# Cable Force Identification of Two Bending Cable Networks with Arbitrary Boundary Conditions
## 4 Application of Cable Force Identification Method
### A Two-Cable Network with Mixed Boundary Conditions  
In this case study, the cable network configurations are similar to those in the previous case study but different in the boundary conditions. The supports of Cables 1 and 2 are fixed-fixed ends and hinged-fixed ends, respectively, as shown in Figure 7. The cable forces and axial stiffness of the cross-tie are assumed to be unknown. The values of the cable forces and the cross-tie axial stiffness shown below are just for verifying this method. Since the boundary conditions of Cable 1 are not hinged-hinged supports, the analytical solutions in Equation (16) and (18) are not suitable for identifying the cable forces and computed frequencies of the cable network. For applying the solutions of the model \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**No.** & \(\mathbf{T_{1}}\) **(kN)** & \(\mathbf{T_{2}}\) **(kN)** & _MAPE (\%)_ & **No.** & \(\mathbf{T_{1}}\) **(kN)** & \(\mathbf{T_{2}}\) **(kN)** & _MAPE (\%)_ \\ \hline
1 & 1784.69 & 1715.42 & 0.95 & 16 & 1784.69 & 1665.43 & 0.75 \\
2 & 1919.67 & 1578.13 & 0.22 & 17 & 1919.67 & 1570.53 & 0.22 \\
3 & 2054.65 & 1454.34 & 1.17 & 18 & 2054.65 & 1482.74 & 1.02 \\
4 & 1816.45 & 1729.70 & 1.17 & 19 & 1784.69 & 1663.36 & 0.75 \\
5 & 1943.49 & 1526.69 & 0.58 & 20 & 1919.67 & 1582.02 & 0.22 \\
6 & 2070.53 & 1426.50 & 1.42 & 21 & 2054.65 & 1502.12 & 1.01 \\
7 & 1784.69 & 1647.56 & 0.83 & 22 & 1784.69 & 1630.08 & 0.98 \\
8 & 1919.67 & 1588.03 & 0.27 & 23 & 1919.67 & 1560.54 & 0.26 \\
9 & 2054.65 & 1524.93 & 1.15 & 24 & 2054.65 & 1504.59 & 1.02 \\
10 & 1784.69 & 1672.68 & 0.76 & 25 & 1784.69 & 1293.34 & 5.95 \\
11 & 1919.67 & 1562.12 & 0.25 & 26 & 1975.25 & 1707.68 & 2.64 \\
12 & 2054.65 & 1465.93 & 1.09 & 27 & 2082.44 & 1655.24 & 3.22 \\
13 & 1836.30 & 1731.93 & 1.35 & 28 & 1784.69 & 1607.81 & 1.20 \\
14 & 1947.46 & 1509.33 & 0.79 & 29 & **1899.82** & **1589.19** & **0.11** \\
15 & 2058.62 & 1362.32 & 2.17 & 30 & 2054.65 & 1564.63 & 1.57 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: _MAPE_ of thirty satisfying combinations. Figure 7: Cable network’s configuration with mixed boundary conditions. with hinged-end supports, the effective vibration length of Cable 1 must be determined beforehand. Then, the actual cable length is replaced with the effective vibration length to transform Cable 1 with fixed-end supports into the cable with hinged-end supports. The effective vibration length of Cable 1 is determined by the technique proposed by [PERSON] et al. [22]. However, since their technique is appropriate for a single cable, the cables of the cable network need to be separated by releasing the cross-tie. Although the cross-tie is required to be removed in this stage, it is necessary to obtain the effective vibration length and other unknown parameters of the cable network, such as the axial stiffness of the cross-tie and the current cable forces. It is noted that the cross-tie is required to be removed one time only. For later measurements such as cable health monitoring, the cable forces can be identified directly without further removing the cross-tie. The effective vibration length of Cable 1 is determined by using the vibration signals at five sensor locations. The cable configurations and sensor locations on Cable 1 are shown in Figure 8. The measured frequency spectra of the single Cable 1 at five sensor locations under free vibration are shown in Figure 9.  
By applying the [PERSON] et al. [22] method, the cable effective vibration lengths and cable tensions of single Cables 1 and 2 at the first five modes are obtained and shown in Table 6. The mean value of effective vibration lengths in five modes is the equivalent lengths of Cable 1 with hinged-hinged supports.