64

S.B. Karavashkin, O.N. Karavashkina

In the second section of line (k i n), according to (12), there form the complex-form standing waves determined by the superposition of two standing waves. Their form also distinguishes from that considered, so that between the summands in the square brackets of (12) the phase shift is absent too. Therefore, with all the vibration pattern complexity, (12) cannot describe the progressive wave. Furthermore, we can see from (12) that the standing waves forming here are not able to exhibit the resonance, and 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 anti-phase damping vibration forms. The regularity of damping is power-type but not exponential. The base of power diminishes with growing 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 refines 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 acts on the heavy part of elastic line. Since this case is important for the practical use, consider it separately.

3.3

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

for i k :

The interchange between the light and heavy parts of a line has led to appearing the anti-phase damping along-the-line vibrations in the first section. The base of power of _1- depending on the external action frequency determines the degree of damping. With raising frequency, the damping grows. The square-brackets expression in the right-hand part of (17) has no effect on the anti-phase pattern of vibrations, as 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 lead to the failure of the system of elastically linked masses, while out of the region of external force effect the vibrations can be practically absent.