# Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation
## 1 Introduction  
Micro and nano electromechanical systems (MEMS and NEMS) have aroused great research interest due to their unique properties and features. These systems have widespread applications in many engineering and modern technology fields such as composites and electronics. During the manufacturing process, producing, assembling, and packaging, some micro and nano-structures may have geometrical irregularities, as variations in the height and the width that may have significant effect on the characteristics and the performance of these structures. For more damage tolerance and vibration control, and for the sake of providing weight reduction, structures with non-uniform cross sections (tapered) are used in several industrial applications. Therefore, it is worthwhile to create reliable models for these non-uniform micro and nano structures in order to predict their dynamical response and characteristics. Additionally, for the sake of increasing the strength with respect to the weight of a structure, functionally graded (FG) materials are found in numerous civil and aerospace structures [1]. Moreover, carbon nanomaterials such as carbon nanotubes and graphene are heterogeneous and non-uniform because of the effect of grid distance or particle size on their characteristics and properties. Double-beam structures are commonly utilized in numerous applications due to their high specific strength, high rigidity, and vibration isolation. However, investigating the vibrations of axially functionally graded (AFG) non-uniform double beams has not been well addressed in the literature. Therefore, the motivation of this research is to fill the gap in modeling and analyzing the vibration characteristics of double AFG nanobeams lying on elastic medium. Experimental statistical investigations and simulations are able to display the effect of the scale parameter on the response of structures at the micro and nano scales; however, these approaches are costly, time consuming, and limited. As the continuum approach for elastic structures at the regular macro scales are not capable of showing the influence of the scale parameter on the behavior of different nano structures; therefore, the local continuum approach is insufficient to expect the dynamical response of these systems [2]. In order to use the continuum theories to study different characteristics of nanostructures, the scale parameter must be taken into consideration. Thus, different theories were proposed, such as the strain gradient theory, the modified coupled stress theory, and the nonlocal elasticity theory [3] that will be applied in this research to carry out the vibrations of axially functionally graded double beams lying on elastic medium. Many researchers applied [PERSON]'s non-local theory to analyze the vibration and buckling behavior of micro and nano structures. For example, [PERSON] et al. [4] examined the natural vibration and the buckling analyses of micro-structured plates utilizing [PERSON]'s nonlocal elasticity model with three different plate theories. Analytical solutions for frequencies and critical loads of these structures were obtained, and it was found that the buckling load and the geometry of the plate have remarkable influence on the small-scale parameter. [PERSON] et al. [5] used the nonlocal theory along with the Kelvin model to study the flexural oscillations of double-viscoelastic-Functionally Graded (FG) nanoplate system resting on a visco-elastic Pasternak medium, and subjected to in-plane loads. It was observed that the frequencies and the buckling loads depend on the structural damping, loading ratio, Pasternak foundation, and the small-scale effect. [PERSON] et al. [6] used [PERSON]'s theory to study the vibrations of orthotropic Kirchhoff nano-plates. The effects of several factors including the scale parameter on the frequencies were conducted. Moreover, [PERSON] and [PERSON] [7] used [PERSON]'s nonlocal theory to examine the natural vibration of defective graphene sheets.