Nonlinear flexural vibration of a circular ring. A single mode approach

In this paper, nonlinear flexural vibrations of nonrotating thin circular rings is investigated. In the first part, a multimode approach formulation, based on Lagrange’s equation, is developed and new formulation is provided. This formulation follows the nonlinear Sander’s theory that consider the same expression for the radial and circumferential displacements. Nonlinear free and forced responses are analyzed in the case of a single degree-of-freedom considering the fourth mode. The backbone curve is obtained and compared with other theoretical and experimental results. In a second part, dynamic and bifurcation of the weakly damped and forced considered mode is examined in the vicinity of the 1:1 resonance. Using the multiple scale technique, the slow flow is derived and then the trivial and nontrivial periodic responses of the mode are obtained. It was shows that the fourth mode of the thin circular ring considered in this study presents a softening behavior and can have two possible stable periodic responses near the 1:1 resonance. It was also shown that the introduction of the circumferential displacement has no significant effect on the forced response of the fourth mode of the circular ring near the principal resonance. In this case, the considered model without in-plane inertia can be adopted for the forced case analysis.