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Research Papers

Free Vibration of Doubly Curved Thin Shells

[+] Author and Article Information
April Bryan

Mem. ASME
No. 7 Jack Trace, Enterprise,
Chaguanas 500234, Trinidad and Tobago
e-mail: aprilbr@gmail.com

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August 12, 2017; final manuscript received November 3, 2017; published online December 20, 2017. Assoc. Editor: Stefano Lenci.

J. Vib. Acoust 140(3), 031003 (Dec 20, 2017) (11 pages) Paper No: VIB-17-1366; doi: 10.1115/1.4038578 History: Received August 12, 2017; Revised November 03, 2017

While several numerical approaches exist for the vibration analysis of thin shells, there is a lack of analytical approaches to address this problem. This is due to complications that arise from coupling between the midsurface and normal coordinates in the transverse differential equation of motion (TDEM) of the shell. In this research, an Uncoupling Theorem for solving the TDEM of doubly curved, thin shells with equivalent radii is introduced. The use of the uncoupling theorem leads to the development of an uncoupled transverse differential of motion for the shells under consideration. Solution of the uncoupled spatial equation results in a general expression for the eigenfrequencies of these shells. The theorem is applied to four shell geometries, and numerical examples are used to demonstrate the influence of material and geometric parameters on the eigenfrequencies of these shells.

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References

Liew, K. M. , Lim, C. W. , and Kitipornchai, S. , 1997, “ Vibration of Shallow Shells: A Review With Bibliography,” ASME Appl. Mech. Rev., 50(8), pp. 431–444. [CrossRef]
Qatu, M. S. , 2002, “ Recent Research Advances in the Dynamic Behavior of Shells: 1989–2000—Part 2: Homogeneous Shells,” ASME Appl. Mech. Rev., 55(5), pp. 415–434. [CrossRef]
Narita, Y. , and Leissa, A. , 1986, “ Vibrations of Completely Free Shallow Shells of Curvilinear Planform,” ASME J. Appl. Mech., 53(3), pp. 647–651. [CrossRef]
Tornabene, F. , and Erasmo, V. , 2008, “ 2-D Solution for Free Vibrations of Parabolic Shells Using Generalized Differential Quadrature Method,” Eur. J. Mech. A: Solids, 27(6), pp. 1001–1025. [CrossRef]
Leissa, A. W. , 1973, Vibration of Shells, Ohio State University, Columbus, OH.
Leissa, A. W. , 1969, Vibration of Plates, Ohio State University, Columbus, OH.
Ventsel, E. , and Krauthammer, T. , 2001, Thin Plates and Shells: Theory, Analysis, and Applications, Marcel Dekker, New York. [CrossRef]
Love, A. E. H. , 1888, “ The Small Free Vibrations and Deformation of a Thin Elastic Shell,” Philos. Trans. R. Soc. London. A, 179, pp. 491–546. [CrossRef]
Soedel, W. , 2005, Vibration of Shells and Plates, Taylor and Francis, Oxfordshire, UK.
Wilkinson, J. P. , and Kalnins, A. , 1965, “ On Nonsymmetric Dynamic Problems of Elastic Spherical Shells,” ASME J. Appl. Mech., 32(3), pp. 525–532. [CrossRef]
Wilkinson, J. P. , 1965, “ Natural Frequencies of Closed Spherical Shells,” J. Acoust. Soc. Am., 38(2), pp. 367–368. [CrossRef]
Naghdi, P. M. , and Kalnins, A. , 1962, “ On Vibrations of Elastic Spherical Shells,” ASME J. Appl. Mech., 29(1), pp. 65–72. [CrossRef]
Kalnins, A. , 1964, “ Effect of Bending on Vibrations of Spherical Shells,” J. Acoust. Soc. Am., 36(1), pp. 74–81. [CrossRef]
Ross, E. W. , 1965, “ Natural Frequencies and Mode Shapes for Axisymmetric Vibration of Deep Spherical Shells,” ASME J. Appl. Mech., 32(3), pp. 553–561. [CrossRef]
Niordson, F. I. , 1984, “ Free Vibrations of Thin Elastic Spherical Shells,” Int. J. Solids Struct., 20(7), pp. 667–687. [CrossRef]
Plaut, R. H. , and Johnson, L. W. , 1984, “ Optimal Forms of Shallow Shells With Circular Boundary,” ASME J. Appl. Mech., 51(3), pp. 526–530. [CrossRef]
Liew, K. M. , Peng, L. X. , and Ng, T. Y. , 2002, “ Three-Dimensional Vibration Analysis of Spherical Shell Panels Subjected to Different Boundary Conditions,” Int. J. Mech. Sci., 44(10), pp. 2103–2117. [CrossRef]
Irie, T. , Yamrada, G. , and Muramoto, Y. , 1985, “ Free Vibration of a Point-Supported Spherical Shell,” ASME J. Appl. Mech., 52(4), pp. 890–896. [CrossRef]
Jiang, S. , Yang, T. , Li, W. L. , and Du, J. , 2013, “ Vibration Analysis of Doubly Curved Shallow Shells With Elastic Edge Restraints,” ASME J. Vib. Acoust., 135(3), p. 034502. [CrossRef]
Shi, P. , Kapania, R. K. , and Dong, C. Y. , 2015, “ Free Vibration of Curvilinearly Stiffened Shallow Shells,” ASME J. Vib. Acoust., 137(3), p. 031006. [CrossRef]
Chakravorty, D. , and Bandyopadhyay, J. N. , 1995, “ On the Free Vibration of Shallow Shells,” J. Sound Vib., 185(4), pp. 673–684. [CrossRef]
Singh, A. V. , 1985, “ Asymmetric Modes and Associated Eigenvalues for Spherical Shells,” ASME J. Pressure Vessel Technol., 107(1), pp. 77–82. [CrossRef]
Singh, A. V. , 1991, “ On Vibrations of Shells of Revolution Using Bezier Polynomials,” ASME J. Pressure Vessel Technol., 113(4), pp. 579–584. [CrossRef]
Petyt, M. , 2010, Introduction to Finite Element Vibration Analysis, Cambridge University Press, New York. [CrossRef]
Choi, S. T. , and Chou, Y. T. , 2003, “ Vibration Analysis of Non-Circular of Curved Panels by the Differential Quadrature Method,” J. Sound Vib., 259(3), pp. 525–539. [CrossRef]
Tornabene, F. , Brischetto, S. , Fantuzzi, N. , and Viola, E. , 2015, “ Numerical and Exact Models for Free Vibration Analysis of Cylindrical and Spherical Shell Panels,” Composites, Part B, 81, pp. 231–250. [CrossRef]
Artioli, E. , and Viola, E. , 2006, “ Free Vibration Analysis of Spherical Caps Using a GDQ Numerical Solution,” ASME J. Pressure Vessel Technol., 128(3), pp. 370–378. [CrossRef]
Bryan, A. , 2017, “ Free Vibration of Thin Shallow Elliptical Shells,” ASME J. Vib. Acoust., 140(1), p. 011004. [CrossRef]
Bryan, A. , 2017, “ Free Vibration of Thin Spherical Shells,” ASME J. Vib. Acoust., 139(6), p. 061020.
Potter, M. C. , and Goldberg, J. , 1995, Mathematical Methods, Great Lakes Press, Okemos, MI.
Kreyszig, E. , 1995, Advanced Engineering Mathematics, Wiley, New York.
Gutiérrez-Vega, J. C. , 2000, “Formal Analysis of the Propagation of Invariant Optical Fields in Elliptic Coordinates,” Ph.D. thesis, INAOE, Monterrey, Mexico.
United States, National Bureau of Standards, 1951, Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors, Columbia University Press, New York.
Young, P. , 2009, “Helmholtz's and Laplace's Equations in Spherical Polar Coordinates: Spherical Harmonics and Spherical Bessel Functions,” Physics 116C Lecture Notes, University of California, Santa Cruz, CA. http://physics.ucsc.edu/~peter/116C/helm_sp.pdf
Anon, N. D. , 2006, “Spherical Harmonics,” Physics 221A Lecture Notes, University of California Berkeley, Berkeley, CA. http://sandman.berkeley.edu/221A/sphericalharmonics.pdf

Figures

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Fig. 1

Geometry of the doubly curved thin shell

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Fig. 2

Stresses acting on a single element of a doubly curved thin shell

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Fig. 3

Geometry of the SC

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Fig. 4

Geometry of the RC

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Fig. 5

Geometry of the EC

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Fig. 6

Geometry of the SS

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Fig. 7

First ten eigenfrequencies for the SC, RC, EC, and SS

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Fig. 8

The effect of radius and thickness on ω for the first and tenth modes: (a) SC, (b) RC, (c) EC, and (d) SS

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Fig. 9

ω′/ω versus radii and thickness for the first and tenth modes: (a) SC, (b) RC, (c) EC, and (d) SS

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Fig. 10

Effect of aspect ratio on ω

Tables

Errata

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