Force driven nonlinear vibrations of a thin plate with 1:1 internal resonance in a fractional viscoelastic medium

Classification of Internal Resonances in Nonlinear Fractionaly Damped Uflyand-Mindlin Plates

“…The fractional derivative standard linear solid model has been utilized in [44] for a viscoelastic layer for active damping of geometrically nonlinear vibrations of smart composite plates using the higher order plate theory and finite element method with discretizing the plate by eight-node isoparametric quadrilateral elements. Recently the approaches suggested in [19,20] for solving the problem on free nonlinear vibrations of elastic plates in a viscoelastic medium, damping features of which are governed by the Riemann-Liouville derivatives of the fractional order, and in [45] for studying the dynamic response of the fractional Duffing oscillator subjected to harmonic loading have been generalized for the case of forced vibrations of a simply-supported nonlinear thin elastic plate under the conditions of different internal resonances, when two or three natural modes corresponding to mutually orthogonal displacements are coupled [46][47][48][49]. In the present paper, the procedure proposed in [20] for solving the problem of free nonlinear vibrations of elastic plates in a fractional derivative viscoelastic medium, when the damped motion is described by a set of three nonlinear equations, has been extended for the case of free vibrations of a simply-supported fractionally damped nonlinear thin elastic plate, the motion of which is described by five equations involving shear deformations and rotary inertia.…”