ORIGAMI BASED FOLDING PATTERNS FOR COMPACT DEPLOYABLE STRUCTURES P.M. Liyanage1 and H.M.Y.C. Mallikarachchi2 1 Department of Civil Engineering, University of Moratuwa, Sri Lanka Telephone: +94 71 6051834 E-mail: [email protected] 2 Department of Civil Engineering, University of Moratuwa, Sri Lanka Telephone: +94 11 2650567 Ext. 2006; Fax: +94 11 2651216 E-mail: [email protected] Abstract A deployable structure should mainly be adequately compact and should fit into any remaining space of the launch vehicle. The main factors which will determine these are the folding pattern,ease of deployment and stresses in the fold lines. Two folding patterns are selected based on extensive literature review to investigate the possibility of using those techniques for a large solar sail mission. It is expensive as well as extremely time consuming to perform experimental investigation under reduced gravity environment for this type of large membranes. Thus developing simulating techniques are quite important. Two models are simulated using Abaqus/Explicit commercial finite element software. Quasi-static conditions and numerical accuracy are verified by comparing strain energy together with kinetic energy and artificial strain energy. It is shown that spiral folding pattern requires less energy for deployment and hence that is preferred. Keywords: deployable structures, folding patterns, origami 1. Introduction Space structures such as solar sails, solar reflectors, and sunshields are essential features for many space exploration missions. These components require a larger surface area in the operational stage which can be even larger than a football field. The size of these space structures have given rise to great difficulties in designing, packaging, ground testing, and transportation. Researchers have come up with four major structural concepts which can be used to overcome the difficulties caused by the larger size of these space structures: Erectable, deployable, adaptive, and gossamer structures (Natori, Katsumata, & Yamakawa, 2010). To realize very large structures, all or a few of these concepts may be combined as necessary. Deployable structures is one of the main, and widely used methods from very early stages of space exploration. Such structures can be defined as arrangements which change its shape and size to achieve convenience in packaging and transportation. This paper will investigate the behaviour of two origami based folding patterns, spiral and circumferential folding, which can be used to fold thin membrane type compact deployable structures. The most suitable idea is selected based on the merits of each pattern. Section 2 presents the two folding patterns considered. Section 3 describes the finite element modelling techniques used and Section 4 presents the results obtained. Section 5 concludes the paper. 2. Folding Patterns The two folding patterns considered in this paper, are simple mechanisms which wraps the membrane around a hub with a polygonal shaped cross section. Some aspects of these two patterns such as, use of embedded inflatable tubes as the deployment mechanism (Natori, Katsumata, & Yamakawa, 2010) and some folding aspects of the patterns (Sakamoto, et al., 2012) have been researched by several authors. Some general information about these two folding patterns is discussed in this section. Here the background of the folding pattern, mechanisms which can be used to understand the behaviour, and other relevant details about the folding patterns are discussed. 2.1. Spiral Folding Pattern Origami Flasher(Lang, 1997) is a very famous Origami folding pattern. This was jointly developed by Chris Palmer and Jeremy, based on the technique “iso-area twist folding” (See Figure 2.1). Many variations of this pattern such as the flasher hat, flasher spinner, etc. exist in the pattern. Figure 2.1 - Origami Flasher In the flasher, if the fully deployed state as shown in Figure 2.1 is closely examined, significant plastic deformations can be observed near the centre of the membrane. Wrinkling of the membrane can be seen, even between the major fold lines. This is not a suitable condition for a deployable structure. These deformations can cause the membrane to be strained after full deployment. Spiral folding pattern utilizes the same folding mechanisms as in Origami flasher, but it is a more orderly pattern which has well defined fold lines and wraps itself around a central hub. This pattern can be called an improved version of the origami flasher, as the wrapping mechanism removes the unfavourable conditions mentioned above. In addition, this pattern undergoes much smoother deployment and folding processes that the flasher. Figure 2.2 - One dimensional folding mechanisms (a) Zigzag folding mechanism (b) Rollup folding mechanism Unlike the origami flasher, spiral folding pattern is not restricted to square modules. It can be used for polygonal modules with any number sides. These polygons do not even have to be regular polygons. The spiral folding pattern uses several folding mechanisms based on one dimensional elements and configuration of knots (points in the membrane where fold lines meet).The folding pattern it is made of a combination of zigzag and rollup folding one dimensional mechanisms(Sakamoto, et al., 2012)as shown in Figure 2.2, which goes in the circumferential and radial directions respectively. Spiral folding consists offour-fold mechanisms (Trautz & Kunstler, 2010)which contains knots which only have four fold lines meeting in them (See Figure 2.3). In these folds, the number of valley folds and the hill folds conform to the condition for inextensional wrapping patterns(Guest & Pellegrion, 1992)as given in Equation 2.1. Another positive aspect of the spiral folding pattern is that several modules can be integrated through tessellated patterns, which can be used to form large membranes. These modules are not just integrated though their shape, but also from the deployment mechanisms. | − | = 2 (2.1) Figure 2.3 - Modules with spiral folding pattern (a) Triangular shape module (b) Square shaped module (c) Hexagonal shaped module 2.2. Circumferential Folding Pattern IKAROS a light-weight membrane type 14.0 m × 14.0 m square shaped solar sail demonstrator (See Figure 2.4), wrapped around a cylindrical body of diameter 1.6 m and height 0.8 m was successfully deployed on June 2010 (Mori, et al., 2009). IKAROS is a gossamer type deployable space structure, i.e. a structure consisting of thin membranes and light weight booms aimed to achieve ultra-light weight structures, uses the circumferential pattern of folding. Figure 2.4 - Solar sail demonstrator IKAROS (Source - http://upload.wikimedia.org/wikipedia/commons/a/af/IKAROS_solar_sail.jpg) As shown in Figure 2.5 the deployment and folding processes of the circumferential folding pattern is a two step process. When considering the deployment of the membrane, first the arms which are wrapped around the hub is deployed and then once the arms are unfolded, the membrane will spread out from the arms to get the fully deployed structure. Similar to spiral folding pattern, circumferential folding pattern can also be extended for modules of any polygonal shape. Even though circumferential folding can be used for modules with irregular polygonal shaped membranes, it is not the most preferred option as it can give rise to additional fold lines which is not a preferred situation in deployable structures. Looking at the mechanisms of the circumferential folding pattern, it too can be deliberated in one dimensional elements and arrangement of knots. The circumferential folding pattern consists of zigzag folding in the radial direction and rollup folding in the circumferential direction (Sakamoto, et al., 2012). These mechanisms are arranged in a more complicated manner than in spiral folding. Looking at the fold lines meeting at knots (See Figure 2.6), it can be seen that this pattern too consists of four-fold mechanisms (Trautz & Kunstler, 2010), which conforms to the requirement for inextensional wrapping (Guest & Pellegrion, 1992) mentioned in Equation 2.1 above. Figure 2.5 - Deployment Process of IKAROS (Source - http://www.aviationweek.com) Figure 2.6 - Modules with circumferentialfolding pattern (a) Triangular shape module (b) Square shaped module (c) Hexagonal shaped module Unlike in the spiral folding pattern, modules folded with the circumferential folding pattern cannot be tessellated to fold large membranes. As a result of this the dual triangular, dual square, dual hexagonal, etc. folding patterns (Natori, Katsumata, & Yamakawa, 2010) have been developed from the circumferential folding pattern which becomes only useful for membranes consisting of tessellated modules. 3. Models to simulate Deployment Process The deployment methodsfor the above mentioned folding patterns are identically reversible processes. Hence simulation of either folding or deployment processes will be adequate to interpret the behaviour of the respective pattern in both ways. Due to the convenience of simulation, the deployment behaviour of the patterns is investigated. Analysis of the results, and their interpretations, from the simulation are given in the next two sections of the paper. For the simulation purposes, the models were taken to be two square modules made from thin sheets made from Mylar, of 10m thickness and 50 mm × 50 mm area, which are wrapped around a square hub of 10 mm × 10 mm. The membranes are folded in the most basic forms of the respective patterns, which consist of all the folding characteristics of the pattern as shown in Figure 3.1 (inclusive of the minor fold lines, which doesn’t have an opening angle of 0° in the completely folded state). During simulations the effect of membrane thickness on the folding patterns is neglected, as it goes beyond the scope of this paper. The density and elastic properties of the material are given in Table 3.1. Figure 3.1 - Models used for simulation (a) Spiral folding pattern (b) Circumferential folding pattern Table 3.1 - Material properties of Mylar Material Properties Elastic Modulus (N/mm2) Poisson’s Ratio Density (kg/m3) Value 5200 0.38 1390 Figure 3.2 - Abaqus model for spiral folding pattern The models were made to simulate a quasi-static deployment process, where they would be pulled from all four corners until the fully deployed configuration is reached. In the model all four corners are provided with displacements at a uniform rate. Each model consist of 3513 nodes and 2900 (S4) elements with the minimum length an element being around 1 mm. Figure 3.2 below shows a snapshot during the deployment process of the spiral folding model. The Abaqus/Standard solver (an implicit type solver) which uses Newton–Raphson time integration struggles in solving problems with significant discontinuities (Mallikarachchi & Pellegrino, 2011). The Abaqus/Explicit solver uses the dynamic equilibrium equations of analysis to provide solutions by finding the state of a system at a later time using the state of the system at current time which makes it capable of solving systems with significant discontinuities. In doing this, it is essential to keep the time increment to a very low value (Mallikarachchi & Pellegrino, 2011). The folding patterns considered here can be subjected to dynamic snapping (a significant change of geometry of the structure within a very small time interval). Therefore the Abaqus/Explicit solver is used for these simulations. In practice these structures are deployed in a quasi-static manner. Thus selecting a suitable time step is quite important in simulating quasi-static conditions accurately. This can be done by maintaining the kinetic energy of the model to be less than 5% of the strain energy during the simulation. However kinetic energy can have a higher percentage when it comes to dynamic snapping. 4. Results of Simulations This section will discuss about the behaviour of the models during the deployment process and the results obtained from the simulation of the above mentioned models, using Abaqus/Explicit. 4.1. Spiral Folding Pattern Analysis of the results of the models can be done considering the aspects, special characteristics of the deployment behaviour, energy variation of the model during deployment, reaction forces developed, and the stress variation of the structure during deployment. As shown in Figure 4.1, the deployment of the spiral folding model shows a smooth deployment process. There is very little intra-plane deformation. The four corners which are used to pull the membrane, shows significant deformation. Figure 4.2 shows the variation of strain energy, artificial strain energy and the kinetic energy during the deployment of the spiral folding model. Note that the kinetic energy isnegligible compared to the strain energy and sudden increase of kinetic energy corresponds to snapping. This phenomenon is called dynamic snapping, where a structure moves to a more stable state while releasing some of itsstored energy. That is the reason forhaving sudden drops in strain energy curve that coincideswith peaks in the kinetic energy curve. Another observation from the energy curve is that artificial strain energy developed in the model is negligible throughout the deployment process (See Figure 4.2). This confirms there are no artificial strains due to numerical integration scheme. Figure 4.3 shows the reaction force developed due to the deployment process. As the simulation is carried out by pulling four corners at a uniform rate, this will give a measure of the loads required for the deployment. This shows a very high variation of load, with a maximum of around 100 N, required during the initial stages of deployment. Later on this reduces to a minute amount. 4.2. Circumferential Folding Pattern Similar to the spiral folding pattern, results of the circumferential pattern can be shown as follows. Deployment of circumferential folding model occurs in two stages. During the first stage the arms which is wrapped around the central hub is extended. Then the membrane which is between the arms is spread out to get the fully deployed configuration. As shown in Figure 4.4, the first stage of deployment occurs smoothly with very little deformation in the panels. This stage which spans from 0 – 0.7 s, has very little energy development when considering the final stage as seen in Figure 4.5. During the second stage of deployment, very high energy is developed in the structure. Here wrinkling of the whole membrane can be seen, with noticeably high stresses as well. (a) (b) (c) (e) (g) (d) (f) (h) Figure 4.1 - Stress distribution of the model for spiral folding pattern during deployment (a) Time step 0s (b) Time step 0.15s (c) Time Step 0.30s (d) Time Step 0.45s (e) Time Step 0.60s (f) Time Step 0.75s (g) Time step 0.90s (h) Time step 1.00s Figure 4.2 - Energy variation of spiral folding pattern Figure 4.3 - Force vs. displacement relationship for the spiral folding model (a) (b) (c) (e) (g) (d) (f) (h) Figure 4.4 - Stress Distribution of the model for circumferential folding pattern during deployment (a) Time step 0 (b) Time step 0.10 (c) Time Step 0.35 (d) Time Step 0.45 (e) Time Step 0.60 (f) Time Step 0.80 (g) Time step 0.90 (h) Time step 1.00 Figure 4.5 - Energy variation of circumferential folding pattern Above figure provided little detail on the behaviour of the structure during the first stage of its deployment. Figure 4.6 provided a better view of the energy variation during this stage. This shows that the kinetic and artificial strain energies are very small when compared with the strain energy during this stage as well. Similar to spiral folding pattern, dynamic snapping has occurred during deployment. Figure 4.6 – Variation of energy during the first stage of deployment of the circumferential folding pattern Similar to all the other characteristics, the reaction forces developed in this pattern is much higher than that of the spiral folding pattern (See Figure 4.7). The force required shows a gradual increase until it reaches a peak value and then a sudden drop in the transition zone of stage one and stage 2 of deployment. Figure 4.7 - Force vs. displacement relationship for the circumferential folding model 5. Discussion and Conclusions Both the folding patterns considered, have many similar characteristics. Both patterns are simple folding patterns which do not require a lot of twisting and turning in the folding and deployment processes, unlike complex patterns like the Mars fold (Giesecke, 2004). For the deployment process of the two patterns many deployable mechanisms like curved tapes (Seffen, You, & S., 2000), inflatable tubes (Kishimoto, Natori, Higuchi, & Ukegawa, 2006; Natori, Katsumata, & Yamakawa, 2010), and self-deployable thin-walled composite booms (Soykasap, 2009) can be integrated to the structure, which will decrease the part count, hence the complexity. In both the models, the artificial strain energy developed is insignificant when compared with the strain energy of the structure. Artificial strain energy is the energy built-up by Abacus/Explicit solver to prevent uncontrolled deformations, by giving the critical elements a virtual stiffness (Lempiäinen). Hence it can be concluded that the simulation shows acceptable results with no uncontrollable deformations, such as shear locking and hourglassing. None of the two simulations can be classified as complete quasi-static processes from the available results, as kinetic energy is not insignificant when compared with the strain energy at all times. But the models show piecewise quasi-static behaviour with the condition for such behaviour is violated only on locations with dynamic snapping. Spiral folding pattern shows a smooth and regular deployment sequence with very small deformations in the panels. During deployment, the stress development and reaction force of the model too shows significantly lower values than the circumferential folding model. The model analysed for circumferential folding pattern wrinkles during the transition of the deployment stages, which develops very large stresses in the structure and hinders the complete deployment of the structure. From the above reasons, the spiral folding pattern proves to be the more preferred case for use in deployable structures. All these models were checked for ideal conditions, with the assumptions of zero-thickness in membranes and no plastic deformation in the fold lines. These are some aspects available for future studies of these two patterns. References Giesecke, K. (2004). Deployable Structures Inspired by the Origami Art. Massachusetts Institute of Technology. Guest, S., & Pellegrion, S. (1992). Inextensional Wrapping of Flat Membranes. Proceedings of International Seminar Structure Morphology, (pp. 203-215). Kishimoto, N., Natori, M., Higuchi, K., & Ukegawa, K. (2006). New Deployable Membrane Structure Models Inspired by Morphological Changes Nature. American Institute of Aeronautics and Astronautics. Lang, R. J. (1997). Origami in Action. MacMillan. Lempiäinen, J. (n.d.). Finite Element Simulation of Roll Forming Of High Strength Steel. Mallikarachchi, H. (2011). 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