# Cable Force Identification of Two Bending Cable Networks with Arbitrary Boundary Conditions
## 3 Methodology
### Identification of Cable Forces
\begin{array}{l}F(T_{1},T_{2},\ f_{m})=\\ \left\{\begin{array}{l}\frac{\phi_{1}\sinh(\beta_{1}l_{11})\sinh(\beta_{1}(L _{1}-l_{11}))}{\beta_{1}}\sin(\phi_{1}L_{1})\sin(\phi_{2}L_{2})\\ -\sin(\phi_{1}l_{11})\sin[\phi_{1}(L_{1}-l_{11})]\sin(\phi_{2}L_{2})\\ -\frac{\phi_{1}\sqrt{T_{1}+16\pi^{2}\overline{m}_{i}E_{1}}}{K\sin^{2}\theta_{ c}}\sin(\phi_{1}L_{1})\sin(\phi_{2}L_{2})\\ +\frac{\phi_{1}\sqrt{T_{1}^{2}+16\pi^{2}\overline{m}_{i}E_{1}}}{\phi_{2}\sqrt {T_{1}^{2}+16\pi^{2}\overline{m}_{i}E_{1}}}\left\{\begin{array}{l}\frac{ \phi_{2}\sinh(\beta_{2}l_{11})\sinh(\beta_{2}(L_{2}-l_{11}))}{\beta_{2}}\sin (\phi_{2}L_{2})\\ -\sin(\phi_{2}l_{21})\sin[\phi_{2}(L_{2}-l_{21})]\sin(\phi_{1}L_{1})\end{array} \right\}\end{array}\right\}=0\\ \end{array} \right\}=0 \tag{16}\]For cables with fixed or partly fixed ends, the cable lengths \(L_{i}\) and segment lengths \(l_{ij}\) are replaced by the cable effective vibration length \(L_{\textit{eff},i}\) and \(L_{\textit{eff},j}\), respectively. The auxiliary wave number parameters are defined as follows:  
\[\beta_{i}=\sqrt{\frac{1}{2}\sqrt{\left(\frac{T_{i}}{EI_{i}}\right)^{2}+\frac{16 \pi^{2}\overline{m_{i}}f_{m}^{2}}{EI_{i}}}+\frac{1}{2}\frac{T_{i}}{EI_{i}}},\ \phi_{i}=\sqrt{\frac{1}{2}\sqrt{\left(\frac{T_{i}}{EI_{i}}\right)^{2}+\frac{16 \pi^{2}\overline{m_{i}}f_{m}^{2}}{EI_{i}}}}-\frac{1}{2}\frac{T_{i}}{EI_{i}}} \tag{17}\]  
Step 6: Substitute the cable forces, i.e., \(T_{1}\) and \(T_{2}\), of the selected combinations (in Step 5) in Equation (18) to compute the cable network frequencies. This study will occupy the mean absolute percentage error (\(MAPE\)) method to select the pair of cable tensions \(T_{1}\) and \(T_{2}\) in which the computed frequencies are the closest to the measured frequencies. If \(T_{1}\) and \(T_{2}\) in any combination can provide the minimum \(MAPE\) between measured and calculated frequencies, they are identified as the cable tensions of the cable network. The minimum \(MAPE\) between measured frequencies \(f_{m}\) and the computed frequencies \(f_{c}\) are defined as in Equation (19). Notably, the \(MAPE\) method is occupied because it is more robust to data with bias measured frequencies. Additionally, the non-zero values of cable network frequencies are suitable for applying the \(MAPE\) method. Moreover, the \(MAPE\) method can provide relative errors between the measured and computed frequencies. The relative errors are instrumental in judging the error size. \[F(f_{c})=\left\{ \begin{array}{l}\left\{\begin{array}{l}\frac{\phi_{1}}{P_ {1}}\frac{\sinh(j_{1}l_{1})\sinh(j_{1}(T_{1}-l_{1}))}{\sinh(j_{1}k_{1})}\sin( \phi_{1}L_{1})\sin(\phi_{2}L_{2})\\ -\frac{\phi_{1}\sqrt{T_{1}^{2}+16\pi^{2}\overline{m_{i}}E_{1}}}{\cosh(j_{1}k_ {1})}\sin(j_{1}(L_{1}-l_{1}))\sin(\phi_{2}L_{2})\\ -\frac{\phi_{1}\sqrt{T_{1}^{2}+16\pi^{2}\overline{m_{i}}E_{1}}}{\phi_{2}\sqrt {T_{1}^{2}+16\pi^{2}\overline{m_{i}}E_{1}}}\sin(j_{1}(L_{1})\sin(\phi_{2}L_{2 })\\ +\frac{\phi_{1}\sqrt{T_{1}^{2}+16\pi^{2}\overline{m_{i}}E_{1}}}{\phi_{2}\sqrt {T_{1}^{2}+16\pi^{2}\overline{m_{i}}E_{1}}}\left(\frac{\phi_{2}}{P_{2}}\frac{ \sinh(j_{1}l_{1})\sinh(j_{1}(L_{2}-l_{1}))}{\sinh(j_{2}k_{2})}\sin(j_{1}k_{2}(L _{2}-l_{2}))}{-\sin(j_{2}k_{2})\sin(j_{2}(L_{2}-l_{2}))\sin(j_{1}(L_{1})} \sin(j_{1}L_{2})\right)\\ -\sin(j_{2}l_{2})\sin(j_{2}(L_{2}-l_{2}))\sin(j_{1}(L_{1})\end{array}\right\} \end{array} \right\}=0 \tag{18}\]  
\[MAPE_{\min}=Min\left[\left(\frac{1}{N}\sum_{k=1}^{N}\left|\frac{f_{m}k-f_{c}k }{f_{m,k}}\right|\times 100\%\right)\right]_{p} \tag{19}\]  
where \(N\) is the number of used modes to calculate the cable forces, \(k\) indicates the mode \(k\)-th, and \(p\) indicates the selected combination \(p\)-th.