Nonlinear dynamics of a boom crane

We study a boom crane modeled as a spherical pendulum undergoing base excitations. We demonstrate how instabilities in the payload motion arise due to a combination of a one-to-one internal resonance and a primary (additive) resonance or a parametric (multiplicative) resonance. The method of multiple scales is used to derive four nonlinear ordinary-differential equations describing the amplitudes and phases of the in-plane and out-of-plane modes. The modulation equations are used to study the equilibrium and dynamic solutions and their stability. The response could be a single-mode (planar) or a two-mode (three-dimensional) motion. We also study the limit cycles arising in the response and ascertain their stability. Numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos, multiple attractors, intermittency of type I, and cyclic-fold bifurcations.