By development of mathematical model the following assumptions have been made:

·   Bearing assemblies of the main shaft are in rather intensive vibrational condition. It testifies to a yielding of bearing pedestals of the engine;

· using a nomenclature of classical mechanics, the main shaft of the engine together with a drum it is possible to assimilate to a (whipping) top with a motionless point in the left half-coupling;

·   in neighbourhoods of the hub the transversal sag of the shaft is less than sag of a support, at least, on two order even at the most crude estimations of its yielding. On this foundation we shall count a (whipping)  top as an absolute rigid body, besides a proper rotation making also characteristic movements, called as a precession and a nutation.

Fig. 1 - The diagram of an action of forces on the main shaft of the engine.

On fig.1 in two projections the calculating diagram of main shaft OA it is shown schematically with the adjoining shaft of motor OD. The main shaft of the engine leans on compliant supports of bearings in points A and B, and the shaft of a rotor leans on an absolutly rigid supports in points C and D. A direction of axes ÎX, ÎY and ÎZ are indicated on fig. 1 and axes ÕÓZ form the right frame of reference.

Bearings in points A and B have stiffness coefficients Ñ_X and Ñ_Y according to directions of axes of coordinates. Forces G'_Á = G_Á/2, where G_Á is a weight of a drum of the engine, distributed equally on two hubs in points of À', Â'. Force G_B is a weight of the main shaft supposed be concentrated in a point O', being middle of span ÀÂ. Forces Ð_Õk and Ð_Yk (k = 1, 2 ..., n) are projections to the corresponding axes of summarized efforts in tight and slack strand k-th cable at their common amount equal n.

For case of a disposition of deviation pulleys on a slack strand they are equal:

where F_íàá and F_ñá are summarized dynamic forces in tight and slack strand of all cables; α₀ - an angle of a girth a cable of a friction pulley; δR_k - a deviation of a k-th radius groove from average radius of (winding) coiling R; A - a (modular) aggregate longitudinal rigidity of a cable; f₁, f₂ dimensionless functions.

Forces F_Õ1, F_Õ2, F_Y1, F_Y2 are projections of forces of action of the corresponding hubs to the main shaft in points of À', Â'.

Reactions of elastic supports in points A and B with use of relations (2) with a condition of equality to null of the moments of forces relatively a point O and on the basis of the fourth assumption concerning of zero lateral deformation of the shaft reactions formulates as follows:

Where it is designated δ" = 2 (Δ + Δ') + (n - 1) δ + δ' - length of the main shaft. From fig.1 it is evident, that

In turn reactions R_X1 and R_Y1 are defined as product of stiffness coefficients Ñ_X, Ñ_Y on the corresponding cross motion of the shaft in a point A (see fig. 1) Õ_À, Y_À, taken with opposite sign, that is R_Õ1 =-Ñ_ÕÕ_À, R_Y1 =-Ñ_YY_À. As a radius-vector ÎÀ in a mobile system has components 0, 0, δ", then components of same vector Õ_À, Y_À, Z_À in motionless axes deduce with the help à_k = À_kià', á'_k = À_ikà_i, having produced the following operations of a squarte matrix on a column matrix:

Hence, Õ_À = δ"βsinα, Y_À =-δ"βcosα, thus, dynamic components of a reaction of the right bearing can be defined as

R_X1=-Ñ_Xδ"βsin(α), R_Y1=Ñ_Yδ"βcos(α)                                                                         (5)

Similarly by virtue of conditions of proportionality (4) similar relations for the right bearing we’ll formulate:

R_X2 =-Ñ_Xδ'βsin(α), R_Y2=Ñ_Yδ'βcos(α).                                                                         (6)

Quasidynamic quantities of a precession and a nutation can be defined with the help of the first and third expressions from (2) as solutions of equations set

R_X1 =-Ñ_Xδ"βsin(α), R_Y1=Ñ_Yδ"βcos(α).                                                                       (7)

By virtue of conditions of proportionality we shall receive

R_X2 =-Ñ_Xδ'βsin(α), R_Y2=Ñ_Yδ'βcos(α).                                                                          (8)

From (7) and (8) it follows:

Where α and β quasidynamic values of dynamic quantities of a precession α and nutations β of the shaft.

Components of an axial vector of the moments of outside forces M concerning a point O (see fig. 1):

Where Ì_Ý - the electrodynamic moment of a motor.

If in a relation (10) to substitute values (7) and (8), we shall receive:

Let's remark, that (11) defines components of the moments of forces in a motionless frame, and it is essential condition for an output of the differential equations of motion of a top. As it has been marked above, the main shaft of the engine with hafted drum represents classical type of a (whipping) top with a motionless point O.

At the simplified approach it is enough to suppose the main shaft a usual rotator in length δ" and mass G_Â/g, and a drum as a hollow-core cylindrical solid in length l_á = 2Δ+(n-1)δ and mass G_Á/g. Then with a sufficient exactitude for the engineering purposes it is possible to write:

The kinetic moment of inertia I_Z is equal:

I_Z=GD2/4g.                                                                                             (13)

So, the obtained formulas can serve initial ones for a development and the further solutions of dynamic equation of a concerned system.