## 2 Fundamentals of Plate Structural Vibration
### Equation of Motion of Cracked Plates  
Based on previous plate theories, the governing equations were developed and utilized to comprehend the behavior of plates under various loading conditions and boundary constraints. [PERSON] [82] and [PERSON] [37] deeply analyzed the governing equation of the RP in its traditional form. They proposed the governing equation of an intact isotropic and orthotropic RP with the UT and VT based on classical plate theory. Their works serve as a reference point for developing the governing equations to identify irregularities caused by structural damage. Building upon these works, [PERSON] et al. (2011) [83] developed a governing equation for RP with UT featuring a central crack (a). They utilized the approximate relationship between nominal tensile and bending stresses at the crack location, as shown in Figure 1, to enhance the model's accuracy, as shown in Equation (3). [PERSON] and [PERSON] (2012) [84] extended [PERSON]'s work by formulating a governing equation that accounts for a crack inclined at an angle on the top surface of the plate, subjected to bending moments and tension, as shown in Equation (4). Compliance coefficients \(a\)tt, \(a\)bb, and \(a\)bt = \(a\)tb are recognized from [PERSON] and [PERSON]'s paper [85]. Meanwhile, the Ctt, Cbb, and Cbt = Ctb coefficients can be found in the paper by [PERSON] and [PERSON] [86]. \[D\Bigg{(}\frac{\partial^{4}w}{\partial x^{4}}+2\frac{ \partial^{4}w}{\partial x^{2}\partial y^{2}}+\frac{\partial^{4}w}{\partial y^{ 4}}\Bigg{)}\] \[=-\rho h\frac{\partial^{2}w}{\partial t^{2}}+n_{x}\frac{ \partial^{2}w}{\partial x^{2}}+P_{z}+\frac{2 aD\Bigg{(}\frac{\partial^{4}w}{ \partial y^{4}}+v\frac{\partial^{4}w}{\partial x^{2}\partial y^{2}}\Bigg{)}}{ 3\Bigg{(}\frac{\alpha_{bt}^{\circ}}{6}+\alpha_{bb}^{\circ}\Big{)}(3+v)(1-v)h +2a} \tag{3}\] \[-\frac{2a}{\big{(}6\alpha_{bb}^{\circ}+\alpha_{tt}^{\circ}\big{)} (1-v^{2})h+2a}n_{\text{rs}}\frac{\partial^{2}w}{\partial y^{2}}\]  
Figure 1: (**a**) In-plane forces acting on the plate, and (**b**) cracked plate deformation and two-sided constraints [83]. \[D\Bigg{(}\frac{\partial^{4}w}{\partial x^{4}}+2\frac{\partial^{4}w}{ \partial x^{2}\partial y^{2}}+\frac{\partial^{4}w}{\partial y^{4}}\Bigg{)} \tag{4}\] \[=-\rho h\frac{\partial^{2}w}{\partial t^{2}}+n_{x}\frac{\partial^{ 2}w}{\partial x^{2}}+q_{z}+\frac{aD(1+cos2\beta)\Bigg{(}\frac{\partial^{4}w}{ \partial y^{4}}+v\frac{\partial^{4}w}{\partial x^{2}\partial y^{2}}\Bigg{)}}{3 \Big{(}\frac{\alpha_{bt}}{6}+\alpha_{bb}\Big{)}(3+v)(1-v)h+2a}\] \[-\frac{a(1+cos2\beta)}{(6\alpha_{tb}+\alpha_{tt})(1-v^{2})h+2a}n_ {\circ}\frac{\partial^{2}w}{\partial y^{2}}+\frac{2 aDsin2\beta\Bigg{(}\frac{ \partial^{4}w}{\partial x\partial y^{3}}+v\frac{\partial^{4}w}{\partial x^{3} \partial y}\Bigg{)}}{3\Big{(}\frac{C_{bt}}{6}+C_{bb}\Big{)}(1+v)h+2a}\] \[-\frac{2 asin2\beta}{(6C_{tb}+C_{tt})(1+v)h+2a}n_{\circ}\frac{ \partial^{4}w}{\partial x\partial y}\]  
[PERSON] et al. (2018) [27] further expanded this area of study by deriving equations of motion for partially cracked isotropic and FGM plates with both linear and parabolically varying thicknesses (LVT and PVT). They developed the crack terms using the line spring model (LSM) for plates with VT, making them dependent on the plate thickness. Their analysis, based on classical plate theory, assumed that the volumes of the plates with the VT and the UT were identical to enable a comparison of dynamic properties. For simplification, they disregarded the normal stress, shear deformation, and rotary inertia, as shown in Equation (5). [PERSON] et al. (2022) [87] continued this research by focusing on cracked orthotropic plates with VT under different boundary conditions and incorporated crack terms using the LSM. The study also explored the impact of in-plane forces on frequency response by analyzing nonlinearity using multiple scales. \[