# Form Finding of Asymmetric Cable Network Reflector with Flexible Ring Truss
###### Abstract  
In this work, the form finding of an asymmetric cable network reflector is considered by taking into account the elastic deformation of the flexible ring truss. Firstly, by treating the boundary nodes as fixed nodes, the balance equations of the structure are transformed into linear equations using the force density method (FDM). Next, the genetic algorithm (GA) is used to find the force density values that satisfy the surface accuracy required to complete the form finding of a cable network reflector. Secondly, a design procedure is presented by incorporating the elastic deformation of the ring truss. Finally, using an iterative process, the coordinates of the boundary nodes are modified to repeat the first step to obtain better surface accuracy. Numerical experiments are presented to demonstrate the effectiveness of the proposed method in comparison with the conventional method. flexible ring truss; asymmetric; force density method; genetic algorithm; boundary condition modification +
Footnote †: journal: Applied  
Recently, many researchers have considered the problem of form finding of deployable mesh reflectors. For example, [PERSON] et al. [8] constructed a small-scale physical model that deploys two identical cable nets (front and rear nets) to demonstrate that a 10 GHz reflector with a 3 m diameter and a focal length-to-diameter ratio of 0.4 can be packaged within an envelope that has a volume of \(~{}0.1~{}\times~{}0.2~{}\times~{}0.8~{}\mathrm{m^{3}}\). Further, [PERSON] et al. [9] used FDM to revisit and examine the tension truss concept in terms of adjustability to different support conditions and pre-stressability. In another study, an inverse iteration algorithm was used for the form-finding design of the creep and recovery behaviors of cable net reflector antennas [10]. A combination of a non-linear finite element method [11] and other optimization methods [12] was used to determine the form-finding process using iterative techniques. However, these methods have drawbacks in terms of slow convergence speed and poor accuracy. The surface accuracy of an antenna reflector impacts its electromagnetic performance [13; 14]. Hence, the design of a large antenna reflector with a high surface accuracy requirement has been the subject of much research [15]. The allowable random surface error in each frequency band was presented by [PERSON] et al. [16]. In the study, the authors assumed that error tolerance was 1/50 th of the wavelength (gain loss is \(-\)0.2 dB) and achieved a higher surface accuracy to obtain Ku-band antenna reflector performance. [PERSON] et al. [17] studied the surface error distribution, root-mean-square error, and far-field electrical performance of an antenna influenced by pillow distortion. Additionally, there are also many other active shape adjustment technologies that are used to improve the surface accuracy. [PERSON] et al. [18; 19] established a finite-element model of a cable net structure with piezoceramic (PZT) actuators and tried to find the desired shape by finding the optimal actuation voltages. [PERSON] et al. [20] proposed a novel self-sensing vibration control method and applied a self-sensing electromagnetic transducer that acts as both an isolator and a velocity sensor to suppress the vibration of a space antenna reflector. [PERSON] et al. [21] proposed  
Figure 1: An illustration of the AstroMesh concept. an active cable structure constructed by incorporating a piezoelectric (PZT) actuator into flexible cables to achieve the active shape control of the reflector surface. The deployable reflector that we studied in this work is a composite structure comprising the supporting truss and the flexible cable net. The supporting truss will be deformed under the pre-stress of the flexible cable net [22]. If the antenna aperture size is small, the deformation of the ring truss can be ignored because of the significant stiffness of the supporting structures.