V.2 No 1

77

Some features of the forced vibrations modelling

The considered matrix method has all above disadvantages, as it works only at the known values that have to be found numerically. But not only the common disadvantages unite these methods. In their essence, they are variations of one and the same approach, as, on one hand, “since the characteristic equation has 2s roots , ()  can be presented as the product

where a is some constant” [7, p. 296]. On the other hand, the introduction of the main coordinates is tantamount to the simultaneous reduction of two quadratic forms T and U to the canonical form [7, p. 266]. We can conclude from it that both matrix methods can predict in analytical form only the fact that “if the attenuation was small, each amplitude _k^e will have s resonance peaks at s frequencies _{c a} ( = 1, 2, ..., s) . These maximums turn into infinity if the energy dissipation is absent, i.e.,  _e 0. In this case

[7, p.297].

The techniques using the Voronoy, Toeplitz and other matrixes also have the essential drawbacks. Despite all attempts to put these matrixes in order and to reduce them to the diagonal form, these techniques work well only if the succession of numbers p₀ > 0, p₁ 0, p₂ 0 , ... has been specified. Then at the condition [17, p. 120] we can try to find the solutions. But the task is just to find these p₀, p₁, p₂, ..., and especially p₀ , which determine the line start reaction to an external affection. To find these parameters, even in case of simple models, as we showed in [2], we may not use a priori some stable phase delay, because even in cases of semi-finite and finite lines the line element parameters under the force affection are different. And these techniques, the same as the integral and classical matrix techniques, see only one regime - under the critical vibrations.

As a counterpoise to this, the exact analytical solutions yielded by the original non-matrix method in [1] show not one but three vibration regimes: periodical ( < ₀) , critical ( = ₀)   and aperiodical ( > ₀) . In particular, for a line having one (right) fixed end, these solutions have the following form:

at  < ₀

Comparing (37)- (39) with (36), we see them clearly determined in relation to the external force parameters, to the elastic line parameters and to the resonance frequencies. With it, when n growing, the complicacy of analysis does not increase. These solutions can be easy extended to a distributed line, which is unrealisable analytically by the matrix methods. Outwardly one can even doubt, whether (37)- (39) are reliable? As [1] shows, (37)- (39) completely satisfy the conventional modelling system of differential equations. And the aperiodical regime is not so much unexpected. The indirect methods, though they give the applicability limitation and incomplete results, in particular cases also corroborate the physical reality of this vibration regime. As an example, consider the original indirect method described by Magnus [4, pp. 282- 285]. It will be the more interesting for us that when presenting his method, Magnus studied a finite line with the fixed right end too, so we can compare the results.

Magnus has drawn his attention to the relation between and in (17); true, in this expression took the discrete values (14). Besides, considering the forced vibrations, he lifted the fixation of the left end of an elastic line (filter) and made it moving periodically as

.

In doing so, he naturally violated one of the boundary conditions that had to cause him the difficulties. Getting them round, he supposed that “with it in the motion equations (8) nothing will change for the individual masses, so we can seek the periodical solution having the same frequency as the excitation, and coming either in phase or in anti-phase with the excitation, supposing