7

Solutions for finite elastic lumped lines

Forced vibrations of a line having the start and end unfixed (Fig. 2). The following system of differential equations

corresponds to this model. The same as for infinite lines, (12) has in general case three basically different solutions depending on the ratio of to 1, where

For finite lines the periodic solution corresponding to < 1 has the form

where

.

This solution describes standing waves, which is determined by the constant phase delay of vibrations for all line elements. But unlike the conventional solutions (6), the exact solution structure has basically other form. With

an infinite resonance will take place in the line. But in other cases, the vibration amplitude is finite and determined by the value sin 2n. With

it will be minimal but not zero, as we used to think for resonant lines. The typical vibration pattern is shown in Fig. 3. But the main that follows from the obtained solutions is that line vibrations are not limited by the critical frequency.

In the aperiodical regime corresponding to >1  the exact solution takes the form

where

The same as in case of infinite lines in [1], we can see the anti-phase vibrations, but damping not so fast as in infinite lines. In infinite lines the damping distributes along the entire line. In [1], when analysing free vibrations, we mentioned the resemblance between vibrations for finite and infinite lines and that in the aperiodical case the damping begins from the point of energy input into a vibrant system. In Fig. 4 one can see just this phenomenon, when the damping passes from the first to the last element, distributing uniformly along the entire line, which confirms the analogy indicated in [1].

For the critical regime corresponding to = 1, the changes are also interesting:

In a finite line these vibrations have a damping pattern, not remaining their amplitude, as in infinite lines. But unlike the aperiodical regime, the damping has not a power-type but linear pattern. We can see these features in Fig. 4.