V.2 No 1

51

On complex resonance vibration systems calculation

Transforming (1) – (3) for the cases of our interest k = 1 and k = n and summing the results with regard to resulting displacement equal to the half-sum of displacements caused by each force, we yield:

for the periodical regime, _s < 1

(4)

for the aperiodical regime, _s > 1

(5)

and for that critical _s = 1

(6)

In (4) – (6) the full symmetry of the studied subsystem reflects, hence the boundary elements displacements will be synchronous in all the vibration regimes. So we can present the subsystem as some rigid body having the measure of inertia M as

(7)

We see from (7) that when the substructure is resonance, the measure of inertia of the main line elements depends on frequency nonlinearly. When n_s = r at r = 0, 1, … [n/2] or (n – 1) _s = (2r + 1) rat r = 0, 1, … [(n – 2)/2] were true, zeroes and poles correspondingly appear in the plot M(), and when conditions

or

(8)

were true, the measure of subsystem inertia becomes negative. Skudrzyk also mentioned that the negative values of mechanical subsystem parameters appear possible: “In case of transient (mechanical) conductivity, it is not necessary, all the equivalent scheme elements to be positive. In the canonical scheme having negative elements, the vibration phase against the excitation force is opposite to the vibration phase in the scheme having positive elements… The energetic principle does not contradict it, because the excitation force work does not relate to the vibration velocity at any other points except the excitation point. In the canonical scheme for the transition (mechanical) conductivity, some shunting circuits will consist of negative elements, and due to it, mutual wave suppression will be possible” [1, p.320]. And vice versa, “the reason of limited pattern of conductivity variation at the excitation point is the requirement, all the elements of equivalent canonical scheme to be positive” [1, p.320].