## 2 Wire Rope Flexural Vibration Modeling  
The flexural vibration of a wire rope under tension is analyzed based on the Euler beam model. The dynamic characteristics of the wire rope under tension are measured through flexural vibration. Figure 1 shows a spring-supported Euler beam model under tension. The ends are supported by the translational spring \(k_{T}\) and the rotational spring \(k_{R}\). The static tension \(T\) is applied to the beam of length \(L\). Force \(F\) is applied at a distance \(L_{f}\) from the left clamp end to excite the beam. The equation of motion for the Euler beam under tension is given as follows:  
\[D\frac{\partial^{4}w}{\partial x^{4}}-T\frac{\partial^{2}w}{\partial x^{2}}+ M_{b}\frac{\partial^{2}w}{\partial t^{2}}=0, \tag{1}\]  
where \(w\) is transverse displacement, \(D\) is bending stiffness, and \(M_{b}\) is mass per unit length. When a force is applied to the middle of the beam, the beam function satisfying Equation (1) is given as follows:  
\[\hat{w}(x)= \begin{cases}\hat{w}_{1}(x)&0\leq x\leq L_{f}\\ \hat{w}_{2}(x)&L_{f}\leq x\leq L\end{cases} ,\] \[\hat{w}_{1}(x)=\hat{A}_{1}\sin\hat{M}x+\hat{A}_{2}\text{cos}\, \hat{M}x+\hat{A}_{3}\text{e}^{\hat{N}x}+\hat{A}_{4}\text{e}^{-\hat{N}x}, \tag{2}\] \[\hat{w}_{2}(x)=\hat{A}_{5}\text{sin}\,\hat{M}\Big{(}x-L_{f}\Big{)} +\hat{A}_{6}\text{cos}\,\hat{M}\Big{(}x-L_{f}\Big{)}+\hat{A}_{7}\text{e}^{\hat {N}(x-L_{f})}+\hat{A}\text{ge}^{-\hat{N}(x-L_{f})},\] \[ \begin{pmatrix}\hat{M}\\ \hat{N}\end{pmatrix} =\left[\mp\hat{t}+\left(\hat{t}^{2}+\hat{b}^{2}\right)^{1 /2}\right]^{1/2},\] (3) \[\hat{t}=T/2\hat{D}= T\hat{b}^{2}/2M_{b}\omega,\ \ \hat{b}=\left(M_{b}\omega^{2}/\hat{D}\right)^{1/2},\ \ \hat{D}=D(1+i\text{n}), \tag{4}\]  
where \(\hat{A}_{1}\)-\(\hat{A}_{8}\) are complex amplitude coefficients, \(\hat{M}\) and \(\hat{N}\) are wavenumbers (\(\hat{M}=M_{r}-iM_{i}\), \(\hat{N}=N_{r}-iN_{i}\)), \(\omega\) is angular frequency, \(\hat{D}\) is complex bending stiffness, \(\eta\) is loss factor, and \(H\) is the Heaviside step function. Figure 1: Spring-supported Euler beam model under tension. To obtain a general solution for the wave equation satisfying Equation (2), it is necessary to solve for the unknown coefficients. Eight boundary conditions are used to determine the coefficients as follows:  
\[ \begin{split}\hat{D}^{\frac{\partial^{2}\phi_{1}(0)}{\partial x^{2}}} =k_{R}\frac{\partial\phi_{1}(0)}{\partial x},\\ \hat{D}^{\frac{\partial^{3}\phi_{1}(0)}{\partial x^{3}}}=-k_{T} \hat{\psi}_{1}(0),\\ \hat{\eta}_{1}(L_{f})=\hat{\eta}_{2}(L_{f}),\\ \frac{\partial\phi_{1}(L_{f})}{\partial x}=\frac{\partial\phi_{2}(L_ {f})}{\partial x},\\ \hat{D}^{\frac{\partial^{2}\phi_{1}\left(L_{f}\right)}{\partial x^{ 3}}}=\hat{D}^{\frac{\partial^{2}\phi_{2}\left(L_{f}\right)}{\partial x^{2}}}, \\ \hat{D}^{\frac{\partial^{2}\phi_{1}\left(L_{f}\right)}{\partial x^{ 3}}}+T\frac{\partial\phi_{1}\left(L_{f}\right)}{\partial x}+T\frac{\partial \phi_{2}\left(L_{f}\right)}{\partial x}-\hat{D}^{\frac{\partial^{2}\phi_{2} \left(L_{f}\right)}{\partial x^{3}}}=F,\\ \hat{D}^{\frac{\partial^{2}\phi_{2}\left(L\right)}{\partial x^{ 3}}}=-k_{R}\frac{\partial\phi_{2}\left(L\right)}{\partial x},\\ \hat{D}^{\frac{\partial^{2}\phi_{2}\left(L\right)}{\partial x^{ 3}}}=k_{T}\hat{\eta}_{2}(L),\end{split}  \tag{5}\]  
The eight equations are rearranged, and the unknown complex amplitude coefficients are solved via symbolic calculation. The transfer function for the tensioned beam displacement is used to measure the wavenumbers, and it is expressed as follows:  
\[\Lambda\text{e}^{i\phi}=\frac{\hat{\psi}(x_{2})}{\hat{\psi}(x_{1})}, \tag{6}\]  
where \(\Lambda\) and \(\phi\) are the amplitude and phase of the measured transfer function, and \(x_{1}\) and \(x_{2}\) are the vibration measurement locations. The unknown frequency parameter \(\hat{b}=b_{r}-ib_{i}\) is calculated using the transfer function method as follows:  
\[ \begin{bmatrix}b_{r}\\ b_{i}\end{bmatrix} _{j+1}= \begin{bmatrix}b_{r}\\ b_{i}\end{bmatrix} _{j}-