Numerical Study of Nonlinear Vibrations of Fractionally Damped Cylindrical Shells Under the Additive Combinational Internal Resonance

In the present chapter, nonlinear damped vibrations of a cylindrical shell embedded into a fractional derivative medium are investigated for the case of the combinational internal resonance, resulting in modal interaction, using two different numerical methods with further comparison of the results obtained. The damping properties of the surrounding medium are described by the fractional derivative Kelvin-Voigt model utilizing the Riemann-Liouville fractional derivatives. Within the first method, the generalized displacements of a coupled set of nonlinear ordinary differential equations of the second order are estimated using numerical solution of nonlinear multi-term fractional differential equations. Adopting the second procedure based on the generalized method of multiple time scales, the governing nonlinear differential equations describing amplitude-and-phase modulations for the case of the combinational internal resonance have been derived, and the envelopes of amplitudes have been constructed.