36

S.B. Karavashkin and O.N. Karavashkina

In its turn, this causes the thrice-fold narrowing of the frequency band of periodical regime for the third harmonic, with retaining the general number of resonance peaks which are located now on the lower one third of the band of the first harmonic. At the frequency higher than that boundary ₀₃, vibrations of the third harmonic correspond to the aperiodical along-line-damping vibration regime and are localised in the regions of application of the equivalent forces. At the same time, the periodical vibration regime of the first harmonic remains and effects on the vibration amplitude of the third harmonic through the value of equivalent forces Q_i3 in accordance with (31)- (33). Due to this, in the present case, despite the non-resonant vibration pattern in the aperiodical vibration regime, at the overcritical range of the third harmonic there arise the resonances, introduced to it from the first harmonic. These resonances are limited by the regions of affection of equivalent forces, because they arise on the background of vibrations of the third harmonic effectively damping in space, which is typical for the aperiodical regime.

To reveal the general regularities of the next harmonics, determine the fourth harmonic.

On the basis of (13), the system of equations for its finding has the following form

is also true. However if (19) was not true, then due to the general structure of (20) and (36) the solution of (36) is similar to (30), with the substitution of Q_i3 into Q_i4 and changing the coefficient at from 3 to 4. With it the parameter ₄ changes, it becomes four times greater than ₁, and the boundary frequency ₀₄ becomes one-fourth as many as ₀₁, too. All features of the resonances described above for the third harmonic remain true for this fourth.

Generalising the investigation of four harmonics, we can state that for all the following harmonics the structure of systems of equations will remain, and we can represent it in the following form: