84

S.B. Karavashkin, O.N. Karavashkina

The limiting process to a distributed line can be effected on the basis of obtained solutions at the condition 0, if the main parameters characterising the line were presented in the form convenient for it, i.e.,

where x₀ is the distance from the reading-point to the point of rest of the considered line element.

Besides, the necessary requirement is, the values , T and x₀ to remain finite in the limiting process. For example, for the solution (50) we yield

where x_0i is the investigated point location in the non-excited state; x_0k is the external force application point location, in the non-excited state too; x_i is the momentary location of investigated point in the excited state; l is the line length, and _i   is the line element shift.

As follows from (73), when transiting to a distributed line, there retains the main distinction caused by the external force application point sectioning the line. Consequently, the vibration structure remains on the whole and, at definite conditions, the vibrations are produced only in a part of a line. With it, when passing to the limit, the expression (50) transforms irreversibly, which makes the reverse transition impossible.

When investigating the conditions of transition to a distributed line, it is interesting to consider, how the solutions for critical and aperiodical regimes transform. Conveniently investigate this not on the basis of particular solutions but at the condition of these regimes realisation. When a 0, noting (53) and (72), we yield

whence it follows that of all three vibration regimes only that periodical can be realised with the limiting process. For two other regimes the conditions of existence will be simply absent.

Thus, analysing the transition to a distributed line, we see disappearing the number of properties inherent namely in lumped lines, which makes non-identical these models description. We can pass from the lumped line description to that distributed, but vice versa we cannot. There is allowable only some particular approximation, when the condition (71) was fulfilled.

7. Conclusions

In the course of the carried out investigation we have revealed that not only the line finiteness, the way of the ends fixation or vibration regime has a great effect on the vibration pattern of an external force application point. This point is able, particularly, to be a border of a quasi-finite section where the standing wave forms, to form the complicated-structure-vibrations and conditions at which the vibration process completely vanishes in one of the sections of line, while other sections retain the vibrations. This all gives the grounds to indicate the feature of heterogeneity which the external force application point causes in the line.

We also have revealed that in lumped lines the progressive wave propagation velocity decreases non-linearly and monotonously, with the growing vibration frequency, reaches the minimal value at _crit and retains it in the aperiodical regime.

Similarly, the wavelength in the line decreases nonlinearly to the double distance between the line elements, and it retains this value in the aperiodical regime.

At the limit passing to the distributed line, the aperiodical and critical vibration regimes become impossible. Besides, at this passing, the main distinctions of the solutions, caused by an external force action on the interior elements, retain. At the same time, when transiting to a distributed line, the solutions transform irreversibly, which makes the reverse transition impossible.

We have established also that a lumped line can be modelled by means of that distributed only when the ratio of between-the-elements distance to the wavelength is considerably less than ^-1 .