SELF

64

S.B. Karavashkin, O.N. Karavashkina

In the second section of a line (k i n), according to (12), there form the complex-form standing waves determined by the superposition of two standing waves. Their form will also differ from that considered, so that between the summands in the square brackets of (12) the phase shift will be absent, too. Therefore, by all complexity of vibration pattern, (12) cannot describe the progressive wave. Furthermore, we can see from (12) that the forming standing waves have not a resonance, - this is one more typical feature of the studied case. The diagrams in Figures 2a and 2b corroborate the said.

As is expected, in the third section the antiphase damping vibrations form. The regularity of damping is power-type, not exponential. The base of power diminishes with the growth of external force frequency, therefore the damping increases. One more distinction of the studied section is, the phase shift between the neighbouring elements vibration never exceeds . It essentially improves the settled opinion concerning the pattern of vibration process in the overcritical band (see e.g. [8], [9]).

Solutions (11)–(13) get some other form, if the external force affected the heavy part of elastic line. As this case is important for the practical use, consider it separately.

When the indicated conditions were realised, the terms containing ₁ and ₁ in (2)–(4) transform. Modifying them and noting (9), we yield:

for i k

for r i n

and for i n + 1

where

The interchange of the light and heavy parts of a line causes in the first section the antiphase damping along-line vibrations. The base of power _{1 -} depending on the frequency of external affection determines the damping degree. With the growing frequency, the damping grows. The square-brackets expression in the right part of (17) has no effect on the antiphase pattern of vibrations, as it was in the item 3.1, but together with the phase ₁ it determines the general delay of vibration process.

At the same time, similarly to the item 3.2, the amplitude in the near of external force application point depends on the value of multiplier . When small difference between m₁ and m₂ , it can take large values, despite the damping. In real models this distinction can destroy the system of elastically constrained masses, while out of the region of external force affection the vibrations can be practically absent.