Improved stiffness modeling of permanent magnetic bearings with relative permeability correction

Analytical stiffness models are most effective in the preliminary design of permanent magnetic bearings (PMBs). While semi-analytical models based on Coulombian and Amperian formulations provide high computational efficiency, they cannot directly account for the finite relative permeability of magnetic materials, potentially leading to an overestimation of bearing stiffness. This paper presents an improved semi-analytical approach that considers the non-unity relative permeability of rare-earth magnets. First, a correction model iteratively estimates the effective magnetization of the magnets, which is then used as input to idealized semi-analytical stiffness models. Two correction methodologies are proposed and compared: (i) the load line correction method (LLCM), based on magnet demagnetization curves, and (ii) the surface charge correction method (SCCM), based on a surface charge density formulation. These models build on established frameworks and are tailored for PMBs with a focus on reducing the computational cost. The improved semi-analytical model is verified against finite element simulations to assess its accuracy for axially magnetized radial PMBs. Results show that the LLCM and SCCM improve radial stiffness and axial force estimation. For single-layer radial PMBs, the SCCM achieves lower modeling errors, while the LLCM offers superior computational efficiency. For multi-layer radial PMBs, the LLCM yields lower modeling errors while preserving its computational advantages for the investigated cases. Overall, the improved model offers a computationally efficient and robust design tool, well-suited for preliminary bearing design, particularly in high-precision applications.