Abstract
In this article, the free vibration characteristics of spherical caps with different thickness distribution subjected to general boundary conditions are investigated using a semi-analytical approach. Based on the theory of thin shell, the theoretical model of spherical cap is established. Spherical caps are partitioned into sections along the meridional orientation. The displacement components of spherical caps along the meridional direction are represented by Jacobi polynomials. Meanwhile, Fourier series are utilized to express displacement components in the circumferential direction. Various boundary conditions can be easily achieved by the penalty method of the spring stiffness technique. The vibration characteristics of spherical caps are derived by means of the Rayleigh–Ritz energy method. Reliability and validity of the current method are verified by convergence studies and numerical verification. The comparison of results between the current method, finite element method, and those published in the literature prove that the current method works well when handling free vibration of spherical caps. More results of spherical caps with different geometric specifications and edge conditions are displayed in the form of table and graphic, which may serve as a reference for future studies.
Highlights
Spherical caps with uniform and stepped thickness have been widely applied in practical engineering applications for their excellent mechanical properties
Where i signifies the ith segment of the spherical cap; eiu and eiu are the normal strains of the spherical cap; eiuu means the shear strains; kui and kui signify the midsurface variation in the curvature direction; kui u means the twist mid-surface variation; ui, vi, and wi represent the displacement components of the spherical cap along u, u, and d orientations, respectively; and A and B correspond to different Lameparameters
The spherical cap was divided into sections along the axial direction
Summary
Spherical caps with uniform and stepped thickness have been widely applied in practical engineering applications for their excellent mechanical properties. Lee[2] investigated the vibration of a spherical cap by means of the pseudospectral method He represented the displacement function by Chebyshev polynomials, and the rotations are expressed as Fourier series. Ye et al.[9,10] investigated the vibration behaviors of functionally graded (FG) and composite laminated spherical shells with arbitrary boundary conditions The results of these papers are compared with those published in the literature and those of the finite element method (FEM). Most studies mentioned above are about the investigation of the vibration characteristic of uniform spherical shells by means of the pseudospectral method, FEM, Fourier series method, and so on. A unified and efficient formulation is necessary and of great significance to establish to analyze free vibration of spherical caps with uniform and stepped thickness subjected to various boundary conditions.
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