This study attempts to explain the motion behavior of the marine riser coupled to drill string when the vortex induced vibration is involved. Vibrations have been reported to have a major effect on drilling performance, affecting the Rate of Penetration (ROP), causing severe damage to drilling tools and also reduces the efficiency of the drilling process. There are two major components of drilling tools that are subjected to vibration, namely the marine riser and the drilling string. Analysis of vibration in the marine riser and drill string are two topical areas that have individually received considerable attention by researchers in the past. While these two subjects are interrelated, borne by the fact that the marine riser encapsulates and protects the drill pipe, they have been little attempt to investigate them together as an assembly. Due to the complexities of the models, simplifying assumptions were made in order to undertake the investigation. Using staggered approach, the results were compared with experimental and simulation data from the open literature. It was found that the maximum displacement with negative damping occurs at low frequency and rotation speed. The addition of vortex shedding forces caused the drill string to deform from its vertical axis and thus increased the risk of failure of drill string and riser.