Materials.Technologies.Tools

16

S.B. Karavashkin

2. Analysis and check of results yielded for forced vibrations

2.1. Semi-finite line

Begin this consideration with the most simple and spectacular model of semi-finite elastic line. Its general form is presented in Fig. 2.

Fig. 2. General form the model of semi-finite elastic lumped line

The system of differential equations describing this model is

As we see, (4) consists of the boundary (first) equation and of an infinite number of equations similar by their structure and describing the pattern of all rest masses. The external affecting force depends on time by the harmonic function

where F₀ is the external force amplitude.

The connective (4)–(5) is known to describe steady vibrations, which seems some limiting the range of study, because excludes from the analysis the transitional processes in the line (though this simplification is usual in the wave physics and is applied to the most problems being solved by its methods). However, taking into account that any complex affection can be presented as a superposition of some simple harmonic vibrations, we see that such limitation does not take place in reality. When yielded, the base analytic solutions show sufficient if we simply account both the spectral characteristics and the initial phase of vibration, thereby having provided the scope to model most structurally complicated processes in the line. But at the stage of study of the base solutions themselves, an additional generalisation can only cumber the exposition and hamper the analysis. Proceeding from this last, in this study we will use (5) as the regularity of affecting force variation, though in the course of analysis we will, of course, bear in mind the above peculiarity related to the steady-state pattern of vibrations.

The considered system has three solutions:

periodical at  < 1

aperiodical at > 1

and critical at = 1

where

This means, the solutions vary in accordance with the value - and consequently, at the given parameters of line, with the value of force frequency connected with through (9).

For periodical regime (6), the amplitude remaining along the line and the phase delay by 2 with the growing number of elements are typical. The typical form of vibration for this regime is shown in Fig. 3. It is basically important that , as it was in conventional solutions (2), but is equal to arcsin (see (10)). With it  when 0 , i.e., when the line by its characteristics tends to that distributed.

Fig. 3. The wave propagation in a semi-finite homogeneous elastic line vibrating in the periodic regime (f = 15 Hz, F₀ = 0,6 N, m = 0,01 kg, s = 100 N/m, a = 0,01 m, f_crit = 31,8 Hz)

Check the correctness of solution (6), substituting it to the first equation of the system. In the left part we yield

Substituting (6) to the right part of the first equation of (4), we yield