An investigation on the improved magnetic stiffness model and characteristic analysis for two cylindrical permanent magnets

As a functional material, permanent magnets (PMs) are widely used in magnetic actuation, magnetic bearings, energy harvesters, and lithography semiconductor industries. The magnetic stiffness determines the stability and performance of the magnetic systems. Therefore, an accurate and efficient magnetic stiffness model tool is needed. This paper newly establishes an improved high precision magnetic stiffness model between two cylindrical PMs for parallel and perpendicular magnetization directions, which takes the PMs' relative permeability, residual magnetic flux density, magnetization direction, dimensions, and relative positions into account. The magnetic stiffness model is solved by a numerical algorithm. Moreover, the magnetic stiffness calculation results are indirectly validated by the finite-element method (FEM) and experimental measurement, and the accuracy and efficacy of the established model are demonstrated. Furthermore, the sensitivity of magnetic stiffness to magnets' size and position parameters is explored, and the influence of these parameters on magnetic stiffness characteristics is studied and discussed. The results indicate that the relative positions between PMs have a significant influence on magnetic stiffness. Thus the accurate control of the relative position is important to design PM devices. In addition, the developed magnetic stiffness model has lower computational efforts than the FEM and lower costs than experimental measurement. The model includes all magnetic properties and relative positions parameters, which can calculate the principal magnetic stiffness and cross-coupling magnetic stiffness and doesn't produce any principle error. The established model makes it easy to design and optimize PM devices that depend on magnetic stiffness, such as magnetic bearings, magnetic vibration isolator, magnetic piezoelectric cantilever beam energy harvester, etc.