12

S.B. Karavashkin

for the left part

and for the right one

As wee see, (48) quite simply satisfies the incomplete differential equation of (45) describing a finite fixed-end line, the same as (32) satisfied the differential equation of the system (29) for a finite unfixed-end line.

In the present paper, the same as in [1], we gave the solutions for two types of finite lines that cover the entire range of frequencies and describe both forced and free vibrations. All given solutions are exact and satisfying the related systems of differential equations. This corroborates the completeness and exactness of the presented solutions, as well as the method ability to investigate vibration processes not only for infinite but also for finite homogeneous elastic lines.

5. Validity to use a distributed line as an analogue

Analysing the exact analytical solutions for two presented finite lines, we practically did not consider the pattern of vibration process, since this is much alike the conventional concept. But there are some distinctions.

For example, one usually thinks that if an elastic line end was free, the end element bunching displacement takes place, and if the end was rigidly fixed, one will see a node, but as for displacement bunching, there will be some difficulties connected with the obtained exact solutions. In the view of the first (i=1) and last (i=n) line elements vibration, from the expression (33) determining the free vibrations amplitude for a line with unfixed ends, we yield the following expressions for the amplitude: for i =1

It means that for the last line element the amplitude is some less than the value corresponding to the bunching displacement. And the difference vanishes as one transits to a distributed line.

Generally, the shown difference between the lumped and distributed lines is typical. Before, in that number in [1], we noted it, when considered the phase delay  being basic in the view of the line vibration pattern exact description. This all evidences that if we strive to describe the processes exactly, then generally it would be incorrect to use a distributed line as the basic model analogous for that lumped. The more that the shown difference is not limited to the phase and amplitude deviations but reflect in the vibration pattern itself.

5. Conclusions

The carried out investigation has revealed the essential drawbacks in the known solutions for 1D elastic lines, because of which one cannot think these solutions covering for finite lines.

The exact analytical solutions obtained by the presented original method show that in general case in a finite ideal elastic line, the same as in that infinite, three vibration regimes take place, and they essentially differ in their properties. The periodical regime typically produces a standing non-damping wave. The aperiodical regime makes the vibration process damping along the entire line in which the neighbouring elements vibrate in anti-phase. In distinction from infinite lines, the critical regime is characterised by vibration process damping along the entire line, with the anti-phase vibration of the neighbouring elements. But unlike the aperiodical regime, the element-to-element damping occurs not as the power-type but as the linear regularity. All revealed vibration regimes are different forms of the general solution, which completely holds true the validity of theorem of uniqueness of solution of differential equation. In finite lines the revealed regimes are inherent only in forced vibrations. Free vibrations can exist only in the band of periodical regime.