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Some features of the forced vibrations modelling

Furthermore, the presented regularity of the phase velocity of wave propagation with respect to frequency much complicates, or rather makes impossible to obtain straight the exact analytical solutions for the acting complex-form pulses, particularly the rectangular pulses, out of their decomposing into a spectrum. This is especially important to note, because many authors try currently to solve the dynamical problems, giving as the statement just rectangular pulses. Hence, with such approach, the researchers make themselves in advance disable to obtain a high-quality solution.

When the frequency tends to the second boundary value of the band related to the periodical regime, i.e., when 1,

Correspondingly, the velocity

It means, when transiting to the critical regime and further to that aperiodical, the wavelength tends to the double distance between the non-excited elements of a line; it corresponds to the anti-phase vibrations which we observed both in [2] and in the present paper. Similarly, the propagation velocity does not vanish with the limiting process, but stabilises at some ultimate value v_crit. To illustrate it, in Fig. 5 the typical regularities () and v()  are presented.

It presents a difficulty to investigate more detailed the group and phase velocities in the frames of an ideal elastic line model, because the fact itself that the elements vibrate in anti-phase in the critical and aperiodical regimes does not evidence yet that the wave propagation velocity in these regimes is absent. We showed it obviously in our paper [18], when analysing the wave propagation processes in a resistant line. Passing to the limit at the vanishing line resistance, we showed that in an ideal line the phase velocity in the aperiodical regime grows linearly with the growing frequency from the minimal value (67). Generally, the group velocity also exists at the overcritical frequency, but in transition to an ideal line its value actually turns into infinity. Just this last caused the incorrect conclusion settled in the literature that in elastic lines the group velocity does not exist at the overcritical frequencies. Up to now the scientists used in modelling some fragmentary results accessible in the absence of the exact analytical solutions, and only for ideal lines (see, e.g., [15]). However the exact analytical solutions show that if a line had a least impedance, the group velocity of the wave propagation will exist, though its value will be very large. More detailed, see in the indicated paper.

6. The limiting process to a distributed line

Investigating the limiting process for the vibration parameters, in view of the practice of modelling, it is important to determine the conditions, when the line elements can be considered as those distributed.

To determine them, consider the parameter . As and v were determined, (53) takes the form

where = m / a is the line elements density, and T = sa is the line tension. Since the transition to a distributed line occurs at 0, we can write the sought condition so:

follows. Noting that, when transiting to a distributed line, the velocity varies slightly, we can simplify (70):

whence

It means that only when the wavelength well exceeds the distance between the line elements, this line parameters can be considered as those distributed.