5. Dynamic stability of the rotor

The use of a control system to stabilise the unstable equilibrium of a magnetic bearing has been explained in the previous section. The rotor must then have a specific length-to-width ratio for a stable equilibrium of torques to be achieved. A requirement to be met by the control system is that it should be sufficiently stable al all frequencies of periodic disturbances. If a magnetically supported rotor - e.g. the one illustrated in fig. 6 - is set in rotation, its speed of revolution cannot be increased indefinitely, however. In this final section we shall consider the effects - other than the mechanical strength of the rotor - that set limits on the speed of revolution.

a
b
c
d
e
Fig. 14. a) Quantities for setting up the equations of motion of a magnetically supported rotor. (x,y,z) rectangular coordinate system. The position of the centre of mass Z of the rotor is determined by the coordinates x and z; The control system keeps Z in the (x,z)-plane. 2l length of the rotor. 2r diameter of a pole piece.  angular velocity of the rotor.  angles defining the direction of the axis of symmetry of the rotor. In the situations shown x = z = 0 and  = 0
b) The outer annular pole pieces of the rotor and stator, seen from Z in the y-direction. The rotor pole piece is shaded. The common region of both pole pieces - with maximum flux density - is shown hatched. C centre of stator pole piece. C' centre of rotor pole piece. rc vector from C to C'. angular velocity corresponding to a rotation of C' about C. Ps (stationary) point of the stator pole piece. Pt (moving) point of the rotor pole piece.
c) The movement of the centres C' of the rotor pole piece relative to the centres C of the stator pole pieces at the limiting speed . The point C' rotate in phase about C at an angular velocity . kr is a function of the loss coefficient of rotor and stator pole pieces.
d) as c) but at the limiting speed . The points C' rotate with a phase difference  and angular velocity  about C. The centre of mass Z does not change position, as it did in c).
e) The limiting speeds  and  as a function of K = I/J, the ratio of the moments of inertia along and perpendicular to the axis of symmetry. In the hatched area the rotor becomes unstable. This region should therefore be avoided in the design of magnetic bearings.

Fig. 14a shows the rotor rotating at an angular velocity  in a coordinate system (x,y,z). This system is fixed with respect to the non-rotating pole pieces, which we shall refer to as the 'stator pole pieces'. If the rotor rotates without deflection, the y-axis coincides with the axis of symmetry through the central points of the rotor pole pieces. We can write the equations of motion for the centre of mass Z of the rotor (neglecting the movement in the y-direction because the control loop is active in this direction):

(21)

and

(22)

where Fx represents the external forces in the x-direction, Fz those in the z-direction, Fg the gravitational force and M the mass of the rotor. Similarly, we can write the equations of motion for small rotations  and , which - together with the positions x and z of the centre of mass - establish the position of the rotor-symmetry axis in the coordinate system (x,y,z); see fig. l4a. Then we have:

(23)

and

(24)

where Tx and Tz are the torques acting on the rotor about the x- and z-axes, J is the moment of inertia about an axis perpendicular to the symmetry axis, and I is the moment of inertia about the axis of symmetry. The final terms of equations (23) and (24) are the 'gyroscopic' moments.

First of all we shall consider a translation of the rotating rotor as described by (21) and (22), i.e. without the rotations  and . The axis of symmetry then remains parallel to the y-axis. The terms Fx and Fz contain a component determined by the static stiffness krad for one half of the bearing as calculated above. Another component of Fx and Fz is a function of the velocity of the rotor pole pieces and the stator pole pieces relative to one another and is due to eddy currents.

Fig. l4b shows a rotor pole piece (shaded) that is in motion relative to a stationary stator pole piece, so that forces Frs and Frr act on the rotor pole piece. Frs is the reaction of the sum of the Lorentz forces on the eddy currents, Frr is the sum of the eddy currents in the rotor pole piece.

The force Frs acts in the opposite direction to the velocity of the centre point C' of the rotor pole piece relative to an observer at a point Ps of the stator, and is approximately proportional to this velocity.

(25)

where  is the derivative of the position for vector for C' from the centre C of the stator pole piece and crs is a loss coefficient for the stator.

Frs is proportional to the velocity of C relative to an observer at a point Pr on the rotating rotor pole piece. This velocity is equal to the sum of the vector  and the vector product . For Frr we can thus write

(26)

where crr is a loss coefficient for the rotor.

There is another way of showing that equation (26) must contain a term in . During one revolution the flux density for part of the surface of the rotor pole pieces varies. A surface element near the point Pt, for example, passes the common region (shown hatched) of the rotor and stator pole piece, where the flux density is at maximum, twice in one revolution. On the other hand, at a surface element of the stator pole pieces near the point Ps, for example, the flux density is constant during a revolution. Since the loss coefficient crs and crr for small values of rc can be taken as constants, Frs and Frr can be compared with the damping forces in a hydrodynamic bearing.

The summated damping force for a half-bearing may be written as

(27)

In general Frs + Frr has the opposite direction to  the velocity of the rotor centre. The summated force can however change direction at large values of , giving rise to oscillations and making the bearing unstable. Let us assume that the centre C' of a rotor pole piece describes a circular path with angular velocity  relative to the point C; see fig. 14b. If we substitute  in (27) we obtain, after some manipulation:

(28)

We see that when  instability occurs, with kr = crs/crr + 1. At the limiting speed  the rotor shaft describes a whirling movement relative to the y-axis. The centres C' of the rotor pole pieces at both ends then move in a circular path around the y-axis at an angular velocity , in the same direction as the rotation  of the rotor shaft; see fig. 14c. Manipulation of the equations of motion (21) and (22) shows that  is equal to the frequency  of the resonance of the rotor mass with the static radial stiffness. If crs = crr, - which does not necessarily correspond to a practical bearing - kr = 2; the whirling movements of the axis of symmetry of the rotor spinning at an angular velocity 2 may be compared with the 'half-omega whirl' that is known to occur in hydrodynamic bearings.

Closer inspection of the equations of motion (23) and (24) reveals that an instability can also occur due to precession of the axis of symmetry of the rotor about the y-axis, with the centre of mass Z remaining in position; see fig. 14a. At the corresponding limiting speed of the rotor the centres C' of the rotor pole pieces also describe circular movements, but now at an angular frequency . However, there is a phase difference  between the movements at opposite ends and the axis of symmetry describes a conical surface; see fig. 14d. The limiting speed depends on the ratio K = I/J, where I and J are the moments of inertia about the axis of symmetry and an axis perpendicular to it. Manipulation of (23) and (24) gives the following equation for the critical speed :

(29)

where

(30)

The quantity G is the (negative) stiffness of the rotor for rotation about an axis perpendicular to the axis of symmetry.

In fig. 14e the limiting speeds  and  are plotted as a function of the ratio K of the moments of inertia.

If  only the limiting speed  of the translation of the rotor determines the stability. For  both critical speeds determine the stability, each in a particular region of K. For K = (1/kr),  goes to infinity in theory. In all the types of bearings we have investigated it has been found that  is greater than , in other words the translation of the axis of symmetry of the rotor determines the stability.

In the derivations described above it has been assumed that the stator pole pieces are connected to the outside world with infinite stiffness. If finite stiffness and damping are taken into account for the mounting of the pole pieces, the limiting speed is not identical with the previously calculated value. By giving the pole-piece mounting an appropriate stiffness and damping, we were able to increase the limiting speed of our bearings. At the same time kr = crs/crr + 1 was increased. This was done by making the loss factor crs larger by introducing short-circuited windings in the concentric grooves of the stator pole pieces. (This was only done for the stator pole pieces in which the magnitude of the flux density is not controlled - the inner pole pieces in fig. 7 - since otherwise the characteristics of the control system would suffer.) In addition the loss factor crr was limited by making the rotor pole pieces of cobalt iron, which has a high electrical resistivity. With the type of bearing shown in fig. 7 we were finally able to achieve a ratio  greater than 4.4. As a result the limiting speed for this bearing is greater than 20 000 revolutions per minute.