## 3 Numerical Parametric Study
### Cable with HDR Damper and TM-HDR-D  
This section focuses a system of a cable with two dampers, and the cable's modal damping is obtained via Equation (12) by numerically solving Equation (10). Figure 4: First modal damping of a cable with a single TM-HDR-D versus the frequency ratio \(\rho\) (\(x_{2}/L=0.5\,,\varphi=0.5\)). Figure 3: Modal damping of a cable with a single TM-HDR-D versus damper location \(x_{2}/L\) (\(\varphi=0.5\)): (**a**) mode 1 (\(\rho=1\)); (**b**) mode 2 (\(\rho=0.5\)). Figure 5 depicts the cable's first modal damping ratio with respect to the HDR damper's stiffness with four TM-HDR-D mass ratios, i.e., \(\mu=0\), 0.005, 0.01, and 0.02. Note that the loss factor \(\varphi\) of both the HDR damper and the TM-HDR-D is set to 0.5. The TM-HDR-D tunes to the cable's first vibration mode; thus, the frequency ratio is determined as \(\rho=1\). In this situation, the location of the TM-HDR-D is 0.5\(L\). Figure 5a,b show the first modal damping ratio curves of the cable with two dampers, where an HDR damper is located at \(x_{1}/L=0.02\) and 0.05, respectively. Figure 5 illustrates that the cable's modal damping enhances greatly with the increase in the HDR damper's installation height. Compared with the HDR damper, the TM-HDR-D can provide a greater damping contribution because it is not limited by its installation position. Increasing the TM-HDR-D's installation mass further improves the modal damping of the cable system. Interestingly, when the TM-HDR-D has a small mass, the corresponding cable damping curves are approximately parallel to that of the HDR damper only, which indicates that the modal damping contribution of the TM-HDR-D and the HDR damper is approximately the linear superposition of their respective damping effects. However, when the mass of the TM-HDR-D is large, e.g., \(\mu=0.02\), the curves are not parallel. In this situation, there is a certain coupling effect between the two dampers, and the combined damping effect is no longer subject to superposition. Figure 6 shows the typical modal shapes of a cable with a single HDR damper and of a cable with an HDR damper and a TM-HDR-D, corresponding to points (i) and (ii) in Figure 5b, respectively. Note that the solid line corresponds to the first modal shape of the cable without any dampers. Figure 6 shows that the cable-end HDR damper causes considerable changes in the modal shape of the cable system. Meanwhile, adding a TM-HDR-D to the cable-HDR damper system will continue to change the system's modal shape, although not significantly. This also leads to the coupling effect of the two dampers, and the effect of the two dampers on the system's damping is not a simple linear superposition. Figure 7 depicts the second modal damping ratio versus the HDR damper stiffness, with \(\mu=0,\ 0.005,\ 0.01,\ \text{and}\ 0.2\). The TM-HDR-D herein is tuned to the second mode, i.e., \(\rho=0.5\), and installed at \(x_{2}/L=0.25\). Figure 7 indicates that, with the TM-HDR-D tuned to the cable's second mode, the cable's damping can be significantly improved. When the TM-HDR-D has a small mass, the combined damping effect of both dampers is approximately the linear superposition of their respective effects, which is in accordance with the analysis results shown in Figure 5.  
Figure 5: First modal damping of a cable with two dampers versus the HDR damper stiffness \(\overline{K}_{1}\) (\(\varphi=0.5\), \(\rho=1,\ x_{2}/L=0.5\)), with (**a**) \(x_{1}/L=0.02\) and (**b**) \(x_{1}/L=0.05\). Figures 8 and 9 investigate the effect of the installation location of the TM-HDR-D on the system's modal damping ratios in the first and second modes, respectively. In Figure 8, the TM-HDR-D is tuned to the cable's first vibration mode, i.e., the frequency ratio is \(\rho=1\). Note that the parameters of the HDR damper are \(\varphi=0.5\) and \(\overline{K}_{1}=40\). Figure 8 demonstrates that adding a TM-HDR-D to a cable-HDR damper system will greatly improve the system's modal damping ratio.