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 Skudrzyks
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. |
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