# Cable Force Identification of Two Bending Cable Networks with Arbitrary Boundary Conditions
## 3 Methodology
### Identification of the Cross-Tie Axial Stiffness  
Since this paper introduces a method to identify the cable forces of the cable network without removing the cross-tie, the axial stiffness of the cross-tie, an important parameter, needs to be determined beforehand. The axial stiffness of the cross-tie can be directly determined from the design document or the relationship among three measured parameters of the cross-tie: the cross-section area, length, and material Young's modulus. Nevertheless, the material Young's modulus is difficult to obtain directly from the field measurement. The cross-tie must be released and sent to a material laboratory to acquire the Young's modulus using a tensile test. This procedure is time-consuming and the tensile test also damages the cross-tie sample. Hence, this paper will introduce a dynamic method to determine the axial stiffness of the cross-tie from the measured frequencies of the cable network. This dynamic method can directly obtain the cross-tie's axial stiffness at the inspection field and does not damage it. The procedure to obtain the axial stiffness of the cross-tie from the dynamic measurement is described as follows:  
The cross-tie is released to transform the cable network into two independent cables. Each cable is excited by an impulse load to obtain acceleration signals under free vibration at five locations. The acceleration signals at five locations are used to determine the measured cable frequency, cable force, cable vibration effective length, and cable bending rigidity. The cross-tie is reconnected to form the cable network. Then, the cable network is excited by an impulse load to obtain an acceleration signal under free vibration at any location on Cable 1 or 2. The acceleration signals are transferred in the time domain to those in the frequency domain by using FFT to obtain the measured frequencies. Since the cable force, cable vibration effective length, and measured cable frequencies are known, the cross-tie axial stiffness can be obtained using Equation (20). For cables with full or partly fixed ends, the cable lengths \(L_{i}\) and segment lengths \(l_{ij}\) are replaced by the cable effective vibration length \(L_{eff,i}\) and \(l_{eff,ij}\), respectively. \[\left.\begin{array}{l}F(K_{c},\ f_{m})=\\ \left\{ \begin{array}{l}\frac{\phi_{1}}{\beta_{1}}\sin(\beta_{1}l_{11})\sinh[ \beta_{1}(L_{1}-l_{11})]}{\sinh(\beta_{1}L_{1})}\sin(\phi_{1}L_{1})\sin(\phi_ {2}L_{2})\\ -\sin(\phi_{1}l_{11})\sin[\phi_{1}(L_{1}-l_{11})]\sin(\phi_{2}L_{2})\\ -\frac{\phi_{1}\sqrt{l_{1}^{2}+16\pi^{2}\frac{\phi_{2}}{2}m_{1}E_{1}}}{\kappa _{1}\sin^{2}\phi_{2}}\sin(\phi_{1}L_{1})\sin(\phi_{2}L_{2})\\ +\frac{\phi_{1}\sqrt{l_{1}^{2}+16\pi^{2}\frac{\phi_{2}}{2}m_{1}E_{1}}E_{1}}{ \phi_{2}\sqrt{l_{1}^{2}+16\pi^{2}\frac{\phi_{2}}{2}m_{2}E_{1}}E_{1}}\left\{ \begin{array}{l}\frac{\phi_{2}}{\beta_{2}}\sinh(\beta_{2}l_{12})\sinh[ \beta_{2}(L_{2}-l_{11})]}{\sinh(\beta_{2}L_{2})}\sin(\phi_{1}L_{1})\sin(\phi_ {2}L_{2})\\ -\sin(\phi_{2}l_{21})\sin[\phi_{2}(L_{2}-l_{21})]\sin(\phi_{1}L_{1})\end{array} \right\}\end{array} \right\}=0 \tag{20}\]  
Equation (20) can be reformed as follows:  
\[ \begin{array}{l}F(K_{c},f_{m})=A(f_{m})-\frac{B(f_{m})}{K_{c}}+C(f_{m})=0\\ K_{c}=\frac{B(f_{m})}{A(f_{m})+C(f_{m})}\end{array}  \tag{21}\]  
where \(A\), \(B\), and \(C\) variables are defined as follows:  
\[\left\{