Nonlinear Torsional Vibration of Size-Dependent Functionally Graded Rods

This study aims to investigate the effect of functionally graded materials (FGMs) on the internal resonances of nanorods in torsional vibration. The von-Kármán type nonlinearity is considered and the governing equation of motion is derived using Hamilton's principle based on the surface elasticity theory. It is assumed that the properties of the functionally graded (FG) nanorod vary through the radius direction based on power-law distribution. Then, the multi-mode Galerkin method is implemented to convert the partial differential equation to an ordinary differential one. In the next step, the method of multiple scales is used to derive the natural frequencies as well as the conditions in which the internal resonances occur. The results are presented for two types of end conditions, fixed-fixed and fixed-free, and the effects of variations of various parameters like length, radius, and amplitude of vibration on natural frequencies are investigated. This research shows that functionally graded materials differ in the state of happening the internal resonances in the presence of the surface energy.