Turning points and caustics in linear problems of thin shell free vibrations and buckling

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies of joined hemispherical–cylindrical shells of revolution with a top opening. Unlike conventional shell theories, which are mathematically two-dimensional (2D), the present method is based upon the 3D dynamic equations of elasticity. Displacement components ur, uθ and uz in the radial, circumferential, and axial directions, respectively, are taken to be periodic in θ and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the joined shells are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Natural frequencies are presented for different boundary conditions. The frequencies from the present 3D method are compared with those from 2D thin shell theories.

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Free vibrations of restrained cylindrical thin shell panel under axial and transverse stresses

Journal of Sound and Vibration ◽

10.1016/0022-460x(88)90312-4 ◽

1988 ◽

Vol 127 (2) ◽

pp. 384-390 ◽

Cited By ~ 2

Author(s):

W.H. Liu ◽

W.C. Chen

Keyword(s):

Thin Shell ◽

Free Vibrations ◽

Shell Panel ◽

Transverse Stresses

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I. The small free vibrations and deformation of a thin elastic shell

Proceedings of the Royal Society of London ◽

10.1098/rspl.1887.0146 ◽

1888 ◽

Vol 43 (258-265) ◽

pp. 352-353 ◽

Cited By ~ 2

Keyword(s):

Potential Energy ◽

Energy Function ◽

Principal Curvature ◽

Thin Shell ◽

Spherical Surface ◽

Elastic Shell ◽

Potential Energy Function ◽

Free Vibrations ◽

Plane Plate ◽

The Difference

In this paper the method employed by Kirchhoff and Clebsch for the treatment of a thin plane plate is applied to the case of a thin shell, or plate of finite curvature. The form of the potential-energy function for the strain in an element of the shell is the same as that obtained by Kirchhoff for a plate, the quantities depending on the curvature of the surface being replaced by the difference of their values in the strained and unstrained states. It is proved that only for an inextensible spherical surface is this function the same function of the changes of principal curvature as for a plane plate.

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Axisymmetric Free Vibrations of Conical Shells

Journal of Applied Mechanics ◽

10.1115/1.3629663 ◽

1964 ◽

Vol 31 (3) ◽

pp. 458-466 ◽

Cited By ~ 47

Author(s):

Hyman Garnet ◽

Joseph Kempner

Keyword(s):

Shear Deformation ◽

Transverse Shear ◽

Thin Shell ◽

Shell Theory ◽

Free Vibrations ◽

Transverse Shear Deformation ◽

Rotatory Inertia ◽

Conical Shells ◽

Thin Shell Theory ◽

Thickness Parameter

The lowest axisymmetric modes of vibration of truncated conical shells are studied by means of a Rayleigh-Ritz procedure. Transverse shear deformation and rotatory inertia effects are accounted for, and the results are compared with those predicted by the classical thin-shell theory. Additionally, the results are compared when either of these theories is formulated in two ways: First, in the manner of Love’s first approximation in the classical thin-shell theory, and then by including the influence of the change of the element of arc length through the thickness. It was found that the Love and the more complex formulation yielded results which differed negligibly in either theory. The results predicted by the shear deformation-rotatory inertia theory differed significantly from those predicted by the classical thin-shell theory within a range of parameters which characterize short thick cones. These differences resulted principally from the influence of the transverse shear deformation. It was also found that within this short-cone range an increase in the shell thickness parameter was accompanied by an increase in the natural frequency. Moreover, the increase in frequency with increasing thickness parameter became less severe as the length-to-mean radius ratio was increased. For the longer cones, the frequency was virtually independent of the thickness.

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