An exact dynamic stiffness formulation for predicting natural frequencies of moderately thick shells of revolution.

*(English)*Zbl 1427.74111Summary: An exact dynamic stiffness formulation is proposed for calculating the natural frequencies of shells of revolution based on the first-order Reissner-Mindlin theory. Equations of motion are reduced to be one-dimensional, should the circumferential wave number is specified, and then are rewritten in Hamilton form. Therefore, a shell of revolution with a moderate thickness is analysed as a special skeletal structure with five degrees of freedom at element ends. Dynamic stiffnesses are constructed directly from the one-dimensional governing vibration equations using classic skeletal theory. Number of natural frequencies below a specific value is counted by applying the Wittrick-Williams (W-W) algorithm, and exact eigenvalues can be simply obtained using bisection method. A solution to the number of clamped-end frequencies \(J_0\) in the W-W algorithm is also proposed and proven to be reliable. Numerical examples on a variety of shells with different boundary conditions are investigated, and
results are well compared and validated. Influences of a variety of shell parameters on natural frequencies of both cylindrical and spherical shells are discussed in detail, demonstrating that the proposed dynamic stiffness formulation is applicable to analyse natural frequencies of moderately thick shells of revolution with high accuracy.

##### MSC:

74K25 | Shells |

74H45 | Vibrations in dynamical problems in solid mechanics |

74S05 | Finite element methods applied to problems in solid mechanics |

##### Software:

COLSYS
PDF
BibTeX
XML
Cite

\textit{X. Chen} and \textit{K. Ye}, Math. Probl. Eng. 2018, Article ID 4156360, 15 p. (2018; Zbl 1427.74111)

Full Text:
DOI

##### References:

[1] | Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, (1944), New York, NY, USA: Dover Publications, New York, NY, USA · Zbl 0063.03651 |

[2] | Arnold, R. N.; Warburton, G. B., The flexural vibrations of thin cylinders, Proceedings of the Institution of Mechanical Engineers, 167, 1, 62-80, (1953) |

[3] | Wang, C.; Lai, J. C. S., Prediction of natural frequencies of finite length circular cylindrical shells, Applied Acoustics, 59, 4, 385-400, (2000) |

[4] | Lamb, H., On the Vibrations of a Spherical SHEll, Proceedings of the London Mathematical Society, 14, 50-56, (1882/83) · JFM 15.0893.01 |

[5] | DiMaggio, F. L.; Silbiger, A., Free extensional torsional vibrations of a prolate spheroidal shell, The Journal of the Acoustical Society of America, 33, 56-58, (1961) |

[6] | Al-Jumaily, A. M.; Najim, F. M., An approximation to the vibrations of oblate spheroidal shells, Journal of Sound and Vibration, 204, 4, 561-572, (1997) |

[7] | Lin, Y. K.; Lee, F. A., Vibrations of thin paraboloidal shells of revolution, Journal of Applied Mechanics, 27, 4, 743-744, (1960) · Zbl 0102.19006 |

[8] | Hoppmann, I.; Cohen, M. I.; Kunukkasseril, V. X., Elastic vibrations of paraboloidal shells of revolution, The Journal of the Acoustical Society of America, 36, 349-353, (1964) |

[9] | Carter L, R.; Robinson, R. A.; Schnobrich, W. C., Free vibration of hyperboloidal shells of revolution, ASCE-Engineering Mechanics, EM5, 1033-1052, (1969) |

[10] | Li, F.-M.; Kishimoto, K.; Huang, W.-H., The calculations of natural frequencies and forced vibration responses of conical shell using the Rayleigh-Ritz method, Mechanics Research Communications, 36, 5, 595-602, (2009) · Zbl 1258.74106 |

[11] | Luah, M. H.; Fan, S. C., General free vibration analysis of shells of revolution using the spline finite element method, Computers & Structures, 33, 5, 1153-1162, (1989) · Zbl 0724.73230 |

[12] | Tan, D.-Y., Free vibration analysis of shells of revolution, Journal of Sound and Vibration, 213, 1, 15-33, (1998) |

[13] | Naghdi, P. M.; Cooper, R. M., Propagation of Elastic Waves in Cylindrical Shells, Including the Effects of Transverse Shear and Rotatory Inertia, The Journal of the Acoustical Society of America, 28, 1, 56-63, (1956) |

[14] | Mirsky, I.; Herrmann, G., Nonaxially symmetric motions of cylindrical shells, The Journal of the Acoustical Society of America, 29, 1116-1123, (1957) |

[15] | Leissa, A. W., Vibration of shells, Report No. 288, (1973), Washington, Wash, USA: National Aeronautics and Space Administration (NASA), Washington, Wash, USA · Zbl 0268.73033 |

[16] | Hosseini-Hashemi, S.; Fadaee, M., On the free vibration of moderately thick spherical shell panel - A new exact closed-form procedure, Journal of Sound and Vibration, 330, 17, 4352-4367, (2011) |

[17] | Su, Z.; Jin, G.; Ye, T., Free vibration analysis of moderately thick functionally graded open shells with general boundary conditions, Composite Structures, 117, 1, 169-186, (2014) |

[18] | Tornabene, F.; Viola, E., 2-D solution for free vibrations of parabolic shells using generalized differential quadrature method, European Journal of Mechanics - A/Solids, 27, 6, 1001-1025, (2008) · Zbl 1151.74360 |

[19] | Kang, J.-H.; Leissa, A. W., Three-dimensional vibration analysis of thick hyperboloidal shells of revolution, Journal of Sound and Vibration, 282, 1-2, 277-296, (2005) · Zbl 1237.74070 |

[20] | Al-Khatib, O. J.; Buchanan, G. R., Free vibration of a paraboloidal shell of revolution including shear deformation and rotary inertia, Thin-Walled Structures, 48, 3, 223-232, (2010) |

[21] | Williams, F. W.; Wittrick, W. H., An automatic computational procedure for calculating natural frequencies of skeletal structures, International Journal of Mechanical Sciences, 12, 9, 781-791, (1970) |

[22] | Wittrick, W. H.; Williams, F. W., A general algorithm for computing natural frequencies of elastic structures, The Quarterly Journal of Mechanics and Applied Mathematics, 24, 263-284, (1971) · Zbl 0227.73022 |

[23] | Yuan, S.; Ye, K.; Xiao, C.; Williams, F. W.; Kennedy, D., Exact dynamic stiffness method for non-uniform Timoshenko beam vibrations and Bernoulli-Euler column buckling, Journal of Sound and Vibration, 303, 3–5, 526-537, (2007) · Zbl 1242.74052 |

[24] | Banerjee, J. R.; Cheung, C. W.; Morishima, R.; Perera, M.; Njuguna, J., Free vibration of a three-layered sandwich beam using the dynamic stiffness method and experiment, International Journal of Solids and Structures, 44, 22-23, 7543-7563, (2007) · Zbl 1166.74367 |

[25] | Su, H.; Banerjee, J. R.; Cheung, C. W., Dynamic stiffness formulation and free vibration analysis of functionally graded beams, Composite Structures, 106, 854-862, (2013) |

[26] | Pagani, A.; Boscolo, M.; Banerjee, J. R.; Carrera, E., Exact dynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structures, Journal of Sound and Vibration, 332, 23, 6104-6127, (2013) |

[27] | El-Kaabazi, N.; Kennedy, D., Calculation of natural frequencies and vibration modes of variable thickness cylindrical shells using the Wittrick-Williams algorithm, Computers & Structures, 104-105, 4-12, (2012) |

[28] | Chen, X., Research on dynamic stiffness method for free vibration of shells of revolution. Master Thesis, Tsinghua University [Master, thesis], (2009), Tsinghua University: Chinese, Tsinghua University |

[29] | Chen, X.; Ye, K., Free vibration analysis for shells of revolution using an exact dynamic stiffness method, Mathematical Problems in Engineering, (2016) · Zbl 1400.74033 |

[30] | Chen, X.; Ye, K., Analysis of free vibration of moderately thick circular cylindrical shells using the dynamic stiffness method. Engineering Mechanics, Analysis of free vibration of moderately thick circular cylindrical shells using the dynamic stiffness method, 40-48, (2016), Engineering Mechanics: Chinese, Engineering Mechanics |

[31] | Chen, X.-D.; Ye, K.-S., Free vibration analysis of moderately thick elliptical shells using the dynamic stiffness method, Zhendong yu Chongji, 35, 6, 85-90, (2016) |

[32] | Chen, X.; Ye, K., Comparison study on the exact dynamic stiffness method for free vibration of thin and moderately thick circular cylindrical shells, Shock and Vibration, 2016, (2016) |

[33] | Khalili, S. M. R.; Davar, A.; Malekzadeh Fard, K., Free vibration analysis of homogeneous isotropic circular cylindrical shells based on a new three-dimensional refined higher-order theory, International Journal of Mechanical Sciences, 56, 1, 1-25, (2012) |

[34] | Gautham, B. P.; Ganesan, N., Free vibration analysis of thick spherical shells, Computers & Structures, 45, 2, 307-313, (1992) · Zbl 0775.73138 |

[35] | Shim, H.-J.; Kang, J.-H., Free vibrations of solid and hollow hemi-ellipsoids of revolution from a three-dimensional theory, International Journal of Engineering Science, 42, 17-18, 1793-1815, (2004) · Zbl 1211.74154 |

[36] | Deb Nath, J. M., Free vibration, stability and ‘non-classical modes’ of cooling tower shells, Journal of Sound and Vibration, 33, 1, 79-101, (1974) |

[37] | Ascher, U.; Christiansen, J.; Russell, R. D., Collocation software for boundary value ODEs, ACM Transactions on Mathematical Software, 7, 2, 209-222, (1981) · Zbl 0455.65067 |

[38] | Ascher, U.; Christiansen, J.; Russell, R. D., Algorithm 569, COLSYS: collocation software for boundary value ODEs [D2], ACM Transactions on Mathematical Software, 7, 2, 223-229, (1981) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.