Non-linear dynamics of a slender beam carrying a lumped mass under principal parametric resonance with three-mode interactions

The non-linear response of a base-excited slender beam carrying a lumped mass subjected to principal parametric resonance is investigated. The attached mass and its location are so adjusted that the system exhibits 1:3:5 internal resonances. Method of multiple scales is used to reduce the second-order temporal differential equation to a set of first-order differential equations which is then solved numerically to obtain the steady-state response and stability of the system. The steady-state response thus obtained is compared with those found by single- and two-mode analyses and very significant differences are observed in the bifurcation and stability of the response curves. The effects of external and internal detuning, amplitude of excitation and damping on the non-linear steady state, periodic, quasi-periodic and chaotic responses of the system are investigated. Funnel-shaped chaotic orbits, fractal orbits, cascade of period-doubling, torus doubling and intermittency routes to chaos are observed in this system. A simple illustration is given to control chaos by changing the system parameters. (C) 2001 Elsevier Science Ltd. All rights reserved.