## 2 Methods
### Rayleigh-Ritz Formulation
Here, the angle \(\theta_{k}\) accounts for both the stacking angle of the lamina and the relative orientations between the clamped boundaries and the material. Furthermore, it is assumed that \(G_{13}=G_{23}=G_{12}\). The assumption, for which the three material properties are equal, is certainly coarse for some composite materials as well as it is exact for some others [18; 19]. Furthermore, the \(G_{23}\) material property cannot be assessed from the measurements of the first resonant frequency. Therefore, it is assumed here that the three properties are equal. As clamped boundaries do not admit an exact solution, the displacements \(u\), \(v\), and \(w\) are approximated through a set of admissible functions. The plate is firstly assumed to be in harmonic motion, which allows writing the displacement vector \(\underline{u}\) as:  
\[ \begin{split} u_{0}&=U(x,y)\sin\omega t,\\ v_{0}&=V(x,y)\sin\omega t,\\ w_{0}&=W(x,y)\sin\omega t,\\ \varphi_{x}&=\Phi_{x}(x,y)\sin\omega t,\\ \varphi_{y}&=\Phi_{y}(x,y)\sin\omega t,\\ \end{split}  \tag{10}\]  
where \(\omega\) is the natural angular frequency. Then, the displacements and rotations are approximated with the following polynomial functions:  
\[ \begin{split} U(x,y)&=\sum_{i=1}^{l_{1}}\sum_{j=1}^ {l_{1}}c_{ij}^{u}\Big{(}\frac{x}{a}\Big{)}^{i}\Big{(}\frac{y}{b}\Big{)}^{j} \Big{(}1-\frac{x}{a}\Big{)}\Big{(}1-\frac{y}{b}\Big{)},\\ V(x,y)&=\sum_{i=1}^{l_{2}}\sum_{j=1}^{l_{2}}c_{ij}^{u} \Big{(}\frac{x}{a}\Big{)}^{i}\Big{(}\frac{y}{b}\Big{)}^{j}\Big{(}1-\frac{x}{a }\Big{)}\Big{(}1-\frac{y}{b}\Big{)},\\ W(x,y)&=\sum_{i=2}^{l_{3}}\sum_{j=2}^{l_{3}}c_{ij}^{u} \Big{(}\frac{x}{a}\Big{)}^{i}\Big{(}\frac{y}{b}\Big{)}^{j}\Big{(}1-\frac{x}{ a}\Big{)}^{2}\Big{(}1-\frac{y}{b}\Big{)}^{2},\\ \Phi_{x}(x,y)&=\sum_{i=1}^{l_{4}}\sum_{j=1}^{l_{4}}c_{ij }^{u_{x}}\Big{(}\frac{x}{a}\Big{)}^{i}\Big{(}\frac{y}{b}\Big{)}^{j}\Big{(}1- \frac{x}{a}\Big{)}\Big{(}1-\frac{y}{b}\Big{)},\\ \Phi_{y}(x,y)&=\sum_{i=1}^{l_{5}}\sum_{j=1}^{l_{5}}c_{ij }^{u_{y}}\Big{(}\frac{x}{a}\Big{)}^{i}\Big{(}\frac{y}{b}\Big{)}^{j}\Big{(}1- \frac{x}{a}\Big{)}\Big{(}1-\frac{y}{b}\Big{)}\\ \end{split}  \tag{11}\]  
Here, the extremities of the summations are considered as \(l_{1}l_{1}=l_{2}l_{2}=(l_{3}-1)(l_{3}-1)\)\(=\)\(I_{4}J_{4}=\)\(I_{5}J_{5}\). These products define the number of terms \(n\) of the polynomials. In particular, parameters I and J represent the grades of the polynomial functions, which are adopted to approximate the modal shape of the resonant frequencies. Finally, substituting Equations (11), (10), (6), and (4) into Equations (1) and (2), the strain and kinetic energies can be written as  
\[U=\frac{1}{2}[C]^{T}[K][C], \tag{12}\]  
\[T=\frac{1}{2}\omega^{2}[C]^{T}[M][C],\]  
where \([K]\) and \([M]\) are the stiffness and mass matrices of dimensions \(R=I_{1}J_{1}+I_{2}J_{2}+(I_{3}-1)(J_{3}-1)+I_{4}J_{4}+I_{5}J_{5}\) and are related to the functions of Equation (11) adopted to describe the displacements field. \([C]\) is the vector of coefficients, i.e.,:\[\left[C\right]^{T}=\left[c_{11}^{u},c_{12}^{u},\dots,c_{l_{1}l_{1}}^{u},\;c_{11}^{ v},c_{12}^{v},\dots,c_{l_{2}l_{2}}^{v},c_{11}^{v},c_{12}^{v},\dots,c_{l_{3}l_{3}}^{v},c_{1 1}^{v},c_{12}^{v},\dots,c_{l_{4}l_{4}}^{v},c_{l_{3}l_{3}}^{v},c_{11}^{v},c_{12}^ {v},\dots,c_{l_{4}l_{4}}^{v}\right] \tag{13}\]  
Finally, equaling the strain and kinetic energy, as prescribed by the Rayleigh-Ritz method, the following eigenvalue problem can be written:  
\[\left([K]-\omega^{2}[M]\right)\left[C\right]=0, \tag{14}\]  
which allows calculating the resonant frequencies by zeroing the determinant of the matrix. For further details on the matrix elements, the reader can refer to [PERSON] et al. (2018).