Studies on vibration control of stay cables in cable-stayed bridges have been keenly interested by researchers and engineers indesigning new bridges and assessing in-service bridges. Mitigation of undesired cable vibration can be achieved by attaching adamping device. A Viscous Mass Damper (VMD) is composed by the arrangement a rotational Viscous Damper (VD) and an inertialmass element in parallel, which has been used to control seismic in many buildings in Japan; however, it has not been studied toapply in field of cable vibration control. This paper proposes the application of VMD to robustness in suppressing cable vibration. Oscillation parameters of the cable-VMD system are investigated in detail using an analytical formulation of the complex eigenvalueproblem and compared with Viscous Damper (VD). Asymptotic formulas to calculate the complex eigenvalue solutions and thedamping ratios of the cable with VMD installed in the proximity of the cable end are proposed, discussed and compared with theexact solutions. Further, the influence of elastic support on the VMD effectiveness of a cable-VMD system is analyzed andexamined. Lastly, a case study is provided to justify the proposed methodology. The results of this study show that VMD canimprove the cable damping ratio more efficiently than VD where the maximum damping ratio of VMD is larger than that of VD, andthe increased damping ratio is higher when the relative modes are higher. The paper also provides necessary insights into thedynamics of the cable-VMD system and a useful tool for selecting optimal parameters of VMD for stay cables.