A New Approach for Studying Nonlinear Dynamic Response of a Thin Fractionally Damped Cylindrical Shell with Internal Resonances of the Order of $$\varepsilon $$

Abstract

Non-linear vibrations of a cylindrical shell embedded in a fractional derivative viscoelastic medium and subjected to the different conditions of the internal resonance of the order of \(\varepsilon \), where \(\varepsilon \) is a small value, are investigated. The displacement functions are determined in terms of eigenfunctions of linear vibrations. The procedure resulting in decoupling linear parts of equations is proposed with the further utilization of the method of multiple scales for solving nonlinear governing equations of motion, in so doing the amplitude functions are expanded into power series in terms of the small parameter and depend on different time scales. The influence of viscosity on the energy exchange mechanism is analyzed. It is shown that each mode is characterized by its damping coefficient connected with the natural frequency by the exponential relationship with a negative fractional exponent. Comparison of the results obtained in this paper for the nonlinear shallow cylindrical shell in the cases of the internal resonance of the order of \(\varepsilon \) with those for a nonlinear plate, the motion of which is described also by three coupled nonlinear equations in terms of three displacements, reveals the fact that the shell equations could produce much more diversified variety of internal resonances, including combinational resonances of the additive and difference types, than the plate equations.