# Robust Structural Damage Detection Using Analysis of the CMSE Residual's Sensitivity to Damage
## 1 Introduction
The method has some advantages, including the incorporation of an adaptive strategy to simultaneously determine \(\xi^{2}\) and \(\mathbf{L}\). Additionally, the ITR method shows good convergence behavior. Over the past decades, numerous techniques for damage identification have been reported. [PERSON] and [PERSON] [25] gave a review of vibration-based condition monitoring that revealed numerous and diverse algorithms using data in the time, frequency, and modal domains. [PERSON] et al. [26] presented a general summary of the state-of-art of intelligent algorithms, and their application prospects in structural damage identification were introduced. Among many reviews of the existing literature concerning damage identification methods, [PERSON] and [PERSON] [27] dealt in particular with the subset of methods related to variations in basic modal properties. Changes or lack of smoothness in mode shape or mode shape curvature, analysis of dynamically measured flexibility, and updating of structural model parameters provide different examples of this family of methods. [PERSON] and [PERSON] [28] also focused on technique processing information about mode shape curvature or strain modes with or without knowledge of baseline data. The general modal strain energy (MSE), as an extension of mode shape analysis, is widely appreciated because of its excellent damage-sensitive features [29; 30]. It is formed by the product of the stiffness matrix and the second power of mode shape:  
\[P_{j,n}^{*}=\left(\mathbf{\phi}_{j}^{*}\right)^{T}\mathbf{K}_{n}^{*}\mathbf{\phi}_{j}^{*} \left(j=1,2,\cdots,N_{j}\right) \tag{4}\]  
where \(N_{j}\) is the number of measured modes; \(\mathbf{\phi}_{j}^{*}\) and \(\mathbf{K}_{n}^{*}\) are the \(j\)-th mode shape and the \(n\)-th stiffness submatrix of the damaged structure, respectively; and \(P_{j,n}^{*}\) is the MSE of the \(n\)-th damaged structural element associated with the \(j\)-th measured mode. When damage occurs, the distribution of strain energy originally stored in the structure will change in a more pronounced manner in the detected areas. Therefore, changes in the modal strain energy distributions of the healthy and damaged structures can be used to detect the existence, location, and extent of the damage. A review of MSE-based methods was given by [PERSON] and [PERSON] [29], and will not be discussed here. In this study, a special MSE method named the CMSE method, proposed by [PERSON] et al. [10] and later improved by [PERSON] et al. [31], is used for damage detection. The adjective \"cross\" here indicates that MSE-like terms are product terms extending over the baseline finite element model (FEM) of the healthy structure and the measured damaged structure, also extending over various modes. The CMSE is defined as  
\[C^{*}_{m,n}=\left(\phi_{i}\right)^{T}\mathbf{K}_{n}\phi^{*}_{j}\left(i=1,2, \cdots,N_{i},\ j=1,2,\cdots,N_{j}\right) \tag{5}\]  
where \(\phi_{i}\) and \(\mathbf{K}_{n}\) are the \(i\)-th analytical mode shape and the \(n\)-th stiffness submatrix of the baseline FEM, respectively, and \(N_{i}\) is the number of analytical modes. In practice, it is easy to obtain the analytical modes of the baseline FEM, but difficult or expensive to extract the measured modes of the damaged structure [10]; therefore, one may select a much larger \(N_{i}\) than \(N_{j}\), i.e., \(N_{i}\gg N_{j}\). It is clear that the CMSE does not require matching modes between the healthy and damaged structures, and thus has more available mode combinations (MCs) than the general MSE, i.e., \(N_{i}\times N_{j}\gg N_{j}\). Like many other damage identification techniques, the CMSE method can establish a linear system of the same number of equations as the available measured mode or MCs. Therefore, a distinct advantage of this method is that it breaks through the constraint that the number of available measured modes is smaller than that of actual damages, i.e., \(N_{j}<N_{d}\).