THE NON-LINEAR SLOW NORMAL MODE AND STOCHASTICITY IN THE DYNAMICS OF A CONSERVATIVE FLEXIBLE ROD/PENDULUM CONFIGURATION

Abstract

The geometric structure has been analyzed of the slow periodic motions of a conservative structural/mechanical system consisting of a stiff linear elastic rod coupled to a non-linear pendulum oscillator. Using the theory of geometric singular perturbations, we have computed analytically a two-dimensional invariant non-linear manifold of motion in phase space, called the slow manifold. Numerical experiments reveal that all motions initiated on the slow manifold are purely slow periodic and share common properties. The slow invariant manifold is the geometric realization of a non-linear normal mode, which consists of master and slaved dynamics. The normal mode is non-classical since it does not satisfy the classical definition of vibrations-in-unison. The analysis reveals that the slow invariant manifold carries a heteroclinic motion. Its existence is verified numerically by showing that all its Lyapunov characteristic exponents are zero. Above some critical coupling between the flexible rod and pendulum, the slow normal mode interacts, first at the energy level of the heteroclinic motion, with the fast dynamics to create stochastic motions. The strength of stochasticity, measured by the Lyapunov characteristic exponents, increases as the coupling increases.