On nonlinear vibrations of an elastic plate on a fractional viscoelastic foundation in a viscoelastic medium in the presence of the one-to-one internal resonance

Abstract

In the present paper, the dynamic response of a nonlinear Kirchhoff–Love plate resting on a viscoelastic foundation in a viscoelastic medium, damping features of which are described by the Kelvin–Voigt fractional derivative model is studied by the generalized method of multiple time scales. The viscoelastic features of the foundation are modelled via the two fractional derivative models: Kelvin–Voigt or standard linear solid model with fractional derivatives. Resolving equations are obtained for the case of the one-to-one internal resonance for determining nonlinear amplitudes and phases in the case of free oscillations, when the eigen frequencies of the two dominant oscillation modes are close to each other, and for the case of forced oscillations, when the frequency of the external harmonic force is close in value to the eigen frequencies of the interacting eigen modes. The resulting system of equations allows one to control the damping properties of the environment and the base by changing the parameters of fractionality, which expands the range of problems of applicability of this solution.