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Abstract

This paper shows a solution for non-linear vibrations in a homogeneous elastic line consisting of three elements connected by non-linear elastic constraints. The solution obtained is a functional spectral series, each harmonic of which is determined analytically on the basis of a system of equations describing the vibration process when the same linear system is acted on by a series of forces. The process forces depend on the degree of elastic constraint non-linearity and on the vibration amplitude of the lowest-order harmonics. We show that the boundary frequency of each harmonic decreases in proportion to the order of harmonic, and the resonance spectrum of harmonics of the dynamic process contains a spectrum of natural frequencies lower than the natural boundary frequency and the spectrum of frequencies of lower-order harmonics located between the natural boundary frequency and the boundary frequency of the first harmonic. It is shown that the method of recurrent determination of the spectrum of a non-linear dynamic process can be extended to models with non-linear resistance and to the case of a complex-spectrum external force.

Keywords

wave physics mathematical physics theoretical physics many-body systems non-linear dynamics spectrum of non-linear dynamical process

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Investigation of Elastic Constraint Non-Linearity

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