On complex resonance vibration systems calculation

S.B. Karavashkin and O.N. Karavashkina, Special Laboratory for Fundamental Elaboration SELF e-mail: selflab@go.com, selflab@mail.ru http://angelfire.lycos.com/la3/SELFlab/index.html http://angelfire.lycos.com/la3/selftrans/index.html ( sb30-1e.zip)

Basing on exact analytical solutions obtained for semi-finite elastic lines with resonance sub-systems 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 vibra-tion 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 ana-lytical 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 mo-lecular chains vibrations [1], of molecules vibration level [4], of lattice oscillations [5], [6], [7], mo-lecular acoustics [8], quantum systems statistical mechanics [9], control problems [10] and so on.
Among the multitude of approaches to the solution of these problems, one can mark out "such well-known methods of vibration theory as perturbation theory methods, averaging method, analyti-cal methods of slow and fast motions separation etc." [10, p.25]. 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 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].
Despite a broad spectrum of approaches, all these methods are qualitative, approximate or nu-merical. "The presence of singular boundaries in most of practical problems does not offer us to construct the analytical solution of differential equations, and numerical methods became the only possible means to yield quite exact and detailed results" [19, p.12]. "Even for a simplest case of hy-drogen molecule Н2 , the exact quantum-mechanical calculation of constant quasi-elastic force is a laborious mathematical problem, and for more complicated cases the force constants calculation is practically unrealisable by means of sequential quantum-mechanical techniques" [4, p.12]. "The other difficulty connected with the collective motions method is, it gives no possibility to determine the collective motion nature, proceeding from the form of Hamiltonian. We have to guess suitable collective variables and then to check, whether the Hamiltonian divides into collective and interior parts" [9, p.120].
Giacagrilia [17], Reiscig [14] and Cherepennikov [16] gave quite complete analysis of problems arising with the conventional approaches to the multiresonance models investigation. Particularly, "the old problem is still open. Up to now no available 'modern' methods make possible to calculate the real frequencies of a nonlinear system. This problem stays unsolved for applications, because in approximations by series, converging or only formal, only finite and, generally speaking, little num-ber of terms can be calculated. We still cannot find a way to express the common term and the sum of these series" [17, p.305]. Furthermore, "to make the series converging, sometimes we have to presume that the differential equations parameters determining the degree of nonlinearity have quite small module. By this reason the indirect technique is often applicable only in the narrow edge do-main of nonlinear mechanics. The other demerit of these techniques is, they enable to obtain quite accurate information about the separate solutions, but give no idea about the structure of solutions family as a whole" [14, p.12]. Giacagrilia confirms this last: "The other problem of a great interest is the better understanding of solution 'in the near, in the far and at resonance conditions'. When we have a real process of resonance lock-in, and which definition of the system resonance is prefer-able?" [17, p.309]. "Exact analytical methods are preferable in the analysis, however obtaining the analytical formulas of solution even for comparatively simple differential equations entails great dif-ficulties sometimes" [16, p.10].
In the light of indicated demerits of conventional methods, Skudrzyk has presented the most exact qualitative pattern. According to his approach, "any homogeneous system, either monolithic or consisting of homogeneous parts and loading masses, can be rigorously presented in the form of canonical scheme, specifically, of infinite number of sequential (mechanical) circuits connected in parallel, one for each form of natural vibrations" [1, p.317]. However Skudrzyk's application of ma-trix methods to solve the systems of differential equations for the systems he modelled disabled him to describe the pattern of processes analytically, since, as is known, for complex elastic systems the matrix method offers only numerical solutions. The analysis in matrix writing of vibration is practi-cally impossible in analytical form. This demerit inherent in the most of conventional methods did not offer Skudrzyk to develop the introduced concept for the case of multiresonance elastic subsys-tems, in which the aggregate of subsystem resonance frequencies is determined not by the ensemble of mechanical resonance circuits, but by the integral multiresonance mechanical subsystem that forms all the gamut of subsystem resonances.
Now having the exact analytical solutions presented in [20]-[23], we have a scope to get over a number of problems in the resonance circuits method and to determine exact analytical solutions for some elastic mechanical systems having multiresonance subsystems.
In this paper we will consider the simplest case - a semi-finite homogeneous 1D system with rigidly fixed end elements of resonance subsystems. Though this problem is quite particular, it is used quite often in the engineering practice. Specifically, the problems of vibrant elastically con-nected rigid blocks containing some substructure of elements elastically connected between them-selves and with the block are reduced to this case. Furthermore, we will suppose that the described method may be extended to the finite and heterogeneous elastic lines with resonance subsystems. The only, we will complicate the subsystem structure, presenting it as an elastic finite line with n masses equivalent to n circuits. Again, we will suppose that this method is easily extended to the case of a number of aforesaid type subsystems connected in parallel. In this way we will reduce the model in its generality to that investigated by Skudrzyk, but with the higher level of resonance sub-system structure.