunc ang (up o 5 H: =17.4 N mm -6-6 =413 1 m d =166 1 m α = α 1 = 13 α = 59 β =.1 1 β 1 = and β = Eignvalu a compad in abl 1 fo paiv and aciv moun. On addiional ignvalu wih a high damping aio i in h aciv mouning m (du o a pol in h paiv pah whil h iolaion m wih pul paiv (and fqunc-indpndn moun ha onl i ignvalu. Obv ha h onan fqunci would diff fom ho obaind whn w mplo paiv ubb moun (wih K ( = + c modl. No ha and θ mod a ignificanl coupld wih θ du o h aciv moun and h coponding onanc how lag chang fom ho fo h paiv mouning m alon. Sinc h ignucu of an aciv iolaion m i dmind b h innal paiv pah ( KP ( boh paiv and aciv pah mu b cafull idnifid bfo an paamic dign udi can b caid ou. Fo h a of illuaion w amin h mod of a V6 dil ngin iolaion m [6]. h inia pop i clo o b mmic wih pc o canhaf ai and fou moun a alo placd in nal mmic locaion. Ral and compl ignoluion a calculad and compad wih maud naual fqunci in Fig. 7. hi anali ugg ha h maud naual fqunc a 1.47 H copond o h ih (and no h fifh mod. B agmn wih maud daa i achivd onl whn h compl ignoluion mhod (wih conidaion of moun damping i applid. hi impli ha h compl ignoluion mhod hould b uilid o anal h al-lif ngin mouning m.

8 18 CG Maud Ral ignoluion Compl ignoluion φ a CG Naual fqunc (H b Fig. 6 Focalid powain mouning m (6-DOF. H a i h pincipal compiv iffn and b i h pincipal ha iffn. abl 1 Compaion of ignvalu fo a powain mouning m (Fig. 6 wih h paiv moun and on aciv iolao (Fig. 5 K( = + c : Paiv K( = K ( : civ moun Dominan mod( ubb moun ω ( H ζ (% Dominan mod( ω ( H ζ (% Moun mod ( θ ( θ ( θ θ K: θ θ θ θ θ θ ( θ ω = naual fqunc; ζ = damping aio. Fig. 8 how fqunc pon fo h anlaion and h oaion; obv ha h analical modl uing Mhod II acl mach wih numical (dic invion mhod ul uing Mhod I. h ffc of j( hamonic aciv diplacmn ( ( = ω + φ on h focalid mouning m moion i amind n. Figu 9 compa h fqunc pon fo h diffn aciv diplacmn inpu givn jω ( = ng wih ng = 1 N m. Obv ha h oll moion ( θ i ignificanl ducd in ovall fqunc ang whn h acuao diplacmn i ou of pha wih h oqu ciaion whil i i amplifid b an inpha acuao inpu. hi indica ha h aciv moun could ac a a oll conol moun vn hough om coupld moion in oh dicion a n. Modal chaaciic do no chang wih aciv foc opaion (a pcd inc Mod Fig. 7 Compaion of calculad (uing boh al and compl ignoluion mhod and maud [6] naual fqunci fo a V6 dil ngin (a Fig. 8 Fqunc pon of an aciv powain mouning m of Fig. 6 givn hamonic oqu wih 1 Nm ampliud. On aciv moun of Fig. 5 i placd a locaion #1 and h a paiv moun. (a ( ω ; (b Y ( ω ; (c Z( ω ; (d θ ( ω ; ( θy ( ω ; (f θz ( ω. K: analical (modal panion mhod; numical (dic invion mhod. h a icl dmind b h paiv pah( of an aciv moun. h ffc of aciv moun oinaion (b (d ( (c (f

9 angl φ a hown in Fig. 6 on h focalid mouning m moion i alo invigad h. Figu 1 compa h fqunc pon in h oll dicion jω fo wo diffn oinaion angl givn ( = ng wih ng = 1 N m. h oll moion ( θ i ignificanl ducd in ovall fqunc ang whn φ = (vical compad o h ca wih φ = 3. hi how ha h oinaion of h aciv moun pla impoan ol in pon ducion b aciv moun. (a (b (c (d ( (f Fig. 1 Effc of oinaion angl on fqunc pon in oll dicion ( θ ( ω of h aciv powain m (Fig. 6 givn hamonic oqu ciaion wih 1 Nm ampliud. On aciv moun of Fig. 5 i placd a locaion #1. K: = mm φ = 3 ; = 1.5mm wih φ = 18 φ = 3 ; = 1.5mm wih φ = 18 φ = (vical. Fig. 9 Effc of aciv diplacmn on fqunc pon of h aciv powain m (Fig. 6 givn hamonic oqu ciaion wih 1 Nm ampliud. On aciv moun of Fig. 5 i placd a locaion # 1 wih an oinaion angl φ =. (a ( ω ; (b Y ( ω ; (c Z( ω ; (d θ ( ω ; ( θ ( ω ; (f θ ( ω. K: Y = mm ; = 1.mm wih φ = ; = 1.5mm wih φ = 18. h oqu oll ai (R could b dcoupld fo a popoionall o non-popoionall dampd m b judicioul lcing moun paam locaion oinaion angl and iffn aio a uggd b Jong and Singh [] and mo cnl Pa and Singh [3]. Evn hough ignifican coupling a plac whn pcall-vaing moun a mplod dcoupling i ill poibl fo a focalid mouning m (wih φ = and = mm Z. Now an aciv moun and a paiv hdaulic moun a placd a locaion #1 and # pcivl fo h focalid m (Fig. 6. o bgin wih aum ha aciv foc i no applid und h oqu ciaion. h R i dcoupld givn φ = and = mm. Moun locaion a illuad in Fig. 11; and Fig. 1 how h uling dcoupld oll mod ( θ ( ω. N appl h aciv foc; h oll mod i now coupld wih Z( ω and θ Y ( ω. hi i pcd inc h conda foc aiing fom h aciv moun inoduc h ciaion in Z( ω θ ( ω and θy ( ω in addiion o h pima ngin oqu. hi ampl clal how ha on hould cafull dign h R mouning chm whil including h ul of conda foc gnad b h aciv moun. CONCLUSION In hi aicl wo mhod fo pning moun wih ( ω and c( ω a ciicall amind in dcibing h ignoluion and fqunc pon of an iolaion m. o ovcom h dficinci of Mhod I (limid o onl h fqunc domain anali Mhod II i dvlopd b mploing anf funcion (in Laplac domain. h analical mhod compa wll wih h dic invion mhod in pdicing h fqunc pon. wo majo conibuion mg. Fi a nw 6-DOF igid bod modl wih a combinaion of aciv and paiv moun i popod. o facilia hi dvlopmn a find anf funcion modl fo

10 (a (b (c powain #1 #3 #4 # Fig. 11 Moun locaion fo coupld and dcoupld powain of Fig. 6 givn hamonic oqu ciaion. On aciv moun (wih inaciv acuao of Fig. 5 i placd a locaion #1 and h a paiv moun; moion a dcoupld b adjuing locaion and oinaion angl of moun #1. K: R dcoupld ( φ = = mm; Coupld ( φ = 15 = 68mm fluid-pion diplacmn p aciv moun i dvlopd and hn i incopoad ino mouning m uling in a pcall-vaing lina iminvaian m fomulaion. Ou modl i paiall vifid b compaion wih numicall obaind fqunc pon funcion; alo compl ignoluion mach wih maud naual fqunci fo on powain ampl. Scond ignucu and muli-dimnional dnamic (pciall moion coupling iu whn cid b hamonic oqu a amind. Fo inanc modal oluion (ha a dicad b h paiv pah a pdicd a wll a h ol of aciv pah. h ffc of moun paam uch a oinaion angl and locaion on moion dcoupling i amind and appopia lcion of h paiv pah wihin aciv moun povid h oqu oll ai dcoupling. Coupling phnomna a illuad b compaing powain moion pca wih and wihou opaion of aciv moun. h moion coupling iu inoducd b h aciv moun a plaind via fqunc pon. Fuu wo includ nion of hi wo o oh aciv iolaion m. Fuh popi of an aciv moun could b pcifid fom h m ppciv (a dcoupld moion onanc conol and ducd anmiibili and hn paiv and aciv pah could b opimid o ild h did pfomanc ov h fqunc ang of in. Fig. 1 Effc of aciv diplacmn on fqunc pon of a dcoupld powain of Fig. 6 givn hamonic oqu ciaion wih 1 Nm ampliud. On aciv moun of Fig. 5 i placd a locaion # 1 wih an oinaion angl φ = and h acuao opa inuoidall wih diffn valu of ampliud and pha φ. (a Z( ω ; (b θ ( ω ; (c θy ( ω ; oh pon (no hown ( ω = Y ( ω = θz ( ω =. K: = mm ; = 1.5mm wih φ = 18. Obv coupling in Z( ω and θ ( ω wih = 1.5mm wih φ = 18 CKNOWLEDGMENS W a gaful o h mmb oganiaion of h Sma Vhicl Concp Cn ( and h Naional Scinc Foundaion Indu/Univi Coopaiv Rach Cn pogam ( fo uppoing hi wo. REFERENCES 1. C.M. Hai Shoc and Vibaion Handboo McGaw- Hill Nw Yo 1995 (Chap 3... Jong and R. Singh nalical mhod of dcoupling h auomoiv ngin oqu oll ai Jounal of Sound and Vibaion (. 3. J. Pa and R. Singh Effc of non-popoional damping on h oqu oll ai dcoupling of an ngin mouning m Jounal of Sound and Vibaion (8. 4. H. hafiuon Dign opimiaion of aicaf ngin moun Sm SME Jounal of Vibaion and couic 115( ( H. hafiuon and C. Naaaj Dnamic anali of ngin-moun m SME Jounal of Vibaion and couic 14( (199. Y

11 6. C.E. Spimann C.J. Radcliff and E.D. Goodman Opimal dign and imulaion of vibaional iolaion m Jounal of Mchanim anmiion and uomaion in Dign ( J.S. ao G.R. Liu and K.Y. Lam Dign opimiaion of main ngin-moun m Jounal of Sound and Vibaion (. 8. R. Singh G. Kim and P.V. Ravinda Lina anali of auomoiv hdo-mchanical moun wih mphai on dcoupl chaaciic Jounal of Sound and Vibaion ( M.L. in and M.. Cuchin Inabilii in a nonlina modl of a paiv damp Jounal of Sound and Vibaion ( E.I. Rivin Paiv Vibaion Iolaion SME P Nw Yo R.. Ibahim Rcn advanc in nonlina paiv vibaion iolao Jounal of Sound and Vibaion (8. 1. Y. Yu N.G. Naganahan and R.V. Duipai Rviw of auomoiv vhicl ngin mouning m Innaional Jounal of Vhicl Dign 4( (. 13. S. H and R. Singh Eimaion of ampliud and fqunc dpndn paam of hdaulic ngin moun givn limid dnamic iffn maumn Noi Conol Engining Jounal 53( ( Jong and R. Singh Incluion of maud fqunc- and ampliud-dpndn moun popi in vhicl o machin modl Jounal of Sound and Vibaion ( Y-W L and C-W L Dnamic anali and conol of an aciv ngin moun m Poc. Iniuion of Mch. Engin Pa D: Jounal of uomobil Eng ( Shibaama K. Io. Gami. Ou Z. Naajima and. Ichiawa civ ngin moun fo a lag ampliud of idling vibaion SE Pap #95198 ( Gnnau nw gnaion of ngin moun SE Pap #95196 ( K. oi. Shiaa Y. Houdou. Hiad and. ihaa pplicaion of an aciv conol moun (EM fo impovd dil ngin vhicl quin SE Pap # ( H. Mauoa. Miaa and H. Nmoo NV counmau chnolog fo a clind-on-dmand ngin Dvlopmn of aciv conol ngin moun SE Pap # (4.. P. Gadonio S.J. Ellio and R.J. Pinningon civ iolaion of ucual vibaion on muli-dg-offdom m pa 1: h dnamic of h m Jounal of Sound and Vibaion ( J-H Kim and C-W L Smi-aciv damping conol of upnion m fo pcifid opaional pon mod Jounal of Sound and Vibaion (3...J. Roon and R. Singh Opimiaion of paiv and aciv non-lina vibaion mouning m bad on vibao pow anmiion Jounal of Sound and Vibaion ( J. Hilli.J.L. Haion and D.P. Son compaion of wo adapiv algoihm fo h conol of aciv ngin moun Jounal of Sound and Vibaion (5. 4. Y. Naaji S. Saoh. Kimua. Hamab Y. au and H. Kawajo Dvlopmn of aciv conol ngin moun m Vhicl Sm Dnamic ( G. Kim and R. Singh ud of paiv and adapiv hdaulic ngin moun m wih mphai on non-lina chaaciic Jounal of Sound and Vibaion ( Gnnau Rach fo nw vibaion iolaion chniqu: Fom hdo-moun o aciv moun SE Pap #93134 (1993. CONC Pofo Rajnda Singh couic and Dnamic Laboao NSF I/UCRC Sma Vhicl Concp Cn h Ohio Sa Univi ingh.3@ou.du Phon: Wbi: