This project pursues the parallel problems of how best to dampen a vibrating body and how to identify, from observations of a vibrating body, the dissipative forces acting within it and upon it. Damping can be achieved through various mechanical means, and for each of these, one seeks ways to distribute material properties to optimally stifle vibrations. Yet optimality can be characterized in several ways, such as the total energy caused by a worst-case vibration, total energy averaged over all vibrations, and the asymptotic rate of energy decay. Each of these different metrics requires different mathematical tools and computational techniques. To address these problems, this project will draw upon, and augment, tools from mechanics, functional analysis, and numerical analysis. The task of finding the best damping is a spectral optimization problem, which is particularly challenging because the mathematical models for these vibrating bodies are non-self-adjoint and have variable coefficients. As such, they are representative of a broad class of important problems in application areas such as fluid dynamics, and thus insights gained from this work may suggest advances on a broad class of problems. In addition to design optimization, this project will also develop techniques for characterizing the material properties of a body based on measurements of vibration. Of particular interest are models comprising several one-dimensional entities vibrating together. At the macroscopic level, such models describe violin strings composed of an elastic core enclosed in a dissipative sheath below an outer elastic winding, while at the microscopic level, similar models have been proposed to model DNA as two intertwined rods glued together elastically.
