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S.B. Karavashkin and O.N. Karavashkina

Sergey B. Karavashkin and Olga N. Karavashkina

Special Laboratory for Fundamental Elaboration SELF

187 apt., 38 bldg., Prospect Gagarina, 38, Kharkov 61140, Ukraine

Phone: +38 (057) 7370624

e-mail: selftrans@yandex.ru , selflab@mail.ru

Abstract

This paper is devoted to finding the solution of nonlinear vibrations in a homogeneous elastic line consisting of three elements connected by non-linear elastic constraints. The obtained solution is a functional spectral series whose each harmonic is determined analytically on the basis of solution for the system of equations describing the vibration process in one and the same linear system under the forces depending on the degree of elastic constraint non-linearity and on the vibration amplitude of the lowest harmonics. The obtained solutions are analysed. We reveal that the boundary frequency of each harmonic drops proportionally to the order of harmonic, and resonance spectrum of harmonics of dynamical process contains the spectrum of natural frequencies lower than the natural boundary frequency and the spectrum of frequencies of lower 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 non-linear dynamical process can be extended to the models with nonlinear resistance and to the case of complex-spectrum external force.

Classification by MSC 2000: 37N15; 70G60; 70G70; 70K40; 70K75; 74H45; 74J30; 93B18

Classification by PASC 2001: 05.45.-a; 05.45.Tp; 45.20.-d; 45.50.Jf; 45.90.+t

Key words: wave physics, mathematical physics, theoretical physics, many-body systems, non-linear dynamics, spectrum of non-linear dynamical process

1. Introduction

Vibrations surround us, beginning with natural vibrations of the Earth (in that number seismic) and internal oscillations of molecules - and up to modern communications and transport, any mechanisms and constructions. The safety of these and other constructions, of our lives and activity to a large extent depends on reliability of methods of calculation of vibration systems. These methods, mainly matrix, are known to be far from perfect, so most of these systems are calculated approximately and numerically. They are highly laborious in processing, especially it concerns the lumped systems and systems which we cannot reduce to some simplest basic models.

Most of real-world models are conventionally modelled for calculations by nonlinear systems; this is the most voluminous, complicated and laborious class of problems. "The circumstance that the non-linearity of general equations of the elasticity theory has a double nature causes the following classification of problems of this theory:

1. Linear problems in which the extensions, shears, rotation angles of separate elements are small in comparison with the unity, being the values of the same order…

2. Geometrically non-linear but physically linear systems where the rotation angles of spatial elements well exceed the extensions and shears, and the values of these last allow to use the Hook law…

3. Physically non-linear but geometrically linear problems where the extensions, shears and rotation angles are small in comparison with the unity and comparable in their values, but the conditions of Hook law are violated…

4. Geometrically and physically non-linear problems" [1, p. 262].

Quite serious problems in solution make necessary so detailed classification of dynamical problems. First of all, we still do not know an unified method to solve nonlinear problems. Sometimes successfully, sometimes not so much, such method is substituted by multitude of scattered particular techniques, and each, along with its advantages, brings to the calculation its own difficulties. It is a luck if we know an explicit formula for a set of solutions (an aggregate of motions), - R. Reiscig et al indicate, but not so often this is possible [2, p. 12]; accordingly, not often we can obtain a direct solution of the problem. And when choosing an indirect technique, researchers find that there still does not exist an unified theory of vibrations of strongly nonlinear systems in absence of small parameter and with appearing 'strange' features, even when considering quite simple model systems [3, p. 7]. The narrow applicability of conventional methods also is a hindrance. In particular, "applicability of the Krylov-Bogolyubov method is practically determined not by the approximations convergence when their number growed, but by the asymptotic properties of series with the fixed number of terms of a series and _r  tending to zero" [4, p. 308]. This limits the applicability of indirect methods to a narrow edge area of nonlinear mechanics [2, p. 12]. Additionally, the indirect methods give uncoordinated solutions, but give no idea of the structure of the set of solutions as the whole [ibidem].

It is still impossible to calculate the real frequencies of nonlinear system. The mathematical tool of the theory of series is cumbersome and still allows to calculate only a small number of terms, but does not give a way to express the common term and the sum of these series [5, p. 305]. The resonance conditions and behaviour of the system in the near and in the far from resonance locations are also a great problem [5, p. 309].