Gennadey. B. Filimonikhin

CONDITIONS of an EQUILIBRATION of a ROTOR by a absolutely rigid BODY with a fixed POINT on an AXIS of the SHAFT

(Reports of the National Academy of Scienses of Ukraine, 2001, N 1, pp. 65-69)

(is submitted by the member - correspondent NAS of Ukraine Mikle. O. Perestuk)

It was obtained conditions imposed on the geometry of mass of a absolutely rigid body, at which fulfillment this body with a fixed point on an axis of the shaft of a rotor can balanced the rotor in a certain plane of correction. The examples of absolute rigid bodies satisfying to obtained conditions are presented.

The passive autobalancers, such as pendulum-type, ring-type, ball-type etc are applied for balancing of the fast rotated rotors on the move. In devices the mobile masses with time come in a position, in which they the most counterbalancing a rotor and at a constant disbalance and the speed of rotation of a rotor move with him as one whole [1]. In this work, by available materials, for the first time, are determined the conditions, at which fulfillment the absolutely rigid body (ARB) with a fixed point on an axis of the shaft can balanced a rotor in a certain plane of correction. The position of the plane of correction and geometry of masses of ARB is determined.

Let in a point O of an axis of the shaft is suspended ARB. We assume, that the following conditions of ideal balancing of a rotor are executed:

1) The motion of a system a rotor - ARB is steady-state, therefore the relative motion ARB is stopped, and the system rotate with a constant angular speed;

2) axis of the shaft is combined with rotation axis;

3) gravity do not noticeably influence on a motion of a system and their operation can be neglected.

In order the ARB could counterbalance a disbalance its center of masses should not coincide with a point of suspend. A distance from a point of suspend up to center of masses of a ARB we shall designate through r. We assume, that the disbalance of a rotor is such limited, that the ARB could it counterbalance.

We shall conduct through a point O of principal axes of inertia of a ARB and we shall designate them through x, h, z. Let concerning them the main axial moments of inertia is J_x, J_h, J_z. The motion of a ARB round a fixed point is set by equations of the Euler [2]:

J_xdw_x/dt+(J_z-J_h)w_hw_z=M_x, J_hdw_h/dt+(J_x-J_z)w_zw_x=M_h, J_zdw _z /dt+(J_h-J_x)w_xw_h=M_z. (1)

Here: w_x, w_h, w_z - projection of angular speed of a ARB on an axes x, h, z; M_x, M_h, M_z - moment of external forces concerning axes x, h, z.

In steady-state motions the derivatives address in a zero. The moment form of external forces gravity, but under the supposition their influence can be neglected. Then, the equations of steady-state motion of a ARB will take a kind

(J_z-J_h)w_hw_z=0, (J_x-J_z)w_zw_x=0, (J_h-J_x)w_xw_h=0. (2)

Let's consider equations (2) as conditions. At their fulfillment the main moment of forces of inertia of a ARB concerning a point O is equal to zero and the forces of inertia do not reject a ARB from a position, in which it counterbalances a rotor. Also forces of inertia are reduced to resultant, which is affixed in a point O. As all elementary forces of inertia are perpendicular axes of the shaft, also their resultant is perpendicular axes. Therefore plane of correction of a ARB passes through a point of suspend O and is perpendicular to the shaft.

We shall conduct through a point O mobile axes x, y, z. Let thus the axis z coincides rotation axis and is directed to a leg of an angular-velocity vector, and the axes x, y rotate with this speed together with a rotor and are directed so, that a system of axes right-hand. Let's remark, that In steady-state motions of an axes x, h, z take under the attitude of axes x, y, z a certain fixed position and rotate together with them with constant angular speed. Let's consider the following in essence various cases.

1. The axes x, h, z can be obtained from axes x, y, z by a turn round an axis z on a angle q (fig. 1, a). Let's remark, that ARB can make any spherical motions. Then an axis z a principal axis of inertia,

w_x=w_h=0, w_z=w, (3)

and the equations (2) are executed identically.

Figure 1. In essence various cases of obtaining of principal axes x, h, z from axes x, y, z.

In this case one ARB can not counterbalance any on size a disbalance. In steady-state motions its center of masses moves in a plane of correction on a circle. Therefore disbalance, which is counterbalanced, is constant on size and can change only direction.

Any disbalance can counterbalanced two ARB, but at fulfillment of a side condition:

m₁h₁=m₂h₂, (4)

- at two ARB are identical a product of mass on a distance from center of masses up to an axis of the shaft.

This condition is necessary that at absence of a disbalance two ARB could each other balance.

Let's assume, that the ARB can turn only round an axis z. Then to case of equal ARB there correspond classical autobalancers, such, as ring-type, ball-type, pendulum-type (with pendulums on the shaft of a rotor) [1]. To this case there correspond also pendulum-type autobalancers with connections [3]. To case of different ARB there correspond row-type ball autobalancer [4].

2. The axes x, h, z can be obtained from axes x, y, z by two series turns: round an axis z on a angle q; round an axis x on a angle j (fig. 1, b). ARB can make any spherical motions. Then

w_x=0, w_h, w_z�0, (5)

and from equations (2) is discovered the following condition

J_h=J_z. (6)

At absence of a disbalance the center of masses of a ARB should be on an axis of the shaft. In view of sequence of rotation of axes it will be possible if the center of masses of a ARB will be in a plane Ohz. By virtue of a condition (6) each axes lying in this plane and passing through a point O is main. Therefore not limiting a generality it is possible to consider, that in principal axes x, h, z the center of masses of a ARB should have coordinates

r_G=(0, 0, -r). (7)

An elementary system satisfying to conditions (6), (7) are two mathematical pendulums, connected under a right angle, (fig. 2, a). From an elementary system it is possible to receive others ARB, satisfying to these conditions. It is two rods, connected under a right angle, or physical pendulum (fig. 2, b, c), semiring and half-disk (fig. 2, d, e). From flat figures it is possible to receive volumetric. Thus on a property of principal axes [2], last should be symmetric concerning a plane of an initial figure (fig. 2, f, g). Satisfy to a condition (5) half-spheres and half-ball (fig. 2, k, l). From volumetric ARB at the least overall dimensions the greatest balance capacity has the half-ball. Among flat ARB such properties are exhibited with a half-disk.

3. The axes x, h, z can be obtained from axes x, y, z by two series turns: round an axis x on a angle j; round an axis h on a angle y (fig. 1, c). Then

w_x, w_h, w _z�0, (8)

And from equations (2) is discovered the following conditions

J_x=J_h=J_z. (9)

Because of a symmetry of a tensor of inertia, not limiting a generality it is possible to suppose, that in principal axes x, h, z the center of masses of a ARB has coordinates

r_G=(0, 0, -r). (10)

The elementary system satisfying to conditions (9), (10), consists of three mutual - perpendicular mathematical pendulums (fig. 2, h). From this system it is possible to receive others ARB, satisfying to conditions (9), (10). It is three mutual - perpendicular rods or physical pendulum (fig. 2, i, j). To these conditions satisfy a half-sphere and half-ball (fig. 2, k, l).

Figure 2. Absolute rigid bodies with a fixed point on an axis of the shaft of a rotor

4. The axes x, h, z can be obtained from axes x, y, z by three series turns: round an axis z on a angle q; round an axis x�, in which the axis x after the first turn on a angle j passes; round an axis h on a angle y (fig. 1, d). This case, as well as previous, results in conditions (9), (10).

In the considered cases it was supposed, that the ARB can make any spherical motions round a point of suspension. For balancing a rotor such quantity of degree of freedoms is excessive. However in spherical motion ARB, installed in viscous environment can simultaneously counterbalance a rotor and to damp its vibrations (angular and torsional oscillations). The imposing of connections on motion of a ARB allows to receive autobalancers synchronously rotated together with a rotor [5]. It possible to use for balancing rotors, which speed of rotation change in time.

From a mathematical point of view for functioning an autobalancer it is necessary, that the steady-state motion of a system, in which the rotor is counterbalanced, was steady. Autobalancers with corrective masses which moved in planes, perpendicular shaft of a rotor now are most investigated. Dynamics of balancing of a rotor of corrective masses moved in planes, passing through an axis of the shaft of a rotor was investigated in [6]. Remaining cases of motion of a ARB as autobalancer round a point on an axis of the shaft of a rotor under the available data on today were not investigated. The research of stability of steady-state motions leaves for frameworks of the present paper.

Has arrived in edition 23.12.1999

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