An algorithm for analyzing a nonlinear system as an augmented linear system is presented. The method uses a nonlinear discrete model of the system and the form of the nonlinearities to create an augmented linear model of the system. A linear modal analysis technique that uses forcing that is known but not prescribed is then used to solve for the modal properties of the augmented linear system after the onset of damage. Due to the specialized form of the augmentation, nonlinear damage causes asymmetric damage in the updated matrices. A generalized minimum rank perturbation theory, which requires knowledge of both right and left eigenvectors, is developed to handle the asymmetric damage scenarios. The damage extent algorithm becomes an iterative process when an incomplete set of right eigenvectors are known. The method is demonstrated using numerical data from nonlinear 3-bay truss structures. Various damage scenarios of the nonlinear systems are used to demonstrate the effectiveness of the augmentation and the generalized minimum rank perturbation theory, and the effect of random noise on the technique. The nonlinearities included in the 3-bay truss are cubic springs.
Damage detection in nonlinear systems using system augmentation and generalized minimum rank perturbation theory