Bifurcations and chaotic forced vibrations of cantilever beams with breathing cracks

•Finite degrees-of-freedom nonlinear dynamical system, which describes the vibrations of the beam with two cracks.•Partial differential equation of the beam motions with one crack.•Analysis of cracked beam linear vibrations using by spline approximation.•Numerical analysis of linear vibrations.•Nume...

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Bibliographic Details
Published in:Engineering fracture mechanics 2019-06, Vol.214, p.289-303
Main Authors: Avramov, K., Malyshev, S.
Format: Article
Language:English
Subjects:
Bifurcations
Breathing
Cantilever beams
Chaos theory
Chaotic vibration
Cracked beam
Cracks
Dynamical systems
Forced vibration
Galerkin method
Liapunov exponents
Lyapunov exponent
Mathematical analysis
Numerical analysis
Period-doubling bifurcation
Sequences
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Summary:•Finite degrees-of-freedom nonlinear dynamical system, which describes the vibrations of the beam with two cracks.•Partial differential equation of the beam motions with one crack.•Analysis of cracked beam linear vibrations using by spline approximation.•Numerical analysis of linear vibrations.•Numerical analysis of bifurcations and chaotic vibrations. 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.
ISSN:0013-7944
1873-7315
DOI:10.1016/j.engfracmech.2019.03.021