Nonlinear Vibrations of Viscoelastic Plates of Fractional Derivative Type: An AEM Solution

ABSTRACT

The nonlinear dynamic response of thin plates made of linear viscoelastic material of fractional derivative type is investigated. The resulting governing equations are three coupled nonlinear fractional partial differential equations in terms of displacements. The solution is achieved using the AEM. According to this method the original equations are converted into three uncoupled linear equations, namely a biharmonic (linear thin plate) equation for the transverse deflection and two Poisson's (linear membrane) equations for the inplane deformation under time dependent fictitious loads. The resulting thus initial value problems for the fictitious loads is a system of nonlinear fractional ordinary differential equations, which is solved using the numerical method developed recently by Katsikadelis for multi-term fractional differential equations. Several plates subjected to various loads and boundary conditions are analyzed and the influence of the viscoelastic character of the material is investigated. Without excluding other viscoelastic models, the viscoelastic material employed herein is described by the generalized Voigt model of fractional order derivative. The numerical results demonstrate the efficiency and validate the accuracy of the solution procedure. Emphasis is given to the resonance response of viscoelastic plates under harmonic excitation, where complicated phenomena, similar to those of Duffing equation occur.

Keywords:

Plates, viscoelasticity, fractional viscoelastic models , nonlinear vibration, nonlinear resonance , boundry elements, analog equation method, fractional partial differential equations, numerical soloution .