V.2 No 1

49

On complex resonance vibration systems calculation

Despite a broad spectrum of approaches, all these methods are qualitative, approximate or numerical. “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 hydrogen molecule H₂ , 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 number 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 boundary domain of nonlinear mechanics. The other demerit of these techniques is, they enable us 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 better understanding the 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 preferable?” [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 difficulties 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 matrix methods to solve the systems of differential equations for the systems he modelled did not offer 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 practically 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 subsystems, in which the assemblage 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 the rigidly fixed end elements of resonance subsystems. Though this problem is particular enough, it is used quite often in the engineering practice. Specifically, the problems of vibrant elastically connected rigid blocks containing some substructure of elements elastically connected between themselves 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 subsystem structure.