Issue 29

E. Grande et alii, Frattura ed Integrità Strutturale, 29 (2014) 325-333; DOI: 10.3221/IGF-ESIS.29.28 326 The studies available in literature show the ability of these indicators to detect the presence, the position, and in some cases, the severity of damage. Nevertheless, the same studies have also underlined some drawbacks generally arising when multiple damages occur or noises/errors affect the identified dynamic properties of the systems. In these cases, a reliable prediction of damage requires a significant number of information particularly in terms of number of modes. Recently, in order to improve the ability of classical damage indicators, data information fusion techniques have been extended to structural damage identification [6, 9-11]. The fusion of information derived from different sources allows, indeed, to improving the ability of damage indicators particularly in detecting the damage position. In this paper, two approaches for damage detection in linear systems based on the use of the rdi and MSECRj indicators combined with the Dempster-Shafer data fusion theory are presented. In particular, while the first approach, denoted in the following DF as Data Fusion, is based on the fusion of the information derived separately from the MSECR j and rdi indicators, considered as separate and independent sources, the second approach is an innovative procedure, denoted in the following MDF as Multi-stage Data Fusion, consisting in a data fusion implemented in a multi-stage process where the sources are based on the same damage indicator, either rdi or MSECR j , but evaluated on the basis of different combinations of modes of vibration. Numerical applications are presented in the paper to assess the reliability of the proposed approach considering different damage scenarios, different sets of modes of vibration and presence of noise. D EMPSTER -S HAFER THEORY empster-Shafer theory [12] represents the first data fusion theory developed by Dempster and Shafer in 1976 and, still, one of the most valuable. Some key definitions of the theory, those used in the approach proposed in the paper, are summarized in the following. Considering a finite set   , , A B C   of mutually exclusive and exhaustive propositions, the corresponding power set 2  is defined as the set of all the subsets of  which also includes the null set. The theory of evidence assigns a basic probability assignment function, named BPA or m(X), to any subset of 2  , defined as: : 2 [0,1] m   (3) being: ( ) 1 and ( m X m O  ) 0 X    (4) In the framework of the Dempster-Shafer theory [12] the BPA can be interpreted as a generalization of the probability concept being the probability assigned not only to one hypothesis but to a set of hypotheses without any information on how it is distributed among the elements of the set itself. The Dempster’s rule provides a method for combining the basic probabilities assignment of different information sources i S . In particular, given 1 S and 2 S two information sources and 1 m and 2 m the BPAs given by the two sources, the fused BPA is given by: 1 2 1 1 1 2 2 1 ( ) ( ) ( ) 1 S S S m S m S m S Q       (5) where Q represents a measure of the degree of conflict between the two sources defined as: 1 2 1 1 2 2 ( ) ( ) S S Q m S m S      (6) DF AND MDF PROPOSED TECHNIQUES FOR DAMAGE DETECTION he approaches presented in this paper for damage identification of linear systems are developed by combining the use of classical damage indicators based on the modal strain energy through the Modal Strain Energy Change Ratio index (MSECR) and on the flexibility matrix through the relative damage indicator (rdi) with the Dempster- Shafer data fusion theory. In particular, two different approaches are presented. The DF approach is simply based on the X D T