Origami-based tunable truss structures for non-volatile mechanical memory operation
OOrigami-based tunable truss structuresfor non-volatile mechanical memory operation
Hiromi Yasuda , Tomohiro Tachi , Mia Lee , and Jinkyu Yang Department of Aeronautics & Astronautics,University of Washington, Seattle, WA 98195-2400, USA Graduate School of Arts and Sciences,University of Tokyo, Tokyo 153-8902, Japan
Abstract
Origami has recently received significant interest from the scientific community as a building blockfor constructing metamaterials [1–9]. However, the primary focus has been placed on their kinematic ap-plications, such as deployable space structures [10–12] and sandwich core materials [13], by leveragingthe compactness and auxeticity of planar origami platforms. Here, we present volumetric origami cells– specifically triangulated cylindrical origami (TCO) [14–20] – with tunable stability and stiffness, anddemonstrate their feasibility as non-volatile mechanical memory storage devices. We show that a pair oforigami cells can develop a double-well potential to store bit information without the need of residualforces. What makes this origami-based approach more appealing is the realization of two-bit mechanicalmemory, in which two pairs of TCO cells are interconnected and one pair acts as a control for the otherpair. Using TCO-based truss structures, we present an experimental demonstration of purely mechanicalone- and two-bit memory storage mechanisms. a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec echanical memory operations can be highly useful not only to mimic electronic/opticalmemory devices, but also to manage the flow of mechanical energy for sound isolation, heatinsulation, and energy harvesting purposes [21–25]. These mechanical devices can be highlyuseful in harsh environments, such as space and nuclear power plants, where extreme thermal,mechanical, and radiation conditions can hinder the operation of electronic devices. The robust-ness of mechanical systems, together with nanoelectromechanical technologies, has indicatedthe possibility and effectiveness of mechanical memory storage and computing devices [26, 27].In previous studies, however, the operations of mechanical memory devices are mostly limitedto an individual one-bit memory level (few attempts on multi-bit memories, e.g., [28]), and thepossibility of cross-linked operations across the neighboring bits has not been fully explored.Here, we study how we can realize a two-bit mechanical memory operation by using origamicells. These origami units can function in a modular way, and they can interact with each otherto demonstrate hierarchical, multi-bit memory operations. For this purpose, tunability of unitcells plays an important role to enable the coupling and bit-flipping behavior between the adja-cent cells. We show that origami-based structures provide an excellent platform to manipulatetheir tunable mechanical characteristics, such as stability and stiffness, in a controllable manner.Origami has been a popular building block to construct mechanical metamaterials [1–9]. Inparticular, an overconstrained quadrangular mesh origami such as Miura-ori pattern [10] hasbeen studied extensively because it offers a single degree of freedom (DOF) mechanism offolding without relying on the elasticity of materials. Such a structure is called rigid foldableorigami, or rigid origami, in which the kinematics of folding governs the behavior of the wholestructure. This can be beneficial for the control of deployable planar structures, such as solarpanels and sails [11, 12].In this paper, we feature volumetric origami as a building block of tunable metamaterialsthat can store information. In contrast to the rigid planar origami, volumetric origami gener-ally inherits a highly nonlinear elastic behavior, at the sacrifice of small deformation of panels.2he coupled behavior of folding and elastic deformation can result in versatile kinematic anddynamic motions. We here investigate the mechanics of volumetric origami, specifically trian-gulated cylindrical origami (TCO), which can develop coupled dynamics of axial and rotationalmotions during folding (Fig. 1a). Despite its simplicity, we find a rich tunability in this struc-ture to enable the design of monostable/bistable, zero-stiffness, and bifurcation structures fromone-parameter family of the initial geometry. By assembling multiple origami modules withdesigned initial conditions, we demonstrate the feasibility of mechanical memory storage unitswith non-volatile, bit-flipping behavior.The TCO consists of repeating triangular arrays, which are characterized by valley creaselines (length a ) and mountain crease lines (length b ) as shown in Fig 1b. Top and bottom surfacesof the TCO unit cell are n -sided polygons (e.g., n = 5 in Fig. 1) with side length c . Sincethe TCO is not a rigid foldable origami, folding/unfolding motions cause the warping of eachfacet, which may result in surface fatigue and damage under repeated usage. To overcome thisissue while preserving the key characteristics of the TCO, we replace its surfaces with purelyelastic truss members, which support tension/compression by using linear springs (Fig. 1c; SeeSupplementary Movie 1 for the comparison between the paper- and truss-based TCO models).If we assume that the top and bottom surfaces always share the same rotational axis duringfolding/unfolding, we can characterize the shape of the unit cell by defining its height ( h ),relative angle between the top and bottom polygons ( θ ), and radius of the circle circumscribingthe polygon ( R ). Note that for the sake of mathematical simplicity, θ is defined as an anglebetween OB and the perpendicular bisector of A a A b as shown in the top view of Fig 1c. Letting h and θ be the initial height and relative angle respectively, we can express deformations ofthe structure by axial displacement u = − ( h − h ) where compression is defined to be positive,and rotational angle ϕ = θ − θ .In this truss model, two crease lines a and b are intersecting at a vertex of the polygon (e.g., B in Fig. 1c). For the fabrication of this truss model, we need to secure space for mechanical3oints. Thus, we modify the geometry of the TCO model, such that the two crease lines avoidintersecting (Fig. 1d). The difference between the original (Fig. 1c) and modified (Fig. 1d)models is characterized by the correction of the relative angle ( θ cal in Fig. 1d). By adoptingthis modified model, we fabricate, test, and analyze four different types of the TCO structuresin various combinations of h and θ . Figure 1e-h show the graphical illustration of these fouroriginal models (top row) and the digital images of their modified physical prototypes (bottomrow): ( h , θ ) = (90 mm, 46 ◦ ), (150 mm, 40 ◦ ), (140 mm, 92 ◦ ), and (119 mm, 0 ◦ ). In thesemodels, we use R = 90 mm and θ cal = 9.7 ◦ . See Methods and Supplementary Information formore details on the experimental configuration.To understand the folding behavior of the TCO-based structure, we calculate the total elas-tic energy ( U ) stored in the TCO cell as a function of u and ϕ (Methods and SupplementaryInformation). The insets of Fig. 2a-d show the surface maps of U for the four models, wheredark colored region indicates the valley of the minimum energy level. This highlighted regionforms a near-curve trajectory in this configuration space, indicating that the TCO-based struc-ture exhibits a near mechanism of single DOF. Thus, simultaneously compressive and rotationalmotions of the TCO will follow this trajectory to satisfy the minimum potential energy princi-ple (Methods). We can also identify the change of the normalized energy ( U/kh where k isthe elastic constant of the linear truss element) under non-dimensionalized axial compression( u/h ) by imposing ∂U/∂ϕ = 0 (see Supplementary Information and Supplementary Movie 2for this uni-axial test). Figure 2a-d shows the energy plots, where solid curves denote analyticalresults predicted by the minimum potential energy trajectory in the inset surface maps. Theexperimental measurements (dashed curves) corroborate these analytical results.Comparing the four plots in Fig. 2a-d, we observe remarkably different trends: monostable,bistable, zero-stiffness, and bifurcation behaviors, respectively. If ( h , θ ) = (90 mm, 46 ◦ ), thestructure possesses only one minimum energy state at u = 0 (Fig. 2a). Therefore, the totalenergy increases monotonically as the TCO cell is compressed, implying a monostable prop-4rty. If ( h , θ ) = (150 mm, 40 ◦ ), there exist two local minimum states along the energy valleyas shown in Fig. 2b, indicating bistability. The TCO-based structure can also exhibit zero tan-gential stiffness, so called zero-stiffness mode, in which the application of axial compressiondoes not create significant axial force or torque at the initial stage. Therefore, the total energyincreases at an extremely low rate around u = 0 (Fig. 2c). The discrepancy between the an-alytical and experimental results attributes to the friction of the mechanical joints in the trusselements. Nonetheless, we observe much smaller stiffness in this model compared to the previ-ous two cases (0.26% and 0.18% in terms of the linearized initial stiffness compared to those ofthe monostable and bistable cases, respectively; see the force-displacement curves in Supple-mentary Information). We analytically find that this zero-stiffness mode can be obtained when θ = π/ . Interestingly, this mode is independent of k and h (mathematical proof in Supple-mentary Information). This zero-stiffness mode can be potentially useful for impact absorptionapplications of origami, while maintaining its reusable and tailorable feature.Lastly, we observe that the TCO-based truss can experience bifurcation if θ = 0 (Fig. 2d).That is, in the initial stage of the folding, the TCO-based structure is axially compressed withoutrotation. However, if it reaches a bifurcation point, there are three branches: one unstable branch(continuing pure compression without rotation as indicated by arrow 1 in Fig. 2d) and two stablebranches (starting to develop twisting motions in one or the other direction as pointed by arrow2 in Fig. 2d). The two different trends in the uni-axial testing verify this bifurcation behavior(Fig. 2d; see Supplementary Information and Supplementary Movie 3 for the specially deviseduni-axial compression setup). Overall, the results from these four prototypes manifest versatiledynamics of the TCO, which can be controlled simply by altering its initial geometry (i.e., h and θ , more details in Supplementary Information).Using this TCO-based truss structure as a unit cell, we further investigate the folding mech-anism of multi-cell structures composed of serially stacked TCO cells. We start with a two-cellstructure that consists of identical monostable TCO units with ( h , θ ) = (90 mm, ◦ ) . They5re linked together by sharing the interfacial polygon (Fig. 3a). Note that the chirality of theTCO cells is important in the multi-cell architectures. In this two-cell level, we arrange thecells in the opposite chirality, i.e., ( h , θ ) = (90 mm, ± ◦ ) , such that they collectively showan interesting coupling motion. To test the dynamics of the combined structure, we fix the rightend of the stacked prototype to the wall and impose precompression u C = 45 mm to the leftend of the system (Fig 3a). Then, the total elastic energy of the system will differ depending onthe rotational angles of the two unit cells, characterized by ϕ and ϕ . Note that these anglesare measured with respect to the initial positions of the left and central polygons, respectively.The inset of Fig. 3b shows the analytical values of U as a function of ϕ and ϕ , where thehighlighted zone represents the valley of the minimum potential energy. We find that the pair ofTCO cells collectively possess two local minimum states: one with the right cell folded and theother with the left cell folded (see the graphical illustrations in the inset of Fig. 3b). The normal-ized elastic energy can be re-plotted as a function of ϕ by imposing ∂U/∂ϕ = 0 . Figure 3bevidently shows a symmetric double-well potential. This demonstrates that a pair of monostableTCO cells can successfully form a bistable system, requiring energy to overcome the potentialbarrier for the transition between the two stable states. Please note that this potential barriercan be manipulated by controlling precompression, i.e., tunable potential barrier. Let ‘1’ be thestate where the first unit cell is folded, and ‘0’ be the state where the second unit cell is folded.Then we can use this system as a TCO-based mechanical memory device, which can store bitinformation (‘1’ or ‘0’) by exploiting the double-well potential. One advantage of this mechan-ical memory is its non-volatility, implying that it can store bit information stably without thenecessity of external residual force. To change the states from ‘0’ to ‘1’ or vice versa, we con-trol only ϕ so that the two-unit cell system can switch its state (see Supplementary Informationand Supplementary Movie 4 for experimental verification).Now we demonstrate a two-bit memory operation. We use two pairs of the TCO cells, i.e.,four identical units of the TCO-based truss elements ( h = 90 mm) in the sequence of θ =646 ◦ , – 46 ◦ , 46 ◦ , – 46 ◦ ]. Similar to the previous setup for the single bit operation, we fix therightmost polygon to the wall in both translational and rotational directions, while we let theother polygons rotate freely. We apply precompression of u C = 50 mm and 47.5 mm to thefirst and second pairs respectively, such that the two pairs maintain the specified compressedstates throughout the operation. Here, we intentionally introduce distinctive u C values to breaksymmetry, thereby inducing controlled coupling behavior between the two bits. (further detailsto be explained later). We first analyze the total elastic energy of the system in relation to thedeformed status of the two single-bit memory units. We represent the deformation of thesetwo pairs by measuring the rotational angles ϕ and ϕ , which represent the twisted angles ofthe first and third polygons respectively with respect to their uncompressed positions (Fig. 4a).The analytical results are shown in Fig. 4b, where we identify four minimum energy statesrepresenting ‘00’, ‘01’, ‘10’, and ‘11’. Here the first and second numbers indicate the first bit(left pair denoted in blue color in the inset of Fig. 4b) and second bit (right pair in red color)bit, respectively. For example, ‘10’ is the state where the first bit shows ‘1’, and the second bitshows ‘0’, following the definition of on and off status from the memory operation as illustratedin Fig. 3b.The next step is to test the operation of the two-bit memory. Unlike operations of conven-tional memories (i.e., manipulation of each bit one by one), we demonstrate a unique operationof the two-bit memory by utilizing controlled coupling behavior between the two bits. In par-ticular, this operation flips the second bit if the first bit is ‘1’, which indicates that the first bitcan control the state of the second bit. In this process, however, the second bit does not affectthe state of the first bit, thus exhibiting a one-directional coupling mechanism. We demonstratethis operation by applying a pulse input to the first bit and measuring the response from thesecond bit. For this, we impose a trapezoid-shaped waveform on ϕ to alternates the conditionof the first bit between ‘1’ and ‘0’ (Supplementary Information). The key point here is to verifythat the onset of the control pulse (i.e., ‘1’) can flip the information stored in the second bit.7his process can be represented by the following two cases: ‘00’ → ‘11’ and ‘01’ → ‘10’. Theformer corresponds to the conversion of the second bit from off to on (i.e, ‘0’ to ‘1’), while thelatter implies the opposite case that the second bit changes from on to off (i.e, ‘1’ to ‘0’) as thefirst bit is turned on.We start with demonstrating the first case (‘00’ → ‘11’). Figure 4c illustrates the conceptualchart of the sequential operation of ϕ and the consequential change of ϕ . The correspondingevolution of the TCO pairs’ states is plotted in Fig. 4d, where the red curve denotes the experi-mental result of ϕ and ϕ measured by non-contact laser Doppler vibrometry (SupplementaryInformation). The system is initially positioned at ‘00’, as denoted by the state at (i) in Figs. 4c-d. As we apply the pulse input to the first bit (i.e., ϕ = − ◦ to ◦ ), the first bit changes itsstate from ‘0’ to ‘1’ in the beginning of the operation. See the transition of red curve from point(i) to point (ii) in Fig. 4d. This onset of the first bit eventually flips the second bit from ‘0’ to‘1’ (i.e., ϕ = − ◦ to ◦ , see the state (iii) in Fig. 4d and Supplementary Movie 5). Thus,we verify the transition from the initial state ‘00’ to the final state ‘11’ without resorting to anydirect excitations applied to the second bit.Next, we move on to demonstrate the second case of the two-bit memory operation (‘01’ → ‘10’). Since the previous operation started from ‘00’, we need to first perform input preparationto change the initial state from ‘00’ to ‘01’. For this, we apply the pulse input directly to thesecond bit to convert it from ‘0’ to ‘1’. This pulse input to the second bit does not affect the firstbit, because ϕ is not constrained. Therefore, ϕ and ϕ rotate in the same direction at the samerate without flipping the status of the first bit (see the input preparation process in Fig. 4e). Nowwe apply the pulse input to the first bit, which manipulates the first bit from ‘0’ to ‘1’ as shownby the trajectory of the red curve from the state (i) to the top right corner (ii). Sequentially, theenergy state moves from the state (ii) to the state (iii), flipping the second bit from ‘1’ to ‘0’.Since distinctive u C values are applied to the first and second bits, the energy slope from ‘11’ to‘10’ is less than that from ‘11’ to ‘01’. Therefore, we observe that the state changes from ‘11’8o ‘10’ instead of ‘11’ to ‘01’. This serial process eventually changes the combined states ofthe first and second bits from ‘01’ to ‘10’, successfully verifying the second case of the two-bitmemory operation (Supplementary Information and Supplementary Movie 6).Although we showed the versatile nature and potential of the TCO elements hierarchicallyfrom a single-cell, double-cell, up to four-cell levels in one dimension, we envision that itcan be further extended to multi-dimensions, e.g., honey-comb like 3D clusters. This willfunction as a layer of mechanical memory storage and computing structures, which can alsoprovide multi-functional features, such as rugged protective layers. Likewise, while this studyexplored only serial connections of origami cells, they can be also connected in parallel orcoaxially, to achieve various functionalities. Moreover, origami can be scaled to miniaturizeddimensions. The versatile nature of the TCO together with nanoelectromechanical systems(NEMS) has great potential to develop robust NEMS actuators and sensing devices. Also,the intrinsic nature of the TCO cells that interweave axial and torsional motions can be furtherexploited for dynamic purposes. Conclusively, the volumetric origami cells can pave a new wayfor designing novel engineering systems for mechanical computing and other purposes relyingon their rich constitutive mechanics in a single-cell level and strong cohesion in a multi-celllevel. Methods
Principle of minimum total potential energy approach.
By using the geometry of theTCO-based unit cell, we calculate the length of crease lines a and b (see Fig 1(c)) as follows: a = (cid:115) ( h − u ) + 4 R (cid:48) sin (cid:18) ϕ θ − π n + θ cal (cid:19) b = (cid:115) ( h − u ) + 4 R (cid:48) sin (cid:18) ϕ θ π n − θ cal (cid:19) (1)where R (cid:48) and θ cal are a modified radius of the circle circumscribing the cross-section and a cal-ibrated angle to compensate for the difference between original and physical prototype models9Please see Supplementary Information for details). Then, the total elastic energy is calculatedas U = nk ( a − a ) + nk ( b − b ) where a and b are initial length of a and b , and k is aspring constant of the truss members. Also, the work is obtained by W = F u + T ϕ where F and T are the external force and torque applied to the TCO cell, respectively. Based on theseexpressions, the total potential energy ( Π ) is Π( u, ϕ ) = U − W = 12 nK ( a − a ) + 12 nK ( b − b ) − F u − T ϕ (2)By applying the principle of minimum total potential energy (i.e., ∂ Π /∂u = 0 and ∂ Π /∂ϕ = 0 ),we obtain the analytical expressions of the two-DOF folding/unfolding motion of the TCO-based structure (Supplementary Information). Prototype fabrication and compression test.
We use acrylic plates tailored by a laser cutterfor the top and bottom polygons, and 3D printed parts made of polylactic acid for universaljoints to attach truss elements to the polygons. Stainless steel shafts (diameter is 3.18 mm) andlinear springs ( k = 3.32 kN/m for monostable, bistable, zero-stiffness models; k = 1.08 kN/mfor bifurcation model) are used to form truss elements that support tension/compression. Toobtain the mechanical properties of the prototypes, we build a customized testing setup wherethe prototype is placed horizontally and its bottom surface is mounted on a fixed wall. The topsurface is supported by a ball bearing and stainless steel shaft, so that it can translate and rotatewith minimal friction (Supplementary Information and Supplemental Movies 2–6). Acknowledgments
We thank Mike Clark at CoMotion at the University of Washington for technical support.We also thank Hansuek Lee at KAIST and Hyochul Kim at Samsung Advanced Institute ofTechnology in Korea for helpful discussions. We are grateful for the support from the ONR(N000141410388) and NSF (CAREER-1553202).10 uthor contributions
H.Y. and M.L. conducted the research and interpreted the results, and J.Y. and T.T. providedguidance throughout the research. H.Y., T.T., and J.Y. prepared the manuscript.
Additional information
The authors declare that they have no competing financial interests. Correspondence andrequests for materials should be addressed to Jinkyu Yang (email: [email protected]). [1] Hawkes, E. et al. Programmable matter by folding.
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Nanotechnol-ogy , 125206 (2007). IG. 1:
Geometry of triangulated cylindrical origami. a , Folding motion of the TCO. b , The flatsheet with crease patterns consisting of mountain crease lines ( a shown as blue solid lines) and valleycrease lines ( b shown as red dashed lines). c , Truss version of the TCO, where all facets are removedand crease lines are replaced by linear springs with a spring constant of k . d , Modified truss structure forthe fabrication of physical prototypes. e-h , Graphical illustrations of four different TCO configurations(upper row) and digital images of their physical prototypes (lower row). Their initial configurations are ( h , θ ) = (90 mm, 46 ◦ ), (150 mm, 40 ◦ ), (140 mm, 92 ◦ ), and (119 mm, 0 ◦ ) from left to right. IG. 2:
Folding motions of the TCO cells. (a-d) , The energy analysis for the TCO-based truss structuresshows remarkably different behaviors: a , Monostability at ( h , θ ) = (90 mm, 46 ◦ ); b , bistability at(150 mm, 40 ◦ ); c , zero-stiffness mode at (140 mm, 92 ◦ ); and d , bifurcation at (119 mm, 0 ◦ ). Thedisplacement is normalized by h , and energy is normalized by kh . Experimental results (mean valueis shown as dashed curves, and standard deviation is represented by colored areas) show qualitativeagreements with the analytical predictions (solid curves). The inset plots show the equi-potential plotsof U/kh as a function of u/h and ϕ , in which highlighted trajectories indicate the valley of minimumpotential energy. In the experimental curves, the range of u/h is restricted by the folding motions of theTCO-based truss prototypes. For example, the highly twisted shape of the zero-stiffness TCO prototype(Fig. 1g) causes the truss elements overlap in the early stage of folding, allowing only ∼
15% of u/h asshown in the panel c . The moderately twisted geometry of the monostable and bistable cases (Figs. 1eand f) permit more compression, allowing approximately 50% folding of the truss structure in terms of u/h as shown in the panels a and b . IG. 3:
A pair of TCO cells’ capability to demonstrate mechanical memory storage. a , Twomonostable TCO-based unit cells are connected horizontally. We fix the distance between leftmost andrightmost polygons by imposing a constant distance between them. Photograph of the correspondingconfiguration is shown in the right panel. b , The normalized elastic energy as a function of ϕ shows thedouble-well potential numerically. The inset shows the surface map of the elastic energy as a functionof both ϕ and ϕ , where the highlighted region denotes the valley of the map corresponding to theminimum potential energy trajectory. There exist two minimum states, and the illustrations show theschematic shapes of the pair of TCO cells at these points. Here, the configuration of the folded right cellrepresents ‘0’, while the one with the folded right cell denotes ‘1’. IG. 4: