**5. Conclusions**

In this paper, the combined damping effect of two discrete IMDs on a stay cable, either on the opposite end or the same end, was theoretically investigated in comparison with a single IMD, especially for the single-mode cable vibration control. Results showed that the maximum supplemental modal damping ratio of a cable provided by two opposite IMDs with small or moderate inertial mass is approximately the sum of contributions from each IMD. However, damping performances of the cable with two opposite IMDs will be reduced when the IMDs adopt relatively large inertial mass, in which the superposition effect of each IMD gets weak. As for a cable with two IMDs at the same end, the maximum modal damping ratio of the cable is smaller than that of a single IMD at the further distance when the IMDs adopt relatively small or large inertial mass. Fortunately, when the inertial mass of the IMD is appropriate, attaching two IMDs at the same end of the cable is able to obtain a larger maximum modal damping ratio than that of a single IMD at a bigger distance, which is even more than the sum of contributions from each IMD. Generally, attaching two opposite IMDs on a cable has shown better control performance than two IMDs at the same end. However, installing a damper at cable-tower anchorage is difficult and inconvenient. As an alternative, attaching two IMDs with appropriate inertial mass at the same end of a cable seems to be more promising for practical application. However, it is still necessary to explore the performance of two IMDs for the multimode cable vibration control, especially for super-long cables, which will be our consideration for further study.

**Author Contributions:** Z.W., F.Y., and H.G. conceived the idea of this research; Z.W. and F.Y. wrote the paper; H.G. reviewed and revised the paper.

**Funding:** The research was financially supported by the National Natural Science Foundation of China (Grant No. 51878274).

**Conflicts of Interest:** The authors declare no conflict of interest.
