Research Article An Improved Fourier Series Method for the Free Vibration Analysis of the Three-Dimensional Coupled Beams

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1 Hdaw Publshg Corporato Shock ad Vbrato Volume 6, Artcle ID 97844, 8 pages Research Artcle A Improved Fourer Seres Method for the Free Vbrato Aalyss of the Three-Dmesoal Coupled Beams Ruze hag, peg Cao, Wepg hag, Hogbo L, ad agme L College of Power ad Eergy Egeerg, Harb Egeerg Uversty, Harb, Cha Hudog-hoghua Shpbuldg (Group) CO., LTD., Shagha, Cha Cha Martme Polce Academy, hejag, Cha Correspodece should be addressed to peg Cao; ypegcao6@6.com Receved 7 July 6; Accepted 7 October 6 Academc Edtor: Br N. Agrawal Copyrght 6 Ruze hag et al. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. Ths paper presets a free vbrato aalyss of three-dmesoal coupled beams wth arbtrary couplg agle usg a mproved Fourer method. The dsplacemet ad rotato of the coupled beams are represeted by the mproved Fourer seres whch cossted of Fourer cose seres ad closed-form auxlary fuctos. The couplg ad boudary codtos are accomplshed by settg couplg ad boudary sprgs ad assgg correspodg stffess values to the sprgs. Modal parameters are determed through the applcato of Raylegh-Rtz procedure to the system eergy formulato. The accuracy ad covergece of the preset method are demostrated by fte elemet method (FEM) result. Ivestgato o vbrato of the propulso shaftg structure shows the extesve applcablty of preset method. The studes o the vbrato suppresso devces are also reported.. Itroducto For the propeller rug the ouform wake flud feld, the logtudal ad lateral vbratos of the propulso shaftgwllbestmulated.ithefollowg,theshaftg logtudal ad lateral vbrato eergy trasfers from shaftg to bearg pedestal, thereby to the structure whch wll cause structure uderwater radated ose. Therefore, t s ecessary to fd a method that ca ot oly calculate the vbrato characterstcs of the propulso shaftg but also cosdertheshellstructurethefollowgstep. There s a vast amout of formato avalable o the logtudal or lateral vbrato of propulso shaftg. hag ad hao [, obtaed the logtudal vbrato modes of the propulso shaftg usg oedmeso elastc wave approach based o ts cotuous model. The coupled logtudal-trasverse dyamcs of the mare propulso shaftg uder prmary ad teral resposes was vestgated by ou et al. [. The coupled logtudal-trasverse dyamc model s establshed by usg the exteded Hamlto prcple ad dscretzed by Galerk method. Huag et al. [4 researched the coupled torsoal-logtudal vbrato of the propulso shaftg. The chages of dsplacemet ad rotato agle caused by coupled torsoal-logtudal vbrato of the propulso shaftgarestudedbythefteelemetmethod.houet al. [5 appled the fte elemet method to calculate the shp propulso shaftg vbrato resposes uder the propeller bearg forces. L et al. [6 studed the dyamc ad acoustc characterstcs of the coupled propeller ad shaftg system by trasfer matrx method. Kae ad McGoldrck [7 studed the dyamc behavor of the propulso shaftg caused by logtudal exctato. Pa et al. [8 used a seres of mass sprg damper system to smulate the propulso shaftg ad performed expermet aalyss, they smplfed propeller as lumpedmass,adthefludloadusedtheaddedmasstobe smulated. Amog those papers, the shaftg s assumed to be a uform cross-secto beam. A wde spectrum of techques has bee developed for the vbratos of beams. Some authors have vestgated the dyamc aalyses of multspa beams whch used moder umercal methods such as the fte elemet method. BshopadJohso[9presetedamodalreceptace approachtothecasesgleadcoupledbeamstructure.

2 Shock ad Vbrato Farag ad Pa [ appled the receptace approach techque to the dyamc respose ad power flower flow of the three-dmesoal coupled beam structure. Lee [ used the assumed mode method to research the trasverse vbrato of a Tmosheko beam whch s acted o by acceleratg mass.wuadhsu[studedfreevbratoaalyssof smply supported beams carryg multple pot mass ad sprg-mass system. L [ 6 proposed a mproved Fourer seres method to establsh the multspa beam model wth the elastc support ad calculated the atural characterstcs ad dyamc respose uder movg load. A aalytcal soluto cosstg of the expaso of the dsplacemet fucto to a Fourer seres, the calculato of the ketc eergy ad the elastc stra eergy, ad the solvg the Lagrage equato s preseted by Hedebrecht [7. The two-dmesoal coupled beams model s used the shaftg vbrato calculato. However, the effect of the torsoal force caused by the propeller ca ot be expressed well. Therefore, order to study the characterstc of shaftg vbrato wth the umercal method better, the threedmesoal coupled beams are selected as the research object ths paper. The dsplacemets ad rotato compoets of the coupled beams are expressed as a mproved Fourer seres whch the closed-form auxlary fuctos troducedtoremoveaypotetaldscotutesadesurethe covergeceofthesoluto.allthecoeffcetswllbesolved drectly from the Raylegh-Rtz procedure. The covergece ad accuracy of the preset method are verfed by umercal examples.. Theoretcal Formulatos.. Theoretcal Model. The three-dmesoal coupled beams model s preseted Fgure, together wth the global ad local coordates employed ths paper. Ths model dcates ample formato cludg geeral boudary codto, arbtrary couplg agles, ad elastcally coupled codto. I order to smulate the geeral boudary codtos ad the elastcally coupled codto, artfcal sprg techque s adopted here. Specfcally, sx sets of stffess-lke sprgs are used at the ed of the beam wth rectagular coordates, cludg three sets of lear sprgs, respectvely, alog the x-, y-, z-drectos, ad three sets of rotatoal sprgs aroud the x-, y-, z-drectos. The geeral boudary codtos ca be acheved by varyg the value of sprgs stffess. The sprgs are assged very hgh stffess values for a clamped edge or zero for a free boudary codto. Smlarly, aother sx sets of sprgs are used to lk the adjacet beams. For a rgd coecto, the stffess of all the sprgs ca be assged a fte valve. Through assgg the sprgs at proper stffess, the solator ca be readly acheved. The form of sprgs s show Table. The couterclockwse s predefed as the postve drecto of the couplg agle whch s varable the rage from π to π. Theagle(α ) of each beam was obtaed by the agle betwee the local ad the global coordate. The arbtrary couplg agle (θ,j ) ca be acheved by varyg the value of the agle (α )ofeachbeam... Eergy Fuctoal of Coupled Beams Structure. The total eergy fucto of the coupled beams s take as the sum of the eergy cotrbutos from the beam compoets, couplg costrats, ad boudary costrats. Thus, the Lagraga eergy fucto of the coupled structure ca be expressed as L V T + V B.C + ( δ ) V C. () δ sthecoeffcet,adf j, δ ;otherwse δ. Also the stra eergy ad the ketc eergy for the coupledbeamscabewrtteas V l [ E (S ( U x ) [ +I,z ( W,y x ) +I,y ( W,z x ) )+ G I p ( Θ x ), T l (ρ S U +ρ S W,y +ρ S W,z +J Θ )dx. The potetal eergy stored the boudary ad couplg sprgs ca be descrbed as where V B.C (UT K B.C U ) x + (UT Kl B.C U ) xl, V C (U U j ) T K c (U U j ), w,y w U,z [μ w,y w,z θ x x e jωt, K B.C dag [kx (,) k y (,) k z (,) K x (,) K y (,) K(,), z K l B.C dag [kx (,) k y (,) k z (,) K x (,) K y (,) K z (,). K c dag [k x k y k z K x K y K z,u R U, R s the trasformato matrx to trasform U to the global coordate system, ad ts form s as follows: T () () (4) cos α s α s α cos α R cos α s α. (5) [ s α cos α [.. Soluto for the Coupled Beams Structure. The dsplacemets of the beam are descrbed by U, W,y, W,z, ad Θ the x-, y-, z-drectos ad rotato agle aroud

3 Shock ad Vbrato k x k y k x Beam k x (,) k x (,) k x (,) k y (,) k z (,) k y (,) k z (,) k z (,) Beam k y (,) k z (,) k x (,) k y (,) Fgure:Allustratoofacoupledbeamssystem. Table : The form of sprgs. The boudary sprgs at the rght eds of beam The couplg sprgs at the jucto of beams ad j The boudary sprgs at the left eds of beam j x-drecto sprg k x (,) k x k x (j,) y-drecto sprg k y (,) k y k y (j,) z-drecto sprg k z (,) k z k z (j,) Aroud x-axs rotatoal sprg K x (,) K x K x (j,) Aroud y-axs rotatoal sprg K y (,) K y K y (j,) Aroud z-axs rotatoal sprg K z (,) K z K z (j,) x-drecto. Ad they ca be wrtte the form of the mproved Fourer seres as follows: M U (x, t) ( A m u cos (λ m x) + α u ξ T u (x))e jωt, W,y (x, t) M m ( A m w,y cos (λ m x) + α w,y ξ T w,y (x))e jωt, m W,z (x, t) M ( A m w,z cos (λ m x) + α w,z ξ T w,z (x))e jωt, m M Θ (x, t) ( A m θ cos (λ m x) + α θ ξ T θ (x))e jωt, m where ω s the agular frequecy, t represets tme, λ m mπ/l, A m u, A m w,y, A m w,z,ada m θ are the Fourer expaded coeffcet, ad α u, α w,y, α w,z,adα θ are the expaded coeffcets of the auxlary fuctos. They ca be solved by (6) the Raylegh-Rtz procedure. ξ(x) are the auxlary fuctos whch ca remove ay potetal dscotutes of the orgal dsplacemets ad effectvely ehace the covergece of the result.ithspaper,theseauxlaryfuctosarespecally selected as ξ u (x) ξ θ (x) {ξ (x),ξ (x)}, ξ w,y (x) ξ w,z (x) {ξ (x),ξ (x),ξ (x),ξ 4 (x)}, ξ (x) ξ ξ (x) [ ξ (x) [ ξ 4 (x) 6Lx L x 6L x L 6L. 5x 4 +6Lx 6L x +8L 4 6L [ 5x 4 L x +7L 4 [ 6L (7)

4 4 Shock ad Vbrato Substtutg ()-() ad (6) together wth the admssble fuctos to () ad the performg the Raylegh-Rtz procedure, the goverg egevalue equato ca be obtaed ad gve as (K ω M) C, (8) where the matrx K s the stffess matrx of the system ad tsformsasfollows: K K K j d K K.. K sym d.. [ K jj [ d whch K represets the total stffess matrx of the th beam.theformsasfollows: K [ [ K K K K K 4 K 44 K 55 K 56 sym K 66 K 67 K 77 K 88 (9). () The matrx K s the couplg stffess matrx caused by the couplg sprg betwee the th ad jth beams, ad ts form s as follows: K [ [ K K K K 4 K K K K 4 K K K K 4 K 4 K 4 K 4 K 44 K 55 K 56 K 57 K 65 K 66 K 67 K 75 K 76 K 77 K 85 K 86 K 87 K 58 K 68 K 78 K 88. () Thedetalformoftheaboveequatocabefoud Appedx A. The system mass matrx M ca be wrtte as M dag [M M, () where M s the mass matrx of the beam ad ca be expressed as M [ [ M M M M M 4 M 44 M 55 M 56 sym M 66 M 77 M 78 M 88. () Thedetalformoftheaboveequatocabefoud Appedx B. The defto of the coeffcet vector C (8) s where C [C C T, (4) C [A u α u A w,y α w,y A w,z α w,z A θ α θ, A u [A u,a u,...,a M u, α u [α u,α u, A w,y [A w,y,a w,y,...,a M w,y, α w,y [α w,y,α w,y,α w,y,α 4 w,y, A w,z [A w,z,a w,z,...,a M w,z, α w,z [α w,z,α w,z,α w,z,α 4 w,z, A θ [A θ,a θ,...,a M θ, α θ [α θ,α θ. (5) All the atural frequeces ad mode shapes of the coupled beams ca be obtaed by solvg (8), whch s a stadard characterstc equato.. Numercal Example ad Dscussos I ths secto, several examples are preseted to verfy the accuracy ad relablty of the preset method. Frstly, the covergece of the preset soluto s checked ad excellet accuracy s valdated through comparso of the results obtaed from the FEM. The coupled beams wth arbtrary couplg agle are studed. Ifluece of couplg sprgs s dscussed as well. The propertes for these coupled beams structures are summarzed Table.

5 Shock ad Vbrato 5 Table : Parameters of each beam. Parameter Ut Beam Beam L m.4.8 S m.9.9 I m e e 8 E Nm.e.e ρ Kgm Beam Beam K K K Fgure : Couplg beam structure wth agle... Covergece ad Valdato Study. Theoretcally, there are fte terms the mproved Fourer seres soluto. However,theseressumercallytrucatedadolyfte terms are couted actual calculatos. The covergece of the preset method wll be checked. As a example, cosder twobeamscoectedtogetherat agle, as show Fgure.Tableshowsthefrsteghtfrequecesofthecoupled beams wth some classcal boudary codtos, cludg C- C, F-F, ad S-S, are derved by dfferet trucato umber M. It s obvous that the atural frequeces coverge quckly wth the creasg trucated umber. It ca be also foud that the results obtaed by preset method coverge well at M. Therefore, the trucated umber wll be uformly selected as M. Ad also, a excellet agreemet of comparsos betwee the results ad those from FEM shows theaccuracyofpresetmethod.someselectedmodeshapes for the C-C supported structure are gve Fgure. It s obvous that the two sets of modes are essetally detcal adthecoupledbeamsarecommolycoupled... Two Coupled Beams wth Arbtrary Agle. I order to llustrate the applcato of the curret method, the studes o free vbrato of the coupled beams wth a 45 ad 9 couplgaglearecarredout,asshowfgure4.the leftedofbeamadtherghtedofbeamareclamped. Table 4 shows the comparso of ature frequeces obtaed by preset method ad FEM. Some selected mode shapes are show Fgures 5 ad 6. It s see that very good agreemet of the result s obtaed. The comparsos Table 4 ad Fgures 5 ad 6 dcated that the curret aalyss for coupled beams wth varous couplg agles s accurate ad relable... Two Coupled Beams wth Elastcally Couplg Codto. I ths secto, the effects of the couplg sprg o the mode shapeofthecoupledbeamsarevestgated.fourtypesof couplg codtos as show Table 5 are cosdered ths part. The fourth mode shape of the coupled beams wth varous couplg codtos s preset Fgure 7. Fgure 7(a) shows that the mode shape s ucouplg ad depedet from each other whe the couplg sprg stffess s K. Whe the couplg sprg stffess chages from e6 to e8, because of the elastc couplg, the dsplacemet o the beams s o loger cotuous across the jucto. Ad there s a tred for the coupled beams from weak-couplg to strog-couplg. Whe the couplg sprg stffess s K E, the coupled beams realzed the rgdty coecto. 4. Applcato of Preset Method to the Propulso Shaftg System I ths secto, the propulso shaftg system wll be take as a example to llustrate the applcato of preset method practce. The shaftg s assumed to be the beam structure wth arbtrary boudary codtos. Thrust bearg s smulated as a beam whch ca be both trasmt force ad bedg momet.thepropellerscosderedtobethelumpedmass ad the mddle bearg ad ster bearg are cosdered to be the lear sprgs. The propulso shaftg s smplfed Fgure 8 ad the propertes for shaftg are summarzed Table Vbrato Characterstc of the Propeller Shaftg System. For propulso shaftg wth a propeller, a lumped mass s adopted, t eeds to cosder the fluece of the lumped mass to the mass matrx of couplg system, ad the eergy of lumped mass s as follows: T M4 ( m 4 ( U 4 t ) + m 4 ( W 4,y ) t + m 4 ( W 4,z ) + t m 4r ( θ 4 t ). )xl 4 (6) Through () to (8), we ca get the vbrato characterstcs of propulso shaftg uder multsupport beargs wth propeller. I Table 7, the ature frequeces of the propulso shaftg are compared to those obtaed by FEM. From Table 7, we ca see that the preset results agree well wth the referetal data. The devato s small ad does ot exceed.5% for the worst case, whch valdates the accuracy ad feasblty of preset method for propulso shaftg system. Some selected mode shapes are show Fgure Vbrato Isolator. Logtudal vbrato solator ca be acheved by assgg the x-drecto sprgs betwee shaftg ad thrust bearg at proper stffess value. I ths secto, the dampg effect of the solator wth dfferet stffess valve s studed. The stffess values were separately take as E9 N/m, 5E8 N/m, E8 N/m, ad 5E7 N/m. From Table8,wecaseethat,wththedecreasgofstffess valve, the two atural frequeces of logtudal vbrato are gradually reduced. The smulato of the solator ca be acheved.

6 6 Shock ad Vbrato ANSS ANSS (a) (b) ANSS ANSS (c) (d) ( ) ( ) (L) (L) (e) (f) Fgure : The modes of coupled beams wth C-C: (a) the frst mode, (b) the secod mode, (c) the thrd mode, (d) the fourth mode, (e) frst rotatoal vbrato, (f) secod rotatoal vbrato.

7 Shock ad Vbrato 7 Table : The atural frequecy for coupled beams wth varous boudary codto f/hz. Boudary codtos Mode M6 M8 M M ANSS Errors/% C-C S-S F-F Table 4: The atural frequeces for the coupled beams wth 9 ad 45 couplg agle (f/hz). Agle Model ANSS Θ 9 Error% Model ANSS Error% Θ ANSS Error% Model ANSS Error% Dyamc Vbrato Absorber. Dyamc vbrato absorbers are commoly desged ad tued to suppress vbratosofoevbratomodeofavbratgstructure.asshow Fgure, the sprg-mass systems whch are attached to propulso shaftg are smulated as the dyamc vbrato absorbers. The mass s M a ad the stffess coeffcet of the sprg betwee the mass ad the beam s K a : M a.5m s, K(πf a ) M a, (7)

8 8 Shock ad Vbrato Couplg agle: 9 Couplg agle: 45 Fgure 4: The coupled beams wth 9 ad 45 couplg agle. Four cases x-drecto sprg Table 5: Four cases of couplg codto for coupled beams. y-drecto sprg z-drecto sprg Aroud x-axs rotatoal sprg Aroud y-axs rotatoal sprg Aroud z-axs rotatoal sprg Case Case e6 e6 e6 e6 e6 e6 Case e8 e8 e8 e8 e8 e8 Case 4 e e e e e e Table 6: Parameters of propulso shaftg. Parameter Ut Beam Beam L m.4.68 S m.4e 4.4e 4 I m e e 9 E N/m.e.e ρ kg/m where m s s the mass of the shaftg ad f a s the frequecy of the vbrato mode whch s suppressed. The dsplacemet of the mass has a form of e jωt. (8) Accordgly, the ketc eergy ad the potetal eergy stored the sprg-mass system ca be descrbed as V a k a (W,y (a) ), (9) T a m a. () Substtutg (8) () to the Lagraga eergy fucto, we ca get the vbrato characterstcs of propulso shaftg wth dyamc vbrato absorbers. I Table 9, we ca see that there are two atural frequeces o both sdes of the absorbecy frequecy after the dyamc vbrato absorber s attached, ad the other atural frequeces rema uchaged. Therefore, the frequecy characterstc of the dyamc vbrato absorbers s met. Accordgly the sprg-mass system ca acheve the smulato of the vbrato absorber. 5. Coclusos A mproved Fourer seres method has bee developed for the free vbrato aalyss of three-dmesoal coupled beams wth arbtrary couplg agle. The dsplacemet ad rotato of the structure are expressed as the Fourer cose seres ad closed-form auxlary fuctos. O the bass of eergy fuctoal of the structure elemets, the solutos are obtaed usg Raylegh-Rtz procedure. By usg the preset method, the three-dmesoal coupled beams wth arbtrary couplg agle ad boudary codtos ca be solved a ufed form. The accuracy ad covergece of the preset method are demostrated by the FEM results. Based o ths, the methodology s expressed to be applcable to vbrato aalyss for the propulso shaftg system ad proved accurate for the FEM result as referece ad obtaed a effcet soluto for the vbrato suppresso devces. Appedx A. Stffess Matrces Varous stffess matrces are represeted as K K l E S φ T u (x) φ u (x) dx + k x (,) φt u () φ u () +k x (,) φt u (l )φ u (l )+ ( δ ) (k x cos α +k y s α )[φ T u (x )φ u (x ), l E S φ T u (x) ξ u (x) dx + k x (,) φt u () ξ u ()

9 Shock ad Vbrato ANSS (a) ANSS (b) ANSS (c) ANSS (d) ANSS (e) ANSS (f) Fgure 5: Cotued.

10 Shock ad Vbrato ANSS (g) ANSS (h) Fgure 5: The frst eght modes of coupled beams wth 9 couplg agle: (a) the frst mode, (b) the secod mode, (c) the thrd mode, (d) the fourth mode, (e) the ffth mode, (f) the sxth mode, (g) the seveth mode, ad (h) the eghth mode. Table 7: The frst te atural frequeces of propulso shaftg f/hz. Model ANSS Error% Model ANSS Error% Table8:Theeffectofthestffessvalueofthesolator. Logtudal vbrato frequecy E9 5E8 E8 5E k x (,) φt u (l )ξ u (l )+ ( δ ) (k x cos α +k y s α )[φ T u (x )ξ u (x ), K ( δ )( k x +ky ) s α cos α [φ T u (x )φ w,y (x ), K 4 ξ w,y (x ), ( δ )( k x +ky ) s α cos α [φ T u (x ) Table 9: The frequecy of the shaftg wth absorber f/hz. Model Org model Absorber model K l E S ξ T u (x) ξ u (x) dx + k x (,) ξt u () ξ u () +k x (,) ξt u (l )ξ u (l )+ ( δ )

11 Shock ad Vbrato ANSS (a) ANSS (b) ANSS (c) ANSS (d) ANSS (e) ANSS (f) Fgure 6: Cotued.

12 Shock ad Vbrato ANSS (g) ANSS (h) Fgure 6: The frst eght modes of coupled beams wth 45 couplg agle: (a) the frst mode, (b) the secod mode, (c) the thrd mode, (d) the fourth mode, (e) the ffth mode, (f) the sxth mode, (g) the seveth mode, ad (h) the eghth mode. (k x cos α +k y s α )[ξ T u (x )ξ u (x ), K ( δ )( k x +ky ) s α cos α [ξ T u (x )φ w,y (x ), K 4 K K 4 ( δ )( k x +ky ) s α cos α [ξ T u (x )ξ w,y (x ), l E I φ T w,y (x) φ w,y (x) dx + k y (,) φt w,y () φ w,y () +k y (,) φt w,y (l )φ w,y (l )+K z (,) φ T w,y () φ w,y () +K z (,) φ T w,y (l )φ w,y (l )+ ( δ ) (k x s α +k y cos α )[φ T w,y (x )φ w,y (x ) + ( δ )K z φ T w,y (x )φ w,y (x ), l E I φ T w,y (x) ξ w,y (x) dx + k y (,) φt w,y () ξ w,y () +k y (,) φt w,y (l )ξ w,y (l )+K z (,) φ T w,y () ξ w,y () +K z (,) φ T w,y (l )ξ w,y (l )+ ( δ ) (k x s α +k y cos α )[φ T w,y (x )ξ w,y (x ) K 44 K 55 K 56 + ( δ )K z φ T w,y (x )ξ w,y (x ), l E I ξ T w,y (x) ξ w,y (x) dx + k x (,) ξt w,y () ξ w,y () +k x (,) ξt w,y (l )ξ w,y (l )+K z (,) ξ T w,y () ξ w,y () +K z (,) ξ T w,y (l )ξ w,y (l )+ ( δ ) (k x s α +k y cos α )[ξ T w,y (x )ξ w,y (x ) + ( δ )K z ξ T w,y (x )ξ w,y (x ), l E I φ T w,z (x) φ w,z (x) dx + k z (,) φt w,z () φ w,z () +k z (,) φt w,z (l )φ w,z (l )+K y (,) φ T w,z () φ w,z () +K y (,) φ T w,z (l )φ w,z (l )+ ( δ ) (K z cos α +K z s α )[φ T w,z (x )φ w,z (x ) + ( δ )k z φt w,z (x )φ w,z (x ), l E I φ T w,z (x) ξ w,z (x) dx + k z (,) φt w,z () ξ w,z () +k z (,) φt w,z (l )ξ w,z (l )+K y (,) φ T w,z ()

13 Shock ad Vbrato Beam Beam (a) Beam Beam (b) Beam Beam (c) Beam Beam (d) Fgure 7: The fourth mode for the coupled beams wth the four cases: (a) case, (b) case, (c) case, ad (d) case 4. K Mddle bearg Ster bearg Propeller M Thrust bearg K Fgure 8: The propulso shaftg wth multsupport beargs. ξ w,z () +K y (,) φ T w,z (l )ξ w,z (l )+ ( δ ) (K z cos α +K z s α )[φ T w,z (x )ξ w,z (x ) + K 57 ( δ )k z φt w,z (x )ξ w,z (x ), ( δ )( k x +ky ) s α

14 4 Shock ad Vbrato ANSS (a) ANSS (b) ANSS (c) ANSS (d) ANSS (e) ANSS (f) Fgure 9: Cotued.

15 Shock ad Vbrato ANSS (g) ANSS (h) ANSS () ANSS (j) Fgure 9: The frst te modes for the shaftg: (a) the frst mode, (b) the secod mode, (c) the thrd mode, (d) the fourth mode, (e) the ffth mode, (f) the sxth mode, (g) the seveth mode, (h) the eghth mode, () the th mode, ad (j) the teth mode. cos α [φ T w,z (x )φ θ (x ), K 58 K 66 ( δ )( k x +ky ) s α cos α [φ T w,z (x )ξ θ (x ), l E I ξ T w,z (x) ξ w,z (x) dx + k z (,) ξt w,z () ξ w,z () +k z (,) ξt w,z (l )ξ w,z (l )+K y (,) ξ T w,z () ξ w,z () +K y (,) ξ T w,z (l )ξ w,z (l )+ ( δ ) (K z cos α +K z s α )[ξ T w,z (x )ξ w,z (x ) + K 67 ( δ )k z ξt w,z (x )ξ w,z (x ), ( δ )( k x +ky ) s α cos α [ξ T w,z (x )φ θ (x ), K 68 K 77 ( δ )( k x +ky ) s α cos α [ξ T w,z (x )ξ θ (x ), l G J φ T θ (x) φ θ (x) dx + K x (,) φt θ () φ θ ()

16 6 Shock ad Vbrato xa (a) The frst lateral vbrato mode f a 7.9Hz Mddle bearg Ster bearg Propeller K M Thrust bearg xa k a M a K (b) The model of the shaftg wth the absorber Fgure : The calculated mode wth absorber. K 78 K 88 +K x (,) φt θ (l )φ θ (l )+ ( δ ) (K x cos α +K y s α )[φ T θ (x )φ θ (x ), l G J φ T θ (x) ξ θ (x) dx + K x (,) φt θ () ξ θ () +K x (,) φt θ (l )ξ θ (l )+ ( δ ) (K x cos α +K y s α )[φ T θ (x )ξ θ (x ), l G J ξ T θ (x) ξ θ (x) dx + K x (,) ξt θ () ξ θ () +K x (,) ξt θ (l )ξ θ (l )+ ( δ ) (K x cos α +K y s α )[ξ T θ (x )ξ θ (x ), K (k x cos α cos α j +k y s α s α j )[φ T u (l ) φ u (), K (k x cos α cos α j +k y s α s α j )[φ T u (l ) ξ u (), (A.) K ( k x cos α s α j +k y s α cos α j )[φ T u (l ) φ wj,y (), K 4 ( k x cos α s α j +k y s α cos α j )[φ T u (l ) ξ wj,y (), K (k x cos α cos α j +k y s α s α j )[ξ T u (l ) ξ u (), K ( k x cos α s α j +k y s α cos α j )[ξ T u (l ) φ wj,y (), K 4 ( k x cos α s α j +k y s α cos α j )[ξ T u (l ) ξ wj,y (), K ( k x s α s α j +k y cos α s α j ) [φ T w j,y (l )φ u (), K ( k x s α s α j +k y cos α s α j ) [φ T w j,y (l )ξ u (),

17 Shock ad Vbrato 7 K ( k x s α s α j +k y cos α s α j ) [φ T w j,y (l )φ wj,y (), K 4 ( k x s α s α j +k y cos α s α j ) [φ T w j,y (l )ξ wj,y (), K 4 ( k x s α s α j +k y cos α s α j ) [ξ T w j,y (l )φ u (), K 4 ( k x s α s α j +k y cos α s α j ) [ξ T w j,y (l )ξ u (), K 4 ( k x s α s α j +k y cos α s α j ) [ξ T w j,y (l )φ wj,y (), K 44 ( k x s α s α j +k y cos α s α j ) [ξ T w j,y (l )ξ wj,y (), K 55 (k x cos α cos α j +k y s α s α j )[φ T w,z (l ) φ w,z (), K 56 (k x cos α cos α j +k y s α s α j )[φ T w,z (l ) ξ u (), K 57 ( k x cos α s α j +k y s α cos α j ) [φ T w,z (l )φ θj (), K 58 ( k x cos α s α j +k y s α cos α j ) [φ T w,z (l )ξ θj (), K 66 (k x cos α cos α j +k y s α s α j )[ξ T w,z (l ) ξ u (), K 67 ( k x cos α s α j +k y s α cos α j ) [ξ T w,z (l )φ wj,y (), K 4 ( k x cos α s α j +k y s α cos α j ) [ξ T w,z (l )ξ θj (), K ( k x s α s α j +k y cos α s α j )[φ T θ (l ) φ w,z (), K ( k x s α s α j +k y cos α s α j )[φ T θ (l ) ξ w,z (), K ( k x s α s α j +k y cos α s α j )[φ T θ (l ) φ θ (), K 4 ( k x s α s α j +k y cos α s α j ) [φ T w j,y (l )ξ wj,y (), K 4 ( k x s α s α j +k y cos α s α j ) [ξ T w j,y (l )φ w,z (), K 4 ( k x s α s α j +k y cos α s α j ) [ξ T w j,y (l )ξ w,z (), K 4 ( k x s α s α j +k y cos α s α j ) [ξ T w j,y (l )φ wj,y (), K 44 ( k x s α s α j +k y cos α s α j ) [ξ T w j,y (l )ξ wj,y (). B. Mass Matrces Varous mass matrces are represeted as M M M M M 4 M 44 M 55 M 56 l ρ S φ T u (x) φ u l ρ S φ T u (x) ξ u l ρ S ξ T u (x) ξ u l ρ S φ T w,y (x) φ w,y l ρ S φ T w,y (x) ξ w,y l ρ S ξ T w,y (x) ξ w,y l ρ S φ T w,z (x) φ w,z l ρ S φ T w,z (x) ξ w,z (A.)

18 8 Shock ad Vbrato M 66 M 77 M 78 M 88 l ρ S ξ T w,z (x) ξ w,z l ρ S φ T θ (x) φ θ l ρ S φ T θ (x) ξ θ l ρ S ξ T θ (x) ξ θ (x) dx. Nomeclature L: Legth of beam ρ : Mass desty S : Cross-sectoalarea J: Rotatoal erta momet E : oug smodulus I : Ierta momet about z λ m : (λ m mπ/l, x l, m,,,...) K: Stffessmatrx M: Massmatrx α : The agle betwee the local ad global coordate system of the beam θ,j : θ,j α α j, the couplg agle betwee the adjacet beams C, S, F: Clamped, smple support, free. Competg Iterests (B.) The authors declare that there s o coflct of terests regardg the publcato of ths paper. Ackowledgmets The research s supported by the Natoal Natural Scece Foudato of Cha (o. 599). Ther facal support s gratefully ackowledged. Refereces Proceedgs of the st Iteratoal Cogress o Soud ad Vbrato (ICSV 4), Beg, Cha, July 4. [6 C. L,. Huag, ad H. Hua, Dyamc modelg ad aalyss of axal vbrato of a coupled propeller ad shaft system, Joural of Mechacal Scece ad Techology,vol.,o.7,pp , 6. [7 J. R. Kae ad R. T. McGoldrck, Logtudal vbratos of mare propulso shaftg systems, Trasactos of the Socety of Naval Archtects ad Mare Egeers, vol.57,pp.9 5, 949. [8J.Pa,N.Farag,T.L,adR.Juper, Propellerduced structural vbrato through the thrust bearg, Proceedgs of Meetgs o Acoustcs, pp. 9 99, November. [9 R. E. Bshop ad D. C. Johso, The Mechacs of Vbrato, Cambrdge Uversty Press, Cambrdge, Uk, 96. [ N. H. Farag ad J. Pa, Dyamc respose ad power flow three-dmesoal coupled beam structures. I. Aalytcal modelg, Joural of the Acoustcal Socety of Amerca,vol., o., pp. 5 5, 997. [ H. P. Lee, Trasverse vbrato of a Tmosheko beam acted o by a acceleratg mass, Appled Acoustcs, vol. 47, o. 4, pp. 9, 996. [ J.-S. Wu ad T.-F. Hsu, Free vbrato aalyses of smply supported beams carryg multple pot masses ad sprgmass systems wth mass of each helcal sprg cosdered, Iteratoal Joural of Mechacal Sceces, vol.49,o.7,pp , 7. [ W. L. L, Free vbratos of beams wth geeral boudary codtos, JouralofSoudadVbrato,vol.7,o.4,pp.79 75,. [4 W. L. L ad H. u, A exact fourer seres method for the vbrato aalyss of multspa beam systems, Joural of Computatoal ad Nolear Dyamcs, vol.4,o.,artcle ID, 9. [5 H. u ad W. L. L, Dyamc behavor of mult-spa brdges uder movg loads wth focusg o the effect of the couplg codtos betwee spas, Joural of Soud ad Vbrato, vol., o. 4-5, pp , 8. [6 W. L. L, M. W. Bolha, ad J. ao, Vbratos of two beams elastcally coupled together at a arbtrary agle, Acta Mechaca Solda Sca,vol.5,o.,pp.6 7,. [7 A. C. Hedebrecht, Vbrato of o-uform smply-supported beams, Joural of the Egeerg Mechacs Dvso,vol. 9,o.,pp. 5,967. [ G. B. hag ad. hao, Reduced-order modelg method for logtudal vbrato cotrol of propulso shaftg, IERI Proceda,vol.,o.,pp.7 8,. [ G. B. hag ad. hao, Propeller exctato of logtudal vbrato characterstcs of mare propulso shaftg system, Shock & Vbrato,vol.4,ArtcleID459,9pages,4. [ D. ou, L. Lu, ad. Rao, Coupled logtudal-trasverse dyamcs of a mare propulso shaftg uder prmary ad teral resoaces, Joural of Soud & Vbrato, vol. 7, pp. 99 6, 6. [4 Q.Huag,J.Lu,adC.hag, Aalyssopropulsoshaftg coupled torsoal-logtudal vbrato uder dfferet appled loads, 4. [5. hou, L. Q, ad. P. og, Vbratoal characterstcs of a mare shaft couplg system excted by propellor force,

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