Forced Dynamic of Elastically Connected Nano-Plates and Nano-Shells in Winkler-Type Elastic Medium

With the continuous advancement of manufacturing technology, the demand for small and lightweight structures for various engineering applications has increased. This study examines forced vibration behaviour of an orthotropic nano-system consisting of an elastically connected nano-plate and a doubly curved shallow nano-shell. Both nano-elements (plate and shell) are simply supported and embedded in a Winkler-type elastic medium. Utilizing the Eringen constitutive elastic relation, Kirchhoff-Love plate theory, and Novozhilov linear shallow shell theory, we derive a system of four coupled nonhomogeneous partial differential equations (PDEs) describing the forced transverse vibrations of the system. For forced vibrations, we employ detailed numerical method to solve the differential equations. The forced vibration analysis is conducted using modal analysis. A key finding of this study is that the upper excited element of the nano-system (nano-plate) exhibits smaller amplitude transverse vibrations when the lower element is curved (nano-shell). This phenomenon is observed by comparing the amplitude of forced transverse responses between an elastically connected system of two nano-plates (ECSTNP) and a system composed of a nano-plate and a nano-shell (ECSNPS). We analyse the effects of the nonlocal parameter, external excitation, damping proportional coefficients, and the radii of curvature of the nano-shell on the ECSNPS in detail. The study reveals that the amplitude of the excited upper nano-plate decreases with increasing nonlocal parameter and decreasing radii of curvature of the nano-shell. Additionally, the damping proportional coefficients and external excitation significantly influence the transverse displacements of both the nano-plate and nano-shell. Specifically, an increase in damping proportional coefficients reduces the transverse displacements, while an increase in external excitation to the upper plate increases them. This study provides measurable data relating the vibration characteristics of the nano-system to its geometric and material properties. The proposed mathematical model of the ECSNPS can be applied in the development of new nano-sensors, which respond to transverse vibrations based on the geometry of the nano-shell element.