# Roll-Out Deployment Process Analysis of a Fiber Reinforced Polymer (FRP) Composite Tape-Spring Boom

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction and Literature Review

## 2. Deployment Process Dynamic Analysis of a Bistable STEM Boom

#### 2.1. Strain Energy Model Establishment

_{x}, N

_{y}, and N

_{xy}are the stretching force per unit length, M

_{x}, M

_{y}, and M

_{xy}are the bending torque per unit length, ε

_{x}, ε

_{y}, and γ

_{xy}are stretching and shearing strain and κ

_{x}, κ

_{y}, and κ

_{xy}are bending and twisting curvatures. Note that κ

_{x}is the curvature along the boom’s longitudinal cross-section, and κ

_{y}is the transversal cross-section curvature (parameters above shown in Figure 2).

**B**in Equation (2) should be zero [9]. Furthermore, D

_{16}and D

_{26}can also be assumed to be zero as the coupling between the boom’s bending and twisting behaviors has less effect on the boom deployment behaviors. Furthermore, κ

_{xy}would be ignored as well because the behavior caused by κ

_{xy}cannot be revealed when the boom is working ideally. Meanwhile, with regard to matrix

**A**, only the stretching behavior along the boom longitudinal direction (A

_{11}) is obvious [16]. Therefore, according to the analysis above, Equation (1) can be simplified as follows:

_{1}is the hub rotation angle from the initial state during the deployment, r

_{i}is the coiled radius of the boom root, e

_{min}is the minimum strain energy per unit length acquired by the minimum energy principle which is illustrated in Ref. [14] in details, and α

_{b}is the hub rotation angle deploying from the start to the end which can be acquired by:

_{b}is the total length of the boom, r

_{h}is the hub radius, and T is the boom’s wall thickness.

#### 2.2. Boom Deployment Process Analysis

_{1}is the deployed length (corresponding with α

_{1}, 0 ≤ l

_{1}≤ l

_{b}, see Figure 3), m

_{s}and m

_{b}are the mass of the hub and the boom coiled/undeployed section, J

_{s}and J

_{b}are the rotational inertia of the hub and the boom coiled section, and v

_{1}is the hub movement velocity with a deployed length l

_{1}. The terms m

_{b}, J

_{s}and J

_{b}in Equation (11) can be found by:

_{b}is the boom linear density, and t

_{h}is the hub wall thickness.

_{1}is the only independent value in this equation.

## 3. Deployment Analysis and Experimental Comparison

_{m}, G

_{m}, and v

_{m}are the elastic modulus, shear modulus, Poisson’s ratio of the matrix, E

_{f}, G

_{f}, and v

_{f}are the elastic modulus, shear modulus, Poisson’s ratio of the fiber, T

_{UD}, V

_{UD}, and Φ

_{UD}are the thickness, volume fraction, porosity of the unidirectional (UD) ply and the T

_{f}, V

_{f}, and Φ

_{f}are the thickness, volume fraction, and porosity of each fabric ply. Note that the introducing method of the material parameters in Table 2 into the equations in Section 2 was commonly used in the mechanics of composite laminate materials, which could also be found in Ref. [1]. No hub was used in the experiment (i.e., m

_{s}= J

_{s}= 0 in Equations (11) and (13)). The two boom samples in the experiment were manufactured with the same geometric and material parameters listed in Table 1 and Table 2, except the laminate layout was assigned as [±45°F/0°/±45°F] (Sample 1) and [±50°F/0°/±50°F] (Sample 2), respectively.

_{x}, were further plotted in Figure 5. Note that the plots in Figure 5 were the energy integrals along the boom cross-sections based on the data in Figure 4 for better viewing. From Figure 4 and Figure 5, it could be found that each boom sample had two minimum energy value points. That was to say that each boom had two energy stabilities, which was a bistable tape-spring boom. One of the stabilities was at a boom’s fully deployed (initial) state (κ

_{x}

_{1}= κ

_{x}

_{2}= 0 and κ

_{y}

_{1}= κ

_{y}

_{2}= 1/R = 50 m

^{−1}), which was called the first stability in this paper for the sake of illustration. Additionally, the curvature of the other/second stabilities of the two samples were κ

_{x}

_{1}= 36.4 m

^{−1}and κ

_{x}

_{2}= 46.4 m

^{−1}, i.e., the curvature radii were r

_{x}

_{1}= 27.5 mm and r

_{x}

_{2}= 21.6 mm, respectively, according to the plots in Figure 5. The areas apart from the stable points in Figure 4 and Figure 5 were unstable regions which were the transition stages (boom deforming process) during the deployment. Since there was no hub introduced in this experiment, the curvature radii of the booms’ second stabilities were regarded as the hub radii, i.e., r

_{h}

_{1}= r

_{x}

_{1}= 27.5 mm and r

_{h}

_{2}= r

_{x}

_{2}= 21.6 mm. According to the experimental experience presented in Ref. [23], for a three-layer fabric laminate boom, the damping factor would be selected as μ = 0.72. By introducing the damping factor μ into the theoretical model in Section 2, the comparison of the experimental and the analytical results can be seen in Figure 6, and the deployment process of boom Sample 1 in the experiment is shown in Figure 7 as a representative, in which the instantaneous velocities and deployed lengths were marked and the time increment of each frame was 1/30 s for illustration. Three repeated tests were carried out for each boom sample in the experiment in order to improve reliability.

_{1}) should be acquired, and this will be investigated in the future work. Nevertheless, the analytical model in Section 2 with a constant μ could still be used to predict the deployment process of a relatively short tape-spring boom (less than 1.5 m according to the experiment, which could cover most booms using CubeSats). For example, in Figure 6, the samples’ velocities were nearly constant near the end of the deployment as the damping forces were approaching the booms’ driving forces during this section.

_{1}) which was a function of the deployment velocity, was needed for further investigation.

## 4. Parametric Study

_{f}whose influences were given in Figure 8, and the boom’s natural cross-section radius R and path length b, whose effects were listed in Figure 9. The parameters which were not marked in Figure 8 and Figure 9 were the same as those listed in Table 1 and Table 2 (fabric plies laminate layout for the plots in Figure 8b and Figure 9a,b was all set as [±45°F/0°/±45°F]).

_{h}because a boom with lower ply angles would have a larger natural coiled radius r

_{n,}and the hub radius should be smaller than the boom’s natural coiled radius to keep the boom coiling on the hub tightly (see Ref. [16] for more details), and this would pump up the folded volume of the mechanism. Further, a smaller natural radius R or shorter path length b could lead to a lower driving force as well, and, meanwhile, this reduction was also able to decrease the boom’s bending stiffness when wholly deployed. Therefore, from the analysis above, the parametric design of the roll-out deployable boom should consider the deployment velocity, the folded volume and the deployed stiffness comprehensively.

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Tape-spring boom deployment modes. (

**a**) Normal deployment (

**b**) Roll-out deployment (with rollers) (

**c**) Roll-out deployment (no rollers necessary).

**Figure 5.**Boom strain energy per unit length under different coiled radii. (

**a**) Boom Sample 1; (

**b**) Boom Sample 2.

R (mm) | b (mm) | l_{b} (m) |
---|---|---|

20 | 50 | 3 |

E_{m} (GPa) | G_{m} (GPa) | v_{m} | E_{f} (GPa) |

4 | 2.7 | 0.35 | 240 |

G_{f} (GPa) | v_{f} | T_{UD} (mm) | V_{UD} (%) |

95 | 0.22 | 0.057 | 31 |

Φ_{UD} (%) | T_{f} (mm) | V_{f} (%) | Φ_{f} (%) |

85 | 0.096 | 53 | 85 |

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**MDPI and ACS Style**

Wang, S.; Xu, S.; Lu, L.; Sun, L. Roll-Out Deployment Process Analysis of a Fiber Reinforced Polymer (FRP) Composite Tape-Spring Boom. *Polymers* **2023**, *15*, 2455.
https://doi.org/10.3390/polym15112455

**AMA Style**

Wang S, Xu S, Lu L, Sun L. Roll-Out Deployment Process Analysis of a Fiber Reinforced Polymer (FRP) Composite Tape-Spring Boom. *Polymers*. 2023; 15(11):2455.
https://doi.org/10.3390/polym15112455

**Chicago/Turabian Style**

Wang, Sicong, Shuhong Xu, Lei Lu, and Lining Sun. 2023. "Roll-Out Deployment Process Analysis of a Fiber Reinforced Polymer (FRP) Composite Tape-Spring Boom" *Polymers* 15, no. 11: 2455.
https://doi.org/10.3390/polym15112455