SELF

54

S.B. Karavashkin, O.N. Karavashkina

For the first subsystem (i = 1) the amplitude varies from F₀ / s _g at _g to infinity at _g 0. For all the following subsystems (i > 1) this variation takes place in limits from infinity to infinity with the minimum at

.      

(14)

It means, with the growing subsystem number, the value _{g min} at which the minimal amplitude is achieved increases. Skudrzyk indicated the resonance peaks bifurcation in complex vibrant systems: “Due to damping, some resonances seem single, but if we cool the studied specimen to diminish the damping, then two maximums are clearly seen” [1, p.244]. He explained this phenomenon so: “If two resonance frequencies are closely spaced and the shape parameters of natural frequencies are opposite in sign, then the transfer impedance of homogeneous systems nearby some of their resonances can resemble the bound circuits. In this case the frequency regularity is alike the band filter characteristic” [1, p.323]. However, as the exact analytical solutions show, the reason of resonance peaks bifurcation are the complex aperiodical regime features, not the circuits connectivity. Note additionally that the complex aperiodical regime existence is new in our knowledge of elastic systems vibration. In simple elastic lines without resonance subsystems this regime is impossible. And as we said in the introduction, complex systems have been currently studied not so well to reveal this regime.

Summing up the aforesaid, we see that the system of solutions (9) – (12) completely describes the vibration pattern in all the range. Thus we have achieved the aim of this item.

3.      Analysis of obtained solutions and comparison with experimental results

To analyse the obtained solutions, conveniently use the following concept of an elastic line transfer function

(15)

and of the input resistance which in case of external force action on the start of line is equal to

.      

(16)

We can easy determine from (9) – (12) that for the studied model

(17)

(18)