1. Introduction
With the increasingly urgent demand for large-scale antennas, solar panels, solar sails and other space structures, the research on deployable mechanisms has started to gather attention in recent years [
1,
2]. Compared with the traditional rigid deployable mechanisms, space membrane deployable mechanisms with significant advantages in system quality and deployment ratio are becoming a creative approach for specific space applications [
3].
Extensive research on space membrane deployable mechanisms has yielded a large number of remarkable engineering applications, ranging from space membrane antennas [
4,
5] and membrane solar cell arrays [
6,
7] to large-area solar sails [
8,
9,
10]. With an increasing demand for high-resolution Earth observation [
11], innovative large-scale deployable membranes antenna have recently been attracting significant interest. The DLR and European Space Agency (ESA) collaborated to develop the deployable Synthetic Aperture Radar (SAR) antenna composed of membranes and two coilable booms [
12]. Due to the advantages of areal density and power-to-mass ratio, the Hubble solar array jointly developed by the National Aeronautics and Space Administration (NASA) and ESA [
13], the International Space Station (ISS) solar array [
14] and Roll Out solar array (ROSA) [
15,
16] all adopted membrane solar cell arrays. In the Interplanetary Kite-craft Accelerated by Radiation Of the Sun (IKAROS) solar sail [
17] and Nanosail-D [
18] solar sail, which have been successfully launched, deployed and carried out in-orbit tests, membrane deployable mechanisms are necessary for deep space exploration. The implementation of membrane in these typical large-scale space structures indicates that the membrane deployable mechanism is the key development field in both engineering and academia.
Despite light weight, large deployment ratio and low cost, the dynamic problems of the membrane deployable mechanism are inevitable and prominent considering the large flexibility of the membrane and complexity of the space environment [
19]. As discussed above, many efforts have been made by scholars on dynamic characteristics [
20] and experimental analysis [
21]. Shen et al. [
22] of the Canadian Space Agency (CSA) established membrane models in simulation software and analyzed the effect of tension forces and damping ratios. The assumed mode method was adopted to conduct membrane analysis by Liu et al. [
23,
24] and compared with nonlinear finite method, which indicated that the former result is smaller. Using the coupling coefficient approach, Fan et al. [
25] studied the coupling dynamics characteristics of a satellite and membrane. Ahmadi et al. [
26] investigated the effects of different parameters on the frequency ratio and nonlinear frequency of a prestressed membrane. As previously stated, numerous discussions have focused on the analysis and modeling of single membranes, while the dynamic characteristics of the boom–cable–membrane coupling overall system have been considered in little research so far. Many researchers have examined noncontact membrane measurement due to the delicate and flexible properties of membranes. Both Zhang et al. [
27] and Chakravarty et al. [
28] adopted laser Doppler vibrometers (LDV) to conduct noncontact vibration tests to measure the membranes, respectively. Moreover, the tie-system calibration [
29] can provide an indication to the experimental setup for the tensioning of membranes. A novel, nondestructive methodology, using vibro-acoustic tests to measure the membrane modal characteristics and mechanical properties was put forward by Lima-Rodriguez et al. [
30] recently. Gaspar et al. [
31] from NASA conducted a noncontact modal test of membranes using the Polytec scanning laser vibrometer and discussed the results obtained at various tension levels and at various excitation locations.
In order to address the conflict between the large scale required of the deployed space membranes and the performance of high stiffness, the configuration design and multiobjective optimization can provide solutions. As for the configuration design, the deployable booms and web-like tensioned membrane scheme are major prerequisites for the implementation of high-stiffness, lightweight design. Deployable membranes coupled with booms have been extensively studied, ranging from Synthetic Aperture Radar (SAR) satellite [
32] and deployable membrane structures with rolled-up booms [
33,
34] to 3U CubeSat OrigamiSat-1 [
35], which focus on conceptual model design and on-orbit experiments. However, little research has derived the mass equation based on the analysis of the cable tension theoretically. The studies on deployable membranes using the multiobjective optimization approach [
36,
37] can provide an indication to tune parameters for the membrane mechanism designed in this work. By the response surface method, the dynamic surrogate model based on the boom–cable–membrane mechanism simulation results is established, and then combined with the derived mass equation to conduct multiobjective optimization, which has been considered in little of the research on overall deployable membrane systems compared with single-component ones so far. In this work, the aim is to explore the configuration design process, including deployable booms and tensioned schemes and multiobjective optimization based on a mathematical surrogate model to produce a deployable membrane mechanism with satisfactory dynamic performance of weight and stiffness.
In
Section 2, the configuration design and analysis of a space membrane deployable mechanism, as well as mass calculation for deployable booms, membranes and cables, respectively, are presented. The boom–cable–membrane dynamic simulation model is given in
Section 3 and verified by the modal test of scaled prototype.
Section 4 provides surrogate model establishment of the fundamental frequency and its application to multiobjective optimization with the mass formula.
2. Configuration Design and Mass Calculation for Space Membrane Deployable Mechanism
To meet the requirements of the properties for deployable structures in aerospace engineering, including light weight, high deployment ratio, high stiffness and large size, the triangular space membrane deployable mechanism based on deployable booms is proposed, as shown in
Figure 1. This mechanism with height
h consisting of an unfolding support mechanism, a membrane and a tensioning system, has the advantages of excellent deployment synchronization and ease of control, fewer deployment units and lower surface density. The membrane is folded according to Miura-ori, and the deployable booms are wrapped into the folded state. During deployment, the deployable booms drive the cables, and the cables drive the membrane to unfold and be tensioned by the web-like tensioned membrane scheme.
2.1. Unfolding Support Mechanism
Compared with the rigid truss, the deployable boom is utilized as the unfolding support mechanism for the triangular space membrane deployable mechanism because of its light weight, high deployment ratio and self-deployable performance, and the prerequisites for the same stowed height and mass are set when selecting sections. As shown in
Figure 2, there are mainly three different sections of deployable booms: the storable tubular extendable member (STEM), the collapsible tube mast (CTM) and the triangular rollable and collapsible (TRAC) boom (orient 90°), so that these three sections are symmetric with respect to the y-axis of the established local coordinate system.
Then, the flexural stiffness
and principal moment of inertia
of the deployable booms can be calculated analytically according to the inertia moment, the static moment and the formula of parallel displacement axis, and the STEM is taken as an example to illustrate the calculation procedure, as shown in Equation (
1). As for the unfolding support mechanism, the flexural stiffness of these three sections is calculated and compared in the case of the same stowed height and mass
, which can provide guidance on the section selection.
where
A represents the cross-sectional area, and the parametric equation for the cross-section curve is
.
Since the Young’s modulus E of the three sections of booms is the same, the principal moment of inertia can be selected to compare the flexural stiffness and assess the sections. The deployable booms are made of carbon fiber with a Young’s modulus of 96 GPa and a density of . Addiitonally, the geometric parameters of the booms are of length L = 31.25 m, = 96 mm and = 220 mm, radius r = 110 mm and = 275 mm, opening angle and boom thickness mm. According to these material and geometric parameters, the calculation results of the three sections can be obtained.
It can be seen from the results listed in
Table 1 that the CTM is a closed-loop cross-section with the smallest inertia. On the precision of the same mass, the TRAC has the largest inertia, but its performance is seriously affected by the machining accuracy, especially the strict bonding requirements.However, the STEM is an integral structure with higher reliability than other forms, is made from a pair of symmetric halves bonded at the edges and has relatively balanced mechanical properties in all directions. Therefore, the STEM is selected as the cross-sections of the deployable booms applied to the triangular space membrane deployable mechanism because of its better properties and ease of manufacture.
2.2. Tensioned Membrane Scheme
In view of the large-size triangular membrane structure, five kinds of tensioned membrane schemes in
Figure 3 are designed, respectively, they are the corner tensioning scheme, conventional catenary design, Miura–Natori tensioning system, shear-compliant border design and the web-like tensioned membrane scheme, varying in the boundary shape of the membranes and the arrangement of the catenaries. The web-like tensioned membrane scheme is selected considering that it effectively reduces the overall mass of the cables and improves the surface accuracy and flatness of the membrane by greatly eliminating wrinkles. The outer perimeter cables can be used to absorb the majority of disturbances emanating from the support points, alleviating the membrane wrinkles as a result.
The membrane is attached to the inner catenary cables across the three curved edges of the triangle, and the inner catenary and outer perimeter cables are coupled by tie cables, as denoted in
Figure 4. Each triangular edge consists of
N arcs with radius of
, then
can be deduced according to the geometric relations illustrated in
Figure 4, as shown in Equation (
2),
where the side length of the membrane
l and the angle
are labeled in
Figure 4.
As plotted in
Figure 5, one arc of the inner catenary cable and part of the membrane are extracted from the web-like tensioned membrane scheme. According to the equilibrium condition, the uniform tension in inner catenary cable
is given as
where
is the membrane uniform stress. Moreover,
is expressed as
where
,
and
represent the Young’s modulus, the cross-sectional area and the uniform strain of the inner catenary cables, respectively. The membrane uniform stress
is assumed to be constant, and can be calculated as
where
,
t,
v and
represent Young’s modulus, thickness, Poisson’s ratio and uniform strain of the membrane, respectively.
With the assumption that the membrane wrinkle is negligible,
is considered equal to
. Then, substituting Equations (
2), (
4) and (
5) to Equation (
3),
is analytically derived based on the equations above as
Combined with the geometric relations, the total length of the inner catenary cables
is expressed as follows:
As shown in
Figure 4,
and
represent the tensions in tie cables. According to the equilibrium condition and Equation (
4),
and
can be expressed as
Considering equilibrium, the tensions in outer perimeter cables
labeled in
Figure 4 can be obtained as
where
describes the gradient and is given as
in which
, and the relevant parameters are annotated in
Figure 4. Combined with Equation (
9) and equilibrium, the cable tension
F can be obtained as
From Equation (
11), the following equation can be calculated.
Substituting Equation (
12) into Equation (
10), the final expression of
can be obtained as
and then the total length of the tie cables
and outer perimeter cables
can be obtained, respectively, as follows:
Similar to Equation (
4),
F can also be expressed as
where
and
are Young’s modulus and the uniform strain of the outer perimeter cables. The cross-sectional area of the outer perimeter cables
is assumed equal to that of the tie cables
. Based on the derivations above, when the material properties of the cables and membrane and these geometric parameters, including
l,
p,
q,
,
and
N, are given, the total length of all the cables
,
and
can be calculated, respectively, and the cross-sectional area of various cables are simplified as a function of
F and
t. Then, the total length and cross-sectional area of the cables are substituted into the mass calculation and the multiobjective optimization of subsequent sections. Meanwhile, these derivations also guide the design of the tensioned scheme applied in the FEA model by calculating the variables, such as
and
.
2.3. Mass Calculation
As mentioned above, the membrane mechanism consists of STEM deployable booms, tensioned cables and membrane, and the total mass equation based on the cross-sectional areas and the total length of the cables gained above is derived here to establish the analytical model on design parameters, which is applied to the multiobjective optimization in a subsequent section. Consequently, a total mass equation can be established:
where
represents the mass of the web-like cables, and the mass of the STEM deployable booms
= 32.54 kg, as listed in
Table 1. The mass of the membrane
is derived as follows:
where
represents the density of the membrane. Moreover, the total mass of cables
is given by
where
is the density of the cables. The material of the membrane is Kapton with density of
, Young’s modulus of 2.5 GPa and Poisson’s ratio of 0.34. Additionally, the cables are Kevlar with a density of
, Young’s modulus of 131GPa and Poisson’s ratio of 0.35. Furthermore, other geometric parameters are
l = 25 m,
,
,
,
and
N = 10. According to these material and geometric parameters, the total mass of all the cables
is only related to the cable tension
F and the membrane thickness
t by substituting Equations (
6), (
7), (
14) and (
15) into Equation (
18), as shown in the following equation.
Substituting the parameters mentioned above into Equation (
17),
can be obtained as
In summary, Equation (
16) can be rewritten as
5. Conclusions
According to the property requirements of space deployable structures, this paper proposes the triangular space membrane deployable mechanism based on deployable booms with lightweight, high deployment ratio, high stiffness and large size. These features are primarily generated by the STEM deployable booms and web-like tensioned membrane scheme, the mechanical property and total mass of which are analyzed and deduced, respectively. Then, the membrane mechanism simulation model, including booms, cables and membrane, is built and verified by the membrane model test, which aims to reveal the relationship between its fundamental frequency and design parameters. The dynamic surrogate model of the design variables in respect to the fundamental frequency of the membrane mechanism is created through the response surface approach. Additionally, based on the genetic algorithm, a comprehensive multiobjective optimization to achieve high stiffness and minimal mass is performed to obtain the optimal dynamic and lightweight design parameters of the membrane mechanism.
This study endorses the exploitation of mass derivation and the mathematical surrogate model to set up a theoretical analysis basis for the membrane mechanism. On the other hand, the configuration design process and the multiobjective optimization modeling presented in this paper pave the way to design a more novel space membrane deployable mechanism with exhilarating features and properties.