Multistable Structures Research Articles

Multistable Composite Laminate Grids as a Design Tool for Soft Reconfigurable Multirotors

Adaptive‐morphology multirotors exhibit superior versatility and task‐specific performance compared to traditional multirotors owing to their functional morphological adaptability. However, a notable challenge lies in the contrasting requirements of locking each morphology for flight controllability and efficiency while permitting low‐energy reconfiguration. A novel design approach is proposed for reconfigurable multirotors utilizing soft multistable composite laminate airframes. These airframes show kinematically determinate morphologies corresponding to multiple minima in their elastic potential energy landscape. By varying design parameters, the methodology allows for tuning the energy landscape characteristics governing each morphology's structural stability and reconfiguration energetics. The airframe, composed of multistable composite laminate grids, is optimized to maximize rigidity under flight loads and minimize reconfiguration work. The 130‐g reconfigurable multirotor design demonstrates self‐locking properties in an open and a folded configuration, enabling a 48% reduction in width‐span without compromising stability during flight. Soft pneumatic actuators, actuated using a tethered compressed air supply, enable reversible reconfiguration on the ground between open and folded configurations. The design resolves the conflicting requirements of high‐stiffness to lock each flight configuration and low‐actuation work for reconfigurability. By exploiting soft yet multistable structures, the approach combines the stability observed in rigid‐linked reconfigurable multirotors with the low‐effort reconfigurability of soft multirotors, offering new methods for designing adaptive‐morphology multirotors.

Synthesis of a highly programmable multistable Kresling origami-inspired unit cell

Multistable origami structures have been exploited for mechanical property tailoring, deployable robotic arms, wave propagation tuning and others, due to its ability to possess multiple stable states with distinct properties. Traditionally these structures are made by assembling bistable unit cells, which results in a significant increase in the size and weight of the system when larger number of stable states are required. Recently, researchers have uncovered a third stable state in the Kresling origami pattern. Although this is an advancement over the bistable unit cell, there is an unexplored opportunity for more systematically expanding the design space of Kresling unit cells to possess many more stable configurations (>>2) and enhance its programmable multistability. In this research, we seek to develop a methodology for the design of a Kresling origami-inspired unit cell that can be easily programmed to achieve up to 10 stable configurations, with the potential to achieve even more. We exploit the rich kinematics of the Kresling origami-inspired unit cell, that arise from its coupled translational and rotational deployment, and propose the strategic integration of tensile elements to realize multiple stable states. Analytically, we study the unstretched length values (defined to be the precise length between the string “slacked” and “tensioned” configurations) of the strings that yield the distinct number of stable states. We present the potential energy profiles with its corresponding force-displacement plots for the bistable, tristable, quadstable, pentastable and decastable unit cells. Moreover, we show that by simply adjusting the unstretched length of the strings we can program and tune the number of stable states of the unit cell. Tristable and pentastable unit cell prototypes are designed and experimentally tested to validate the proposed methodology. Lastly, a study is performed on the mechanical property tailoring capabilities of two unit cells assembled in series. The results show that the decastable unit cell assembly can achieve up to 55 discrete values of equivalent stiffness, while the bistable one can only achieve 3. For the bistable unit cell assembly to match this number, it will require 54 unit cells in series, which will significantly increase the size and weight of the structural system. These findings show that the modular structure will have more programmable capabilities, while maintaining its size and weight at a minimum, as the number of stable states per unit cell is increased.

A new triangular multistable structure composed of transition element and deformation elements

A novel triangular multistable structure composed of transition and deformable elements is proposed. In order to investigate the effects of the transition element on structures, a design with an opening in the transition element is proposed, and models with varying hole sizes are prepared based on this design. The experimental results of stable states and transformation loads for the structure are well matched with the simulations. These results show that larger hole corresponds to flatter surface, lower snap-through and higher snap-back loads, thereby effectively enhancing the stability of structures. This study can provide insights for active control of multistable structures.

Programmable multi-stability of curved-crease origami structures with travelling folds

Multi-stable structures capable of rapid switching among different stable states have seen applications in various fields such as energy absorption, mechanical computing, and soft actuators. Curved-crease origami, which naturally involves simultaneous deformation of creases and facets, shows great potential in the development of multi-stable structures. However, upon loading, existing curved-crease origami structures tend to follow the same deformation mode as in the curved surface formation process which involves facets elastic bending, flattening, and reverse bending, thus leading to only two stable states. To achieve multi-stability, here we propose a series of curved-crease origami structures composed of planar facets and curved ones. Through a combination of experiments, numerical simulations, and analytical modelling, we demonstrate that the planar facets forming a Sarrus linkage guide the deformation mode of the curved ones, leading to the initiation and propagation of travelling folds in the curved facets. By transforming the travelling folds at specific positions into actual creases, we can achieve multiple stable states in a single structure. In addition, the number and positions of the stable states, as well as the initial peak force, can be programmed by varying the geometric parameters. Consequently, this work opens a new pathway for the development of generic multi-stable structures with programmable mechanical properties.

Intrinsically Multi-Stable Spatial Linkages.

Multi-stable structures can be reconfigured with fewer, lightweight, and less accurate actuators. This is because the attraction domain in the multi-stable energy landscape provides both reconfiguration guidance and shape accuracy. Additionally, such structures can generate impulsive motion due to structural instability. Most multi-stable units are planar structures, while spatial linkages can generate complex 3D motion and hold a more promising potential for applications. This study proposes a generalized approach to design a type of intrinsically multi-stable spatial (IMSS) linkages with multiple prescriptible configurations, which are structurally compatible, and naturally stable at these states. It reveals that all over-constrained mechanisms can be transformed into multi-stable structures with the same design method. Single-loop bi-stable 4R and quadra-stable 6R spatial linkages modules with intrinsic non-symmetric stable states, which are transformed from fundamental kinematic linkage mechanisms unit such as Bennett and Bricard linkages, are designed to illustrate the basic idea and the superiority over the ordinary methods. Multi-loop assembly by these IMSS linkage modules shows potential for practical applications that are required for the deployability and impulsivity of reconfiguration. Two preliminary design cases of a deployable tube and an impulsive gripper are experimentally presented to validate this applicability. Further promisingly, this design method of IMSS linkages paves the way for morphing platforms with lightweight actuation, high shape accuracy, high stiffness, and prescribed impulsive 3Dmotion.

Novel 4D-printed multi-stable metamaterials: programmability of force-displacement behaviour and deformation sequence.

The unique properties of metamaterials are determined by the configuration and spatial arrangement of artificially designed unit structures. However, the configuration and mechanical properties of conventional metamaterials are challenging to reverse and adjust. Based on curved beams, two types of novel three-dimensional (3D) multi-stable metamaterials with reconfigurable deformation and tunable mechanical properties are designed and fabricated using a four-dimensional (4D) printing method. The effects of temperature and curved-beam thickness on the force-displacement curves and multi-stable snapping sequence of the 3D multi-stable metamaterials are investigated by finite-element analysis (FEA) and experiments. In addition, based on the designed four-branch multi-stable metamaterials, three- and six-branched multi-stable structures are designed by changing the number of curved-beam branches. It is shown that, owing to shape memory effects, the 3D multi-stable metamaterials can realize mechanical programmability, and the multi-stable deformation sequence can be precisely regulated by varying the temperature and curved-beam thickness. These 4D-printed multi-stable metamaterials provide valuable contributions to the design of programmable multi-stable metamaterials and their applications in soft robots and intelligent structures. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

Multistable Soft Robots Assembled from Bistable Auxetic Building Blocks

Soft robotics have significantly influenced both scientific and industrial domains over the past decades. However, their inherent limitations of material or structural softness pose a persistent challenge in terms of shape retention and load capacity for various applications. To address this challenge, bistable and multistable structures with two or more mechanically stable states and snap‐through switchability emerge as a promising solution. Herein, a multistable soft robot design assembled from bistable auxetic building blocks with negative Poisson's ratio and a large volumetric change is presented. The bistable mechanical behavior and optimal structural parameters have been investigated by mechanical modeling, finite element analysis, and experimental testing. To demonstrate the shape retention ability and enhanced load capacity of our robotic design, a multistable manipulator with a low‐complexity closed‐loop control system is presented, capable of maintaining 216 discrete stable states even when the pneumatic pressure supply is removed. To validate the multifunctionality of design, a multistable tube crawler with a notable load capacity of approximately five times its own body weight is developed.

Multistability of segmented rings by programming natural curvature

Multistable structures have widespread applications in the design of deployable aerospace systems, mechanical metamaterials, flexible electronics, and multimodal soft robotics due to their capability of shape reconfiguration between multiple stable states. Recently, the snap-folding of rings, often in the form of circles or polygons, has shown the capability of inducing diverse stable configurations. The natural curvature of the rod segment (curvature in its stress-free state) plays an important role in the elastic stability of these rings, determining the number and form of their stable configurations during folding. Here, we develop a general theoretical framework for the elastic stability analysis of segmented rings (e.g., polygons) based on an energy variational approach. Combining this framework with finite element simulations, we map out all planar stable configurations of various segmented rings and determine the natural curvature ranges of their multistable states. The theoretical and numerical results are validated through experiments, which demonstrate that a segmented ring with a rectangular cross-section can show up to six distinct planar stable states. The results also reveal that, by rationally designing the segment number and natural curvature of the segmented ring, its one- or multiloop configuration can store more strain energy than a circular ring of the same total length. We envision that the proposed strategy for achieving multistability in the current work will aid in the design of multifunctional, reconfigurable, and deployable structures.

The whole-process tracking method of stable state transformation for multistable tensegrity based on Levenberg-Marquardt method

The tensegrity structure is prestressed spatial structure composed of rods and cables. Its multistable phenomenon brings great research potential in energy storage and robotic. In order to completely analyze the changes of configurations and mechanical property in stable state transformation (SST) processes of multistable tensegrity structures, a whole-process tracking (WPT) method based on Levenberg-Marquardt method is proposed. Three numerical examples are calculated, and the WPT is successfully realized, which proves the feasibility and universality of proposed WPT method. According to the calculation results, the variation of the driving force and elastic strain energy during SST are further summarized. Alternatively, different types of driving-paths are investigated, the results show that mechanical property during SST could be adjusted effectively by modifying the driving-path. The proposed WPT method and analysis results provides insights for designing and implementing multistable tensegrity structures.

Reprogrammable and reconfigurable mechanical computing metastructures with stable and high-density memory.

Mechanical computing encodes information in deformed states of mechanical systems, such as multistable structures. However, achieving stable mechanical memory in most multistable systems remains challenging and often limited to binary information. Here, we report leveraging coupling kinematic bifurcation in rigid cube-based mechanisms with elasticity to create transformable, multistable mechanical computing metastructures with stable, high-density mechanical memory. Simply stretching the planar metastructure forms a multistable corrugated platform. It allows for independent mechanical or magnetic actuation of individual bistable element, serving as pop-up voxels for display or binary units for various tasks such as information writing, erasing, reading, encryption, and mechanologic computing. Releasing the pre-stretched strain stabilizes the prescribed information, resistant to external mechanical or magnetic perturbations, whereas re-stretching enables editable mechanical memory, akin to selective zones or disk formatting for information erasure and rewriting. Moreover, the platform can be reprogrammed and transformed into a multilayer configuration to achieve high-density memory.

Data-driven modeling of multi-stable origami structures: Extracting the global governing equation and exploring the complex dynamics

In recent years, multi-stable origami structures have garnered increasing attention for their applications in dynamic scenarios such as robotic arm motions, impact energy absorption, and spectrum gap regulation. Understanding the intricate working mechanisms and exploring the rich dynamics of these structures necessitate the development of dynamic models. However, existing dynamic modeling methods for origami structures are often cumbersome, and the resulting dynamic models often lack interpretability. To overcome these limitations, we propose a novel data-driven dynamic modeling approach based on B-spline Galerkin method and subset selection strategy. This approach directly captures the dynamics of multi-stable origami structures using measured data, eliminating the need for reliance on empirical or prior knowledge. To validate the effectiveness of the proposed approach, we first evaluate it on the Duffing system, which has explicit expressions, successfully reconstructing the governing equation. Subsequently, we apply this method to the dynamic modeling of the origami ball structure with tri-stability and the multi-cell stacked Miura-origami (SMO) structure with high dimensionality through simulation, showing favorable results. Finally, using experimental data collected from a bi-stable SMO structure prototype, we employ the proposed method to obtain a global model that can accurately predict different dynamic behaviors over a broad range of excitation frequencies, including intra-well periodic vibrations, inter-well periodic vibrations, and inter-well chaotic vibrations. Overall, our method showcases outstanding efficacy in formulating interpretable, manageable, and comprehensive dynamic models. It plays a pivotal role in delving into the intricate dynamics of multi-stable origami structures.

Mechanical Computing with Transmissive Snapping of Kirigami Shells

AbstractContinuum shape‐morphing structures with the capability to encode memory and execute logic operations have garnered significant interest for the development of mechanical systems with embodied intelligence and soft robots. Achieving the integration of memory and computing within a mechanical system necessitates building blocks that possess a range of tunable, metastable states. Prior efforts have been dedicated to constructing mechanical memory and logic through the exploitation of snap‐through instabilities in multistable structures. Typically, the creation of each logic gate demands a distinct structural design. Here, presents an unconventional design strategy that leverages a single kirigami architecture to perform and switch between multiple fundamental logic operations. By utilizing the kirigami architecture as the fundamental element, mechanical signal transmission is demonstrated and half‐adder computations are performed. It is envisioned that this design strategy can be applied to a wide range of materials and structures, and reduce the complexity of developing materials systems with embodied intelligence.

Topological edge states in reconfigurable multi-stable mechanical metamaterials

Multi-stable mechanical structures find cutting-edge applications across various domains due to their reconfigurability, which offers innovative possibilities for engineering and technology advancements. This study explores the emergence of topological states in a one-dimensional chain-like multi-stable mechanical metamaterial composed of bistable units through a combination of mechanical and optical experiments. Drawing inspiration from the SSH (Su-Schrieffer-Heeger) model in condensed matter physics, we leverage the unique mechanical properties of the reconfigurable ligament-oscillator metamaterial to engineer a system with coexisting topological phases. Based on the one-dimensional periodic bistable chain, there is an exponential decay diffusion of elastic energy from both end boundaries towards the interior of the body. Experimental characterizations demonstrate the existence of stable topological phases within the reconfigurable multi-stable mechanical metamaterial. The findings underscore the potential of reconfigurable mechanical metamaterials as versatile platforms for flexibly exploring and manipulating topological phenomena, with applications ranging from impact resistance to energy harvesting and information processing.

Easy-to-actuate multi-compatible truss structures with prescribed reconfiguration

Multi-stable structures attract great interest because they possess special energy landscapes with domains of attraction around the stable states. Consequently, multi-stable structures have the potential to achieve prescribed reconfiguration with only a few lightweight actuators (such as shape-memory alloy springs), and do not need constant actuation to be locked at a stable state. However, most existing multi-stability designs are based on assembling bi-stable unit cells, which contain multitudes of distractive stable states, diminishing the feasibility of reconfiguration actuation. Another type is by introducing prestress together with kinematic symmetry or nonlinearity to achieve multi-stability, but the resultant structure often suffers the lack of stiffness. To help address these challenges, we firstly introduce the constraints that a truss structure is simultaneously compatible at multiple (more than two) prescribed states. Then, we solve for the design of multi-stable truss structures, named multi-compatible structures in this paper, where redundant stable states are limited. Secondly, we explore minimum energy paths connecting the designed stable states, and compute for a simple and inaccurate pulling actuation guiding the structure to transform along the computed paths. Finally, we fabricated four prototypes to demonstrate that prescribed reconfigurations with easy-actuation have been achieved and applied a quadra-stable structure to the design of a variable stiffness gripper. Altogether, our full-cycle design approach contains multi-stability design, stiffness design, minimum-energy-path finding, and pulling actuation design, which highlights the potential for designing morphing structures with lightweight actuation for practical applications.

Multistable compliant linkages with multiple kinematic paths separated by energy barriers

Multistable morphing structures can reconfigure between different stable states that are separated by energy barriers, and one-degree-of-freedom (1-DOF) mechanisms have many merits, like simple actuation. This paper combines the two and proposes a new family of reconfigurable compliant linkages with many (2−6) 1-DOF kinematic paths that are separated by energy barriers. This new type of design is an extension of multistable structures, where each stable state corresponds to not just one configuration but a 1-DOF configuration space, i.e., a kinematic path. Components of the linkages are made elastically compliant, therefore enabling the switch between two isolated compatible paths with multi-stability. A generation-selection hybrid design algorithm to follow prescribed reconfigurable paths is proposed, and a minimum energy path (MEP) finding method to guide actuation to switch between different kinematic paths is developed. Four design examples with 2–3 reconfigurable paths and their experiments are presented, and the effectiveness of this method is verified. This work provides a fresh perspective to design the single-DOF reconfigurable mechanisms with larger design space, more reconfigurable kinematic paths, and easier reconfiguration actuation.

Nonlinear dynamics of elastically connected multistable structures

Nonlinear dynamics of elastically connected multistable structures

Programmable multistable kirigami chain: Decoupling energy barrier and snapping force/displacement in a unified topology

Arraying bistable cells in a chain results in a multistable mechanical structure that is capable of sequential deformation paths and structural reconfigurability. However, to customize the multistability, for example, a prescribed energy barrier with variable snapping force/displacement, is constrained by the difficulty of decoupling energy from kinematic characteristics since they are always governed by the same geometric parameters. This paper proposes a bistable kirigami cut topology to realize decoupling. Cut parameters are classified into independent groups, corresponding to objective energy barrier or force/displacement, which are consequently independent also. On this basis, a machine learning (ML) and genetic algorithm (GA) based approach is presented to guide the inverse design of the nonlinear responses. Optimal individual kirigami cells that meet decoupled multi-objectives such as energy barrier, snapping force and stable displacement are obtained to construct kirigami chains. The conceivable deformation sequences, configurations and energy/force-displacement curves are validated by both the simulation and experimental results. The proposed mechanism decoupled strategy and inverse design framework open up new horizons for programming energy landscape and provide a general recipe for tailoring multistability in metamaterials.

Directed Instability as a Mechanism for Fabricating Multistable Twisting Microstructure

Herein, The study of multistable structures, particularly those that can twist, has attracted significant attention in recent years. This ability to transition between multiple stable geometries of these structures paves the way for advances in diverse applications, such as morphing structures and robot actuation mechanisms. Conventional methods of designing and fabricating these structures often involve complex and resource‐intensive fabrication processes, which restrict their widespread adoption and limit their miniaturization. Here, we present a novel inflatable multistable twisting structure, based on helical folds of an elastic tube. Our fabrication approach utilizes directed mechanical instability as a method for a rapid fabrication, which is readily implemented at various length scales. We developed a theoretical model for the deformation of the bistable helical elements comprising the twisting structure, and compared the theoretical results to the experimental data. Furthermore, we demonstrate our fabrication methodology using a variety of polymers, including medical grade polymers, as well as various inner radii ranging from 5 mm to 44 μm and thicknesses of the tubes’ walls ranging from 250 μm to 19 μm.

Theoretical Analysis of the Power Performance of a Monostable Galloping-Based Piezoelectric Energy Harvester

In the past few years, different kinds of structural nonlinearities, such as external magnetic interaction and multistable structures, have been introduced into galloping energy harvesters to enhance energy harvesting efficiency. However, since the galloping-based piezoelectric energy harvester (GPEH) is a self-excited system, the conventional impedance matching method that is widely used in vibration energy harvesters is no longer valid for it. Therefore, the power characteristics of the nonlinear GPEH are still an open question to be solved. To this end, this paper is motivated to derive the approximate analytical solution of a monostable galloping-based piezoelectric energy harvester through the harmonic balance method and impedance matching theory. The analytical maximum power, power limit, and critical electromechanical coupling are studied from the perspective of the influence of structural nonlinearity on the power performance. Firstly, the approximate analytical solutions of output power, power limit, optimal resistance, and critical electromechanical coupling coefficient are derived. The accuracy of the analytical solution is verified by numerical simulation. It is found that the influence of structural nonlinearity on the power characteristics of the nonlinear GPEH is quite different under different wind speeds. After that, the power characteristics of the system under different wind speeds and different coupling conditions are investigated. The results showed that the maximum power of the system can be increased by introducing stiffness nonlinearity under low wind speed and weakly coupled configuration. Even more, the system can be shifted into a strongly coupled system when the nonlinearity is enhanced to a certain level. Reasonable design of stiffness nonlinearity can effectively reduce the critical electromechanical coupling, which indicates that stiffness nonlinearity is a feasible and effective way to improve the power performance of low-speed wind energy harvesting.

Tuning the Properties of Multi-Stable Structures Post-Fabrication Via the Two-Way Shape Memory Polymer Effect.

Multi-stable elements are commonly employed to design reconfigurable and adaptive structures, because they enable large and reversible shape changes in response to changing loads, while simultaneously allowing self-locking capabilities. However, existing multi-stable structures have properties that depend on their initial design and cannot be tailored post-fabrication. Here, a novel design approach is presented that combines multi-stable structures with two-way shape memory polymers. By leveraging both the one-way and two-way shape memory effect under bi-axial strain conditions, the structures can re-program their 3D shape, bear loads, and self-actuate. Results demonstrate that the structures' shape and stiffness can be tuned post-fabrication at the user's need and the multi-stability can be suppressed or activated on command. The control of multi-stability prevents undesired snapping of the structures and enables higher load-bearing capability, compared to conventional multi-stable systems. The proposed approach offers the possibility to augment the functionality of existing multi-stable concepts, showing potential for the realization of highly adaptable mechanical structures that can reversibly switch between being mono and multi-stable and that can undergo shape changes in response to a change in temperature.