Shape optimization of pressurized air bearings

Use of externally pressurized air bearings allows for the design of mechanical systems requiring extreme precision in positioning. One application is the fine control for the positioning of mirrors in large-scale optical telescopes. Other examples come from applications in robotics and computer hard-drive manufacturing. Pressurized bearings maintain a finite separation between mechanical components by virtue of the presence of a pressurized flow of air through the gap between the components. An everyday example is an air hockey table, where a puck is levitated above the table by an array of vertical jets of air. Using pressurized bearings there is no contact between “moving parts” and hence there is no friction and no wear of sensitive components. This workshop project is focused on the problem of designing optimal static air bearings [15] subject to given engineering constraints. Recent numerical computations of this problem, done at IBM by Robert and Hendriks [11, 12], suggest that near-optimal designs can have unexpected complicated and intricate structures. We will use analytical approaches to shed some light on this situation and to offer some guides for the design process. In Section 2 the design problem is stated and formulated as an optimization problem for an elliptic boundary value problem. In Section 3 the general problem is specialized to bearings with rectangular bases. Section 4 addresses the solutions of this problem that can be obtained using variational formulations of the problem. Analysis showing the sensitive dependence to perturbations (in numerical computations or manufacturing constraints) of near-optimal designs is given in Section 5. In Section 6, a restricted class of “groove network” designs motivated by the original results of Robert and Hendriks is examined. Finally, in Section 7, we consider the design problem for circular axisymmetric air bearings.