# Refined Study on Free Vibration of a Cable with an Inertial Mass Damper
## 4 Case Studies
However, the optimal first modal damping ratios of the cable begin to increase when the inertial mass shown in Figure 8d is adopted. Compared to the cable sag, the flexural rigidity has almost the same effect on all concerned cable modes. For the case of fixed-fixed boundary condition of the cable, the flexural rigidity tends to reduce the maximum modal damping ratios. Moreover, the adverse effect of the cable flexural rigidity on damper efficiency will be amplified with the increase of the inertial mass of the IMD. While for the case of pinned-pinned boundary condition of the cable, unlike the results of the fixed-fixed cable, the flexural rigidity can increase the maximum achievable damping ratios of the cable for the IMD with small inertial mass (\(\overline{m}_{\text{e},n}\leq 0.3\)). However, the maximum achievable damping ratios begin to decrease when medium inertial mass (\(0.6\leq\overline{m}_{\text{e},n}\leq 0.9\)) is adopted. It is also found that the flexural rigidity will result in larger optimal damping coefficients whether the boundary conditions of the medium cable are fixed or pinned. In general, considering combined effects of the cable sag and the flexural rigidity, the maximum achievable damping ratio of the fixed-fixed cable with the IMD will be the lower bound, while the corresponding optimal damping coefficient will be the upper bound. Figures 12-15 present the supplemental modal damping ratios in the first four modes of the long cable versus damping coefficient for various inertial masses of the IMD. Similarly to the medium cable, both the cable sag and the cable flexural rigidity mainly have adverse influences on the IMD effectiveness for cable vibration control. Moreover, the maximum achievable damping ratio of the cable-IMD system considering the influence of the cable sag and the cable flexural rigidity in the case of fixed boundary conditions provides relatively conservative predictions. Unlike the medium cable, the cable sag plays a dominant role in the dynamic behavior of an IMD to suppress cable vibrations for the lower modes of interest, especially for the first mode, while the presence of the cable flexural rigidity has fewer effects on the performance efficiency of the long cable with the IMD. Figure 8: The first supplemental modal damping ratio of the medium cable versus damping coefficient for various inertial masses: (**a**) \(\overline{m}_{\text{e},1}=0\); (**b**) \(\overline{m}_{\text{e},1}=0.3\); (**c**) \(\overline{m}_{\text{e},1}=0.6\); (**d**) \(\overline{m}_{\text{e},1}=0.9\). Figure 10: The third supplemental modal damping ratio of the medium cable versus damping coefficient for various inertial masses: (**a**) \(\overline{m}_{\text{e,3}}=0\); (**b**) \(\overline{m}_{\text{e,3}}=0.3\); (**c**) \(\overline{m}_{\text{e,3}}=0.6\); (**d**) \(\overline{m}_{\text{e,3}}=0.9\). Figure 9: The second supplemental modal damping ratio of the medium cable versus damping coefficient for various inertial masses: (**a**) \(\overline{m}_{\text{e,2}}=0\); (**b**) \(\overline{m}_{\text{e,2}}=0.3\); (**c**) \(\overline{m}_{\text{e,2}}=0.6\); (**d**) \(\overline{m}_{\text{e,2}}=0.9\). Figure 11: The fourth supplemental modal damping ratio of the medium cable versus damping coefficient for various inertial masses: (**a**) \(\overline{m}_{\text{e},4}=0\); (**b**) \(\overline{m}_{\text{e},4}=0.3\); (**c**) \(\overline{m}_{\text{e},4}=0.6\); (**d**) \(\overline{m}_{\text{e},4}=0.9\). Figure 12: The first supplemental modal damping ratio of the long cable versus damping coefficient for various inertial masses: (**a**) \(\overline{m}_{\text{e},1}=0\); (**b**) \(\overline{m}_{\text{e},1}=0.3\); (**c**) \(\overline{m}_{\text{e},1}=0.6\); (**d**) \(\overline{m}_{\text{e},1}=0.9\). Figure 14: The third supplemental modal damping ratio of the long cable versus damping coefficient for various inertial masses: (**a**) \(\overline{m}_{\text{e,3}}=0\); (**b**) \(\overline{m}_{\text{e,3}}=0.3\); (**c**) \(\overline{m}_{\text{e,3}}=0.6\); (**d**) \(\overline{m}_{\text{e,3}}=0.9\).