# Free Vibration of a Taut Cable with Two Discrete Inertial Mass Dampers
## 2 Formulation of the Cable-IMD System  
A taut cable with two transversely attached inertial mass dampers is shown in Figure 1. The length, the mass per unit length, and the tension of the cable are \(L\), \(m\), and \(T\), respectively. The coordinate system defines that the \(x\)-axis and the \(v\)-axis are along the cable chord and the transverse direction, respectively. \(x^{*}=L-x\) represents the coordinate from the right end of the cable. Two discrete IMDs are respectively installed at distances \(x_{1}\) and \(x_{2}\) from the left end of the cable (\(x_{2}\geq x_{1}\)). The distance between the right IMD and the right end is denoted as \(x_{2}^{*}=L-x_{2}\). The damping coefficient and inertial mass of the \(j^{th}\) IMD are denoted as \(c_{j}\) and \(b_{j}\) (\(j=1,2\)), respectively. The equation of motion of the cable-IMD system is given by:  
\[T\frac{\partial^{2}v}{\partial x^{2}}-m\frac{\partial^{2}v}{\partial t^{2}}= \sum_{j=1}^{2}F_{\text{IMD}j}(t)\delta(x-x_{j}), \tag{1}\]  
where \(v(x,t)\) is the cable transverse displacement and \(\delta(\cdot)\) is delta function to specify the location of the damping force \(F_{\text{IMD}j}\) at \(x=x_{j}\). Under free vibration, applying separation of variables, the cable transverse displacement and the IMD force can be respectively expressed as:  
\[v(x,t)=\overline{v}(x)e^{i\omega t},F_{\text{IMD}j}(t)=\overline{F}_{\text{ IMD}j}e^{i\omega t} \tag{2}\]  
where \(i^{2}=-1\), \(\omega\) is a complex natural frequency of the cable, and \(\overline{v}(x)\) is the corresponding complex mode shape. To find \(\overline{v}(x)\), the cable can be considered as a multispan structure connected at the damper locations [77]. Substituting Equation (2) into Equation (1), \(\overline{v}(x)\) in each span needs to satisfy a homogeneous equation [77]:  
\[\frac{d^{2}\overline{v}}{dx^{2}}+\beta^{2}\overline{v}=0\left\{ \begin{array}[ ]{l}0\leq x\leq x_{1}\\ x_{1}\leq x\leq x2\\ 0\leq x^{*}\leq x_{2}^{*}\end{array} \right., \tag{3}\]  
where \(\beta=\omega\sqrt{m/T}\) refers to the wavenumber. Figure 1: The taut cable with two discrete inertial mass dampers. Applying boundary conditions at cable ends, i.e., \(\widetilde{\upsilon}(0)=\widetilde{\upsilon}(L)=0\), and the transverse displacement compatibility conditions at damper locations, i.e., \(\widetilde{\upsilon}(x_{1})=\widetilde{\upsilon}_{1}\), \(\widetilde{\upsilon}(x_{2}^{*})=\widetilde{\upsilon_{2}}\), the general solution of Equation (3) can be further written in the form [77]:  
\[\widetilde{\upsilon}(x)= \begin{cases}\widetilde{\upsilon}_{1}\frac{\sin \beta x}{\sin\beta x_{1}}&0\leq x\leq x_{1}\\ \widetilde{\upsilon}_{1}\frac{\sin\beta(x_{2}-x)}{\sin\beta(x_{2}-x_{1})}+ \widetilde{\upsilon}_{2}\frac{\sin\beta(x-x_{1})}{\sin\beta(x_{2}-x_{1})}&x_ {1}\leq x\leq x2\,,\\ \widetilde{\upsilon}_{2}\frac{\sin\beta^{\prime}}{\sin\beta x_{2}^{*}}&0\leq x ^{*}\leq x_{2}^{*}\end{cases}  \tag{4}\]  
where \(\widetilde{\upsilon}_{j}\) is the mode shape amplitude at the \(j^{th}\) damper location. At damper locations, there is:  
\[T(\frac{\mathrm{d}\upsilon}{\mathrm{d}x}\Big{|}_{j}^{x_{j}^{*}}-\frac{\mathrm{ d}\upsilon}{\mathrm{d}x}\Big{|}_{j}^{x_{j}^{*}})=\widetilde{F}_{\mathrm{IMDj}}(t). \tag{5}\]  
Substituting Equation (4) into Equation (5), it yields:  
\[\left\{ \begin{array}{ll}\cot\beta x_{1}+\cot(x_{2}-x_{1})-\frac{\widetilde{ \upsilon}_{2}}{\widetilde{\upsilon}_{1}}\frac{1}{\sin\beta(x_{2}-x_{1})}=- \frac{\widetilde{F}_{\mathrm{IMDj}}/T}{\beta\widetilde{\upsilon}_{1}}\\ -\frac{\widetilde{\upsilon}_{2}}{\widetilde{\upsilon}_{1}}\frac{1}{\sin\beta( x_{2}-x_{1})}+\cot\beta(x_{2}-x_{1})+\cot\beta x_{2}^{*}=-\frac{\widetilde{F}_{ \mathrm{IMDj}}/T}{\beta\widetilde{\upsilon}_{1}}\end{array} \right.. \tag{6}\]  
Substituting \(\widetilde{\upsilon}_{1}/\widetilde{\upsilon}_{2}\) from the second one into the first one of Equation (6) and rearranging, the characteristic equation of the wavenumber \(\beta\) is derived as:  
\[