Dynamic Response of the Mechanical System Involving a Nonlinear Plate on a Viscoelastic Foundation and a Moving Sprung Load with Finite Viscosity Utilizing the Fractional Derivative Models

The dynamic behavior of a nonlinear elastic plate on a viscoelastic foundation under the action of a moving oscillating load is studied in the case of the internal resonance accompanied by the external resonance. The damping properties of the viscoelastic foundation are described by the generalized Fuss-Winkler model with the damping term given via the standard linear solid model with the Riemann–Liouville fractional derivatives. The external load is presented by a linear viscoelastic oscillator governed by the constitutive equation based on the fractional derivative Kelvin-Voigt model. The process of vibrations of such a system is described by a set of nonlinear ordinary differential equations of the second order in time with respect to generalized displacements. The method of multiple time scales is used for solving the obtained set of equations in combination with the method of expansion of the fractional derivative in a Taylor series. The assumption of viscosity of the oscillator to be a finite value leads to the solution of the characteristic equation for defining the behavior of complex conjugate roots. Resolving equations for determining the nonlinear amplitudes and phases of force driven vibrations of the plate is obtained. Comparative analysis of the numerical studies is carried out for oscillators with fractional derivatives in damping description for the cases of small and finite viscosity. The influence of the fractional parameters of the environment and the foundation on the amplitudes of nonlinear vibrations is also shown.