# Refined Study on Free Vibration of a Cable with an Inertial Mass Damper
## 2 Formulation of the Cable-IMD System  
Consider an inclined cable with sag and flexural rigidity as shown in Figure 1. The coordinate system is defined such that the x-axis and the y-axis are along the cable chord OO' and the transverse direction, respectively. An IMD is attached at the location of \(x_{\text{d}}\) away from the cable lower end. After introducing the assumptions that: (1) the sag-to-span ratio is sufficiently small; (2) the cable vibrates only in the x y-plane and its motion in the x-direction is negligible; (3) the cable has a uniform cross-section along its length; (4) the static profile of the cable is a second-order parabola [64]. The equation of motion of the cable-IMD system shown in Figure 1 can be expressed as  
\[EI\frac{\partial^{4}y(x,t)}{\partial x^{4}}-T\frac{\partial^{2}y(x,t)}{ \partial x^{2}}+\frac{\lambda^{2}T}{l^{3}}\int_{0}^{l}y(x,t)dx+c\frac{ \partial y(x,t)}{\partial t}+m\frac{\partial^{2}y(x,t)}{\partial t^{2}}+F_{ \text{IMD}}\delta(x-x_{\text{d}})=0 \tag{1}\]  
where \(EI\) is the flexural rigidity of the cable, \(y(x,t)\) is the in-plane transverse cable motion due to vibration at location \(x\) and time \(t\), \(T\) is the tension force along the chord OO', \(l\) is the length of the cable, \(c\) is the viscous damping per unit length, \(m\) is the cable mass per unit length, \(\delta(\cdot)\) is the Dirac delta function. \(F_{\text{IMD}}\) is the damper force of the IMD with the expression as  
\[F_{\text{IMD}}=m_{\text{e}}\frac{\partial^{2}y(x_{\text{d}},t)}{\partial t^{2 }}+c_{\text{d}}\frac{\partial y(x_{\text{d}},t)}{\partial t} \tag{2}\]  
where \(m_{\text{e}}\) and \(c_{\text{d}}\) denote the inertial mass and the damping coefficient of the IMD, respectively. \(\lambda^{2}\) is the non-dimensional parameter for sag extensibility  
\[\lambda^{2}=\frac{\left(mgl\cos\theta\right)^{2}}{T^{3}}\frac{EA}{Le/l} \tag{3}\]  
where \(g\) is the gravity acceleration, \(EA\) is the extensional rigidity of cable, \(\theta\) is the inclination angle, and \(L_{\text{e}}\) is the static (stretched) length of the cable  
\[Le\approx l(1+\frac{\left(mgl\cos\theta/T\right)^{2}}{8}) \tag{4}\]  
The Equation (1) can be discretized using the finite difference method given by [PERSON] and [PERSON] [59] and further considering the corresponding boundary conditions (i.e., \(y(0,t)=y(l,t)=y(0,t)^{\prime\prime}=y(l,t)^{\prime\prime}=0\) for pinned ends or \(y(0,t)=y(l,t)=y(0,t)^{\prime}=y(l,t)^{\prime}=0\) for fixed ends. The ( )' and ( )\" denote the first and second derivatives with respect to \(x\), respectively.). For a discretized  
Figure 1: Analysis model of an inclined cable with an inertial mass damper (IMD) (**a**) pinned-pinned ends (**b**) fixed-fixed ends. cable with \(N\) equal elements and \(n=N\)-1 internal nodes between two cable ends, the transverse displacement of the cable can be expressed as  
\[y(x,t)=\mathbf{w}(x)e^{pt} \tag{5}\]  
where \(p=-\omega_{n}(\xi_{n}\pm i\sqrt{1-\xi_{n}^{2}})\) is the complex eigenvalue pair and \(\mathbf{w}=\left[ \begin{array}{cccc}w_{1}&w_{2}&\cdots&w_{n}\end{array}  \right]^{\mathrm{T}}\) is the corresponding eigenvector. \(\omega_{n}\) is modal frequency, \(\xi_{n}\) is modal damping ratio and \(w_{i}\) is displacement of the \(i\)th internal node. Therefore, the discretized equation in the matrix form can be derived as  
\[\mathbf{Kw}+p\mathbf{Cw}+p^{2}\mathbf{Mw}=0 \tag{6}\]  
where \(\mathbf{K}=\left[k_{ij}\right]_{m\times m^{\prime}}\), \(\mathbf{C}=\left[c_{ij}\right]_{m\times m^{\prime}}\), \(\mathbf{M}=\left[m_{ij}\right]_{m\times m}\) are the stiffness matrix, the damping matrix and the mass matrix, respectively. According to the reference [59], the stiffness matrix is defined as  
\[\mathbf{K}=\mathbf{K}_{1}+\frac{\lambda^{2}Ta}{l^{3}}\mathbf{B} \tag{7}\]  
\[\mathbf{K}_{1}=\left[ \begin{array}{ccccc}Q&U&W&&&0\\ &S&U&W&&\\ &&S&U&\ddots&\\ &&&\ddots&\ddots&W\\ \mathrm{sym}&&&S&U\\ &&&Q\end{array}