# Refined Study on Free Vibration of a Cable with an Inertial Mass Damper
## 4 Case Studies  
Previous studies often introduce a dimensionless parameter \(\overline{\xi}=L\sqrt{T/EI}\) for flexural rigidity to investigate the individual effects of flexural rigidity or the combined effects of sag and flexural rigidity on the dynamic properties of a stay cable with a damper [57; 58; 59; 60; 61]. However, the dynamic properties of the stay cable may have obvious differences even that the dimensionless bending-stiffness \(\overline{\xi}\) for various cables are the same. Alternatively, three typical cables (a short cable, a medium cable and a \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{3}{*}{**Case**} & \multirow{3}{*}{\begin{tabular}{c} **Inertial mass** \\ **(kg)** \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} **Damping** \\ **Coefficient (Ns/m)** \\ \end{tabular} } & \multirow{3}{*}{\
\begin{tabular}{c} **Experiment** \\ **(\%)** \\ \end{tabular} } & \multicolumn{3}{c}{**Theory (\%)**} \\ \cline{4-6}  & & & &
\begin{tabular}{c} _EI = 0,_ \\ \(\lambda^{2}\) = 0_ \\ \end{tabular} } & \multicolumn{2}{c}{**With _EI_ and \(\lambda^{2}\)**} \\ \hline \multirow{3}{*}{1} & \multirow{3}{*}{102.6} & 4659 & 0.23 & 0.37 & 0.27 & 0.13 \\  & & 3117 & 0.18 & 0.26 & 0.19 & 0.09 \\  & & 4778 & 0.28 & 0.40 & 0.30 & 0.14 \\
2 & 140.5 & 3356 & 0.21 & 0.30 & 0.22 & 0.10 \\  & & 5488 & 0.31 & 0.54 & 0.42 & 0.18 \\
3 & 259.7 & 4636 & 0.35 & 0.49 & 0.38 & 0.16 \\  & & 4968 & 0.48 & 0.71 & 0.58 & 0.22 \\
4 & 422.0 & 3197 & 0.41 & 0.53 & 0.44 & 0.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The first supplemental modal damping ratio obtained in the theoretical calculation and experimental tests [65]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{3}{*}{**Case**} & \multirow{3}{*}{\begin{tabular}{c} **Inertial** \\ **Modass (kg)** \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} **Damping** \\ **Coefficient (Ns/m)** \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} **Experiment** \\ **(\%)** \\ \end{tabular} } & \multicolumn{3}{c}{**Theory (\%)**} \\ \cline{4-6}  & & & \
\begin{tabular}{c} _EI = 0,_ \\ \(\lambda^{2}\) = 0_ \\ \end{tabular} } & \multicolumn{2}{c}{**With _EI_ and \(\lambda^{2}\)**} \\ \cline{4-6}  & & &
\begin{tabular}{c} _EI = 0,_ \\ \(\lambda^{2}\) = 0_ \\ \end{tabular} } & \multicolumn{2}{c}{**With _EI_ and \(\lambda^{2}\)**} \\ \hline \multirow{3}{*}{1} & \multirow{3}{*}{102.6} & 2884 & 0.65 & 0.74 & 0.70 & 0.29 \\  & & 4160 & 0.61 & 0.81 & 0.84 & 0.39 \\
2 & 140.5 & 2921 & 0.82 & 1.02 & 0.92 & 0.35 \\  & & 4536 & 0.77 & 1.00 & 1.06 & 0.48 \\
3 & 259.7 & 1847 & 2.01 & 5.02 & 2.28 & 0.46 \\  & & 3704 & 1.23 & 1.98 & 2.35 & 0.77 \\  & & 3298 & 2.01 & 0.81 & 1.76 & 2.40 \\
4 & 422.0 & 4015 & 1.90 & 0.84 & 1.74 & 2.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The second supplemental modal damping ratio obtained in the theoretical calculation and experimental tests [65]. long cable) in actual cable-stayed bridges are selected as example cables to explore the influence of the sag and the flexural rigidity with different boundary conditions on the control performance of the cable with the IMD in this section. It should be pointed out that the lengths of short, medium and long cables are relative values and their definitions are mainly referred to the Xu and Yu [54]. The properties of three typical cables in cable-stayed bridges are shown in Table 4. An IMD is assumed to be attached to each cable at a distance of the 1% length from its lower end without considering practical application especially for the long cable. The modal damping ratios in the first four modes of each cable are obtained using the finite difference method above.