Natural frequencies and mode shapes for axisymmetric vibrations of shells in turning-point range

In the present paper, the natural frequencies and modes in turning-point frequency range where the governing equations have turning points are studied for revolution shells under various boundary conditions by applying the uniformly valid solutions obtained in a previous paper. Due to the presence of the turning points in this frequency range, several novel features emerge from the analytic and computational results for vibration of shells, in which the most basic nature is the coupling of bending and membrane solutions for frequencies and modes. And simple expressions for the bending-edge-condition effect and frequency spacing are presented.