# Forced Dynamics of Elastically Connected Nano-Plates and Nano-Shells in Winkler-Type Elastic Medium
## 5 Formulation of Model ECSNPS
\begin{array}{l}\rho h\frac{\partial^{2}w_{2}}{\partial t^{2}}-\mu\rho h \Big{(}\frac{\partial^{4}w_{2}}{\partial x^{2}\partial t^{2}}+\frac{\partial ^{4}w_{2}}{\partial y^{2}\partial t^{2}}\Big{)}+\Big{(}1\frac{a}{R_{1}}-\frac{ A_{12}}{R_{2}}+\frac{D_{11}+D_{12}}{R_{1}^{2}R_{2}}\Big{)}\frac{\partial^{2} w_{2}}{\partial x^{2}}+\big{(}D_{12}+2D_{6}\big{)}\frac{\partial^{4}w_{2}}{ \partial x^{2}\partial y}\\ +\Big{(}\frac{A_{11}}{R_{1}}+\frac{A_{12}}{R_{2}}\Big{)}\frac{\partial^{2}w_{2} }{\partial x^{2}}-\Big{(}4C_{66}+\frac{D_{12}}{R_{1}}+\frac{D_{22}}{R_{1}^{2} }\Big{)}\frac{\partial^{4}w_{2}}{\partial x^{2}}-\Big{(}\frac{D_{11}}{R_{1}}+ \frac{D_{12}}{R_{1}^{2}}\Big{)}\frac{\partial^{3}w_{2}}{\partial x^{2}}+\big{(} \frac{A_{22}}{R_{1}}+\frac{A_{12}}{R_{1}}\Big{)}\frac{\partial w_{2}}{\partial y }\\ -\Big{(}\frac{D_{11}}{R_{1}}+\frac{D_{12}}{R_{2}}-4C_{66}\Big{)}\frac{\partial^{ 2}w_{2}}{\partial x^{2}\partial y}-\Big{(}\frac{D_{12}}{R_{1}}+\frac{D_{22}}{R_ {1}^{2}R_{2}}\Big{)}\frac{\partial^{3}w_{2}}{\partial y^{2}}-\Big{(}\frac{D_{1 1}}{R_{1}}+\frac{D_{12}}{R_{2}}\Big{)}\frac{\partial^{3}w_{2}}{\partial x^{2}} -\Big{(}\frac{D_{12}}{R_{1}}+\frac{D_{12}}{R_{1}^{2}R_{2}}\Big{)}\frac{\partial^ {3}w_{2}}{\partial x^{2}}-\Big{(}\frac{D_{12}}{R_{1}}+\frac{D_{12}}{R_{1}} \Big{)}\frac{\partial^{3}w_{2}}{\partial x^{2}}-\Big{(}\frac{D_{12}}{R_{1}R_{1} }\Big{)}\frac{\partial^{3}w_{2}}{\partial x^{2}}-\Big{(}\frac{D_{12}}{R_{1}R_{1} }+\frac{D_{12}}{R_{2}}\Big{)}\frac{\partial^{3}w_{2}}{\partial x^{2}}-\Big{(} \frac{D_{12}}{R_{1}R_{1}}\Big{)}\frac{\partial^{3}w_{2}}{\partial x^{2}}-\Big{(} \frac{D_{12}}{R_{1}}+\frac{D_{12}}{R_{1}}\Big{)}\frac{\partial^{3}w_{2}}{ \partial x^{2}}-\Big{(}\frac{D_{12}}{R_{1}}+\frac{D_{12}}{R_{1}^{2}R_{2}} \Big{)}\frac{\partial^{3}w_{2}}{\partial x^{2}}-\Big{(}\frac{D_{12}}{R_{1}R_{1} }+\frac{D_{12}}{R_{1}^{2}R_{2}}\Big{)}\frac{\partial^{3}w_{2}}{\partial x^{2}} \\ +\Big{(}\frac{A_{11}}{R_{1}}+2\frac{A_{12}}{R_{1}R_{2}}+\frac{A_{22}}{R_{1}^{2} }\Big{)}w_{2}-\ k(w_{1}-w_{2})+k\mu\Big{[}\Big{(}\frac{\partial^{2}w_{1}}{ \partial x^{2}}-\frac{\partial^{2}w_{2}}{\partial x^{2}}\Big{)}+\Big{(}\frac{ \partial^{2}w_{1}}{\partial y^{2}}-\frac{\partial^{2}w_{2}}{\partial y^{2}}\Big{)} \Big{]}=0\end{array}  \tag{52}\]  
The transverse displacements of the midplane of the nanoplate are described by \(w_{1}\) in Equation (49). In Equations (50)-(52), it can be observed that the displacement components in all three directions, \(u_{2}\), \(v_{2}\), and \(w_{2}\), appear. Therefore, all equations must be considered when describing the small transverse oscillations of an orthotropic doubly curved shallow nano-shell. The mathematical analysis was conducted using the Cartesian coordinate reference system. The boundary conditions of a simply supported nanoplate and nano-shell, of length \(a\) and width \(b\), are  
\[w_{i}(x,0,t)=w_{i}(x,b,t)=\ 0,w_{i}(0,y,t)=w_{i}(a,y,t)=\ 0,\ i=\ 1,2, \tag{53}\]  
\[ \begin{array}{l}M_{xxi}(0,y,t)=M_{xxi}(a,y,t)=\ 0,M_{yyi}(x,0,t)=\ M_{yyi}(x,b,t)=\ 0,\ i=\ 1,2,\end{array}  \tag{54}\]  
\[\begin{array}{l}N_{xxi}(0,y,t)=N_{xxi}(a,y,t)=\coincide only with the simply supported boundary conditions when using both forms of [PERSON]'s equation (integral and differential) [37; 38; 39]. [PERSON]'s constitutive relation in integral form should be applied to other boundary conditions in future publications, and it is worthwhile to compare the results with those from the paper [40]. For this reason, due to the detailed methodology presented in this paper, using [PERSON]'s differential form, only simply supported boundary conditions of the system are considered.