Effects of non-uniform Winkler foundation and non-homogeneity on the free vibration of an orthotropic elliptical cylindrical shell

Flügge's shell theory and solution for the vibration analysis of a non-homogeneous orthotropic elliptical cylindrical shell resting on a non-uniform Winkler foundation are presented. The theoretical analysis of the governing equations of the shell is formulated to overcome the mathematical difficulties of mode coupling of variable curvature and homogeneity of shell. Using the transfer matrix of the shell, the vibration equations based on the variable Winkler foundation are written in a matrix differential equation of first order in the circumferential coordinate and solved numerically. The proposed model is applied to get the vibration frequencies and the corresponding mode shapes of the symmetrical and antisymmetrical vibration modes. The sensitivity of the vibration behavior and bending deformations to the non-uniform Winkler foundation moduli, homogeneity variation, elliptical and orthotropy of the shell is studied for different type-modes of vibrations.