Although large cryogenic space telescopes may provide a means of answering compelling astrophysics questions, the required increase in the primary mirror diameter presents technical challenges. Larger primaries are more flexible, and cryogenic mirrors are typically very lightly damped-the material damping is negligible, and common damping methods break down. To address these challenges, we propose placing flux-pinning mechanisms along the edges of adjacent mirror segments. These mechanisms consist of a collection of magnets and superconductors, and like flexures, they preferentially allow motion in specific degrees of freedom. Motion in nonpreferred degrees of freedom is resisted by a force analogous to a damped spring force, and the stiffness and damping can be adjusted independently. As an example, we consider simple mechanisms consisting of an inexpensive magnet and a single superconductor. These mechanisms provide increasing resistance as the magnet and superconductor-or mirror segments attached to each-come closer to colliding. These mechanisms, with typical stiffness and damping values on the order of 5000 N∕m and 5 kg∕s, respectively, also provide modest improvements to the mirror performance. Greater gains can be achieved by using stronger magnets or smaller separations, or by placing nonmagnetic conductive materials near the mechanism.