  We consider frequency-weighted damping optimization for vibrating systems
described by a second-order differential equation. The goal is to determine
viscosity values such that eigenvalues are kept away from certain undesirable
areas on the imaginary axis. To this end, we present two complementary
techniques. First, we propose new frameworks using nonsmooth constrained
optimization problems, whose solutions both damp undesirable frequency bands
and maintain stability of the system. These frameworks also allow us to weight
which frequency bands are the most important to damp. Second, we also propose a
fast new eigensolver for the structured quadratic eigenvalue problems that
appear in such vibrating systems. In order to be efficient, our new eigensolver
exploits special properties of diagonal-plus-rank-one complex symmetric
matrices, which we leverage by showing how each quadratic eigenvalue problem
can be transformed into a short sequence of such linear eigenvalue problems.
The result is an eigensolver that is substantially faster than standard
techniques. By combining this new solver with our new optimization frameworks,
we obtain our overall algorithm for fast computation of optimal viscosities.
The efficiency and performance of our new methods are verified and illustrated
on several numerical examples.
