Efficient Sensitivity Analysis of Structures with Local Modifications. I: Time Domain Responses

Abstract

The time evolution of the parameter sensitivity of a nonlinear dynamical system is essential to its successful design, control, and identification. For an important class of systems, the governing equations of motion can be expressed as a linear system of ordinary differential equations augmented with a low-rank parameter-dependent (generally nonlinear) vector function. This paper builds on previous efforts by the authors to devise an approach for computing not only the responses of such systems but also the sensitivities of their responses to changes in design parameters in a computationally efficient manner. The proposed solution procedure first segregates the degrees of freedom appearing in the augmented functions from those that involve only linear deterministic operations. Second, the responses and sensitivities of the augmented terms are determined by solving the Volterra integral equations of reduced dimension. Finally, the responses and sensitivities of all remaining desired outputs are computed through a convolution approach using the fast Fourier transform to further increase the computational efficiency. The proposed method is applied to two examples, one simple linear model where exact solutions can be computed, and a second, more realistic, example of a wind-excited 20-story building augmented with nonlinear tuned mass dampers on its roof. The proposed method is shown to possess accuracy levels similar to those obtained through the time integration of the corresponding analytical sensitivity equations; however, it is several orders of magnitude faster than conventional sensitivity analysis approaches.