Rigid Folding Research Articles

Analysis of the Rigid Foldability of Origami Patterns Based on Spatial Positions of Creases

Abstract Rigid foldability is a special property of rigid origami patterns, where each origami plane remains undeformed during continuous movement along the predetermined crease. Current research on the rigid foldability of origami patterns mainly focuses on kinematics, while less attention is paid to factors that cause deformation of the folding plane. Whether the relative spatial position of adjacent creases has been changed is a critical factor that influences the state (rigid or deformed) of the folding plane between the two adjacent creases during the folding process. This study considered two factors (linear relationship and Euclidean distance) to measure the changes in the spatial positions of creases, explored the relationship between the two factors and rigid folding, and identified deformation forms that affect rigid foldability. First, the origami pattern was regarded as a linkage mechanism, and the linear relationship between creases was determined from the single-vertex origami unit forming this origami structure. Then, the geometric parameters of the origami pattern were used to calculate the Euclidean distance between two points on adjacent creases during the folding process. If the linear relationship and Euclidean distance always remain the same, the origami pattern has rigid foldability. Based on changes in the Euclidean distance, this method can also help determine the main deformation of non-rigidly foldable origami patterns. In addition, it can be applied to origami patterns with four or five vertices and multiple loops, and it further provides a novel approach for determining the layout of the crease position and the judgment of rigid foldability during origami-inspired mechanism design.

Simulation and design of isostatic thick origami structures

Thick origami structures are considered here as assemblies of polygonal panels hinged to each other along their edges according to a corresponding origami crease pattern. The determination of the internal actions in equilibrium with the external loads in such structures is not an easy task, owing to their high degree of static indeterminacy, and the likelihood of unwanted self-balanced internal actions induced by manufacturing imperfections. Here, we present a method for reducing the degree of static indeterminacy which can be applied to several thick origami structures to make them isostatic. The method utilizes sliding hinges, which allow relative translation along the hinge axis, to replace conventional hinges. After giving the analytical description of both types of hinges and describing a rigid folding simulation procedure based on the integration of the exponential map, we present the static analysis of a series of noteworthy examples based on the Miura-ori pattern, the Yoshimura pattern, and the Kresling pattern. Our method, based on kinematic-static duality, provides a novel design paradigm that can be applied for the design and realization of thick origami structures with adequate strength to resist external actions.

Interactive design of discrete Voss nets and simulation of their rigid foldings

Voss nets are surface parametrizations whose parameter lines follow a conjugate network of geodesics. Their discrete counterparts, so called V-hedra, are flexible quadrilateral meshes with planar faces such that opposite angles at every vertex are equal; by replacing this equality condition with the closely related constraint that opposite angles are supplementary, we get so-called anti-V-hedra. In this paper, we study the problem of constructing and manipulating (anti-)V-hedra. First, we present a V-hedra generator that constructs a modifiable (anti-)V-hedron in a geometrically exact way, from a set of simple conditions already proposed by Sauer in 1970; our generator can compute and visualize the flexion of the (anti-)V-hedron in real time. Second, we present an algorithm for the design and interactive exploration of V-hedra using a handle-based deformation approach; this tool is capable of simulating the one-parametric isometric deformation of an imperfect V-hedron via a quad soup approach. Moreover, we evaluate the performance and accuracy of our tools by applying the V-hedra generator to constraints obtained by numerical optimization. In particular, we use example surfaces that originate from one-dimensional families of smooth Voss surfaces – each spanned by two isothermal conjugate nets – for which an explicit parametrization is given. This allows us to compare the isometric deformation of the smooth target surface with the rigid folding of the optimized (imperfect) V-hedron.

Experimental and numerical study on energy absorption performance of truncated origami materials

In this study, a truncated origami structure, originated from the Miura-ori structure, was designed. The compressive behaviors of this structure subjected to x2-direction (in-plane) and x3-direction (out-of-plane) compression were studied both experimentally and numerically. The structure with a smaller truncated ratio value, e, was more similar to a dense solid. Representative specimens of the Miura-ori and the truncated origami designs were fabricated and tested under both quasi-static and dynamic (v = 7 m/s) conditions. Compared to the Miura-ori structure, the truncated origami showed higher specific energy absorption (SEA) and less load drop between the initial peak force and mean crushing force. In both loading directions, the SEA increased as the truncated ratio decreased for the truncated origami specimens, which was comparable under both quasi-static and dynamic loading. The truncated origami was extrapolated to a numerical model with 5 × 5 × 5 cells to mimic a block of material, and the effects of truncated ratio and loading velocity were investigated. As the e decreased, the deformation mode gradually changed from rigid folding and progressive deformation to plate buckling mode under quasi-static compression. This change was accompanied by an increase in both SEA and overall strength. The truncated origami exhibited the best energy absorption performance when e = 0.4. Under quasi-static compression in the x2- and x3-direction, the SEA of truncated origami is 3.2 and 1.6 times that of Miura-ori metamaterial, respectively. The truncated origami materials with a truncated ratio of 0.8 remained velocity-insensitive up to 25 m/s and displayed similar strength and deformation modes as under quasi-static loading. At a speed of 50 m/s the deformation became more localized, and the SEA increased significantly. The auxetic and expansion behaviors were significantly diminished under higher impact velocities.

A secreted bacterial protein protects bacteria from cationic antimicrobial peptides by entrapment in phase-separated droplets.

Mammalian hosts combat bacterial infections through the production of defensive cationic antimicrobial peptides (CAPs). These immune factors are capable of directly killing bacterial invaders; however, many pathogens have evolved resistance evasion mechanisms such as cell surface modification, CAP sequestration, degradation, or efflux. We have discovered that several pathogenic and commensal proteobacteria, including the urgent human threat Neisseria gonorrhoeae, secrete a protein (lactoferrin-binding protein B, LbpB) that contains a low-complexity anionic domain capable of inhibiting the antimicrobial activity of host CAPs. This study focuses on a cattle pathogen, Moraxella bovis, that expresses the largest anionic domain of the LbpB homologs. We used an exhaustive biophysical approach employing circular dichroism, biolayer interferometry, cross-linking mass spectrometry, microscopy, size-exclusion chromatography with multi-angle light scattering coupled to small-angle X-ray scattering (SEC-MALS-SAXS), and NMR to understand the mechanisms of LbpB-mediated protection against CAPs. We found that the anionic domain of this LbpB displays an α-helical secondary structure but lacks a rigid tertiary fold. The addition of antimicrobial peptides derived from lactoferrin (i.e. lactoferricin) to the anionic domain of LbpB or full-length LbpB results in the formation of phase-separated droplets of LbpB together with the antimicrobial peptides. The droplets displayed a low rate of diffusion, suggesting that CAPs become trapped inside and are no longer able to kill bacteria. Our data suggest that pathogens, like M. bovis, leverage anionic intrinsically disordered domains for the broad recognition and neutralization of antimicrobials via the formation of biomolecular condensates.

Folding solar panels in space: Miura-ori and its kinematic behavior

Solar panels are made by absorbing Sunlight , which will Solar radiation energy through Photovoltaic effects or Photochemical effects directly or indirectly into Electrical energy to a device that is the central part of a solar power system and is often used in spacecraft. Spacecraft allow for large energy requirements, and solar panels require a larger area to meet the spacecraft's needs. Origami is an ingenious solution to this problem by reducing the size of solar panels needed for launch by specific folding methods, such as Miura-ori, which is a rigid origami paper in which each parallelogram remains unbent and flat during the smooth unfolding of the paper. Therefore, this paper investigates the geometric configuration and kinematic behaviour of Miura-ori based on Miura-ori units with rigid folding properties.

Modification and oxidative degradation of β-lactoglobulin by UVB irradiation.

Ultraviolet (UV) B irradiation induces protein modification, especially the conformational rearrangement of proteins, and is therefore promising as a non-thermal and non-chemical functionalization technique. Nevertheless, UVB irradiation introduces radicals and oxidizes side chains resulting in the loss of food quality. Thus, assessing the UVB irradiation-based functionalization of β-lactoglobulin (BLG) versus its oxidative degradation is of interest. UVB irradiation of up to 8h was successfully applied to loosen the rigid folding of BLG and increase its flexibility. Thereby, the cysteine at position 121 and hydrophobic regions became surface-exposed as indicated by the increase in accessible thiol groups and increased surface hydrophobicity. Furthermore, we demonstrated the cleavage of the "outer" disulfide bond C66-C160 by LC-MS/MS after tryptic digestion of BLG. The 2-h-irradiated BLG showed adequate conformational rearrangement for protein functionalization while being minimally oxidized.

Hard-Soft Cooperativity of an Allosteric Tweezer.

Drawing inspiration from allosteric proteins, a zigzag-shaped π-conjugation was structurally engineered into a tweezer-like ionophore having multiple disparate binding sites. When a soft metal ion binds to the central tridentate ligand motif, the rigid backbone folds, bringing two macrocyclic arms into close proximity. Stabilized by a coordinating anion, the tweezer-like conformation of the resulting metalloligand recruits a hard cation to form a sandwich-like complex with a remarkably enhanced binding affinity and selectivity.

Nonlinear wrap-folding of membranes with predefined creases and seams

The investigation of folding processes of aerospace membrane structures holds significant importance due to the limited volume of launch vehicles. This paper presents a simulation system for membrane wrap-folding with predefined creases and seams. To transform non-rigid foldable configurations into rigid foldable ones, the methods of adding creases and cutting strategies are introduced. The formulation of constraint equations for the hinged rod system model enables the analysis of the rigid folding motion. A two-stage folding strategy is also implemented for fully folding, consisting of an initial rigid folding stage followed by an additional displacement folding stage. By incorporating the crease strategy and employing a two-stage driving mode, the stress levels throughout the folding process are minimized. Furthermore, accounts for the presence of seam areas, placing the seam at the crease creates wrinkles smaller when folded. The findings contribute to a deeper understanding of membrane folding behavior, leading to advancements in their design and application.

Experimental Demonstration of Low‐Energy First‐Order Hybridized Plasmon Resonances in Origami Metashells

AbstractMetallic nanoshells have emerged as promising optical cavities to steer hybridized plasmonic resonances that show the high spectral tunability and a large surface area of localized fields. Despite various optical and photochemical applications, plasmonic resonances in shells have rarely been explored at low frequencies due to a nonlinear increase in particle size and a rapid degeneration in field localization. Here, this work reports the experimental realization of hybridized plasmon resonances in origami metashells that can be excited by low‐energy photons ≈7 GHz, corresponding to a level below 10−4 eV. Metasurfaces built of interconnected meshes act as thin Drude metallic films that support symmetric spoof surface plasmons. Rigid folding of the metasurface leads to diverse 3D hollow cavities that exhibit strong plasmonic responses, large surface areas, and low relative densities. Experimental measurements demonstrate three typical behaviors in Drude metallic shells: shielding to quasi‐static fields, strong scattering at first‐order hybridized plasmon resonances, and transparency to high frequency radiation. The relative density is remarkably reduced owing to the void cavity enabled by 3D folding, and an extra degree of freedom in anisotropic load capacity is provided by customizing crease patterns. These results open a door for extremely low‐energy plasmons with low‐density mechanics.

Energy absorption analysis of origami structures based on small number of samples using conditional GAN

Origami-inspired metamaterials are increasingly being applied to the fields of science and engineering owing to their unique mechanical characteristics. Geometric and material arrangements of the panels and creases can lead to different mechanical properties of origami structures, such as energy absorption and multi-stability. The actual folding processes of the origami structures usually include crease rotation and panel deformation. However, the traditional analytical or empirical solutions based on rigid folding assumptions cannot sufficiently reflect the folding process precisely. Finite element analysis can provide detailed origami folding information, but it is a highly time-consuming cycle, including modeling and computing. Typically, conventional data-driven approaches necessitate a considerable amount of data samples to analyze the energy absorption of origami structures. The unavailability of extensive datasets poses a major impediment in employing data-driven methods to explore the authentic behavior of origami structures. Based on the situations above, a new data-driven framework with conditional generative adversarial networks with tabular data (CTGAN) is proposed in this study. To verify the feasibility of the framework, two study cases are presented herein: a Miura-ori structure in-plane quasi-static compression and the square-twist (Type 1) folding process. Energy absorption properties are predicted accurately and efficiently by the framework based on small number of samples, which are highly consistent with predicted results based on a large dataset. The framework not only provides a promising solution for energy absorption analysis of origami structures but also overcomes the bottlenecks (small number of sample datasets and a mixture of data types) associated with machine learning methods in origami structure area.

Algorithmic Spatial Form-Finding of Four-Fold Origami Structures Based on Mountain-Valley Assignments

AbstractOrigami has attracted tremendous attention in recent years owing to its capability of inspiring and enabling the design and development of reconfigurable structures and mechanisms for applications in various fields such as robotics and biomedical engineering. The vast majority of origami structures are folded starting from an initial two-dimensional crease pattern. However, in general, the planar configuration of such a crease pattern is in a singular state when the origami starts to fold. Such a singular state results in different motion possibilities of rigid or non-rigid folding. Thus, planar origami patterns cannot act as reliable initial configurations for further kinematic or structural analyses. To avoid the singularities of planar states and achieve reliable structural configurations during folding, we introduce a nonlinear prediction–correction method and present a spatial form-finding algorithm for four-fold origami. In this approach, first, initial nodal displacements are predicted based on the mountain-valley assignments of the given origami pattern, which are applied to vertices to form an initial spatial and defective origami model. Subsequently, corrections of nodal displacements are iteratively performed on the defective model until a satisfactory nonplanar configuration is obtained. Numerical experiments demonstrate the performance of the proposed algorithm in the form-finding of both trivial and non-trivial four-fold origami tessellations. The obtained configurations can be effectively utilized for further kinematic and structural analyses. Additionally, it has been verified that corrected and nonplanar configurations are superior to initial configurations in terms of matrix distribution and structural stiffness.

Collision course: Bagnara crumbling without care?

Bagnara - Castelvolturno - Photo by Mario Ferrara «Collision is the crash, the impact between two bodies in motion» (Treccani, “Collisione”). In the landscape, movement characterises both the evolution of built parts and the mutability of natural parts, and new balances or new imbalances are defined starting from the possibility of recognising stable or unstable lasting arrangements. The intermediate areas between city and nature are the place of contact between the different modes and speeds of transformation of the natural and artificial components of the landscape. The contact between these components can be defined as an encounter or a collision, depending on the greater or lesser level of acceptance or rejection that is produced by the intersection. The collision is therefore identified as an extreme figure for interpreting the relationship between different phenomena that is difficult to translate, with the same power, into a design theme. «In the passage from analysis to design, it seems that the term collision tends to lose its precision to some extent. Its descriptive power rarely goes beyond the level of metaphorical evocation to become an operational design tool» (Corbellini 2015, p.69). In Bagnara - Castelvolturno, indiscriminate construction has come into collision with the uncontrolled power of the sea. In this coastal landscape of southern Italy between the Volturno river to the south and the Agnene canal to the north, the unstable balance between land and water has created a narrow sandy strip dotted with houses' ruins and inhabited by bathers. The lack of breakwaters leaves the sea currents free to erode the beaches; the presence of houses built illegally on state-owned land put the inner linear city beyond the boundaries of stability. The current landscape of Bagnara consists on the one hand of a sandy shore, with unique characteristics on the Campania coast, which acts as an unstable base for crumbling ruins, and on the other, the water swallows up shreds of buildings that become artificial rocks. The sinuous curves of the sandy and watery nature of Bagnara are juxtaposed with the rigid folds of buildings broken by the slow catastrophe. A social and aesthetic practice, photography, understood with Olivier Lugon as a documentary art, explores this landscape, becoming tangible evidence of a reality in oblivion that presents a multifaceted and complex pathological dimension. Illuminating, revealing and disseminating the state of health of Bagnara is the only cure currently in place for this landscape. The photos, like a “caress on the world” of this unstable strip of land, provide a therapeutic practice and a poetic vision of the crumbling reality of Bagnara.

Geometry design and in-plane compression performance of novel origami honeycomb material

Origami is widely used for honeycombs to improve mechanical behavior. This research presents an origami-honeycomb material based on the assembly of Miura patterns. Firstly, the geometry stacking scheme and rigid folding mechanism are discussed. Secondly, the in-plane compression behavior of the proposed origami-honeycomb is experimentally and numerically investigated. Its performance is then compared with a hexagonal honeycomb. Both results show that the origami-honeycomb outperforms the conventional honeycomb in terms of in-plane compression strength and energy absorption behavior. Subsequently, deformation mode and energy efficiency ratio are obtained to illustrate the mechanical behavior. A parametric analysis is also conducted by employing finite element simulations. Finally, the analytical model of the proposed origami honeycomb is built to estimate the gather strain and plateau stress.

Bending-active kirigami

Kirigami techniques use prescribed arrangements of cuts and fold lines to transform 2D sheet materials into complex 3D forms. These sheet transformations occur via elastic deformations, rigid folding motions, and/or pop-up mechanisms; design techniques for kirigami patterns are correspondingly based on mechanics, kinematics, and physical prototyping. This paper proposes a new family of pop-up kirigami patterns termed ‘bending-active kirigami’, which transform flat sheets into approximated double-curved surfaces using a simple 1-DOF actuation mechanism. A pattern design technique is first developed from a combination of the well-known ‘elastica’ solution for the large elastic deformation of slender rods, with the kinematic constraints imposed by a kirigami pop-up mechanism. This combination creates a rapid and intuitive design method for specification of a formed surface and its constituent kirigami pattern. An extended family of patterns are then developed based on other elastica solutions and pop-up mechanism types. Finally, a bending-active kirigami pattern is applied to the design of a deployable structure, with measurements of a physical prototype used to verify the developed design method.

Rigid folding equations of degree-6 origami vertices.

Rigid origami, with applications ranging from nano-robots to unfolding solar sails in space, describes when a material is folded along straight crease line segments while keeping the regions between the creases planar. Prior work has found explicit equations for the folding angles of a flat-foldable degree-4 origami vertex and some cases of degree-6 vertices. We extend this work to generalized symmetries of the degree-6 vertex where all sector angles equal . We enumerate the different viable rigid folding modes of these degree-6 crease patterns and then use second-order Taylor expansions and prior rigid folding techniques to find algebraic folding angle relationships between the creases. This allows us to explicitly compute the configuration space of these degree-6 vertices, and in the process we uncover new explanations for the effectiveness of Weierstrass substitutions in modelling rigid origami. These results expand the toolbox of rigid origami mechanisms that engineers and materials scientists may use in origami-inspired designs.

Deployment Dynamics of Miura Origami Sheets

AbstractOrigami has great potential for creating deployable structures, however, most studies have focused on their static or kinematic features, while the complex and yet important dynamic behaviors of the origami deployment process have remained largely unexplored. In this research, we construct a dynamic model of a Miura origami sheet that captures the combined panel inertial and flexibility effects, which are otherwise ignored in rigid folding kinematic models but are critical in describing the dynamics of origami deployment. Results show that by considering these effects, the dynamic deployment behavior would substantially deviate from a nominal kinematic unfolding path. Additionally, the pattern geometries influence the effective structural stiffness, and it is shown that subtle changes can result in qualitatively different dynamic deployment behaviors. These differences are due to the multistability of the Miura origami sheet, where the structure may snap between its stable equilibria during the transient deployment process. Lastly, we show that varying the deployment rate can affect the dynamic deployment configuration. These observations are original and these phenomena have not and cannot be derived using traditional approaches. The tools and outcomes developed from this research enable a deeper understanding of the physics behind origami deployment that will pave the way for better designs of origami-based deployable structures, as well as extend our fundamental knowledge and expand our comfort zone beyond current practice.

Unsteady Aeroelastic Analysis of the Semi Aeroelastic Hinge Including Local Geometric Nonlinearities

A recent consideration in aircraft design is the use of folding wing tips, with the aim of enabling higher aspect ratio wings with less induced drag but also meeting airport gate limitations. Of particular interest is the concept of using in flight free-floating wing tips in order to reduce aircraft gust and maneuver loads. This study investigates the effects of local geometric nonlinearities on the dynamic stability, postflutter behavior and gust response of floating wing tips. A multibody formulation is introduced to account for finite rotations of rigid folding wing tips attached through hinges on a flexible airframe structure including also aerodynamic follower forces for the folding wing tip components. It is found that the postflutter behavior is characterized by super-critical limit cycle oscillations and no subcritical instability was observed for the analyzed cases. Moreover, the wing tip gust response can vary significantly when geometric nonlinearities are accounted for whereas a small impact was observed on the main airframe structure.

The Mechanical Behavior of Polyline Crease Origami Self-Locking Structure under Quasi-Static Compressive Loading

The unique mechanical behavior of polyline crease (PLC) origami structures inspired by the Tachi-Miura polyhedron (TMP) is investigated. We explore the potential usage as mechanical metamaterials that exhibit negative Poisson’s ratio, self-locking mechanism, and load capacity simultaneously. Poisson’s ratio, locked folding ratio, dimensionless force and folding ratio relationship, and elastic-plastic response during its compressive process (rigid folding motion or plastic bending/buckling) are investigated analytically and experimentally, respectively. The effect of original folding angle and the length-to-height ratio on negative Poisson’s ratio, locked folding ratio, dimensionless force, deformation modes, and the mean compression force are also considered. Based on the experiments, an approximate theoretical study has also been conducted, which shows a reasonable correlation with the experimental results. The results are significant to guide the engineering applications of PLC self-locking origami structures.

Quasi-static compressive behaviour of 3D-printed origami-inspired cellular structure: experimental, numerical and theoretical studies

ABSTRACT This paper proposed a novel periodic cellular structure by introducing re-entrant hexagon-shaped honeycombs and Miura origami geometry. The large-scale multi-layer metallic specimens were manufactured by a selective laser melting (SLM) system and tested for the first time. The compression deformation and energy absorption characteristics under quasi-static load were investigated thoroughly, considering three loading directions and various geometric parameters. The results show that this novel structure has balanced mechanical properties in the three loading directions. Among them, the z-direction is the best bearing direction. Moreover, this structure exhibits an obvious three-dimensional negative Poisson's ratio effect during the compression process. Besides the experimental study, the numerical investigation base on ANSYS/LS-DYNA was carried out to provide an accurate and effective simulation method. Predictive formulas for the critical energy absorption indexes were derived by theoretical analysis based on the rigid folding assumption, validated to be well-matched with the experimental results.