Computer analysis of asymmetric free vibrations of ring-stiffened orthotropic shells of revolution.

A method of iteration, analogous to Stodola's method for beams, is presented for the determination of the mode shapes and frequencies of free vibration of composite axisymmetric structures consisting of or tho tropic heterogeneous shells of revolution and elastic rings. Numerical results are presented for 1) a spherical shell previously studied analytically (using Legendre functions) and also by means of a competitive numerical method; 2) a conical shell previously studied analytically (using the Galerkin technique) and also experimentally; and 3) a complete entry vehicle modeled as a continuous layered shell-ring-rigid mass structure, for which no previous results exist. In the case of the first example it is shown that what were previously reported to be the first three modes of axisymmetric vibration are, in reality, the first, second, and fourth modes. The missing third mode, which is presented in this paper, has, unexpectedly, no interior nodes of normal deflection. The second example serves to confirm and sharpen the qualitative result that edge constraint of circumferential deflection, in contrast to edge constraint of normal deflection, suppresses the tendency for vibration modes with a small number of circumferential waves to be essentially inextensional in the shell interior, thereby causing a large increase in frequency. At the same time it illustrates the deficiencies of the Donnell-type assumptions made in the Galerkin analysis for predicting frequencies of modes with a small number of circumferential waves.