AN INTERNAL NONLINEAR RESONANCE IN AN OSCILLATION SYSTEM WITH TWO DEGREES OF FREEDOM

DOI: https://doi.org/10.15407/rpra27.01.017

Yu. V. Kornienko, L. V. Stulova, D. S. Masalov

Abstract


Subject and Purpose. The paper is concerned with the behavior of a nonlinear dynamic system that has two degrees of freedom and whose joint nonlinearity is established by all the nonlinear coupling between the degrees of freedom. The purpose is to find out if the Krylov—Bogolyubov—Mitropolsky (KBM) method is applicable to a system of partial differential equations.

Methods and Methodology. The consideration of the problem is by the Krylov—Bogolyubov—Mitropolsky method in the first approximation. Then the results are treated using numerical methods.

Results. An electromechanical system with two degrees of freedom and a known parametric resonance has been studied using the Krylov—Bogolyubov—Mitropolsky method in the first approximation. The phase space of the system has been described. It has been shown that the obtained solution covers an energy periodic transfer between the two degrees of freedom. The difference between the considered oscillation system and its analogs discussed in the literature lies in that the considered circuit is parametrically excited by an internal force rather than external one. In a similar system of two circuits connected through a diode, the coupling includes a linear component. In the system of present concern, the coupling is all-nonlinear.

Conclusion. The obtained results are of interest for the research into internal nonlinear resonances between degrees of freedom in an oscillation system that has two degrees of freedom and whose joint nonlinearity is due to all the nonlinear coupling between the degrees of freedom. The considered system can serve a test example in the development of programs implementing the Krylov—Bo- golyubov—Mitropolsky method as applied to an oscillation system with numerous degrees of freedom and a small nonlinearity.

 

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Keywords


nonlinear dynamic system with two degrees of freedom, internal nonlinear resonance, Krylov—Bogolyubov—Mitropolsky method, nonlinear oscillations

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