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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 (0572) 276624; e-mail: selflab@go.com, selflab@mail.ru

Abstract

This paper is devoted to finding the solution of non-linear 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 are revealing 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 non-linear 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

"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].

Due to the major difficulties in their solution, the dynamical problems should be classified in details. In [2]- [9] we have proved that the exact and complete analytical solutions still were not obtained even for linear models. This naturally has an effect on solving the problems of non-linear dynamics. "The practical issues are solved relatively easy, if an explicit formula for a set of solutions (of the assemblage of motions) is known. However such possibility is very rare, so in most cases we inevitably have to refuse applying the explicit formula in solving quite important issues. For example, this takes place in the problems that are crucial in studying the behaviour of the considered dynamical system" [10, p. 12]. "The main problem lies in the absence of general vibration theory of essentially non-linear systems in the small parameter absence and in the "strange" features appearing even in consideration of rather simple modelling systems, such as attractor or chaos" [11, p. 7]. In particular, the Krylov-Bogolyubov method applicability is practically determined not by the approximations convergence with their number growing, but by the asymptotic properties of a series with the fixed number of terms of a series and _r tending to zero" [12, p. 308]. "For this reason the indirect method is often applicable only in a narrow edge domain of non-linear mechanics. The other shortcoming of these methods is that they offer quite accurate information of the separate solutions, but give no idea on the structure of an assemblage of solutions as the whole" [10, p. 12].