V.2 No 2

21

Compression waves in a rod

3. The limiting process from the lumped elastic line to that distributed

Realising the succession to find the solution presented in the item 2 and describing the vibrations in a finite-cross-section rod, we will first use the results obtained in [14] in studying an ideal elastic lumped line, and on this basis we will determine the solutions for an infinitely thin distributed elastic line (rod).

In [14] we presented two blocks of solutions for a semi-finite elastic lumped line for forced and free vibrations correspondingly. For the present problem we are interesting in forced vibrations, because just this regime relates to the process of propagation of longitudinal compression waves. While free vibrations relate to the standing waves taking place in case of non-zero energy density in a line, as it was shown in [14].

In their turn, for forced vibrations in [14] three solutions were presented, dependently on relation between the parameter Image1123.gif (1040 bytes) and the unity (where omegacut.gif (838 bytes) is the frequency of external force, m  is the line elements mass, s is the stiffness coefficient of a line). Three vibration regimes correspond to these three solutions: periodical (betacut.gif (852 bytes) < 1), aperiodical (betacut.gif (852 bytes) > 1) and critical (betacut.gif (852 bytes) = 1).

To process the limit passing to a distributed line, according to [15] we will be interesting in the periodical regime, because the critical and aperiodical regimes are impossible in a distributed line.

Thus, for the case betacut.gif (852 bytes) < 1 the exact analytical solution for a semi-finite ideal elastic lumped line will be the following [14]:

(12)

where F0  is the amplitude of the external force, deltabig.gif (843 bytes)n is the absolute shift of the nth body from the state of rest, taucut.gif (827 bytes) = arcsin betacut.gif (852 bytes), and n = 1, 2, 3, ... .

To make the limit passing to the solution for a distributed line, by analogy with [15] we have to introduce the correspondence between the parameters m, sdeltabig.gif (843 bytes)n, n  determining the lumped line and the following parameters: the line density rocut.gif (841 bytes), stiffness T , location of the studied point in an excited state x_cap.gif (850 bytes) and in the state of rest xo.gif (847 bytes), which determine the processes in a distributed line. Conveniently do it by the following simple way, introducing

(13)

(14)

where a is the distance between the line elements in the state of rest.

With taking into account (13) and (14), the expression (12) takes the form

(15)
where

(16)

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