SELF |
48 |
S.B. Karavashkin, O.N.
Karavashkina |
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On complex resonance
vibration systems calculation Sergey B. Karavashkin and Olga N. Karavashkina Special Laboratory for Fundamental Elaboration SELF E-mail: selftrans@yandex.ru , selflab@mail.ru Basing on exact analytical solutions obtained for semi-finite elastic lines
with resonance subsystems having the form of linear elastic lines with rigidly connected
end elements, we will analyse the vibration pattern in systems having such structure. We
will find that between the first boundary frequency for the system as a whole and that for
the subsystem, the resonance peaks arise, and their number is equal to the integer part of
[(n 1)/2] , where n is the number of subsystem elements. These
resonance peaks arise at the bound between the aperiodical and complex aperiodical
vibration regimes. This last regime is inherent namely in elastic systems having resonance
subsystems and impossible in simple elastic lines. We will explain the reasons of
resonance peaks bifurcation. We will show that the phenomenon of negative measure of
subsystems inertia arising in such type of lines agrees with the conservation laws. So we
will corroborate and substantiate Professor Skudrzyks concept. We
will obtain a good qualitative agreement of our theoretical results with Professor
Skudrzyks experimental results. Keywords:
Many-body theory, Wave physics, Complex resonance systems, ODE. 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 The classical vibration theory is based on solving the differential
equations and on joining the solutions for different parts of a system regarding
continuity conditions. Any negligible change of the system form makes necessary to
calculate it all anew. But out of any relation to the calculation difficulty, one should
note that the high accuracy of classical theory is illusory. Materials never are
absolutely homogeneous or isotropic, and natural frequencies and vibration distributions
usually perceptibly differ from those what the theory gives, especially at high
frequencies [1,
p.317]. At the
same time, multifrequent resonance systems are interesting by
their applications to analytical and celestial mechanics, to Hamiltonian dynamics,
theoretical and mathematical physics [2, p.173]. Some of these problems
are the problem of discrete-continual elastic system [3], of long molecular chains vibrations
[1], of molecules vibration level [4],
of lattice oscillations [5], [6], [7], molecular acoustics [8], quantum systems statistical
mechanics [9], control problems [10] and so on. Among
the multitude of approaches to these problems solution, one can mark out such well-known methods of vibration theory as perturbation theory
methods, averaging method, analytical methods of slow and fast motions separation
etc. [10, p.45]. Each of them has an ample literary basis. Particularly, the
investigation by Tong Kin [11] is
devoted to pure matrix methods; by Kukhta and others [3] to finding the recursive
relationships; by Atkinson [12]
to differential methods; by Palis and de Melo [13] to geometrical methods, by
Reiscig and others [14] to
qualitative theory, etc. Mitropolsky and Homa [15] and Cherepennikov [16] gave good surveys of
solutions obtained with asymptotic methods. Methods based on the perturbation theory are
well stated by Giacagrilia [17]
and Dymentberg [18]. Approaches
based on an elastic model presented by mechanical resonance circuits were described quite
completely by Skudrzyk [1]. |
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