Bifurcations and chaotic forced vibrations of cantilever beams with breathing cracks

Abstract

Finite-degrees-of-freedom nonlinear dynamical system, which describes the forced vibrations of the beams with two breathing cracks, is derived. The cracks are spaced at the opposite sides of the beam. The Galerkin method is applied to derive the nonlinear dynamical system. The infinite sequences of the period-doubling bifurcations, which cause the chaotic vibrations, are observed at the principle and the second order subharmonic resonances. The Poincare sections and spectral densities are calculated to analyze the properties of chaotic vibrations. Moreover, the Lyapunov exponents are calculated to validate the chaotic behavior. As follows from the numerical analysis, the chaotic vibrations originate due to the nonlinear interaction between the cracks.