© 2022. Techno-Press, Ltd.In the current work work, static and free torsional vibration of functionally graded (FG) nanorods are investigated using Fourier sine series series. The boundary conditions are described by the two elastic torsional springs at the ends ends. The distribution of functionally graded material i s considered using a power power-law rule rule. The systems of equations of the mechanical response of nanorods subjected to deformable boundary conditions are achieved by using the modified couple stress theory (MCST) and taking the effects of torsional springs into accountaccount. The idea of the study is to construct an eigen value problem involving the torsional spring parameters with small scale parameter and functionally graded index index. This article investigates the size dependent free torsional vibration based on the MCST of functionally graded nano/micro rods with deformable boundary conditions using a Fourier sine series solution for the first time time. The eigen value problem is construc ted using the Stokes’ transform to deformable boundary conditions and also the convergence and accuracy of the present methodology are discussed in various numerical examplesexamples. The small size coefficient influence on the free torsional vibration characteris tics is studied from the point of different parameters for both deformable and rigid boundary conditions conditions. It shows that the torsional vibrational response of functionally graded nanorods are effected by geometry geometry, small size effects effects, boundary conditions and material composition composition. FurthermoreFurthermore, for all deformable boundary conditions in the event of nano nano-sized FG nanorods nanorods, the incrementing of the small size parameters lead leads to increas the torsional frequencies frequencies.