and structures exploring a modular adaptive metastructure ......skeletal muscle is a remarkable,...

Original Article

Journal of Intelligent Material Systemsand Structures1–14� The Author(s) 2015Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1045389X15586451jim.sagepub.com

Exploring a modular adaptivemetastructure concept inspired bymuscle’s cross-bridge

Z Wu, RL Harne and KW Wang

AbstractThe multifunctionality and versatility of skeletal muscle is a worthy inspiration towards the development of engineeredadaptive structures and material systems. Recent mechanical modeling of muscle suggests that some of muscle’s intri-guing macroscale, passive adaptivity results from the assembly of nanoscale, cross-bridge constituents that maintain mul-tiple metastable configurations. Inspired by the new observations, this research explores a concept of creating modular,engineered structures from the assembly of mechanical, metastable modules, defined as modules that exhibit coexistentmetastable states. The proposed integrated systems are termed metastructures: modular, engineered structures exhibit-ing unprecedented characteristics resulting from a synergy of the constituents. Analytical and experimental resultsdemonstrate that when modular metastructures are prescribed a global shape/topology, the systems may yield significantand valuable properties adaptivity including variation in reaction force magnitude and direction, numerous globally stabletopologies, and orders of magnitude change in stiffness. The influences of important parameters on tailoring the displace-ment range of coexistent metastable states are investigated to provide insight of how the assembly strategy governs theintriguing versatility and functionality which may be harnessed.

KeywordsModular structure, metastructure, adaptive properties, shape change, muscle inspired

Introduction and motivation

Skeletal muscle is a remarkable, natural system com-bining adaptability, durability, robustness, and ener-getic versatility (Lindstedt et al., 2001; Lombardi et al.,1992; Shimamoto et al., 2009). These characteristics aredesirable for engineered structural/material systems toachieve high performance, safety, and sustainability inmany mechanical, aerospace, and civil applications(Gomez and Garcia, 2011; Kuder et al., 2013; Peraza-Hernandez et al., 2014). Recent study on a mechanicalmodel, developed by Caruel et al. (2013), found thatthe nanoscale constituents of muscle, termed cross-bridges, are a critical source for muscle’s intriguing pas-sive, macroscopic functionalities once assembled intomicroscale assemblies called sarcomeres. These charac-teristics include means to ‘‘amplify interactions, ensurestrong feedback, and achieve considerable robustnessin front of random perturbations’’ (Caruel et al., 2013).In the mechanical representation, the cross-bridge wasshown to exhibit coexistent metastable states for thesame sarcomere length (Caruel et al., 2013; Marcucciand Truskinovsky, 2010). The metastable configura-tions represent the cross-bridge in either pre- or post-

power stroke during the sarcomere contraction cyclewhich is the origin of macroscopic force (Tortora andDerrickson, 2006). The study modeled the passivecross-bridge as bistable and linear springs in series(Caruel et al., 2013), thus identifying equivalent struc-tural constituents that emulate the essential ingredientsof muscle’s manifestation of metastability.

Inspired by the intriguing versatility elucidated in therecent studies of cross-bridges and sarcomeres, the goalof this research is to develop and explore a new class ofadaptive structures possessed with muscle-like versati-lity and multifunctionality. By the development andstrategic assembly of mechanical metastable modules,this research examines a novel concept for engineeringpassive-adaptive metastructures. The term ‘‘metastablemodule’’ denotes that the module exhibits coexistent

Department of Mechanical Engineering, University of Michigan, Ann

Arbor, MI, USA

Corresponding author:

RL Harne, Department of Mechanical Engineering, University of Michigan,

Ann Arbor, MI 48109-2125, USA.

Email: [email protected]

at UNIV OF MICHIGAN on May 25, 2015jim.sagepub.comDownloaded from

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metastable states for the same topology/shape—like thecross-bridge pre- and post-power stroke configurationsfor a prescribed sarcomere length. ‘‘Metastructure’’denotes that the modular structural/material system isinvested with unprecedented characteristics derivedfrom a synergy of the assembled members. Through thebiologically inspired constituent and assembly strategy,the aim is to uncover and elucidate the advanced func-tionalities and versatility invested in passive-adaptiveengineered systems when a metastable building block istranslated upwards from the nanoscale.

The following sections present analytical and experi-mental studies on the design and assembly of meta-stable modules into assembled metastructures.Investigations first illustrate the basic concept anduncover the single-module-level adaptivity generateddue to coexisting metastable states. The studies thenclosely explore the opportunities produced in conse-quence to modular assembly for a dual-module metas-tructure. Analyses and experiments explore howtailoring individual module design parameters governsthe adaptability features of the assembled metastruc-ture, which gives insights on means to effectively guidemodule development to achieve desired metastructureversatility.

Mathematical formulation for metastablemodules and assembled metastructures

In agreement with the previous cross-bridge modeldeveloped by Caruel et al. (2013), the one-dimensional,metastable modules created in this research are eachcomposed using bistable and linear springs in series(Figure 1). The parallel assembly of modules is then thebasis for the metastructures examined here, similar tothe sarcomere composition of cross-bridges

A model is developed to explore the mechanicalproperties of metastable modules and assembled metas-tructures. A schematic of an assembled metastructureis shown in Figure 1. For this system, n modules arearranged in parallel with respect to a common globalend displacement z. Within each module, a bistablespring with restoring force

FiB(xi)= kiBxi(xi � ai)(xi � 2ai) ð1Þ

is connected in series to a linear spring having restoringforce FiL = kiL(z� xi). Here, xi is the internal displace-ment representing the deformation of the bistablespring of the ith module from the static equilibriumxi = 0; ai is the ith bistable spring span which denotesthe distance between the unstable and stable equili-brium positions of the bistable spring; and kiB and kiL

are the stiffnesses. Note that the bistable spring forceexpression indicates there is a symmetry between thestatically-stable equilibria of the bistable spring and thecentral unstable position (when disconnected from the

linear spring). In contrast to the previous model of thecross-bridge which utilized a bilinear/bistable spring(Caruel et al., 2013; Marcucci and Truskinovsky, 2010),the metastable module employs a continuous bistablespring characterized using a cubic polynomial for thespring restoring force to more accurately reflect theexperimental bistable element behavior. The totalpotential energy of the metastructure is expressed by

U =Xn

i= 1

1

2(2a2

i kiB + kiL)x2i � aikiBx3

i +1

4kiBx4

i � kiLzxi +1

2kiLz2

� �ð2Þ

which is a function of internal coordinate positions xi

as well as the global end displacement z. A metastableinternal state/configuration for a fixed global topologyz of the metastructure satisfies ∂U=∂xi = 0 and∂2U=(∂xi∂xj).0, where the latter indicates that theHessian matrix is positive definite (Puglisi andTruskinovsky, 2000). The former constraint leads to aseries of force balance equations, one for each modulein the assembly

kiBx3i � 3aikiBx2

i +(2a2i kiB + kiL)xi

� kiLz= 0; 8 i= 1, 2, . . . , nð3Þ

The solutions to equation (3), that is, the roots ofthe cubic polynomials in terms of parameter xi, are sub-stituted into the Hessian matrix to evaluate for positivedefiniteness. Since the system is assembled from mod-ules in parallel, the modules are statically decoupled for

Figure 1. Schematic of the modular metastructure with nmetastable modules assembled in parallel with respect to acommon global end displacement z.

2 Journal of Intelligent Material Systems and Structures



prescribed end displacements z. Therefore, the determi-nation of positive definiteness is equivalent to inspect-ing ∂2U=∂x2

i .0 8 i= 1, 2, . . . , n. From equation (3),the cubic polynomial may have at most three stationarypoints, and at most two of them are stable via∂2U=∂x2

i .0. Thus, for each module in the metastruc-ture, there are at most two configurations satisfyingboth constraints. The two real roots of the ith cubicpolynomial require that the respective discriminants ofequation (3) satisfy

Di = 54k2iBaikiL(2a2

i kiB + kiL)z� 108k3iBa3

i kiLz

+ 9k2iBa2

i (2a2i kiB + kiL)

2

� 4ki(2a2i kiB + kiL)

3 � 27k2iBk2

iLz2 � 0

ð4Þ

Equation (4) can be rearranged into a quadraticpolynomial inequality in terms of global end displace-ment z

Ci2z2 +Ci1z+Ci0� 0 ð5Þ

Here, the coefficients Ci2 = 27k2iLk2

iB, Ci1 = �54aik2iLk2

iB,and Ci0 = kiB(4kiL � a2

i kiB)(kiL + 2a2i kiB) are functions

of spring stiffnesses and the bistable springs’ spans ai.Therefore, the end displacements leading to multiplecoexistent metastable states for each module canbe explicitly derived as a function of systemparameters from equation (5) and satisfy the followingcriterion

�Ci1 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2

i1 � 4Ci2Ci0

p2Ci2

� z� �Ci1 +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2

i1 � 4Ci2Ci0

p2Ci2

;

8 i= 1, 2, . . . , n ð6Þ

Consequently, the extent of the metastability rangefor a metastructure assembled from n modules is deter-mined using

[ni= 1

�Ci1 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2

i1 � 4Ci2Ci0

p2Ci2

,�Ci1 +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2

i1 � 4Ci2Ci0

p2Ci2

" #

ð7Þ

Having determined the metastable states of themetastructure, the total, static end (reaction) force forprescribed global topology z is found usingFend =

Pni= 1 kiL(z� xi). The stiffness of the metastruc-

ture at each global end displacement is obtained byHooke’s law

ka =∂Fend

∂z=Xn

i= 1

kiL 1� ∂xi

∂z

� �ð8Þ

where ∂xi=∂z is determined implicitly from equation (3).These static properties of metastable modules and

assembled metastructures are then utilized in a dynami-cal model formulation for the case when the systems are

acted upon by slow, periodic end displacements, suchthat the frequency of the excitation is significantly lessthan the linearized natural frequency of any given mod-ule. Inertial influences in the metastable modules areaccounted for via including masses mi that interface thebistable and linear springs. Additionally, damper ele-ments are included in parallel with the bistable springsto take into account dissipative effects. Since the mod-ules are assembled in parallel, the equations of motionfor the internal masses mi are decoupled under enddisplacement–controlled motions, leading to

mi€xi + bi _xi +FiB(xi)= kiL(z� xi); 8 i= 1, 2, . . . , n ð9Þ

with FiB(xi) defined in equation (1). Based on a pre-scribed, periodic end displacement z(t), the instanta-neous total end force is then

Fend(t)=Xn

i= 1

kiL z(t)� xi(t)ð Þ ð10Þ

In this study, the area enclosed in hysteresis loopsinduced in consequence to the periodic end displace-ment actuations is used to evaluate the energy dissipa-tion capacity of modules and metastructures

E =

ðz(T )z(0)

Fend(t)dz(t) ð11Þ

where T is the end displacement excitation period.

Properties of a proof-of-concept cross-bridge inspired metastable module

Mechanical properties

To illustrate the adaptable mechanical properties of themetastable module similar to the characteristics of thecoexisting pre- and post-power stroke states of muscle’scross-bridge constituent, the following study examines asingle module. For analytical insight, the model devel-oped in the previous section is employed using the para-meters k1B = 1N=m3, a1 = 1m, and k1L = 0:3N=macross a range of static end displacements. The para-meters are arbitrarily selected, at this stage in develop-ment, for qualitative comparison between analyticalpredictions and experimental findings. Figure 2(a) pre-sents the potential energy profile of the individual mod-ule as the end displacement varies. Throughout thiswork, in the profiling of potential energy, force, and stiff-ness as end displacement is varied, the distinct line stylesrepresent the mechanical properties due to different com-binations of metastable internal configurations and thusare representative of the different metastable states. Theresults in Figure 2(a) show clearly that coexisting meta-stable states are induced across a certain range of enddisplacement values. In the figure, these displacements

Wu et al. 3



are denoted as the metastability range, defined as therange of global displacements z within which multiplemetastable configurations coexist, and are featured inFigure 2 using the light background shading. For someprescribed end displacements, the coexistent metastablestates exhibit different values of potential energy. Thedistinct potential energy values are of particular interestsince this is the mechanism by which cross-bridges in sar-comeres achieve such remarkable characteristics as theefficient switching back-and-forth between pre- andpost-power stroke states (Caruel et al., 2013). Indeed,recent studies on adaptive machines (Allahverdyan andWang, 2013), metamaterials (Klatt and Haberman,2013; Nicolaou and Motter, 2012), and sensory adapta-tion systems (Lan et al., 2012) provide complementaryevidence that adaptive or non-natural characteristicsmay be induced when non-unique potential energies sat-isfy shared external constraints. Comparable, intriguingpotential energy landscapes produced due to coexistingmetastable states in a parallel ‘‘bundle of bistable snap-springs’’ have recently been shown by Caruel et al.(2015) to be distinct with respect to displacement-controlled and force-controlled loading scenarios repre-sentative of isometric and isotonic loading, respectively,on skeletal muscle networks. Yet, in spite of the differ-ences, it was shown that coexisting metastable states areobserved in both loading scenarios (Caruel et al., 2015).As a result, to focus the attention of this research on theproperties adaptation enabled by leveraging the coexistingmetastable states, the following studies consider the caseof displacement-controlled loading for insights obtainedthrough a channeled and systematic exploration.

For the following experimental explorations andstudies, proof-of-concept metastable modules are fabri-cated using frame-mounted bistable springs, realizedusing compressed and buckled spring steel beams fixedinto supporting poly(methyl methacrylate) (PMMA)frames. The bistable springs are then connected in seriesto a linear compression coil spring, aligned to translateuniaxially (Figure 2(d)). The coil spring is attached tothe center point of the buckled beam and is acted uponby prescribed global topologies/displacements z on theopposing end. For experimentation with a single mod-ule, the opposing coil spring end is attached to a loadcell which is attached to a quasi-statically actuated elec-trodynamic shaker. The shaker displacement (the glo-bal topology/displacement z) is measured using a laserinterferometer. Stiffness is determined by the derivativeof the force–displacement profiles. The supportingframes measure 0.15 m 3 0.15 m 3 6.5 mm with aninterior cavity of 0.1 m 3 0.1 m. Thus, 0.1 m is thespring steel beam length prior to applying axial com-pression to buckle the beam. The spring steel beams are0.127 m wide 3 0.18 mm thick. Axial beam compres-sion is accomplished via micrometer adjustmentsagainst a sliding end of the beam clamping fixtureinside the frame. The straightforward design of the fab-ricated metastable modules captures the essential, pas-sive mechanical ingredients of the cross-bridge and,indeed, is advantageous so that investigations can befocused on the new phenomenological behavior withoutpotential misdirection by nonessential elements. InFigure 2(d), the physical configurations of two coexis-tent metastable states are illustrated for the

(a) (b) (c)

(d) (e) (f)

Figure 2. Single-module analytical results of (a) potential energy, (b) reaction force, and (c) stiffness as functions of globaldisplacement. (d) Single-module experimental setup and measurements of (e) reaction force and (f) stiffness. Markers regardingcoexistent forces/displacements in (e) correspond to coexistent metastable configurations in (d). Light shading denotes metastabilityrange where structural adaptation is realized, and each line indicates a metastable state.




experimental system, showing that for one prescribedglobal end displacement z, the internal state (coordi-nate) x may rest at one of two metastable configura-tions. Such coexistence of metastable states for a singlemodule is the key feature derived from the cross-bridgeinspiration.

Due to the non-uniqueness in the potential energy,the metastable module exhibits adaptivity in themechanical properties across the metastability range.The analytical and experimental results (Figure 2(b)and (e), respectively) reveal multiple static equilibriatopologies, indicated by the global displacements satis-fying zero force. Moreover, the results in Figure 2(b)and (e) show that magnitude and direction adaptationof module reaction forces is possible via switchingbetween coexistent metastable states for certain fixedvalues of end displacement. The connections betweenmetastable state realization and corresponding reactionforce behaviors are exemplified experimentally by themarkers labeled 1 and 2 in Figure 2(d) and (e).

In fact, a large body of research has consideredexploiting the stable configurations of bistable struc-tures for adaptivity and functionality enhancements.Among many recent examples, Johnson et al. (2013)investigated energy dissipation of a bistable oscillatorshowing drastic passive-adaptive power dissipationcapability depending on the harmonic excitation char-acteristics. Pontecorvo et al. (2013) examined the bis-table behavior of an arch for morphing applicationsand provided detailed design guidelines to exploit bist-ability for shape change and load support. Schultz(2008) investigated a new concept of bistable twistingplates for adaptive airfoil morphing enabled by the stra-tegic fabrication of composite laminates. Inspired by afolding mechanism observed in nature, Daynes et al.(2014) developed an elastomeric bistable cell geometrycapable of reversibly unfolding from a flat configura-tion to a highly textured configuration which couldpotentially be used as a means of adaptive camouflage.Wu et al. (2015) elucidated the parameter sensitivitiesof harmonically excited bistable structures to selectivelyachieve desired dynamic displacement amplitudes andmean response values, which serve as operating guide-lines for a wide range of applications that strategicallyutilize several of the dynamic behaviors. These possibili-ties for adaptivity enhancements are due to the natureof the force–displacement profiles for bistable struc-tures which are bi-valued in displacement for the sameforce (Kovacic and Brennan, 2011). Yet, in great con-trast, the unique adaptability of reaction force for afixed global topology is only possible owing to the factthat the metastable module is not only bi-valued in dis-placement for constant force but also bi-valued in forcefor the same global end displacement. Thus, eventhough the metastable module includes a bistable ele-ment, the capacity for multiple coexistent metastablestates sharply differentiates the metastable module from

a strictly bistable system. In fact, the metastable moduleis more analogous to a phase-transition material sincethe shape remains constant while its reaction to theenvironment drastically alters via change between coex-istent metastable states.

In addition to adaptation of the reaction force, byHooke’s law the metastable module may exhibit one oftwo stiffnesses corresponding to the respective metastablestates. Between these two states, orders of magnitude instiffness variation may be observed experimentally andanalytically (Figure 2(e) and (f)). The global topologiesfor which stiffness adaptations are most dramatic areseen to occur near the extremes for which the respectivemetastable configurations occur. For example, measure-ments reveal more than four orders of magnitude stiff-ness change by switching between metastableconfigurations when the global displacement is fixed atz = 10 mm (86.71 N/m to 0.02799 N/m), Figure 2(f).Moreover, the analytical and experimental results findthat the module may realize negative stiffness at theextremities of the metastability range (Figure 2(e) and(f)). Due to the global end displacement constraint forthe module, it has been shown that negative stiffnessmay in fact be ‘‘stabilized’’ (Lakes and Drugan, 2002).

Energy dissipation for low-frequency dynamicexcitation

A dynamic study using the model of a single metastablemodule is conducted to examine how the significantadaptation in the mechanical properties is translated toenergy dissipation characteristics. In addition to theprevious module parameters, this dynamic studyemploys values of internal mass m1 = 1kg and damperconstant b1 = 0:5N=(m=s). The global end displace-ment z is prescribed to be a triangular wave with peak-to-peak amplitude A = 5 m and period T = 2 3 104 s,as shown in Figure 3(a), and the displacement begins atone of the statically-stable equilibrium, marked by thecircle in Figure 3(b). The prescribed dynamic end dis-placement is such that it encompasses the full range ofstatic end displacements considered in Figure 2. Fromnumerically integrating equation (9) under these condi-tions, the instantaneous reaction force, computed usingequation (10), is depicted in Figure 3(b) with arrowsindicating the trajectory of the force over time. Asshown in Figure 3(b), when the global end displace-ment approaches the extremes of the metastabilityrange, sudden dramatic changes in both force level anddirection are observed due to the change in the internalconfiguration from one metastable state to the other.Such phase transition-like phenomena can lead to largehysteresis. Since the area of the hysteresis loop corre-sponds to energy dissipation over the course of anactuation cycle (Lakes, 2009), it is found that for thisparticular module design, the energy dissipation is

Wu et al. 5



0.73 J, according to equation (11). Although thisenergy dissipation mechanism is not a particular prop-erty exploited by cross-bridges in muscle (Caruel et al.,2013), leveraging sudden switches among metastablestates in the metastable modules created and exploredin this research is a novel approach to damping enabledas a result of the biologically inspired design. In fact,for material systems, similar means to achieve largedamping has also been explored when the compositesare composed of negative stiffness inclusions withinpositive stiffness elastic matrices, creating a materialsystem that features metastable states for certain valuesof temperature (Lakes et al., 2001).

Analytical and experimental case studiesof a dual-module metastructure

To explore how the intriguing mechanical and dynami-cal properties adaptivity of the metastable modules istransformed when modules are assembled together, thissection investigates a dual-module metastructure forfirst insights.

Analytical case study for a dual-modulemetastructure

To begin, an analytical study of a dual-module metas-tructure is undertaken where the two modules are

different in design, and hence in structural features.The specific parameters used in the analysis arek1B = 1N=m3, a1 = 1m, k1L = 0:3N=m, andk2B = 16=81N=m3, a2 = 3=2m, and k2L = 8=45N=m.The parameter values indicate that the stable equilibriaof the two bistable springs of the modules are locatedat (0, 2) m and (0, 3) m, respectively. The parametersare chosen arbitrarily and are thus used for representa-tive purposes in deriving first insights pertaining to theprimary features of the metastructures. The global endforces and stiffnesses, as computed by the analyticalmodel, are plotted in Figure 4. Here, the label AA indi-cates that the attachments between linear and bistablesprings of both modules remain at positions with

(a)

(b)

Figure 3. (a) Prescribed displacement profile for dynamicsimulations. (b) Single-module analytical result of reaction forcechange as global displacement is slowly and continuously varied,with circle marker indicating the starting and final states. Arrowsindicate direction of changes induced via the controlled changein end displacement.

(a)

(b)

(c)

Figure 4. Analytical profiles for dual-module metastructure asglobal end displacement is varied: (a) end force and (b) stiffness.Different line styles denote different metastable configurations.(c) Reaction force change as global displacement is slowly andcontinuously varied, with circle marker indicating the startingand final states. Arrows indicate direction of changes induced viathe controlled change in end displacement.




displacements greater than the unstable static equili-bria, that is, in this case study satisfying x1.1m andx2.1:5m, respectively. Conversely, the label BB meansthat these spring connections remain at positions withdisplacements less than the unstable static equilibria,and thus satisfy x1\1m and x2\1:5m, respectively.The mixed labels AB and BA denote the correspondingmixed cases of internal module configurations.

Figure 4(a) shows that the dual-module metastruc-ture exhibits multiple distinct force profiles, one repre-senting each metastable configuration set of the system.The plot also reveals four global displacements yieldingzero reaction force which are statically-stable equili-brium positions. As shown in Figure 4, for this repre-sentative case, the metastability range spans fromapproximately z= 0:25m to z= 2:17m. As shown inthe force profile, Figure 4(a), when the end displace-ment z is prescribed, at most four different reactionforces can be realized for this metastructure. In otherwords, the total number of coexisting metastable statesmultiplies from two to four as the two modules areassembled together. According to the model, by theparallel assembly of the metastable modules acted uponby end displacement constraints, the number of totalpossible coexisting metastable states is 2n for an n-mod-ule metastructure, plainly a synergistic product of themetastructure development. In addition to changes inreaction force, by Hooke’s law, Figure 4(b) shows thatthe dual-module metastructure exhibits large adaptabil-ity in stiffness when the global shape/displacement isheld constant. In fact, orders of magnitude of stiffnessvariation may be achieved by switching among theinternal states when global end displacement is fixed,for example, at z’1:7m. The large adaptation of stiff-ness afforded by the metastable module is an attractivecapability to leverage for advancing structural function-ality such as for shape morphing (Shan et al., 2009),wave-guiding (Fok et al., 2008), and vibration control(Bonello et al., 2005), just to name a few applications.

Similar to the dynamic study of the single metastablemodule, Figure 4(c) presents the instantaneous reactionforce profile of the dual-module metastructure when itis acted upon by the slow, periodic end displacementexcitation shown in Figure 3(a). The numerical integra-tion of the equation system (9) for this metastructureemploys the additional system parametersm1 =m2 = 1kg and b1 = b2 = 0:5N=(m=s). The circlepoint in Figure 4(c) denotes the starting position whilethe arrows indicate the instantaneous trajectory ofreaction force. As shown in Figure 4(c), the suddenchange in force amplitude occurs not only at theextremes of the metastability range, similar to the indi-vidual metastable module, but also occurs inside themetastability range. Each sudden transition in the forceamplitude qualitatively corresponds to a change ininternal metastable configuration. For this case, theenergy loss per cycle is determined to be 1.27 J. In

other words, by the assembly of two different meta-stable modules, the net energy dissipation capacity ofthe metastructure is increased by more than 70%. Themore intricate features of how energy dissipation prop-erties scale and are modulated by the modular assemblywill be the focus of future research efforts.

Experimental case study for a dual-modulemetastructure

Following the analytical case study, the proof-of-concept experimental system is employed to verify theprimary trends of the analysis regarding the mechanicalcharacteristics empowered by assembling two meta-stable modules together to form a metastructure.Figure 5 shows the experimental setup of a dual-modulemetastructure, where the modules are labeled I and II.To realize the parallel assembly strategy, the other endsof the coil springs of each module are attached to acommon rigid connecting bar that is connected to thecontrolled electrodynamic shaker to prescribe uniaxialmotions. To evaluate all possible coexisting metastablestates (if they occur for a given system design), theexperiments are conducted to reconstruct the profilessatisfying the conditions that: (AA) both modules’ bis-table beam displacements remain to the right of theunstable equilibrium positions with respect to a central,vertical, pre-buckled configuration; (BB) both bistablebeam displacements remain to the left, as shown inFigure 5; (AB) the bistable spring displacement of mod-ule I remains right and that of module II remains left;and (BA) the bistable spring displacement of module Iremains left and that of module II remains right.

Figure 6(a) presents the measured force profiles of adual-module metastructure as end displacement varies.In agreement with the trends predicted analytically in

Figure 5. Photograph of a dual-module metastructure.

Wu et al. 7



Figure 4(a), the measurements shown in Figure 6(a)reveal four distinct force profiles depending on themetastable configurations exhibited within the metas-tructure. In this case, each profile exhibits one globaltopology/displacement satisfying zero reaction force;thus, each metastable configuration set provides for adifferent statically-stable global topology. The mea-surements show that when the metastructure is pre-scribed a global displacement, switching amonginternal configurations may change amplitude anddirection of the reaction force, which is also in agree-ment with analytical predictions. In addition, theexperimental results in Figure 6(b) show that extremechange in stiffness may be realized by switching amongthe metastable configurations while the global end dis-placement is fixed, notably when the end displacementis held near the limits of stability for a given metastablestate. Collectively, the results of Figure 6(a) and (b)verify the primary trends of the analysis and clearlydemonstrate that the metastructures can yield dramaticmechanical properties adaptivity for constant globaltopology and induce a multiplying number of stati-cally-stable equilibria across a broad metastabilityrange of global displacement.

The analytical and experimental results suggest thatswitches among the metastable states provide

immediately reversible properties or change in globalmetastructure shape. To experimentally explore thereversible topology changes using metastable stateswitching, another dual-module metastructure is exam-ined using dynamic transitions among the metastableconfigurations. First, Figure 7(a) plots the force pro-files of the system, once again uncovering the charac-teristic multitude of unique mechanical propertieswhich may be realized depending on the metastableconfiguration set of the assembled modules. The fourstatically-stable equilibria are labeled 1–4 in Figure 7(a)and highlighted via the circle points. Then, Figure 7(b)presents the time history of the global end displacementwhile the various metastable configurations are chan-ged via fast manual actuations upon the attachmentsbetween module bistable and linear springs. In theexperiment, the global end displacement is free to moveaccording to its one-dimensional axis of motion asguided by an unconstrained motion of the shaker upon

(b)

(a)

Figure 6. (a) Force and (b) corresponding stiffness profiles ofthe dual-module metastructure as global end displacement isvaried.

(a)

(b)

Figure 7. (a) Force profiles of the dual-module metastructureas end displacement is varied. Circle points indicate staticallystable topologies, correspondingly labeled as (1)–(4). (b) Timehistory of global end displacement as metastable configuration ischanged to induce changes in global topology.




the low-friction shaker guide rails. Both modules startat positions to the right of the unstable equilibria of thebistable beams (AA) with respect to the pre-buckledconfigurations, such that the initial global displacementis the statically stable topology labeled 1 in Figure 7(a).By switching the internal coordinate of module I to theleft of the bistable beam unstable equilibrium, realizingthe BA configuration, the global end position immedi-ately changes to the topology labeled 2, which isanother statically stable metastructure topology, asshown in Figure 7(a). The corresponding time historyof global displacement at the point of the metastablestate switch is shown in Figure 7(b) at time around 6 s,and the reversible switch back to topology 1occurs around time 15 s. This process is repeated forthe changes in topology 1 to 3 to 1, and 1 to 3 to 4 to1. The experimental demonstration of metastablestate transitions verifies the reversibility of switchingamong the metastable configuration sets as means torealize repeatable and consistent changes in statically-stable global topologies. Likewise, reversible adapta-tion in force amplitude, and potentially force direction,can be achieved within the metastability range byswitching among internal configurations when globalend displacement z is fixed. Moreover, according to theanalytical model, the number of potential coexistingmetastable states increases exponentially, a findingfrom which one could hypothesize empowers a signifi-cant opportunity for reversible properties adaptivitydue to the assembly of many modules in themetastructure.

Parametric study for a dual-modulemetastructure

The unique adaptations in shape and mechanical prop-erties observed from the case studies of the previoussections illustrate that the metastructures may providehigh potential for engineering applications in whichsuch versatility is advantageous, including morphingaircraft (Barbarino et al., 2011), energy absorption(Tolman et al., 2014), and load impedance control(Philen, 2009), to name a few. To develop an under-standing of how to take the greatest advantage of thesefeatures, the sensitivities of metastructure properties areevaluated as key design parameters are changed. In thisresearch, except the number of modules in the parallelassembly, all the design parameters are at the modulelevel. In the following sections, as individual metastablemodule design parameters are varied, the changes inoverall metastructure characteristics are examined.

Analytical study of change in bistable spring span a2

In this section, the analytical model is employed toinvestigate the influence on metastructure propertiesdue to changing span between the statically-stable equi-libria of the bistable spring of the second module,denoted by parameter a2. The other system parametersfor the following study are k1B = k2B = 1N=m3,k1L = k2L = 0:5N=m, a1 = 1m, and a2 = ½0:8; 1:2;1:6�m. The top row of Figure 8 plots the global endforce as a function of global displacement, while the

(a) (b) (c)

(d) (e) (f )

Figure 8. Profiles of (a, b, c) end force and (d, e, f) stiffness as global end displacement is varied for the dual-module metastructure.From left to right for both rows, the bistable spring span a2 of the second module increases.

Wu et al. 9



bottom row of Figure 8 depicts the corresponding stiff-ness profiles. The labeling convention for the meta-stable configurations is the same as employed in thecase studies. From left to right, the bistable spring spana2 increases from 0.8 to 1.2 to 1.6 m.

As clearly shown in Figure 8, the metastability rangeincreases as the bistable spring span a2 increases. Thisis due to the fact that as the bistable spring span of thesecond module increases, according to equation (1) thecorresponding peak force amplitudes of the secondmodule bistable spring are increased at the displace-ments that transition from positive to negative stiffnessvalues (for reference, the feature here discussed is expli-citly observable in Figure 10 in the following experi-mental studies). This in turn makes it possible tostabilize the metastable states over progressivelyincreasing ranges of global displacement because largerforces are required to deform the linear spring (andhence extend the range of end displacement) in order tocounterbalance the increasing bistable spring forces.

As shown in Figure 8(d) to (f), it is observed that thechange in bistable spring span a2 has little effect on thestiffness profiles at very small and large displacementsof the dual-module metastructure. On the other hand,increases in the second module bistable spring spanparameter are seen to gradually increase the location atwhich the static equilibrium position of the AA meta-stable configuration occurs. Finally, it is observed thatas the bistable spring span a2 increases, considering thecolumns of Figure 8 from left to right, the ranges ofdisplacements over which the metastable configurationsAB and BA occur also increase. This effect is especiallyprominent for configuration AB, where the range isalmost non-existent considering the case of shortest bis-table spring span (Figure 8(a) and (d)), yet is found toextend over a majority of the total metastability rangefor the largest bistable spring span (Figure 8(c) and (f)).The latter finding is important because each metastablestate provides for a unique force and stiffness profile,which the metastructure may favorably harness foradaptation of the mechanical properties. Thus, a widerrange of displacements across which each unique meta-stable configuration profile exists provide for a broaderrange of global topologies that are empowered withadvantageous adaptivity.

The previous sections found that dynamic end dis-placements may trigger sudden changes in the internalmetastable configuration set of the metastructure whichleads to large periodic hysteresis and hence energy dis-sipation. To investigate the influence of the bistablespring span a2 on the energy dissipation capacity, thedynamic model of the dual-module metastructure isnumerically integrated for the case that the end displa-cement is periodically actuated in the style shown inFigure 3(a); the remaining system parameters for theslow dynamic model evaluation are m1 =m2 = 1 kgand b1 = b2 = 0:5N=(m=s). Following equation (11),

the energy dissipation per cycle of the metastructureincreases from 0.42 to 1.70 to 6.65 J as the bistablespring span a2 increases. Such significant increase inenergy dissipation per actuation cycle (more than anorder of magnitude) serves as further evidence of theextreme adaptability of metastructures composed fromthe metastable modules.

Analytical study of change in linear spring stiffnessk2L

The analytical model is utilized to explore how charac-teristics of the dual-module assembly are influenced dueto changes in linear spring stiffness k2L. The parametersfor the study are k1B = k2B = 1N=m3, k1L = 0:5N=m,a1 = 1m, a2 = 1:25m, and k2L = ½0:4; 0:8; 1:2�N=m.Similar to the plotting convention in Figure 8, the toprow of Figure 9 presents the end force profiles while thebottom row shows the corresponding stiffness profilesfor the dual-module metastructure. In both rows ofFigure 9 from the left-most column to the right-mostcolumn, the linear spring stiffness k2L increases from 0.4to 0.8 to 1.2 N/m. As shown in Figure 9(a) to (c), forthe range of the linear spring stiffnesses k2L employed inthis parametric study, all three cases preserve the char-acteristic four distinct force profile paths per dual-module assembly. Two primary trends are observed forincreasing values of linear spring stiffness k2L. First, themetastructure stiffnesses for very small or very largeend displacements are seen to increase (Figure 9(d) to(f)); this is in sharp contrast to the observations in theprior section that metastructure stiffness variation isnegligibly influenced for the extreme end displacementsas bistable spring span is changed. Yet, for displace-ments within the metastability range, the trends of glo-bal stiffness change are more intricate as linear springstiffness is tailored. For example, as the linear springstiffness k2L increases, the metastable configurationslabeled as AB exhibit broader ranges of end displace-ment having negative or very small positive stiffnessvalues, as shown in Figure 9(d) to (f) by considering theplots from left to right. These trends may be explainedby considering that the upper bound in effective stiff-ness of two springs in series is determined by the weakerlink. Therefore, as the linear spring stiffness k2L

increases, the ‘‘negative stiffness’’ characteristic of thebistable spring constituent has greater influence on themodule-level stiffness profiles, which is reflected in theprofile for the dual-module metastructure.

The second primary trend is that the metastabilityrange decreases as linear spring stiffness k2L increases.This finding is intuitive when one considers the limitingcase in which the linear spring k2L becomes nearly rigid.In this situation, the global end displacement is effec-tively connected directly to the bistable spring internalcoordinate of the second module. Consequently, the




metastability range for the second module will be 0,although there is still the possibility of a finite metast-ability range of the dual-module metastructure but thisdepends on the design parameters of the first moduleaccording to equation (7).

Again using the dynamic model of the dual-modulemetastructure, the energy dissipation per cycle due toslow actuations of the metastructure is seen to decreasefrom 2.37 to 1.25 to 0.60 J as linear spring stiffnessincreases. Since the previous dynamic modeling resultsfound that large hysteresis is a consequence of a largemetastability range, the reduction in the metastabilityrange due to the increase in the linear spring stiffnessplainly reduces the potential for considerable energydissipation.

Experimental parametric study

An experimental dual-module metastructure isemployed to examine the influences of changing theparameters that were investigated in the prior sections:the linear spring stiffness and bistable spring span ofone of the modules. The experimental studies areintended to verify the primary analytical observationsas well as to gather meaningful insight about practicalfactors involved in assembling metastructures from thecurrent proof-of-concept modules.

To begin, a series of experiments are conducted byvarying the bistable spring span of module II while allother parameters of the metastructure remain constant.The span is changed by adjusting the compression dis-tance on the buckled beam in module II. The measuredforce profiles for the four cases of bistable springs used

bistable in module II are presented in Figure 10, show-ing that from case 1 to case 4, the span graduallyincreases. Due to the slight asymmetry observed in theforce–displacement profiles (Figure 10), the experimen-tally determined bistable spring span is quantified asthe average of the two distances from each respectivestable equilibria to the central unstable equilibrium.For completeness, Figure 10 also shows the relation ofthe spring span for module I. In addition, note the fea-ture that the peak force amplitude gradually increasesas the static equilibria span of the bistable spring ofmodule II is increased, as discussed in the analyticalparametric study.

In Figure 11, the end force (top row) and stiffness(bottom row) profiles of the dual-module metastructureare presented for all four cases of bistable spring span,as it is gradually increased from case 1 to case 4

(a) (b) (c)

(d) (e) (f)

Figure 9. Profiles of (a, b, c) end force and (d, e, f) stiffness as global end displacement is varied for the dual-module metastructure.From left to right for both rows, the linear spring stiffness k2L increases.

Figure 10. Force profiles of the bistable springs of modules Iand II as the bistable spring displacement is varied. The dashedcurve corresponds to the bistable spring force profile formodule I, whereas the remaining curves are those employed formodule II. The spans of the bistable springs used for module IIincrease from 1 to 4 as labeled in the plot.

Wu et al. 11



considering the figure columns from left to right. Inagreement with the analytical trends shown in Figure 8,it is experimentally observed that as the bistable springspan of module II increases from cases 1 to 4, the rangeof existence for the metastable configuration AB notice-ably increases. Due to the current experimental con-straints that measurements begin from a statically-stable (zero force) position for each combination ofinternal configurations, the total absence of the ABconfiguration force profiles in Figure 11(a) and (b) maybe caused by the fact that these force profiles do notinclude a statically-stable (zero force) position andtherefore could not be identified experimentally, ratherthan the fact that the entire AB metastable configura-tion does not exist. Yet considering Figure 11 further,also in agreement with analytical findings, it is foundthat the change in bistable spring span does not appre-ciably affect the stiffness of the metastructure for verysmall or very large end displacements, as seen in Figure11(e) to (h): the maximum stiffness for such displace-ments remains around 300 N/m for all cases. However,the increase in the bistable spring span is observed toshift the location of the statically-stable equilibrium ofconfiguration AA to higher values of displacement,from approximately 7.5 to 10 mm as shown in Figure11. Each of these findings is in very good agreementwith the observations from the analytical parametricstudy by employing the mathematical model.

The experiments are then performed to investigatethe influence of changing the linear spring stiffness ofmodule II using the coil spring stiffnesses of 54, 77, 98,and 122 N/m, denoted here as cases 1–4, respectively,while all other metastructure parameters remain thesame. For comparison, the linear spring stiffness ofmodule I is 122 N/m throughout the following evalua-tions. Figure 12 quantifies the trends observed in themeasurements on how the metastability range of thedual-module metastructure is affected as linear coilspring stiffness (solid bars/blue) and bistable springspan (dashed bars/red) of module II are varied. Cases

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 11. Measured profiles of (a, b, c, d) end force and (d, e, f, h) stiffness as global end displacement is varied for the dual-module metastructure. From left to right for both rows, the bistable spring span of module II increases from example 1 (far left) toexample 4 (far right).

Figure 12. Metastability range as linear coil spring stiffness(solid/blue and left-most vertical axis measures) and bistablespring span (dashed/red and right-most vertical axis measures)varies.




1–4 correspond to either increase in linear coil springstiffness or increase in bistable spring span, where thelatter results were considered in full detail viaFigure 11. As depicted in Figure 12, the general trendis that the metastability range of a dual-module metas-tructure increases as the linear spring stiffness decreasesor as the bistable spring span increases. These findingsare in accord with the analytical observations describedin the analytical and experimental parametric studiesand examined in Figures 8 and 9. To maximally har-ness the mechanical properties adaptivity of the metas-tructure, a multitude of coexistent metastable statesmust be realized over a broad range of end displace-ments. The results of the parametric studies indicatethat the linear spring stiffness and the span betweenbistable spring static equilibria are critical toward gov-erning the breadth of the metastability range.Moreover, the findings indicate that even one moduleof a multi-module assembly may play a lead role indetermining the metastructure properties by the adjust-ment of a single-module-level constituent.

Conclusion

To achieve significant adaptivity and multifunctionalityin engineered structures, this research explored a newconcept to create engineered metastructures from theassembly of mechanical, metastable modules. The ideais inspired by recent mechanical study of the micro-and nanoscale composition of skeletal muscle, whichgave evidence that the nanoscale, metastable cross-bridge constituents and their assembly are a criticalsource of the intriguing and favorable properties adap-tivity evident in muscle on the macroscale. The criticalfeatures of the modular platform are the exploitation ofmultiple coexistent metastable states engendered via thestrategic module design, and the adaptable propertiesthat are produced in consequence to switching amongstates. Through analysis and experimentation, it isfound that metastructures are endowed with immedi-ately reversible adaptation of global shape/topologyand the mechanical properties. By the theoretical devel-opment, the potential for adaptivity is seen to growsynergistically according to the increasing number ofassembled modules. The versatile characteristics includeorders of magnitude stiffness change, variable reactionforce amplitude and direction, and tunable energy dissi-pation capacity. Analytical and experimental para-metric studies show that the design parameters of linearspring stiffness and bistable spring span of even onemodule may play a key role toward influencing themodular metastructure global properties adaptivity.

Declaration of conflicting interests

The authors declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Funding

This research was supported by the Air Force Office ofScientific Research (FA9550-13-1-0122) under the administra-tion of Dr David Stargel and by the University of MichiganCollegiate Professorship.

References

Allahverdyan AE and Wang QA (2013) Adaptive machine

and its thermodynamic costs. Physical Review E 87:

032139.Barbarino S, Bilgen O, Ajaj RM, et al. (2011) A review of

morphing aircraft. Journal of Intelligent Material Systems

and Structures 22: 823–877.Bonello P, Brennan MJ and Elliott SJ (2005) Vibration con-

trol using an adaptive tuned vibration absorber with a

variable curvature stiffness element. Smart Materials and

Structures 14: 1055.Caruel M, Allain JM and Truskinovsky L (2013) Muscle as a

metamaterial operating near a critical point. Physical

Review Letters 110: 248103.Caruel M, Allain JM and Truskinovsky L (2015) Mechanics

of collective unfolding. Journal of the Mechanics and Phy-

sics of Solids 76: 237–259.Daynes S, Grisdale A, Seddon A, et al. (2014) Morphing

structures using soft polymers for active deployment.

Smart Materials and Structures 23: 012001.Fok L, Ambati M and Zhang X (2008) Acoustic metamater-

ials. MRS Bulletin 33: 931–934.Gomez JC and Garcia E (2011) Morphing unmanned aerial

vehicles. Smart Materials and Structures 20: 103001.Johnson DR, Thota M, Semperlotti F, et al. (2013) On

achieving high and adaptable damping via a bistable oscil-

lator. Smart Materials and Structures 22: 115027.Klatt T and Haberman MR (2013) A nonlinear negative

stiffness metamaterial unit cell and small-on-large multi-

scale material model. Journal of Applied Physics 114:

033503.I Kovacic and MJ Brennan (eds) (2011) The Duffing Equation:

Nonlinear Oscillators and their Behaviour. Chichester: John

Wiley & Sons.Kuder IK, Arrieta AF, Rathier WE, et al. (2013) Variable

stiffness material and structural concepts for morphing

applications. Progress in Aerospace Sciences 63: 33–55.Lakes RS (2009) Viscoelastic Materials. Cambridge: Cam-

bridge University Press.Lakes RS and Drugan WJ (2002) Dramatically stiffer elastic

composite materials due to a negative stiffness phase? Jour-

nal of the Mechanics and Physics of Solids 50: 979–1009.Lakes RS, Lee T, Bersie A, et al. (2001) Extreme damping in

composite materials with negative-stiffness inclusions.

Nature 410: 565–567.Lan G, Sartori P, Neumann S, et al. (2012) The energy-speed-

accuracy trade-off in sensory adaptation. Nature Physics 8:

422–428.Lindstedt SL, LaStayo PC and Reich TE (2001) When active

muscles lengthen: properties and consequences of eccentric

contractions. News in Physiological Sciences 16: 256–261.Lombardi V, Piazzesi G and Linari M (1992) Rapid regenera-

tion of the actin-myosin power stroke in contracting

muscle. Nature 355: 638–641.

Wu et al. 13



Marcucci L and Truskinovsky L (2010) Mechanics of thepower stroke in myosin II. Physical Review E 81: 051915.

Nicolaou ZG and Motter AE (2012) Mechanical metamater-ials with negative compressibility transitions. Nature Mate-

rials 11: 608–613.Peraza-Hernandez EA, Hartl DJ, Malak RJJr, et al. (2014)

Origami-inspired active structures: a synthesis and review.Smart Materials and Structures 23: 094001.

Philen M (2009) On the applicability of fluidic flexible matrixcomposite variable impedance materials for prosthetic andorthotic devices. Smart Materials and Structures 18:104023.

Pontecorvo ME, Barbarino S, Murray GJ, et al. (2013) Bis-table arches for morphing applications. Journal of Intelli-gent Material Systems and Structures 24: 274–286.

Puglisi G and Truskinovsky L (2000) Mechanics of a discretechain with bi-stable elements. Journal of the Mechanics and

Physics of Solids 48: 1–27.Schultz MR (2008) A concept for airfoil-like active bistable

twisting structures. Journal of Intelligent Material Systems

and Structures 19: 157–169.

Shan Y, Philen M, Lotfi A, et al. (2009) Variable stiffness

structures utilizing fluidic flexible matrix composites. Jour-

nal of Intelligent Material Systems and Structures 20:

443–456.Shimamoto Y, Suzuki M, Mikhailenko SV, et al. (2009) Inter-

sarcomere coordination in muscle revealed through indi-

vidual sarcomere response to quick stretch. Proceedings of

the National Academy of Sciences of the United States of

America 106: 11954–11959.Tolman SS, Delimont IL, Howell LL, et al. (2014) Material

selection for elastic energy absorption in origami-inspired

compliant corrugations. Smart Materials and Structures

23: 094010.Tortora GJ and Derrickson BH (2006) Principles of Anatomy

and Physiology. New York: John Wiley & Sons, Inc.Wu Z, Harne RL and Wang KW (2015) Excitation-induced

stability in a bistable Duffing oscillator: analysis and

experiment. Journal of Computational and Nonlinear

Dynamics 10: 011016.




and structures exploring a modular adaptive metastructure ......skeletal muscle is a remarkable,...

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