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S.B. Karavashkin, O.N. Karavashkina

Sergey B. Karavashkin and Olga N. Karavashkina

Special Laboratory for Fundamental Elaboration SELF

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

Phone: +38 0572 276624; e-mail: sbkarav@altavista.com

In this paper we will present some results of investigation of an infinite 1D elastic lumped line having one section of heterogeneity. We have obtained these results, using the original non-matrix method to find exact analytical solutions of the infinite system of differential equations. We will present few features important for the practical use, conditioned by the transition of an elastic line section to the anti-phase damping regime. We also will consider the conditions of solutions transformation, when transiting to the models related to basic, as well as to the related elastic distributed line. The results of this investigation can be extended to the rotary vibrations of lumped and distributed elastic lines, as well as with the help of original dynamical electromechanical analogy (DEMA) they can be applied to the electrical filters calculation.

Keywords: Mathematical physics; Wave physics; Nonlinear dynamics; ODE; Many-body theory; Heterogeneous elastic lines; Finite deformation; Oscillation theory; Dynamical systems

Classification by MSC 2000: 34A34; 34C15; 37N05; 37N15; 70E55; 70K30; 70K40; 70K75; 70J40; 74H45.

Classification by PASC 2001: 02.60.Lj; 46.25.Cc; 46.15.-x; 46.40.Fr

1. Introduction

In [1] – [4], on the basis of ideal elastic finite and infinite, lumped and distributed lines, we showed that the vibration process is not limited by an undercritical frequency band (lower than critical frequency ₀). At the overcritical band we can see the aperiodical regime of damping anti-phase vibrations; with them the mechanical line behaves as a natural damper.

In a heterogeneous line, the influence of aperiodical process on the vibration pattern substantially complicates, when the frequency of external excitation exceeded the local critical frequency _0i  for some sections of a line, while for the others the undercritical vibration regime retains. It leads to the complex amplitude and phase transitions whose pattern depends both on the local parameters of section and on the features of particular line as the whole. In this case, to analyse completely a heterogeneous line, it is deficient to know the natural frequencies and eigenfunctions. It appears important to know an integral pattern of vibration process running with the amplitude and phase vibration characteristics at the undercritical and overcritical bands determined in the analytical form.

These features cannot be completely described by the conventional methods. ''In the vibration theory, the solution of boundary vibrations reduces basically to the determination of eigenvalues connected with natural frequencies or other parameters of the studied system and to finding the eigenfunctions (of vibration forms). If the eigenvalues and eigenfunctions have been found, we can think the boundary problem solved... At present a great amount of approximate methods has been developed for finding the eigenvalues, but they all are quite laborious, give only the first eigenvalues, and the main, do not unite the studying of systems with discrete and continuous mass distribution'' [5, pp.3 – 4]. Only in some particular cases we have incomplete, limitedly applicable solutions or mentionings that the overcritical vibrations are possible (see, e.g., [6, pp.272 – 275], [7, p.294], [8, p.109]). However they also are deficient to analyse the system completely, especially in heterogeneous lines.

In this paper we will partly fill this gap in. We will present some results of the analysis carried out for the particular case of an ideal elastic line with one heterogeneity transition. With it we will suppose that the main regularities of these phenomena may be extended to the more complex heterogeneous elastic lines, when using the method on whose basis we have obtained the solutions analysed in this paper.