# Forced Dynamics of Elastically Connected Nano-Plates and Nano-Shells in Winkler-Type Elastic Medium
## 7 Forced Vibration of the Damped ECSNPS  
In this section, the forced vibrations of the damped system for the considered nano-system are analyzed. Damping is implemented using the assumption of Rayleigh-type proportional damping, defined by the coefficients \(\alpha\) and \(\beta\). Specifically, the damping matrix is a linear combination of the mass matrix \(\beta\mathbf{M}\) and the stiffness matrix \(\alpha\mathbf{K}\); see [PERSON]'s study [41]. Considering the PDEs of the nanoplate (49) and nano-shell (50)-(52), the PDEs for the time modes of the transverse vibrations, which describe the forced vibrations of the damped ECSNPS, can be presented in the following matrix form, as shown in [PERSON]'s study [41]:  
\[\mathbf{M}\Big{\{}\tilde{S}\Big{\}}+(\alpha\mathbf{K}+\beta\mathbf{M})\Big{\{} \hat{S}\Big{\}}+\mathbf{K}\{\mathrm{S}\}=\{\mathrm{F}\}, \tag{73}\]where **M**, **K**, \(\left\{\mathbb{S}\right\}\), and \(\left\{\mathbb{S}\right\}\) are presented in expressions ((66) and (67)) and  
\[\left\{F\right\}= \begin{cases}f\ -\ \mu\Big{(}\frac{\partial^{2}f}{\partial x^{2 }}+\frac{\partial^{2}f}{\partial y^{2}}\Big{)}\\ 0\\ 0\\ 0\end{cases} . \tag{74}\]  
To determine the forced vibration response of the presented system (73), the standard procedure of modal analysis was used based on the principal coordinates from Kelly [41]. For the natural frequencies, \(\omega_{1 mm}\leq\omega_{2 mm}\leq\omega_{3 mm}\leq\omega_{4 mm}\), obtained from expression (72), the matrix \(\mathbb{P}\) presents the system's modal matrix, a matrix whose columns are normalized mode shapes:  
\[\mathbb{P}=\Big{\{}X_{1}\quad X_{2}\quad X_{3}\quad X_{4}\Big{\}}. \tag{75}\]  
The system of n-differential equations corresponding to the generalized coordinates \(S(t)\) can be transformed into a system with the principal coordinates \(p_{j}(t),\ j=1,2,3,4\) by introducing the modal matrix of the system, \(\mathbb{P}\). The columns of this matrix represent the normalized mode shapes. The transformation is given by  
\[\mathbb{S}(t)=\sum_{j=1}^{4}p_{j}(t)X_{j}, \tag{76}\]  
which is equivalent to a linear transformation between the generalized and the principal coordinates of the system:  
\[\mathbb{S}(t)=\mathbb{P}(t). \tag{77}\]  
Substituting of Equation (39) into Equation (36) yields  
\[\mathbb{M}\sum_{j=1}^{4}\ddot{p}_{j}(t)X_{j}+(\alpha\mathbb{K}+\beta\mathbb{M })\sum_{j=1}^{4}\dot{p}_{j}(t)X_{j}+\mathbb{K}\sum_{j=1}^{4}p_{j}(t)X_{j}=\{ \mathbb{F}\}. \tag{78}\]  
By applying the mode-shape orthogonality condition and multiplying Equation (78) by a scalar \(X_{r}\), for an arbitrary value of \(r=1,2,3,4\), we find that only one term in each of the sums is non-zero, specifically when \(r=j\). With the mode shapes normalized, Equation (78) can be expressed as follows:  
\[ \begin{array}{l}\sum\limits_{r=1}^{4}\ddot{p}_{r}(t)(X_{r}X_{r})_{M}+\sum \limits_{j=1}^{4}\dot{p}_{r}(t)\big{[}\alpha(X_{r}X_{r})_{K}+\beta(X_{r}X_{r}) _{M}\big{]}\\ +\sum_{r=1}^{4}p_{r}(t)(X_{r}X_{r})_{K}\ =(X_{r}F).\end{array}  \tag{79}\]  
Equation (79) can also be written in the form  
\[\ddot{p}_{r}(t)+2\zeta_{r}\dot{p}_{r}(t)+\omega_{mm}^{2}p_{r}(t)=g_{r}(t),r= 1,\ 2,\ 3,\ 4, \tag{80}\]  
where  
\[\zeta_{r}=\frac{1}{2}\Big{(}\alpha\omega_{mm}^{2}+\beta\Big{)}, \tag{81}\]  
For load \(\left\{F\right\}\) in expression (74), we take the uniformly distributed surface harmonic load in the following form:  
\[f(x,y,t)=F_{0}cos(\Omega t). \tag{82}\]By substituting (82) into expression (74), we obtain  
\[\text{F}=\{\text{F}\}cos(\Omega t)=\left\{ \begin{matrix}f-\mu\left(\frac{\partial^{ 2}f}{\partial x^{2}}+\frac{\partial^{2}f}{\partial y^{2}}\right)\\ 0\\ 0\\ 0\end{matrix}