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 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 (BVP) for the magnetomotive force (MMF) 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 BVP 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.