# Cable Force Identification of Two Bending Cable Networks with Arbitrary Boundary Conditions
## 2 Formulation of Analytical Solution
### Governing Equation of Free Vibration System  
To describe the governing equation of the two-bending cable network interconnected using cross-ties, five assumptions are adopted similar to those in research promoted by [PERSON] et al. (2018). By applying [PERSON]'s principle to the total energy of the cable system, the governing differential equations of the motion for the axially tensioned beams ([PERSON] _et al._, 2018) under free vibration can be obtained:  
\[\overline{m}_{i}\frac{\partial^{2}v_{ij}\big{(}x_{ij},t\big{)}}{\partial t^{2}} +EI_{i}\frac{\partial^{4}v_{ij}\big{(}x_{ij},t\big{)}}{\partial x_{ij}^{4}}-T_{ i}\frac{\partial^{2}v_{ij}\big{(}x_{ij},t\big{)}}{\partial x_{ij}^{2}}=0;\ i,j=1,2 \tag{1}\]  
The solutions of the system can be assumed by using the separation of the variable method as:  
\[v_{ij}\big{(}x_{ij},t\big{)}=\overline{v}_{ij}\big{(}x_{ij})h(t);\ i,j=1,2 \tag{2}\]  
where \(\overline{v}_{ij}\big{(}x_{ij}\big{)}\) represents the local mode shape of the \(j\)-th segment of the \(i\)-th cable and \(h(t)\) is the time function that indicates the vibration type. The governing equations for the local mode shape functions can be obtained:  
\[EI_{i}\frac{d^{4}\overline{v}_{ij}\big{(}x_{ij}\big{)}}{dx_{ij}^{4}}-T_{i} \frac{d^{2}\overline{v}_{ij}\big{(}x_{ij}\big{)}}{dx_{ij}^{2}}-\overline{m}_{ i}\omega^{2}\overline{v}_{ij}\big{(}x_{ij}\big{)}=0;\ i,\ j=1,2. \tag{3}\]  
where \(\omega\) is also the natural frequency of the cable networks. By solving Equation (3), the local mode shapes of cable segment \(i\)-\(j\) can be obtained and shown in Equation (4). \[\overline{v}_{ij}\big{(}x_{ij}\big{)}=C_{ij,1}\cosh\big{(}\beta_{i}x_{ij} \big{)}+C_{ij,2}\text{sinh}\big{(}\beta_{i}x_{ij}\big{)}+C_{ij,2}\cos\big{(} \phi_{i}x_{ij}\big{)}+C_{ij,4}\sin\big{(}\phi_{i}x_{ij}\big{)} \tag{4}\]  
where \(C_{ij,k}\) (\(i,j=1,2;k=1,2,\ldots,4\)) are constants and the auxiliary wave number parameters are defined as follows:  
\[\beta_{i}=\sqrt{\frac{1}{2}\sqrt{\left(\frac{T_{i}}{EI_{i}}\right)^{2}+\frac{ 4\overline{m}_{i}\omega^{2}}{EI_{i}}}+\frac{1}{2}\frac{T_{i}}{EI_{i}}};\ \phi_{i}=\sqrt{\frac{1}{2}\sqrt{\left(\frac{T_{i}}{EI_{i}}\right)^{2}+\frac{4 \overline{m}_{i}\omega^{2}}{EI_{i}}}-\frac{1}{2}\frac{T_{i}}{EI_{i}}}\]  
Equation (4) has 16 unknown coefficients. Thus, 16 equations of boundary conditions, continuities, and force equilibriums are acquired to determine these unknowns. The boundary conditions of the vertical displacements and curvatures of the cables at the hinged supports are equal to zero. \[\overline{v}_{ij}(0)=0\ \text{and}\ \overline{v}^{\prime}{}_{ij}(0)=0\quad i,j=1,2 \tag{5}\]  
Figure 3: Network cables modeled as two simply supported cables and a flexible cross-tie. By applying the boundary condition in Equation (5), the constants \(C_{ij,1}\) and \(C_{ij,3}\) can be determined to equal zero. Hence, 16 unknown constants reduce to 8 unknowns. Then, the local mode shapes of cable segment \(i\)-\(j\) are rewritten as  
\[\overline{v}_{ij}\big{(}x_{ij}\big{)}=C_{ij,2}\text{sinh}\big{(}\beta_{i}x_{ij} \big{)}+C_{ij,4}\sin\big{(}\phi_{i}x_{ij}\big{)} \tag{6}\]  
For continuity conditions, the vertical displacements and curvatures of the left-side segments equal those of the right-side segments at the cross-tie location. However, the cable slopes are opposite because the positive directions of the \(x_{i1}\) and \(x_{i2}\) (\(i=1,2\)) axes are reversed. Moreover, the difference between the lateral displacements of Segment 2-1 at \(x_{21}=l_{21}\) and Segment 1-1 at \(x_{11}=l_{11}\) equals the vertically projected axial deformation in the cross-tie, as shown in Equation (8). Noticeably, the details of the derivation of Equation (8) may be found in Appendix A. \[\left\{ \begin{array}{l}\overline{v}_{i1}(l_{i1})=\overline{v}_{i2}(l_{i2}) \\ \overline{v}^{\prime}{}_{i1}(l_{i1})=-\overline{v}^{\prime}{}_{i2}(l_{i2})\\ \overline{v}^{\prime}{}_{i1}(l_{i1})=\overline{v}^{\prime}{}_{i2}(l_{i2})\\ \end{array} \right.