Mechanics Design of Conical Spiral Structure for Flexible Coilable Antenna Array

Limited by the e ﬀ ective launch capacity of a rocket, the deployable antenna is very important in the design of spaceborne antenna array. Compared to traditional deployable antenna, ﬂ exible coilable antenna array has higher surface precision and better vibration control and therefore is more suitable for high frequency communication. In order to minimize the weight of satellite and reduce cost of its launch, a design guideline to the geometry parameters of ﬂ exible coilable antenna array is crucial. Existing models cannot be directly applied to interaction and large deformation between coilable membrane and conical spiral antenna in the ﬂ exible coilable antenna array. Hence, the geometry parameters of the conical spiral structure and the thickness of the coilable membrane in the ﬂ exible coilable antenna array have not been optimized yet. In this paper, the interaction between the coilable membrane and the concial spiral antenna is analyzed in the antenna array. A concise formula is derived to predict the critical force that ﬂ attens the conical spiral antenna by a coiling scroll. Combined with a theoretical model to predict the deformation of the membrane, the model provides an important theoretical support for the lightweight design and mechanical design of ﬂ exible coilable antenna array, such as the thickness of the coilable membrane. The proposed design is validated by experiments. The above ﬁ ndings have potential applications in the e ﬀ ective reduction of antenna array weight and satellite launch costs.


Introduction
Antenna is an important terminal in information transmission system [1][2][3][4]. Limited by the effective launching capacity of the rocket, the existing large-scale spaceborne antenna array is usually folded to reduce the space occupied at the launching stage [5][6][7][8] and then deployed in orbit and responsible for communication transmission. For example, tensegrity-membrane and umbrella antenna have been put into use on the satellite [9][10][11]. Due to folding traces on the antenna surface, these folded antennas have problems such as large overall weight and poor shape accuracy [12][13][14][15]. Hence, it is difficult to apply in highfrequency communication which requires strict control of membrane deformation and vibration [16][17][18]. Inspired by serpentine design with low strain in two-dimensional (2D) flexible electronics [19][20][21][22], flexible coilable antenna array with 2D patch antennas can overcome the above problems [23]. However, the 2D patch antenna behaving linear polarization cannot handle the ionosphere of the atmosphere compared with 3D spiral antennas behaving circular polarization [24]. Hence, we propose a new concept of a spaceborne antenna array, i.e., a flexible coilable array with resilient spiral antennas which behave linear polarization and has a low strain under the compression, as shown in Figure 1. This coilable array has advantages such as high shape accuracy and deployment and small impact and vibration, which has not been used on satellites at present.
When the flexible coilable antenna array is coiling, the conical spiral antenna would be compressed and combined with coilable membrane. There may be two extremely unfavorable deformation conditions at this time. When the thickness of the coilable membrane is too large, the membrane is too rigid to roll and also overweight. On the other hand, when the thickness of the membrane is too small, the membrane is too compliable to compress conical spiral antennas on the surface of the scroll. Therefore, a theoretical model is essential to determine the minimum thickness required for the coilable membrane to fully compress the conical spiral antenna and to achieve lightweight design of membrane to optimize the launching cost. However, during the coiling process, the conical spiral antenna from the outer ring to the inner ring gradually contacts the bottom surface of membrane. The conical spiral antenna has a complex, large deformation and displacement, exhibiting significant nonlinearity [25][26][27][28]. It remains challenging to model the mechanics design of conical spiral antenna for flexible coilable antenna array. The widely used close-coiled spring theory and sparse-coiled spring theory are only suitable for small deformation in the linear elastic regime [29][30][31][32][33]. In addition, there are some nonlinear theories in consideration of the change of the helix angle [34][35][36][37]. However, the constraint of the underlying membrane on the deformation of the spring structure is not considered, such that these models cannot be applied to the coiling process. Finite element models were adopted to obtain the deformation process of a general conical spiral antenna [38][39][40][41] but could not lead to any analytical solution or scaling law [42][43][44]. It is of great significance to the lightweight design and mechanical design of the flexible coilable antenna array. This paper is arranged as follows: the coiling behavior of the flexible coilable antenna array with conical spiral antennas is studied. The deformation between the coilable membrane and the spiral antenna is deduced during the coiling process. The finite element method and the phototype are used to verify the accuracy of the theoretical derivation. This provides modeling and numerical support and theoretical guidance for the design of conical spiral antenna and the selection of membrane thickness.

Structural Model of the Flexible Coilable
Antenna Array Figure 1 shows a schematic diagram of flexible coilable antenna array with three-dimensional (3D) conical spiral antennas during the coiling process. The coilable antenna array is mainly composed of a feed, a flexible coilable membrane, and several conical spiral antennas. A large number of conical spiral antennas are uniformly distributed on the coilable membrane. One end of the conical spiral antenna is fixedly connected with the coilable membrane, and the other end is traction-free. The membrane is rolled up by a drive motor to compress the coilable antenna array and press it on the scroll, as shown in Figure 1(b). After the coilable antenna array enters its orbit, the coilable membrane rotates in the opposite direction and unfolds such that the antennae return to the original 3D state due to elasticity, as shown in Figure 1(d).
For ease of analyzing the mechanical behavior of the coilable antenna array, this antenna model would be simplified as coilable membrane with conical spiral antenna. During the coiling process, the straight portion of the coilable membrane (see Figure 1(d)) has rigid-body displacement in the horizontal direction without deformation. The coiled portion presses and deforms the spiral antenna. The conical spiral antenna is a tubular structure fabricated by highly elastic material. The inner cavity of the conical spiral tube (typical diameter is greater than 1 mm) is arranged with a metal wire as a radiating element (typical diameter is less than 0.2 mm). The mechanical behavior of the spiral tube is considered at this time since the metal wire is thin.
At the moment of contact, the height H of the antenna and the radius R r of the scroll are related geometrically via (see Figure 2(a)) To ensure that the conical spiral antenna can be flattened rather than tipped during the coiling process, the contract angle α between the antenna and the membrane should be close to zero for a vertical pressing. Given the height of the conical spiral antenna, this requires the radius of the scroll to be sufficiently large, so that the height of the conical spiral antenna is much smaller than the radius of the scroll. Therefore, the interaction between the coilable membrane and one conical spiral antenna can be regarded as the vertical force applied by two planar membranes.
Figures 2(b) and 2(c) show the cross section of the conical spiral antenna, which define geometric parameters of the flexible coilable antenna array. Thickness of the coilable membrane is defined as b m . R r represents the radius of the scroll. Pitch of the conical spiral antenna is defined as t. R n and R 1 are the upper and lower radius of the conical spiral antenna. From the biggest layer to the smallest layer for the radius of the conical spiral antenna, it numbers as 1, 2, i, and n, successively. d is the diameter of the conical spiral antenna. b is the ratio of the inner and outer diameter of the conical spiral antenna. L is the total length of the conical spiral antenna, which is equal to nπðR 1 + R n Þ. The radius of the ith layer is

Mechanical Behavior of Conical Spiral Antenna
For the spiral antenna to be fully flattened in the coiling process, the membrane rigidity needs to be much larger than the antenna rigidity, such that the membrane deformation is much smaller than the antenna deformation. Therefore, the mechanical behaviors of the conical spiral antenna can be analyzed without consideration of the membrane Because the conical spiral antenna has a small helix angle, the deformation is dominated by torsion. It is different for the force required to flatten each layer of the conical spiral antenna. Initially, for a small deformation, the displacement of the antenna at the contact position, induced by the contact force, can be obtained analytically as where a polar angle θ of the conical spiral antenna goes downward from the smallest to the biggest for the radius.  3 International Journal of Aerospace Engineering is the polar moment of inertia of the hollow circular section with respect to center of circle for the conical spiral antenna. λ = D/d is the ratio of inside and outside diameters of the tube.
The stiffness in the ith layer for the conical spiral antenna can be expressed as k i = GI p /πR 3 i based on the torsional theory with small deformation. When the force exceeds a critical value, The first layer before all others of the spiral antenna is fully flattened on the membrane and no longer has displacement. At this moment, the displacement at the contact position with the membrane is t. When the pressing force further increases, the model can be regarded as a spring with outer radius R 2 replacing R 1 in the previous analysis. In general, the displacement can be obtained similarly as The maximum force required to flatten all layers of the conical spiral antenna is Based on Equation (6), the normalized displacement nt − δ/nt can be expressed in terms of normalized force F/F 0 as Equation (6) illustrates that the maximum compressive force is closely related to the third power of the minimum radius of the conical spiral antenna. Equation (7) gives the quantitative relationship between normalized force and displacement during compression. At the initial stage of compression, the force on the conical spiral antenna is linearly proportional to its displacement. When F approaches F 0 , the normalized force increases rapidly with the normalized displacement. The stiffness of the conical spiral antenna increases during the entire compressive process.
Numerical simulation is performed to validate the above analytical model. Several 3D geometric models of the conical spiral antenna are established in different geometric parameters (G, t, d, R 1 , R n , etc.) by using a modeling software UG. The finite element model of the conical spiral antenna is established by using the commercial software ABAQUS. The upper and lower membranes are defined by analytical rigid body, as shown in Figures 2(b) and 2(c). And there is a friction between the top of the antenna and the upper membrane. The beam element is applied to the coilable antenna model. Hence, there is a complex nonlinear contact between the conical spiral antenna and the membrane in the numerical model, making it difficult to predict the maximum force. When the conical spiral is fully flattened, it has a great effect on the convergence and accuracy of the numerical results. Therefore, the corresponding force extracted as a comparison, when the normalized displacement is 60-90%. As shown in Figure 3, the force and displacement in the theoretical solution are validated by finite element analysis (FEA) without any parameter fitting, for the conical spiral antenna with polypropylene (PP) material. The baseline for the geometrical and material parameters of conical spiral antenna are E = 890 MPa, d = 1 mm, λ = 0:2, R 1 = 2:5 mm, R n = 14 mm, t = 4 mm, and n = 5. The parameter ð1 − δ/ntÞ ðR 1 /R n − 1Þ has a large of range.
It can be seen that there is a high nonlinearity between the normalized force and displacement in the conical spiral antenna. It is shown that, through the numerical analysis, the changes of geometric parameters for the conical spiral antenna have no effect on the nonlinear relationship. It is proved that numerical results are highly consistent with the theoretical results. Therefore, this derived theory can be used to estimate the compressive force and displacement. In turn, it can be used for the design of conical spiral antenna structure yet.

Thickness of Coilable Membrane
The foregoing studies have investigated the deformation behavior of the conical spiral antenna under the force applied by two parallel membranes. Dozens of conical spiral antennas are periodically arrayed on the coilable membrane in both the circumferential and the axial directions, where the spacing between adjacent antennas is defined as w 1 and w 2 , respectively. As the spacing is much smaller than the membrane size, the period force F applied by the conical spiral antenna on the rolled membrane is equivalent to a uniform pressure P = F/w 1 ⋅ w 2 in both circumferential and axial directions. The larger the diameter of the scroll, the greater the thickness (b m ) of the membrane can be wound. The increasing thickness magnifies the rigidity of the membrane itself, and the total weight of the antenna increases at the same time. To optimize the flight cost of the spacecraft, it is necessary to determine the minimum thickness required by the coilable membrane to fully compress the antenna.
For the pressure P applied by the conical spiral antenna much smaller than the elastic modulus (E m ) of membrane, i.e., P ≪ E m , the induced membrane deformation (ΔR r ) is small and linearly elastic, as shown in Figure 4. The Poisson's ratio of the membrane is defined as ν m . The stress in circumferential direction of the membrane can be obtained from force balance as Noticing that the strain in radial direction is negligible for a membrane with small thickness, the membrane deformation is 4 International Journal of Aerospace Engineering When the conical spiral antennas are fully flattened, the membrane deformation is equal to the cross-sectional diameter of the conical spiral structure, i.e., ΔR = d, and a pressure applied by the membrane with thickness b m is E m db m ð1 − ν m Þ/πR 2 r ð1 − ν m − 2ν m 2 Þ. According to Equation (6), the pressure needs to be larger than tGI p /2πw 1 ⋅ w 2 R 3 n for the membrane to fully flatten the antenna, i.e., Hence, with the other parameters given, the required thickness of the membrane is The minimum thickness b min of the membrane can be calculated, which is related to the geometrical and material parameters and the arrangement density of the conical spiral antenna. Taking the C-band conical spiral antenna with polymer as an example, the geometrical and material parameters of conical spiral antenna and membrane and the calculated minimum thickness b min are shown in Table 1. Finally, one prototype for flexible coilable antenna array with conical spiral antennas is fabricated as shown in Figure 5. The   Twelve conical spiral antennas are periodically arrayed on the coilable membrane in both the circumferential direction with 50 mm and the axial direction with 50 mm. The radius of scroll is 75 mm. Under the action of the membrane, the conical spiral antennas can be fully flattened and resiled during the coiling and outspreading processes, separately, as shown in Figure 5.

Conclusion
The flexible coilable antenna array with conical spiral antenna is a brand new spaceborne array. This paper investigates mechanical behavior of the flexible coilable antenna array during the coiling process. A theoretical relationship of the conical spiral antenna is given between normalized force and displacement, which exhibits a high nonlinear behavior. FEA is performed to validate the above theoretical model. Furthermore, the thickness of the coilable membrane to fully flatten the antenna can be predicted by an analytical solution about the geometrical and material parameters between the coilable membrane and the conical spiral antenna. This research provides important theoretical support for the design and fabrication of the flexible coilable antenna array with conical spiral antennas.

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.