Branch switching at Hopf bifurcation analysis via asymptotic numerical method: Application to nonlinear free vibrations of rotating beams

•Nonlinear free vibrations of rotating isotropic and anisotropic beams are treated.•The partial differential equations are transformed to algebraic equations.•The ANM continuation method is used to calculate branches.•The branches bifurcate from a Hopf bifurcation point.•This paper has the exception...

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Bibliographic Details
Published in:Communications in nonlinear science & numerical simulation 2015-05, Vol.22 (1-3), p.716-730
Main Authors: Bekhoucha, Ferhat, Rechak, Said, Duigou, Laëtitia, Cadou, Jean-Marc
Format: Article
Language:English
Subjects:
Asymptotic properties
Beams (radiation)
Bifurcations
Branch switching
Engineering Sciences
Frequency domain
Galerkin method
Hopf bifurcation
Mathematical analysis
Mathematical models
Mathematics
Mechanics
Nonlinear free vibration
Nonlinearity
Numerical analysis
Physics
Rotating
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Summary:•Nonlinear free vibrations of rotating isotropic and anisotropic beams are treated.•The partial differential equations are transformed to algebraic equations.•The ANM continuation method is used to calculate branches.•The branches bifurcate from a Hopf bifurcation point.•This paper has the exception that this problem has never been solved before. This paper deals with the computation of backbone curves bifurcated from a Hopf bifurcation point in the framework of nonlinear free vibrations of a rotating flexible beams. The intrinsic and geometrical equations of motion for anisotropic beams subjected to large displacements are used and transformed with Galerkin and harmonic balance methods to one quadratic algebraic equation involving one parameter, the pulsation. The latter is treated with the asymptotic numerical method using Padé approximants. An algorithm, equivalent to the Lyapunov–Schmidt reduction is proposed, to compute the bifurcated branches accurately from a Hopf bifurcation point, with singularity of co-rank 2, related to a conservative and gyroscopic dynamical system steady state, toward a nonlinear periodic state. Numerical tests dealing with clamped, isotropic and composite, rotating beams show the reliability of the proposed method reinforced by accurate results.
ISSN:1007-5704
1878-7274
DOI:10.1016/j.cnsns.2014.09.001