Tape springs are being used with increasing frequency in today’s space industry to deploy small satellite aerials and array areas. However, to accurately model the deployment of an appendage mounted with tape spring hinges, it is necessary to accurately model the opening moments produced from the material strains in the tape spring fold. These moments are primarily a function of curvature. This paper uses a photographic method to study the post buckling curvatures at the fold location for both two and three dimensional tape spring folds. The results are finally compared to determine the overall data trends

Aglietti, Guglielmo S.

Walker, Scott J.I.

Southampton (e-Prints Soton)

An Investigation of Tape Spring Fold Curvature  Scott J.I. Walker, Guglielmo S. Aglietti School of Engineering Sciences, University of Southampton, UK  Abstract  Tape springs are being used with increasing frequency in today’s space industry to deploy small satellite aerials and array areas. However, to accurately model the deployment of an appendage mounted with tape spring hinges, it is necessary to accurately model the opening moments produced from the material strains in the tape spring fold. These moments are primarily a function of curvature. This paper uses a photographic method to study the post buckling curvatures at the fold location for both two and three dimensional tape spring folds. The results are finally compared to determine the overall data trends.  Introduction  Over recent decades there has been growing research and development in the miniaturisation of electronic and mechanical technologies [1][2]. It is now possible to construct very low mass satellites [3], thereby dramatically reducing the cost of access to space. However, in order to produce efficient and reliable small satellite deployable structures, it is clear that new technologies need to be developed.   The miniaturisation of larger systems does not result in efficient designs due to the complexity of small mechanisms and electric motors. One attractive structural solution is tape springs (more commonly known as carpenter tape). Tape springs are generally defined as thin metallic strips with a curved cross section. They have the key property that they are continuous (i.e. contains no mechanical hinges or other folding devices) and yet they can still be folded elastically. Therefore they require no moving parts, lubrication or locking devices and once deployed have good structural properties.   Tape springs have recently been incorporated into various hinge designs such as the Tape Spring Rolling (TSR) hinges developed at the Deployable Structures Laboratory (DSL) in the United Kingdom (UK) [4] and the three tape hinge developed by Metravib in France [5]. These designs incorporate tape springs folded in two dimensions (where the fold line is perpendicular to the lengthwise direction, Figure 1a. However, new deployment designs are under development that incorporates three dimensional tape spring folds (where the fold line does not remain perpendicular to the lengthwise direction, Figure 1b.   Figure 1: Fold lines overlaid on a tape spring  In order to model the deployment dynamics of a tape spring mounted appendage, it is first necessary to determine the static opening moment of the tape fold. This is calculated using shell theory mathematics, as a function of the surface curvature at the fold [6]. Therefore, to accurately model the opening moment the relationship between the fold angle and the fold curvature needs to be investigated.   Background  Before the curvature investigation can be discussed it is first necessary to introduce some general properties of tape springs, both statically and when subjected to applied moments. The general co-ordinate system used throughout this study is shown in Figure 2. The x-axis is defined as running along the longitudinal length of the tape spring, the y-axis lies horizontally in the sectional plane of the tape spring and the z-axis is perpendicular to the flat plane of the strip.   Figure 2: General tape spring co-ordinate system   Figure 3: General tape spring parameters The standard tape spring geometry for theoretical analysis is defined by four initial parameters (three in the sectional plane and one defining the length). These can be seen in Figure 3a and 3b. When a tape spring is subjected to two equal and opposite applied moments M (acting about the y direction), the tape spring initially deflects, then buckles to form a localised fold as shown in Figure 4.    Figure 4: Tape spring rotated by θ in an equal sense bend  Depending on the direction of the applied moments, the tape spring will buckle producing a longitudinal radius of curvature either on the same side, or the opposing side to the initial transverse radius of curvature. These folds are known as two-dimensional equal and opposite sense bends respectively. The general relationship between M and θ can be seen in Figure 5, where opposite and equal sense buckling occurs at points A and B respectively.   Figure 5: General illustration of the moment rotation relationship for a two dimensional tape spring fold  The general sign convention for this relationship is that a positive moment (M) produces an opposite sense bend [7]. Therefore Figure 4 displays negative moments and a rotational displacement in the negative direction of θ (forming an equal sense bend). From Figure 5 it should be noted that there are four key moment values identified for two-dimensional bends. These are the peak buckling moments, M+max and M-max, for opposite and equal sense bends respectively and the post buckling moments (known as steady state moments) M+* and M-*, for opposite and equal sense bends respectively. When the peak moments are studied, it is found that opposite sense bends buckle via a snap through mode, whereas equal sense bends buckle via a torsional mode [8]. This results in higher peak moment magnitudes for opposite sense bends than for equal sense bends. This research focuses on the localised curvature of post buckled tape spring folds.  Two dimensional investigation  Previous equations derived to calculate the opening moment of a tape spring fold [9] are primarily functions of the curvature at the fold location. The basic steady state moment equations, [8] have been derived and simplified because of the assumption that the longitudinal curvature (κx) at the fold location is the same as the initial transverse curvature (κy,0). However equations have also previously been used to model the steady state moment based on the assumption that the opening moment calculated as a function of curvature is equivalent to the opening moment calculated as a function of bend angle. These two assumptions fundamentally contrast with each other. Does the curvature of a buckled tape spring fold remain constant or does it increase with the bend angle and if the curvature at the fold location does remain constant, can it be assumed to be equal to the initial transverse curvature?  In order to investigate these problems, a tape spring cross section and longitudinal folds were photographed along with a scale marker and overlaid with perfect curves. To ensure the optics of the camera was accurate enough for these experiments, some perfect curves and ellipses of known proportions were photographed and printed. All dimensions were accurate to within 1%.    Figure 6: Curvature graph for 2D tape spring folds Both fold directions were analysed at five and six different bend angles for equal and opposite sense bends respectively. Circular curves were overlaid to match the curvature at both the centre and the edges of the tape section to determine the maximum and minimum curvature at the fold. These two values were averaged to give a single curvature value. A summary of the data is displayed in Figure 6.   It can be seen that the opposite sense bend curvatures correspond closely to κy,0 and are very stable with a low variation between maximum and minimum curvatures. In contrast the equal sense bend curvatures are much more unstable with a larger variation of results, however, with increasing bend angle the curvature approaches κy,0. The low angle high curvature values are primarily due to sharp curvatures at the tape edges, which significantly increase the average curvature magnitude. It can be seen that the opposite and equal sense bend curvatures tend to be less than and greater than κy,0  respectively. The fundamental conclusion of this analysis is that the curvature does not increase with the bend angle θ. The assumption of constant curvature is closer to the real situation. As κy,0  is centrally located between the bend curvatures, it is a good single approximation for all bending results. The assumption of constant curvature equating to κy,0 therefore provides a basis for further analytical studies.  Three dimensional investigation  Three-dimensional folds (as shown previously in Figure 1) occur when the fold line is not perpendicular to the lengthwise direction of the tape. This causes the tape to fold out of the two-dimensional plane. However, all three-dimensional folds can be modelled as a combination of bending and twisting moments in a single ‘theoretical’ plane, as shown in Figure 7. The relative twist angle of the tape ends is denoted by γ.   Figure 7: Theoretical (x-z) plane of tape fold  From analytical studies of three dimensional surface curvature it is known that a three dimensional surface is defined by two lines of principle curvature, which are perpendicular to each other [10], as shown in Figure 8a. These are the lines of maximum and minimum curvature of the surface. The mathematics state that if a twisting curvature is applied to the surface then the lines of principle curvature rotate as displayed in Figure 8b. This can be seen experimentally as shown in Figure 9. The angle of rotation is denoted by φ. The tape fold curvatures used in the theoretical equations are all elements of the curvature matrix for the surface as shown in equation 1.  The rotation of φ modifies this surface curvature matrix as shown in equation 2, where λ1 and λ2 are the maximum and minimum curvatures of the surface respectively.    Figure 8: The lines of principle curvature     This study uses the photographic method to investigate the variation of λ1 as γ increases. If λ1 can be proven to be constant as γ (and hence φ) increases then the analytical expression that calculates M simplifies to becomes a function of only one variable, φ. A full summary of the average curvature data for both bend directions is shown in Figures 10 and 11.   It can be seen from Figure 10a that when γ = 0o the curvature rises towards a value of 0.065 as the bend angle increases. This result confirms the conclusions of the two dimensional study. It is also found that as γ increases, the maximum curvature of the surface increases. These trends remain consistent over the three figures, however the overall rise in λ1 reduces as the length of tape increases. In fact it can be seen from Figure 10c that it could be assumed λ1 remains constant for all values of γ.   Similar trends are displayed in Figure 11. In this case as the bend angle increases, the curvature decreases towards 0.065. It can be seen from Figure 11a that an increase in γ results in a significant rise in λ1. However, when the sample length increases to 433mm, this trend is only noticeable at the higher bend angles. The final figure, corresponding to a sample length of 567mm, shows no rise in curvature, resulting in λ1 being constant for all values of γ. (1) (2)  Figure 9: Rotation of principle curvature lines for an equal sense bend       Figure 10: Opposite sense bend relationship between λ1 and γ       Figure 11: Equal sense bend relationship between λ1 and γ  Conclusions  This investigation has studied the curvature properties of tape springs folded in both two and three dimensions as a preliminary step to predicting the opening moment of a three dimensional tape spring fold. From previous studies of two dimensional tape spring folds [6], it has been concluded that the length of the tape spring and the bend angle of the fold (after buckling has occurred) only has a minor effect on the opening moment of the tape. However, as soon as twist is added to the theoretical system, the length of the tape spring becomes critical. It directly influences the rotation of the principle curvature lines, which has a varying impact on the curvature of the fold. These results suggest that a constant curvature model could be used to predict the opening moment for a three dimensional fold if the length and the bend angle of the specimen were above a tolerance level. However, for the short tape lengths commonly used in space applications, variable curvature models would be required.  References  [1] Liebe, C.C. and Mobasser, S., Mems Based Sun Sensor, Proceedings of the IEEE Aerospace Conference, 31565-31572, 2001 [2] Paine, M.D. and Gabriel, S.B., A Micro-Fabricated Colloidal Thruster Array, Proceedings of the 37th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Salt Lake City, UT, July 8-11, 2001, 2001-3329  [3] Sweeting, M.N., 25 Years of Space at Surrey – Pioneering Modern Microsatellites, Acta Astronautica, 49(12), 681-691 [4] Watt, A.M. and Pellegrino, S., Tape Spring Rolling Hinges, Proceedings of the 36th Aerospace Mechanisms Symposium, Glen Research Centre, 2002 [5] Givois, D., Sicre, J. and Mazoyer, T., A Low Cost Hinge for Appendicies Deployment: Design, Test and Applications, Proceedings of the 9th European Space Mechanisms and Tribology Symposium, ESA/ESTEC, 2001 [6] Mansfield, E.H., Large-Deflection Torsion and Flexure of Initially Curved Strips, Proceedings of the Royal Society London, A. 334, 279-298 [7] Seffen, K.A., You, Z. and Pellegrino, S., Folding and Deployment of Curved Tape Springs, International Journal of Mechanical Sciences, 42, 2055-2073 [8] Seffen, K.A. and Pellegrino, S., Deployment Dynamics of Tape Springs, Proceedings of the Royal Society London, A 455, 1003-1048 [9] Walker, S.J.I and Aglietti, G.S., A Study into the Dynamics of Three Dimensional Tape Spring Folds, American Institute of Aeronautics and Astronautics Journal, V42, N. 4, April 2004, pp.  850-856 [10] Gray, A., Modern Differential Geometry of Curves and Surfaces, CRC Press, 1993 

An investigation of tape spring fold curvature

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