V.2 No 1

83

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 yielding directly the exact analytical solutions for the affecting complex-form pulses, in particular the rectangular pulses, without their decomposing into a spectrum. This is especially important to account, because many authors currently try to solve the dynamical problems, giving as the statement just rectangular pulses of affection. Hence, with such approach, the researchers disable themselves in advance yielding 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,

The related velocity is

It means, when passing to the critical regime and further to that aperiodical, the wavelength tends to the double distance between the line elements at rest; it corresponds to the anti-phase vibrations which we saw 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 shown.

Fig. 5. The plots v() and (). The elastic line parameters are m = 0,01 kg ; s = 100 N/m ; a = 0,01 m ; _crit = 200 s ^-1 ;  _crit = 0,02 m

It is difficult to study in more details the group and phase velocities in frames of an ideal elastic line model, because the fact itself that the elements vibrate in anti-phase in the critical and aperiodical regimes is yet not an evidence that the wave propagation velocity in these regimes is absent. We clearly showed it in [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 of (67). Generally, the group velocity also exists at the overcritical frequency, but in passing 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. The matter is, up to now the scientists used in modelling some fragmentary results accessible in the absence of exact analytical solutions, and only for ideal lines (see e.g. [15]). However, the exact analytical solutions show that if a line had at least an infinitesimal impedance, the group velocity of the wave propagation will exist, though its value will be very large. In more details, see in [18].

6. The limiting process to a distributed line

Studying the limiting process for the vibration parameters, in view of practical 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):

where from

It means that only when the wavelength well exceeded the distance between the line elements, we can consider the parameters of this line as those distributed.