Numerical analysis of nonlinear vibrations of a plate on a viscoelastic foundation under the action of a moving oscillating load based on models with fractional derivatives

Aim. In the present paper, nonlinear vibrations of an elastic simply supported plate on a viscoelastic foundation under the action of a moving oscillating load are studied in the case of the internal resonance $1{:}1$ accompanied by the external resonance. The properties of the viscoelastic foundation are given via the generalized Fuss–Winkler model with the damping term described by the standard linear solid model with the Riemann–Liouville fractional derivatives. The external load is presented by linear viscoelastic oscillator based on the Kelvin–Voigt model with a fractional derivative in the case when the viscosity of the oscillator is considered to be small value. The dynamic behavior of the plate is described by a set of nonlinear ordinary differential equations of the second order in time with respect to generalized displacements. Methods. To solve the resulting set of equations, the method of multiple time scales is used in combination with the method of expansion of the fractional derivative in a Taylor series. Results. Resolving equations for determining of the nonlinear amplitudes and phases of force driven vibrations of the plate are obtained. The governing set of equations allows one to control not only the damping properties of the environment and the foundation by changing the fractional parameters, but also to control the damping parameters of the external load. Conclusion. Numerical analysis has shown that in the system “a plate on a viscoelastic foundation + a moving oscillating load”, energy transfer between the interacting vibration modes is observed. A comparison of the results of numerical studies for various values of the external load is presented, and the dependence of the amplitudes of nonlinear vibrations on the values of the fractional parameters of the environment and the foundation is also shown.