Calculation of the Shaping of a Space Umbrella Antenna During Strong Bending of Radial Rods Connected Along Parallels by Tensile Cables

— A cyclically symmetric umbrella antenna is considered, the frame of which consists of flexible inextensible radial rods connected in nodes along parallels by tensile cables. In the initial transport position, the multilink rods are packed in packages oriented in the direction of the system axis. After the packing ties are removed, the rods are deployed in radial planes under the action of elastic springs connecting the links, and are fixed in rectilinear positions at a given angle with respect to the axis, at which all cables connecting the same type of rod nodes take the form of regular polygons, while remaining loose. Further, under the action of the force of a damping hydraulic cylinder with pre-compressed springs, the root parts of all rods are slowly turned to the stops. In the final position, the radial rods, connected at the nodes by tensioned cables, take a curved shape. The tensile stiffnesses of the cables are determined so that the radial and axial coordinates of the nodes of the curved rods coincide with the coordinates of the points of the given surface of revolution. A model of strong bending of a flexible inextensible rod is constructed taking into account the unknown radial reactions of tensioned cables acting on it at the nodes. The links of the rod are considered as “cantilever” elements connected in series at the nodes in local coordinate systems, which can make large displacements and turns. The bending of each element is described by two specified functions, the shrinkage of the element due to bending is taken into account in a quadratic approximation. The obtained nonlinear deformation equations of the system, taking into account the geometric connections at the nodes, are solved by the method of successive approximations with respect to the unknown reactions of the cables. The obtained values of the reactions are then used to determine the required tensile stiffness of the cables at the given coordinates of the nodes. As an example of the calculation, a parabolic antenna is considered for various numbers of radial rods and components of links. The estimates of the accuracy of the proposed computational model of the antenna shaping are carried out.