SELF

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S.B. Karavashkin, O.N. Karavashkina

The typical form of regularity M() plotted for odd and even numbers of subsystem elements n on the basis (7) is shown in Fig. 3. It fully corroborates the above analysis. We can see from the construction that this regularity is the succession of resonance peaks whose density increases with frequency, and the peaks width decreases with it. With the transition to the aperiodical regime, the subsystem inertia gains the monotonously increasing pattern with respect to frequency. We should especially note the pattern of this regularity at low frequencies. In the region preceding the first resonance peak, the subsystem inertia increases monotonously. In the considerable part of the band it is approximately equal to the subsystem elements total mass, and at 0 the value M nm. It is easy to check it, noting the values _s and _s , when finding the limiting value for the first expression of (7).

On the basis of subsystem inertia value, we can easy determine the exact analytical solution for the whole elastic system shown in Fig. 1. We will use the fact that the subsystem inertia in (7) depends not on time but only on the external action frequency. So, under external harmonic force action, the measure of subsystems inertia may be regarded as constant for each frequency, and the features caused by the subsystems resonances reveal only in the external action frequency variation.

Thus we can use directly the solutions presented in [20] for a semi-finite elastic lumped line. The same as in case of solutions for a finite elastic line, their form depends on the relationship between the parameter and the unity. We should especially mark that despite the vibration pattern of the elements of an elastic line as a whole and of subsystem elements depends on parameters _g and _s having similar functional regularity, with respect to frequency they behave essentially different. The parameter _s depends on the subsystem elements masses that are constant for a specific line, while _g depends on the measure of inertia of subsystem (7) which depends on frequency nonlinearly, and at definite values _s becomes negative. With it _g becomes complex, which is impossible for _s. In this connection, the features appear in the vibration of elements within the subsystem and of the subsystems as elements of general system. For the subsystems, the same as for simple elastic lines, it is typical a clear division of the range into the periodical, aperiodical and critical vibration regimes with the single boundary frequency corresponding to the critical vibration regime. For the elastic line as a whole it is typical some other range division. At low frequencies, before the first resonance peak of subsystem, the considered elastic line behaves the same as a simple elastic line without any resonance subsystems. The boundary of this range is the first boundary frequency _0g , which is close to the similar frequency of a simple elastic line whose element masses are equal to the total static mass of the subsystem. Naturally, this boundary frequency is lower than that of subsystem _0s . Higher than _0g , in a simple elastic line there takes place the aperiodical regime of antiphase vibrations damping along the line. In the line having the resonance subsystems, there reveals the influence of subsystem’s measure of inertia dependently on frequency, which determines the vibration pattern up to the critical frequency _0s for the resonance subsystem. Due to this feature, further we will distinguish the concepts of subsystem element and the elastic line element being the resonance subsystem.