Nonlinear dynamics of a three-beam structure with attached mass and three-mode interactions

Abstract

Nonlinear dynamics of elastic structures with two-mode interactions have been extensively studied in the literature. In this work, nonlinear forced response of elastic structures with essential inertial nonlinearities undergoing three-mode interactions is studied. More specifically, a three-beam structural system with attached mass is considered, and its multidegree-of-freedom discretized model for the structure undergoing planar motions is carefully studied. Linear modal characteristics of the structure with uniform beams depend on the length ratios of the three beams, the mass of the particle relative to that of the structure, and the location of the mass particle along the beams. The discretized model is studied for both external and parametric resonances for parameter combinations resulting in three-mode interactions. For the external excitation case, focus is on the system with 1:2:3 internal resonances with the external excitation frequency near the middle natural frequency. For the case of the structure with 1:2:5 internal resonances, the problem involving simultaneous principal parametric resonance of the middle mode and a combination resonance between the lowest and the highest modal frequencies is investigated. This case requires a higher-order approximation in the method of multiple time scales. For both cases, equilibrium and bifurcating solutions of the slow-flow equations are studied in detail. Many pitchfork, saddle-node, and Hopf bifurcations appear in the amplitude response of the three-beam structure, thus resulting in complex multimode responses in different parameter regions.