Vibration and Buckling Approximation of an Axially Loaded Cylindrical Shell with a Three Lobed Cross Section Having Varying Thickness

Author(s)
Mousa Khalifa Ahmed

ABSTRACT

On the basis of the thin-shell theory and on the use of the transfer matrix approach, this paper presents the vibrational response and buckling analysis of three-lobed cross-section cylindrical shells, with circumferentially varying thickness, subjected to uniform axial membrane loads. A Fourier approach is used to separate the variables, and the governing equations of the shell are formulated in terms of eight first-order differential equations in the circumferential coordinate, and by using the transfer matrix of the shell, these equations are written in a matrix differential equation. The transfer matrix is derived from the non-linear differential equations of the cylindrical shells with variable thickness by introducing the trigonometric series in the longitudinal direction and applying a numerical integration in the circumferential direction. The natural frequencies and critical loads beside the mode shapes are calculated numerically in terms of the transfer matrix elements for the symmetrical and antisymmetrical vibration modes. The influences of the thickness variation of cross- section and radius v

On the basis of the thin-shell theory and on the use of the transfer matrix approach, this paper presents the vibrational response and buckling analysis of three-lobed cross-section cylindrical shells, with circumferentially varying thickness, subjected to uniform axial membrane loads. A Fourier approach is used to separate the variables, and the governing equations of the shell are formulated in terms of eight first-order differential equations in the circumferential coordinate, and by using the transfer matrix of the shell, these equations are written in a matrix differential equation. The transfer matrix is derived from the non-linear differential equations of the cylindrical shells with variable thickness by introducing the trigonometric series in the longitudinal direction and applying a numerical integration in the circumferential direction. The natural frequencies and critical loads beside the mode shapes are calculated numerically in terms of the transfer matrix elements for the symmetrical and antisymmetrical vibration modes. The influences of the thickness variation of cross- section and radius v

KEYWORDS

Free Vibration, Buckling, Vibration of Continuous System, Noncircular Cylindrical Shell, Transfer Matrix Approach, Variable Thickness, Axial Loads

Free Vibration, Buckling, Vibration of Continuous System, Noncircular Cylindrical Shell, Transfer Matrix Approach, Variable Thickness, Axial Loads

Cite this paper

nullM. Ahmed, "Vibration and Buckling Approximation of an Axially Loaded Cylindrical Shell with a Three Lobed Cross Section Having Varying Thickness,"*Applied Mathematics*, Vol. 2 No. 3, 2011, pp. 329-342. doi: 10.4236/am.2011.23039.

nullM. Ahmed, "Vibration and Buckling Approximation of an Axially Loaded Cylindrical Shell with a Three Lobed Cross Section Having Varying Thickness,"

References

[1] W. Flügge, “Die Stabilit?t der Kreiszylinderschale,” Ingenieur Archiv, Vol. 3, No. 5, 1932, pp. 463-506.

[2] L. H. Donnell, “Stability of Thin-Walled Tubes under Torsion,” NACA Report, No. NACA TR-479, 1933.

[3] H. Becker and G. Gerard, “Elastic Stability of Orthotropic Shells,” Journal of Aerospace Science, Vol. 29, No. 5, 1962, pp. 505-512.

[4] G. Gerard, “Compressive Stability of Orthotropic Cylinders,” Journal of Aerospace Science, Vol. 29, 1962, pp. 1171-1179.

[5] S. Cheng and B. Ho, “Stability of Heterogeneous Aeolotropic Cylindrical Shells under Combined Loading,” AIAA Journal, Vol. 1, No. 40, 1963, pp. 892-898.

[6] R. M. Jones, “Buckling of Circular Cylindrical Shells with Multiple Orthotropic Layers and Eccentric Stiffeners,” AIAA Journal, Vol. 6, No. 12, 1968, pp. 2301-2305. doi:10.2514/3.4986

[7] Y. Stavsky and S. Friedl, “Stability of Heterogeneous Orthotropic Cylindrical Shells in Axial Compression,” Israel Journal of Technology, Vol. 7, 1969, pp. 111-119.

[8] M. M. Lei and S. Cheng, “Buckling of Composite and Homogeneous Isotropic Cylindrical Shells under Axial and Radial Loading,” Journal of Applied Mechanics, Vol. 36, No. 4, 1969, pp. 791-798.

[9] J. B. Greenberg and Y. Stavsky, “Buckling and Vibration of Orthotropic Composite Cylindrical Shells,” Acta Mechanica, Vol. 36, No. 12, 1980, pp. 15-29. doi:10.1007/BF01178233

[10] J. B. Greenberg and Y. Stavsky, “Vibrations of Axially Compressed Laminated Orthotropic Cylindrical Shells, Including Transverse Shear Deformation,” Acta Mechanica, Vol. 37, No. 1-2, 1980, pp. 13-28. doi:10.1007/BF01441240

[11] J. B. Greenberg and Y. Stavsky, “Stability and Vibrations of Compressed Aeolotropic Composite Cylindrical Shells,” Journal of Applied Mechanics, Vol. 49, No. 4, 1982, pp. 843-848. doi:10.1115/1.3162625

[12] J. B. Greenberg and Y. Stavsky, “Vibrations and Buckling of Composite Orthotropic Cylindrical Shells with Nonuniform Axial Loads,” Composites Part B: Engineering, Vol. 29, No. 6, 1998, pp. 695-702. doi:10.1016/S1359-8368(98)00029-8

[13] A. Rosen and J. Singer, “Vibrations of Axially Loaded Stiffened Cylindrical Shells,” Journal of Sound and Vibration, Vol. 34, No. 3, 1974, pp. 357-378. doi:10.1016/S0022-460X(74)80317-2

[14] G. Yamada, T. Irie and M. Tsushima, “Vibration and Stability of Orthotropic Circular Cylindrical Shells Subjected to Axial Load,” Journal of Acoustical Society of America, Vol. 75, No. 3, 1984, pp. 842-848. doi:10.1121/1.390594

[15] G. Yamada, T. Irie and Y. Tagawa, “Free Vibration of Non-Circular Cylindrical Shells with Variable Circumferential Profile,” Journal of Sound and Vibration, Vol. 95, No. 1, 1984, pp. 117-126. doi:10.1016/0022-460X(84)90264-5

[16] K. Suzuki and A. W. Leissa, “Free Vibrations of Noncircular Cylindrical Shells Having Circumferentially Varying Thickness,” Journal of Applied Mechanics, Vol. 52, No. 1, 1985, pp. 149-154. doi:10.1115/1.3168986

[17] V. Kumar and A. V. Singh, “Approximate Vibrational Analysis of Noncircular Cylinders Having Varying Thi- ckness,” AIAA Journal, Vol. 30, No. 7, 1991, pp. 1929-1931. doi:10.2514/3.11161

[18] O. Mitao, T. Hideki and S. Tsunemi, “Vibration Analysis of Curved Panels with Variable Thickness,” Engineering Computations, Vol. 13, No. 2, 1996, pp. 226-239. doi:10.1108/02644409610114549

[19] R. F. Tonin and D. A. Bies, “Free Vibration of Circular Cylinders of Variable Thickness,” Journal of Sound and Vibration, Vol. 62, No. 2, 1979, pp. 165-180. doi:10.1016/0022-460X(79)90019-1

[20] R. M. Bergman, S. A. Sidorin and P. E. Tovstik, “Construction of Solutions of the Equations for Free Vibration of a Cylindrical Shell of Variable Thickness along the Directrix,” Mechanics of Solids, Vol. 14, No. 4, 1979, pp. 127-134.

[21] T. Irie, G. Yamada and Y. Kaneko, “Free Vibration of a Conical Shell with Variable Thickness,” Journal of Sound and Vibration, Vol. 82, No. 1, 1982, pp. 83-94. doi:10.1016/0022-460X(82)90544-2

[22] S. Takahashi, K. Suzuki and T. Kosawada, “Vibrations of Conical Shells with Varying Thickness,” Japan Society of Mechanical Engineering, Vol. 28, No. 235, 1985, pp. 117-123.

[23] W. I. Koiter, I. Elishakoff, Y. W. Li and J. H. Starness, “Buckling of an Axially Compressed Cylindrical Shell of Variable Thickness,” International Journal of Solids and Structures, Vol. 31, No. 6, 1994, pp. 797-805. doi:10.1016/0020-7683(94)90078-7

[24] H. Abdullah and H. Erdem, “The Stability of Non-Homogenous Elastic Cylindrical Thin Shells with Variable Thickness under a Dynamic External Pressure,” Turkish Journal of Engineering and Environmental Sciences, In Turkish, Vol. 26, No. 2, 2002, pp. 155-164.

[25] S. L. Eliseeva and S. B. Filippov, “Buckling and Vibrations of Cylindrical Shell of Variable Thickness with Santed Edge,” Vestnik Sankt-Peterskogo Universiteta, In Russian, No. 3, 2003, pp. 84-91.

[26] S. B. Filippov, D. N. Ivanov and N. V. Naumova, “Free Vibrations and Buckling of a Thin Cylindrical Shell of Variable Thickness with Curelinear Edge,” Technische Mechanik, Vol. 25, No. 1, 2005, pp. 1-8.

[27] A. L. Goldenveizer, “Theory of Thin Shells,” Pergamon Press, New York, 1961.

[28] V. V. Novozhilov, “The Theory of Thin Elastic Shells,” P. Noordhoff Ltd., Groningen, 1964.

[29] R. Uhrig, “Elastostatik und Elastokinetik in Matrizenschreibweise,” Springer-Verlag, Berlin, 1973.

[30] A. Tesar and L. Fillo, “Transfer Matrix Method,” Kluwer Academic, Dordrecht, 1988.

[31] M. Khalifa, “A Study of Free Vibration of a Circumferentially Non-Uniform Cylindrical Shell with a Four Lobed Cross Section,” In Press, Journal of Vibration and Control, 2010.

[1] W. Flügge, “Die Stabilit?t der Kreiszylinderschale,” Ingenieur Archiv, Vol. 3, No. 5, 1932, pp. 463-506.

[2] L. H. Donnell, “Stability of Thin-Walled Tubes under Torsion,” NACA Report, No. NACA TR-479, 1933.

[3] H. Becker and G. Gerard, “Elastic Stability of Orthotropic Shells,” Journal of Aerospace Science, Vol. 29, No. 5, 1962, pp. 505-512.

[4] G. Gerard, “Compressive Stability of Orthotropic Cylinders,” Journal of Aerospace Science, Vol. 29, 1962, pp. 1171-1179.

[5] S. Cheng and B. Ho, “Stability of Heterogeneous Aeolotropic Cylindrical Shells under Combined Loading,” AIAA Journal, Vol. 1, No. 40, 1963, pp. 892-898.

[6] R. M. Jones, “Buckling of Circular Cylindrical Shells with Multiple Orthotropic Layers and Eccentric Stiffeners,” AIAA Journal, Vol. 6, No. 12, 1968, pp. 2301-2305. doi:10.2514/3.4986

[7] Y. Stavsky and S. Friedl, “Stability of Heterogeneous Orthotropic Cylindrical Shells in Axial Compression,” Israel Journal of Technology, Vol. 7, 1969, pp. 111-119.

[8] M. M. Lei and S. Cheng, “Buckling of Composite and Homogeneous Isotropic Cylindrical Shells under Axial and Radial Loading,” Journal of Applied Mechanics, Vol. 36, No. 4, 1969, pp. 791-798.

[9] J. B. Greenberg and Y. Stavsky, “Buckling and Vibration of Orthotropic Composite Cylindrical Shells,” Acta Mechanica, Vol. 36, No. 12, 1980, pp. 15-29. doi:10.1007/BF01178233

[10] J. B. Greenberg and Y. Stavsky, “Vibrations of Axially Compressed Laminated Orthotropic Cylindrical Shells, Including Transverse Shear Deformation,” Acta Mechanica, Vol. 37, No. 1-2, 1980, pp. 13-28. doi:10.1007/BF01441240

[11] J. B. Greenberg and Y. Stavsky, “Stability and Vibrations of Compressed Aeolotropic Composite Cylindrical Shells,” Journal of Applied Mechanics, Vol. 49, No. 4, 1982, pp. 843-848. doi:10.1115/1.3162625

[12] J. B. Greenberg and Y. Stavsky, “Vibrations and Buckling of Composite Orthotropic Cylindrical Shells with Nonuniform Axial Loads,” Composites Part B: Engineering, Vol. 29, No. 6, 1998, pp. 695-702. doi:10.1016/S1359-8368(98)00029-8

[13] A. Rosen and J. Singer, “Vibrations of Axially Loaded Stiffened Cylindrical Shells,” Journal of Sound and Vibration, Vol. 34, No. 3, 1974, pp. 357-378. doi:10.1016/S0022-460X(74)80317-2

[14] G. Yamada, T. Irie and M. Tsushima, “Vibration and Stability of Orthotropic Circular Cylindrical Shells Subjected to Axial Load,” Journal of Acoustical Society of America, Vol. 75, No. 3, 1984, pp. 842-848. doi:10.1121/1.390594

[15] G. Yamada, T. Irie and Y. Tagawa, “Free Vibration of Non-Circular Cylindrical Shells with Variable Circumferential Profile,” Journal of Sound and Vibration, Vol. 95, No. 1, 1984, pp. 117-126. doi:10.1016/0022-460X(84)90264-5

[16] K. Suzuki and A. W. Leissa, “Free Vibrations of Noncircular Cylindrical Shells Having Circumferentially Varying Thickness,” Journal of Applied Mechanics, Vol. 52, No. 1, 1985, pp. 149-154. doi:10.1115/1.3168986

[17] V. Kumar and A. V. Singh, “Approximate Vibrational Analysis of Noncircular Cylinders Having Varying Thi- ckness,” AIAA Journal, Vol. 30, No. 7, 1991, pp. 1929-1931. doi:10.2514/3.11161

[18] O. Mitao, T. Hideki and S. Tsunemi, “Vibration Analysis of Curved Panels with Variable Thickness,” Engineering Computations, Vol. 13, No. 2, 1996, pp. 226-239. doi:10.1108/02644409610114549

[19] R. F. Tonin and D. A. Bies, “Free Vibration of Circular Cylinders of Variable Thickness,” Journal of Sound and Vibration, Vol. 62, No. 2, 1979, pp. 165-180. doi:10.1016/0022-460X(79)90019-1

[20] R. M. Bergman, S. A. Sidorin and P. E. Tovstik, “Construction of Solutions of the Equations for Free Vibration of a Cylindrical Shell of Variable Thickness along the Directrix,” Mechanics of Solids, Vol. 14, No. 4, 1979, pp. 127-134.

[21] T. Irie, G. Yamada and Y. Kaneko, “Free Vibration of a Conical Shell with Variable Thickness,” Journal of Sound and Vibration, Vol. 82, No. 1, 1982, pp. 83-94. doi:10.1016/0022-460X(82)90544-2

[22] S. Takahashi, K. Suzuki and T. Kosawada, “Vibrations of Conical Shells with Varying Thickness,” Japan Society of Mechanical Engineering, Vol. 28, No. 235, 1985, pp. 117-123.

[23] W. I. Koiter, I. Elishakoff, Y. W. Li and J. H. Starness, “Buckling of an Axially Compressed Cylindrical Shell of Variable Thickness,” International Journal of Solids and Structures, Vol. 31, No. 6, 1994, pp. 797-805. doi:10.1016/0020-7683(94)90078-7

[24] H. Abdullah and H. Erdem, “The Stability of Non-Homogenous Elastic Cylindrical Thin Shells with Variable Thickness under a Dynamic External Pressure,” Turkish Journal of Engineering and Environmental Sciences, In Turkish, Vol. 26, No. 2, 2002, pp. 155-164.

[25] S. L. Eliseeva and S. B. Filippov, “Buckling and Vibrations of Cylindrical Shell of Variable Thickness with Santed Edge,” Vestnik Sankt-Peterskogo Universiteta, In Russian, No. 3, 2003, pp. 84-91.

[26] S. B. Filippov, D. N. Ivanov and N. V. Naumova, “Free Vibrations and Buckling of a Thin Cylindrical Shell of Variable Thickness with Curelinear Edge,” Technische Mechanik, Vol. 25, No. 1, 2005, pp. 1-8.

[27] A. L. Goldenveizer, “Theory of Thin Shells,” Pergamon Press, New York, 1961.

[28] V. V. Novozhilov, “The Theory of Thin Elastic Shells,” P. Noordhoff Ltd., Groningen, 1964.

[29] R. Uhrig, “Elastostatik und Elastokinetik in Matrizenschreibweise,” Springer-Verlag, Berlin, 1973.

[30] A. Tesar and L. Fillo, “Transfer Matrix Method,” Kluwer Academic, Dordrecht, 1988.

[31] M. Khalifa, “A Study of Free Vibration of a Circumferentially Non-Uniform Cylindrical Shell with a Four Lobed Cross Section,” In Press, Journal of Vibration and Control, 2010.