Free vibration of spherical cap subjected to various boundary conditions

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.