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Brief analysis
of the state of the art The principal feature of vibration processes is a
great diversity of their forms with not so large number of the effecting
parameters – such as the mass of system elements, stiffness of constraints,
type of dissipative forces and so on. To outline the difference between the
conventional approaches and results that we present in [1], [2], and [3], and that we have obtained by our new original
non-matrix method, let us briefly analyse the state-of-the-art of this branch
of the wave physics. From all the diversity we can pick a few basic
conventional methods on which the entire spectrum of variations is built. To construct the mathematical model itself, two
methods are used mainly: Lagrange method based on the energy balance
and the method of forces balance at the elastic system elements. The
first method advantage is a direct possibility to state the problem on the generalised
co-ordinates for systems having many degrees of freedom. It makes this method
profitable in the basic research of complex vibration processes. While the
second method is more visual and simple when constructing the specific models
of elastic systems, so it is used more often in applied developments. Both
methods give similar results, so usually there are not considerable
difficulties in their use at the stage of mathematical modelling. The
difficulties arise when seeking the solutions. The conventional methods to seek the analytical
solutions are based, as a rule, on one of three approaches to the solving the
problem of a vibrant elastic system: ·
With the matrix techniques direct use (see, e.g., [4] – [10]); ·
Using the techniques to solve the integral equations (see, e.g., [11] – [13]); ·
Using the indirect techniques based on the revelation of the
regularities in specific modelling systems of differential equations (see,
e.g., [4], [14], [15]). The solving of forced vibration problems is
conventionally based on the studying of a homogeneous system of differential
equations corresponding to free vibrations in an elastic model. The reason is
the coincidence of the resonance frequencies for forced and free vibrations.
So begin our brief analysis with such type of methods. Though the matrix method is thought
conventionally analytical, in fact it is numerical, since the main vibration
parameters can be obtained in general case only in numerical form. “The
solving of these equations requires to find first the related determinants;
in their written form and any considerable number of lines it is practically
impossible or requires tremendous work” [6, p. 98]. “It goes without
saying that a direct finding the determinant of 4n+4 elements (4n
conjugation conditions plus 4 boundary conditions, – authors) would be
impossible and helpless, as its root values might be calculated only
numerically, not literally” [6, pp.278 – 282]. These computations are so
laborious that going this way, one has to use all computers of his department
at once to realise the parallel computing method developed just for this
purpose. Furthermore, this demerit essentially limits the matrix techniques
applicability to calculate the elastic systems having a finite number of
elements. It also causes the development of such indirect techniques as
described, e.g., in [6, p.161], [14, pp.95 – 99]. These techniques are
grounded on some regularities revealed for the specific mathematical models,
so they are essentially less applicable. None the less, for example, K.
Magnus, when studying an infinite elastic lumped line, has obtained a
solution not only for the subcritical band, which can be revealed by the
matrix techniques, but also for the overcritical band, which cannot be
revealed by the matrix techniques, because the natural frequencies are absent
at this band. However, to find the natural frequencies by these techniques,
one has to solve the power-type equation, for which one cannot conventionally
yield analytically the solutions higher than of the fourth power. True, for today there is widely used a practice to
extend the boundary conditions being true for distributed lines to those
lumped. In particular, this procedure is used in [6, pp.149 – 150] and [11,
pp.48 – 49], when studying a shaft having n fitted disks and a beam
having n lumped masses relatively. None the less, as we proved in [1],
when seeking the analytical solutions for a lumped line, one has to take into
account the different boundary conditions for distributed and lumped lines. Almost nobody of the researches paid a due attention
to the problem of the boundary conditions, but just this is the reason
of many difficulties in the wave physics. Particularly, to write the
modelling system of differential equations on generalised co-ordinates
jointly with the initial and boundary conditions is an alphabet of the
mathematical modelling. At the same time, applying this system of
differential equations to any specific lumped model, we see that the system
itself has already taken into account both initial and boundary conditions.
It means that one reiterates the statement of problem. Should this
reiteration be a duplication, there would be no problem. But as we showed in
[1], e.g., for a finite line with unfixed ends, the boundary element
vibration amplitude will be not maximal, as we used to think it for
distributed lines. The vibrations of this element will delay by some phase
depending on the action frequency of an external force and on the line
discreteness. At the limit passing to a distributed line, this phase actually
vanishes. But using the extreme condition to solve the problem with discrete
elements, we automatically introduce to the solution an error increasing with
the growing frequency. In case of more complicated boundary conditions, the
solution errors will considerably increase, as generally the solution is not
determined by the superposition of the direct and reflected waves, but as a
rule is a result of multiple complex reflections from the boundaries. One
cannot take it into account by a simple summing, the more that there will be
particular features for a closed line, for a line having a heterogeneity
transition they will be other, for a kinked elastic lumped line having
inequal longitudinal and transversal stiffness coefficients they will be
third, and so on. One more problem arises when using the matrix
techniques. “The problem is solvable, if the forces and moments have been stated,
but for a rotating shaft neither forces nor moments are stated, only their
expressions through the shaft masses and moments of inertia and their
deflections and inclinations at the fixation places” [6, p.167]. In this
case, considering the system of equations composed for the boundaries of all
homogeneous sections of a shaft, Krylov comes to the following conclusion:
the resulting “expression will be so complicated that it will be possible to
solve this equation only by way of sequential approximations” [6, p.173].
This demerit is inherent in the indirect techniques too. In most cases the
solutions can be found only at the given amplitude and phase values – usually
the boundary elements of a line. But in practical cases to give the external
force and moments determining the initial vibration parameters is more
important. And this problem is solvable in an exact analytical form only for
some simple cases and under all above limitations. Regarding the conventional methods describing the
infinite models, note that this area is even less clear. The solutions are
mostly limited by finding the phase delay along an infinite elastic line and
by the dispersion characteristics (see, e.g., [14, p.169], [15, pp.106 – 107]). At the
same time Pain writes: “The most ultrasonic frequency produced at present is
about 10 times less than that critical. In the band from 5.1012 to
1.1013 Hz one can expect many interesting experimental results”
[14, p.169]. Actually, the results presented in [2] show that in
infinite elastic systems both forced and free vibrations are possible. Note
that they can exist not only at the subcritical band (lower than critical
vibration frequency), as it takes place in finite lines, but also at the
overcritical band, when all the vibration energy localises in a small
vicinity of the external force application point, not distributed along an
elastic line. In this case the energy accumulated in an infinite line can be
finite. These conclusions are especially important in the view that many
physical processes, e.g., in crystals, can be modelled just by an ideal
infinite model. Moreover, this modelling is valid also for finite elastic
lines loaded on the wave impedance. When using the method to solve the integral
equations, the results are the same. “If an integral equation of the vibrant
system with a continually-discrete mass distribution was constructed, then,
using Fredholm method, one can find the eigenvalues. Actually, to find the
eigenvalues, one needs to construct prematurely Green functions and to
determine the complicated coefficients Cn, especially when
finding the high-order eigenvalues. Note also that there is a whole class of
the boundary problems for which one cannot construct so named generalised
Green function” [11,
p.38]. Using this method, Kukhta and Kravtchenko have reduced a specific
problem on forced vibrations to the numerical modelling. “To find the roots
(frequencies of the considered problem), their lower boundary is estimated,
and then, combining the chord technique and choise technique conveniently
realisable on computer in case of both simple and multiple roots, a necessary
quantity of frequencies in an order of their increase is determined” [11,
p.47]. Just as in matrix techniques, in this technique all
above demerits are inherent, as it “works” only at the known eigenvalues
which one must determine numerically. But not only demerits join these
techniques. In essence, they are variations of the same approach. As a
result, they both are able only predict in an analytical form that “if the
damping is small, each amplitude will have s resonance peaks at s
frequencies A special place is given at present to asymptotic
techniques, qualitative techniques, differential techniques etc. Although
they are approximate, for today they are the only enabling to obtain solutions
for the models with resonance subsystems, nonlinear models and others. “The
classical vibration theory is based on the solution of differential equations
and on joining the solutions for different parts of the system, basing on the
continuity conditions. Any negligible change of the system form causes the
necessity to compute all anew. But out of any connection with the greatly
laborious computations, note that the high accuracy of the classical theory
is illusory. Materials never are absolutely homogeneous or isotropic, and
natural frequencies and vibration distributions usually considerably differ
from those which theory gives, especially at high frequencies” [17, p.317]. At the same time, “multifrequent resonance systems
are interesting by their applications to analytical and celestial mechanics,
in Hamiltonian dynamics, in theoretical and mathematical physics” [18, 173]. In
particular, the problem of discrete-continual elastic system vibration [19], of long molecular
chains vibrations [17, p.317], of molecules vibration levels [20], of crystal
lattice oscillations [21],
[15], [22], problems in molecular
acoustics [23], in quantum systems statistical mechanics [24], control problems [25] etc., etc. From the multitudinous approaches to the solving
these problems, one can emphasise “such widely known vibration theory
techniques as the perturbation theory, averaging technique, analytical
techniques to separate slow and fast motions and so on” [25, p.45]. Each of
these techniques has an extensive basis of sources. Tong Kin [26, p.45] devotes his
investigation to pure matrix techniques; Kukhta and others [19] – to the
solving by finding the recurrent relations; Atkinson [27] – to differential
techniques; Palis and de Melo [28]
– to geometric techniques; Reiscig and others [29] – to the qualitative
theory. A good survey of solutions obtained by asymptotic techniques
Mitropolsky and Homa [30]
and Cherepennikov [31]
gave. The techniques based on the perturbation theory are stated well by
Giacagrilia [32]
and Dymentberg [33].
The approaches based on the elastic model presentation by mechanical
resonance circuits are described quite completely by Skudrzyk [17]. Despite the wide scope of approaches to the studied
problem, all these techniques are qualitative, approximative or numerical.
“The presence of irregular bounds in the most of practical problems disables
us to construct an analytical solution of differential equations, and
numerical techniques became the only possible means to obtain quite accurate
and detailed results” [34,
p.12]. “Even for the simplest case of hydrogen molecule H2 the
exact quantum-mechanical calculation of the constant quasi-elastic force is a
laborious mathematical problem, and for more complex cases the force
constants calculation with the help of sequential quantum-mechanical
techniques is practically unrealisable at all” [20, p.12]. “Another
difficulty connected with the collective motions technique is that it does
not allow to determine the collective motion nature, proceeding from the form
of Hamiltonian. We must guess the proper collective variables and then check,
does the Hamiltonian divide into the collective and interior parts” [24, p.120]. Enough complete analysis of the problems arising in
the conventional approaches to the studying the multiresonance systems is
given by Giacagrilia [32], Reiscig [29] and Cherepennikov [31]. In
particular, “an old problem retains open. Up to now no available “modern”
techniques enable us to compute the real frequencies of a nonlinear system.
For applications this problem retains unsolved, since in the approximations
by series, converging or only formal, only finite and, generally speaking,
small number of terms can be calculated. We cannot yet find a way to express
the general term and the sum of these series” [32, p.305].
Furthermore, “to provide the series converging, sometimes we must presume
that the parameters of differential equations determining the power of
nonlinearity have quite small module. By this reason the indirect technique
is often applicable only in a narrow boundary area of the nonlinear
mechanics. Another demerit of these techniques is that they enable one to
obtain quite accurate information about the separate solutions, but give no
idea concerning the structure of the family of solutions as a whole” [29, p.12]. Giacagrilia
confirms it: “Another problem of a great interest is the item of better understanding
the solution “far, near and at the resonance conditions”. When we will really
have a process of capture into the resonance, and which definition of the
resonance is preferable?” [32, p.309]. “Exact analytical techniques are
preferable in the analysis, however obtaining the analytical formulas even
for comparatively simple differential equations is sometimes connected with
great difficulties” [31,
p.10]. In the light of above demerits of conventional
methods, the most exact qualitative pattern of the vibration process was
presented by Skudrzyk. According to his approach, “any heterogeneous system,
monolith or consisting of homogeneous parts and loading masses, can be
rigorously presented as a canonical diagram, namely by a parallel connection
of infinitely large number of sequentially connected (mechanical) circuits,
one for each form of natural vibrations” [17, p.317]. But Skudrzyk
applied the matrix techniques to solve the systems of differential equations
in the systems he modelled. This disabled him describing the pattern of
processes analytically, because for complex elastic systems, as is known, the
matrix techniques allow only the numerical solutions. And in the analytical
form the vibration pattern written in the matrix form is practically
non-describable. This demerit (inherent in most of conventional methods)
disabled Skudrzyk also to extend the introduced concept to the case of
multiresonance elastic systems, in which the assemblage of resonance
frequencies is determined not by a set of mechanical resonance circuits but
by an integral multiresonance mechanical subsystem forming the entire
spectrum of the subsystem resonances. With the exact analytical
solutions presented in [1]
– [3] and [46] – [50] we obtain the
scope to overcome the problems inherent in the conventional approaches and to
find the exact analytical solutions for a broad gamut of elastic mechanical
and electrical systems. |
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State of the art |
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Applications to
the mismatched ladder filters, transmission
lines and networks |
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c 
= 1, 2, …, s).
These maximums turn into infinity, if the energy dissipation was absent. As a
counterpoise, the exact analytical solutions obtained by the original
non-matrix method in [1], [
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