Bifurcation and Modal Interaction in a Simplified Model of Bending-Torsion Vibrations of the Thin Elastica

Abstract

This paper presents a numerical study of bifurcation and modal interaction in a system of partial differential equations first proposed as a simplified model for bending-torsion vibrations of a thin elastic beam. A system of seven ordinary differential equations obtained using the first six bending and first torsional normal modes is studied, and Floquet theory is used to locate regions in the forcing frequency, forcing amplitude parameter plane where “planar” (i.e., zero torsion) motions are unstable. Numerical branch following and symmetry considerations show that the initial instability arises from a subcritical pitchfork bifurcation. The subsequent nonplanar chaotic attractor is part of a branch of 2-frequency quasiperiodic orbits which undergoes torus-doubling bifurcations. A new statistical technique which identifies interacting modes and the average stability properties of the associated subspaces is presented. The technique employs the Lyapunov vectors used in the calculation of the Lyapunov exponents. We show how this method can be used to split the modes into active and passive sets: active modes interact to contain the attractor, whereas passive modes behave like isolated driven oscillators. In particular, large amplitude modes may simply serve as conduits through which energy is supplied to the active modes.