Longitudinal vibration analysis of FG nanorod restrained with axial springs using doublet mechanics

In the current paper, the free longitudinal vibration response of axially restrained functionally graded nanorods is presented for the first time based on the doublet mechanics theory. Size dependent nanorod is considered to be made of functionally graded material consist of ceramic and metal constituents. It is assumed that the material properties of the functionally graded nanorod are assumed to vary in the radial direction. The aim of this study is that to investigate the influences of various parameters such as functionally graded index, small size parameter, length of the nanorod, mode number and spring stiffness on vibration behaviors of functionally graded nanorod restrained with axial springs at both ends. For this purpose, Fourier sine series are used to define the axial deflection of the functionally graded nanorod. Then, an eigenvalue approach is established for longitudinal vibrational frequencies thanks to Stokes' transformation to deformable axial springs. Thus, the presented eigenvalue solution method is attributed to both rigid and deformable boundary conditions for the axial vibration of the functionally graded nanorod. With the help of the results obtained with the presented eigenvalue problem, it is observed that the parameters examined cause significant changes in the frequencies of the functionally graded nanorod.