# Refined Study on Free Vibration of a Cable with an Inertial Mass Damper
## 4 Case Studies
The damping coefficient \(c_{\text{d}}\) and the inertial mass \(m_{\text{e}}\) of the IMD can be normalized as,  
\[\overline{c}_{\text{d},n}=\frac{c_{\text{d}}}{T/\left(x_{\text{d}}(\omega_{n} ^{0})\right)}=n\pi\frac{c_{\text{d}}}{\sqrt{Tm}}\frac{x_{d}}{l},\overline{m}_{ \text{e},n}=\frac{m_{\text{e}}}{T/\left(x_{\text{d}}(\omega_{n}^{0})\right)^{2 }},\omega_{n}^{0}=\frac{n\pi}{l}\sqrt{\frac{T}{m}} \tag{12}\]  
where \(\overline{c}_{\text{d},n}\) and \(\overline{m}_{\text{e},n}\) are the dimensionless damping coefficient and the dimensionless inertial mass of the IMD, respectively; \(\omega_{n}^{0}\) is the \(n\)th modal frequency of an undamped taut cable. Figures 4-7 present the supplemental modal damping ratios in the first four modes of the short cable versus damping coefficients for various inertial masses of the IMD. The single impact of cable sag or cable flexural rigidity and their combined effects on the performance of the cable with the IMD are given. For the convenience of comparisons, theoretical results with the taut cable model are also shown. All curves for the first four modal damping ratios of the short cable almost coincide with each other when the dimensionless inertial mass is the same, indicating that presences of the cable sag, the cable flexural rigidity and the boundary condition have negligible effects on the control performance of the IMD in mitigating short cable vibration. It is mainly because the short cable with low cable sag and low flexural rigidity used here is close to the taut cable model assumption. Hence, the taut cable model is still suitable for the short cable to rapidly design an optimum IMD and evaluate corresponding control performance. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{1}{c}{\begin{tabular}{c} **Short Cable** \\ **(Dongting Lake Bridge)** \\ \end{tabular} } & \multicolumn{1}{c}{\
\begin{tabular}{c} **Medium Cable** \\ **(Stonecutters Bridge)** \\ \end{tabular} } & \multicolumn{1}{c}{
\begin{tabular}{c} **Long Cable** \\ **(Sutong Bridge)** \\ \end{tabular} } \\ \hline \hline Cable length \(l\) (m) & 114.7 & 306.7 & 576.8 \\ Mass per unit length \(m\) (kg/m) & 51.8 & 98.6 & 100.8 \\ Flexural rigidity \(EI\) (N/m\({}^{2}\)) & \(3.842\times 10^{3}\) & \(5.525\times 10^{6}\) & \(2.309\times 10^{6}\) \\ Axial stiffness \(EA\)(N) & \(1.255\times 10^{9}\) & \(2.429\times 10^{9}\) & \(2.409\times 10^{9}\) \\ Inclination angle \(\theta\) (\({}^{\circ}\)) & 37.0 & 30.5 & 22.5 \\ Sag parameter \(\lambda^{2}\) & 0.0915 & 0.9365 & 2.2101 \\ Tension force \(T\) (kN) & 3095 & 5530 & 6708 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Properties of three typical cables in cable-stayed bridges. Figure 4: The first supplemental modal damping ratio of the short 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 5: The second supplemental modal damping ratio of the short 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 7: The fourth supplemental modal damping ratio of the short 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**) \(\overline{m}_{\text{e,4}}=0.9\). Figures 8-11 present the supplemental modal damping ratios in the first four modes of the medium cable versus damping coefficient for various inertial masses of the IMD. The maximum achievable  
damping ratios in the first mode are highly sensitive to the cable sag while those in the higher modes are hardly affected. Generally, the cable sag reduces the optimal modal damping ratios.