## 3 Crack Detection in Plates Using Dynamic Response (DR)
### Inverse Problem for the DR of the Cracked Plate  
Employing the DR measurements to identify the crack location and size is known as an inverse problem. The primary theory of vibration-based damage detection is based on  
Figure 4: The correlation between the frequency parameter and the index of parabolic varying thickness (PVT) contains 3 cracks, where (c) crack length, (d) distance between cracks, and (a and b) plate length and width, respectively [167]. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{**Thickness Variation**} & \multirow{3}{*}{**Gradient Index ’n’**} & \multirow{3}{*}{**Crack Length ’a’**} & \multicolumn{4}{c}{**Fundamental Frequency \(\omega\) (rad/s)**} \\ \cline{5-8}  & & & & \multicolumn{3}{c}{**Taper Parameter**} \\ \hline \multirow{6}{*}{LVT} & & & & \(\alpha\) = \(\beta\) = 0 & \multicolumn{3}{c}{\(\alpha\) (1 D)} & \multicolumn{3}{c}{\(\alpha\) = \(\beta\) (2 D)} \\ \cline{5-8}  & & & & **0.5** & **1** & **0.5** & **1** \\ \hline \multirow{6}{*}{LVT} & & & 0 & 719.70 & 731.55 & 752.15 & 736.10 & 756.10 \\  & & 0.01 & 700.11 &  
the dependency of modal parameters on the physical properties of the structure under inspection. Damage-caused changes in the structure's physical properties lead to clear changes in modal parameters (including natural frequencies, modal damping, and mode shapes). Research effectively substantiates the application of DR measurements for damage detection in plate structures; however, compared with the forward problem, there is much less research on the inverse problem. The study of this type of inverse problem began in 1978, when [PERSON] et al. [168] introduced a methodology to identify damage in a one-dimensional structure by analyzing the changes in its longitudinal vibration frequencies. [PERSON] et al. (1997, 2001, 2003) [169; 170; 171] developed a series of crack models and damage detection techniques for plate structures. Their initial crack model, introduced in 1997, used the strain energy equivalence principle (SEEP) to detect damage in elliptical through-cracks in 2D and 3D elastic solids [169]. In 2001, they developed a novel structural damage identification method for plate structures, utilizing frequency response functions (FRFs) as its foundation. This method, rooted in the principles of the dynamic equation of motion, employed a damage distribution function to effectively characterize the extent and location of damage within the structure [170]. In 2003, they developed a new damage identification theory for thin plates with line micro-cracks using the effective stiffness model and the FRF. However, this theory, limited to micro-cracked plates not close to boundaries, requires experimental validation, as it has only been demonstrated numerically [171]. [PERSON] et al. (1999) [172] explored the effectiveness of strain energy-based methods for damage detection in plate-like structures through a comprehensive research program encompassing both experimental and numerical studies. These methods efficiently localized areas of reduced stiffness down to 10% accuracy with the use of limited modes. This technique only needed the comparison of the structure's mode shapes pre- and post-damage. [PERSON] et al. (2002) [173] implemented the strain mode technique through numerical simulations and experiments to pinpoint damage locations in plate-like structures, utilizing an aluminum plate with free boundary conditions as their test subject. Their findings, detailed in Figure 5, revealed a consistent correlation between the simulation and experimental results. This was particularly evident in identifying areas of damage based on the bending moment index \(\bar{\beta}_{r,x}\). This index proved to be a reliable indicator for locating damage, as evidenced by the congruence of the major maximum of residual strain \(\Delta\bar{x}_{r,x}\) at the damaged area and minor maxima near it.