V.2 No 1

63

On solution for an infinite heteroheneous line

3.2

When ₂ exceeds the unit value, in the heavy part of line the aperiodical vibration regime settles and all terms in the right-hand parts of the system (2)–(4) transform as according to (8), and the system takes the following form:

for  i k

for  k i n

for  i n + 1

where

In the first section corresponding to i k , there has remained the periodical process of progressive wave propagation with the phase delay [2(n - i) +1] ₁  and the amplitude depending on the parameters of both sections of the line. But the general phase delay has changed. If in (2) the vibration process delayed by in relation to the external force phase, then in (11) the general delay is determined by the parameter (- ₂). At  ₂ +1 we will have ₂ , and at ₂ the angle will be , not zero.

The less is difference in the element masses m₁ and m₂  the more is vibration amplitude. Basically, at m₁ m₂   the amplitude _i also tends to infinity, which is determined by the multipliers and cos ₁ . But the amplitude will not reach the infinity, since at m₁ = m₂  in the light section the aperiodical vibration regime will also settle, and the solution (11) will transform in the corresponding way.

When the expression in square brackets in (11) becomes zero, i.e. at

the vibration amplitude in the first section vanishes, as it is visual in Fig. 2a. And in the case considered before, when all sections vibrated periodically, such phenomenon is impossible, because of phase shift in the summands in the braces of (2). In (14) the phase shift vanishes, as the heavy section passes to the aperiodical regime.

Fig. 2. Vibration diagrams in a heterogeneous elastic line at 23,0515 Hz (a) and 22,5 Hz (b). The external force frequency is between the boundary frequencies for the light and heavy parts of line;  F₀ = 0,6 N ; s = 100 N / m ; m₁ = 0,01 kg ;  ₀₁ = 200 s ^{- 1} ; m₂ = 0,02 kg ; ₀₂ = 141,42  s ^{- 1}