SELF

60

S.B. Karavashkin, O.N. Karavashkina

Sergey B. Karavashkin and Olga N. Karavashkina

Special Laboratory for Fundamental Elaboration SELF

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

We will present some results of study of an infinite 1D elastic lumped line with one section of heterogeneity. We yielded these results, using the original non-matrix method to find exact analytical solutions for an infinite system of differential equations. We will present few features important for practical use, caused by the transition of an elastic line section to the antiphase damping regime. We will show the conditions of solutions transformation, when transiting to the models related to this basic, as well as to the related elastic distributed line. The results of this study can be extended to the rotary vibrations of elastic lumped or distributed lines, as well as with the help of original dynamical electromechanical analogy (DEMA) they can be applied to the calculation of electrical filters.

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 lines with lumped and distributed parameters, we showed that a vibration process is not limited by undercritical frequency band (lower than the boundary frequency ₀). Across the overcritical band we can see the aperiodical regime of damping antiphase vibrations; with them the mechanical line behaves as a natural damper.

In a heterogeneous line, the influence of aperiodical process on the vibration pattern much complicates, when the frequency of external excitation exceeded the local boundary frequency _0i  for some sections of a line, while for others the undercritical vibration regime will remain. 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 a 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 at the amplitude and phase vibration characteristics across the undercritical and overcritical bands determined in the analytical form.

These features cannot be completely described by conventional methods. “In conventional methods, the solution of boundary problems is reduced, basically, to determining of the eigenvalues related to the natural frequencies or other parameters of 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 link the systems studying with discrete and continuous mass distribution” [5, pp. 3- 4]. Only in some particular cases we have incomplete, limitedly applicable solutions or mentioning that overcritical vibrations are possible (see e.g. [6, pp. 272- 275], [7, p. 294], [8, p. 109], but they also do not offer the complete analysis of a system, especially in heterogeneous lines.

In this paper we will partly fill in this gap. We will present some results of the analysis carried out for a particular case of an ideal elastic line with one transition of heterogeneity, expecting that, using the method on whose basis we yielded the solutions analysed in this paper, we will be able to extend the main regularities of phenomena to the more complex heterogeneous elastic lines.