## 2 Fundamentals of Plate Structural Vibration
### Plate Theories  
The study of plate vibrations began with the development of linear models. These models have gradually progressed to address the complexities of real-world applications. The initial work by [PERSON] and [PERSON] in 1766 and 1789 provided the foundation by modeling plates as two-dimensional membranes. This work led to the derivation of the first differential equations describing the vibrational behavior of thin plates, as illustrated in Equation (1) [36]. These models serve as a reference point for identifying irregularities caused by structural damage.  
\[\frac{\partial^{2}w}{\partial t^{2}}=A\frac{\partial^{2}w}{\partial x^{2}}+B \frac{\partial^{2}w}{\partial y^{2}} \tag{1}\]  
After that, [PERSON] [37] formulated the first true equation governing the free vibration of plates by integrating the effects of bending stiffness into the vibration equations. [PERSON] 1823, further enhanced the theory by considering the molecular structure of the plate. This approach enabled the derivation of a comprehensive differential equation for lateral loading, as shown in Equation (2) [36].  
\[D\Bigg{(}\frac{\partial^{4}w}{\partial x^{4}}+2\frac{\partial^{4}w}{\partial x^{ 2}\partial y^{2}}+\frac{\partial^{4}w}{\partial y^{4}}\Bigg{)}=q \tag{2}\]  
Later, in 1887, [PERSON] re-evaluated this equation using an alternative energy method [36]. However, significant developments were made between 1877 and 1945 by [PERSON] [38; 39] and [PERSON] [40]. These improvements included the effects of transverse shear deformation and rotary inertia, expanding the usefulness of classical theories to higher frequencies. These led to studies on rectangular plate (RP) vibrations using the Rayleigh-Ritz method (RRM) [41; 42; 43].  
Subsequent developments came from [PERSON] (1945) [44] and [PERSON] (1951) [45] by introducing the first-order shear deformation theory (FSDT). This theory is used when a shear correction factor is required to reconcile the actual stress state with the assumed normal stress state [46].  
Building upon [PERSON]'s work, higher-order shear deformation theories (HSDTs) were developed in the 1960s and 1970s. These theories incorporate higher-order terms in the displacement fields, leading to better accuracy for thick plates without the need for a shear correction factor [47; 48; 49]. [PERSON] (1984) [50] made significant contributions in this area by developing a third-order theory that accurately predicted displacements and stresses while ensuring zero transverse shear stress on the plate surfaces.  
More recently, [PERSON] et al. (2012) [51] developed a quasi-3D sinusoidal shear deformation theory for functionally graded plates (FGPs) based on [PERSON]'s unified formulation (CUF). This theory improves the accuracy of static and free vibration analyses by accounting for thickness stretching. Building on these advancements, [PERSON] et al. (2024) [52] further refined these theories, increasing the accuracy of plate analysis, especially for thick and complex structures. This improvement was achieved by enhancing the displacement fields and eliminating the need for shear correction factors.