A Rayleigh-Ritz method based on improved Fourier series for vibration analysis of cylindrical shell coupled with elastic beams

Transcription

1 A Raleigh-Rit method based on improved Fourier series for vibration analsis of lindrial shell oupled with elasti beams Zhang Rune ; Cao Yipeng ; Li liaouan College of Power and Energ Engineering, Harbin Engineering Universit, Harbin, P.R. China ABSTRACT A finite irular lindrial shell model oupled with elasti beams is built in this paper. The beam-shell struture is onneted with linear s and rotational s. Considering different strutural oupling onditions, the Raleigh-Rit approah based on improved Fourier series method is emploed to anale the free vibration properties of the alulation model. The improved Fourier series method is used to desribe the displaements of the oupling struture, while the Raleigh-Rit method is utilied to solve the oeffiients. The important advantage of this approah is that it an be universall applied to the oupling struture with a variet of different boundar and oupling onditions. The results are verified b the finite element method. It's shown that the proposed approah is a onvenient, effiient and aurate one for determining the modal behavior of suh a omple struture sstem. Kewords: beam-shell struture, oupling onditions, Raleigh-Rit approah, improved Fourier series method. INTRODUCTION The vibration studies of a finite irular lindrial shell model oupled with elasti beams have separatel reeived a great deal of attention. When the two strutures are oupled together the behavior of the resulting struture is more omple and, onsequentl, there is less literature available on the topi. However, the stud of these ombined strutures is of great importane in man engineering appliations, suh as in the design of aeronautial or spae strutures, submarine struture and industrial vessels. This paper investigates the free vibration harateristis of one variant of these strutures. In reent ears, man efforts have been made on vibration analsis with relativel simple strutures suh as beams and lindrial shells. As far as elasti beams are onerned, a large bod of literature eists. Hsien-Yuan Lin, used numerial assembl method to stud the free and fored vibration of multi-span beam with various intermediate unit (inluding the middle support, lumped mass, mass sstem, et.). W.L.Li 3, 4, 5 proposed an improved Fourier series method to establish the multi-span beam model with the elasti support and alulated the natural harateristis and dnami response under moving load. Cao Yipeng 6 used the Raleigh-Rit method based on an improved Fourier series to solve the oupled three-dimensional beam. For irular lindrial shells, Zhou Haun 7 solved the free vibration of lindrial shell under elasti support boundar ondition with flutuation analsis method, in whih he analed the influene of boundar stiffness to natural frequen of lindrial shell. Dai Lu 8 applied eat Fourier series as displaement admissible funtion and substituted it into governing equation to solve the free vibration harateristi of lindrial shell under an boundar onditions, of whih the results show that the aura is higher. As to ombine shell-plate strutures, less literature is available. Missaoui 9 established variation, equation of oupled struture b simulating the oupling sstem via introduing. Guo foused on the influene of internal struture to struture vibration. He applied the displaement admittane theor to obtain the effet of fore and moment between plate and shell, thus he made great ontribution to the influene of oupling between internal and shell struture. rune_wuang@ahoo.om Inter-noise 4 Page of

2 Page of Inter-noise 4. Theoretial model. Desription of the model The o-ordinate sstem and various diagrams illustrating the parameters used in the model are shown in Figure. The struture onsists of a finite irular lindrial shell oupled with elasti beams at middle of the shell. The shell has a length L and is assumed to be thin; that is, its wall thiness h is muh smaller than its radius R. Therefore, the onventional assumptions of flűgge's shell equation are adopted. The shell displaement is represented b U, V and W, whih are, respetivel, aial displaement, tangential displaement and radial displaement. (a) (b) () (d) Figure - Struture diagram of the struture : (a) beam-shell struture; (b) lindrial shell; () beam; (d) beam-shell interfae at =L/ For the beam, the beam i has a length L Bi, and its ross-setion is square, the length of the square is b whih is muh smaller than the beams' length, therefore, It is assumed that beam is Euler Bernoulli beam. The beam displaement is represented b U i, W i,, W i, and θ i, whih are respetivel, aial, vertial and horiontal displaement and rotation angle around -ais of the ith beam. The beam-shell joint at =L/ an be seen in figures (d). Boundar and oupling of the beam-shell struture are onstrained b linear s and rotational s as shown in Figure. B hanging the stiffness of s from ero to infinite, it is eas to obtain the lamped, simpl supported and free boundar onditions. The form of s is shown in Table. Table -The form of s K i aial-diretion tangential -diretion radial -diretion torsion K 3 4 K L diretion -diretion -diretion around -ais around -ais around -ais K i K torsion torsion torsion K Page of Inter-noise 4

3 Inter-noise 4 Page 3 of. Series representations of the displaement funtions.. Series representations of the displaement funtions Aial, tangential and radial displaement an be respetivel represented as follows: jwt U,, t u, e jwt V,, t v, e () jwt W,, t w, e Where, ω is the angular frequen. The displaement admissible funtion of lindrial shell an be represented as improved Fourier series form. The improved Fourier series epansion an be written as: u u mn m ln l u, a os os n os n m n l n v v mn m ln l v, a os sinn sinn m n l n 4 w w, os os os mn m ln l w a n n m n l n Where, ξ is the additional auiliar funtion to mae the boundar ontinuous. The are ontinuous and derivable funtions whih satisf the boundar onditions. Auiliar polnomial funtion form is given here: 6L L 3 6L 3 L 6L (3) 3 5 6L 6L 8L 4 36L L 7L 36L.. Series representations of the displaement funtions Aial, vertial and horiontal displaement and rotation angle around -ais of the ith beam an be respetivel represented as follows: jωt jωt Ui(, t) ui( )e Wi, (, t) wi, ( )e (4) jωt jωt Wi, (, t) wi, ( )e i(, t) i( )e Where, the subsript i is on behalf of the beam i. ω is the angular frequen. Aording to the literature 4, the displaement admissible funtion of beam i an be represented as improved Fourier series form. ф represent u i, w i,, w i, or θ i the improved Fourier series epansion an be written as: M m T os i i u A i u im u l m l M 4 m T w A os i, wi, im wi, l m l M 4 m T w A os i, wi, im wi, l m l M m T A os i i i im l m l.3 Series representations of the displaement funtions Aording to the oupling beam sstem shown in Figure, the Lagrange funtion an be epressed as: LC VS -TS -QS VB -TB -QB V B V (6) () (5) Inter-noise 4 Page 3 of

4 Page 4 of Inter-noise 4 Where Vs and V B are respetivel, the potential energ of the beam-shell struture, inluding their strain energ and the potential energ stored in the boundar and the potential energ assoiated with the oupling s. T S and T B are their ineti energ. Q S and Q B are the wor done b eternal fores. V C represents the potential energ stored in the oupling s between the beam and lindrial shell. These terms are V V S Eh l U V W Rdd R R Eh l U V W Rdd R Eh l U V Rdd Eh l h W W V Rdd R Eh l h W W V Rdd R Eh l h W V Rdd R W u v w 3 4 Rd W u 5 v 6 w 7 8 R d l (7) L U V W TS h Rd d t t t (8) l S W QS F,,,,,,,, u tufv tvfw twmw t dd (9) V B Q B l i U Wi, W i i, i E Bi Si Ii, I i, GI i p i () l i T B ( BiSU i i BiSW i i, BiSW i i, Ji i d ) () i l i li li F (, t) U d F (, t) W d F (, t) W d Ui i Wi, i, Wi, i, i i i W W M t d M t d M t d l i li i, li i, (, ) (, ) (, ) i i Wi, Wi, i i i l l l = W U U W W W,,,, W, W, W, W, l l l () (3) Page 4 of Inter-noise 4

5 Inter-noise 4 Page 5 of V,, C U W, W U R R, V W R, V W W U V, i R R R R, R, W W, R, Where, ρ, μ and E are, respetivel, the densit, Poisson ratio, and Young s modulus of the shell material. The same smbols, with the subsript B added, are used to denote the equivalent harateristis of the beam material. Substituting Eq. (), Eq. (5) and Eqs.(7-4) into Eq.(6) and using Raleigh-Rit method to solve the Fourier and auiliar polnomial oeffiients. We an get the dnami equilibrium equation of the oupling sstems: K ω MC F (5) Where the matri K is the stiffness matri of the sstem, the matri M is the mass matri of the sstem, the vetor C is the oeffiient vetor, the vetor F is the outside fore vetor One the stiffness and mass matries are alulated, the responds of the oupling sstem aused b the eternal fore an be alulated form Eq.(5). It is noteworth that the modal properties an also be obtained from Eq. (5) b setting the eternal fore vetor to ero and solving a simple matri eigenvalue problem. The modal frequenies for the struture are diretl related to the eigenvalues. The atual modes, however, will have to be determined b substituting the eigenvetors into Eq.(,) and Eq.(4,5), beause eah of the eigenvetor ontains all the Fourier oeffiients for the orresponding mode. 3. Disussion As shown in Figure, The end of the shell is fied onto the ground, orresponding stiffness are en/m. The left end of the vertial beam (Beam ) is rigid onneted with the shell, The left end of the vertial beam (Beam ) is rigid onneted with Horiontal beam (Beam ) through the oupling s. The oupling stiffness of the struture is en/m. The rest stiffness is.the properties for this beam-shell struture are summaried in Table. For model validation, this problem is solved using both the urrent method and the ommerial finite element ode, ANSYS. Table -Parameters of eah beam. Parameter Unit Beam Beam L B m.4.8 S B m.9.9 I B m e e-8 E B Nm -.e.e p B Kgm (4) Inter-noise 4 Page 5 of

6 Page 6 of Inter-noise 4 Table 3-Parameters of shell. Parameter Unit Shell R m.47 h m.6 E N/m.e ρ g/m Table 4-Natural frequen omparison of oupling beam with vertial angle. f/h Model Current FEA error% From Table 4, an eellent agreement is observed between these two sets of solutions with a maimum differene less than 5.4%. It illustrates the high preision and orretness of this method to solve natural frequenies of shafting. From Figure, it an be seen that the two sets of modes are essentiall idential. Page 6 of Inter-noise 4

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9 Inter-noise 4 Page 9 of (e) Figure - Comparison of the shafting model shapes: (a) third; (b) fourth; () fifth; (d) seventh; (e) tenth; From figure, modes whih are obtained b the two methods are the same. Compare two inds of method; the result mathes ver good, showing that the urrent method an be used to alulate the free vibration properties of the beam-shell struture. 4. CONCLUSIONS A general analtial method has been developed for the Beam-Shell struture. Basing on the improved Fourier series method, the free and fored vibration harateristis are alulated. The results are ompared with that b using finite element methods. The onlusions are as follows:. Based on the improved Fourier series method, displaement admissible funtion of beam and shell vibration is established. Raleigh-Rit method is utilied to solve the Fourier oeffiient and auiliar polnomial oeffiients. The results show that the alulation program is simple and aurate whih an be used to simulate the struture vibration properties with arbitrar boundar onditions onsidering different boundar stiffness.. The free and fored vibrations for propulsion shafting are investigated. The results indiate this method is aurate and reliable b omparing with the FEM, and an be used in the aquisition of shafting vibration. ACKNOWLEDGEMENTS The researh is supported b the National Natural Siene Foundation of China (No. 5993). Their finanial support is gratefull anowledged. Inter-noise 4 Page 9 of

10 Page of Inter-noise 4 REFERENCES Hsien-Yuan Lin, Free vibration analsis of a uniform multi-span beam arring multiple mass sstems, Journal of Sound and Vibration, 3(3), , (7). Hsien-Yuan Lin, Dnami analsis of a multi-span uniform beam arring a number of various onentrated elements, Journal of Sound and Vibration, 39(-), 6-75, (8). 3 W.L.Li, Free vibrations of beams with general boundar onditions, Journal of Sound and Vibration, 37(4), 79-75(). 4 W.L.Li, Vibrations of two beams elastiall oupled together at an arbitrar angle, Ata Mehania Solida Sinia, 5(), 6-7, (). 5 Hongan Xu, W.L.Li, Dnami behavior of multi-span bridges under moving loads with fousing on the effet of the oupling onditions between spans, Journal of Sound and Vibration, 3(4-5), , (8). 6 Cao Yipeng, Zhang Rune, The Raleigh-Rit method based on eat Fourier series for propulsion shafting sstem vibration analsis, ICSV 4, Being China 4. 7 Haun Zhou. Free vibrations of lindrial shells with elasti-support boundar onditions [J].Applied Aoustis, (73) : Lu Dai, Tiejun Yang. An eat series solution for the vibration analsis of lindrial shells with arbitrar boundar onditions [J]. Applied Aoustis, 3(74): Missaoui,L.Cheng,M.J.Rihard,Free and fored vibration of a lindrial shell with a floor partion[j].journal of Sound and Vibration,996,9(): Y.P.Guo,Aousti radiation from lindrial shells due to internal foring[j].journal of the Aoustial Soiet of Ameria,996,99(3): Y.P.Guo, Aousti aattering from lindrial shells with de-tpe internal plate at oblique inidene[j].journal of the Aoustial Soiet of Ameria,996,99(5):7-73 Page of Inter-noise 4