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S.B. Karavashkin, O.N. Karavashkina

Thus the solutions obtained for the heterogeneous elastic distributed line show that the full pattern of the process in a line is not limited by a simple superposition of the direct and reverse waves. The reason is, in the first section the wave reflected from the heterogeneity is shifted relatively to the wave produced directly by the external force, by the angle depending on the second section length x_k and on the complex conditions of reflection from the heterogeneity. When transiting to the section having another density, the wave process also transforms dependently on the parameters of this section. Basically, this method reveals the indicated distinctions, however it needs the additional investigation being out of frames of present paper.

The results presented here essentially extend the understanding of variety of vibration process distinctions in the heterogeneous elastic lines, if noting the overcritical regime, and establish the most important and typical transformations of the vibration pattern under external and internal factors action. This was just the target of presented investigation.

5.  Conclusions

In this paper we have revealed that the aperiodical regime of anti-phase damping vibrations, having been not taken into consideration before, essentially effects on the pattern and parameters of vibration process running in a heterogeneous elastic lumped line.

We have observed that each section of a heterogeneous line has its own boundary frequency ₀. Dependently on frequency and parameters of each section, three vibration regimes may occur in it:

• the periodical regime of non-damping vibrations;
• the aperiodical regime of vibrations, fast-damping along the line;
• the transient, critical regime.

In the semi-finite sections of line there arise the progressive waves having specific phase delays and vibration amplitudes. These phase delays depend nonlinearly on the frequency and parameters of corresponding sections of an elastic line. In the region between the application point of external force and the heterogeneity transition, there form the complex-form standing waves being the result of superposition of the waves having different amplitudes and delay phases. When the hard section of a line vibrated aperiodically, in the light section of line, at certain frequencies, the amplitude of progressive wave can vanish. But when the hard section vibrates in the periodical regime, the vanishing is possible at no conditions.

We showed that the presented solutions are base for a number of models analogous by structure of their lines with lumped parameters and easy transform into solutions for the lines with distributed parameters. The obtained vibration pattern essentially differs from the conventional concept. Particularly, in a distributed line, when reflecting from the heterogeneity transition, the wave amplitude and phase gain the dependence on the external force frequency. In the section between the external force application point and the heterogeneity transition, the standing complex-structure waves settle.

The results of this investigation can be extended to the torsion vibrations of elastic rods, and with the help of dynamical electromechanical analogy DEMA they can be applied to find the solutions for the electric filters.

 References:

1. Karavashkin, S.B. Exact analytical solutions on infinite one-dimensional elastic lumped-parameters line vibration. Materials, Technologies, Tools. Journal of National Academy of Sciences of Belarus, 4 (1999), 3, pp.15-23 (Russian)

2. Karavashkin, S.B. Exact analytical solutions on finite one-dimensional elastic lumped-parameters line vibration. Materials, Technologies, Tools. Journal of National Academy of Sciences of Belarus, 4 (1999), 4, pp.5-14 (Russian)

3. Karavashkin, S.B. Peculiarities of modelling of forced vibrations in homogeneous elastic lumped-parameters lines. Materials, Technologies, Tools. Journal of National Academy of Sciences of Belarus, v.5, #3, 2000, pp.14-19 (Russian)

4. Karavashkin, S.B. Peculiarities of inclined force action upon one-dimensional homogeneous elastic lumped line. arXiv, Los Alamos, \#math-ph/0006028.

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10. Karavashkin, S.B. Refined method of electric long lumped-parameters lines calculation on the basis of exact analytical solutions for mechanical elastic lines; in: Transactions of international conference Control of Oscillations and Chaos (COC 2000, July 2000, Russia), 1, p.154 (English).

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12. Karavashkin, S.B. and Karavashkina, O.N. The features of oscillation pattern in mismatched finite electric ladder filters. SELF Transactions, 2 (2002), 1, pp.35 - 47