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Some features of the forced vibrations modelling

As we see from this consideration, the matrix method, though it is thought analytical, is numerical in its essence, since in general case all main vibration parameters can be obtained only as the numbers. ''The solution of these equations requires first to find the proper determinants. Their form and any sizeable number of lines make it practically impossible or extremely laborious'' [6, p.98]. ''It stands to reason that we could not find the determinant of (4n +4)² elements directly (4n integration conditions plus 4 boundary conditions), and it would be useless, since its root values might be calculated only numerically, not with a literal presentation'' [6, p.191]. Even to find the frequency and shape of the first main oscillation of a system, one uses different approximate methods, since the exact solution of a problem is impossible at a large number of degree of freedom'' [9, p.162]. Above all, when calculating the finite elastic systems, this demerit strongly limits the matrix methods applicability and causes the indirect analytical methods to calculate the vibration processes main parameters. As we mentioned before, these methods are based on some regularities revealed for the specific mathematical models, so their applicability is narrower.

One of the mostly used indirect methods is described by K. Magnus. According to this method, ''to calculate the natural vibrations, consider an infinite 1D elastic line. At equal masses m_p =m and spring stiffnesses c_p =c we obtain

We will seek the known to be existing main vibration of the pth mass in the form

Substituting this solution into (8), we will yield

[[4, pp.278-282]. After some modification, (9) reduces to

where

''From this we can sequentially calculate all _p, since the boundary conditions, i.e. the conditions at both ends of a chain, are known'' [4, p.279]. Further the author limits himself, considering the case when both ends of an elastic line are fixed rigidly, i.e.

Then (10) lead us to the following result:

These functions ''are often used to calculate the natural frequencies of vibrant chains (particularly, [16]). The natural frequencies can be calculated as zero values of (n+1)th frequency function (12). It means, for the unitless natural frequency _q

is true. Thus, the natural frequencies distinguish by the feature that for them the boundary condition _n+1 = 0 specified at the line end is true automatically. The zero tables for the frequency functions up to n=11 are given in [16, vol. II, chapter XIII]'' [4, p.279].

From this brief description we can see the demerits of the method. To find the natural frequencies, we have to solve the power equation all the same; as is known, the equations higher than the 4th power (in this case of 8^th power) cannot be solved exactly in the analytical form. The author gives also the analytical method. ''We can express the natural frequency also in the explicit form. In order to do so, try to find the solution of