# Form Finding of Asymmetric Cable Network Reflector with Flexible Ring Truss
###### Abstract
However, with the increase in the aperture size, the ring truss should be lightweight to decrease the weight of the antenna structure, which could increase the deformation of the supporting structures. Hence, it is important to consider the flexibility of the supporting structures in the form-finding of the reflector. There have been several research works focusing on the design of cable network structures by considering the flexibility of the ring truss. [PERSON] et al. [23] extends the traditional force density method for the form-finding analysis of an asymmetric offset antenna. [PERSON] et al. [24] optimized the pre-stress distribution of the boundary cables connected directly to the supporting truss by considering the elastic deformation of the antenna structure. [PERSON] et al. [25] searched for the desired mesh configurations with an iterative strategy by incorporating the standard FEM (finite element method) with the traditional FDM (force density method). [PERSON] et al. [26] tried to minimize the reflector surface error by adjusting the tension-force distribution of the boundary cables and by considering deformations of the supporting truss. However, the above works all treated the reflectors as identical paraboloid-shaped nets. The advantage of the asymmetric paraboloid-shaped net will be shown in the following. Since only the reflective mesh is useful during the process of transmitting or receiving messages, the focal length-to-diameter ratio of the rear net can be reduced to minimize the total mass. Figure 2 shows the identical paraboloid-shaped nets model, where \(L\) is the depth of the reflector, \(H_{c}\) is the length between the front and rear reflectors, and \(R\) is the radius of the reflector. Using these notations, the depth of the truss \(H\) can be expressed as:  
\[H=H_{c}+2L \tag{1}\]  
An asymmetric paraboloid-shaped net model is shown in Figure 3. In the figure, \(H_{2}\) is the depth of the truss, \(H_{c}\) is the length between the front and rear reflectors, and \(R\) is the radius of the reflector. The depth of the front and rear reflectors are \(L\) and \(\frac{L}{5}\), respectively. The depth of the truss for the asymmetric paraboloid-shaped net model can be expressed as:  
\[H_{2}=H_{c}+\frac{6}{5}L \tag{2}\]  
From Equations (1) and (2), it can be noted that the depth of the truss for the asymmetric reflector structure is lower than the identical one. Hence, in this work, an asymmetric cable network reflector is designed to reduce the size of the hoop truss structure. The pre-stress distribution in cables has a significant effect on the deformation of the supporting structure. The aim of this work is to obtain an even distribution of pre-stress to  
Figure 2: Identical paraboloid-shaped net model. form a high-accuracy mesh reflector. In order to solve the non-linear equations, the force density method is utilized, which has been frequently used in the form finding of cable and membrane structures. The rest of the paper is organized as follows. The form finding for an asymmetrical cable network reflector using FDM and GA is presented in Section 2. In Section 3, the proposed form-finding method that involves considering the deformation of the ring truss is discussed. Numerical results are presented to illustrate the effectiveness of the proposed approach in Section 4. Some conclusions are summarized in Section 5.