# Cable Force Identification of Two Bending Cable Networks with Arbitrary Boundary Conditions
## 1 Introduction
In this research, the analytical solutions of cable networks with fixed-end boundary conditions [10; 11; 12; 13; 14; 15; 16; 17; 18], dampers [10; 11; 17; 18], distributed cross-ties [16], and flexible or rigid cross-ties [12; 13; 14; 15] are used to calculate the networks' natural frequencies from given cable tensions. These analytical solutions are based on string theory and do not consider the cable bending rigidity, which may limit their application to cables with high bending  
Figure 1: An example of cable network configuration. Figure 2: Comparison of modal frequency spectrum between network cable and single cable. stiffness. [PERSON] et al. [20] and [PERSON] [21] first presented the effect of cable bending stiffness on the behavior of the cable network. The cables were modeled as axially tensioned beams with fixed-end boundary conditions in their cable network models. However, since the boundary conditions of cables are fixed-end, their models may not be appropriate to estimate the internal cable forces of an existing cable network where its boundary conditions are rarely fixed-ends. Hence, it is necessary to develop a method to identify cable forces of cable networks considering cable bending rigidity and actual boundary conditions. Many previous studies introduced methods to estimate the cable forces of a single cable with partly fixed boundary conditions. [PERSON] et al. [22], [PERSON] [23], and [PERSON] et al. [24] introduced a method to determine the cable forces of a cable stayed-bridge based on the effective vibration length with a two-frequency approach. The effective vibration length is the equivalent length that leads the dynamic behaviors of a cable with non-hinged-end supports to be similar to those of a cable with hinged-end supports. Thus, cable force can be obtained from a cable with hinged-end supports and the effective vibration length instead of the cable with partly fixed-end supports and the actual length. In the real application, [PERSON] et al. [22] suggested that the rubber-to-rubber length can be treated as the effective vibration length if multiple measurements are difficult to be conducted. When multiple measurements can be performed, [PERSON] et al. [22] suggested using the vibration signals at five sensor locations to determine the effective vibration length of the cables. However, their technique is appropriate for a single cable. [PERSON] et al. [25] studied the dynamic behaviors of two bending cable networks with hinged-end boundary conditions. They combined their analytical solution and the ideas of cable vibration effective length [22] to determine the natural frequencies and mode shapes of two bending cable networks with partly fixed-end boundary conditions, which are similar to those of actual cables. This study will follow the combined method promoted by [PERSON] et al. [25] to develop an inverse analysis to identify the cable forces of cable networks with actual boundary conditions, i.e., partly fixed ends, through measured frequencies. In conclusion, there are many techniques to determine the cable forces of single cables, but few researchers study methods to identify cable forces of cable networks while considering cable bending rigidity and arbitrary boundary conditions. Hence, this study will develop a technique to determine the cable forces of two-bending cable networks with actual boundary conditions without repeated removals of cross-ties. This method is necessary because it can preserve the accuracy of cable-force identification and reduce bridge health monitoring costs.