"IIS-RT"-2002. Collection №30-1
On complex resonance vibration systems calculation
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
"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"
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
, of long mo-lecular chains vibrations , of molecules vibration
level , of lattice oscillations , , , mo-lecular acoustics
, quantum systems statistical mechanics , control problems 
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  is devoted to
pure matrix methods; by Kukhta and others  - to finding the recursive
relationships; by Atkinson  - to differential methods; by Palis and
de Melo  - to geometrical methods, by Reiscig and others  - to
qualitative theory, etc. Mitropolsky and Homa  and Cherepennikov 
gave good surveys of solutions obtained with asymptotic methods. Methods
based on perturbation theory are well stated by Giacagrilia  and Dymentberg
. Approaches based on an elastic model presented by mechanical resonance
circuits were described quite completely by Skudrzyk .
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 , Reiscig  and Cherepennikov  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 -, 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.