Analysis of Non-linear Vibrations of a Fractionally Damped Cylindrical Shell Under the Conditions of Combinational Internal Resonance

  • Yury Rossikhin
  • Marina ShitikovaEmail author
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 343)


Non-linear damped vibrations of a cylindrical shell subjected to the different conditions of the combinational internal resonance are investigated. Its viscous properties are described by Riemann-Liouville fractional derivative. 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 time 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 a circular cylindrical shell the internal additive and difference combinational resonances are always present. 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.


Cylindrical shell Free nonlinear damped vibrations Combinational internal resonance Fractional derivative Method of multiple time scales 



This research was made possible by the Grant No. 7.22.2014/K as a Government task from the Ministry of Education and Science of the Russian Federation.


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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Research Center for Wave Dynamics of Solids and StructuresVoronezh State University of Architecture and Civil EngineeringVoronezhRussia

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