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In the present paper, the forced driven nonlinear vibrations of an elastic plate in a viscoelastic medium and resting on a viscoelastic Winkler-type foundation are studied. The damping features of the surrounding medium and foundation are described by the Kelvin-Voigt model and standard linear solid model with fractional derivatives, respectively. The dynamic response of the plate is described by the set of nonlinear differential equations with due account for the fact that the plate is being under the conditions of the internal resonance accompanied by the external resonance. The expressions for the stress function and nonlinear coefficients for different types of boundary conditions are presented.

In the world literature there exists a wide variety of papers devoted to linear viscoelastic models. This work was initiated by the absence of a single generally accepted classification of viscoelastic models. We focused on the basic mechanical models, namely, the Kelvin-Voigt, Maxwell, standard linear solid and Jeffreys models. All other models are different combinations of basic elements connected in series or in parallel. The classification also includes viscoelastic models with fractional derivatives and fractional operators.