## 2 Methods
### Rayleigh-Ritz Formulation  
A Rayleigh-Ritz formulation is here presented in order to calculate the first resonant frequency of clamped plates made of laminated composites and loaded by a concentrated mass. A rectangular plate of dimensions \(a\) and \(b\) and constituted by \(N\) orthotropic layers for a total thickness \(h\) is considered, as shown in Figure 1.  
The displacements in \(x\), \(y\), and \(z\) directions are denoted with \(u\), \(v\), and \(w\), respectively. Thus, the strain energy for the entire laminate is expressed as:  
\[U=\frac{1}{2}\sum_{k=1}^{N}\int_{V_{k}}\frac{\epsilon_{k}^{T}}{k}\frac{ \sigma_{k}}{\sigma_{k}}dV_{k}, \tag{1}\]  
Figure 1: Scheme of the clamped plates made of laminated composites. where subscript \(k\) refers to \(k\)-th lamina, \(N\) is the total number of layers, \(V_{k}\) is the volume of each lamina, and \(\underline{\underline{\epsilon}_{k}}\) and \(\underline{\underline{\sigma}_{k}}\) are the strain and stress vectors at the lamina level. Due to the presence of the concentrated mass, the kinetic energy is given by the following sum:  
\[T=\frac{1}{2}\sum_{k=1}^{N}\int_{V_{k}}\rho_{k}\Bigg{[}\bigg{(}\frac{\delta u} {\delta t}\bigg{)}^{2}+\bigg{(}\frac{\delta v}{\delta t}\bigg{)}^{2}+\bigg{(} \frac{\delta w}{\delta t}\bigg{)}^{2}\Bigg{]}dV_{k}+\frac{M}{2}\Bigg{[}\bigg{(} \frac{\delta u_{lm}}{\delta t}\bigg{)}^{2}+\bigg{(}\frac{\delta v_{lm}}{\delta t }\bigg{)}^{2}+\bigg{(}\frac{\delta w_{lm}}{\delta t}\bigg{)}^{2}\Bigg{]} \tag{2}\]  
where \(\rho_{k}\) is the mass density of the \(k\)-th lamina and \(M\) is the concentrated mass located at \(x=l\) and \(y=m\). Therefore, for example, \(u_{lm}\) indicates the displacement in the x direction in correspondence of the mass location. Here, the effect of the concentrated mass on rotary inertia is neglected, and it is assumed for simplicity that the mass is located on the mid-plane, i.e., at \(z=0\). The Rayleigh-Ritz formulation is here based on the higher-order shear deformation theory for laminated composites proposed by [PERSON] [17]. According to the [PERSON]'s theory, the displacements \(u\), \(v\), and \(w\) of an arbitrary point of the plate are represented as  
\[ \begin{split} u(x,y,z,t)&=u_{0}(x,y,t)+z\varphi_{x} (x,y,t)-\frac{4\cdot z^{3}}{3\cdot h^{2}}\bigg{(}\varphi_{x}(x,y,t)-\frac{ \delta w(x,y,t)}{\delta x}\bigg{)},\\ v(x,y,z,t)&=v_{0}(x,y,t)+z\varphi_{y}(x,y,t)-\frac{ 4\cdot z^{3}}{3\cdot h^{2}}\bigg{(}\varphi_{y}(x,y,t)-\frac{\delta w(x,y,t)}{ \delta y}\bigg{)},\\ w(x,y,z,t)&=w_{0}(x,y,t),\end{split}  \tag{3}\]  
where \(u_{0}\), \(v_{0}\), and \(w_{0}\) are the displacements of the mid-plane, and \(\varphi_{x}\) and \(\varphi_{y}\) are the rotations around the \(x\) and \(y\) axes, respectively. Hence, the stress and strain vectors of Equation (1) are \(\underline{\underline{\epsilon}_{k}}=\big{[}\sigma_{x},\varphi_{y},\sigma_{ xz},\varphi_{yz},\sigma_{xy}\big{]}^{T}\) and \(\underline{\underline{\epsilon}_{k}}=\big{[}\underline{\epsilon}_{x},\underline {\epsilon}_{y},\gamma_{xz},\gamma_{yz},\gamma_{xy}\big{]}^{T}\), respectively. The congruent equations for each lamina can be written as follows:  
\[\underline{\underline{\epsilon}_{k}}=B\cdot\underline{u}, \tag{4}\]  
where \(\underline{u}=\big{[}u_{0},v_{0},w_{0},\varphi_{0x},\varphi_{0y}\big{]}\) and the matrix \(B\) is defined as  
\[B=\left[\begin{array}{cccc}\frac{\delta}{\delta x}&0&-\frac{4\cdot x^{3} \delta^{2}}{3h^{2}\delta x^{2}}&\left(z-\frac{4x^{3}}{3  
and  
\[Q=\left[ \begin{array}{ccccc}\frac{E_{11}}{(1-v_{12}v_{21})}&\frac{v_{12}\cdot E _{22}}{(1-v_{12}v_{21})}&0&0&0\\ \frac{v_{12}\cdot E_{22}}{(1-v_{12}v_{21})}&\frac{E_{22}}{(1-v_{12}v_{21})}&0&0 &0\\ 0&0&G_{23}&0&0\\ 0&0&0&G_{13}&0\\ 0&0&0&0&G_{12}\end{array} \right]. \tag{9}\]  
where, as usual, \(E_{11}\) is the [PERSON]'s modulus in the longitudinal direction, \(E_{22}\) is the [PERSON]'s modulus in the transverse direction, \(G_{12}\), \(G_{13}\), and \(G_{23}\) are the shear moduli, and \(v_{12}\) is the [PERSON]'s ratio.