Nonlinear natural responses of a rotating circular string

This study investigates the nonlinear dynamics of a rotating circular string subjected to a spring force fixed in space. The governing equation for out-of-plane vibration is developed using Hamilton's principle. The nonlinearities of the string deformation and the spring stiffness are considered in the governing equation. Applying Galerkin's method, the governing equation is transformed from a nonlinear partial differential equation into a set of coupled nonlinear ordinary differential equations through orthogonal trigonometric shape functions. Butenin's method is adopted to develop a closed-form analytical solution for single-mode oscillations of the system. Comparison shows that the closed-form solution is in a good agreement with the numerical results over a wide range of the nonlinearities. Multi-mode responses of the string are investigated by means of numerical integration. Based on the results, the nonlinear dynamics of the string are discussed.