## 4 Discussion
Moreover, the governing equations for plates of UT are relatively simple and can be derived using classical plate theories, while those for plates of VT are more complex and require additional considerations. This is primarily due to the presence of thickness variation in VT plates, which leads to more intricate behavior and necessitates the incorporation of a thickness function into the governing equations. The analysis of VT plates is consequently more challenging than that of UT plates. The forward problem shows that an increase in crack length, size, depth, and angle typically leads to a decrease in the natural frequencies, a common observation in vibrationcharacteristics. The growth or reorientation of a crack reduces the plate's overall stiffness, which in turn lowers the natural frequencies, causing the plate to vibrate at slower speeds for the same mode shapes. Moreover, the presence of cracks can alter the modal shapes, introducing localized flexibility and leading to more significant vibrations in certain areas, resulting in distortions or asymmetries compared to an uncracked plate. The location of a crack also significantly affects the vibration characteristics. Internal cracks, for example, have a more pronounced impact on natural frequencies than surface cracks. Because the internal cracks cut deeper into the plate, effectively removing material and reducing its overall stiffness, this drop in stiffness significantly lowers the natural frequencies, as the plate vibrates slower with less rigidity. Additionally, the shape of the plate influences its frequencies, which is a complex interplay of various factors. For instance, the CP under SSSS and CCCC boundary conditions with UT generally exhibited the lowest frequency compared to other shapes [216]. Plates with higher aspect ratios also tend to have higher fundamental frequencies due to their reduced bending capacity. The boundary conditions and material properties of the cracked plate are crucial in determining the vibrational characteristics, with different boundary conditions and materials significantly affecting the natural frequencies. Finally, the type of excitation condition can significantly influence the frequency response of a vibrating system. The nature of vibration affects the plate's frequency response by determining which natural frequencies are excited and to what extent based on the frequency content, duration, and damping of the excitation. In the field of inverse damage detection, numerous DR techniques have been utilized. Each of these techniques comes with its own set of advantages and disadvantages. For instance, vibration-based monitoring is a promising technique for structural damage detection due to its ability to identify damage by analyzing changes in the dynamic properties of a structure [217]. These dynamic properties, such as natural frequencies, mode shapes, and damping ratios, are sensitive to changes in stiffness caused by structural damage. Various methods have been developed to detect damage based on these properties including methods based on changes in natural frequencies [218], mode shapes [219; 220], and their derivatives [221; 222; 223; 224; 225; 226]. Additionally, non-modal methods, such as the FRF method [227] and guided waves approach [228], can also be used for damage detection but often require additional algorithms to enhance their effectiveness. Furthermore, the DRSBFEMU method has shown promise in real-world applications [188]. This was compared to methods used in [229; 230; 231] and found to be more accurate in identifying simulated damage in a plate. The DRSBFEMU method was particularly close to the true values in its identification results and is a promising tool for practical damage identification applications. However, it is important to note that none of these methods can predict the exact crack size. The DLV method is a valuable tool for the SHM and has been widely used in various research projects [190; 232; 233; 234].