A robust singular value decomposition for damage detection under changing operating conditions and structural uncertainties

Abstract

A technique is proposed to detect damage in structures from measurements taken under different conditions (i.e. different operational excitation levels, geometrical uncertainties and surface treatments of the structure). The method is based on a robust singular value decomposition (RSVD) which will be introduced in this article. Using the RSVD the distance of an observation to the subspace spanned by the intact measurements can be computed. Furthermore, from statistics, a threshold can be determined to automatically decide, based on the observation's distance to the subspace, if the observation comes from a damaged or intact sample. The proposed RSVD method is compared with an existing method based on the classical least-squares (LS) SVD. The damage detection method is validated on an aluminium beam with different damage scenarios (a saw cut and a fatigue crack), under several conditions (different beams with small dimensional changes, beams covered with damping material and different operating levels).

Introduction

In recent decades many techniques have been proposed for detecting damage based on changes of modal parameters. Unfortunately, the modal parameter changes due to varying operating conditions and structural uncertainties (e.g. due to inter-product variability) are often much more important, and hence they mask the information on the damage state of the structure. Therefore, most vibration-based damage detection techniques only give good results in well-controlled laboratory conditions.

During the last decades many techniques were proposed to detect damage based on changes of modal parameters (see the literature survey in Ref. [1]). A common property of these methods—based on changes in modal parameters—is that they are very sensitive to changes in: (a) the environment, (b) the operational conditions, and (c) structural uncertainties. Recently, a few publications appeared that aim at removing the influence of these variations. In Ref. [2] a technique is proposed to eliminate the influence of operational conditions. The technique is based on a singular value decomposition (SVD) of a frequency response function (FRF) matrix $\mathbf{H}=[H_1,\dots ,H_N]$, where $H_1,\dots ,H_N$ are FRFs at N conditions. In Ref. [3] another matrix decomposition (principal component analysis or PCA) of a matrix of estimated features (i.e. resonance frequencies) is used to eliminate the influence of environmental changes. Both authors have shown that matrix decompositions can help in removing the influence of varying conditions of the structure/measurement. However, when damaged features (FRFs or estimated parameters) are included in the matrix $\mathbf{H}$, the decomposition largely deviates from the one based on merely features from intact structures. This is true since damaged features are outliers, as was recognized by Worden in Ref. [4] (in the paper [4] the fact that a damaged feature is an outlier with respect to the features of intact samples is used to separate damaged and intact observations). This means that existing SVD damage detection techniques [5], [3] only work well when the SVD of the data matrix from intact observations can be computed (in general, however, it is not known beforehand if an observation is intact or damaged).

The purpose of this article is to present a robust SVD which computes the SVD of the intact structures from a set of observations of both intact and damaged structures, which are possibly measured in different conditions. In order to guarantee the success of the proposed technique, it is assumed that there are at least as many intact as damaged observations (FRFs, response spectra or estimated parameters) are present in the observation matrix $\mathbf{H}$ which has to be decomposed.

In the next section the theory behind the method will be described. Firstly, in Section 2.2 a review of damage detection using the least-squares SVD will be given. Next, in Section 2.3, a somewhat more robust iterative SVD (ISVD) is introduced. The proposed robust SVD (RSVD) is described in Section 2.4. All three methods are validated on an extensive experimental case study of an aluminium beam measured under various conditions. The results of the experimental validation can be found in Section 4. Finally conclusions will be drawn in Section 5.