5. C onclusions
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 nonlinearity of an elastic
constraint and the vibration amplitude of lower harmonics affects 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 boundary frequency, and
the spectrum of introduced resonances of lower harmonics located between the natural
boundary frequency and 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, in order
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
nonlinear resistance and for the case of external force having a complex spectral
composition.
Acknowledgements
We are grateful to Dr. Yuri V. Mikhlin of Kharkov
Polytechnic University (Ukraine) and Dr. Yuri L. Bolotin of Kharkov Physical-Engineering
Institute (Ukraine) for their valuable comments 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 thank deeply Mrs. Lena North and Mr. Adam
North, Oxford (UK) for their great help in improving the English language of this paper.
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