# Forced Dynamics of Elastically Connected Nano-Plates and Nano-Shells in Winkler-Type Elastic Medium
## 7 Forced Vibration of the Damped ECSNPS
\right\}cos(\Omega t), \tag{83}\]  
Equation (80) can be written in the form  
\[\ddot{p}_{r}(t)+2\zeta_{r}\dot{p}_{r}(t)+\omega_{rmn}^{2}p_{r}(t)=h_{r}cos( \Omega t),\ k=1,\ 2,\ 3,\ 4, \tag{84}\]  
where  
\[\text{h}_{r}=(\textbf{X}_{r}\textbf{F}_{0}). \tag{85}\]  
The general solution of Equation (80) can be developed as a sum of homogeneous, \(p_{rh}(t)\), and particular, \(p_{rp}(t)\), solutions in the form  
\[p_{r}(t)=p_{rh}(t)+p_{rp}(t). \tag{86}\]  
Each nonhomogeneous version of Equation (80) has a corresponding homogeneous equation:  
\[\ddot{p}_{r}(t)+2\zeta_{r}\dot{p}_{r}(t)+\omega_{r}^{2}p_{r}(t)=0,r=1,\ 2,\ 3,\ \ldots,n. \tag{87}\]  
The homogeneous solution is assumed to be in the form  
\[p_{rh}=(Acos\theta t+Bsin\theta t)e^{-\zeta_{r}t}, \tag{88}\]  
where \(A\) and \(B\) are the arbitrary constants obtained from the initial conditions and  
\[\vartheta=\sqrt{\omega_{rmn}^{2}-\zeta_{r}^{2}}. \tag{89}\]  
Based on the nonhomogeneous term \(h_{r}\cos(\Omega t)\) given in Equation (84), we can predict the type of function that the particular solution \(p_{rp}\) would be, given as  
\[p_{rp}=Cos(\Omega t)+Dsin(\Omega t), \tag{90}\]  
where \(C\) and \(D\) are coefficients that are easy to determine. By substituting \(\dot{p}_{rp}\) and \(\ddot{p}_{rp}\) into Equation (84), we obtain two equations:  
\[\left(\omega_{rmn}^{2}-K\Omega^{2}\right)C+2\zeta_{r}\Omega D=h_{r}, \tag{91}\]  
\[-2\zeta_{r}\Omega C+\left(\omega_{rmn}^{2}-\Omega^{2}\right)D=0. \tag{92}\]  
From the system of Equations (91) and (92), the coefficients are determined in the following forms:  
\[C=\frac{h_{r}\left(\omega_{rmn}^{2}-\Omega^{2}\right)}{\left(\omega_{rmn}^{ 2}-\Omega^{2}\right)^{2}+4\zeta_{r}^{2}\Omega^{2}}=Ncos\varphi, \tag{93}\]  
\[\text{h}_{r}=(\textbf{X}_{r}\textbf{F}_{0}). \tag{85}\]0  
where \(N\) is the amplitude of the vibration. From Equations (93) and (94), it is easy to obtain the unknowns \(N\) and \(\varphi\) in the following form:  
\[\text{h}_{r}=(\textbf{X}_{r}\textbf{F}_{0}). \tag{85}\]1  
Now, the particular and general solutions have the following forms:  
\[\text{h}_{r}=(\textbf{X}_{r}\textbf{F}_{0}). \tag{85}\]2  
\[\text{h}_{r}=(\textbf{X}_{r}\textbf{F}_{0}). \tag{85}\]3  
For the arbitrary initial condition \(p_{r}(0)=p_{r0}\) and \(\dot{p}_{r}(0)=\dot{p}_{r0}\), it is possible to obtain the constants \(A\) and \(B\) in the following form:  
\[\text{h}_{r}=(\textbf{X}_{r}\textbf{F}_{0}). \tag{85}\]4  
Finally, the general solution of Equation (84) is obtained:  
\[ \begin{array}{c}p_{r}(t)=\left(p_{r0}cos\theta t\ +\ \frac{\dot{p}_{r0}+\zeta_{r}p_{r0}}{ \theta}sin\theta t\right)e^{-\zeta_{r}t}\\ -\left[cos\varphi cos\theta t\ +\ \frac{(\zeta_{r}cos\varphi+\Omega sin \varphi)}{\theta}sin\theta t\right]Ne^{-\zeta_{r}t}+\ Ncos(\Omega t-\varphi). \end{array}  \tag{99}\]  
When the solution from Equation (99) is resubstituted into Equation (77), we obtain  
\[ \begin{array}{c}\text{S(t)}=\left\{\begin{array}{c}W_{1 mm}(t)\\ U_{2 mm}(t)\\ V_{2 mm}(t)\\ W_{2 mm}(t)\end{array}\right\}=\left\{\begin{array}{c}X_{11}(\omega_{1 mm})p_{1 }(t)+X_{12}(\omega_{2 mm})p_{2}(t)+X_{13}(\omega_{3 mm})p_{3}(t)+X_{14}(\omega_ {4 mm})p_{4}(t)\\ X_{21}(\omega_{1 mm})p_{1}(t)+X_{22}(\omega_{2 mm})p_{2}(t)+X_{23}(\omega_{3 mm})p_ {3}(t)+X_{24}(\omega_{4 mm})p_{4}(t)\\ X_{31}(\omega_{1 mm})p_{1}(t)+X_{32}(\omega_{2 mm})p_{2}(t)+X_{33}(\omega_{3 mm})p_ {3}(t)+X_{34}(\omega_{4 mm})p_{4}(t)\\ X_{41}(\omega_{1 mm})p_{1}(t)+X_{42}(\omega_{2 mm})p_{2}(t)+X_{43}(\omega_{3 mm}) p_{3}(t)+X_{44}(\omega_{4 mm})p_{4}(t)\\ m=1,2,3\ldots,\infty.n=1,2,3,\ldots,\infty.\end{array}\right.\end{array}  \tag{100}\]  
Finally, the solutions of Equation (73) for the damped ECSNPS are given in the following forms:  
\[ \begin{array}{c}w_{1}(x,y,t)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^ {\infty}sin(\alpha_{m}x)sin(\beta_{n}y)\\ [X_{11}(\omega_{1 mm})p_{1}(t)+X_{12}(\omega_{2 mm})p_{2}(t)+X_{13}(\omega_{3 mm} )p_{3}(t)+X_{14}(\omega_{4 mm})p_{4}(t)],\end{array}  \tag{101}\]  
\[