New approach for the vibrations of inhomogeneous orthotropic cylindrical shells resting on step-wise Winkler foundations

Abstract

In this paper, Flügge’s shell theory and solution for the vibration analysis of an inhomogeneous, orthotropic cylindrical shell resting on a circumferentially stepped Winkler foundation are carried out. The stepwise foundation consists of two regions having different stiffnesses and both regions are symmetric about the centrelines of the shell. The governing equations of the shell are formulated as a three-dimensional boundary-value problem and their solutions are obtained by a numerical–analytical approach. The trigonometric functions are used with Fourier’s approach to approximate the solution in the longitudinal direction and also to reduce the two-dimensional problem to one-dimensional one. Using the transfer matrix approach and based on the method of initial parameters, these equations can be written in a matrix differential equation of first order in the circumferential coordinate and solved numerically as an initial-value problem. The proposed model is applied to get the inverse vibration frequencies and the corresponding mode shapes of the shell vibrations. The sensitivity of the vibration behaviour and bending deformations to the geometry of Winkler foundation, homogeneity variation and shell orthotropy is investigated for different type-modes of vibration.