Semi-analytical Expression of Force and Stiffness of Perpendicular Polarized Ring Magnets for Nonlinear Dynamic Analysis

Magnetic coupling arrays composed of ring permanent magnets have been widely applied in industrial occasions for achieving nonlinearity, such as passive magnetic bearings, quasi-zero stiffness isolators and multi-stable energy harvesters. These magnetic couplings can be divided into basic configurations including axial magnetization, radial magnetization, and perpendicular magnetization. For the purpose of structure design and parameter optimization, the semi-analytical expressions of first two configurations have been analyzed to obtain high accuracy and low computational cost in previous literatures, while the semi-analytical calculation of perpendicular magnetization has not still been investigated. Therefore, the semi-analytical expressions of magnetic force and stiffness for perpendicular polarized ring magnets are proposed. Then, the magnetic forces calculated by the proposed method, numerical simulation, and COMSOL software under different parameters are obtained. The results show that the proposed semi-analytical calculation has higher accuracy and less computational time than numerical simulation. Moreover, the influence of structural parameters on magnetic stiffness is analyzed. It can be demonstrated that with the increase of air gap, the decrease of the width of axial magnetized magnet, and the decrease of the height of axial magnetized magnet, the magnetic force and magnetic stiffness are both reduced. In general, the proposed semi-expression model can be applied for the design and optimization in the practical applications of ring permanent magnets.