Vibration Analysis of Geometrically Nonlinear and Fractional Viscoelastic Cantilever Beams

We investigate the nonlinear vibration of a fractional viscoelastic cantilever beam, subject to base excitation, where the viscoelasticity takes the general form of a distributed-order fractional model, and the beam curvature introduces geometric nonlinearity into the governing equation. We utilize...

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Bibliographic Details
Published in:arXiv.org 2019-09
Main Authors: Varghaei, Pegah, Kharazmi, Ehsan, Suzuki, Jorge L, Zayernouri, Mohsen
Format: Article
Language:English
Subjects:
Amplitudes
Bifurcations
Cantilever beams
Curvature
Decay rate
Distribution functions
Equations of motion
Free vibration
Frequency response
Geometric nonlinearity
Hamilton's principle
Mathematical models
Multiscale analysis
Nonlinear analysis
Nonlinear equations
Numerical integration
Parameter sensitivity
Perturbation methods
Time integration
Vibration analysis
Viscoelasticity
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Summary:We investigate the nonlinear vibration of a fractional viscoelastic cantilever beam, subject to base excitation, where the viscoelasticity takes the general form of a distributed-order fractional model, and the beam curvature introduces geometric nonlinearity into the governing equation. We utilize the extended Hamilton principle to derive the governing equation of motion for specific material distribution functions that lead to fractional Kelvin-Voigt viscoelastic model. By spectral decomposition in space, the resulting governing fractional PDE reduces to nonlinear time-fractional ODEs. We use direct numerical integration in the decoupled system, in which we observe the anomalous power-law decay rate of amplitude in the linearized model. We further develop a semi-analytical scheme to solve the nonlinear equations, using method of multiple scales as a perturbation technique. We replace the expensive numerical time integration with a cubic algebraic equation to solve for frequency response of the system. We observe the super sensitivity of response amplitude to the fractional model parameters at free vibration, and bifurcation in steady-state amplitude at primary resonance.
ISSN:2331-8422