V.2 No 1

57

On complex resonance vibration systems calculation

Basing on _K () and on results presented in [22], we can determine the phase velocity v_f of the wave propagation along the ideal line

,

(20)

where a is the distance between the elastic line non-excited elements, and _K is the i th element phase delay in (17). The typical form of the phase velocity dependence on frequency is shown in Fig. 7.

We can see from Fig. 7 and (20) that in the complex aperiodical regime the phase velocity turns to infinity, as at these bands _K is zero (see Fig. 4). At the same time, the phase delay vanishing evidences that the standing wave has produced in the line, despite its limited length. This is the consequence of the steady-state pattern of the studied wave process. If the line impedance was present, the phase _K does not vanish. Similarly, if along the line the wave process propagated whose spectrum is wider than one band of complex aperiodical regime, then the resulting phase delay also does not vanish. Thus, the obtained result is the idealisation too and can be corrected automatically when solving the practical models having the impedance.

Going on analysing (20), we see that the phase velocity achieves its minimal value in aperiodical regime, since at these bands _K is maximal. At these bands the phase velocity dependence on frequency is linear, as the phase delay is constant and equal to . With it, all bands of aperiodical regime are located on one line coinciding with the linear regularity, after which the phase velocity would increase in the absence of resonance substructure. It corroborates additionally the above statement that the resonance peaks of the subsystem arise in the region between the periodical and aperiodical regimes for the system as a whole.

Furthermore, it is typical that despite the sections having negative measure of inertia appear, the transfer function phase retains delaying always, and this also is in full accordance with the above Skudrzyk’s statement [1] that the negative measure of inertia of line elements fully corresponds to the conservation laws.

This negative measure of inertia, which we used to think strongly associated with the mass, does not mean a negative mass introduction. In this case, there reacts not a separate mass but a complex system of elastically connected masses being the parts of a general elastic system. So we have to identify just this reaction with the negative measure of inertia of the subsystem. We see that the pattern of subsystem reaction to the external action changes. With it the pattern of process also changes. And the phase with regard to the external action retains negative. Thus, introducing the idea of negative measure of inertia, we do not contradict the laws by Newton who considered an accelerated body as an entire rigid system.