Vortex-induced vibration (VIV) is a common fluid-structure interaction (FSI) phenomenon in the field of wind engineering and marine engineering. The large-amplitude VIV has a marked impact on the slender structure in fluids, at times even destructive. To study how the VIV can be controlled, the dynamics of a rigid cylinder attached to a rotational nonlinear energy sink (R-NES) is investigated in this paper. This is done using a two degrees of freedom (2-DOF) Van der Pol wake oscillator model adapted to consider a coupled vibration in cross-flow and streamwise directions. The governing equation of R-NES are coupled to the wake oscillator model, hence a flow-cylinder-NES coupled system is established. While exploring the dynamics of the cylinders with different mass ratios under the action of R-NES, it was found that the R-NES deliver better performance in suppressing the VIV of a cylinder with high mass ratios than that of a low mass ratios cylinder. The effect of the distinct parameters of R-NES on VIV response was also systematically investigated in this study. The results indicate that higher mass parameter and rotation radius can lead to improved performance, while the effect of the damping parameter is complex, and appears to be linked to the mass ratio of the column structure.