Frequencies and mode shapes for thick truncated hollow cones

The finite element method of analysis is employed to study the vibration of truncated thick hollow cones. The three-dimensional strain–displacement equations are used to formulate the finite element in the conical coordinate system. Axial symmetry is assumed and the formulation is reduced to two dimensions while maintaining the three-dimensional character of the analysis. The accuracy of the analysis is established by comparing results for free–free boundary conditions with existing solutions. The effects of different boundary conditions, including fixed–free and fixed–fixed, are studied to determine their effects on the frequency of vibration. Frequency results are given for an additional boundary condition corresponding to a layer attached to the exterior of a rigid cone. Nondimensional results for an isotropic material with Poissonʹs ratio of 0.3 are presented in terms of tables, plots of frequency versus cone apex angle and plots of selected mode shapes.