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On solution for an infinite heteroheneous line

In the second section of elastic line, the pattern of aperiodical process is much more complex. This is conditioned by the influence of the square-brackets expression in the right-hand part of solution (18). The presence of phase shifts ₂  departs from the rigorous anti-phase pattern of neighbouring elements of section and the regularity of the frequency damping degree will complicate.

In the third section the progressive wave propagates. Its amplitude is proportional to , i.e. to the damping degree in the heavy part of line. As we see in Fig. 3, the vibration amplitude in this section can be well less than the vibration amplitude in the region of external force action, despite when plotting, we chose m₁ = m₂ , so the external force acting point was closed to the utmost to the heterogeneity transition. It evidences the residual pattern of vibration process in the third section of line.

As the next typical transformation of basic solutions, consider the case when condition (8) is true for the entire line.

3.4.

In this case we should apply the transformation (9) both to the light and heavy parts of line. The result will be the following:

for i k

for k i n

and for i n + 1

We see from (21) – (23) that in all three sections of an elastic line the aperiodical regime settles. None the less, in each section it has its distinctions. In the first section the damping degree along the line is determined by the multiplier , in the second section – by the square-brackets expression in the right-hand part of solution (22), and in the third section – by the multiplier . In each case this influence reflects not only on the base of power but also on its index and even on the function type.

The common salient feature of solutions (21) – (23) in comparison with, e.g., [1] and [2] is the vibration phase shift by (-/2) in the first and second sections; the unit imaginary number in the right-hand parts of (21) and (22) determines it.

Thus, having studied the transformation of solutions (2) – (4) dependently on complete or partial satisfying the condition (8), we see that if taking into consideration the aperiodical regime in case of elastic infinite line, it allows to study much more completely the entire variety of vibration pattern possible in a line. It helps analyse better the possible transformations of vibration pattern dependently on the parameters of elastic system and external force, as well as to reveal more completely the most dangerous and stressed sections.