Anomalous Nonlinear Dynamics Behavior of Fractional Viscoelastic Beams

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

Automated summary

The study examines how fractional models influence the nonlinear dynamics of mechanical systems, particularly in the vibration of a geometrically nonlinear viscoelastic cantilever beam. By employing numerical methods and a semi-analytical approach, the research reveals the impact of model parameters on dynamic characteristics.