Special Issue Article

Journal of Intelligent Material Systemsand Structures2015, Vol. 26(13) 1622–1632� The Author(s) 2014Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1045389X14546781jim.sagepub.com

Passive measurement of progressivemass change via bifurcation sensingwith a multistable micromechanicalsystem

RL Harne and KW Wang

AbstractMass sensing using the onset and crossing of a dynamic bifurcation of a micromechanical system has been shown toreduce the mass threshold which may be detected and exhibited improved robustness to damping and noise comparedto traditional tracking of resonant frequency shift. Previous investigations demonstrated that mass measurement overtime via actively controlled strategies or explored passively operated threshold-type methods to indicate a pre-determined mass was adsorbed. Recently, an alternative idea integrating aspect of frequency shift- and bifurcation-basedmass sensors and methods was proposed, providing initial illustration of a noteworthy ability to passively quantify pro-gressive mass adsorption due to sequentially activated bifurcations. To advance the state of the art, this research pro-vides a thorough investigation of this new sensing concept in terms of its dynamic characteristics and devises guidelinesfor effective, reliable operations. The conceptual foundation of the sensing method and an experimentally validated sen-sor model are reviewed. The results of numerous simulated operational trials and parametric investigations are detailedto reach important conclusions on sensor operations and versatility, and to uncover the influences of key operationalconditions upon detection metrics. Finally, suitable microscale sensor architectures and fabrications are described toexemplify the flexibility of successfully realizing the mass sensing strategy.

KeywordsMass sensing, nonlinear, bifurcation, micromechanical system

Introduction

The opportunities enabled by resonant micro/nanome-chanical systems have stimulated a large body ofresearch aimed at applications as encompassing asmagnetic field detection, gyroscopic orientation, andthe monitoring of mass and force changes. The latterfield of study has various outlets including biologicaland chemical analyte detection (Thundat et al., 1995)and the probing of quantum constituents (Li et al.,2007; Schwab and Roukes, 2005). Changing mass andforce in the measurement environment may be reflectedby a change in the microresonator structural character-istics which thereafter induce a shift in the observednatural frequency, often the fundamental mode (Ekinciet al., 2004). The translation between shift in naturalfrequency and consequent monitored parameter changeis ideally straightforward, but is frequently complicatedby a high degree of sensitivity to noise sources anddamping (Cleland, 2005; Vig and Kim, 1999), require-ment for tracking hardware and interrelated resolution

constraints, and difficulty to directly equate the moni-tored shift to the measured quantity due to responsenonlinearities (Lifshitz and Cross, 2008).

To circumvent the challenges of mass detectionbased upon tracking the resonant frequency peak,recent studies have sought to directly utilize stronglynonlinear phenomena of microscale sensors by exploit-ing bifurcations. Due to the loss of stability of a givenresponse type following small parameter change, a sig-nificant and unambiguous qualitative change in sensorresponse is used to denote a change in sensor mass.The ultimate limits of bifurcation-based sensing sensi-tivity are hinged upon thermomechanical random

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: rharne@umich.edu

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motion, allowing for a theoretical detection resolutionapproaching individual quantum events (Zorin, 1996).For microscale mass sensing, parametric resonance (Liet al., 2014; Turner et al., 2011; Zhang and Turner,2004, 2005) and characteristics of monostable softeningDuffing oscillators (Kumar et al., 2011; Younis andAlsaleem, 2009) have been utilized to activate bifurca-tions. These sensors and sensing strategies have demon-strated reduced sensitivity to noise and dampingcompared to direct peak detection-based approaches(Turner et al., 2011; Zhang and Turner, 2004, 2005)and the opportunity to eliminate tracking hardwareusing one-time threshold-based switches (Kumar et al.,2011; Younis and Alsaleem, 2009). Yet challenges stillremain including an inability to passively control theprobability distribution of triggering thresholdsencountered during mass accumulation sweeps, poten-tial for adverse nonlinear phenomena encountered withprolonged excitation approaching a bifurcation fre-quency (Baesens, 1991; Berglund and Gentz, 2000; Luand Evan-Iwanowski, 1994) which inhibits reliabledetection, and mass measurement over time relies onadditional active hardware. These initial explorationsof bifurcation-based mass sensing indicate great prom-ise, but much remains to be uncovered and improvedupon to fully capitalize on the potential.

To address the existing limitations and enhance sen-sor versatility, the authors recently introduced an alter-native sensor platform and corresponding sensingstrategies that integrate aspects of frequency shift-basedand bifurcation-based mass sensors and methods(Harne and Wang, 2014). The mass sensing strategiesincluded a passive approach and a controlled frequencysweeping approach, each providing important serviceand functionality. A model of the sensor platform wasformulated and validated by experimentation with aproof-of-concept device. By combination of sensordevelopment and sensing strategy, the ability to pas-sively quantify mass adsorption over time was initiallyillustrated. It was conceptually stated that, by the pas-sive mass measurement approach, one can tailor thesystem to address application constraints and meet pre-ferred detection sensitivities. However, in the scope ofthe past research, the passive mass measurement abilitywas presented only via the fundamental illustrationsand not explored to further depth. Numerous experi-mental investigations using the controlled frequencysweeping strategy exemplified its robustness for accu-rate detection of accumulating mass as compared totracking the shifting resonance frequency peak. Finally,using a stochastic model, insight into the influences ofadditive noise interferences on the first jump events wasprovided through direct numerical simulations to assessnon-deterministic sensitivities of the sensing methods.

From the initial evidence and conceptual foundation(Harne and Wang, 2014), it is clear that a significantadvancement in ease of implementation for microscale

mass sensing may be realized via the proposed sensorand sensing strategy that enable passive measurementof progressive mass adsorption. However, apart fromthe broad methodology overview and first exampledemonstrating its feasibility in the authors’ prior study,the potential of the passive mass sensing approachremains otherwise unexplored. Indeed, critical factorsinfluencing the passive mass measurement strategy inits ideal, deterministic operation—including operationalparameter influences, initial condition sensitivities, andpreferred employment schemes and guidelines—werenot evaluated in the prior research. Therefore, toadvance the understanding and realizable achievementof the new mass measurement approach, the aims ofthis research are to provide a thorough investigation ofits characteristics, explore the method’s operational sen-sitivities, and to devise guidelines for its effective andreliable implementation.

In this paper, a brief review of the proposed sensorplatform, the passive operational strategy, and theirmodeling is first presented. These include the key ratio-nales dictating architecture and sensing strategy devel-opment, which are necessary background informationto clarify prior to presentation and discussion of thenew investigations. Then, the results of numerous sen-sing trial simulations are evaluated to uncover keytrends regarding mass detection sensitivity and trialrepeatability with the passive mass measurementapproach. A wide range of operational conditions anddesign parameters are considered to thoroughly probethe sensor and method. From the parametric studies,useful implementation guidelines are presented forensuring the system’s effectiveness and reliability topassively measure mass adsorption over time. Finally,to complement this article’s focus on sensing strategy,discussion is provided on several potential sensor archi-tecture designs and fabrication protocols to successfullyrealize the system on the microscale.

Rationale for sensor architectureand sensing strategy

In this section, for the sake of completeness, a reviewof the new sensor platform and passive mass measure-ment strategy is provided to summarize the conceptualfoundation developed in the previous study (Harne andWang, 2014). As compared to existing bifurcation-based sensing strategies that use mass adsorption toshift a bifurcation frequency such that it coincides withthe excitation frequency, the proposed system uniquelyleads to an excitation level sweep for the bifurcatingcomponent. The change in effective sweeping styleenables a critical ability to activate bifurcations insequence for passive measurement of progressive accu-mulation of mass on the sensor. Although only concep-tually described in the previous work, as will be

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apparent in the later detailed discussions of the findingsof this research, the sensor and method may be favor-ably employed to strategically govern the manner andrate of triggering the bifurcations. This helps to directlytailor the probability distribution of jump events, andtherefore the detection reliability, regardless of massaccumulation rate.

To achieve this flexibility of operation, the sensorarchitecture is a 2-degree-of-freedom (DOF) systemwhere the excited element is a host linear resonatorupon or into which is attached a small, bistable non-linear inclusion. In the sensor embodiment shown inFigure 1(a), the host resonator is an excited microcanti-lever having a buckled inset beam at the host beam freeend. In the authors’ recent study, the inset bistablebeam of a sensor prototype was buckled via repulsivemagnetic forces (Harne and Wang, 2014). Alternativecoupled linear–bistable sensor realizations may be envi-sioned, and greater details regarding suitable micro-scale platforms and fabrication methods are given inthe final section of this paper. However, the essentialcomponents of design require that the host be signifi-cantly more massive than the small bistable inclusionand that the linear natural frequency of the bistableinclusion be much greater than the host structure natu-ral frequency such that the second linear mode naturalfrequency of the coupled system is much greater thanthe fundamental mode resonance.

In contrast to parametric resonators or softeningDuffing oscillators employed by the past studies, a bis-table element is selected as the strongly nonlinear com-ponent due to an increased assurance of bifurcationsacross large range of damping level (Kovacic andBrennan, 2011) and the opportunity to utilize the criti-cal events sequentially. The advantage of the 2-DOFsensor platform is the dynamic coupling between thehost linear resonator and smaller bistable inclusion,which serves to crucially transform mass adsorptionupon the host into an effective excitation level sweep forthe bistable inclusion. Such an effective sweep for thebistable inclusion is necessary to enable sequential acti-vation of bifurcations for passive measurement ofadsorbed mass over time, that is, when the whole sen-sor is driven by unchanging excitations like the baseexcitation in Figure 1(a).

The ability to passively activate sequential bifurca-tions in consequence to steady mass adsorption wasrecently demonstrated by direct numerical simulationusing a coupled linear–bistable sensor architecture formicroscale mass sensing (Harne and Wang, 2014) andby experimentation with a coupled mechanical–electrical oscillator system for structural monitoringapplications (Harne and Wang, 2013). To achieve thisunique sensing functionality, the sensor excitation isdriven at a constant frequency that is less than the fun-damental mode natural frequency prior to mass accu-mulation. The sensor excitation level is selected suchthat the bistable inclusion initially oscillates with smallamplitude intrawell oscillations around one of the sta-ble equilibria. Because the operational strategy onlyconcerns the bandwidth immediately around the funda-mental mode, both bodies of the 2-DOF sensor vibratein phase. Thus, from the perspective of the small bis-table inclusion, the influence of the base excitation leveland host oscillation is comparable to a variable gainexcitation source. The sensor initialization point is rep-resented by point 1 in Figure 2, showing the fundamen-tal mode response transfer function amplitude,H = f½k � v2m(t)�2 + ½vb�2g�1=2, as a function of exci-tation frequency v and increasing modal massm(t)=m+ dm(t) due to accumulating mass dm(t). Thered solid curve in Figure 2 is therefore representative ofthe excitation level up and down sweep for the bistableinclusion induced via steady mass accumulation on thesensor.

As mass adsorbs on the larger host resonator, theeffect is to reduce the fundamental natural frequency,thus gradually amplifying sensor response levelsbecause the resonance approaches the constant sensorexcitation frequency. Eventually, a time is reached atwhich the amplified level is great enough to trigger thebifurcation of the bistable inclusion that activates theunmistakable switch from low amplitude intrawell tohigh amplitude interwell vibrations, represented bypoint 2 in Figure 2. The fine degree of mass adsorption

Figure 1. Schematic of coupled linear–bistablemicromechanical sensor architecture as adapted and redrawn inthe style of (Harne and Wang, 2014): (a) realization via hostmicrocantilever and inset buckled cantilever, (b) schematicshowing coordinate convention used in model, and (c) lumpedparameter representation of (b).

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necessary to push the bistable element across the criticalpoint is the trademark of high sensitivity bifurcation-based mass sensing. However, further steady mass accu-mulation on the sensor may still be monitored by theproposed 2-DOF system. Continuing with the example,eventually mass adsorption reaches the level that thereduced fundamental natural frequency coincides withthe excitation frequency, and effective amplificationgain to the bistable inclusion is maximized, as in point 3in Figure 2. Thereafter, further mass accumulation indi-cates the system is excited above resonance and thusreduces the effective excitation level working on the bis-table element, leading to the bifurcation drop down inresponse amplitude (inter- to intrawell oscillations), asin point 4. Conceptually, the sequential activation ofbifurcations is straightforward when considering theinfluence of the varying fundamental mode responseamplitude as an effective excitation level sweep up andthen down for the bistable inclusion. Such dynamicinfluence is comparable to recent vibration energy har-vesting studies that investigated the addition of‘‘dynamic magnifier’’ resonators to amplify externalexcitation levels to enhance energy harvesting perfor-mance for the attached electromechanical oscillators(Aldraihem and Baz, 2011; Wu et al., 2012; Wu et al.,2014; Zhou et al., 2012). In the sensing context, twobifurcation events observed in sequence are needed toquantify mass over time in a passive manner as will be

detailed in a later section of this paper. This abilitynotably contrasts previous bifurcation-based microscalemass sensing studies that used actively controlled strate-gies to measure progressive mass accumulation (Zhangand Turner, 2005) or employed passive methods thatonly indicate a threshold amount of mass was adsorbed(Kumar et al., 2011; Younis and Alsaleem, 2009). Tobest capture the importance of the novel utility enabledby the proposed sensor, a model of the sensor architec-ture was developed and validated (Harne and Wang,2014). The model formulation is reviewed in the nextsection.

Modeling and governing equations

The governing equations of the sensor platform arederived according to lumped parameter assumptions,which are appropriate in light of the intended operationaround the fundamental mode resonance and theprincipal vibration response of many micromechanicalsystems (Lifshitz and Cross, 2008). The sensor systemis shown in Figure 1(a) indicating the host structuredisplacement relative to base motion, x, and the relativedisplacement between the bistable element and hoststructure, v, as the response coordinates. Thegoverning equations of the lumped parameter systemare derived as

m1(€x+€z)+ b1 _x+ k1x� b2 _v� L9(v)= 0 ð1aÞ

m2(€v+€x+€z)+ b2 _v+L9(v)= 0 ð1bÞ

where mi, bi, and ki are effective mass, damping, andlinear stiffnesses of the sub-systems, respectively, andi = 1,2; L(v) is the potential energy of the bistableelement as function of relative displacement coordinatev; and operators ( � ) and ( )9 denote time and spatialderivatives, respectively. In this study, the double-wellpotential energy profile is utilized for the bistable inclu-sion which represents a variety of microscale bistablesystem potentials when considering fundamental buck-ling mode response (Harne and Wang, 2014; Krylovet al., 2011; Ruzziconi et al., 2013)

L(v)= � 1

2k2Lv2 +

1

4k2NLv4 ð2Þ

The spatial derivative of the potential energy,L9= ∂L=∂v, is the restoring force from which may becomputed the fixed points of the bistable sub-system:L9(v)= 0! v�=6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2L=k2NL

p. Here, v� are the

stable equilibria that are symmetric with respect to acentral unstable configuration. A coordinate transfor-mation is then applied by defining y= v� v� such thatthe bistable inclusion response coordinate y is 0 at oneof the stable positions, as indicated in schematics ofFigure 1(b) and (c). Substitution of the transformation

Figure 2. Concept of sequential bifurcation activation inconsequence to fundamental mode response transfer functionamplitude shifting due to mass adsorbing over time, adapted andredrawn in the style of Harne and Wang (2014). At a constantexcitation frequency, the response amplitude H changes becausethe natural frequency reduces due to progressive massaccumulation. Thus, at constant excitation frequency, the sensorresponse is amplified to varying levels depending on the amountof adsorbed mass.

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into equation system (1) and following simplificationleads to equation system (3)

m1€x+ b1 _x+ k1x� b2 _y� k2y½1+ay+by2�= �m1€z

ð3aÞ

m2€y+ ð1+ ðm2=m1ÞÞ½b2 _y+ k2y(1+ay+by2)�

� m2

m1

� �(b1 _x+ k1x)= 0

ð3bÞ

Here, k2 = 2k2L, while a= 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2NL=k2L

pand

b= k2NL=k2L are stiffness proportionality constants,and �€z= Za sinOt is the harmonic base acceleration ofamplitude Za and frequency O. In the event of practicaland oftentimes unavoidable bistable inclusionasymmetries (Harne and Wang, 2014; Krylov et al.,2011; Ruzziconi et al., 2013), proportionality constantsa and b may be modified from the prior definitionssuch that the equilibria are positioned atv�= ½�a6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � 4b

p�=2b and at v�= 0.

Having reviewed the sensor platform composition,the dynamic concept behind the passive sensing strat-egy, and the modeling formulation, the following sec-tions present the new investigations performed toadvance the understanding of the proposed passivemass measurement system through explorations of itssensitivities. From these new insights, preferred designand implementation guidelines may be devised, whichassist in latter discussions regarding realizable sensorarchitecture fabrications.

System operation studies

In this section, simulated demonstrations of the sensorimplementation are provided to observe key operationalparameter influences and overall functionalities. In the fol-lowing simulations, for simplicity it is assumed that massis adsorbed only upon the larger host structure such thatthe time-varying mass of the host is �m1(t)=m1 + dm(t).Figure 3 plots computed velocity responses for variousconstant excitation conditions as mass accumulates at alinear rate of 0.001 mass ratio (mr) per second, where mris the ratio of adsorbed mass dm(t) to primary linear struc-ture mass m1. The results are calculated by numericallyintegrating equation system (3) in MATLAB. Simulationsin rows (a, b) and (c, d) are conducted using base accelera-tion levels of 5 and 7 m/s2, respectively. In column (a, c),the system is excited at 99.4% of the baseline fundamentalmode natural frequency f1 prior to mass accumulation,while in column (b, d) the excitation frequency is 99.5%of f1. Other parameters used in simulation are given inTable 1.

The unique elements and dynamic sensing eventsdescribed conceptually earlier in this work are all evi-dent in the simulation results shown in Figure 3: pro-gressive amplification and then reduction in sensorresponse due to the fundamental resonance shifting

downward in frequency because of mass adsorption,the activation of a low-to-high amplitude (intra- tointerwell) response bifurcation for the bistable inclu-sion, and the later activation of an inter-to-intrawellbifurcation down in amplitude. When the highamplitude response is activated, there is a sudden andapparent increase in bistable inclusion velocity. Duringthe period when the energetic oscillations are activated,both DOF of the sensor exhibit highly perturbedresponses due to the strongly nonlinear interwell vibra-tions of the bistable inclusion. In comparing Figure3(a) to (b) and (c) to (d), change in constant excitationfrequency is seen to primarily shift the points in time atwhich the bifurcations occur but does not notablychange the span of time elapsed between jumps. In con-trast, changing overall sensor excitation level Za, asshown comparing Figure 3(a) to (c) and (b) to (d),reveals that the span of time between jumps is adjusted;specifically, as level increases, the time elapsed betweenbifurcations increases, and vice versa.

Because two bifurcation events occur in sequence,net mass accumulation may be determined usingknowledge of time elapsed between jumps and theknown mass accumulation rate. In Figure 3, thesequantities are provided as Dmr values. Outside of con-trolled laboratory experimentation, accumulation ratewould likely be unknown, and thus a bifurcation analy-sis of equation system (3) is needed to derive closed-form expressions for the critical mass quantitiesrequired, using a given excitation parameter set, to acti-vate each bifurcation event. Comparable bifurcationanalyses were utilized by prior bifurcation-based sen-sing studies for mass detection (Zhang et al., 2003;Zhang and Turner, 2005) as well as force measurement(Hassanpour et al., 2011) to determine expressionsrelated to their representative sensor architectures.Likewise, for the present sensor platform, such analysisrepresents a 2-DOF example of the 1-DOF derivationconducted in a prior study by the authors in the contextof vibration energy harvesting with bistable systems(Harne et al., 2013). With derived expressions for thecritical mass values to induce the bifurcations, theoccurrence of both jumps in sequence passively indi-cates an amount of mass was adsorbed in the span oftime which elapsed such that both bifurcations weretriggered. As shown in studying trends of Figure 3,selection of constant excitation frequency is the meansto control the initial nearness to bifurcation (and hencethe ultimate sensitivity to a pre-determined change insensor mass), while the constant sensor excitation leveltailors the time elapsed between jumps (and hence theamount of mass that is measured). Since the excitationconditions are selected by the user or designer of thesensor, these critical mass sensing factors may bedirectly adjusted for the specific application of interest.These insights were not obtainable through the investi-gations of the past study by Harne and Wang (2014)

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and suggest further comprehensive explorations mayuncover additional critical abilities and sensitivities ofthe passive mass measurement strategy that may lead tothe development of valuable design guidelines.

Investigations of operational parameterinfluences

Observations from the examples in Figure 3 suggestgeneral trends related to changing sensor excitation fre-quency and level. However, a more deliberate assess-ment is required to develop well-supported conclusions

regarding such dependencies and to determine usefulguidelines for effective passive measurements of pro-gressive mass adsorption. To achieve these objectives,the following studies assess the critical influences ofexcitation conditions in addition to the accumulationrate of the monitored analyte mass.

Sensor excitation frequency and level

Figure 4 presents results of changing the constant exci-tation frequency in terms of the final amount of addedmass ratio that is measured, for a wide range of sensorexcitation levels and fixed mass ratio adsorption rate of0.001. The minimum excitation level was selected toensure that sequential bifurcations were activated in allsimulated trials. Each data point represents the meanabsorbed mass ratio value determined from 40 simu-lated trials, while bars are first standard deviations.Random initial conditions are employed for each trial.Both trends observed in Figure 3 regarding constantexcitation frequency and level are consistently sup-ported throughout the results of Figure 4. Change in

Figure 3. Velocity responses due to steadily increasing mass accumulation upon sensor host structure with excitation at (a, c)99.4% of the baseline fundamental mode resonance f1 prior to mass accumulation, and (b, d) 99.5% of f1. Row (a, b) at Za = 5 m/s2

base acceleration level, and row (c, d) at Za = 7 m/s2.

Table 1. Simulation parameters.

m1 (ng) m2 (ng) b1 (nN s/m) b2 (nN s/m)

5 0.1 471 18.2

k1 (N/m) k2 (N/m) a (1/m) b (1/m2)

7.11 0.299 4.57 3 105 4.64 3 1010

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excitation frequency has little influence in modifyingthe amount of added mass measured, whereas the exci-tation level plays a clear role in adjusting the value.However, the results show deviation around a meanvalue, and the adsorbed mass amount is not entirelyinsensitive to change in excitation frequency.

These findings reflect the initial condition depen-dence of strongly nonlinear systems and the non-stationarity of the system (mass changing over time),which lead to non-uniform activation of the bifurca-tions from trial to trial. The deviation around meanadsorbed mass values is seen to increase as the excita-tion level increases. This result is intuitive when consid-ering the conceptual framework of activating thebifurcations using the 2-DOF sensor architecture. Asillustrated in Figure 2, the shifting fundamental reso-nance leads to amplification and then reduction in thesensor response at a fixed excitation frequency. Sincemass accumulates at a linear rate in the simulatedtrials, every time instant corresponds to a specific valueof ∂H=∂m according to the slope of the fundamentalmode resonance, as shown in Figure 2. By increasingsensor excitation levels, the representative bifurcationpoints 2 and 4 in Figure 2 effectively move down thesurface contour, which means that smaller rates ∂H=∂m

activate the critical events. Because prolonged steady-state oscillation close to a bifurcation may lead to spur-ious switching events due to the initial conditionsensitivity of strongly nonlinear systems (Baesens,1991; Berglund and Gentz, 2000; Lu and Evan-Iwanowski, 1994) and because the saddle-node bifurca-tion here utilized may exhibit fractal basin of attraction

erosion as the system passes through bifurcation(Soliman and Thompson, 1989), the ‘‘shallower’’ slopes∂H=∂m caused by too high of excitation levels are notconducive to repeatable triggering. Thus, Figure 4shows that increasing excitation level produces largerdeviations of quantified mass value as compared to thesmaller excitation levels which take advantage of the‘‘steeper’’ slopes of the shifting fundamental resonanceto more reliably activate bifurcations from the sameincrement of adsorbed mass over time. This findingindicates a clear means to directly ensure that the sen-sing strategy maintains viability by modification of thesensor excitation level. This is a versatility for passivebifurcation-based microscale mass sensing otherwiseachieved only by employing actively controlledapproaches to date (Burgner et al., 2010).

Mass adsorption rate

Because mass accumulates at a finite rate in practicalmass sensing contexts, the sensor response is non-sta-tionary. This factor is important because the specificparameters of periodic non-stationary excitations havea strong influence on the outcomes of both nonlinearand linear system responses (Cronin, 1966; Lewis, 1932;Lu and Evan-Iwanowski, 1994). Consequently, sensorsensitivity to change in adsorption rate must be charac-terized for important design and operating guidelines.Change in adsorption rate represents practical micro-scale mass sensing scenarios such as rapid diffusion of anoxious gas or slow particle accumulation to evaluateultimate detection sensitivity in the laboratory. Thus, awide range of rates should be considered to thoroughlyassess the proposed sensor architecture and sensingstrategy.

Figure 5 plots results of the measured adsorbed massratio as function of the accumulation rate. Each datapoint is the mean value for a given excitation level com-puted from 40 simulated trials across a range of fiveexcitation frequencies: thus each data point is the meanfrom the results of 200 simulations. The bars are firststandard deviations. Ideally, variation in adsorptionrate should not influence the ultimate amount of massthat accumulates on the sensor spanning the sequentialbifurcations. However, Figure 5 indicates that acrossthe range of excitation levels, there is some degree ofsensitivity to adsorption rate changes. In general, thefaster the rates, the greater the deviation of results.These trends become more apparent for increased exci-tation levels. Non-stationary passage through bifurca-tion is a process prone to variability, but as higherexcitation levels drive the sensor to greater degrees ofstrongly nonlinear response, the sensitivity to the ulti-mate point at which the bifurcations occur is likewiseheightened. The trend for higher excitation levels andfaster adsorption rates is an increase in the adsorbedmass that is measured. Recall that from the perspective

Figure 4. Measured adsorbed mass ratio as function ofconstant excitation frequency and level. Mass ratio accumulationrate for all trials is 0.001. Data points are mean values across 40simulated trials; bars are first standard deviations.

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of the small bistable inclusion, the effect of mass accu-mulation is like an effective excitation level sweep upand down to sequentially induce the bistable elementbifurcations. The prior trend of increasing measuredmass as excitation level and absorption rate bothincrease is explained because the bifurcation dropdownward from inter- to intrawell oscillations is moresensitive to initial conditions during non-stationaryexcitation level sweeps, leading to delayed or deferredbifurcations, than is the opposite bifurcation upward inresponse amplitude (Lu and Evan-Iwanowski, 1994).Thus, results of Figure 5 reveal that for the highest exci-tation levels, the sensor remains activated in the inter-well response for too long and consequently quantifiesan erroneous increasing amount of accumulated mass.Uncovering these important trends helps to determinesensor excitation levels desirable to minimize adverseinfluence of non-stationarity on the reliability of thepassive mass measurement strategy.

Feasibility of detection

A final evaluation is valuable in light of sensor datareadout constraints. Depending on the fabrication pro-cess selected for the microscale sensor, it could be chal-lenging to electrically isolate the bistable inclusion fromthe host so as to strictly measure the response of thesmaller inclusion. Thus, it may be necessary to rely onresponse measurements acquired solely from the hostresonator. Therefore, this section evaluates the feasibil-ity of monitoring only the host linear resonator velocityresponse, which, it is assumed, will be measured via

one of many common transduction methods propor-tional to velocity, for example, optical, piezoresistivity,etc. Specifically, the metric of performance used here isthe ratio of variance of the host structure squared velo-city during the energetic interwell vibration period tothat measured before and after the jumps (the intrawelloscillations). Thus, larger values of this metric indicatethat detection of the bifurcation events is more easilyobserved.

Figure 6 presents results of the mean values of var-iance ratio determined from five constant excitationfrequencies using 40 simulated runs each, across arange of excitation levels and mass adsorption rates.The results show an undesired variance ratio minimiza-tion for excitation level of 6.5 m/s2, an explanation ofwhich is not immediately apparent. However, moreimportantly, the smaller excitation levels yield thegreatest variance ratios beneficial for bifurcation detec-tion, particularly for the slower mass ratio adsorptionrates. The variance ratios for the smaller excitation lev-els reduce as mass adsorption rate increases but stillremain greater than that provided by higher excitationlevels.

Recalling that the smallest excitation level in thesimulated studies was selected because it ensured thatboth bifurcations were activated at some point duringthe sensing trials, the results of Figures 4 to 6 suggestthat the smallest excitation level usable to activate bothjump events is the most beneficial for the proposed sen-sor architecture and sensing strategy. Through theparametric investigations, it is seen that the smallestexcitation level (a) leads to the least deviation aroundthe mean value for the amount of adsorbed mass that

Figure 5. Measured adsorbed mass ratio as function of massratio adsorption rate and excitation level. Data points are meanvalues across five sets of excitation frequency each using 40simulated trials; bars are first standard deviations.

Figure 6. Ratio of host oscillator velocity2j _xj2 variance duringenergetic response period to before/after as function of massratio adsorption rate and excitation level. Data points are meanvalues across five sets of excitation frequency each using 40simulated trials.

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is measured, (b) shows reduced sensitivity to adversechange in measured mass as accumulation ratechanges, and moreover (c) leads to the most useful hostlinear resonator variance ratio for mass detection fromthe more easily acquired measurements of thelarger host resonator. The results, therefore, set valu-able guidelines for implementation of the proposed sen-sor to ensure its operation remains effective andreliable.

Sensor embodiments and fabricationstrategies

The coupled linear–bistable sensor in Figure 1(a) wasrealized on the mesoscale by the authors to provideexperimental proof-of-concept demonstrations andmodel validations (Harne and Wang, 2014). While theprimary aims of this study have been to thoroughlyinvestigate the passive mass sensing strategy and deviseguidelines for its effective and reliable operation, it isvaluable to provide complementary discussion onmicroscale sensor embodiments and fabrication meth-ods that may be employed to satisfy the general archi-tecture requirements.

Developments in two fields—atomic force micro-scopy (AFM) and vibration energy harvesting—together demonstrate a suite of opportunities to realizethe coupled linear–bistable sensor platform shown inFigure 1(a) using coupled microcantilevers. With theaims of enhancing AFM sensitivity and topographicalreconstruction resolution, microcantilever-in-cantileversensors have been fabricated where the oscillator inclu-sions have been machined facing, opposing, and ortho-gonal to the free end of the host cantilever (Loganathanand Bristow, 2014; Sarioglu et al., 2012; Shaik et al.,2014). Such designs would meet the elastic and geo-metric requirements of the proposed linear–bistablesensor in Figure 1(a), but magnet deposition would alsobe needed to induce bistability. Recent studies in vibra-tion energy harvesting using energetic bistable systemshave provided detailed fabrication protocols anddemonstrated successful deposition of repulsive perma-nent magnets on microcantilever ends (Ando et al.,2010; Emery, 2014). Together, these sensor designs andfabrication methods meet the architecture requirementsof the proposed sensor in Figure 1(a).

Alternative sensor platforms may be envisioned,which satisfy the coupled linear–bistable design frame-work. For example, Figure 7 is a schematic for coupledin-plane oscillators, where an electrostatically actuatedhost oscillator contains an inset curved beam, and thus,bistability is realized by the geometry itself rather thanelectrostatic, magnetic, or otherwise multi-physicsinteractions. Such configuration would employ compa-rable etching procedures as a recently realized large-stroke actuator, which in fact utilized serially connected

bistable curved beams with an inset, actuated mass(Gerson et al., 2012). In spite of the opposite configura-tional order, the success of fabricating microscale, seri-ally connected linear and bistable in-plane oscillatorssupports the feasibility of the proposed schematic inFigure 7. Moreover, by utilization of a geometricallybistable inclusion—that is, bistability by mechanics—one may alleviate intricate coupling concerns related toachieving bistability via electromechanical factors(Abu-Salih and Elata, 2006; Elata and Abu-Salih,2005; Hassanpour et al., 2011) or other multi-physicsmethods generally. These examples provide evidencethat successfully realizing the passive mass sensingstrategy on the microscale is not constrained to onespecific sensor architecture and may be employedaccording to fabrication expertise and packaging pre-ferences, be they for out-of-plane cantilever vibrationsor in-plane shuttle oscillator dynamics.

Conclusion

A 2-DOF sensor architecture and sensing strategy havebeen proposed to advance the performance and versati-lity of microscale bifurcation-based mass sensing.Together, the sensor and sensing approach exploit thesequential activation of a bistable element’s bifurca-tions, thus enabling a novel ability to passively measureadsorbed mass over time. The research reported in thispaper provides a thorough investigation of this impor-tant ability and devises guidelines for its effective andreliable operation. Through the studies, it is found thatthe sensor and sensing strategy provide unique meansto adjust critical sensor sensitivities for maintainingmeasurement consistency regardless of mass adsorptionrate. By tailoring operating conditions, the specificamount of accumulated mass measured over time maybe directly adjusted and effective sensor readout usingonly the larger host structure may be ensured.Moreover, the sensor architecture requirements are metby multiple platform designs, exemplifying flexibilityin implementing the sensing strategy based upon

Figure 7. Schematic of coupled linear–bistable sensorarchitecture employing in-plane dynamics of electrostaticallyactuated, host comb-drive shuttle oscillator and inset curvedbeam.

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fabrication expertise and preference. The high degreeof operational versatility evidenced by the investiga-tions of this research sets the proposed system apartfrom previous bifurcation-based microscale mass sen-sing approaches.

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 in part by the University ofMichigan Collegiate Professorship fund.

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