## 3 Crack Detection in Plates Using Dynamic Response (DR)
### Forward Problem for the DR of the Cracked Plate
#### 3.1.1 Plate of the Uniform Thickness (UT)  
Numerous researchers have focused on studying the dynamic behaviors of UT plates with cracks, developing various analytical approaches. Before the widespread use of the FE method, mathematical techniques were essential for understanding the vibration dynamics of these plates [88]. Early significant contributions to this field, dating back to 1967 by [PERSON] and [PERSON] [89], involved using [PERSON]'s function to formulate a Fredholm integral equation of the first kind to analyze vibrations in cracked RP. This equation was designed for mixed edge conditions influenced by the crack's presence. The Krylov and Bogoliubov method was later used to solve this equation, particularly for cases involving narrow cracks and plates with fully supported boundaries. The outcomes of this solution were observed as frequency changes corresponding to different crack sizes. [PERSON] and [PERSON] (1972) [90] studied the vibrations and stability of cracked RP using dual-series equations, resulting in a second kind of homogeneous Fredholm integral equation. [PERSON] et al. (1988) used the LSM to analyze part-through cracks in shells under mixed loading, and found that shell curvature minimally affected the stress intensity factors for superficial cracks [91]. [PERSON] and [PERSON] (1989) investigated the free vibration dynamics of the RP with straight slits, using holographic interferometry to study the effect of slit dimensions, positions, and angles on natural frequencies and related modal shapes. Their experiments included different slit positions and angles, showing a decrease in natural frequencies with increasing slit dimensions and angles [92]. [PERSON] et al. (1994) [93] aimed to understand the vibration characteristics of cracked RP by employing the domain decomposition method. Their study was divided into two specific cases, cracks originating from the edge and cracks located in the center. Their findings revealed that the presence of cracks reduced the plates' flexural rigidity, resulting in lower vibrational frequencies in both cases. [PERSON] et al. (2000) [94] created and utilized modified comparison functions to analyze vibrations in the SS, cracked RP. This method provided benefits like not being restricted by specific types of functions, having higher accuracy, and being easier to use compared to eigenfunctions. The researchers used these modified functions to determine the natural frequencies of cracked plates via the RRM, with variations in frequencies based on crack depth, length, and position. [PERSON] et al. (2002) [95] used the scanning laser vibrometer technique to study the dynamics of damaged CP and successfully identified minor defects through the analysis of modal shapes and frequencies. [PERSON] et al. (2005) [96] utilized the incremental harmonic balance (IHB) method to examine the dynamic instability and nonlinear responses of cracked plates under periodic forces, illustrating instability regions and exploring the effects of different parameters on the instability and nonlinear behaviors of these plates. [PERSON] et al. (2006) [97] used both experimental and analytical techniques to study the effects of patch layers and loading frequencies on the fatigue fracture behaviors of aluminum plates, focusing on variations in key fracture mechanics parameters. In the same year, [PERSON] et al. (2006) [98] presented approximate solutions for plate vibrations related to cracks, developing an analytical equation for the dynamics of edge-loaded plates with various boundary conditions. They later expanded the applications of their research, providing relationship information between forces and moments at crack tips and plate edges, significantly advancing the study of vibrations in plates with partial cracks. Their findings highlighted the impact of cracks on natural frequencies, where the frequency of damaged plate dropped, as shown in Table 1[7].