Dynamics of a bistable Miura-origami structure

Evaluating dynamic models for rigid-foldable origami: unveiling intricate bistable dynamics of stacked-Miura-origami structures as a case study

Recent advances in origami science and engineering have particularly focused on the challenges of dynamics. While research has primarily focused on statics and kinematics, the need for effective and processable dynamic models has become apparent. This paper evaluates various dynamic modelling techniques for rigid-foldable origami, particularly focusing on their ability to capture nonlinear dynamic behaviours. Two primary methods, the lumped mass-spring-damper approach and the energy-based method, are examined using a bistable stacked Miura-origami (SMO) structure as a case study. Through systematic dynamic experiments, we analyse the effectiveness of these models in predicting bistable dynamic responses, including intra- and interwell oscillations, in different loading conditions. Our findings reveal that the energy-based approach, which considers the structure's inertia and utilizes dynamic experimental data for parameter identification, outperforms other models in terms of validity and accuracy. This model effectively predicts the dynamic response types, the rich and complex nonlinear characteristics and the critical frequency where interwell oscillations occur. Despite its relatively increased complexity in model derivation, it maintains computational efficiency and shows promise for broader applications in origami dynamics. By comparing model predictions with experimental results, this study enhances our understanding of origami dynamics and contributes valuable insights for future research and applications. This article is part of the theme issue 'Origami/Kirigami-inspired structures: from fundamentals to applications'.

Improving mass lumping and stiffness parameters of bar and hinge model for accurate modal dynamics of origami structures

The bar and hinge framework uses truss elements and rotational springs to efficiently model the structural behaviour of origami. The framework is especially useful to investigate origami metamaterials as they have repeating geometry, which makes conventional finite element simulations very expensive due to a large number of degrees of freedom. This work proposes improvements to the parameters of bar and hinge model within the context of structural dynamics, specifically modal analysis under small deformations, which has not been carried out previously in the literature. A range of low-frequency modes involving origami folding and panel bending deformations that can be accurately captured by the bar and hinge framework are identified. Within this range, bar and hinge parameters like the lumped masses and the rotational spring stiffness values are derived using conservation laws and finite element tests. The best among the proposed schemes is found to predict natural frequencies of the considered origami structures to within 10% maximum error, improving the accuracy by more than three times from existing schemes. In most cases, the errors in natural frequencies are less than 5%. This article is part of the theme issue 'Origami/Kirigami-inspired structures: from fundamentals to applications'.

A Soft Amphibious Voxel-Type Quadruped Robot Based on Origami Flexiball of Rhombic Dodecahedron

The research work presents a novel voxel-type soft amphibious robot based on an assembly of origami flexiballs. The geometric and elastic constitutive models of the origami flexiball are theoretically established to elucidate its intricate deformation mechanism. Especially, the zero-energy storage phenomenon and the quasi-zero-stiffness characteristic are revealed to prove that the origami flexiball is suitable for serving as soft robotic components. As a proof of concept, fourteen origami flexiballs are interconnected to form a quadruped robot capable of walking or crawling in both underwater and terrestrial environments, including flat surfaces and sandy terrain. Its adaptability across multiple environments is enhanced by the origami polyhedra-inspired hollow structure, which naturally adjusts to underwater conditions such as hydrostatic pressure and currents, improving stability and performance. Other advantages of the voxel-type soft amphibious quadruped robot include its ease of manufacture using 3D printing with accessible soft elastic materials, ensuring rapid and cost-effective fabrication. We anticipate its potentially versatile applications, including underwater pipeline inspections, offshore maintenance, seabed exploration, ecological monitoring, and marine sample collection. By leveraging metamaterial features embodied in the origami polyhedra, the presented voxel-type soft robot exemplifies an innovative approach to achieving complex functionalities in soft robotics.

Reconfigurable origami-inspired multistable metamorphous structures

Origami-inspired metamorphous structures can adjust their shapes and mechanical behaviors according to operational requirements. However, they are typically composed of nonrigid origami, where required facet deformation complicates actuation and makes them highly material dependent. In this study, we present a type of origami metamorphous structure composed of modular bistable units, each of which is a rigid origami. The elasticity within the origami creases and switching of mountain and valley crease lines enable it to have bistability. The resultant metamorphous structure has multistability, allowing it to switch among multifarious configurations with programmable profiles. This concept was validated by potential energy analysis and experiments. Using this concept, we developed a robotic limb capable of both lifting and gripping through configuration changes. Furthermore, we used the origami units to construct a metamaterial whose properties could change with the variation of configurations. These examples demonstrate the concept's remarkable versatility and potential for many applications.

Connecting the branches of multistable non-Euclidean origami by crease stretching

Non-Euclidean origami is a promising technique for designing multistable deployable structures folded from nonplanar developable surfaces. The impossibility of flat foldability inherent to non-Euclidean origami results in two disconnected solution branches each with the same angular deficiency but opposite handedness. We show that these regions can be connected via "crease stretching," wherein the creases exhibit extensibility in addition to torsional stiffness. We further reveal that crease stretching acts as an energy storage method capable of passive deployment and control. Specifically, we show that in a Miura-Ori system with a single stretchable crease, this is achieved via two unique, easy to realize, equilibrium folding pathways for a certain wide set of parameters. In particular, we demonstrate that this connection mostly preserves the stable states of the non-Euclidean system, while resulting in a third stable state enabled only by the interaction of crease torsion and stretching. Finally, we show that this simplified model can be used as an efficient and robust tool for inverse design of multistable origami based on closed-form predictions that yield the system parameters required to attain multiple, desired stable shapes. This facilitates the implementation of multistable origami for applications in architecture materials, robotics, and deployable structures.

Three-dimensional ori-kirigami metamaterials with multistability

Ori-kirigami structures offer a good avenue for designing mechanical metamaterials due to their unique advantage of being independent of material properties and scale limitations. Recently, the scientific community has been greatly interested in exploiting the complex energy landscape of ori-kirigami structures to construct multistable systems and play their valuable role in different applications. Here, we present three-dimensional ori-kirigami structures based on generalized waterbomb units, a cylindrical ori-kirigami structure based on waterbomb units, and a conical ori-kirigami structure based on trapezoidal waterbomb units. We investigate the inherent relationships between the unique kinematics and mechanical properties of these three-dimensional ori-kirigami structures and explore their potential usage as mechanical metamaterials that exhibit negative stiffness, snap-through, hysteresis effects, and multistability. What makes the structures even more attractive is their massive folding stroke, where the conical ori-kirigami structure can obtain a huge folding stroke of more than twice its initial height through penetration of its upper and lower boundaries. This study forms the foundation for designing and constructing three-dimensional ori-kirigami metamaterials based on generalized waterbomb units for various engineering applications.

Leaf-Like Origami with Bistability for Self-Adaptive Grasping Motions

The leaf-like origami structure is inspired by geometric patterns found in nature, exhibiting unique transitions between open and closed shapes. With a bistable energy landscape, leaf-like origami is able to replicate the autonomous grasping of objects observed in biological systems such as the Venus flytrap. We show uniform grasping motions of the leaf-like origami, as well as various nonuniform grasping motions that arise from its multitransformable nature. Grasping motions can be triggered with high tunability due to the structure's bistable energy landscape. We demonstrate the self-adaptive grasping motion by dropping a target object onto our paper prototype, which does not require an external power source to retain the capture of the object. We also explore the nonuniform grasping motions of the leaf-like structure by selectively controlling the creases, which reveals various unique grasping configurations that can be exploited for versatile, autonomous, and self-adaptive robotic operations.

Novel Deployable Panel Structure Integrated with Thick Origami and Morphing Bistable Composite Structures

This paper proposes a novel deployable panel structure integrated with a bistable composite structure and thick panel based on the thick origami technique. To overcome the interference effects between thick panels, the axis shift method is used in this deployable structure design. Bistable composite structures are employed as hinges for morphing characteristics. The trigger force and load-displacement curves of the structure are obtained by experiments and numerical simulations. The factors that affect the coverage area-to-package volume ratio and trigger force are discussed. The experimental and numerical results verify that the structure has two stable configurations and a large coverage area-to-package volume ratio.

Mechanical computing

Mechanical mechanisms have been used to process information for millennia, with famous examples ranging from the Antikythera mechanism of the Ancient Greeks to the analytical machines of Charles Babbage. More recently, electronic forms of computation and information processing have overtaken these mechanical forms, owing to better potential for miniaturization and integration. However, several unconventional computing approaches have recently been introduced, which blend ideas of information processing, materials science and robotics. This has raised the possibility of new mechanical computing systems that augment traditional electronic computing by interacting with and adapting to their environment. Here we discuss the use of mechanical mechanisms, and associated nonlinearities, as a means of processing information, with a view towards a framework in which adaptable materials and structures act as a distributed information processing network, even enabling information processing to be viewed as a material property, alongside traditional material properties such as strength and stiffness. We focus on approaches to abstract digital logic in mechanical systems, discuss how these systems differ from traditional electronic computing, and highlight the challenges and opportunities that they present.

Yoshimura-origami Based Earthworm-like Robot With 3-dimensional Locomotion Capability

Earthworm-like robots have received great attention due to their prominent locomotion abilities in various environments. In this research, by exploiting the extraordinary three-dimensional (3D) deformability of the Yoshimura-origami structure, the state of the art of earthworm-like robots is significantly advanced by enhancing the locomotion capability from 2D to 3D. Specifically, by introducing into the virtual creases, kinematics of the non-rigid-foldable Yoshimura-ori structure is systematically analyzed. In addition to exhibiting large axial deformation, the Yoshimura-ori structure could also bend toward different directions, which, therefore, significantly expands the reachable workspace and makes it possible for the robot to perform turning and rising motions. Based on prototypes made of PETE film, mechanical properties of the Yoshimura-ori structure are also evaluated experimentally, which provides useful guidelines for robot design. With the Yoshimura-ori structure as the skeleton of the robot, a hybrid actuation mechanism consisting of SMA springs, pneumatic balloons, and electromagnets is then proposed and embedded into the robot: the SMA springs are used to bend the origami segments for turning and rising motion, the pneumatic balloons are employed for extending and contracting the origami segments, and the electromagnets serve as anchoring devices. Learning from the earthworm’s locomotion mechanism--retrograde peristalsis wave, locomotion gaits are designed for controlling the robot. Experimental tests indicate that the robot could achieve effective rectilinear, turning, and rising locomotion, thus demonstrating the unique 3D locomotion capability.

CFRP Origami Metamaterial with Tunable Buckling Loads: A Numerical Study

Origami has played an increasingly central role in designing a broad range of novel structures due to its simple concept and its lightweight and extraordinary mechanical properties. Nonetheless, most of the research focuses on mechanical responses by using homogeneous materials and limited studies involving buckling loads. In this study, we have designed a carbon fiber reinforced plastic (CFRP) origami metamaterial based on the classical Miura sheet and composite material. The finite element (FE) modelling process's accuracy is first proved by utilizing a CFRP plate that has an analytical solution of the buckling load. Based on the validated FE modelling process, we then thoroughly study the buckling resistance ability of the proposed CFRP origami metamaterial numerically by varying the folding angle, layer order, and material properties, finding that the buckling loads can be tuned to as large as approximately 2.5 times for mode 5 by altering the folding angle from 10° to 130°. With the identical rate of increase, the shear modulus has a more significant influence on the buckling load than Young's modulus. Outcomes reported reveal that tunable buckling loads can be achieved in two ways, i.e., origami technique and the CFRP material with fruitful design freedoms. This study provides an easy way of merely adjusting and controlling the buckling load of lightweight structures for practical engineering.

Foundations for Soft, Smart Matter by Active Mechanical Metamaterials

Emerging interest to synthesize active, engineered matter suggests a future where smart material systems and structures operate autonomously around people, serving diverse roles in engineering, medical, and scientific applications. Similar to biological organisms, a realization of active, engineered matter necessitates functionality culminating from a combination of sensory and control mechanisms in a versatile material frame. Recently, metamaterial platforms with integrated sensing and control have been exploited, so that outstanding non-natural material behaviors are empowered by synergistic microstructures and controlled by smart materials and systems. This emerging body of science around active mechanical metamaterials offers a first glimpse at future foundations for autonomous engineered systems referred to here as soft, smart matter. Using natural inspirations, synergy across disciplines, and exploiting multiple length scales as well as multiple physics, researchers are devising compelling exemplars of actively controlled metamaterials, inspiring concepts for autonomous engineered matter. While scientific breakthroughs multiply in these fields, future technical challenges remain to be overcome to fulfill the vision of soft, smart matter. This Review surveys the intrinsically multidisciplinary body of science targeted to realize soft, smart matter via innovations in active mechanical metamaterials and proposes ongoing research targets that may deliver the promise of autonomous, engineered matter to full fruition.

Programmable stopbands and supratransmission effects in a stacked Miura-origami metastructure

Origami-based mechanical metamaterials and metastructure have been demonstrated to exhibit unique properties originating from their intricate geometries of folding. This research aims to extend the current investigation level from quasistatics to dynamics. In detail, this research focuses on the wave dynamics of a metastructure composed of stacked Miura-origami (SMO) units. The SMO unit could possess two stable configurations, endowing the metastructure with rich possibilities in the layout of its periodic repeating cell. Through linear dispersion analyses and numerical studies, we show that the long-desired stopband tunability and programmability of the metastructure along the three principal directions can be acquired by strategically programming the layout of the periodic cell. Based upon that, we further discover that energy supratransmission through the metastructure is possible within the stopband by increasing the driving amplitude. Through numerical means, the amplitude threshold of supratransmission is obtained. We demonstrate that the fundamental mechanism that triggers the supratransmission phenomenon is the transition of the responses from the low-energy intrawell oscillations to the high-energy interwell oscillations. Numerical studies also indicate that the supratransmission threshold can be effectively tailored by adjusting the periodic cell layout. The results of this research provide a wealth of fundamental insights into the origami wave dynamics and offer useful guidelines for developing origami metastructures with tunable and programmable dynamic characteristics.

Theory and practice of origami in science

No longer just the purview of artists and enthusiasts, origami engineering has emerged as a potentially powerful tool to create three dimensional structures on disparate scales. Whether origami (and the closely related kirigami) engineering can emerge as a useful technology will depend crucially on both fundamental theoretical advances as well as the development of further fabrication tools.

Microdroplet-guided intercalation and deterministic delamination towards intelligent rolling origami

Three-dimensional microstructures fabricated by origami, including folding, rolling and buckling, gain great interests in mechanics, optics and electronics. We propose a general strategy on on-demand and spontaneous rolling origami for artificial microstructures aiming at massive and intelligent production. Deposited nanomembranes are rolled-up in great amount triggered by the intercalation of tiny droplet, taking advantage of a creative design of van der Waals interaction with substrate. The rolling of nanomembranes delaminated by liquid permits a wide choice in materials as well as precise manipulation in rolling direction by controlling the motion of microdroplet, resulting in intelligent construction of rolling microstructures with designable geometries. Moreover, this liquid-triggered delamination phenomenon and constructed microstructures are demonstrated in the applications among vapor sensing, microresonators, micromotors, and microactuators. This investigation offers a simple, massive, low-cost, versatile and designable construction of rolling microstructures for fundamental research and practical applications.

Rolling microstructures have great potential among optics and micromechanics but on-demand construction with versatile materials remains challenging. Here Xu et al. precisely construct 3D rolling microstructures by liquid-triggered delamination in predictable manner and demonstrate their applications.

Solitary waves in a nonintegrable chain with double-well potentials

We study solitary waves in a one-dimensional lattice of identical masses that are connected in series by nonlinear springs. The potential of each spring is nonconvex, where two disjoint convex regions, phase I and phase II, are separated by a concave, spinodal region. Consequently, the force-strain relation of the spring is nonmonotonous, which gives rise to a bistable behavior. Based on analytical treatment, with some approximations, combined with extensive numerical simulations, we are able to reveal important insights. For example, we find that the solitary-wave solution is indifferent to the energy barrier that separates the two energy wells associated with phase I and phase II, and that the shape of the wave can be described by means of merely two scalar properties of the potential of the springs, namely, the ratio of stiffness in phase II and phase I, and the ratio between the Maxwell's force and corresponding transition strain. The latter ratio provides a useful measure for the significance of the spinodal region. Linear stability of the solitary-wave solution is studied analytically using the Vakhitov-Kolokolov criterion applied to the approximate solutions obtained in the first part. These results are validated by numerical simulations. We find that the solitary-wave solution is stable provided that its velocity is higher than some critical value. It is shown that, practically, the solitary waves are stable for almost the entire range of possible wave velocities. This is also manifested in the interaction between two solitary waves or between a solitary wave and a wall (rigid boundary). Such interaction results in a minor change of height and shape of the solitary wave along with the formation of a trail of small undulations that follow the wave, as expected in a nonintegrable system. Even after a significant number of interactions the changes in the wave height and shape are minor, suggesting that the bistable chain may be a useful platform for delivering information over long distances, even concurrently with additional information (other solitary waves) passing through the chain.

Architected Origami Materials: How Folding Creates Sophisticated Mechanical Properties

Origami, the ancient Japanese art of paper folding, is not only an inspiring technique to create sophisticated shapes, but also a surprisingly powerful method to induce nonlinear mechanical properties. Over the last decade, advances in crease design, mechanics modeling, and scalable fabrication have fostered the rapid emergence of architected origami materials. These materials typically consist of folded origami sheets or modules with intricate 3D geometries, and feature many unique and desirable material properties like auxetics, tunable nonlinear stiffness, multistability, and impact absorption. Rich designs in origami offer great freedom to design the performance of such origami materials, and folding offers a unique opportunity to efficiently fabricate these materials at vastly different sizes. Here, recent studies on the different aspects of origami materials-geometric design, mechanics analysis, achieved properties, and fabrication techniques-are highlighted and the challenges ahead discussed. The synergies between these different aspects will continue to mature and flourish this promising field.

Transformation Dynamics in Origami

A single origami crease pattern can be folded into many different structures from the flat-unfolded state, effectively transforming in shape and function. This makes origami engineering a promising approach to transforming machines, but predicting and controlling this transformation is difficult because the fundamental dynamics of origami systems at the flat-unfolded state are not well understood. Working with Miura-inspired mechanisms, we identify and validate a model to predict configuration switching in mechanical origami systems. The model incorporates a hidden degree of freedom introduced by material compliance in the Miura mechanism. We characterize this pseudojoint statically and dynamically to identify its lumped stiffness and inertia and use it to create a new dynamic model. This model can be used to predict which configuration an origami mechanism will settle in by balancing the kinetic and potential energy of the system. We apply this model to design a branching origami structure with 17 distinct configurations controlled by a single actuator and demonstrate reliable switching between these configurations with tailored dynamic inputs. Given the fact that origami can replicate almost any shape, we expect that this framework will be applicable to transformation in arbitrary structures and mechanisms.

Programmable Self-Locking Origami Mechanical Metamaterials

Developing mechanical metamaterials with programmable properties is an emerging topic receiving wide attention. While the programmability mainly originates from structural multistability in previously designed metamaterials, here it is shown that nonflat-foldable origami provides a new platform to achieve programmability via its intrinsic self-locking and reconfiguration capabilities. Working with the single-collinear degree-4 vertex origami tessellation, it is found that each unit cell can self-lock at a nonflat configuration and, therefore, possesses wide design space to program its foldability and relative density. Experiments and numerical analyses are combined to demonstrate that by switching the deformation modes of the constituent cell from prelocking folding to postlocking pressing, its stiffness experiences a sudden jump, implying a limiting-stopper effect. Such a stiffness jump is generalized to a multisegment piecewise stiffness profile in a multilayer model. Furthermore, it is revealed that via strategically switching the constituent cells' deformation modes through passive or active means, the n-layer metamaterial's stiffness is controllable among 2ⁿ target stiffness values. Additionally, the piecewise stiffness can also trigger bistable responses dynamically under harmonic excitations, highlighting the metamaterial's rich dynamic performance. These unique characteristics of self-locking origami present new paths for creating programmable mechanical metamaterials with in situ controllable mechanical properties.