Dynamic stiffness analysis of toroidal shells

Abstract

The free vibration of a toroidal shell is studied using the dynamic stiffness method. The dynamic stiffness method eliminates both spatial discretization error and mesh generation. Moreover, with a finite number of degrees of freedom, the dynamic stiffness method can predict an infinite number of natural frequencies. The dynamic behavior of the toroidal shell is modeled by DMV (Donnell-Mushtari-Vlasov) linear thin shell theory in the present paper. However, the procedure can be adapted to be used with any other linear thin shell theory without difficulty. Since a close form solution of toroidal shell using DMV theory is not (yet) possible, in order to obtain the desired dynamic stiffness matrix, a finite number of Fourier's series terms are taken in the circumferential direction and the unknown longitudinal displacements are then solved from the reduced governing equations exactly. The solution obtained from the dynamic stiffness method can be regarded as semi-analytical due to the Fourier approximation. With the dynamic stiffness matrices in hands, a toroidal shell with different boundary conditions and connections (to other toroidal shells) can be analyzed. This paper presents the procedure and assumption made in order to obtain the dynamic stiffness matrix of a toroidal shell in harmonic oscillation. Also some numerical examples will be given and discussed.