10

S.B. Karavashkin

Thus we see that the exact analytical solutions (32) essentially differ from the conventional solutions.

As before, we can easy check that the expressions (32) really are the solutions, substituting them immediately into the first, i th and last equations of the modelling system (29). For the left part of the first equation

and for that right

consequently,

For the left part of the i th equation

and for that right

For the left part of the last equation

and for the right one

Proceeding from the described above and considering the theorem of uniqueness of a solution of differential equation, we can surely say that the presented solutions are exact, which substantially changes the concept of the vibration processes pattern in finite lines. The same as in infinite lines considered in [1], the solutions transform in correspondence with the line features, but remaining the solution block structure. We can make it sure, considering a finite line having an unfixed start and fixed end.

The forced and free vibrations of an elastic line having unfixed start and fixed end under an external harmonic force F(t) action is presented in Fig. 5. The following system of differential equations

corresponds to this model. Three solutions correspond to this system:

for the periodical regime at <1