## 2 Formulation of the Cable--HDR Damper--TM-HDR-D System  
A taut cable with an HDR damper and a TM-HDR-D is depicted in Figure 1. The cable length is denoted by \(L\), the cable mass per unit length is denoted by \(m\), and the cable tension is denoted by \(T\). \(K_{1}\) indicates the HDR damper's stiffness, and \(\varphi\) denotes the HDR damper's loss factor. For the TM-HDR-D, \(K_{2}\), \(\varphi\), and \(M\) denote its stiffness, loss factor, and mass, respectively. The coordinate system is defined as shown in Figure 1, where the position coordinate axis \(x\) starts from the left cable end and points horizontally to the right end of the cable, and the direction of the \(x^{*}\) axis is opposite to the \(x\) axis; \(v(x,t)\) and \(u_{\text{tmd}}\) are the vertical dynamic displacement of the cable and the mass of the TM-HDR-D, respectively. The HDR damper and TM-HDR-D are at locations \(x_{1}\) and \(x_{2}\), respectively, and \(x_{2}^{*}=L-x_{2}\). The cable sag, bending stiffness, and inherent damping are not considered in the model, due to their influence being so small that they can be ignored [32]. The motion equation of the cable damper system can be given by the following formula [32]:  
\[T\frac{\partial^{2}v}{\partial x^{2}}-m\frac{\partial^{2}v}{\partial t^{2}}= f_{\text{hdr}}(t)\delta(x-x_{1})+f_{\text{tmd}}(t)\delta(x-x_{2}) \tag{1}\]  
where \(\delta(\cdot)\) denotes the delta function--that is, the two concentrated damping forces provided by the dampers acting on the cable, where \(f_{\text{hdr}}(t)\) located at \(x=x_{1}\) and \(f_{\text{tmd}}(t)\) located at \(x=x_{2}\). Considering the free vibration of the cable system, the transverse displacement of the cable and the damping forces of two dampers have the following expressions:  
\[v(x,t)=\widetilde{v}(x)\dot{e}^{\text{i}\omega t},\ f_{\text{hdr}}(t)= \widetilde{f}_{\text{hdr}}\dot{e}^{\text{i}\omega t}\text{ and }f_{\text{tmd}}(t)=\widetilde{f}_{\text{tmd}}\dot{e}^{\text{i} \omega t} \tag{2}\]  
where \(\text{i}=\sqrt{-1}\), \(\omega\) denotes the cable's complex angular frequency, and \(\widetilde{v}\), \(\widetilde{f}_{\text{hdr}}\), and \(\widetilde{f}_{\text{tmd}}\) denote the amplitude of the corresponding time-dependent variables. For a cable with two dampers, the cable displacement \(\widetilde{v}\) can be written as follows [25]:  
\[\widetilde{v}(x)= \begin{cases}\widetilde{v}_{\text{hdr}}\frac{\sin(\beta x)}{ \sin(\beta x)},\ \ 0\leq x\leq x_{1}\\ \widetilde{v}_{\text{hdr}}\frac{\sin[\beta(x_{2}-x_{1})]}{\sin[\beta(x_{2}-x_ {1})]}+\widetilde{v}_{\text{tmd}}\frac{\sin[\beta(x-x_{1})]}{\sin[\beta(x_{2}- x_{1})]},\ \ \ \ x_{1}\leq x_{2}\\ \widetilde{v}_{\text{tmd}}\frac{\sin\left(\beta x_{2}^{*}\right)}{\sin(\beta x ^{*})},\ \ \ \ 0\leq x^{*}\leq x_{2}^{*}\end{cases}  \tag{3}\]  
where \(\beta=\omega\sqrt{m/T}\) = the complex wavenumber; \(\ \widetilde{v}_{\text{hdr}}\) and \(\widetilde{v}_{\text{tmd}}\) denote the cable amplitude at the HDR damper and the TM-HDR-D installation points, respectively. According to the vertical force equilibrium condition at the damper installation position, one finds  
\[T\left(\frac{\partial v}{\partial x}|x_{j}^{+}-\frac{\partial v}{\partial x}| x_{j}^{-}\right)=f_{j}(t),\ \ j=1,2 \tag{4}\]  
Note that \(f_{1}(t)=f_{\text{hdr}}(t)\) and \(f_{2}(t)=f_{\text{tmd}}(t)\). Figure 1: Model diagram of a taut cable with an HDR damper and a TM-HDR-D.  
Then, substituting Equation (3) into Equation (4), one obtains  
\[\left\{ \begin{array}{l}\cot(\beta x_{1})+\cot[\beta(x_{2}-x_{1})]-\frac{\widetilde {v}_{\rm{tmd}}}{\widetilde{v}_{\rm{hdr}}}\frac{1}{\sin[\beta(x_{2}-x_{1})]}=- \frac{\widetilde{f}_{\rm{hdr}}/T}{\beta\widetilde{v}_{\rm{hdr}}}\\ -\frac{\widetilde{v}_{\rm{hdr}}}{\widetilde{v}_{\rm{tmd}}}\frac{1}{\sin[\beta( x_{2}-x_{1})]}+\cot[\beta(x_{2}-x_{1})]+\cot(\beta x_{2}^{s})=-\frac{\widetilde{f}_{ \rm{tmd}}/T}{\beta\widetilde{v}_{\rm{tmd}}}\end{array} \right.