This paper was published first in the journal of National Academy of Sciences of Belarus "Materials. Technologies. Tools", 4 (1999), 4, pp.5-13 (in Russian). This is the second of a large cycle devoted to the new method to obtain the exact analytical solutions for vibrant elastic systems, both lumped and distributed. Though the material stated in it is quite simple, as well as the investigated problems, you can make sure from the presented solutions that in the exact analytical form and in all their completeness these solutions were not obtained up to now. It is important that we give the checks here that these solutions are exact and complete, and the connection between the lumped and distributed systems as well. Of a great importance is that in vibration systems we can pass from the lumped to distributed systems but in no case vice versa, as is conventional and in which the investigators are absolutely sure. Also important is that these results are basic for the further investigations which we carry out for much more complicated models. There we often refer to this introducing paper, so we decided to load it also to the Archive, to make its English version available for the English-language readers. You can read some of the further papers, visiting for free our electronic journal "SELF Transaction" at the URL 
http://angelfire.lycos.com/la3/selftrans . 
We will be thankful if you express us your opinion and the more grateful for your wish to go on with us in the investigation of this huge and prospective theme.