An analysis of the free vibration of thick-walled isotropic toroidal shells

The equations of motion and strain–displacement equations for toroidal shells are derived using toroidal coordinates and dyadic methods of elasticity. Suitable assumptions are proposed that reduce the governing equations to axisymmetric format while maintaining the three-dimensional character of the solution. The free vibration analysis of toroidal shells is carried out using numerical solutions of the governing equations. A nine-node Lagrangian finite element is formulated in the toroidal coordinate system and solutions are obtained for the case where an axis of symmetry can be assumed at the center of the torus. Results for frequency of vibration are tabulated for solid and thick-walled shells. Mode shapes are shown for a representative thick-walled torus.