# Forced Dynamics of Elastically Connected Nano-Plates and Nano-Shells in Winkler-Type Elastic Medium
## 1 Introduction  
Developing a suitable mathematical model is crucial to obtain the accurate response for the sandwich nano-structures. Generally, an appropriate type of coupling layer is responsible for the stiffness of the sandwich nano-system. For this reason, this study aims to demonstrate how geometric and material parameters can affect the stiffness of nano-systems by analyzing the amplitude values of forced vibrations of nano-systems. In recent years, nonlocal elastic theory has attracted much attention because of the necessity for modeling and analyzing very small mechanical structures in the development of nanotechnologies [1, 2]. It is known that the size effect of nano-structures is important fortheir mechanical behavior because their dimensions are small and comparable to molecular distances. There are various higher-order continuum theories involving material length scale parameters. Among them are strain gradient elasticity theory [3] and nonlocal theory [4; 5; 6]. In these theories, the phenomenon known as material size effects arises because internal material properties, such as inhomogeneity and interatomic interactions, significantly influence elastic behaviors. Due to the expanding applications of nano-sized structures, this consideration has gained increasing interest in fields such as sandwich nano-structures, which have been widely used in various applications in aerospace, biomedicine, and mechanical and civil engineering [7; 8]. Many researchers have shown that the advantage of these structures is their high strength and stiffness-to-weight ratios [9; 10; 11; 12]. The mechanical and geometrical properties of the nano-sandwich system significantly influence the system's behavior [13; 14; 15; 16]. For this reason, this study aims to show how the stiffness of the system would change if one element (a nanoplate) in the nano-system were replaced with another corresponding element of the same surface area, only with a radius of curvature (nano-shell). Additionally, researchers have previously questioned the effect of curvature upon the vibration frequencies of rectangular shallow shells [17]. Open shallow shells have large radii of curvatures compared to other shell parameters. A shallow shell can be doubly curved, and more applications can be found in engineering practice. [PERSON] [18] investigated the free vibration analysis of homogeneous and laminated doubly curved shells on a rectangular planform using three-dimensional elasticity equations. In a paper by [PERSON], [PERSON] et al. [19], an extensive investigation into the deformation, shear, and normal stress values of sandwich structures with lattice cores of varying aspect ratios is presented. Their findings suggest potential for optimization in lightweight structures, which could lead to innovative advancements in design and manufacturing processes within the aerospace and automotive sectors. The nonlocal electro-elastic bending analysis of a doubly curved nano-shell was studied in a paper by [PERSON] [20] based on nonlocal elasticity theory and first-order shear deformation theory. In a paper by [PERSON] and [PERSON] [21], the forced resonate vibration of nano-shells, which included four different geometries of shells--spherical, elliptical, hyperbolic, and cylindrical--was studied. The comprehensive free vibration analysis of doubly curved shallow shells, which are made of an orthotropic material, was presented in a paper by [PERSON] and [PERSON] [22]. In a paper by [PERSON] et al. [23], the influences of changing geometrical parameters and scale parameters on the natural frequency of the micro-/nano-spherical shell based on first-order shear deformation theory were investigated. The increased use of orthotropic shells in the design of various devices, including their mechanical behavior, has attracted the attention of many researchers. As fundamental elements, shells are used in many fields of modern engineering technologies because they provide dynamic stability. A study by [PERSON], [PERSON] et al.