V.2 No 1

19

Homogeneous 1d resistant line

After the conventional elimination of the time dependence

where _n  is the amplitude of nth element shift, and substitution

the system (3) takes the form

We can easily see that (6) became fully identical to the conventional modelling system of differential equations for an ideal semi-infinite elastic line, for which in [10] the exact analytical time-dependent solutions have been obtained. They had the following form:

for the periodical regime  ( < ₀,   < 1)

for the aperiodical regime ( > ₀,   > 1)

and for the critical regime ( = ₀,   = 1)

where ₀ is the initial phase of external force affection, and ,  ,  _- are the parameters of an elastic lumped line:

Applying (7)–(9) to (6) as the solutions, we have to take into account that after we substituted (5), parameter in (10) became a complex value:

It means that in a resistant line only the periodical regime (7) can exist. The critical and aperiodical regimes are possible only when transiting to an ideal line, i.e. at a real value realisable at r = 0, as we can see it in (13).