Magnetic bearings offer high speed and low power losses as compared to film riding and rolling element bearings. Significant efforts are underway to apply magnetic bearings to gas turbines and jet aircraft engines. Negative stiffness coefficients for magnetic actuators can have a significant impact on shaft rotordynamics. These coefficients are typically computed as the sensitivity of a magnetic force expression derived from a lumped parameter reluctance network. However, as the complexity of magnetic actuator designs increases, the reluctance network method may become impractical for, or even incapable of, coefficient determination. In this paper, an alternative method is presented for determination of negative stiffness coefficients for a large class of magnetic actuators. The method solves the Dirichlet boundary value problem for the magnetomotive force in the actuator air gap, subject to periodic boundary conditions that can be represented by Fourier series. A conformal transformation to bipolar coordinates is used that results in a boundary value problem that is solvable using separation of variables. Negative stiffness coefficients are presented and the method is benchmarked against well-known solutions using the reluctance network method.

1.
Schultz, R. R., Bornstein, K. R., Jayawant, R. C., and Leung, R., 1995, “Applications of Active Magnetic Bearings to High Speed Turbomachinery with Aerodynamic Rotor Disturbance,” Proceedings of Mag ’95, Technomic, Lancaster, PA, pp. 14–23.
2.
Iannello, V., 1995, “Magnetic Bearing Systems for Gas Turbine Engines,” Proceedings of Mag ’95, Technomic, Lancaster, PA, 1995, pp. 77–86.
3.
Kelleher, W. P., and Kondoleon, A. S., 1997, “A Magnetic Bearing Suspension System for High Temperature Gas Turbine Applications,” Proceedings of MAG ’97, Technomic, Lancaster, PA, pp. 15–24.
4.
Maslen
,
E.
et al.
,
1989
, “
Practical Limits to the Performance of Magnetic Bearings: Peak Force, Slew Rate, and Displacement Sensitivity
,”
ASME J. Tribol.
,
111
, pp.
331
336
.
5.
Okada
,
Y.
,
Miyamoto
,
S.
, and
Ohishi
,
T.
,
1996
, “
Levitation and Torque Control of Internal Permanent Magnet Type Bearingless Motor
,”
IEEE Trans. on Control Systems Technology
,
4
, No.
5
, pp.
565
571
.
6.
Bischel, J., 1991, “The Bearingless Electrical Machine,” Proceedings of the International Symposium on Magnetic Suspension Technology ’91, NASA Langley Research Center, p. 561.
7.
Maslen
,
E.
et al.
,
1996
, “
Magnetic Bearing Design for Reduced Power Consumption
,”
ASME J. Tribol.
,
118
, pp.
839
845
.
8.
Meeks, C., 1993, “Magnetic Bearing Structure Providing Radial, Axial, and Moment Load Bearing Support for a Rotating Shaft,” U.S. Patent No. 5,216,308.
9.
Sortore, C., Allaire, P., Maslen, E., Humphris, R., and Studer, P., 1990, “Permanent Magnet Biased Magnetic Bearings—Design, Construction, and Testing,” Proceedings of 2nd International Symposium on Magnetic Bearings, Tokyo, July 12–14, University of Tokyo, Tokyo.
10.
Knospe
,
C. R.
, and
Stephens
,
L. S.
,
1996
, “
Side-Pull and Stiffness of Magnetic Bearing Radial Flux Return Paths
,”
ASME J. Tribol.
,
118
, pp.
98
101
.
11.
Walowit
,
J. A.
, and
Pinkus
,
O.
,
1982
, “
Analytical and Experimental Investigation of Magnetic Support Systems. Part 1: Analysis
,”
ASME J. Lubr. Technol.
,
104
, pp.
418
428
.
12.
Happel, J., and Brenner, H., 1986, Low Reynolds Number Hydrodynamics, Matrinus Nijhoff, Boston, pp. 474–499.
You do not currently have access to this content.