Eigenvalue problems of rotor system with uncertain parameters

Transcription

1 Jounal of Mechanical Science and echnology 6 () () ~ DOI.7/s Eigenvalue obles of oo syse wih unceain aaees Bao-Guo Liu * Insiue of Mechaonic Engineeing, Henan Univesiy of echnology, Lianhua See, Zhengzhou 45, China (Manusci Received Febuay, ; Revised July 7, ; Acceed Seebe 8, ) Absac A geneal ehod fo invesigaing he eigenvalue obles of a oo syse wih unceain aaees is esened in his ae. he ecuence eubaion foulas based on he Riccai ansfe aix ehod ae deived and used fo calculaing he fis- and secondode eubaions of eigenvalues and hei esecive eigenvecos fo he oo syse wih unceain aaees. In addiion, hese foulas can be used fo invesigaing he indeenden, and eeaed, as well as he colex eigenvalue obles. he geneal ehod is called he Riccai eubaion ansfe aix ehod (Riccai-PMM). he foulas fo calculaing he ean value, vaiance, and covaiance of he eigenvalues and eigenvecos of he oo syse wih ando aaees ae also given. Riccai-PMM is used fo calculaing he ando eigenvalues of a sily suoed ioshenko bea and a es oo suoed by wo oil beaings. he esuls show ha he ehod is accuae and efficien. Keywods: Roodynaics; Unceain aaee; Eigenvalue oble; Peubaion; Riccai ansfe aix ehod Inoducion Sucual syses wih unceain aaees exis widely in engineeing. Coehensive eviews on he dynaic obles of hese syses have been esened by Ibahi [], Oden e al. [], and Sanchez [3]. he eubaion finie eleen ehod (PFEM) lays an ioan ole in he invesigaion of he dynaic chaaceisics of hese sucual syses [, 3]. he coesonding eseach subecs include he eubaion ehod of he indeenden eigenvalue and is eigenveco [4-8], he eubaion ehod of he eeaed eigenvalue and is eigenvecos [9-4] and he calculaion ehod of he ando eigenvalue and is eigenvecos [5, 6], aong ohes. Howeve, since hese ehods ae based on PFEM, he eos caused by uncaing he high odals canno be avoided [4-6]. On he ohe hand, siila o FEM, which is no he os efficien ehod fo solving oodynaic obles [7, 8], PFEM is also no he os efficien ehod fo analyzing oodynaic syses wih unceain aaees. Roodynaic syses wih unceain aaees also exis widely in engineeing as a secial kind of sucue. In ecen yeas, oe and oe scholas have focused on hei dynaic obles [9-]. Howeve, he dynaic odel ha os eseaches have sudied is he Jeffco oo. he cuen sudy aes o invesigae a colicaed oodynaic odel and his ae was ecoended fo ublicaion in evised fo by Associae Edio Ohseo Song * Coesonding auho. el.: , Fax.: E-ail addess: bguoliu@hau.edu.cn; bgliu978@sina.co.cn KSME & Singe esen a geneal ehod o analyze is ando eigenvalues and eigenvecos. he ansfe aix ehods (MM) ae vey useful and efficien in undeaking he dynaic analysis of oodynaic syses wih deeinisic aaees [7, 8, 3-6]. hee ae wo kinds of MMs: he Myklesad-Pohl ansfe aix ehod (MP-MM) [7, 3] and he Riccai ansfe aix ehod (Riccai-MM) [8]. MP-MM has sile aheaical foulas and is easy o oga. Howeve, nueical difficulies can aise when highe fequencies ae calculaed and/o when hee ae oo any degees of feedo. Riccai- MM, on he ohe hand, also has sile aheaical foulas and is easy o oga. In addiion, hee ae no nueical difficulies [8] involved in using his ehod, which is why i is oe ofen used in engineeing. In his ae, he ecuence eubaion foulas based on he Riccai-MM ae deived o calculae he fis- and second-ode eubaion soluions of he eigenvalues and hei eigenvecos fo oo syses wih unceain aaees. he eubaion fequency equaions foulaed o analyze he indeenden and eeaed eigenvalues ae also educed. Moeove, he ehods used fo evaluaing he eubaion soluions of eigenvecos ha coesond o he indeenden and eeaed eigenvalues ae also given. hus, by no dealing wih he obles of uncaing odals in he ocess of deiving he foulas, he eos caused by uncaing he high odals ae avoided. Fuheoe, his ehod can be alied o evaluae he eubaion soluions of eal eigenvalues and hei eigenvecos as well as evaluae he eubaion solu-

2 B.-G. Liu / Jounal of Mechanical Science and echnology 6 () () ~ ions of colex eigenvalues and hei eigenvecos. his eubaion ehod is called he Riccai eubaion ansfe aix ehod (Riccai-PMM). In he esen sudy, based on he Riccai-PMM, he foulas ae given fo calculaing he ean value, vaiance, as well as covaiance of he eigenvalues and eigenvecos fo he oo syse wih ando aaees. he foulas ae used fo calculaing ando eigenvalues of a sily suoed ioshenko bea and a es oo suoed by wo oil beaings. he esuls show ha he ehod is accuae and efficien in coaison wih he siulaed esuls.. Recuence foulas of he Riccai eubaion ansfe aix ehod o solve eigenvalue obles, he ansfe and ecuence foulas of he Riccai-MM ae wien as [8]: { } { } f = i S i e () i S = + + = [ ] [ ] i+ u S u u S u i S i u S i u () i {} e = u S + u {} e = [ S u]{} e (3) i i i+ i+ whee {f} i and {e} i ae he sae vecos wih eleens a secion i ha saisfy he bounday condiions {f} = {} and {e} {}, esecively; [u ] i, [u ] i, [u ] i, and [u ] i ae he aixes wih eleens defined by he ansfe elaions of sae vecos beween secions i and i+ on he eleen i; [S] i is he so-called Riccai ansfe aix; [ S u] = i u S + u ; i and [ S u] = i u S + u. i Suose b ( =,, ) ae he unceain aaees of he oo syse, hey ae calculaed using he following exession: ( ε ) b = b + ( =, L, ) (4) whee b is he iniial value of he unceain aaee b, and ε < is a sall aaee. Since he aaees ae unceain, he eigenvalue β of he syse is also unceain. Is eubaion exession wih second-ode accuacy is saed as: β = β + β ε + β εε,,, k = = k= (5) whee β, is he iniial eigenvalue of he syse, while b = b ( =,, ); β, and β, ae he fis- and second-ode eubaions of β, esecively. he eubaion exessions of all he ohe aixes and vecos ha ae no lised hee ae siila o hose of β. By subsiuing he eubaion exessions ino Eqs. ()- (3) and allowing he coefficiens of ε wih he sae owe equal yields he following equaions: { } { } f = i, S i e (6a), i, { } = { } + { } f S e S e (6b) i, i, i, i, i, { f } = S { e } + S { e } i, i, i, i, i, k ( δ ) { } { } i, + S e + S e (6c) ik, i, i, [ ] [ ] i, i, [ ] [ ] [ ] [ ] i, i, i, i, [ ] [ ] [ ] [ ] i, i, i, i, k ( δ )[ u] [ u] [ u] [ u] S = S S (7a) i+, u u S = S S + S S (7b) i+, u u u u S = S S + S S i+, u u u u + S S + S S (7c) { } = [ ] { } i, u i, i+, i, k i, i, i, e S e (8a) { } = [ ] { } + [ ] { } e S e S e (8b) i, u i, i+, u i, i+, { e} = [ S ] { } [ ] { }, u e + S i i, i+, u e i, i+, k + ( )[ ] { } + [ ] { } S e S e (8c) δ u ik, i+, u i, i+, whee δ is Konecke dela. Eqs. (6)-(8) consiue he ecuence eubaion foulas of he Riccai-MM. By subsiuing he eubaion exessions ino he a- ' ixes [S u ] i and [S u ] i and allowing [ ] Su = S u, he i, i, foulas fo he calculaion of he aixes [S u ] i,, [S u ] i,, and [S u ] i, as well as [S u ] i,, [S u ] i,, and [S u ] i, ae esecively given as follows: [ ] [ ] [ ] S = u S + u (9a) u i, i, S = u S + u S + u (9b) u i, i, i, i, i, i, S = u S + u S u i, i, i, i, i, k ( δ ) + u S + u S + u (9c) i, k i, i, i, i, S = u S + u (a) ' u i, i, S = u S + u S + u (b) ' u,,,, i, i i i i i, S = u S + u S ' u,, i, i i i, i, k ( δ ) + u S + u S + u (c) i, k i, i, i, i, [ ] ' u = i, u i, S S (a) [ ] ' ' ' u = i, u u u i, i, i, S S S S (b) [ ] ' ' ' ' S = S S S S S ' [ S ] u i, u i, u i, u i, u i, u i, u ik, ' ' ( δ ) u u [ u ]. i, i, k i, S S S (c) If he bounday condiions a boh ends of he oo syse ae he sae (i.e., {f} = {} and {e} {}), hen he

3 B.-G. Liu / Jounal of Mechanical Science and echnology 6 () () ~ 3 following equaion us be esablished: { } = { } = { } = { } f f f. (),,, By subsiuing Eq. () ino Eq. (6a), he fequency equaion fo β, is obained as: Δ= S = (3) N +, whee S, is he deeinan of aix [S],. Since β, is obained fo he above equaion, he aix [S], us be singula. Conducing singula value decoosiion yields: = N +, S U D V (4) whee he aixes [U] and [V] ae uniay aixes, and [D] is a diagonal aix. 3. Indeenden eigenvalue oble If he eigenvalue β, is an indeenden oo, he las eleen of aix [D] us be zeo as shown in he equaion below: D = diag d, L, d,. (5) by he aix U, obains he following saeen: {} % + {} = {} D e., U S e (),, Since he las eleen of he aix [D] is zeo, he condiion ha akes Eq. () equal is: { {}, } ˆ = U S e =. () e N +, N +, Eq. () is he fequency equaion fo he fis-ode eubaion β, ( =,, ) of he eigenvalue β, whee he sybol { } denoes he h eleen of he veco { }. Afe β, is obained, subsiuing he esuls ino Eq. () yields: e% = e ˆ ( =, L, ) (a),, d e % + = (b) N, whee e % N +, is he h eleen of he veco { %e } N +,. Accodingly, Eq. () is subsiued ino Eq. (9) and eulilied by he aix [V]. he esuling equaion becoes: { } = { } e V e %. (3) N +,, By subsiuing Eqs. (4) and () ino Eq. (6a), he esuling exession is: {} {} U D V e =. (6) N +, he nonzeo soluion of he veco {e}, is: { } = { } e V (7) N +, whee {V } is a veco cobined by he eleens of he las colun of aix [V]. 3. he fis-ode eubaion of indeenden eigenvalue and is eigenveco Subsiuing Eqs. (4) and () ino Eq. (6b) yields: {} {} {} U D V e + =, S e (8),, whee {} {} %e =. N +, V e (9), Subsiuing Eq. (9) ino Eq. (8), which is hen eulilied 3. he second-ode eubaion of indeenden eigenvalue and is eigenveco Siilaly, by subsiuing Eqs. (4) and () ino Eq. (6c), he fequency equaion is deived fo he second-ode eubaion β, ( =,, ; k=,, ) of he eigenvalue β, which is exessed as: { ( {} ( δ ) {}, = U S e +,, k S e, k, eˆ, {} N +, )} + S e =. (4) he soluion of he veco {e}, is as follows: { } = { } e V e %. (5) N +, N +, he eleens of he veco {%e }, ae: e% = e ˆ ( =, L, ) (6a),, d e % + = (6b) N,, whee e % N +, is he h eleen of he veco {} e N +,, e + is defined by Eq. (4). ˆ N, ; and

4 4 B.-G. Liu / Jounal of Mechanical Science and echnology 6 () () ~ Eqs. (5)-(6) ae he given foulas o solve indeenden eigenvalue obles. Using he fequency Eqs (3), (), and (4), he iniial eigenvale β, and is fis-ode eubaions β, ( =,, ) as well as is second-ode eubaions β, ( =,, ; k =,, ) can be seached, esecively. By subsiuing Eqs. (7), (3), and (5) ino he ecuence foulas (8) and (6), he coesonding eigenveco and is fis- and second-ode eubaions can be calculaed. 4. Reeaed eigenvalue oble If he eigenvalue β, is a eeaed oo wih s odes, he las s diagonal eleens of he aix [D] in Eq. (4) us be zeoes. hee us be s nonzeo soluions of veco {e}, in Eq. (6). hese s nonzeo vecos ae deived by: {} { } e = V ( =, L, s) (7) N +, + whee {V -+ } is a veco cobined by he eleens of he (+)h colun of he aix; and {} e is he h soluion N + of he veco {e},,. 4. he fis-ode eubaions of eeaed eigenvalue and is eigenvecos By subsiuing Eqs. (4) and () as well as he s soluions of he veco {e}, ino Eq. (6b), he s equaions below ae obained: {} {} {} U D V e + = ( =,, s)., S e L,, (8) Mulilied by s nonzeo consans, he above s equaions ae ewien in he following aix fo: { } { } { } U D V e u + =, S, e u (9), whee {} e N +, is he h soluion of he veco {e}, ; s = {},,{} s e N +, e L e ;,, {} {} e =,,, e L e ;,, and {u} is a colun veco cobined by he s nonzeo consans. Define %e =. N +, V e (3), Subsiuing Eq. (3) ino Eq. (9) and e-ulilied by he aix [ U ] yields he following: { } + { } = { } D e % u., U S, e u (3), Since he las s diagonal eleens of he aix [D] ae zeoes, he condiion ha akes Eq. (3) equal is: W u (3) { } = { } whee [W] is a aix ade u by he las s eleen ows of he aix U S, e. Obviously, if he deeinan of aix [W] is equal o zeo, Eq. (3) has a nonzeo, soluion. heefoe, he fequency equaion fo he fis-ode eubaion β, ( =,, ) of he eigenvalue β is: W =. (33) Eq. (33) has s oos which coesond o he s fis-ode eubaions of he eigenvalue β. By subsiuing he s oos ino Eq. (8), he s soluions of he veco {e}, ae obained as follows: {} {} e = V e % ( =, L, s). (34),, %e ae: he eleens of he veco { } N +, e% = e ˆ ( =, L, s ; =, L, s ) (35a),,,, d, N ( ) e% +, = =, L, s; = s+, L, (35b), whee e % N +, is he h eleen of he veco {} e ; and N +,, ˆ e N +, is he h eleen of he veco U S, {} e N +,. 4. he second-ode eubaions of eeaed eigenvalue and is eigenvecos By subsiuing Eqs. (4) and (), as well as he soluions of he vecos {e}, and {e}, ino Eq. (6c), he fequency equaion is deived fo he second-ode eubaion β, ( =,, ; k=,, ) of he eigenvalue β as shown below: W =. (36) he aix [W] is ade u by he las s eleen ows of he following aix [W]: ( ( δ,, k ), k,,, ) W = U S e + S e + S e whee {}, L, {} (37) s e =, e e N +, N +,. Eq. (36) has s oos ha coesond o he s second-ode eubaions of he eigenvalue β. he s soluions of he veco {e}, ae deived as follows: {} {} e = V e % ( =, L, s). (38),, he eleens of he veco { } N +,,,,, d %e ae: e% = e ˆ ( =, L, s ; =, L, s ) (39a)

5 B.-G. Liu / Jounal of Mechanical Science and echnology 6 () () ~ 5 ( ), e%, = =, L, s; = s+, L, (39b), whee e % is he h eleen of he veco {} N +, %e ; and N +,, e ˆ N +, is he eleen of aix [W] a he h ow and h colun. Eqs. (7)-(39) ae he deived foulas fo he eeaed eigenvalue obles. Using he fequency Eqs. (3), (33) and (36), he iniial eeaed eigenvalue β, wih s odes along wih hei fis-ode eubaions β, ( =,, ) and second-ode eubaions β, ( =,, ; k=,, ) can be seached, esecively. hus, by subsiuing Eqs. (7), (34), and (38) ino he ecuence foulas (8) and (6), he coesonding eigenvecos and hei fis- and second-ode eubaions can be calculaed. 5. Rando eigenvalue oble If β is he h eigenvalue of a oo syse, Eq. (5) is ewien as: ) (4) β = β + β ε + β εε whee,,, k = = k= ) ) β, = β, k= β, ( k). he coesonding eigenveco is: ) { } = { } + { } ε + { },,, k = = k= (4) ψ ψ ψ ψ εε. (4) If ε ( =,, ) ae ando aaees in Eq. (4), hei ean values ae zeoes, and hei sandad deviaions ae σ ( =,, ). hus, he elaion beween he vaiaion coefficien ν of he aaee b and he sandad deviaions σ of he aaee ε is obained as follows: ν = E / σ = σ ( =, L, ) (43) b b whee E b and σ b ae he ean values and he sandad deviaion of aaee b, esecively. Fo Eq. (4), he exessions of he ean value and vaiance of he eigenvalue β ae deived as: E ) = + ( β ) β, β, E( εε k) = k= (44) ( β ) ( ) = β β β, β, ke( ε εk) β ), β, kle( ε εkεl) D E E = + = k= = k= l= ) = k= l= s= ( ) ( ) ( ) + β, β ls E εεεε k l s E εε k E εε l s. (45) In addiion, fo Eq. (4), he esecive exessions of he ean value and covaiance of he eigenveco {ψ} ae deived as follows: E ) ({ } ) = { } + { }, E, ( ε ε k ) ψ ψ ψ (46) = k= ({ },{ } ) { } { } ( ) ( ( )) { } ({ } ) = Cov ψ ψ E ψ E ψ ψ E ψ { } { } E ( ε ε k ) = ψ ψ,, k = k=,, kl, kl, = k= l= ψ ) ψ ) E εεεε,, ls k l s E εε k E εε l s = k= l= s= ) ) { ψ} { ψ} { ψ} { ψ} E ( ε εε k l) + + { } { } ( ) ( ) ( ) +. (47) If ε ( =,, ) ae subec o he noal disibuion, and indeenden, Eqs. (44)-(47) ae silified as follows: ) = + ( ),, E β β β σ = ) D β β σ β σ E ( ) ( 4 =, +, ) = ) ({ } ) = { } + { }, σ, = (48) (49) ψ ψ ψ (5) ({ } { } ) ) ψ ψ ) = { ψ} { ψ} + { ψ} { ψ } 4 Cov, σ σ.,,,, = (5) 6. Nueical exales 6. he sily suoed ioshenko bea he accuacy of he ehod esened in his ae was fis esed in calculaing he ando eigenvalues of a sily suoed ioshenko bea. Suose ha he ass densiy ρ and he secion diaee d of he bea ae indeenden ando vaiables o follow noal disibuion, he ean value of he ass densiy is E ρ = 78 kg/ 3, and he ean value of he diaee is E d =.. he ohe aaees ae ceain, such as he lengh L = and he Young s odulus E =. N/. When he vaiaion coefficien of he ass densiy ν ρ is he sae as ha of he secion diaee ν d, he ean value cuves of he fis naual fequency vesus he vaiaion coefficiens ae shown in Fig. (a). Based on he analyical foula of naual fequency of he sily suoed ioshenko bea, he solid cuve is illion ies he ando siulaed esuls. he doed cuve is he fis-ode ando

6 6 B.-G. Liu / Jounal of Mechanical Science and echnology 6 () () ~ Naual feq. ω(ad/s) s-ode RP Vaiaion coef. ν, ρ ν d Naual feq. ω (ad/s) ν d =.5 s-ode RP Vaiaion coef. of ass densiy ν ρ (a) Mean values vs. vaiaion coefficien ν ρ and ν d (a) Mean values vs. vaiaion coefficien ν ρ Sandad deviaion σ(ad/s) s-ode RP nd-ode R Sandad deviaion σ (ad/s) ν d =.5 s-ode RP Vaiaion coef. ν, ρ ν d (b) Sandad deviaions vs. vaiaion coefficien ν ρ and ν d. Fig.. Rando eubaion and siulaion esuls of he fis naual fequency of he ioshenko bea as ν ρ = ν d Vaiaion coef. of ass densiy ν ρ (b) Sandad deviaions vs. vaiaion coefficien ν ρ Fig.. Rando eubaion and siulaion esuls of he fis naual fequency of he ioshenko bea as ν d =.5. eubaion esuls. Meanwhile, he dash-do cuve eesens he second-ode ando eubaion esuls. Fig. (a) shows ha he ean values of he second-ode ando eubaion ae in ageeen wih hose of he Mone Calo siulaion esuls, and he eos ae less han.% as he vaiaion coefficiens ν ρ = ν d <=.. Howeve, he eos of he fis-ode ando eubaion ae less han.7% in he sae case. I is aaen ha he second-ode ean values ae oe accuae han hose of he fis-ode. he cuve of he fis-ode ean values is a hoizonal line, and is value is he naual fequency wih vaiaion coefficiens ν ρ = ν d = fo Eq. (44). Fig. (b) los he sandad deviaion cuves of he fis naual fequency. he figue shows ha he accuacies of he fis- and second-ode sandad deviaions ae close a less han 4.4% and 3.9%, esecively, as he vaiaion coefficiens ν ρ = ν d <=.. Wih he vaiaion coefficien of he secion diaee ν d =.5, Fig. los he ean value and he sandad deviaion cuves of he fis naual fequency vesus he vaiaion coefficien of he ass densiy ν ρ. he eos of he fis-ode ean values and sandad deviaions ae less han.7% and 9.7%, esecively, as ν d =.5 and ν ρ <=.; and he eos of he second-ode ean values and sandad deviaions ae less han.% and 8.%, esecively, in he sae case. Figs. and show ha he second-ode ando eubaion is oe accuae han he fis-ode ando eubaion. he esuls obained by he ando eubaion ehod esened in his ae ae in ageeen wih he esuls of he Mone Calo siulaion ehod wihin a lage ange. Howeve, he eos end o incease as he vaiaion coefficiens incease. he ando analyzed esuls of he second, hid, and fouh Fig. 3. A es oo wih wo oil beaings: he disance beween he wo beaings is 4.5, he oal lengh is 5.35, and he oal ass is 68.6 kg. naual fequencies have he sae accuacies as well. Howeve, he elaed cuves and daa ae no lised hee. Since he heoeical vibaion odes of he sily suoed ioshenko bea ae no elaed wih he ass densiy ρ and he secion diaee d, he ando vaiaions of hese wo aaees do no influence vibaion odes. he ando eubaion esuls ae siila o hose ha have been esened eviously. 6. he oo suoed by wo oil beaings Fig. 3 shows a es oo suoed by wo oil beaings. he woking seed of he oo was. he siffness and daing aixes of he beaings ae esecively given as: kxx kxy K = = N / kyx k yy cxx cxy C = = N s/. cyx c yy

7 B.-G. Liu / Jounal of Mechanical Science and echnology 6 () () ~ Real a of naual feq. β (ad/s) s-ode RP Iage a of naual feq. β s-ode RP Vaiaion coef. of siffness ν kxx (a) Real as of ean values of he fis naual fequency Vaiaion coef. of siffness ν kxx (b) Iage as of ean values of he fis naual fequency Real a of sandad deviaion σ β (ad/s) s-ode RP Iage a of sandad deviaion σ β s-ode RP Vaiaion coef. of siffness ν kxx Vaiaion coef. of siffness ν kxx (c) Real as of sandad deviaions of he fis naual fequency (d) Iage as of sandad deviaions of he fis naual fequency Real a of naual feq. β (ad/s) s-ode RP Vaiaion coef. of siffness ν kxx (e) Real as of ean values of he second naual fequency Iage a of naual feq. β s-ode RP Vaiaion coef. of siffness ν kxx (f) Iage as of ean values of he second naual fequency Real a of sandad deviaion σ β (ad/s) s-ode RP Iage a of sandad deviaion σ β s-ode RP Vaiaion coef. of siffness ν kxx Vaiaion coef. of siffness ν kxx (g) Real as of sandad deviaions of he second naual fequency (h) Iage as of sandad deviaions of he second naual fequency Fig. 4. Rando eubaion and siulaion cuves of he naual fequencies of he oo wih wo oil beaings vesus he vaiaion coefficien ν kxx. he oo was segened o 7 secions accoding o he diaee, lengh, and ass of he secions. Pola oens and ansvese oens of ineia wee lued a boh ends of he secions. Suose iaily ha he siffness k xx of he wo oil beaings ae indeenden ando vaiables o follow noal disibuion, hei vaiaion coefficien values ay vay synchonously o faciliae he dawing. his is a colex eigenvalue oble. Fig. 4 los he ealand iage-a cuves of he ean value and he sandad deviaion of he fis and second naual fequencies vesus he vaiaion coefficien of he siffness k xx. he solid cuves ae, ies he ando siulaed esuls based on he Riccai-MM. Fig. 4 shows ha he esuls of he ando eubaions and he siulaions ae in ageeen wih each

8 8 B.-G. Liu / Jounal of Mechanical Science and echnology 6 () () ~ ohe wihin a lage ange. Howeve, o colee a calculaion on he do of he cuve, he fis- and second-ode ando eubaions ake only.9 and.6 second, esecively. Meanwhile, he siulaion ehod needs 8,668 seconds on he sae coue. Wih he vaiaion coefficien of he siffness ν kxx =.5, he siulaion and he ando eubaion esuls of he fis vibaion ode in X-diecion ae esened in he aendix. he fis- and second-ode eos of he no of ean values of he vibaion ode ae.73% and.%, esecively; he coesonding eos of sandad deviaions ae 4.65% and.74%. he ohe vibaion odes ae no lised in he cuen ae due o liied lengh. 7. Conclusions he Riccai-MM has been used widely in he field of engineeing because i is highly useful and efficien in conducing a dynaic analysis of he oo syse. Fo his ehod, he Riccai-PMM is develoed fo he eubaion analysis of he eigenvalue obles of he oo syse wih unceain o ando aaees. he ecuence eubaion foulas esened in his ae can be used fo he fis- and secondode eubaion analyses of he indeenden, eeaed, as well as colex eigenvalues and hei eigenvecos. Consequenly, he foulas fo calculaing he ean value, vaiance, and covaiance of he eigenvalues and eigenvecos of he oo syse wih ando aaees ae given. he Riccai-PMM is used fo he ando laeal eigenvalue oble analyses of a sily suoed ioshenko bea and a es oo suoed by wo oil beaings. he fis nueical exale eveals ha he ehod is highly accuae wihin a lage ange. he second exale, on he ohe hand, deonsaes ha he ehod is efficien. Boh exales show ha he second-ode ando eubaion esuls ae oe accuae han hose of he fis-ode. Howeve, hese ecision ioveens ae no obvious. he second-ode ando eubaion analysis is uch oe colex. Hence, he fis-ode ando eubaion analysis of Riccai-PMM is a useful choice fo enginees. he calculaion ie of he fis-ode eubaion analysis is also shoe han ha of he second-ode. Fo concee obles, as long as he eleen ansfe aix and he eleen eubaion exessions ae given, he ehod can be used fo longiudinal, laeal, and/o osional vibaion eigenvalue obles of he oo syse wih unceain o ando aaees. heefoe, he ehod esened in his ae can be used geneally in ohe alicaions. Acknowledgen he auho would like o hank he Coiee of Naional Naual Science Foundaion of China (No and No. 79) and he Zhengzhou Bueau of Science and echnology (No. LJRC87) fo he financial suo. Refeences [] R. A. Ibahi, Sucual dynaics wih aaee unceainies, Alied Mechanics Review, 4 (3) (987) [] J.. Oden,. Belyschko, I. Babuska and. J. R. Hughes, Reseach diecions in couaional echanics, Coue Mehods in Alied Mechanics and Engineeing, 9 (7-8) (3) 93-. [3] E. N. Sanchez, A view o he new eubaion echnique valid fo lage aaees, Jounal of Sound and Vibaion, 8 (3-5) (5) [4] L. C. Roges, Deivaives of eigenvalues and eigenvecos, Aeican Insiue of Aeonauics and Asonauics Jounal, 5 (5) (977) [5] J. C. Chen and B. K. Wada, Maix eubaion fo sucual dynaics, Aeican Insiue of Aeonauics and Asonauics Jounal, 7 (6) (979) 95-. [6] R. B. Nelson, Silified calculaion of eigenveco deivaives, Aeican Insiue of Aeonauics and Asonauics Jounal, 4 (9) (976) -5. [7] B. B. Willia, An ioved couaional echnique fo eubaions of he genealized syeic linea algebaic eigenvalue oble, Inenaional Jounal of Nueical Mehods in Engineeing, 4 (3) (987) [8] B. P. Wang, Ioved aoxiae ehod fo couing eigenvalue deivaives in sucual dynaics, Aeican Insiue of Aeonauics and Asonauics Jounal, 9 (6) (99) 8-9. [9] E. J. Haug and B. Roussele, Design sensiiviy analysis in sucual dynaics II, Eigenvalue vaiaions, Jounal of Sucual Mechanics, 8 () (98) [] K. B. Li, J. N. Juang and P. Ghaeaghai, Eigenveco deivaives of eeaed eigenvalues using singula value decoosiion, Jounal of Guidance - Conol and Dynaics, () (989) [] W. C. Mills-Cuan, Calculaion of eigenveco deivaives fo sucues wih eeaed eigenvalues, Aeican Insiue of Aeonauics and Asonauics Jounal, 6 (7) (988) [] R. L. Dailey, Eigenveco deivaives wih eeaed eigenvalues, Aeican Insiue of Aeonauics and Asonauics Jounal, 7 (4) (989) [3] J. Shaw and S. Jayasuiya, Modal sensiiviies fo eeaed eigenvalues and eigenvecos deivaives, Aeican Insiue of Aeonauics and Asonauics Jounal, 3 (3) (99) [4] G. J. W. Hou and S. P. Kenny, Eigenvalue and eigenveco aoxiae analysis fo eeaed eigenvalues obles, Aeican Insiue of Aeonauics and Asonauics Jounal, 3 (9) (99) [5] W. Gao, Naual fequency and ode shae analysis of sucues wih unceainy, Mechanical Syses and Signal Pocessing, () (7) [6] J. Dai, W. Gao, N. Zhang and N. G. Liu, Seisic ando vibaion analysis of shea beas wih ando sucual aaees, Jounal of Mechanical Science and echnology, 4 () ()

9 B.-G. Liu / Jounal of Mechanical Science and echnology 6 () () ~ 9 [7] M. A. Pohl, A geneal ehod fo calculaing ciical seeds of flexible oos, Jounal of Alied Mechanics - ansacions of he ASME, (3) (945) [8] G. C. Hone and W. D. Pillkey, he Riccai ansfe aix ehod, Jounal of Mechanical Design-ansacions of he ASME, (4) (978) [9] Y. M. Zhang, B. C. Wen and Q. L. Liu, Unceain esonses of oo-sao syses wih ubbing, Mechanical Syses, Machine Eleens and Manufacuing - Inenaional Jounal of he JSME, 46 () (3) [] M. F. Dienbeg, Vibaion of a oaing shaf wih andoly vaying inenal daing. Jounal of Sound and Vibaion, 85 (3) (5) [] M. F. Dienbeg, ansvese vibaions of oaing shafs: obabiliy densiy and fis-assage ie of whil adius, Inenaional Jounal of Non-Linea Mechanics, 4 () (5) [] M. F. Dienbeg, D. V. Iouchenko and A. Naess, Coheence funcion of ansvese ando vibaions of a oaing shaf, Jounal of Sound and Vibaion, 9 (3-5) (6) [3] N. O. Myklesad, A new ehod fo calculaing naual odes of uncouled bending vibaion of ailane wings and ohe yes of beas, Jounal of he Aeonauical Sciences, () (944) [4] S. C. Hsieh, J. H. Chen and A. C. Lee, A odified ansfe aix ehod fo he couling laeal and osional vibaions of syeic oo-beaing syses, Jounal of Sound and Vibaion, 89 (-) (6) [5] Y. Kang, A. C. Lee and Y. P. Shih, A odified ansfe aix ehod fo asyeic oo-beaing syses, Jounal of Vibaion and Acousics, 6 (3) (994) [6] S. J. Oh, Influence coefficiens on oo having hick shaf eleens and esilien beaings, Jounal of Sound and Vibaion, 7 (3-5) (4) Aendix: Daa of he fis vibaion ode of he oo wih wo beaings siulaion Mean values in X-diecion Fis-ode ando eubaion Second-ode ando eubaion siulaion Sandad deviaions in X-diecion Fis-ode ando eubaion Second-ode ando eubaion i i i i.46 +.i i i i i i.4 +.4i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i

10 B.-G. Liu / Jounal of Mechanical Science and echnology 6 () () ~ i i i i i i i i i i i i i i i i i i i i i i i i i i i i i.5 +.5i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i.5 +.5i i i i i i i i i i i i i i i i i i i i i i i i.5 +.9i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i. +.69i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i.99 +.i i i.35 +.i i i Bao-Guo Liu was bon in Henan, China in 96. He eceived his B.E. a he Henan Univesiy of echnology, China in 98, M.E. a he Dalian Univesiy of echnology, China in 986, and his Ph.D a he Chongqing Univesiy, China, in. He woked fo Zheng-zhou Reseach Insiue of Mechanical Engineeing (986-), and woks cuenly as a ofesso in Henan Univesiy of echnology. His eseach focuses on oodynaics, ando aaee vibaion and faul diagnosis of oaional achiney.