Efficient Sensitivity Analysis of Structures with Local Modifications. II: Transfer Functions and Spectral Densities

The deterministic frequency response analysis and the stationary random vibration of large, linear structural models are often computationally intensive tasks. The computational expense is only exacerbated when stiffness and damping variabilities are present in the system’s description. This paper proposes an approach for the highly computationally efficient and accurate computation of the sensitivities, to changes in design parameters that are spatially localized within some small portion of the computational model, of the transfer functions (TFs), power spectral densities, and mean-square responses of a structural system. The proposed method is derived, first for system TFs and then their sensitivities; the corresponding spectral densities are then derived using the TF approach. Finally, the proposed method is applied to two examples, one simple model where exact solutions can be computed, and a second, more realistic, building example with a high-order model that causes significant difficulties for standard state-space (SS) approaches in MATLAB as a result of numerical inaccuracies in the computations for very high-order SS models. The proposed method is shown to be highly accurate, with relative errors on the order of ${10}^{-10}$, but is 4–6 orders of magnitude faster than conventional approaches.