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S.B. Karavashkin, O.N. Karavashkina

On complex resonance vibration systems calculation

S.B. Karavashkin and O.N. Karavashkina

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

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

 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 Skudrzyk’s concept.

We will obtain a good qualitative agreement of our theoretical results with Professor Skudrzyk’s 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].