A vibration-based damage detection method is presented, which updates a finite element model from measured eigenfrequencies and mode shapes. Changes in stiffness of individual elements are interpreted as damage. The update problem is ill-posed and needs therefore regularization. Tikhonov regularization is a well known regularization method for linear problems. However, the update problem is nonlinear and there are different ways to use Tikhonov regularization in the nonlinear case. The usual way is to linearize the update problem with the sensitivity matrix and then apply regularization in each iteration. This approach has the disadvantage that the regularization effect depends on the number of iterations and may get lost as the number of iterations increases. The problem has been discussed in the mathematical literature but is not widely recognized in the engineering community. The alternative way is to regularize the nonlinear problem and then linearize it. This results in an additional term in the update equation that guarantees that regularization is independent of the number of iterations.
Both ways are applied to a two-story frame with simulated measurements. Generalized cross-validation, L-curves, and errors of the update parameters are compared. The simulations confirm that the algorithm characterized by "regularize, then linearize" gives superior results. This algorithm is finally applied to a real frame recently tested in the lab. The frame is part of a full-size structure that has been damaged by pseudo-dynamic tests simulating different earthquake levels.