Nonlinear damped vibrations of such thin-walled structures as plates and shells subjected to the different conditions of the internal resonance are investigated. Its viscous properties are described by Riemann-Liouville fractional derivative. The displacement functions are determined in terms of eigen functions 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. It is shown that the phenomenon of internal resonance can be very critical, since in thin plates and shells internal resonance of the type 2:1, 1:1, 3:1 and additive and difference combinational resonances are always present. The influence of viscosity on the energy exchange mechanism is analyzed. It is shown that viscosity may have a twofold effect on the system: a destabilizing influence producing unsteady energy exchange, and a stabilizing influence resulting in damping of the energy exchange mechanism.
This research is supported by the Grant No. 7.22.2014/K as a Government task from the Ministry of Education and Science of the Russian Federation.