Materials.Technologies.Tools |
8 |
| S.B. Karavashkin | |
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The presented solutions emphasise again a vast diversity of vibration forms and patterns dependently on the line type and parameters. Check the correspondence of presented solutions (13), (16) and (17) to (12). It will be sufficient to substitute them sequentially into the first, arbitrary ith and the last equations of this system. For the periodic regime on the left part of the first equation of (12) we yield |
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(18) |
and on the right part |
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(19) |
For the left part of the ith equation |
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(20) |
and for the right part |
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(21) |
For the left part of the last equation of the system |
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(22) |
and for the right part |
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(23) |
Thus, the periodic solution satisfies (12) completely. For the aperiodical regime it is sufficient to check, whether the solutions satisfy any equation of the system, because, as we showed in [1], this regime, the same as that critical, is one of the forms of the common solution, and if the periodical solution satisfied the system, then the aperiodical and critical ones satisfy, too. So let us make a control check - for example, for the first, the most complicated equation of the system. For its left part we yield |
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(24) |
and for the right one |
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(25) |
Similarly, on the left part of the first equation for critical regime |
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(26) |
and on the right part |
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(27) |
As we can see, if the solutions were complete and exact, they satisfy as the general equation of the system as the starting and finishing equations as the equation containing the external force. With it, we can see from the pattern of reduction of equations that the way to reduce is different in each case, because in the solutions themselves their own specific multipliers corresponding to each equation of the modelling system are present; in this way they provide the identity in each case. This is one more interesting property of the yielded solutions from which it follows that one and the same solution satisfies few types of differential equations at the same time. This was the main complication and main reason of failures when many generations of scientists sought the exact solutions. |
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