Non-linear dynamics of a slender beam carrying a lumped mass with principal parametric and internal resonances

The non-linear behaviour of a slender beam carrying a lumped mass subjected to principal parametric base excitation is investigated. The dimension of the beam–mass system and the position of the attached mass are so adjusted that the system exhibits 3 : 1 internal resonance. Multi-mode discretization of the governing equation which retains the cubic non-linearities of geometrical and inertial type is carried out using Galerkin’s method. The 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 the stability of the system. The linear first-order perturbation results show new zones of instability due to the presence of internal resonance. For low amplitude of excitation and damping Hopf bifurcations are observed in the trivial steady-state response. The multi-branched non-trivial response curves show turning point, pitch-fork and Hopf bifurcations. Cascade of period and torus doubling, crises as well as the Shilnikov mechanism for chaos are observed. This is the first natural physical system exhibiting a countable infinity of horseshoes in a neighbourhood of the homoclinic orbit.