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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.
|𝑛𝑉𝑎𝑙𝑙𝑒𝑦 − 𝑛𝐻𝑖𝑙𝑙 | = 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 10m 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.
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