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On solution for an infinite heteroheneous line

4.3. m₂ =

By the same principle we can obtain the solution for a semi-finite line with a fixed end. First of all we should note that when the masses m₂ turn into infinity, vibrations in the section containing these masses turn into periodical regime. Besides, it follows from this condition that at m₂ 

To transform (2) – (4), conveniently use the system (11) – (13) where the transition of the third section vibrations into aperiodical regime was taken into account. Substituting (33) sequentially to the expressions of this system, we will obtain:

for i k

for k i n

and for i n + 1

In the first section of a studied line there propagates the progressive wave whose amplitude depends in a complex way on the external force frequency and the line parameters. In the second section the standing wave with some phase delay 2(n – k)₁ has formed. In that third, just as it was expected, the vibration amplitude is zero.

If we continue transforming the solutions (34) – (36), taking k = n , we will obtain the solution for a semi-infinite elastic line on whose first after fixation element the external force acts. With it only one solution of three will remain:

Comparing (37) with (32), we see that at the same external force parameters the vibration amplitude in a line with fixed ends is less at low frequencies, when the condition

or

is valid. At the frequencies higher than indicated in (38), the vibration amplitude in the line with fixed end will be higher than in a line with unfixed end.

Thus we see that the basic solutions transform easy into solutions for the models generalised by the basic model. This is a very important property of the complete analytical solutions. Should the initial base solutions be incomplete, or should it be presented only numerically, such transformation would be impossible.

4.4. The limit passing to a distributed line

To make a complete analysis, trace the transformation of solutions (2) – (4) at the limit passing to a distributed line. To transform the base solutions for this case, present the parameters of elastic line in the form, corresponding to a distributed line. To do so, introduce

where T is the stiffness of a line with lumped parameters, a is the distance between the unexcited elements in a lumped line, ₀₁ and ₀₂ are the densities of relative sections of heterogeneous unexcited line with lumped parameters.