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Study on the criterion to determine the bottom deployment modes of a coilable mast
Acta Astronautica 141 (2017) 89–97
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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Study on the criterion to determine the bottom deployment modes of a coilable mast Haibo Ma a, **, Hai Huang a, *, Jianbin Han b, Wei Zhang c, Xinsheng Wang a a b c
School of Astronautics, Beihang University, Beijing, 100191, China DFH Satellite Co., Ltd., CAST, Beijing, 100094, China The Institute of Manned Space System Engineering, CAST, Beijing 100094, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Bottom deployment Bottom deformation analysis Initial bottom helical angle Micro-gravity deployment test Dynamic simulation
A practical design criterion that allows the coilable mast bottom to deploy in local coil mode was proposed. The criterion was defined with initial bottom helical angle and obtained by bottom deformation analyses. Discretizing the longerons into short rods, analyses were conducted based on the cylinder assumption and Kirchhoff's kinetic analogy theory. Then, iterative calculations aiming at the bottom four rods were carried out. A critical bottom helical angle was obtained while the angle changing rate equaled to zero. The critical value was defined as a criterion for judgement of bottom deployment mode. Subsequently, micro-gravity deployment tests were carried out and bottom deployment simulations based on finite element method were developed. Through comparisons of bottom helical angles in critical state, the proposed criterion was evaluated and modified, that is, an initial bottom helical angle less than critical value with a design margin of 13.7% could ensure the mast bottom deploying in local coil mode, and further determine a successful local coil deployment of entire coilable mast.
1. Introduction Coilable masts are typical one-dimension deployable structures with high packing factor and strength-to-weight ratio which have been widely applied in spacecraft [1]. A coilable mast consists of three consecutive longerons and a series of transverse battens and diagonals, as shown in Fig. 1. The longerons and battens maintain strength and stiffness with diagonals enhanced by preload. The longerons are braced by battens and diagonals in the corners, where all three parts are connected via hinges [2]. Lanyard-deployed method is commonly used because of its low power requirement and simple mechanical configuration. A coilable mast is fixed at the bottom and deployed by paying out the lanyard while the mast top rotates during deployment [3]. Lanyard-deployed coilable mast is extensively applied in space missions, such as ST8 Sailmast [4,5], GOES Astromast [6], and Akebono Satellite Simplex Mast [7]. Fig. 2 illustrates two main deployment deformation modes of lanyarddeployed coilable mast, namely, helix mode and local coil mode. Mast deployed in local coil mode can be divided into three zones, namely, deployed, transition, and coiled zones from bottom to top, as shown in Fig. 1. Mast with a deployed zone stays in a local instability, and has good stiffness and stability to bear lateral load during deployment.
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (H. Ma), [email protected] (H. Huang). https://doi.org/10.1016/j.actaastro.2017.09.035 Received 26 May 2017; Received in revised form 29 July 2017; Accepted 25 September 2017 Available online 28 September 2017 0094-5765/© 2017 IAA. Published by Elsevier Ltd. All rights reserved.
Nevertheless, mast in helix mode is entirely unstable, and cannot provide the required lateral stiffness and stability. Hence, local coil mode is preferable for various applications. The local coil mode requires that the mast is deployed from the fixed bottom first, and the longerons subsequently uncoil from bottom to top. During the actual experiment, the bottom deployment performance has great influence on the entire mast deployment. When the mast is lanyarddeployed from a fully coiled state, a helix deployment was randomly carried out from any sections. When the bottom first one or more sections were deployed in advance, the entire mast could be successfully deployed in local coil mode. Thus, the bottom successful local coil deployment would ensure the entire mast deploying in local coil mode. The experiment results indicated that the longeron boundary condition, especially the initial bottom helical angle θini, affects the mast bottom deployment mode. Hence, study on the criterion to determine the mast bottom deployment modes would be focused on initial bottom helical angle. The obtained criterion would further determine the entire mast deployment modes. Takayuki and Han Jianbin studied Y-section hingless mast and triangle-section coilable mast, respectively, which are two kinds of coilable masts [8,9]. The influence of material characteristics and
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an initial bottom helical angle and obtained by bottom deformation analyses based on the mast technical parameters. By discretizing the longerons into short rods, theoretical results of elastic rod constrained to a cylinder were applied for each rod equilibrium state analysis. The critical bottom helical angle was subsequently obtained while the bottom helical angle changing rate equaled to zero. Micro-gravity deployment tests were developed, and bottom deployment simulations based on finite element model were performed. Finally, bottom deployment mode validations were conducted based on the test and simulation results. Furthermore, a practical margin of initial bottom helical angle was obtained to design the mast deployment in local coil mode. 2. Criterion determination method For convenience, the consecutive longerons are discretized into short rods. As shown in Fig. 3, from the bottom, the sections are defined as Section 1, Section 2, …, and the corresponding short rods are Rod 1, Rod 2, …. The transverse batten is defined as Batten 1, Batten 2, …. In this work, the section number of transition zone in local coil mode is set to 4; thus, the longeron helical angle changes from zero to θcoil (helical angle of coiling zone) in four sections. When Section 5 is just starting to deploy, namely, the contact force between Battens 5 and 6 decreases to zero, the four-section transition zone reaches an equilibrium state under the lanyard tension and boundary conditions. Coilable mast at this moment is defined as the mast critical state, and the critical bottom helical angle is defined similarly. Because of assembling hinges with transverse battens, the longerons can rotate freely at the bottom of the coilable mast along the radial direction of circumscribed circle of mast cross section. The relevant bottom external torque is zero. Thus, the bottom helical angle changing rate along the longeron curvilinear coordinate s, (dθ/ds)bottom, equals to zero when the mast is in critical state. If an external torque M is applied at the longeron bottom as shown in Fig. 3, longeron deformation of four-section transition zone in critical state variates accordingly, and the relevant angle changing rate would not equal to zero. The signs of angle changing rate are depending on the torque directions. Only if the bottom external torque is along the direction as shown in Fig. 3, bottom helical angle could be decrease from θini to zero. The relevant angle changing rate is required to be greater than zero, that is,
Fig. 1. Coilable mast in local coil mode.
Fig. 2. Deployment deformation modes of coilable mast.
mechanical configurations on the deformation modes was analyzed. The analysis indicated that properly selecting the values of batten pitch and stiffness ratio of longerons to battens resulted in the desired local coil mode. However, both studies required that the mast was deformed with deployed, transition, and coiled zones. When coilable mast bottom deploys, the mast has not achieved an obvious deployed zone. Hence, the results proposed by Takayuki and Han Jianbin were not suitable for analysis of coilable mast bottom deployment. By virtue of elastic rod theory based on kinetic analogy, coilable mast longerons during the deployment can be analyzed as twisted elastic rods constrained to a cylinder. Seemann and Van studied the deformation and analyzed the stability and equilibrium of thin elastic rods constrained to a cylinder [10,11]. Benham and Le established the elastic model of duplex DNA in the 1970's. Bernard discussed the self-contact of an elastic rod model for DNA based on the assumption that the rod under consideration obeys Kirchhoff's theory [12]. Liu studied the Kirchhoff kinetic analogy theory and proposed the mathematical equilibrium equations of a circular cross section elastic rod constrained to a cylinder [13]. Based on Liu's research, Li and Zhao built the physics-based mode for a cable harness [14]. However, the mentioned mechanical models were analyzed with ideal boundary conditions at both ends only. For coilable mast, elastic rod theory cannot be directly applied because of lateral forces caused by transverse battens. To solve this problem, the longerons were discretized to obtain a series of thin elastic rods with ideal boundary conditions at both ends only. Then, elastic rod theory is suitable for longeron deformation analysis and bottom helical angle calculation. The criterion that allows the mast bottom to deploy in local coil mode can be obtained. In this paper, a practical design criterion that allows the mast bottom to deploy in local coil mode was proposed. The criterion was defined with
dθ ds
>0
(1)
bottom
Through the aforementioned analyses, different angle changing rates
Fig. 3. Longeron discretization and mast definition. 90
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the coordinate system P-x1y1z1 by θ around the Px1 axis and obtain the coordinate system P-x2y2z2. All mentioned coordinates are shown in Fig. 4. Taking φ as the rotating angle around the axis Pz2, angles ψ, θ, and φ are selected to define the dimensional orientation of the rod cross section. Angle variations along the arc length dψ/ds, dθ/ds, and dφ/ds are used to define the rod deformation. dφ/ds defines the relative rotating angle between two adjacent cross sections, whereas dψ/ds and dθ/ds indicate the dimensional orientation of the normal line of the rod cross section. Considering each short rod after discretization, rod deformation satisfies Equation (2) based on Liu's research [13], where i ¼ 1, 2, 3, 4, corresponding to different rods in Fig. 3.
could be calculated via adjusting the bottom helical angles. And a critical value could be obtained while the angle changing rate equals to zero. Judging with Equation (1), a criterion that ensure the mast bottom deploying in local coil mode could be proposed. 3. Bottom deformation analysis Ignoring the mast deployment dynamic effects and the coiling diameter decrease in the transition zone, a cylinder assumption is proposed, namely, the deploying longerons can be regarded as twisted elastic rods constrained to a cylinder. A mathematical bottom deformation analysis is applied for criterion determination based on Liu's elastic
8 2 d θi 1 pi 2 cos θi sin2 θi > > > ðl þ sin θi ¼ cos θ m cos 2θ Þ þ 0i i i i > > R 2 ds2 R2 > > > > > > R sin3 θi M0i > > < FYi þ mi cos θi þ ¼ l0i ¼ A R A > > > C dψ dφ > > mi ¼ ω30i ω30i ¼ i cos θi þ i > > A ds ds > > > > > 2FZi > : pi ¼ A
ði ¼ 1; 2; 3; 4Þ
(2)
In Equation (2), θi represents the rod helical angle of a cross section; s is rod curvilinear coordinate; pi, l0i, mi are the integration constants concerned with rod boundary conditions; A and C are cross section bending rigidity and torsion rigidity, respectively; M0i is the cross section external torque along the Oζ axis; R is the coiling radius; FYi and FZi are the cross section external forces along the PY and PZ axes, respectively. And ψ i(s) and θi(s) satisfy the following equation.
rod research under the cylinder assumption. The coilable mast is a spaceaxisymmetric structure. Thus, deformation analysis focuses on the discretized rods of one single longeron.
3.1. Rod deformation equation After the discretization, each rod is considered as ideally linearly elastic, homogeneous, one-dimension body, and is described by its centerline in a curvilinear coordinate system. The rotation angle between two adjacent cross sections is a continuous function of curvilinear coordinate s. Similar to Liu's research [13], a rectangular coordinate system O-ξηζ is introduced firstly, with the cylinder central axis as Oζ axis and Point O as the coordinate origin. And the Oξ axis is defined randomly. Rotate the coordinate system O-ξηζ by Ψ around the Oζ axis and obtain the coordinate system O-XYZ. Then translate the coordinate system O-XYZ from Point O to Point P, central point of the researched rod cross section. Rotate the coordinate system P-XYZ by 180 around the PZ axis and obtain the coordinate system P-x1y1z1, where ψ ¼ Ψ þ π. Finally, rotate
dψ i ¼
ds⋅sin θi R
(3)
ω30i is rod torsional curvature. Longerons are assembled with transverse battens via hinges, which allow the longerons to rotate along the PX axis merely. However, rotate along the Pz2 axis is limited; thus, the integral of dφi/ds in the interval [0, t] is zero, where t is batten pitch length. Thus, the integral of ω30i in the interval [0, t] is
t
t
∫ 0 ω30i ds ¼∫ 0 t
¼ ∫0
dψ i dφ t dψ i t dφ cos θi þ i ds ¼ ∫ 0 cos θi ds þ ∫ 0 i ds ds ds ds ds
sin θi cos θi ds R (4)
Torsional curvature remains constant while the rod was constrained at both ends only [13], then t
∫0
sin θi cos θi ds ¼ ω30i t R
(5)
3.2. Rod integral conditions and initial values Taking Rod i of Fig. 3 as analysis objects and disregarding the
Fig. 4. Different coordinate definition.
Fig. 5. Cross Section External Forces and Torques of the Top End of Rod i. 91
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dynamic effects, boundary force analyses are conducted to determine FYi, FZi, and M0i. Cross section external forces and torques of the top end of Rod i (s ¼ 0) are shown in Fig. 5, where θi(0) is the top helical angle of Rod i. Among them, MZi(0) could be synthesized with two rectangular components My2i(0) and Mz2i(0) along the axes of Py2 and Pz2.
8 2 < M ð0Þ ¼ M ð0Þ þ M ð0Þ ¼ A sin θi ð0Þ sin θ ð0Þ þ Cω cos θ ð0Þ Zi y2i z2i i 30i i R : M0i ¼ MZi ð0Þ þ FYi ð0ÞR (6)
Fig. 7. Infinitesimal balance analysis.
The coiling zone and top plate, together with the sections above Rod i, are considered as a unity T. Thus, T is in a balance state with lanyard tension (FL), rod reactive forces and torques, and the diagonal tension shown in Fig. 6. Fri, Fvi, and Fhi are three components of diagonal tension in P-XYZ. The coilable mast is a space-axisymmetric structure. Thus, three longerons exert the same reactive forces and torques on the unity T. Through the balance analysis along the Oζ axis, MZi(0), FL, FYi(0), FZi(0), Fvi, and Fhi would satisfy the Equation (7).
FL þ 3FZi ð0Þ þ 3Fvi ¼ 0 3ðMZi ð0Þ þ FYi ð0ÞR þ Fhi RÞ ¼ 0
(7)
Considering an infinitesimal of intersection point between the adjacent rods as the analysis object, a balance state is illustrated as shown in Fig. 7 and a force analysis along the PX and PZ axis is described in Equation (8).
FYi ð0Þ þ Fhi ¼ FYiþ1 ðtÞ þ Fhiþ1 FZi ð0Þ þ Fvi ¼ FZiþ1 ðtÞ þ Fviþ1
Fig. 8. Section deformation analysis.
Combining the equations from (6) to (9), the required integral conditions, FYi, FZi, and M0i, are obtained for each rod. Integration constants, pi, l0i, and mi, are subsequently applied for rod deformation analyses. Since the diagonals are not always tensioned in several sections of the transition zones, further settings are conducted on three components of diagonal tension. As observed in experiments, the diagonals are slack in Sections 4 and 3. Thus, for i ¼ 3, 4, Fvi, and Fhi equal to zero, then the relevant mathematic expressions of FYi, FZi, and M0i are simplified without solving Equation (9). The analyses are carried out from Section 4 to 1. Since, the longeron is continuously deformed on the top of Rod 4, θ4(0) equals to θcoil. Based on Han's research results [9], the rod helical angle of equilibrium state in Sections 2–4 exhibits an approximate linear change, and the rod helical angle in Section 1 remains nearly invariant. Then
(8)
where FYiþ1(t) and FZiþ1(t) could be solved from FYiþ1(0) and FZiþ1(0) based on the mentioned integration constants piþ1 and l0iþ1 in Equation (2). The components of diagonal tension could be obtained based on the section deformation analysis as shown in Fig. 8. The diagonal length is assumed as l; then, its three components in P-XYZ are lri, lvi, and lhi. The components of diagonal length and tension share the same geometry relationship. Thus, Fri, Fvi, and Fhi could be calculated based on the following Equation (9).
8 > > > > > > > > > > > > > > > > > > <
Fli Fhi Fvi Fri ¼ ¼ ¼ l lhi lvi lri
8 <
θ4 ð0Þ ¼ θcoil
: dθ4 ¼ θbottom θcoil ds s¼0 3t
lvi ¼ ∫ t cos θi ds
sin θi ds αi ¼ ∫ t R π α π α > i i > > > lhi ¼ 2R cos 3 þ 2 sin 3 þ 2 > > > π α > > i > > lri ¼ 2R sin2 þ > > 3 2 > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > > : l¼ l2vi þ l2hi þ l2ri
(9)
(10)
where θbottom is the bottom helical angle of Rod 1, namely, the longeron bottom helical angle. For Section 3 to 1, longeron is similarly continuously deformed. And the helical angle changing rate dθi/ds satisfies the equation below according to Liu's research [13].
8 > <
dθi
> :
ds
s¼0
θi ð0Þ ¼ θiþ1 ðtÞ
3 dθiþ1
3 sin 2θi ð0Þ ¼ sin 2θ mi m ðtÞ iþ1 iþ1 2R 2R ds s¼t (11)
3.3. Bottom deformation analyses The bottom deformation analyses are carried out with successive integrals from Section 4 to 1. The values θbottom and FL are assumed initially. Different conditions and initial values are obtained, and then and are subsequently gained, where s changes from zero to t. Changing the value of FL, an iterative calculation from Rods 4 to 1 is
Fig. 6. Balance state analysis of the unity T. 92
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Fig. 10. Longeron deformations in equilibrium state.
Fig. 9. Analysis procedure of bottom deformation.
Table 1 Technical parameters of the coilable mast. Parameter
Value
Coiling Radius (R) Batten Pitch Length (t) Longeron and Batten Material Longeron and Batten Elasticity Modulus (E) Longeron and Batten Poisson's Ratio (σ) Longeron Radius (r) Batten Radius Hinge Material Hinge Elasticity Modulus Hinge Poisson's Ratio
75 mm 95 mm Titanium-Nickel Alloy 83 GPa 0.31 1 mm 0.4 mm Aluminum Alloy 70.6 GPa 0.33
Fig. 11. Helical angle variations of the four-section longeron.
performed until θ1(t) ¼ θbottom. Then, another iteration is conducted via adjusting θbottom to obtain dθ1(t)/ds ¼ 0. The final obtained θbottom is the critical bottom helical angle, and relevant FL is the critical lanyard tension. The complete bottom deformation analysis procedure is shown in Fig. 9. A lanyard-deployed coilable mast was developed for a microsatellite as a semi-rigid gravity gradient boom to obtain a primary attitude stabilizing state. Coilable mast technical specifications, such as stiffness, strength, packing factor, and dimension, were restricted by the stabilization requirement. The present coilable mast technical parameters are shown in Table 1. Based on Han’ research, technical parameters of material characteristics and mechanical configurations satisfy the requirement of deployment in local coil mode. A successful bottom deployment would ensure the entire coilable mast deploying in local coil mode. Other initial parameters were required for bottom deformation analyses. The distance between the adjacent battens in the coiling zone is 8 mm, and the required θcoil ¼ arccos (8/95) ¼ 85.17 .
A¼ C¼
Eπr 4 4 Gπr 4 2
E 2ð1 þ σÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ 3R2 þ t 2
Table 2 Performance of different initial bottom helical angles.
G¼
Initial Bottom Helical Angle ( )
Changing Rate ( /mm)
5 10 15 20 23.164 25 30 35 40 45 50 55 60 65 70 75 80
0.18208 0.13648 0.08807 0.03871 0.00097 0.01065 0.05933 0.10667 0.15208 0.19494 0.23444 0.26946 0.29875 0.32000 0.33007 0.32266 0.28187
critical bottom helical angle is 23.164 .
(12) 3.4. Criterion determination
Bottom deformation analyses are carried out and the results represent longeron deformation in critical state as shown in Fig. 10. The obtained
Adjusting the initial bottom helical angles from 5 to 80 , a series of results is obtained as shown in Fig. 11. And the relevant angle changing 93
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rates along the curvilinear coordinate s are listed in Table 2, which represents the required external torques. Judging with Equation (1), a criterion is obtained; thus, an initial bottom helical angle that is less than the critical value, 23.164 , can ensure mast bottom deploying in local coil mode. In addition, the qualitative kinetic analysis results are consistent with the judgement based on the obtained criterion. When the bottom helical angle is greater than the critical angle, the required external torque is along the –PX axis, which would compress the longeron to the coiling state. This is unexpected during the coilable mast deployment. By contrast, bottom helical angle that is less than the critical value requires the external torque along the þPX axis, which will deploy the coilable mast. Fig. 13. Bottom variable hinge connection.
4. Micro-gravity deployment test and dynamic simulation
The data collection equipment included two sensors, namely, the tension dynamometer for the axial lanyard tension and the MEMS gyroscope for the rotating angular velocity of the mast top plate. When the mast was deployed, both sensors acquired the measurements sequentially (five targets per second).
Micro-gravity deployment test and dynamic simulation were conducted. Different initial bottom helical angles were applied in the tested and simulated models. Bottom deployment performances were applied to further validate the criterion effectiveness.
4.1.2. Bottom helical angle restriction The bottom helical angle should be restricted to achieve different bottom boundary conditions. A variable hinge connection was designed and illustrated in Fig. 13. By adjusting the locations of restriction pin, a series of rotation angle ranges were restricted, and the relevant initial bottom helical angles were obtained while being compressed. Fig. 13 indicates that the initial bottom helical angles were restricted as 0 , 15 , 20 , 30 , 45 , 60 , and 75 . To further validate the result, a constant deployment velocity of 15 mm/s was selected. The contact force between Battens 5 and 6 was difficult to measure. Thus, the critical state of the four-section deployment was defined as the time when Batten 6 started to separate from Batten 5. A typical longeron deformation in critical state is illustrated in the following results validation. When the coilable mast was purely coiled, (i.e., the initial bottom helical angle was 87.04 ), the deployment was randomly performed from any section. Hence, deployment tests focused at the initial bottom helical angle from 0 to 75 .
4.1. Micro-gravity deployment test 4.1.1. Micro-gravity deployment facility A one-dimensional deployment facility was developed for microgravity deployment test, as shown schematically in Fig. 12. The facility consisted of a truss base, linear track, synchronizing system, and data collection equipment. The truss base supported the entire facility and formed the effective deployment space with horizontal linear track. The synchronizing system consisted of a coaxial lanyard bobbin, linear bearing, and balance weight. The coaxial lanyard bobbin released the coilable mast and linear bearing via a stepper motor. Balance weight was connected to the linear bearing to ensure the bearing's synchronous movement with the mast top plate. The key to the micro-gravity deployment test was to ensure the gravity compensation effect. By adjusting the vertical angle of suspended rope, linear bearing should be located right above the top plate before test, and the error between the suspended rope and plumb line was restricted within 3 . Furthermore, all the lanyards should be wrapped around the coaxial bobbin in the same order to ensure the releasing synchronism. Balance weight was determined based on the bearing frictions via actual releasing test.
4.2. Dynamic simulation The dynamic simulations were performed using the software
Fig. 12. Micro-gravity deployment facility. 94
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Fig. 14. Coilable mast finite element model. Table 3 Results of simulation and test for bottom helical angle.
MSC.ADAMS, which is widely used in dynamic design of aeronautics and astronautics. Elastic parts such as the longerons and transverse battens were discretized into a series of rigid parts while connecting with the elements of Discrete Flexible Link. The diagonals, with limited dimension and mass, are simplified to a pair of action and reaction along the diagonal direction. And the pair of action and reaction is set with a step function changing with the distance between two ends of action and reaction. All the finite element parts were defined with the characteristics of the actual coilable mast mentioned in Table 1. A 19-section coilable mast model was established, as shown in Fig. 14. Initial bottom helical angles were individually defined for different simulation cases. Torsion springs were added at the longeron bottom. After compression, the required longeron bottom helical angle was restricted with the step-functional spring stiffness. The coilable mast was deployed with a constant velocity of 15 mm/s, consistent with the tests. To further validate the result, the critical state of the four-section deployment was defined similarly as the mathematical calculations, namely, when the contact force between Battens 5 and 6 decreased to zero. Typical longeron deformations in critical state are illustrated in the following result validation.
Initial Bottom Helical Angle ( )
Simulated Bottom Helical Angle ( )
Tested Bottom Helical Angle ( )
0 15 20 30 45 60 75
0 1.8348 1.9529 2.6207 36.4866 36.5609 36.4865
9.8 9.2 4.1 30 42.2 42.8 42.6
Local Coil Local Coil Local Coil Local Coil Helix Helix Helix
Local Coil Local Coil Local Coil Helix Helix Helix Helix
5. Result validation and discussion The obtained criterion that ensure the coilable mast bottom deploying in local coil mode was validated by comparisons of bottom helical angle in critical state. A practical design margin of initial bottom helical angle was provided for criterion modification. Furthermore, the section number of transition zone was discussed, and the relevant bottom deployment design was conducted. 5.1. Bottom deployment performance validation Fig. 15. Tested and simulated deformation with initial bottom helical angle of 15 .
The initial bottom helical angle determines whether the mast bottom could be deployed in local coil mode. During the coilable mast deployment, the bottom helical angle decreased from the initial value to zero, and the coiled rod in Section 1 became straight. Bottom deployment, namely, Section 1 deployment, could be assessed based on whether the bottom helical angle was approximately equal to zero. Both tests and simulations were conducted with different initial helical angles of 0 , 15 , 20 , 30 , 45 , 60 , and 75 . Table 3 presents the tested and simulated results with different initial helical angles in critical state. When the initial value was not greater than 20 , both the tested and simulated bottom helical angles were less than 3 in critical state, that is, the mast bottom had been deployed in critical state. Deployment modes were verified based on the results in Table 3, and the presented simulation and tested results were consistent with the criterion described in Chapter 3.4. The two data sets of tested and simulated longeron deformations are evaluated as shown in Fig. 15 and Fig. 16. With an initial bottom helical angle of 15 , both longeron deformations indicated that Section 1 was fully deployed and the transition zone included four sections at most. Deformations satisfied the local coil mode definition. However, with an initial bottom helical angle of 60 , both longeron deformations showed that Section 1 was not fully deployed. Transition zone would include at least five sections, which indicated that deformations did not satisfy the local coil mode definition. For the simulated results, another method would be applied to assess the bottom deployment mode. The time when the bottom helical angle
Fig. 16. Tested and simulated deformation with initial bottom helical angle of 60 .
first decreased to less than 3 is defined as t1, and the time when the contact force between Battens 5 and 6 decreased to zero is set to t2. If t1 95
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Acta Astronautica 141 (2017) 89–97 Table 5 Calculation results with different section numbers. Section Number
Critical Bottom Helical Angle ( )
Practical Bottom Helical Angle with Margin ( )
3 4 5
18.677 23.164 31.505
16.12 20 27.19
13.7% could be effectively applied to determine the bottom deployment mode. That is, an initial helical angle less than the critical value with a 13.7% margin could ensure mast bottom deploying in local coil mode. Moreover, the obtained criterion could further determine a successful local coil deployment of entire coilable mast [9].
5.2. Section number discussion of transition zone in local coil mode Fig. 17. Variations of the bottom helical angle and contact force between battens 5 and 6.
All the calculations and validations were based on the definition that the section number of transition zone in local coil mode was 4. Adjusting the section number, analysis objective of bottom deformation could be changed, and the calculation results with different section numbers are presented in Table 5. Table 5 showed that the obtained critical bottom helical angle varied with different section numbers of transition zone. More sections contained in the transition zone required a greater critical bottom helical angle. The result could be explained based on the mast deployment motion similar with Fig. 17. Along with mast deployment in local coil mode, sections separated with each other successively from Section 1, and the bottom helical angle decreased from the initial value to zero. With the same initial bottom helical angle, a transition zone with less sections could cause the relevant contact force to decrease to zero earlier. Therefore, to shorten the duration of the bottom helical angle decreasing to zero, a small initial angle was required. Vice versa, a transition zone with more sections required a greater initial bottom helical angle. In actual applications, a transition zone with more sections would lower the lateral bending stiffness and stability, which are undesirable in the mast design. According to the aforementioned discussion, the initial bottom helical angle should be set as small as possible to obtain less sections of transition zone and achieve higher lateral bending stiffness and stability.
Table 4 Time comparisons and deployment mode determination. Initial Bottom Helical Angle ( )
t1 (s)
t2 (s)
Deployment Mode
0 15 20 30 45 60 75 87.04 (Pure Coiling)
0 8.2 11.3 14.9 18.6 19.4 20.7 21.4
15 15.4 15.6 16.1 13.5 14.3 15.5 16.2
Local Coil Local Coil Local Coil Local Coil Helix Helix Helix Helix
t2, at most four sections exist in the transition zone, and the coilable mast would be deployed from the bottom in the local coil mode. Fig. 17 represents the variations of the bottom helical angle and contact force between Battens 5 and 6 when the initial helical angle was 30 . Time comparisons were performed for all eight data sets, and deployment modes were determined as shown in Table 4. The influence of initial helical angle on the deployment mode was validated to be consist with the obtained criterion. The simulation results in Table 3 indicate that mast bottom was deployed in local coil mode with the initial bottom helical angle of 30 . However, the tested results presented a helix mode. Analysing the difference of calculation results and test, the error was mainly resulted from the model difference. It can be divided into three parts:
6. Conclusion A practical design criterion that allows the coilable mast bottom to deploy in local coil mode is proposed in this study. With the bottom deformation analyses based on the mast technical parameters, the criterion was defined that an initial bottom helical angle less than the critical value with a design margin of 13.7% could ensure mast bottom deploying in local coil mode, and further determine a successful local coil deployment of entire coilable mast. The criterion with a design margin is evaluated by the micro-gravity deployment tests and dynamic simulations. With existed elastic rod theory, longeron discretization and successive integrals presents a practical mathematical analysis method for longeron deformation. And the calculated results are consistent with the tested and simulated results. In addition, the initial bottom helical angle should be set as small as possible during the actual mast design. Then the transition zone with less sections could be obtained, and lateral bending stiffness and stability of the coilable mast would be improved.
● Mathematic calculation was conducted under the cylinder assumption, ignoring the coiling radius decrease. However, the radius decrease is indispensable to the mast deployment. The assumption introduction could bring in the calculation error, which is inevitable. ● There were several frictional damping existed in the tested mast model, which had not been considered accurately in the mathematic calculations and simulations. The frictional damping were mainly existed in the mast hinges, and consumed some energy that was used for mast deployment. ● Mathematic calculations and simulations were conducted with uniform mast configuration and material characteristics. And the obtained results were ideal. However, the tested model had to be faced with the mast assembly error and the inevitable material inhomogeneity, which could affect the final deployment performance obviously. Considering these, the difference between two sets of results was acceptable, and a conservative design margin was required to reduce the model error. The initial value of 20 could ensure both tested and simulated mast bottoms deploying in local coil mode. Thus, a practical design margin was defined with (20 23.164 )/23.164 100% ¼ 13.7%. Therefore, the obtained mathematical result with a design margin of
Acknowledgements This research work is supported by the National Natural Science Foundation of China (Grant No.11672016), which the authors gratefully acknowledge. 96
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Acta Astronautica 141 (2017) 89–97 [7] T. Kitamura, K. Okazaki, M. Natori, et al., Development of a “hingeless mast” and its applications, Acta Astronaut. 17 (1988) 341–346. [8] T. Kitamura, K. Okazaki, K. Yamashiro, Developments of Extendible Beams for Space Applications, Aerospace Design Conference, Irvine, CA, 1993. [9] J. Han, X. Wang, H. Ma, et al., Mechanical principle of the deploying mode for coilable mast, J. Beijing Univ. Aeronaut. Aeronaut. 39 (2013) 1168–1173. [10] W. Seemann, Deformation of an elastic helix in contact with a rigid cylinder, Arch. Appl. Mech. 67 (1996) 117–139. [11] G.H.M. Van Der, The static deformation of a twisted elastic rod constrained to lie on a cylinder, Proc. R. Soc. A 457 (2001) 695–715. [12] Bernard D. Coleman, David Swigon, Theory of self-contact in Kirchhoff rods with applications to supercoiling of knotted and unknotted DNA plasmids, Phil. Trans. R. Soc. A Math. Phys. Eng. Sci. 362 (2004) 1281–1299. [13] Y. Liu, Nonlinear Mechanics of Thin Elastic Rod: Theoretical Basis of Mechanical Model of DNA, Tsinghua University Press, Beijing, 2006, pp. 80–83. [14] J. Liu, Z. Tao, R. Ning, et al., Physics-based modeling and simulation for motional cable harness design, Chin. J. Mech. Eng. 27 (2014) 1075–1082.
References [1] Johnson L, Alexander L, Fabisinski L, et al, Multiple NEO Rendezvous Using Solar Sail Propulsion, Global Space Exploration Conference. Washington DC. 2012. [2] D.M. Murphy, Validation of a scalable solar sailcraft system, J. Spacecr. Rockets 44 (2015) 797–808. [3] L. Puig, A. Barton, N. Rando, A review on large deployable structures for astrophysics missions, Acta Astronaut. 67 (2010) 12–26. [4] David M Murphy, Michael E McEachen, Brian D, Macy. Demonstration of a 20-m Solar Sail System. 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference. Austin, Texas. 2005. [5] Michael E. McEachen, Validation of SAILMAST Technology and Modeling by Ground Testing of a Full-Scale Flight Article. 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition. Orlando, Florida. 2010. [6] M. Eiden, O. Brunner, C. Stavrinidis, Deployment analysis of the olympus Astromast and comparison with test measurements, J. Spacecr. Rockets 24 (1987) 63–68.
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Contents lists available at ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Study on the criterion to determine the bottom deployment modes of a coilable mast Haibo Ma a, **, Hai Huang a, *, Jianbin Han b, Wei Zhang c, Xinsheng Wang a a b c
School of Astronautics, Beihang University, Beijing, 100191, China DFH Satellite Co., Ltd., CAST, Beijing, 100094, China The Institute of Manned Space System Engineering, CAST, Beijing 100094, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Bottom deployment Bottom deformation analysis Initial bottom helical angle Micro-gravity deployment test Dynamic simulation
A practical design criterion that allows the coilable mast bottom to deploy in local coil mode was proposed. The criterion was defined with initial bottom helical angle and obtained by bottom deformation analyses. Discretizing the longerons into short rods, analyses were conducted based on the cylinder assumption and Kirchhoff's kinetic analogy theory. Then, iterative calculations aiming at the bottom four rods were carried out. A critical bottom helical angle was obtained while the angle changing rate equaled to zero. The critical value was defined as a criterion for judgement of bottom deployment mode. Subsequently, micro-gravity deployment tests were carried out and bottom deployment simulations based on finite element method were developed. Through comparisons of bottom helical angles in critical state, the proposed criterion was evaluated and modified, that is, an initial bottom helical angle less than critical value with a design margin of 13.7% could ensure the mast bottom deploying in local coil mode, and further determine a successful local coil deployment of entire coilable mast.
1. Introduction Coilable masts are typical one-dimension deployable structures with high packing factor and strength-to-weight ratio which have been widely applied in spacecraft [1]. A coilable mast consists of three consecutive longerons and a series of transverse battens and diagonals, as shown in Fig. 1. The longerons and battens maintain strength and stiffness with diagonals enhanced by preload. The longerons are braced by battens and diagonals in the corners, where all three parts are connected via hinges [2]. Lanyard-deployed method is commonly used because of its low power requirement and simple mechanical configuration. A coilable mast is fixed at the bottom and deployed by paying out the lanyard while the mast top rotates during deployment [3]. Lanyard-deployed coilable mast is extensively applied in space missions, such as ST8 Sailmast [4,5], GOES Astromast [6], and Akebono Satellite Simplex Mast [7]. Fig. 2 illustrates two main deployment deformation modes of lanyarddeployed coilable mast, namely, helix mode and local coil mode. Mast deployed in local coil mode can be divided into three zones, namely, deployed, transition, and coiled zones from bottom to top, as shown in Fig. 1. Mast with a deployed zone stays in a local instability, and has good stiffness and stability to bear lateral load during deployment.
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (H. Ma), [email protected] (H. Huang). https://doi.org/10.1016/j.actaastro.2017.09.035 Received 26 May 2017; Received in revised form 29 July 2017; Accepted 25 September 2017 Available online 28 September 2017 0094-5765/© 2017 IAA. Published by Elsevier Ltd. All rights reserved.
Nevertheless, mast in helix mode is entirely unstable, and cannot provide the required lateral stiffness and stability. Hence, local coil mode is preferable for various applications. The local coil mode requires that the mast is deployed from the fixed bottom first, and the longerons subsequently uncoil from bottom to top. During the actual experiment, the bottom deployment performance has great influence on the entire mast deployment. When the mast is lanyarddeployed from a fully coiled state, a helix deployment was randomly carried out from any sections. When the bottom first one or more sections were deployed in advance, the entire mast could be successfully deployed in local coil mode. Thus, the bottom successful local coil deployment would ensure the entire mast deploying in local coil mode. The experiment results indicated that the longeron boundary condition, especially the initial bottom helical angle θini, affects the mast bottom deployment mode. Hence, study on the criterion to determine the mast bottom deployment modes would be focused on initial bottom helical angle. The obtained criterion would further determine the entire mast deployment modes. Takayuki and Han Jianbin studied Y-section hingless mast and triangle-section coilable mast, respectively, which are two kinds of coilable masts [8,9]. The influence of material characteristics and
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an initial bottom helical angle and obtained by bottom deformation analyses based on the mast technical parameters. By discretizing the longerons into short rods, theoretical results of elastic rod constrained to a cylinder were applied for each rod equilibrium state analysis. The critical bottom helical angle was subsequently obtained while the bottom helical angle changing rate equaled to zero. Micro-gravity deployment tests were developed, and bottom deployment simulations based on finite element model were performed. Finally, bottom deployment mode validations were conducted based on the test and simulation results. Furthermore, a practical margin of initial bottom helical angle was obtained to design the mast deployment in local coil mode. 2. Criterion determination method For convenience, the consecutive longerons are discretized into short rods. As shown in Fig. 3, from the bottom, the sections are defined as Section 1, Section 2, …, and the corresponding short rods are Rod 1, Rod 2, …. The transverse batten is defined as Batten 1, Batten 2, …. In this work, the section number of transition zone in local coil mode is set to 4; thus, the longeron helical angle changes from zero to θcoil (helical angle of coiling zone) in four sections. When Section 5 is just starting to deploy, namely, the contact force between Battens 5 and 6 decreases to zero, the four-section transition zone reaches an equilibrium state under the lanyard tension and boundary conditions. Coilable mast at this moment is defined as the mast critical state, and the critical bottom helical angle is defined similarly. Because of assembling hinges with transverse battens, the longerons can rotate freely at the bottom of the coilable mast along the radial direction of circumscribed circle of mast cross section. The relevant bottom external torque is zero. Thus, the bottom helical angle changing rate along the longeron curvilinear coordinate s, (dθ/ds)bottom, equals to zero when the mast is in critical state. If an external torque M is applied at the longeron bottom as shown in Fig. 3, longeron deformation of four-section transition zone in critical state variates accordingly, and the relevant angle changing rate would not equal to zero. The signs of angle changing rate are depending on the torque directions. Only if the bottom external torque is along the direction as shown in Fig. 3, bottom helical angle could be decrease from θini to zero. The relevant angle changing rate is required to be greater than zero, that is,
Fig. 1. Coilable mast in local coil mode.
Fig. 2. Deployment deformation modes of coilable mast.
mechanical configurations on the deformation modes was analyzed. The analysis indicated that properly selecting the values of batten pitch and stiffness ratio of longerons to battens resulted in the desired local coil mode. However, both studies required that the mast was deformed with deployed, transition, and coiled zones. When coilable mast bottom deploys, the mast has not achieved an obvious deployed zone. Hence, the results proposed by Takayuki and Han Jianbin were not suitable for analysis of coilable mast bottom deployment. By virtue of elastic rod theory based on kinetic analogy, coilable mast longerons during the deployment can be analyzed as twisted elastic rods constrained to a cylinder. Seemann and Van studied the deformation and analyzed the stability and equilibrium of thin elastic rods constrained to a cylinder [10,11]. Benham and Le established the elastic model of duplex DNA in the 1970's. Bernard discussed the self-contact of an elastic rod model for DNA based on the assumption that the rod under consideration obeys Kirchhoff's theory [12]. Liu studied the Kirchhoff kinetic analogy theory and proposed the mathematical equilibrium equations of a circular cross section elastic rod constrained to a cylinder [13]. Based on Liu's research, Li and Zhao built the physics-based mode for a cable harness [14]. However, the mentioned mechanical models were analyzed with ideal boundary conditions at both ends only. For coilable mast, elastic rod theory cannot be directly applied because of lateral forces caused by transverse battens. To solve this problem, the longerons were discretized to obtain a series of thin elastic rods with ideal boundary conditions at both ends only. Then, elastic rod theory is suitable for longeron deformation analysis and bottom helical angle calculation. The criterion that allows the mast bottom to deploy in local coil mode can be obtained. In this paper, a practical design criterion that allows the mast bottom to deploy in local coil mode was proposed. The criterion was defined with
dθ ds
>0
(1)
bottom
Through the aforementioned analyses, different angle changing rates
Fig. 3. Longeron discretization and mast definition. 90
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the coordinate system P-x1y1z1 by θ around the Px1 axis and obtain the coordinate system P-x2y2z2. All mentioned coordinates are shown in Fig. 4. Taking φ as the rotating angle around the axis Pz2, angles ψ, θ, and φ are selected to define the dimensional orientation of the rod cross section. Angle variations along the arc length dψ/ds, dθ/ds, and dφ/ds are used to define the rod deformation. dφ/ds defines the relative rotating angle between two adjacent cross sections, whereas dψ/ds and dθ/ds indicate the dimensional orientation of the normal line of the rod cross section. Considering each short rod after discretization, rod deformation satisfies Equation (2) based on Liu's research [13], where i ¼ 1, 2, 3, 4, corresponding to different rods in Fig. 3.
could be calculated via adjusting the bottom helical angles. And a critical value could be obtained while the angle changing rate equals to zero. Judging with Equation (1), a criterion that ensure the mast bottom deploying in local coil mode could be proposed. 3. Bottom deformation analysis Ignoring the mast deployment dynamic effects and the coiling diameter decrease in the transition zone, a cylinder assumption is proposed, namely, the deploying longerons can be regarded as twisted elastic rods constrained to a cylinder. A mathematical bottom deformation analysis is applied for criterion determination based on Liu's elastic
8 2 d θi 1 pi 2 cos θi sin2 θi > > > ðl þ sin θi ¼ cos θ m cos 2θ Þ þ 0i i i i > > R 2 ds2 R2 > > > > > > R sin3 θi M0i > > < FYi þ mi cos θi þ ¼ l0i ¼ A R A > > > C dψ dφ > > mi ¼ ω30i ω30i ¼ i cos θi þ i > > A ds ds > > > > > 2FZi > : pi ¼ A
ði ¼ 1; 2; 3; 4Þ
(2)
In Equation (2), θi represents the rod helical angle of a cross section; s is rod curvilinear coordinate; pi, l0i, mi are the integration constants concerned with rod boundary conditions; A and C are cross section bending rigidity and torsion rigidity, respectively; M0i is the cross section external torque along the Oζ axis; R is the coiling radius; FYi and FZi are the cross section external forces along the PY and PZ axes, respectively. And ψ i(s) and θi(s) satisfy the following equation.
rod research under the cylinder assumption. The coilable mast is a spaceaxisymmetric structure. Thus, deformation analysis focuses on the discretized rods of one single longeron.
3.1. Rod deformation equation After the discretization, each rod is considered as ideally linearly elastic, homogeneous, one-dimension body, and is described by its centerline in a curvilinear coordinate system. The rotation angle between two adjacent cross sections is a continuous function of curvilinear coordinate s. Similar to Liu's research [13], a rectangular coordinate system O-ξηζ is introduced firstly, with the cylinder central axis as Oζ axis and Point O as the coordinate origin. And the Oξ axis is defined randomly. Rotate the coordinate system O-ξηζ by Ψ around the Oζ axis and obtain the coordinate system O-XYZ. Then translate the coordinate system O-XYZ from Point O to Point P, central point of the researched rod cross section. Rotate the coordinate system P-XYZ by 180 around the PZ axis and obtain the coordinate system P-x1y1z1, where ψ ¼ Ψ þ π. Finally, rotate
dψ i ¼
ds⋅sin θi R
(3)
ω30i is rod torsional curvature. Longerons are assembled with transverse battens via hinges, which allow the longerons to rotate along the PX axis merely. However, rotate along the Pz2 axis is limited; thus, the integral of dφi/ds in the interval [0, t] is zero, where t is batten pitch length. Thus, the integral of ω30i in the interval [0, t] is
t
t
∫ 0 ω30i ds ¼∫ 0 t
¼ ∫0
dψ i dφ t dψ i t dφ cos θi þ i ds ¼ ∫ 0 cos θi ds þ ∫ 0 i ds ds ds ds ds
sin θi cos θi ds R (4)
Torsional curvature remains constant while the rod was constrained at both ends only [13], then t
∫0
sin θi cos θi ds ¼ ω30i t R
(5)
3.2. Rod integral conditions and initial values Taking Rod i of Fig. 3 as analysis objects and disregarding the
Fig. 4. Different coordinate definition.
Fig. 5. Cross Section External Forces and Torques of the Top End of Rod i. 91
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dynamic effects, boundary force analyses are conducted to determine FYi, FZi, and M0i. Cross section external forces and torques of the top end of Rod i (s ¼ 0) are shown in Fig. 5, where θi(0) is the top helical angle of Rod i. Among them, MZi(0) could be synthesized with two rectangular components My2i(0) and Mz2i(0) along the axes of Py2 and Pz2.
8 2 < M ð0Þ ¼ M ð0Þ þ M ð0Þ ¼ A sin θi ð0Þ sin θ ð0Þ þ Cω cos θ ð0Þ Zi y2i z2i i 30i i R : M0i ¼ MZi ð0Þ þ FYi ð0ÞR (6)
Fig. 7. Infinitesimal balance analysis.
The coiling zone and top plate, together with the sections above Rod i, are considered as a unity T. Thus, T is in a balance state with lanyard tension (FL), rod reactive forces and torques, and the diagonal tension shown in Fig. 6. Fri, Fvi, and Fhi are three components of diagonal tension in P-XYZ. The coilable mast is a space-axisymmetric structure. Thus, three longerons exert the same reactive forces and torques on the unity T. Through the balance analysis along the Oζ axis, MZi(0), FL, FYi(0), FZi(0), Fvi, and Fhi would satisfy the Equation (7).
FL þ 3FZi ð0Þ þ 3Fvi ¼ 0 3ðMZi ð0Þ þ FYi ð0ÞR þ Fhi RÞ ¼ 0
(7)
Considering an infinitesimal of intersection point between the adjacent rods as the analysis object, a balance state is illustrated as shown in Fig. 7 and a force analysis along the PX and PZ axis is described in Equation (8).
FYi ð0Þ þ Fhi ¼ FYiþ1 ðtÞ þ Fhiþ1 FZi ð0Þ þ Fvi ¼ FZiþ1 ðtÞ þ Fviþ1
Fig. 8. Section deformation analysis.
Combining the equations from (6) to (9), the required integral conditions, FYi, FZi, and M0i, are obtained for each rod. Integration constants, pi, l0i, and mi, are subsequently applied for rod deformation analyses. Since the diagonals are not always tensioned in several sections of the transition zones, further settings are conducted on three components of diagonal tension. As observed in experiments, the diagonals are slack in Sections 4 and 3. Thus, for i ¼ 3, 4, Fvi, and Fhi equal to zero, then the relevant mathematic expressions of FYi, FZi, and M0i are simplified without solving Equation (9). The analyses are carried out from Section 4 to 1. Since, the longeron is continuously deformed on the top of Rod 4, θ4(0) equals to θcoil. Based on Han's research results [9], the rod helical angle of equilibrium state in Sections 2–4 exhibits an approximate linear change, and the rod helical angle in Section 1 remains nearly invariant. Then
(8)
where FYiþ1(t) and FZiþ1(t) could be solved from FYiþ1(0) and FZiþ1(0) based on the mentioned integration constants piþ1 and l0iþ1 in Equation (2). The components of diagonal tension could be obtained based on the section deformation analysis as shown in Fig. 8. The diagonal length is assumed as l; then, its three components in P-XYZ are lri, lvi, and lhi. The components of diagonal length and tension share the same geometry relationship. Thus, Fri, Fvi, and Fhi could be calculated based on the following Equation (9).
8 > > > > > > > > > > > > > > > > > > <
Fli Fhi Fvi Fri ¼ ¼ ¼ l lhi lvi lri
8 <
θ4 ð0Þ ¼ θcoil
: dθ4 ¼ θbottom θcoil ds s¼0 3t
lvi ¼ ∫ t cos θi ds
sin θi ds αi ¼ ∫ t R π α π α > i i > > > lhi ¼ 2R cos 3 þ 2 sin 3 þ 2 > > > π α > > i > > lri ¼ 2R sin2 þ > > 3 2 > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > > : l¼ l2vi þ l2hi þ l2ri
(9)
(10)
where θbottom is the bottom helical angle of Rod 1, namely, the longeron bottom helical angle. For Section 3 to 1, longeron is similarly continuously deformed. And the helical angle changing rate dθi/ds satisfies the equation below according to Liu's research [13].
8 > <
dθi
> :
ds
s¼0
θi ð0Þ ¼ θiþ1 ðtÞ
3 dθiþ1
3 sin 2θi ð0Þ ¼ sin 2θ mi m ðtÞ iþ1 iþ1 2R 2R ds s¼t (11)
3.3. Bottom deformation analyses The bottom deformation analyses are carried out with successive integrals from Section 4 to 1. The values θbottom and FL are assumed initially. Different conditions and initial values are obtained, and then and are subsequently gained, where s changes from zero to t. Changing the value of FL, an iterative calculation from Rods 4 to 1 is
Fig. 6. Balance state analysis of the unity T. 92
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Fig. 10. Longeron deformations in equilibrium state.
Fig. 9. Analysis procedure of bottom deformation.
Table 1 Technical parameters of the coilable mast. Parameter
Value
Coiling Radius (R) Batten Pitch Length (t) Longeron and Batten Material Longeron and Batten Elasticity Modulus (E) Longeron and Batten Poisson's Ratio (σ) Longeron Radius (r) Batten Radius Hinge Material Hinge Elasticity Modulus Hinge Poisson's Ratio
75 mm 95 mm Titanium-Nickel Alloy 83 GPa 0.31 1 mm 0.4 mm Aluminum Alloy 70.6 GPa 0.33
Fig. 11. Helical angle variations of the four-section longeron.
performed until θ1(t) ¼ θbottom. Then, another iteration is conducted via adjusting θbottom to obtain dθ1(t)/ds ¼ 0. The final obtained θbottom is the critical bottom helical angle, and relevant FL is the critical lanyard tension. The complete bottom deformation analysis procedure is shown in Fig. 9. A lanyard-deployed coilable mast was developed for a microsatellite as a semi-rigid gravity gradient boom to obtain a primary attitude stabilizing state. Coilable mast technical specifications, such as stiffness, strength, packing factor, and dimension, were restricted by the stabilization requirement. The present coilable mast technical parameters are shown in Table 1. Based on Han’ research, technical parameters of material characteristics and mechanical configurations satisfy the requirement of deployment in local coil mode. A successful bottom deployment would ensure the entire coilable mast deploying in local coil mode. Other initial parameters were required for bottom deformation analyses. The distance between the adjacent battens in the coiling zone is 8 mm, and the required θcoil ¼ arccos (8/95) ¼ 85.17 .
A¼ C¼
Eπr 4 4 Gπr 4 2
E 2ð1 þ σÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ 3R2 þ t 2
Table 2 Performance of different initial bottom helical angles.
G¼
Initial Bottom Helical Angle ( )
Changing Rate ( /mm)
5 10 15 20 23.164 25 30 35 40 45 50 55 60 65 70 75 80
0.18208 0.13648 0.08807 0.03871 0.00097 0.01065 0.05933 0.10667 0.15208 0.19494 0.23444 0.26946 0.29875 0.32000 0.33007 0.32266 0.28187
critical bottom helical angle is 23.164 .
(12) 3.4. Criterion determination
Bottom deformation analyses are carried out and the results represent longeron deformation in critical state as shown in Fig. 10. The obtained
Adjusting the initial bottom helical angles from 5 to 80 , a series of results is obtained as shown in Fig. 11. And the relevant angle changing 93
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rates along the curvilinear coordinate s are listed in Table 2, which represents the required external torques. Judging with Equation (1), a criterion is obtained; thus, an initial bottom helical angle that is less than the critical value, 23.164 , can ensure mast bottom deploying in local coil mode. In addition, the qualitative kinetic analysis results are consistent with the judgement based on the obtained criterion. When the bottom helical angle is greater than the critical angle, the required external torque is along the –PX axis, which would compress the longeron to the coiling state. This is unexpected during the coilable mast deployment. By contrast, bottom helical angle that is less than the critical value requires the external torque along the þPX axis, which will deploy the coilable mast. Fig. 13. Bottom variable hinge connection.
4. Micro-gravity deployment test and dynamic simulation
The data collection equipment included two sensors, namely, the tension dynamometer for the axial lanyard tension and the MEMS gyroscope for the rotating angular velocity of the mast top plate. When the mast was deployed, both sensors acquired the measurements sequentially (five targets per second).
Micro-gravity deployment test and dynamic simulation were conducted. Different initial bottom helical angles were applied in the tested and simulated models. Bottom deployment performances were applied to further validate the criterion effectiveness.
4.1.2. Bottom helical angle restriction The bottom helical angle should be restricted to achieve different bottom boundary conditions. A variable hinge connection was designed and illustrated in Fig. 13. By adjusting the locations of restriction pin, a series of rotation angle ranges were restricted, and the relevant initial bottom helical angles were obtained while being compressed. Fig. 13 indicates that the initial bottom helical angles were restricted as 0 , 15 , 20 , 30 , 45 , 60 , and 75 . To further validate the result, a constant deployment velocity of 15 mm/s was selected. The contact force between Battens 5 and 6 was difficult to measure. Thus, the critical state of the four-section deployment was defined as the time when Batten 6 started to separate from Batten 5. A typical longeron deformation in critical state is illustrated in the following results validation. When the coilable mast was purely coiled, (i.e., the initial bottom helical angle was 87.04 ), the deployment was randomly performed from any section. Hence, deployment tests focused at the initial bottom helical angle from 0 to 75 .
4.1. Micro-gravity deployment test 4.1.1. Micro-gravity deployment facility A one-dimensional deployment facility was developed for microgravity deployment test, as shown schematically in Fig. 12. The facility consisted of a truss base, linear track, synchronizing system, and data collection equipment. The truss base supported the entire facility and formed the effective deployment space with horizontal linear track. The synchronizing system consisted of a coaxial lanyard bobbin, linear bearing, and balance weight. The coaxial lanyard bobbin released the coilable mast and linear bearing via a stepper motor. Balance weight was connected to the linear bearing to ensure the bearing's synchronous movement with the mast top plate. The key to the micro-gravity deployment test was to ensure the gravity compensation effect. By adjusting the vertical angle of suspended rope, linear bearing should be located right above the top plate before test, and the error between the suspended rope and plumb line was restricted within 3 . Furthermore, all the lanyards should be wrapped around the coaxial bobbin in the same order to ensure the releasing synchronism. Balance weight was determined based on the bearing frictions via actual releasing test.
4.2. Dynamic simulation The dynamic simulations were performed using the software
Fig. 12. Micro-gravity deployment facility. 94
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Fig. 14. Coilable mast finite element model. Table 3 Results of simulation and test for bottom helical angle.
MSC.ADAMS, which is widely used in dynamic design of aeronautics and astronautics. Elastic parts such as the longerons and transverse battens were discretized into a series of rigid parts while connecting with the elements of Discrete Flexible Link. The diagonals, with limited dimension and mass, are simplified to a pair of action and reaction along the diagonal direction. And the pair of action and reaction is set with a step function changing with the distance between two ends of action and reaction. All the finite element parts were defined with the characteristics of the actual coilable mast mentioned in Table 1. A 19-section coilable mast model was established, as shown in Fig. 14. Initial bottom helical angles were individually defined for different simulation cases. Torsion springs were added at the longeron bottom. After compression, the required longeron bottom helical angle was restricted with the step-functional spring stiffness. The coilable mast was deployed with a constant velocity of 15 mm/s, consistent with the tests. To further validate the result, the critical state of the four-section deployment was defined similarly as the mathematical calculations, namely, when the contact force between Battens 5 and 6 decreased to zero. Typical longeron deformations in critical state are illustrated in the following result validation.
Initial Bottom Helical Angle ( )
Simulated Bottom Helical Angle ( )
Tested Bottom Helical Angle ( )
0 15 20 30 45 60 75
0 1.8348 1.9529 2.6207 36.4866 36.5609 36.4865
9.8 9.2 4.1 30 42.2 42.8 42.6
Local Coil Local Coil Local Coil Local Coil Helix Helix Helix
Local Coil Local Coil Local Coil Helix Helix Helix Helix
5. Result validation and discussion The obtained criterion that ensure the coilable mast bottom deploying in local coil mode was validated by comparisons of bottom helical angle in critical state. A practical design margin of initial bottom helical angle was provided for criterion modification. Furthermore, the section number of transition zone was discussed, and the relevant bottom deployment design was conducted. 5.1. Bottom deployment performance validation Fig. 15. Tested and simulated deformation with initial bottom helical angle of 15 .
The initial bottom helical angle determines whether the mast bottom could be deployed in local coil mode. During the coilable mast deployment, the bottom helical angle decreased from the initial value to zero, and the coiled rod in Section 1 became straight. Bottom deployment, namely, Section 1 deployment, could be assessed based on whether the bottom helical angle was approximately equal to zero. Both tests and simulations were conducted with different initial helical angles of 0 , 15 , 20 , 30 , 45 , 60 , and 75 . Table 3 presents the tested and simulated results with different initial helical angles in critical state. When the initial value was not greater than 20 , both the tested and simulated bottom helical angles were less than 3 in critical state, that is, the mast bottom had been deployed in critical state. Deployment modes were verified based on the results in Table 3, and the presented simulation and tested results were consistent with the criterion described in Chapter 3.4. The two data sets of tested and simulated longeron deformations are evaluated as shown in Fig. 15 and Fig. 16. With an initial bottom helical angle of 15 , both longeron deformations indicated that Section 1 was fully deployed and the transition zone included four sections at most. Deformations satisfied the local coil mode definition. However, with an initial bottom helical angle of 60 , both longeron deformations showed that Section 1 was not fully deployed. Transition zone would include at least five sections, which indicated that deformations did not satisfy the local coil mode definition. For the simulated results, another method would be applied to assess the bottom deployment mode. The time when the bottom helical angle
Fig. 16. Tested and simulated deformation with initial bottom helical angle of 60 .
first decreased to less than 3 is defined as t1, and the time when the contact force between Battens 5 and 6 decreased to zero is set to t2. If t1 95
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Acta Astronautica 141 (2017) 89–97 Table 5 Calculation results with different section numbers. Section Number
Critical Bottom Helical Angle ( )
Practical Bottom Helical Angle with Margin ( )
3 4 5
18.677 23.164 31.505
16.12 20 27.19
13.7% could be effectively applied to determine the bottom deployment mode. That is, an initial helical angle less than the critical value with a 13.7% margin could ensure mast bottom deploying in local coil mode. Moreover, the obtained criterion could further determine a successful local coil deployment of entire coilable mast [9].
5.2. Section number discussion of transition zone in local coil mode Fig. 17. Variations of the bottom helical angle and contact force between battens 5 and 6.
All the calculations and validations were based on the definition that the section number of transition zone in local coil mode was 4. Adjusting the section number, analysis objective of bottom deformation could be changed, and the calculation results with different section numbers are presented in Table 5. Table 5 showed that the obtained critical bottom helical angle varied with different section numbers of transition zone. More sections contained in the transition zone required a greater critical bottom helical angle. The result could be explained based on the mast deployment motion similar with Fig. 17. Along with mast deployment in local coil mode, sections separated with each other successively from Section 1, and the bottom helical angle decreased from the initial value to zero. With the same initial bottom helical angle, a transition zone with less sections could cause the relevant contact force to decrease to zero earlier. Therefore, to shorten the duration of the bottom helical angle decreasing to zero, a small initial angle was required. Vice versa, a transition zone with more sections required a greater initial bottom helical angle. In actual applications, a transition zone with more sections would lower the lateral bending stiffness and stability, which are undesirable in the mast design. According to the aforementioned discussion, the initial bottom helical angle should be set as small as possible to obtain less sections of transition zone and achieve higher lateral bending stiffness and stability.
Table 4 Time comparisons and deployment mode determination. Initial Bottom Helical Angle ( )
t1 (s)
t2 (s)
Deployment Mode
0 15 20 30 45 60 75 87.04 (Pure Coiling)
0 8.2 11.3 14.9 18.6 19.4 20.7 21.4
15 15.4 15.6 16.1 13.5 14.3 15.5 16.2
Local Coil Local Coil Local Coil Local Coil Helix Helix Helix Helix
t2, at most four sections exist in the transition zone, and the coilable mast would be deployed from the bottom in the local coil mode. Fig. 17 represents the variations of the bottom helical angle and contact force between Battens 5 and 6 when the initial helical angle was 30 . Time comparisons were performed for all eight data sets, and deployment modes were determined as shown in Table 4. The influence of initial helical angle on the deployment mode was validated to be consist with the obtained criterion. The simulation results in Table 3 indicate that mast bottom was deployed in local coil mode with the initial bottom helical angle of 30 . However, the tested results presented a helix mode. Analysing the difference of calculation results and test, the error was mainly resulted from the model difference. It can be divided into three parts:
6. Conclusion A practical design criterion that allows the coilable mast bottom to deploy in local coil mode is proposed in this study. With the bottom deformation analyses based on the mast technical parameters, the criterion was defined that an initial bottom helical angle less than the critical value with a design margin of 13.7% could ensure mast bottom deploying in local coil mode, and further determine a successful local coil deployment of entire coilable mast. The criterion with a design margin is evaluated by the micro-gravity deployment tests and dynamic simulations. With existed elastic rod theory, longeron discretization and successive integrals presents a practical mathematical analysis method for longeron deformation. And the calculated results are consistent with the tested and simulated results. In addition, the initial bottom helical angle should be set as small as possible during the actual mast design. Then the transition zone with less sections could be obtained, and lateral bending stiffness and stability of the coilable mast would be improved.
● Mathematic calculation was conducted under the cylinder assumption, ignoring the coiling radius decrease. However, the radius decrease is indispensable to the mast deployment. The assumption introduction could bring in the calculation error, which is inevitable. ● There were several frictional damping existed in the tested mast model, which had not been considered accurately in the mathematic calculations and simulations. The frictional damping were mainly existed in the mast hinges, and consumed some energy that was used for mast deployment. ● Mathematic calculations and simulations were conducted with uniform mast configuration and material characteristics. And the obtained results were ideal. However, the tested model had to be faced with the mast assembly error and the inevitable material inhomogeneity, which could affect the final deployment performance obviously. Considering these, the difference between two sets of results was acceptable, and a conservative design margin was required to reduce the model error. The initial value of 20 could ensure both tested and simulated mast bottoms deploying in local coil mode. Thus, a practical design margin was defined with (20 23.164 )/23.164 100% ¼ 13.7%. Therefore, the obtained mathematical result with a design margin of
Acknowledgements This research work is supported by the National Natural Science Foundation of China (Grant No.11672016), which the authors gratefully acknowledge. 96
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