Paulo B. Gonçalves. Zenon J. G. N. del Prado. Introduction

Transcription

1 Transient Stability of Empty and Fluid-Filled Cylindrical Sells aulo B. Gonçalves and Frederico M. A. da Silva ontifical University of Rio de Janeiro - UC Rio Civil Engineering Department Rio de Janeiro, RJ. Brazil silvafma@yaoo.com.br Zenon J. G. N. del rado Federal University of Goiás Civil Engineering Department raça Universitaria Goiânia, GO. Brazil. zenon@eec.ufg.br Transient Stability of Empty and Fluid-Filled Cylindrical Sells In te present work a qualitatively accurate low dimensional model is used to study te non-linear dynamic beavior of sallow cylindrical sells under axial loading. Te dynamic version of te Donnell non-linear sallow sell equations are discretized by te Galerkin metod. Te sell is considered to be initially at rest, in a position corresponding to a pre-buckling configuration. Ten, a armonic excitation is applied and conditions to escape from tis configuration are sougt. By defining steady state and transient stability boundaries, frequency regimes of instability may be identified suc tat tey may be avoided in design. Initially a steady state analysis is performed; resonance response curves in te forcing plane are presented and te main instabilities are identified. Finally, te global transient response of te system is investigated in order to quantify te degree of safety of te sell in te presence of small perturbations. Since te initial conditions, or even te sell parameters, may vary widely, and indeed are often unknown, attention is given to all possible transient motions. As parameters are varied, transient basins of attraction can undergo quantitative and qualitative canges; ence a stability analysis wic only considers te steady-state and neglects tis global transient beavior, may be seriously non-conservative. Keywords: Cylindrical sells, fluid-structure interaction, parametric instability, nonlinear vibrations Introduction Tin-walled cylindrical sells are widely used in many industries. Due to te increasing use of ig-strengt materials, sopisticated numerical tecniques and optimization metods in analysis, te design of suc sells is often buckling-critical. In many circumstances tese sells are subjected not only to static loads but also to dynamic disturbances and filled wit internal fluid. However, tin-walled cylindrical sells wen subjected to axial compressive loads often exibit a igly nonlinear beavior wit a ig imperfection sensitivity and may loose stability at loads levels as low as a fraction of te material strengt. Many studies are concerned wit te analysis of sells vibrating in vacuum; far fewer are focused on te analysis of te nonlinear vibrations of cylindrical sells in contact wit a dense fluid. One of te first studies on vibrations of circular cylindrical sells in contact wit a dense fluid considering sell nonlinearity is due to Ramacandran (979). He studied te large-amplitude vibrations of circular cylindrical sells aving circumferentially varying tickness and immersed in a quiescent, non-viscous and incompressible fluid, using te Donnell s sell teory. Boyarsina (984, 988) studied teoretically te nonlinear free and forced vibrations and stability of a circular cylindrical tank partially filled wit a liquid and aving a free surface. Here, nonlinearity is attributed to te interaction of free surface waves and elastic flexural vibrations of te sell. Gonçalves and Batista (988) considered simply supported circular cylindrical sells filled wit incompressible fluid. To model te sell, Sanders nonlinear sell teory and a novel mode expansion tat includes two terms in te radial direction (te asymmetric and te axisymmetric ones) and ten terms to describe te in-plane displacements were used. Numerical results were obtained concerning te effect of te liquid on te nonlinear beavior of sells. It was found tat te presence of a dense fluid increases te softening caracteristics of te frequency-amplitude resented at XI DINAME International Symposium on Dynamic roblems of Mecanics, February 8t - Marc 4t, 5, Ouro reto. MG. Brazil. aper accepted: June, 5. Tecnical Editor: José Roberto de França Arruda. relation wen compared wit te results for te same sell in vacuum. Ciba (993) studied experimentally large-amplitude vibrations of two vertical cantilevered circular cylindrical sells made of polyester seets partially filled wit water to different levels. He observed tat for bulging modes wit te same axial wave number, te weakest degree of softening nonlinearity can be attributed to te mode aving te minimum natural frequency, as observed for te same empty sells. He also found tat sorter tanks ave a larger softening nonlinearity tan taller ones, as in vacuum. Te tank wit a lower liquid eigt as a stronger softening nonlinearity tan te same tank wit a iger liquid level. Traveling wave modes and coupling between two bulging modes (and between two slosing modes) were also observed. Amabili et al (998) studied te nonlinear free and forced vibrations of a simply supported, circular cylindrical sell in contact wit an incompressible and non-viscous, quiescent dense fluid. Donnell s nonlinear sallow-sell teory is used, so tat moderately large vibrations can be analyzed. Te boundary conditions on radial displacement and continuity of circumferential displacement are exactly satisfied, wile te axial constraint is satisfied on te average. Te problem is reduced to a system of ordinary differential equations by means of te Galerkin metod, assuming an appropriate deflection sape. Te mode sape is expanded by using two asymmetric modes (driven and companion modes) plus te axisymmetric mode. In te present study, a low dimensional model wic retains te essential nonlinear terms is used to study te nonlinear oscillations and instabilities of te sell. Here te interest is focused on a pivotal interaction between non-symmetric and axisymmetric modes wic allows te escape from te pre-buckling configuration. To discretize te sell, Donnell sallow sell equations, modified wit te transverse inertia force, are used togeter wit Galerkin metod to derive a set of coupled ordinary differential equations of motion. Tese equations are integrated numerically using te fourt order Runge-Kutta metod. In order to study te nonlinear beavior of te sell, several numerical strategies were used to obtain time responses, oincaré maps and bifurcation diagrams. Te interested reader will find a description of te relevant numerical algoritms in Del rado (). It is considered a non-viscous and incompressible fluid and an irrotational flow. As a result, it can be caracterized by a velocity J. of te Braz. Soc. of Mec. Sci. & Eng. Copyrigt 6 by ABCM July-September 6, Vol. XXVIII, No. 3 / 33

2 aulo B. Gonçalves et al potencial. Te solution for te velocity potencial is taken as a sum of suitable functions, were te unknown parameters are determined by te kinetic condition along te sell wetted surface (Batista and Gonçalves, 988). Steady state and transient stability boundaries are presented and special attention is devoted to te determination of te critical load conditions. From tis teoretical analysis, dynamic buckling criteria can be property establised wic may constitute a consistent and rational basis for design of tese sell structures under armonic loading. roblem Formulation Sell Equations Consider a perfect tin-walled fluid-filled circular cylindrical sell of radius R, lengt L, and tickness. Te sell is assumed to be made of an elastic, omogeneous, and isotropic material wit Young s modulus E, oisson ratio ν, and mass per unit area M. Te axial, circumferential and radial co-ordinates are denoted by, respectively, x, y and z, and te corresponding displacements on te sell surface are in turn denoted by U, V, and W, as sown in Fig.. R L z, (w) y, (v) R x, (u) Figure. Sell geometry and coordinate system. Te sell is subjected to a uniformly distributed axial load given by: ( t) cos( ω t ) () + were is te uniform static load applied along te edges x, L, is te magnitude of te armonic load, t is time and ω is te forcing frequency. Te nonlinear equations of motion based on te Von Karmán- Donnell sallow sell teory, in terms of a stress function f and te transversal displacement w are given by: were: 4 Mwɺɺ + β wɺ + β w p R + F + F,xx w,yy + F R,xy w,xy,yy w,xx L () 4 f w, xx w, xx w, yy + w, xy (3) E R F f F + f f F y and p is te fluid pressure, 4 is te biarmonic operator, β and β are damping coefficients and D is te flexural rigidity defined as: 3 E D (4) v ( ) In te foregoing, te following non-dimensional parameters will be introduced: W w τ ω t o Γ o cr Ω R x ξ L ω R f ω E L 3 o ( ν ) R 3 ( ν ) E o f θ y R Γ cr E were ω o is te lowest natural frequency of te empty sell. Modal Analysis Te numerical model is developed by expanding te transversal displacement component w in series in te circumferential and axial variables. From previous investigations on modal solutions for te non-linear analysis of cylindrical sells under axial loads (Hunt et al. 986; Gonçalves and Batista, 988; Gonçalves and Del rado, ) it is observed tat, in order to obtain a consistent modeling wit a limited number of modes, te sum of sape functions for te displacements must express te non-linear coupling between te modes and describe consistently te unstable post-buckling response of te sell as well as te correct frequency-amplitude relation. Te dimensionless lateral deflection W can be generally described as (Gonçalves and Batista, 988): W i, 3, 5 k,, 4 l,, 4 Wij cos( i nθ ) sin ( j mπ ξ ) + j, 3, 5 W cos ij ( k nθ ) sin ( l mπ ξ ) were n is te number of waves in te circumferential direction of te basic buckling or vibration mode, m is te number of alf-waves in te axial direction, θ y / R and ξ x / L. Tese modes represent bot te symmetric and asymmetric components of te sell deflection pattern. Te first double series represents te unsymmetrical modes wit odd multiples of te basic wave numbers m and n. Te second double series represents, besides te asymmetric modes wic contains an even multiple of te basic wave numbers m and n and rosette modes, te axysimmetric modes wic play an important role in te non-linear modal coupling and loss of stability of te sell. revious studies on buckling of cylindrical sells ave sown tat te most important modes are te basic buckling or vibration mode and te axi-symmetric mode wit twice te number of alf waves in te axial direction as te basic mode, tat is: W ζ + ζ ( τ ) cos( nθ ) sen( mπ ξ ) ( τ ) cos( mπ ξ ) Te relevance of tese modes from a pysical point of view was explained by Croll and Batista (98) and, from symmetry and catastrope teory arguments, by Hunt et al. (986). Tese modes are enoug to describe te initial post-buckling beavior of te sell as well as te topology of te pre-buckling configuration and te (5) (6) (7) 33 / Vol. XXVIII, No. 3, July-September 6 ABCM

3 Transient Stability of Empty and Fluid-Filled Cylindrical Sells potential barrier connected wit te unstable equilibrium positions lying on te initial post-buckling pat. Substituting te assumed form of te lateral deflection, Eq. (7), on te rigt-and side of te compatibility Eq. (3), tis equation can be solved to obtain te stress function f in terms of w togeter wit te relevant boundary and continuity conditions. Upon substituting te modal expressions for f and w into Eq. () and applying te Galerkin metod, a set of non-linear ordinary differential equations is obtained in terms of modal amplitudes ζ(τ) ij. Fluid Mecanics Equations Te sell is assumed to be completely fluid-filled. Te irrotational motion of an incompressible and non-viscous fluid can be described by a velocity potential φ(x, r, θ, t). Tis potential function must satisfy te Laplace equation wic can be written in dimensionless form as: were: φ, ξξ + φ, κ + φ, +, θθ φ κκ (8) κ κ / κ r R φ γ φ R γ [ ρ s R ( ν )/ E] Te dynamic fluid pressure acting on te sell surface is obtained from te Bernoulli equation: p ρ F ρ S 4δ γ φ,t were ρ F is te density of te fluid and ρ S is te sell material density. At te sell-fluid interface, te fluid velocity normal to te sell surface must be equal to te sell velocity in tis direction, tat is: ( w t) (9) φ κ γ / (), δ were δ / R. Furter, for a fluid-filled sell, te following restriction must be imposed at κ : φ, κ () Solving equations (8) to (), one obtains te following expressions for te idrodynamic fluid pressure: ( nθ ) sin ( π ξ ) p ζ ττ m cos m (), a were m a is te added mass due to te fluid contained in te sell, wic is given by: m ( ρ R ) ( mπ ξ ) ( mπ ξ ) n a F mπ ξ (3) I n mπ ξ I were I n- and I n are Bessel functions. Results To ceck te validity and accuracy of te present metodology for te determination of te natural frequencies, a key point in any non-linear dynamic analysis, empty and fluid-filled cylindrical n sells are analyzed and te results are compared wit experimental and oter numerical values found in literature. As a first example, te lowest natural frequencies of a simply supported empty cylinder are compared wit te analytical solution derived by Dym (973) using Sanders sell teory and te experimental results obtained by Gasser (987). Te results are sown in Table. For te same sell, te present results for a water filled sell are compared wit tose obtained experimentally by Gasser (987) and te numerical results obtained by Gonçalves and Batista (987) in Table. In bot cases, tere is an excellent agreement between all results. Table. Comparison of natural frequencies (Hz) for an empty cylindrical sell.(m, L.4 m, R.35 m,. m, E.x 8 kn/m², ν.3, ρ 785 kg/m³). n Gasser (987) Dym (973) resent work Table. Comparison of natural frequencies (Hz) for a cylinder filled wit water. (m, L.4 m, R.35 m,. m, E.x 8 kn/m², ν.3, ρ 785 kg/m³, ρ F kg/m³). n Gasser (987) Gonçalves and Batista (987) resent work Consider a tin-walled cylindrical sell wit.m, R. m, L.4 m, E.x 8 kn/m, ν.3, β εmω, wit ε.3 (fluid-filled sell) and ε.8 (empty sell) (opov et al. 998), and β ηd wit η.. Te sell and fluid densities are: ρ s 785kg/m 3 and ρ F kg/m 3. For tis sell geometry te lowest natural frequency occurs for (n,m)(5, ). Now te parametric instability and escape from te pre-buckling configuration of te fluid-filled cylinder under axial armonic forcing, as described by Eq. (), will be considered. In te following, te constant part of te loading (Γ ) is assumed to be between te upper and lower static critical load of te sell. In tese circumstances, te sell potential energy exibits tree wells, one associated wit te fundamental pre-buckling configuration and two wells associated wit te two possible post-buckling configurations. If te cylinder is subjected to a periodic axial load, it will undergo te familiar longitudinal forced vibration, exibiting a uniform transversal motion, due to te effect of oisson s ratio, also known as breating mode. However, at certain critical values, te longitudinal motion becomes unstable and te cylinder executes transverse bending vibrations. Figure sows some representative time istories for Γ.4. Here Ωω/ω and ω is te lowest natural frequency of te unloaded sell. A projection of te pase space and oincaré section are also sown in tese figures. Tese figures were obtained by numerically integrating te equation of motion wit te Runge-Kutta metod. In Fig..a, for a forcing amplitude lower tan a critical value (Γ.45 and Ω.), after a finite initial disturbance, te amplitude of te response decreases rapidly converging to te trivial solution. If te J. of te Braz. Soc. of Mec. Sci. & Eng. Copyrigt 6 by ABCM July-September 6, Vol. XXVIII, No. 3 / 333

4 aulo B. Gonçalves et al control parameter Γ is increased beyond a critical value, te sell exibits initially an exponential growt of te amplitude, as sown in Fig..b, converging to a limit cycle witin te pre-buckling well. In tis case, te trivial solution becomes unstable (parametric instability) and te system converges to a period-two stable solution. If Γ is increased to a iger value, for example Γ.3, te sell escapes from te pre-buckling well (snap-troug buckling) and exibits large cross-well caotic motions, as sown in Fig..c, or small amplitude oscillation around a post-buckling configuration. Figure 3 sows te numerically obtained parametric instability boundary as well as te transient and permanent escape boundaries for te fluid-filled sell and te same sell in vacuum, in (frequency of excitation x amplitude of excitation) control space for Γ.4, Γ.6 and Γ.8. Te lower stability boundary corresponds to parameter values for wic small perturbations from te trivial solution will result in an initial growt in te oscillations; terefore it defines te parametric instability boundary. Te second limit corresponds to escape from te pre-buckling potential well in a slowly evolving environment. Tese curves were obtained by increasing slowly te amplitude wile olding te frequency constant. As one can observe, te parametric stability boundary is composed of various curves, eac one associated wit a particular bifurcation event. Te deepest well is associated wit te principal instability region at ω, wile te second well to te left is te secondary instability region occurring around ω and te oter smaller wells to te left are connected wit super-armonic resonances. Te orizontal dotted line corresponds to te static critical load of tis sell. Comparing Figures 3.a, 3.b and 3.c, one can conclude tat te static pre-loading as te effect of lowering te stability boundaries, of enlarging te widt of te instability regions and of sifting te instability regions to te left. In bot cases te instability boundaries can be muc lower tan te static critical load. Te fluid as a similar influence on te stability boundaries. Tis is expected since te influence of te fluid is to increase te effective mass of te system, decreasing consequently te natural frequencies. For te region between te parametric instability limit and te transient escape limit, te sell exibits vibrations in te prebuckling potential well during bot permanent and transient states. Wen comparing te permanent and transient boundaries, one can observe tat te transient escape limit is lower tan te permanent one. Tis means tat te sell may exibit large amplitude vibrations during te transient state but converge to a low amplitude solution witin te pre-buckling well wen te steady-state response is reaced. A structure may display in a nonlinear regime long transients, but teir lengts can not be known a priori. So, in order to avoid any damage due to large amplitude vibrations te transient response of te sell must be analyzed in detail ζ dζ dζ Time ζ.5 (a) Γ ζ ζ dζ dζ Time ζ 8 (b) Γ ζ ζ dζ dζ Time ζ (c) Γ ζ Figure. Time response, pase plane and oincaré section for Γ.4 and Ω.. Fluid-filled circular cylindrical sell. 334 / Vol. XXVIII, No. 3, July-September 6 ABCM

5 Transient Stability of Empty and Fluid-Filled Cylindrical Sells Γ + Γ.5.5 Γ + Γ (a.) Sell in vacuum. Γ (a.) Fluid-filled sell. Γ Γ + Γ (b.) Sell in vacuum. Γ Γ + Γ Γ + Γ (b.) Fluid-filled sell. Γ Γ + Γ (c.) Sell in vacuum. Γ.8 (c.) Fluid-filled sell. Γ.8 Figure 3. Instability boundaries in control space for different values of static load. Figure 4 sows typical bifurcation diagrams connected wit te principal instability region for te fluid-filled sell as a function of te forcing amplitude Γ, for different values of te forcing frequency Ω. Tese bifurcation diagrams were obtained by numerical continuation tecniques (Del rado, ). In tese diagrams a dotted line means unstable solutions and a continuous line means stable solutions. Te bifurcation diagram depicted in Fig. 4.a is typical of te left descending branc of te principal region of parametric instability. Te system exibits a sub-critical bifurcation, tat is, te fundamental solution looses its stability, giving rise to a T unstable periodic motion. In tis case, any increase in Γ beyond te critical value leads to a jump to anoter stable solution tat may exist witin te pre-buckling well or around a post-buckling J. of te Braz. Soc. of Mec. Sci. & Eng. Copyrigt 6 by ABCM July-September 6, Vol. XXVIII, No. 3 / 335

6 aulo B. Gonçalves et al configuration. Also, te T solution exibits a stable branc between two unstable brances. So, for load levels lower tan te critical value te sell may display different types of beavior witin te pre-buckling well. As observed in Fig. 4.a, tis non-trivial stable region corresponds to forcing values lower tan te critical load. Tis leaves a regime were tere is no attractor witin te prebuckling well after te critical point is reaced and ence an unavoidable jump to escape under increasing forcing occurs. Tis explains wy in tis region te numerically obtained parametric instability boundary practically coincides wit te transient and permanent escape boundaries ζ ζ ζ (a) Ω.6 (b) Ω.78 (c) Ω. Figure 4. Bifurcation diagrams of te oincaré map. rincipal instability region for fluid-filled sell, Γ.4. (a) Γ.4 (b) Γ.6 (c) Γ.8 (d) Γ. Figure 5. Cross sections of te basins of attraction, in transient state, for increasing values of te forcing amplitude Γ of te fluid-filled cylindrical sell. Evolution of te basin for Γ.4 and Ω.. load / Vol. XXVIII, No. 3, July-September 6 ABCM

7 Transient Stability of Empty and Fluid-Filled Cylindrical Sells (a) Γ.4 (b) Γ.6 (c) Γ.8 (d) Γ. Figure 6. Cross sections of te basins of attraction, in permanent state, for increasing values of te forcing amplitude Γ of te fluid-filled cylindrical sell. Evolution of te basin for Γ.4 and Ω.. load.55. In Figure 4.b, te jump is indeterminate. Te bifurcation is subcritical, but te stable small-amplitude non-trivial solution subsists for forcing values iger tan te critical load. So, wen te fundamental trivial solution becomes unstable, te response may restabilize witin te pre-buckling well or jump to a remote attractor. Te response tat is attained pysically depends on te initial conditions. Te bifurcation diagram sown in Fig. 4.c is typical of te rigt ascending branc of te stability boundary. Wen Γ is lower tan te critical value, te only possible steady state solution witin te pre-buckling well is te trivial one, wic is stable. Consequently, te response is trivial. Wen Γ is greater tan a critical value, tere are two possible steady state solutions: (a) te trivial one, wic is unstable; and (b) a finite amplitude period-two (T) solution, wic is stable. In tis case, since disturbances are always present, te response is non-trivial. Also, tese figures sow tat as Γ increases from zero, te response consists of te trivial solution. As Γ exceeds te critical value, ζ begins to increase slowly wit increasing Γ. Te critical value in tis case is a supercritical bifurcation. As te amplitude of te forcing increases, te amplitude of te response increases until te escape boundary is reaced. Before escape occurs, te period-two solution may also become unstable, being followed by a period doubling cascade, eventually reacing a narrow zone of caotic motion. In order to evaluate te safety of te structure one sould analyze te beavior of te basins of attraction of te solutions in bot transient and permanent states. Figure 5 sows te evolution of te transient basin of attraction for increasing values of te forcing amplitude Γ, Ω. and Γ.4. Here te ζ ζ cross-sections ɺ ζ ζɺ are sown for of te four dimensional pase space ( ). increasing values of te forcing amplitude. Figure 6 sows te evolution of te permanent basin of attraction for increasing values of te forcing amplitude Γ, Ω. and Γ.4. Bot figures are associated wit te bifurcation diagram of Fig. 4.c and cover te same set of initial conditions. In Figure 5 te gray area is associated wit te escape during te transient response and te wite area corresponds to te fundamental trivial and period-two stable solutions witin te prebuckling well. As Γ increases te region associated wit te escape increases and after a certain critical value, it covers completely te analyzed region, sowing tat escape occurs for any set of initial conditions during te transient response, well before te critical escape load displayed in te bifurcation diagram of Fig. 4.c is reaced. In Figure 6 te black area corresponds to te fundamental trivial solution, te gray areas correspond to te period-two stable solution witin te pre-buckling well and te wite area corresponds to te escape. For Γ lower tan te critical point, te response for initial conditions witin te analyzed area converges to te trivial solution or to escape. Of course, escape can only occur for large perturbations. After te critical point, te black region suddenly disappears and te response for te majority of initial conditions converges to te period-two stable solutions witin te pre-buckling well. As Γ increases, te region associated wit tis solution J. of te Braz. Soc. of Mec. Sci. & Eng. Copyrigt 6 by ABCM July-September 6, Vol. XXVIII, No. 3 / 337

8 aulo B. Gonçalves et al decreases and a rapid erosion is observed. Also, after a certain critical value te wole basin of attraction becomes fractal. In tis case te response becomes very sensitive to te initial conditions and te steady state response, unpredictable. Comparing te trivial and period-two areas of Fig. 5 and Fig. 6, one can observe tat te basin area occupied by te transient response is smaller tan te area occupied by te permanent response. So, a practical design criterion must be based on te transient analysis rater tan on te steady state response of te system. Also, te critical loads obtained from te bifurcation diagrams are not enoug to evaluate te robustness of te structure in te presence of unavoidable disturbances occurring during its construction or service life. Te analysis of size and structure of te basin of attraction must be taken into account in order to specify allowable disturbances in a dynamic environment. A detailed parametric analysis of te basin evolution considering empty and fluid-filled sells can be found in Silva (4). Concluding Remarks Based on Donnell s sallow sell equations, an accurate lowdimensional model is derived and applied to te study of te nonlinear vibrations of an axially loaded fluid-filled circular cylindrical sell in transient and permanent states. Te results sow te influence of te modal coupling on te post-buckling response and on te nonlinear dynamic beavior of fluid-filled circular cylindrical sells. Also te influence of a static compressive loading on te dynamic caracteristics is investigated wit empasis on te parametric instability and escape from te pre-buckling configuration. Te most dangerous region in parameter space is obtained and te triggering mecanisms associated wit te stability boundaries are identified. Also te evolution of transient and permanent basin boundaries is analyzed in detail and teir importance in evaluating te degree of safety of a structural system is discussed. It is sown tat critical bifurcation loads and permanent basins do not offer enoug information for design. Only a detailed analysis of te transient response can lead to safe lower bounds of escape (dynamic buckling) loads in te design of fluidfilled cylindrical sells under axial time- dependent loads. Acknowledgements Tis work was made possible by te financial support of te Brazilian Researc Council CNq. References Amabili, M., ellicano, F. and aïdoussiss, M., 998, Nonlinear Vibrations of Simply Supported Circular Cylindrical Sells, Coupled to Quiescente Fluid, Journal of Fluids and Structures, Vol., pp Amabili, M., ellicano, F. and aïdoussis, M..,, Nonlinear vibrations of fluid-filled, simply supported circular cylindrical sells: teory and experiments, Nonlinear Dynamics lates and Sells; AMD,. New York: ASME, Vol. 38, pp Amabili, M., ellicano, F. and aïdoussis, M..,, Nonlinear supersonic flutter of circular cylindrical sells, AIAA Journal, Vol. 39, pp Boyarsina, L. G., 984, Resonace effects in te nonlinear vibrations of cylindrical sells containing a liquid, Soviet Applied Mecanics, Vol., pp Boyarsina, L. G., 988, Nonlinear wave modes of an elastic cylindrical sell partially filled wit a liquid under conditions of resonance, Soviet Applied Mecanics, Vol. 4, pp Ciba, M., 993, Non-Linear Hydroelastic Vibration of a cantilever Cylindrical Tank, International Journal of Non-Linear Mecanics, Vol. 8, pp Croll, J. G. A. and Batista, R. C., 98, Explicit Lower Bounds for te Buckling of Axially Loaded Cylinders, International Journal of Mecanical Science, Vol. 3, pp Del rado, Z.J.G.N.,, Modal coupling and interaction in te dynamic instability of cylindrical sells (in ortuguese) D. Sc. Tesis, Civil Engineering Department, Catolic University, UC-Rio. Rio de Janeiro, RJ, Brazil Dym, C. L., 973, Some new results for te vibrations of circular cylinders. Journal of Sound an Vibration, Vol. 9, pp Gasser, L. F. F, 987, Free vibrations of tin cylindrical sells containing fluid (in ortuguese). Master s Tesis, EC-COE, Federal University of Rio de Janeiro. Rio de Janeiro, RJ, Brazil. Gonçalves,. B. and Batista, R. C, 987, Frequency response of cylindrical sells partially submerged or filled wit liquid. Journal of Sound and Vibration, Vol. 3, pp Gonçalves,. B. and Batista, R. C., 988, Non-Linear Vibration Analysis of Fluid-Filled Cylindrical Sells, Journal of Sound and Vibration, Vol. 7, pp Gonçalves,. B. and Del rado, Z. J. G. N.,, Te Role of Modal Coupling on te Non-linear Response of Cylindrical Sells Subjected to Dynamic Axial Loads, Nonlinear Dynamics of lates and Sells; AMD Vol. 38, pp New York: ASME. Gonçalves,. B. and Del rado, Z. J. G. N.,, Non-Linear Oscillations and Stability of arametrically Excited Cylindrical Sells, Meccanica, Vol. 37, pp Hunt, G. W., Williams, K. A. J. and Cowell, R. G., 986, Hidden Symmetry Concepts in te Elastic Buckling of Axially Loaded Cylinders, International Journal of Solid and Structures, Vol., pp opov, A. A., Tompson, J. M. T. e McRobie, F. A., 998, Low dimensional models of sell vibration. arametrically ecited vibrations of cylindrical sells. Journal of Sound and Vibration, Vol. 9, no, pp Ramacandran, J., 979, Nonlinear Vibrations of Cylindrical Sells of Varying Tickness in an Incompressible Fluid, Journal of Sound and Vibration, Vol. 64, pp: Silva, F. M. A. Instability dinamics analisys of cylindrical fluid-filled sells (in ortuguese). 4. Master s Tesis, Federal University of Goiás, Goiânia, GO, Brazil, 4. Yamaki, N., 984, Elastic Stability of Circular Cylindrical Sells, Ed. Amsterdam: Nort Holland. 338 / Vol. XXVIII, No. 3, July-September 6 ABCM