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Design and Experiment of Triangular Prism Mast with TapeSpring Hyperelastic Hinges
Chinese Journal of Mechanical Engineering volumeÂ 31, ArticleÂ number:Â 33 (2018)
Abstract
Because of the limited space of the launch rockets, deployable mechanisms are always used to solve the phenomenon. One dimensional deployable mast can deploy and support antenna, solar sail and space optical camera. Tapespring hyperelastic hinges can be folded and extended into a rod like configuration. It utilizes the strain energy to realize selfdeploying and drive the other structures. One kind of triangular prism mast with tapespring hyperelastic hinges is proposed and developed. Stretching and compression stiffness theoretical model are established with considering the tapespring hyperelastic hinges based on static theory. The finite element model of tenmodule triangular prism mast is set up by ABAQUS with the tapespring hyperelastic hinge and parameter study is performed to investigate the influence of thickness, section angle and radius. Twomodule TPM is processed and tested the compression stiffness by the laser displacement sensor, deploying repeat accuracy by the high speed camera, modal shape and fundamental frequency at cantilever position by LMS multichannel vibration test and analysis system, which are used to verify precision of the theoretical and finite element models of tenmodule triangular prism mast with the tapespring hyperelastic hinges. This research proposes an innovative one dimensional triangular prism with tapespring hyperelastic hinge which has great application value to the space deployable mechanisms.
1 Introduction
Conventional articulated truss structures are composed of mechanical hinges which can meet accuracy and stiffness requires of space mission. But those structures have some disadvantages, such as large weight, high friction and energywasting features. Tapespring hyperelastic (TSH) hinges, which are folded elastically can selfdeploy by releasing stored strain energy, which consist of a fewer component parts, can be manufactured conveniently [1, 2]. Flexible hinges have several advantages for space applications, including a low masstodeployedstiffness ratio, cost, and selflatch [3]. With the increasing demand, flexible hinges have been widely used as folding and deployment mechanisms in deployable structures, such as synthetic aperture radars (SARs) [4,5,6], solar arrays and antenna booms. Tapespring hinges have been used in the Japanese Mars orbiter PLANTB for barlike deployment structures of the thermal plasma analyzer [7,8,9].
The U.S. Air Force Research Laboratory (AFRL) [10] used TSH hinge in the main trussâ€™s folding longeron elements, which provided considerable snapthrough force to drive and lock the main truss. Imperial College London Santer [11] proposed a concertinafolded magnetometer boom with TSH hinges for CubeSat use. Watt et al. [12] proposed a TSH hinge with two sets of wheels held together by wires wrapped around them, and deploying impact is reduced for the added damp. The Mars Advanced Radar Express spacecraft [13, 14] consisted of two 20Â m dipoles and a 7Â m monopole which were slotted at certain intervals to stow them in a much small size. Silver et al. [15] proposed an integral folding hinge to deploy camera and investigated axial loading, bending induced buckling response. Schioler et al. and Seffen et al. [16, 17] analyzed buckling properties of single layer TSH hinge based on Timoshenko theory. Seffen et al. [18] got sample points by finite element method and obtained fitting nonlinear mechanical models by single value decomposition method of TSH hinge. Guan et al. [19, 20] designed a TSH hinge for solar sail and investigated its buckling properties by finite element method. Bai et al. [21], Yan et al. [22], Wang et al. [23] investigated geometrical and mechanical properties of ultrathinwalled lenticular collapsible composite tube in fold deformation. Yang et al. [24, 25] optimized the geometric parameters of TSH hinge to improve driving moment and reduce deploying impact, and established two different theoretical models to analyze the stability of deployment status for the TSH hinges. However, there are still some engineering problems for the TSH hinge applying to a deployable mechanism.
This paper proposed a new tenmodule triangular prism mast (TPM) with TSH hinges. Static bending stiffness and compression stiffness theoretical models are established. The compression stiffness tests are performed to verify the accuracy of the static theoretical models of the TPM. Finite element model of tenmodule TPM are establish by ABAQUS and geometrical parameter study are analyzed for bending and twisted modal frequencies. Twomodule TPM is developed to test the fundamental frequencies and related modal shapes which are used to verify the accuracy of the tenmodule TPMÂ (Additional file 1).
2 Design and Static Analysis
2.1 Structures Design
One module TPM with tapespring hinges is shown in FigureÂ 1. Adjacent twomodule is connected by triangular frame. The length of triangular frame is 469Â mm, stowed and deployed longitudinal lengths of tenmodule prism mast are 475.2Â mm and 5278Â mm respectively. All components except standard parts are manufactured by duralumin 2A12. Each modulus TPM with TSH hinge contains three longitudinal links and two triangular frames. The TSH hinges are used to drive and lock TPM. Lateral area of each modulus has a series of flexible Kevlar rope with tensile force 30Â N which ensures high stiffness of the deployed prism mast and reduces deploying impact. There is one resinbased carbon fiber drive pipe one each end of the tensile lock which prevent involvement during deploying process.
The TSH hinge folded with 180Â°, the two triangular frame close to each other, and three longitudinal links folded between the two triangular frames when the mast stowed. The tapespring drive the mast to deploy. After deploying, the TSH hinge restore original shape, the Kevlar ropes tensile and the TPM was rigidified to a structure.
2.2 Static Analysis
Bending stiffness analysis has been analyzed in Ref. [8]. Thus, stretching stiffness and compression stiffness will be derived in this paper.
2.2.1 Stretching Stiffness
Axial force F_{0} is applied on each point A, B and C. Circumcircle radius of cross section is \(R_{1} = {{l_{b} } \mathord{\left/ {\vphantom {{l_{b} } {\sqrt 3 }}} \right. \kern0pt} {\sqrt 3 }}\). Stretch force diagram is shown as FigureÂ 2.
By resolving the forces in the xdirection at point A, the equation is obtained as follows.
where F_{ l } is axial force of longitudinal link, F_{r0} is initial pretension of the rope, l_{ b } is the cross link length, Î²_{0} is the angle between line GD and line ED.
When the TPM is under stretching, deformation of longitudinal link and diagonal rope has following related equation based on geometric deformation condition.
where Î´ ^{i}_{ r } is rope deformation in each modulus, Î´ ^{i}_{ l } is longitudinal stretch deformation of each modulus, l_{ r } is rope length, F_{ r } is the force in rope, E_{ r } is the material Young modulus for rope, A_{ r } is cross section area of the rope, l_{1} is the length of longitudinal rigid link, l_{2} is the length of the TSH hinge, F_{ l } is the force in the longitudinal link, E_{1} is the material Young modulus for longitudinal link, A_{1} is the crosssection area of the longitudinal link, n_{1} is the number of the tape spring, a_{11} is the unit stretching stiffness of the TSH hinge.
Stretch deformation Î´_{l} of the TPM is
where n_{2} is the module number of the TPM, EA is the stretching stiffness of the TPM.
Simultaneous Eqs.Â (1), (2) and (3), stretching stiffness of the TPM is obtained as follows:
If the TPM has no rape, the stretching stiffness is rearranged as follows:
2.2.2 Compression Stiffness
One end of the TPM is fixed and a compressive force 3F_{0} is applied on the other end. Due to initial tension F_{r0} of rope initial deformation is Î´_{r0}â€‰=â€‰F_{r0}l_{ r }/(E_{ r }A_{ r }), which leads to a critical compress value 3F_{0}â€² on the end. Based on the geometric deformation conditions longitudinal link deformation of each module is written as follows:
When compressive force is applied at point A, equilibrium equation is gotten as follows:
Simultaneous Eq.Â (2c), Eq.Â (5) and Eq.Â (6), critical unloaded compressive force is expressed as follows:
If axial compressive force is no less than the critical unloaded compressive force, unloaded compressive stiffness equals to sum total compression stiffness of longitudinal rigid links and hyperelastic tapespring hinges, that is
If axial compressive force is more than the critical unloaded compressive force, the total deformation of the rope is
where Î´ ^{â€²}_{ r } is deformation of the rope with only axial compress force, Î´_{r0} is deformation of the rope under initial pretension force.
Single module longitudinal link and rope deformation meets the geometric deformation conditions. When 3F_{0} is lower than Fâ€², simultaneous Eq.Â (2), Eq.Â (3), Eq.Â (6) and Eq.Â (9) equivalent compression stiffness of the TPM is
It is found that when 3F_{0} is more than Fâ€² the compression stiffness is related to the compression stiffness of the longitudinal link and the TSH hinges; when 3F_{0} is lower than Fâ€², the compression stiffness changed with axial load. If the end compress load 3F_{0} is much small, that is
The compression stiffness can be written as follows
The compression stiffness is only related to the rope stiffness E_{ r }A_{ r } and initial angle Î²_{0}.
3 Deploying State Modal Analysis
3.1 Modal Analysis
Due to nonlinear characteristics of the TSH hinge modal analysis of the TPM is performed by ABAQUS. In finite element model xaxis is along direction of transverse link, yaxis is along longitudinal link and zaxis points from section center of transverse links to point of the other two transverse links. Materials of the TSH hinge, rigid link, transverse link and rope are Ni36CrTiAl, stainless steel, aluminum alloy and Kevlar respectively. Longitudinal link, transverse link and the TSH hinge are set up with four nodes that are fully integrated to reduce shell elements (S4R). Rope is modeled by two nodes and three dimensional elements (T3D2). Weld is defined to model connection between ropes and transverse links. Reference point (RP) is established at each joint which are given mass and inertial properties. Multipoint coupling is applied to model the connection of transverse links. The joint hinges are modeled by defining Hinge connection. Contact of tape springs are modeled by defining Tie constraint.
Six reference points at the root of the TPM restrain three translational Degree of Freedoms (DOFs) and the TPM is in the state of cantilever. Firstly, the TPM is performed on static analysis with 30Â N pretension force and deformation is got. Then, modal analysis is done by subspace method with rope prestress deformation. Total length for tenmodule TPM is 5.26Â m. FigureÂ 3 shows the FEM of the tenmodule TPM with TSH hinges.
The five order modal shapes of ten modules TPM with TSH hinge are shown in FigureÂ 4. The five order mode frequencies and mode shape description are listed in TableÂ 1.
3.2 Parameter Study
Compared to conventional rigid deployable mechanism the TSH hinge introduces flexible influence. Thus, it is necessary to analyze the effect of geometric parameters, such as thickness t, crosssection radius R, center angle Ï† and separation distance s, to mode frequency and propose method of increasing stiffness. TableÂ 2 lists frequencies of tenmodule TPM under different thickness. It is shown that bend frequency increases 3.695%â€“11.84% and twist frequency increases 0.186%â€“0.221% when the thickness changes from 0.12Â mm to 0.14Â mm.
TableÂ 3 lists tenmodule mast frequencies under different tape central angles. It can be seen that the bend frequency increases by 3.654%â€“6.156% and twist frequency increases by 0.393%â€“0.434% when the central angle changes from 80Â° to 100Â°.
Frequencies of the tenmodule TPM under different radiuses are listed in TableÂ 4. It can be seen that bend frequencies increase by 2.674%â€“5.343% and twist frequencies increase by 0.372%â€“0.41% when radius change from 18Â mm to 22Â mm.
Frequencies of tenmodule TPM under different separations are listed in TableÂ 5. It can be seen that bend frequencies decrease by 0.0428%â€“0.0919% and twist frequencies increase by 0.134%â€“0.141% when the separation changes from 16Â mm to 20Â mm.
It can be concluded that geometric parameters have greater influence on bend stiffness than twist stiffness. Sensitivity of the geometric parameter is from large to small as follows: thickness, central angle, radius and separation. Whatâ€™s more, front three parameters have enhanced effect on bend stiffness and the last one has induced effect.
4 Experiment Investigation
4.1 Twomodule TPM
To validate the precision of static stiffness theoretical model, two units triangular prism mast is developed. FigureÂ 5 is folding and deployment configurations of triangular prism mast with hyperelastic hinges.
Two adjacent units triangular prism mast are closed to each other by locating pins and fastened to a work holder by a rope when it is folded. At this time, the longitudinal links are stowed into the prism frames and tension ropes are located to grooves. After releasing the tension ropes, the triangular prism mast is deploying by the driving of the TSH hinges. Spherical wheels support the mast to reduce the influence of gravity.
4.2 Compression Stiffness Test
FigureÂ 6 is compression stiffness test diagram of the TPM. Weights are applied at the end of the TPM for 10Â kg at each time. Compress load and displacement for the twomodulus TPM are listed in TableÂ 6.
It can be calculated that experimental compression stiffness is 2.324â€‰Ã—â€‰10^{6}Â N/m and theoretical value is 2.167â€‰Ã—â€‰10^{6}Â N/m. Relative error between the experimental and theoretical value is 7.08%. The main reason for the phenomenon is that equivalent stiffness of the TSH hinges is much smaller, contact stiffness between the tape springs should be considered.
4.3 Deploying Repeat Accuracy Test
FigureÂ 7 shows deploying configuration for the two modulus TPM. Highspeed camera is employed to capture deploying process of the TPM. Spherical hinges and elastic rope are used to eliminate the effect of weight and friction. FigureÂ 8 shows longitudinal displacementtime relationship for five times deploying processes. TableÂ 7 lists compress load and displacement for the twomodulus TPM.
Based on longitudinal displacement for five times deploying test standard deviation of the experimental value is 1.688Â mm. Thus, deploying repeat accuracy is 1.688Â mm.
4.4 Modal Analysis of Cantilever Position
LMS multichannel vibration test and analysis system is employed to measure fundamental frequencies and mode shapes which consists of exciting hammer, acceleration sensor which can test three directional accelerations, dynamic signal acquisition system and data processor. The longitudinal direction of the TPM is set as xaxis, outer normal direction of cross section is set as yaxis and vertical downward direction is set as zaxis which is selected as force hammer stimulating direction. The measurements of three acceleration sensors, which are located at three nodes on the crossbeam, are divided into three groups. Then, integral modal superposition is carried out. The exciting point is set at one end of the crossbeam. Test apparatus and geometry diagram of cantilever position are shown in FigureÂ 9. Three vertexes on the bottom are constrained points. Alphabets a and b stand for the order of two times measurement. Location of point b_{3} is closest to the exciting point. Vibration test curves of six nodes for twomodule TPM at cantilever position are shown in FigureÂ 10. Modal shapes for the TPM at cantilever position are shown in FigureÂ 11. Modal test results for the twomodulus TPM at cantilever position are listed in TableÂ 8. It can be seen that acceleration sensor at point b_{3} has a larger response at initial phase; response curves of two group acceleration sensors are concentrated on middle and low frequency. The first order frequency of the TPM at cantilever position is 13.02Â Hz and corresponding mode shape is bending.
Then, modal analysis of the twomodule TPM at cantilever position and the modeling method is identical to the tenmodule TPM by ABAQUS. Modal shape simulation results for the twomodule TPM at cantilever position are shown in FigureÂ 12.
The model with black lines is undeformed configuration and the model with colored lines is deformed configuration. Comparison between modal test and simulation results for the twomodulus TPM at cantilever position is listed in TableÂ 9. It can be seen that the mode shapes between simulated and experimental results are consistent compared to FiguresÂ 11, 12 and TableÂ 9. The relative errors between simulated and experimental results for frequencies are no more than 5.501%. It indicates that the finite element model for the tenmodule TPM is accurate.
5 Conclusions
Structure design and deploying modal analysis of the tenmodulus TPM are performed. The twomodulus TPM is processed and its mechanical properties are investigated by experiment.

(1)
The experimental and theoretical compression stiffness static models are 2.324â€‰Ã—â€‰10^{6}Â N/m and theoretical value is 2.167â€‰Ã—â€‰10^{6}Â N/m. It verifies the accuracy of the static theoretical models.

(2)
Geometric parameters have greater influence on bend stiffness than twist stiffness. Sensitivity of the geometric parameter is from large to small as follows: thickness, central angle, radius and separation.

(3)
The deploying repeat accuracy of the twomodule TPM is 1.688Â mm which is tested by the highspeed camera.

(4)
The veracity of the finite element model of the tenmodule TPM at cantilever position is validated by modal test of the twomodule TPM. The first fundamental frequency of the tenmodule TPM is 2.561Â Hz and the corresponding mode shape is bending.
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Authorsâ€™ Contributions
HWG and RQL was in charge of the whole trial; HY wrote the manuscript; YW and ML assisted with sampling and laboratory analysis. All authors have read and approved the final manuscript.
Authorsâ€™ Information
Hui Yang, born in 1986, is currently a lecturer at College of Electrical Engineering and Automation, Anhui University, Hefei, China. She got the doctor degree from Harbin Institute of Technology, China, in 2015. Her research interest includes deployable mechanism, membrane antenna, triangular rollable and collapsible boom, tapespring hyperelastic hinge, multiobjective optimization design, deployment dynamics and finite element analysis. Email: huiyang_0431@163.com.
HongWei Guo, born in 1980, is currently an associate professor at Harbin Institute of Technology, China. His main research interest includes space manipulator system vibration control, large deployable structure and energy absorber optimization. Email: guohw@hit.edu.cn.
Yan Wang, born in 1986, is currently a senior engineer at China Electronics Technology Group Corporation No.38 Research, Hefei, China. His research interest includes configuration synthesis and design of deployable truss structures. Email: wangyan_597@163.com.
RongQiang Liu, born in 1965, is currently a professor at Harbin Institute of Technology, China. His main research interest includes wearing robot, military civil power robot and large deployable structure research. Email: liurq@hit.edu.cn.
Meng Li, born in 1985, is currently a senior engineer at Qian Xuesen Laboratory of Space Technology, China Academy of Space Technology. He got the doctor degree from Harbin Institute of Technology in 2013. His main research interest includes the optimization design of energyabsorber structures, impact dynamics and finite element method. Email: limeng@qxslab.cn.
Competing Interests
The authors declare that they have no competing interests.
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Funding
Supported by National Natural Science Foundation of China (Grant No. 51605001), Joint Funds of the National Natural Science Foundation of China (Grant No. U1637207), and Anhui University Research Foundation for Doctor (Grant No. J01003222).
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Yang, H., Guo, HW., Wang, Y. et al. Design and Experiment of Triangular Prism Mast with TapeSpring Hyperelastic Hinges. Chin. J. Mech. Eng. 31, 33 (2018). https://doi.org/10.1186/s1003301802425
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DOI: https://doi.org/10.1186/s1003301802425