## 3 Theoretical Formulations
### Solution Procedure  
In this section, the nonlinear equations of motion are solved numerically, in order to obtain the linear and nonlinear frequency equations. In this regard, the nonlinear partial differential equations (PDEs) of motion (32)-(34) are discretized as ordinary differential equations by employing the Galerkin method. Afterwards, the multiple timescales approach is used to obtain the nonlinear frequency equation. Here, we assume that the GPL-reinforced sandwich beam is simply supported at both ends with movable supports. Based on these assumptions, the displacement and rotation field of the beam can be defined as expansions of the free vibration mode shapes, namely,  
\[u(x,t)=\sum_{n=1}^{\infty}U_{n}(t)\chi_{n}(x),\quad\chi_{n}(x)=\text{Cos}\Big{(} \frac{n\pi x}{L}\Big{)}, \tag{35}\]  
\[\phi(x,t)=\sum_{n=1}^{\infty}\varphi_{n}(t)\psi_{n}(x),\quad\psi_{n}(x)=\text{ Cos}\Big{(}\frac{n\pi x}{L}\Big{)}, \tag{36}\]  
\[w(x,t)=\sum_{n=1}^{\infty}W_{n}(t)\lambda_{n}(x),\quad\lambda_{n}(x)=\text{Sin} \Big{(}\frac{n\pi x}{L}\Big{)}, \tag{37}\]  
\(U_{n}\), \(\varphi_{n}\) and \(W_{n}\) being the unknown generalized coordinates which stand for the amplitude of the vibration. Moreover, the following functions \(\chi_{n}(x)\), \(\psi_{n}(x)\), and \(\lambda_{n}(x)\) are introduced to satisfy all boundary conditions of the system. By considering a single mode approximate solution and by substitution of Equations (35)-(37) into Equations (32)-(34), after multiplying the results by \(\chi\), \(\psi\), and \(\lambda\) and after their integration over the domain of the system, the following nonlinear differential equations of motion are obtained  
\[g_{11}U_{n}+g_{12}W_{n}+g_{13}{W_{n}}^{2}+g_{14}\varphi_{n}=0, \tag{38}\]  
\[g_{21}U_{n}+g_{22}W_{n}+g_{23}{W_{n}}^{2}+g_{24}\varphi_{n}=0, \tag{39}\]  
\[g_{31}U_{n}+g_{32}W_{n}+g_{33}U_{n}W_{n}+g_{34}{W_{n}}^{2}+g_{35}{W_{n}}^{3}+g_ {36}\varphi_{n}+g_{37}W_{n}\varphi_{n}+g_{38}U_{n^{\prime\prime}}+g_{39}{W_{n }}^{\prime\prime}+g_{310}\varphi_{n^{\prime\prime}}=0, \tag{40}\]  
where coefficients \(g_{11}\), \(g_{12}\),..., \(g_{310}\) are detailed in Appendix A. The ordinary differential equation of transverse motion can be obtained by solving \(U_{n}\) and \(\varphi_{n}\) in terms of \(W_{n}\) from Equations (38) and (39) and substituting the results in Equation (40). Thus, we get  
\[{W_{n}}^{\prime\prime}+\alpha_{1}W_{n}+\alpha_{2}{W_{n}}^{3}=0, \tag{41}\]  
where  
\[\alpha_{1}=\omega_{L}^{2}=\frac{g_{14}(g_{22}g_{31}-g_{21}g_{32})+g_{12}(-g_ {24}g_{31}+g_{21}g_{36})+g_{11}(g_{24}g_{32}-g_{22}g_{36})}{g_{24}(-g_{12}g_{38 }+g_{11}g_{39})+g_{14}(g_{22}g_{38}-g_{21}g_{39})+(g_{12}g_{21}-g_{11}g_{22})g _{310}}, \tag{42}\]  
\[\alpha_{2}=\frac{g_{14}(-g_{23}g_{33}+g_{21}g_{35})+g_{13}(g_{24}g_{33}-g_{21} g_{37})+g_{11}(-g_{24}g_{35}+g_{23}g_{37})}{g_{24}(g_{12}g_{38}-g_{11}g_{39})+g_{14}(-g_ {22}g_{38}+g_{21}g_{39})+(-g_{12}g_{21}+g_{11}g_{22})g_{310}}, \tag{43}\]  
and \(\omega_{L}\) is the natural frequency of the nanocomposite sandwich beam. According to the multiple timescale approach [57; 58], we approximate the solution of Equation (41) by means of the following expansion,  
\[W(t,\epsilon)=\epsilon W_{1}(T_{0},T_{1},T_{2},\ldots)+\epsilon^{2}W_{2}(T_{0},T_{1},T_{2},\ldots)+\epsilon^{3}W_{3}(T_{0},T_{1},T_{2},\ldots)+\ldots \tag{44}\]  
where \(\epsilon\) is a small perturbation parameter and \(T_{n}=\epsilon^{n}t\) refers to the independent variables for \(n=0,1,2,\ldots\), whose derivatives with respect to \(t\) are defined as follows:  
\[ \begin{array}{c}\frac{d}{dt}=\frac{dT_{0}}{dt}\frac{\partial}{dT_{0}}+\frac{ dT_{1}}{dt}\frac{\partial}{\partial T_{1}}+\ldots=D_{0}+\epsilon D_{1}+\ldots\\ \frac{d^{2}}{dt^{2}}=D_{0}^{2}+2\epsilon D_{0}D_{1}+\epsilon^{2}\Big{(}D_{1}^{2 }+2D_{0}D_{2}\Big{)}+2\epsilon^{3}D_{1}D_{2}+\epsilon^{4}D_{2}^{2}+\ldots\end{array}  \tag{45}\]where \(D_{n}=\partial/\partial T_{n}\). In our case, we apply the expansion up to \(O\Big{(}\epsilon^{3}\Big{)}\), such that we need \(T_{0},\;\;T_{1}\) and \(T_{2}\).