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): |
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(14) |
When substituting (6) to the right part, yield |
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(15) |
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
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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, fcrit = 31,8 Hz)
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The aperiodical solutions are convenient to be checked, considering the reduction formulas given in the Table 1. |
Table 1. Reduction formulas where |
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(16) |
Substituting (7) to the left part of the first equation of (4), we yield |
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(17) |
For the right part |
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(18) |
which means, the solution (7) satisfies the first equation of system (4). For the left part of the nth equation of this system |
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(19) |
for the right part of the nth equation |
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(20) |
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. |