Frequency domain analysis of micro-electro-mechanical systems (MEMS) is a great source of unusual linear algebra problems. We use eigenvalue analysis and model reduction methods to build compact approximations to the frequency response that we can use for design of these devices. Because we care about the details of damping, the eigenvalue problems that arise in these computations are non-self-adjoint and sometimes nonlinear. But though the problems lack a nice self-adjoint structure, they often have more specific structure: complex symmetry in problems with radiating boundary conditions, weak coupling between thermal and mechanical fields in thermoelastic damping calculations, etc. Our work exploits this structure as the basis of eigenvalue and model reduction algorithms that are faster and more accurate than more general-purpose alternatives.