Free vibration analysis of a hermetic capsule by pseudospectral method
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1 Journal of Mechancal Scence and echnology 6 (4) (0) 0~05 wwwsprngerlnkcom/content/78-494x DOI 0007/s y Free vbraton analyss of a hermetc capsule by pseudospectral method Jnhee ee * Department of Mechancal and Desgn Engneerng HongkUnversty Chochwon Choongnam 9-70 Korea (Manuscrpt Receved August 5 0; Revsed November 6 0; Accepted January 6 0) Abstract A pseudospectral method s appled to the axsymmetrc free vbraton analyss of a hermetc capsule he dsplacements and the rotaton of a hermetc capsule are expressed by the Chebyshev expansons he equatons of moton are collocated to yeld the system of equatons n the hemsphercal regons and the cylndrcal regon separately he numbers of collocaton ponts are chosen to be less than those of the expanson terms he contnuty condtons of deformatons and stress resultants at the junctons serve as the constrants of the expanson coeffcents he set of algebrac equatons s condensed so that the total number of the expanson terms matches the total number of degrees of freedom of the problem Present method mght be useful n the analyses of composte shell structures where dfferent types of shells are joned together Keywords: Chebyshev expanson; Composte shell structure; Free vbraton; Hermetc capsule; Pseudospectral method Introducton * Correspondng author el: Fax: E-mal address: jnhlee@hongkackr Recommended by Edtor Yeon June Kang KSME & Sprnger 0 A hermetc capsule whch usually conssts of a cylndrcal shell closed at both ends wth hemsphercal shells s used n many engneerng applcatons such as pressure vessels storage tanks and compressor covers o avod resonance fracture of a hermetc capsule t s necessary to predct ts vbraton behavor avakol and Sngh [] proposed a substructure synthess method based on the state space method for the free vbraton of hermetc shell structures Structural elements were formulated from ove s shell theory and a system of eght coupled frst order dfferental equatons was solved for each shell substructure usng Padé method and the substructures were then joned by matchng the dsplacement and force boundary varables Wong and Sze [] appled a matched asymptotc expanson method to study the axsymmetrc vbraton of a hermetc capsule hey employed the membrane approxmaton for the sphercal and cylndrcal shells whch resulted n fnte jumps n the normal dsplacement across the juncton jonng the hemsphercal shell and the cylndrcal shell Shang [] developed an analytcal soluton based on the Naghd- Ressner shell theory for the axsymmetrc vbraton of a hermetc capsule he egendre functon was employed as the bass functon of the hemsphercal shell whch satsfed the boundary condtons at the pole automatcally he coeffcents as well as the degree of egendre functon were determned by the contnuty condtons at the junctons he condtons of the juncton were gven by the contnuty of the deformaton and the equlbrum he egenvalues were obtaned when the determnant of the system of equatons of order sx vanshed he fnte element method has been one of the most favored dscretzaton methods n the analyses of joned shell structures Özakça and Hnton [4] developed axsymmetrc fnte elements based on the Mndln-Ressner theory for the free vbraton analyss of axsymmetrc shell structures Buchanan [5] presented fnte element solutons of a hermetc capsule where three-dmensonal models based on the elastcty theory were developed orsonal frequences of a hermetc capsule were found that had not been prevously reported Even though t was not a hermetc capsule ee et al [6] analyzed the free vbraton of a joned shell structure where a cylndrcal shell s joned by a sphercal shell at one end he hemsphercal shell and the cylndrcal shell were assumed to have free boundary condton and the smply supported boundary condton respectvely at the juncton and the effects of the sphercal shell was assumed to be smaller than the cylndrcal shell to the free vbraton of joned structure Redekop [7] also reported a study on the free vbraton of toruscylnder shell assembly usng dfferental quadrature method Hemsphercal shells A schematc dagram of a hermetc capsule s gven n Fg whch s dvded nto three subregons Ω represent the hemsphercal shells s the cylndrcal sec-
2 0 J ee / Journal of Mechancal Scence and echnology 6 (4) (0) 0~05 are expressed by the expansons as follows: K j k k = K j k k = K j ck k = U = a ξ (a) W = b ξ (b) Ψ = (c) ( ) j = Fg Geometry and coordnate systems of hermetc capsule ton he subregons are connected through the junctons Γ =Ω Ω and Γ =Ω Ω Assumng smple harmonc motons the equatons of moton of the axsymmetrc vbraton of hemsphercal regons Ω wth the effects of transverse shear and rotary nerta taken nto account are derved from Soedel [8] as C U U W U ν ( ν) R φ tanφ φ tan φ φ U W Ψ =ωρhu R R R φ Ψ Ψ R φ tanφ U U W W R φ tanφ φ tanφ φ C U U ( ν) W =ω ρhw R φ tanφ D Ψ Ψ ν Ψ R φ tanφ φ tan φ U W ρh Ψ =ω Ψ R R φ D (a) (b) (c) U and W are the dsplacements n the colattudnal ( φ ) and the normal drectons Ψ s the total angular rotatons about the axs that are mutually perpendcular to φ drecton and the normal drecton he stffness and the bendng stffness of the shell are defned as C = Eh ( ν ) and D = Eh ( ν ) E G R h ν are Young s modulus the shear modulus the radus of the hemsphere the thckness of the shell and Posson s rato respectvely κ s the shear correcton factor and ω s the natural frequency n radan per second For a hemsphere the colattude φ ranges from 0 to and t s convenent to use the normalzed form [ ] ξ = φ 0 () ee [9] showed that the dsplacements and the rotaton n an axsymmetrc vbraton analyss of the hemsphercal shells could be represented by the Chebyshev expansons In present study the dsplacements and rotaton n the hemsphercal regons Ω a b and c are the expanson coeffcents he subscrpt j refers the subregon number and K s the number of collocaton ponts n the colattudnal drecton he Chebyshev polynomals of the frst knd k are defned recursvely as ( ξ) = ξ ( ξ) ( ξ) ( ) 0 = ξ = ξ ξ k k k k Expansons of Eqs (a)-(c) satsfy the boundary condtons of the axsymmetrc free vbraton at the pole of the hemsphere Expansons of (a-c) are substtuted nto Eqs (a)-(c) whch are then collocated at the one-sded Gauss-Chebyshev ponts ( ) ξ = cos ( = K) (5) 4K to yeld the collocated equatons as follows: R C ν k( ξ) tan φ b K 4C C a k k k = tanφ { C( ν K ) } k R k = K ck ( ξ ) R k = K ωρhak ( ξ ) k = K k ξ k k = tanφ = 4 R C( ν) k ( ξ)} ( ν) κ K ωρhbk k = ( ξ) (4) (6a) K C Gh k c k a k = R R tanφ = 4D D D ν k( ξ) R tan φ K c k k k = R R tanφ (6b)
3 J ee / Journal of Mechancal Scence and echnology 6 (4) (0) 0~05 0 K a b R k = ρh =ω k K ( = K) ( j = ) c k k k = (6c) wth φ = ξ he notaton ( ' ) stands for the dfferentaton wth respect to ξ Cylndrcal shell he equatons of moton of the axsymmetrc vbraton of cylndrcal regon Ω wth the effects of transverse shear and rotary nerta taken nto account are derved from Soedel [8] as M 4 b c m m m m M νm η m η C am bm R R = 4D M ωρhbmm ( η) M cm m Ghm κ M bm m ρh =ω ( M) = M c m m (b) (c) U ν W C R x =ωρhu ν C hw x R R x Ψ W ρh D ω Ψ = Ψ x W Ψ W U =ω ρ (7a) (7b) (7c) assumng smple harmonc motons he axal dstance x ranges from to where s the length of the cylnder and t s normalzed as [ ] η = x (8) he dsplacements and the rotaton of regon Ω are approxmated as: M m m M m m M cmm U = a η (9a) W = b η (9b) Ψ = (9c) M s the number of collocaton ponts n the axal drecton Expansons of (Eqs 9(a)-(c)) are substtuted nto Eqs 7(a)-(c) whch are then collocated at the Gauss-Chebyshev ponts ( ) η = cos ( = M) (0) M he notaton ( ) n Eqs (a)-(c) stands for the dfferentaton wth respect to η he collocated Eqs 6(a)-(c) and (a)-(c) can be rearranged n the matrx form [ ]{ δ} [ ]{ χ} = ω ([ ]{ δ} [ ]{ χ} ) A B C D () where the vectors n Eq () are defned as { δ} { = a a b b c c a a K K K K b b K c c K c c M} χ a K b K c K a K b K c K a M a M b M b M c M c M } { } = { Matrces [ A ] and [ ] matrces [ B ] and [ ] () C are of order 6KM and the sze of D s (6KM) he total number of equatons n Eqs 6(a)-(c) and (a)-(c) s 6KM whereas the total number of expanson coeffcents n Eqs (a)-(c) and 9(a)- (c) s 6KM he remanng twelve equatons are obtaned from the contnuty condtons at the junctons Γ and Γ 4 Contnuty condtons at the junctons he contnuty condtons of deformatons and stress resultants at Γ where the local coordnates are ξ = and η = are U = U W = W ξ = η= ξ= η= Ψ =Ψ N = N ξ = η= φφ ξ = xx η= M = M Q = Q φφ ξ = xx η= φ ξ= x η= (4) to result n the collocated equatons as follows: and the contnuty condtons at Γ ( ξ = and η = ) are ν M 4 C a b m m m m = M ωρhamm ( η) R (a) U = U W = W ξ = η= ξ= η= Ψ =Ψ N = N ξ = η= φφ ξ = xx η= M = M Q =Q φφ ξ = xx η= φ ξ= x η= (5)
4 04 J ee / Journal of Mechancal Scence and echnology 6 (4) (0) 0~05 In the hemsphercal regons Ω the stress resultants Nφφ Mφφ Qφ are defned as U ν ν Nφφ = C U W (6a) R φ Rtanφ R Ψ ν Mφφ = D Ψ (6b) R φ Rtan φ U W Qφ = kgh Ψ (6c) R R φ In the cylndrcal regon Ω the stress resultants N xx Mxx Q x are defned as U ν N xx = C W R (7a) Ψ Mxx = D (7b) x W Qx = kgh Ψ (7c) he contnuty condtons (4) are expressed as: K M akk() amm ( ) = 0 (8a) k= K M b k k() bmm ( ) = 0 k= K M ckk() cmm ( ) = 0 k= K K k k ak bk k= R ξ k= R M m am = 0 η K M k m c k cm k= R (8b) (8c) (8d) = 0 (8e) ξ η K K k k ak bk k= R k= R ξ M m b m = 0 η (8f) usng the expanson relatonshps of Eqs (a)-(c) and 9(a)-(c) he contnuty condtons (5) are also expressed as: K M akk() amm () = 0 (9a) k= K M bkk() bmm () = 0 k= K M ckk() cmm () = 0 k= K K k k ak bk k= R ξ k= R M m () am = 0 η (9b) (9c) (9d) K M k m ck cm = 0 (9e) k= R ξ η K K k k ak bk k= R k= R ξ M m () b m = 0 η (9f) he contnuty condton set 8(a)-(f) and 9(a)-(f) can be rearranged n the matrx form [ ]{ δ} [ ]{ χ} = { 0} F G (0) G s a ma- χ n Eq (0) can be expressed as he sze of matrx [ F ] s (6KM) and [ ] trx of order Snce { } { χ} [ ] [ ]{ δ} = G F () Eq () can be reformulated as ([ ] [ ][ ] [ ]){ } ( δ = ω [ ] [ ][ ] [ ]){ δ} A B G F C D G F () In ths way the contnuty condtons at the junctons are merged nto the governng equatons of moton and the soluton of Eq () yelds the estmates for ω and { δ } 5 Numercal examples he algebrac equaton of Eq () s solved for the natural frequences usng Matlab Consder an example of a hermetc capsule where the geometrc constants and materal propertes are the same as used n the lterature [ 5] as shown n able he natural frequences of the hermetc capsule are computed and are gven n able whch shows good agreement wth the exstng lterature he numbers of collocaton ponts are K = 0 and M = 0 respectvely Also assume a specal case that the length of the cylnder becomes zero ( = 0) whch makes the hermetc capsule a full sphercal shell In ths case the system of equatons s represented by Eqs 6(a)-(c) only he coeffcent vectors of Eq () are redefned as { δ} = { a a K b b K c c K a a K b b K c c K} { } = { a b c a b c } χ K K K K K K he contnuty condtons of Eqs (4)-(5) are reduced to U = U W = W ξ= ξ = ξ= ξ= Ψ =Ψ N = N ξ= ξ= φφ ξ= φφ ξ= M = M Q =Q φφ ξ= φφ ξ= φ ξ= φ ξ= () (4) Eq (4) can be expressed n the matrx form usng the ex-
5 J ee / Journal of Mechancal Scence and echnology 6 (4) (0) 0~05 05 R h E ρ ν κ 04 m 04 m 0000 m 07 GPa 7800 kg/m 0 5/6 able Comparson of frequences of hermetc capsule n Hz wth exstng lterature Symmetrc able Geometrc constants and materal propertes of hermetc capsule Antsymmetrc avakol Sngh [] Shang [] Buchanan [5] Present study able Comparson of frequences of a full sphercal shell of axsymmetrc modes n Hz wth exstng lterature m avakol and Sngh [] Present study pansons of (a)-(c) and 9(a)-(c) and are merged nto the set of the governng equatons as explaned n the procedure of Eqs (0)-() he natural frequences of a sample problem are computed and are gven n able he number of collocaton ponts K s 40 he computed frequences are practcally dentcal to those of avakol and Sngh [] n the axsymmetrc modes 6 Conclusons A pseudospectral method s appled to the axsymmetrc free vbraton analyss of a hermetc capsule A hermetc capsule s a typcal composte shell structure however few analytcal solutons of a hermetc capsule are avalable due to the dffculty to match the condtons at the junctons between dfferent types of shell substructures Researchers preferred to employ egendre functon for the hemsphercal sectons whle other types of functons were used for the cylndrcal shell segment In the present study Chebyshev expansons are used to represent the dsplacements and the rotaton n every part of the substructures whch helps retan the consstency and conceptual smplcty n dervng the system of equatons and the contnuty condtons he equatons of moton whch are based on the Mndln theory are collocated to buld the system of equatons n the hemsphercal regons and the cylndrcal regon separately o handle the contnuty condtons of deformatons and stress resultants at the junctons the number of collocaton ponts s chosen to be less than the number of the expanson terms he contnuty condtons at the junctons are used as the constrants of the expansons and the set of algebrac equatons s condensed so that the number of the expanson coeffcents matches that the number of degrees of freedom of the problem Numercal examples are provded for a hermetc capsule and a full sphercal shell he results show that present method are n good agreement wth those of exstng lterature whch suggests that the present method may be utlzed effectvely n the analyses of shell structures where dfferent types of shells are joned together Acknowledgment hs work was supported by 00 Hongk Unversty Research Fund References [] M S avakol and R Sngh Egensolutons of joned/hermetc shell structures usng the state space method Journal of Sound and Vbraton 0 () (989) 97- [] S K Wong and K Y Sze Applcaton of matched asymptotc expansons to the free vbraton of a hermetc shell Journal of Sound and Vbraton 09 (4) (998) [] X Shang Exact soluton for free vbraton of a hermetc capsule Mechancs Research Communcatons 8 () (00) 8-88 [4] M Özakça and E Hnton Free vbraton analyss and optmzaton of axsymmetrc plates and shells- I fnte element formulaton Computers & Structures 5 (6) (994) 8-97 [5] G R Buchanan An analyss of the free vbraton of a hermetc capsule Journal of Sound and Vbraton 59 () (00) [6] Y S ee M S Yang H S Km and J H Km A study on the joned cylndrcal-sphercal shell structures Computers and Structures 80 (00) [7] D Redekop Vbraton analyss of a torus-cylnder shell assembly Journal of Sound and Vbraton 77 (004) [8] W Soedel On the vbraton of shells wth moshenko- Mndln type shear deflectons and rotary nerta Journal of Sound and Vbraton 8 () (98) [9] J ee Free vbraton analyss of sphercal caps by the pseudospectral method Journal of Mechancal Scence and echnology () (009) -8 Jnhee ee receved hs BS and MS degrees from Seoul Natonal Unversty and KAIS n 98 and 984 respectvely He receved hs PhD degree from Unversty of Mchgan Ann Arbor n 99 and joned Dept of Mechancal and Desgn Engneerng of Hongk Unversty n Choongnam Korea Hs research nterests nclude nverse problems pseudospectral method vbraton and dynamc systems
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