The influence of gravity load on the accuracy of mesh reflector antenna

Space large deployable mesh antennas are the focus of research in the aerospace field. There is a high requirement for the reflector surface accuracy of the mesh antenna in orbit. Due to the low stiffness of the truss cable net system, the deformation of the flexible mesh surface under the ground gravity load cannot be ignored. Therefore, it is the key technology to ensure the accuracy of on orbit cable network reflector that accurately predict the difference of reflector under gravity load and microgravity load. This study establishes the mechanical model of the cable-truss system. The Newton Iterative Method is used to solve the nodal nonlinear equilibrium equation in the gravity load, and the deformation of the antenna mesh surface under the gravity load is obtained.


Introduction
In recent years, the development of large deployable mesh reflector antennas [1,2] , represented by ring truss mesh antenna, has received great attention of aerospace researchers from all over the world. And it is regarded as the key technology to improve the level of communication satellites and earth observation satellites. As shown in Fig. 1, the ring truss mesh reflector is composed of cable network system (including front cable network, rear cable network and tie force cable) and truss. The metal mesh is attached to the front cable network to reflect electromagnetic signals, there is a high accuracy requirement for the cable network reflector in orbit microgravity environment [1] . The processing, assembly, adjustment and test of the antenna are all under the ground gravity environment [3] , the stiffness of the truss cable net system along the vertical direction and the radial direction of the ring truss is low, and the deformation of the whole reflector caused by the gravity of the metal net and the cable net cannot be ignored [4,5] . Therefore, how to eliminate the influence of gravity and ensure the on-orbit microgravity surface accuracy of the cable network has become a major problem in the process adjustment and surface accuracy measurement stage of large cable network. . Typical structure of a mesh reflector This study intends to establish the mechanical model of the cable-truss system in the gravity environment. Based on the result of the geometric method of form find the cable network, the Newton Iterative Method is used to solve the balanced position of the node cable network in the gravity load. The rest of this Note is organized in the following way. Section 2 establishes the mechanical model of cable net in gravity environment by analyzing the geometric and mechanical characteristics of cable truss structure, and the Newton iteration method is proposed to solve the position of the nodes under the gravity load. Section 3 applies this method in gravity form problem problems for frontfed mesh reflectors. And Section 4 is the summary of the paper.

Compatibility equation
When applied gravity load, the displacements of node i and node j are   Therefore, the change of the rope length is,

Equilibrium equations
The expressions of cosine cosθ x , cosθ y and cosθ z of the angles between the cable and the axes of the design coordinate system are [7] :   (15) Where C is the set of connection topologies of the cable-truss system. Since the Newton method recalculates the Jacobian matrix according after each iteration, the calculation cost of this process is relatively large for cable-truss antennas with a large number of cables and trusses. In order to improve the calculation efficiency, you can use the initial Instead of the Jacobi matrix of the nth iteration, this is the simplified Newton method. The initial Jacobian matrix can be used to replace the Jacobian matrix of the n-th iteration, which is the simplified Newton method.

Gravity load deformation analysis example of Mesh Reflectors
The case in this article takes the result of geometric method form finding [6] as input to calculate the gravitational load deformation. The frontfed mesh reflectors has a circular aperture of D =12 m and a focal length of f= 7.5 m. The reflection surface is meshed with 30 border nodes and 121 inner nodes on the paraboloid. In engineerin, in order to reduce the height of the cable net antenna, the arch height of the rear cable network is usually reduced, as shown in Fig.2, the front and rear cable network are not completely symmetrical. The commonly method to reduce the arch height of the rear cable network is to extend the focal length of the paraboloid of the rear cable network. In this example, the arch-to-height ratio of the front and rear cable network is 4, that is, the rear cable network is a standard parabola with a focal length of 30m. The same geometric method is used to calculate the balance pretension of rear cable network, the front and rear network balance pretension is shown in the Fig.3. The cable network form-finding design is the tension in the orbit without gravity, and the original length of the rope is calculated by substituting the above material parameters. The original length of the cables is calculated by substituting the above material parameters. The simplified Newton Iterative Method is used to solve the cable net deformation under gravity load, and the cable tension of form finding design is used as the initial iterative tension. The relationship between the number of iterations and the maximum resultant force of cable net nodes is shown in Fig.4. The iteration termination condition is that the maximum resultant force of nodes is 10 -6 N. Solve the displacements of the nodes when the reflector port of the front network is placed upward under the gravity load. Take the ring truss plane of the front network as the xy plane, the ring center as the origin, the line connecting the ring center and the 122 nodes as the positive x direction, and the direction of the aperture surface perpendicular to the focal length is the z axis to establish a Cartesian coordinates,. The displacement of the front cable network nodes under gravity load is shown in Fig. 5. The displacement in the xy direction of the node is small, less than ±1mm, and the displacement in the z direction of the node is larger, the average displacement of the node the z direction is 4mm, and the maximum displacement reaches 5.35mm.

Conclusions
In this paper, by analyzing the geometric nonlinear geometrical nonlinear characteristics of the space cable truss system, the mechanical nonlinear model of deployable cable truss antenna under gravity load is established, The simplified Newton iterative algorithm is used to solve the displacement of the node. For the frontfed cable network form-finding design case, the displacement of the nodes is solved when the front cable network is placed upward under the gravity load. After calculation, under gravity load mainly along the z direction, and the displacement in the xy direction of the node is small. This article provides a method for solving the deformation of the cable-truss system subjected to external force loads.