# Forced Dynamics of Elastically Connected Nano-Plates and Nano-Shells in Winkler-Type Elastic Medium
## 8 Results and Discussion
### Forced Vibration of the ECSNPS  
The main objective of this research is to observe the expected occurrence of dynamic absorption or decreased amplitudes in the excited upper nanoplate of the ECSNPS in comparison with the results for the ECSTNP. Figure 3 presents the two analyzed nano-systems. This section also examines the material and geometrical influences on the ECSNPS for the first three pairs of forced vibration modes. In this section, the forced transverse vibration response at the mid-points of the nano-systems is analyzed. The ECSNPS is excited with a periodic force in the form F\({}_{0}\)cos (\(\Omega\)t), where F\({}_{0}=10\) [nN] and \(\Omega=0.8\omega_{11}\). The value of the nonlocal parameter is chosen to be \(e_{0}\widetilde{\omega}=1\) [nm] in an analysis that does not include the influence of this parameter. The values of the damping proportional coefficients are chosen to be \(\alpha=\beta=1\) nNs/m. All the parameters for the material and geometrical characteristics used are given in Table 1. The varied parameters are listed in the corresponding figures. The model developed in this paper has not been sufficiently researched by other authors, and it could only be concretely compared with the plate-plate nano-model if we used the material and geometric characteristics from [11]. Assuming that the radii of curvature of the doubly curved shallow shell tends to infinity, i.e., \(R_{1}=R_{2}=\infty\) in Equations (49)-(52), the equations are reduced to two-plate equations, as in the paper by [PERSON] et al. [11]. In this case, the transverse displacements would match those reported in the mentioned paper. As already mentioned during the frequency analysis, assuming that the nonlocal parameter is equal to zero, \(\mu=0\), and the radii of curvature of the doubly curved shallow shell tend to infinity, i.e., \(R_{1}=R_{2}=\infty\) in Equations (49)-(52), the equations are reduced to two-plate equations, as in the paper by [PERSON] [43]. Similarly, in that case, the transverse displacements would match those reported in the paper by [PERSON] [43]. The main novelty and goal of this research are the development of a model and the proof that the upper excited element of the nano-system (nanoplate) exhibits smaller amplitude transverse vibrations when the lower element is curved (nano-shell). A comparison of the amplitude values of the transverse vibrations of the ECSNPS and ECSTNP nano-systems is shown in Figure 4. The transverse displacements as a function of time for the plate-plate (ECSTNP) system are shown in gray, while those for the plate-shell system (ECSNPs) are shown in blue. Figure 4a presents a comparison of the amplitude of the forced transverse vibrations of the upper nanoplate in the ECSNPS with that of the upper nanoplate in the ECSTNP. The most important observation is that the amplitude values of the upper elements in the plate-shell system (ECSNPs) are lower than those in the plate-plate system (ECSTNP), as indicated by the blue line. This decrease in amplitude  
Figure 3: The nano-systems coupled by a Winkler-type elastic layer: (**a**) elastically connected system composed of nanoplate and nano-shell (ECSNPs); (**b**) elastically connected system from two nanoplates (ECSTNP). values in the ECSNPS is one of the primary findings of this study. In Figure 4, the radius of curvature values for the nano-shell are R\({}_{1}\) = \(\infty\) and R\({}_{2}\) = 450 [nm]. Additionally, Figure 4b compares the amplitude values of the forced transverse vibrations of the nano-shell in the ECSNPS with those of the lower nanoplate in the ECSTNP. It was observed that the amplitude value is significantly smaller for the nano-shell in the ECSNPS compared to the vibration amplitude of the lower nanoplate in the ECSTNP. It can be concluded that, with a change in geometric shape, the transverse response decreases, and consequently, the stiffness of the nano-system increases.