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S.B. Karavashkin and O.N. Karavashkina

5. Conclusions

We have revealed that generally the sought solution of the considered problem can be presented as a spectral functional series whose each harmonic is the solution of a linear system of equations for an elastic line with the stiffness coefficient equal to the linear term of expansion of this parameter into a power series in the amplitude of constraint deformation. The degree of non-linearity of an elastic constraint and the vibration amplitude of lower harmonics effects the vibration pattern of each harmonic. With the growing number of the harmonic its boundary frequency diminishes proportionally to the harmonic number. The resonance frequency spectrum of each harmonic contains the spectrum of natural frequencies located lower than the boundary frequency, and the spectrum of the introduced resonances of lower harmonics located between the natural boundary frequency and the boundary frequency of the first harmonic.

We have ascertained that out of the resonance band the harmonics amplitude decreases as its number grows, but in the case of ideal systems it does not effect the resonance amplitude. Noting that in the general solution the density of resonance frequencies grows up to infinity with the frequency of process tending to zero, it will be more efficient to take the line resistance into account at once, to describe the dynamical processes more accurately. The reason is that with the finite resonance amplitudes being typical for a resistant line, the growth of density of resonance frequencies is compensated by the fast decrease of their amplitude.

We have showed that this technique of the recurrent finding of the harmonics of a dynamical process can be extended to the models with a non-linear resistance and for the case of the external force having a complex spectral composition.

Acknowledgements

We are grateful to Dr. Yuri V. Michlin of Kharkov Polytechnic University (Ukraine) and Dr. Yuri L. Bolotin of Kharkov Physical-Engineering Institute (Ukraine) for their valuable comment of the basic aspects of the method presented in this paper, in particular, of the possibility to extend this method to the models described by the Duffing equation, during the workshop at the Kharkov Polytechnic University in June 2002.

We would like to express our thanks to Dr. John Harrold who kindly provided us the reference [22].

We wish to thank deeply Mrs. Elena North and Mr. Adam North, Oxford (UK) for their great help in improving the English language of this paper.

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