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.


INTRODUCTION
Large composite space antennas are launched into space in a folded state and deployed under conditions of vacuum and zero gravity.
The design and calculation of folding umbrella-type antennas are discussed in the book [1]. An overview of modern transformable antenna designs is presented in [2,3]. Application of the finite element method and commercial software systems for numerical modeling of statics and dynamics of large space antennas is considered in [4][5][6][7][8]. Nonlinear equations of the dynamics of the deployment of a flat system of flexible inextensible rods connected in series by elastic hinges with stops, folded in the initial state into a packing, were obtained in [9,10], where analytical expressions are given in the form of equations for all coefficients of equations in generalized coordinates.
The predetermined deformed shape of the antenna after deployment and damping of vibrations is obtained when designing a structure as a statically indeterminate system by determining the internal forces required for stable equilibrium of this form of force. For this, it is required to solve the inverse geometrically nonlinear problem of deformation of the system with constraints and to determine the necessary geometric and stiffness parameters of some of its adjustable elements. In [11], an algorithm was developed for the numerical solution of a geometrically nonlinear problem of the space umbrella antenna shaping in the form of a shallow parabolic surface of revolution with cyclically symmetrically arranged flexible radial rods connected in parallel by tensile tether elements. In this paper, this approach is generalized for an antenna with a non-sloping surface of revolution. In this case, when solving the problem of shaping due to deformation of flexible radial rods, their strong bending with large displacements and angles of rotation of the elements is considered.

STATEMENT OF THE PROBLEM
The diagram of the proposed cyclically symmetric space antenna of the umbrella type with n radial rods 1, each of which consists of m links, is shown in Fig.1 in the folded state (a) and in the final deformed state (b). The opening and shaping of the antenna is as follows. On the signal, the holding connection between the body of the damping hydraulic cylinder 2 and the rod 3 is removed and a slow movement of the rod begins under the action of the pre-compressed springs 4, due to which, with the help of cables 5 and levers 6, the packing of rods 1 is rotated in radial planes. With some deviation of the packages, the bonds 7 are broken and the multi-link rods with elastic hinged joints are deployed with fixation on the stops in a straight-line position. It is believed that in some given deflected position (1', Fig. 1b) of the straightened rods (at an angle between the X axis and the rod axis) due to the choice of the initial lengths of the cable sections 8 connecting in the parallel planes the corresponding nodes k = 1, 2, …, m rods, these sections become rectilinear (without sagging), but still not tensioned. In this case, all cables k = 1, 2, …, m will have the form of regular n-gons. It should be noted that after fast (dynamic) opening of the packs of elastically connected links, each of the n rods will become rectilinear in a deflected angle position with some spread in time and angles . The deflections of the rods at a slow (quasi-static) stroke of the rod 3 of the damping hydraulic cylinder will be leveled out under the action of the cables connecting them k = 1, 2, …, m, since in all n elements of each of these cables the forces must be the same with the cyclic symmetry of the system.
When turning the levers 6 to the stop, the cables will stretch, and the rods will bend in the radial planes under the action of the reactions of the cables at the nodes k = 1, 2, …, m. The final curved shape 1 of the rod with stretched cables 8 is shown in Fig. 1b. The antenna canvas is connected to the rods at separate points, including the nodes; when folded, it is located in the space between the packs of the rods. When opening and shaping the composite structure of the antenna, the reactions of the canvas are not taken into account.  The axisymmetric surface of the antenna has a given shape , Fig. 1b. When designing and calculating a compound cyclic symmetric antenna structure, it is required that the coordinates of the nodal points k = 1, 2, …, m of curved radial rods coincide with the corresponding coordinates of the antenna surface , , k = 0, 1, …, m. The central section C-0 after turning the lever 6 to the stop is considered absolutely rigid and profiled in the form ; at point C and , , at point 0. The elastic part of the rod consists of links of approximately the same length , k = 1, 2, …, m (for the convenience of folding them into packings). For given values of a k , the coordinates of the nodes of the curved inextensible rod X k , are calculated sequentially from the relations using the iteration method. Flexural rigidity of each link within its length a k is considered constant.
To describe the strong bending of the rod, each of its links k = 1, 2, …, m will be considered as a finite element (FE) in the local coordinate system xy, which at the node k -1 is rigidly connected with the (k -1)th FE, Fig.2. Coordinates and the angle of inclination of the rod axis at the node k -1 are equal to , , . The relative transverse displacement and the rotation angle of the k-th FE as a cantilever rod of length a k are written in the form [11] (2.1) where the prime denotes the derivative with respect to the x coordinate. Longitudinal displacement of the FE end, due to strong bending provided that the rod is inextensible ( = 0), taking into account (2.1), is written in the form The displacements and angles of rotation at the nodes of the rod k = 1, 2, …, m are determined as follows The variation of the work of forces R k on displacements ( Fig. 2) is written in the form (2.5) where, taking into account (2.2) and (2.4), the following notations are introduced: The double and triple sums in (2.5) transform as Then we obtain where is the Kronecker symbol ( for , for k = m).
The equilibrium equations of a curved rod in generalized coordinates and , k = 1, 2, …, m, are obtained on the basis of the principle of possible displacements and, taking into account (2.3), and (2.6), are written in the form Let us reduce Eqs. (2.7) to dimensionless form. For this, we introduce the following notation: , , where the stiffness matrix G and nonlinear matrices A and B of order have the form: 3. ALGORITHM FOR SOLVING THE PROBLEM The shaping of the surface of revolution (or, in this case, a curved radial rod of the frame, on which a soft shell is superimposed with some tension) according to the equation can be carried out within the considered elements by choosing the reactions of tensioned cables located in the planes of parallels, or due to bending stiffnesses of the elements of radial rods, or by changing both at the same time. The solution to such a nonlinear problem is not unique, and in some areas of variation of the indicated parameters, it may not exist. Here, we consider the case when the equation of a given form is satisfied at points k = 1, 2, …, m due to the controlled reactions of the cables only along the coordinates and , and the tilt angles at these points remain free, i.e. condition is not required. The stiffnesses of the elements of the bending rod are considered given. The equivalent displacement of the curved rod, which exactly corresponds to the given shape along the X k , coordinates, is determined from the kinematic relations: is a vector (the right side of Eq. (3.2)) depending on , , …, . In the initial (zero) approximation at r = 0 for the angles we will use the exact values of , corresponding to the given shape of the curved rod at points k, which are defined as at , .
After obtaining a solution converging with a given accuracy for the vector , the vector of the reactions of the cables is calculated from the equation that is obtained from system (2.10) by eliminating the vector : In the initial state, after the system is opened, the radial rod 1' is rectilinear and deflected with respect to the X axis by a certain angle β 0 , Fig. 1b. The coordinates of the point of the rod in the initial state are equal to , where is the distance between the hinge connecting the hydraulic cylinder 2 with the pivot arm 6 (Fig. 1a), and point . The length of the k-th cable in the form of a regular n-gon connecting the k-th points of all n radial rods in parallel is chosen so that in the initial state it is unstretched and has no sagging. After turning the bracket 6 to the stop (the angle of inclination of the rod at the root point 0 becomes equal to ), the k-th cable will stretch with a force , where is the cross-sectional area, is the tensile stress in the -th cable. The relative elongation of the k-th cable in the final tensioned state will be where is the known coordinate of node k in the final deformed state (i.e., the coordinate of a given shape), and is the coordinate of node in the initial undeformed position of the rod. The reaction of the k-th cable at the k-th node of the rod is determined from the equilibrium equation of the node (3.5) In the case of linear elastic deformation of the cables at and at , where is the modulus of elasticity of the cable material. From the found values of R k , it is possible to determine the required areas F k or tensile stiffness of the cables to obtain the shape of curved radial rods.

STRONG BENDING OF THE CANTILEVER ROD
To verify the developed FE-model of strong bending of a rod described by Eqs. (2.9), let us consider a rod of length l with constant bending stiffness EI, which is fixed at the end s = 0 and loaded at the end s = l by a longitudinal compressive force exceeding the critical loss force stability (Euler elastica). The differential equations of bending of an inextensible rod (Fig. 3) are written in the form (4.1) where Initial conditions for s = 0: , , . Comparisons of solutions of differential equations (4.1) with solutions of the system of algebraic equations (2.9) for the FE model for q three different values of the parameter : 2.5, 3.0, 3.5. We used the following values of the initial (zero) approximations for "zeroing in" the numerical solutions of Eqs. (4.1) by the method of successive approximations in combination with the Adams method: and at ; and at ; and at (two initial values of for one lead to the same result). The results of solutions converging with an accuracy of 10 -9 for the rotation angles at points are given in Table 1 in lines I.   (2.9) show that the FE model of strong bending of the rod has a sufficiently high accuracy.

CALCULATION EXAMPLES
Let us consider a parabolic antenna with a shape at three values of the parameter : 1/15, 1/10, 1/5; the number of radial rods n = 24. The angles of inclination of the rods in the initial rectilinear position (Fig. 1b), in which the sections of the cables connecting the k-th nodes of the rods are also rectilinear and non-tensioned, are taken equal to ° at and , = 70° for . The length of the central non-deformable section is m ( ). The maximum antenna diameter is about 20 m.
Calculations of the strong bending of radial antenna rods under the action of reactions of stretched cables in the final deformed state of a cyclically symmetric system are performed. The results were obtained by solving nonlinear algebraic equations (3.2), and (3.3) by the method of successive approximations with an accuracy of 10 -9 relative to unknowns , taking into account the coupling conditions (3.1), which ensure the exact fulfillment of the given antenna shape at the nodes along the coordinates ,  , k = 1, 2, …, m. In this case, the values of of a given shape were used as initial approximations for , i.e., . Two design cases are considered for each of the 3 antennas differing in the value of : (1) The radial rod of the antenna is divided into 4 elastic links (m = 4, k = 1, 2, 3, 4) m long; given bending stiffnesses of these links , Pa · m 4 and design angles of inclination at nodes of a given shape are presented in Table 2. The shapes of the radial rod in the initial undeformed state (dashed line) and in the required final deformable state (solid line) for m = 4 at are shown in Fig. 5a-5c, respectively, in coordinates X, Y, m.
(2) The radial rod is divided into 6 elastic links ( , k = 1, 2, …, 6) m long; for this case, the values of , Pa · m 4 and angles are presented in Table 3. The shapes of the radial rod in the initial and required final states at are shown in coordinates X, Y, m in Figs. 6a, b, and c, respectively.
The obtained values of , , N, as well as the relative tensile deformations of the cables and the cable stiffness in tension (Pa · m 2 ) required to obtain a given antenna shape, which are determined by the equation , are given in Table 2 and Table 3, respectively, for cases m = 4 and m = 6, . Table 4 and Table 5 for the cases m = 4 and m = 6, respectively, compare the accuracy of the solution by comparing the bending moments , N · m, , at the nodes of the rod in the final

CONCLUSIONS
A constructive scheme is proposed and a mathematical model is developed for calculating the formation of a cyclically symmetric space umbrella antenna formed by a system of flexible multi-link radial rods connected in parallel at certain nodal points by tensile cables. The bending of the rods in the radial plane, taking into account the reactions of the cables, is created by turning the root parts of the rods by the force   The model of strong bending of an inextensible antenna rod is built using "cantilever" finite elements (links) that allow large displacements and turns as rigid bodies and relative elastic displacements with a two-term approximation along the length of the element, taking into account its shrinkage due to bending in a quadratic approximation. The obtained nonlinear equations are solved by the method of successive approximations with respect to the unknown reactions of the cables at the given values of the bending stiffnesses of the rod elements and the given radial and axial coordinates of the nodes, which exactly correspond to the shape of the simulated antenna. Based on the found values of the reactions of the cables, taking into account their known displacements in the nodes, the tensile stiffnesses of the cables necessary to ensure the given antenna shape are determined.
The accuracy of the developed model is estimated by comparison with the numerical solution of the problem of strong bending of a cantilever rod under the action of a longitudinal compressive force exceeding the critical buckling force of the rod.
As an example, calculations were performed for a parabolic antenna with n = 24 radial rods and with m = 4 and m = 8 of their constituent links (respectively, cables).

FUNDING
The research was carried out within the framework of the state assignment of the Institute of Applied Mechanics of RAS (state registration number of the topic AAAA-A19-119012290118-3).

OPEN ACCESS
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