From the nano scale to the macro scale, large models are used to simulate and predict the responses of dynamic systems. The construction and evaluation of such models, often in the form of finite element models, require tremendous computational resources and time. Due to this large computational endeavor, it is paramount to learn as much as possible from the models and their solutions. In this work, analyses and methods for efficiently deriving significant knowledge of damped systems from models and their solutions are presented. Of primary interest to this work is the analysis of damped structures. Damping, the means by which energy is dissipated, often adds an additional layer of complexity to finite element models and any subsequent analyses. This added complexity is due to the relative complexity of many damping models and their accompanying computational burden. Furthermore, on the micro and nano scale, a variety of damping mechanisms, each with their own unique set of physics, may be present. The research presented in this work is organized in two parts. The first part presents methods for deriving knowledge from models and their solutions. Here, the developed methods perform approximate yet highly efficient analysis on the matrices and solution vectors of finite element models. In this work, methods utilizing the Neumann series approximation are presented. These methods efficiently predict how the response of a structure depends on its damping or any other input model parameter. Additionally, a method for analyzing the spatial dependence of damping with the use of loss factor images is presented. Research presented in the second part derives knowledge solely from solutions of models. In this part, it is assumed that the matrices of the models are not available, and therefore analysis is restricted to the solution itself. Here, research is focused on the analyses of structures on the micro and nano scale. Specifically, micro and nano beams surrounded by a viscous compressible fluid are analyzed. The dynamic responses of the structure and the surrounding fluid are analyzed to determine the prominent damping mechanisms. Here, results from 2--Dimensional analytical models and 3--Dimensional finite element models are complemented by experimental measurements to analyze damping due to viscous dissipation and acoustic radiation.