Performance of closed-form equations for force between cylindrical magnets over wide range of volume, aspect ratio, and force

Four types of magnets were used in this study: neodymium NdFeB (grade N35 and N52), ferrite (Y10), and samarium-cobalt SmCo (XG30 2:17). They were chosen to represent a wide range of volumes from 0.035 to 19 cm³ (540 times), radius R from 1.5 to 12.5 mm (8 ×), length L from 0.5 to 40 mm (80 ×), aspect ratio L/R from 0.051 to 17 (330 ×), and contact forces from 0.2 to 250 N (over 1000 ×). The study shows that previously reported closed-form equations are valid only at large distances (small forces). At short distances (large forces) the calculated force diverges to infinity or the accuracy depends on the aspect ratio, and some equations fail more than others. A new equation is proposed as a small modification of a previously known function, which provides reasonable behaviour over the whole studied range. However, the accuracy is unknown in a general practical case, because theoretical calculations do not take into account imperfections of real magnets, so there is no single absolute reference.