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Under crease-pattern parameters (α, β, N, M) = (45°, 34.6154°, 24, 100), plots of sequences (x m , y m ) m=0,1,...,M−1 (0 < x m , y m < sin α) under different initial values are shown above. Each sequence is computed numerically and plotted in different colors while the parts of corresponding waterbomb tubes are placed on the left side of the plot. Also, the first three points (x 0 , y 0 ), (x 1 , y 1 ) and (x 2 , y 2 ) of each sequence are highlighted in triangle, square, and pentagon markers, respectively. The disk region indicates the set of initial values that yield solutions for any m.
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Folded surfaces of origami tessellations have attracted much attention because they sometimes exhibit non-trivial behaviors. It is known that cylindrical folded surfaces of waterbomb tessellation called waterbomb tube can transform into wave-like surfaces, which is a unique phenomenon not observed on other tessellations. However, the theoretical re...
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... configurations as the sequence of points in x, y-plane, i.e., the phase space. The sequence {(x m , y m ) = F m (x 0 , y 0 )|m ∈ Z} is called orbit of (x 0 , y 0 ) and the plot of them is called phase diagram. Here, we fix the crease-pattern parameters (α, β, N, M ) = (45°, 34.6154°, 24, 100) and show the phase diagram under this parameter in Fig. 7. Under this parameter, we can observe all possible types of solutions we explain ...
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... Cylinder Solution. The top and bottom of waterbomb tube in Fig. 7, left, forms a constant-radius cylinder that corresponds to fixed two points shown in black on the phase space. In this type of Each sequence is computed numerically and plotted in different colors while the parts of corresponding waterbomb tubes are placed on the left side of the plot. Also, the first three points (x 0 , y 0 ), (x 1 , ...
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... we call (w, w) satisfying Eq. (6) cylinder solution of crease pattern under parameters (α, β, N, M). Remarkably, cylinder solutions defined by Eq. (6) are called symmetric fixed points of the system (3) which is invariant under the map F and G. Cylinder solutions shown in black in Fig. 7 are numerically calculated by solving Eq. (6) under crease-pattern parameter (α, β, N, M) = (45°, 34.6154°, 24, 100), which two solutions exist and each have different ...
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... Wave-Like Solution. The second case corresponds to third to fifth from the top of waterbomb tube shown in Fig. 7, left. The wave-like folded state corresponds to the sequence of points on the phase space moving in a clockwise direction. We call such solution a wave-like solution. In a wave-like solution, the point continues rotating around the same closed curve without divergence or convergence. Note that the points along the closed curve does ...
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... m = x m . The symmetry of the orbits is the result of the reversibility of the system. Generally, the orbit of reversible systems initiated from x 0 = (x 0 , y 0 ), that is defined as the set {x = F m (x 0 ) | m ∈ Z}, is invariant under the symmetry G when the orbit has a point which is invariant under G [21]. For this reason, the orbits shown in Fig. 7, which initial terms x 0 are invariant under G, are actually symmetric about the graph of y m = x m ...
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... Finite Solution. The second waterbomb tube from the top in Fig. 7, left, corresponds to the third type, where its points are plotted just a little outside of the above-mentioned concentric plots. The reason plots stop at the middle is that, at some index m, (x m , y m ) deviate from the region (0, sin α) × (0, sinα), that is, there is no state of modules corresponding to parameter (x m , y m ). In ...
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... (0, sin α) × (0, sinα), that is, there is no state of modules corresponding to parameter (x m , y m ). In other words, finite solution appears in the case that the intersection of three spheres become empty at some step, when computing vertices of modules as shown in Fig. 6. Specifically, in the sequence corresponding to second waterbomb tube in Fig. 7, the numerical value of ninth term (x 8 , y 8 ) is (0.689573, 0.724059), which y 8 is greater than sin α = sin 45° ≈ 0.707107. So, these solution terminates at some point, and only a finite portion of the paper can be folded along this ...
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... Kinematics. From this visualization, we can observe a single disk region in the phase space as the set of initial values yielding solutions for any m (the gray region in Fig. 7). The region is the union of all wave-like solutions and the smaller cylinder solution. As the region is the configuration space of the mechanism with m → ∞, there exists a 2-DOF rigid folding motion. The motion can be represented by the "amplitude" and "phase" of wave-like surfaces. As the initial configuration gets closer to the ...
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... folding motion. The motion can be represented by the "amplitude" and "phase" of wave-like surfaces. As the initial configuration gets closer to the cylinder solution, the amplitude of wave shapes gets smaller. We can change the phase by rotating the initial configuration along the closed curve (Supporting Movie). 1 In the example state shown in Fig. 7, each initial value is taken along x 0 = y 0 , so the left-most module forms the "valley" of the wave. Fig. 7, we found that the system behaves differently around the different cylinder solutions. In Sec. 4, we perform a stability analysis to classify the symmetric fixed points, i.e., cylinder solutions, by the behavior of the system ...
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... configuration gets closer to the cylinder solution, the amplitude of wave shapes gets smaller. We can change the phase by rotating the initial configuration along the closed curve (Supporting Movie). 1 In the example state shown in Fig. 7, each initial value is taken along x 0 = y 0 , so the left-most module forms the "valley" of the wave. Fig. 7, we found that the system behaves differently around the different cylinder solutions. In Sec. 4, we perform a stability analysis to classify the symmetric fixed points, i.e., cylinder solutions, by the behavior of the system around them. We also investigate how the cylinder solutions emerge and disappear and how the stability changes ...
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... of x m+1 and y m+1 . Next, we derive x m+1 and y m+1 , i.e., the functions f and g. Here, we take the coordinate system shown in Fig. 17 and represent the coordinates of each vertex as the functions of x m and y m . Then, we obtain x m+1 and y m+1 , i.e., the functions f and g, by using the coordinates of vertices. In Figs. 17 and 18, the coordinate system is taken so that the rotation axis of symmetry of the waterbomb tube is the X-axis and one of the modules belonging ...
Citations
... The waterbomb base pattern is selected as a case study to describe the methodology developed herein to improve the mechanical performance of bistable origami structures. This choice is motivated by its simple geometry, ease of fabrication, and extensive researches on its kinematics [25], bistability [22], and potential applications to tune acoustic waves [11], create logic gates [44], build mechanical metamaterials [10], and develop innovative origami-based robots [18]. The geometrical description of the waterbomb base is presented in Fig. 1. ...
Bistable mechanical systems exhibit two stable configurations where the elastic energy is locally minimized. To realize such systems, origami techniques have been proposed as a versatile platform to design deployable structures with both compact and functional stable states. Conceptually, a bistable origami motif is composed of two-dimensional surfaces connected by one-dimensional fold lines. This leads to stable configurations exhibiting zero-energy local minima. Physically, origami-inspired structures are three-dimensional, comprising facets and hinges fabricated in a distinct stable state where residual stresses are minimized. This leads to the dominance of one stable state over the other. To improve mechanical performance, one can solve the constrained optimization problem of maximizing the bistability of origami structures, defined as the amount of elastic energy required to switch between stable states, while ensuring materials used for the facets and hinges remain within their elastic regime. In this study, the Mesh Adaptive Direct Search (MADS) algorithm, a blackbox optimization technique, is used to solve the constrained optimization problem. The bistable waterbomb-base origami motif is selected as a case-study to present the methodology. The elastic energy of this origami pattern under deployment is calculated via Finite Element simulations which serve as the blackbox in the MADS optimization loop. To validate the results, optimized waterbomb-base geometries are built via Fused Filament Fabrication and their response under loading is characterized experimentally on a Uniaxial Test Machine. Ultimately, our method offers a general framework for optimizing bistability in mechanical systems, presenting opportunities for advancement across various engineering applications.
... By visualizing the configuration of the waterbomb tube under the proposed kinematics model, solutions are divided into three categories: cylindrical, wave-like, and finite solutions. The existence of wave-like solutions is proved using the discrete theorem [30]. ...
... Imada and Tachi revealed part of the mathematical structure behind the wave-like solution of the waterbomb tessellation [30]. Through computation and visualization of the folded states of the waterbomb tube using the recurrence relation, they observed that the solutions fall into three types: cylinder, wave-like, and finite solutions. ...
The basic principle of origami is to use two-dimensional flat materials to obtain various three-dimensional target shapes by folding crease patterns. Among them, the study of waterbomb tessellations inspires the design and functionality implementation of engineering structures. However, there are some optimization spaces when building three-dimensional structures based on waterbomb origami. In this paper, we propose modeling methods to construct cylindrical and axisymmetric three-dimensional waterbomb tessellations with multi-objective optimization. Our methods aim to solve two optimization problems: (I) Constructing waterbomb tessellations to approximate the target surface more accurately. (II) Unifying waterbomb units to construct flat-foldable three-dimensional waterbomb tessellations. In addition, we present waterbomb approximations, the performance of optimization, rigid folding sequences, comparison of crease patterns, and physical origami fabrications to demonstrate the validity of our methods. Our work can expand the exploration of cylindrical and axisymmetric origami-inspired structures, such as foldable roofs, tubular materials, etc.
Graphical Abstract
... Recently, the authors have proposed a novel mathematical model of nonuniform-folding, dynamical systems of origami tessellations 26 . We focused on an origami tessellation called waterbomb tube [27][28][29][30] and formulated the coupling folding motion of the modules as the recurrence relation, i.e., the discrete dynamical system, by solving the geometric constraints. ...
... In the figures, we can observe the same structure as the waterbomb tube described in the previous research 26 , that is, there are nested closed cyclic plots, namely, quasiperiodic solutions around the elliptic fixed point, which neither converge/diverge to/from the fixed point. Each solution of F corresponds to the certain folded state of the crease pattern. ...
... In particular, the folded state corresponding to the fixed point [d,ρ] T forms the cylinder-like shape in which the folded stats of rings are identical, i.e., uniform folded state. The orbit in the empty regions outside of the quasiperiodic solutions results in the "finite solution" observed in the previous research 26 , where the three-sphere-intersection has no solution at some time steps. ...
Origami tessellations, origami whose crease pattern has translational symmetries, have attracted significant attention in designing the mechanical properties of objects. Previous origami-based engineering applications have been designed based on the “uniform-folding” of origami tessellations, where the folding of each unit cell is identical. Although “nonuniform-folding” allows for nonlinear phenomena that are impossible through uniform-folding, there is no universal model for nonuniform-folding, and the underlying mathematics for some observed phenomena remains unclear.
Wavy folded states that can be achieved through nonuniform-folding of the tubular origami tessellation called waterbomb tube are an example. Recently, the authors formulated the kinematic coupled motion of unit cells within waterbomb tube as the discrete dynamical system and identified a correspondence between its quasiperiodic solutions and wavy folded states. Here, we show that the wavy folded state is a universal phenomenon that can occur in the family of rotationally symmetric tubular origami tessellations. We represent their dynamical system as the composition of the two 2D mappings: taking the intersection of three spheres and crease pattern transformation. We show the universality of the wavy folded state through numerical calculation of phase diagrams and geometric proof of the system’s conservativeness. Additionally, we present a non-conservative tubular origami tessellation, whose crease pattern includes scaling. The result demonstrates the potential of the dynamical system model as a universal model for nonuniform-folding or a tool for designing metamaterials.
... The Japanese paper-folding art of origami has been inspiring engineers and scientists throughout the past decades, and it was found to have useful applications in many fields such as architecture, mechanical engineering, medicine and aerospace. Waterbomb is an origami pattern with six facets meeting at one vertex point in the center, which offers significant form flexibility and structural properties when tessellated and folded [5,13,9,8]. This provides an attractive design space for textile-reinforced concrete (TRC) shells, which were found to be a thin, durable and high performance alternative to the conventional steel-reinforced shells [11,7,3,10]. ...
The development of lightweight, high-performance cementitious composite materials such as carbon concrete has triggered the need for innovative design and construction methods for thin-walled structural elements with an optimal material utilization. Combining the material efficiency of carbon concrete with the form flexibility provided by parameterized origami patterns, such as a waterbomb tessellation, opens up new options for innovative design of high-performance lightweight shell structures. Indeed, the possibility to fold concrete shells in a fresh state of concrete matrix from a planar configuration to a spatial form can be exploited to develop an efficient and customizable construction method that eliminates an expensive, single-use spatial formwork. A crucial question posed in this context is, how to achieve ductile structural behavior with a composite material consisting solely of brittle and quasi-brittle components, i.e. carbon, glass, basalt reinforcement and concrete matrix. Since these materials do not provide any source of ductility on their own, substantially different design approaches compared to traditional steel-reinforced concrete are required, that exploit the stress redistribution effects at the structural level due to debonding and multiple cracking of the composite. To contribute to the understanding of these stress redistribution effects in the context of folded thin-walled spatial shell structures, this paper presents preliminary numerical and experimental studies investigating the structural behavior of shells folded using waterbomb tessellations with varied geometrical parameters. The conducted experiments on folded waterbomb shells made of carbon concrete and produced using the fold-in-fresh method [12] are presented showing the potential of this structural concept with a maximum ultimate load 45 times larger than the self weight of the shell. This development serves the aim to optimize thin-walled geometrical forms to reach a high-performance carbon concrete elements with high material utilization in service state, high ductility before failure and high load-to-weight ratio.
... Moreover, the crease buckling can break up the topological arrangement of crease lines, opening up different deformation modes for tubes that will otherwise only elongate or shorten under pressurization ( Fig. 1.3H). [45], bendy straw [46], conical Kresling tube, cylindrical Kresling tower [8], Yoshimura tube [5], and waterbomb tube [47]. ...
Thin-walled origami-inspired tubes can be used as lightweight systems for various functional applications in engineering. Folding motions can allow for deployment, reconfiguration, and compact storage of the systems, while buckling of the thin walls can be used to tune the system properties or achieve secondary functions such as energy absorption. This thesis aims to explore the stability of morphing tubes and harness buckling for functional applications. The dissertation first explores a deployable design where origami tubes extend, lock, and absorb energy through crushing (buckling and plasticity). Numerical and experimental studies investigate the tunable stiffness and energy absorption behaviors of these systems under static and dynamic scenarios. The stiffness, peak crushing force, and total energy absorption of these origami tubes can be changed through reconfiguration. These deployable systems can increase the crushing distance between impacting bodies and can allow for on-demand energy absorption characteristics. Next, the bending stability that allows for morphing in corrugated tubes is explored (bending in drinking straws). Finite element models and a reduced-order elastic simulation package can capture the nonlinear multi-stable behaviors. Modified cross-sections for the corrugated tubes are introduced and explored to identify how geometry affects bending stability, energy barriers, and stable configurations. Results show that thinner shells, steeper cones, and weaker creases are required to achieve bending bi-stability. A bar and hinge simulation model is then used to identify and capture a unique pop-up mechanism in Kresling origami that enables shape-morphing and stiffness tuning. By buckling the valley creases, the conical Kresling will pop into a dome-like shape and the crease network will be distorted. As a result, the flexible twisting motion via crease folding is prohibited, and the cone stiffness can be increased by up-to-four orders of magnitude. Parametric studies revealed that a shallower and more twisted Kresling unit will have more significant stiffness tuning. Experimental tests were used to verify the numerical predictions of tunable stiffness. This thesis explores how buckling in thin-walled origami tubes can be harnessed for functional purposes. The mechanics of three different tubular designs are explored to give insight on how geometry, sheet thickness, and material properties affect the buckling and multi-stable behaviors. These findings can inform future designs of tubular origami for shape-morphing and other functional uses.
Investigations of origami tessellations as effective media reveal the ability to program the components of their elasticity tensor. However, existing efforts focus on crease patterns that are composed of parallelogram faces where the parallel lines constrain the quasi-static elastic response. In this work, crease patterns composed of more general trapezoid faces are considered and their low-energy linear response is explored. Deformations of such origami tessellations are modeled as linear isometries that do not stretch individual panels at the small scale yet map to non-isometric changes of coarse-grained fundamental forms that quantify how the effective medium strains and curves at the large scale. Two distinct mode shapes, a rigid breathing mode and a nonrigid shearing mode, are identified in the continuum model. A specific example, called Morph-derivative trapezoid-based origami, is presented with analytical expressions for its deformations in both the discrete and continuous models. A developable specimen is fabricated and tested to validate the analytical predictions. This work advances the continuum modeling of origami tessellations as effective media with the incorporation of more generic faces and ground states, thereby enabling the investigation of novel designs and applications.
Origami has emerged as a powerful mechanism for designing functional foldable and deployable structures. Among various origami patterns, a large class of origami exhibits rotational symmetry, which possesses the advantages of elegant geometric shapes, axisymmetric contraction/expansion, and omnidirectional deployability, etc. Due to these merits, origami with rotational symmetry has found widespread applications in various engineering fields such as foldable emergency shelters, deformable wheels, deployable medical stents, and deployable solar panels. To guide the rational design of origami-based deployable structures and functional devices, numerous works in recent years have been devoted to understanding the geometric designs and mechanical behaviors of rotationally symmetric origami. In this review, we classify origami structures with rotational symmetry into three categories according to the dimensional transitions between their deployed and folded states as three-dimensional to three-dimensional, three-dimensional to two-dimensional, and two-dimensional to two-dimensional. Based on these three categories, we systematically review the geometric designs of their origami patterns and the mechanical behaviors during their folding motions. We summarize the existing theories and numerical methods for analyzing and designing these origami structures. Also, potential directions and future challenges of rotationally symmetric origami mechanics and applications are discussed. This review can provide guidelines for origami with rotational symmetry to achieve more functional applications across a wide range of length scales.
