Volume 4 (1999), No 3, pp. 15-23

17

Exact analytic solution for 1D infinite vibrant elastic lumped line

As we see, the solution (6) satisfies the first equation of system.

Now check it for the nth equation of (4). For the left part, the check is similar to (12):

When substituting (6) to the right part, yield

Satisfying the first and nth equation, (6) satisfies the whole system (4).

  The distinction of aperiodical regime (7) is that all neighbouring elements vibrate in antiphase and (-1)ⁿ before the coefficient points it. Besides, the amplitude falls abruptly with the growing number of element, because _- < 1 always. The more is (i.e., accordingly to (9), the more and m are or less s is) the more abruptly amplitude damps, because _- 0 monotonously at . Hence, the ideal linear system behaves at the given conditions as an effective damper in which the damping itself follows not as a consequence of the energy dissipation but results from along-the-line redistribution of the elements vibration energy. The typical appearance of these vibrations is given in Fig. 4.

Fig. 4. The wave propagation in a semi-finite homogeneous elastic line vibrating in the aperiodical regime (f = 35 Hz, F₀ = 2 N, m = 0,01 kg, s = 100 N/m, a = 0,01 m, f_crit = 31,8 Hz)

The aperiodical solutions are convenient to be checked, considering the reduction formulas given in the Table 1.

Table 1. Reduction formulas

where

Substituting (7) to the left part of the first equation of (4), we yield

For the right part

which means, the solution (7) satisfies the first equation of system (4).

  For the left part of the nth equation of this system

for the right part of the nth equation

Thus, the solution (7) satisfies (4) completely.

For the critical case, (8), antiphase vibrations of elements with amplitude constant along the line are typical. In the infinite line it is practically impossible to realise it experimentally, because these solutions describe steady vibrations, while in reality we observe along-the-line propagating step-like effect. With it, in the given signal spectrum there are present the frequencies of as critical as periodical as aperiodical regimes. None the less, in this case also, the presented solution is important, because the base mode of vibration will be related just to the critical case.