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.
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 F_{x} represents the external forces in the x-direction, F_{z} those in the z-direction, F_{g} 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 T_{x} and T_{z} 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 F_{x} and F_{z} contain a component determined by the static stiffness k_{rad} for one half of the bearing as calculated above. Another component of F_{x} and F_{z} 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 F_{rs} and F_{rr} act on the rotor pole piece. F_{rs} is the reaction of the sum of the Lorentz forces on the eddy currents, F_{rr} is the sum of the eddy currents in the rotor pole piece.
The force F_{rs} 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 P_{s} 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 c_{rs} is a loss coefficient for the stator.
F_{rs} is proportional to the velocity of C relative to an observer at a point P_{r} on the rotating rotor pole piece. This velocity is equal to the sum of the vector and the vector product . For F_{rr} we can thus write
(26) |
where c_{rr} 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 P_{t}, 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 P_{s}, for example, the flux density is constant during a revolution. Since the loss coefficient c_{rs} and c_{rr} for small values of r_{c} can be taken as constants, F_{rs} and F_{rr} 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 F_{rs} + F_{rr} 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 k_{r} = c_{rs}/c_{rr} + 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 c_{rs} = c_{rr}, - which does not necessarily correspond to a practical bearing - k_{r} = 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/k_{r}), 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 k_{r} = c_{rs}/c_{rr} + 1 was increased. This was done by making the loss factor c_{rs} 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 c_{rr} 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.