  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.
