On calculation of preliminary design parameters for ... - SAGE Journals

377

On calculation of preliminary design parameters for lenticular booms F Hakkak1 and S Khoddam2 1 Iran Telecom Research Center, Tehran, Iran 2 Mechanical Engineering Department, Clayton Campus, Monash University, Australia The manuscript was received on 29 July 2006 and was accepted after revision for publication on 8 January 2007. DOI: 10.1243/09544100JAERO138

Abstract: Deployable booms play an important role in holding instruments away from the satellites, and the more rigid they are, the greater the attitude stability of the satellite and instruments will be. Several design concepts for the deployable booms have been proposed and implemented, among which the lenticular boom is one of the most rigid ones. In the present paper, a method for estimation of the design parameters of lenticular booms is proposed. The design parameters investigated in this paper are bending and torsional equivalent moments of area, and elastic energy and maximum stresses developed in the flattened booms. The proposed method has been validated against finite-element solutions and is shown to be sufficiently accurate to be safely used for preliminary design of lenticular booms. Keywords: boom, lenticular, moment of area, flattening, elastic strain energy, stress

1

INTRODUCTION

Deployable mechanisms are a very important class of space mechanisms, which have been widely used in space applications. Deployable booms are among the simplest and most frequently used deployable mechanisms which hold antennas, sensors, and other instruments at a given distance from the satellite. As well, deployable booms with their tip masses are frequently used in small satellites pointing the earth (nadir) to stabilize their attitude using gravity gradient moments. Deployable booms may be classified according to their deployment mechanism, their boom-element cross-section and the number of tape-measure-like structures used in their boom-element [1]. The lenticular boom is one of the boom-element types that provide higher bending and torsional stiffnesses, when compared with those of the open-section boom-element types, such as storable tubular elastic member (STEM) [1], Bi-STEM [1], STACER (spiral

∗ Corresponding

author: Iran Telecom Research Center, End of

North Kargar Street, Tehran, Iran. email: [email protected]

JAERO138 © IMechE 2007

tube & actuator for controlled extension/retraction) [2], etc. This type of boom has been termed as foldable tube [3], welded seam mast [1], bi-convex foldable elastic tube [4], collapsible tube mast [5], lentiltype [6], lentiform [7], lenticular [8], etc. Throughout this paper such structures will be referred to as ‘lenticular booms’. The lenticular boom can be flattened and coiled on a spool in a relatively small volume (Fig. 1). This type of tube was first developed and studied at NASA Lewis Research Center in the mid-1960s [4] and was reported in reference [3]. Several studies have been done on the lenticular tube [3–5, 9], some of which have focused on the manufacture of composite booms instead of the classical metallic ones [5, 9]. Recently, Yuzhnoye SDO, Ukraine has developed a lenticular gravity gradient boom that is commercially available [7]. In the process of designing a boom, the design constraints imposed by the requirements have to be met. These constraints and parameters include: weight, size (stowed and deployed), natural frequency, bending stiffness, torsional stiffness, temperature range, etc. Preliminary design utilizes simple formulas and models to provide good initial guess for iterations required during critical design. For this,

Proc. IMechE Vol. 221 Part G: J. Aerospace Engineering

378

F Hakkak and S Khoddam

Fig. 1

Fig. 2

Lenticular boom coiled on a spool

exact and simple formulas are needed, to perform the design process with maximum exactitude and minimum iterations. As the lenticular boom is not of standard shape, no standard structural design formulas can be directly used. In this paper, the authors propose a simple method which can be used to reach a good initial guess for the design of a lenticular boom. The method which is derived by using the methods of the basic mechanics of materials, estimates bending and rotational moments of area of the boom cross-section (Ixx , Iyy , and J ), elastic strain energy of the flattened boom, and maximum stresses developed in the boom after flattening and coiling around a spool. Several finiteelement solutions will be used then for validating the proposed method. 2

BENDING AND TORSIONAL MOMENTS OF AREA

In order to calculate the bending stiffness (EI ) of lenticular booms, the moments of area of the lenticular cross-section have to be estimated, where E and I are Young’s modulus and moment of area, respectively. Generally, thin-walled tubes, such as lenticular booms, do not follow the same laws which have been established for deformation of solid beams [10], but as the analytical equations for the thin-walled lenticular tube would be extremely complicated, the ordinary moments of area will be used. The extent of validity for such assumption will be investigated next using several finite-element solutions. As the section has two perpendicular axes of symmetry, it is sufficient to calculate the moments of area for only a quarter of the section, i.e. the region x, y > 0 (Fig. 2), and multiply the results by four. Ixx can be calculated as

Ixx = 4 y 2 dA = 4 y 2 t dl = 4 y 2 t 1 +

dy dx

where t is the thickness of the section. Neglecting the straight part, the curves are circular arcs and are generally of the form (x − x0,i )2 + (y − y0,i )2 = ri2 , i = 1, 2

2 dx

(2)

where x0,1 = 0 and x0,2 = x2 . By differentiation one obtains dy x − x0 =− dx y − y0

(3)

Combining equations (1), (2), and (3) yields 2 r12 − x 2 + y0,1 =4 tr1 dx r12 − x 2 0

2 x2 − r22 − (x − x2 )2 + y0,2 +4 tr2 dx x1 r22 − (x − x2 )2 x1

Ixx

(4)

Iyy can be found in a similar manner. Here the share of the straight part is significant and cannot be neglected. Also because dx/dy becomes infinite in some parts, dy/dx can be used instead 2 Iyy = 4 x dA = 4 x 2 t dl 2 dy = 4 x2t 1 + dx (5) dx Combining equations (2), (3) and (5), one obtains x1 Iyy = 4

tr1 0

x2

+4

x2 r12 − x 2

tr2 x1

(1) Proc. IMechE Vol. 221 Part G: J. Aerospace Engineering

Cross-section for a quarter of the lenticular boom

dx x3

x2 r22 − (x − x2 )2

dx + β

tx 2 dx (6) x2

Ixx and Iyy may be calculated by numerical integration of the terms in equations (4) and (6). Alternatively, analytical integration of equations (4) and (6) yields JAERO138 © IMechE 2007

Preliminary design parameters for lenticular booms

379

the following expressions for the bending moments of area of the lenticular cross-section

x1 2 Ixx = 2tr1 x1 r12 − x12 + (r12 + 2y01 ) sin−1 r1

2 2 + 2tr2 (x2 − x1 ) r2 − (x2 − x1 ) − 4r2

Fig. 3 The area enclosed by the boom wall (excluding the straight part of the section)

x2 − x 1 + 6tr23 sin−1 + 8tr1 x1 y01 (7) r2 x1 βt 3 Iyy = 2tr1 r12 sin−1 (x − x23 ) − x1 r12 − x12 + 3 3 r1

2 2 + 2tr2 (3x2 + x1 ) r2 − (x2 − x1 ) − 4x2 r2 + 2tr2 (2x22 + r22 ) sin−1

x2 − x 1 r2

4A2 t=const 4A2 t −−−−→ J = P dP/t

J=

x1 2 2 +y dx r − x 0,1 1 0

2 x2 2 2 + x1 − r2 − (x − x2 ) + y0,2 dx x1 r /( r12− x 2 ) dx 0 1 x2 + x1 r2 /[ r22 − (x − x2 )2 ]dx

3

(10)

ELASTIC STRAIN ENERGY OF THE FLATTENED BOOM

The elastic strain energy stored in the flattened boom is required to derive the dynamic equations of the boom deployment [14]. It may also be needed for other estimations during preliminary design of the boom, such as power rating of the deploying/retracting motor of the boom assembly [14]. In order to estimate the elastic strain energy (U ), the theory of curved beams with small initial curvature will be used [12]. Since the ratio of the initial radius of curvature to the height of the section is sufficiently large, the assumption of small initial curvature may be used. The half-section of the lenticular boom shown in Fig. 4 is assumed to be a curved beam of length L (L = L1 + 2L2 + 2L3 ) and rectangular cross-section, where Li ’s are lengths of different curved segments. In the model of Fig. 4, the height and width of the cross-section are represented by symbols t and h, respectively. In this way, the flattening process of a lenticular boom-element of wall thickness t and length h may be analysed using a two-dimensional beam model.

(9)

where according to Fig. 3, A is the area enclosed by the centre-line of the boom wall (neglecting the straight part of the section), and P is the perimeter of this area. JAERO138 © IMechE 2007

16t

(8)

An important section of the lenticular boom is the flat edge of the section which is bounded between x2 and x3 , as shown in Fig. 2. This part of the lenticular section is used during the manufacturing of the boom to join two similar lenticular sections symmetrically. In equations (6) and (8), the parameter β is used instead of the expected fixed number ‘4’. This parameterization of the equation is to consider the different thicknesses that may be achieved in the production phase. In fact, if the joining of the two halfsections is done through seam welding, the plates may be pressed against each other, and as a result β may become less than four. Similarly, if the straight part of the section is perforated to facilitate engagement with the deployment/retraction gears (as the case described in reference [11]), a number smaller than four may be applicable for the parameter β to estimate the average Iyy over the whole boom length. In general, a proper value of β should be used such that β = (2 × total edge thickness after joining)/t. In the design process, it is important to know the minimum moment of area and the related principal direction in the boom cross-section. Since the lenticular section is symmetric, the principal moments of area are the calculated Ixx and Iyy themselves [12]. Therefore, one of these two is the minimum moment of area of the section, and must be greater than the required moment specified as the design constraint. As the boom is a thin-walled hollow tube, the equivalent torsional moment of area can be found using the formula [13] J=

Equation (9) can also be written in terms of the boom section parameters as

Fig. 4 Two-dimensional thin curved beam with rectangular cross-section


380


The elastic strain energy of the beam of Fig. 4 can be estimated as L Uhalf−section = 0

M2 dl 2EI

(11)

The bending moment (M ) needed to bend the halfsection shown in Fig. 4 can be calculated as M = EI

1 1 − R R

(12) Fig. 5

where R and R are the general symbols for the radii of curvature before and after applying the bending moment M , respectively [12]. In the model of Fig. 4, the initial radii of curvature (R) for the curved segments are r1 and r2 , which by completion of the flattening process approach infinity (R = ∞). Thus, combining equations (11) and (12) yields L Uhalf−section = 0

EI EI dl = 2R 2 2

L2 L1 +2 2 r12 r2

(13)

where moment of area of the beam, I is equal to ht 3 /12. For a unit length of the full lenticular boom, rewriting equation (13) by substituting h = 1 yields Et 3 U= 12

L1 L2 +2 2 r12 r2

(14)

Total strain energy comprises of the energy calculated by equation (14) and that needed for bending of the flattened boom around a spool of radius Rs .

written as t/2r. Assuming E as the Young’s modulus of the shell, the maximum strain and stress will be σmax =

MAXIMUM STRESSES

In this section, the strains and stresses in a thin-walled long strip with transverse radius r will be calculated. The strip is first flattened and then coiled around a spool of radius Rs in the longitudinal direction. The derivation is based on the Euler–Bernoulli beam theory which suffers from some errors in large deformations. As the ratio of the radius of curvature to the beam thickness in the lenticular boom is considerably high, the deformation may be assumed to be small. The formulas that will be derived here have been in use in the profession (e.g. [5, 15]) without addressing derivations and validations of the formulas. With flattening of the shell, the neutral axis remains unstrained therefore the length of the flattened shell will be equal to rθ (Fig. 5), but the length of the upper layer before flattening is (r + t/2)θ, and the change of length is tθ/2. Therefore, the strain in the upper layer is t/(2r + t). Since 2r + t ≈ 2r, the strain can be Proc. IMechE Vol. 221 Part G: J. Aerospace Engineering

Et t , εmax = 2r 2r

(15)

which are the maximum stress and strain due to flattening of a long thin strip with transverse radius of r. If r is a variable in the cross-section or when several radii exist in transverse direction (as shown in Fig. 4), then rmin should be used in equation (15). When after flattening in transverse direction, the boom is tightly coiled around a spool of radius Rs , some additional stress will arise which will be perpendicular to the stress developed during flattening. In this case, the total thickness of the model strip will be 2t and the neutral axis will be on the inner surface of the boom element. Thus, the additional maximum stress in the uppermost fibres will be equal to Et/Rs and the maximum additional strain and stress after coiling will be σmax =

4

Cross-section of a thin long strip

Et t , εmax = Rs Rs

(16)

Assuming these stresses represent principal stresses, Mohr’s circle may be used next to combine them and to find the maximum normal and shear stresses in the coiled boom using equations (15) and (16) σmax = max

Et Et Et 1 1 , τmax = (17) , + 2r Rs 2 2r Rs

Equations (16) and (17) are left without verification in this paper. However, the same equations have been mentioned and used in the literature [5], and they seem to be sufficiently accurate. 5 VERIFICATION METHODS In order to validate the proposed method, 12 different cases were analysed using the proposed formulas. The results were compared then with those of FEM JAERO138 © IMechE 2007


Table 1

r1 (mm) r2 (mm) y0,1 (mm) x1 (mm) x0,2 (mm) y0,2 (mm) x3 (mm) L1 (mm) L2 (mm) L3 (mm) A (mm2 ) P (mm)

Geometric parameters of the four model cross-sections M1

M2

M3

M4

12.0 12.0 2.0 10.9 21.8 12.0 25.8 27.4 13.7 4.0 611 110

10.0 15.0 2.0 8.5 21.4 15.0 25.4 20.5 15.4 4.0 470 102

10.0 20.0 −2.0 6.8 20.4 20.0 24.4 15.0 15.0 4.0 286 90

14.0 10.0 −7.0 9.9 16.9 10.0 20.9 21.9 7.8 4.0 252 75

Fig. 6

solutions. These 12 cases included four model crosssections, summarized in Table 1, each with three different wall thicknesses of 0.05, 0.10, and 0.15 mm. It was assumed that in all cases, the behaviour of material could be represented with a perfectly elastic solid for which Young’s modulus, Poisson’s ratio, and shear modulus was E = 200 GPa, ν = 0.3, and G = 76.92 GPa, respectively. Ansys Structural 9.0, a product of Ansys Inc., Pennsylvania, USA, was used for finite-element analysis of the cases. The FEM model used to find the bending and torsional moments of area was threedimensional, whereas the one used to find the elastic strain energy and maximum flattening stress was twodimensional. The details of these two models are discussed in the following subsections. FEM model 1 – bending and torsional moments of area

In order to evaluate the moments of area (Ixx , Iyy , and J ) and compare them with the results of equations (7) to (9) a three-dimensional model was used. The model comprises a three-dimensional 30 cm cantilever boom with a rigid plate attached to its tip. The primary FEM results showed negligible connection between boom length and Ixx , Iyy , and J , therefore short booms of 30 cm length were selected to reduce the computational efforts. A tip plate was added to the loading end of the boom to prevent distortion of the section at the tip during its bending. To simulate pure bending of the boom, two parallel forces were symmetrically applied to the tip plate, whereas for pure torsion a force-couple was applied (Fig. 6). The tip plate was left with no restrictions to its degrees of freedom. The element type used to mesh the whole models was SHELL63, which is a four-node shell element [16]. The analysis type was small-deformation static analysis. All the models were meshed with a large number of elements (around 13 000–18 000 elements) to ensure accuracy. The choice of element size JAERO138 © IMechE 2007

FEM model 1 (M1) with a couple-force applied to simulate pure torsion

was made by decreasing the element dimensions until the results differed in the second decimal place. Also proper aspect ratio of the elements was taken into consideration. The FEM analyses for Iyy were carried out with β = 2, as in the models the straight edges of the half-sections were merged and the resulting total edge thicknesses were t (not 2t). Because it was not possible to obtain the moments of area directly from the FEM solutions, the displacement and rotation of the tip plate ( and θ ) were used as inputs from FEM solutions, and the following formulas were used to find the required moments of area. 1.

5.1

381

2.

FL 3 FL 3 = −→ I = (bending) 3EI 3E TL TL = θ −→ J = (torsion) GJ Gθ

where the applied force (F ) and applied torque (T ) were chosen arbitrarily (since the analysis type was ‘small-deformation’ and the results were linear functions of the loads). 5.2

FEM model 2 – flattening elastic energy and maximum stresses

In order to evaluate elastic strain energy of the flattened boom (U ) and maximum axial stresses developed in the flattened boom (σ ), a two-dimensional FEM model was used (Fig. 7). In this model, the maximum tensile/compressive stresses developed in the beam were obtained for the points around the inflection points (I.P. in Fig. 7) as well as the points away from them (O.P. or ‘ordinary points’). The reason for this distinction will be discussed in section 6.1. To find the consumed energy for flattening the boom using FEM, the area under the force– displacement curve of the top plate was used, which Proc. IMechE Vol. 221 Part G: J. Aerospace Engineering

382


tension, compression, and bending capabilities [16]. The contact elements were TARGE169 and CONTA171 with zero friction. Analysis type was ‘largedeformation static’. In all the models, the half-lentil beams were meshed with a large number of elements (400–600 elements) and the element density around the inflection points was higher than other parts of the beams.

Fig. 7 The meshed model for the flattening of the half-lentil

represents the energy consumed in the flattening process of a half-section boom of unit length (1 m). Twice this energy is the elastic energy stored in unit length of the full lenticular boom. Similar to the method described in section 3, the boom was modelled as a two-dimensional curved beam according to the geometries described in Table 1. Two rigid flat plates were also utilized in the models. The lower plate represented the ground, and the upper one facilitated flattening of the section during pressing phase before rolling around the spool. The boundary conditions applied to the model were (Fig. 7):

6

The results of FEM analyses for 12 cases are summarized in Table 2. These results include bending and torsional moments of area of the lenticular crosssection, elastic strain energy of a unit length of the flattened lenticular boom (U ), and maximum axial stresses developed in the flattened boom both at ordinary points (σOP ) and inflection points (σIP ). The same parameters calculated using the formulas derived in sections 2 and 3, as well as their errors against the FEM results are listed in Table 2. On studying Table 2 and comparing the results of the proposed formulas with the results of the finiteelement analysis, following conclusions may be made for the investigated cases.

(a) zero displacement in all directions for elements of the lower flat plate; (b) downward one-dimensional rigid body motion for elements of the upper flat plate to flatten the beam; (c) zero vertical displacement and zero rotation in the end points of the beam (points P1 and P2) to simulate the effect of the other half of the lenticular section; (d) zero horizontal displacement and rotation in the middle point of the beam (point P3), in order to ensure symmetry.

1. Maximum error in the results of equation (5) for Ixx is 7.6 per cent for the third model (M3) with t = 0.05 and 0.15 mm. 2. Maximum error in the results of equation (6) for Iyy is 3.2 per cent for model M2 with t = 0.05 mm. 3. Maximum error using equation (9) for the equivalent torsional moment of area is 6.5 per cent for model M2 with t = 0.05.

The element type used for the beam was BEAM3, which is a two-dimensional uniaxial element with Table 2 M1

Ixx (10−10 m4 ) FEM Equation Error (%) Iyy (10−10 m4 ) FEM Equation Error (%) FEM J (10−10 m4 ) Equation Error (%) U (J/m) FEM Equation Error (%) FEM σOP (MPa) Equation Error (%) FEM σIP (MPa) Equation Error (%)

FEM results compared to the results of the proposed formulas M2

M3

M4

M1

t = 0.05 mm

Method 4 3.9 2.5 10.3 (6) 10.6 2.9 7.1 (9) 6.8 4.2 0.82 (14) 0.79 2.9 417 (15) 417 0.0 452 (15) 417 7.7 (5)

RESULTS AND DISCUSSIONS

2.51 2.4 4.4 9.3 9.6 3.2 4.6 4.3 6.5 0.68 0.71 4.3 500 500 0.0 532 500 6.0

0.92 0.85 7.6 7.9 8 1.3 1.88 1.82 3.2 0.49 0.47 3.6 500 500 0.0 529 500 5.5


M2

M3

M4

M1

t = 0.10 mm 0.74 0.74 0.0 5 5 0.0 1.69 1.69 0.0 0.59 0.56 4.9 500 500 0.0 529 500 5.5

7.9 7.8 1.3 20.7 21.2 2.4 14.2 13.6 4.2 6.48 6.34 2.1 833 833 0.0 893 833 6.7

5 4.8 4.0 18.6 19.1 2.7 9 8.7 3.3 5.78 5.70 1.5 1000 1000 0.0 1061 1000 5.7

1.84 1.7 7.6 15.7 16 1.9 3.7 3.6 2.7 3.85 3.75 2.5 1000 1000 0.0 1052 1000 4.9

M2

M3

M4

t = 0.15 mm 1.48 1.47 0.7 9.9 10 1.0 3.4 3.4 0.0 4.49 4.46 0.6 1000 1000 0.0 1057 1000 5.4

11.9 11.8 0.8 31.1 31.8 2.3 21.2 20.4 3.8 21.40 21.41 0.0 1250 1250 0.0 1336 1250 6.4

7.5 7.2 4.0 27.9 28.7 2.9 13.5 13 3.7 19.13 19.23 0.5 1500 1500 0.0 1586 1500 5.4

2.76 2.55 7.6 23.6 24 1.7 5.6 5.5 1.8 12.61 12.66 0.4 1500 1500 0.0 1577 1500 4.9

2.23 2.21 0.9 14.9 15 0.7 5.1 5.1 0.0 14.89 15.06 1.2 1500 1500 0.0 1586 1500 5.4



4. Maximum error in equation (14) for the elastic energy of the flattened boom is 4.9 per cent for the model M4 with t = 0.05. 5. Equation (15) for calculation of the maximum flattening stress at ordinary points (away from the inflection points) shows excellent agreement in all the models. 6. Maximum error in equation (15) for calculation of the maximum flattening stress at inflection points is 7.7 per cent for the model M1 with t = 0.05. The above results show that the proposed formulas are sufficiently accurate for preliminary design of a lenticular boom. This will eliminate the need of using sophisticated analytical tools such as FEM software in early stages of the design of lenticular booms.

6.1

Discussions for FEM model 2

The FEM analyses made during this investigation show a typical force–displacement curve as illustrated in Fig. 8. This figure in two different scales illustrates approximately the linear behaviour in the beginning steps of the flattening process (Fig. 8(a)) followed by an unstable, cusp-type behaviour towards the end of the process (Fig. 8(b)). Also in the FEM analyses made during this investigation, it was observed that in the flattened booms, elements around the inflection points (I.P.) (Fig. 7) show an irregular stress pattern, typically shown in Fig. 9. The typical irregular pattern of Fig. 9 was observed in a neighbourhood of about 2 mm radius around the inflection points. This irregular stress pattern can be explained in light of the fact that at the inflection point (I.P.), the sign of the section axial stresses (which are mainly because of bending moments) changes and theoretically there is a sudden shift in the stress function. However, in reality this sudden stress shift cannot be handled by the beam as an ideal step function and as a result the typical pattern shown in Fig. 9 is formed around the inflection points. It is to be noted that this irregular pattern is not related to mesh size, since mesh refining did not change the situation in the

383

Fig. 9 Typical axial stress variations around inflection points

FEM analyses and this phenomenon persisted despite fine meshing around the inflection points. In contrast to the stresses at ordinary points, the overshoot stresses at the inflection points cannot be easily calculated. Therefore, an investigation has to be made regarding the existence of any correlation between the stresses at the ordinary and inflection points. Also, the results obtained around the inflection points suggest that special attention must be paid to these points as the critical design points of the lenticular cross-section. Further, analytical and experimental analyses are needed to explain this irregular stress pattern. 7

CONCLUSIONS

In this paper, several formulas were proposed for estimation of the bending and torsional equivalent moments of area of lenticular booms, as well as the elastic energy and the maximum stresses developed in the flattening process of such booms. Using the finiteelement models, it was shown that the proposed formulas are sufficiently accurate and can be safely used to perform the preliminary design of lenticular booms. After establishing the preliminary design, modal analysis, buckling analysis, and other important design analyses must be done to validate and refine the estimated parameters. Finding appropriate design formulas for the modal and buckling analyses of lenticular booms could be the subject of further research in this area. REFERENCES

Fig. 8

Force–displacement curve of the FEM solution for the half-lentil of model M1 with t = 0.05 mm, (a) zoomed view of the beginning of the curve and (b) overall view of the curve


1 Herzl, G. G., Walker, W. W., and Ferrera, J. D. Tubular spacecraft booms (extendible, reel stored). NASA Space Vehicle Design Criteria (Guidance and Control); NASA SP-8065, 1971. 2 Jensen, F. and Pellegrino, S. Arm development – review of existing technologies. CUED/D-STRUCT/TR 198, Cambridge University Engineering Department, 2001, 42 pp. 3 Gertsma, L. W., Dunn, J. H., and Kempke, E. E. Jr Evaluation of one type of foldable tube. NASA technical memorandum TM X-1187, 1965.


384


4 Jones, I. W., Boateng, C., and Williams, C. D. Study of foldable elastic tubes for large space structure – phase 1 final report. NASA N80-16083, 1981. 5 Aguirre-Martinez, M., Bowen, D. H., Davidson, R., Lee, R. J., and Thorpe, T. The development of a continuous manufacturing method for a deployable satellite mast in CFRP. British Plastics Congress, September 1986, pp. 107–110. 6 Moroka, O. A. and Denisov,V. P. Some problems of engineering of gravitational stabilizator for microsatellite magnetic-gravitational sub-system of orientation and stabilization. 5th IAA Symposium on Small Satellites for Earth Observation, Berlin, Germany, April 2005 paper IAA-B5-0509P. 7 Boom catalogue. Yuzhnoye State Design Office, Ukraine, 2005. 8 Bowden, M. L. Deployment devices. In Space vehicle mechanisms: elements of successful design (Ed. P. L. Conley), 1998, pp. 495–542 (John Wiley & Sons, New York). 9 Sickinger, C. and Herbeck, L. Deployment strategies, analyses and tests for the CFRP booms of a solar sail. European Conference on Spacecraft Structures, Materials and Mechanical Testing, Toulouse, France, 2002. 10 Goldenveizer, A. L. Theory of thin-walled rods. Technical memorandum 1322, National Advisory Committee for Aeronautics, Washington, 1951. 11 Del Campo, F. and Ruiz Urien, J. I. Collapsible tube mast – technology demonstration program. Space Technol. 1993, 13(1), 61–76. 12 Kumar, K. and Ghai, R. C. Advanced mechanics of materials, 6th edition, 1997 (Khanna Publishers, Delhi). 13 Den Hartog, J. P. Advanced strength of materials, 1952 (McGraw-Hill Book Company, New York). 14 Hakkak, F. and Khoddam, S. Deployment dynamics of a folded lenticular boom, in preparation. 15 Michael, S. K. On-orbit space shuttle inspection system utilizing an extendable boom. PhD Thesis, Depart-


ment of Aerospace Engineering, University of Maryland, 2004. 16 ANSYS Release 9.0 Documentation. Ansys Inc., Pennsylvania, 2004.

APPENDIX Notation A E F G h Ixx Iyy J L M P r R R Rs t T U

area Young’s modulus force shear modulus depth, length moment of area about x-axis moment of area about y-axis torsional moment of area length bending moment perimeter radius of curvature radius of curvature before deformation radius of curvature after deformation radius of spool thickness torque elastic strain energy

β ε θ ν σ σI.P. σO.P.

edge thickness coefficient displacement strain rotation angle Poisson’s ratio stress maximum stress around inflection points maximum stress around ordinary points
