NUMERICAL ANALYSIS OF ROTATIONALLY SYMMETRIC SHELLS BY THE MODAL SUPERPOSITION METHOD

Abstract

The system of differential equations governing the analysis of rotationally symmetric shells under time-dependent or static surface loadings is formulated with the transverse, meridional, and circumferential displacements as the dependent variables. The thickness of the shell may vary, and four homogeneous boundary conditions may be prescribed at each boundary edge of the shell. The governing differential equations for each Fourier harmonic are obtained by use of Fourier series in the circumferential direction of the shell. Influence coefficients at each of the node points along the shell meridian are obtained for each Fourier component by employing ordinary finite difference representations for the meridional co-ordinate derivatives. With these influence coefficients, a set of homogeneous flexibility equations governing the free vibration characteristics of the shell is obtained and solved for the frequencies and mode shapes for each Fourier component. The solutions under time-dependent or static surface loadings and due to initial displacements and velocities are then obtained by expanding the solutions in terms of the modes of free vibration of the shell for each Fourier component. The solution for the total shell response is obtained by Fourier series summation. Solutions for typical shells have been found to be in excellent agreement with solutions by the method of temporal and spatial finite differences. Solutions for a parabolic shell are presented as an example. The solution is also presented for a published cylindrical shell example and is seen to be in excellent agreement with the published finite element method results.