SELF

62

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 whole vibration process is shifted by (-) ; it is determined by the imaginary unity in the right 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 of (3). In this connection the vibration pattern in the second section of a line will become more complicated. As we will show further, this will be the cause, why (3) describes not only the standing but also progressive waves.

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

arose. For the whole line, these frequencies determine two conditions transitive to the overcritical band, when the external affection frequency of each value in (7) exceeded. 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 passes to the aperiodical regime first, while in the second section the periodical regime of non-damping vibrations remains until (8) is true.

To analyse the solution transformation pattern when (8) is true for some section, it will be sufficient to transform 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. The 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 untrue for all sections.

At these conditions both sections vibrate periodically and (2)–(4) remain 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 towards each other and having the amplitudes proportional to sin (₁ + ₂)  and sin (_{1 - 2})  relatively. At the same time, the expression in braces can be presented otherwise:

In accordance 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.

Fig.1. Vibration diagrams in a heterogeneous elastic line at 8 Hz (a) and 22 Hz (b). The external force frequency is lower than that boundary for both parts of line;  F₀ = 0,6 N; s = 100 N/m; m₁ = 0,01 kg; ₀₁ = 200 sec^-1; m₂ = 0,02 kg; ₀₂ = 141,42 sec^-1.

The typical vibration pattern is shown in Fig.1 for two different values of the external force frequency. In the first case, in Fig. 1a, the frequency is small, and in the middle section we see the progressive wave that propagates to 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 boundary frequency  ₀₂ . With it in the middle section we see practically standing waves; in the heavy section they are almost antiphase but not damping. This last corroborates the explanation of the vibration damping, when passing to the aperiodical regime given by K. Magnus – true, only in case of two elastically constrained masses. “The vibration damping phenomenon can be explained as follows. With a correct tuning, the second mass vibrates in antiphase to the excitation and has namely such amplitude that the force of the second spring affection 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 antiphase pattern of neighbouring masses, but due to the combination of vibration amplitude and opposite phase. Rather, not amplitude itself but the degree of element mass response 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.