Non-Resonant and Resonant Chaotic Dynamics in Externally Excited Cyclic Systems

Abstract

The present work studies the vibratory response of geometrically nonlinear cyclic structures to harmonic excitations. These cyclic structure, in their linear approximations, possess pairwise double degenerate natural frequencies with orthogonal normal modes. The dynamic response of such systems, when excited near primary resonance, is studied using the method of averaging. The averaged equations, representing the evolution of the amplitudes and phases of the interacting normal modes, are known to exhibit complex dynamics including period-doubling bifurcations and Silnikov type chaos. The higher-dimensional Melnikov method and its extensions are utilized to detect the transversal intersection of stable and unstable manifolds of periodic orbits. Such intersections imply the existence of Smale horseshoes and, hence, chaotic dynamics for the averaged system.