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P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January-February 2013, pp.181-207 A Review on Non-Linear Vibrations of Thin Shells P.Kiran Kumar 1, J.V.Subrahmanyam 2, P.RamaLakshmi 3 1,3 Assistant Professor, Department of Mechanical Engineering, Chaitanya Bharathi Institute of Technology, Gandipet, Hyderabad – 500 075, 2 Professor, Department of Mechanical Engineering, Vasavi College of Engineering, Ibrahimbagh, Hyderabad – 500 0075,Abstract This review paper aims to present an computational facilities have enabled researchers toupdated review of papers, conference papers, abandon the linear theories in favor of non-linearbooks, dissertations dealing with nonlinear methods of solutions.vibrations of circular cylindrical thin shells. Thispaper surveyed mathematically, experimentally, 2. Literature Reviewanalytically, numerically analyzed vibrations of 2.1. GEOMETRICALLY NONLINEAR SHELLcylindrical shells. This includes shells of open type, THEORIESclosed type, with and without fluid interactions; A short overview of some theories forshells subjected to free and forced vibrations, geometrically nonlinear shells will now be given.radial harmonic excitations, seismicallyexcitations; perfect and imperfect shell structuresof various materials with different boundaryconditions. This paper presented 210 referencepapers in alphabetical order. This paper is presented as Geometricallynonlinear shell theories, Free and forcedvibrations under radial harmonic excitation,Imperfect shells, shells subjected to seismicexcitations, References.1. INTRODUCTION Most of the structural components are Figure 2.1 shows a circular cylindrical shell with thegenerally subjected to dynamic loadings in their co-ordinate system and the displacements of theworking life. Very often these components may have middle surface.to perform in severe dynamic environment where inthe maximum damage results from the resonant Shell geometry and coordinate systemvibrations. Susceptibility to fracture of materials due Donnell (1934) established the nonlinearto vibration is determined from stress and frequency. theory of circular cylindrical shells under theMaximum amplitude of the vibration must be in the simplifying shallow-shell hypothesis. Due to itslimited for the safety of the structure. Hence vibration relative simplicity and practical accuracy, this theoryanalysis has become very important in designing a has been widely used. The most frequently used formstructure to know in advance its response and to take of Donnell‟s nonlinear shallowshell theory (alsonecessary steps to control the structural vibrations referred as Donnell-Mushtari-Vlasov theory)and its amplitude.The non-linear or large amplitude introduces a stress function in order to combine thevibration of plates has received considerable attention three equations of equilibrium involving the shellin recent years because of the great importance and displacements in the radial, circumferential and axialinterest attached to the structures of low flexural directions into two equations involving only therigidity. These easily deformable structures vibrate at radial displacement w and the stress function F:large amplitudes. The solution obtained based on the The fundamental investigation on thelineage models provide no more than a first stability of circular cylindrical shells is due to Vonapproximation to the actual solutions. The increasing Karman and Tsien (1941), who analyzed the staticdemand for more realistic models to predict the stability (buckling) and the postcritical behavior ofresponses of elastic bodies combined with the axially loaded shells. In this study, it was clarifiedavailability of super that discrepancies between forecasts of linear models and experimental results were due to the intrinsic 181 | P a g e
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P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January-February 2013, pp.181-207simplifications of linear models; indeed, linear hardening or softening type, depending on theanalyses are not able to predict the actual buckling geometry of the lobe. Almost at the same time,phenomenon observed in experiments; conversely, Grigolyuk (1955) studied large-amplitude freenonlinear analyses show that the bifurcation path is vibrations of circular cylindrical panels simplystrongly subcritical, therefore, safe design supported at all four edges. He used the same shellinformation can be obtained with a nonlinear theory as Reissner (1955) and a two-mode expansionanalyses only. After this important contribution, for the flexural displacement involving the first andmany other studies have been published on static and third longitudinal modes. He also developed a singledynamic stability of shells. mode approach. Results show a hardening type Mushtari and Galimov (1957) presented nonlinearity. Chu (1961) continued with Reissner‟snonlinear theories for moderate and large work, extending the analysis to closed cylindricaldeformations of thin elastic shells in their book. In shells. He found that nonlinearity in this case alwaysthe book of Vorovich (1999) the nonlinear theory of leads to hardening type characteristics, which, inshallow shells is also discussed. some cases, can become quite strong. Sanders (1963) and Koiter (1966)developed a more refined nonlinear theory of shells, Cummings (1964) confirmed Reissner‟sexpressed in tensorial form; the same equations were analysis for circular cylindrical panels simplyobtained by them around the same period, leading to supported at the four edges; he also investigated thethe designation of these equations as the Sanders- transient response to impulsive and step functions, asKoiter equations. Later, this theory has been well as dynamic buckling. Nowinski (1963)reformulated in lines-of-curvature coordinates, i.e. in confirmed Chu‟s results for closed circular shells. Hea form that can be more suitable for applications; see used a single-degree-of-freedom expansion for theBudiansky (1968). According to the Sanders-Koiter radial displacement using the linear mode excited,theory, all three displacements are used in the corrected by a uniform displacement that wasequations of motion. introduced to satisfy the continuity of the Changes in curvature and torsion are linear circumferential displacement. All of the aboveaccording to both the Donnell and the Sanders-Koiter expansions for the description of shell deformation,nonlinear theories (Yamaki 1984). The Sanders- except for Grigolyuk‟s, employed a single modeKoiter theory gives accurate results for vibration based on the linear analysis of shell vibrations.amplitudes significantly larger than the shellthickness for thin shells. Evensen (1963) proved that Nowinski‟s analysis was not accurate, because it did not maintain Details on the above-mentioned nonlinear a zero transverse deflection at the ends of the shell.shell theories may be found in Yamaki‟s (1984) Furthermore, Evensen found that Reissner‟s andbook, with an introduction to another accurate Chu‟s theories did not satisfy the continuity of in-theory, called the modified Flügge nonlinear theory plane circumferential displacement for closed circularof shells, also referred to as the Flügge-Lur‟e-Byrne shells. Evensen (1963) noted in his experiments thatnonlinear shell theory (Ginsberg 1973). The Flügge- the nonlinearity of closed shells is of the softeningLur‟e-Byrne theory is close to the general large type and weak, as also observed by Olsondeflection theory of thin shells developed by (1965). Indeed, Olson (1965) observed a slightNovozhilov (1953) and differs only in terms for nonlinearity of the softening type in the experimentalchange in curvature and torsion. Additional nonlinear response of a thin seamless shell made of copper; theshell theories were formulated by Naghdi and measured change in resonance frequency was onlyNordgren (1963),using the Kirchhoff hypotheses, and about 0.75 %, for a vibration amplitude equal to 2.5Libai and Simmonds (1988). times the shell thickness. The shell ends were attached to a ring; this arrangement for the boundary 2.2. Free and Forced (Radial Harmonic conditions gave some kind of constraint to the axialExcitation) Vibrations of Shells displacement and rotation. Kaña et al. (1966; see also The first study on vibrations of circular Kaña 1966) also found experimentally a weakcylindrical shells is attributed to Reissner (1955), softening type response for a simply supported thinwho isolated a single half-wave (lobe) of the circular cylindrical shell.vibration mode and analyzed it for simply supportedshells; this analysis is therefore only suitable for To reconcile this most importantcircular panels. By using Donnell‟s nonlinear discrepancy between theory and experiments,shallow-shell theory for thin-walled shells, Reissner Evensen (1967) used Donnell‟s nonlinear shallow-found that the nonlinearity could be either of the shell theory but with a different form for the assumed 182 | P a g e
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P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January-February 2013, pp.181-207flexural displacement w, involving more modes. approach and shows that nonperiodic (moreSpecifically, he included the companion mode in the specifically, quasiperiodic) motion is obtained foranalysis, as well as an axisymmetric contraction free nonlinear vibrations. In their analysis based onhaving twice the frequency of the mode excited. Donnell‟s theory, the same expansion introduced by Evensen, without the companion mode, was initially Evensen‟s assumed modes are not moment- applied; the backbone curves (pertaining to freefree at the ends of the shell, as they should be for vibrations) of Evensen were confirmed almostclassical simply supported shells, and the exactly. A second expansion with an additionalhomogeneous solution for the stress function is degree of freedom was also applied for shells withneglected; however, the continuity of the many axial waves; finally, an expansion with morecircumferential displacement is exactly satisfied. axisymmetric terms was introduced, but theEvensen studied the free vibrations and the response corresponding backbone curves were not reported. Into a modal excitation without considering damping their analysis based on the Sanders-Koiter theory, anand discussed the stability of the response curves. His original expansion for the three shell displacementsresults are in agreement with Olson‟s (1965) (i.e. radial, circumferential and axial) was used,experiments. Evensen‟s study is an extension of his involving 7 degrees of freedom. However, only onevery well-known study on the vibration of rings term was used for the flexural displacement, and the(1964, 1965, 1966), wherein he proved theoretically axisymmetric radial contraction was neglected;and experimentally that thin circular rings display a axisymmetric terms were considered for the in-planesoftening-type nonlinearity. The other classical work displacements. The authors found that the backboneon the vibration of circular rings is due to Dowell curves for a mode with two circumferential waves(1967a) who confirmed Evensen‟s results and predict a hardening-type nonlinearity which increasesremoved the assumption of zero mid-surface with shell thickness.circumferential strain. Evensen (1968) also extendedhis work to infinitely long shells vibrating in a mode Matsuzaki and Kobayashi (1969a, b) studiedin which the generating lines of the cylindrical theoretically simply supported circular cylindricalsurface remain straight and parallel, by using the shells (1969a), then studied theoretically andmethod of harmonic balance and without considering experimentally clamped circular cylindrical shellsthe companion mode. Evensen (2000) published a (1969b). Matsuzaki and Kobayashi (1969a) basedpaper, in which he studied the influence of pressure their analysis on Donnell‟s nonlinear shallow-shelland axial loading on large-amplitude vibrations of theory and used the same approach and modecircular cylindrical shells by an approach similar to expansion as Evensen (1967), with the opposite signthe one that he used in his previous papers, but to the axisymmetric term because they assumedneglecting the companion mode; consequently only positive deflection outwards. In addition, theybackbone curves, pertaining to free vibrations, were discussed the effect of structural damping and studiedcomputed. in detail travelling-wave oscillations. In the paper, considering clamped shells, Matsuzaki and It is important to note that, in most studies, Kobayashi (1969b) modified the mode expansion inthe assumed mode shapes (those used by Evensen, order to satisfy the different boundary conditions andfor example) are derived in agreement with the retained both the particular and the homogeneousexperimental observation that, in large amplitude solutions for the stress function. The analysis foundvibrations, (i) the shell does not spend equal time- softening type nonlinearity also for clamped shells, inintervals deflected outwards and deflected inwards, agreement with their own experimental results. Theyand (ii) inwards maximum deflections, measured also found amplitude-modulated response close tofrom equilibrium, are larger than outwards ones. resonance and identified it as a beating phenomenonHowever, it is important to say that these differences due to frequencies very close to the excitationare very small if compared to the vibration amplitude. frequency.Hence, the original idea for mode expansion was toadd to asymmetric linear modes an axisymmetric Dowell and Ventres (1968) used a differentterm (mode) giving a contraction to the shell. expansion and approach in order to satisfy exactly the out-of-plane boundary conditions and to satisfy “on Mayers and Wrenn (1967) analyzed free the average” the in-plane boundary conditions. Theyvibrations of thin, complete circular cylindrical studied shells with restrained in-plane displacementshells. They used both Donnell‟s nonlinear shallow- at the ends and obtained the particular and theshell theory and the Sanders-Koiter nonlinear theory homogeneous solutions for the stress function. Theirof shells. Their analysis is based on the energy interesting approach was followed by Atluri (1972) 183 | P a g e
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P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January-February 2013, pp.181-207who found that some terms were missing in one of and Cummings (1964); the converse was found for thethe equations used by Dowell and Ventres; Dowell et hyperbolic paraboloid. The analysis shows that for allal. (1998) corrected these omissions. The boundary shells, excluding the hyperbolic paraboloid but includingconditions assumed both by Dowell and Ventres and the circular cylindrical panel, a softening nonlinearity isby Atluri constrain the axial displacement at the shell expected, changing to hardening for very largeextremities to be zero, so that they are different from deflections (in particular for vibration amplitudethe classical constraints of a simply supported shell equal to 15 times the shell thickness for a circular(zero axial force at both ends). Atluri (1972) found cylindrical panel).that the nonlinearity is of the hardening type for aclosed circular cylindrical shell, in contrast to what El-Zaouk and Dym (1973) investigated freewas found in experiments. Varadan et al. (1989) and forced vibrations of closed circular shells havingshowed that hardening-type results based on the a curvature of the generating lines by using antheory of Dowell and Ventres and Atluri are due to extended form of Donnell‟s nonlinear shallow-shellthe choice of the axisymmetric term. More recently, theory to take into account curvature of theAmabili et al. (1999b, 2000a) showed that at least the generating lines and orthotropy. They also obtainedfirst and thirdaxisymmetric modes (axisymmetric numerical results for circular cylindrical shells; for thismodes with an even number of longitudinal half- special case, their study is almost identical to Evensen‟s (1967), but the additional effects of internal pressure andwaves do not give any contribution) must be included orthotropy are taken into account. Their numericalin the mode expansion (for modes with a single results showed a softening type nonlinearity, whichlongitudinal half-wave), as well as using both the becomes hardening only for very largedriven and companion modes, to correctly predict the displacements, at about 30 times the shell thickness.trend of nonlinearity with sufficiently good accuracy. Stability of the free and forced responses was also investigated. Sun and Lu (1968) studied the dynamic Ginsberg used a different approach to solvebehaviour of conical and cylindrical shells under the problem of asymmetric (Ginsberg 1973) andsinusoidal and ramp-type temperature loads. The axisymmetric (Ginsberg 1974) large-amplitudeequations of motion are obtained by starting with vibrations of circular cylindrical shells. In fact, hestrain-displacement relationships that, for cylindrical avoided the assumptions of mode shapes andshells, are those of Donnell‟s nonlinear shallow-shell employed instead an asymptotic analysis to solve thetheory. All three displacements were used, instead of nonlinear boundary conditions for a simply supportedintroducing the stress function F. Therefore, the three shell. The solution is reached by using the energy ofdisplacements were expanded by using globally six the system to obtain the Lagrange equations, which areterms, without considering axisymmetric terms in the then solved via the harmonic balance method. Bothradial displacement; these six terms were related softening and hardening nonlinearities were found,together to give a single-degree-of-freedom system depending on some system parameters. Moreover,with only cubic nonlinearity, and the shell thus Ginsberg used the more accurate Flügge-Lur‟e-Byrneexhibited hardening type nonlinearity. nonlinear shell theory, instead of Donnell‟s nonlinear Leissa and Kadi (1971) studied linear and shallow-shell theory which was used in all the worknonlinear free vibrations of shallow shell panels, discussed in the foregoing (excluding the two paperssimply supported at the four edges without in-plane allowing for curvature of the generating lines). Therestraints. Panels with two different. curvatures in Flügge-Lur‟e-Byrne theory is the same referred as theorthogonal directions were studied. This is the first modified Flügge theory by Yamaki (1984). Ginsbergstudy on the effect of curvature of the generating (1973) studied the damped response to an excitationlines on large-amplitude vibrations of shallow shells. and its stability, considering both the driven andDonnell‟s nonlinear shallow-shell theory was used in companion modes.a slightly modified version to take into account themeridional curvature. A single mode expansion of the Chen and Babcock (1975) used thetransverse displacement was used. The Galerkin perturbation method to solve the nonlinear equationsmethod was used; the compatibility equation was obtained by Donnell‟s nonlinear shallow-shell theory,exactly satisfied, and in-plane boundary conditions without selecting a particular deflection solution.were satisfied on the average. Results were obtained They solved the classical simply supported case andby numerical integration and non-simple harmonic studied the driven mode response, the companionoscillations were found. For cylindrical and spherical mode participation, and the appearance of a travellingpanels, phase plots show that, during vibrations, wave. A damped response to an external excitationinward deflections are larger that outward was found. The solution involved a sophisticateddeflections, as previously found by Reissner (1955) 184 | P a g e
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P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January-February 2013, pp.181-207mode expansion, including boundary layer terms in Raju and Rao (1976) employed the Sanders-order to satisfy the boundary conditions. They also Koiter theory and used the finite element method topresented experimental results in good agreement study free vibrations of shells of revolution. Theywith their theory, showing a softening nonlinearity. found hardening-type results for a closed circularRegions with amplitude-modulated response were cylindrical shell, in contradiction with allalso experimentally detected. experiments available. Their paper was discussed by Evensen (1977), Prathap (1978a, b), and then again It is important to state that Ginsberg‟s by Evensen (1978a, b). In particular, Evensen (1977)(1973) numerical results and Chen and Babcock‟s commented that the authors ignored the physics of(1975) numerical and experimental results constitute the problem: i.e., that thin shells bend more readilyfundamental contributions to the study of the than they stretch.influence of the companion mode on the nonlinearforced response of circular cylindrical shells. Ueda (1979) studied nonlinear freeMoreover, in these two studies, a multi-mode vibrations of conical shells by using a theoryapproach involving also modes with 2n equivalent to Donnell‟s shallow-shell theory and acircumferential waves was used. However, Ginsberg finite element approach. Each shell element was(1973) and Chen and Babcock (1975), in order to reduced to a nonlinear system of two degrees ofobtain numerical results, condensed their models to a freedom by using an expansion similar to Evensen‟stwo-degree-of-freedom one by using a perturbation (1967) analysis but without companion modeapproach with some truncation error. participation. Numerical results were carried out also for complete circular cylindrical shells and are in Vol‟mir et al. (1973) studied nonlinear good agreement with those obtained by Olson (1965)oscillations of simply supported, circular cylindrical and Evensen (1967).panels and plates subjected to an initial deviationfrom the equilibrium position (response of the panel Yamaki (1982) performed experiments forto initial conditions) by using Donnell‟s nonlinear large-amplitude vibrations of a clamped shell madeshallow-shell theory. Results were calculated by of a polyester film. The shell response of the firstnumerical integration of the equations of motion eight modes is given and the nonlinearity is ofobtained by Galerkin projection, retaining three or softening type for all these modes with a decrease offive modes in the expansion. The results reported do natural frequencies of 0.6, 0.7, 0.9 and 1.3 % fornot show the trend of nonlinearity but only the time modes with n = 7, 8, 9 and 10 circumferential wavesresponse. (and one longitudinal half-wave), respectively, for vibration amplitude equal to the shell thickness (rms Radwan and Genin (1975) derived nonlinear amplitude equal to 0.7 times the shell thickness). Hemodal equations by using the Sanders- Koiter also proposed two methods of solution of thenonlinear theory of shells and the Lagrange equations problem by using the Donnell‟s nonlinear shallow-of motion, taking into account imperfections. shell theory: one is a Galerkin method and the secondHowever, the equations of motion were derived only is a perturbation approach. No numerical applicationfor perfect, closed shells, simply supported at the was performed.ends. The nonlinear coupling between the linearmodes, that are the basis for the expansion of the Maewal (1986a, b) studied large-amplitudeshell displacements, was neglected. As previously vibrations of circular cylindrical and axisymmetricobserved, this singlemode approach gives the wrong shells by using the Sanders-Koiter nonlinear shelltrend of nonlinearity for a closed circular shell. The theory. The mode expansion used contains the drivennumerical results give only the coefficients of the and companion modes, axisymmetric modes andDuffing equation, obtained while solving the terms with twice the number of circumferentialproblem. Radwan and Genin (1976) extended their waves of the driven modes. However, it seems thatprevious work to include the nonlinear coupling with axisymmetric modes with three longitudinal half-axisymmetric modes; however, only an approximate waves were neglected; as previously discussed, thesecoupling was proposed. Most of the numerical results modes are essential for predicting accurately thestill indicated a hardening type nonlinearity for a trend of nonlinearity. It is strongly believed that theclosed circular shell; however, a large reduction of difference between the numerical results of Maewalthe hardening nonlinearity was obtained relative to (1986b) and those obtained by Ginsberg (1973) maythe case without coupling. be attributed to the exclusion of the axisymmetric modes with three longitudinal half-waves in the analysis. Results were obtained by asymptotic 185 | P a g e
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P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January-February 2013, pp.181-207analysis and by a speciallydeveloped finite element however, here forced vibrations were studied. Itmethod. should be recognized that travelling-wave responses were also obtained in all the other previous work that Nayfeh and Raouf (1987a, b) studied included the companion mode; however, thevibrations of closed shells by using plane-strain expansion used was different.theory of shells and a perturbation analysis; thus,their study is suitable for rings but not for supported Andrianov and Kholod (1993) used anshells of finite length. They investigated the response analytical solution for shallow cylindrical shells andwhen the frequency of the axisymmetric mode is plates based on Bolotin‟s asymptotic method, butapproximately twice that of the asymmetric mode they obtained results only for a rectangular plate. A(two-to-one internal resonance). The phenomenon of modified version of the nonlinear Donnell shallow-saturation of the response of the directly excited shell equations was used. Large amplitude vibrationsmode was observed. Raouf and Nayfeh (1990) of thin, closed circular cylindrical shells with wafer,studied the response of the shell by using the same stringer or ring stiffening were studied by Andrianovshell theory previously used by Nayfeh and Raouf et al. (1996) by using the Sanders-Koiter nonlinear(1987a, b), retaining both the driven and companion shell theory. The solution was obtained by using anmodes in the expansion, finding amplitude- asymptotic procedure and boundary layer terms tomodulated and chaotic solutions. Only two degrees of satisfy the shell boundary conditions. Only the freefreedom were used in this study; an axisymmetric vibrations (backbone curve) were investigated. Theterm and terms with twice the number of single numerical case investigated showed acircumferential waves of the driven and companion softening type nonlinearity.modes were obtained by perturbation analysis andadded to the solution without independent degrees of Chiba (1993a) studied experimentally large-freedom. The method of multiple scales was applied amplitude vibrations of two cantilevered circularto obtain a perturbation solution from the equations cylindrical shells made of polyester sheet. He foundof motion. By using a similar approach, Nayfeh et al. that almost all responses display a softening(1991) investigated the behaviour of shells, nonlinearity. He observed that for modes with theconsidering the presence of a two-to-one internal same axial wave number, the weakest degree ofresonance between the axisymmetric and asymmetric softening nonlinearity can be attributed to the modemodes for the problem previously studied by Nayfeh having the minimum natural frequency. He alsoand Raouf (1987a, b). found that shorter shells have a larger softening nonlinearity than longer ones. Travelling wave modes Gonçalves and Batista (1988) conducted an were also observed.interesting study on fluid-filled, complete circularcylindrical shells. A mode expansion that can be Koval‟chuk and Lakiza (1995) investigatedconsidered a simple generalisation of Evensen‟s experimentally forced vibrations of large amplitude(1967) was introduced by Varadan et al. (1989), in fiberglass shells of revolution. The boundaryalong with Donnell‟s nonlinear shallow-shell theory, conditions at the shell bottom simulated a clampedin their brief note on shell vibrations. This expansion end, while the top end was free (cantilevered shell).is the same as that used by Watawala and Nash One of the tested shells was circular cylindrical. A(1983); as previously observed, it is not moment-free weak softening nonlinearity was found, excluding theat the ends of the shell. The results were compared beam-bending mode for which a hardeningwith those obtained by using the mode expansion nonlinearity was measured. Detailed responses forproposed by Dowell and Ventres (1968) and Atluri three different excitation levels were obtained for the(1972). Varadan et al. (1989) showed that the mode with four circumferential waves (second modeexpansion of Dowell and Ventres and Atluri gives of the shell). Travelling wave response was observedhardening-type results, as previously discussed; the around resonance, as well as the expected weakresults obtained with the expansion of Watawala and softening type nonlinearity.Nash correctly display a softening type nonlinearity. Kobayashi and Leissa (1995) studied free Koval‟chuk and Podchasov (1988) vibrations of doubly curved thick shallow panels;introduced a travelling-wave representation of the they used the nonlinear first-order shear deformationradial deflection, in order to allow the nodal lines to theory of shells in order to study thick shells. Thetravel in the circumferential direction over time. This rectangular boundaries of the panel were assumed toexpansion is exactly the same as that introduced by be simply supported at the four edges without in-Kubenko et al. (1982) to study free vibrations; plane restraints, as previously assumed by Leissa and 186 | P a g e
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P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January-February 2013, pp.181-207Kadi (1971). A single mode expansion was used for Evensen (1977) on Nowinski‟s and Kanaka Raju andeach of the three displacements and two rotations Venkateswara Rao‟s results should be recalled toinvolved in the theory; in-plane and rotational inertia explain the hardening type of nonlinearity. Lakis etwere neglected. The problem was then reduced to one al. (1998) presented a similar solution for closedof a single degree of freedom describing the radial circular cylindrical shells.displacement. Numerical results were obtained forcircular cylindrical, spherical and paraboloidal A finite element approach to nonlinearpanels. Except for hyperbolic paraboloid shells, a dynamics of shells was developed by Sansour et al.softening behaviour was found, becoming hardening (1997, 2002). They implemented a finite shellfor vibration amplitudes of the order of the shell element based on a specifically developed nonlinearthickness. However, increasing the radius of shell theory. In this shell theory, a linear distributioncurvature, i.e. approaching a flat plate, the behaviour of the transverse normal strains was assumed, givingchanged and became hardening. The effect of the rise to a quadratic distribution of the displacementshell thickness was also investigated. field over the shell thickness. They developed a time integration scheme for large numbers of degrees of Thompson and de Souza (1996) studied the freedom; in fact, problems can arise in finite elementphenomenon of escape from a potential well for two- formulations of nonlinear problems with a largedegree-of-freedom systems and used the case of number of degrees of freedom due to the instabilityforced vibrations of axially compressed shells as an of integrators. Chaotic behaviour was found for aexample. A very complete bifurcation analysis was circular cylindrical panel simply supported on theperformed. straight edges, free on the curved edges and loaded by a point excitation having a constant value plus a Ganapathi and Varadan (1996) used the harmonic component. The constant value of the loadfinite element method to study large-amplitude was assumed to give three equilibrium points (onevibrations of doubly- curved composite shells. unstable) in the static case.Numerical results were given for isotropic circularcylindrical shells. They showed the effect of Amabili et al. (1998) investigated theincluding the axisymmetric contraction mode with nonlinear free and forced vibrations of a simplythe asymmetric linear modes, confirming the supported, complete circular cylindrical shell, emptyeffectiveness of the mode expansions used by many or fluid-filled. Donnell‟s nonlinear shallowshellauthors, as discussed in the foregoing. Only free theory was used. The boundary conditions on thevibrations were investigated in the paper, using radial displacement and the continuity ofNovozhilov‟s theory of shells. A four-node finite circumferential displacement were exactly satisfied,element was developed with five degrees of freedom while the axial constraint was satisfied on thefor each node. Ganapathi and Varadan also pointed average. Galerkin projection was used and the modeout problems in the finite element analysis of closed shape was expanded by using three degrees ofshells that are not present in open shells. The same freedom; specifically, two asymmetric modes (drivenapproach was used to study numerically laminated and companion modes), plus an axisymmetric termcomposite circular cylindrical shells (Ganapathi and involving the first and third axisymmetric modesVaradan 1995). (reduced to a single term by an artificial constraint), were employed. The time dependence of each term of Selmane and Lakis (1997a) applied the finite the expansion was general. Different tangentialelement method to study free vibrations of open and constraints were imposed at the shell ends. Anclosed orthotropic cylindrical shells. Their method is inviscid fluid was considered. Solution was obtaineda hybrid of the classical finite element method and both numerically and by the method of normal forms.shell theory. They used the refined Sanders-Koiter Numerical results were obtained for both free andnonlinear theory of shells. The formulation was forced vibrations of empty and water-filled shells.initially general but in the end, to simplify the Some additions to this paper were given by Amabilisolution, only a single linear mode was retained. As et al. (1999a). Results showed a softening typepreviously discussed, this approximation gives nonlinearity and travelling-wave response close toerroneous results for a complete circular shell. In resonance. Numerical results are in quantitativefact, numerical results for free vibrations of the same agreement with those of Evensen (1967), Olsonclosed circular cylindrical shell, simply supported at (1965), Chen and Babcock (1975) and Gonçalves andthe ends, investigated by Nowinski (1963) and Batista (1988).Kanaka Raju and Venkateswara Rao (1976) showed ahardening type nonlinearity. The criticism raised by 187 | P a g e
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P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January-February 2013, pp.181-207 In a series of four papers, Amabili, Pellicano number of modes can be retained. Numerical resultsand Païdoussis (1999b, c, 2000a, b) studied the with up to 23 modes were obtained and thenonlinear stability and nonlinear forced vibrations of convergence of the solution discussed. In particular,a simply supported circular cylindrical shell with and it is shown that the model developed by Amabili etwithout flow by using Donnell‟s nonlinear shallow- al. (2000a) is close to convergence. In that study ashell theory. The Amabili et al. (1999c, 2000a) map is also given, for the first time, showing whetherpapers deal with large-amplitude vibrations of empty the nonlinearity is softening or hardening as aand fluid-filled circular shells, also investigated by function of the shell geometry.Amabili et al. (1998), but use an improved model forthe solution expansion. In Amabili et al. (1999c) If the shell is not very short (L/R > 0.5) andthree independent axisymmetric modes with an odd not thick (h/R < 0.045) the nonlinearity is of thenumber of longitudinal half-waves were added to the softening type, excluding the case of long (L/R > 5)driven and companion modes. Therefore, the model thin shells. It is interesting to note that the boundarycan be considered to be an extension of the three- between softening and hardening regions has beendegree-of-freedom one developed by Amabili et al. found to be related to internal resonances (i.e. an(1998), in which the artificial kinematic constraint integer relationship between natural frequencies)between the first and third axisymmetric modes, between the driven mode and axisymmetric modes.previously used to reduce the number of degrees offreedom, was removed. Results showed that the first Large-amplitude (geometrically nonlinear)and third axisymmetric modes are fundamental for vibrations of circular cylindrical shells with differentpredicting accurately the trend of nonlinearity and boundary conditions and subjected to radial harmonicthat the fifth axisymmetric mode only gives a small excitation in the spectral neighbourhood of the lowestcontribution. resonances have been investigated by Amabili (2003b). In particular, simply supported shells with Driven modes with one and two longitudinal allowed and constrained axial displacements at thehalf-waves were numerically computed. Periodic edges have been studied; in both cases the radial andsolutions were obtained by a continuation technique circumferential displacements at the shell edges arebased on the collocation method. Amabili et al. constrained. Elastic rotational constraints have been(2000a) added, to the expansion of the radial assumed; they allow simulating any condition fromdisplacement of the shell, modes with twice the simply supported to perfectly clamped, by varyingnumber of circumferential waves vis-à-vis the driven the stiffness of this elastic constraint. The Lagrangemode and up to three longitudinal half-waves, in equations of motion have been obtained by energyorder to check the convergence of the solution with approach, retaining damping through the Rayleigh‟sdifferent expansions. Results for empty and water- dissipation function. Two different nonlinear shellfilled shells were compared, showing that the theories, namely Donnell‟s and Novozhilov‟scontained dense fluid largely enhances the weak theories, have been used to calculate the elastic strainsoftening type nonlinearity of empty shells; a similar energy. The formulation is valid also for orthotropicconclusion was previously obtained by Gonçalves and symmetric cross-ply laminated composite shells;and Batista (1988). Experiments on a water-filled geometric imperfections have been taken intocircular cylindrical shell were also performed and account. The large-amplitude response of circularsuccessfully compared to the theoretical results, cylindrical shells to harmonic excitation in thevalidating the model developed. Experiments showed spectral neighbourhood of the lowest naturala reduction of the resonance frequency by about 2 % frequency has been computed for three differentfor a vibration amplitude equal to the shell thickness. boundary conditions. Circular cylindrical shell withThese experiments are described in detail in Amabili, restrained axial displacement at the shell edgesGarziera and Negri (2002), where additional display stronger softening type nonlinearity thanexperimental details and results are given. It is simply supported shells. Both empty and fluid-filledimportant to note that, in this series of papers shells have been investigated by using a potential(Amabili et al. 1999b, c; 2000a, b), the continuity of fluid model.the circumferential displacements and the boundary A problem of internal resonance (one-to-conditions for the radial deflection were exactly one-to-one-to-two) was studied by Amabili, Pellicanosatisfied. and Vakakis (2000c) and Pellicano et al. (2000) for a water-filled shell. In these papers, both modal and Pellicano, Amabili and Païdoussis (2002) point excitations were used.extended the previous studies on forced vibrations ofshells by using a mode expansion where a generic 188 | P a g e
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P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January-February 2013, pp.181-207 Autoparametric resonances for free flexural (elastic support), becoming hardening for a vibrationvibrations of infinitely long, closed shells were amplitude of the order of the shell thickness; in-planeinvestigated by McRobie et al. (1999) and Popov et constraints reduce the softening nonlinearity, whichal. (2001) by using the energy formulation previously turns to hardening for smaller vibration amplitudes.developed by the same authors (Popov et al. 1998a). The objective of this study was to investigate regionsA simple two-mode expansion, derived from Evensen of chaotic motion; these regions were identified by(1967) but excluding the companion mode, was used. means of Poincare maps and Lyapunov exponents. ItAn energy approach was applied to obtain the was found that, when approaching the staticequations of motion, which were transformed into instability point (due to the constant accelerationaction-angle co-ordinates and studied by averaging. load), chaotic shell behaviour may be observed. In aDynamics and stability were extensively investigated previous study, Nagai and Yamaguchi (1995)by using the methods of Hamiltonian dynamics, and investigated shallow cylindrical panels without in-chaotic motion was detected. Laing et al. (1999) plane elastic support, via an approach similar to thatstudied the nonlinear vibrations of a circular used by Yamaguchi and Nagai (1997). Also in thiscylindrical panel under radial point forcing by using case, an accurate investigation of the regions ofthe standard Galerkin method, the nonlinear Galerkin chaotic motion was performed.method and the post-processed Galerkin method. Thesystem was discretized by the finite-difference Using a similar approach, Yamaguchi andmethod, similarly to the study of Foale et al. (1998). Nagai (2000) studied oscillations of a circularThe authors conclude that the post-processed cylindrical panel, simply supported at the four edges,Galerkin method is often more efficient than either having an elastic radial spring at the center. Thethe standard Galerkin or nonlinear Galerkin panel was excited by an acceleration with a constantmethods. term plus a harmonic component, and regions of chaotic motion were identified. Nagai et al. (2001) Kubenko and Koval‟chuk (2000b) used also studied theoretically and experimentally chaoticDonnell‟s nonlinear shallow-shell theory with vibrations of a simply supported circular cylindricalGalerkin projection to study nonlinear vibrations of panel carrying a mass at its center. A single-modesimply supported circular cylindrical shells. Driven approximation was used by Shin (1997) to study freeand companion modes were included, but vibration of doubly-curved, simply supported panels.axisymmetric terms were neglected, giving a Backbone curves and response to initial conditionshardening type response. Interaction between two were studied. Free vibrations of doubly-curved,modes with different numbers of circumferential laminated, clamped shallow panels, including circularwaves was discussed; this kind of interaction arises in cylindrical panels, were studied by Abe et al. (2000).correspondence to internal resonances. Results show Both first-order shear deformation theory andinteresting interactions between these modes. classical shell theory (analogous to Donnell‟s theory)Another study considering the interaction between were used. Results obtained neglecting in-plane andtwo modes with different numbers of circumferential rotary inertia are very close to those obtainedwaves was developed by Amiro and Prokopenko retaining these effects. Only two modes were(1999), who also did not consider axisymmetric considered to interact in the nonlinear analysis. Freeterms in the mode expansion. vibrations and parametric resonance of complete, simply supported circular cylindrical shells were Yamaguchi and Nagai (1997) studied studied by Mao and Williams (1998a, b). The shellvibrations of shallow cylindrical panels with a theory used is similar to Sanders‟, but the axialrectangular boundary, simply supported for inertia is neglected. A single-mode expansion wastransverse deflection and with in-plane elastic used without considering axisymmetric contraction.support at the boundary. The shell was excited by an Han et al. (1999) studied the axisymmetric motion ofacceleration having a constant value plus a harmonic circular cylindrical shells under axial compressioncomponent. Donnell‟s nonlinear shallow-shell theory and radial excitation. A region of chaotic responsewas utilized with the Galerkin projection along with was detected.a multi-mode expansion of flexural displacement.Initial deflection (imperfection) was taken into Extensive experiments on large-amplitudeaccount in the theoretical formulation but not in the vibrations of clamped, circular cylindrical shells werecalculations. The harmonic balance method and performed by Gunawan (1998). Two almost perfectdirect integration were used. The response of the aluminum shells and two with axisymmetricpanel was of the softening type over the whole range imperfection were tested. Contact-free acousticof possible stiffness values of the in-plane springs excitation and vibration measurement were used. 189 | P a g e
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P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January-February 2013, pp.181-207Mostly of the experiments were performed for the Numerical results display a softening type responsemode with 11 and 9 circumferential waves for perfect with traveling wave in the vicinity of resonance. Theand imperfect shells, respectively. These modes were narrow frequency range in which there are amplitudeselected in order to have a good frequency separation modulations in the response was deeply investigated.from other modes and avoid nonlinear interaction In another paper, Jansen (2001) presented aamong different modes. The effect of axial load on perturbation approach for both the frequency and thethe shell response was investigated. Softening type shell coordinates. The resulting boundary valuenonlinearity and traveling wave response were problem was numerically approached by using aobserved for all the shells. Nonstationary responses parallel shooting method. Results are in goodappeared at relatively large vibration amplitudes. agreement with those obtained by the same author (Jansen 2002) via Galerkin‟s method. Mikhlin (2000) studied vibrations of circularcylindrical shells under a radial excitation and an Amabili et al. (2003) experimentally studiedaxial static load, using Donnell‟s nonlinear shallow- large-amplitude vibrations of a stainless steelshell theory with Galerkin projection and two circular cylindrical panels supported at the fourdifferent two-mode expansions. One included the edges. The nonlinear response to harmonic excitationfirst circumferential harmonic of the driven mode and of different magnitude in the neighbourhood of threethe other was derived from Evensen‟s (1967) resonances was investigated. Experiments showedexpansion; in both cases, the companion mode was that the curved panel tested exhibited a relativelynot considered. Stability and energy “pumping” from strong geometric nonlinearity of softening type. Inone mode to the other were discussed. Lee and Kim particular, for the fundamental mode, the resonance(1999) used a single-mode approach to study was reached at a frequency 5 % lower than thenonlinear free vibration of rotating cylindrical shells. natural (linear) frequency, when the vibrationThe equation of motion shows only a cubic amplitude was equal to 0.55 times the shell thickness.nonlinearity, thus explaining the appearance of This is particularly interesting when these results arehardening nonlinearity. Moussaoui et al. (2000) compared to the nonlinearity of the fundamentalstudied free, large-amplitude vibrations of infinitely mode of complete (closed around the circumference),long, closed circular cylindrical shells, neglecting the simply supported, circular cylindrical shells ofmotion in the longitudinal direction and assuming similar length and radius, which display muchthat the generating lines of the shell remain straight weaker nonlinearity.after deformation. Thus, their model is suitable forrings but it is not adequate for real shells of finite Finite-amplitude vibrations of long shells inlength. The system was discretised by using a multi- beam-bending mode were studied by Hu and Kirmsermode expansion that excluded all axisymmetric (1971). Birman and Bert (1987) and Birman andterms, which are fundamental. Therefore, an Twinprawate (1988) extended the study to includeerroneous hardening type nonlinearity was obtained. elastic axial constraints, shells on an elasticNayfeh and Rivieccio (2000) used a single-mode foundation, and uniformly distributed static loads.expansion of the displacements to study forced Single-mode and two-mode approximations werevibrations of simply supported, composite, complete used. Results show a softening nonlinearity for longcircular cylindrical shells. This approach, as shells without restrained axial displacements at thepreviously discussed, is inadequate for predicting the ends, which becomes hardening for long shells withcorrect trend of nonlinearity; in fact, hardening type restrained displacements; other boundary conditionsresults were erroneously obtained and only a cubic are those of a clamped shell.nonlinearity was found in the equation of motion.However, due to the simple expansion, the equation Circular cylindrical shells carrying aof motion was obtained in closed-form. concentrated mass were investigated by Likhoded (1976). Large-amplitude free vibrations of non- Jansen (2002) studied forced vibrations of circular cylindrical shells on Pasternak foundationssimply supported, circular cylindrical shells under were studied by Paliwal and Bhalla (1993) by usingharmonic modal excitation and static axial load. the approach developed by Sinharay and BanerjeeDonnell‟s shallow-shell theory was used, but the in- (1985, 1986).plane inertia of axisymmetric modes was taken intoaccount in an approximate way. In-plane boundary 2.3 Imperfect shellsconditions were satisfied on the average. A four- A point that was still waiting for an answerdegree-of-freedom expansion of the radial was the accuracy of the different nonlinear shelldisplacement was used with Galerkin‟s method. theories available. For this reason, Amabili (2003a) 190 | P a g e
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P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January-February 2013, pp.181-207computed the large-amplitude response of perfect and mode expansion involving two degrees of freedom:imperfect, simply supported circular cylindrical the driven mode and an axisymmetric term havingshells subjected to harmonic excitation in the spectral the same spatial form of the one introduced byneighbourhood of the lowest natural frequency, by Evensen. Imperfections having the same shape as thisusing five different nonlinear shell theories: (i) expansion were assumed. Donnell‟s nonlinearDonnell‟s shallow-shell, (ii) Donnell‟s with in-plane shallow-shell theory was used. Softening andinertia, (iii) Sanders-Koiter, (iv) Flügge-Lur‟e-Byrne hardening nonlinearity were found depending onand (v) Novozhilov‟s theories. Except for the first shell geometry. Results show, in contrast with othertheory, the Lagrange equations of motion were studies, that initial imperfections change the linearobtained by an energy approach, retaining damping frequency only in the presence of an axial load.through Rayleigh‟s dissipation function. The Koval‟chuk and Krasnopol‟skaia (1980) studiedformulation is also valid for orthotropic and forced vibrations of closed, simply supported circularsymmetric cross-ply laminated composite shells. cylindrical shells using Donnell‟s nonlinear shallow-Both empty and fluid-filled shells were investigated shell theory. The same expansion of the radialby using a potential fluid model. The effect of radial displacement introduced by Evensen (1967) waspressure and axial load was also studied. Results used. Geometric imperfections having the same shapefrom the Sanders-Koiter, Flügge-Lur‟e-Byrne and as the driven mode were considered. Both radial andNovozhilov‟s theories were extremely close, for both longitudinal (parametric) excitations were examined.empty and water-filled shells. For the thin shell Some unstable zones with nonstationary vibrationsnumerically investigated by Amabili (2003a), for and travelling waves were detected, due towhich h/R 288, there is almost no difference differences between the driven- and companion-modeamong them. Appreciable difference, but not natural frequencies, caused by the geometricparticularly large, was observed between the previous imperfections. The authors found softening radialthree theories and Donnell‟s theory with in-plane responses. Kubenko et al. (1982) used a two-modeinertia. On the other hand, Donnell‟s nonlinear travellingwave expansion, taking into accountshallow-shell theory turned out to be the least axisymmetric and asymmetric geometricaccurate among the five theories compared. It gives imperfections; the rest of the model for the completeexcessive softening type nonlinearity for empty circular shell was similar to those proposed byshells. However, for water-filled shells, it gives Koval‟chuk and Krasnopol‟skaia (1980). Only freesufficiently precise results, also for quite large vibrations were studied, but the effect of the numbervibration amplitude. The different accuracy of the of circumferential waves on the nonlinear behaviourDonnell‟s nonlinear shallow-shell theory for empty was investigated. In particular, the results showedand water-filled shell can easily be explained by the that the nonlinearity is of the softening type and thatfact that the in-plane inertia, which is neglected in it increases with the number of circumferentialDonnell‟s nonlinear shallowshell theory, is much less waves. In the studies of Koval‟chuk andimportant for a water-filled shell, which has a large Krasnopol‟skaia (1980) and Kubenko et al. (1982)radial inertia due to the liquid, than for an empty the effect of axisymmetric and asymmetricshell. Contained liquid, compressive axial loads and imperfections was to increase the natural frequency;external pressure increase the softening type this is in contrast with results obtained in othernonlinearity of the shell. A minimum mode studies.expansion necessary to capture the nonlinearresponse of the shell in the neighbourhood of a Watawala and Nash (1983) studied the freeresonance was determined and convergence of the and forced conservative vibrations of closed circularsolution was numerically investigated. cylindrical shells using the Donnell‟s nonlinear shallow-shell theory. Empty and liquidfilled shells Geometric imperfections of the shell with a free-surface and a rigid bottom were studied.geometry (e.g. out-of-roundness) were considered in A mode expansion that may be considered as abuckling problems since the end of the 1950s; e.g., simple generalization of Evensen‟s (1967) wassee Agamirov and Vol‟mir (1959), Koiter (1963) and introduced; therefore, it is not moment-free at theHutchinson (1965). However, there is no trace of ends of the shell. A single-term was used to describetheir inclusion in studies of largeamplitude vibrations the shell geometric imperfections. Cases of (i) theof shells until the beginning of the ‟70s. Research on circumferential wave pattern of the shell responsethis topic was probably introduced by Vol‟mir being the same as that of the imperfection and (ii) the(1972). Kil‟dibekov (1977) studied free vibrations of circumferential wave pattern of the response beingimperfect, simply supported, complete circular shells different from that of the imperfection were analyzed.under pressure and axial loading by using a simple Numerical results showed softening type 191 | P a g e
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P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January-February 2013, pp.181-207nonlinearity. The imperfections lowered the linear softening nonlinearity, which becomes hardeningfrequency of vibrations when the circumferential only for very large vibrations, as expected. Thewave pattern of shell response was the same as that equations of motion are obtained by Gakerkin‟sof the imperfection, affecting the observed method and are studied by using the harmonicnonlinearity. Results for forced vibrations were balance method. Only the backbone curves are given.obtained only for beam-bending modes, for which Iu and Chia (1988a) studied antisymmetricallyDonnell‟s nonlinear shallow-shell theory is not laminated cross-ply circular cylindrical panels byappropriate; these results indicated a hardening ype using the Timoshenko-Mindlin kinematic hypothesis,nonlinearity. Hui (1984) studied the influence of which is an extension of Donnell‟s nonlinearimperfections on free vibrations of simply supported shallow-shell theory. Effects of transverse shearcircular cylindrical panels; he used Donnell‟s deformation, rotary inertia and geometricalnonlinear shallow-shell theory and a single-mode imperfections were included in the analysis. Theexpansion; the imperfection was assumed to have the solution was obtained by the harmonic balancesame shape as his single-mode expansion. These method after Galerkin projection. Fu and Chia (1989)imperfections changed the linear vibration extended this analysis to include a multi-modefrequency (increase or decrease according with the approach. Iu and Chia (1988b) used Donnell‟snumber of circumferential half-waves) and nonlinear shallow-shell theory to study freeinfluenced the nonlinearity of the panel. vibrations and post-buckling of clamped and simply supported, asymmetrically laminated cross-ply Jansen (1992) studied large-amplitude circular cylindrical shells. A multi-mode expansionvibrations of simply supported, laminated, complete was used without considering companion mode, ascircular cylindrical shells with imperfections. He only free vibrations were investigated; radialused Donnell‟s nonlinear shallowshell theory and the geometric imperfections were taken into account. Thesame mode expansion as Watawala and Nash (1983); homogeneous solution of the stress function wasthe boundary conditions at the shell ends were not retained, but the dependence on the axial coordinatefully satisfied. The results showed a softening-type was neglected, differently to what was done by Funonlinearity becoming hardening only for very large and Chia (1989). The equations of motion wereamplitude of vibrations (generally larger than ten obtained by using the Galerkin method and weretimes the shell thickness). Imperfections having the studied by the harmonic balance method. Threesame shape as the asymmetric mode analyzed gave a asymmetric and three axisymmetric modes were usedless pronounced softening behaviour, changing to in the numerical calculations. Numerical results werehardening for smaller amplitudes. Moreover, the compared to those obtained by El-Zaouk and Dymlinear frequency of the imperfect shell may be (1973) for a simply supported, glass-epoxyconsiderably lower than the frequency of the perfect orthotropic circular cylindrical shell, showing someshell. In a subsequent study Jansen (2002) developed differences. In a later paper, Fu and Chia (1993)his model further by using a four-degree-of-freedom included in their model nonuniform boundaryexpansion. However, numerical results are presented conditions around the edges. Softening or hardeningonly for a perfect shell. In a third paper by the same type nonlinearity was found, depending on theauthor (Jansen 2001), results for composite shells radius-to-thickness ratio. Only undamped freewith axisymmetric and asymmetric imperfections are vibrations and buckling were investigated in all thisgiven. series of studies. Chia (1987a, b) studied nonlinear free Elishakoff et al. (1987) conducted avibrations and postbuckling of symmetrically and nonlinear analysis of small vibrations of an imperfectasymmetrically laminated circular cylindrical panels cylindrical panel around a static equilibrium position.with imperfections and different boundary Librescu and Chang (1993) investigatedconditions. Donnell‟s nonlinear shallow-shell theory geometrically imperfect, doubly-curved, undamped,was used. A single-mode analysis was carried out, laminated composite panels. The nonlinear theory ofand the results showed a hardening nonlinearity. In a shear-deformable shallow panels was used. Thesubsequent study Chia (1988b) also investigated nonlinearity was due to finite deformations of thedoubly-curved panels with rectangular base by using panel due to in-plane loads and imperfections. Onlya similar shell theory, and a single-mode expansion in small-amplitude free vibrations superimposed on thisall the numerical calculations, for both vibration finite initial deformation were studied. A single-modeshape and initial imperfection. However, in this expansion was used to describe the free vibrationsstudy, numerical results for circular cylindrical panels and the initial imperfections. The authors found thatand doubly curved shallow shells generally show a imperfections of the panels, of the same shape as the 192 | P a g e
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P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January-February 2013, pp.181-207mode investigated, lowered the vibration frequency manufacturing imperfections, changes the travelingsignificantly. They also described accurately the post- wave response. Good agreement betweentheoreticalbuckling stability. In fact, curved panels are and experimental results was obtained. All the modescharacterized by an unstable post-buckling investigated show a softening type nonlinearity,behaviour, in the sense that they are subject to a snap- which is much more accentuated for the water-filledthrough-type instability. Librescu et al. (1996), shell, except in a case displaying internal resonanceLibrescu and Lin (1997a) and Hause et al. (1998) (one-to-one) among modes with different number ofextended this work to include thermomechanical circumferential waves. Travelling wave andloads and nonlinear elastic foundations (Librescu and amplitude-modulated responses were observed in theLin 1997b, 1999). Cheikh et al. (1996) studied experiments.nonlinear vibrations of an infinite circular cylindrical Results for imperfect shells obtained byshell with geometric imperfection of the same shape Donnell‟s nonlinear shell theory retainingas the mode excited. Plane-strain theory of shells was inplaneinertia and Sanders-Koiter nonlinear shellassumed, which is suitable only for rings (generating theory are given by Amabili (2003a).lines remain straight and parallel to the shell axis)and a three-degree-of-freedom model, subsequently Kubenko and Koval‟chuk (1998) publishedreduced to a single-degree-of-freedom one, was an interesting review on nonlinear problems of shells,developed. Results exhibit a hardening type where several results were reported about parametricnonlinearity, which is not reasonable for the shell vibrations; in such review the limitations of reducedgeometry investigated. order models were pointed out. Babich and Khoroshun (2001) presented results obtained at the S.P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine over 20 years of research; the authors focused the attention on the variational–difference methods; more than 100 Amabili (2003c) included geometric papers were cited.imperfections in the model previously developed byPellicano et al. (2002) and performed in depth Kubenko and Koval‟chuk (2004) analyzedexperimental investigations on large-amplitude the stability and nonlinear dynamics of shells,vibrations of an empty and water-filled, simply following the historical advancements in this field,supported circular cylindrical shell subject to about 190 papers were deeply commented; theyharmonic excitations. The effect of geometric suggested, among the others, the effect ofimperfections on natural frequencies was imperfections as an important issue to be furtherinvestigated. In particular, it was found that: (i) investigated.axisymmetric imperfections do not split thedouble natural frequencies associated with each The fundamental investigation on thecouple of asymmetric modes; outward (with respect stability of circular cylindrical shells is due to Vonwith the center of curvature) axisymmetric Karman and Tsien (1941), who analyzed the staticimperfections increase natural frequencies; small stability (buckling) and the postcritical behavior ofinward (with respect with the center of curvature) axially loaded shells. In this study, it was clarifiedaxisymmetric imperfections decrease natural that discrepancies between forecasts of linear modelsfrequencies; (ii) ovalisation has a small effect on and experimental results were due to the intrinsicnatural frequencies of modes with several simplifications of linear models; indeed, linearcircumferential waves and it does not split the double analyses are not able to predict the actual bucklingnatural frequencies; (iii)imperfections of the same phenomenon observed in experiments; conversely,shape as the resonant mode decrease both nonlinear analyses show that the bifurcation path isfrequencies, but much more the frequency of the strongly subcritical, therefore, safe designmode with the same angular orientation; (iv) information can be obtained with a nonlinearimperfections with the twice the number of analyses only. After this important contribution,circumferential waves of the resonant mode decrease many other studies have been published on static andthe frequency of the mode with the same angular dynamic stability of shells.orientation and increase the frequency of the othermode; they have a larger effect on natural frequencies 2.4 Shells Subjected to Periodic axial loadsthan imperfections of the same shape as the resonant Koval (1974) used the Donnell‟s nonlinearmode. The split of the double natural frequencies, shallow-shell theory to study the effect of awhich is present in almost all real shells due to longitudinal resonance in the parametric transversal 193 | P a g e
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P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January-February 2013, pp.181-207instability of a circular shell. He found that, the dynamic stability of layered anisotropic circularcombined parametric resonances give rise to complex cylindrical shells.regions of parametric instability. However, this kindof phenomenon was found at very high frequency Popov et al. (1998) analyzed the parametricand no damping was included in the analysis. stability and thepostcritical behavior of an infinitely Hsu (1974) used the Donnell‟s linear long circular cylindrical shell, dropping the boundaryshallow-shell theory to analyzethe parametric conditions. A three-mode expansion was used,instability of a circular cylindrical shell: a uniform without the inclusion of the companion mode.pressure load and an axial dynamic load were Membrane theory was used to evaluate the in-planeconsidered, the former was added in order to stresses due to the axial load. The effect of internaleliminate the Poisson‟s effect in the in-plane stresses. resonances between asymmetric modes was analyzedThe same problem was studied by Nagai and Yamaki in detail.(1978) using the Donnell‟s linear shallow-shelltheory, considering different boundary conditions; in Goncalves and Del Prado (2000, 2002)this work, the effect of axisymmetric bending analyzed the dynamic buckling of a perfect circularvibrations, induced by the axial load and essentially cylindrical shell under axial static and dynamic loads.due to the Poisson‟s effect, was considered. The The Donnell‟s nonlinear shallow-shell theory wasclassical membrane approach for the in-plane stresses used and the membrane theory was considered towas found inaccurate when the vibration amplitude of evaluate the in-plane stresses. The partial differentialaxisymmetric modes is not negligible. operator was discretized through the Galerkin technique, using a relatively large modal expansion. Koval‟chuck and Krasnopol‟skaya (1979) However, no companion mode participation wasconsidered the role of imperfections in the resonances considered and the boundary conditions wereof shells subjected to combined lateral and axial dropped by assuming an infinitely long shell. Escapeloads. They used the Donnell‟s nonlinear shallow from potential well was analyzed in detail and ashell theory and a three terms expansion for correlation of this phenomenon with the parametricinvestigating the onset of travelling waves and resonance was given.nonstationary response; moreover, they gave aninitial explanation about discrepancies between In Pellicano and Amabili (2003), parametrictheoretical stability boundaries and experiments, instabilities of circular shells (containing an internaladdressing such discrepancies to the geometrical fluid) due to combined static and dynamic axial loadsimperfections. A further contribution toward such were studied by means of the Donnell‟s nonlineardiscrepancies is due to Koval‟chuck et al. (1982) shallow shell theory for the shell and potential flowusing the same theory and similar modeling; they theory for the fluid. Among the other findings it wasclaimed that: the theoretical analyses give narrower found that the membrane stress assumptions can leadinstability region with respect to experiments if to wrong forecasting of the dynamics when theimperfections are not considered. Imperfections dynamics of axisymmetric modes cannot because splitting of conjugate mode frequencies and neglected (this is in agreement with Nagai andgive rise to doubling of the instability regions. Yamaki (1978)), the presence of an internal fluid greatly changes the dynamic response. Bert and Birman (1988) studied theparametric instability of thick circular shells, they In Catellani et al. (2004) a multimodedeveloped a special version of the Sanders– Koiter modelling of shells subjected to axial static andthin-shell theory for thick shells. A single mode periodic loads was proposed considering theanalysis, giving rise to a Mathieu type equation of Donnell‟s nonlinear shallow shell theory. The effectmotion, allowed to determine the instability regions. of combined static and periodic loads was analyzed in detail considering geometric imperfections. It was Argento (1993) used the Donnell‟s linear found a great influence of imperfections on thetheory and the classical lamination theory to study instability boundaries and low influence on thethe dynamic stability of composite circular clamped– postcritical dynamic behaviour for moderateclamped shells under axial and torsional loading. The imperfections.linear equations of motion, obtained fromdiscretization, were analyzed by means of the Goncalves and Del Prado (2005) presented aharmonic balance method and the linear instability low order model based on the Donnell‟s nonlinearregions were found. Argento and Scott (1993) studied shallow shell theory, the nonlinear dynamics and stability of circular cylindrical shells was analyzed. 194 | P a g e
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P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January-February 2013, pp.181-207They pointed out that suitable reduced order models lead the shell to large amplitude of vibration and thecould be able to reproduce correctly the shell response is highly sensitive to geometricresponse. An analytical simplified approach was imperfections. In Pellicano (2009) the dynamicdeveloped by Jansen (2005), in order to simulate stability of strongly compressed shells, subjected todynamic step and periodic axial loads acting on periodic axial loads, having geometric imperfections,isotropic and anisotropic shells; he showed that was investigated using the nonlinear Sanders–Koitersimple periodic responses can be simulated through theory, a multimode expansion was considered in thelow dimensional models. modelling. Such paper clarifies that, the role of imperfections becomes important in the case of The effect of in-plane inertia was included combined axial compression and periodic loads;in the Donnell-type equations: it was found that, indeed, dynamic instabilities interacts with the staticneglecting the in-plane inertia gives rise to a potential well. Kochurov and Avramov (2010)moderate underestimation of the instability region. presented an analytical study based on the Donnell‟sThe nonlinear dynamics and chaos of shells subjected nonlinear shallow shell theory, a low order Galerkinto an axial load with or without the presence of a approach and a perturbation procedure. They provedcontained fluid was presented in Pellicano and the coexistence of nonlinear normal modes andAmabili (2006); both nonlinear Donnell‟s shallow travelling waves. They pointed out the high modalshell and nonlinear Sanders–Koiter theories were density of shell like structures and the need of multi-used; a multimode expansion, able to model both mode models for an accurate analysis.postcritical buckling and nonlinear vibrations, was Ilyasov (2010) developed a theoreticalconsidered for reducing the original nonlinear partial model for analyzing the dynamic stability of shellsdifferential equations (PDE) to a set of ordinary having viscoelastic behavior. This kind of models candifferential equations (ODE). The effect of a be particularly useful in the case of polymericcontained fluid and as analyzed and an accurate materials.analysis of the postcritical dynamics was carried out,including a deep study on the chaotic properties of 2.5 Shells Subjected to Seismic excitationthe response. Vijayarachavan and Evan-Iwanowski (1967) analyzed, both analytically and experimentally, the Goncalves et al. (2007) studied steady state parametric instabilities of a circular shell underand transient instabilityof circular shells under seismic excitation. The cylinder position was verticalperiodic axial loads using the Donnell‟s nonlinear and the base was axially excited by using a shaker. Inshallow shell theory and a low dimensional Galerkin this problem, the in-plane inertia is variable along themodel. The study clarified the complexity of the shell axis and, when the base is harmonically excited,basins of attraction of low vibration responses and it gives rise to a parametric excitation. Instabilitythe escape from the potential well due to the regions were found analytically and compared withcombination of periodic and static compressive loads; experimental results.moreover, they pointed out that shells are highly Bondarenko and Galaka (1977) investigatedsensitive to geometric imperfections. the parametric instabilities of a composite shell under base seismic motion and free top end. They identified Darabi et al. (2008) analyzed the dynamic principal instability regions of several modes and alsostability of functionally graded circular cylindrical secondary regions; they observed that, the transitionshells using Donnell shallow shell theory. They used from stable to unstable regions is accompanied by aa displacement expansion without axisymmetric „„bang‟‟ that can lead the shell to the collapse.modes, the Galerkin approach for reducing the PDE Bondarenko and Telalov (1982) studiedto ODE and the Bolotin method for solving the latter experimentally the dynamic instability and nonlinearone. Only hardening results were obtained, probably oscillations of shells; the frequency response wasdue to neglect axisymmetric modes. A detailed hardening in the region of the main parametricbibliographic analysis can be found in such paper. resonance for circumferential wave number n = 2 and softening for n > 2. Del Prado et al. (2009) analyzed the Trotsenko and Trotsenko (2004), studiedinstabilities occurring to a shell conveying an internal vibrations of circular cylindrical shells, with attachedheavy fluid and loaded with compressive and rigid bodies, by means of a mixed expansion basedperiodic axial loads. The Donnell‟s nonlinear shallow on trigonometric functions and Legendreshell theory was used in combination with a low polynomials; they considered only linear vibrations.order Galerkin approach. They showed that the Pellicano (2005) presented experimentalcombination of a flowing fluid with axial loads can results about violent vibration phenomena appearing 195 | P a g e
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P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January-February 2013, pp.181-207in a shell with base harmonic excitation and carrying found, the conjecture made by such scientists wasa rigid mass on the top. When the first axisymmetric that the disagreement was due to shells geometricmode is in resonance conditions the top mass imperfections. Such paper summarizes some ofundergoes to large amplitude of vibration and a huge experimental results published on a previous bookout of plane shell vibration is detected (more than (Kubenko et al., 1984)2000g), with a relatively low base excitation (about Mallon et al. (2010) studied circular10g). cylindrical shells made of orthotropic material, the Pellicano and Avramov (2007) published a Donnell‟s nonlinear shallow shell theory was usedpaper concerning the nonlinear dynamics of a shell with a multimode expansion for discretization (PDEwith base excitation and a top disk. The work was to ODE). They presented also experimental results.mainly theoretical, i.e. only some experimental The theoretical model considered also the shaker-results concerning the linear dynamics were shell interaction (it is to note that one of the firstpresented; the shell was modelled using the nonlinear works concerning the interaction between anSanders–Koiter theory and a reduced order model electromechanical shaker and a mechanical system iswas used; the analysis was mainly focused on the due to Krasnopol‟skaya (1976)). Mallon et al. (2010)principal parametric resonance due to high frequency investigated the imperfections sensitivity and found aexcitations.A similar analysis was presented also in reduction of the instability threshold. No goodAvramov and Pellicano (2006). quantitative match between theory and experiments was found: saturation phenomena, beating and chaos Pellicano (2007) developed a new method, were found numerically. However, this can bebased on the nonlinear Sanders–Koiter theory, considered a seminal work due to the intuition thatsuitable for handling complex boundary conditions of some complex phenomena can be due to the shakercircular cylindrical shells and large amplitude of shell interaction.vibrations. The method was based on a mixed In Pellicano(2011), experiments are carriedexpansions considering orthogonal polynomials and out on a circular cylindrical shell, made of aharmonic functions. Among the others, the method polymeric material (P.E.T.) and clamped at the baseshowed good accuracy also in the case of a shell by gluing its bottom to a rigid support. The axis ofconnected with a rigid body; this method is the the cylinder is vertical and a rigid disk is connectedstarting point for the model developed in the present to the shell top end. In Pellicano (2007) this problemresearch. was fully analyzed from a linear Point ofview. Here nonlinear phenomena are investigated by exciting the Mallon et al. (2008) studied isotropic shell using a shaking table and a sine excitation.circular cylindrical shells under harmonic base Shaking the shell from the bottom induces a verticalexcitation; they used the Donnell‟s nonlinear shallow motion of the top disk that causes axial loads due toshell theory; a multimode expansion, suitable for inertia forces. Such axial loads generally give rise toanalyzing the nonlinear resonance and dynamic axial-symmetric deformations; however, in someinstability of a specific mode, was used; they conditions it is observed experimentally that a violentpresented comparisons between the semianalytical resonant phenomenon takes place, with a strongprocedure and a FEM model in the case of linear energy transfer from low to high frequencies andvibrations or buckling. Using the nonlinear semi- huge amplitude of vibration. Moreover, an interestinganalytical model they found beating and chaotic saturation phenomenon is observed: the response ofresponses, severe out of plane vibration and the top disk was completely flat as the excitationsensitivity to imperfections. frequency was changed around the first axisymmetric mode resonance. Kubenko and Koval‟chuk (2009) published Farshidianfar A. and Farshidianfaran experimental work focused on shells made of M.H.(2011) were carried out both theoretical andcomposite materials; they pointed out the experimental analyses on a long circular cylindricalinadequateness of the linear viscous damping models. shell. A total of 18 modes, consisting of all the threeAxial loads (base excitation, and free top end of the main mode groups (axisymmetric, beam-like andshell) and combined loads were considered. The asymmetric) were found under a frequency range ofanalysis of the principal parametric instability was 0–1000 Hz, by only applying acoustical excitation.carried out (probably with sub harmonic response, Acoustical excitation results were compared withbut no information are give about). Dynamic those obtained from contact excitation. It wasinstability regions were determined experimentally: a discovered that if one uses contact methods, severaldisagreement between previous theoretical models exciting points are required to obtain all modes;(narrower region) and experiments (wider) was whereas with acoustical excitation only one 196 | P a g e
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