# Forced Dynamics of Elastically Connected Nano-Plates and Nano-Shells in Winkler-Type Elastic Medium
## 9 Conclusions  
This work presents a detailed analytical and numerical analysis of the forced vibration behavior of an orthotropic nano-system composed of an elastically connected nanoplate and a doubly curved shallow nano-shell. The main novelty is the discovery that the upper excited element of the nano-system (the nanoplate) exhibits smaller amplitude transverse vibrations when the lower element is curved (the nano-shell). This study demonstrates how the stiffness in such a nano-system changes based on variations in geometric and material parameters. Both nano-elements are simply supported and embedded into a Winkler-type elastic medium. Using [PERSON]'s constitutive elastic relation, [PERSON] plate theory, and [PERSON]'s linear shallow shell theory, a system of four coupled nonhomogeneous PDEs for the transverse vibration of the system was derived. The numerical method for solving these PDEs is presented in detail, with forced vibration analysis solved using modal analysis. The effects of the nonlocal parameter, external excitation, proportional damping coefficients, and radii of curvature of the nano-shell on the ECSNPS were examined in detail. The amplitude decrease of the excited upper nanoplate was related to the increasing nonlocal parameter and the decreasing radii of curvature of the nano-shell. The phenomenon of smaller amplitude vibrations was observed by comparing the amplitude of forced transverse responses in a system composed of two nanoplates (ECSTNP) to a system composed of a nanoplate and a nano-shell (ECSNPs). An increase in proportional damping coefficients results in decreased transverse displacements of the nanoplate and nano-shell. An increase in external excitation to the upper plate results in increased transverse displacements. The proposed ECSNPS model aids in developing nano-sensors that respond to transverse vibrations based on nano-shell geometry. The results can be used for the development of Nano-Electromechanical Devices (NEMDs), providing engineers with better insight into the behavior of the ECSNPS under the influence of the considered parameters. In particular, adaptable NEMDs that require specific geometrical shapes with different curvatures could benefit from such an analysis. Additionally, it would also be possible and useful to extend such an analysis to other configurations, such as a shell-shell elastically coupled system, which will be the subject of our future research. This paper provides a solid framework for future research to extend the analysis to similar elastically coupled nano-configurations, to compare the results with our numerical findings, and to conduct experimental measurements to validate our numerical results. Conceptualization, M.S.A., I.R.P., J.S., C.B., A.P. and R.P. ; methodology, M.S.A. ; software, M.S.A. ; formal analysis, M.S.A., J.S., I.R.P., C.B., A.P. and R.P. ; writing--original draft, M.S.A. ; writing--review and editing, M.S.A., J.S. and I.R.P. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was financially supported by the ERASMUS+Partnerships for cooperation and exchanges of practices (KA220)--\"Biomedical Innovations through Digital Transformation of Additive Technologies and Knowledge Exchange-BIOMEDIX\"--Project No. 2024-1-LV01-KA220-HED-000255929. Also, this research was financially supported by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia (Contract No. 451-03-136/2025-03/200109). **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The data presented in this study, including the review protocol, are available upon request from the corresponding author. **Acknowledgments:** This research is a revised and expanded version of a doctoral dissertation titled \"Stability and Forced Vibration of Coupled Nano-Structures\" by the corresponding author M.S.A., which was defended at the Faculty of Mechanical Engineering, University of Nis, on 4 September 2023.