Sage Journals: Discover world-class research

Abstract

Successful unfolding in the space is the first step for a satellite antenna to work normally, so that it is important to study the movement reliability of a large deployable satellite antenna. By studying the movement principium, a mechanical analysis model of the deployment mechanism and calculating formulae for the internal forces of trusses were presented. From the performance function of the truss unfolding to meet, a movement reliability analysis model of the mechanism was also given. Synthetically considering the dimension errors and the impact of space environmental factors, we treated the mechanism's movement as a function of random variables and derived the reliability calculation formula by using the method of second-order moment. Finally, the movement probability of the deployment mechanism was simulated by the computer within whole spreading process. The trend of the computing results agrees with the truss model experiments, which shows that the method is reasonable and effective.

1. Introduction

Along with people's exploratory work to the space, a variety of satellite antennas have come out in succession. As a result of the launch vehicle platform size's limit, a large satellite antenna is usually puckered up during the launching stage and will be unfolded under remote control when the satellite has entered into its orbit. Successful unfolding in the space is the first step for a satellite antenna to work normally, and some starting malfunctions of antenna or solar panels have occurred in the aerospace history and led to launch failure. So it is important to study the movement reliability of a large deployable satellite antenna.

The mechanical parts of the deployment equipments on a satellite antenna are usually moveable mechanisms. For changeable structural forms and different failure types, the reliability analysis of a moveable mechanism is usually more complicated and difficult than that of a fixed structure. The movement reliability of a mechanism is calculated mainly by using the perform functions or the security allowances. At present the reliability analysis of a mechanism is not mature in the theory and the method, and the reliability of the mechanism movement function is seldom considered in the previous researches [1–3]. Most research of the deployable satellite antenna focuses on the structural style design and launch dynamics analysis [4–9]. The deployment mechanism design of a large meshed space antenna was introduced in [4], and the analysis of its principle of work was also presented. The kinetic characteristic of deployment mechanisms and the vibratory characteristic of an antenna were investigated in [5]. The work in [6] discussed the role of environmental testing to enhance the reliability of spacecraft as well as the relationship between environmental test and reliability test. In [8], two concepts of deployable truss structures were compared in view of the required deployment force against the mesh surface tension. The comparison was performed using the flexible multibody dynamics computer code that the author developed. The work is [9] gave the kinetic analysis of deployable scissor structures.

We investigated the blue print of a large truss-board satellite antenna and tentatively forecasted the antenna unfolding probability, with emphasis on the analysis and prediction of the mechanism reliability by using movement function.

2. Principium of the Hoop-Truss Deployment

This paper deals with a large deployable satellite antenna with hooptruss. The antenna has a paraboloidal mesh reflector, the long-axis diameter of which is about 17 m. The reflecting surface of it consists of a metallic web and some tensed cords. It is unfolded by using a hoop-truss deployment mechanism. Figure 1 shows the axonometric drawing of the antenna.

Figure 1:

Axonometric drawing of the antenna.

The hooptruss has 48 structural units, and each of which has a rectangular framework and a shrinkable bar along one diagonal line of the rectangular. Some expandable outriggers are fitted at the endpoints of the rectangular to improve the structural stiffness. The truss bars are connected to each other by node joints and tight cords. A sketch map of two units of the hooptruss is given in Figure 2. It shows the structural composition and connection of hoop trusses, where 1,3,5 are four-bar nodes, 2,4,6 are eight-bar nodes, 7 is a two-bar node, 8 ≾ 17 are truss bars, and 18 ≾ 29 are tight cords.

Figure 2:

Two units of the hooptruss.

The whole unfolding process of a satellite antenna involves three stages: unlocking, expansion, and orientation.

At the first stage, the binding cordages around the antenna structure desquamate to make the antenna in a nonrestraint condition. Under the force of torque spring, each joint truss opens to a slight angle.

At the second stage, the fixed electromotor starts driving a wheel to enwind a hauling rope around it. The shrinkable bar along one diagonal line of the structural unit is made hollow for the hauling rope to pass through. As the hauling rope enwinds around the rolling wheel, the length of the shrinkable bar shortens and the shape of each truss unit changes from a slender rhombus to a rectangle.

At the last or orientation stage, the electromotor shuts down when the length of each shrinkable bar has shortened to the minimum. Under the forces of knuckle blocks and tight cords, the antenna structure expands to the maximum extent and maintains lock status.

The principium of the deployment mechanism is shown in Figure 3, and a schematic drawing of the deployment process is given in Figure 4.

Figure 3:

Principium of the deployment mechanism.

Figure 4:

Schematic drawing of the deployment process.

3. Mechanical Analysis of the Hoop-Truss Deployment

The mechanical analysis chart of two hoop-truss units is shown in Figure 5, where T is the tension force of the hauling rope. The deployment of a large antenna structure is so slowly that it will take about an hour to spread out. Thus the antenna is approximately in a static equilibrium at each moment of the deployment. Being in weightlessness of the space environment, the deadweight of the whole antenna structure can be ignored. At the expansion stage, the cords between truss bars are in looseness, and so the tension forces of cords are also neglected.

Figure 5:

Configuration of two truss units.

3.1. Spatial Position of Truss Bars

From the triangle relationship shown in Figure 5 we find that

\begin{matrix} l_{4} & = \sqrt{{l_{1}}^{2} + {l_{3}}^{2} + 2 l_{1} l_{3} \cos θ}, \\ α & = arc sin (\frac{l_{3} \sin θ}{l_{4}}), \\ γ & = arc cos (\frac{l_{4}}{2 l_{2}}), \\ l_{2} & = \frac{l_{1} + l_{3}}{2}, \end{matrix}

(1)

in which l_i(i=1,…,4) means the length of L_i bar, α is the included angle between bars of L₁ and L₄, γ is the angle between L₂ and L₄, and θ denotes the angle between L₁ and L₃.

We view β as the projected angle of L₃ bar of two units on the X-O-Z coordinate plane. The projection of circumferential bars of 48 units on it is a regular polygon, so that

β = 23 π / 24 .

(2)

3.2. Internal Force of Each Bar

From Figure 5, according to symmetry, the equations of force projection are given as follows.

At node B, along the y-axis,

2 N_{3} \cos (θ) - N_{1} = 0 .

(3)

At node A, along the y-axis,

2 T \cos α + N_{1} + 2 N_{2} \cos γ \cos α + 2 N_{3} \cos (θ) = 0,

(4)

and along the Z-axis,

\begin{matrix} 2 T \sin α \cos (\frac{β}{2}) - 2 N_{2} \sin γ \sin (\frac{β}{2}) \\ + 2 N_{2} \cos γ \sin α \cos (\frac{β}{2}) + 2 N_{3} \sin θ \cos (\frac{β}{2}) = 0 . \end{matrix}

(5)

Solving the simultaneous equations from (3) to (5), we can get the internal force of each bar:

\begin{matrix} N_{1} \\ = \frac{- 2 T \cos θ \cos α \sin γ \tan (β / 2)}{\sin θ \cos α \cos γ - 2 \cos θ \sin α \cos γ + 2 \cos θ \sin γ \tan (β / 2)}, \\ N_{2} \\ = \frac{T (2 \cos θ \sin α - \sin θ \cos α)}{\sin θ \cos α \cos γ - 2 \cos θ \sin α \cos γ + 2 \cos θ \sin γ \tan (β / 2)}, \\ N_{3} \\ = \frac{- T \cos α \sin γ \tan (β / 2)}{\sin θ \cos α \cos γ - 2 \cos θ \sin α \cos γ + 2 \cos θ \sin γ \tan (β / 2)}, \end{matrix}

(6)

where N_i(i=1,2,3) is the internal force of L_i bar. We suppose that each bar is under pressure, which is defined as a positive internal force. It has been proved that the composite force along Z-axis is not equal to zero at nodes B and D. It is just the composite driving force for the hoop-truss to spread.

4. Probability Analysis of the Hoop-Truss Deployment

4.1. Building the Probabilistic Model

In the deployment mechanism, a trussbar is connected to others by some knuckles, and the unfolding power comes from an electromotor. The resistances are calculated in moments, which come from frictional resistances caused by knuckles and high-frequency cables and from thermal deformation of the antenna structure. The frictional moment of resistances in knuckles is calculated by the normal pressure on shaft axles multiplied by the friction coefficient between the knuckle materials. The total resistance moment can be estimated approximately as one that is k_M times larger than the frictional moment, where k_M is a synthetic influence coefficient.

If the hooptruss can be unfolded successfully, one of the following performance functions should be satisfied.

The driving moment is always larger than the resistance moment during deployment.

Such a performance function of movement is corresponding to a limiting state that

Z_{M} (θ) = M_{a} (θ) - M_{r} (θ) = 0 (θ_{0} \leq θ \leq θ_{m}),

(7)

where Z_M(θ) is a performance function expressed in moments, M_a(θ) is the total driving moment, and M_r(θ) is the total resistance moment. θ₀ and θ_m are, respectively, the start- and end-angle of deployment; Z_M(θ), M_a(θ), and M_r(θ) are functions varying with the unfolding angle θ.

Let the radius of a knuckle shaft be R, and let the friction coefficient between the shaft and muff be f. Considering the effect of dimensional errors, assembling errors and random factors in the space environment, we can treat actual values of the towing force T, the radius R, and the friction coefficient f as random variables. So the movement function Z_M(θ) is a function of random variables T, R, and f.

By using the second-moment method in probability theory [10], the fiduciary level β_M and the movement probability P_{r
M} of Z_M(θ) can be deduced:

\begin{gathered} β_{M} = \frac{μ_{Z_{M}}}{σ_{Z_{M}}} = \frac{μ_{M_{a}} - μ_{M_{r}}}{\sqrt{σ_{M_{a}}^{2} + σ_{M_{r}}^{2}}}, \end{gathered}

(8)

\begin{gathered} P_{r M} = P (M_{a} (θ) - M_{r} (θ) > 0) = Φ (β_{M}), \end{gathered}

(9)

where μ_(^*) is the mean value of a random variable (^*), σ_(^*) is the mean square deviation of it, and Φ_(^*) is the standard normal distribution function.

(ii)

The driving moment does more work than the resistance moment during deployment.

After deployment starting, if the accumulative work done by the driving moment is larger than that done by the resistance moment, the unfolding process will continue. The equations of limiting state, the fiduciary level, and the movement probability of such a performance function are, respectively, given as follows:

\begin{gathered} Z_{W} (θ) = W_{a} (θ) - W_{r} (θ) = 0 (θ_{0} \leq θ \leq θ_{m}), \end{gathered}

(10)

\begin{matrix} β_{W} = \frac{μ_{Z_{W}}}{σ_{Z_{W}}} = \frac{μ_{W_{a}} - μ_{W_{r}}}{\sqrt{σ_{W_{a}}^{2} + σ_{W_{r}}^{2}}} \end{matrix},

(11)

\begin{gathered} P_{r W} = P (W_{a} (θ) - W_{r} (θ) > 0) = Φ (β_{W}), \end{gathered}

(12)

where Z_W(θ) is a performance function expressed in works, W_a(θ) is the accumulative work done by the driving moment, and W_r(θ) is that by the resistance moment. Z_W(θ), W_a(θ), and W_r(θ) are functions varying with the unfolding angle θ, and they are also functions of random variables T, r, and f.

If either of the above two performance functions can be satisfied, the hooptruss will be able to deploy.

4.2. Derivation of Reliability Formulae

4.2.1. Calculating the Movement Probability in Moment Form

When the hooptruss unfolds, the maximum of the total resistance moment is (refer to Figure 5)

\begin{matrix} M_{r} (θ) & = 48 k_{M} \\ \times (4 | N_{3} f_{1} r_{1} | + 3 | N_{2} f_{1} r_{1} | \\ + 2 | N_{2} f_{1} r_{1} \cos γ | + 2 | T f_{2} r_{2} \cos α |), \end{matrix}

(13)

where f₁ is the friction coefficient in truss knuckles and f₂ is that in pulley wheels; r₁ and r₂ are, respectively, the shaft radius in truss knuckles and in pulley wheels. The driving moment is offered by the towing rope:

M_{a} (θ) = 24 \times | T L_{4} \sin 2 α | .

(14)

Substituting (13) and (14) into (7), we arrange the limiting condition function as

Z_{M} = M_{a} (θ) - M_{r} (θ) = T * (A (θ) - B (θ) f_{1} r_{1} - C (θ) f_{2} r_{2}),

(15)

where A(θ),B(θ), and C(θ) are deterministic functions about the bar-length l_i and the unfolding angle θ. By using the second-moment method [10] of a random function, the mean value and the square deviation of Z_M can be given as follows:

\begin{matrix} μ_{Z_{M}} & = μ_{T} * [A (θ) - B (θ) μ_{f_{1}} μ_{r_{1}} - C (θ) μ_{f_{2}} μ_{r_{2}}], \\ σ_{Z_{M}}^{2} & = {[A (θ) - B (θ) μ_{f_{1}} μ_{r_{1}} - C (θ) μ_{f_{2}} μ_{r_{2}}]}^{2} σ_{T}^{2} \\ + {[B (θ) μ_{T} μ_{r_{1}}]}^{2} σ_{f_{1}}^{2} + {[B (θ) μ_{T} μ_{f_{1}}]}^{2} σ_{r_{1}}^{2} \\ + {[C (θ) μ_{T} μ_{r_{2}}]}^{2} σ_{f_{2}}^{2} + {[C (θ) μ_{T} μ_{f_{2}}]}^{2} σ_{r_{2}}^{2} . \end{matrix}

(16)

Substituting the right parts of the above two equations into (8) and (9), the fiduciary level β_M and the movement probability P_{r
M} can be calculated, respectively.

4.2.2. Calculating the Movement Probability in Work Form

If the included angle θ has a little change of dθ, the differential of the work done by resistance moments (refer to Figure 5) is

\begin{matrix} d W_{r} & = 96 k_{M} \\ \times [2 | N_{3} f_{1} r_{1} d θ | + 2 | N_{2} f_{1} r_{1} d γ | \\ + | N_{2} f_{1} r_{1} \cos γ d α | + | T f_{2} r_{2} \cos α d (\frac{L_{4}}{r_{3}}) |] \\ = \bar{M_{r}} (θ) d θ, \end{matrix}

(17)

and the differential of the work done by the driving moment (refer to Figure 2) is

d W_{a} = 48 \times | T d L_{4} | = \bar{M_{a}} d θ .

(18)

When the truss structure has unfolded from angle θ₀ to θ, the total accumulative work done by resistance moments is

W_{r} (θ) = \int_{θ_{0}}^{θ} \bar{M_{r}} (θ) d θ .

(19)

On the other hand, the total accumulative work done by the driving moment is

W_{a} (θ) = \int_{θ_{0}}^{θ} \bar{M_{a}} (θ) d θ .

(20)

The limiting condition function Z_W(θ) is arranged as

Z_{W} = W_{a} (θ) - W_{r} (θ) = T * (E (θ) - F (θ) f_{1} r_{1} - G (θ) f_{2} r_{2}) .

(21)

In the previous formulae, r₃ is the radius of pulley wheels in the deployment mechanism, which is a determinant parameter; E(θ),F(θ), and G(θ) are determinant functions about bar-length l_i and unfolding angle θ.

The mean value and the square deviation of Z_W are given, respectively:

\begin{matrix} μ_{Z_{W}} & = μ_{T} * [E (θ) - F (θ) μ_{f_{1}} μ_{r_{1}} - G (θ) μ_{f_{2}} μ_{r_{2}}], \\ σ_{Z_{W}}^{2} & = {[E (θ) - F (θ) μ_{f_{1}} μ_{r_{1}} - G (θ) μ_{f_{2}} μ_{r_{2}}]}^{2} σ_{T}^{2} \\ + {[F (θ) μ_{T} μ_{r_{1}}]}^{2} σ_{f_{1}}^{2} + {[F (θ) μ_{T} μ_{f_{1}}]}^{2} σ_{r_{1}}^{2} \\ + {[G (θ) μ_{T} μ_{r_{2}}]}^{2} σ_{f_{2}}^{2} + {[G (θ) μ_{T} μ_{f_{2}}]}^{2} σ_{r_{2}}^{2} . \end{matrix}

(22)

Substituting the right parts of the above two equations into (11) and (12), the fiduciary level β_W and the movement probability P_{r
W} can be calculated.

5. Calculating the Deployment Probability of Hoop-Truss

The data of the deployment mechanism are given as follows.

The lengths of trussbar are l₁ = 1400 mm, l₃ = 1050 mm; the diameter of knuckle shaft is D₁ = Φ12_{− 0.1}⁰ mm, and that of pulley shaft is D₂ = Φ16_{− 0.1}⁰ mm; the diameter of the pulley wheel is D₃ = Φ30 mm; the mean value of the friction coefficient at the truss nodes is μ_{f
₁} = 0.15, and at the pulley wheels is μ_{f
₂} = 0.2, which has the same coefficient of variation v_{f
₁} = v_{f
₂} = 0.1; the mean value of tension force in the towing rope is μ_T = 150 N, which has the coefficient of variation v_T = 0.1; the variation range of the unfolding angle θ is $[1^{°}, 8 9^{°}]$ .

Based on the method of this paper, with the synthetic influence coefficient k_M = 2.5, the bounds of truss-bar internal forces are N₁∊ $[- 148.59, - 47.72] (N)$ , N₂∊ $[- 125.82, - 1.41] (N)$ , N₃∊ $[- 1367.3, - 74.3] (N)$ . N₁ N₂ and N₃ are all negative values during the whole deployment, which shows that each truss bar bears compression force.

From curves shown in Figure 6 we can see that the mean value of the total damping moment M_r∊ $[90.46,736.32] (Nm)$ , the mean value of the driving moment $M_{a} ∊ [131.9,6246] (Nm)$ , and the fiduciary level $β_{M} ∊ [2.683,9.998]$ . At the beginning of deployment, the hooptruss is nearly in self-locking, so the antenna structure is difficult to unfold and the fiduciary level β_M is low at the middle stage of deployment, the hooptruss has broken away from self-locking, so the antenna structure becomes easy to outspread and the fiduciary level β_M keeps in a high value at the last five degrees of angle θ with the descent of the driving moment, and the increment of the resistance moment, β_M starts to decrease.

Figure 6:

Curves of moments and the fiduciary level.

From curves shown in Figure 7, the mean value of the work done by the driving moment is W_a $∊ [0,4.934 \times 1 0^{3}] (Nm)$ , the mean value of the work done by the damping moment is W_r $∊ [0,2.479 \times 1 0^{3}] (Nm)$ , and the fiduciary level β_W $∊ [0,7.188]$ . β_W increases from zero to about six at the first ten degrees of angle θ and then goes up slowly.

Figure 7:

Curves of works and the fiduciary level.

If calculated in moment form, the movement probability of the deployment mechanism is P_{r
M}≥Φ(2.683) = 0.996351; if calculated in work form, the movement probability of the deployment mechanism is P_{r
W}≥Φ(0) = 0.5. It is ought to consider both the two factors; therefore we use a logic model [10] of parallelconnection to get the deployment probability P_r = 1 − (1 − P_{r
M})(1 − P_{r
W})≥0.998176.

6. Conclusion

(1) Models of mechanical analysis and movement probability of the hooptruss are presented in this paper, which provides some theoretic methods for the study on the deployment mechanism of a large satellite antenna.

(2) The movement reliability of the deployment mechanism during the whole spreading is predicted. The trend of the computing results agrees with model experiment process, which shows that the method of this paper is reasonable and efficient.

Footnotes

Nomenclature

Acknowledgment

This research work has been sponsored by the National High-tech Research and Development Program (863 Program), funded project (2006AA04Z402), China.

References

Feng

Y. S.

, “The development of a theory of mechanism reliability,” Reliability Engineering and System Safety, vol. 41, pp. 95–99, 1993.

Sandler

B. Z.

, Probabilistic Approach to Mechanisms, Science Press, Beijing, China, 1991.

Székely

G. S.

Teichert

W. H.

Brenner

C. E.

Pradlwarter

H. J.

Klein

, and Schuëller

G. I.

, “Practical procedures for reliability estimation of spacecraft structures and their components,” AIAA Journal, vol. 36, no. 8, pp. 1509–1515, 1998.

Feng

D. W.

and Zhao

R. J.

, “The research of large-size space mesh antenna deploying mechanism,” Chinese Space Science and Technology, vol. 17, no. 1, pp. 64–70, 1997 (Chinese).

Masayoshi

Tetsuo

, and Shojiro

, “Analytical and experimental investigations for satellite antenna deployment mechanisms,” Journal of Spacecraft and Rockets, vol. 26, no. 3, pp. 181–187, 1989.

Jing

, “On environmental test and reliability of spacecraft,” Chinese Space Science and Technology, vol. 17, no. 5, pp. 33–40, 1997 (Chinese).

Liu

M.-Z.

and Gao

G. F.

, “Advances in the study on structure for space deployable antenna,” Journal of Astronautics, vol. 24, no. 1, pp. 82–87, 2003 (Chinese).

Mitsugi

, “Comparative analysis of deployable truss structures for mesh antenna reflectors,” AIAA Journal, vol. 36, no. 8, pp. 1546–1548, 1998.

Langbecker

, “Kinematic analysis of deployable scissor structures,” International Journal of Space Structures, vol. 14, no. 1, pp. 1–15, 1999.

10.

Chen

J. J.

and Duan

B. Y.

, Reliability Aspects in Dynamic & Structural Optimization, Gordon & Breach, Amsterdam, The Netherlands, 1999.