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Abstract

Space antennas with high gain and high directivity are in great demand for future communication and observation applications. Deployable cable-mesh reflector antennas are required to be tensioned in a self-equilibrated state through form-finding design. In order to ensure the cable-mesh reflector antennas’ high performances, both surface accuracy requirements and tension uniformity should be considered in the form-finding design process. To effectively implement the form-finding design for asymmetric cable-mesh antennas, a two-step uniform-tension form-finding approach is presented. In step 1, with the cable tension and membrane stress being considered simultaneously, an iterative design technique which combines force density method and surface stress density method is presented. In step 2, considering the asymmetry between the rear and front nets, the nodal z-coordinates and cable tensions of the rear net are designed with the combination of the equilibrium matrix method and force density method. Finally, an offset AstroMesh antenna is designed using the proposed method. For the obtained antenna, the cable tension and membrane stress of the front net are completely uniform, and the maximum tension ratio of the rear cable net is 1.06, which are very satisfactory.

Keywords

Uniform-tension form-finding asymmetric cable-mesh antennas force density method surface stress density method

Introduction

Large cable-mesh deployable antennas are widely used for space applications.^1–5 The AstroMesh antenna, one popular kind of cable-mesh antennas, has the advantages of light mass, high packing efficiency, and good thermal stability. An AstroMesh antenna⁵ mainly consists of a deployable rim truss, two curved nets placed back-to-back across the truss, some vertical tension ties, and a reflective mesh stretched across the convex side of the front net.

AstroMesh antennas must be tensioned in equilibrated state to achieve on-orbit task, so form-finding design is indispensable. Moreover, on-orbit antennas are affected by time-varying thermal environment, and the stability of their surface shape is dependent on the tension uniformity of the cables. Therefore, study on AstroMesh antennas, which considers both surface accuracy requirements and tension uniformity, becomes a hot issue in recent years.^6–9 Currently, there are mainly two research approaches to address this problem: approach I, “tension-finding for given shape”^6,7 and approach II, “shape-finding for given tension.”⁸ Based on approach I, the initial tensions of AstroMesh antennas were designed with the geometry unchanged using the equilibrium matrix method (EMM) in Yang and Shi⁶ and Yang and Duan,⁷ but the maximum tension ratio is relatively large for asymmetric offset AstroMesh antennas. To improve the tension uniformity, employing force density method (FDM), an iterative form-finding technique was presented in Morterolle et al.,⁸ on the basis of approach II. In this study, cable tensions of the front and rear nets were designed to be completely uniform, under the assumption that the two nets were symmetric. However, in practical engineering, the front and rear nets of AstroMesh antennas are always asymmetric, which results in that the method in Morterolle et al.⁸ cannot implement the form-finding of the rear net. In addition, as the reflective mesh (taken as membrane structure) is an indispensable part of AstroMesh antennas, the influence of membrane stress on the stability of surface shape cannot be ignored. Thus, membrane stress uniformity of the AstroMesh should be considered in the form-finding design. Unfortunately, the existent form-finding methods for AstroMesh antennas only considered the tensions of the cables, with the stress of the reflective membranes not taken into account.

To implement the form-finding design of asymmetric AstroMesh antennas, a two-step uniform-tension form-finding method is presented with the combination of “tension-finding and shape-finding.” Step 1 is for the front net: to consider both cable tension and membrane stress simultaneously, an iterative design technique which combines FDM and surface stress density method (SSDM) is presented based on the works of Morterolle et al.⁸ and Maurin and Motro,¹⁰ through which both the cable tension and membrane stress are designed to be uniform. In step 2, the rear net of asymmetric AstroMesh antennas is considered. EMM and FDM are combined to implement the form-finding process of the rear net as follows: first, the horizontal components of cable tensions are designed using EMM; second, the cable force densities are determined according to the horizontal components of cable tensions and cable lengths; then, the nodal z-coordinates and cable tensions are obtained based on FDM.

Proposed form-finding method

In this section, equations and design process of the two-step form-finding method are presented. The proposed method is implemented under the following assumptions: (1) all the front net nodes should be on the ideal paraboloid to meet the reflector’s accuracy requirements, (2) there is no accuracy requirement on the rear net, and (3) all the vertical tension ties must be kept vertical throughout the two-step form-finding process.

Form-finding of the front net

The front net is a cable-membrane structure, of which the cables are taken as two-node liner elements and the meshes are taken as three-node triangular plane stress membrane elements.^10–12

As shown in Figure 1, a free node c of the front net is connected by $c_{s}$ cables and $c_{m}$ membranes, and it is forced by both cables and membranes. Cable element i connects nodes c and $k_{i}$ , and nodes c, a_j, and b_j are the vertexes of membrane element j. The resultant internal forces on node c by both the cables and membranes are expressed as

F_{c} = \sum N_{ci} + \sum P_{cj} = \sum_{i = 1}^{c_{s}} q_{li} \vec{c k_{i}} + \sum_{j = 1}^{c_{m}} \frac{t}{4} W_{j}^{2} q_{S_{j}} \vec{c h_{j}}

(1)

where $q_{l_{i}} = T_{l_{i}} / L_{l_{i}}$ is force density of the ith cable element; $q_{S_{j}} = σ_{0 j} / S_{j}$ is the surface stress density of membrane element j; $\vec{c k_{i}}$ is the vector connected from point c to point k_i; $\vec{c h_{j}}$ is the vector connected from point c to point h_j, where $\vec{c h_{j}}$ is orthogonal to the side a_jb_j and h_j is the pedal point; W_j is the length of side a_jb_j; t, $σ_{0 j}$ , and S_j are the thickness, stress, and area of element j, respectively.

Figure 1.

Internal forces at node c.

By expressing the vectors $\vec{c k_{i}}$ and $\vec{c h_{j}}$ as the coordinate differences between node c and node k_i or h_j in equation (1), the components of F _c can be written as

{\begin{matrix} F_{cx} = R_{cx} - D_{c} x_{c} \\ F_{cy} = R_{cy} - D_{c} y_{c} \\ F_{cz} = R_{cz} - D_{c} z_{c} \end{matrix}

(2)

where

{\begin{matrix} R_{cx} = \sum_{i = 1}^{c_{s}} q_{l_{i}} x_{k_{i}} + \sum_{j = 1}^{c_{m}} \frac{t}{4} W_{j}^{2} q_{S_{j}} x_{h_{j}} \\ R_{cy} = \sum_{i = 1}^{c_{s}} q_{l_{i}} y_{k_{i}} + \sum_{j = 1}^{c_{m}} \frac{t}{4} W_{j}^{2} q_{S_{j}} y_{h_{j}} \\ R_{cz} = \sum_{i = 1}^{c_{s}} q_{l_{i}} z_{k_{i}} + \sum_{j = 1}^{c_{m}} \frac{t}{4} W_{j}^{2} q_{S_{j}} z_{h_{j}} \\ D_{c} = \sum_{i = 1}^{c_{s}} q_{l_{i}} + \sum_{j = 1}^{c_{m}} \frac{t}{4} W_{j}^{2} q_{S_{j}} \end{matrix}

(3)

For the front net of AstroMesh antennas, there is no external loads at node c in x- and y-directions, so $F_{cx} = F_{cy} = 0$ . Consequently, x- and y-coordinates of node c can be figured out as

{\begin{matrix} x_{c} = \frac{R_{cx}}{D_{c}} \\ y_{c} = \frac{R_{cy}}{D_{c}} \end{matrix}

(4)

It should be noted that all the front net nodes should be on the ideal paraboloid to meet the reflector’s accuracy requirements, so the z-coordinate of node c must be modified according to the paraboloid equation in the form-finding process.⁸ Then, the required force in the vertical tension tie connected to node c can be calculated as

T_{v_{c}} = - (R_{cz} - D_{c} z_{c})

(5)

To obtain uniform cable tension and membrane stress, iterative strategy is adopted. Denoting the required uniform cable tension and membrane stress by T_u and $σ_{u}$ , the force density of cable i and the surface stress density of membrane j after the gth iteration step can be updated as

{\begin{matrix} q_{l_{i}}^{(g + 1)} = \frac{T_{u}}{L_{i}^{(g)}} \\ q_{s_{j}}^{(g + 1)} = \frac{σ_{u}}{S_{j}^{(g)}} \end{matrix}

(6)

where $L_{i}^{(g)}$ and $S_{j}^{(g)}$ are the length of cable i and the area of membrane j after the gth iteration step, respectively.

In the design process, force density and surface stress density of the front net cables and membranes are iterated according to equation (6) until it converges. The iteration design process of the front net is illustrated in Figure 2, and the detailed design process is as follows:

Step 1. Give the required uniform cable tension T_u and membrane stress $σ_{u}$ .

Step 2. Give the initial geometry topology and node coordinates of the front net.

Step 3. With the uniform cable tension T_u and membrane stress $σ_{u}$ exerted on the front net, calculate the x- and y-coordinates of the front net nodes through equation (4).

Step 4. According to the ideal paraboloid equation, calculate the z-coordinates of the front net nodes.

Step 5. Calculate the tensions of the vertical tension ties by equation (5).

Step 6. Update the node coordinates of the front net, and calculate the lengths and tensions of the front net cables.

Step 7. Judge whether the coordinates of the front net nodes in the equilibrium state do not vary in the iteration process. If yes, the iteration process has converged, and the current form is the required form under the given uniform cable tension and membrane stress. Otherwise, modify the force density and surface stress density by equation (6) and return to Step 3 until the iteration process converges.

Figure 2.

Iteration design flow chart of the front net.

Form-finding of the rear net

After the form-finding process in section “Form-finding of the front net,” the front net node coordinates $(x^{front}, y^{front}, z^{front})^{T}$ and the vertical cable tensions $T_{v}$ are determined. Considering the assumption that all the tension ties must be kept vertical, x- and y-coordinates of the rear net nodes are the same as those of the front ones, so we obtain that $(x^{rear}, y^{rear})^{T} = (x^{front}, y^{front})^{T}$ .

The equilibrium equations of the rear net node i in x- and y-directions are written as

\sum_{j} \frac{T_{ij}^{rear}}{L_{ij}^{rear}} (x_{i}^{rear} - x_{j}^{rear}) = f_{ix}^{rear}

(7)

\sum_{j} \frac{T_{ij}^{rear}}{L_{ij}^{rear}} (y_{i}^{rear} - y_{j}^{rear}) = f_{iy}^{rear}

(8)

where cable element ij connects node i and node j, and $T_{ij}$ and $L_{ij}$ are the tension and length of element ij. For free node i, one can get that $f_{i x}^{rear} = 0$ and $f_{i y}^{rear} = 0$ .

According to the geometric relationship shown in Figure 3, the relationship between the axial and horizontal components of the tension and length of cable ij can be obtained as

\frac{T_{ij}^{rear}}{L_{ij}^{rear}} = \frac{{T_{ij}^{rear}}_{xy}}{{L_{ij}^{rear}}_{xy}}

(9)

Figure 3.

The axial and horizontal components of the tension and length of cable ij.

Then, the equivalent equilibrium equations of (7) and (8) can be expressed as

\sum_{j} \frac{{T_{ij}^{rear}}_{xy}}{{L_{ij}^{rear}}_{xy}} (x_{i}^{rear} - x_{j}^{rear}) = f_{ix}^{rear}

(10)

\sum_{j} \frac{{T_{ij}^{rear}}_{xy}}{{L_{ij}^{rear}}_{xy}} (y_{i}^{rear} - y_{j}^{rear}) = f_{iy}^{rear}

(11)

where $T^{rear}$ and $L^{rear}$ are the horizontal components of $T_{ij}^{rear}$ and $L_{ij}^{rear}$ .

In the horizontal projection plane oxy, the geometry of the rear net is determined by the front one. According to equations (10) and (11), the equilibrium equations of all the rear net nodes in oxy plane can be written in matrix form as

B_{xy}^{rear} T_{xy}^{rear} = 0

(12)

where $B_{xy}^{rear}$ with dimension of $2 p \times m$ is the coefficient matrix, $T_{xy}^{rear}$ is horizontal components of the rear net cable tensions, p is the sum number of the rear net free nodes, and m is the sum number of the rear net cables.

For equation (12), the rank of matrix $B_{xy}^{rear}$ satisfies the condition that

r_{B} \leq \min {2 p, m}

(13)

There two cases for the solution of equation (12). If $r_{B} = m$ , there is only zero solution for equation (12); else if $r_{B} < m$ , there will be multiple solutions for equation (12).

For cable-mesh antennas structures, it is always satisfied that

r_{B} \leq 2 p < m

(14)

Thus, there are always multiple solutions for equation (12), and it can be solved through optimization method to make the cable tensions of the rear net as uniform as possible. The optimization model is established⁷ as

Find T_{xy}^{rear} = {(T_{1 xy}^{rear}, T_{2 xy}^{rear}, \dots, T_{m xy}^{rear})}^{T}

(15)

Min f = ‖ T_{xy}^{rear} - {T_{xy}^{rear}}^{(0)} ‖^{2}

(16)

S . t . B_{xy}^{rear} T_{xy}^{rear} = 0

(17)

a^{T} T_{xy}^{rear} = {\bar{T}}_{xy}^{rear}

(18)

T_{j xy}^{rear} > 0, j = 1 ~ m

(19)

where $T_{xy}^{(0)} = ({\bar{T}}_{xy}^{rear}, {\bar{T}}_{xy}^{rear}, \dots, {\bar{T}}_{xy}^{rear})_{m \times 1}^{T}$ , $a = (1 / m, 1 / m, \dots, 1 / m)_{m \times 1}^{T}$ , and ${\bar{T}}_{xy}^{rear}$ is a given mean value of $T_{xy}^{rear}$ .

The optimum solution $T_{xy}$ of the optimization model can be obtained using minimum norm method.⁷ As equation (12) is homogeneous linear equation, horizontal components of the rear net cable tensions can be expressed as

T_{xy}^{rear} = γ T_{xy}^{rear (*)}

(20)

where $γ (γ > 0)$ is a scaling factor, which affects the height and nodal z-coordinates of the rear net and can be determined through the height requirement of the rear net.

According to FDM, when cable force densities are given, z-coordinates of the rear net free nodes can be obtained as¹³

z_{f}^{rear} = {(C_{f}^{T} Q^{rear} C_{f})}^{- 1} (f_{z}^{rear} - C_{f}^{T} Q^{rear} C_{b} z_{b}^{rear})

(21)

where $z_{f}^{rear}$ and $z_{b}^{rear}$ are the z-coordinates of free and fixed nodes of the rear net; $C_{f}$ and $C_{b}$ are the topology matrixes; $Q^{rear} = diag (q^{rear})$ is the force density matrix and $q^{rear}$ is the force density vector of rear net cables; and $f_{z}^{rear} = T_{v}$ .

To calculate $z_{f}^{rear}$ , $Q^{rear}$ should be pre-determined. According to the geometric relationship shown in Figure 3, the force density of element ij can be obtained from the horizontal components of its tension and length, it holds that

q_{ij}^{rear} = \frac{T_{ij}^{rear}}{L_{ij}^{rear}} = \frac{T_{ij xy}^{rear}}{L_{ij xy}^{rear}} = γ \frac{T_{ij xy}^{rear (*)}}{L_{ij xy}^{rear}} = γ q_{ij}^{rear (*)}

(22)

where $q_{ij}^{rear}$ is the force density of element ij, and $T^{rear}$ and $L^{rear}$ are the horizontal components of tension and length of element ij.

Thus, one can get $Q^{rear} = γ Q^{rear (*)}$ . Then, equation (21) is written as

z_{f}^{rear} = \frac{1}{γ} {(C_{f}^{T} Q^{rear (*)} C_{f})}^{- 1} f_{z}^{rear} - {(C_{f}^{T} Q^{rear (*)} C_{f})}^{- 1} C_{f}^{T} Q^{rear (*)} C_{b} z_{b}^{rear}

(23)

Since ${T^{rear}}_{xy}^{(*)}$ is obtained from the optimization solution of (12) and $L_{xy}^{rear}$ is determined by the geometry of the front net, $Q^{rear (*)}$ can be figured out through (22). Therefore, $z_{f}^{rear}$ is a monotonous function regarding to $γ$ .

In practical engineering, although there is no accuracy requirement on the rear net, AstroMesh antennas always have height restrictions to reduce stowed volume, and z-coordinates of the rear net nodes are constrained by the antenna height. Here, the height of the rear net $H^{rear}$ is required to satisfy $H^{rear} \leq {\bar{H}}^{rear}$ , where ${\bar{H}}^{rear}$ is the maximum allowable value of $H^{rear}$ . Moreover, $H^{rear}$ is expressed as

H^{rear} = \max (z_{f}^{rear}) - z_{b}^{rear}

(24)

Obviously, $H^{rear}$ is also a monotonous function of $γ$ . Thus, $γ$ can be figured out through the condition that $H^{rear} = {\bar{H}}^{rear}$ .

Correspondingly, according to the obtained rear net node coordinates and force densities, cable tension vector $T^{rear}$ is calculated as

T^{rear} = Q^{rear} L^{rear}

(25)

where $L^{rear}$ is the vector of cable lengths in the equilibrium state.

Numerical simulations

An offset AstroMesh antenna is employed to demonstrate the effectiveness of the proposed method. The initial geometry configuration is shown in Figures 4(a) and 5(a), whose specifications are as follows: the optical aperture is 10 m, the focal length and offset distance of the front net are both 6 m, and the initial rear net is symmetric with the front one. In this example, for the front net, T_u is 40 N and $σ_{u}$ is 50 KPa; for the rear net, the height condition is $H^{rear} \leq 0.375 m$ , which is not satisfied for the initial geometry configuration in Figure 5(a).

Figure 4.

Comparison of the top view of the AstroMesh antenna (a) before and (b) after form-finding design.

Figure 5.

Comparison of the front view of the AstroMesh antenna (a) before and (b) after form-finding design.

First, the form-finding design of the front net is implemented using the proposed iteration method. The top views of the AstroMesh antenna before and after form-finding design are compared in Figure 4. It can be seen that x- and y-coordinates of the front net nodes deviate from the initial ones to achieve the uniform cable tension (40 N) and membrane stress (50 KPa). Meanwhile, the maximum and minimum tensions of the vertical cables are 12.15 and 6.52 N, respectively.

Then, according to the results of the front net, nodal z-coordinates and cable tensions of the rear net are designed. In this example, ${\bar{T}}_{xy}^{rear} = 1$ is set to get $T_{xy}$ , and ${\bar{H}}^{rear} = 0.375 m$ is given to restrict the antenna height. The front view of the designed AstroMesh antenna is compared with the initial one in Figure 5. After the form-finding design, $H^{rear} = 0.375 m$ is achieved with γ = 96.89, so we can obtain that the height of rear net is greatly reduced compared with the initial height of 0.926 m. The maximum and minimum tensions of the rear net cables are 106.42 and 100.87 N, respectively. The maximum tension ratio of the rear net cables is 1.06, not equal to 1. This is because there is no membrane on the rear net and the two cable nets are asymmetric. The detailed information of cable tension distribution is shown in Table 1.

Table 1.

Cable tension distribution after form-finding design.

Items	Tension values (N)		Maximum/minimum
Items	Minimum	Maximum	Maximum/minimum
Front net	40	40	1.00
Rear net	100.87	106.42	1.06
Tension ties	6.52	12.15	1.86

Conclusion

In this study, a uniform-tension form-finding method is presented for AstroMesh antennas. First, the front net is designed iteratively with the combination of FDM and SSDM, by which the front net is obtained with uniform cable tension and membrane stress. Then, according to the results of the front net and the relationship between the front and rear nets, form-finding of the rear net is implemented through the combination of EMM and FDM, from which the obtained tension uniformity of the rear net is satisfactory and the height of the rear net is effectively reduced. Numerical simulations validate the effectiveness of the proposed method. In this article, the proposed method is focused on the initial form-finding design of the cable-mesh antennas, in which the uncertain errors in the actual manufacturing process are not taken into consideration. There are always two major aspects of uncertain error sources: one is the manufacturing error, and the other is the assemble error. Thus, the influence of the uncertain errors on the reflector surface accuracy and cable tensions will be further studied in our future work.

Footnotes

Academic Editor: Filippo Berto

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National 973 Program under Grant No. 2015 CB857100 and the National Natural Science Foundation of China under Grant Nos 51490660 and 51475349.

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