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NUMERICAL SOLUTION OF NON-LINEAR VIBRATIONS OF A FRACTIONALLY DAMPED CYLINDRICAL SHELL UNDER THE CONDITIONS OF COMBINATIONAL INTERNAL RESONANCE
Last modified: 2017-12-05
Abstract
Yury A. Rossikhin1, Marina V. Shitikova1 and Basem Ajarmah1,2*
1: Voronezh State Technical University
Research Center on Dynamics of Solids and Structures
20-letija Oktjabrja Street 84, 394006 Voronezh, RUSSIA
2:
Correspondent author: e-mail: shitikova@vmail.ru
Keywords: cylindrical shell, free nonlinear damped vibrations, combinational internal resonance, method of multiple time scales, fractional differential equations
ABSTRACT
Non-linear 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 by the procedure based on the reduction of the problem to a system of fractional differential equations. According to the second method, the amplitudes and phases of nonlinear vibrations are estimated from the governing nonlinear differential equations describing amplitude-and-phase modulations for the case of the combinational internal resonance. A good agreement in results is declared. It is also shown that the phenomenon of the internal resonance could 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 mechanisms is analyzed. It is shown that each mode is characterized by its damping coefficient dependent on the natural frequency by the exponential relationship with a negative fractional exponent.
Acknowledgements: This research was made possible by the Grant No. 9.5138.2017/BP as a Government task from the Ministry of Education and Science of the Russian Federation.
1: Voronezh State Technical University
Research Center on Dynamics of Solids and Structures
20-letija Oktjabrja Street 84, 394006 Voronezh, RUSSIA
2:
Correspondent author: e-mail: shitikova@vmail.ru
Keywords: cylindrical shell, free nonlinear damped vibrations, combinational internal resonance, method of multiple time scales, fractional differential equations
ABSTRACT
Non-linear 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 by the procedure based on the reduction of the problem to a system of fractional differential equations. According to the second method, the amplitudes and phases of nonlinear vibrations are estimated from the governing nonlinear differential equations describing amplitude-and-phase modulations for the case of the combinational internal resonance. A good agreement in results is declared. It is also shown that the phenomenon of the internal resonance could 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 mechanisms is analyzed. It is shown that each mode is characterized by its damping coefficient dependent on the natural frequency by the exponential relationship with a negative fractional exponent.
Acknowledgements: This research was made possible by the Grant No. 9.5138.2017/BP as a Government task from the Ministry of Education and Science of the Russian Federation.