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2nd WSEAS Int. Conf. On NON-LINEAR ANALYSIS, NON-LINEAR SYSTEMS and CHAOS (WSEAS NOLASC 2003) Vouliagmeni, Athens, Greece, December 29-31, 2003 Hybrid Finite Element in Non-Linear Structural Dynamics of Anisotropic Tubes Conveying Axial Flow M.H. Toorani1 and A.A. Lakis2 1. Nuclear Engineering Department, Babcock & Wilcox Canada P.O. Box 310, Cambridge, Ontario, Canada N1R 5V3 2. Mechanical Engineering Department, Ecole Polytechnique of Montreal C.P. 6079, Station Centre-ville, Montreal (QC), Canada H3C 3A7 Tel.:(514) 340-4711, x4906, Fax: (514) 340-4176 Abstract: A semi-analytical approach has been developed in the present theory to determine the geometrical non-linearity effects on the natural frequencies of anisotropic cylindrical tubes conveying axial flow. Particular important in this study is to obtain the natural frequencies of the coupled system of the fluid- structure, taking the geometrical non-linearity of the structure into consideration and estimating the critical flow velocity at which the structure loses its stability. The displacement functions, mass and stiffness matrices, linear and non-linear ones, of the structure are obtained by exact analytical integration over a hybrid element developed in this work. Linear potential flow theory is applied to describe the fluid effect that leads to the inertial, centrifugal and Coriolis forces. Stability of tubes subjected to the flowing fluid is also discussed. Numerical results are given and compared with those of experiment and other theories to demonstrate the practical application of the present method. Key-Words: Non-Linear, Dynamics, Flowing Fluid, Anisotropic, Cylindrical Tubes 1 Introduction instability can be illustrated as a feedback Component failures due to excessive flow- mechanism between structural motion and the induced vibrations continue to affect the resulting fluid forces. A small structural performance and reliability of nuclear displacement due to fluid forces or whatever alerts components, piping system and tube heat the flow pattern, inducing a change in the fluid exchangers. Fluid-elastic vibrations have been forces; this in turn leads to further displacement, recognized as a major cause of failure in shell- and so on. When the flow velocity becomes and-tube-type heat exchangers. Fluid-elastic higher, the vibration amplitude becomes larger vibrations result from coupling between fluid- and the impact phenomenon occurs that can lead induced dynamic forces and the motion of to unacceptable tube damage due to fatigue and structure. Depending on the boundary conditions, /or fretting wear in critical process equipment. static (buckling) and dynamic (flutter) instabilities Therefore, the evaluation of complex vibrational are possible in the structures at sufficiently high behavior of these structures is highly desirable in flow velocities. The nature of fluid-elastic nuclear industry to avoid such problems. 2 Corresponding author 1 2.1 Structure Model While several studies have been conducted on the The equations of motion of anisotropic cylindrical dynamic stability of circular cylindrical shells shells in terms of U, V and W (axial, tangential conveying fluid, but in contrast, non-linear studies and radial displacements of the mean surface of of shells subjected to either internal or external the shell), x and (the rotations of the axial flow are few. Particularly interesting, in case normal about the coordinates of the reference of internal axial flow, are the studies of Païdoussis surface), see Fig. 1, and in terms of Pij’s elements &Dennis [1], Lakis & Païdoussis [2, 3], Weaver are written as follows: & Unny [4], Pettigrew [5], Selmane &Lakis [6] and Amabili et al. [7]. These works are performed based on the classical shell theory by neglecting Lm U , V , W , x , , Pij 0. m 1,...,5 (1) the shear deformation effect while this later plays i, j 1,2,3,...,10 a very important role in reducing the effective where anisotropic elasticity matrix (Pij’s flexural stiffness of composite shells and also the elements), and five linear differential operators moderately thick structures. The present work (Lm), are fully given in [8]. addresses the question of stability of anisotropic cylindrical shells, based on a shearable shell The finite element developed is shown in Fig. 1. theory, subjected to internal and external axial It is a cylindrical panel segment defined by two flow. The non-linearities due to large amplitude nodal lines i and j. Each node has five degrees of shell motion are taken into account, by using the freedom, three displacements and two rotations. modal coefficient approach, while the amplitude of shell displacement remains within the linear 2 i j 1 range from the fluid point of view. 1 m N Wmi 1mi 2 Mathematical Model Vmi m i The analytical solution involves the following 2mi steps: Umi j a) The strain-displacement relations expressed in an arbitrary orthogonal curvilinear coordinate are (A) inserted into the equations of motion, obtained based on shearable shell theory, of anisotropic cylindrical shells. The mass and linear stiffness (B) matrices are determined for an empty finite Figure 1: (A) Finite element discretization. element, Fig. 1, and assemble the matrices for the (B) Nodal displacements at node i. complete shell. The displacement functions associated with the b) The coefficients of the modal equations are axial wave number are assumed to be: derived using the non-linear part of the kinematics m m relations. U ( x, ) ACos xe ; x ( x, ) DCos xe L L m m (2) c) A finite fluid element bounded by two nodal V ( x, ) BSin xe ; ( x, ) ESin xe L L lines, Fig. 1-B, is considered to account the effect m of the fluid on the structure. W ( x, ) CSin xe L d) The linear and non-linear natural vibration where m is the axial mode, and is a complex frequencies are then obtained and compared with the available results. number. A system of five homogeneous linear functions is obtained by substituting (2) into equations of motion (1). For the solution to be non-trivial, the determinant of this system must be 2 equal to zero. This brings us to the following equation (6). The global matrices [Ms] and [Ks(L)] characteristic equation (see References [8, 9] for may be obtained, respectively, by superimposing more detail): the mass and stiffness matrices for each individual panel finite element [9]. DetH f10 10 f 8 8 f 6 6 f 4 4 f 2 2 f 0 (3) 2.1.2 Non-Linear Stiffness Matrices of Each roots of this equation yields a solution to the Structure linear equations of motion (1). The complete The exact Green strain-displacement relations are solution is obtained by adding the ten solutions used in order to describe the non-linear behavior, independently. After carrying out the some including large displacements and large rotations, intermediate manipulations, that are not displayed of anisotropic cylindrical shells. In common with here (see Reference [9]), the following equations linear theory, it is based on refined shell theory in are obtained : which the shear deformations and rotary inertia U ( x, ) effects are taken into account. The approach V ( x, ) developed by Radwan and Genin [10] is used with i (4) particular attention to geometric non-linearities. W ( x, ) N ( x, ) j x The coefficients of the modal equations are ( x, ) obtained through the Lagrange method. Thus, the where is the displacement vector at the non-linear stiffness matrices of second and third boundaries and [N] represents the displacement order are then calculated by precise analytical function matrix. The constitutive relation between integration and superimposed on the linear part of the stress and deformation vector of cylindrical equations to establish the non-linear modal shells is given as [9]: equations. The main steps of this method are as follow: N xx N x Qxx N Nx Q M xx M x M M x 2.1.2.a Shell displacements are expressed as P B i (5) generalized product of coordinate sums and j spatial functions: The matrices [P], as a function of geometrical and mechanical parameters of anisotropic cylindrical shells, and [B] are given in [9]. u qi (t ) Ui x, x qi (t ) xi x, i i (7) v qi (t ) Vi x, qi (t ) i x, i i 2.1.1 Mass and Linear Stiffness Matrices w qi (t ) Wi x, Using the procedure of the classical finite i element, the mass and stiffness matrices are then calculated. For one finite element, they may be where the qi (t)’s functions are the generalized written as follows: coordinates and the spatial functions U, V, W, x L and are given by equation (2). m s h N T N dA 00 L (6) 2.1.2.b The deformation vector is written as a k L B T PB dA function of the generalised coordinates by 00 separating the linear part from non-linear one: where s is the density of the shell, h its L NL T (8) thickness, dA a surface element, [P] the elasticity matrix and the [N] and [B] are derived from equations (4, 5). The matrices [ms] and [ks(L)] are This vector is given in [11]. The subscripts “L” obtained analytically by carrying out the and “NL” mean “linear” and “non-linear”, necessary matrix operations over x and in 3 respectively. In general, these terms can be carrying out a large number of the intermediate expressed in the following form: mathematical operations, while are not given here x a j q j AA jk q j q k o due to the complexity of the manipulations, the j j k following non-linear modal equations are x b j q j BB jk q j q k o obtained. These non-linear modal equations are j j k used to study the dynamic behavior of an empty anisotropic cylindrical shell. c j q j CC jk q j q k o x (9-a) j j k mij j kij j kijk j k ( L) ( NL 2 ) o d j q j DD jk q j q k j j j k (12) j j k kijks ( NL 3) j k s Qi i 1,2,... o e j q j EE jk q j q k j k s j j k Where mij, kij(L) are the terms of mass and linear stiffness matrices given by equation (6). The o g j q j GG jk q j q k terms of kijk(NL2) and kijks(NL3) , which represent the j j k second and third-order non-linear stiffness x n j q j NN jk q j q k matrices, are given by the following integrals in j j k the case of anisotropic laminated cylindrical shell x p j q j PPjk q j q k (9-b) based on the refined shell theory in which the j j k shear deformation and rotary inertia effects are s j q j SS jk q j q k considered: j j k P AAijk P22 BBijk P33CCijk t j q j TT jk q j q k 11 j j k P44 DDijk P55 EEijk (13-a) P66GGijk P77 NN ijk P88 PP ijk dA Note: AAij=AAji, BBij=BBji and etc. kijk ( NL 2) P99 SSijk P 1010TTijk Pmn AUX ijk I AUX ijk J 2.1.2.c Lagrange’s equations of motion in the generalized coordinates qi (t) is defined as: P36 AUX ijk 57 AUX ijk 58 and d T T V P AAijks P22 BBijks P33CCijks q q Qi (10) 11 dt qi i i P44 DDijks P55 EEijks (13-b) P66GGijks P77 NN ijks P88 PP ijks kijks( NL3) P99 SSijks P1010TTijks dA Where T is the total kinetic energy, V the total elastic strain energy of deformation and the Qi’s Pmn AUXijksI AUXijksJ are the generalized forces. Assuming AUX 57 AUX 58 NL 1 , 2 ,..., 10 T , the strain energy V can P36 ijks ijks be defined as follow: Where dA=R dx d and: m =1,2,...,9 I=1,3,5,...,55 and m, n3,6 Pij i j Pkl k l R d x d a L V (11) n=m+1 to 10 J=I+1 2 00 The Pij’s are the terms of the elasticity matrix [P] Where: and the terms AAijk, BBijk, ..., AUXijk58 and AAijks, a =1 if i = j or k = l (i, j =1,2,...,10), (k,l=3,6) BBijks,..., AUXijks58 represent the coefficients of the and i , j 3,6 modal equations in step (2.1.2.d). Follow are the a =2 if i j or k l (i ,j=1,2…,10) (k,l=3,6) expressions for the coefficients ai, AAij, AAijk and AAjkrs, the others coefficients are obtained in the 2.1.2.d After developing the total kinetic energy same way, details are given in [11]. and strain energy, using definitions (9), and then substituting into the Lagrange equation (10) and 4 U i 2.2.1 Dynamic Pressure ai Based on the previous hypothesis, the potential x function must satisfy the Laplace equation. This 1 U i U j Vi V j Wi W j relation is expressed in the cylindrical coordinate AAij 2 x x x x x x system by: 2 1 r ,r ,r 12 , , xx 0. (16) AAijk ai AA jk a j AAki ak AAij (14) r r AAijks 2 AAis AA jk is the potential function that represents the velocity potential. The components of the flow where U,V and W are spatial functions velocity are given by: determined by equations (2). In equation (14), the 1 subscript “i,j”, “i,j,k” and “i,j,k,s” represent the Vx U xu , x ; V , ; Vr , r (17) R coupling between two; three and four mode, respectively. Substituting equation (2) into where Vx ,V and Vr are respectively the axial, equations (14), we obtain tangential and radial components of the fluid velocity; Uxu is the velocity of the liquid through ai Ci i mSinmx e i the shell section. The Bernoulli equation is given m 2 Sin 2 mx by: i j ( i j ) (15) AAij Ci C je (1 )m Cos 2 mx 2 i j 2 , t V P 0. (18) m m / L 2 f r where i ( i =1,...,10) are the roots of A full definition of the flow requires that a characteristic equation (3) and m is the axial mode condition be applied to the structure-fluid number. The same definitions, as relation (15), are interface. The impermeability condition ensures obtained for other parameters and given in [11]. contact between the shell and the fluid. This The constants Ci(i=1,...10) can be determined should be: using ten boundary conditions for each element. The axial, tangential and radial displacements as Vr r R , r r R W,t U xW, x r R (19) well as the rotations have to be specified for each node. From the theory of shells, we have: Substituting these definitions into equation (13) m and then integrating over x and , the two j 1 W x, , t C j exp j it sin 10 L x (20) expressions for the second- and third-order non- linear matrices are obtained, as given in equation (12). Assuming then, x, , r , t R j (r ) S j x, , t 10 (21) 2.2 Fluid Model j 1 Linear potential flow theory is applied to describe the fluid effects that lead to the inertial, The function S j ( x, , t ) is explicitly determined centrifugal and Coriolis forces. The mathematical by applying the impermeability condition (19) and model is based on the following hypothesis: i) the using the radial displacement (20). Substituting fluid flow is potential; ii) the fluid is irrotational, the assumed function into equation (16) leads to incompressible and non-viscous. the following differential Bessel equation: 5 Substituting relation (24) into (18), we obtain the 2 d R j (r ) dR j (r ) equation for the pressure on the shell wall. r2 2 r (22) dr dr 0. 10 Pu f u Zuj W j ,tt 2U xuW j , xt U xuW j , xx (27) 2 R j (r ) i i j 2 2 2 mk r 2 j 1 where “i” is the complex number, i2=-1 and j is By introducing the displacement function (20) the complex solution of the characteristic equation into pressure expression (27), performing the for the empty shell. The general solution of matrix operation and thereafter integration over equation (22) is given by: the fluid element required by the finite element method, the linear matrices (mass [mf], damping im im R j r A J i j r B Yi j r (23) [cf] and stiffness [kf]) of moving fluid are L L obtained. Finally, the global linear matrices [Mf], [Cf] and [Kf] may be obtained, respectively, by where Ji j and Yi j are, respectively, the Bessel superimposing the different matrices for each individual fluid finite element. functions of the first and second kind of complex order “ i j ”. For inside flow, the solution (23) must be finite on the axis of shell (r=0); this means we have to set the constant “B” equal to 3 Non-Linear Differential zero. For outside flow ( r ); this means that Relations the constant “A” is equal to zero. When the shell The structural and fluid mass and stiffness is simultaneously subjected to internal and matrices, either linear or non-linear, as well as the external flow, we have to take the complete fluid damping matrix, obtained in the previous solution (23). We carry the Bessel equation sections, are only determined for one element. solution back into (21) to obtain the final The global mass, stiffness and damping matrices expression of velocity potential evaluated at the are obtained by assembling the matrices for each shell wall: element. Assembling is done in such way that all imRu )W j ,t U xuW j , x 10 the equations of motion and the continuity of (r , x, , t ) Z uj ( (24) j 1 L displacements at each node are satisfied. These where matrices are designated as [M], [K] and [C], respectively. mRu Ru ) if u i (25) (M s M f ) ( K s K f ) Z uj ( ( L) L imRu J i j 1 (imRu / L) (28) i j J i j (imRu / L) C K K 0 ( NL 2 ) 2 ( NL 3) 3 L f s s and Z uj ( mRu ) Ru if u e (26) where { } is the displacement vector and [Ms], L imRu Yi j 1 (imRu / L) [KsL], [KsNL2] and [KsNL3] are, respectively the i j L Yi j (imRu / L) mass, linear and second- and third-order non- linear stiffness matrices of the structure, where j ( j 1,...,10) are the roots of the respectively, and [Mf], [Kf] and [Cf] are the characteristic equation of the empty shell; inertial, centrifugal and Coriolis forces, J i j and Yi j are, respectively, the Bessel respectively, due to the fluid effect. functions of the first and second kind of order Setting: “ i j ”; “m” is the axial mode number; “R” is the mean radius of the shell; “L” its length; the subscript “u” is equal to “i” for internal flow and q (r ) q j (t ) j j (t ) (29) is equal to “e” for external flow. j (0) 1 and j (0) 0. 6 where represents the square matrix for in this work is a hybrid finite element based on a eigenvectors of the linear system and qis a time combination of refined shell theory, modal expansion approach and potential flow theory. related vector. Numerical solution of the coupled This method is capable of obtaining the high as system (28) is difficult and costly. Here, we limit well as low frequencies with high accuracy. The ourselves to solving the uncoupled system. In this values of shear correction factors used in the case, equation (28) is reduced to the following equation: calculation have been taken 2 / 12 . 90 i i i i i (i / h) o 2 2 Fibre 90 Non-Dimensional Frequency i i 80 (30) i (i / h 2 ) i3 0. x 70 y o Fibre 0 60 where Sanders' Theory 50 Present NL 2 C f ii k ii k i ; i2 ; i sii 40 h R/h=50 L/R=5 mii mii mii 30 m=4 NL 3 m=3 k 20 i h ; mii msii m fii sii 2 (31) mii 10 m=1 and k ii k k f ii L sii 0 0 1 2 3 4 5 6 7 8 9 10 where “h” represents the shell thickness. The Circumferential Wave Number (n) square root of coefficient k ii / mii represents the ith linear vibration frequency of system. The solution Figure 2: Variation of non-dimensional natural frequencies in conjunction with variation of m. i (t ) of the non-linear differential equations (30), which satisfies the conditions in (29), is calculated a) Linear Vibrations of empty and liquid filled by a fourth order Runge-Kuta numerical method. isotropic and anisotropic cylindrical shells- It The linear and non-linear natural frequencies are should be noted that in the two first examples, the evaluated by a systematic search for the natural frequencies of the structures are also i (t ) roots as a function of time. The NL / L obtained using Sanders’ theory (non-shearable ratio of linear and non-linear frequency is shell theory), by authors. expressed as a function of non-dimensional ratio i / h where i is the vibration amplitude. In the first example, the different longitudinal vibration modes ( o R 2 ( / E2 ) / h ) as a function of the circumferential wave number are drawn in Fig. 2. This figure shows the results for 4 Numerical Results and four symmetric layers cross-ply (0o/90o/90o/0o) Discussions laminated shell whose mechanical properties are given as: This research work is focused on the shear deformation and geometrically non-linear effects E1=25E2; G23=0.2E2; G13=G12=0.5E2, ν12=0.25; on the dynamic behavior of anisotropic cylindrical ρ=1 shells conveying fluid. Non-linearity effects produce either hardening or softening behavior in 3 circular cylindrical shells. Considering the shear deformation effects leads to reducing the flexural Present 2.5 ensional Freque/ncy stiffness of the structures. The developed method Sanders' Theory m=5 2 7 1.5 Figure 4: Natural frequencies of a simply supported cylindrical shell. 0.7 0.6 Non-Dimensional Frequency 0.5 [Ref. 14] 0.4 Present R/h=20 Figure 3: Variation of non-dimensional natural m=1 0.3 frequencies in terms of m & n. n=1 0.2 In the next example (Fig. 3), the effect of axial R/h=100 mode number on the non-dimensional natural 0.1 frequencies ( o R( (1 2 ) / E )1 / 2 ) of an R/h=300 isotropic cylindrical shell is studied and the results 0 1 2 3 4 5 6 7 8 9 10 are compared with the obtained corresponding values based on the Sanders’ theory. Length-to-Radius Ratio L/R Fig. 4 shows the natural frequencies computed for Figure 5: Frequency distribution of a fluid-filled closed simply supported, circular cylindrical shell cylindrical shell. for m=2 and compared with the experimental results, given in [13]. To determine natural The fluid depth effect is also studied for the half- frequencies with the developed program, based on filled cylindrical shell in Fig. 4. the present theory, only 10 elements are required to provide acceptable accuracy. As can be seen, Fig. 5 is carried out for a simply supported, there is good agreement between the present isotropic circular cylindrical shell completely theoretical results and those of experimental. filled with liquid. The frequency parameter, Dimensions and material properties are given as ( o R ( (1 2 ) / E )1 / 2 ), is shown for follow: different values of R/h and L/R and is compared R 0.175(m) L 0.664(m) t 1(mm) with provided results in Ref. [14]. E 206(GPa) 0.3 7680(kg / m3 ) b) Stability of the shells subjected to flowing fluid- 200 0 The influence of the flow velocity on the Present, Empty 180 0 Present, Full frequency parameter of cylindrical shells is studied through Figs 6 and 7 for different values Natural Frequency (Hz) 160 Experiment [13], 0 Empty Half-Filled Present, of R/h, L/R, axial and circumferential wave 140 0 numbers. The obtained results, in Fig. 6, are 120 compared with those of theory [4]. 0 100 0 25 80 0 Present nsional Frequency m=2 60 0 [Ref.4] 20 40 0 m=2 L/R=2 20 R/h=100 0 8 0 15 f/s=0.128 0 2 4 6 8 10 12 Circumferential Wave Number (n) n=5 10 Figure 7: Stability of a cylindrical shell as a function of flow velocity. In Fig. 6, the first frequency becomes negative imaginary at U=2.96, indicating static divergence instability in the first axial mode, and reappeared and coalesced at U=3.36 with that of the second axial mode to produce mode flutter. Fig. 7 shows the divergence instability phenomenon for an Figure 6: Stability of a simply supported isotropic simply supported cylindrical shell. cylindrical shell as a function of internal flow velocity. c) Linear vibration of submerged cylindrical shells- Fig. 8 shows the non-dimensional, In these two figures, the following parameters are ( o R( s (1 2 ) / E )1 / 2 ) ,frequency variation defined: as a function of circumferential wave number for U u / u o ; / o ; o u o / L three different cases, shell in air, fluid-filled shell K 1/ 2 2 Eh3 and shell immersed in fluid and are compared uo 2 ( ) ;K with those of theory [15]. The two theories give L sh 12(1 2 ) nearly identical results. The u and ω are, respectively, the velocity of the d) Non-linear vibrations of empty and submerged flowing fluid and the natural frequency. As the cylindrical shells- The influence of non-linearities flow velocity increases, Fig. 6, the two theories on the frequencies of a simply supported generate significantly different results. This might cylindrical shell, along with corresponding values be attributed to i) not considering the influence of given in References [16 and 17] is shown in Fig. transverse shear deformation in Ref.[4] and ii) 9. The given results in Ref. [16] were obtained limitations of the theory (Ref.[4]) associated with based on Donnell’s simplified non-linear method. the use of too few terms in the application of Raju and Rao [17] used the finite element method Galerkin’s method. based on an energy formulation. 30 L/R=2 n=3 R/h=100 1 f/s=.128 Non-Dimensional Frequency 25 d b Non-Dimensional Frequency n=4 20 Present n=5 [Ref.15] 0.1 m=1 15 - - - -m=2 Shell in Air n=3 Fluid Filled 10 n=4 n=5 m=1 0.01 L / R=4 5 R / h=400 Shell Immersed in b /d=1.0 Fluid /=.132 f s 0 9 0 1 2 3 4 5 Dimensionless Velocity 0.001 0 2 4 6 8 10 12 Figure 10: Non-linearity effect on the frequency Figure 8: Frequency variation of empty, fluid- ratio of a submerged cylindrical shell. filled and immersed in fluid shell with respect to (n). 5 Conclusion This paper deals with some of the problems that Fig. 10 shows the non-linearity effect on the arise when considering geometric non linearities, frequency ratio of a steel open (φ=100o) shear deformation, rotary inertia and flowing fluid cylindrical shell totally submerged in fluid. The effects in the study of dynamic and stability shell is simply supported. The following data are behavior of elastic, anisotropic and isotropic considered into calculations: cylindrical shells. An efficient hybrid finite element method, modal expansion approach, R=450mm; h=1.5mm; L=1350mm, ρf/ρs=.128 shearable shell theory and linear potential flow 1.3 one have been used to develop the non-linear Present E=200GPa Raju and Rao [17] dynamic equations of the coupled fluid-structure =0.3 Relative Frequency NL/L =7800kg / m 3 Nowinski [16] system. The shell equations are used in full for the 1.2 R=2.54 m=1 determination of the displacement functions. It is L=40 cm h=0.0254 cm n=4 believed that the refined shear deformation theory cm and effects of geometric non-linearities of the structures presented here are essential for 1.1 predicting an accurate response for anisotropic shell structures. A full implementation of the non- linear dynamic equations is conducted to show the 1 reliability and effectiveness of the present formulation that gives a very good description of geometrical non-linear and shear deformation effects on the dynamic and stability behavior of 0.9 0 0.5 1 1.5 2 2.5 3 the cylindrical shells subjected to flowing fluid. Amplitude to Thickness Ratio j/h The natural frequencies of the coupled fluid- structure are lower than the corresponding values Figure 9: Relative frequency of simply supported of empty shells due to increased kinetic energy cylindrical shell versus relative amplitude. without a corresponding increase in the strain energy. In the case of flowing fluid, the centrifugal and Coriolis terms generate complex 2.1 eigenvalue problems, non-self-adjoint differential 1.9 equations. Therefore, the system may experience n=1 n=2 static (buckling) and dynamic (flutter) 1.7 instabilities. As long as the effective stiffness of the system remains positive as flow velocity Relative Frequency 1.5 increases, the system will oscillate asymptotically 1.3 about its neutral equilibrium position; otherwise it will diverge to a new equilibrium position, 1.1 different from neutral (buckling). As long as the ωNl/ωL 0.9 0.7 10 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Amplitude to Thickness Ratio Γ /h effective fluid damping of the system remains Quiescent Fluid, Journal of Fluid and Structures, positive as flow velocity is increased, vibrations No. 12, 1998, pp. 883-918. will be damped; otherwise they will be amplified (flutter). [8] M.H. Toorani and A.A. Lakis, General Equations of Anisotropic Plates and Shells It is shown that the non-linearities associated with Including Transverse Shear Deformations, Rotary fluid, under no-flow conditions, have no or very Inertia and Initial Curvature Effects, Journal of little effect on the natural frequency of a Sound and Vibration, 237 (4), 2000, pp. 561-615. cylindrical shell for amplitudes up to two times the shell thickness [6]. Under flow conditions, [9] M.H. Toorani and A.A. Lakis, Shear non-linear effects were found to increase with Deformation Theory in Dynamic Analysis of flow rate increasing but the importance of the Anisotropic Laminated Open Cylindrical Shells contribution of flow-non-linearities to this overall Filled With or Subjected to a Flowing Fluid, trend has yet to be determined. Attempting to this Computer Methods in Applied Mechanics and work is left to future investigations. Engineering, 190, 2001, pp. 4929-4966. [10] H. Radwan and J. Genin J., Non-Linear References: Modal Equations for Thin Elastic Shells, International Journal of Non-Linear Mechanics, [1] M. P. Païdoussis and J.P. Denis, Flutter of 10, 1975, pp. 15-29. Thin Cylindrical Shells Conveying Fluid, Journal of Sound and Vibration, No. 20, 1972, pp. 9-26. [11] M.H. Toorani and A.A. Lakis, Geometrically Non-Linear Dynamics of Anisotropic Open [2] A.A. Lakis and M.P. Païdoussis, Free Cylindrical Shells with a Refined Shell Theory, Vibration of Cylindrical Shells Partially Filled Technical Report, Polytechnique of Montreal, with Liquid, Journal of Sound and Vibration, No. EPM-RT-01-07, 2002. 19, 1971, pp. 1-15. [12] J.L. Sanders J.L., Nonlinear Theories for [3] A.A. Lakis and M.P. Païdoussis, Shell Natural Thin Shells, Quarterly of Applied Mathematics, Frequencies of the Pickering Steam Generator, 21, 1963, pp.21-36. Atomic Energy of Canada Ltd., AECL, Report No. 4362, 1973. [13] M. Amabili and G. Dalpiaz, Breathing Vibrations of a Horizontal Circular Cylindrical [4] D.S. Weaver and T.E. Unny, On the Dynamic Tank Shell, Partially Filled with Liquid, Journal Stability of Fluid-Conveying Pipes, Journal of of Vibration and Acoustics, 117, 1995, pp. 187- Applied Mechanics, No. 40, 1973, pp. 48-52. 191. [5] M.J. Pettigrew, Flow-Induced Vibration [14] A.A. Lakis and M. Sinno, Free Vibration of Technology: Application to Steam Generators, Axisymmetric and Beam-Like Cylindrical Shells Lecture Series Presented at Babcock & Wilcox Partially Filled with Liquid, Int. J. of Num. Meth. Canada, November 2000. In Eng., 33, 1992, pp. 235-268. [6] A. Selmane and A.A. Lakis, Non-Linear [15] P.B. Gonçalves and R.C. Batista, Frequency Dynamic Analysis of Orthotropic Open Response of Cylindrical Shells Partially Cylindrical Shells Subjected to a Flowing Fluid, Submerged or Filled with Liquid, Journal of Journal of Sound and Vibration, No. 202, 1997, Sound and Vibration, 113(1), 1987, pp. 59-70. pp. 67-93. [16] J.L. Nowinski J.L., Non-Linear Transverse [7] M. Amabili M., F. Pellicano and M.P. Vibrations of Orthotropic Cylindrical Shells, Païdoussis, Non-Linear Vibrations of Simply AIAA Journal, 1, 1963, pp. 617-620. Supported Circular Cylindrical Shells, Coupled to 11 [17] K.K. Raju and G.V. Rao, Large Amplitude Shells of Revolution, Journal of Sound and Asymmetric Vibrations of Some Thin Elastic Vibration, 44, 1976, pp. 327-333. 12

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