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S.B. Karavashkin, O.N. Karavashkina

As a merit of Magnus method we can mark, he did not follow the conventional way, introducing the boundary conditions for an unfixed end, but suggested a complex way to get over the discrepancy of solutions. ''We can now write the cited expression (13) for the amplitude X_p as

The introduced constant gives us the scope to 'adjust' the solution to the boundary condition in the line beginning. The value makes it impossible, because when substituting (42) into the amplitude relationship (9), we see that the relation between and   is determined by (16). The distinction from the previous item relation is, when studying the natural vibrations, we have first to find as a relative natural frequency, and vice versa, for the forced vibrations the relative frequency of excitation is known'' [4, p.282]. On this grounds Magnus has replaced the discrete relation between and   (14) by the continuous relation ().

Further, ''at the boundary condition given here, taking into account (42), the following requirements are imposed on the amplitudes:

[4, p.283]. The result is

To compare (44) with (37), note that X_e corresponds to the amplitude of the first element (40), 0 p n +1, while 1 i n . Noting these features, on the basis of (37), we obtain

Noting moreover that according to (16) and (19)

and substituting (45) into (37), we obtain

which fully corresponds to (44).

The demerits of Magnus method are seen from the consideration. If lifting the fixation from the second end of a line, the condition (43) will be violated and the whole method will not work. Besides, the Magnus method has established the relationship between the vibration amplitudes of the pth and the first body, but did not establish the relation between this body vibration and the external force parameters. As we see from (45), this relationship is quite complex. Basically, this demerit reflects the impossibility to specify vibrations exactly at the free end of a line with the help of boundary conditions, since, as it follows from (45), this amplitude vitally depends on the external force parameters. None the less, despite these demerits, the Magnus method completely corroborates (37)-(39) validity in the particular case of a finite line having one end free. And not only in the band of periodical regime. Basing on (44), Magnus considered further also aperiodical (overcritical) regime. ''First of all we see that for all frequencies > 2₀, i.e. for all  * = - i, the signs before the amplification coefficients alternate, so the chain masses always vibrate in anti-phase with the neighbouring masses'' [4, p.284]. ''It follows from the hyperbolic sine function behaviour that in the most general case for each mass, with the growing *, the more this mass is remote from the chain start the more the amplification coefficient value decreases. For the last mass of a chain (p = n) the amplification coefficient is

At quite large n this function decreases so much with the growing frequency that practically we can say, the frequency higher than that boundary is cut off. The chain does not pass the frequencies > 2₀, it works as a low-frequency filter'' [4, p.285]. Blakemore [15] in his calculation also obtains the anti-phase vibrations in critical regime for an infinite 1D crystalline lattice. But he considers neither critical nor aperiodical regimes, thinking, due to the incompleteness of his solutions, that the phase delay at the overcritical domain will exceed . And due to the strong absorption, ''the waves having angular frequency exceeding _m = 2v₀ / a  cannot exist in an imaginary 1D crystal'' [15, p.110]. None the less, in many problems of the applied mechanics, solid physics etc., not only the energy transmission by an elastic line but also the process of energy accumulation and redistribution within the line is important. The local accumulation and redistribution of the vibration energy is inherent in the aperiodical regime. This is just the case when, for example, under an external force the reaction at the support is absent, even under the dynamical load in the excitation region being critical for the elastic constraints. This is a very important aspect, when studying the fatigue processes in elastic systems. We should note here, at the periodical regime the neighbouring elements vibrate in the anti-phase - it means that the constraints are loaded maximally. In this view the case is important when the external force acted on the line interior elements. With it both supports will not experience the load, while in the excitation region the critical vibrations can take place, crushing the internal constraints of an elastic line. And when the external force "action radius" dependence on the frequency and elastic system parameters is complicated, it is important to obtain the solutions of modelling system equations in the analytical form. These indicate the most exactly the measure of each factor effect and allow choosing qualitatively the elastic line parameters dependently on the external force type.