Analysis of forces in rectangular-pole geometries using numerical integration techniques

The force and stiffness characteristics for rectangular-pole geometries are derived as a function of the parameters gap width, pole width and pole displacement. The Schwarz-Christoffel transformation technique is employed, with the simpler cases giving directly integrable solutions, or solutions in terms of elliptic integrals. For the most general case, numerical integration is used to determine the transformation and the force characteristics. The analysis shows that the restoring force and initial stiffness for the finite-width geometry are bounded by those for the infinite-width and line-potential geometries, and that the unbalance force and stiffness vary linearly with pole-width.