# Cable Force Identification of Two Bending Cable Networks with Arbitrary Boundary Conditions
## 4 Application of Cable Force Identification Method
### A Two-Cable Network with Hinged-End Supports
The cable forces out of the detected ranges are removed, and the cable forces inside the detected ranges are shown in Table 3  
Finally, the cable forces of Cable 1 are predicted from 1784.69 kN to 2181.28 kN (1982.98 kN \(\pm\) 10%), and those of Cable 2 are predicted from 1281.69 kN to 1734.05 kN (1507.87 kN \(\pm\) 10%). Step 5: In this step, the measured frequencies of the cable network from mode 1 to mode 10 and the cable forces \(T_{1}\) and \(T_{2}\) varying from 1784.69 kN to 2181.28 kN and from 1281.69 kN to 1734.05 kN are used to create numerous combinations, such as (1784.69 kN, 1715.42 kN, 1.3278 Hz), (1919.67 kN, 1588.03 kN, 2.6883 Hz), (1784.69 kN, 1293.34 kN, 6.6917 Hz), etc. These combinations are substituted into Equation (16) to find the combinations that satisfy the condition \(F(T_{1},T_{2}f_{m})\) = 0. Thirty examples in numerous satisfied combinations are listed in Table 4 for reference. Step 6: For identifying the most appropriate cable tensions, \(T_{1}\) and \(T_{2}\) in numerous selected combinations obtained in Step 5 are introduced in Equation (18) to calculate the cable network frequencies. The _MAPE_ values of thirty satisfying combinations are shown in Table 5 for reference. The combination with the minimum _MAPE_ of computed frequencies, as defined in Equation (19), provides the most accurate cable tensions \(T_{1}\) and \(T_{2}\). The combination with \(T_{1}\) = 1899.82 kN and \(T_{2}\) = 1589.19 kN has the minimum _MAPE_ of computed frequencies at 0.11%. Hence, the cable tensions, i.e., \(T_{1}\) = 1899.82 kN and \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**No.** & \(T_{1}\) **(kN)** & \(T_{2}\) **(kN)** & \(f_{m}\) **(Hz)** & **No.** & \(T_{1}\) **(kN)** & \(T_{2}\) **(kN)** & \(f_{m}\) **(Hz)** \\ \hline
1 & 1784.69 & 1715.42 & 1.3278 & 16 & 1784.69 & 1665.43 & 4.6050 \\
2 & 1919.67 & 1578.13 & 1.3278 & 17 & 1919.67 & 1570.53 & 4.6050 \\
3 & 2054.65 & 1454.34 & 1.3278 & 18 & 2054.65 & 1482.74 & 4.6050 \\
4 & 1816.45 & 1729.70 & 2.2564 & 19 & 1784.69 & 1663.36 & 5.4582 \\
5 & 1943.49 & 1526.69 & 2.2564 & 20 & 1919.67 & 1582.02 & 5.4582 \\
6 & 2070.53 & 1426.50 & 2.2564 & 21 & 2054.65 & 1502.12 & 5.4582 \\
7 & 1784.69 & 1647.56 & 2.6883 & 22 & 1784.69 & 1630.08 & 6.5048 \\
8 & 1919.67 & 1588.03 & 2.6883 & 23 & 1919.67 & 1560.54 & 6.5048 \\
9 & 2054.65 & 1524.93 & 2.6883 & 24 & 2054.65 & 1504.59 & 6.5048 \\
10 & 1784.69 & 1672.68 & 3.2030 & 25 & 1784.69 & 1293.34 & 6.6917 \\
11 & 1919.67 & 1562.12 & 3.2030 & 26 & 1975.25 & 1707.68 & 6.6917 \\
12 & 2054.65 & 1465.93 & 3.2030 & 27 & 2082.44 & 1655.24 & 6.6917 \\
13 & 1836.30 & 1731.93 & 4.0023 & 28 & 1784.69 & 1607.81 & 7.2340 \\
14 & 1947.46 & 1509.33 & 4.0023 & 29 & 1899.82 & 1589.19 & 7.2340 \\
15 & 2058.62 & 1362.32 & 4.0023 & 30 & 2054.65 & 1564.63 & 7.2340 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Thirty combinations in numerous satisfied combinations. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Cable Network** & \multicolumn{2}{c}{**Single Cable 1**} & \multicolumn{2}{c}{**Single Cable 2**} \\ \hline
**Mode** & **Measured** & **Mode** & \(-\) & **Mode** & \(-\) \\
**Number** & **Frequency (Hz)** & **Number** & \(T_{1}\) **(kN)** & **Number** & \(T_{2}\) **(kN)** \\ \hline
1 & 1.3278 & 1 & 1974.93 & 1 & 1525.94 \\
2 & 2.2564 & - & - & 2 & 1082.39 \\
3 & 2.6883 & 2 & 2010.34 & 2 & 1546.27 \\
4 & 3.2030 & - & - & - & - \\
5 & 4.0023 & 3 & 1957.38 & 3 & 1493.50 \\
6 & 4.6050 & - & - & 3 & 1994.30 \\
7 & 5.4582 & 4 & 2017.81 & 4 & 1523.76 \\
8 & 6.5048 & 5 & 1786.58 & 5 & 1323.60 \\
9 & 6.6917 & 5 & 1897.34 & 5 & 1409.32 \\
10 & 7.2340 & 5 & 2236.50 & 5 & 1671.78 \\ Average & & 1982.98 & & 1507.87 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The refinement of the predicted values of cable forces. \(T_{2}=1589.19\) kN with errors of 0.00% and 0.55%, respectively, are identified as the cable forces of the cable network. Noticeably, the relative errors are calculated by comparing the obtained cable forces in this stage and the known cable forces in Step 1.