Journal of Sound and Vibration (1997) 207(5), 647669ON THE FREE VIBRATION OF COMPLETELYFREE, OPEN, CYLINDRICALLY CURVED,ISOTROPIC SHELL PANELSN. S. B:rbrii, J. M. DiNsboN :Nb R. S. L:Ncir.DepartmentofAeronauticsandAstronautics, UniversityofSouthampton, Higheld,SouthamptonSO171BJ, England(Received17May1996, andinnal form14May1997)Theh-pversionoftheniteelementmethodisusedtofurnishadetailedstudyofthevibrationcharacteristics of completely free, open, cylindrically curved, isotropic shellpanels. Convergence studies have been carried out to ascertain the most appropriate mixofh-renementandp-enrichmentnecessarytogiveareasonablesolutionaccuracy; thisapproach has been corroborated by comparing results from the present methodology withthe workof other investigators andthe results forthcomingfromaproprietaryniteelement package. Extensive results have been presented for panels with aspect ratios of 05,1and2, thecurvatureof whichisgraduallyincreasedfromthat of aat plateat oneextreme, to a complete, although unjoined, cylinder at the other. Curves of frequency versusincluded angle (curvature) have been obtained for the rst 20 modes of each of these panels,and the following observations have been made: (i) the majority of the low frequency modesare virtually independent of shell curvature; and (ii) the modes that are heavily inuencedby curvature eects involve motion that is conned to the lengthwise edge regions of thepanel.7 1997AcademicPressLimited1. INTRODUCTIONThevibrationcharacteristicsof completelyfree, open, cylindricallycurvedshell panelshave received scant attention in the classical literature, because, like most other problemswith unrestrained edges, no exact analytical solution is available. Resort has to be madetoapproximatesolutions, andtheearliest of thesethat (indirectly) addressedthetitleproblem can be traced to Gontkevitch [1, 2]. He derived frequency determinants for openshellsusingtheRitzmethodwiththeassumeddisplacementeldbasedonproductsofconventional beam functions and curved-beam functions; however, no explicit numericalresultswerereported. ThisworkremainedtheonlyextantliteraturesourceonthetitleproblemthatLeissa[3]wasabletoreferenceinhiscomprehensivemonographof1973.It was not until 1984 that Leissa and Narita [4] provided the rst known set ofcomprehensive results for a completely free shallow shell, by using the Ritz method withpolynomial assumeddisplacement functions. Inasequel publishedtwoyearslater[5],NaritaandLeissasolvedthesameproblem, but nowwiththeshapeof theplatformboundaries generalized to accommodate curvilinear, rather than rectangular, edges.Subsequent literature searches [6] failed to reveal any further work on this particular topicprior to 1992; however, a number of relevant articles by Liew and Lim [710] have recentlyappeared in the literature devoted to the vibration of shallow shells. None of these paperscontains results for freely suspended cylindrical shell panels, although it is clear from theWestland Lecturer in Helicopter Engineering.0022460X/97/450647 +23 $25.00/0/sv971115 7 1997 Academic Press LimitedN. s. n:rbrii r1 :i. 648methodologyemployedbythese authors that their pb-2Ritz methodologywouldbecapableofanalyzingsuchproblems.Notwithstandingthis,itcanbeconcludedthatthecompletely free edge condition still remains one of the few largely unreported cases of the18 496[3]distinctlydierentpanelboundaryconditionsthatcanexistforthisproblem;furtherstudyisclearlydesirable.Inall theprecedingwork, withtheexceptionofGonkevitchs, theshell panelswereassumed to be shallow, and the resulting analyses were based on a simplied set of shellequations [4]. In what follows, no such restrictions are made, and it is possible to modelan arbitarily deep panel of constant curvature by using Loves full shell equations [11] withthe modication introduced by Arnold and Warburton [12]. (A full discussion concerningthecomparativemeritsanddemeritsofusingshallowanddeepshell theorieshasbeenpresentedbyLim, Kitipornchai andLiew[10].)The layout of this paper naturally divides into two major sections. In the rst part, thetheoretical model is developed, andsomeconvergencetests areconductedagainst theLeissa and Narita results [4]. In the second part, results are presented for three thin shellswithaspectratios(a/b)of05,1and2,thecurvaturesofwhicharegraduallyincreasedfromnothing(i.e., aat plate) until acompletealthoughunconnectedcylinder isobtained. Plotsarepresentedofmodal frequencyversusincludedangleovertherange0 EfE360, which clearly show the eect curvature has on the frequencies and modesof these panels. Some discussion, related to various features of the plots just mentioned,isgivenregardingtheunderlyingphysicsoftheproblem.The twofold purpose of this paper is, therefore, to provide a solution of the title problemandtoprovidesomeuseful benchmarkresultsforthoseengagedeitherinthedynamicanalysis of restrained cylindrical shell panels, where it is customary to solve the completelyfree(unrestrained)problemrstinordertocheckthattheeigensolverisreturningthecorrect numberof zerofrequencyrigidbodymodes, orthoseengagedinexperimentalwork, for which completely free edge conditions are the easiest to simulate in thelaboratory.2. METHODOFANALYSISAn h-p version of the nite element method is utilized in what follows, because it aordstheuserarobust,exible,andeconomicanalysistoolthatcanaccommodategeometricand material irregularities with greater facility than a conventional Ritz type offormulation. (If no such irregularities are present, which is the case for the title problem,then it can be shown that a single super-element with a high p-boostwhich is eectivelyaRayleighRitzmethodprovidesthemostecientsolution, andingwhichconcurswiththeworkofotherinvestigators[7, 9].)Theh-pmethodmayberegardedasthemarriageoftheconventional h-versionandp-version,inwhichconvergenceisnowsoughtbysimultaneouslyreningthemeshandincreasing the degree of the elements [1316]. For the type of problem under considerationhere, in which the motions in all three co-ordinate directions are coupled, it isadvantageous to represent both the out-of-plane displacements and the in-planedisplacements by the same set of assumed modes. There are two good reasons for doingthis: (i) it greatly reduces the computational eort required to calculate the element stinessandmassmatrices; and(ii)itsimpliestheelementassemblyprocess.An ascending hierarchy of K-orthogonal polynomials, used in conjunction with Hermitecubics, will furnishacomplete set of admissible displacement functions withfull C1continuitythat is, one that is capable of satisfying both displacement and slopecontinuity, inall threeco-ordinatedirections, across anelement boundary(this set isvinr:1ioN or rrrr, cirvrb snrii i:Nris 649T:nir1ThersttenC1shapefunctionsusedinthisworkHermitecubicshapefunctionsf1(j) =1/2 3/4j +1/4j3f2(j) =(1/8 1/8j 1/8j2+1/8j3)LEf3(j) =1/2 +3/4j 1/4j3f4(j) =(1/8 1/8j +1/8j2+1/8j3)LEK-orthogonal polynomial shapefunctionsf5(j) =1/8 1/4j2+1/8j4f6(j) =1/8j 1/4j3+1/8j5f7(j) =1/48 +3/16j25/16j4+7/48j6f8(j) =1/16j +5/16j37/16j5+3/16j7f9(j) =1/128 5/32j2+35/64j421/32j6+33/128j8f10(j) =5/128j 35/96j3+63/64j533/32j7+143/384j9summarizedinTable1). TheHermitecubicsareusedtodenethedisplacementsandrotations at the four nodes, and the hierarchical functions are used to provide additionaldegrees of freedom to the edges and interior of the element. (It is noted that an in-planerotation can be regarded as a drilling degree of freedom, and as such represents an in-planeshear strainrather than a displacementat a node.) The nodal displacements/rotationsand the amplitudes of the hierarchical functions along the edges and in the interior of thepanel element constitute the generalized co-ordinates of the problem. Interelementcompatibilityis achievedsimplybymatchingthegeneralizedco-ordinates at commonelementnodesandalongcommonedges.Considerathin, cylindricallycurved, shell element, ofuniformthicknessh, boundedalong its edges by the lines x =0, x =a, y =0 and y =b. (See Figure 1). Upon introducingthe element-local non-dimensional co-ordinates j, hwhichare relatedtothe elementCartesianco-ordinatesthroughj =2x/a 1andh =2y/b 1,thestrainenergyofthepanel [11, 12] isgivenbyU=12 g+11 g+11 g+h/2h/2{o}T{s} ab4dh dj, (1)Figure1. Thecurvedpanel element.N. s. n:rbrii r1 :i. 650wherethestraincomponentsareox2a11j0 z4a2121j2u(j, h)oy= 02b 01 zR111h1Rz4b2121h2v(j, h) ,GGGGGKkGGGGGLlGGGGGGGKkGGGGGGGLlGGGGGKkGGGGGLl gxy2b11h2a 01 2zR111jz8ab121j 1hw(j, h)i.e., {o} =[D]{d}, (2)andthematerial constitutiverelationshipisgivenbysx1 n 0 oxsy=E(1 n2)n 1 0 oy, i.e., {s} =[D]{o}. (3)GGGKkGGGLlGGGKkGGGLlGGGKkGGGLltxy0 0(1 n)2gxyBoth the in-plane displacement eldsu andv, and the out-of-plane displacement eldw, may approximately be represented by a nite series of the previously assumed modesinthej-andh-directions; namely,u(j, h) =spuxrx =1spuysx =1Xrx,sx frx (j)fsx (h), v(j, h) =spvxry =1spvysy =1Yry,sy fry (j)fsy (h), (4a, b)andw(j, h) =spwxrz =1spwysz =1Zrz,sz frz (j)fsz (h). (4c)Xrx,sx, Yry,sy and Zrz,sz are the [unknown] generalized co-ordinates of the problem, and eachsummationistakenover(anynumberof) passumedmodes. (Therst subscript onpdenotes thetypeof displacement eld; theseconddenotes whether it is inthe x- ory-direction.)Equation(4)canbewrittenmoresuccinctlyinmatrixnotationas{d} =[N]{q}, (5)where d ={u(j, h), v(j, h), w(j, h)}T, q ={Xrx,sx, Yry,sy, Zrz,sz}TandNis a rectangularmatrix containing the shape functions. Hence, substituting equations (2, 3, 5) into equation(1)yieldsU=12 {q}T$ab4 g+11 g+11 g+h/2h/2([D] [N])T[D] [D] [N] dh dj dz%{q}. (6)TheterminthesquarebracketsisclearlytheplateelementstinessmatrixKE.Likewise, inelement-local co-ordinates, thekineticenergyoftheplateisgivenbyT=12 rg+11 g+11 g+h/2h/2{d }T{d } ab4dh dj dz. (7)vinr:1ioN or rrrr, cirvrb snrii i:Nris 651Substitutingequation(5)intoequation(7)yieldsT=12 {q }$r ab4 g+11 g+11 g+h/2h/2[N]T[N] dh dj dz%{q }. (8)Evidently, theterminthesquarebracketsistheplateelementmassmatrixME.Both KEand MEare comprised of nine sub-matrices; namely, one direct bending, twodirect stretching, four coupled bendingstretching and two coupled stretchingstretching.Equation(4)givestheoverall orderas(puxpuy) +(pvxpvy) +(pwxpwy).The matrix multiplication and integration required to evaluate the element stiness andmass matrices shown in equations (6) and (8) was performed symbolically (for a maximumof ten assumed in-plane and out-of-plane modes in both the x- and y-directions) by usingthe computer algebra package MAPLE [17]. This enables all the entries in KEand MEtobedeterminedinexact, fractional, algebraicformat, andthendownloadedtothemainanalysis programwheretheyarestoredat afamilial level. Inthis way, nonumericalrounding errors are introducedinthe calculationof the element stiness andmassmatrices; also, onlyonesignicantcomputational expenseiseverincurred, because, bykeepingKEandMEcompletelygeneric, theintegrationhasonlytobeperformedonce.Theh-pmethodof assemblingtheelementstoobtaintheglobal stinessandmassmatricesisslightlydierent fromthestandardh-versiontechnique, onaccount of theinternal polynomial enrichment that is available in each element. The formulation of theelement stiness and mass matrices is in accordance with the theory espoused previously,with each entry in the matrix corresponding to a particular generalized co-ordinate. Therst stage in formulating an assembly process is to re-order these matrices such that thegeneralized co-ordinates q appear in the following sequence: nodal (i.e., corner), edge andpurelyinternal; thisisillustratedschematicallybyKEor ME=&[N.N][E.N][I.N][N.E.][E.E][I.E][N.I][E.I][I.I]'. (9)There are always the following: 16 w-wise, eight u-wise and eight v-wise nodal degrees offreedom( 0Ninthenotationofequation(9),whichcorrespondtotheuseofthefourHermite cubics; 4(pwx4) +4(pwy4) w-wise, 4(puy4) +8 +2(pux4) u-wise and4(pvx4) +8 +2(pvy4) v-wise edge degrees of freedom (0E in the notation of equation(9)), which correspond to the products of the Hermite cubics with the hierarchicalmodes; (pwx4) (pwy4) w-wise, 2(pux4) +(pux4) (puy4) u-wise and2(pvy4) +(pvx4) (pvy4) v-wise purely internal degrees of freedom(0I in thenotation of equation (9)), which correspond to the products of only the hierarchical modes.Itisnowpossibletoformtheglobal stinessandmassmatrices, byidentifying, andthenaddingtogether, all theliketermsfromanynumberof adjacent elementswhichcorrespond to common nodal and edge degrees of freedom along their interface. Note thefollowing: (i) thepurelyinternal modes fromoneelement cannot aect thepurelyinternal modes from any other element, and so these contributions to the global stinessand mass matrices remain unaected by the assembly processin fact, [I.I]Gwill be blockdiagonal; (ii) all edge-to-edge interfaces are fully conforming; and (iii) since the in-planemotioninthe shell (or plate) is governedbytwo, coupled, secondorder dierentialequations, it is onlynecessarytomatchthedisplacements uandvacross anelementinterface to ensureC0 continuity;rstderivative continuity (i.e., 1u/1h, 1v/1h, 1u/1j and1v/1j, whicheectivelydescribesthein-planeshearanddirectstrain/stressatapoint)N. s. n:rbrii r1 :i. 652across the interface will approximately be satised as a consequence of the energy methodused.The nal structure of KGand MGis similar to the partitioned format shown in equation(9), except that now the dimensions of the sub-matrices are signicantly larger than, andtheircontentsignicantlydierentfrom, thoseforanindividual element.Althoughavarietyofdierentboundaryconditionsmaybeappliedtothepanel bynullifyingtheappropriategeneralizedco-ordinates, itisnotnecessarytodothisinthecurrentwork, andKGandMGremaincompletelyintact.By assuming simple harmonic motion and the absence of any external forcing agency,the governing equations of motion can be derived by substituting the total potential energyexpression into Lagranges equation. This yields a standard matrix-eigenvalue problem oftheform[[K] V2[M]]{q} =0, (10)wherethenon-dimensional frequencyis renderednot interms of theusual shell ringfrequencybut ratherintermsof aat platefrequencyV2=rhv2b4/D, whereDistheexural rigidity of the panel Eh3/12(1 n2). The reasonfor choosing this frequencymeasure is two-fold: (i) it permits a direct comparison to be made with the work on shallowshells reported by other investigators; and (ii) it is the best way of quantifying the eecton the radian (dimensional) frequency of varying parameters such as the panels radius ofcurvature. The solution of equation (10) yields the natural frequencies. The rst six of theseare always zero, correspondingtorigid-bodymodes; these will be disregardedinthenumbering oftheelasticmodes. Correspondingtoeacheigenvalueisaneigenvector,which may be used in conjunction with equation (4) to recover the associated displacementof each element in the model, and hence the complete mode of the panel underconsideration.3. CONVERGENCESTUDYA convergence study was undertaken using three dierent h meshes, each with dierentamounts ofp-enrichment, for a completely free, square planform, shallow shell with thefollowinggeometry: a/h =100, a/b =101, theincludedanglef=2896, andPoissonration =03. The reason such unusual parameters have been chosen is simply to enableastrict,like-for-likecomparisontobeconductedagainsttheworkofLeissaandNarita[4]*; inthismanner, theirresultscanbeusedasabenchmarksolutionforthepresentmethod. The frequencies of the rst four elastic modes are presented in Table 2. As wouldbeexpected, thefrequencyparametersareseentoconvergemonotonicallyfromabove,yieldingupperboundstotheexactvalues. Thefour-andeight-elementmeshesprovidemarginallymoreaccurateresultsthanthesinglesuper-element, aswouldbeexpected,albeit at avastlyinatedcost consideringthenumber of degrees of freedom(dof)involved. However, it is worth noting that for a given number of degrees of freedom, thesolution arising from the use of a single super-element mesh is always the more accurate.Independent verication has been obtained from a standard nite element analysis usingthe proprietary package ANSYS [18]. The frequency results arising from dierent meshesconstructed fromthe four-noded SHELL-63 element are included in Table 2 forcompleteness. The results fromboth the h-p methodology and ANSYS agree to aremarkable degree; however, some dierences are noticed between these sets of results andthoseofLeissaandNarita. Therearetwolikelysourcesoferrorthatcanaccountfor* LeissaandNaritadenedtheaspectratioa/bintermsoftheprojectedlengthofthecurvededge, ratherthantheactual lengthofthecurvededgeadoptedhere.vinr:1ioN or rrrr, cirvrb snrii i:Nris 653T:nir2Convergencestudybasedonthreedifferent meshdesigns: therst four non-dimensionalfrequenciesof acompletelyfreecylindrical shell (a/h =100, a/b =101, f=2895andn =03)Currentmethod V1 V2 V3 V4 dofh =1 p =6 13.423 21.678 34.709 49.815 108p =8 13.403 21.475 34.148 48.917 192p =10 13.403 21.473 34.147 48.908 300h =4 p =6 13.403 21.475 34.151 48.926 320p =8 13.403 21.473 34.147 48.907 616p =10 13.403 21.473 34.147 48.906 1 008h =8 p =6 13.403 21.474 34.148 48.910 588p =8 13.403 21.473 34.147 48.907 1 160p =10 13.403 21.473 34.147 48.906 1 924ANSYS30 30 13.407 21.476 34.165 48.963 5 766ANSYS50 50 13.404 21.474 34.153 48.926 15 606ANSYS100 100 13.403 21.473 34.148 48.913 61 206LeissaandNarita[4] 13.508 22.073 34.868 48.703 75this dierence: (i) the dierent types of shell theory employed: and (ii) theacknowledgedlevelof(incomplete)convergenceinLeissaandNaritaswork.However,sucient agreement has been demonstrated to inspire condence in the currentmethodology.4. RESULTSRatherthanpresent aseriesof adhocresultsforarbitrarypanel geometries, it wasdecided that a more eective strategy would be to x the panel planform and thickness,and then vary the radius of curvature such that a at plate is realized at one extreme, anda full, although unjoined, cylinder is obtained at the other. By taking panel aspect ratiosa/b of 05, 1 and 2, and xing the shell thickness* h =b/2000 and Poisson ration =03,a large range and combination of geometric parameters can be studied without renderingthisarticleundulyprolix.In Figures 2, 3 and 4 are presented plots of frequency versus included angle for the rst20elastic modes of the aformentionedpanels withaspect ratios a/bof 05, 1and2respectively. All of these results have been generated using a xed mesh design comprisingfourelementsaroundtheshell circumference; thelevelsof p-enrichment intheh- andj-directionswereeight andtenrespectively, forall threecomponentsof displacement,within each element. This level of mesh renementwhich amounts to an 810 dofproblemensuredtheresultsfortherst20modes(25%ofthetotalnumberyielded)were fully converged. A logarithmic scale was chosen for the abscissa (included anglef)simply because the most complicated modal behaviour takes place within the rst 10, anditwasfeltimportanttoshowthisclearly.* For all three shells, the length of the curved edge, b, is xed, and the length of the straight edge, a, is variedtogivetheappropriateaspectratio.Hencebisusedasthebasisforthenon-dimensionalfrequencyfactorinequation (10), and the non-dimensional thickness-to-length ratio. The latter value of h/b =1/2000 has been usedsimplytopreservethethinshell criterion.N. s. n:rbrii r1 :i. 654On each gure, the shell ring frequency vR=(E/r)1/2/R, which becomesVR=f(b/h) [12(1 n2)]1/2ontheplatenon-dimensional frequencyscale, isplottedasadashed line. It is evident from each gure, but especially so on Figure 2 (a/b =05) that,for agivenshell, the modal density(number of modes per unit frequency) increasessignicantlyintheregionof theringfrequency, whichaccords withexpectation[19].In order to quantify further the vibrational behaviour of these panels, the rst 12 elasticmodes of the square planform shell with included angles of 0, 1, 10, 100 and 360 arepresented in Figures 5, 6, 7, 8 and 9 respectively. The modes for the other two rectangularpanels exhibit most, if not all, of the features observed for the square panel, and do notthereforewarrantinclusionhere.It should be noted that the lengthwise edges of the full, unjoined, cylinder cantheoretically overlap on part of their cycle, as typied by caseV3 in Figure 9. The resultsare not intended to suggest such a slit tube would constitute a viable structurerather,they complete the set of results presented here, and provide a stark contrast to the modesthatwouldexistinafreelysuspendedclosedcylindrical tube.4.1. crNrr:i onsrrv:1ioNsWhentheradiusof curvatureisinnite(correspondingtozeroincludedangle), thefrequenciesof acompletelyfreeat plateareobtained. (Note, however, that it isnotpossible to show these when using logarithmic axes.) It is well-known [20] that for platesFigure 2. Frequency variation as a function of included angle of curvature for the rst 20 modes of a curvedpanel (a/b =05, b/h =2000andn =03).vinr:1ioN or rrrr, cirvrb snrii i:Nris 655Figure 3. Frequency variation as a function of included angle of curvature for the rst 20 modes of a curvedpanel (a/b =10, b/h =2000andn =03).with certain aspect ratios, some of their modes are degenerate. For example, a completelyfreesquareplatewill havethefollowing(elastic) degeneratepairs: modes 4and5atV=34801, modes 6 and 7 atV=61093, modes 11 and 12 atV=105461, and modes15 and 16 at V=131472. Reference to Figure 3 makes it clear that such degeneracies aredestroyedforall fq0; theapparentdisplayofdegeneratepairsuptof=01arisesas aconsequenceof therelativelylargescaleusedonthefrequencyaxis. Thereasondegenerate pairs separate upon the introduction of curvature is due to the now dierentexural stinesses in thej- andh-directionsa curved shell is stier in a direction alongitsgeneratorthaninadirectionarounditscircumference. Thiseectivelyrendersthestinessofthepanel orthotropic, andhencedegeneratemodesforthesquareatplatebecomedistinctlydierentwhenthepanel iscurvedslightly.For all three shells considered here, the frequency of the fundamental mode, and someof the higher modes, is more or less independent of the amount of curvature. Anexplanationcanbefoundbyexaminingtherstandsecondmodesofvibrationofthesquarepanel showninFigures59, casesV1andV2.Therstmodeconsistsofapredominantlytwistingactionaboutthej-andh-axes,leading to an antisymmetricantisymmetric mode of vibration. The torsional rigidity abouteither axis is unaected by curvature, but the polar second moment of area (which dictatesthe mass inertia), and the separation between the centre of mass (bending) and the shearN. s. n:rbrii r1 :i. 656centre (twisting), will change withincreasingcurvature. These latter eects aect thefrequencyofthismodeonlywhentheangleofcurvatureexceeds100.FortherangeofanglesshowninFigure3, thesecondmodeinvolvespredominantlyout-of-plane bendinginthe h-direction, withlittle or noin-plane extensional motion(middlesurfacestretching).(Acontourplottakenatz =0wouldshownodallinesthatarenearlyparallel tothegeneratorofthecylinder, becomingincreasinglystraightwithincreasing curvature.) Because Ih does not change with increasing curvature, the frequencyagain will remain constant as a function of curvature until elastic coupling eects betweenthein-planeandout-of-planemotionsbecomesignicant. ThesendingsareconsistentwiththeresultsgivenbyLeissaandNarita[4].(ItcanbenotedfromFigure5thatthesecond mode of the at plate is distinctly dierent from, and occurs at a signicantly lowerfrequency than, the results reported for panels withfe1. Again, this accords with thendingsofLeissaandNarita[4].)Also, the greater the value of a/b, the greater the tendency for the lower frequency modestoinvolvebendingbotharoundandalongthelengthofthepanel.Forthisreason,therst four modes of the a/b =05 panel consist only of bending motions around the shellcircumference, whereas for the a/b =2 panel, the corresponding four modes areinterleavedwithothermodesinvolvingbendinginbothdirections.It canbeseeninFigures2, 3and4that alot of complicatedlowfrequencymodere-orderingtakesplacewithintherst10ofincludedangle,afterwhichamoresettledFigure 4. Frequency variation as a function of included angle of curvature for the rst 20 modes of a curvedpanel (a/b =20, b/h =2000andn =03).vinr:1ioN or rrrr, cirvrb snrii i:Nris 657trend appears. For the square panel, this trend may best be described between10 EfE90 as a series of pairs of horizontal lines, traversed by (in this case) three steeplyascending curves (denoted A, B and C in Figure 3). For included angles in excess of 90,a general reduction in the frequencies of all but the rst three modes is apparent. (Althoughthis latter range is somewhat compressedinFigures 2, 3and4due tothe use of alogarithmic scale, it nonetheless represents three-quarters of the includedangle rangeconsideredhere.) The observedreductioninfrequency occurs beyondthe previouslyquoted range of validity of shallow shell theory (this range has been indicated on Figure 3).To conclude this general overview, it is interesing to note that the frequencies of someof the early modese.g., modes 7 and 8 of the a/b =2 panel (see Figure 2)can increaseby as much as 50% when the panel is curved by some 5. Hence investigators should beaware of the diculties that maybe experiencedinobtainingconsistent results withexperimental workanyslight handlingdamage that introduces aminute amount ofcurvature would render useless any measurements taken on a supposedly at plate. (Notethat, forsmall angles, theratioof thepanel thicknesstothemid-spanriseabovethehorizontal can be shown to be 8h/bf; for the current shell, withh/b =1/2000 and 5 ofincluded angle, this ratio equals 0046. In other words, the mid-span rise is approximately22timesthepanelthickness,which,althoughsmall,isnonethelessanoticeableamountofcurvature.)4.2. sirciric onsrrv:1ioNs ror 1nr sqi:rr ii:Nrorx i:NriIntheprecedingsectionitwasexplainedwhythefrequenciesofthersttwomodesremained more or less independent of the amount of shell curvature. Likewise, the generaltrend of the modal frequency curves to the right of the ring frequency curve were described.However, itisworthconsideringthismodal behaviouringreaterdetail, sincetherearesomeinterestingmanifestationsthat hithertohavenot beenreported. WiththeaidofFigures 59, it can be seen that the series of near horizontal pairs of lines shown in Figure 3to the right of the ring frequency curve correspond to modes exhibiting increasing numbersof half-waves in the (h) circumferential direction, with near linear symmetric andantisymmetricdisplacementsinthelength-wise(j)direction.Foragivenpair, themodeexhibitingsymmetryabouttheh-axisalwaysoccursatalowerfrequencythanitscompanionmodeexhibitingasymmetrythatis, straightedgetwistingabout the h-axis. A view directly on to the curved end of the panel would conrmthatthesemodesarethepanelequivalentofthefreefreemodesofacurvedbeam.Tothe left of the ring frequency, this dichotomous behaviour is less obvious, although it canstill be identied, albeit in a less distinct fashion. The frequencies of this family of modesare largely independent of the amount of curvature present since they all involve bendingaroundthecircumference, withnodal linesparallel withashell generator.Perhaps of greater interest is the signicance of the three steeplyascendingcurves(denotedA,BandCinFigure3);actuallytherearetwooverlappinglinesconstitutingeach of these curves. Consider curve A: (almost) degenerate modes 10 and 11 occur atfrequencies of V=274314and274612respectivelyfor the100 shell; closelyspacedmodes6and7occuratfrequenciesofV=103053and105859respectivelyforthe10shell.FromFigure8(casesV10andV11)andFigure7(casesV6andV7),itcanbeseenthat these modes involve elastic motion which is restricted to the panel edges ath =21,with the rest of the shell remaining motionless. These panel edge modes always occurin symmetric/antisymmetric pairs about thej-axis, and exhibit one clear half wavelengthof motion along the length. In eect, the lengthwise edges are behaving as uncoupled, highaspect ratio, cantilevered shell panels. Curve A thus describes the pair of fundamentalpanel edge modes. It can also be seen from Figures 7 and 8 that increasing the curvatureN. s. n:rbrii r1 :i. 658vinr:1ioN or rrrr, cirvrb snrii i:Nris 659Figure5.Therst12modesofvibrationofasquareplanformpanel(f=0).N. s. n:rbrii r1 :i. 660vinr:1ioN or rrrr, cirvrb snrii i:Nris 661Figure6.Therst12modesofvibrationofasquareplanformpanel(f=1).N. s. n:rbrii r1 :i. 662vinr:1ioN or rrrr, cirvrb snrii i:Nris 663Figure7.Therst12modesofvibrationofasquareplanformpanel(f=10).N. s. n:rbrii r1 :i. 664vinr:1ioN or rrrr, cirvrb snrii i:Nris 665Figure8.Therst12modesofvibrationofasquareplanformpanel(f=100).N. s. n:rbrii r1 :i. 666vinr:1ioN or rrrr, cirvrb snrii i:Nris 667Figure9.Therst12modesofvibrationofasquareplanformpanel(f=360).N. s. n:rbrii r1 :i. 668of the panel decreases the width of the curved edge capable of motion and thus eectivelyraises the aspect ratio of the cantilevered shell edge and hence increases the frequencyof its edge modes. (A similar eect has been noted by Lee, Leissa and Wang [21] concerninglocalized bending along the free edges of a cantilevered shell panel, suggesting thismechanismmanifestsitselfinstructuresotherthancompletelyfreeones.)The rst overtone of this pair of edge modes can be found on curve B in Figure 3,andvisualizedfrommodes10and11(V=184875and186059)inFigure7(casesV10andV11) for the 10 shell. Twohalf-wavelengths nowexist alongbothh-wise edges.Likewise, curve C describes the second overtone of the panel edge modes in this series,withthreehalf-wavelengthsalongeachedge, andsoon.Where an edge mode curve crosses a closely spaced pair of beam mode curves,forexampleintheregionmarkedbyadottedcircleonFigure3,acomplexinteractionand exchange of modal characteristics takes place. Whilst the nature of this quid pro quoremains uncertain, what can be said is that the identity of modes 10 and 11 to the left ofcurve B is retained by modes 8 and 9 to the right of curve B; also, the nature of the twoedge modes denoted on curve A by the letter X are identical to the two edge modesdenoted further up the curve A by the letter Y. In other words, crossovers of the type justdescribed do not alter the nature of any of the modes either side of the crossing.* Clearly,theexistenceofedgemodesisverymuchaconsequenceofthecompletelyfreepanelboundaryconditions.5. CONCLUSIONSA detailed study of the vibration characteristics of completely free, open, cylindricallycurved,isotropicshellpanelshasbeenundertakenbyusingtheh-pversionoftheniteelement method. The convergence properties of this shell element have been established,therebyrenderingit suitable for use inthe parameter studies that follow. Curves offrequency versus included angle have been obtained for the rst 20 modes of panels rangingfromaatplateatoneextreme,toafull,althoughunjoined,cylinderattheother.Byxing the panel aspect ratios a/b at 05, 1 and 2 respectively, and maintaining a constantthickness-to-length ratioh/b of 1/2000, results have been generated for a large family ofshell panels.The curves of frequency versus included angle illustrate (i) that the majority of the lowfrequency modes of completely free, open, curved panels, are virtually independent of shellcurvature, and(ii) that the relativelyfewmodes that are most heavilyinuencedbycurvature eects all involve motion conned to the lengthwise edges of the panel. Similarmodes have been observed for cantilevered cylindrical panels [21], and it can be concludedthat the fewer the constraints on a curved panel are, the more strongly these edge modesmanifestthemselves.ACKNOWLEDGMENTSThe nancial assistance of EPSRC, contract number GR/J 06306, is gratefullyacknowledged. TheANSYSresultswerecomputedcourtesyofMrG. S. Aglietti.REFERENCES1. V. S. GoN1irvi1cn 1962 Transactions Akademii Nauk USSR, Kiev, LaboratoriyaHidraulichnykhMashyn.SbornikTrudou,No.10,2727.Freevibrationsofshallowshells(inRussian).* Withinthecrossoverregionitself, somemodere-structuringclearlywill takeplace.vinr:1ioN or rrrr, cirvrb snrii i:Nris 6692. V. S. GoN1irvi1cn1964Natural VibrationsofPlatesandShells. Kiev: NaukDumka; A. P.Filipov, editor. (TranslatedbyLockheedMissilesandSpaceCo.)3. A. W. Lriss:1973VibrationofShells(NASASP-288). Washington, D.C.: U.S. GovernmentPrintingOce.4. 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