Negative stiffness coefficients for magnetic actuators using Laplace's equation

Magnetic bearings offer high speed and low power losses as compared to film riding and rolling element bearings. Significant efforts are underway to a pply magnetic bearings to gas turbines and jet aircraft engines. Negative s tiffness coefficients for magnetic actuators can have a significant impact on shaft rotordynamics. These coefficients are typically computed as the se nsitivity of a magnetic force expression derived from a lumped parameter re luctance network. However, as the complexity of magnetic actuator designs i ncreases, the reluctance network method may become tn impractical for, or e ven incapable of coefficient determination. In this paper an alternative me thod is presented for determination of negative stiffness coefficients for a Inr-se class of magnetic actuators. The method solves the Dirichlet bound ary value problem far the,magnetomotive force in the actuator air gap, subj ect to periodic boundary conditions that can be represented by Fourier seri es. A conformal transformation to bipolar coordinates is used that results lit a boundary value problem that is solvable using separation of variables . Negative stiffness coefficients ar-e presented and the method is benchmar ked against well-known solutions using the reluctance network method.