Damage Detection of an Aluminum Truss Using Tikhonov Regularization

A damage detection algorithm is presented based on updating a finite element model with measured eigenfrequencies and mode shapes. The update needs regularization because it is an ill-posed problem. Tikhonov regularization and truncated singular value decomposition are commonly used regularization techniques for linear problems. These techniques can also be used for nonlinear updating problems, provided some additional details are incorporated. We present an iterative Gauss-Newton algorithm with Tikhonov regularization that includes line search and constraints on the update parameters to improve convergence. The algorithm is applied to a truss model with actual measured dynamic properties. A first step is the calibration of the undamaged model. Several refinements of the finite element model were necessary to find a model that matches the measured dynamic properties. These include modeling of joint eccentricities and the base flexibility. Damage is simulated by removing individual members. Results show that the algorithm is capable of detecting location and extend of the simulated damage. INTRODUCTION One of the classical damage identification techniques is to determine stiffness changes due to damage by model updating methods. However, a precise model update does not necessarily give good results for the model parameters. This is because the update problem is ill-posed and small measurement errors can leads to large distortion of the model parameters. The classical method of solving these kinds of problems is by regularization. Regularization techniques have been successfully applied in a number of fields such as astronomical and medical imaging or parameter estimation in geophysics. Theoretical background and mathematical algorithms for linear illposed problems are given in [1,2]. One of the few treatments of nonlinear ill-posed problems is [3]. Regularization has also been used for model updating [4]. However, in most cases, the algorithms have not been rigorously adapted to the nonlinear updating problem, the regularization parameter has been selected in an ad hoc manner, and more advanced techniques from numerical optimization have not been included. In this article, we describe an algorithm for nonlinear model updating with Tikhonov regularization. The regularization parameter is found with generalized cross-validation. The algorithm includes techniques from optimization such as line search and bound constraints of the update parameters. MODEL UPDATE Damage detection is performed by updating a finite element model with respect to measured eigenvalues and mode shapes. The finite element model is parameterized as = +∑ (0) ( ) j j a K K K (1) where denotes the constant part of the stiffness matrix, representing the undamaged structure, and (0) K ( ) j K are submatrices related to the update parameters j a . Only stiffness changes are considered; the mass is assumed to remain constant. Updating is performed by minimizing the weighted residual, that is, the difference between measured and calculated eigenvalues and mode shapes: