## 2 Formulation and Methodology  
As illustrated in Figure 1, a rectangular orthotropic thin plate is considered herein, which has two opposite edges free (i.e., \(y=0\) and \(y=b\)) and the others rotationally restrained (i.e., \(x=0\) and \(x=a\)). The differential equation for the free transverse vibration of orthotropic plates is [1]  
\[D_{x}\frac{\partial^{4}w}{\partial x^{4}}+2H\frac{\partial^{4}w}{\partial x^{2 }\partial y^{2}}+D_{y}\frac{\partial^{4}w}{\partial y^{4}}+\rho h\frac{ \partial^{2}w}{\partial t^{2}}=0 \tag{1}\]  
in which \(w(x,y,t)\) is the displacement function; \(\rho\) is the density of the plate; \(D_{x}\) and \(D_{y}\) are the flexural rigidity in the \(x\)-direction and \(y\)-direction, respectively; and \(H\) is effective torsional rigidity. The frequency of sinusoidal oscillations is denoted by \(\omega\). Then, the displacement function of the plate \(w(x,y,t)\) can be given by  
\[w(x,y,t)=W(x,y)e^{i\omega t} \tag{2}\]  
Substituting Equation (2) into Equation (1), it can be obtained  
\[D_{x}\frac{\partial^{4}W}{\partial x^{4}}+2H\frac{\partial^{4}W}{\partial x^{ 2}\partial y^{2}}+D_{y}\frac{\partial^{4}W}{\partial y^{4}}-\omega^{2}\rho hW=0 \tag{3}\]For simplicity, the partial differentiation was denoted by a comma (e.g., \(\frac{dw}{dx}=w_{x}\)). The boundary conditions of the plate can be written as  
\[w=0,\ M_{x}=-D_{x}\left(w_{,xx}+v_{y}w_{,yy}\right)=-R_{x0}w_{,x} \ at\ x=0 \tag{4a}\] \[w=0,\ M_{x}=-D_{x}\left(w_{,xx}+v_{y}w_{,yy}\right)=R_{xa}w_{,x} \ at\ x=a\] (4b) \[M_{y}=-D_{y}\left(w_{,yy}+v_{x}w_{,xx}\right)=0\ at\ y=0,\ b\] (4c) \[V_{y}=-D_{y}w_{,yyy}-\left(H+2D_{xy}\right)w_{,yxx}=0\ at\ y=0,\ b \tag{4d}\]  
where \(M_{x}\) and \(M_{y}\) are the bending moments, \(V_{y}\) is the effective shear force, and \(R_{x0}\) and \(R_{xa}\) are rotational stiffness as shown in Figure 1. In order to describe the rotational stiffness regarding the flexural stiffness of the plate, a rotational fixity factor \(r\) was developed by [PERSON] and [PERSON] [21] as  
\[r_{x0}=\frac{1}{1+3\frac{D_{x}}{R_{x0}a}} \tag{5a}\] \[r_{xa}=\frac{1}{1+3\frac{D_{x}}{R_{xa}a}} \tag{5b}\]  
Rotational fixity factors have a range from 0 to 1. Then, a general boundary condition can be modelled by different values. For instance, the limiting cases, simply supported and fully clamped boundary conditions, will be treated as \(r=0\) and \(r=1\), respectively. Thus, it can be obtained from Equations (5) that  
\[w(x,y,t)=W(x,y)e^{i\omega t} \tag{2}\]0 \[w(x,y,t)=W(x,y)e^{i\omega t} \tag{2}\]1  
\[w(x,y,t)=W(x,y)e^{i\omega t} \tag{2}\]2 Orthotropic plate with opposite rotationally restrained and free edges (R–F–R–F). At first, the boundary conditions were rearranged by separately taking the finite cosine transform (FCT) of Equations (4a) and (4b) with respect to \(y\) and the finite sine transform (FST) of Equations (4c) and (4d) with respect to \(x\). As a result, the boundary conditions can be expressed as  
\[w(x,y,t)=W(x,y)e^{i\omega t} \tag{2}\]3 \[w(x,y,t)=W(x,y)e^{i\omega t} \tag{2}\]4 (7b) \[w(x,y,t)=W(x,y)e^{i\omega t} \tag{2}\]5 (7c) \[w(x,y,t)=W(x,y)e^{i\omega t} \tag{2}\]6  
Subsequently, the governing equation of Equation (3) is solved by using the method of finite integral transform. In this paper, the joint finite integral transform is defined by applying FST with respect to \(x\) with \(m\) as the subsidiary variable and the FCT in regard to \(y\) with \(n\). \[w(x,y,t)=W(x,y)e^{i\omega t} \tag{2}\]7  
where  
\[w(x,y,t)=W(x,y)e^{i\omega t} \tag{2}\]8  
The transform inversion of Equation (8) can be obtained as [24]  
\[w(x,y,t)=W(x,y)e^{i\omega t} \tag{2}\]9  
where  
\[\varepsilon_{n}=\left\{ \begin{array}{cc}1/2,&n=0\\ 1,&n\
eq 0\end{array} \right.