Exact analysis of resonance frequency and mode shapes of isotropic and laminated composite cylindrical shells; Part I: analytical studies

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1 Joural of Mechaical Sciece ad Techology ( (0 ~ DOI 0.00/s Eact aalysis of resoace frequecy ad mode shapes of isotropic ad lamiated composite cylidrical shells; Part I: aalytical studies Pouria Oliazadeh,*, Mohammad H. Farshidiafar ad Aooshirava Farshidiafar Mechaical Egieerig Departmet, Ferdowsi Uiversity of Mashhad, Mashhad, Ira Mechaical Egieerig Departmet, Uiversity of Waterloo, Waterloo, Caada (Mauscript Received Jauary, 0; Revised Jue, 0; Accepted July, Abstract I order to study the free vibratio of simply supported circular cylidrical shells, a eact aalytical procedure is developed ad discussed i detail. Part I presets a geeral approach for eact aalysis of atural frequecies ad mode shapes of circular cylidrical shells. The validity of the eact techique is verified usig four differet shell theories Soedel, Flugge, Morley-Koiter ad Doell. The eact procedure is compared favorably with eperimetal results ad those obtaied usig a umerical fiite elemet method. A literature review reveals that beam fuctios are used etesively as a approimatio for simply supported boudary coditios. The accuracy of the resoace frequecies obtaied usig the approimate method are also ivestigated by comparig results with those of the eact aalysis. Part II presets effects of differet parameters o mode shapes ad atural frequecies of circular cylidrical shells. Keywords: Circular cylidrical shell; Eact approach; Numerical method; Natural frequecy Itroductio Similar to beams ad plates, cylidrical shells are the practical elemets of various egieerig structures such as pipes ad ducts, car bodies, space shuttles, aircraft fuselages, ship hulls, submaries, ad buildigs. However, aalyzig the dyamic characteristics of cylidrical shells is more complicated tha aalyzig those of beams ad plates. This situatio occurs maily because the equatios of motio i a cylidrical shell are more complicated. Moreover, i may cases boudary coditio effects are hard to apply. A comprehesive summary ad discussio of shell theories, icludig atural frequecies ad mode shape idetificatio, has bee doe by iessa [] i. More recetly, adoptig a o-liear poit of view, Amabili ad Paidoussis [], Amabili [] ad Kurylov ad Amabili [] have preseted oteworthy reviews o cylidrical shell vibratio. May researchers such as Flugge [] have followed the pioeerig work of ove [], usig his first approimatio theory. The Flugge theory is based o the Kirchhoff-ove hypothesis for thi elastic shells. Usig this theory, the strai-displacemet relatios ad chages of curvature i the middle surface of a cylidrical shell ca be obtaied. The simplified Doell s theory is achieved by eglectig a few terms i the Flugge * Correspodig author. Tel.: + 0, Fa.: + 0 address: pouria.oliazadeh@gmail.com Recommeded by Editor Yeo Jue Kag KSME & Spriger 0 equatios. ivaov [] applied love s assumptio ad used displacemet fuctios to solve the problem of aisymmetrical vibratios of simply supported cylidrical shells. Riehart ad Wag [] ivestigated the vibratio of simply supported cylidrical shells stiffeed by discrete logitudial stiffeers usig Doell s approimate theory, Flugge s eact theory, ad ove s assumptio for logitudial wave umbers. These theories are ot oly suitable for simply supported ed coditios, but they ca also be applied to other cylidrical shell boudary coditios such as fied free [, 0], clampedclamped [] ad ifiite legth oes []. I most reports available i the literature, beam fuctios are used as approimatios for the boudary coditio effects. Thus, resoace frequecies are obtaied through a approimate procedure rather tha a eact aalysis. The mai reaso for these approimatios is the difficulties posed whe actual boudary coditios are applied. The first part of the preset study proposes a eact aalytical approach to ivestigate the free vibratio of simply supported cylidrical shells. As metioed, i covetioal aalysis, beam fuctios with similar boudary coditios are used to approimate wave umbers i the aial directio. This approach is cosidered to be approimate; however, the degree of ucertaity i these approimatios is ukow. A eact method is developed, which will cosider the actual effects of shell boudaries i order to obtai eact resoace

2 P. Oliazadeh et al. / Joural of Mechaical Sciece ad Techology ( (0 ~ Fig.. Circular cylidrical shell: coordiate system ad dimesios. frequecies of a cylidrical shell. The eact method is used to obtai the atural frequecies based o four differet shell theories (Soedel, Flugge, Morley-Koiter ad Doell. This method is compared with the approimate method, to study the validity of beam fuctio approimatios. The results are also compared with eperimetal ad umerical oes ad show good agreemet.. Theoretical aalysis As metioed i the literature, circular cylidrical shells with both eds closed have wide applicatios i the aviatio idustry. Moreover, hemispherical shells closed at both eds are broadly used for pressure vessels ad storage taks []. A schematic diagram of a circular cylidrical shell closed at both eds is give i Fig.. The cylidrical shell uder cosideratio has a costat thickess h, mea radius R, aial legth, Poisso s ratiou, desity r ad Youg s modulus of elasticity E. Here, the respective displacemets i the aial, circumferetial ad radial directios are deoted by u(, q, t, v(, q, t ad w(, q, t. I order to study free vibratio of a cylidrical shell, the equatios of motio ca be writte i matri form as follows: ( q ( q ( q ìu,, t ü ì0ü ï ï ï ï ív,, t ý = í0ý - - ï w,, t ï ï 0 ï î þ î þ where ij ( i, j =,, are differetial operators with respect to, q ad t. Differet systems of equatios are used to model the vibratio behavior of circular cylidrical shells. I this paper four of the most commo theories, amely: Soedel [], Flugge [], Morley-Koiter [] ad Doell s theory [], are used to fid atural frequecies. It is coveiet to defie the followig differetial operators for these theories, respectively: ( ( r - E t R + =, = R R r æ - ö = - ( - + ( + k + E t R ø æ ö = - k + R R ø r æ ö = - ( k R + E t R R q ø ( ( k ( ( k ( r - + E t R + = R q æ - ö = + k - R R R ø r æ - ö E t + ø R æ - ö = - k R ø r = - - E t æ æ ö - ö + k + R + + R R ø R ø ( ( r - E t R + =, = R R r - E t R ( r =, = - - R E t ì ï æ ö æ ö - í + k R R R q R q ï ø ø î ( ( r - E t R + =, = R R r - E t R = R r æ ö = - ( k R + E t R R q ø ( ( ü ï ý ï þ ( (

3 P. Oliazadeh et al. / Joural of Mechaical Sciece ad Techology ( (0 ~ The operators i Eqs. (-( ca be treated as the sum of two operators as doe by eissa []: = + k ( D MOD where D is the differetial operator accordig to the Doell theory. MOD is a modifyig operator that differs for each theory ad is preseted i Appedi A. Moreover, k is the odimetioal thickess parameter defied by h k =. ( R The first attempt at solvig Eq. ( ivolves the assumptio of a sychroous motio: (, q, (, q ( (, q, = (, q ( (, q, (, q ( ì u t = U f t ï í v t V f t ï î w t = W f t where ( the mode shapes U (, q, V (, q ad (, f t is the scalar model coordiate correspodig to W q. The et step is to use the separatio of variables method i order to separate the spatial depedece of the modal shape betwee logitudial ad circumferetial directios. Hece, the aial, tagetial ad radial displacemets of the wall vary accordig to: ( ( ( ì lm u, q, t = Ae si( q cos( wt ï lm ív, q, t = Be cos( q cos( wt ï lm w, q, t = C e si( q cos( wt ïî i which l m ad are the aial waveumber ad the circumferetial wave parameter, respectively. A, B ad C are the udetermied costats, ad w is the circular frequecy of the atural vibratio. Substitutig Eq. ( ito Eq. (, usig ay of the shell theories give by Eqs. (-(, leads to a set of homogeous equatios havig the followig matri form: C C C ìaü ì0ü ï ï ï ï - C C C íbý = í0ý -C C C ï C ï ï 0 ï î þ î þ ( ( (0 i which C ij ( i, j =,, are fuctios of, l m ad a frequecy parameter W that is defied as follows: ( -u r W = w R. ( E As a eample, for Doell s theory, C ij ca be writte i matri form as i Eq. (. The coefficiet matri for other shell theories is defied i Appedi B. - + W + lm - - l m l m + - l m W + lm - -l m W - + k - ( lm. ( For a otrivial solutio, the determiat of the coefficiet matri i Eq. (0 must be zero: ( C ij det = 0 ; i, j =,,. ( The epasio of Eq. ( will give the followig two eigevalue problems: For a give value of l m, there eists oe or more proper values for w so that Eq. ( vaishes. For a give value of w, there eists oe or more proper values for l m so that Eq. ( vaishes. Solvig Eq. ( leads to a cubic equatio i terms of the o-dimesioal frequecy parameter W. Thus, for a fied value of ad l m, three positive roots ad three egative roots are yielded for the o-dimesioal frequecy. The three positive roots are the atural frequecies of the cylidrical shell that ca be classified as primarily aial, circumferetial or radial. The lowest frequecy is usually associated with a motio that is primarily radial (or fleural.. The approimate beam fuctio method I geeral, solvig the roots of the characteristic equatio of Eq. ( for l m is ot possible i closed form. Hece, researchers have ofte favored towards usig approimate techiques. Accordig to previous studies, beam fuctios are widely used to obtai atural frequecies ad approimate displacemets for closed circular cylidrical shells. Accordig to the approimate method, for a shell simply supported at both eds, the ature of the aial mode ca be defied as: R lm = mp -. ( Substitutig Eq. ( ito Eq. (, the oly ukow of the characteristic equatio will be the frequecy parameter W for a fied combiatio of m ad. For cylidrical shells with simply supported boudary coditios, typical fleural, logitudial ad circumferetial odal patters accordig to differet m ad combiatios are show i Fig.. Although this approach is straightforward, it is a approimatio for boudary coditios of a simply supported circular cylidrical shell. However, sice cylidrical shell vibratio is totally differet compared to that of beams, it is importat to check the accuracy of this approimatio. Thus, a eact aalytical techique that uses the boudary

4 P. Oliazadeh et al. / Joural of Mechaical Sciece ad Techology ( (0 ~ (a (b M w w - w üï = - í ý - ïî R R R ïþ ( u æ ö - w üï N = w k í + R + - q ý - ïî ø R ïþ M N w w üï = - í + ý - ïî R ïþ ( Eh ì u æ öü = w. í + - ý - î R øþ ( ( coditio equatios to obtai resoace frequecies is proposed. Results of both the approimate ad eact methods are compared with eperimetal ad umerical data.. The eact method This sectio presets a eact aalysis of the vibratio of cylidrical shells. The method ca be applied to ay of the theories discussed. At each ed of the cylidrical shell, four boudary coditios must be specified. For the simply supported shell the followig boudary coditios are imposed: w = 0, v = 0, M = 0, N = 0 at = 0,. ( These coditios restrai the v ad w compoets of shell displacemets at their mutual boudaries ad will cause egligible iteral bedig momet M ad membrae ormal force N y i the shell as the shell deforms. Usig the approach metioed i Ref. [] ad elimiatig the oliear terms, the boudary coditio equatios for a simply supported shell are obtaied as follows for Soedel [], Flugge [], Morley-Koiter [] ad Doell s theory [], respectively: M N M w æ w öüï = - í + - ý - R ïî øïþ ( Eh ì u æ öü = w í + + ý R - î øþ w æ w ö u üï = - í ý - R q q ïî ø R ïþ ( (c Fig.. Mode shapes of a cylidrical shell: (a circumferetial mode shapes; (b logitudial ad radial mode shapes; (c odal arragemet of a cylidrical shell for =, m =. u æ ö w üï N = w kr í + R + - ý q - ïî ø ïþ ( ( Substitutig the modal displacemets ito these costraits, leads to a set of eight homogeous equatios, epressed as follows: { } = { 0} H b (0 i which ω, λ m ad (,..., bi i = are the te ukows. For a otrivial solutio of Eq. (0 oe requires det H = 0. ( The drivig frequecies are obtaied by simultaeously solvig both the characteristic equatios for the displacemet coefficiet matri, Eq. (, ad the boudary coditio determiat, Eq. (.. Results ad discussio. Validatio of the eact aalysis A Matlab program was writte to obtai the resoace frequecies of a shell, usig the eact method described i the previous sectio. First, eact resoace frequecies were obtaied based o the Soedel theory. Cosider a cylidrical shell with dimesios of /R =, h/r = /0 ad ν = /. I order to yield the eact frequecies, oe has to solve Eq. ( i terms of the o-dimesioal frequecy parameter, W. I Figs. -, the determiat of the boudary coefficiet matri (Eq. ( is calculated for costat values of circumferetial wave parameter ( =,,,,. As poited out i sectio., to obtai a otrivial solutio, the boudary coefficiet determiat should be equal to zero at the resoace frequecy, W. However, as it ca be see by Figs. -, oe of the determiats reach zero. This fact is completely eplaiable cosiderig the sesitivity of Eq. ( to W. To obtai the determiat of the boudary coefficiet matri i Figs. -, a frequecy sweep was carried out usig steps of DW = O the other had, Eq. ( is very sesitive to frequecy variatios. Thus, although the determiats represeted i Figs. - should actually equal to zero at resoace frequecies, sice the frequecy steps are ot small eough, they may ot make the determiat eactly zero. I Figs. -, the determiats are ot eactly zero at ay frequecies, but several miimum

5 P. Oliazadeh et al. / Joural of Mechaical Sciece ad Techology ( (0 ~ Fig.. Determiat of the boudary coefficiet matri versus frequecy parameter for =. Fig.. Determiat of the boudary coefficiet matri versus frequecy parameter for =. Fig.. Determiat of the boudary coefficiet matri versus frequecy parameter for =. Fig.. Determiat of the boudary coefficiet matri versus frequecy parameter for =. Fig.. Determiat of the boudary coefficiet matri versus frequecy parameter for =. poits are observed. These miimum poits are actual represetatios of the resoace frequecies of the shell, at which the determiats are zero. However, as discussed i sectio., the miimum poits should also satisfy Eq. ( i order to be the resoace frequecy of the system. Thus, i Figs. -, the umarked miimum poits before did ot satisfy Eq. ( ad are ot a resoace frequecy. Accordigly, at higher frequecies, higher aial wave parameters (m occur. Figs. - are graphical represetatios of how the eact resoace frequecies are obtaied usig the method described i this paper. Accordig to the eact method, each miimum poit i the diagrams represets a resoace frequecy with a specific mode shape. Net, let us ivestigate the accuracy of usig beam fuctios as a alterative to the eact method. To study such a approimatio, atural frequecies were calculated usig two methods: the approimate method (i which λ m = mπ(r/ - ad the eact method (Eq. (. I Tables -, the values of the o-dimesioal frequecy parameter W are compared for a simply supported cylidrical shell. Four diverse theories have bee applied to evaluate their accuracy. All o-dimesioal frequecies are calculated up to the fourth digit. As it ca be see, the approimate beam fuctio aalysis yields close results with the least errors. I Tables -, mode shapes at which the approimate method obtais errors are marked by a (* sig. As oe ca see from the comparisos of Tables -, there are some miute discrepacies, which ca be classified ito two groups for all theories: first, those related to the m = modes, ad secod, the differeces related to the m ³ modes. I m = modes, the values of the errors are betwee percet. However, for the m ³ modes, the errors are betwee percet, which are maily the result of a roudig error to the fourth digit. Moreover, accordig to Tables -, as the aial wave parameter m icreases, the umber of resoace frequecies cotaiig a error decrease. Hece, the approimate method is similar to the eact aalysis for m ³ modes ad for low mode umbers of m = with ³. O the other had, for low mode umbers of m = with, the approimate method yields high

6 0 P. Oliazadeh et al. / Joural of Mechaical Sciece ad Techology ( (0 ~ Table. Frequecy parameter for the Soedel s shell theory by the approimatio method with /R =, h/r = /0 ad ν = /. Table. Frequecy parameter for the Morley-Koiter s shell theory by the approimatio method with /R =, h/r = /0 ad ν = /. 0 m = m = m = m = m = m = m = m = m = m = Table. Frequecy parameter for the Soedel s shell theory by the eact method with /R =, h/r = /0 ad ν = /. Table. Frequecy parameter for the Morley-Koiter s shell theory by the eact method with /R =, h/r = /0 ad ν = /. 0 m = 0.0 * 0. * 0.0 * * *.. m = 0.0 * * 0. * *.0 *.0 m = *.. * m = *.0 *. m = * *. 0 m = 0. * 0.0 * 0. * 0. * 0. * *.0. m = 0.0 * * 0.0 * 0. * m = * m = * 0.0 * *.0. m = * *. *. * *.0 *. *..0 *. * *.0 Table. Frequecy parameter for the Flugge s shell theory by the approimatio method with /R =, h/r = /0 ad ν = /. Table. Frequecy parameter for the Doell s shell theory by the approimatio method with /R =, h/r = /0 ad ν = /. 0 m = m = m = m = m = m = m = m = m = m = Table. Frequecy parameter for the Flugge s shell theory by the eact method with /R =, h/r = /0 ad ν = /. Table. Frequecy parameter for the Doell s shell theory by the eact method with /R =, h/r = /0 ad ν = /. N 0 m = 0. * 0.00 * 0. * 0. * 0.0 * *.. m = 0.00 * * m = *. m = * *. * m = *. * *. *.0 * *..0 *. 0 m = 0. * 0. * 0. * 0. * 0. * *. m = * 0. * 0.0 * 0. * * m = * *. * m = * *. * m = * *. *. * *.0 *...0 *.0.00 *.0.. *.0 *... *.. *. errors ad so is ot accurate for low mode umbers. Thus, it should oly be used for high mode umbers with high resoace frequecies. Comparig the differet approimate theories listed i Tables -, the approimate Soedel theory is the most accurate. I cotrast, the approimate Morley-Koiter theory yields the

7 P. Oliazadeh et al. / Joural of Mechaical Sciece ad Techology ( (0 ~ highest errors. However, for all theories as the frequecy ad mode umbers icrease, the errors of the approimatio decrease. The eact ad approimate methods were applied to the four theories discussed: Soedel, Flugge, Morley- Koiter ad Doell. I Table, the eact ad approimate results accordig to these theories are compared to a eperimet held by Farshidiafar et al. []. The simply supported circular cylidrical shell ivestigated i Table is made of alumium with material properties of E =. GPa, r = 00 Kg m ad u = 0.. The dimesios of the shell are =. m, R = 0.0 m ad h = 0.00 m. Simulatios were carried out usig a fiite elemet model (FEM. The fiite elemet aalysis was modeled i the commercial fiite elemet aalysis (FEA software ANSYS. The FEA was developed i the followig order: Pre-processig: icludes defiig the costructig mesh, elastic ad stregth properties, elemet type (Solid for liear isotropic shell ad Shell for liear orthotropic shell, layer orietatio, thickess for orthotropic shell, ad geometrical modelig. Processig: i this sectio, the fiite elemet model of the modal aalysis is proposed ad costructed. Also, boudary coditio effects (such as v = w = 0 are applied to the boudary odes. Post-processig: this is the fial step that presets the results accordig to the type of FEM aalysis. First, let us study the accuracy of the Soedel eact theory, sice it is the mai focus of this research. As show i Table, the eact method applied to the Soedel theory is foud be much more accurate tha the approimate method. Iterestigly, the eact method calculates the fudametal frequecy ( m, = {(, } with early o errors. Moreover, it is remarkable that the eact aalysis predicts five resoace modes, ( m, = {(,,(,,(,,(,,(,}, with errors of equal or less tha oe percet. Although the errors of the eact method are higher tha those of the approimate at some resoace frequecies, the differece i the errors are small ad egligible. However, the eact aalysis has reduced the errors of most resoace frequecies dramatically. For eample, at mode shapes of ( m, = {(,,(,}, the percet error of the approimate theory is reduced to less tha percet usig the eact aalysis. O the other had, some errors eist at low mode umbers for both the eact ad approimate methods, especially for ( m, = {(,}. However, as the frequecy icreases, the errors decrease. Thus, at high mode umbers, the eact method of the Soedel theory is i complete agreemet with eperimetal ad umerical results. Such a tred is also observed for the approimate method; however, with higher errors at low mode umbers. Accordig to Tables ad 0, the Flugge theory is also very similar to the Soedel theory, both i terms of accuracy ad errors. Although at low mode umbers the errors of the Flugge theory are higher tha those of the Soedel s, the eact results of this theory at high mode umbers have less errors. For eample, i modes Table. Compariso betwee eact ad approimatio aalysis with eperimetal ad umerical data (FEM. m Eperimet(Hz[] Soedel (Eact(Hz 0 Soedel (App.(Hz { } Flugge (Eact(Hz Flugge (App.(Hz ( m, = (,,(,, the eact Flugge theory obtais less errors compared to the Soedel theory. Moreover, comparig results of the eact ad approimate methods i the Flugge theory with eperimetal results, it is observed that the eact method is more accurate i early all frequecy rages. Thus, the eact method is a good choice whe applied to the Flugge theory. O the other had, a rather differet behavior is observed for the Morley-Koiter ad Doell theories. The eact method of the Morley-Koiter theory has high errors i most mode shapes, especially at low mode umbers. Hece, a eact aalysis of this theory may ot be suitable. The eact results of the Doell theory are more accurate tha the approimate method; however, there are a few modes at which the errors icrease slightly. The oly high error is yielded at mode ( m, = {(, }, i which the error is icreased dramatically to. percet. I coclusio, the eact method im- Morley-Koiter(Eact(Hz 0 0 Morley-Koiter (App.(Hz 0 0 Doell (Eact(Hz 0 0 Doell (App.(Hz 0 FEM(Hz 0 0 Table 0. Errors of the eact ad approimate methods with respect to eperimetal data. Morley Morley Soedel Soedel Flugge Flugge Doell -Koiter -Koiter m (Eact (App. (Eact (App. (Eact (% (% (% (% (% (Eact (% (App. (% Doell (App. (% FEM (%

8 P. Oliazadeh et al. / Joural of Mechaical Sciece ad Techology ( (0 ~ ogitudial Circumferetial mode shape mode shape m = = (a m = = D mode shape m =, = m =, = It was observed that, for low mode umbers, the approimate method yields differet results tha the eact method does, whereas, for high mode umbers, o sigificat discrepacies were oticed. Moreover, the approimate method based o the Soedel theory obtaied better results tha the other theories. To check the accuracy of the eact method, eperimetal ad umerical results were compared. Accordig to this compariso, the eact aalysis predicted most of the resoace frequecies with errors of less tha oe percet. I cotrast, the approimate method yielded high errors for some mode shapes. proved results of the approimate techique compared to eperimetal ad umerical data. This statemet is correct for all theories, ecept for the Morley-Koiter. The accuracy of the eact method icreased at high mode umbers. Geerally, the Soedel ad Flugge theories yielded the most accurate results compared to eperimetal data, predictig accurate resoace frequecies i low ad high mode umbers. The FEM results are foud to be i strog agreemet with the eperimetal oes as well. Of ote, at some mode shapes the FEM is eve more accurate tha the other theories discussed. I Fig., mode shapes are reported from the fiite elemet aalysis for ( m, = {(,,(,,(,} modes. Overall, the validity of the eact method was approved with compariso to eperimetal, umerical ad theoretical results. The approimate method was show to be imprecise at low mode umbers. However, at high mode umbers the approimate techique was i good agreemet with eperimetal ad eact results.. Coclusios (b m = = (c m =, = Fig.. Mode shapes of the circular cylidrical shell: (a for m =, = ; (b for m =, = ; (c for m =, =. The free vibratio of circular cylidrical shells with simply supported boudary coditios has bee studied usig four differet thi shell theories: Soedel, Flugge, Morley-Koiter ad Doell. The scope of the ivestigatio was focused upo the eact aalysis of atural frequecies. The approimate beam fuctio method was also evaluated. First, a graphical represetatio of the eact aalysis was preseted i order to fid the atural frequecies of a shell. Net, eact results of the four theories were compared to approimate calculatios. Refereces [] W. eissa, Vibratio of shells, US Govermet Pritig Office, Washigto DC, USA (. [] M. Amabili ad M. P. Paidoussis, Review of studies o geometrically oliear vibratios ad dyamics of circular cylidrical shells ad paels, with ad without fluid structure iteractio, Applied Mechaics Reviews, (00 -. [] M. Amabili, Noliear vibratios ad stability of shells ad plates, Cambridge Uiversity Press, New York, USA (00. [] Y. Kurylov ad M. Amabili, Polyomial versus trigoometric epasios for oliear vibratios of circular cylidrical shells with differet boudary coditios, Joural of Soud ad Vibratio, (00 -. [] W. Flugge, Stresses i shells, Spriger, New York, USA (. [] A. E. H. ove, O the small free vibratios ad deformatios of thi shells, Philosophical Trasactios of the Royal Society, A ( -. [] K. K. ivaov, Aisymmeric vibratios of simply supported cylidrical shells, PMM, ( -. [] S. A. Riehart ad J. T. S. Wag, Vibratio of simply supported cylidrical shells with logitudial stiffeers, Joural of Soud ad Vibratio, ( -. [] G. B. Warburto ad J. Higgs, Natural frequecies of thi catilever cylidrical shells, Joural of Soud ad Vibratio, (0 -. [0] C. B. Sharma, Calculatio of atural frequecies of fiedfree circular cylidrical shells, Joural of Soud ad Vibratio, ( -. [] J. Callaha ad H. Baruh, A closed-form solutio procedure for circular cylidrical shell vibratios, Iteratioal Joural of Solids ad Structures, ( -0. [] F. Moussaoui, R. Beamar ad R. G. White, The effects of large vibratio amplitudes o the mode shapes ad atural frequecies of thi elastic shells, Part I: Coupled trasversecircumferetial mode shapes of isotropic circular cylidrical shells of ifiite legth, Joural of Soud ad Vibratio, ( [] J. ee, Free vibratio aalysis of a hermetic capsule by pseudospectral method, Joural of Mechaical Sciece ad Techology, ( ( [] W. Soedel, Vibratios of shells ad plates, rd ed., Marcel

9 P. Oliazadeh et al. / Joural of Mechaical Sciece ad Techology ( (0 ~ Dekker, New York, USA (00. [] J. Callaha, Cylidrical shell vibratios: Closed-form aalysis ad measuremet via piezoelectric films, PHD Dissertatio, New Bruswick Rutgers, The State Uiversity of New Jersey, USA (. [] A. Farshidiafar, M. H. Farshidiafar, M. J. Crocker ad W. O. Smith, The vibratio aalysis of log cylidrical shells usig acoustical ecitatio, Joural of Soud ad Vibratio, (00 -. Appedi A A. Differetial operators accordig to the Doell theory D r - ( - E t + R R R r - ( - + E t = R - R + + R r - ( - E t - - R R q æ ö - + k R + R R ø A. Modifyig differetial operators accordig to the Soedel theory MOD = R R R A. Modifyig differetial operators accordig to the Flugge theory MOD R R R ( - æ - ö = 0 - ø - æ - ö æ ö - + R R q - + ø R ø A. Modifyig differetial operators accordig to the Morley-Koiter theory MOD = æ ö R q ø Appedi B B. Coefficiet matri accordig to the Soedel theory - + W + lm - - l m l m + æ - ö l m W + ( + k lm k ( lm ø - l m + k ( - lm W - + k ( lm - B. Coefficiet matri accordig to the Flugge theory æ - ö + æ - ö W + lm - ( + k - l m l m- klm + lm ø ø + æ - ö æ - ö l m W + ( + k lm klm - - ø ø æ - ö æ - ö - l m+ klm + lm - klm W - + k + k ( lm - - k ø ø B. Coefficiet matri accordig to the Morley-Koiter theory æ - ö + W + lm - - l m l m ø + æ - ö l m W + lm - ø -l m W - + k + k ( lm - ( lm - + Pouria Oliazadeh received his B.S. ad M.S. degrees from Ferdowsi Uiversity of Mashhad i 00 ad 0, respectively ad ow he is the studet of Ph.D. i Ferdowsi Uiversity of Mashhad. His research iterests iclude: vibratio, dyamics, acoustics ad composite structures. Mohammad H. Farshidiafar received the B.S. degree i mechaical egieerig from Ferdowsi Uiversity of Mashhad, Mashhad, Ira i 0. He is curretly pursuig the M.A.Sc. degree i mechaical egieerig at Uiversity of Waterloo, Waterloo, Caada. His research iterests iclude: vibratio, dyamics, acoustics ad cotrol. Mr. Farshidiafar has bee the subject of several awards. Aooshirava Farshidiafar received the B.S. ad M.S. degrees i mechaical egieerig from Uiversity of Tehra, Tehra, Ira. He received the Ph.D. i mechaical egieerig from Uiversity of Bradford, Bradford, Eglad, i 000. He is curretly a Professor at Ferdowsi Uiversity of Mashhad, Mashhad, Ira. The curret research iterests i Professor Farshidiafar s group iclude: vibratio, dyamics, acoustics ad ao techology. Professor Farshidiafar has authored ad co-authored over 0 publicatios.