# Forced Dynamics of Elastically Connected Nano-Plates and Nano-Shells in Winkler-Type Elastic Medium
## 1 Introduction
[24] deals with the free vibration and buckling responses of porous orthotropic doubly curved shallow shells subjected to edge compressions that are non-uniformly distributed. In this paper, parametric studies were developed to discuss the influence of various factors, such as different porosity coefficients, aspect ratios, arc length-to-thickness ratios, radius-to-arc length ratios, orthotropy ratios, and shell types, on the free vibration and buckling behavior of porous orthotropic doubly curved shallow shells. The free vibration of composite sandwich plates and cylindrical shells, composed of two composite laminated faces and an ideally orthotropic elastic core, was considered by [PERSON] [25]. Based on several different numerical mathematical methods, the natural frequencies of composite sandwich plates and cylindrical shells [25] were obtained. It was shown that the values of the natural frequencies of such a system agree well with each other. In the present study, the authors analyze in detail a nano-system composed of an elastically connected nanoplate and nano-shell. The nanoplate and the doubly curved shallow nano-shell are made of orthotropic materials. Both nano-elements (the plate and shell) are simply supported and connected by an elastic layer, which is approximated by the Winkler model of discretely distributed springs of linear stiffness, \(k\), acting on the surface of the nanoplate and nano-shell. In this paper, the forced vibration analysis was conducted for an elastically connected system composed of a nanoplate and nano-shell (ECSNPS). Based on [PERSON]'s constitutive elastic relation, [PERSON] plate theory [26; 27; 28], and [PERSON]'s linear shallow shell theory [29], a system of four coupled nonhomogeneous PDEs for the transverse vibrations of an ECSNPS was derived. The forced transverse vibration analysis was solved by modal analysis. The effect of the nonlocal parameter on the values of the amplitudes of the forced vibration of the ECSNPS is presented. Additionally, the effects of external excitations, damping proportional coefficients, and radius curvatures of nano-shell on the ECSNPS are analyzed in detail. We rigorously examine the effects of nano-shell curvature on nano-systems, with a particular focus on its role in reducing the amplitude of transverse vibrations in the upper nanoplate. The advanced mathematical model of the sandwich nano-structure proposed here holds significant promise for the dynamic analysis of nano-sensors. These applications leverage the unique geometry of the nano-shell element to precisely tune and optimize transverse responses, thereby driving significant advancements in the field of nanotechnology. The model developed in this paper is not sufficiently researched by other authors. For this reason, the authors provide insights into the nano-system of an elastically connected nanoplate and a doubly curved shallow nano-shell and give a detailed analysis of the solution of partial differential equations describing transverse oscillations. It also provides clear insight into natural frequency analysis and how the nonlocal parameter affects them. Also, the analysis of the impact of changing the radius of curvature of the lower element (nano-shell) in this model yields significant results.