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Analysis of tape spring hinges
ARTICLE IN PRESS
International Journal of Mechanical Sciences 49 (2007) 853–860 www.elsevier.com/locate/ijmecsci
Analysis of tape spring hinges O¨mer Soykasap Department of Mechanics, Faculty of Technical Education, Afyon Kocatepe University, Gazlıgol Yolu, 03200 Afyonkarahisar, Turkey Received 15 May 2006; received in revised form 20 November 2006; accepted 30 November 2006 Available online 16 January 2007
Abstract In this paper, the performance of four different tape spring hinges are studied. The tape spring hinges consist of pairs of short-length hardened-steel tape springs side by side but mounted in different ways, and can be used to fold and deploy structural elements on spacecraft. Behaviour of a tape spring subject to two- and three-dimensional folds is investigated by both analytical and non-linear finite element methods. The moment–rotation profiles of the hinges are obtained experimentally. It is observed that two hinge configurations yield negative moments in their moment–rotation profiles during deployment. The results of finite element analysis are compared with the experimental measurements, and are in good agreement. r 2006 Elsevier Ltd. All rights reserved. Keywords: Elastic spring; Deployable structures; Solar panel; Shell structures
1. Introduction Present and future structures on spacecraft demand low mass, stowage in a small volume and low cost. A robust deployment mechanism is needed in order to deploy the structural elements reliably. Tape spring hinges were used for deployable space structures as a deployment mechanism. A tape spring, widely known as carpenter tape, is straight strip section of a cylindrical shell. Its moment–rotation behaviour is a linear elastic for small rotations and constant moment for large rotations. Since peak moments occur at the end of linear elastic range, the tape spring latches the components attached by in deployed configuration. The tape can be bent longitudinally either in the sense opposite to the transverse curvature or in the same sense (see Fig. 1). The maximum moment in opposite sense bending is much bigger than that in equal sense bending. Therefore, they are used in pairs when there is a concern in post-latching behaviour. In a pair, one tape is subjected to equal sense bending while the other is subjected to opposite sense bending during deployment and vice versa after postlatching. Behaviour of a metallic tape spring subject to the opposite sense bending was first studied by Wuest [1]. Later Tel.: +90 272 2281311x353; fax: +90 272 2281319.
E-mail address: [email protected]. 0020-7403/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2006.11.013
Mansfield studied large-deflection torsional and flexural behaviour of tape springs [2]. Deployment dynamics of a tape spring was then studied by Seffen and Pellegrino [3]. Recently, carbon fibre reinforced plastic (CFRP) tape springs were developed in order to provide high stiffness to mass ratio for deployable structures [4]. Although moment–rotation behaviour of a tape spring for two-dimensional (2D) [5] or three-dimensional (3D) folds [6] is known, the behaviour of one pair of tape springs subject to 2D or 3D folds is quite complex because twisting moments appear in addition to bending moments. Negative moments in moment–rotation profile might appear during deployment. This paper discusses the performance of four tape-spring hinges, comprising pairs of short-length elastic tape-springs side by side but mounted in different ways, which can be used to fold and deploy structural elements on spacecraft. First the folding behaviour of a tape spring is investigated. Analytical expressions for moment versus curvature of the tape spring are given; limits on the moment versus curvature are discussed. Then behaviour of the tape spring hinges during deployment is studied experimentally. Their moment–rotation profiles are obtained experimentally, and compared with analytical and numerical solutions. It was observed that some hinge configurations yield undesired negative moments in their moment–rotation profiles during deployment. Finally
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Soykasap / International Journal of Mechanical Sciences 49 (2007) 853–860
M M +max
M +* O M −∗ Fig. 1. Folding of a tape spring (a) equal sense bending, (b) opposite sense bending. Fig. 2. Schematic moment–rotation behavior of a tape spring.
geometrically non-linear finite element analyses are carried out in order to further investigate the hinge behaviour. 2. Analysis of tape spring hinges 2.1. Analysis of a tape spring during folding 2-D behaviour of a tape spring can be described for two types of fold; equal (or same) sense and opposite sense as seen in Fig. 1. For opposite-sense bending, the tape spring initially shows high stiffness, followed by the snap-through formation of a localised elastic fold accompanied by constant-moment behaviour. For equal sense bending, the tape spring is much less stiff as, even for very small moments, lateral torsional buckling occurs which causes a number of sharp edge kinks to form. These kinks gradually coalesce into a single elastic fold with almost uniform longitudinal curvature as in the case of opposite sense bending. Simplified schematic moment–rotation behaviour for a tape spring is shown in Fig. 2. Starting from the unloaded, straight configuration, for opposite sense bending the tape behaves linearly up to a maximum moment, M max þ , then it snaps and behaves as a constant moment spring, carrying M þ while it behaves linearly down to a minimum moment, M and again behaves as a constant moment spring, carrying M for equal sense bending. Opposite-sense moment versus curvature of a tape spring made of isotropic material was first obtained by Wuest [1]. Wuest obtained the moment–curvature relationship of a tape spring subject to equal and opposite end moments, considering a slightly distorted axi-symmetric cylindrical shell. The end moment was obtained by integrating the moments about the transverse axis for the whole cross section of the tape spring Z s=2 n 1 M¼ þ nkl F 1 ðM l N l wÞ dy ¼ sD kl þ n R R s=2 2 ! 1 1 þ nkl F 2 , þ ð1Þ kl R
where s ¼ 2R sinðy=2Þ is the width of the tape spring; M l and N l are the bending moment per unit length and the axial force per unit length, respectively; w is out-of-plane deflection, the y-axis corresponds to the longitudinal direction; D ¼ Et3 =12ð1 n2 Þ is bending stiffness, in which E is the Young’ modulus; t is thickness of the tape spring, n is Poisson’s ratio; R is initial transverse radius of the tape; y is initial subtended angle of the tape spring; kl is longitudinal curvature. F1 and F2 are calculated from 2 cosh l cos l , l sinh l þ sin l F1 sinh l sin l F2 ¼ , ð2Þ 4 ðsinh l þ sin lÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi is then obtained by where l ¼ 4 3ð1 n2 Þs= t=kl . M max þ maximizing Eq. (1). F1 ¼
2.2. Folded behaviour of a tape spring Steady moments M þ and M are calculated assuming that the fold region is approximately cylindrical, i.e. the transverse radius of curvature in the fold region is zero. Fold radius in both equal sense and opposite sense bending equals to the initial transverse radius 1=kt ¼ r ¼ R [4]. The bending moment per unit length is related to the curvature changes in longitudinal and transverse directions as follows: m ¼ DðDkl þ nDkt Þ.
(3)
The folded tape undergoes curvature changes: Dkl ¼ 1=R in longitudinal direction, the positive and negative signs correspond to opposite sense and equal sense bending respectively, and Dkt ¼ 1=R in transverse direction. The steady moments are obtained by substituting the curvature changes in Eq. (3) and multiplying by the arc length of the tape, Ry M þ ¼ ð1 þ nÞDy, M ¼ ð1 nÞDy.
ð4Þ
The folding behaviour of the tape spring depends on the geometrical and material parameters. Fig. 3 shows the
ARTICLE IN PRESS Soykasap / International Journal of Mechanical Sciences 49 (2007) 853–860
400
300 40 deg 60 deg 80 deg 100 deg flat strip
300
250 Moment (Nmm)
350 Moment (Nmm)
855
250 200 150
80 deg lower bound initial stiffness
200 150 100
100 50 50 0
0 0
0.01
0.02 Curvature (1/mm)
0.03
0
0.04
0.01
0.02 Curvature (1/mm)
0.03
0.04
Fig. 4. Bounds on the moment versus curvature for y ¼ 801.
Fig. 3. Effect of subtended angle on moment versus curvature.
effect of subtended angle on moment versus curvature for a steel tape spring with R ¼ 16:91 mm; t ¼ 0:115 mm, E ¼ 210 GPa, and a constant arc length of Ry ¼ 25:55 mmrad, using Eq. (1). Increasing subtended angle for a constant arc length yields higher maximum moments but the steady moments are asymptotical to the bending moment of an initially flat strip with the same thickness t and a width of Ry. A lower bound is then obtained from the bending stiffness of the initially flat strip as follows:
follows:
t3 , (5) 12 where I l is the second moment of area of the cross section of the flat strip. The initial bending stiffness of the tape spring is important for estimating the stiffness of the structural elements in the deployed configuration. For small rotations, the tape bends into a smooth curve. The initial flexural stiffness of the tape spring can be estimated from R3 t 2 2 y EI i ¼ E y sin y þ y 8 sin , (6) 2y 2
where kl ; kt ; klt are longitudinal, transverse and twisting curvatures; l1 and l2 are maximum and minimum curvatures; a is the angle between the line of the maximum curvature and longitudinal direction, related to applied twist. Note that for a 2D fold a ¼ 0, l1 ¼ kl ¼ 1=R, and l2 ¼ kt ¼ 0 since the fold region is approximately cylindrical. For small twist angles in folded configuration, l1 and l2 are assumed to be constant, the curvature matrix reduces to ! ! 2 kl klt 1=2 sin 2a 1 cos a k¼ . ¼ klt kt R 1=2 sin 2a sin2 a
EI l ¼ ERy
where I i is the second moment of area of the cross section of the tape spring about its neutral axis. The initial stiffness and lower bound for moment versus curvature of the steel tape with a subtended angle of 801 are given in Fig. 4 along with the values obtained from Eq. (1). Note that the stiffness of the tape spring approaches to the lower bound for large curvatures. Behaviour of a 3D fold for a tape spring subject to equal sense bending was recently studied by Walker et al. [6]. However, the expressions for bending and twisting moments were based on the empirical data. To understand behaviour of a general fold let us consider a 3D surface. A 3D surface can be defined by two perpendicular lines of principal curvatures, representing directions of maximum and minimum curvature of the surface. The curvature matrix of the surface for a direction in the tangent plane of the surface is written in terms of the principal curvatures as
k¼
kl klt
klt kt
!
cos a sin a ¼ sin a cos a
"
l1 0
0 l2
#
cos a sin a , sin a cos a
(7a) k¼
! l1 cos2 a þ l2 sin2 a 1=2 sin 2aðl2 l1 Þ , 1=2 sin 2aðl2 l1 Þ l1 sin2 a þ l2 cos2 a
(7b)
(8) The steady bending moments, M þ and M , and corresponding twisting moments M Tþ and M T , are obtained for the new curvature changes as follows M þ ¼ Dy½cos2 a þ nð1 þ sin2 aÞ, M ¼ Dyð1 nÞcos2 a,
ð9aÞ
Dy sinð2aÞ, 2 Dy M T ¼ þ sinð2aÞ, ð9bÞ 2 where subscripts (+) and () correspond to opposite sense and equal sense bending, respectively. Note that the steady bending moments in Eq. (9a) reduce to those in Eq. (4) and the twisting moments become zero if a ¼ 0. In a 3D fold, the tape is subject to off axis bending moment M h . The M Tþ ¼
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Eq. (1) is used for the tapes subject to opposite sense bending whereas M h in Eq. (11) is used for the tapes subject to equal sense bending. The steady hinge moment is calculated using Eq. (11) as follows: M H ¼ 2ðM hþ M h Þ.
(12)
3. Tape spring hinge configurations and experiments
Fig. 5. Inter panel hinge configuration (option 5).
hinge moment around fold line is the sum of resolved bending and twisting moments as follows: M h ¼ M cos b þ M T sin b,
(10)
where b is off axis angle as seen in Fig. 5. Substituting the steady moments in Eq. (9a) into Eq. (10), the steady offaxis bending moment is calculated from M hþ ¼ Dy½cos2 a þ nð1 þ sin2 aÞ cos b
Dy sinð2aÞ sin b, 2
Dy sinð2aÞ sin b. ð11Þ 2 Experimental observations have shown that the magnitude of a is approximately equal to the off-axis angle b when b is small—as in the hinge configurations studied in this paper.
The tape spring hinges consist of two pairs of short-length hardened steel tape springs and are used to connect deployable solar array panels as seen in Fig. 5. A pair consists of two 140-mm-long tape springs with a separation distance of 25 mm, measured as the distance between the inner surfaces of the tape springs. The tapes are made of hardened steel, with a radius of 16.91 mm, a subtended angle of 86.551, a thickness of 0.115 mm, Young’s modulus of 210 GPa, Poisson’s ratio of 0.3, a density of 7850 kg/m3, a yield stress of 2.75 GPa. The ends of tape springs are held by rigid clamps with a width of 8 mm and a length of 21 mm. Fig. 6 shows four different hinge configurations considered: each configuration uses two pairs of the same tape springs with different arrangements, all facing each other as in Fig. 5 except in option 2b.
M h ¼ Dyð1 nÞcos2 a cos b þ
2.3. Behaviour of tape spring hinges Because the maximum moment of a tape spring in opposite sense bending is much bigger than that of in equal sense bending, they are used in pairs when there is a concern in post latching behaviour. In a pair, the tape springs are mounted side by side with different arrangements so that one of tape is subject to equal sense whereas the other is subject to opposite sense bending. The tape spring hinges consist of two pairs of the tape springs, which are symmetrically arranged, i.e. the tapes are at 7b in a pair whereas the tapes are at 7b in the other pair. See the following section for the details of hinge configurations considered, in which the tapes are straight or aligned, and contacting or non-contacting each other during folding. A simplistic approach to estimate the behaviour of the hinges is to ignore the contact, and to use superposition method. The hinge moment is taken to be the sum of the moments of the P individual tape springs around the fold line as M H ¼ M h . When calculating the hinge moment,
Option 1: In one pair, the tapes are rotated by 12.51 from horizontal axis, and have 17 mm offset between inner and outer tapes. Option 2: Three different configurations are studied. Both inner and outer tape springs straight: the tapes are mounted opposite in 2a, the tapes are flipped over in 2b, both inner and outer tape springs are straight with an offset in 2c. Option 3: Inner tapes are at 12.51 and outer tapes straight. Option 4: Each pair consists of tape springs rotated 712.51, with a 28.6 mm offset as in Fig. 5. Inner tapes are subject to opposite sense bending during folding. During deployment tapes interfere with full-width.
Moment–rotation measurements are carried out on a special hinge-bending rig as shown in Fig. 7. The end moments of the tape spring hinges subject to equal and opposite end rotations are measured by torque cells. The rotations up to 901 are applied by two knobs manually, and transferred to the hinge via gearboxes connected to the knobs. One of the gearboxes is fixed, and allows only rotation whereas the other is mounted on ball bearings so that the axial force on the hinge is zero during measurements. The test procedure is as follows: first the rotation of one side of the hinge is set, next the rotation of the other side is adjusted until the moments on both sides are approximately equal, and then the moment and the rotation values are recorded. The procedure is repeated for a new rotation. The moments are measured from fully folded configuration to unfolded configuration, corresponding to 1801 and
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Fig. 6. Hinge configurations tested.
700
Moment (Nmm)
600 analytical option 2a option 2b steady moment
500 400 300 200 100 0 0
20
40
60
80 100 120 Rotation (deg)
140
160
180
Fig. 8. Moment–rotation profile for options 2a and 2b. Fig. 7. Hinge-bending rig.
01 rotations, respectively. Measured results along with analytical estimates are given in Figs. 8–10. The behaviour of the hinges is highly non-linear. The hinge moment varies with the rotation angle linearly for small rotations as the tape springs bend without contacting each other. The hinge moment reaches a maximum for a larger rotation, then the tapes snap through and the fold localises near the ends while the moment decreases. As the rotation increases, the tapes in pair contact each other as in case of Options 1, 2a, 2b, and 4, and the friction between the tapes prevents the deformation from developing into a single fold in the
middle hence the moment increases. A further increase in rotation results the slip of the contacting tapes, and the deformation develops in a single fold in the middle with almost uniform hinge moment. Options 1, 2a, 2b, and 4 give positive torque, and the tapes in a pair interfere each other during deployment whereas 2c and 3 give negative torque and the tapes do not interfere. In option 1, around 1/3 of the width of the inner tape interfere the outer tape during deployment that gets rid of the torsional instability. In option 2a, the tapes have interference in full width; inner tape bends opposite sense
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700
Moment (Nmm)
600 analytical option 1 option 4 steady moment
500 400 300 200 100 0 0
20
40
60
80 100 120 Rotation (deg)
140
160
180
Fig. 9. Moment–rotation profile for options 1 and 4. Fig. 11. Failed hinge configuration (option 3).
500 400 option 2c option 3
Moment (Nmm)
300 200 100 0 0
20
40
60
80
100
120
140
160
180
-100 -200 -300 Rotation (deg) Fig. 10. Moment–rotation profile for options 2c and 3.
while outer tape bends equal sense. In option 2b, the tapes have interference in full width; the inner tape bends equal sense while outer tape bends opposite sense. When compared option 2a with option 2b, option 2b is stabler and has almost uniform moment for the rotation angle ranging 401 and 1601. In option 2c, both inner and outer tape springs are straight with an offset so that there is no interference during deployment. Inner tapes are subject to opposite sense bending during folding. This option yields negative torque between 1101 and 1351. In option 3, there is no interference during deployment, yielding negative torque between 1251 and 1401. In option 4, the tapes interfere with full width during deployment. In option 1, tapes interfere with only a small portion of their width, would probably damage the tapes for repeated deployments; therefore this is not preferable for the design point of view. When compared with options 2a and 2b with straight tapes, option 4 can be packed in a more compact volume; hence it is better. However, options 2c and 3 yield negative torque during deployment, which is not desirable. For both options, there is no interference during deploy-
ment due to their arrangement. Fig. 11 shows the folded hinge of option 3, in which the hinge stays in equilibrium without any constraints during deployment, having zero moment at this point. The analytical estimates for the hinge moment are in agreement with the measured results, particularly for the hinges with straight tapes as in Fig. 8, and yield a lower hinge moment with the tapes at an angle as in Fig. 9. Note that interaction between tapes is ignored for the analytical estimates. The simplified fold model captures the hinge behaviour adequately except in the region in which the tapes first start to contact and then stick each other. However, there is a disagreement for the hinges with an offset as in options 2c and 3, the unexpected negative hinge moments appear in Fig. 10. This will be investigated in detail in the following section. The steady hinge moments obtained using Eq. (12) are given in Figs. 8 and 9. The hinges with straight tapes, options 2a and 2b, have 33% higher the steady moment than those of the hinges with the tapes at an angle, options 1 and 4. The steady moment estimate could be used in an early design phase of the hinges without the need of elaborate calculations. 3.1. Finite element analysis To understand the hinge behaviour with offset configuration, one pair of the hinge in option 2c is modelled in Abaqus [7] finite element program, using 4-node doubly curved general-purpose shell elements (S4R). Isometric and side views of the model are seen in Fig. 12. Axis-1, -2, and -3 are fixed global axis, aligned with transverse, normal, and longitudinal direction of the hinge in unfolded configuration. The ends of tape springs are held by rigid clamps with a width of 8 mm and a length of 21 mm. There are two reference points under clamps in mid plane of the hinge. All the nodes within clamp region at one end are connected to the same reference point using multi point
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600
Moment (Nmm)
500
2
400
Test
300
Abaqus
200 100 0 -100
3
1
-200
Isometric view
-300 0
RP2
RP1
20
40
60
×
×
80 100 120 Rotation (deg)
140
160
180
Fig. 13. Comparison of moment–rotation profiles.
2 Side view Fig. 12. Finite element model of option 2c.
constraints. Each tape spring is meshed with 3640 elements. For folding simulation geometrically non-linear static analysis is carried out by applying prescribed conditions at the reference points RP1 and RP2. The reference points are allowed to move along longitudinal direction, axis-3, and are not allowed to move along axis-1 and -2 or to rotate about axis-2 and -3. RP1 is subject to p/2 rotation about axis-1 while RP2 is subject to p/2 rotation about axis-1. The hinge folding is a geometrically non-linear problem, the solution of which is found by breaking the simulation into increments, aiming at finding the approximate equilibrium configuration at each increment. Instabilities such as local buckling of the tape might occur during simulation; hence a converged solution might not be found. Therefore, an artificial damping is added to control the solution using a stabilization parameter. The stabilization parameter is the ratio of the extrapolated dissipated energy to the extrapolated strain energy, from which the damping factor is determined. The large amount of damping can yield an undesirable solution. Therefore, the viscous forces are compared with expected nodal forces so that the viscous forces do not dominate solution. The solution is achieved with a stabilization parameter of 1 106 along with discontinuous option of the analysis type in Abaqus. Fig. 13 compares the moment versus rotation profile of the finite element analysis with the experimental profile, showing an agreement. The hinge moment is taken to be twice the moment of one pair. Starting from straight position, the tapes snap through first in the middle, and the hinge moment decreases. For larger rotations the deformation of the tape subject to opposite sense bending develops into uniform curvature in the middle whereas the deformation of the other tape progress to both ends and develops into two folds near the ends. According to the
300 Reaction Moment (Nmm)
3
M1
200
M2
100
M3
0 -100 -200 -300 -400 0
20
40
60
80 100 120 Rotation (deg)
140
160
180
Fig. 14. Reaction moments versus rotation angle. (a) Mises stress contours at rotation angle of 1351. (b) Mises stress contours at rotation angle of 1801.
simulation, significant twisting moments occur due to the arrangement of the tapes during folding, causing the bending moment of the hinge being negative at folding angles from 123 to 1351. The hinge moment becomes positive at 1351, and increases further non-linearly up to 143.2 N mm at 1801. A comparison of reaction moments of the model is given in Fig. 14. The moments M1, M2, and M3 are the reaction moments of the reference point RP2, which are directed parallel to the global axis-1, -2, and -3, respectively. M1, M2, M3 correspond longitudinal bending, transverse bending, and twisting moments for the tape springs in unfolded position. Note that M1 is equal to half of the hinge moment. During folding M3 is negligible, and M2 is always negative. Although M2 corresponds to transverse bending moment of the tape spring for small folding angles, it corresponds to twisting moment of the tape spring for large folding angles. M1 and M2 are related to each other; M1 deceases whereas M2 increases up to 851, and both have similar profile afterwards. Although zero twisting moment is desirable for the tape springs, apparent twisting affects bending moments during deployment.
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experimentally. Analytical expressions for the hinge moment are obtained ignoring the interaction of the tape springs, the results of which agree with experiments except for the tapes with offset and no contacting arrangement. The experiments show that the tape hinges with offset configuration yield negative moment during deployment, which is against the deployment of the structural elements connected by the hinges, and hence is not desirable. During deployment interaction between tape springs in a pair is found useful. The hinge configuration in which the tapes are rotated 712.51 provides more compact volume in folded configuration and no negative moments occur in moment–rotation profile. The finite element analysis shows that significant twisting moments occur for the hinges with offset configuration.
2 3
1
[1] Wuest W. Einige anwendungen der theorie der zylinderschale. Zeitschrift fur angewandte Mathematik und Mechanik 1954;34: 444–54. [2] Mansfield EH. Large-deflection torsion and flexure of initially curved strips. Proceedings of the Royal Society London 1973;A334:279–98. [3] Seffen KA, Pellegrino S. Deployment dynamics of tape springs. Proceedings of the Royal Society of London 1999;A455:1003–48. [4] Yee JCH, Soykasap O, Pellegrino S. Carbon fiber reinforced plastic tape springs. In: Proceeding of the 45th AIAA/ASME/ASCE/AHS/ ASC structures, structural dynamics and materials conference, Palm Springs CA, 19–22 April 2004. AIAA paper 2004–1819. [5] Seffen KA, You Z, Pellegrino S. Folding and deployment of curved tape springs. International Journal of Mechanical Sciences 1999; 42(2000):2055–73. [6] Walker SJI, Aglietti G. Study of the dynamics of three-dimensional tape spring folds. AIAA Journal 2004;42(4):850–6. [7] Hibbit, Karlsson and Sorensen, Inc. ABAQUS/standard user’s manual. version 6.4. Pawtucket, RI, USA, 2004.
2 3
References
1
Fig. 15. (a) Mises stress contours at rotation angle of 1351. (b) Mises stress contours at rotation angle of 1801.
Mises stress contours at 1351 and 1801 rotation angle are given in Fig. 15. Note that at 1351 rotation angle the hinge moment is zero, and peak Mises stresses occur near clamp region of the tape subject to equal sense bending and in the middle fold region of the other tape subject to opposite sense bending. The maximum stresses are 1.879 GPa at 1351and 1.738 GPa at 1801, and show that the material is safe. 4. Conclusions Moment–rotation behaviour of four different tape spring hinges is studied analytically, numerically and
Dr Soykasap is an Assistant Professor at the Department of Mechanics, Afyon Kocatepe University. He graduated from Istanbul Technical University with B.Sc. degree in Aeronautical Engineering in 1989. Then he became a research assistant at the Department of Aeronautical Engineering. He took M.Sc. degree at the same department in 1992. He started Ph.D. programme at School of Aerospace Engineering, Georgia Institute of Technology in USA, having been awarded a scholarship from The Undersecretariat for Defence Industries of the Turkish Republic in 1996. He completed his Ph.D. at Georgia Tech. in 1999. After returning his country, he was appointed to a University Lectureship at the Department of Mechanics, Afyon Kocatepe University in 1999, and promoted to an Assistant Professorship at the same department in 1999. He worked as a Research Associate and then as a Senior Research Associate at Engineering Department, University of Cambridge between 2002 and 2005. Dr Soykasap is a member of the American Institute of Aeronautics and Astronautics, a member of The Chamber of Mechanical Engineering of Turkey, and a senior member of Wolfson College, Cambridge.
International Journal of Mechanical Sciences 49 (2007) 853–860 www.elsevier.com/locate/ijmecsci
Analysis of tape spring hinges O¨mer Soykasap Department of Mechanics, Faculty of Technical Education, Afyon Kocatepe University, Gazlıgol Yolu, 03200 Afyonkarahisar, Turkey Received 15 May 2006; received in revised form 20 November 2006; accepted 30 November 2006 Available online 16 January 2007
Abstract In this paper, the performance of four different tape spring hinges are studied. The tape spring hinges consist of pairs of short-length hardened-steel tape springs side by side but mounted in different ways, and can be used to fold and deploy structural elements on spacecraft. Behaviour of a tape spring subject to two- and three-dimensional folds is investigated by both analytical and non-linear finite element methods. The moment–rotation profiles of the hinges are obtained experimentally. It is observed that two hinge configurations yield negative moments in their moment–rotation profiles during deployment. The results of finite element analysis are compared with the experimental measurements, and are in good agreement. r 2006 Elsevier Ltd. All rights reserved. Keywords: Elastic spring; Deployable structures; Solar panel; Shell structures
1. Introduction Present and future structures on spacecraft demand low mass, stowage in a small volume and low cost. A robust deployment mechanism is needed in order to deploy the structural elements reliably. Tape spring hinges were used for deployable space structures as a deployment mechanism. A tape spring, widely known as carpenter tape, is straight strip section of a cylindrical shell. Its moment–rotation behaviour is a linear elastic for small rotations and constant moment for large rotations. Since peak moments occur at the end of linear elastic range, the tape spring latches the components attached by in deployed configuration. The tape can be bent longitudinally either in the sense opposite to the transverse curvature or in the same sense (see Fig. 1). The maximum moment in opposite sense bending is much bigger than that in equal sense bending. Therefore, they are used in pairs when there is a concern in post-latching behaviour. In a pair, one tape is subjected to equal sense bending while the other is subjected to opposite sense bending during deployment and vice versa after postlatching. Behaviour of a metallic tape spring subject to the opposite sense bending was first studied by Wuest [1]. Later Tel.: +90 272 2281311x353; fax: +90 272 2281319.
E-mail address: [email protected]. 0020-7403/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2006.11.013
Mansfield studied large-deflection torsional and flexural behaviour of tape springs [2]. Deployment dynamics of a tape spring was then studied by Seffen and Pellegrino [3]. Recently, carbon fibre reinforced plastic (CFRP) tape springs were developed in order to provide high stiffness to mass ratio for deployable structures [4]. Although moment–rotation behaviour of a tape spring for two-dimensional (2D) [5] or three-dimensional (3D) folds [6] is known, the behaviour of one pair of tape springs subject to 2D or 3D folds is quite complex because twisting moments appear in addition to bending moments. Negative moments in moment–rotation profile might appear during deployment. This paper discusses the performance of four tape-spring hinges, comprising pairs of short-length elastic tape-springs side by side but mounted in different ways, which can be used to fold and deploy structural elements on spacecraft. First the folding behaviour of a tape spring is investigated. Analytical expressions for moment versus curvature of the tape spring are given; limits on the moment versus curvature are discussed. Then behaviour of the tape spring hinges during deployment is studied experimentally. Their moment–rotation profiles are obtained experimentally, and compared with analytical and numerical solutions. It was observed that some hinge configurations yield undesired negative moments in their moment–rotation profiles during deployment. Finally
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M M +max
M +* O M −∗ Fig. 1. Folding of a tape spring (a) equal sense bending, (b) opposite sense bending. Fig. 2. Schematic moment–rotation behavior of a tape spring.
geometrically non-linear finite element analyses are carried out in order to further investigate the hinge behaviour. 2. Analysis of tape spring hinges 2.1. Analysis of a tape spring during folding 2-D behaviour of a tape spring can be described for two types of fold; equal (or same) sense and opposite sense as seen in Fig. 1. For opposite-sense bending, the tape spring initially shows high stiffness, followed by the snap-through formation of a localised elastic fold accompanied by constant-moment behaviour. For equal sense bending, the tape spring is much less stiff as, even for very small moments, lateral torsional buckling occurs which causes a number of sharp edge kinks to form. These kinks gradually coalesce into a single elastic fold with almost uniform longitudinal curvature as in the case of opposite sense bending. Simplified schematic moment–rotation behaviour for a tape spring is shown in Fig. 2. Starting from the unloaded, straight configuration, for opposite sense bending the tape behaves linearly up to a maximum moment, M max þ , then it snaps and behaves as a constant moment spring, carrying M þ while it behaves linearly down to a minimum moment, M and again behaves as a constant moment spring, carrying M for equal sense bending. Opposite-sense moment versus curvature of a tape spring made of isotropic material was first obtained by Wuest [1]. Wuest obtained the moment–curvature relationship of a tape spring subject to equal and opposite end moments, considering a slightly distorted axi-symmetric cylindrical shell. The end moment was obtained by integrating the moments about the transverse axis for the whole cross section of the tape spring Z s=2 n 1 M¼ þ nkl F 1 ðM l N l wÞ dy ¼ sD kl þ n R R s=2 2 ! 1 1 þ nkl F 2 , þ ð1Þ kl R
where s ¼ 2R sinðy=2Þ is the width of the tape spring; M l and N l are the bending moment per unit length and the axial force per unit length, respectively; w is out-of-plane deflection, the y-axis corresponds to the longitudinal direction; D ¼ Et3 =12ð1 n2 Þ is bending stiffness, in which E is the Young’ modulus; t is thickness of the tape spring, n is Poisson’s ratio; R is initial transverse radius of the tape; y is initial subtended angle of the tape spring; kl is longitudinal curvature. F1 and F2 are calculated from 2 cosh l cos l , l sinh l þ sin l F1 sinh l sin l F2 ¼ , ð2Þ 4 ðsinh l þ sin lÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi is then obtained by where l ¼ 4 3ð1 n2 Þs= t=kl . M max þ maximizing Eq. (1). F1 ¼
2.2. Folded behaviour of a tape spring Steady moments M þ and M are calculated assuming that the fold region is approximately cylindrical, i.e. the transverse radius of curvature in the fold region is zero. Fold radius in both equal sense and opposite sense bending equals to the initial transverse radius 1=kt ¼ r ¼ R [4]. The bending moment per unit length is related to the curvature changes in longitudinal and transverse directions as follows: m ¼ DðDkl þ nDkt Þ.
(3)
The folded tape undergoes curvature changes: Dkl ¼ 1=R in longitudinal direction, the positive and negative signs correspond to opposite sense and equal sense bending respectively, and Dkt ¼ 1=R in transverse direction. The steady moments are obtained by substituting the curvature changes in Eq. (3) and multiplying by the arc length of the tape, Ry M þ ¼ ð1 þ nÞDy, M ¼ ð1 nÞDy.
ð4Þ
The folding behaviour of the tape spring depends on the geometrical and material parameters. Fig. 3 shows the
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300 40 deg 60 deg 80 deg 100 deg flat strip
300
250 Moment (Nmm)
350 Moment (Nmm)
855
250 200 150
80 deg lower bound initial stiffness
200 150 100
100 50 50 0
0 0
0.01
0.02 Curvature (1/mm)
0.03
0
0.04
0.01
0.02 Curvature (1/mm)
0.03
0.04
Fig. 4. Bounds on the moment versus curvature for y ¼ 801.
Fig. 3. Effect of subtended angle on moment versus curvature.
effect of subtended angle on moment versus curvature for a steel tape spring with R ¼ 16:91 mm; t ¼ 0:115 mm, E ¼ 210 GPa, and a constant arc length of Ry ¼ 25:55 mmrad, using Eq. (1). Increasing subtended angle for a constant arc length yields higher maximum moments but the steady moments are asymptotical to the bending moment of an initially flat strip with the same thickness t and a width of Ry. A lower bound is then obtained from the bending stiffness of the initially flat strip as follows:
follows:
t3 , (5) 12 where I l is the second moment of area of the cross section of the flat strip. The initial bending stiffness of the tape spring is important for estimating the stiffness of the structural elements in the deployed configuration. For small rotations, the tape bends into a smooth curve. The initial flexural stiffness of the tape spring can be estimated from R3 t 2 2 y EI i ¼ E y sin y þ y 8 sin , (6) 2y 2
where kl ; kt ; klt are longitudinal, transverse and twisting curvatures; l1 and l2 are maximum and minimum curvatures; a is the angle between the line of the maximum curvature and longitudinal direction, related to applied twist. Note that for a 2D fold a ¼ 0, l1 ¼ kl ¼ 1=R, and l2 ¼ kt ¼ 0 since the fold region is approximately cylindrical. For small twist angles in folded configuration, l1 and l2 are assumed to be constant, the curvature matrix reduces to ! ! 2 kl klt 1=2 sin 2a 1 cos a k¼ . ¼ klt kt R 1=2 sin 2a sin2 a
EI l ¼ ERy
where I i is the second moment of area of the cross section of the tape spring about its neutral axis. The initial stiffness and lower bound for moment versus curvature of the steel tape with a subtended angle of 801 are given in Fig. 4 along with the values obtained from Eq. (1). Note that the stiffness of the tape spring approaches to the lower bound for large curvatures. Behaviour of a 3D fold for a tape spring subject to equal sense bending was recently studied by Walker et al. [6]. However, the expressions for bending and twisting moments were based on the empirical data. To understand behaviour of a general fold let us consider a 3D surface. A 3D surface can be defined by two perpendicular lines of principal curvatures, representing directions of maximum and minimum curvature of the surface. The curvature matrix of the surface for a direction in the tangent plane of the surface is written in terms of the principal curvatures as
k¼
kl klt
klt kt
!
cos a sin a ¼ sin a cos a
"
l1 0
0 l2
#
cos a sin a , sin a cos a
(7a) k¼
! l1 cos2 a þ l2 sin2 a 1=2 sin 2aðl2 l1 Þ , 1=2 sin 2aðl2 l1 Þ l1 sin2 a þ l2 cos2 a
(7b)
(8) The steady bending moments, M þ and M , and corresponding twisting moments M Tþ and M T , are obtained for the new curvature changes as follows M þ ¼ Dy½cos2 a þ nð1 þ sin2 aÞ, M ¼ Dyð1 nÞcos2 a,
ð9aÞ
Dy sinð2aÞ, 2 Dy M T ¼ þ sinð2aÞ, ð9bÞ 2 where subscripts (+) and () correspond to opposite sense and equal sense bending, respectively. Note that the steady bending moments in Eq. (9a) reduce to those in Eq. (4) and the twisting moments become zero if a ¼ 0. In a 3D fold, the tape is subject to off axis bending moment M h . The M Tþ ¼
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Eq. (1) is used for the tapes subject to opposite sense bending whereas M h in Eq. (11) is used for the tapes subject to equal sense bending. The steady hinge moment is calculated using Eq. (11) as follows: M H ¼ 2ðM hþ M h Þ.
(12)
3. Tape spring hinge configurations and experiments
Fig. 5. Inter panel hinge configuration (option 5).
hinge moment around fold line is the sum of resolved bending and twisting moments as follows: M h ¼ M cos b þ M T sin b,
(10)
where b is off axis angle as seen in Fig. 5. Substituting the steady moments in Eq. (9a) into Eq. (10), the steady offaxis bending moment is calculated from M hþ ¼ Dy½cos2 a þ nð1 þ sin2 aÞ cos b
Dy sinð2aÞ sin b, 2
Dy sinð2aÞ sin b. ð11Þ 2 Experimental observations have shown that the magnitude of a is approximately equal to the off-axis angle b when b is small—as in the hinge configurations studied in this paper.
The tape spring hinges consist of two pairs of short-length hardened steel tape springs and are used to connect deployable solar array panels as seen in Fig. 5. A pair consists of two 140-mm-long tape springs with a separation distance of 25 mm, measured as the distance between the inner surfaces of the tape springs. The tapes are made of hardened steel, with a radius of 16.91 mm, a subtended angle of 86.551, a thickness of 0.115 mm, Young’s modulus of 210 GPa, Poisson’s ratio of 0.3, a density of 7850 kg/m3, a yield stress of 2.75 GPa. The ends of tape springs are held by rigid clamps with a width of 8 mm and a length of 21 mm. Fig. 6 shows four different hinge configurations considered: each configuration uses two pairs of the same tape springs with different arrangements, all facing each other as in Fig. 5 except in option 2b.
M h ¼ Dyð1 nÞcos2 a cos b þ
2.3. Behaviour of tape spring hinges Because the maximum moment of a tape spring in opposite sense bending is much bigger than that of in equal sense bending, they are used in pairs when there is a concern in post latching behaviour. In a pair, the tape springs are mounted side by side with different arrangements so that one of tape is subject to equal sense whereas the other is subject to opposite sense bending. The tape spring hinges consist of two pairs of the tape springs, which are symmetrically arranged, i.e. the tapes are at 7b in a pair whereas the tapes are at 7b in the other pair. See the following section for the details of hinge configurations considered, in which the tapes are straight or aligned, and contacting or non-contacting each other during folding. A simplistic approach to estimate the behaviour of the hinges is to ignore the contact, and to use superposition method. The hinge moment is taken to be the sum of the moments of the P individual tape springs around the fold line as M H ¼ M h . When calculating the hinge moment,
Option 1: In one pair, the tapes are rotated by 12.51 from horizontal axis, and have 17 mm offset between inner and outer tapes. Option 2: Three different configurations are studied. Both inner and outer tape springs straight: the tapes are mounted opposite in 2a, the tapes are flipped over in 2b, both inner and outer tape springs are straight with an offset in 2c. Option 3: Inner tapes are at 12.51 and outer tapes straight. Option 4: Each pair consists of tape springs rotated 712.51, with a 28.6 mm offset as in Fig. 5. Inner tapes are subject to opposite sense bending during folding. During deployment tapes interfere with full-width.
Moment–rotation measurements are carried out on a special hinge-bending rig as shown in Fig. 7. The end moments of the tape spring hinges subject to equal and opposite end rotations are measured by torque cells. The rotations up to 901 are applied by two knobs manually, and transferred to the hinge via gearboxes connected to the knobs. One of the gearboxes is fixed, and allows only rotation whereas the other is mounted on ball bearings so that the axial force on the hinge is zero during measurements. The test procedure is as follows: first the rotation of one side of the hinge is set, next the rotation of the other side is adjusted until the moments on both sides are approximately equal, and then the moment and the rotation values are recorded. The procedure is repeated for a new rotation. The moments are measured from fully folded configuration to unfolded configuration, corresponding to 1801 and
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Fig. 6. Hinge configurations tested.
700
Moment (Nmm)
600 analytical option 2a option 2b steady moment
500 400 300 200 100 0 0
20
40
60
80 100 120 Rotation (deg)
140
160
180
Fig. 8. Moment–rotation profile for options 2a and 2b. Fig. 7. Hinge-bending rig.
01 rotations, respectively. Measured results along with analytical estimates are given in Figs. 8–10. The behaviour of the hinges is highly non-linear. The hinge moment varies with the rotation angle linearly for small rotations as the tape springs bend without contacting each other. The hinge moment reaches a maximum for a larger rotation, then the tapes snap through and the fold localises near the ends while the moment decreases. As the rotation increases, the tapes in pair contact each other as in case of Options 1, 2a, 2b, and 4, and the friction between the tapes prevents the deformation from developing into a single fold in the
middle hence the moment increases. A further increase in rotation results the slip of the contacting tapes, and the deformation develops in a single fold in the middle with almost uniform hinge moment. Options 1, 2a, 2b, and 4 give positive torque, and the tapes in a pair interfere each other during deployment whereas 2c and 3 give negative torque and the tapes do not interfere. In option 1, around 1/3 of the width of the inner tape interfere the outer tape during deployment that gets rid of the torsional instability. In option 2a, the tapes have interference in full width; inner tape bends opposite sense
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Moment (Nmm)
600 analytical option 1 option 4 steady moment
500 400 300 200 100 0 0
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40
60
80 100 120 Rotation (deg)
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160
180
Fig. 9. Moment–rotation profile for options 1 and 4. Fig. 11. Failed hinge configuration (option 3).
500 400 option 2c option 3
Moment (Nmm)
300 200 100 0 0
20
40
60
80
100
120
140
160
180
-100 -200 -300 Rotation (deg) Fig. 10. Moment–rotation profile for options 2c and 3.
while outer tape bends equal sense. In option 2b, the tapes have interference in full width; the inner tape bends equal sense while outer tape bends opposite sense. When compared option 2a with option 2b, option 2b is stabler and has almost uniform moment for the rotation angle ranging 401 and 1601. In option 2c, both inner and outer tape springs are straight with an offset so that there is no interference during deployment. Inner tapes are subject to opposite sense bending during folding. This option yields negative torque between 1101 and 1351. In option 3, there is no interference during deployment, yielding negative torque between 1251 and 1401. In option 4, the tapes interfere with full width during deployment. In option 1, tapes interfere with only a small portion of their width, would probably damage the tapes for repeated deployments; therefore this is not preferable for the design point of view. When compared with options 2a and 2b with straight tapes, option 4 can be packed in a more compact volume; hence it is better. However, options 2c and 3 yield negative torque during deployment, which is not desirable. For both options, there is no interference during deploy-
ment due to their arrangement. Fig. 11 shows the folded hinge of option 3, in which the hinge stays in equilibrium without any constraints during deployment, having zero moment at this point. The analytical estimates for the hinge moment are in agreement with the measured results, particularly for the hinges with straight tapes as in Fig. 8, and yield a lower hinge moment with the tapes at an angle as in Fig. 9. Note that interaction between tapes is ignored for the analytical estimates. The simplified fold model captures the hinge behaviour adequately except in the region in which the tapes first start to contact and then stick each other. However, there is a disagreement for the hinges with an offset as in options 2c and 3, the unexpected negative hinge moments appear in Fig. 10. This will be investigated in detail in the following section. The steady hinge moments obtained using Eq. (12) are given in Figs. 8 and 9. The hinges with straight tapes, options 2a and 2b, have 33% higher the steady moment than those of the hinges with the tapes at an angle, options 1 and 4. The steady moment estimate could be used in an early design phase of the hinges without the need of elaborate calculations. 3.1. Finite element analysis To understand the hinge behaviour with offset configuration, one pair of the hinge in option 2c is modelled in Abaqus [7] finite element program, using 4-node doubly curved general-purpose shell elements (S4R). Isometric and side views of the model are seen in Fig. 12. Axis-1, -2, and -3 are fixed global axis, aligned with transverse, normal, and longitudinal direction of the hinge in unfolded configuration. The ends of tape springs are held by rigid clamps with a width of 8 mm and a length of 21 mm. There are two reference points under clamps in mid plane of the hinge. All the nodes within clamp region at one end are connected to the same reference point using multi point
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Moment (Nmm)
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2
400
Test
300
Abaqus
200 100 0 -100
3
1
-200
Isometric view
-300 0
RP2
RP1
20
40
60
×
×
80 100 120 Rotation (deg)
140
160
180
Fig. 13. Comparison of moment–rotation profiles.
2 Side view Fig. 12. Finite element model of option 2c.
constraints. Each tape spring is meshed with 3640 elements. For folding simulation geometrically non-linear static analysis is carried out by applying prescribed conditions at the reference points RP1 and RP2. The reference points are allowed to move along longitudinal direction, axis-3, and are not allowed to move along axis-1 and -2 or to rotate about axis-2 and -3. RP1 is subject to p/2 rotation about axis-1 while RP2 is subject to p/2 rotation about axis-1. The hinge folding is a geometrically non-linear problem, the solution of which is found by breaking the simulation into increments, aiming at finding the approximate equilibrium configuration at each increment. Instabilities such as local buckling of the tape might occur during simulation; hence a converged solution might not be found. Therefore, an artificial damping is added to control the solution using a stabilization parameter. The stabilization parameter is the ratio of the extrapolated dissipated energy to the extrapolated strain energy, from which the damping factor is determined. The large amount of damping can yield an undesirable solution. Therefore, the viscous forces are compared with expected nodal forces so that the viscous forces do not dominate solution. The solution is achieved with a stabilization parameter of 1 106 along with discontinuous option of the analysis type in Abaqus. Fig. 13 compares the moment versus rotation profile of the finite element analysis with the experimental profile, showing an agreement. The hinge moment is taken to be twice the moment of one pair. Starting from straight position, the tapes snap through first in the middle, and the hinge moment decreases. For larger rotations the deformation of the tape subject to opposite sense bending develops into uniform curvature in the middle whereas the deformation of the other tape progress to both ends and develops into two folds near the ends. According to the
300 Reaction Moment (Nmm)
3
M1
200
M2
100
M3
0 -100 -200 -300 -400 0
20
40
60
80 100 120 Rotation (deg)
140
160
180
Fig. 14. Reaction moments versus rotation angle. (a) Mises stress contours at rotation angle of 1351. (b) Mises stress contours at rotation angle of 1801.
simulation, significant twisting moments occur due to the arrangement of the tapes during folding, causing the bending moment of the hinge being negative at folding angles from 123 to 1351. The hinge moment becomes positive at 1351, and increases further non-linearly up to 143.2 N mm at 1801. A comparison of reaction moments of the model is given in Fig. 14. The moments M1, M2, and M3 are the reaction moments of the reference point RP2, which are directed parallel to the global axis-1, -2, and -3, respectively. M1, M2, M3 correspond longitudinal bending, transverse bending, and twisting moments for the tape springs in unfolded position. Note that M1 is equal to half of the hinge moment. During folding M3 is negligible, and M2 is always negative. Although M2 corresponds to transverse bending moment of the tape spring for small folding angles, it corresponds to twisting moment of the tape spring for large folding angles. M1 and M2 are related to each other; M1 deceases whereas M2 increases up to 851, and both have similar profile afterwards. Although zero twisting moment is desirable for the tape springs, apparent twisting affects bending moments during deployment.
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experimentally. Analytical expressions for the hinge moment are obtained ignoring the interaction of the tape springs, the results of which agree with experiments except for the tapes with offset and no contacting arrangement. The experiments show that the tape hinges with offset configuration yield negative moment during deployment, which is against the deployment of the structural elements connected by the hinges, and hence is not desirable. During deployment interaction between tape springs in a pair is found useful. The hinge configuration in which the tapes are rotated 712.51 provides more compact volume in folded configuration and no negative moments occur in moment–rotation profile. The finite element analysis shows that significant twisting moments occur for the hinges with offset configuration.
2 3
1
[1] Wuest W. Einige anwendungen der theorie der zylinderschale. Zeitschrift fur angewandte Mathematik und Mechanik 1954;34: 444–54. [2] Mansfield EH. Large-deflection torsion and flexure of initially curved strips. Proceedings of the Royal Society London 1973;A334:279–98. [3] Seffen KA, Pellegrino S. Deployment dynamics of tape springs. Proceedings of the Royal Society of London 1999;A455:1003–48. [4] Yee JCH, Soykasap O, Pellegrino S. Carbon fiber reinforced plastic tape springs. In: Proceeding of the 45th AIAA/ASME/ASCE/AHS/ ASC structures, structural dynamics and materials conference, Palm Springs CA, 19–22 April 2004. AIAA paper 2004–1819. [5] Seffen KA, You Z, Pellegrino S. Folding and deployment of curved tape springs. International Journal of Mechanical Sciences 1999; 42(2000):2055–73. [6] Walker SJI, Aglietti G. Study of the dynamics of three-dimensional tape spring folds. AIAA Journal 2004;42(4):850–6. [7] Hibbit, Karlsson and Sorensen, Inc. ABAQUS/standard user’s manual. version 6.4. Pawtucket, RI, USA, 2004.
2 3
References
1
Fig. 15. (a) Mises stress contours at rotation angle of 1351. (b) Mises stress contours at rotation angle of 1801.
Mises stress contours at 1351 and 1801 rotation angle are given in Fig. 15. Note that at 1351 rotation angle the hinge moment is zero, and peak Mises stresses occur near clamp region of the tape subject to equal sense bending and in the middle fold region of the other tape subject to opposite sense bending. The maximum stresses are 1.879 GPa at 1351and 1.738 GPa at 1801, and show that the material is safe. 4. Conclusions Moment–rotation behaviour of four different tape spring hinges is studied analytically, numerically and
Dr Soykasap is an Assistant Professor at the Department of Mechanics, Afyon Kocatepe University. He graduated from Istanbul Technical University with B.Sc. degree in Aeronautical Engineering in 1989. Then he became a research assistant at the Department of Aeronautical Engineering. He took M.Sc. degree at the same department in 1992. He started Ph.D. programme at School of Aerospace Engineering, Georgia Institute of Technology in USA, having been awarded a scholarship from The Undersecretariat for Defence Industries of the Turkish Republic in 1996. He completed his Ph.D. at Georgia Tech. in 1999. After returning his country, he was appointed to a University Lectureship at the Department of Mechanics, Afyon Kocatepe University in 1999, and promoted to an Assistant Professorship at the same department in 1999. He worked as a Research Associate and then as a Senior Research Associate at Engineering Department, University of Cambridge between 2002 and 2005. Dr Soykasap is a member of the American Institute of Aeronautics and Astronautics, a member of The Chamber of Mechanical Engineering of Turkey, and a senior member of Wolfson College, Cambridge.