## 2 Modal Based Damage Detection: Development & Challenges
### Development of Modal-Based Damage Sensitive Features
#### 2.1.6 Modal Flexibility  
Modal flexibility is another popular modal parameter that has been used for structural damage detection. First proposed by [PERSON] & [PERSON] [63] and subsequently applied to bridges by [PERSON] & [PERSON] [64], the modal flexibility method (MFM) defines the flexibility matrix as the inverse of the stiffness matrix (see Equation (5)). It is deemed that this format can be adequately defined with fewer modes than is required for stiffness matrices. This allows for greater damage sensitivity to be attributed to the more easily extracted lower modes. Furthermore, the MFM combines the information of many frequency modes into a single damage feature, which increases damage sensitivity. [PERSON] et al. [65] confirmed that MFM advanced damage sensitivity when compared to other modal-based damage indicators; while [PERSON] et al. [66] demonstrated MFM's damage localisation capabilities in beam and plate structures when tested via a dynamic computer simulation.  
One reason for MFM's decent level of performance stems from its use of mass-normalised mode shapes, which allow the flexibility matrix to portray the displacement pattern of the structure per unit force applied. This allows damage events (increases in flexibility/decreases in stiffness) to be assessed uniformly throughout the structure and also enhances localisation. However, MFM's performance suffers when subjected to ambient or unknown conditions, as mass-normalised mode shapes require the load effect to be known in advance. Some studies have attempted to mass-normalise dynamic mode shapes extracted from ambient vibration in an effort to utilise MFM's superior damage detection and localisation sensitivity [67; 68], but these methods carry a considerable level of uncertainty through having no prior knowledge of the force input data.  
Equation (5) portrays the stiffness/flexibility relationship, where \(\{f\}\) and \(\{y\}\) are the load vector and displacement vector, respectively, and \([k]\) and \([G]\) are the stiffness matrix and flexibility matrix, respectively, that have been mass normalised. Equation (6) furthers portrays how the flexibility matrix is computed, where \([\psi]\) is the mass-normalised mode shape matrix, \([\Lambda]\) is the measured eigenvalue matrix, \(\omega_{i}\) is the \(i\)th modal frequency, and \(\{\psi\}_{i}\) is the \(i\)th mode shape vector.  
\[\{f\}=[k]\{y\}\ \rightarrow\ \{y\}=[k]^{-1}\{f\}\ =\ [G]\{f\} \tag{5}\]  
\[[G]\approx\ [\psi][\Lambda]^{-1}[\psi]^{T}\ \approx\ \sum_{i=1}^{m}\frac{1}{ \omega_{i}^{2}}\{\psi\}_{i}\{\psi\}_{i}^{T} \tag{6}\]  
Further examples of modal flexibility used in Structural Health Monitoring (SHM can be found in [PERSON] & [PERSON] [69], who incorporated the use of the modal curvature method to monitor changes in flexibility. They proposed that damage events that reduce stiffness and increase flexibility would also increase local curvature in the same location, and by combining both methods, damage sensitivity would be increased. [PERSON] et al. [70] tested the MFM and its curvature hybrid technique on a beam and determined that modal flexibility is very sensitive to local damage; however, in cases where numerous damage scenarios were introduced, the resulting flexibility peaks merged together, making localisation difficult. Furthermore, they noted that multiple damage events of varying magnitudes resulted in flexibility changes in locations other than the damage location. However, they found that the hybrid flexibility curvature method produced superior results in locating closely distributed damage and discerning between different magnitudes of damage events.