Dynamic Sensitivity Analysis Based on Sherman–Morrison–Woodbury Formula

Abstract

The sensitivity analysis plays a significant role in structural optimization, and repetitive calculations are necessary for the analysis procedure. A reanalysis approach of frequency response functions (FRFs) is proposed based on the Sherman–Morrison–Woodbury (SMW) formula. Multiple perturbations are considered as the change of elements in the dynamic stiffness matrix, and the SMW formula is used as an efficient mathematical tool for solving the problem of repetitive calculations. The perturbed FRFs are directly calculated by the SWM formula, and then the sensitivity of the FRFs is obtained by using the finite difference procedure. The basic theory of the proposed method is illustrated by adopting a mass–spring system, which is extended to truss structures. A five-bay truss and a cantilever plate are adopted to verify the method. The lumped mass, the damping, and the cross-sectional area can be regarded as design variables. Results show that the FRF sensitivity is obtained efficiently by using the initial FRFs and parameter perturbations to avoid repeated finite element analyses, and the FRF sensitivity of simultaneously perturbed multiparameters can be determined by using the proposed method.