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Research Papers

Free Vibration of Thin Shallow Elliptical 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 July 3, 2017; final manuscript received July 11, 2017; published online August 17, 2017. Editor: I. Y. (Steve) Shen.

J. Vib. Acoust 140(1), 011004 (Aug 17, 2017) (9 pages) Paper No: VIB-17-1294; doi: 10.1115/1.4037300 History: Received July 03, 2017; Revised July 11, 2017

This research presents a study of the free vibration of thin, shallow elliptical shells. The equations of motion for the elliptical shell, which are developed from Love's equations, are coupled and nonlinear. In this research, a new approach is introduced to uncouple the transverse motion of the shallow elliptical shell from the surface coordinates. Through the substitution of the strain-compatibility equation into the differential equations of motion in terms of strain, an explicit relationship between the curvilinear surface strains and transverse strain is determined. This latter relationship is then utilized to uncouple the spatial differential equation for transverse motion from that of the surface coordinates. The approach introduced provides a more explicit relationship between the surface and transverse coordinates than could be obtained through use of the Airy stress function. Angular and radial Mathieu equations are used to obtain solutions to the spatial differential equation of motion. Since the recursive relationships that are derived from the Mathieu equations lead to an infinite number of roots, not all of which are physically meaningful, the solution to the eigenvalue problem is used to determine the mode shapes and eigenfrequencies of the shallow elliptical shell. The results of examples demonstrate that the eigenfrequencies of the thin shallow elliptical shell are directly proportional to the curvature of the shell and inversely proportional to the shell's eccentricity.

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Figures

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Fig. 1

Geometry of the elliptical shell

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Fig. 2

Geometry of an ellipse showing elliptical coordinates

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Fig. 3

Mode shapes with corresponding eigenfrequencies (rad/s) for the elliptical shell with midsurface tensions

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Fig. 4

Mode shapes and eigenfrequencies (rad/s) shells with various aspect ratios: the first and second values are ω (rad/s) with and without the consideration of midsurface tensions, respectively

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