# Forced Dynamics of Elastically Connected Nano-Plates and Nano-Shells in Winkler-Type Elastic Medium
## 6 Solution Methodology for Determining the Natural Frequencies of the ECSNPS  
Governing Equations (49)-(52) can be solved by assuming displacements in the following forms:  
\[w_{1}(x,y,t)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}W_{1 mn}(t)sin(\alpha_{m}x) sin(\beta_{n}y), \tag{56}\]  
\[u_{2}(x,y,t)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}U_{2 mn}(t)cos(\alpha_{m}x) sin(\beta_{n}y), \tag{57}\]  
\[v_{2}(x,y,t)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}V_{2 mn}(t)sin(\alpha_{m}x) cos(\beta_{n}y), \tag{58}\]  
\[w_{2}(x,y,t)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}W_{2 mn}(t)sin(\alpha_{m}x) sin(\beta_{n}y), \tag{59}\]  
where \(\alpha_{m}=\frac{m\pi}{a}\) and \(\beta_{n}=\frac{m\pi}{b}\), \(m=1\),\(2\),\(3\),\(\ldots\); \(n=1\),\(2\),\(3\),\(\ldots\). Unknown time functions are marked by \(\mathrm{W}_{1 mn}(t)\),\(\mathrm{U}_{2 mn}(t)\),\(\mathrm{V}_{2 mn}(t)\), and \(\mathrm{W}_{2 mn}(t)\). By substituting the assumed solutions (56)-(59) into Equations (49)-(52) and applying the orthogonality conditions, a system of ordinary differential equations (ODEs) of free vibrations is obtained in the following form:  
\[m_{1}\ddot{W}_{1 mn}(t)+s_{1}W_{1 mn}(t)+\widetilde{\kappa}_{w}W_{2 mn}(t)=0, \tag{60}\]  
\[m_{1}\ddot{U}_{2 mn}(t)+s_{2}\;U_{2 mn}(t)+s_{3}\;V_{2 mn}(t)+s_{4}\;W_{2 mn}(t)=0, \tag{61}\]  
\[m_{1}\ddot{V}_{2 mn}(t)+s_{5}\;V_{2 mn}(t)+s_{6}\;U_{2 mn}(t)+s_{7}\;W_{2 mn}(t)=0, \tag{62}\]  
\[m_{1}\ddot{W}_{2 mn}(t)+s_{8}W_{2 mn}(t)+\widetilde{\kappa}_{w}W_{1 nm}(t)+s_{9} \;U_{2 mn}(t)+s_{10}\;V_{2 mn}(t)=0, \tag{63}\]  
where \(m_{1},\widetilde{\kappa}_{w}\), and \(s_{i}(i=1,\;2\ldots 10)\) are the mass and extensional and bending stiffness constants presented in Appendix A. Taking into account the presented differential Equations (49)-(52), the ODEs (60)-(63) of the time modes of the transverse oscillations that describe the free vibration of the undamped observed ECSNPS have the following matrix form:  
\[ \begin{bmatrix}m_{1}&0&0&0\\ 0&m_{1}&0&0\\ 0&0&m_{1}&0\\ 0&0&0&m_{1}\end{bmatrix}\begin{Bmatrix}\ddot{W}_{1 nm}(t)\\ \ddot{U}_{2 mn}(t)\\ \ddot{V}_{2 mn}(t)\\ \widetilde{\kappa}_{w}\end{Bmatrix} + \begin{bmatrix}s_{1}&0&0&\widetilde{\kappa }_{w}\\ 0&s_{2}&s_{3}&s_{4}\\ 0&s_{6}&s_{5}&s_{7}\\ \widetilde{\kappa}_{w}&s_{9}&s_{10}&s_{8}\end{bmatrix}\begin{Bmatrix}W_{1 mn}(t )\\ U_{2 mn}(t)\\ V_{2 mn}(t)\\ W_{2 mn}(t)\end{Bmatrix} =0, \tag{64}\]  
or for a more concise form, see [PERSON] [41]:  
\[\mathbf{M}\Big{\{}\tilde{\mathrm{S}}\Big{\}}+\mathbf{K}\{\mathrm{S}\}=0, \tag{65}\]  
where  
\[\Big{\{}\tilde{\mathrm{S}}\Big{\}}= \begin{Bmatrix}\ddot{W}_{1 mn}(t)\\ \ddot{U}_{2 mn}(t)\\ \ddot{V}_{2 mn}(t)\\ \ddot{W}_{2 mn}(t)\end{Bmatrix} ,\{\mathrm{S}\}= \begin{Bmatrix}W_{1 mn}(t)\\ U_{2 mn}(t)\\ V_{2 mn}(t)\\ W_{2 mn}(t)\end{Bmatrix} , \tag{66}\]\[\mathbf{M}= \begin{bmatrix}m_{1}&0&0&0\\ 0&m_{1}&0&0\\ 0&0&m_{1}&0\\ 0&0&0&m_{1}\end{bmatrix} \text{and }\mathbf{K}= \begin{bmatrix}s_{1}&0&0& \overset{\sim}{\kappa}_{w}\\ 0&s_{2}&s_{3}&s_{4}\\ 0&s_{6}&s_{5}&s_{7}\\ \overset{\sim}{\kappa}_{w}&s_{9}&s_{10}&s_{8}\end{bmatrix} . \tag{67}\]  
The free vibrations of the ECSNPS described by Equation (65) are periodic. Thus, when free vibrations at a single frequency are initiated for a particular system, the ratio of any two dependent variables is independent of time. These assumptions lead to hypothesizing the normal-mode solution of Equation (65) in the form  
\[\mathbf{S}(t)=\mathbf{Q}e^{i(\omega_{rm})t},m= 1,2,3,\ldots,n= 1,2,3\ldots,r= 1,2,3,4, \tag{68}\]  
where \(\mathbf{Q}\) is the \(m\)-dimensional vector of constants (according to mode shapes) and \(\omega_{rm}\) represents the natural frequencies of the ECSNPS. Substituting Equation (68) into Equation (65) gives  
\[\Big{(}-\omega_{rm}^{2}\mathbf{M}\mathbf{Q}+\mathbf{K}\mathbf{Q}\Big{)}e^{i( \omega_{rm})t}=0. \tag{69}\]  
Since \(e^{i(\omega_{rm})t}\) is valid for any real value of time \(t\),  
\[-\omega_{rm}^{2}\mathbf{M}\mathbf{Q}+\mathbf{K}\mathbf{Q}=0. \tag{70}\]  
The mass matrix is nonsingular, and thus \(\mathbf{M}^{-1}\) exists.