# Refined Study on Free Vibration of a Cable with an Inertial Mass Damper
## 3 Verification of Theoretical Results  
To further verify the accuracy of the finite difference method for the cable-IMD system, the experimental data from the free vibration tests of the model cable with an electromagnetic inertial mass damper (EIMD) were used. The detailed description of the experiment can be referenced to [PERSON] et al. [65]. The main properties of the model cable are shown in Table 1. An EIMD, which mainly consists of a rotary generator, a ball screw, a liner guide way and a flywheel, was attached transversely to the model cable at 0.114 m (i.e., 1% of the cable length) away from the anchorage. In the experiment, the equivalent damping coefficient and the inertial mass of the EIMD were adjusted by changing the load resistance of the generator and the size of the flywheel, respectively. The equivalent damping coefficient and inertial mass of the EIMD were identified by the least square method according to the measured displacement and the force of the EIMD. Modal damping ratios of the model cable were identified through fitting the envelope curve of the free decay cable responses with an exponential function. In the theoretical calculation, the cable is divided into 200 equally spaced segments with 199 internal nodes (\(N=199\)), and the EIMD locates at the second internal node. The theoretical and experimental supplemental modal damping ratios in the first two modes of the cable are compared in Figures 2 and 3. For the convenience of comparison, theoretical results for the taut cable model are also shown. It is noteworthy that a taut-cable model generally overestimates the supplemental modal damping ratios of the cable in most cases. Nevertheless, the supplemental modal damping ratios theoretically predicted by the refined model of the cable-IMD system agree well with those identified from experimental results. Furthermore, experimental damping ratios always fall between theoretical results with fixed ends or pinned ends for each case, which implies the cable anchorages in the experiment were not ideally fixed or pinned. It is also observed that the difference of the optimum supplemental modal damping ratios and the corresponding damping coefficients between the cable with fixed and the cable with pinned increases with the increase of the inertial mass of the IMD. This demonstrated that the effect of boundary conditions would be amplified with the increase of inertial mass of the IMD. Hence, the results clearly indicate that considering the effects of the cable sag, the cable flexural rigidity, and the boundary condition is essential to give a more accurate prediction in evaluating the vibration control performance of a stay cable with an IMD and designing an optimum IMD for cable vibration mitigation. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \hline Cable length \(l\) (m) & 11.4 \\ Cable cross-section area \(A\) (cm\({}^{2}\)) & 1.374 \\ Mass per unit length \(m\) (kg/m) & 9.5 \\ Elastic modulus \(\mathrm{\varepsilon}\) (GPa) & 200 \\ Flexural rigidity \(EI\) (N/m\({}^{2}\)) & 42.95 \\ Static tension \(T\) (kN) & 19.2 \\ Inclination angle \(\theta\) (\({}^{\circ}\)) & 0 \\ Sag parameter \(\lambda^{2}\) & 4.513 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the model cable. Tables 2 and 3 summarize the first two supplemental modal damping ratios obtained in the theoretical calculation and experimental tests. As expected, the discrepancies between experimental  
Figure 3: Comparison between the theoretical and experimental supplemental modal damping ratios in the second mode of the cable: (**a**) \(m_{\text{e}}=102.6\) kg; (**b**) \(m_{\text{e}}=140.5\) kg; (**c**) \(m_{\text{e}}\)=259.7 kg (**d**) \(m_{\text{e}}=422.0\) kg. Figure 2: Comparison between the theoretical and experimental supplemental modal damping ratios in the first mode of the cable: (**a**) \(m_{\text{e}}=102.6\) kg; (**b**) \(m_{\text{e}}=140.5\) kg; (**c**) \(m_{\text{e}}\)= 259.7 kg (**d**) \(m_{\text{e}}=422.0\) kg.