# Development of a Self-Powered Magnetorheological Damper System for Cable Vibration Control
## 4 Experimental Results and Discussion
### The Stay Cable with Electromagnetic Damping Alone  
To investigate the effect of electromagnetic damping alone on the vibration mitigation performance of the cable, two circuits of the generator were considered. One is a short circuit, and the other is loaded with an MR damper. However, the MR damper herein is not attached to the cable. The main difference between them is that the resulting equivalent damping coefficients for the case of the short circuit are larger than those for the other case. The test results are summarized in Table 3, where the supplemental modal damping ratios are derived from the subtraction between the identified value with the electromagnetic damping and  
Figure 10: Second mode acceleration time histories of the cable controlled by a passive MR damper with different constant voltages: (**a**) 0 V; (**b**) 0.5 V; and (**c**) 1 V.  
Figure 9: Damping ratios in the first four modes of the cable versus constant voltage of the MR damper. the uncontrolled case. It is noted that the supplemental damping ratio of the cable controlled by the electromagnetic damping alone is pretty limited, and the supplemental modal damping ratios of the same mode in the short circuit case are always larger than those in the circuit loaded with the MR damper. In addition, the supplemental modal damping ratio continuously increases with the increase of the mode order for both of the circuits. However, the supplemental modal damping ratio of the cable with the electromagnetic damping alone is much smaller than the theoretical maximum damping ratio induced by an ideal linear viscous damper (3%). This is mainly because the damping coefficient of the generator is much smaller than the corresponding optimum value. Therefore, the test results above imply that the generator can serve as an alternative damper for cable vibration control, and its efficiency can be further enhanced through optimum design of the generator system. The relationship between the damping force and the linear displacement of the generator is shown in Figure 11. The damping characteristics of the generator are very similar to those of a combination of a traditional linear viscous damper and a negative stiffness device in the higher-order mode of the cable. It is noteworthy that the negative stiffness component of the generator is not visible in the 1 st mode, which is mainly due to the fact that the negative stiffness of the generator for the 1 st mode of the cable is smaller than its internal stiffness. In fact, the total force \(F_{\text{g}}\) of the rotary DC generator can be divided into the inertial force \(F_{I}\) and the electromagnetic damping force \(F_{\text{d}}\)[35]. This relationship can be expressed as  
\[F_{\text{g}}=F_{I}+F_{\text{d}}. \tag{7}\]  
Figure 11: The damping force versus the linear displacement of the DC generator. (**a**) First mode; (**b**) Second mode; (**c**) Third mode; and (**d**) Fourth mode. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Mode Order**} & \multicolumn{2}{c}{**Short Circuit**} & \multicolumn{2}{c}{**Circuit Loaded with an MR Damper**} \\ \cline{2-5}  & **Damping Ratio (\%)** & **Supplemental Damping Ratio (\%)** & **Damping Ratio (\%)** & **Supplemental Damping Ratio (\%)** \\ \hline
1 & 0.71 & 0.12 & 0.68 & 0.09 \\
2 & 0.59 & 0.44 & 0.54 & 0.39 \\
3 & 0.81 & 0.70 & 0.65 & 0.54 \\
4 & 1.05 & 0.93 & 0.74 & 0.62 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Modal damping ratios of the cable in the first four modes with electromagnetic damping alone. When the rotary DC generator is applied to mitigate cable vibration, we have  
\[F_{I}=k_{\text{g}}u=-m_{\text{e}}\omega_{i}^{2}u,\ F_{\text{d}}=c_{\text{eq}}\dot{u} \tag{8}\]  
where \(k_{\text{g}}\), \(m_{\text{e}}\), and \(c_{\text{eq}}\) represent the equivalent stiffness, the inertial mass, and the damping coefficient of the generator, respectively; \(u\) and \(\omega_{i}\) denote the displacement and the natural frequency of the cable in the \(i\)th-order mode, respectively.