Analysis of natural vibration of truncated conical shells partially filled with fluid

Abstract

In this paper, we study the vibrational behavior of shells in the form of truncated cones containing an ideal compressible fluid. The sloshing effect on the free surface of the fluid is neglected. The dynamic behavior of the elastic structure is investigated based on the classical shell theory, the constitutive relations of which represent a system of ordinary differential equations written for new unknowns. Small fluid vibrations are described in terms of acoustic approximation using the wave equation for hydrodynamic pressure written in spherical coordinates. Its transformation into the system of ordinary differential equations is carried out by applying the generalized differential quadrature method. The formulated boundary value problem is solved by Godunov’s orthogonal sweep method. Natural frequencies of shell vibrations are calculated using the stepwise procedure and the Muller method. The accuracy and reliability of the obtained results are estimated by making a comparison with the known numerical and analytical solutions. The dependencies of the lowest frequency on the fluid level and cone angle of shells under different combinations of boundary conditions (simply supported, rigidly clamped, and cantilevered shells) have been studied comprehensively. For conical straight and inverted shells, a numerical analysis has been performed to estimate the possibility of finding configurations at which the lowest natural frequencies exceed the corresponding values of the equivalent cylindrical shell.

Keywords

classical shell theory / conical shell / Godunov’s orthogonal sweep method / ideal compressible fluid / generalized differential quadrature method / natural vibrations

References

[1] Shmakov VP. Selected Works on Hydroelasticity and the Dynamics of Elastic Structures. MGTU;2011. (in Russian).

[2] Paknejad R, Ghasemi FA, Fard KM. Natural frequency analysis of multilayer truncated conical shells containing quiescent fluid on elastic foundation with different boundary conditions. Int J Appl Mech. 2021;13(7):2150075.

[3] Bochkarev SA, Lekomtsev SV, Matveenko VP. Natural vibrations of truncated conical shells containing fluid. Mech Solids. 2022;57(8):1971-1986.

[4] Gorbunov YA, Novokhatskaya LM, Shmakov VP. Theoretical and experimental study of the spectrum of natural axisymmetric vibrations of a conical shell containing a fluid in the presence of internal pressure. In: Bogoryad IB, ed. Dynamics of Elastic and Solid Bodies Interacting with a Fluid. Tomsk State University;1975:47-52.(in Russian).

[5] Lakis AA, Dyke PV, Ouriche H. Dynamic analysis of anisotropic fluidfilled conical shells. J Fluids Struct. 1992;6(2):135-162.

[6] Zhu HZ, Wu JH. Free vibration of partially fluid-filled or fluidsurrounded composite shells using the dynamic stiffness method. Acta Mech. 2020;231(9):3961-3978.

[7] Pozhalostin AA, Kamensky OA, Kulikov VZ. Experimental determination of mode shapes and frequencies for axisymmetric vibrations of a conical tank partially filled with liquid. In: Axelrad EL, ed. Proceedings of the X All-Union Conference on the Theory of Shells and Plates. Metsniereba;1975:178-180. (in Russian).

[8] Godunov SK. Ordinary Differential Equations with Constant Coefficients. American Mathematical Society;1997.

[9] Shu C. Differential Quadrature and Its Application in Engineering. Springer;2000.

[10] Amabili M. Eigenvalue problems for vibrating structures coupled with quiescent fluids with free surface. J Sound Vib. 2000;231(1):79-97.

[11] Gorshkov AG, Morozov VI, Ponomarev VI, Shklyarchuk FN. Aerohydroelasticity of Structures. Fizmatlit;2000. (in Russian).

[12] Kumar DS, Ganesan N. Dynamic analysis of conical shells conveying fluid. J Sound Vib. 2008;310(1-2):38-57.

[13] Zhu Y, Yu J, Guan W, Tang W, Yue L, Zhang J. Effects of the geometrical shapes on buckling of conical shells under external pressure. Int J Pressure Vessels Piping. 2022;196:104624.

[14] Ambartsumyan S. Theory of Anisotropic Shells. NASA TTF-118;1964.

[15] Karmishin AV, Lyaskovets VA, Myachenkov VI, Frolov AN. The Statics and Dynamics of Thin-walled Shell Structures. Mashinostroyeniye;1975. (in Russian).

[16] Averbukh AZ, Veitsman RI, Genkin MD. Vibration of Structural Elements in Fluid. Nauka;1987 (in Russian).

[17] Païdoussis MP. Fluid-Structure Interactions: Slender Structures and Axial Flow. 1st ed. Elsevier Academic Press;2004.

[18] Amabili M. Free vibration of partially filled, horizontal cylindrical shells. J Sound Vib. 1996;191(5):757-780.

[19] Muller DE. A method for solving algebraic equations using an automatic computer. Math Comput. 1956;10(56):208-215.

[20] Wu X. Improved Muller method and bisection method with global and asymptotic superlinear convergence of both point and interval for solving nonlinear equations. Appl Math Comput. 2005;166(2):299-311.

[21] Bochkarev SA. Free vibrations of a cylindrical shell partially resting on elastic foundation. J Appl MechTech Phys. 2018;59(7):1242-1250.

[22] Bochkarev SA. Natural vibrations of truncated conical shells of variable thickness. J Appl Mech Tech Phys. 2021;62(7):1222-1233.

[23] Bochkarev SA. Study of natural vibrations of fluid-containing composite cylindrical shells resting on elastic foundation. Mech Compos Mater Struct. 2023;29(2):149-166. (in Russian).

[24] Bochkarev SA, Lekomtsev SV, Matveenko VP. Natural vibrations of composite cylindrical shells partially filled with fluid. Vestn St-Petersb Univ: Math. 2023;56(4):435-445.

[25] Bochkarev SA, Matveenko VP. Free vibration analysis of a cylindrical shell of variable thickness partially filled with fluid. Proc Steklov Inst Math. 2023;321(Suppl 1):S20-S32.

[26] ANSYS. Release 2022 R1 Documentation;2022.

[27] Rahmanian M, Firouz-Abadi RD, Cigeroglu E. Free vibrations of moderately thick truncated conical shells filled with quiescent fluid. J Fluids Struct. 2016;63:280-301.

[28] Kerboua Y, Lakis AA, Hmila M. Vibration analysis of truncated conical shells subjected to flowing fluid. Appl Math Modell. 2010;34(3):791-809.

[29] Norrie DH, De Vries G. An Introduction to Finite Element Analysis. Academic Press;1978.

[30] Bochkarev SA, Matveenko VP. Finite-element analysis of natural vibrations of a cylindrical shell conveying fluid. In: Andreichenko VN, ed. Computational Mechanics. Collection of Scientific Papers. No. 4 Perm State Technical University;2006:3-12. (in Russian).

[31] Bochkarev S, Matveenko V. Natural vibrations and stability of shells of revolution interacting with an internal fluid flow. J Sound Vib. 2011;330(13):3084-3101.

[32] Thambiratnam DP, Zhuge Y. Axisymmetric free vibration analysis of conial shells. Eng Struct. 1993;15(2):83-89.

[33] Buchanan GR, Wong FI. Frequencies and mode shapes for thick truncated hollow cones. Int J Mech Sci. 2001;43(12):2815-2832.

[34] Shu C. An efficient approach for free vibration analysis of conical shells. Int J Mech Sci. 1996;38(8-9):935-949.

[35] Irie T, Yamada G, Tanaka K. Natural frequencies of truncated conical shells. J Sound Vib. 1984;92(3):447-453.

[36] Narita Y, Ohta Y, Yamada G, Kobayashi Y. Analytical method for vibration of angle-ply cylindrical shells having arbitrary edges. AIAA J. 1992;30(3):790-796.

[37] Musa A, Damatty AAE. Capacity of liquid steel conical tanks under hydrodynamic pressure due to horizontal ground excitations. Thin-Walled Struct. 2016;103:157-170.

[38] Musa A, Damatty AAE. Capacity of liquid-filled steel conical tanks under vertical excitation. Thin-Walled Struct. 2016;103:199-210.

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2024 The Authors. International Journal of Mechanical System Dynamics published by John Wiley & Sons Australia, Ltd on behalf of Nanjing University of Science and Technology.