Normal modes and global dynamics of a two-degree-of-freedom non-linear system—II. High energies

Abstract

The high-energy global dynamics of an undamped, strongly non-linear, two-degree-of-freedom system are considered. As shown in an earlier work [A.F. Vakakis and R.H. Rand, Int. J. Non-Linear Mech. 27, 861–874 (1992)], the oscillator under consideration contains “similar” non-linear normal modes and at certain values of its structural parameters a mode bifurcation is possible. For low energies, the mode bifurcation gives rise to a homoclinic orbit in the Poincaré map of the system. For high energies, large- and low-scale chaotic motions are detected, resulting from transverse intersections of the stable and unstable manifolds of an unstable antisymmetric normal mode, and from the breakdown of invariant KAM-tori. The creation of additional free subharmonic motions is studied by a subharmonic Melnikov analysis, and the stability of the subharmonic motions is examined by an averaging methodology. The main conclusion of this work is that the bifurcation of similar normal modes results in a class of large-scale free chaotic motions, which do not exist in the system before the bifurcation.