Guest, SD

He, Z

English

Apollo (Cambridge)

Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019Form and Force7–10 October 2019, Barcelona, SpainC. Lázaro, K.-U. Bletzinger, E. Oñate (eds.)Design of Novel Origami Shell StructuresZeyuan He∗, Simon D. Guest∗ Department of Engineering, University of CambridgeTrumpington Street, Cambridge CB2 1PZ, UKzh299@cam.ac.ukAbstractA developable quadrilateral creased paper is the union of a planar region (the paper) and aquadrilateral mesh embedded on this paper. In this article we will present some design methodsfor constructing a shell structure based on three different types of rigid-foldable and developablequadrilateral creased papers. Each type mentioned in this article can approximate differenttypes of surfaces. The first two types of quadrilateral creased papers also have the “self-locking"property – the rigid folding motion halts at the final rigidly folded states due to clashing ofpanels, and the final rigidly folded state forms the approximation. Each of the examples has aone degree of freedom unique rigid folding motion.Keywords: rigid origami, rigid-foldability, approximation1 IntroductionThis paper will present some design methods for constructing a shell structure, which are basedon different types of rigid-foldable and developable quadrilateral creased papers. A developablequadrilateral creased paper is the union of a planar region (the paper) and a quadrilateral meshembedded on this paper.Deployable shell structures are often considered for temporary shelters [1]. For such applicationsthe shell structure needs to have the following properties: (1) enough structural strength andlow self-weight (preferably portable by one person); (2) small packaged volume; (3) easily de-ployable (without the use of heavy lifting devices). It has been shown that origami has greatpotential for designing structures with those properties [1, 2], since (1) the rigid folded plates ofthese structures provide enhanced structural performance in the deployed form; (2) origami foldspermit easy packaging in a small volume. Considering the design of an origami-inspired shelter,[3, 4] provide multi-objective shape optimization methodologies which balance the priorities ofstructural performance and energy efficiency, including dimension control and material selection.Some other shape design methods and shelter examples can be seen in [5, 6]. Based on the shapeprovided by origami design, it will then be possible for soft wall (canvas) and rigid wall solutionsto be packaged, assembled, deployed and fixed to become a practical shelter.In this paper we will enlarge the range of possible design of such shell structures by applyingthe techniques developed in [7] and considering new types of rigid-foldable quadrilateral creasedpaper. Here three design examples for a shell structure, each based on one of these types, willCopyright c© 2019 by [Zeyuan He, Simon D. Guest]Published by the International Association for Shell and Spatial Structures (IASS) with permission.1Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019Form and Forcebe discussed in detail. The first example is the developable case of the “parallel repeating” type,which is generated with rows of parallel inner creases; the second one is the developable caseof the “orthodiagonal” type, which is generated with several parallel straight line segments; andthe third one is the developable case of the “longitudinal linear repeating” type, which is newlydiscovered [8] and has less regular rigid folding motion than the other two types.2 Design Method with the Parallel Repeating TypeIn this section we will present the design method for the parallel repeating type of rigid-foldablequadrilateral creased paper. This method can generate an approximation to a surface where aplanar target curve f is scanned along a planar datum curve g. The angle between the planecontaining f and the tangent vector of g, can be adjusted. With given f and g, an example ofthe parallel repeating type is shown in Figure 1(a). Here all the rows of inner creases are parallel,and all the columns are straight lines. If the number of inner vertices in the longitudinal andtransverse directions are m (m is odd) and n, there are m + 2 parallel rows and n + 2 straightline columns in the crease pattern. In Figure 1(b), the second left column (colored green) ap-proximates f at the final rigidly folded state, and the middle row (colored red) approximates gat the final rigidly folded state. Note that the first and last column of panels are designed tohalt the rigid folding motion by clashing of panels. Then the geometric parameters of the creasepattern are the sector angles on the middle row, and the length of all the inner creases coloredred and green.The algorithm to determine the above geometric parameters are listed below.(1) From the creased paper we choose, the target curve f : I 3 t → (x(t), y(t)) ∈ R2 isapproximated by a polyline, and the angle ξ between adjacent line segments is constant(Figure 1(c)). In order to be approximated, f should satisfy the following condition: thereexists a rotation θ ∈ [0, 2pi) and a shear transformation of magnitude pi/2 − ξ, ξ ∈ [0, pi],s.t. after the affine transformation f → f described below, f is monotone decreasing.[x(t)y(t)]=[1 −1/ tan ξ0 1/ sin ξ] [cos θ sin θ− sin θ cos θ] [x(t)y(t)](1)The proof of this condition and the approximation of f based on the above shear trans-formation are provided in Proposition 2 of [7]. With ξ chosen, the length of inner creasescolored green di (1 ≤ i ≤ m+ 1) can be obtained.(2) Approximate g by a polyline (Figure 1(d)), then the lengths of inner creases colored redli (1 ≤ i ≤ n − 1) can be obtained. Consider the projection of the plane containing eachcolumn of inner creases, the angles between these projections and each line segment ofthe polyline (measured from the same side) are named δi (1 ≤ i ≤ 2n − 2), which canbe calculated directly. In Figure 1(d) all the projections are perpendicular to the tangentvector of g, but other choices are also possible.2Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019Form and Force(b)(c)(a)straight linesin total m+2parallel rowsto approximatethe target curveto approximatethe datum curvesector angles on the middle row:α1α2 α4α3α5α6 α8α7...... α4n-3α4n-2 α4nα4n-1the approximationof target curve the approximationof datum curve(d)ξδ1δ2δ3......δ2n-2δ2n-4δ2n-3Figure 1: The approximation of the parallel repeating type. (a) shows the crease pattern. Here all therows of inner creases are parallel, and all the columns are straight lines. There are m = 9 rows and n = 11columns of inner vertices. The second left column (colored green) approximates the target curve f = −x2(−1 ≤ x ≤ 1) at the final rigidly folded state, and the middle row (colored red) approximates the datumcurve g = 5 sin(pix/10) (0 ≤ x ≤ 20) at the final rigidly folded state. (b) shows the approximation, wherethe first and last column of panels are designed to halt the rigid folding motion by clashing of panels.This figure is plotted by Freeform Origami [2], where the mountain and valley creases are colored red andblue. (c) is the approximation of the target curve (colored green). Here the angle ξ between adjacent linesegments is constant. (d) is the approximation of the datum curve (colored red). Consider the projectionof the plane containing each column of inner creases, the angles between these projections and each linesegment of the polyline (measured from the same side) are named δi (1 ≤ i ≤ 2n− 2).3Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019Form and Force(3) Calculate all the sector angles on the middle row αi (1 ≤ i ≤ 4n):α3 =pi − ξ2| cosα4| = cos δ1 cosα3tanα4i+3tan δ2i=tanα4itan δ2i−1(1 ≤ i ≤ n− 1)cosα4i+4cos δ2i+1=cosα4i+3cos δ2i(1 ≤ i ≤ n− 2)| cosα4n| = cosα4n−1cos δ2n−2α4i−3 = pi − α4i (1 ≤ i ≤ n)α4i−2 = pi − α4i−1 (1 ≤ i ≤ n)(2)(4) With αi (1 ≤ i ≤ 4n), di (1 ≤ i ≤ m+ 1) and li (1 ≤ i ≤ n− 1), draw the creased paper.From direct geometric analysis, at the jth (2 ≤ j ≤ n) column of inner creases, the angle betweenadjacent inner creases is not ξ. However, we can still write the curve approximated by the nthcolumn as an affine transformation of f , which is given in [7]. If m,n→∞, the approximationfrom each column will become f , hence this type can approximate a surface where a planartarget curve f is scanned along a planar datum curve g.3 Design Method with the Orthodiagonal TypeIn this section we will present the design method for the orthodiagonal type. This method canapproximate a cylindrical developable surface where a planar target curve f is scanned alonga planar datum curve g. Here each column or row of inner creases are co-planar, and eachplane containing a column of inner creases is orthogonal to each plane containing a row of innercreases. With given f and g, an example of the orthodiagonal type is shown in Figure 2(a).All the columns are straight lines and parallel to each other, and the sector angles in differentrows satisfy certain equations. If the number of inner vertices in the longitudinal and transversedirections are m (m is odd) and n, there are m+ 2 rows and n+ 2 straight line columns in thecrease pattern. In Figure 2(b), the second left column (colored green) approximates f at thefinal rigidly folded state, and the middle row (colored red) approximates g at the final rigidlyfolded state. Note that the first and last column of panels are designed to halt the rigid foldingmotion by clashing of panels. Then the geometric parameters of the crease pattern are all of thesector angles, and the length of all the inner creases colored red and green.The algorithm to determine the above geometric parameters is listed below.(1) Approximate f by a polyline (Figure 1(c)) which turns left and right alternately. Thus thelengths of inner creases colored green di (1 ≤ i ≤ m+1) can be obtained. Then calculate theangle between adjacent line segments βi (measured from the alternating sides, 1 ≤ i ≤ m).(2) Approximate g by a polyline, and find the lengths of inner creases colored red li (1 ≤ i ≤n − 1). Calculate the angle between adjacent line segments γi (measured from the sameside, 1 ≤ i ≤ n− 2).4Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019Form and Force(b)(a) straight linesin total n+2straight line columnsto approximatethe target curveto approximatethe datum curvesector angles on the middle row:α1α2α4α3the approximationof target curvethe approximationof datum curveα5α6 α8α7 ...... α4n-3α4n-2α4nα4n-1β2β1 βm............γ1γ2γn-2(c) (d)Figure 2: The approximation of the orthodiagonal type. (a) shows the crease pattern. Here each columnor row of inner creases are co-planar, and each plane containing a column of inner creases is orthogonal toeach plane containing a row of inner creases. There are m = 9 rows and n = 13 columns of inner vertices.The second left column (colored green) approximates the target curve f = −x2 (−1 ≤ x ≤ 1) at the finalrigidly folded state, and the middle row (colored red) approximates the datum curve g = 0.25 sin(pix/2)(0 ≤ x ≤ 4) at the final rigidly folded state. (b) shows the approximation, where the first and lastcolumn of panels are designed to halt the rigid folding motion by clashing of panels. This figure is plottedby Freeform Origami [2], where the mountain and valley creases are colored red and blue. (c) is theapproximation of the target curve (colored green). The angles between adjacent line segments are βi(1 ≤ i ≤ m), which are measured from the alternating sides. (d) is the approximation of the datum curve(colored red). The angles between adjacent line segments are γi (1 ≤ i ≤ n − 2), which are measuredfrom the same side.5Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019Form and Force(3) Choose α0 s.t. pi/2 < α0 < (pi + β(m+1)/2)/2, then calculate all the sector angles αij(1 ≤ i ≤ m, 1 ≤ j ≤ 4n), α0 controls the angle between the tangent vector of g and theplane containing f :αi,3 = αi,4n =pi + βi2(1 ≤ i ≤ m)α(m+1)/2,4 = α0α(m+1)/2,4j−1 = pi − α(m+1)/2,4j−4 (2 ≤ j ≤ n)cosα(m+1)/2,4j| cosα(m+1)/2,3|= cos(γj−1 − arccoscosα(m+1)/2,4j−1| cosα(m+1)/2,3|) (2 ≤ j ≤ n− 1)tanαi,4jtanαi+1,4j=tanαi,4j−1tanαi+1,4j−1(m− 12≥ i ≥ 1, 1 ≤ j ≤ n− 1)αi,4j+3 = pi − αi,4j (m− 12≥ i ≥ 1, 1 ≤ j ≤ n− 1)tanαi,4jtanαi−1,4j=tanαi,4j−1tanαi−1,4j−1(m+ 32≤ i ≤ m, 1 ≤ j ≤ n− 1)αi,4j+3 = pi − αi,4j (m+ 32≤ i ≤ m, 1 ≤ j ≤ n− 1)αi,4j−3 = pi − αi,4j (1 ≤ i ≤ m, 1 ≤ j ≤ n)αi,4j−2 = pi − αi,4j−1 (1 ≤ i ≤ m, 1 ≤ j ≤ n)(3)(4) With αi (1 ≤ i ≤ 4n), di (1 ≤ i ≤ m+ 1) and li (1 ≤ i ≤ n− 1), draw the creased paper.From direct geometric analysis, at the jth (2 ≤ j ≤ n) column of inner creases, the angle betweenadjacent inner creases is still β. We can also write the curve approximated by the nth columnas an affine transformation of f , which is given in [7]. If m,n → ∞, the approximation fromeach column will become f , and since the planes containing each column are parallel, this typecan approximate a cylindrical developable surface that a planar target curve f is scanned alonga planar datum curve g.4 Design Method with the Longitudinal Linear Repeating TypeIn this section we will introduce the design example for the new type of rigid-foldable quadrilat-eral creased paper, as shown in Figure 3(a). Here the input sector angles are α1, β1, γ1, δ1 ∈ (0, pi)(colored red). We can solve α2, β2, γ2, δ2 numerically from the four equations in (4), and henceall the sector angles in Figure 3(a) are known. These equations are derived in [9].The next step is to tessellate the sector angles in Figure 3(a) to generate a infinitely extendablerigid-foldable quadrilateral creased paper. It is possible to adjust the length of a row (coloredred) and a column (colored green) of inner creases. These lengths are required to fully determinethe creased paper once the input sector angles are given. In each row, the sector angles repeatevery four columns of inner vertices; in each column, the sector angles repeat every two rows ofinner vertices. Note that each row of inner creases are co-planar thoughout the folding, whileeach column of inner creases are not co-planar. For this new type of crease pattern, it is unclearhow we can describe the surface that can be approximated. One possibility is to approximatea target curve f scanning along a line g, where the curvature of f will increase or decreasemonotonously. We provide an example in Figure 3(b), where f is approximately an arc.6Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019Form and Force(a) (b)α1β1γ1δ1α2β2γ2δ2π-α1 π-β1 π-α2 π-β2π-γ1π-δ1 π-γ2π-δ2α1β1γ1δ1α2β2γ2δ2π-α1 π-β1 π-α2π-β2π-γ1π-δ1π-γ2 π-δ2α1β1γ1δ1α2β2γ2δ2π-α1 π-β1 π-α2 π-β2π-γ1π-δ1 π-γ2π-δ2α1β1γ1δ1γ2δ2π-α1 π-β1π-α2 π-β2π-γ1π-δ1π-γ2π-δ2α2β2Figure 3: The longitudinal linear repeating type. (a) illustrates the relation among the sector angles,where the input sector angles α1, β1, γ1, δ1 are colored red, and α2, β2, γ2, δ2 are solved numerically. Allthe sector angles and the length of a row (colored red) and a column (colored green) of inner creasesdetermine the entire creased paper. In each row, the sector angles repeat every four columns of innervertices; in each column, the sector angles repeat every two rows of inner vertices. (b) is an approximationof a circular shell, where the radius of the cross section r mototonously increases to 1.67r. Here the sectorangles are α1 = 85◦, β1 = 80◦, γ1 = 95◦, δ1 = 100◦; α2 = 91.23◦, β2 = 86.23◦, γ2 = 106.23◦, δ2 = 76.23◦.The length of inner creases on the second left column are set to be equal. The length of inner creaseson the middle row is randomly chosen. (b) is plotted by Freeform Origami [2], where the mountain andvalley creases are colored red and blue.β1 + α2 − γ1 − δ2 = 0α1 + β2 − γ1 − δ2 = 0sin γ1 sin γ2 − sin δ1 sin δ2 = 0The last equation can be chosen fromc =sinα1 − β12sinβ2 + α22sinα1 + β12sinβ2 − α22= k√sin(δ2 + γ2) sin(γ1 − δ1)sin(δ2 − γ2) sin(γ1 + δ1)orc =cosα1 − β12cosβ2 + α22cosα1 + β12cosβ2 − α22= k√sin(δ2 + γ2) sin(γ1 − δ1)sin(δ2 − γ2) sin(γ1 + δ1)k = 1 if (γ1 + δ1 − pi)(γ2 + δ2 − pi) > 0k = −1 if (γ1 + δ1 − pi)(γ2 + δ2 − pi) < 0(4)7Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019Form and ForceRemark 1. The design example shown in Figure 3(a) is a periodical solution of the longitudinallinear repeating type in [8], but is not the only case. It is possible to get some non-perodicalsolutions numerically, and the rigidly folded states can reach some other shapes. Since equation(4) is highly non-linear, there might be no solution or only degenerated solutions for a given setof input sector angles.5 ConclusionThis paper demonstrates three methods to approximate different types of surfaces based on thedevelopable case of three types of rigid-foldable quadrilateral creased papers. For the first twotypes, the first and last column of panels are designed to halt the rigid folding motion by clashingof panels. All the quadrilateral creased papers used in this paper have a unique one degree offreedom rigid folding motion.References[1] A. P. Thrall and C. P. Quaglia, “Accordion shelters: A historical review of origami-likedeployable shelters developed by the US military,” Engineering structures, 59:686–692, 2014.[2] Tomohiro Tachi, “Freeform rigid-foldable structure using bidirectionally flat-foldable planarquadrilateral mesh,” Advances in architectural geometry, pages 87–102, 2010.[3] F. J. Martínez-Martín and A. P. Thrall, “Honeycomb core sandwich panels for origami-inspired deployable shelters: Multi-objective optimization for minimum weight and maxi-mum energy efficiency,” Engineering Structures, 69:158–167, 2014.[4] C. P. Quaglia, N. Yu, A. P. Thrall, and S. Paolucci, “Balancing energy efficiency and struc-tural performance through multi-objective shape optimization: Case study of a rapidlydeployable origami-inspired shelter,” Energy and Buildings, 82:733–745, October 2014.[5] Ting-Uei Lee and Joseph M. Gattas, “Geometric design and construction of structurallystabilized accordion shelters,” Journal of Mechanisms and Robotics, 8(3):031009, 2016.[6] Niels De Temmerman, Marijke Mollaert, Tom Van Mele, and Lars De Laet, “Design andanalysis of a foldable mobile shelter system,” International Journal of Space Structures,22(3):161–168, 2007.[7] Zeyuan He and Simon D. Guest, “Approximating a Target Surface with 1-DOF Rigid Origa-mi,” in Origami 7: The proceedings from the seventh meeting of Origami, Science, Mathe-matics and Education, volume 2, pages 505–520. Tarquin, 2018.[8] Zeyuan He and Simon D. Guest, “On Rigid Origami II: Quadrilateral Creased Papers,”arXiv preprint arXiv:1804.06483, 2018.[9] Ivan Izmestiev, “Classification of flexible Kokotsakis polyhedra with quadrangular base,”International Mathematics Research Notices, 2017(3):715–808, 2017.8

Design of novel origami shell structures

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Design of novel origami shell structures

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