Nonlinear vibration of fractional viscoelastic plate: Primary, subharmonic, and superharmonic response

Nonlinear vibrations of fractional viscoelastic plate with Kelvin-Voigt fractional order constitutive relationship is investigated in this paper. Based on the Kirchhoff hypothesis for thin shells and von Karman's assumption, the structural dynamic of the plate is modeled by using the Newton's second law. The nonlinear coupled equations of motion are obtained by introducing the Airy stress function and the Galerkin method is used to discretize the partial differential equations. Analytical solutions for fully simply supported and clamped plate are obtained by using the method of multiple scales and finally the equations of amplitude-frequency and phase-frequency are obtained for primary, super-harmonic and sub-harmonic resonance. The obtained amplitude-frequency and phase-frequency equations are used for studying the effects of excitation, fractional parameters and nonlinearity on the frequency responses of the fractional viscoelastic plate and finally some conclusions are outlined.