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We investigate the nonlinear vibration of a fractional viscoelasticcantilever beam, subject to base excitation, where the viscoelasticity takesthe general form of a distributed-order fractional model, and the beamcurvature introduces geometric nonlinearity into the governing equation. Weutilize the extended Hamilton principle to derive the governing equation ofmotion for specific material distribution functions that lead to fractionalKelvin-Voigt viscoelastic model. By spectral decomposition in space, theresulting governing fractional PDE reduces to nonlinear time-fractional ODEs.We use direct numerical integration in the decoupled system, in which weobserve the anomalous power-law decay rate of amplitude in the linearizedmodel. We further develop a semi-analytical scheme to solve the nonlinearequations, using method of multiple scales as a perturbation technique. Wereplace the expensive numerical time integration with a cubic algebraicequation to solve for frequency response of the system. We observe the supersensitivity of response amplitude to the fractional model parameters at freevibration, and bifurcation in steady-state amplitude at primary resonance.

Vibration Analysis of Geometrically Nonlinear and Fractional Viscoelastic Cantilever Beams