V.2 No 1

53

On complex resonance vibration systems calculation

To analyse the vibration process arising in the elastic line having the resonance subsystems which is presented in Fig. 1, write the solutions in their general form. In case of forced vibrations they have the following form:

for the periodical regime at _g < 1

(9)

for the aperiodical regime at _g > 1

(10)

and for the critical regime at _g = 1

(11)

where

, , ;

 _i is the momentary displacement of the ith subsystem and ₀ is the initial phase of external action.

The presence of the resonance pattern of subsystems naturally leads to the features appearing in solutions (9) – (11). First of all, the presence of resonance peaks in the regularity M() makes impossible to divide exactly the frequency band into subcritical and overcritical regimes. The elastic line will multiply undergo one of these regimes dependently on the value M. The more, the possibility of negative M causes the additional, fourth vibration regime corresponding to the complex value of _g. The solution for this regime, which we will name the complex aperiodical regime, will be the following:

(12)

where

, .

As we can see from (12), in the complex aperiodical regime the vibration amplitude

(13)

varies differently, dependently on the subsystem number i.