# Free Vibration of a Taut Cable with Two Discrete Inertial Mass Dampers
## 3 Two Opposite IMDs
### Asymptotic Solution  
The following assumptions are introduced [12; 17; 77]: (1) The locations of IMDs are very close to the ends, i.e., \(x_{1},x_{2}^{*}\ll L\); (2) the wavenumber \(\beta_{n}\) of each mode \(n\) (\(n=1,2,\dots\) ) has a small perturbations \(\Delta\beta_{n}=\beta_{n}-\beta_{n}^{0}\) from the undamped value \(\beta_{n}^{0}=n\pi/L\). The assumptions above lead to the following approximations:  
\[\tan(\beta_{n}L)\cong\beta_{n}L-n\pi\,\sin(\beta_{n}x_{1})\cong\beta_{n}^{0}x _{1}\,\sin(\beta_{n}x_{2}^{*})\cong\beta_{n}^{0}x_{2}^{*}\,\cos(\beta_{n}x_{1} )\cong\cos(\beta_{n}x_{2}^{*})=1. \tag{10}\]  
The asymptotic formula for the wavenumber \(\beta_{n}\) takes the form:  
\[\beta_{n}L\cong n\pi+\beta_{n}^{0}\frac{E_{1}+iF_{1}}{G_{1}+iH_{1}}, \tag{11}\]  
\[E_{1}=-\overline{b_{1}}x_{1}-\overline{b_{2}}x_{2}^{*}+(\overline{b_{1}b_{2} }-\overline{c_{1}c_{2}})(x_{1}+x_{2}^{*}), \tag{12a}\] \[F_{1}=\overline{c_{1}}x_{1}+\overline{c_{2}}x_{2}^{*}-(\overline{b_{1}c_{2} }+\overline{b_{2}c_{1}})(x_{1}+x_{2}^{*}),\] (12b) \[G_{1}=1-\overline{b_{1}}-\overline{b_{2}}+(\overline{b_{1}b_{2}} -\overline{c_{1}c_{2}})(x_{1}+x_{2}^{*}),\] (12c) \[H_{1}=\overline{c_{1}}+\overline{c_{2}}-(\overline{b_{1}\overline {c_{2}}}+\overline{b_{2}\overline{c_{1}}})(x_{1}+x_{2}^{*}), \tag{12d}\]  
where \(\overline{c_{1}}=\eta_{1}\beta_{n}^{0}x_{1}\) and \(\overline{c_{2}}=\eta_{1}\beta_{n}^{0}x_{2}^{*}\) represent dimensionless damping coefficient groups, while \(\overline{b_{1}}=\chi_{1}\beta_{n}^{0}x_{1}\) and \(\overline{b_{2}}=\chi_{2}\beta_{n}^{0}x_{2}^{*}\) represent dimensionless inertial mass groups.  
The complex eigen-frequency corresponding to the wavenumber \(\beta_{n}\) is denoted as \(\omega_{n}\). The \(n^{th}\) supplemental modal damping radio of a cable \(\xi_{n}\) can be obtained by [17]:  
\[\xi_{n}=\frac{\text{Im}[\omega_{n}]}{|\omega_{n}|}=\frac{\text{Im}[\beta_{n} ]}{\left|\beta_{n}\right|}\cong\frac{\text{Im}[\Delta\beta_{n}]}{\left|\beta_ {n}^{0}\right|}. \tag{13}\]  
Substituting Equation (11) into Equation (13), the asymptotic supplemental modal damping ratio of the cable is finally derived as:  
\[\xi_{n}\cong\frac{\overline{c_{1}}}{\left(1-\overline{b_{1}}\right)^{2}+ \left(\overline{c_{1}}\right)^{2}}\frac{x_{1}}{L}+\frac{\overline{c_{2}}}{ \left(1-\overline{b_{2}}\right)^{2}+\left(\overline{c_{2}}\right)^{2}}\frac{x _{2}}{L}. \tag{14}\]  
If two identical IMDs are symmetrically installed at the cable for practical implementation, some simplifications in the notation can be introduced, i.e., \(x=x_{1}=L-x_{2}=x_{2}^{*}\), \(b_{1}=b_{2}=b\), \(\chi_{1}=\chi_{2}=\chi\), \(\overline{b_{1}}=\overline{b_{2}}=\overline{b}\), \(c_{1}=c_{2}=c\), \(\eta_{1}=\eta_{2}=\eta\), and \(\overline{c_{1}}=\overline{c_{2}}=\overline{c}\). Equation (14) can be further simplified as:  
\[\xi_{n}=\frac{2\overline{c}}{\[\overline{A}_{1}=-\chi_{1}\sin^{2}\beta_{n}^{k}x_{1}-\chi_{2}\sin^{2} \beta_{n}^{k}x_{2}^{*}+(\chi_{1}\chi_{2}-\eta_{1}\eta_{2})\sin\beta_{n}^{k}x_{1} \sin\beta_{n}^{k}x_{2}^{*}\sin\beta_{n}^{k}(x_{1}+x_{2}^{*}), \tag{15a}\] \[\overline{B}_{1}=\eta_{1}\sin^{2}\beta_{n}^{k}x_{1}+\eta_{2}\sin ^{2}\beta_{n}^{k}x_{2}^{*}-(\chi_{1}\eta_{2}+\chi_{2}\eta_{1})\sin\beta_{n}^{k }x_{1}\sin\beta_{n}^{k}x_{2}^{*}\sin\beta_{n}^{k}(x_{1}+x_{2}^{*}),\] (15b) \[\overline{C}_{1}=1-\chi_{1}\sin\beta_{n}^{k}x_{1}\cos\beta_{n}^{k }x_{1}-\chi_{2}\sin\beta_{n}^{k}x_{2}^{*}\cos\beta_{n}^{k}x_{2}^{*}+\] (15c) \[(\chi_{1}\chi_{2}-\eta_{1}\eta_{2})\sin\beta_{n}^{k}x_{1}\sin \beta_{n}^{k}x_{2}^{*}\cos\beta_{n}^{k}(x_{1}+x_{2}^{*})\] (15d) \[\overline{D}_{1}=\eta_{1}\sin\beta_{n}^{k}x_{1}\cos\beta_{n}^{k}x_{1} +\eta_{2}\sin\beta_{n}^{k}x_{2}^{*}\cos\beta_{n}^{k}x_{2}^{*}-\] \[(\chi_{1}\eta_{2}+\chi_{2}\eta_{1})\sin\beta_{n}^{k}x_{1}\sin \beta_{n}^{k}x_{2}^{*}\cos\beta_{n}^{k}(x_{1}+x_{2}^{*})\]  
Finally, the supplemental modal damping ratio of a cable with two opposite IMDs can be calculated by Equation (12) after solving the wavenumber \(\beta_{n}\).