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S.B. Karavashkin, O.N. Karavashkina

As this solution shows, in the first and third sections the progressive wave propagates. It has its specific phase delays depending on the parameters of both sections. Besides, the entire vibration process shifts by (-); it is determined by the unit imaginary number in the right-hand parts of both solutions. In the second section there forms the standing wave with common phase delay [2(n - k)+1]₁ and additional element-to-element phase delay determined by the expression in braces in (3). In this connection the vibration pattern in the second section of a line complicates. As we will show further, this is the cause, why (3) describes not only standing but also progressive waves.

Furthermore, we see from (5) that in the line two boundary frequencies

have appeared. These frequencies determine two conditions transitive to the overcritical band. Due to it, the frequency range divides into three bands; in each the solutions (2) – (4) transform dependently on the conditions

With it the section whose masses are heavier transits to the aperiodical regime first, while in the second section the periodical regime of non-damping vibrations retains until (8) is true.

To analyse the solution transformation pattern when (8) was satisfied for some section, it is sufficient to transform the parameters in (2) – (4) relevant to this section, using an evident relation following from (6):

Below we will consider the possible transformations of (2) – (4) dependently on parameters m₁, m₂ and and analyse the most typical distinctions of vibration process accompanying them.

3. Analysis of typical pattern of vibration process in the infinite heterogeneous elastic lumped line

Begin with the most simple and evident case, when (8) is not true for all sections.

At these conditions both sections vibrate periodically and (2) – (4) retain unchanged. The process in the second section is especially interesting. According to the expression in the braces in (3), the vibration pattern in this section is the superposition of two progressive waves propagating to each other and having the amplitudes proportional to sin(₁ + ₂) and sin(_{1 - 2}) correspondingly. At the same time, the expression in braces can be presented otherwise:

In accord with (10), the process in the middle section can be presented as the superposition of two standing waves having different amplitudes and phase shifts. These two presentations are equivalent.

The typical vibration pattern is shown in Fig. 1 for two different values of the external force frequency. In Fig. 1a, in case of less frequency, we see the progressive wave in the middle section; it propagates into both directions from the external force application point. In the second case shown in Fig. 1b, the external force frequency is some less than the critical frequency ₀₂ . With it in the middle section we see practically standing waves, but in the heavy section they are almost anti-phase but not damping. This last corroborates the explanation of the vibration damping, when transiting to the aperiodical regime given by K. Magnus – true, only in case of two elastically connected masses. “The damping phenomenon can be explained in the following way. At the correct tuning, the second mass vibrates in anti-phase to the excitation and has namely such amplitude that the force of the second spring acting on the first mass counterpoises the excitation force transmitting through the first spring” [6, p.268]. It means, the vibrations damp not due to the anti-phase pattern of neighbouring masses, but due to the combination of the anti-phasing and vibration amplitude. Rather, not amplitude itself but the degree of element mass responses to the elastic link excitation. In absence of this correspondence, the damping will be absent, as we can see in Fig. 1b.

Now raise the frequency so that the condition (8) was violated for the heavy part of a line, and consider the following typical pattern.