The
typical form of regularity M( ) plotted for odd and even numbers of subsystem elements n on
the basis (7) is shown in Fig. 3. It fully corroborates the above analysis. We can see
from the construction that this regularity is the succession of resonance peaks whose
density increases with frequency, and the peaks width decreases with it. With the
transition to the aperiodical regime, the subsystem inertia gains the monotonously
increasing pattern with respect to frequency. We should especially note the pattern of
this regularity at low frequencies. In the region preceding the first resonance peak, the
subsystem inertia increases monotonously. In the considerable part of the band it is
approximately equal to the subsystem elements total mass, and at  0 the value M nm. It is easy to check it, noting the values  s and  s , when
finding the limiting value for the first expression of (7).
On the
basis of subsystem inertia value, we can easy determine the exact analytical solution for
the whole elastic system shown in Fig. 1.
We will use the fact that the subsystem inertia in (7) depends not on time but only on the
external action frequency. So, under external harmonic force action, the measure of
subsystems inertia may be regarded as constant for each frequency, and the features caused
by the subsystems resonances reveal only in the external action frequency variation.
Thus we
can use directly the solutions presented in [20] for a semi-finite elastic lumped line.
The same as in case of solutions for a finite elastic line, their form depends on the
relationship between the parameter g = (2M/4sg)1/2 and the unity. We should
especially mark that despite the vibration pattern of the elements of an elastic line as a
whole and of subsystem elements depends on parameters g and s having similar functional
regularity, with respect to frequency they behave essentially different. The parameter s depends on
the subsystem elements masses that are constant for a specific line, while g depends on the
measure of inertia of subsystem (7) which depends on frequency nonlinearly, and at
definite values s
becomes negative. With it g
becomes complex, which is impossible for s. In this connection, the features appear in the
vibration of elements within the subsystem and of the subsystems as elements of general
system. For the subsystems, the same as for simple elastic lines, it is typical a clear
division of the range into the periodical, aperiodical and critical vibration regimes with
the single boundary frequency corresponding to the critical vibration regime. For the
elastic line as a whole it is typical some other range division. At low frequencies,
before the first resonance peak of subsystem, the considered elastic line behaves the same
as a simple elastic line without any resonance subsystems. The boundary of this range is
the first boundary frequency 0g , which is close to the similar frequency of a
simple elastic line whose element masses are equal to the total static mass of the
subsystem. Naturally, this boundary frequency is lower than that of subsystem 0s . Higher
than 0g ,
in a simple elastic line there takes place the aperiodical regime of antiphase vibrations
damping along the line. In the line having the resonance subsystems, there reveals the
influence of subsystem’s measure of inertia dependently on frequency, which
determines the vibration pattern up to the critical frequency 0s for the resonance subsystem. Due
to this feature, further we will distinguish the concepts of subsystem element and the
elastic line element being the resonance subsystem. |
