# Cable Force Identification of Two Bending Cable Networks with Arbitrary Boundary Conditions
## Appendix B  
This section derives the analytical solution of the two-cable network. Considering the boundary conditions, continuity conditions and force equilibrium are substituted in Equation (6) to determine the 8 unknown constants, that is \(C_{ij,k}\) (\(i,j=1,2\); \(k=2,4\)). The mode shape function of the cable segment is rewritten as:  
\[\overline{v}_{ij}=C_{ij,2}\text{sinh}\big{(}\beta_{i}x_{ij}\big{)}+C_{ij,4} \sin\big{(}\phi_{i}x_{ij}\big{)};\ i,\ j=1,2 \tag{11}\]  
The continuity condition of the lateral displacements at \(x_{ij}=l_{ij}\), namely \(\overline{v}_{i1}(l_{i1})=\overline{v}_{i2}(l_{i2})\), is introduced to Equation (11) to obtain:  
\[C_{i1,2}\text{sinh}(\beta_{i}l_{i1})+C_{i1,4}\sin(\phi_{i}l_{i1})-C_{i2,2} \text{sinh}(\beta_{i}l_{i2})-C_{i2,4}\sin(\phi_{i}l_{i2})=0;\ i,\ j=1,2 \tag{12}\]  
The continuity condition of the cable slopes at \(x_{ij}=l_{ij}\), namely \(\overline{v}^{\prime}{}_{i1}(l_{i1})=-\overline{v}^{\prime}{}_{i2}(l_{i2})\), is introduced to Equation (11) to obtain:  
\[ \begin{array}{l}C_{i1,2}\beta_{i}\cosh(\beta_{i}l_{i1})+C_{i1,4}\phi_{i} \cos(\phi_{i}l_{i1})\\ +C_{i2,2}\beta_{i}\cosh(\beta_{i}l_{i2})+C_{i2,4}\phi_{i}\cos(\phi_{i}l_{i2})= 0\end{array} ;\ i,\ j=1,2 \tag{13}\]  
The continuity condition of the cable curvatures at \(x_{ij}=l_{ij}\), namely \(\overline{v}^{\prime}{}_{i1}(l_{i1})=\overline{v}^{\prime}{}_{i2}(l_{i2})\), is introduced to Equation (11) to obtain:  
\[ \begin{array}{l}C_{i1,2}\beta_{i}^{2}\text{sinh}(\beta_{i}l_{i1})-C_{i1,4} \phi_{i}^{2}\sin(\phi_{i}l_{i1})\\ -C_{i2,2}\beta_{i}^{2}\text{sinh}(\beta_{i}l_{i2})+C_{i2,4}\phi_{i}^{2}\sin( \phi_{i}l_{i2})=0\end{array} ;\ i,j=1,2 \tag{14}\]  
From the force equilibrium in the vertical direction at node N\({}_{1}\) in Equation (9) and the condition in Equation (8), we have:  
\[ \begin{array}{l}EI_{1}\psi^{\prime\prime}{}_{11}(l_{11})+EI_{1}\psi^{\prime \prime}{}_{12}(l_{12})=-[\overline{v}_{21}(l_{21})-\overline{v}_{11}(l_{11})] K_{c}\sin^{2}\theta_{c}\end{array}  \tag{15}\]  
Differentiating the mode shapes in Equation (11) and introducing them into Equation (15) to obtain:  
\[ \begin{array}{l}EI_{1}\big{[}C_{11,2}\beta_{i}^{3}\cosh(\beta_{1}l_{11})-C_{ 11,4}\phi_{i}^{3}\cos\big{(}\phi_{i}l_{11}\big{)}\big{]}\\ +EI_{1}\big{[}C_{12,2}\beta_{i}^{3}\cosh(\beta_{1}l_{12})-C_{12,4}\phi_{i}^{3} \cos(\phi_{i}l_{12})\big{]}=\\ -\left[\begin{array}{l}C_{21,2}\text{sinh}(\beta_{2}l_{21})+C_{21,4}\sin( \phi_{i}l_{21})\\ -C_{11,2}\text{sinh}\big{(}\beta_{1}l_{11})-C_{11,4}\sin\big{(}\phi_{i}l_{11} \big{)}\end{array}\right]K_{c}\sin^{2}\theta_{c}\end{array}  \tag{16}\]  
\[ \begin{array}{l}\left[\frac{EI_{1}\beta_{i}^{3}}{K_{c}\sin^{2}\theta_{c}} \cosh(\beta_{1}l_{11})-\sinh(\beta_{1}l_{11})\right]C_{11,2}\\ -\left[\frac{EI_{1}\phi_{i}^{3}}{K_{c}\sin^{2}\theta_{c}}\cos(\phi_{i}l_{11})+ \sin(\phi_{1}l_{11})\right]C_{11,4}\\ +\frac{EI_{1}\beta_{i}^{3}}{K_{c}\sin^{2}\theta_{c}}\cosh(\beta_{1}l_{12})C_{1 2,2}-\frac{EI_{1}\phi_{i}^{3}}{K_{c}\sin^{2}\theta_{c}}\cos(\phi_{i}l_{12})C_ {12,4}\\ +C_{21,2}\text{sinh}(\beta_{2}l_{21})+C_{21,4}\sin(\phi_{i}l_{21})=0\end{array}  \tag{17}\]  
From the force equilibrium in the vertical direction at node N\({}_{2}\) in Equation (10) and the condition in Equation(8), we have:  
\[ \begin{array}{l}EI_{2}\psi^{\prime\prime}{}_{21}(l_{21})+EI_{2}\psi^{\prime \prime}{}_{22}(l_{22})=[\overline{v}_{21}(l_{21})-\overline{v}_{11}(l_{11})] K_{c}\sin^{2}\theta_{c}\end{array}  \tag{18}\]  
Differentiating the mode shapes in Equation (11) and introducing them into Equation (18) to obtain:  
\[ \begin{array}{l}EI_{2}\big{[}C_{21,2}\beta_{2}^{3}\cosh(\beta_{21}l_{21})-C_ {21,4}\phi_{2}^{3}\cos(\phi_{j}l_{21})\big{]}\\ +EI_{2}\big{[}C_{22,2}\beta_{2}^{3}\cosh(\beta_{2}l_{22})-C_{22,4}\phi_{2}^{3} \cos(\phi_{j}l_{22})\big{]}=\\ \big{[}C_{21,2}\text{sinh}(\beta_{2}l_{21})+C_{21,4}\sin(\phi_{j}l_{21})-C_{ 11,2}\text{sinh}\big{(}\beta_{i}l_{11})-C_{11,4}\sin(\phi_{i}l_{11})\big{]}K_{c} \sin^{2}\theta_{c}\end{array}