V.2 No 1

73

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 yielded 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 and 4 boundary conditions), and it would be useless, since its root values might be calculated only numerically, not with a literal presentation” [6, p. 161]. “Even to find the frequency and shape of the first main oscillation of a system, we use different approximate methods, since the exact solution of a problem is impossible at a large number of freedom degrees” [9, p. 162]. Besides, when calculating the finite elastic systems, this disadvantage strongly limits the matrix methods applicability and causes the scientists to develop the indirect analytical methods to calculate the main parameters of vibration processes. As we said above, these methods are based on some regularities revealed for the specific mathematical models, so their area of applicability is narrower.

One of the mostly used indirect methods is described by K. Magnus [4, pp. 278- 282]. 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 yield

We will seek the main vibration of the pth mass (that we know to be existing) in the form

Substituting this solution into (8), we will yield

After some modification, (9) will be reduced to

where

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

Then (10) leads us to the following result:

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

is true. Thus, the natural frequencies differ in the property that for them the boundary condition _n+1 = 0 specified at the line end is true automatically. The tables of zero 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 disadvantages of the method. To find the natural frequencies, we all the same have to solve the power equation; 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. True, the author gives also the analytical method. “We can express the natural frequency also in the explicit form. To do so, try to find the solution of (9), taking