Keywords

1 Introduction

Extendable thin-walled tape-spring boom is a new kind of deployable mechanism which has been widely used for many applications especially for aerospace technologies. This should thank to its advantages such as light-weight, high folded-ratio and self deployable property. The boom is usually used for combining with flexible membranes forming large-scale boom-membrane structures after wholly deployed, such as membrane antennas, solar arrays, solar sails, space-telescope star-shaders, etc.

A diagram is presented in Fig. 1 to illustrate the deployment process of the boom. The boom looks like a carpenter’s tape, which is like a slit boom when fully deployed. Before the deployment, the boom is coiled and locked onto a cylindrical hub, and thus some strain-energy is restored in the boom material in the process. During the deployment, either the boom tip or the hub is selected to be fixed (depending on working conditions), and the other side would be released and deployed under the drive of the restored strain-energy. In some conditions the boom is needed to be placed into a restraint mechanism for a better control (usually with a motor), while sometimes it can be self-deployed [1, 2].

The boom’s first famous practical use in space was on Hubble Telescope as the supporting structures for the solar arrays. In this mission, the booms deployed and worked on orbit successfully at the beginning and were unfortunately out of work before long because of the bending and buckling failures [3]. Apart from the semi-circular style boom as shown in Fig. 1 (which was commonly called Storable Tubular Extendable Member, STEM), the boom with a lenticular cross-section was also used for engineering on a solar sail by DLR, Germany [4]. The lenticular boom (commonly called Collapsible Tubular Mast, CTM) was able to obtain relatively higher torsional stiffness because of its closed cross-section configuration. Besides, a boom with a triangular cross-section (commonly called Triangular Rollable And Collapsible, TRAC) was also invented and launched in NanoSail-D mission by NASA. The triangular boom could acquire higher bending stiffness and buckling load as its cross-section geometry was more dispersed, however, the torsional stiffness was relatively low just because of this [5, 10, 12].

The existing research concerning tape-spring booms was mainly concentrated on analyzing the boom’s moving behaviors during deployment process, and upgrading or optimizing the boom’s cross-section configurations for acquiring better mechanical properties. However, since the deployment process was accompanied by large-scale and complex deformations on the boom materials, the booms were easily to be damaged by extremely high stress caused by stress concentration [6]. Therefore, finding the stress concentration point which was the most vulnerable to be damaged during the whole deployment process was necessary to be investigated. As the interaction of the boom infinitesimals was too complicated to be analyzed, establishing a numerical model was considered for use. Section 2 built a finite element model of a tape-spring boom for analyzing the stress concentration during the deployment. For giving some verification and providing some support on the corresponding parametric study, the analytical method for driving force calculation was also carried out. Sec. 3 focused on investigating the mechanical properties of the boom’s deployed state based on both numerical and theoretical methods. According to the analysis in Sects. 2 and 3, Sect. 4 studied the parametric impact on the boom’s deploying and deployed states, and Sect. 5 concluded the whole paper. The research in this paper would provide more guidance on the design of extendable tape-spring booms and the parametric selections.

Fig. 1.
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Tape-spring boom diagram (STEM)

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Fig. 2.
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Numerical model of a tip-spring boom

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2 Deployment Process Analysis

Deployment process was one of the most important modes of a tape-spring boom, which directly determined the working success of a boom and even the corresponding membrane structures. From the experimental study previously, tearing-failure caused by stress concentration was easily to be found on a boom material because of the large deformations happening in this process [7, 8]. Therefore, finding the concentration points and acquiring the highest stress a boom needed to sustain were necessary for boom designs and material selections.

Since a boom’s infinitesimal deformations were too complicated to study through analytical methods, a numerical model was considered to be used for the analysis. Meanwhile, the deployment process could be regarded as the inverse process of coiling at low moving speed, which was easier to be analyzed. From above, a numerical model of a tape-spring boom with a central hub was established in Abaqus which was presented in Fig. 2. The parameters used in the model were selected to mimic those used in InflateSail CubeSat mission which was launched in 2015 where the definitions and values of the parameters were listed in Table 1 [9], and the corresponding geometric diagram was shown in Fig. 3 (in which O1 presented the boom’s circular center, and Ix2 marked the neutral surface of the cross-section). Based on the simulation experience, the S4R shell elements were selected and the rectangle meshes were used [9]. According to the element refinement, twenty meshes along the boom’s cross-section were proved to be appropriate through considering calculation amount and simulation precision comprehensively, as the model with forty meshes output very similar results. For the sake of universality of the analysis, the isotropic material was introduced into the model. Furthermore, the material property of the central hub was set as rigid body since the hub deformation was able to be ignored during the deployment, while the mesh size was selected to be appropriate for the boom meshes. For obtaining the stress distribution caused by boom deformation accurately, the quasi-static analysis was used for all the steps in the simulation. The simulation steps and analysis details were shown as follows:

Table 1. Boom parameters
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Fig. 3.
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Parameter diagram of the boom cross-section

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Fig. 4.
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Definition for outer, middle and inner longitudinal cross-sections of a coiled boom (1 turn)

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Step 1 introduced equal edge loads on the boom’s both edges for making the boom flat; Step 2 added a surface pressure acting on the boom root making the root attached and fixed on the central hub; Step 3 released the edge loads and the root pressure, and spun the hub for coiling the boom on the hub (during this step, a slight concentrate force was introduced on the boom tip pointing at the opposite side of the hub for preventing the boom’s uncoiled section rotating around the hub, and the force was removed at the end of the step). The stress distribution of the boom during the coiling process (also regarded as the deployment process) was obtained and recorded in Step 3.

2.1 Stress Distribution During Deployment

Based on the numerical model established in Abaqus, the stress distribution on the boom during coiling (deploying) could be acquired. Since the boom’s configuration was symmetric during the whole process, the results from one side of the symmetry were recorded from the analysis. The forth strength theory was selected for analysis and Von Mises stress was acquired from the model, because plastic deformation of the materials was strictly prohibited during the whole process. Figure 4 showed the definition of the outer, middle and inner (longitudinal) cross-sections, while Fig. 5 presented Von Mises stress along the cross-sections respectively. The results listed in Fig. 5 were the distribution after the boom was coiled for one turn as a representative. In this figure, x-axis showed the actual path-length along the longitudinal cross-section, and 1 m boom-length from the root was present since this section with sharp stress change was easier to be failed.

Fig. 5.
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Stress distribution along the boom’s longitudinal cross-sections

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Fig. 6.
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Stress distribution on boom root when coiling (1–3 turns)

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Fig. 7.
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Simulation contour

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According to the plots shown in Fig. 5, it could be found that the sharp stress change happened at the boom’s root region and the connecting region between the coiled section and the transitional section, while the plots away from the two regions changed smoothly. Meanwhile, the highest stress of the inner cross-section was higher than that of the other two cross-sections. Moreover, the highest stress occurred at the inner cross-section near the boom root. From above, the highest stress point would appear on the middle of the boom root when coiling.

Furthermore, for checking the stress variation trend during coiling, the stress of the boom’s transversal cross-section on the boom root was acquired from the model. Figure 6 showed the stress on the root when the boom was coiled for one turn to three turns as a representative for finding the changing regularity, while the simulation contour was also presented in Fig. 7 for providing more clarification. The definition of x-axis in Fig. 6 was the path-length along the boom’s transversal cross-section, and the stress presented in Fig. 6 and Fig. 7 were both Von Mises stress.

From Fig. 6 and Fig. 7, it could be observed that the stress along the transversal cross-section rose with the number of the coiled turns, while the plots from different turns looked fairly similar. Also, all the plots were generally symmetric around x = 0 because of the boom’s configuration. The stress plots in Fig. 6 ought to be continuously along the cross-section, while the broken lines listed was because of the mesh limit. Nevertheless, the plots in this figure was precise enough to acquire the variation regularity of the stress.

To sum up, the boom’s most vulnerable point during deployment was always staying at the middle of the transversal cross-section on the boom root, and the value of this point rose gradually following the growing of the number of the coiled turns.

2.2 Driving Torque Analysis

Based on the numerical model established, the boom’s deployment torque was also able to be obtained through adding another analysis step after Step 3 in the simulation as follows.

Step 4 released the restraints acted on the central hub for making the boom deploy automatically, and the driving torque on the hub was recorded during the deployment process (3 turns for deployment).

In order to avoid calculation abortion, explicit analysis was used for this step, and the corresponding numerical results were been given in Fig. 8. According to the simulation plots in this figure, the torque overall increased with the deployment progress. This was because the bending curvatures of the inner coil layers was smaller than those on the outer coil layers, which gave the boom higher spring-back ability. Meanwhile, the fluctuation of the plots was mainly caused by the boom’s untight coiling resulted from mesh limit of the finite element models (see Fig. 9 for clarification), and the troughs appearing at the starting point of each turn were caused by the hump on the connecting part between the boom and the central hub.

For the sake of giving some verification for the numerical results and providing some support on the parametric study in the following sections, an analytical model for driving torque analysis was also established.

When coiled on the central hub, the boom’s longitudinal cross-section could be approximately regarded as an Archimedes spiral (see Fig. 10 where ɑ1 presents the boom’s coiling angle around the hub)[10,11,12,13]. According to the geometric analysis from Fig. 10, the boom’s bending/coiling curvature when coiled at ɑ1 can be expressed as

$$ \kappa_{{\alpha_{1} }} = \frac{1}{{R + \frac{t}{2} + \frac{t}{2\pi }\alpha_{1} }} $$
(1)

while, the boom’s coiled length is given as:

$$ l_{1} = \int_{0}^{{\alpha_{1} }} {\left[ {\left( {R + \frac{t}{2}} \right) + \frac{t}{2\pi }\alpha } \right]} d\alpha $$
(2)

Based on the elastic mechanical theory, the boom’s driving torque when deploying at the point with the angle ɑ1 is shown as M = κα1EIz where

$$ I_{z} = \frac{{\left( {r - \frac{t}{2}} \right) \cdot \left( {\pi - \frac{\theta }{2}} \right) \cdot t^{3} }}{6} $$
(3)

which is given through material mechanical method.

By introducing the results calculated from the theoretical analysis into Fig. 8, it could be found that the numerical results were generally scattering around the corresponding analytical results. Therefore, the numerical model was available for analyzing the mechanical properties of the boom during coiling/deploying process.

Fig. 8.
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Driving torque during deployment

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Fig. 9.
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Untight coiling caused by mesh limit

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Fig. 10.
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Diagram of longitudinal cross-section

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3 Deployed State Analysis

Bending stiffness and buckling load are a boom’s critical properties when at its deployed state. The numerical model built in Sect. 2 is able to be applied for deployed state analysis as well, while an analytical model will also be established for providing some verification and for the further parametric study.

3.1 Bending Stiffness Analysis

Based on the numerical model established in Sect. 2, the bending stiffness around x-axis and y-axis (axis definitions shown in Fig. 3.) could be acquired respectively. The numerical results were listed in Table 2.

For giving more verification and providing support on parametric study, the theoretical analysis for bending stiffness was presented as follows.

According to the material mechanical methods [11], the moment of inertia around x-axis Ix can be given as:

$$ I_{x} = \int_{A} {y^{2} } dA = 2\int_{r - t}^{r} \gamma d\gamma \int_{{ - \frac{\pi }{2}}}^{{\frac{\pi }{2} - \frac{\theta }{2}}} {r^{2} \sin^{2} \vartheta } d\vartheta $$
(4)

where A is the area of the boom’s transversal cross-section which can be expressed as:

$$ A = \pi \left[ {r^{2} - \left( {r - t} \right)^{2} } \right]\frac{2\pi - \theta }{{2\pi }} $$
(5)
Table 2. Numerical and analytical results comparison
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Based on Eq. (5) and geometric analysis in Fig. 3, the boom’s inertia moment (deployed state) around x1-axis can be shown as:

$$ I_{{x_{1} }} = r^{3} t\left[ {1 - \frac{3}{2}\left( \frac{t}{r} \right) + \left( \frac{t}{r} \right)^{2} - \frac{1}{4}\left( \frac{t}{r} \right)^{3} } \right]\left( {\pi - \frac{\theta }{2} - \frac{\sin \theta }{2}} \right) $$
(6)

Because t <  < r, i.e. t/r → 0, Eq. (6) can be simplified written as:

$$ I_{{x_{1} }} = r^{3} t\left( {\pi - \frac{\theta }{2} - \frac{\sin \theta }{2}} \right) $$
(7)

Meanwhile, the distance between x1-axis (circular center) and x2-axis (neutral surface) d is able to be found through:

$$ d = \overline{y} = \frac{{\iint_{A} yd\Lambda }}{A} = \frac{{2\int_{r - t}^{r} \gamma d\gamma \int_{{ - \frac{\pi }{2}}}^{{\frac{\pi - \theta }{2}}} {r\sin \vartheta } d\vartheta }}{A} $$
(8)

According to the parallel-axis theorem, the inertia moment around x2-axis is presented as:

$$ I_{{x_{2} }} = I_{{x_{1} }} - d^{2} A = I_{{x_{1} }} - \overline{y}^{2} A $$
(9)

Therefore, the boom’s (deployed state) bending stiffness around x-axis is acquired through Ux = EIx2.

In the similar way, the bending stiffness around y-axis can be shown by analogizing with Eqs. (6) and (9) as:

$$ U_{y} = E \cdot I_{y} = Er^{3} t\left( {\pi - \frac{\theta }{2} + \frac{\sin \theta }{2}} \right) $$
(10)

where y-axis is the boom’s symmetric surface and neutral surface simultaneously in this case.

According to Eqs. (9) and (10), the boom’s (deployed) bending stiffness around x-axis and y-axis are also listed in Table 2. By comparing the results from numerical model and theoretical analysis, it can be observed that the errors are relatively low (around 5%), and the results generally match with each other well. Besides, the analytical result is slightly higher that that from the numerical model, and this is because the boom’s cross-sections are totally non-deformable in the theoretical analysis.

3.2 Buckling Load Analysis

Buckling is also a critical mechanical property for a boom’s deployed state, which is researched by numerical analysis and analytical method in this subsection. For numerical analysis, a reference point was introduced at the circular center of the boom tip, which was rigidly connected with the tip edge for mimicking a closure head commonly used on the tip (especially for boom-membrane structures). The load was acted on the reference point aiming at the boom root which had been fully fixed before the analysis. By using linear perturbation step in Abaqus, the boom’s first buckling load Pcr was acquired which was listed in Table 3. Similarly with the bending stiffness analysis, analytical method by Euler formula as Eq. (13) was also used to provide some analytical verification, whose result was presented in Table 3.

Table 3. Buckling load results comparison
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According to the comparison in Table 3, the error between the numerical and analytical results was relatively high (up to 30%) and the analytical value was higher the numerical one. This was because the boom was not only bent but also twisted under the the pressure of the concentration force in the simulation, while the Euler formula was only used for calculating the cases under ideal conditions (flexural buckling). Therefore, the theoretical method was only able to be used for sketchy calculating the boom’s buckling load in this case, and the numerical method was more precise for predicting.

4 Parametric Study

For the sake of providing some guidance for a boom design, based on the analytical methods in Sects. 2 and 3, a parametric study was carried out in this section.

4.1 Driving Torque Study

Based on the analytical method in Sec. 2, the impact of the boom radius (r), hub radius (R), boom wall-thickness (t) and half opening-angle (θ/2) on the boom’s driving torque was investigated (shown in Fig. 11). The figure was plotted when the boom was coiled for 2 turns as a representative. On account of the plots in Fig. 11, the driving torque rose with the increasing of boom radius and decreasing of hub radius. This was because higher boom radius made the boom have longer cross-section path-length, and the lower hub radius led to higher coiling curvature which provided higher strain-energy for drive during the deployment. Moreover, the driving torque also went up when the boom wall-thickness was rising or the opening angle was reducing with the similar reasons. Meanwhile, the plots changed non-linearly in the figures, and increasing the values of the boom wall-thickness and the hub radius was more efficient for improving the boom’s driving torque.

4.2 Bending Stiffness Study

Furthermore, the parametric study on the boom’s bending stiffness was also investigated based on the theoretical analysis in Sect. 3. The influences of boom wall-thickness (t), hub radius (R), boom wall-thickness (t) and half opening-angle (θ/2) on the bending stiffness around x-axis (easier to be failed according to the study in Subsect. 3.1) was presented in Fig. 12.

Fig. 11.
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Driving torque parametric study

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Fig. 12.
figure 12

Parametric study of boom bending stiffness around x-axis

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According to the plots shown in Fig. 12, the bending stiffness rose linearly with the increasing of the boom wall-thickness, and the stiffness went up non-linearly with the deceasing of the half opening-angle and the increasing of the boom radius. Further, from the figure, changing the boom radius was the most efficient way to improve the boom’s bending stiffness around x-axis.

5 Conclusion

Extendable tape-spring boom is a new kind of deployable mechanism which is widely using for many applications especially for aerospace technologies in recent years. As the boom is a large-deformation flexible mechanism during folding or deployment process, failure caused by stress concentration is one of the most common ways for boom damage. Therefore, acquiring the point with the highest stress on the boom during the whole process is necessary to be carried out. Since the interactions of the infinitesimals were fairly complicated during the deformation process, the numerical method was considered to be introduced into the investigation.

This paper established a numerical model of an isotropic tape-spring boom. Through analyzing the stress distributions along the boom’s longitudinal and transversal cross-sections, the point with the highest stress during the deployment process was found. Meanwhile, the boom’s deployment driving force was also obtained from the simulation. For the sake of verifying the results from numerical analysis and providing support for further parametric study, the theoretical model was also built for acquiring the boom’s driving force during deployment. From the comparison, the results from the two methods generally matched well. Furthermore, the bending stiffness and buckling load at the boom’s deployed state were analyzed through both numerical model and theoretical analysis as well. Afterwards, a parametric study was carried out to find the most efficient method for improving the boom’s driving torque and bending stiffness. From the investigation, increasing the boom wall-thickness was the most efficient way to improve the driving force, and increasing the boom radius was the best mode for promoting the bending stiffness. The research from this paper will provide more guidance on tape-spring boom designs and the corresponding parameter selections.

The authors would like to acknowledge that the research presented in this paper was carried out with the aid of the National Natural Science Foundation of China (Grant No. 52205027) and the Science and Technology Project of Jiangsu Province (Grant No. BK20220496).