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JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2011 157

Manipulating Vibration EnergyConfinement in Electrically Coupled

Microelectromechanical Resonator ArraysPradyumna Thiruvenkatanathan, Student Member, IEEE, Jim Woodhouse,

Jize Yan, Member, IEEE, and Ashwin A. Seshia, Senior Member, IEEE

Abstract—This paper reports the first detailed experimentalevidence of the phenomena of eigenvalue loci veering and vibra-tion mode localization in microelectromechanical resonator arrayssubjected to weak electroelastic coupling. A rapid but continuousinterchange of the eigenfunctions associated with the eigenvaluesis experimentally observed during veering as the variations in theeigenvalues are studied for induced stiffness variations on one ofthe coupled resonators. It is also noticed that the electrical tunabil-ity of the coupling spring constant in such microsystems enables amanipulation of the severity of modal interchange during veeringand in consequence, the extent of energy confinement within thesystem. These results, while experimentally confirming the elasticbehavior of such electrical coupling elements, also suggest thatsuch microsystems provide a unique platform for investigatingthe general nature and properties of these dynamic phenomenaunder significantly weaker tunable coupling spring constants thatare very difficult to implement in corresponding “macroscopic”systems. [2010-0105]

Index Terms—Coupled resonators, curve veering, MEMSfilters, mode-localization, mode-localized sensors.

I. INTRODUCTION

I T IS WELL known that weakly coupled nearly identicalresonators exhibit an abrupt divergence of the loci of their

eigenvalues when these are plotted against a parameter repre-senting the symmetry-breaking disorder in the system [1], [2].This phenomenon is often referred to as “eigenvalue curve veer-ing” and was first observed by Leissa [3]. The eigenfunctionsassociated with the eigenvalues on each locus swap trajectoriesduring veering, resulting in strong mode localization underconditions of weak internal coupling between the resonators[1]–[3]. It is also known that in an array of weakly couplednearly identical resonators, the presence of even very smallperiodicity-breaking structural irregularities inhibits the long-

Manuscript received April 14, 2010; revised July 15, 2010; acceptedOctober 7, 2010. Date of publication December 10, 2010; date of currentversion February 2, 2011. This work was supported in part by the BritishCouncil under UKIERI Grant SA06–250, in part by Churchill College, and inpart by the Cambridge Commonwealth Trust. Subject Editor D. Elata.

P. Thiruvenkatanathan, J. Yan, and A. A. Seshia are with the Department ofEngineering, University of Cambridge, Cambridge, CB2 1PZ, U.K., and alsowith the Nanoscience Center, University of Cambridge, Cambridge, CB3 0FF,U.K. (e-mail: [email protected]).

J. Woodhouse is with the Department of Engineering, University ofCambridge, Cambridge, CB2 1PZ, U.K.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JMEMS.2010.2090501

Fig. 1. Discrete-element model of two coupled oscillators.

range propagation of vibration, leading to confinement of vi-bration energy to small geometric regions. This phenomenonwas first predicted and analyzed in the field of solid statephysics [4] and later applied to the areas of structural dynamicsand vibration acoustics [5], [6]. It has been shown that theexistence of close eigenvalues in a coupled vibratory system islikely to cause the occurrence of both curve veering and modelocalization suggesting that they are closely related phenomena[1], [7]–[10].

This paper reports the first experimental evidence of the veer-ing theory in nearly identical microscopic mechanical resonatorarrays, which unlike any previous studies, are coupled throughelectroelastic coupling springs. Such coupling elements, unliketheir mechanical counterparts, possess voltage-tunable negativeeffective spring constants that allow the veering zone to beconstructed by plotting the eigenvalues as a function of aninduced periodicity-breaking structural disorder. This also en-ables manipulation of the rapidity of eigenfunction interchangeduring veering and in consequence, control over the extent ofenergy confinement within the system for a given structuralperturbation. Such enhanced tunability in coupled dynamicsoffers significant engineering benefits in the design of weaklycoupled resonator configurations that find applicability as nar-row bandwidth mechanical filters for signal processing [11]and as mode-localized sensors for highly sensitive monitoringof small induced structural perturbations in micromechanicalresonators [12]. Furthermore, these electrically coupled micro-electromechanical (MEM) resonators may also be employed asa unique testing ground for studying the fundamental limits ofthe veering theory under very weak elastic coupling which isdifficult to implement in larger (macroscopic) systems.

II. THEORY

In order to understand the underlying physics behind veering,consider a system consisting of two undamped coupled oscilla-tors, the lumped-element model of which is shown in Fig. 1.

1057-7157/$26.00 © 2010 IEEE

158 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2011

Fig. 2. Loci of the dimensionless eigenvalues of the two coupled oscillatorsin terms of the disorder δ.

In Fig. 1, m represents the mass of both resonators 1 and 2;k1 and k2(= k1(1 + δ)) represent their respective stiffnesses;kc is the internal coupling spring constant; and x1 and x2 aretheir displacements. The two most important parameters are thenondimensionalized coupling factor (κ = kc/k1) between thetwo oscillators and the nondimensionalized stiffness disorderof resonator 2 relative to resonator 1 (δ). The variations in theeigenvalues and eigenstates of the system are studied relative tothe disorder δ. The eigenvalue problem of the system shown inFig. 1 may be expressed as∣∣∣∣ β1 − λ −κ

−κ β2 − λ

∣∣∣∣ = 0 (1)

where β1 = 1 + κ; β2 = 1 + κ + δ and λ = mω2/k1. Theeigenvalues are

λ =12

{β1 + β2 ±

√(β1 − β2)2 + 4κ2

}(2)

with corresponding eigenvector components given by

x2

x1=

1 + κ − λ

κ. (3)

When δ = 0; (β1 = β2), the system is tuned or ordered andthe eigenvalues and the corresponding eigenstates of the systemmay be expressed as

λ01 = 1,x2

x1= +1; λ02 = 1 + 2κ,

x2

x1= −1. (4)

Plotting the loci of the two eigenvalues of the system (λ)versus δ under conditions of weak internal coupling (κ � 1)yields Fig. 2.

As the loci of the two eigenvalues approach each other, theydo not cross but rather diverge abruptly. This is because theeigenvalue problem represented by (1) does not allow degener-ate modes so long as there exists a spring coupling betweenthe two resonators, which consequently results in a mutualrepulsion of their eigenvalue loci as they converge. This pointof divergence or “veering” corresponds to the point at which thesystem is perfectly symmetric. When (β1 − β2)2 � (4κ2), thevibration energy gets spatially confined leading to a localizationof the mode shapes as seen from (3) and (2) and shown inFig. 2. Weaker coupling springs result in higher local curvaturesand narrower veering zones as shown in Fig. 3, indicating an

Fig. 3. Loci of the dimensionless eigenvalues of the two coupled oscillatorsin terms of the disorder δ at varying strengths of internal coupling.

increase in the severity of the modal parameter interchangeduring veering for the same symmetry-breaking perturbation instiffness.

The description so far has assumed the coupling to havea spring constant that is positive as would be the case fora purely mechanical coupling element, but the microsystemsconsidered in this paper enable the realization of structuresthat may be electrically coupled [11], [13], [14]. When twomicromechanical resonators separated by a small coupling gapare subjected to different dc polarization voltages, the dis-placement dependent component of the attractive electrostaticforce between the two resonators should result in a spring-like behavior [11]. However, the direction of this displacement-dependent component is opposite to the force and hence shouldresult in an electroelastic spring with a negative effective springconstant [11]. If ΔV represents the difference in dc polarizationvoltages of two closely spaced resonators and C the couplingcapacitance between the two elements, the electrostatic forceof attraction may be expressed as

F =∂(P.E.)

∂x≈ (ΔV 2)

2C0

g

[1 + 2

x

g

](5)

where C0 = ε0A/g represents the initial parallel plate capaci-tance between the resonators; P.E. = C(ΔV 2)/2 denotes theelectrical potential energy stored in the capacitor and x, a smalldisplacement along the coupling gap g. Hence, by analogy withHooke’s law, the electroelastic stiffness of this coupling elementbetween the two resonators may be expressed as

kc = − (ΔV 2)ε0A

g3. (6)

The negative value of the coupling element should signif-icantly alter the coupled dynamics of the system as it wouldinvert the modal behavior of the system at the tuned condition[refer to (4)]. Nevertheless, the phenomena of frequency veer-ing and mode localization should still occur in a very similarmanner. However, due to the significantly weaker couplingspring constant that may be realized using such electroelasticcoupling springs, the local curvature of the eigenvalue lociduring veering could be made considerably higher based on(2), resulting in very narrow eigenvalue veering zone widthsand correspondingly strong sensitivity to the magnitude of

THIRUVENKATANATHAN et al.: MANIPULATING VIBRATION ENERGY CONFINEMENT IN MEM RESONATOR ARRAYS 159

Fig. 4. SEM image of the electrically coupled DETF resonators. Ports 0 and3 serve as the drive and tuning ports, respectively, while ports 1 and 2 areemployed to apply bias voltages on the two resonators in order to induce anelectrical coupling spring between the resonators.

mistuning. Furthermore, since the spring constant of such cou-pling elements may be tuned by simply altering the potentialdifference between the coupled microstructures [refer to (6)],they should allow “voltage-tunable electromechanical veeringzone widths” that enable control over the sensitivity of thesystem. This is very difficult to realize in resonators coupledmechanically.

III. EXPERIMENT

To demonstrate the concepts described earlier, coupledelectromechanical resonators were fabricated in a commercialfoundry process through MEMSCAP Inc., USA. The chosendevice topology consists of two nearly identical double-endedtuning fork (DETF) MEM resonators that are separated by asmall capacitive coupling gap of approximately 2.5 μm. Eachof the tines in both tuning forks was designed to be 10 μm thick,300 μm long, and 6 μm wide. A SEM image of the device isshown in Fig. 4.

There are four electrical ports in the system considered (referFig. 4). Ports 1 and 2 may be utilized to apply different dcpolarization voltages on resonators 2 and 1, respectively. Anydifference between the applied dc voltages would result in apotential difference across the coupling gap thereby enablingthe realization of an electroelastic spring as explained in theprevious section. The resonators were driven and sensed usingcapacitive transduction. Actuation was achieved using parallelplates of equal dimensions (260 μm long, 6 μm wide, and10 μm thick), attached to either side of each resonator, as shownin Fig. 4. Each of the tuning forks was actuated in the out-of-phase mode of vibration as the motion in this mode of theresonance serves to minimize the stresses at the anchors therebyenhancing the quality factor of resonance. The fabricated de-vices were tested under partial vacuum (≈10 mtorr) in a customvacuum chamber. Fig. 5 shows a schematic of the experimentalmeasurement setup.

Initially, an input source power of −30 dBm was applied onport 0 along with a dc polarization voltage of +8 and −8 Von resonators 1 and 2 through ports 2 and 1, respectively,

Fig. 5. Schematic of measurement setup used for the characterization of theelectrically coupled resonator pairs.

while maintaining a dc bias voltage of 0 V on both ports 0and 3. Fig. 6 shows the transfer function responses (extractedafter parasitic capacitance cancellation using the procedureexplained in [15]) measured from the two resonators. Themeasured response follows that of a coupled two-degree-of-freedom oscillator configuration, indicating the presence of aweak coupling spring between the two resonators. A closerlook at the modal dynamics indicates an inversion of the eigen-functions at the two eigenvalues as predicted by (6) (relativeto systems coupled with a mechanical coupling element) asthe out-of-phase mode occurs at a lower frequency than thein-phase mode, as shown by the Nyquist circles formed byplotting the real versus imaginary parts of the transfer functionsobserved at the two modes of vibration (refer to Fig. 7) [16].

The applied potential difference of 16 V between thetwo resonators should result in the formation of a couplingspring with a negative effective spring constant (of magnitude−0.3762 N/m) [refer to (6)] without altering the structuralperiodicity of the system. In order to further confirm the elasticbehavior of this electrical coupling element, the stiffness ofresonator 1 was altered relative to that of resonator 2 using theelectrostatic spring softening effect (by changing the dc biasvoltage on port 3 while maintaining that on port 0 at 0 V).Plotting the variations in the two eigenvalues as a function ofthe induced stiffness perturbations on resonator 1 resulted inFig. 8. Veering behavior is clearly seen, confirming the presenceof an equivalent electroelastic coupling spring. Furthermore,as the potential difference between the two resonators waslowered, the width of this transition zone or the “veeringneck” was observed to get narrower as shown in Fig. 8. Thisresult demonstrates the tunability of the effective electroelasticcoupling in such elements.

It may be noticed that (4) may be utilized to experimentallyquantify the strength of internal coupling for different magni-tudes of induced potential difference between the resonatorsand test the theoretical estimate of the effective stiffness (thisstiffness being very difficult to measure accurately otherwise).Comparing the spring constant hence estimated with theoret-ical predictions obtained from (6) yielded Fig. 9. The pro-portional disparity observed between the measured coupling


Fig. 6. Extracted transfer funtion measurements from resonators (a) 2 and (b) 1, respectively.

Fig. 7. Nyquist circles obtained by plotting the real part of the extracted transfer function versus the imaginary part at the two fundamental modes of vibrationof the coupled vibratory system. (a) indicates the out-of-phase mode occuring at the first eigenvalue and (b) indicates the occurrence of an in-phase mode at thesecond eigenvalue.

Fig. 8. Experimentally measured variations in the eigenvalues for inducedstiffness variations in resonator 1.

spring constant and the theoretical estimates remains nearlyconstant under varying strengths of internal coupling betweenthe resonators, indicating that the error probably arises frommanufacturing tolerances.

The voltage tunability of the strength of electroelastic cou-pling elements in such systems suggests that the spring constantmay be tuned to very low values consequently enabling the

Fig. 9. Comparison between spring constant estimated derived theoreticallyand those estimated from the veering zone widths.

investigation of the ultimate limits of the veering theory. Areduction in the coupling spring constant would, however,significantly impact the bandwidth of the coupled response asweaker coupling should result in narrower bandwidths basedon (4). This was experimentally studied by investigating thecoupled dynamics of the system under varying electrical cou-pling spring constants as shown in Fig. 10(a) and (b). Suchhigh tunability offers significant benefits in weakly coupled


Fig. 10. Variations in the bandwidth of the coupled response for induced coupling spring constant variations.

Fig. 11. Improvement in signal-to-noise by symmetrically increasing theactuation force on both resonators through the application of higher dc voltageson ports 0 and 3 (+7 and −7 V, respectively).

resonator configurations that find applicability in the design ofmechanical narrow bandwidth filters [11] or sensors. However,it is to be noted that lowering the bias voltages on the resonatorsalso reduces the effective signal-to-noise due to the reducedstrength of electromechanical transduction, resulting in noisiersignals at lower voltages.

Applying an ac drive voltage superimposed on higher dcvoltages at both ports 0 and 3, while maintaining the potentialdifference between the resonators (through ports 1 and 2),allows improved electromechanical transduction of resonatormotion without altering the effective strength of coupling be-tween the mechanical resonators. Applying equal magnitudesbut opposite polarities of dc polarization voltages throughports 0 and 3 would also allow for the spring constant of the twomechanical resonators to be softened to the same extent to firstorder thereby conserving the initial periodicity and modal be-havior of the system. This was experimentally verified by com-paring the response of the system when applying 7 and −7 Von ports 0 and 3, and when applying 0 V on the two ports, whilecoupling the two resonators with a potential difference of 4 Vas shown in Fig. 11.

In order to investigate the existence of curve veering andmode localization under very weak coupling, a potential dif-

Fig. 12. Experimentally observed variations in the eigenvalues plotted as afunction of the electrostatically induced stiffness disorder on resonator 2. Asthe loci of the eigenvalues approach each other, they never intersect but veerabruptly.

ference between the resonators corresponding to 4 V (+2 Von port 1 and −2 V on port 2) was applied between the tworesonators with an ac drive on both ports 0 and 3. A study ofthe variations in the eigenvalues for electrostatically inducedstiffness perturbations on resonator 2 (through dc voltage tuningin port 0) indicated the presence of curve veering even undersuch weak coupling conditions as shown in Fig. 12.

A complete interchange of the eigenfunctions associatedwith each of the eigenvalue loci was observed at the transitionzone of the veering spectrum [as predicted by (3) and (2)and shown in Fig. 2]. This is shown in Fig. 13, wherein themeasured variations in the scattering (S21) parameter responsesmeasured from the two resonators along the veering spectrumare plotted as a function of the induced stiffness disorder onresonator 2. The transmission responses from resonator 1 forincreasing magnitudes of induced stiffness perturbation (onresonator 2) are further elaborated in Fig. 14. An extractedtransfer function response measured from resonator 2 after par-asitic feedthrough capacitance cancellation using the procedureexplained in [15] is shown in Fig. 15.


Fig. 13. Observed variation in the scattering (S21) parameter measured from resonators 1 and 2 along the veering spectrum.

Fig. 14. Experimentally measured (S21) parameter response from resonators 1 for increasing magnitudes of electrostatic stiffness perturbation on resonator 2along the veering spectrum.


Fig. 15. Extracted transfer function response of resonator 2 for varying magnitudes of induced stiffness disorder. The variation in amplitudes at the two resonancesindicates the mutual interchange of modal dynamics during veering

TABLE IEXPERIMENTALLY OBSERVED RAPID BUT CONTINUOUS INTERCHANGE OF THE EIGENSTATES FOR DIFFERENT

MAGNITUDES OF INDUCED STIFFNESS PERTURBATION ON RESONATOR 2 ALONG THE VEERING SPECTRUM

IV. DISCUSSION

The variation in the normalized mode shapes of the systemat the first eigenvalue, extracted after cancelling the effects offeedthrough and normalizing to the applied dc bias voltages[12] for different magnitudes of induced stiffness perturbationsare elaborated in Table I. Even minute stiffness perturbationson resonator 2 (spanning merely 0.3 N/m in this case) breakthe periodicity of the nearly symmetric system, causing severelocalization of the vibration modes. Such rapid transitions ofthe eigenfunctions not only yield a quantitative insight intothe extent of vibration energy confinement along the veeringspectrum within such electrically coupled nearly periodic mi-crosystems, but also provide valuable information regarding thesensitivity of the coupled dynamics on the two most importantparameters—the strength of internal coupling between the tworesonators and the magnitude of the structural disparity betweenthe resonators.

Standard perturbation techniques may be employed toanalytically quantify the extent of vibration confinement forvarying magnitudes of induced disorder within the system asdetailed in [12]. However, these predictions hold true onlyso long as the induced structural disorder is smaller than thestrength of internal coupling [12], [17]. When the inducedstiffness variation is comparable or even larger than the strengthof the electroelastic coupling element, a singular perturba-tion technique or the Rayleigh approach may be employed topredict more accurately the eigenstate variations within thesystem [18]. In Figs. 8 and 15 and (2) and (3), it is clearthat tuning the potential difference between the resonators

allows one for altering the effective strength of internal cou-pling consequently enabling the manipulation of the severityof eigenfunction interchange and vibration energy confinementin such microsystems. Such manipulations are very difficult toimplement/observe in corresponding macroscopic systems dueto the difficulty in realizing such tunable coupling elements atlarger length scales.

These results experimentally demonstrate the existence ofboth mode localization and curve veering in micromechanicalresonators subjected to weak electroelastic coupling and showthat the manipulation of the coupling spring can tune the degreeof mode localization in these systems. Furthermore, owing tothe considerably weaker effective coupling spring constantsand the negative effective value of such electrical couplingelements, such microsystems may also be utilized to investigatethe general nature, properties and the fundamental limits ofthese dynamic phenomena.

Manipulation of the modal dynamics in such coupledmicrosystems would have significant benefits in the designof micromechanical narrow bandwidth filters used in signalprocessing applications, as they could be used as simple yeteffective techniques of tuning the ripple and bandwidths ofmicromechanical radio frequency filters, consequently allowingfor enhanced flexibility in their design and operation. Manipu-lating the strength of the internal coupling is also useful in tun-ing and enhancing the parametric sensitivity of mode-localizedsensors to detect small perturbation in mass [17] and stiffness[12] using weakly coupled micro/nanomechanical resonatorarrays. Sensitivity enhancements as high as four to five orders of


magnitude greater than the more conventional resonant fre-quency shift based sensing approach can potentially be obtainedby controlling the coupling spring constant and in effect, theextent of vibration energy confinement [12].

V. CONCLUSION

In conclusion, this paper has presented the first experimen-tal evidence of both curve veering and mode localization innearly identical mechanical resonators coupled through veryweak negative electrical springs. Furthermore, a manipulationof the extent of energy confinement due to mode localizationand in consequence, the severity of eigenfunction interchangeduring veering is also experimentally demonstrated by tuningthe potential difference between coupled micromechanical res-onators. An experimental verification of previous theoreticalstudies of the nature and strength of the coupling behavior ofelectroelastic springs is also presented. These phenomena findapplicability in deducing small structural perturbations in mi-cromechanical sensors and in the manipulation of the effectivebandwidth and ripple in MEM filters for signal processing.

ACKNOWLEDGMENT

The authors would like to thank R. Louca for his assistancewith using the SEM.

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[14] R. Baskaran and K. L. Turner, “Study of electrostatic interactions inMicroelectromechanical resonant oscillators,” Proc. SPIE, vol. 4593,pp. 42–53, 2001.

[15] J. Yan, A. A. Seshia, K. L. Phan, P. G. Steeneken, and J. T. M. van Beek,“Narrow bandwidth single-resonator MEMS tuning fork filter,” in Proc.IEEE Int. Freq. Control Symp., 2007, pp. 1366–1369.

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Pradyumna Thiruvenkatanathan (S’09) receivedthe B.E. degree in electrical and electronics engi-neering from Anna University, Chennai, India, in2006, and the M.Phil. degree in micro- and nano-technology enterprise from the University ofCambridge, Cambridge, U.K., in 2007, where he iscurrently working toward the Ph.D. degree in theDepartment of Engineering.

He has been a member of Churchill College,Cambridge, U.K., since 2006. He is also affiliatedwith the Cambridge Nanoscience Center. His current

research interests include investigating methods of enhancing the signal-to-noise performance in microelectromechanical systems for biomedical andinertial sensor applications.

Jim Woodhouse received the B.A. degree in math-ematics and the Ph.D. degree from the Universityof Cambridge, Cambridge, U.K., in 1972 and 1977,respectively.

He pursued postdoctoral studies related to theacoustics of the violin. After working for a consul-tancy company in 1985, he took up a post in the De-partment of Engineering, University of Cambridge,and has been successively a Lecturer, Reader, andProfessor there. His research interests cover a rangeof topics in structural vibration, namely, friction-

excited vibration, such as vehicle brake squeal, acoustics of musical instru-ments, modeling of structural damping mechanisms, and vibration predictionmethods for complex structures.

Jize Yan (S’06–M’09) received the B.S. degree fromTsinghua University, Beijing, China, 2003, and thePh.D. degree from the University of Cambridge,Cambridge, U.K., 2007.

He is currently a Research Associate in the De-partment of Engineering and Nanoscience Center,University of Cambridge. His research interests in-clude radio-frequency microelectromechanical sys-tems, micro/nanoelectromechanical system sensors,power harvesting, and micro/nanofabrication.

Ashwin A. Seshia (S’98–M’02–SM’10) receivedthe B.Tech. degree in engineering physics from theIndian Institute of Technology Bombay, Mumbai,India, in 1996, the M.S. and Ph.D. degrees in elec-trical engineering and computer sciences from theUniversity of California, Berkeley, in 1999 and 2002,respectively, and the M.A. degree from the Univer-sity of Cambridge, Cambridge, U.K., in 2008.

During his time at the University of California,Berkeley, he was affiliated with the Berkeley Sensorand Actuator Center. He joined the faculty of the

Department of Engineering, University of Cambridge, in October 2002, wherehe is currently a Lecturer in microelectromechanical systems, a Fellow ofQueens’ College, and affiliated with the Nanoscience Center. His researchinterests include the design and fabrication of micro- and nanoscale sensorysystems with applications to the monitoring and study of the natural and builtenvironment.

Dr. Seshia was appointed as a Fellow of the ERA Foundation in 2008.

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