Modal Analysis of Self-Deployable Tape Spring Booms: A Reverse Engineering Approach Open Access

Tape spring booms have demonstrated significant potential as morphing structures due to their low mass and high packing efficiency. This has led to their extensive use in various aerospace engineering domains, including on-orbit deployment of spacecraft components like sensitive science instruments [1–3] and solar sails [4,5] for small satellites. Over the past two decades, multiple advanced configurations of deployable booms have been developed, incorporating various structural and material innovations to optimize deployment efficiency and functional performance that can be tuned to the mission [6–10]. Spacecraft applications of these booms, such as high-precision optics and communication missions, demand a high level of pointing accuracy, which is governed by the stiffness and modes of the structure holding the instruments. Therefore, the ability to accurately model the structural dynamics of tape spring booms are a growing area of interest. Circular cross section booms have been the standard in the past, and conic cross sections have been proposed to exhibit better stiffness characteristics in the extended configuration [11–13]. The recent introduction of composites for manufacturing deployable booms has enabled bistability [14] in tape spring booms. This allows the tape spring to be stable in either the stowed or the fully deployed configuration by virtue of its geometry and material properties. Bistability enables passive deployment and eliminates the need for extra restraining mechanisms in the boom deployer, thereby reducing the mass and complexity of the spacecraft.

This article focuses on using a novel method to develop a correlated finite element analysis (FEA) model of parabolic tape spring booms to enable dynamic characterization of the boom in terms of its modal parameters. The methodology is compared to two conventional modeling methods and experimental modal analysis. The boom utilized in the tests was produced at NASA Langley, and an identical boom will be deployed aboard Ut ProSat-1 (UPS-1), a 3U CubeSat designed and built by students at Virginia Tech [15]. The primary mission of UPS-1 is to repeatedly perform a self-deployment of the 4 ft boom and then retract it using a stepper motor while measuring the satellite’s response in the stowed configuration, the deployed configuration, and during deployment. The mission will deploy and retract the boom under multiple conditions across its life to serve as a verification test for the boom. The mission will enable assessment of the effect of temperature and fatigue on its deployment dynamics. The responses are measured using accelerometers located in two inertial measurement units (IMUs) at its tip and near its root inside the CubeSat chassis, respectively. The boom-tip IMU is embedded on a flexible circuit. The deployment experiment will use a custom deployer design involving a camera shutter mechanism. This mechanism uses a servo motor to hold the spool, preventing the boom from deploying in its stowed state [16]. The fully deployed configuration of the boom with the deployer is shown in Fig. 1.

The modal behavior of the boom is highly sensitive to small variations in geometry and, therefore, any manufacturing or assembly defects. Initial conventional modeling efforts were attempted to perform modal analysis but the sensitivity of this system to geometric error required the development of a new methodology to remove those geometric modeling errors. Previous efforts have worked to remove modeling uncertainty in FEA to predict the modal analysis with higher accuracy, but none provide sufficient geometric accuracy across the full length of the boom. Kamatha et al. [17] and other studies, such as Refs. [18,19], explored the impact of geometry and geometric imperfections on the natural frequencies and dynamic properties of complex systems. However, their approaches fail to capture the complexities of bistable tape springs. Shore et al. [20] presented an analytical solution for the natural frequency of a drum-deployed tape spring using a simple Euler-Bernoulli beam model that was validated using experimental data and FEA. To match the root constraints, the deformed geometry for the FEA was obtained by prestressing the base geometry. This method was not selected in this case due to this boom’s boundary conditions and the prestress not being the primary source of error. Shepenkov [21] did not capture the transition zone where the cross section of the boom changes from the flat to the curved state. Recently, Yao et al. [22] modeled a boom with an idealized geometry with a constant cross section, and the manufacturing defects were not taken into account. Previous models have not captured the exact shape of the deployed boom, including transition zones and the correct boundary conditions imposed by the deployer.

To address this problem, we adopted a systematic methodology for generating high-fidelity Point Cloud Models from 3D-scanned geometry and integrating them into modal analysis using standard design and simulation tools. While previous studies [23–26] have integrated 3D point cloud data into the modal analysis of civil structures using single metrology techniques, our work addresses the need for a much higher point cloud density to accurately capture the geometry of a lightweight deployable boom for space applications, where small geometric defects significantly affect structural dynamics. Unlike earlier efforts using less accurate terrestrial laser scanners [27–29], we combine high-resolution 3D scanning and laser vibrometry to generate a dense, spatially varying point cloud of a complex geometry. This enables improved accuracy in modal analysis, which we validate through comparisons of three FEA models with varying geometric fidelity. The novelty of this work lies in adapting and demonstrating a workflow capable of handling the high point cloud density required for thin, flexible structures, addressing data management challenges and 3D smoothing requirements.

The article is organized as follows: the boom geometry and material properties are presented in Sec. 2. It is followed by a discussion about the setup for experimental modal analysis using a scanning laser doppler vibrometer in Sec. 3. Section 4 talks about the different methods for generating the computer-aided design (CAD) models of the boom and the analysis setup for the finite element model. Section 5 includes the modeling assumptions. We then present results and discuss the lingering uncertainties in Secs. 6 and 7, respectively. Finally, a conclusion on the work and next steps are discussed in Sec. 8.

The boom specimen under study is 4 ft long and has a flat width of 70 mm. The total thickness of the boom is measured to be 0.17 mm with a parabolic cross section. The composite laminate of the tape spring has a [45PW/0UD/45PW] layup. This layup provides reliable self-deployment even after viscoelastic relaxation and creep effects occur due to extended stowage. It consists of three plies: two outer carbon fiber/epoxy (M30S/PMT-F7) plain weave (PW) plies oriented at $±45∘$ and an inner unidirectional (UD) carbon fiber/epoxy (MR60H/PMT-F7) ply aligned along the 0-deg longitudinal axis of the boom. The inner axial ply plays a crucial role in storing the strain energy necessary for the boom’s self-extension. The material properties of the layup (longitudinal Young’s modulus, $E1$⁠; transverse Young’s modulus, $E2$⁠; major Poisson’s ratio, $ν12$⁠, and in-plane shear modulus, $G12$⁠) have been presented in Table 1. The boom under study was measured to be slightly thicker than the nominal thickness of 0.1564 mm. The additional thickness might be due to gaps introduced during layup and curing of the laminate, errors in measuring using vernier calipers or manufacturing defects of the individual plies.

The boom is an open-section composite tape spring and exhibits bistability due to its unique geometry and material characteristics, making it self-deployable. The bistability is designed such that it can be coiled around a spool with a diameter of 70 mm and stowed as shown in Fig. 2(a). The second bistable state is the fully extended configuration. A bi-arc spline interpolation algorithm was used by Lee and Fernandez [30] to approximate the parabolic cross section of the boom and the appropriate curve was selected based on strain energy minimization. Finally, the boom was imparted the designed curvature by curing it over a carbon foam mold [31] during its manufacturing process.

The boom is designed to be deployed on orbit and collect crucial vibration data for verification with ground tests. The boom is equipped with 16 44-AWG Copper wires that are colocated with the middle ply of the laminate and run along the length of the boom as shown in Fig. 2(b). The wires transmit power and data to and from the boom-tip circuit [22]. They are extremely light and add negligible stiffness to the boom and are, therefore, not taken into account in the FEA in this article.

The experimental setup (shown in Fig. 3) consists of the satellite chassis fixed to a vibration isolation table using 3D-printed nylon tabs. To minimize any influence on the boundary conditions, an impulse hammer was employed to excite the boom at a specified location near its root. A Polytec PSV-400 laser vibrometer measured the velocity response at 21 discrete locations along the boom (shown in Fig. 3). A torsion spring attached to the boom tip adds an additional boundary condition, and it is held in place by two tip tabs. The tabs and the spring are designed to ensure proper boom deployment by inducing curvature at the center of the tip. Additionally, the tabs flatten the arms of the parabola at the tip, minimizing flexure and protecting the IMU circuit located at the tip. The boom is attached to the deployer using epoxy along a small portion of its base edge. The deployer has two guiding rods that constrain the direction of boom deployment. However, without additional support, the boom was observed to shift vertically between the rods during ground-based deployment tests. To mitigate this, two epoxy-attached latches near the boom base engage with the upper guiding rod at full deployment. These latches provide secondary support, stabilizing the deployed boom and defining the final deployed configuration. A roller bearing is also used behind the boom to support and guide its movement, ensuring straight deployment. Apart from the physical restraints, a slight tilt was introduced in the boom during its assembly inside the deployer. This 1.38-deg tilt with the vertical seems to be an artifact of several probable reasons, like manual misalignment in attaching the boom to the deployer, defects in the boom, defects in the deployer or both. This tilt, along with the internal constraints on the boom, has been shown in Fig. 4. These boundary conditions are discussed in more detail in subsequent sections.

The averaged velocity/force-frequency response function for the out-of-plane axis, based on velocity measurements at 21 surface locations, is presented in Fig. 5. The first mode is a torsional mode at 4.14 Hz, which was expected due to the limited constraint at the root arc. The second mode, occurring at 5.39 Hz, is a bending mode. The third mode at 11.17 Hz is another torsion mode with a node at the tip, though some in-plane motion was not measured by the single-axis laser vibrometer. The fourth mode at 20.55 Hz is a combined twisting and bending mode, with the node shifting to approximately one-third of the distance from the boom tip.

The first step to create an FEA model in abaqus is to build a 3D CAD model of the boom. Previous methods did not take into account boom or assembly manufacturing defects and assumed an idealized constant cross section geometry [21,22]. Here, we improve the model’s geometric accuracy using two methods: (1) measure the specimen at certain discrete points along the length and create an approximate CAD model and (2) reverse engineering via 3D scanning. Both methods remove geometric uncertainty by more accurately capturing the as-built geometry.

The next phase of the study involved initiating the finite element modeling process. The first step in building the FEA model was to physically measure the fully deployed boom’s parabola at multiple locations along its length to capture its shape with improved accuracy. Defining a parabolic curve can be done by measuring the height (⁠$h$⁠) and width (⁠$w$⁠), as shown in Fig. 6. However, the width of the flattened boom is manufactured with high accuracy, enabling the use of the width as the parabolic arc length (⁠$L$⁠). This enables only the width to be measured at 50-mm intervals, starting from the tip. The corresponding heights were calculated based on the arc length of the parabolic segments [32]. The arc length of a segment can be defined by evaluating the integral $∫1+(y′)2dx$ over the interval $[−w/2,w/2]$⁠. For the parabola, it is equal to $∫1+βx2dx$ where $β=64h2/w4$⁠. This equation can be simplified to obtain Eq. (1) which can then be used to obtain the height $h$⁠. A total of 22 cross sections (CS) between the tip and the “stem” regions were measured, where the stem is defined as the region from the root up to the side latches. Three width measurements were made at each cross section, and averaging was done to minimize measurement errors. To determine the number of measured cross sections between the tip and the root that would be sufficient to accurately capture the geometry of the boom, three different FEA models were created with 6, 17, and 28 cross sections, referred to as 6-CS, 17-CS, and 28-CS. Parabolic splines were created in SolidWorks using geometry measurements. These splines were copied and offset by the boom thickness to create closed cross sections. Then, the Solid Loft feature was applied to join these cross sections to create a 3D CAD model. The surface, which was created by the original splines and had the correct boom curvature, was then exported to abaqus. Geometric deviations reduced with increasing resolution but still remained large enough to warrant a more accurate shape modeling approach. The body compare tool in SolidWorks was used to quantify the geometric error of these discrete measurement models (DMM) (6-CS, 17-CS, 28-CS) and the idealized constant cross section model and compare them with the 3D laser scanner-developed Point Cloud Model as shown in Fig. 7. The boom in question was measured to have a 1.38-deg tilt. This tilt is not addressed in the 28-CS model as can be seen in the relatively large deviation near the tip region.

The error in the position of the boom tip and the parabolic shape, relative to the Point Cloud Model, is calculated for the various models and has been presented in Table 2. A large positional error of 22.69% and $−16.69%$ can be seen for the idealized constant CS in the height and width respectively. The tip distance between the Point Cloud Model and the other models is 2.97 mm along the X-axis. The DMM 28-CS model also shows significant improvement (⁠$∼16%$⁠) over the idealized constant CS model in terms of the error in the height and width of the tip parabola. The calculated height of the DMM 28-CS model has an error of 6.62% due to the parabola assumption. This difference underscores the need for the higher-fidelity Point Cloud Model. Figure 8(a) shows the tip region of the Point Cloud Model. The constant CS and DMM 28-CS models have been superimposed on the Point Cloud in Figs. 8(b) and 8(c), respectively, highlighting the tilt in the Point Cloud Model.

The high-fidelity Point Cloud Model was generated using an Artec Space Spider 3D scanner with a spatial resolution of 0.1 mm. To ensure sufficient return signal strength, a vanishing spray was applied to the boom to increase the optical response. However, the 3D scan cannot be directly inserted into FEA, necessitating further mesh processing. The process of generating an accurate CAD model from a 3D scan point cloud involved converting a scanned physical boom into a point cloud, refining it through mesh reduction, cleaning, surface fitting, and finally obtaining the CAD Model through slicing and parametric trimming. Additionally, a portion of the boom near the root (shown in Fig. 9) was located inside the CubeSat chassis and, therefore, not visible to the 3D scanner. The full process workflow is shown in Fig. 10.

The first step was to generate a mesh file by aligning the point cloud and establishing associations using the 3D scanner software. With a very fine resolution of 0.1 mm, the resulting mesh was dense and computationally heavy. For ease of postprocessing, the mesh was reduced by 53.12% of its original size from 6.4 million to 3 million triangular elements using the Windows 3D Builder application. The reduced mesh effectively reduced the number of elements without compromising the model’s level of detail. The simplified model was then imported into Meshmixer [33] for further editing and refinement. First, the mesh was aligned to the global coordinate system. Then, the stray mesh points were deleted. The edges and the surface were cleaned up using the smoothing tool, and the results can be seen in Fig. 11. Finally, the top and bottom edges were cropped using planes to have a straight datum.

Next, a nonuniform rational B-spline (NURBS) surface was fitted onto the cleaned and reduced model. A number of methods have been used to perform these types of fitting. Tutsch et al. [34] use parametric modeling methods such as the basic geometric template, Lofting, and the least square method to assess their relative surface qualities. Gao and Huang [35] used an alternate approach to define a free-form surface area using the discrete smooth interpolation method, but this led to small gaps on the surface, which needed further approximation through blending. To avoid the gaps, this effort uses a commercially available reverse engineering tool called Mesh2Surface which is available as an add-on to SolidWorks. Once the surface fit was achieved, it was extrapolated to cover the part of the boom inside the cubesat chassis which is not captured by the 3D scanner, shown in Fig. 9. The final NURBS surface is shown in Fig. 12. The resulting surface maintained 93% points within 0.2 mm of the measured locations.

The extrapolated surface was then trimmed to length and width to match the geometry of the boom, using the slicing tool in SolidWorks. The slicing tool creates curvature-defining general splines by intersecting the model with cutting planes. These splines were parametrically cut to ensure that their lengths were 70 mm which is also the flat width of the boom, shown in Fig. 12. Then, splines were drawn on the boom surface parallel to the length of the boom connecting the end points of the cross section splines. The extra material outside these lines was trimmed off on both the side edges of the boom to ensure the accuracy and completeness of the model.

Finite element models were generated in abaqus standard by importing the two types of CAD models as discussed above. The boom was modeled using 3D deformable 4-node reduced integration shell (S4R) elements. A mesh convergence study was performed to verify sufficient mesh density and an approximate global size of 1 mm was selected. The boom surface was then partitioned to mark regions for applying the various boundary conditions shown in Fig. 13. The 1.38-deg tilt in the boom, measured using the Point Cloud Model, is not captured in the discrete measurement models. The ABD matrix of the composite laminate was computed from its ply properties (Table 1) based on the classical lamination theory [36] and directly used to define a general shell stiffness section. The boom was weighed and a mass of 20.4 g [22] was used to calculate the surface density (mass per unit surface area) of the section for all the finite element models. The natural frequencies of the tape spring structure were set to be obtained via a linear perturbation step, employing the Lanczos solver in abaqus. The mass of the plastic tabs at the tip, shown in Fig. 13, was represented using a nonstructural mass spread over the tip area. In addition to this, a SPRING2 element was used to represent the torsion spring at the boom tip. A custom local coordinate system was created in the orientation of the tip parabola for the spring to define its line of action. Multiple spring stiffness values were used to ensure a high-quality correlation with the experimental results. In the end, a stiffness of 5 N/mm was applied in the X and Z directions of the spring coordinate system (see Fig. 13). Spring stiffness was also tested along the Y-axis, but it did not improve the correlation, so it was omitted. Three regions were defined for applying the boundary conditions: the root, the stopper tabs at the stem, and the roller-bearing contact region. The central portion of the root area was clamped down in the deployer and was therefore represented with an Encastre constraint. Additionally, all degrees of freedom except the X translation and X rotation were restricted at the two stopper latches at the stem as well as the roller-bearing contact region.

The study is based on several key assumptions. First, the cross section heights of the DMM are calculated based on the assumption that they form perfect parabolas. The torsion spring is simplified as a linear spring acting along two directions. It is assumed that the surface of the boom is free of defects and prestress. Spline extrapolation is assumed to be accurate, ensuring a reliable geometric representation of the part invisible to the 3D scanner. The material properties of the composite are assumed to remain unchanged and their temperature dependence is ignored. Trimming at discrete splines is deemed sufficient to capture the relevant geometry. Furthermore, the root clamp and base structure are assumed to be perfectly rigid during experimental testing, and the boom is considered to remain perfectly vertical. Gravity effects are also ignored.

The modal analysis of the tape spring booms was conducted using laser vibrometry and FEA. The outcomes from each approach were evaluated to determine the natural frequencies and mode shapes of the booms. A detailed comparison of the results is presented, highlighting the convergence and discrepancies between the techniques. The modal parameters identified provide critical insights into the dynamic behavior of the tape spring booms under complex boundary conditions, enabling a comprehensive understanding of their performance in spacecraft applications.

A summary of the results has been presented in Table 3 comparing natural frequencies between the highest fidelity 3D-scanned FEA model and the experimental model. Additionally, modal analysis results for an idealized constant cross section model from Yao et al. [22] have been included in the table for comparison. It can be observed how the additional boundary conditions, and hence the change in shape, included in this model, shift the natural frequencies. The percentage error (⁠$εF−X$⁠) between FEA and experimental data has been presented in Table 4. The first, second, and fourth modes show close agreement (within 5%), but a discrepancy is observed in the third mode. This discrepancy may be attributed to several uncertainties, which are discussed in Sec. 7.

The displacement mode shapes from the high-fidelity Point Cloud Model have been shown in Fig. 14. They agree with the experimental results. The first four FEA modes for the Point Cloud Model have the following frequencies: 4.09 Hz, 5.59 Hz, 14.59 Hz, and 21.42 Hz. The first mode is a torsion mode, with a twist at the boom tip. The second mode is a bending mode. Then, the third mode is a second torsion mode. This mode, although similar in shape to the experimental results, has a slightly different frequency value. Finally, the fourth mode is a third torsion mode.

As the laser vibrometer only measures the response in the normal direction (Z-axis), the Z-displacement mode shapes were plotted as shown in Fig. 15 for comparing the three different kinds of models. The modal displacements have been normalized by the maximum value in that model. In the higher-resolution DMM 28-CS model, the fourth mode exhibited a sweep mode with significant lateral motion in conjunction with rotation along the longitudinal axis (see Fig. 15(d)). The mode shapes obtained from the 3D scanning show the effect of the tilt incorporated into the Point Cloud Model. Notably, the fourth mode closely resembles the experimental mode, displaying a twisting nature with a node at approximately one-third of the distance from the boom tip.

Despite the increased accuracy achieved using a Point Cloud Model, several limitations limit the accuracy of the results. These limitations stem from approximations in boundary conditions and unaccounted factors such as surface defects and internal structural complexities. First of all, the geometric representation, while close, is not perfectly accurate, introducing small deviations that can affect dynamic behavior. Complex boundary conditions further complicate the model, as they are challenging to replicate precisely in simulations. The torsion spring at the tip was manually bent by hand to follow the curvature of the parabolic cross section at that location. Therefore, the spring stiffness had to be approximated for better correlation with experimental results. The mass of the tip spring and the tip flex circuit together was estimated using CAD as they were fixed to the boom and were not weighed separately prior to installation. Additionally, stiffness augmentation due to the copper traces inside the boom was neglected due to their extremely small diameter. Moreover, it has been shown that small changes in the length of the clamped portion of the root would have significant effects on the natural frequency of the boom [40]. Therefore, there is a need for a better understanding of the root and stem boundary conditions. Also, smoothing errors were ignored since their impact is difficult to measure. Finally, this study assumes no manufacturing defects and prestress. Damage observed after testing in the form of surface defects and deeper cracks in the boom, as shown in Fig. 17, contribute to the uncertainty and are challenging to quantify. Surface defects were captured by the 3D scanner but were diminished during the smoothing and fitting process. The internal defects, on the other hand, could not be captured by the Point Cloud Model.

However, this work marks significant progress in modeling bistable tape springs by leveraging point cloud data and reverse engineering for an improved overall geometric accuracy of 0.2 mm with respect to the actual boom. Approximations in boundary conditions, along with manual adjustments like spring stiffness and tip mass estimates, highlight the method’s adaptability. The study also provides a solid foundation for future advancements in deployable space structure design by identifying key areas for improvement, such as sensitivity to boundary conditions.

As deployable structures become ubiquitous in space applications, understanding the structural dynamics becomes even more important. The dynamic behavior of tape spring booms for small satellites has been studied. In this article, geometric error of the DMM (6-CS, 17-CS, 28-CS) and the idealized constant cross section model was obtained, comparing them to the 3D laser scanner-derived Point Cloud Model. The constant cross section geometry and the 28-CS model showed tip-position errors of 13.04 mm and 7.05 mm, respectively, whereas the Point Cloud Model had a negligible error of 0.1 mm. The first four resonant modes of a bistable tape spring boom were obtained and studied using three different techniques: experiment (laser vibrometry), discrete measurement, and 3D scanning. The cross-MAC was used to compare the experimental and FEA mode shapes. The modal analysis results were also compared to the idealized constant cross section model [22]. The reverse engineering approach using 3D scanning had the best correlation in both the natural frequencies and the mode shapes. The first mode for the idealized constant CS and DMM 28-CS models had an error of 27.05% and 7.69%, respectively, while the Point Cloud Model had a very small error of 1.21%. Overall, a less-than 5% difference was observed between the point cloud and experimental model frequencies for the first, second, and fourth modes. However, even with the higher accuracy of a 3D scanner, accurate FEA modeling is nontrivial, as seen from the $∼30%$ error in the third mode. The discrepancies in the results have been attributed to boundary condition errors, strain energy, geometric and material uncertainties, as well as boom defects. Closer inspection of the mode shapes reveals that mode 3 exhibits more deformation near the base of the boom, making it particularly sensitive to how the boundary conditions are modeled. This suggests that any slight inaccuracy in representing the stiffness or constraint of the latching mechanism could disproportionately affect this mode. Furthermore, the geometry near the base may not have been fully captured during the 3D-scanning process, as a portion of the boom was enclosed within the deployer and thus obscured from the scanner’s view, potentially leading to extrapolation errors in that region. Even with these challenges, this method shows that reducing geometric uncertainties is crucial for improving the accuracy of finite element models, as even small deviations can significantly impact the predicted dynamic behavior.

Future work would address modeling errors with a specific focus on the 3rd mode. Alternative non-FEA methods such as data-driven modeling techniques may better align experimental and numerical results. Additionally, it was observed that these composite booms undergo viscoelastic relaxation at higher temperatures [41]. Temperature dependency could also be addressed in future work to have higher confidence in the predictive models. These models would then be validated with actual on-orbit data from the UPS-1 mission.

We gratefully acknowledge the support from the Center for Space Science and Engineering Research at Virginia Tech (Space@VT) and VT Mechanical Engineering Department faculty Dr. Suyi Li for letting us use their lab space and equipment for the experimental work. We would also like to thank NASA Langley Research Center, Dr. Juan M. Fernandez and Dr. Matthew Chamberlain, for providing the boom samples for testing and flight and for consultations in designing the tests.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.