Materials.Technologies.Tools

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.

4. Feasibility 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 like the conventional conception. But there are some distinctions.

For example, it is accustomed that if a free-end elastic line, we will see the ending element’s shift bunched, and if the end was rigidly fixed, we see a node. This last statement is indisputable, but as for shift bunching, there will be some difficulties connected with the yielded exact solutions. In the view of vibration of the first (i = 1) and last (i = n) line elements, from the expression (33) determining the free vibrations amplitude for an unfixed-ends line, we yield the following expressions for the amplitude:

for i =1

which is quite natural, proceeding from the statement of the given problem, and for i = n

It means that for the last line element, the amplitude is some less than the value related to the bunching displacement. And the difference vanishes when passing to a distributed line.

On the whole, 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 that was the matter of principle in the view of accuracy when describing the line vibration pattern. This all evidences that if we strive to describe the processes exactly, we generally may not use a distributed line as the basic model – analogue 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. Completeness of the yielded solutions

In this paper, the same as in [1], we gave the solutions for two types of finite lines that cover the whole 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 ability of this method to study vibration processes not only for infinite but also for finite homogeneous elastic lines.

6. Conclusions

We revealed in this study a number of essential drawbacks in the known solutions for 1D elastic lines. This disables us thinking these solutions covering for finite lines.

The exact analytic solutions yielded 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 much differ in their properties.

The periodical regime typically produces a standing non-damping wave.

The aperiodical regime makes the vibration process damping along the whole 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 whole 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 regularity but linearly.

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.