1 INTRODUCTION
A thinwalled cylindrical shell is defined by a solid with curved surface and small thickness compared to its other dimensions. Structural dynamic analysis of shells is elaborate due to the occurrence of simultaneous flexural and inplane vibrations. Many concomitant factors influence the dynamic response of a cylindrical shell, like the contained material type (fluid or solid), boundary conditions, the presence of ring support, the influence of the connection between the shell walls and the bottom and the lid, among others.
The study of free vibration in cylindrical shells received and still is a topic of active interest by many researchers. Based on the classic works of ^{Love (1888}), ^{Donnell (1933}), ^{Flugge (1934}), ^{Mushtari (1938}), ^{Reissner (1941}), ^{Vlasov (1951}), ^{Arnold and Warburton (1953}), ^{Timoshenko and Woinowsky (1959}), ^{Sanders (1959}), ^{Naghdi and Berry (1964}), many other theories were developed and improved upon to simulate the dynamic behavior of a shell. These analytical techniques differ from each other by the strains and curvatures assumed for the shell, as well as by the solution methods applied to the problem. ^{Leissa (1973}) presents the different classical theories mentioned and many simulation results.
Based on classical theories, advances were made in analytical studies towards the calculation of vibration properties of cylindrical shells. Some of these studies comprise the direct solution of the equations of movement, whereas others are based on variational methods (energy based). Examples are given by the works of ^{Sharma and Johns (1971}), ^{Chung (1981}), ^{Soedel (2005}), ^{Farshindianfar and Oliazadeh (2012}), ^{Chen et al. (2013}), ^{Dai et al. (2013}), ^{Sun et al. (2013}), ^{Ataabadi et al. (2014}), ^{Ameijeiras and Godoy (2016}), and ^{Qin et al. (2017}). These works admit different shell theories to simulate deformations, as well as different displacement assumptions, such as: beam modal functions, characteristic orthogonal polynomials, Chebyshev polynomials, or exact Fourier series solution. These provide efficient computational methodologies to obtain the dynamic behavior of shells with classic or arbitrary boundary conditions. However, the proposed solutions are often extensive and complex for practical design applications.
On the other hand, among the different analytical methods applied to the free vibration analysis of cylindrical shells coupled with fluids, it is worth mentioning the works of ^{Lakis and Paidoussis (1971}), ^{Gonçalves and Batista (1987}), ^{Fernholz and Robinson (1990}), ^{Sarkar and Sonti (2009}), Gonçalves et al. (2010), ^{Mendes et al. (2014}), ^{Lopez (2014}), ^{França Júnior et al. (2017}) and ^{Ji et al. (2019}). In these studies, the FluidStructure Interaction (FSI) is conceived by adding the fluid hydrodynamic pressures in the shell movement equations, together with the impenetrability condition at the shellliquid interface. In the abovecited works, the calculation procedure is discussed and no explicit equation that expresses the frequency of the coupled fluidshell system is presented  i.e., computational implementation is required for problemsolving.
Although analytical solutions are quite effective, theoreticalexperimental studies were carried out in order to analyze the dynamic behavior of uncoupled and coupled cylindrical shells, as can be highlighted in the works of ^{Haroun (1983}), ^{Fortuny Gasser (1987}), ^{Pedroso et. al. (1994}), ^{Amabili and Dalpiaz (1995}) and ^{Li et al. (2018}). In these studies, experimental research shows that performing the FSI by adding the virtual mass of the fluid in the movement equations of the shell is satisfactory and yields excellent results.
In an overview, it can be said that both for empty and fluidfilled shells, the literature presents elaborate and extensive analytical solutions that require computational methods to solve the governing equations, which, in most cases, do not result in simple closedform solutions. In this sense, a simplified solution that yields good results could be applied for validation purposes and in practical design analysis.
With that in mind, this work presents an innovative, compact and simplified analytical solution to calculate the dynamic behavior of simply supported cylindrical shells, uncoupled and coupled with liquids. The vibration characteristics of (i) the uncoupled shell, (ii) the uncoupled fluid and (iii) the coupled shellliquid system (where the virtual mass of the fluid is added in the shell equations of motion) are discussed. Simplifications in the shell strains and curvature were essential to find the closedform solution, which provides explicit expressions for natural frequencies, that can be easily programmed in a spreadsheet.
The results obtained with the proposed procedure were compared with values obtained using the finite element method through ANSYS software, and with experimental results described by ^{Amabili and Dalpiaz (1995}). The natural frequencies and mode shapes were studied, and the results obtained are consistent with analytical, numerical and experimental results.
2 CYLINDRICAL SHELL (STRUCTURAL DOMAIN)
The thinwalled cylindrical shell (Figure 1) is defined with length
Based on ^{Flugge (1934}) linear theory, the shell strains and curvatures are given by:
where
where
The equations of motion can be obtained by the RayleighRitz procedure using the Lagrange function
where
where
The equations for
where
2.1 Simplifications proposed by the authors (governing hypotheses)
In the frequency domain, the mode shapes are given by:
where
the number of longitudinal half waves is
An extensive investigation of the complete equations/formulation was done to identify possible simplifications in both the total kinetic and total strain energy expressions. This strategy is based on evaluation of relevant terms for computation of the first set of eigenvalues.
The total kinetic energy fraction associated with the axial direction was discarded, due to the displacement amplitudes in the radial and circumferential directions dominating over it. Similarly, the product between the axial and radial deformation, as well as the curvature in the axial direction tend to zero in the total strain energy. Under these hypotheses the simplified equations are given by:
After simplifying the energy terms, the RayleighRitz method applied to the Lagrange function
Thus, a linear system of equations is obtained. These equations can be written in a compact matrix notation, with symmetrical terms, resulting in the following eigenvalue problem:
Where
The proposed procedure relies in the abovementioned hypotheses for construction of a simplified equation for computation of the eigenvalues in (24). After some algebraic manipulations and computational assistance using MAPLE software, explicit expressions for indexes of
The simplified mass matrix
where:
The simplified stiffness matrix
where:
The coefficients
where:
Three roots of the characteristic polynomial are discarded for being imaginary, and the only real root is shown as the uncoupled cylindrical shell natural frequency equation:
where:
This analytical approach can be applied in practice by simply using equation (39), which can be easily programmed in a spreadsheet. Once the desired modes
3 ACOUSTIC CAVITY (FLUID DOMAIN)
The problem is defined by a cylindrical cavity with rigid boundaries containing an homogenous fluid, where the fluid free vibrations are governed by the wave equation in cylindrical coordinates:
where:
where
Using separation of variables for the acoustic pressures:
and substituting this expression in (44):
which leads to the Helmholtz equation for acoustic pressures of frequency
The following boundary conditions are applied to (47):
defining and openopen boundary condition in
where
4 FLUIDSTRUCTURE INTERACTION
In the shellcylindrical cavity fluidstructure interaction (coupled problem), two boundary conditions are present: the radial pressure at the shell surface is equal to the pressure on the cavity boundary, and the radial velocity at the shell surface is equal to the velocity on the acoustic cavity surface (impenetrability condition). As a simplification, the coupled shell modes of vibration are assumed to be equal to the invacuum structural modes shapes.
Using the simplified analytical method of Section 2, the coupling of a simply supported cylindrical shell is done through an added fluid mass in the shell equation terms that correspond to the structure mass in the radial direction. The added mass equation ratio for a partially or filled shell is presented by ^{Gonçalves and Batista (1987}):
where
The added mass increases the inertial load in the shell radial direction. The shell displacements are relevant in this direction and including the added mass in this way yields more realistic results. The hydrodynamic fluid pressure is transformed into an added mass that is summed to the shell structural mass. The modified mass matrix is given by:
The shell stiffness matrix is considered as the same as in (28). Once the determinant of
where:
Three roots of the characteristic polynomial are discarded for being imaginary and the real root, which represents the coupled natural frequencies, is given by:
with:
Similarly to the uncoupled problem, equation (59) can be applied in practical design problems by selecting the desired vibration modes
5 APPLICATIONS
5.1 Geometry and physical parameters
The following material properties were adopted in the models: the shell is made of steel with
FEM modal and harmonic analyses were performed in order to evaluate eigenvalues, eigenvectors and the coupled shell dominating displacements. A harmonic point load described by
5.2 Numerical model
The numerical analysis (Figure 3) was performed using finite elements with ANSYS software. The element SHELL63 was applied to the cylindrical shell (solid domain), whereas the element FLUID30 was applied to the acoustic cavity (fluid domain). The modules Block Lanczos and Unsymmetric were selected for the solution of eigenvalues and eigenvectors. The Full method was applied for computation of the structural response under the harmonic loads. The mesh independence analysis was performed and can be verified in ^{França Júnior (2018}). The number of elements for the final model is given by 6000 SHELL63 elements for the uncoupled shell, 13000 FLUID30 elements for the uncoupled fluid, and 21880 elements (6000 SHELL63 and 15880 FLUID30) for the coupled shell. The numerical formulation used in these simulations is based on the structure displacement and fluid pressure (UP), given by:
where
5.3 Results and validation
In this section, the results obtained using the analytical method and the numerical modeling for free and forced vibrations are presented and compared. Initially, the uncoupled cylindrical shell is analyzed, then the uncoupled fluid and, finally, the coupled shell.
5.3.1 Cylindrical shell in free vibrations
The first six natural frequencies obtained with the analytical technique and the numerical method are presented and compared in Table 1. The index





Diff. (%) 

Diff. (%) 

1  1  4  225,98  221,17  2,17  220,00  2,72 
2  1  5  232,80  230,43  1,03  230,00  1,22 
3  1  6  299,01  297,96  0,35  295,00  1,36 
4  1  3  322,70  315,92  2,15  315,00  2,44 
5  2  6  396,64  396,07  0,14  388,00  2,23 
6  1  7  445,62  441,67  0,89  440,00  1,28 
Based on Table 1, it can be noticed that the analytical and numerical results are almost the same, validating the simplified procedure. Since the difference between the results is very small for these natural frequencies, it can be concluded that the shell discretization (mesh refinement) simulates the cylindrical shell behavior for different mode shapes properly. The values obtained in the experiment by ^{Amabili and Dalpiaz (1995}) also validate the proposed formulation and the numerical method. Higher frequency results (m=2,3,4...) are shown in Figure 4.
Based on the plot in Figure 4, the cylindrical shell lower frequencies do not necessarily correspond to the lowest values of circumferential half waves, and for higher values of
With the simplified formulation validated, a comparison between the results obtained using this formulation and the ones using the complete formulation is presented in Table 2. The analytical frequencies obtained with the complete formulation, including the kinetic energy (15), and strain energy (16) is defined as Analytical [A]. The analytical frequencies using the proposed method is defined as Analytical [B] and is based on reduced equations (21) and (22).








1  1  4  221,24  225,98  221,17  220,00 
2  1  5  230,46  232,80  230,43  230,00 
3  1  6  297,99  299,01  297,96  295,00 
4  1  3  315,83  322,70  315,92  315,00 
5  2  6  396,16  396,64  396,07  388,00 
6  1  7  441,28  445,62  441,67  440,00 
Based on Table 2, it can be noticed that the neglected terms do not influence the results significantly. However, it is worth noting that discarding the terms of equations (15) and (16) was essential to obtain a much simpler closedform solution for the problem.
5.3.2 Acoustic cavity in free vibrations
In this section, the results of the analysis of a cylindrical acoustic cavity with rigid boundaries and openopen boundary condition in






Diff. (%) 

1  0  1  0  1129,52  1130,30  0,07 
2  0  2  0  2259,04  2265,40  0,28 
3  0  1  1  2754,02  2757,20  0,12 
4  0  2  1  3378,18  3385,00  0,20 
5  0  3  0  3388,55  3409,70  0,62 
6  0  3  1  4217,95  4237,30  0,46 
Based on table 3, the analytical and numerical results for the acoustic cavity natural frequencies are very close to each other for the different acoustic modes. Therefore, the numerical model simulates the fluid properly. Figure 6 presents some of the acoustic cavity modes of vibration obtained through finite element method using ANSYS software.
The uncoupled cavity natural frequencies and modes of vibration are essential to identify the dominant modes when the structure is coupled to the fluid, as it will be shown in the next section.
5.3.3 Fluidstructure coupled system in free and forced vibrations
Table 4 shows the fluidstructure coupled problem natural frequencies: simply supported cylindrical shell and openopen acoustic cavity. The natural frequencies obtained through the proposed analytical methodology (adding the virtual mass term), the numerical simulation and the experimental results obtained by ^{Amabili and Dalpiaz (1995}) are compared.






Diff. (%) 

Diff. (%) 

1  1  1  4  99,31  91,77  8,22  92,00  7,95 
2  1  1  5  112,67  104,91  7,40  104,00  8,34 
3  1  1  3  127,08  117,70  7,97  119,00  6,79 
4  1  1  6  155,46  147,08  5,70  147,00  5,76 
5  1  1  7  218,19  210,07  3,87  206,00  5,92 
The coupled cylindrical shell analytical natural frequencies presented in Table 4 are in agreement with FEM and experimental results. This validates the simplified procedure for coupled cylindrical shells in free vibration.
It is worth noting that, in the coupled problem, it was necessary to analyze the system dominating modes, because in the coupled vibration, there are: (i) structure dominating modes, where the fluid accompanies the structural displacement; (ii) acoustic cavity dominating modes, where the structure follows the fluid displacement; and (iii) mixed modes, combining results of the structure and the fluid. Thus, the coupled natural frequencies and mode shapes were compared to the uncoupled results for both the shell and the acoustic cavity.
The first twenty coupled natural frequencies are dominated by the structure with the added mass. This supports the use of the methodology presented herein. Having the formulation and modeling validated, the analysis of the coupled shell was extended to higher modes of vibration. Figure 7 presents these results.
Based on Figure 7 it can be noticed that the coupled shell natural frequencies behavior is similar to the uncoupled shell. According to ^{Lakis and Paidoussis (1971}), the displacement amplitudes of a simply supported shell filled with liquid are smaller if compared to the empty shell due to the fluid influence. This behavior was observed herein, showing that the fluid presence reduces the natural frequencies and reproduce the same tendencies observed in the uncoupled shell.
In the coupled lowfrequency range
To evaluate the dominating displacements in the fluidfilled cylindrical shell, a harmonic point load was applied in the radial direction
Figure 9 presents the radial and circumferential displacements of a single point in the shell (Fig. 9a), the radial displacement in a crosssection at different angles (Fig. 9b), and the radial displacement across the longitudinal direction (Fig. 9c). Based on the frequency spectrum presented (Fig. 9a), the peaks with more significant displacements occur at frequencies that are mainly associated with the radial and circumferential displacements. That is, the forced vibration does not mobilize the axial displacement modes, and the displacement amplitude in this direction is negligible for the first mode shapes. This is the reason why this component was not included in the plot. This fact supports the strategy of discarding the axial displacement term in the total kinetic and total strain energy equations.
6 CONCLUSIONS
This work addressed the aspects of fluidstructure interaction in the dynamic analysis of shells. The proposed procedure is based on an innovative, compact and simplified analytical method for computation of eigenvalues of simply supported cylindrical shells, uncoupled and coupled with liquids.
The assembly of shell mass and stiffness matrices is extremely simple and consist only of order three matrices, derived from the RayleighRitz method. Computation of the matrix indexes is straightforward and can be performed using any basic spreadsheet since explicit expressions are provided for all the involved terms. Explicit expressions for the roots of the characteristic polynomial are also available and the involved coefficients are listed in Appendix A for any given mode shape.
For fluidfilled shells, an added mass assumption (in the radial direction) proved as an extension of the uncoupled shell procedure, enabling the solution of FSI problems. This problem is extremely complicated when treated exactly, with algebraic operations that require numerical solvers. The proposed procedure is an efficient alternative since the simplicity of the invacuum shell computation is maintained, requiring only the inclusion of an added mass ratio in the last term of the main diagonal of the mass matrix.
The analysis of the acoustic cavity provided the comprehension of the uncoupled fluid dynamic behavior. This contributed to the analysis of the dominating modes in the coupled problem because the fluid and the structure vibrate at different frequency ranges.
In the fluidstructure coupled analysis, it was observed that despite influencing the system dynamic behavior, the fluidfilled shell reproduced the structure dominant mode shapes (same as the invacuum shell) in a significant range of analyzed frequencies. In this sense, the study of analytical techniques through the added mass criteria is significant because it applies to practical design situations in a simplified way.
Finally, it is concluded that the proposed procedure is effective whenever the governing assumptions are met, and these include (i) agreement of invacuum and coupled shell mode shapes; and (ii) neglecting energy terms with low representation in the Lagrange function. Under these assumptions, the procedure is reliable, with differences inferior to 10% when compared to FEM results.