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Fractional models and their parameters are sensitive to changes in the intrinsic micro-structures of anomalous materials. We investigate how such physics-informed models propagate the evolving anomalous rheology to the nonlinear dynamics of mechanical systems. In particular, we analyze the vibration of a fractional, geometrically nonlinear viscoelastic cantilever beam, under base excitation and free vibration, where the viscoelastic response is general through a distributed-order fractional model. We employ Hamilton's principle to obtain the corresponding equation of motion with the choice of specific material distribution functions that recover a fractional Kelvin-Voigt viscoelastic model of order $α$. Through spectral decomposition in space, the resulting time-fractional partial differential equation reduces to a nonlinear time-fractional ordinary differential equation, in which the linear counterpart is numerically integrated by employing a direct L1-difference scheme. We further develop a semi-analytical scheme to solve the nonlinear system through a method of multiple scales, which yields a cubic algebraic equation in terms of the frequency. Our numerical results suggest a set of $α$-dependent anomalous dynamic qualities, such as far-from-equilibrium power-law amplitude decay rates, super-sensitivity of amplitude response at free vibration, and bifurcation in steady-state amplitude at primary resonance.