Free Vibration Analysis of Circular Cylindrical Shells: Comparison of Different Shell Theories

In order to study the free vib ration of simply supported circular cy lindrical shells, a semi-analytical p rocedure is discussed in detail. In this technique, beam function is used as an approximat ion for simply supported boundary conditions. A literature review reveals that beam functions are used extensively in predicting natural frequencies of shells. Since this method does not involve with boundary condition equations, there is no need to deal with intense calculations. Hence, it is important to check the accuracy of this approximate technique. So this method was applied to ten different shell theories: 1) Donnell-Mushtari, 2) Love-Timoshenko, 3) Arno ld-Warburton, 4) Houghton-Johns, 5) Flugge-Byrne-Lur’ye, 6) Reissner-Naghdi-Berry, 7) Sanders, 8) Vlasov, 9) Kennard-Simplified and 10) Soedel. The approximate procedure was compared favorably with experimental results. Finally, variations and influences of length, radius and thickness were studied on amplitude ratios.


Introduction
Similar to beams and plates, in many branches of engineering, cylindrical shells are the practical elements of various engineering structures such as pipes and ducts, bodies of cars, space shuttles, aircraft fuselages, ship hulls, submarines and construction buildings. However, analysing the dynamic characteristics of cylindrical shells is mo re complicated than that of beams and plates. This is main ly because the equations of motion of cylindrical shells combined with boundary conditions are more co mplex.
A comprehensive summary and discussion of shell theories including natural frequencies and mode shape informat ion has been done by Liessa [1] in 1973. More recently, Amabili and Paidoussis [2], A mabili [3] and Kurylov and A mabili [4] have presented noteworthy reviews with a non-linear point of v iew. Many investigations followed the p ioneering work of Love [5] using his first approximation theory, such as Flugge [6]. The Flugge theory is based on Kirchhoff-Love hypothesis for thin elastic shells. By using this theory, the strain-displacement relat ions and changes of curvatures of the middle surface of a cylindrical shell can be obtained. The simp lified Donnell's theory would be achieved by neglecting few terms in Flugge equations Livanov [7] applied love's assumption and usedd is p lace ment functions to solve the p roblem o f axisy mmet rical vibrations of simply supported cylindrical shells. Rinehart and Wang [8] investigated the vibration of simply supported cylindrical shells stiffened by discrete longitudinal stiffeners using Donnell's appro ximate theory, Flugge's more exact theory and Love's assumption for longitudinal wave numbers. Thesis theories are not only concerned with simp ly supported end conditions, but they have also applied other boundaries, such as cantilever cylindrical shells [9], fixed free circular cy lindrical shells [10], clamped-clamped shells [11] and infin ite length shells [12]. Most researchers and those cited above, use beam function as an approximation for the simply supported boundary conditions and find natural frequencies of vibration by the approximate method. This appro ximation is also useful for fin ite element analysis of cylindrical shells by using Hermitain polynomial of beam function type [13]. In addition to the approximate method, there are other approaches to find natural frequencies, like the computer based numerical method [14], [15] to avoid cu mberso me computational effort and the wave propagation technique [16]. More recently Farshidianfar etl. [17] used the advantage of acoustical excitation to find natural frequency of long cylindrical shells.
Unlike beams or plates which are normally one or two dimensional structures, shells can freely vibrate in three directions. This has caused complicated the motions of the shell at resonance frequencies. Thus, apart fro m frequency behaviour, modal identification of cy lindrical shells with their amplitudes has always been of great importance. Fields such as engineering design, acoustics and sound radiation are very much dependent on the amplitude rat ios of cylindrical shells. Previous studies [18] have been made in order to identify amp litude ratios of the mode shapes, however, these studies are not complete.
In the present study, a semi-analytical approach is proposed to investigate the free vibration of simp ly supported cylindrical shells. As cited above, in tradit ional analysis, beam functions with similar boundary conditions are used to approximate wave nu mbers in the axial direction. This method is considered as an approximate technique. The approximate method is used to obtain the natural frequencies based on ten different shell theories (Donnell-Mushtari, Love-Timoshenko, Arnold-Warburton, Houghton-Johns, Flugge-Byrne-Lu r'ye, Reissner-Naghdi-B erry, Sanders, Vlasov, Kennard-Simp lified and Soedel). The results are also compared with experimental results and show good agreement. Finally, the behaviour of the circular shells with various aspect and thickness ratios was analysed. It was observed that by varying these parameters the dominant motion of the shell could change to either longitudinal, tangential or radial.

Theoretical Analysis
The cylindrical shell under consideration is with constant thickness h, mean radius R, axial length L, Poisson's ratio υ , density ρ and Young's modulus of elasticity E . Here the respective displacements in the axial, circu mferential and radial d irections are denoted by ( ) , ,θ as shown in Figure 1. In order to study free vibrat ion of a cylindrical shell, the equations of motion can be written in matrix form as fo llo ws: are differential operators with respect to x , θ and t.  [19], are used to find natural frequencies for various boundary conditions. The first attempt in solving (1) is the assumption of a synchronous motion: (2) where ( ) t f is the scalar model coordinate corresponding The next step is to use the separation of variables method in order to separate the spatial dependence of the modal shape between longitudinal and circu mferential direct ions. Hence the axial, tangential and rad ial displacements of the wall vary accord ing to: (3) in which m λ and n are the axial wavenumber and the circu mferential wave parameter, respectively.

B
A, and C are the undetermined constants, and ω is the circular frequency of the natural vibration.
Substituting (3) into (1), using any of the shell theories, leads to a set of homogenous equations having the following matrix form: , and a frequency parameter Ω that is defined as follows: for the ten shell theories is obtained as follo ws: Donnell-Mushtari  (7) Arnold-Warburton (8) Houghton-Johns (9) Flugge-By rne-Lur'ye (10) Reissner-Naghdi-Berry (11) Sanders (12) Vlasov (13) Kennard-Simplified (14) For nontrivial solution the determinant of the coefficient matrix in (4) must be zero : ; (15) The expansion of (15) will give the following two eigenvalue problems: • For a given value of m λ there exists one or more proper values for ω so that the (15) vanishes.
• For a given value of ω there exists one or more proper values for m λ so that the (15) vanishes. Solving (15) leads to a cubic equation in terms of the nondimentioanl frequency parameter 2 Ω . Thus for a fixed value of n and m λ , three positive roots and three negative roots are yield for the nondimensional frequency. The three positive roots are the natural frequencies of the cy lindrical shell that can be classified as primarily axial, circu mferential or radial. The lowest frequency is usually associated with a motion that is primarily radial (or flextural).    In general, solving the roots of the characteristic equation of (15) for m λ is not possible in closed form. Hence, researchers have often leaned to use approximate techniques. Beam functions can be used to obtain natural frequencies and approximate d isplacements for closed circu lar cy lindrical shells. This method is an assimilat ion of the flextural vibration of cy lindrical shell with a transversely vibrating beam of the same boundary conditions. According to the approximate method, for a simp ly supported shell at both ends, the nature of the axial mode can be defined as: (16) By substituting (16) into (15), the only unknown of the characteristic equation will be the frequency parameter 2 Ω for a fixed co mbination of m and n .

Result and Discussion
Since the beam function method is an approximation to obtain natural frequencies for thin circular cy lindrical shells, it is important to check the accuracy of this method. Hence, the natural frequency for simp ly supported boundary conditions, calculated by using the beam functions via ten common theories of cylindrical shells.
In Table 1, results calculated by the approximate method according to the ten theories, are co mpared with an experiment held by Farshidianfar et l. [17] for simp ly supported circular cylindrical shell. The shell investigated in Table 1 is made of alu min iu m with material p roperties;  Table 2. It is observed that, the beam function method yields close results compared to the experiment as well. It is also concluded that some theories (Love-Timoshenko, Arnold-Warburton, Flugge Byrne-Lur'ye, Reissner-Naghdi-Berry, Sanders, Vlasov, and Soedel) reveal same results. Kennard-Simp lified, Reissner-Naghdi-Berry and Soedel are mo re accurate than the other theories and Dunnell-Mushtari and Houghton-Johns theory are not precise compared to other theories.

E=68.2GPa,
Let us now study the effects of length, radius and thickness on amplitude ratios and motions of the mode shapes of the shell. In Figures. 2-5 amp litude rat ios are plotted as functions of the thickness ratio h/R, for shells with two aspect ratios L/R= 3 and 20.
According to Figurs.2-5 for L/R=3, nearly all modes up to m=3 have dominantly radial motions for all values of n, for shells of h/R<0.25. Th is is true except for , which also has an equally axial motion. It is observed that for shells of h/R>0.25 the axial and tangential motions become stronger, however, these type of shells are categorized into thick-walled shells. According to Love's first approximation for thin-walled shells, only shells of small thickness to radius and length are categorized into thin-walled theories. Thus, it may not be exact to calcu late shells of h/R>0.25 with thin-walled theories. However the general trend could be shown. Although, one would expect do minant radial motions for higher mode numbers of m, but, it is interesting that for m>3 modes and L/R=3, both amp litude ratios posses a quasi-sinusoidal and irregular behaviour. Therefore, shells with small L/R ratios have a complete irregular behaviour for different thicknesses at high mode numbers.
On the other hand, for L/R=20 a comp lete different pattern is observed compared to L/R=3. As reported in Figurs.2-5, for shells of L/R=20, at lo w mode nu mbers the tangential ratio behaves similar to a quasi-sinusoidal wave. However, the magnitude of the tangential and axial ratios does not exceed unity. Thus, the motions are always dominantly radial for long shells with small radius, regardless of the thickness.
Hence it is concluded fro m Figurs.2-5 that the thickness is a crucial parameter when dealing with short shells of small radius. This is due to the fact that, these types of shells behave similar to a ring rather than a beam.

Conclusions
The free vibration of circular cylindrical shells with simp ly supported boundary conditions has been studied using ten different thin shell theories: Donnell-Mushtari, Love-Timoshenko, Arnold-Warburton, Houghton-Johns, Flugge-By rne-Lur'ye, Reissner-Naghdi-Berry, Sanders, Vlasov, Kennard-Simp lified and Soedel. The scope of the investigation was focused upon using the beam function as an approximat ion fo r boundary condition to find the natural frequencies of a shell. Next , in order to check the accuracy of the theories, a comparison was carried out with experimental results and it shows good agreement. Moreover, the approxi mate method based on the Soedel and Kennard-Simplified theories reviled better results compared to other theories.
Modal amplitudes which are important parameter in acoustical engineering were also analysed. The effects of length, radius and thickness are completely studied on amp litude rat ios and mode shapes. Results show that both the aspect and thickness ratios are crucial parameters wh ich can change the behaviour of a shell fro m beam type motion to ring type.