Volume 4 (1999), No 3, pp. 15-23

19

Exact analytic solution for 1D infinite vibrant elastic lumped line

Check, how these solutions satisfy the modelling system. It is sufficient to check them for the kth and the arbitrary ith equations of (23).

Substitute the periodical solution (24) into the kth equation of (23). For the left side we yield

for the right side

This means, (24) satisfies the kth equation of system.

Now substitute (24) into the ith equation. On the left side we yield

and on the right one 

Thus, the periodical solution satisfies (23) completely.

  Now check the aperiodical solution (25). For the left side of the kth equation we yield

for that right 

Finally, for the left side of ith equation:

and for the right one

As we see, solutions (24) and (25) satisfy the modelling system (23) completely. But, as it was pointed before, by contrast to the solutions for semi-finite line, the given model has not a final solution for the critical regime. And this is easy to prove by checking the completeness of the amount of solutions (24)–(25).

  According to the above conditions of critical regime, it is an intermediate between those periodical and aperiodical; consequently, if in the aperiodical solution we direct 1, we can yield the solution for critical regime. In equation (25) at 0 the value of multiplier (² - 1) 0, which corresponds to an infinite value _n in critical regime. By contrast, in aperiodical regime (7) for a semi-finite line it does not occur:

i.e. the solutions (24)–(25) are same complete as (6)–(8) are.

  At the same time we see that the aperiodical solution easily transforms into the critical one; we can prove it also for the relation between the periodical and aperiodical regimes. Basically, all three yielded solutions are three forms of one solution transforming in accordance with the value in relation to 1. With it the basically different types of vibration processes form in the line. To understand this aspect is basically important as in the view of physics of vibration processes as in the view, the theorem of unique solution of differential equation to remain its validity.