Free Vibrations of a Cylindrical Shell Partially Resting on Elastic Foundation

Abstract

The free vibrations of a circular cylindrical shell resting on a two-parameter Pasternak elastic foundation are investigated. The elastic medium is inhomogeneous along the shell length, and the inhomogeneity represents an alternation of areas with and without the medium. The behavior of the shell is considered in the framework of the classical shell theory based on the Kirchhoff–Love hypotheses. The corresponding geometric and physical relations, together with the equations of motion, are reduced to a system of eight ordinary differential equations for new unknowns. The problem is solved by applying the Godunov orthogonal sweep method, and the differential equations are integrated using the fourth-order Runge–Kutta method. The natural frequencies are calculated by applying a stepwise iterative procedure, followed by a further refinement based on the bisection method. The results are validated by comparing them with available numerical-analytical solutions. For simply supported, clamped-clamped, and clamped-free cylindrical shells, the numerical results reveal that the lowest vibration frequencies depend on the elastic medium characteristics and the type of the inhomogeneity. It is shown that a violation in the smoothness of the curves is caused by variations in the lowest frequency mode, the ratio of the size of the elastic foundation to the total length of the shell, and its stiffness, as well as by a combination of the boundary conditions specified at the ends of the shell.