SELF

78

S.B. Karavashkin, O.N. Karavashkina

As an advantage of Magnus method we can mark, he did not follow the conventional way, introducing the boundary conditions for an unfixed end as is usually done, but suggested a complicated way to get over the discrepancy of solutions. “We can now write the above 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 relationship in previous item 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 of line (40),  0 p n +1, while 1 i n . Noting these features, on the basis of (37) we yield

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

and substituting (45) into (37), yield

which fully corresponds to (44).

The disadvantages 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’s vibration and the external force amplitude. As we see from (45), this relationship is quite complex. Basically, this disadvantage reflects the impossibility to specify exactly the vibrations at the free end of a line through the boundary conditions, since, as it follows from (45), this amplitude vitally depends on the external force parameters. None the less, despite these disadvantages, the Magnus method fully corroborates the validity of (37)- (39) in particular case of a finite line having one end free. And not only in the band of periodical regime. Basing on (44), Magnus considers also the 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 behaviour of hyperbolic sine function that in the most general case for each mass, with the growing amplification coefficient *, the more this mass is remote from the chain start the more 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 yields 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 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 affection the support reaction 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, the constraints are loaded maximally. In this view the case is important when the external force affected 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 “affection radius” dependence on the frequency and elastic system parameters was complicated, it is important to find the solutions of modelling system equations in the analytical form, as this form most exhaustively shows the measure of each factor’s affection and allows choosing qualitatively the elastic line parameters dependently on the type of external force.