Nonlinear dynamics of unidirectional, fiber-reinforced tori

The symbolic manipulator Mathematica is used to model the nonlinear dynamic behavior of closed, elastic toroidal shells. Transverse shears are neglected and the nonlinearities are of the Von Karman type. Two fiber-reinforcing schemes are considered: reinforcement with fibers along the major direction of the torus, and reinforcement with fibers along the minor direction of the torus. These schemes result in orthotropic material characteristics. Differential geometry is used to derive the nonlinear kinematic relationships, and a combination of the Rayleigh-Ritz technique and the method of harmonic balance is used to approximate the nonlinear natural frequencies of the tori. Numerical examples show that the linear natural frequency increases as the fiber volume fraction increases for any radii ratio. On the other hand, the nonlinear analysis of some reinforcing schemes shows a competition between the geometric and material parameters of the tori. This competition has a significant effect on the qualitative behavior of the torus and demarks the borders separating shell-like behavior and ring like behavior.