# Cable Force Identification of Two Bending Cable Networks with Arbitrary Boundary Conditions
## 3 Methodology
### Identification of Cable Forces  
The procedure to identify the cable forces of the cable network system is summarized as shown in Figure 5. The procedure contains six steps that are explained as follows:  
Figure 5: The cable force identification procedure by using measured vibration data. Step 1: Collect all necessary information about the cable network, such as material properties (e.g., Young's modulus \(E\) and mass density \(\gamma\)), section properties (e.g., cross-section area \(A\), inertia moment \(I\)), the network configuration (e.g., the lengths of two cables \(L_{i}\), four segments \(l_{ij}\) (\(i\), \(j=1\), \(2\)) and cross-tie \(L_{c}\), the inclined angle of the cross-tie \(\theta_{c}\)), cable boundary conditions, and initial cable tensile forces (if possible) of two cables from design documents or field inspection. When the cable boundary conditions differ for hinged-hinged ends, the effective vibration lengths \(L_{e\!f}\) should be determined forehand. Step 2: Record the acceleration signal at an arbitrary location on the cables of a cable network system under ambient vibration from field measurement. Nevertheless, this paper will generate acceleration signals from a finite element model with known cable internal forces for study purposes. Hence, the accuracy and effectiveness of this method can be verified easily by comparing the obtained cable tensile forces and the known cable tensile forces. Step 3: Transfer the acceleration signal obtained in Step 2 in the time domain to those in the frequency domain by using the Fast Fourier Transform (FFT) to obtain the measured frequencies \(f_{m}\) of the cable networks. For illustration, this study uses ABAQUS code to create a cable network model and obtain the measured frequencies of the cable network. Step 4: Substitute the measured frequencies in Step 3 and corresponding mode orders into Equation (15) to calculate the mean cable forces. Notably, Equation (15) is the equation to determine the cable force of a single cable with hinged-hinged supports. Nevertheless, if the design documents can provide the initial cable forces of the cable network, this step can be ignored. Then, the predicted ranges of the cable network tensile forces \(T_{1}\) and \(T_{2}\) equal the mean cable forces \(\pm\) 10% or the designed cable tensions \(\pm\) 10%. \[\overline{T}_{i}=\frac{1}{N_{i}}\sum_{n_{i}=1}^{N_{i}}\left[4\overline{m}_{i} L_{i}^{2}\left(\frac{f_{m}^{n}}{n_{i}}\right)^{2}-\frac{EI_{y,i}}{L_{i}^{2}}(n_{i} \pi)^{2}\right],\ i=1,2;n=1,2,3,\ldots, \tag{15}\]  
where, \(\overline{m}_{i}\), \(L_{i}\), and \(EI_{i}\) are the average mass per unit length, cable length, and cable bending rigidity of cable \(i\) (\(i=1\),\(2\)), respectively. \(n\)-th is the mode order related to the vibration mode of separated cables. \(N\) is the number of modes used to calculate the cable forces of separated cables. \(f_{m}^{n}\) is the measured frequency of the cable network obtained from the field measurement or numerical simulation at mode \(n\)-th. For cables with full or partly fixed ends, the cable length \(L\) is replaced by the effective vibration length \(L_{e\!f}\). Step 5: Create numerous combinations of \(T_{1}\), \(T_{2}\), and the measured frequencies (in Step 3). Then, these combinations are substituted in Equation (16) to identify the cable tensions. Only the combinations that satisfy the condition \(F(T_{1}\), \(T_{2}\), \(f_{m})=0\) are selected. If the cable forces of the selected combinations are in the predicted range of the cable force (in Step 4), they could be the cable forces (i.e., \(T_{1}\) and \(T_{2}\)) of the cable network. Notably, the values \(T_{1}\) and \(T_{2}\) should vary in a wide range of cable forces to cover the possible cable forces. Nonetheless, the existing cable forces are expected in the predicted ranges. The ranges of \(T_{1}\) and \(T_{2}\) can be, therefore, reduced to the predicted ranges to save computational time. \[\left.