## 3 Crack Detection in Plates Using Dynamic Response (DR)
### Inverse Problem for the DR of the Cracked Plate
They applied stochastic subspace identification (SSI), the mode shape curvature method (MSCM), and FEM updating to detect and assess structural damage. The study found that while some methods successfully identified damage, others struggled due to noise and sensitivity to geometric changes. They concluded that a combination of methods provided the most reliable results, emphasizing the need for robust techniques in early damage detection for historical structures. [PERSON] (2010) [186] introduced a new technique for detecting cracks in different structures including metallic and reinforced concrete under real-world conditions. The study compared the DLV and Maia FRF curvature difference (MFCD) methods. While MFCD, working solely with FRF, was specialized for beams, the DLV offered a more comprehensive application but required a dynamic stiffness matrix. [PERSON] et al. (2013) [187] presented a technique for detecting plate damage using two-dimensional Gaussian wavelets, laser-scanned deflection shapes, and focused on harmonic excitation-induced vibrations, with data collected via a scanning laser vibrometer. Furthermore, [PERSON] et al. (2013) [188] introduced a novel damage detection technique for plate structures using dynamic response sensitivity-based finite element model updating (DRSBFEMU). While this method effectively identified local damage in plates from a small number of acceleration measurements, it became computationally expensive and less accurate in scenarios with large finite element discretization. To address this limitation, [PERSON] et al. (2016) [189] proposed a two-stage damage identification method. The first stage utilizes the modal strain energy change ratio (MSERC) to identify potential damage locations, along with a method to minimize false alarms caused by the \"vicinity effect\". These identified elements are then considered suspicious for further investigation. In the second stage, the DRSBFEMU method quantifies the extent of damage for these suspicious elements, reducing the number of unknowns and significantly decreasing the computational time. The effectiveness and efficiency of the proposed method were demonstrated using a cantilevered plate and a two-span continuous plate. The results confirm the method's ability to identify damage in plates from a few acceleration measurements. However, as shown in Figure 9, noise interference can lead to false alarms in the damaged locations identified using the improved MSERC method. Recently, [PERSON] et al. (2019) [190] explored damage detection in composite plates through a theoretical and experimental study using the DLV method. The FEM was employed for theoretical simulation of a damaged plate, followed by a shaker test on a centrally damaged plate under two different boundary conditions. The DLV method successfully identified intentionally damaged elements in the plate, with primary data peaks pinpointing the damage and secondary peaks showing affected neighboring nodes, though the impact on these was less significant. The study concluded that the DLV method could effectively locate damage. [PERSON] et al. (2019) [191] investigated the geometric features of phase diagrams for cantilever beams with breathing cracks and proposed a novel method for crack identification based on these features. The study emphasized how the presence of a crack influenced the dynamics of the system, particularly under primary and super-harmonic resonance conditions. The authors introduced multiple identification indices, such as quasi-standard deviation and vertical segmentation methods, which are used to analyze changes in the phase diagram's area due to variations in crack depth and location. Their findings suggest that phase diagrams, especially under super-harmonic resonance, are highly sensitive to crack parameters, making the proposed method both feasible and effective to accurately determine crack characteristics in engineering structures. [PERSON] et al.