## 3 Crack Detection in Plates Using Dynamic Response (DR)
### Inverse Problem for the DR of the Cracked Plate
After that, [PERSON] and [PERSON] (2004-2005) [174; 175; 176; 177; 178; 179] made significant contributions to the field of damage identification in plate structures and developed various methods for locating and quantifying damage using vibration measurements. The authors conducted a comprehensive comparison of various damage localization techniques. Their findings revealed that uniform load surface (ULS) curvature surpassed other methods in terms of damage localization sensitivity, particularly under conditions of incomplete and noisy measurements, as demonstrated in Figure 6 [174]. Additionally, they proposed an anisotropic damage model to detect edge-parallel cracks in thin RP and compared their results with [PERSON]'s 1997 [169] model [175]. Additionally, the researchers proposed a novel damage localization technique for two-dimensional plate structures, employing changes in ULS curvature as the primary indicator [176]. Additionally, they developed a new method for damage localization in two-dimensional plate structures based on changes in the ULS curvature [177]. Subsequently, they improved their previous study to identify the specific location and measure the severity of damage in two-dimensional plate structures using a sensitivity-based method with ULS curvature, effectively showing the damage location and magnitude, as seen in Figure 7 [178]. Their work was further extended by proposing a new anisotropic damage model for detecting cracks in plates with edge-parallel cracks [179]. Their work has been experimentally and numerically validated, and their methods have demonstrated their ability to identify a variety of damage types including cracks, delamination, and stiffness reductions. In 2004, [PERSON] et al. [180] developed a strategy for detecting damage in structures using vibration analysis on a honeycomb sandwich plate and identified future research areas despite experimental errors. Subsequently, in 2005, [PERSON] et al. examined damage in composite plates with internal cracks and delamination, using piezoelectric devices for measurements. This study found a relationship between damage characteristics and changes in modal parameters, with delamination locations identifiable through damping and strain energy changes [181]. The technique by [PERSON] et al. (2004) [182] used non-destructive vibration tests for damage identification in plate structures. By employing the FEM data of an intact structure and its FRF, they formulated a damage location vector (DLV). This DLV method effectively located and assessed the damage, even with noisy and incomplete data. Furthermore, in 2005 [183], the authors proposed another method for early-stage damage detection where there was no substantial change in mass or damping. The method related alterations in FRFs to shifts in the structural stiffness matrix, with the DLV evaluation showcasing this relationship. An experiment was conducted where element 11 was altered to mimic internal damage. Analysis of the FRF data via the DLV identified element 11 and its neighboring nodes as damaged, with the impact on neighboring nodes being significantly lesser and decreasing with distance from element 11, as shown in Figure 8.  
Moreover, [PERSON] and [PERSON] (2008) [184] developed a novel method that combined static/dynamic approaches to improve damage detection in composite plates. This technique, identifying changes in the DR due to damage, used two actuator-sensor systems to measure the DR and utilized both experimental and numerical mode shapes to detect delamination. The researchers successfully identified, located, and quantified the extent of delamination and the effect of static load on the DR. The researchers concluded that this method effectively enhanced damage detection by amplifying the damage's impact. [PERSON] et al. (2010) [185] explored dynamic-based methods for damage identification in arched structures using ambient and random impact vibrations.