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| S.B. Karavashkin, O.N. Karavashkina | |
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Thus, the solutions yielded for the distributed heterogeneous elastic line show that the full pattern of process in a line is not limited by a simple superposition of the direct and inverse waves. The cause is, in particular, that in the first section, the wave reflected from the heterogeneity is shifted in relation to the wave produced directly by the external force, by the angle depending on the second section length xk and on the complex conditions of reflection from the heterogeneity. The same in the third section. When passing to the section having another density, the wave process also transforms dependently on the parameters of this section. Basically, the method as such reveals these distinctions, but it needs the additional study being out of frames of present paper. The results presented here considerably 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 affection of external and internal factors. This was the target of presented study. Conclusions We have revealed that the aperiodical regime of antiphase damping vibrations having been not taken into consideration before has a great effect on the pattern and parameters of vibration process running in an heterogeneous elastic line with lumped parameters. We have found that each section of a heterogeneous line
has its own boundary frequency
In the semi-finite sections of line, there arise the progressive waves having specific phase delays and vibration amplitudes. These phase delays depend non-linearly on the frequency and parameters of related sections of elastic line. In the domain between the external force application point 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 line vibrated aperiodically, in the light section of line, at certain frequencies, the amplitude of progressive wave can vanish. But when the hard section vibrated in the periodical regime, the vanishing is possible at no conditions. We showed that the presented solutions are basic for a number of models similar in the structure of their lumped lines and can be easily transformed into solutions for the related distributed lines. The yielded vibration pattern much differs from the conventional understanding. In particular, 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 region between the external force application point and the heterogeneity transition, the standing complex-structure waves settle. The results of this study 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 solution on infinite one-dimensional elastic lumped-parameters line vibration. IJMEE, 30 (2002), 2, 138 – 154. 2. Karavashkin, S.B. Exact analytical solutions on finite 1-D elastic lumped line vibration. Materials, Technologies, Tools. Journal of National Academy of Sciences of Belarus, 4 (1999), 4, 5-14 (Russian) 3. Karavashkin, S.B. Peculiarities of modelling of forced vibrations in homogeneous elastic lumped lines. Materials, Technologies, Tools. Journal of National Academy of Sciences of Belarus, 5 (2000), 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. 5. Kukhta, K.Ya. and Kravtchenko, V.P. Normal fundamental systems in the vibration theory problems. Naukova Dumka, Kiev, 1973 (Russian) 6. Magnus, K. Vibrations. Introduction to the vibration systems study. Mir, Moscow, 1982 (Russian) 7. Olkhovski, I.I. Theoretical mechanics for the physicists. Nauka, Moscow, 1970 (Russian) 8. Blakemore, J.S. Solid state physics. Metallurgia, Moscow, 1972 (Russian; original edition: Blakemore, J.S. Solid state physics. W.B. Sounders Company, Philadelphia- London- Toronto, 1970) 9. Born, M. and Goeppert, M. The dynamical theory of crystalline lattice. - In: Born, M. The theory of solid body. The Principal Editorial of Technical and Theoretical Literature, Moscow- Leningrad, 1938 (Russian; translated from German, ed. 1915) 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: Control of Oscillations and Chaos (COC 2000, July 2000, Russia), Transactions, 1, 154 (English) 11. Pain, H.J. The Physics of Vibrations and Waves. Mir, Moscow, 1979 (Russian; original edition: John Wiley and Sons, Ltd. London–New York–Sydney–Toronto, 1976) 12. Karavashkin, S.B. and Karavashkina, O.N. The features of oscillation pattern in mismatched finite electric ladder filters. SELF Transactions, 2 (2002), 1, 35-47 |
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0 . Dependently on frequency and parameters of
each section, three vibration regimes may occur in it: 