Summary
The present work studies the vibratory response of geometrically nonlinear cyclic structures to harmonic excitations. These cyclic structure, in their linear approximations, possesses 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-doupling bifurcations and Silnikov type chaos. The higher-dimensional Melnikov method and its extensions by Kovacic and Wiggins, and Haller and Wiggings, to dissipative and nondissipative singular systems 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.
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Samaranayake, G., Samaranayake, S. & Bajaj, A.K. Non-resonant and resonant chaotic dynamics in externally excited cyclic systems. Acta Mechanica 150, 139–160 (2001). https://doi.org/10.1007/BF01181808
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DOI: https://doi.org/10.1007/BF01181808